Physico-chemical microstructure of matter - atoms and atomic nuclei

AstroNuclPhysics ® Nuclear Physics - Astrophysics - Cosmology - Philosophy Physics and nuclear medicine

1. Nuclear and radiation physics
1.0. Physics - fundamental natural science
1.1. Atoms and atomic nuclei
1.2. Radioactivity
1.3. Nuclear reactions and nuclear energy
1.4. Radionuclides
1.5. Elementary particles and accelerators
1.6. Ionizing radiation

1.1. Atoms and atomic nuclei

Substance, Fields, Particles, Interactions
In the introduction to our treatise on atoms, atomic nuclei, and the physics of the microworld, we make a few preliminary remarks about the basic building blocks of matter and the nature of the forces that govern their behavior. All these findings, which are only outlined here, will always be substantially expanded and specified in the appropriate places during the interpretation .
  In the physical study of nature, we divide the whole material world into two basic forms of matter :

Modern physics shows that this division is to some extent conventional - the two forms change with each other; particles of matter can be interpreted as quantum states of specific fields (unitary field and particle theory ) and physical fields can be described using quantum - particles (see "Quantum field theory" below).

Discrete particle and continuous field model of matter
We model the structure and behavior of matter in physics by the two different ways mentioned above (forms of matter) :
¨ Discrete particle model , according to which all bodies and material environments consist of a large number of small spatially localized objects - particles. Particles have a certain non-zero mass or own energy, have a certain position in space and time, a certain speed, momentum and kinetic energy. The motion of particles is governed by the universal laws of mechanics (idealized "material point") - classical mechanics (Newton's laws of motion) or relativistic kinematics and dynamics, and quantum mechanics.
¨ Continuous field model , describing the structure and behavior of matter by quantities continuously distributed in space. In modern physics, this field description is used for forces - interactions - between particles of matter. It is perfectly elaborated especially for electromagnetic phenomena between charged bodies and particles - Faraday-Maxwell electrodynamics (see below "Electromagnetic field and radiation").
However, the field description can be used (and especially often used earlier) in continuum physics for the study of liquids, gases and partly also solids. The movement of liquids and gases is internally caused by the movements of their atoms and molecules - a perfectly proven kinetic theory . However, for the macroscopic description of the behavior of gases and liquids, the motion of individual atoms or molecules is not investigated, but the collective motions of a set of particles. "Averaged" quantities are used, which are continuously distributed throughout the volume of gas or liquid. Individual data on the position of individual atoms are replaced by the average spatial distribution of their number - the density distribution. In the macroscopic description , the speed of the disordered motion of individual atoms is replaced by temperature, and the ordered motion by flow rate (momentum transfer) in various places of the material environment. Collisions and forces between atoms or molecules inside gases and liquids are expressed by the distribution of pressure. Important equations of state apply to the interdependencies between these quantities . The relationship between the mechanical characteristics of particles (atoms and molecules) and field state quantities in continuum physics is derived by the methods of statistical physics. All properties of material environments and the events observed in them are an integral manifestation of a number of chaotic or coordinated simpler movements of the building blocks of the respective substance.

Basic building particles of matter
With ever deeper penetration into the microworld of building matter, physics discovers that atoms (previously considered indivisible) are composed of particles that can no longer be decomposed into simpler objects capable of independent existence. These smallest particles, which are no longer indivisible, are called elementary particles and can be considered as the basic "building blocks" of matter. However, these elementary particles are not static and immutable, but can undergo mutual changes and some of them may have a certain internal structure. In the study of the structure of atoms, we encounter mainly the three most important particles - the electron, the proton and the neutron *). In the study of excitations and radiation of atoms and atomic nuclei then with a photon - a quantum of electromagnetic radiation, with radioactivity also with a neutrino and a positron (antiparticle to an electron)
- §1.2, part "Radioactivity beta". The properties of these and many other particles are more fully discussed in §1.5 "Elementary particle and accelerators" dedicated elementary particle physics, where it is administered also systematics of elementary particles (neutrinos is moreover closely discussed in §1.2, section "Neutrinos - "ghosts" between particles").
*) The reason why the observed matter is composed only of electrons, protons and neutrons is that all other material particles are very unstable.

Four basic physical interactions
The interaction between particles of matter can be explained by four basic physical interactions. At the level of atomic nuclei and elementary particles, two short-range interactions are dominant :
¨ Strong interaction , important especially by holding atomic nuclei together (see "Atomic nucleus" below). It primarily combines quarks into protons and neutrons, mesons and other hadrons . The inherent strong interaction between quarks, mediated by gluons, has a long range, but the nuclear strong interaction, as its "residual manifestation", is short-range (see "Strong interaction" below).
¨ Weak interaction , which is applied in mutual transformations of neutrons and protons with the participation of neutrinos, in practice mainly in radioactivity b (§1.2, part "Radioactivity beta", passage "Mechanism of weak interactions"). It is also short-range.
Certain types of particles, which we call electrically charged, show a force interaction described by an electromagnetic interaction. When these electrically charged particles are at rest, an attractive or repulsive electric force acts between them according to Coulomb's law, when they are in motion, there is also a magnetic force, in the case of uneven movements of charges, then there is also the emission of electromagnetic waves - photon radiation (see the section "Electromagnetic fields and radiation" below). The electromagnetic interaction has a long range (more precisely, the range is infinite).
¨ The fourth interaction, also of long range, is the gravitational interaction, which acts universally between all particles, is attractive and has a significant effect on high-mass bodies. Its force manifestations are described in classical physics by Newton's law of gravitation, in relativistic physics by Einstein's equations of the gravitational field - see the book "Gravity, black holes and space-time physics", §1.2 "Newton's law of gravitation" and §2.5 "Einstein's equations of the gravitational field".
The biggest and most difficult task of contemporary theoretical physics is to find the so-called unitary field theory, which would unify the 4 basic interactions and explain them as special cases of a single general interaction - see the section "Unitary theory of fields and elementary particles" in §1.5, more details "Unitary field theory and quantum gravity" of the above-mentioned monograph "Gravity, black holes and space-time physics".
  The magnitudes of the force effect of these basic interactions are diametrically different and decisively depend on the distances of the interacting particles. For distances of the order of 10
-13 cm corresponding to the dimensions of atomic nuclei, the relative ratio (or rather, "disproportion") of force effect strong, electromagnetic, weak and gravitational interaction is about 1 : 10-(2-3) : 10-15 : 10-40. At distances of the order of 10-8 cm, corresponding to the dimensions of the atomic shell, short-range strong and weak interactions are practically no longer act, and the electromagnetic interaction has a decisive influence.
  In our treatise on nuclear and radiation physics, we will not deal with the gravitational interaction, which is more pronounced in macroscopic bodies and acquires a dominant character in bodies of cosmic dimensions and masses. The strong and weak interaction will be discussed in more detail below in the relevant passages on the atomic nucleus
("Atomic Nucleus") and in §1.5 on elementary particles (section "Four types of interactions"). We will say some basic information about the electromagnetic interaction here (below "Electromagnetic fields and radiation"), because we will need it first to explain it - already in the science of atoms.

Classical and quantum models in the microworld
In atomic and nuclear physics, we study objects and processes whose behavior is beyond our imagination based on experience from the macroscopic world - the behavior of objects composed of a large set of atoms. Even in the microworld, controlled by quantum laws (see below), we can sometimes help out by use illustrative mechanical comparisons to macroscopic systems known to us. For example we imagine electrons in atoms as light negatively charged "globules" orbiting a heavy positively charged small "sphere" - the nucleus of an atom. Or other times we imagine the particles as waves or a wave pack. However, we must always keep in mind that these are just models, expressing only some selected properties of these microsystems, not their actual material structure in the usual sense! They are all just our human models, how to at least roughly understand the phenomena, that are very foreign to our daily experience. Importantly, it works in a theory-experiment relationship; and we believe that it will also help us to understand the internal mechanisms..?..
  An important difference compared to classical physics is the stochastic (probabilistic) character of quantum phenomena in the microworld. For individual processes, we cannot determine exactly when they will occur, but only their probability. The individual causality of particle behavior is lost, but a new kind of stochastic regularity emerges. Chaotic randomness (apparent or principled?) in the behavior of individual particles results in a regularity for the statistical set of these particles as a whole (not for its individual elements). These aspects of quantum physics will be briefly discussed below
("The Quantum Nature of the Microworld").
  From the philosophical-scientific point of view, the relations between the macroworld, the microworld and the megaworld are discussed in §1.0 "
Physics - fundamental natural science".

Vacuum - emptiness - nothingness ?
In fundamental physics, phenomena occurring with bodies, particles and fields are mostly studied in a vacuum. Vacuum in classical physics means empty space
(lat. Vacuus = empty), approximately achieved in terrestrial conditions in closed vessels by exhausting air so that the gas pressure is significantly lower than at normal atmospheric pressure. An ideal or perfect vacuum is a state of space in which no particles of matter (such as electrons, protons, etc.) nor radiation (photons) are present. Creating such a perfect vacuum is very difficult, even impossible in practice (it is impossible to get rid of, for example, the ubiquitous neutrinos or weakly interacting massive WIMP particles forming hidden matter in space - §1.5). Even if it succeeds, it will not be an empty space, where there is nothing and nothing happens - they can reach a physical fields such as electromagnetic and gravitational (gravitational fields cannot be shielded). Any vacuum is not actually empty - according to quantum field theory, there are many processes of quantum fluctuations, virtual pairs of particles and antiparticles are constantly formed (see "Quantum field theory" below).
  And in any case no means can a vacuum (even the "perfect") be considered as "nothingness"! Nothingness means the absence of anything - matter, energy, even space and time; it is therefore a synonym for "non-existence" - a fictitious philosophical concept without physical content.

From a philosophical point of view, a physical vacuum is not a state of pure nothingness, but contains potentiality of all forms of the world of particles (cf. "Anthropic principle or cosmic God"). Vacuum is a "living void" pulsating in the infinite rhythm of formation and extinction of structures, virtual and real particles ...
Vacuum energy

In classical (non-quantum) physics is the energy density of itself vacuum (without fields) zero. A completely marginal exception here is (non-quantum) relativistic cosmology, some models of which introduce the so - called cosmological constant, which generates a certain immanent fundamental density of vacuum energy in space
(§5.2, part "Cosmological constant" in the above - mentioned book "Gravity, black holes and space - time physics").
  According to quantum field theory, however, countless processes of spontaneous quantum fluctuations take place everywhere and constantly in a vacuum - virtual pairs of particles and antiparticles are constantly being created and destroyed. The duration of these fluctuations is too short for us to directly detect these particles, so they are called virtual. Quantum field fluctuations have different intensities and spatial dimensions and interfere with each other. The result of this wave interference is averaged over time. If the contributions of individual field fluctuations are canceled on average, the mean energy of the vacuum will be zero - this is the so-called "true vacuum". However, if such a canceled does not occur, the mean energy of the vacuum will be non-zero - such a state is called "false vacuum".
  According to current cosmology, a "strongly false" high - energy vacuum could have been the driving force behind the rapid inflationary expansion of the very early universe (
§5.5 "Microphysics and cosmology. Inflationary universe." in the book "Gravity, the black hole ..."). The present value of the vacuum energy is very close to zero, less than about 10-9 J/m3, which corresponds to the mass density of approximately 10-26 kg/m3. Attempts have been made explain the vacuum energy through quantum field theory - as a consequence of quantum fluctuations of vacuum. A straightforward computation (resp. dimensional estimation), encompassing all vibrational modes of energy with a wavelength greater than the Planck length (10-35 meters), can be in and yet incredibly high density vacuum energy, corresponding to a mass density of about 1096 kg/m3..!.. In order for the vacuum to look like an empty space, far-reaching compensations must be applied between the vacuum fluctuations of the different fields, which cancel out the vast majority of the fluctuations. This "scandalous discrepancy" of the 120 orders has not yet been satisfactorily explained; perhaps the unitary field theories promise some hope (§B-.6 "Unification of fundamental interactions. Supergravity. Superstrings." in the above-mentioned book "Gravity, Black Holes ...") .

Electromagnetic fields and radiation
Before we begin to focus on the structure of atoms and the phenomena taking place inside, it will be useful to say a few words about one of the most important phenomena in nature - electromagnetic action and electromagnetic radiation. This is because all events in atoms and their nuclei are closely connected with electromagnetic interaction.
  Each electric charge Q excites around it an electric field of intensity E, proportional (according to Coulomb's law) to the magnitude of the charge Q and inversely proportional to the square of the distance r : E = r
o . k. Q/r2, where ro  is the unit vector extending from the charge Q to the test site and k is the coefficient expressed in SI in terms of vacuum permittivity eo : k = 1/4p eo . If the charge does not move (in the given reference system) , it is an electrostatic field. This electric field causes force effects F = q.E for every other charge q that enters this space. The electric field is generally springs, its source is the electric charges from which it emanates ("springs") and into which the electric field lines enter. However, even in the absence of electric charges, if the electric field is excited by electromagnetic induction with time changes of the magnetic field (as mentioned below), the electric field may be source-free.
What is the strongest electric field can be ?  
In classical (non-quantum) physics, the electric field in a vacuum can be arbitrarily strong, almost to infinity
(in the material environment, however, this is limited by the electrical stability of the dielectric). From the point of view of quantum electrodynamics, however, even in a vacuum there is a fundamental limitation caused by the existence of mutual antiparticles of electron and positron : it is not possible to create an electric field with an intensity stronger than Ee-e+ = me2c3/e.h = 1,32.1016 V/cm, where me is the rest mass of the electron or positron. When this intensity is exceeded, the potential gradient is higher than the threshold energy 2me c2 and a pair of electrons and positrons is formed, which automatically reduces the intensity of the electric field. Such a strong electric field has not yet been created, with conventional electronics this is not possible; strong impulses from extremely powerful lasers could be a certain possibility in the future ...
  If the charge Q moves (electric current), in addition to the electric, a magnetic field also excites around itself. The moving charges, forming a current I in the longitudinal element dl, excite at a distance r a magnetic field of intensity B (unfortunately called magnetic induction for historical reasons) according to the Biot-Savart-Laplace law : dB = k .I .[dl´ro]/r2, where ro is the unit directional vector of the measured point to the current element and k a proportionality constant expressed in SI units called via permeability of vacuum mo: k = mo/4p. The magnetic field shows force effects on each electric charge q moving at a speed v : F = q. (B x v); this so-called Lorentz force acts perpendicular to the direction of movement of the charge. The magnetic field is (unlike the electric field) always source-free, the magnetic field lines are closed curves - there are no so-called magnetic monopolies (magnetic "charges", similar to electric charges).
  During movement or time changes in the magnetic field, it arises according to Faraday's law of electromagnetic induction the electric field - in the form of a kind of "vortex", a rotating electric field around a variable magnetic field. An induced electric field can cause the movement of charges, eg electrons in a conductor - induced electric current. And the time changes of the electric field, in turn, cause a magnetic field
(as if the so-called Maxwell shear current flowed), again of a vortex character. This dialectical unity of electric and magnetic fields finds its application in the concept of electromagnetic field, whose special manifestations are electric and magnetic fields. This field is governed by Maxwell's equations of the electromagnetic field, which were created by merging and generalizing all the laws of electricity and magnetism. The combined science of electricity and magnetism, including the dynamics of charge motions and the time variability of fields, is called electrodynamics.
Note: Details on the theory of the electromagnetic field can be found, for example, in §1.5 "Electromagnetic field. Maxwell's equations" of the book "Gravity, Black Holes and the Physics of Spacetime".
  Below, in the section "
Atomic structure of matter" we will see that electromagnetic forces are decisive for the structure of atoms and for their properties - they are determining significance for the structure of matter at the microscopic and macroscopic level, including all chemical phenomena. Along with strong interactions, electric forces also play an important role in the structure of atomic nuclei (as we will see in the section "Structure of the atomic nucleus") and in the excitations and deexcitation of their excited energy states.

The electromagnetic waves
Maxwell's equations have a number of remarkable properties, but the following regularity is important to us here: The disturbance (change) in the electromagnetic field propagates in space at a finite speed equal to the speed of light. When electric charges move at a variable speed (with acceleration), they create a time-varying electromagnetic field around them, which leads to the formation of electromagnetic waves, which detach from their source and carry some of its energy into space. The electromagnetic field further propagates through space already independently of the source electric charges and currents in the form of a free electromagnetic wave
- it is derived in §1.5, part "Electromagnetic waves" already mentioned monography "Gravity, black holes, and physics of space-time".
  From Maxwell's equations, by a suitable modification can be obtained two partial differential equations for the vectors E and B :
2E/x2 + 2E/y2 + 2E/z2 = e.m .¶2E/t22B/x2 + 2B/y2 + 2B/z2 = e.m .¶2B/t2 ,
which are wave equations describing the propagation of a time-varying electric and magnetic field in space at speeds c =
Ö(1/em), where e is the electrical permittivity and m is the magnetic permeability of the given medium: E(x, y, z, t) = f(t - x/c) and analogously for B, if we consider for simplicity the waves propagating in the direction of the x-axis. The most commonly considered is the harmonic (sine or cosine) time dependence: E(x,y,z,t) = Eo.cos(w.(t - x/c)) and analogously for B, where w = 2p.f is the circular frequency; this is because waves are often caused by periodic oscillating movements of electric charges (eg in antennas powered by a high-frequency signal of frequency f); even in cases where this is not the case (eg braking radiation), the resulting waves can be decomposed by Fourier into harmonic components of different frequencies and phases.
  The highest speed is reached by electromagnetic waves in a vacuum, where c
o = 1/Ö( = 2,998.108 m/s @ 300 000 km/s. In a material environment whose permittivity and permeability are greater than for a vacuum, the speed of electromagnetic waves is somewhat lower - in light this leads to known optical phenomena of refraction of light rays when light passes between substances with different optical densities (see below "Electromagnetic and optical optical properties substances").
  Thus, according to Maxwell's equations of electrodynamics, electromagnetic waves are transverse waves of electric and magnetic fields (mutually excited by their variability), where the vector E of electric intensity and vector B of magnetic induction oscillate with amplitude A constantly perpendicular to each other and perpendicular to the direction of wave propagation (see upper part of Fig.1.1.1), which in a vacuum travels at the speed of light c = 300,000 km/s. The electromagnetic wave periodically exerts a force on electrically charged particles - it sets electrons in motion in conductors and induces alternating electric current in them; the reception of electromagnetic waves by the antenna is based on this. Periodicity in space is given by wavelength, periodicity in time by frequency. The intensity (power) of an electromagnetic wave is given by the amplitude of the oscillating electric intensity E and the magnetic induction B, energy transfer by the so-called Poynting vector. There are simple relations between the speed of light c , the frequency of oscillation
n and the wavelength l : l = c/n, n = c/l, l.n = c. The higher the oscillation frequency of the electromagnetic field, the shorter the wavelength. And it is on this frequency or wavelength that the properties of electromagnetic waves depend significantly.
Note: Wave propagation in material environments and especially physical fields is a general fundamental natural phenomenon - it is analyzed in the introductory part of §2.7 "
Wave propagation - a general natural phenomenon" of the already mentioned book "Gravity, black holes and space - time physics".
Electromagnetic waves in atomic and nuclear physics
The general regularity of electrodynamics, that the temporal changes of electric and magnetic fields are capable to propagate in space as electromagnetic waves transmitting energy, play an important role in atomic, nuclear and radiation physics. First of all, it is the electromagnetic radiation of atoms during the jumps of electrons between the energy levels in the electric field of the nucleus (see below "Radiation of atoms"). Furthermore, it is the braking radiation generated generally during the accelerated motion of electric charges, in radiation physics especially during the impact of fast electrons on matter and their rapid braking during interaction with atoms of matter (§1.6, section "Interaction of charged particles"). The more subtle radiation effects are Cherenkov radiation and transient radiation, arising during the passage of fast charged particles through the material environment (§1.6, passage "Cherenkov radiation"). In the field of atomic nuclei, it is the deexcitation of nuclear levels by the emission of electromagnetic radiation - quantum gamma radiation (§1.2, part "Gamma radiation").

Types of electromagnetic radiation
According to wavelength or frequency, we divide electromagnetic waves into several groups :

   The last two types of shortwave radiation, X and gamma, partially intersect with their spectra (wavelengths or energies) and there are sometimes terminological ambiguities. In the mentioned §1.2, part "Gamma radiation", there is a terminological agreement on the division of shortwave electromagnetic radiation according to its origin - gamma radiation comes from the nucleus, X radiation from other regions of the atom outside the nucleus.

Units of energy, mass and charge in atomic and nuclear physics
In most areas of physics and natural sciences, a system of SI units is used, in which the basic units are: meter [m] as a unit of length, second [s] as a unit of time and kilogram [kg] for mass; decimal multiples are often used - centimeter or gram, etc. The basic unit of work and energy is the joule [J], the unit of electric charge coulomb [C].
  In atomic and nuclear physics, which examines phenomena at small spatial scales and very small values of absolute mass, energy and charge, some somewhat different habits have been established in the units of mass, energy and charge used. These alternative units are better "tailored" to the phenomena studied in the microworld than SI units derived from macroscopic phenomena.
   The unit of time second is to keep, the unit of length, meter or centimeter, is usually also to keep (of course using decimal fractions 10
-xx); sometimes the unit angstrom is used : 1A° = 10-10 m = 10-8 cm (in atomic physics it is a typical dimension of an atom), or fermi : 1fm = 10-15 m = 10-13 cm ( femtometer, in nuclear physics it is a characteristic dimension of the nucleus).
   As a unit of energy in atomic physics it is not used too large 1 Joule, but 1 electron volt, which is the kinetic energy obtained by the charge of one electron in the electric field at accelerating potential difference of one volt: 1eV = 1.602
x10-19 J. In nuclear physics, where there are higher energies and energy differences, then decimal multiples - kiloelenktronvolt (1keV = 103 eV), megaelectronvolt (1MeV = 106 eV) and gigaelectronvolt (1GeV = 109 eV).
   Also the usual unit of weight, kilogram or gram, is impractically large for atomic and nuclear physics. In nuclear physics, mass is usually understood as the rest mass of particles and it is customary to express it in energy units based on the Einstein relation E = m.c
2 equivalence of mass and energy, ie also in electron volts : 1eV = 1.783x10-33 grams; and of course in their decimal multiples. The rest mass of an electron can therefore be expressed as: me = 9.1x10-28 g = 511 keV. In addition to [MeV], the mass of heavier elementary particles is sometimes expressed in multiples of the mass of the electron me - eg the mass of a proton can be expressed in three different ways: mp = 1.673x10-24 g = 938 MeV = 1836 me .  
  For an electric charge, instead of an oversized unit Coulomb, as natural basic units of charge, the electron charge e is use (resp. the same size but the opposite charge of the proton), which is the elementary electric charge: e = 1.602
x10-19 Coulomb.  
  However, the currently used units of dosimetric quantities, characterizing the effects of ionizing radiation on matter and living tissue, are based on the SI system. Because these are cumulative effects of a macroscopic nature. The basic quantity here is the absorbed radiation dose, the unit of which is 1Gray = 1J/1kg
(for more details see §5.1 "Effects of radiation on the substance. Basic quantities of dosimetry.").
A note on quantities and units in nuclear physics 
The terminology, quantities and units related to atoms, nuclei, radioactivity and radioactive radiation have undergone a long and complex development, which has left some illogicalities and ambiguities - to be specified below. After all, similar gnoseological inconsistencies also occur in other physical fields due to historical development. Recall, for example, the unfortunate introduction of an electric current as a basic quantity and its SI unit 1Ampere (using "the force of two infinite parallel conductors ..."), while the physically primary electric charge (and the Coulomb unit) is introduced as derived from current. Or in magnetism the terminological illogicality of the names "magnetic field intensity" and "magnetic induction" (for an electric field it is fine) ...

Excursion to high velocities - a special theory of relativity
Microparticles, of which matter is composed, usually move at very high velocities during processes inside atoms, atomic nuclei and mutual interactions, often approaching the speed of light. In experiments with these high velocities, it was found that the usual laws of classical Newtonian mechanics no longer apply exactly here. Albert Einstein in his research at the beginning of the 20th century, followed up on Galileo and Newton's classical mechanics, Maxwell's electrodynamics and the research of his predecessors (Lorentz, Michelson-Morley, ...) and created a new mechanics - the so-called special theory of relativity, generalizing classical mechanics to movements at high speeds close to the speed of light. A systematic interpretation of this certainly interesting theory is not possible here; it can be found in a number of book publications (on these pages it is eg §1.6 "Four dimensional spacetime and special theory of relativity" in the book "Gravity, black holes and physics of spacetime"). Here we will only briefly recall some basic phenomena of the special theory of relativity, which are of fundamental importance in nuclear processes and interactions of elementary particles.
  The special theory of relativity (STR) is based on two basic postulates :

Relativistic kinematics
From these two experimentally perfectly verified principles it follows, that the relationships between positional coordinates and time intervals of events in different inertial frames of reference with the laws of classical kinematics control only at low velocities, while in general they control so-called Lorentz transformations
     x´ = (x - V.t)/
Ö(1-V2/c2) , y´ = y , z´ = z , t´ = (t - x.(V/c2))/Ö(1-V2/c2) ,
indicating the relationship between the spatial coordinates x,y,z and the time t in the inertial system S and in the system S´ moving with respect to S speed V in the direction of the x- axis.

Note: In non-relativistic physics, the relationship between these coordinates is given by a simple Galileo transformation x´ = x -V.t, y´ = y, z´ = z, t´ = t (time t´ here, of course, flows here as fast as t !).
  Important kinematic effects of the special theory of relativity follow from Lorentz transformations :
Contraction of lengths :
The dimension l of each body of (own) length l
o, which moves with velocity v, appears shortened in the direction of motion compared to its rest dimension lo: l = lo.Ö(1-v2/c2).
Time dilation :
The time on a moving body flows with respect to the time of the external resting observer the slower, the faster the body moves:
Dt = Dt .Ö(1-v2/c2). Here Dt is the time measured by the external rest clock, Dt is the actual time measured by the clock moving together with the body at velocity v .
Einstein's law of velocity addition :
If one body moves with velocity v
1 and the other body with respect to it velocity v2 in the same direction, then with respect to the initial inertial frame of reference, the result of the composition of both velocities will be v = (v1+v2)/(1+v1.v2/c2), and not v1+v2 , as would be was in classical mechanics.
  Of these kinematic effects of special theory of relativity is considerable importance for nuclear and particle physics, especially the dilation of time, thanks to which particles with a short lifetime can live many times longer if they move at a speed close to the speed of light. Thanks to this effect, for example, m - mesones (with a lifetime of 2.10-6 sec) created by the interaction of cosmic rays in the high layers of the atmosphere, it is enough to reach the surface of the earth, where we can observe them. Or we can exploit the mesons p+, p-, created during interactions of high-energy protons from the accelerator, to come out its in form of the beams and study their interactions for a time many times longer than their rest time of life 2.6x10-8 sec.

Relativistic dynamics
Combining relativistic kinematics STR and (Newton) dynamics of body motion arises relativistic dynamics, the basic new finding of which is that the (inertial) mass of bodies m is not constant, but the mass depends on the velocity of the body v according to an important relation
m = m 0 / Ö ( 1-v 2 / c 2 ) ,
where m
0 is the rest mass of the body *), which it has in the inertial frame of reference in which it is at rest. The weight of the body therefore increases with speed, especially when the speed approaches the speed of light - then the mass of the body increases theoretically to infinity: limv® c m = ¥ .
  Another important result of relativistic dynamics is the relation for the total energy E of a body of rest mass mo moving at a velocity v :
     E = m
o .c 2 / Ö (1-v 2 / c 2 )
and the resulting findings of the equivalence of mass and energy expressed by the famous by Einstein's relation E = m.c
2 ; resp. DE = D m . c2.   
  Both these relations of the dependence of mass on velocity and the equivalence of changes in mass and energy play a cardinal role in nuclear and particle physics, where there are mutual transformations of energies and particles moving at high velocities.
*) This relation cannot be used directly for particles with zero rest mass (m
o = 0) moving at the speed of light v = c - such particles are mainly quantum of electromagnetic waves - photons. The photon has energy E = h. n , given by its frequency n and can be attributed to the (relativistic) inertial mass m = E/c2 = h.n/c2 .

General Theory of Relativity
In addition to the special theory of relativity, Einstein also developed a general theory of relativity, which is a unified relativistic physics of gravity and spacetime. We will not deal with this here, because in atomic and nuclear physics the gravitational interaction does not manifest (if we omit the unitary field theory...). This very interesting theory is explained in detail in the monograph "
Gravity, Black Holes and the Physics of Spacetime", especially in Chapter 2 "General Theory of Relativity - Physics of Gravity ", along with its implications in astrophysics and cosmology - Chapter 4 "Black Holes" and Chapter 5. "Relativistic Cosmology".

Corpuscular-wave dualism
In classical physics and in everyday life, we observe a diametrical difference between discrete particles or bodies with their motions described by mechanics, and between continuous waves propagating in a certain environment. However, in a microworld dominated by the laws of quantum physics, this difference is blurred in certain circumstances !

Corpuscular properties of waves
At the turn of the 19th and 20th centuries, physics explained all natural phenomena either using particles, or by means of an electromagnetic field and its waves - electromagnetic radiation, a special kind of which is light. Virtually all the properties of light known in optics at that time (laws of propagation, reflection, refraction, difraction of light, interference) could be very well explained by the wave concept. Huyghens's wave approach to radiation seemed to triumph over Newton's corpuscular notion. However, some of the properties of radiation that were recently discovered at the time could not be fully satisfactorily explained by the pure wave concept.
Black body radiation 
The first such phenomenon was the spectrum of radiation of a heated
("absolutely") black body *), which was examined in detail by M.Planck in 1900. To explain the observed shape of the black body's radiation spectrum as a function of its temperature, Planck hypothesized that the emission (and absorption) of electromagnetic radiation by individual atoms in the body does not occur smoothly and continuously, but after certain small precise doses - quanta of energy. Sources of electromagnetic radiation can be considered as oscillators that cannot oscillate with any value of frequency and energy, but radiate or absorb energy only in certain quantities. The magnitude of the energy E of these quanta depends only on the frequency of the radiation n and Planck established for it the relation E = h. n , where the proportionality constant h @ 6.626x10-34 J/s was called Planck's constant. Planck himself initially considered this assumption only as an ad hoc working hypothesis (a kind of temporary "emergency trick" to explain spectrum discrepancies), which should later be replaced by a more acceptable explanation. In reality, however, this hypothesis proved correct and became the beginning of a new conception of the microworld - quantum physics.
*) Each body (composed of a substance of any state), heated to a temperature higher than absolute zero, emits electromagnetic radiation - thermal radiation, arising from oscillations and collisions of electrons, atoms and molecules due to their thermal movements. This radiation carries away part of the thermal energy supplied to the body from the outside or generated inside the body. For the model study of thermal radiation, a so-called absolutely black body is introduced, which absorbs all the radiation that falls on them. It can be realized with a closed box with heated inner walls provided with a small opening through which thermal radiation escapes into the outer space.
In 1879, Stefan and Boltznan discovered the radiation law for the intensity of black body radiation as a function of temperature: I =
s . T4, where s = 5.67.10-8 Wm-2.K-4 is the Stefan-Boltzman constant. However, a satisfactory and uniform law could not be found to determine the radiated spectrum of thermal radiation. Two laws were formulated for the radiated spectrum, which, however, only partially agreed with the experimentally measured spectral curve: Rayleigh-Jeans's law well described the spectrum in the long wavelength region, but did not agree (even diverged) in the short wavelength region; Wien's law behaved the other way around. Unify both spectral regions managed to M. Planck, who discovered a new radiation law that was in full agreement with experiments in all spectral regions.
Photoelectric effect 
Another phenomenon that resisted satisfactory explanation by the wave nature of light was the photoelectric effect, abbreviated as photoeffect. This phenomenon, first observed in the late 80s of the 19th century A.Stoletov (in experiments with electric arc radiation) and H.Hertz (in famous spark experiments demonstrating electromagnetic waves), consist in the fact that when certain substances, especially metals, light (or electromagnetic radiation in general) of sufficient frequency falls on it, electrons are released from its surface *).
*) We distinguish two types of photo effect, external and internal. Here we deal with the external photo effect, when the action of radiation releases electrons, which escape through the surface from the substance into the surrounding space - occurs electron photoemission. This phenomenon is used in special tubes - photons tubes and photomultipliers. During the internal photoeffect, the released electrons remain inside the irradiated material and contribute to its electrical conductivity (it is used mainly in semiconductor optoelectric components - photoresistor, photodiode). In §1.6, part "Interaction of gamma and X-rays", Fig.1.6.3, we will deal with a special type of photoeffect, where a high-energy quantum of X-rays or g- rays eject electrons from the inner shells of the atomic envelope; and mention also the so-called nuclear photoeffect or photonuclear reaction.

Photoelectric effect
Left: Experimental setup for the study of the photo effect. Top right: Irradiation with strong long-wave radiation does not lead to a photo effect, while irradiation even with weak short-wave radiation causes a photo effect.
Bottom right: Quantum mechanism of a photoeffect by absorbing photons of incident radiation and transferring their energy to electrons.

Detailed experimental tracking (using electron tubes for the left picture - a prototype of so-called photon tube) showed that the photoeffect has certain specific properties, some of which canot be explained by classical wave concept of electromagnetic radiation :
¨ 1. For each metal, there is some threshold minimum frequency nmin , in which a photo effect occurs; if n < nmin , the photo effect does not occur even at the highest radiation intensity. On the contrary, even weak radiation with a higher frequency will cause a photo effect (even if the number of emitted electrons is lower), and immediately; according to the wave idea, the electron would have to "wait" until a weak wave gradually brought it enough energy to release. It follows that if an electron is released, it cannot receive energy gradually and continuously, but must receive the necessary energy at once.
¨ 2. The number of emitted electrons is directly proportional to the intensity of the incident radiation (provided, however, that a photoeffect occurs).
3. The kinetic energy (velocity) of the emitted electrons does not depend on the intensity of the incident radiation. It depends somewhat on the irradiated material and is directly proportional to the frequency of the incident radiation.
    The classical wave concept failed to satisfactorily explain the independence of the energy of the emitted electrons on the intensity of the incident radiation and, conversely, its dependence (even direct proportionality) on the frequency. In 1905, A.Einstein studied in detail the properties of the photo effect and explained all the experimentally established facts by assuming that the absorption of radiant energy takes place not continuously, but in certain small doses, quantum
. The electromagnetic wave of the frequency n and the wavelength l = c/n, during the photoefect behaves as a set of particles - light quanta with a certain energy E and momentum p : E = h.n, p = E/c = h.n/c = h/l. Thus, electromagnetic radiation (including light) not only radiates, but also propagates and interacts (absorbs) in individual quantities.
  The electron on the surface of the plate receives just the energy E
f = h. n of one light quantum - photon. Part of this energy is consumed for the work needed to release the electron from the metal (the output work is equal to the binding energy Ev the electron in the metal, which is relatively small - units of electron volts). The residue is converted into kinetic energy Ek = (1/2) me. v2 emitted electrons of mass me, flying away at speed v. The law of conservation of energy then leads to Einstein's photoelectric equation h. n = Ek + Ev, which quantitatively describes the properties of the photoelectric effect in perfect agreement with the experiment. At longer wavelengths, ie lower frequencies, the energy of the photon is insufficient for the electron to be released from the bond in the metal (or in the atom) - no photo effect occurs.
Compton scattering
Particle nature of shortwave electromagnetic X and gamma radiation is indirectly reflected in some of their interactions such as Compton scattering of this radiation on electrons. The experiment shows that the higher the change in the direction of electromagnetic radiation after scattering on an electron, the lower its frequency. This dependence of frequency on the scattering angle is difficult to explain by the electromagnetic interaction of a plane wave with an electron. On the other hand, the idea that the interaction occurs by the mechanism of a photon collision with energy E = h.n with an electron, similar to the elastic collision of two bodies (such as "billiard balls"), in which the redistribution of directions of motion, velocities and energies (and thus wavelengths and frequencies) is governed by simple laws of classical mechanics of mass points, explains the observed results very well Compton scattering - is analyzed in §1.6, section "Interaction of gamma radiation and X", the passage"Compton Scattering".

The corpuscular-wave dualism of electromagnetic waves
is illustrated in Fig.1.1.1. At the top of the figure, a common electromagnetic wave of lower and higher frequency (i.e., larger and smaller wavelengths) is schematically shown first. If we increase the frequency
n of electromagnetic waves, according to classical physics, nothing happens other than that the wavelength (l = c/n) will be reduced proportionally. However, at very high frequencies (of the order of 1014 Hz, ie 10-7 m) we will observe that the wave will no longer have a constant amplitude, but its amplitude will fluctuate. This tendency will increase with increasing frequency and decreasing wavelength. At extremely high frequencies 1018 Hz (already corresponding to radiation g) we finally find that the wave in the classical sense has disappeared - the radiation will be emitted and propagated in short doses - quanta (Fig.1.1.1 below) particle character, among which are relatively long irregular "gaps".

Fig.1.1.1. Schematic representation of corpuscular-wave dualism in an electromagnetic wave. The upper part shows an electromagnetic wave with a longer and shorter wavelength, the lower part shows a quantum image of the propagation of radiation in quantum - photons.

The quantum of electromagnetic waves are called photons (this name was proposed by American chemist G.N.Lewis) - we can imagine them as a kind of small "packages" or "balls" of electromagnetic waves of a certain frequency, which move at the speed of light c (lower part of Fig.1.1.1). Each photon contains a certain amount of energy E, which is greater the greater the frequency n : E = h.n, where h is the Planck's constant (h = 6.6251x10-34 Js). This constant plays a fundamental role in all phenomena in the microworld. In quantum mechanics, the "crossed out" Planck's constant h = h/2p is often used. The photon is the basic object of the microworld, which has both particle and wave properties, but strictly speaking, it is neither a particle nor a wave.
  In general, it is observed that in the long-wave region of the spectrum, wave properties (bending, interference, scattering, refraction) are more pronounced, while in the short-wave part of the spectrum, particle properties are more pronounced (photo effect, Compton scattering, formation of new particles during interactions). Radiation
g, even though it is inherently an electromagnetic wave, will only behave like a stream of particles - photons, and we will not prove its wave properties by any macroscopic experiment; only if we became "little green dwarfs" in the imaginary experiment, managed to reduce to the dimensions of the order of picometers and "entered" inside the photon, we would find out that the photon is actually an electromagnetic wave inside...
Physical processes of emission Þ wave or photon character of radiation
The wave or photon character of electromagnetic radiation is closely related to the mechanisms and spatial and temporal scales of the physical processes in which this radiation is emitted. In electrical circuits (LC oscillators and antennas) of dimensions millimeters - meters - hundreds of meters, electromagnetic oscillations of frequencies of the order of gigahertz, hundreds of MHz or kHz occur, which leads to the emission of continuous electromagnetic waves (wavelengths of millimeters, meters or hundreds of meters), in which the photon character does not manifest. Light created by deexcitation of the outer electron shells of atoms with dimensions of ~10-8 cm, already bears significant traces of the quantum character of transitions between electron levels; it behaves like a wave as well as a stream of photons. And gamma radiation, which occurs during very fast quantum deexcitations in atomic nuclei of ~10-13 cm in size, is already completely photonic in nature.
Detection of low-intensity high-energy radiation Þ manifestation of corpuscular character
How can we most easily, in normal laboratory conditions, prove the corpuscular character of electromagnetic radiation? Above all, instead of ordinary light, it is appropriate to use high-energy gamma radiation and to register this radiation with a sufficiently sensitive one-photon electronic detector - GM or scintillation detector
(§2.3 "Geiger-Muller detectors", §2.4 "Scintillation detectors"). When measured in a high intensity beam, the signal at the output of the detector will be stable and continuous, in accordance with the idea of a continuous field of radiation. When the radiation intensity decreases, the output signal fluctuates - statistical fluctuations (they are analyzed in §2.11 "Statistical variance and measurement errors"). If we reduce the intensity of radiation even more significantly, the detector will register time-separated discrete pulses - responses to the passage of individual particles through the detector.
  Another possibility is the detection of radiation using a sufficiently sensitive photographic emulsion
(§2.2 "Photographic detection of ionizing radiation"). At high radiation intensities, the film will be continuously blackened after development according to the degree of total exposure. However, the low intensity of the radiation will not cause a continuously distributed response in the volume of the photographic emulsion, but they will see individual separate traces, as if left by individual flying particles...

Wave properties of particles
So we see that electromagnetic waves can behave like a stream of particles - this is one side of corpuscular-wave dualism. But what about the behavior of (real) particles? According to classical physics, particles behave like discrete "pieces of matter" in all circumstances. However, experiments with the passage of electrons, which are typical particles in atomic physics, through fine grids *) showed that the electrons exhibited bending and interference similar to waves - as if the electron had "bifurcated", passed simultaneously through two adjacent grating holes at the same time, and then after bending, the two components interfered with each other, as is common with waves. Bending interference phenomena were also observed in other types of particles (corpuscular radiation). At the same time, these interference phenomena do not depend on the intensity of the particle flux - the pattern does not change, even if the intensity of the electron flux is so small that one electron passes through the system after another!
*) Davisson-Germer experiment
A suitable structure for performing diffraction and interference experiments in the microworld is a crystal lattice, the nodes of which are individual crystal atoms with typical mutual distances of about 10
-7 cm. Such electron diffraction measurements were first performed by C.J.Davisson, L.H.Germer, and J.J.Thomson in 1927. A beam of electrons accelerated by a voltage of U»50V was routed on the surface of a nickel crystal. The electrons reflected from the surface layer of the crystal atoms were registered by means of a detector which was adjustable at different angles ("goniometer"). It was observed that the electrons did not scatter approximately evenly in all directions, but significantly more scattered in some directions, significantly fewer scattered in other directions, and the minima and maxima of the scattered electrons alternated in the scattering pattern. When measuring the dependence of the intensity of scattered electrons on the direction of scattering, they observed distinct minima and maxima, corresponding to the Bragg condition of interference, the same as in the diffraction of X-rays on a crystal grating - that the path angle of two rays is an integer multiple of the wavelength l: n.l = 2.d.sin J; n = 1,2,3, .. (order of interference maximum), d is the distance between two adjacent nodes of the crystal (lattice constant), J is the scattering angle. The wave radiation scattered on the individual atoms of the crystal lattice is amplified by interference in directions in which the path difference of the waves scattered on the individual atoms is equal to an integer multiple of the wavelength. And the electrons for which the "wavelength" l » h/Ö(2me .e.U) was obtained, behaved in the same way, which corresponds to the so-called Broglie wavelength (accelerating voltage U gives the electrons of charge e and mass me kinetic energy Ek = e.U and momentum p = Ö[2me .e.U], so l = h/p).

Fig.1.1.2. The wave properties of the particles are manifested in the thought experiment of electron diffraction at the slit by the formation of interference patterns.

The essence of these experimental facts is clearly illustrated in Fig.1.1.2 in an imaginary experiment, which generalizes the results of many real experiments. The beam of parallel flying electrons impinges on an impenetrable wall (screen) with two slits, behind which a photographic plate is placed. If the electrons were classical particles with rectilinear propagation, two dark stripes would be displayed on the photographic plate after development as shadow images of both slits (Fig.1.1.2 on the left). In reality, however, we obtain an interference pattern - the alternation of light and darker stripes (Fig.1.1.1 on the right), exactly as it would occur when a plane wave with wavelength l = h/p passes, where p = me.v is the momentum of the electron. The interference pattern does not depend on the intensity of the incident beam, so it is not due to the interaction of electrons in the beam. If we attenuated the flow of electrons to such an extent that there would be only one electron in the system at each pass, each such passed electron would create its local ("spot") blackening on the plate. The resulting image, which is the sum of the spots caused by the individual electrons, would still have the character shown in Fig.1.1.2 on the right. Thus, the phenomenon occurs even when the individual electrons fall on the screen one after the other: as if a single electron passed through both holes of the screen at the same time - as if a wave was connected to each individual electron, which interferes with the two holes. We will return to this experiment briefly below in the paragraph on quantum mechanics, for the essence of which is of key importance.
  The analysis of these experiments (real and imaginary) led to the conclusion that any microparticle of mass m moving at velocity v , can behave as a wave of wavelength
l = h/m.v = h/p, where p is the momentum of the particle (this wavelength is sometimes called the Broglie-Compton wavelength). And this is the other side of corpuscular-wave dualism. The acquired knowledge can be summarized in the following way :

Corpuscular-wave dualism

Waves with a frequency n may act as a stream of particles (quant - photons) of energy E = h. n .
A particle moving with momentum
p can behave like a wave of wavelength l = h / p .

Gateway to Understanding Quantum Physics
The duality between waves and particles can be a "gateway" to understanding quantum physics ! If some contemporary quantum physicists question this, it is a misunderstanding in the spirit of the proverb "Everyone is a general after a battle". Without the exploration of corpuscular-wave dualism, many newer concepts of quantum physics would not have emerged, or would have emerged much later, and would have been difficult to understand. Almost all phenomena in the microworld can be clearly explained with the help of this dualism. Particle-wave dualism substantially alleviates that unpleasant "incomprehensible-incomprehensibility" of quantum physics, discussed below in the section "Interpretation of Quantum Physics".

The quantum nature of the microworld
Classical mechanics, extended and generalized by Einstein's special theory of relativity, together with classical Maxwell's electrodynamics, can explain almost all the phenomena observed in the macroworld of our experience. Classical physics (especially Newtonian mechanics) is a generalization of our everyday experience, according to which material objects exist independently of the observer, have certain positions and velocities, and move along precisely defined paths.
  However, as we learned in the paragraph on corpuscular-wave dualism, and as we well'see even more in the next chapters on atoms, atomic nuclei, nuclear reactions, radioactivity, elementary particles and their interactions, the deeper we go into microworld of the structure of matter, the more the experimental behavior of microsystems differs from the laws of classical physics. To understand and describe the atomic and subatomic processes that take place in very small parts of space and in which particles with very small masses participate, it was necessary to fundamentally change or supplement the basic classical ideas and laws - to build new microworld physics, quantum physics.

Note: This "new physics" cannot be imagined in such a way that it would perhaps refute and destroy "old" classical (non-quantum) physics. In science (and in physics in particular) the continuity of scientific knowledge applies. Quantum physics does not refute, but complements, refines, and generalizes classical physics into phenomena that it is no longer able to explain; it contains classical physics as a limit case. The relationship between classical and quantum physics is formulated as the so-called principle of correspondence: In the limit of large quantum numbers, the difference between quantum and classical physics is blurred, quantum physics becomes classical. Or for large quantum numbers, quantum physics gives the same results as classical physics (will be shown below). Thus, although the atoms and subatomic particles that make up everything are governed by quantum physics, their large arrays of macroscopic bodies (including us) follow classical Newtonian mechanics with great precision. The larger the object, the less clearly its quantum nature...
Randomness and Probability in Quantum Physics
As mentioned above
(passage "Classical and quantum models in the microworld"), the basic specific feature of the microworld is stochastic (probabilistic) character of quantum phenomena. The motion of particles and all other phenomena in the microworld show quantum fluctuations - chaotic variability in the positions of particles and their velocities, field intensities, energy values and other quantities. These physical quantities fluctuate around their mean values; the magnitude of quantum fluctuations is limited by the so-called uncertainty relations mentioned below. During quantum fluctuations, the classical law of conservation of energy and momentum, as well as other classical laws that apply exactly in classical physics, can be violated for a brief moment, or rather interrupted.
  Quantum physics is unable to predict with certainty the specific results of a measurement. It offers only individual alternatives of values and their probabilities. And the state of the studied system will change after the measurement.
  According to quantum physics, the result of a physical process in a given system cannot be accurately and unambiguously predicted - regardless of how exactly we know the initial state of the system and how exactly we can solve the relevant equations of system dynamics. The development of the system, as well as the result of the experiment, cannot be determined unambiguously, there are only a number of different possible results, each of which has a certain probability *). When we repeat a certain experiment many times, the frequency of different results corresponds to the probabilities predicted by quantum physics.
*) A.Einstein metaphorically likened it to the situation, as if God always threw a dice, and only according to the result that falls will he decide how the experiment will turn out. Einstein never came to terms with this idea ...
  Quantum theory thus points to a new form of determinism at a deeper microscopic level: if we know the state of the system at a given moment, the laws of physics do not unambiguously determine the future (or the reconstructed past), but only the probabilities of different futures (or pasts).
  The probabilities and randomities that occur in everyday life are a reflection of inaccuracies in the knowledge of initial conditions and other, often complex, influences. When we shoot an air rifle at a target, shots with different probabilities hit different places on the target around the center, depending on the dexterity of the shooter. This probability is not a property of the shots moving towards the target, but is caused by ignorance and variability of shooting circumstances. If we fastened the airgun in the vice, the shots would fall into one precisely focused place. However, probabilities in quantum physics express a principled randomness, that is inherent in the very nature of phenomena. The microparticles targeted and transmitted under the same initial conditions will always fall to a slightly different location around the center of the target.
  To the seemingly trivial question "Where something is ?" classical science responds in such a way that "Every thing is in one particular place", as our common experience in classical physics based on Newton's foundations shows. In the microworld, it turns out to be different: subatomic particles can be "simultaneously in several places". Their exact coordinates are subject to quantum uncertainty relations (they fluctuate) and obtain their specific values only at the moment when we start measuring them (at the moment of interaction). It's as if something only starts to exist the moment we "look at" it (discussed below in the section "Observations and Measurements in the Microworld") ...

Is quantum stochasticity fundamental, or is it caused by hidden unknown parameters ?
Anyone who begins to learn about quantum mechanics, upon deeper reflection, is surprised by a certain "inappropriateness" of the behavior of particles and systems, contrary to common sense and natural experience. Is this surprising and incomprehensible behavior really fundamental, or are there some other hidden and unknown influences behind it - hidden parameters ? In such a case, standard quantum physics would be an incomplete theory, and its stochastic character would be the result of ignorance of some hidden (yet unavailable to us) parameters that take on different values. The behavior of these parameters should be described by some future more "fundamental" theory, probably within the unitary "theory of everything".
  Let's go back for a moment to the simple example of shooting an air rifle. If we were to shoot in open space at a target approx. 50 meters away, we would observe a relatively large dispersion of hits. If we could not measure the variable speed and direction of the wind, small variations in the size and shape of the shots and other possible influences, we could in principle empirically build a kind of "quantum mechanics of the movement of the missiles" that would show the fundamentally stochastic nature of the trajectory of the shots..?..
  No such additional hidden parameters have been found in the quantum mechanics of microworld particles. System analysis of the possibility of influencing measurable quantities in quantum mechanics by hidden parameters was dealt with by J.Bell in 1964, who established limiting inequalities for the results of measuring the polarization of two photons from a quantum "entangled" pair. Experimental measurements were carried out in the 80s on photon pairs during cascade transitions in calcium atoms with correlated polarizations. The experimental results somewhat exceeded the values of ±2 in Bell's inequalities, which is some indication against the concept of hidden parameters...
  Current quantum physics is generally inclined to the opinion that the quantum state represents a complete description of systems, and quantum stochasticity and uncertainty relations have a fundamental character that cannot be derived from anything..?.. So it seems that randomness is somehow incomporated in the deepest essence of our world ?
  Other unusual phenomena related to quantum entanglement are discussed below in the passage "Quantum entanglement and teleportation. Quantum computers.".

Statistical fluctuations and noise in imaging
The emission of radiation quantum, as well as its interaction with atoms of the material environment (and thus the mechanisms of radiation detection) takes place at the microscopic level through events governed not by deterministic laws of classical physics, but by the laws of quantum mechanics. These quantum regularities are in principle stochastic, probabilistic. The transitions of electrons in atoms or the transformation of radioactive atoms is therefore largely a random process and the resulting radiation is emitted randomly, uncorrelated, incoherently *). Therefore, the radiation flow is not smooth, but fluctuating. The response will be just as fluctuating of any device detecting and displaying this radiation - these are fluctuations which cannot be eliminated by any improvement of the device or method, these fluctuations have their origin in the very essence of the measured phenomena.
*) The fact that LASER emits coherent photons is due to the fact that it is not a spontaneous, but stimulated production of photons.
  Statistical fluctuations (noise) in measurements and images are therefore a general phenomenon. They are also hidden in ordinary light during optical vision and photography, where we do not observe them due to the large number of photons (of the order of 10
9), which are available here. At the top of the image is a photographic portrait exposed with varying numbers of photons of light. We see that if the image consists of less than 103 photons, we do not recognize anything at all in the image except scattered clusters of dots. With the increasing number of photons, the quality of the image improves (above 105 photons we begin to recognize the basic motif) and at about 108 photons we get the usual photographic image with all the details, without noticeable noise.

Influence of registered number of photons on image quality in terms of statistical fluctuations (noise) - image quality improves with increasing number of photons.
Above: Photographic portrait exposed with different number of photons of light
(computer image processing performed by Ing.J.Juryšek) .
Bottom: Gammagraphic image of a phantom
(Jasczak, filled with 99mTc radionuclide) accumulated by a scintillation camera with different numbers of g- photons in the image.

In practice, statistical fluctuations are unfavorably applied wherever we do not have enough number of quanta (photons) displaying radiation. This is especially true for radiation detection, spectrometric and imaging measurements. The influence of statistical fluctuations on the results of these measurements can be expressed simply (but concisely) by the following rule: If we measure N pulses on a radiation detector, we actually measured N ± Ö(N) pulses. These statistical fluctuations are reflected in all cells of the image and the only way to reduce them is to increase the accumulated number of pulses - the number of "useful" photons g, which results in a response in the image. An image is sharp and clear when it is created by at least 1 million photons/cm2. This is difficult to achieve with gamma images, in practice we often have to settle for about 500-1000 photons/cm2. Therefore, scintigraphic images tend to be quite "noisy".
  Statistical fluctuations degrade image quality mainly due to the loss of useful structural details. This can be seen at the bottom of the figure in the images of the phantom modeling "cold" lesions of various sizes in a solution of
g- radionuclide 99mTc. If the image consists of less than about 104 photons g, no structure is seen, only statistical fluctuations. With a larger number of registered photons g of about 105, larger lesions are visible, and only with 107-8 photons, even the finest structures with small lesions are shown.
  In addition to gammagraphic images, statistical fluctuations are also reflected in astronomical images of distant objects, from which very little light falls on us.

The essence and interpretation of quantum physics
A systematic interpretation of quantum physics is outside the thematic scope of this treatise
*) and would also take up an enormous amount of space (reference can be made to standard textbooks and monographs, eg Landau, Lifšic: Quantum Mechanics). We will only take a brief excursion into the ideas and laws of quantum physics to outline some basic common principles applied decisively in processes with atoms, atomic nuclei and elementary particles.
*) After all, fully understanding the essence of quantum laws and internally identifying with them is not at all easy - if not impossible! It is said that the theory of relativity is "understandable - incomprehensible", but quantum physics is "incomprehensible - incomprehensible"..!.. Even the best professional specialists in quantum physics (including its builders such as Bohr, Schrodinger, Pauli, Feynman, Fermi, Dirac, Heisenberg, ...) admit, that they can very accurately analyze the results of quantum processes in atoms, nuclei or in particle interactions, but when someone asks them why it behaves like that, they don't know..

Quantum physics in the micro world - but also in the macro world ?
Quantum physics arose during the study of phenomena in the microworld - at the level of atoms, atomic nuclei, elementary particles. And it also has its main application there... Quantum mechanics, however, claims universal validity not only in the microworld, but also in the macroworld
(even in the megaworld, see e.g. §5.5 "Microphysics and Cosmology. Inflationary Universe." in the book " Gravitation, black holes and the physics of spacetime"). The usual idea is that the application of quantum physics is determined by the size of the investigated objects: small objects behave quantumly, large objects classically. In reality, however, it doesn't always have to be this way. What makes an object quantum or classical is not so much the geometric size, but the number of degrees of freedom available to the object. With a small number of degrees of freedom, the object behaves quantumly. Small objects such as elementary particles (with a simple structure, or without structure) have a small number of degrees of freedom and behave uniquely quantum. Larger objects, composed of a large number of particles, have a large number of degrees of freedom and behave classically because the values of their state variables are averaged over many degrees of freedom.
  Under certain circumstances, even relatively large macroscopic objects, such as a laser beam of coherent photons, a laboratory container with superfluid helium, or a stream of Cooper electron pairs in a superconducting wire can have a small number of degrees of freedom. An extreme case of long-range quantum behavior can be states of entangled particles (see below "
Quantum entanglement and teleportation. Quantum computers."). The specific situation is with quantum processes involving black holes (it is discussed in detail in §4.7 "Quantum radiation and thermodynamics of black holes" of the mentioned monograph).
  For ordinary bodies of macroscopic masses and dimensions, quantum effects are completely tiny and unmeasurable. However, in advanced sensitive experiments, quantum properties can be observed in increasingly large objects (such as macromolecules)...

Wave stochastic description of particles and systems
In classical (non-quantum) physics, the state of a physical system is described using directly measurable quantities, such as positions and momentum of particles. In quantum physics, the state of a particle is described by a wave function
(see below), the square of which indicates the probability that the particle is in a particular state. The wave function is not a directly measurable physical quantity, but rather a model idea. When a particle interacts with another particle, there is a decoherence (loss of mutual spatial and temporal connection of phase and amplitude) of the wave function, leading to the particles acquiring a specific measurable state as in classical physics. However, this is only a probability that we measure a certain value of a state quantity at some place and time. It is not possible to predict in advance which value will be measured in a specific case, it is only possible to determine the probability distribution of the occurrence of different measured values with a larger number of measurements (under the same conditions).
  From a gnoseological point of view, however, it must be kept in mind that all these are only mathematical models, enabling the description and quantification of phenomena in the microworld. Their physical nature is probably hidden somewhere deeper ..?..
  Simultaneously with the building of its own quantum mechanics and its mathematical formalism, have been created several ways of interpretation quantum laws and heuristic methods of creating a chain of conceptual structure of quantum physics. We will usually follow an inductive procedure based on a gradual analysis of experimentally established facts, leading to the so-called Copenhagen interpretation quantum mechanics, developed by a physics group led by Niels Bohr
(we will conceive it physically, without questionable philosophical speculations). During the measurement or interaction, the wave function "collapses" (originally describing the superposition of possible states) - to reduce quantum alternatives. This forces the particle to probabilistically "choose" only one final state and position (which was uncertain until then), as it is in classical physics. During this collapse, the original information is not preserved, which can no longer be detected - the particle is described by a completely new wave function from the moment of measurement or interaction.
  In the passage "
Mystical Quantum Physics? ", we mention H.Everett's somewhat bizarre multi-cosmic interpretation, according to which the measurement or interaction does not collapse the wave function, but creates parallel realities ("universes") in which all possible states of the resulting particles exist; these particles can then no longer interact with each other. At the end of this general part, we will briefly mention Feynman's approach of quantizing "path integrals", which gives some opportunity to understand the internal causes of quantum behavior.

Wave function
Let's go back to corpuscular-wave dualism (Fig.1.1.1 and 1.1.2), which is an important characteristic feature of the quantum understanding of the microworld - it suggests that the division of matter into waves and particles is only formal; in general, we must consider corpuscular and wave properties simultaneously. The particle does not move along a fixed localized path, but as if it "waves" along a blurred path, it behaves like a Broglie wave.
  What is the physical significance of Broglie waves associated with particle motion? The first straightforward notion that the particles themselves are waveforms does not hold up, because in some processes, especially scattering, we could in principle register "parts of the waves" as "fragments" of the particle, contrary to experimentation. Even the opposite idea that waves are formations composed of particles is unsatisfactory
(no particles originating from the wave were observed, only the wave can behave as a quantum with the properties of the particle when interacting). A more adequate idea of the relationship between waves and particle motion can be obtained by studying the diffraction of electrons, which we register on photographic film (Fig.1.1.2). If only a small number of electrons pass, we get an irregular image scattered on the film, but after passing a large number of electrons, we get a smooth regular pattern analogous to the diffraction patterns of light waves. This fact leads to a statistical interpretation of Broglie waves: that the intensity of Broglie waves at any place in space is proportional to the probability of the particle occurring at that place. The classical trajectory of a particle is replaced by a kind of "probability cloud", representing a set of places where a particle occurs with different probabilities.
  In quantum mechanics, the state of a particle (or a set of particles and generally every physical system) is described by the so-called wave function
y(x, y, z) (in the simplest case of an isolated particle, this wave function is identical to the Broglie wave). The physical meaning of the wave function is that the square of the modulus of the wave function ú yú 2 determines the probability dW that the particle at a given time t is in the element of volume dV = around the point (x, y, z): dW = ú yú 2 And the mean value of any physical quantity F(x, y, z), which is a function of the coordinates x, y, z, is then given by the relation ` F(x, y, z) =ò F(x, y, z).ú yú 2, where it integrates over the whole field of variables x, y, z.
Note: The wave function y is generally introduced as a complex function (containing both real and imaginary components), so the square of the modulus ú yú2 = y . y*, where y* is a complex associated function by y. For the simplest case of a free particle moving in the direction of the x- axis with momentum px , the wave function is written in the form y = exp[- i/h (E.t - px.x)], representing a plane harmonic wave.

Observation and measurement in the microworld
Things can be observed without disturbing them" - this is an experience of everyday life, especially from visual observation by an "uninvolved observer". However, the process of observation or measurement *) in the microworld differs diametrically in its nature and consequences from the processes of measurement and observation in classical physics describing the macroworld.
*) The terms "observation" and "measurement" are often not distinguished: quantitative observation is a measurement.
  In the physics of classical systems of the macroworld, it is tacitly assumed that the process of observation (measurement) does not disrupt significantly their movement or evolution. The relevant physical quantities can be measured with sufficient accuracy without disturbing their values or disturbing the development of the observed system. Or we assume that any failure caused by the measurement can be accurately corrected, at least in principle.

  E.g. when measuring the voltage in the electrical circuit, we use either a voltmeter with a sufficiently large input resistance, which practically does not affect the measured value, or if this is not possible, we can know the impedances in the circuit and the voltmeter to accurately correct the voltage change. However, experienced electronics experts know that when measuring extremely weak electrical signals (in which only a few hundred electrons participate), unremovable noise and fluctuations are applied, and all correction methods already fail here.

  The most common way to examine the position of an object is to visually observe it: we illuminate the observed object with light
(unless it is itself a source of light) and our eyes register reflected photons of light. If the observed body has a macroscopic size and mass (such as an apple or a stone), the incident and reflected photons of light do not appreciably affect the position of the body and the basic premise of a "non-participating observer" is met. However, if the body is of microscopic size and mass, the impact of each photon can significantly affect its position and velocity - all the more so the more accurately we try to determine the position (accurate measurement of the position of a particle can completely "throw off" its momentum!). For a more accurate localization of the particle position, the wavelength of the irradiating wave must be short enough, ie the energy and momentum of the quanta is correspondingly higher - it causes a more appreciable disturbance of the observed system (position and velocity of the particle).
  Here we will no longer observe directly with the eyes, but through an instrument (eg microscope incl. electron microscope, particle detector, radiometer, spectrometer), while for observation of particles of very small dimensions it is necessary to use radiation with a correspondingly short effective wavelength. The subjective role of the observer is often overestimated in connection with quantum mechanics (it comes from the time of the origin of quantum physics) and sometimes even the objective reality as such is questioned. This is a misunderstanding! Natural processes with innumerable interactions of particles and fields are constantly taking place in nature and their results are independent of us. Only our occasional probes into the events of the microworld are burdened by fundamental quantum uncertainties. However, this is not a consequence of our subjective intervention as a conscious observer, but the influence of the interaction of objectively existing particles with sensors and instruments used for observation or measurement (it is briefly discussed below in the section "Mystical Quantum Physics?").
  The oft-cited claim that "observation creates reality" is misleading and erroneous: this reality has been there already before, the observation has only "illuminated" it for us - and may have changed it, due to quantum influence at the microscopic level.
  So in order to "observe" and measure a microparticle, we must let it bounce off it some other particle or quantum of radiation and observe only the result of this reflection - more generally the result of the interaction. The inevitable consequence of such a process is that the collision or interaction irretrievably changes the state of the monitored particle - it deviates it, changes its velocity, or even the internal structure. In general: in order to observe an object or system, we must interact with it. In this sense, reality in the microworld can be influenced by mere "observation" !
  Thus, the operations (processes) of observation or measurement necessarily affect the physical system (disrupt its evolution), while for small systems
*) this disruption is considerable and irreversible, it has a principled character and cannot be eliminated or corrected in any way; and by no improved method - it is the very essence of things and processes themselves! Quantum mechanics deals with the behavior of such systems and the processes of measuring their physical quantities.
*) In the microworld, the term "small" loses its usual relative character and becomes an objective absolute attribute determining the quantum behavior of a given system.
Mystical quantum physics ?
Quantum physics is from a philosophically point of view often falsely interpreted. From the above fact that reality in the (micro) world can be influenced by observation, mystical claims are drawn that "the human mind creates reality", or "quantum mechanics connects the human mind with the universe", or "quantum physics creates the unity of human and cosmic consciousness", is the essence of "freedom of will" and the like. The basic mistake or misunderstanding here again results from the above-mentioned overestimation and misunderstanding of the role of the subjective "conscious" observer. It is not our mind that makes observations, but the very interactions of the basic particles (objectively existing). They change the quantum state of systems and transmit information, our mind only registers and processes it. In fact, it is an objective reality, whose laws of which (including quantum ones) determine the behavior of our minds and the functioning of the entire universe. This approach corresponds to all the results of our observations so far, it is fully consistent with them.
  The finding that causality and determinism are disrupted in some phenomena in the microworld led to a hypothetical connection between quantum physics and free will. We have a certain legitimate sense of free will - that we can decide what we will do today or what we will plan for tomorrow; that it is mainly up to us, it is not only external circumstances that decide. However, from the point of view of science, real free will is only an illusion. Above all, we cannot do something that is contrary to the laws of nature; and other things prevented by certain other circumstances... "Freedom of will" emerges from an inexhaustible number of interactions and their results, we do not need quantum physics for it (see also the discussion in the section "Determinism-chance-chaos?" §3.3 in the book "Gravity, black holes and space-time physics") ..?..
Infinitely many parallel universes ? 
The stochasticity of the behavior of particles and systems according to quantum physics also gave rise to H.Everett 's somewhat fantastic hypothesis of an infinite number of universes : with each attempt to discover reality - with every interaction of particles - the whole universe splits, branches or "duplicates" into two or more "universes" in which the individual possible results of the interaction take place
(with the appropriate probability). This creates a constantly infinite number of "parallel worlds", in which all possible alternative futures (and pasts) are "real"..?..
  In some such universes, for example, an asteroid would have missed our planet 60 million years ago, and dynosaurs would still rule the Earth (or even create a civilization instead of humans) ..?..
  However, available for us (ie "real") is only the the universe in which we are at the moment; observations can only be made in "our world". Alternative events in parallel universes can only be imagined ...
  There have also been sci-fi hypotheses that parallel universes can be influenced at the quantum micro-level (and therefore also affect our world). Their quantum (particles) can "seep" between individual universes and evoke some "bizarre" effects of quantum mechanics..?.. They are all just unsubstantiated assumptions ...
  On the astronomical and philosophical context of several universes - multiverse - see, for example, the work "Anthropic Principle or Cosmic God", or §5.7 "Anthropic Principle and Existence of Multiple Universes" monograph "Gravity, Black Holes and the Physics of Spacetime".
  In the microworld, the order of measurements is important. E.g. on whether we first measure the position of the particle (thereby disturbing the momentum) and then only its existing momentum, or vice versa (by measuring the momentum we first disturb the position). The more accurately we measure the position, the less accurately we know the momentum of the particle - and vice versa. This leads to the principle non-commutativity of quantum mechanics, expressed in quantum uncertainty relations (see below).
  The term "state" of a physical system means a situation where this system is in a configuration (state) with a certain value of a given physical quantity. In classical physics, for example, the state of a particle is described by entering the position and velocity resp. momentum (as a function of time). In quantum physics, the situation is more complicated, the so-called state vector, denoted by
|y>, is introduced here, where y only symbolically denotes a state quantity, which, however, does not have a certain value, but can be a superposition of several states. E.g. the electron in terms of spin (see below) may be in the spin state |1> with spin-oriented "up" (z-component spin projection has a value of + 1/2 h ), or in the state |2> with spin - 1/2 h. However, it can also be in a more general state |y>, which is a "mixed" superposition of "pure" states |1> and |2>, which is written in vector: |y> = a1 . |1> + a2 . |2>. This state |y> means that the probability a12 we measure a value of + 1/2 h and the probability of a22 we can measure the value of - 1/2 h. In general, a superimposed state can be made up of multiple components: |y> = a1.|1> + a2.|2> + ... + ai.|i> + ... General |y> is therefore a state where a given physical quantity does not have a certain value, but only the probability ai2 measurements of individual potential values "i". Quantum physics is a mathematical algorithm (computational scheme) that can determine these coefficients ai - but does not specify the internal physical nature of these phenomena. In the state |y>, before the measurement, the system has the value of the given physical quantity indeterminate (potentially different values are possible) and only the measurement concretizes this value. Even under the same initial conditions, we always measure different values of physical quantities, statistically divided around the mean values determined by probability coefficients ai2.

Operators. Uncertainty relation.
Observation or measurement operations are modeled in quantum mechanics using so-called operators. Each physical quantity A is assigned in quantum mechanics the operator A
^, which satisfies certain mathematical conditions (it is linear and Hermitian). By ^A we mean a rule that assigns to each function u(x) some other function v(x) - symbolically we write v = ^A u. The operator ^x assigned to the coordinate x is a simple multiplication of x, while the momentum operator ^p is given by the derivative according to x coordinates :
^x ® x, ^p ® - i h . / x.
  The Planck constant h got here to hold the relationship between the momentum of a particle and the corresponding frequency of the Broglie wave in corpuscular-wave dualism. Other physical quantities - energy and momentum - will be discussed below.
  For operators in quantum mechanics, it is important that the sequential application of two operators does not have to be commutative, ie it may depend on the order. For two operators ^A and ^B, the so-called commutator is defined by the relation [^A, ^B] = ^A ^B -  ^B ^A, i.e. the difference between the application of the operator ^A and then ^B, minus the same operators applied in reverse order. This difference is generally not zero as in classical physics, as each observation (measurement) in the microworld can cause a system violation and thus affect the result of the second observation (measurement), so that the two procedures can provide different results. The coordinate and momentum operators satisfy an important commutation relation [^x, ^p] = i . h .
  This commutation relation is related to the key principle of quantum mechanics, the so-called Heisenberg quantum uncertainty principle, which states that the position x and momentum p particles cannot be determined exactly at the same time *), but that the uncertainties of these two (complementary) quantities are given by the relation Dx . Dp ³ h. Every measurement of a particle's position irreversibly perturbs its momentum - and vice versa. Therefore, it is not possible to investigate the exact specific path, along which the particle moves...
*) Quantum "blur", implied by uncertainty relations, is mostly negligible and unobservable in the macroscopic world, but on an atomic and subatomic scale it becomes absolutely decisive!
  This quantum uncertainty is an expression of the basic property of observation and measurement: that it is always an interaction, irreversibly affecting the parameters of the measured system. The same uncertainty applies sessions between other dynamically coupled quantities, eg. between time t and energy E : DE . Dt ³ h , further between potential and kinetic energy, etc. This complementarity, whose "prototype" is corpuscular-wave dualism, is characteristic of quantum physics.

Characteristic equations. Discrete values of physical quantities.
When applying operators to wave functions, are especially important cases where the result of the operator
^A applied to the function y(x) results again are the same function y(x) multiplied by a certain number a : ^A y(x) = a. y(x). In general, to each operator ^A belongs a set of numbers an and a set of functions yn, for which the so-called characteristic equation applies
^A yn (x) = a n . y n (x) .
Numbers (coefficients) a
n are called proper (characteristic) values and yn corresponding own (characteristic) function of the operator ^A. Eigenvalues an of operator ^A then represent possible values, which may be a physical quantity a corresponding operator ^A take. Said equation is a differential equation for the wave function of the state in which the quantity represented by the operator ^A has the value a. Eigenvalues satisfying this equation generally do not take on all possible values, but only certain ones discrete values, in accordance with experimental knowledge about discrete (quantum) values of physical quantities in the microworld - energy of atoms, magnetic moments, spins... It turns out that energetically (field) bound particles in the microworld belong to discrete values of energy, momentum and other quantities - we call them quantum physical quantities. These discrete characteristic values, expressed as multiples of their respective elementary value (usually Planck's constant h), are called quantum numbers.

Quantum energy. Schrödinger equation.
Similarly to classical mechanics, also in quantum mechanics the key concept is energy E. Energy E (consisting of potential energy U and kinetic energy T: E = T + U) is assigned in quantum mechanics an energy operator called Hamilton's operator, which for the simplest the case of a particle of mass m has the form
^ H = - ( h 2 / 2m). D + U,
D s 2/x2 + 2/y2 + 2/z2 is the so-called Laplace differential operator. The proper (characteristic) equation of the Hamiltonian operator
^ H y n  =   E n . y n
is called the stationary Schrödinger equation. Its solution for a particle is the wave functions of the stationary states of the particle in the potential field, in which the particle acquires discrete energy values E
n (for continuous and discrete energy values, see the note below).   
  The time evolution (motion) of the quantum state of a microparticle is then described by the nonstationary Schrödinger equation
^ H y   =   i h . ¶y / t  ,
which contains the time derivative of the wave function.   
  Solutions of the stationary Schrödinger equation indicates what possible stationary physical states might particles in a given force field to acquire, via nonstationary Schrödinger equation can in principle determine the probability with which the particles pass from one quantum state to another. It can be said that Schrödinger's equation has a similar position in quantum mechanics as Newton's laws in classical mechanics. Among other things, all the quantum properties of the structure of atoms flow from it, which will be discussed below (especially discrete energy levels).
Continuous and discrete energy - quantized
In classical physics, energy can acquire all possible values continuously; by doing work, the energy of bodies in a certain system can be changed arbitrarily. In quantum physics, the situation is more complicated. The energy values are the solution of the (stationary) Schrödinger equation given above. In the simplest case of a free particle (U = 0), this equation has the form ( h2/2m). Dy + E.y = 0 and its solution are wave functions of the form y = const. e i/h(E.t-p.r), for any energy values E, where E = p2/2m. Each such function (plane wave) describes a state in which a particle acquires a certain value of energy E and momentum p , where the frequency of such a wave is E/h and the wavelength l = 2ph/p is the Broglie wavelength of the particle. The energy spectrum of a free-moving particle is therefore continuous, the energy can take values from 0 to ¥ - the energy of a free particle is not quantized.
  If the particle is in the potential field U(x, y, z), then the motion of the bound particle with energy E <0 has a discrete spectrum energy levels, while for positive energies the particle is not bound and its energy can take on a continuous spectrum. A typical model case of quantum motion of a bound particle is the motion of a particle in a potential well - in the simplest case a one-dimensional motion bound to a line of length L between two perpendicular walls (infinitely high), from which the particle reflects perfectly elastically. Such a line-segment motion of the particle corresponds the Broglie wave, which is reflected on the walls, while the superposition of the waves reflected from both walls creates a "standing wave". Thus, an integral number of half-waves of standing Broglie waves is formed on the line segment of length L, ie L = n.
l/2, where n = 1,2,3, ... The motion of a particle in a potential well therefore correspond only to certain discrete values of the wavelengths of Broglie waves ln = 2.L/n , n=1,2,3,... The Broglie wavelength is related to the momentum of the particle, l = h/p, so that the momentum of the bound particle pn = h/ln = n.h/L and its energy En = pn2/2m = n2 .h2/8m.L2 will have discrete values *). The state of a particle in a potential field, which corresponds to a standing Broglie wave ln , represents a certain stationary state of the particle. It is a state with a certain energy En - energy level of particles in a potential field in a given steady state. The number n is then called the quantum number of this steady state. The state corresponding to n = 1 is called the ground state and corresponds to the lowest energy level of the particle bound in the potential field. The change in the energy of a particle is associated with the transition (jump) to another stationary state, which is accompanied by the emission or absorption of a quantum (photon) with energy equal to the difference of energies of both stationary states (energy levels). These laws find their application below in Bohr's model of the atom.
) At large values of quantum numbers n, the energy differences of individual quantum states with quantum numbers n+1 and n are small - the ratio En+1/En = [(n+1)2-n2]/n2 is close to 1. Thus, the energy changes at individual higher quantum levels energies are negligible - energy can be considered continuous here; the results of quantum mechanics at higher quantum numbers basically correspond to the results of classical mechanics - the principle of correspondence .
  The actual energy spectrum of particles in the microworld can be discrete or continuous, depending on the process by which the particles form, gain energy and are emitted. The continuous energy spectrum has eg braking radiation, radiation
b, Compton scattered radiation g. Other spectra are discrete, quantized, line - eg radiation a and g, radiation spectra of excited atoms, characteristic X-rays, conversion or Auger electrons. A certain intermediate species between continuous and discrete spectra are band spectra, where individual quantum states are separated only by very small energy intervals and the resulting spectrum appears to be continuous within the resolution of spectrometric instruments. The specific mechanisms of particle emission and energy recovery will be discussed in more detail in the description of the radiation of atoms and atomic nuclei in radioactivity and other processes.

Permitted and forbidden transitions
From the point of view of classical physics, only such events can take place in which the laws of conservation of energy, momentum, angular momentum are fulfilled at all stages. Of course, such processes, called permissible transitions, can also take place according to quantum physics in the microworld. They take place very "willingly", their speed (effective duration of transition, transformation, interaction) depends on the type of force that causes it. The fastest are transitions caused by strong interaction, followed by electromagnetic processes and finally transformations caused by weak interaction.
  However, some such events can take place in the microworld, in which these classical laws of conservation are violated at some stages - the so-called forbidden transitions. We can simply imagine that a particle
(within its permanent quantum oscillations and fluctuations) constantly "tries" it again and again until it manages to "break through" the barrier of prohibition. The wave function of the particle is spread in the phase space and can partially interfere with areas where the transition can "bypass" the violation of the law of conservation. Prohibited transitions can in principle take place, but with less probability. A typical case is the tunneling phenomenon of the passage of a particle through an energy barrier described below, or forbidden transitions between the energy levels of electrons in the envelope (see "Excitation and radiation of atoms" below) or between the energy levels of nucleons in the nucleus (§1.2, part "Radioactivity gamma", passage "Nuclear isomerism and metastability") due to higher differences in angular momentum (multipolarity) than the emitted photon is able to carry.

Quantum angular momentum. Spin. Magnetic moment.
One of the important physical characteristics of the motion of material bodies in space is the angular momentum - a vector quantity quantifying mainly the rotational motion of bodies. The law of conservation of angular momentum *) provides a number of useful data on the properties of motion.
*) The law of conservation of angular momentum is a consequence of the invariance of physical laws (Hamiltonian of system) to spatial rotation by any angle - isotropy of space. This property applies not only in free space without fields, but also when moving in a centrally symmetric field, where, however, the invariance to rotation applies to rotation around the center of the field. The angular momentum therefore plays an important role in observation the motion of planets and in analyzing the motion of electrons around the atomic nucleus in its central field.
  The angular momentum of a particle (mass point) in classical mechanics is a vector quantity L, which is defined as the vector product of the position vector r and the momentum vector p : L = [r
x p], or in components in the direction of the x, y, z axis: Lx= y.pz -, Ly= z.px - x.pz, Lz= - y.px. By replacing the components of coordinates and momentum with the above mentioned operators, we obtain the operators of the angular momentum components: ^Lx= h/i (y./z - z./y), ^Ly= h/i (z./x - x./z), ^Lz= h/i (x./y - y./x). In vector it can be written ^ L = [^r x ^p] = -i h [r x Ñ], where Ñ is the vector form of the Hamiltonian differential operator. The characteristic equation for the angular momentum is customary (without prejudice to generality) to investigate for the component z : ^Lz y = lz . y, in spherical coordinates r, J, j. Her solution is (we cannot break down mathematical details here) : y = f(r,J). ei. lz.j, where f(r, J) is an arbitrary function of the radius r and the angle J. In order for the characteristic function y to be unambiguous, it must be periodic with respect to j with a period of 2p, so it must be:
z  =  m. h   ,  where m = 0, ± 1, ± 2, ....
The eigenvalues of the angular momentum l
z are therefore quantized- can be equal to positive and negative integer multiples of the Planck constant h including zero. This result is important in that it quantum-mechanically justifies Bohr's basic postulate model of the atom, which is discussed below ("Bohr's model of the atom").
  In addition to the components of the angular momentum L, its absolute magnitude L º |L| = Ö(L2) is also important in mechanics. The characteristic values K of the angular momentum square are determined by the equation ^L2 y = K.y. A relatively complex and lengthy mathematical analysis (again using here e.g. the requirement of uniqueness of the characteristic function y leading to the periodicity 2p) can be used to obtain the formula for the characteristic values of the square of the angular momentum
          K = 
h 2 . l (l + 1)  ,  l = 0, 1, 2, ...
Characteristic values of the operator of the absolute magnitude of the angular momentum |L| then they are :
          | L | = 
h . Ö [l (l + 1)]  ,  l = 0, 1, 2, ...
At a given value of the number L, the angular momentum component L
z can take the values Lz = L, L-1, L-2, ..., 0, -1, ..., -L, ie a total of 2.L + 1 of different values, corresponding to different orientations of the angular momentum in space. All these rules apply, among other things, in the structure of the electron shell of the atom, where the energy level corresponding to the angular momentum L is (2.L+1)-times degenerate; in conjunction with Pauli's principle, this implies occupancy rules for electron levels, as described below.
S p i n
In classical mechanics, in addition to the mutual angular momentum of moving bodies, or the angular momentum of a body with respect to a given point, is also applied the own (internal) angular momentum caused by rotation bodies around its own axis. In quantum mechanics, the angular momentum determines the symmetry of the state of the system with respect to rotation in space, ie the way in which wave functions corresponding to different values of the angular momentum projection are mutually transformed when the coordinate system is rotated. The origin of the angular momentum no longer matters here. The analysis of the properties of particles shows that in quantum mechanics we must also attribute a certain intrinsic angular momentum to an elementary particle, which is not related to its motion in space. Own angular momentum of a particle is called spin and is denoted by s, while the angular momentum associated with a particle's motion in space is called the orbital moment (usually denoted by L or l ). This property of elementary particles has a specific quantum nature and cannot be completely explained by classical mechanical concepts (spin cannot be quantitatively explained, for example, by the rotation of a particle around its own axis!) *). In the quantum description of a particle with spin, the wave function must determine not only the probability of different positions of its occurrence in space, but also the probability of different spin orientations. Wave function must therefore depend not only on the three spatial coordinates, but also on the spin variable that indicates the value of the projection of the spins in a particular direction in space (selected axis z ), and acquires a limited number of discrete values.
Difference of spin from angular momentum
The spin of quantum particles differs considerably in some of their properties from the usual orbital or intrinsic rotational angular momentum of bodies :
-> It is quantized, takes on certain discrete values (mentioned below); however, this is generally consistent with quantum physics (so it's not surprising).
-> For a given type of elementary particle, spin has a precisely given value (in multiples of the Planck constant); the particle cannot be "forced" to rotate faster or slower. The spin value depends only on the type of particle and cannot be changed in any way (unlike the orbital angular momentum or spin direction) ...
-> The spin of quantum particles can take values 0, 1/2, 1, 3/2, 2, ... - a multiple of the Planck constant. In contrast, the orbital angular momentum can only take on the integer value of a multiple of the Planck constant.
  Like the angular momentum in general, spin is quantized. The eigenvalues of the square of the spin are equal s2 = h2. s(s + 1), where the spin number s can be an integer (including zero) or a half number; is an intrinsic characteristic of a given type of particle. At a given s, the spin projection can take values sz = -s, -s + 1, ...., s-1, s, so a total of 2.s+1 values. In §1.5 "Elementary particles", passage "Indistinguishability of particles" - "Spin, symmetry of the wave function and statistical behavior of particles", we will see that there are two main groups of particles according to spin s : particles with half-number spin (most of them - electrons, protons, neutrons, muons, etc.) and with integer spin (photons, p and K mesons, hypothetical gravitons and others). This circumstance is closely related to the quantum behavior of sets of particles - the particles behave like fermions or bosons (§1.5, passage "Fermios-Bosons").
Magnetic moment
Every electrically charged body
("charge", "charged particle") generates an electric field in the surrounding space according to Coulomb's law (if the charged particle is at rest with respect to the reference system, it is an electrostatic field). When the charged body moves evenly in a straight line, it also generates a magnetic field according to Biot-Savart-Laplace's law. And if the charge moves unevenly - accelerated or along a curved path, it excites a time-varying electromagnetic field around itself, part of which propagates through space as electromagnetic waves. These basic findings of the unified science of electricity and magnetism - electrodynamics (see eg §1.5 "Electromagnetic Field. Maxwell's Equations." in the book "Gravity, Black Holes, and the Physics of Spacetime") work perfectly not only in classical, but also in relativistic and quantum physics.
  Leaving aside the translational motion (irrelevant here) and the emission of waves (which we will discuss below), the main mechanism of excitation of a magnetic field by charged particles is their rotational motion. The motion of a charged particle in a circular orbit generates a magnetic field, the direction of which is perpendicular to the plane of circulation and whose intensity (magnetic induction) is proportional to the charge of the particle and the angular momentum of its circulation. This magnetic field behaves like a fictitious magnetic dipole- miniature "bar magnet". Its force is quantified by the vector quantity magnetic moment m, expressing the moment of a pair of forces f, which would act on this magnetic dipole in the external homogeneous magnetic field B : f = [m x B]. A particle with charge q and rest mass m , which rotates with the angular momentum L, generates a magnetic dipole moment m = (q/2m).L according to classical electrodynamics. When exciting a magnetic field by the rotational motion of charged bodies, the so-called gyromagnetic ratio g is often introduced, which is the ratio of the excited magnetic moment and the mechanical angular momentum of the rotating body. For a classically charged rotating body, g = q/2m.
  Rotational motion of a charged particle excitation magnetic moment, can be of two kinds :
  - Orbital (circular) motion of charged particles in the field of bonding forces with other particles. This is the case of electrons orbiting an atomic nucleus. The electron of rest mass m
e , orbiting with angular momentum L, behaves like a magnetic dipole with moment m = g . (- e/me). L. However, it is usually expressed in the form m = -g . mB .L/h, where mB = e. h /me is the so-called Bohr magneton. The dimensionless correction factor g indicates the relationship between the actually observed magnetic moment of the particle and the theoretical value of the Bohr magneton.
  - The proper rotational motion of a charged particle, rotating around its axis - the above-mentioned spin of particle. The above classical relation
m = (q/2m).L in principle applies to the excitation of a magnetic moment even if the angular momentum L it is created by the rotation of a body with an equally distributed density of mass and electric charge around its own axis of symmetry. The spin magnetic moment of an electron can be expressed as: ms = -gs.mB . S/h, where S is the spin momentum of the electron (+, - 1/2 h), the g- factor is approximately equal to 2.
  The  more complicated situation is with nucleons - protons and neutrons. By a straight analogy with electrons, we would obtain the relation mp = gp.mN . S/h for the magnetic moment of the proton, where mN is the so-called nuclear magneton mN = e. h/mp. However, the correction factor g here has a relatively high value of gp = 5.58; this suggests that the determination of the magnetic moment of a proton based on its spin is problematic (see below). Thus, the proton has a magnetic moment mp = 1,41.10-26 J/T and a gyromagnetic ratio gp = 2,675.108 rad.s-1.T-1. The gyromagnetic ratio also indicates the frequency Larmor precession of the magnetic moment of particles in an external magnetic field; for protons, the Larmor frequency is 42.577 MHz/T - nuclear magnetic resonance is used in the analytical and imaging method (see §3.4, section "Nuclear magnetic resonance"). A neutron, as an electrically neutral (uncharged) particle, should have no magnetic moment, it should be mn = 0. In reality, however, a neutron has a non-zero magnetic moment mn = -0,97.10-26 J/T, which is only slightly smaller than that of a proton (and has the opposite sign). How is it possible?
  The origin of the magnetic moment of nucleons does not lie in their rotational angular momentum (spin), but comes from their internal structure - that they are composed of quarks "u" and "d" (§1.5, passage "Quark structure of hadrons"). For hypothetical or model quarks, the magnetic moment m"u" and m"d" is assumed, which in the first approximation can be modeled in a similar way as a nuclear magneton: mq = qq . h/mq. The magnetic moment of a nucleon can then be considered to be composed of the vector sum of the magnetic moments of three charged quarks and the orbital magnetic moments caused by the motion of these charged quarks in the nucleon. With quark model magnetic moment of the proton (composed of two quark "u" of charge +2/3e and 1 quark "d" having a charge of -(1/3)e) we can then be approximated as mp = 4/3m"u"-1/3m"d" = 2,8.mN = 1,41.10-26 J/T. And the magnetic moment of a neutron (composed of 2 quarks "d" with charges -1 / 3e and 1 quark "u" with charge +2/3e) then it is mn = 4/3m"d"-1/3m"u" = -1,9.mN = -0,97.10-26 J/T. Quantum chromodynamics seeks a more detailed analysis, including gluon fields and virtual particles inside nucleons (not yet completed).

Quantum field theory
We have so far dealt with the quantum behavior of the microworld from the point of view of quantum mechanics of microparticle motion: probability waves (forming fields) are assigned to mass particles and specific quantum properties of particle motion are obtained by solving relevant wave equations, including discrete values of energy and other physical quantities. Quantization of this kind is sometimes referred to as "primary".
  In addition to particles, the main subject of the scientific description is the physical field. The physical field, which carries energy, momentum and other physical parameters, as well as particles, must also have a quantum character in the microworld. In the quantum description of fields, sometimes called secondary quantization, on the other hand, the field is expressed using particles - quantum excitations in the field. The transition from classical to quantum field theory consists of two basic stages :

  1. The field equations are converted to a wave equation, so that the field in a given region of space can be expressed as a superposition of plane waves. Thus, the field is described by a discrete series of variables - amplitudes and frequencies of waves, the so-called harmonic oscillators of the field. 
  2. In the quantization procedure itself, discrete quanta of energy are assigned to the field, corresponding according to the laws of quantum mechanics to the individual possible energy states of the harmonic oscillators on which the field was decomposed. At the same time, this quantum of energy is considered to be those particles, that mediate the interaction of the sources of a given field.

  The application of this method of quantization to the electromagnetic field is the basis of quantum electrodynamics (QED) and leads to the idea of the electromagnetic field as a set of particles - photons, each of which has energy h.w and momentum h.w/c; the rest mass of photons is zero, their spin (intrinsic angular momentum) is equal to 1 (resp. 1.h). At the same time, these electromagnetic quanta (photons) are interpreted as particles mediating the interaction of electrically charged particles. Radiation and absorption of photons by electric charges (especially electrons) is expressed by means of so-called creation and annihilation operators, that generate or take photons in a certain energy state in an electromagnetic field.
New - quantum concept of force: intermediate exchangeable particles
In classical physics, each kind of interaction of bodies is assigned a corresponding field - a space in which certain forces act on particles. In classical physics, it is an electric, magnetic, gravitational field. The magnitude of the field action at each point in space is expressed by the field intensity (force acting on the "unit test particle") or by its potential 
(work associated with the transfer of particles to a given place). The changes ("commotion") in this field propagate at a finite speed from place to place, which is accompanied by the transfer of energy, momentum, and other physical quantities. From the point of view of classical physics, these quantities, such as energy and momentum, are transmitted continuously during field changes. In quantum physics, it turns out that during changes (disturbances) in the field, physical quantities are transmitted discontinuously over certain "portions" - quanta.
  Quantum field theory, in its concept of secondary quantization, leads to a new concept of the field as a set of particles - quantum fields. And the interaction of particles (interactions) is caused not by the field force, but by the mutual exchange of these quantum field particles - the exchange of intermediate particles. The particles are constantly receiving and emitting quantums of fields, which causes them to interact with each other. This exchange mediating (intermediate) quantums, we interpret as a quantum particles - carriers of interactions. This introduces a new concept of force and interaction in quantum field theory. This concept plays a key role in the interactions of "elementary" particles
- it is discussed in more detail in §1.5 "Elementary particles and accelerators", section "Interactions of elementary particles".
Virtual or real particles ?
Are these intermediate particles mediating the interaction real? The answer is yes and no ! Let's briefly discuss this problem in the electromagnetic field - quantum electrodynamics. According to it (as outlined above), the photons are quantum of the electromagnetic field, and the electric force between two charged particles is caused by the constant exchange of photons. However, if we looked at the space between two stationary charges, we would not register any flow of flying photons. It's just a model, those intermediate photons are virtual here ! In quantum electrodynamics, the force is only modeled by using photons: the static field is artificially decomposed into a superposition of waves (harmonic oscillators), these are quantized and the resulting photons are designated as the quantum of the field that mediates the interaction. Rather physically substantiated is the claim that "photons are a quantum of the electromagnetic wave", not than "quantum of the electromagnetic field". In the static case, nothing is radiated physically! The actual radiation associated with the transfer of energy and momentum - with the flow of photons - occurs only in the dynamic case - during the accelerated motion of the charges. Then the virtual photons turn into real ones. The mutual interactions (collisions, scattering) of particles in the microworld are always dynamic processes (often at high energies), in which the virtual intermediate particles, hidden in the vacuum are "liberated", transformed into real particles and actively participate in the interaction.
Note: An interesting exception to the radiation of virtual particles even in the static case is the so-called Hawking radiation of quantum evaporation of a black hole. It is created
that one of the two particles is pulled below the horizon of the black hole and absorbed and the other particle thus becomes a real particle and can be emitted (it is analyzed in detail in §4.7 "Quantum radiation and thermodynamics of black holes", passage "Mechanism of quantum evaporation" monograph "Gravity, black holes and space - time physics").

Quantum fluctuations of fields
One basic postulate
of quantum mechanics is known Heisenberg uncertainty principle Dx. Dp ³ h , where h º h/2p @ 1,05.10 -27 g cm3/s is the Planck constant. The relation of uncertainty applies not only between position and momentum in quantum mechanics, but between every two dynamically coupled -conjugated quantities, ie also in quantum field theory. If we observe, for example, a magnetic field in a small spatial region characterized by the dimension L, there will be energy proportional to B2.L3 and the time required to measure the field will be L/c; uncertainty relation DE. Dt ³ h then gives (DB)2.L4 l h.c, or DB l hc/L2. Thus, it can be said that the quantum fluctuations of the electromagnetic field in size L are of equal order: DE ~ DB ~ Ö(h.c) / L2.
Thus, the field is constantly "oscillating" between configurations whose fluctuation range is greater the smaller the spatial areas we observe. The influence of these quantum fluctuations on the motion of an electron around the atomic nucleus (these quantum fluctuations "overlap" over Broglie waves in Bohr's model of the atom - see the passage "Bohr's model of the atom", Fig.1.1.6) is schematically shown in the figure :

Schematic representation of the motion of an electron around an atomic nucleus. A closer look at the Kepler trajectory of the electron would reveal small chaotic irregularities caused by quantum fluctuations in the electric field. The mean deviation from the global trajectory is zero, but the standard deviation leads to a small shift in the energy level of the electron. This shift was actually measured as part of the so-called Lamb-Rutheford shift.

In the above-mentioned quantum field description - secondary quantization - these quantum field fluctuations can be considered as quantum - particles. The vacuum thus becomes a highly dynamic environment in which virtual particles are constantly created and destroyed. These particles have an immeasurably short duration, they are undetectable, we say they are virtual. In addition, quantum fluctuations and virtual particles arise everywhere with the same density and impinge on matter from all directions, so that their force effects are balanced and canceled on a macroscopic scale. Under certain circumstances, however, the cumulative effects of a large number of these particles may still have a slight macroscopic effect.
  In addition to the afore mentioned Lamb shift we managed to measure the so-called Casimir effect :
We place two horizontal plates (electrically uncharged) right next to each other in a vacuum. In the gap between the boards may arise quantums - virtual particles with only short wavelengths
(whose integer multiple is given by the width of the gap), while in the space outside the plates they can take on any wavelengths. The total density of the particles is thus lower in the gap and the pressure of the particles from the outside prevails on the plates. The plates are thus attracted to each other by a force which is greater the narrower the gap between the plates.
  Furthermore, the observed dipole magnetic moment of an electron is formed, in addition to the basic electrodynamic component, also by an anomalous magnetic moment, created by the interactions of the electron with virtual photons in quantum electrodynamics.
  Assuming the universal validity of the quantum principle of uncertainty, a similar situation should occur in the general theory of relativity as the physics of gravity and spacetime: should quantum fluctuations in the geometry of spacetime occur..?.. - §B4 "Quantum geometrodynamics".

Feynman quantization of path integrals
At the beginning of our fleeting excursion into quantum physics, we mentioned that it is by no means easy to understand the intrinsic causes of quantum behavior of microsystems based on our experience with classical macroworld behavior. For example, how is it possible that in the famous double-slit experiment (Fig.
1.1.2 ) a particle can pass through both holes at the same time and then interfere "with itself"?
  Feynman's formulation of quantum theory is characterized by a very close relationship to classical physics *) expressed by the principle of least action. In classical physics (mechanics, electrodynamics, GTR), between a given initial x1 and final x2 state, of the investigated systen always make only such a movement for which the integral of the action S = x1òx2 L dt is extreme. On the other hand, in quantum physics, as is well known, such processes also take place that do not comply with this principle and are impossible according to classical physics - for example, the tunneling phenomenon.
*) The transition from classical to quantum physics is so elegant and straightforward here, that J.A.Wheeler used this approach to persuade A.Einstein to revise his opposition to the stochastic principles of quantum mechanics. But to no avail: "I don't believe that God would play dice with the world ", Einsten persistently objected...
  In Feynman's approach, all trajectories leading from the initial state x1 to the final state x2, are considered equiventally and simultaneously, regardless of whether they are permissible or not according to classical physics. As if the particle were moving along each imaginary trajectory at the same time as it traveled between the two states - it is the set of all virtual trajectories ("history"). If the integral x1nx2Ldt is calculated for each trajectory, the probability of transition of the system from the initial state x1 to the final state x2 will be given by the square of the quantity


obtained as sum and taken through all trajectories - the sum through all possible "histories". It is evident that the largest contribution to this sum is made by those trajectories that have a phase coefficient (i/h) ò Ldt almost the same (exponents add up), while for trajectories with large differences in (i/h) ò Ldt the exponents in the sum cancel each other out. The most probable trajectory (corresponding to close values of ò Ldt) will therefore be a classical trajectory with extreme behavior of the integral of the action. Trajectory here means "path" in the space of the given configurations systems; if it is a complex system described by a large number of parameters, it will be a trajectory in multidimensional space. Feynman showed that this formulation is equivalent to the usual Schrödinger and Heisenberg concept of quantum mechanics. Similarly to about the classic principle of least action, in practice in not immediately seek extreme integral ò Ldt, but derive Lagrange equation of motion, even when using Feynman method, the total sum over all trajectories is not directly calculated. Feynman's procedure is rather used as a means for deriving and elaborating quantum theories, as well as their physical interpretation.
..........- add, edit

Gnoseological questions :
Is quantum physics a major obstacle to the knowledge and use of nature ?

We physicists believe in the recognizability of the world, and it is our professional duty to work for the best possible knowledge of the mechanisms and laws according to which nature "works". From this point of view, we are "horrified" that randomness, stochasticity, statistics, which are a reflection of our ignorance of the exact conditions and states in complex sets of many interacting particles, are "dabbling" into fundamental physics !
  Quantum physics is often considered a theory of fundamental constraints, according to which our observations and measurements are inevitably inaccurate, natural phenomena are ruled by chance, and we should give up hope that science can accurately describe our world. Quantum mechanics is often considered an insurmountable obstacle to the knowledge of the deepest microworld or the practical use of microscopic phenomena (eg further miniaturization of electronic circuits). Already in the early periods of the development of quantum physics, it turned out that corpuscular-wave dualism, the randomness of phenomena and their superposition, discretion and especially the quantum relation of uncertainty, fundamentally prevent us understand and use nature in such a way and to such an extent as we were used to in classical physics (mechanics, electrodynamics, ...). This somewhat misleading view has its roots in a time when physicists have developed, perfected and confronted quantum mechanics with classical theories and philosophical concepts.
  In recent decades, however, a different perspective has become increasingly common. What does the uncertainty principle say and what does it not say? They merely claim that not all observed quantities of a physical system can take on certain ("sharp") values at the same time. Not all quantum measurements are limited by the uncertainty principle. Although the position or velocity is indeterminate and "blurred", other properties can be quite "sharply" defined - for example, blurred electrons in an atom produce a well-defined energy of a given orbital. In some cases, we can ingeniously circumvent this dreaded obstacle, and at the quantum level we can use the special properties of microsystems in new advanced devices such as lasers, integrated circuits, nanotechnology, new possibilities in informatics and computers
(see below "Quantum computers").
........ see also the discussion in "
Natural laws, models and physical theories".................
At the deepest level, is the world discrete or continuous ?
This is an important gnoseological question on which opinions of physicists differ. In modern terminology, this question could be paraphrased: “Is the physical world essentially analog or digital ?”. The physical relationship between the continuous (fluent, smooth) and discrete nature of the microworld can be essentially twofold :
Discretion is basic : ® secondarily generates continuity (apparent)
This is the most common approach in modern physics, based on atomistics, thermodynamics and statistical physics. From the point of view of atomic physics, all substances are composed of discrete ones atoms that have a specific integer number of protons in the nucleus and electrons in the envelope. And the spectrometry of the radiation emitted and absorbed by the atoms shows that the electrons in the atom occupy discrete energy levels, determined by integers. Bohr's model of the atom is based on this
(see "Bohr's model of the atom" below). In sets of large numbers of atoms and molecules, the methods of statistical physics make it possible to derive the laws of gas kinetics and thermodynamics, which are continuous. However, the basic "input values" of the theory are discrete integers. The continuity "emerges" from the averaged a large number of discrete events. The water in the glass appears as a continuous medium, but if we look at it with high magnification, we will see the molecules and atoms of which it is composed. These atoms also have the internal structure of electrons, protons and neutrons.
At the most basic level of current knowledge, matter is made up of fundamental leptons and quarks
(§1.5, part "Standard model - uniform understanding of elementary particles"), which are considered as discrete particles that can in principle be "counted down one after the other" *).
*) Gnoseological note: The role of integers is called metaphorically in classical mathematics that "God created only whole numbers, everything else in mathematics is the invention of humans". However, for modeling nature, mathematics introduced more general sets, especially real numbers (see §3.1 "
Geometric-topological properties of spacetime", section "Sets and representations") for which it created extensive apparatus of differential and integral calculus.
Continuity is essential : ® secondarily generates discretions (again only apparent?)
Quantum physics microworld based on a particle-wave ideas. The wave equation of quantum mechanics (Schrodinger equation) contain only continuous quantities. An illustrative example of this concept is Broglie's wave explanation of Bohr's quantum electron paths around the nucleus of an atom - see below, Fig.
AtomBroglie.gif. Here, the discretion of electron paths is formed by the continuity of the wave functions of electrons - their "wave continuity".
  In the standard particle model, it is taught that the basic building blocks of matter are the basic discrete particles - leptons (especially electrons) and quarks. However, on a more fundamental level, in unitary field theory
(§B.1 "The process of unification in physics" in the book "Gravity, black holes ...."), the basic building block of physical theories is  field - a continuous "fluid" substance distributed in space (the best known example is the electric and magnetic field). From the point of view of unitary field theory, "fundamental particles" are not fundamental, but are composed of continuous fields (and their waves or quanta) - see below. Particles are "precipitates" of a unitary field.
  Nature is probably a true continuum, in which we do not find any building elements that are no longer indivisible at any level of magnification. Physical quantities are generally not integers, but continuous real numbers, for which the number of decimal places is constantly increasing with the gradual refinement of the measurement. Integers retain only the significance of the number of types of significant particles in terms of the type of their interactions
(e.g. 3 types of neutrinos, 6 types of quarks), the expression of the number of electrons in atomic shells, the number of protons and neutrons in nuclei, or order and number of excited states.
Is space and time continuous or discrete ?
In most disciplines of classical and quantum physics, space and time are considered to be a continuous, infinitely divisible continuum - a kind of "stage or arena", against the background of which physical processes, interactions of particles and fields take place. What if, however, the continuity of space-time is the same illusion as it was until the 19th century continuity of matter? As modern physics learns about the discrete quantum structure of matter, it is hypothesized that space-time is also quantized - it consists of a huge but countable number of very small already indivisible elementary "cells", a kind of "space-time dust". If these hypothetical "quantum geometries" are small enough, e.g size of the order of the Planck length 10-33 cm, spacetime appears to be completely continuous, as no physical processes studied so far can distinguish finer distances than about 10-15 cm.
  Thus, there is a possibility that continuous quantities could in fact be discrete in a closer (enlarged) view: they may lie on a dense grid of individual separate points, which in the view available to us gives the illusion of a continuum. It is similar to the pixels on a computer screen observed at basic magnification and zoom.

It is interesting that a discretized version of quantum fields - a lattice field - was developed in quantum physics, where the continuous space - time is replaced (modeled) by an evenly arranged set of points, only in which the quantities of the fields are determined. However, it is only a model that facilitates quantum calculations, it does not follow that this is in fact the case.

  Thus, in addition to the generally accepted concept of continuous space, it is possible to alternatively postulate or axiomatically introduce the primary discretion of space : space is formed by individual separate "cells", only in which field values (potentials, intensities) are defined. These fields are therefore also discrete in terms of spatial distribution. If the spatial lattice (matrix) is sufficiently dense or fine (even in the dimensions of Planck lengths 10
-33 cm), space seems to us "illusory" as a continuum. However deepest mikromìøítcích the space could primarily be discreet - "pixellated" or "voxellated"..?..
  General relativity conceives gravitational field as a curved spacetime
(see §2.2 "Versatility - a basic property and the key to understanding the nature of gravity" in the book "Gravity, Black Holes and the Physics of Spacetime"). If we want to quantize gravity, it is necessary to "quantize" spacetime. The combination of the general theory of relativity and quantum physics thus reveals (or postulates) a discrete structure in space-time itself, whether fundamental or induced - see §B4 "Quantum geometrdynamics" and §B5 "Quantization of the gravitational field", part "Loop theory of gravity " in the already mentioned monograph).
  Reflection of continuous versus discrete aspects of nature is probably necessary in creating a unitary theory of physics - the theory of everything - TOE (§B.6 "Unification of fundamental interactions . supergravity. Superstrings." monograph "Gravity, black holes and spacetime physics").
Author's note:

Personally, I slightly prefer the opinion that fundamental is a continuity (perhaps even causal?), which induces an apparent discretion (and perhaps also quantum stochasticity)..?.. However, even the concept of a discrete super-dense space-time lattice could perhaps be the main idea for understanding the microworld..?..
Is the world recognizable ?
This basic gnoseological question is often discussed from a variety of philosophical perspectives. From the scientific point of view, the cognition of our world can be reflected in principle on three levels :

Phenomenological cognition
The study of the specific course of individual natural processes is the basis of scientific knowledge. The accuracy of this knowledge is given by the level (resolution) of physical instrumentation, optical observation systems, chemical-analytical methods. The principal limitations in phenomenological cognition are imposed on us in the microworld by quantum relations of uncertainty
(see, for example, the section "Quantum physics" above), in the macro- and megasworld then horizons of events of relativistic astrophysics (§3.3 "Cauchy's problem, causality and horizons" in the monograph "Gravity, black holes ....").
2. Knowledge of internal causes, mechanisms, laws
This is the main content of advanced scientific research. From a detailed analysis of the course of natural processes
(phenomenological) under different conditions and comparisons from other processes, general natural laws are formulated, if possible with universal validity for a wider class of phenomena. This makes it possible to understand the functioning of nature (it is discussed in the section "Natural laws, models and physical theories" §1.1 in the already mentioned book "Gravity, black holes ...").
3. Absolute deterministic recognizability
The maximalist requirement of complete recognizability of the world would require that for all elementary particles, atoms, molecules and other structures we can predict their exact spatial positions at all times, as well as predict accurate values of fields (potentials, intensities) in all places of space. Our current knowledge shows that this is not possible ! In addition to technical impracticability, quantum relations of uncertainty prevent this at the microscopic level, and at all levels special irregularities in the behavior of sets of particles, called "deterministic chaos", which generate chance - is discussed in more detail in the section "
Determinism-chance-chaos?" §3.3 in the book "Gravity, black holes and the physics of space-time". Furthermore, if space is continuous (see the discussion above "Is space and time continuous or discrete?"), an infinite and even innumerable set of points would require an infinite amount of data for each, even the smallest, district of the system under study!
  However, to deduce from the negative message of level 3 a categorical statement about the unknowability of the world is not substantiated and can be misleading *)! The world is basically recognizable in the sense that we understand the mechanisms of its functioning, on the basis of which we can often predict the behavior of many important systems in nature and space in the long run and with great accuracy. E.g. based on Newton's and Kepler's laws (gravity and mechanics), astronomers can predict the motions of planets in the solar system with high accuracy many centuries to come. In the horizon of millions to billions of years, however, minor gravitational disturbances will eventually result in chaotic deviations, which will significantly change the motions of the planets (some of them may even escape from the system ...). So it is not apt to say that "the world is unknowable", but that "the knowability of the world has its limitations".
*) Such a sharp statement that "the world is unknowable"could provoke skepticism, nihilism, agnosticism. It would also record various "alternatives" and charlatans who downplay the impressive achievements of serious scientific knowledge and claim that only they, thanks to their "miraculous abilities", have the gift of "true knowledge" and can control the world (or rather some trusting people...).

Some unusual and paradoxical consequences of quantum physics
Quantum tunneling phenomenon
If a particle moves in a certain force field, the law of conservation of energy - the sum of the kinetic energy of a particle and its potential energy in a given field - is fulfilled at each point of the trajectory when moving according to classical physics. An interesting case of particle motion is the motion in a force field, whose potential has the shape of the potential barrier - in the simplest case of movement in the direction of the X axis, such forces act on the particle that its potential energy E
p is everywhere zero, except for the area x1 <x <x2, where the value Ep = Vo. If the kinetic energy of the particle Ekin is less than the height Vo potential barrier, according to classical physics the particle should bounce when moving from the place of the barrier and move back against the original direction of motion - the particle is never able to overcome the potential barrier. The particle can overcome the potential barrier only if it has a sufficiently large kinetic energy Ekin>Vo.
  However, in quantum mechanics, where a particle is described by a wave function (according to corpuscular-wave dualism, it is a Broglie wave), there is a non-zero probability that the wave will "seep through" the barrier and the particle may suddenly be on the other side of the barrier. Waves, unlike conventional particles, can get behind the obstacle due to bending and then continue to move through space. Analysis of the wave function using Schrödonger's equation shows that a plane wave incident on the barrier wall partially bounces off it (and interferes with the original wave) and partially penetrates inside the barrier. If the width d = x
2 - x1 of the barrier is small enough compared to the depth of penetration of the wave, the Broglie wave reaches the second wall of the barrier, where the potential suddenly decreases - the wave enters free space and continues to move away from the barrier as a plane wave, with a lower amplitude (expressing the probability of the particle passing to the other side of the barrier). The particle has passed potential barrier even if, according to classical physics, the energy of a particle is insufficient to overcome the barrier!

Symbolic representation of the quantum tunneling phenomenon.
Left: A ball rolling with the kinetic energy of E
kin against an elevated terrain wave (hill - gravitational potential barrier of height Vo) can overcome it only if it has sufficient energy Ekin > Vo. Middle: If a tunnel is pierced in a terrain obstacle, the body can overcome it even at a significantly lower energy than the potential height Vo. Right: Simplified representation of a rectangular potential barrier of height Vo, through which particles (~waves) can pass with a certain probability even at a lower kinetic energy than Vo - as if "hidden tunnels" led across the barrier.

From an energy point of view, the described phenomenon can be explained by the quantum uncertainty relation DE.Dt ³ h between energy and time (discussed above). The value of the instantaneous energy of a particle fluctuates in its short time around its mean size, up and down. Once a particle reaches a potential barrier, energy can (coincidentally) be momentarily increased for a short time, allowing it to cross the barrier. During the act of self-penetration, within the uncertainty relation, the law of conservation of energy may not apply. The shorter the moment of fluctuation Dt, the greater its possible range DE. If the particle does not manage to reach the other side of the barrier during this duration of sufficiently large fluctuation, it will return - it will bounce off; this, of course, happens even if the energy fluctuation is negative at the moment the barrier is reached. The wider and higher the potential barrier, the less likely the particle is to penetrate successfully; most of the particles penetrate the barrier only partially and eventually bounce off the potential wall. With a sufficiently narrow barrier, the particle is more likely to successfully overcome it due to a sufficiently high short-term energy fluctuation.
  This effect, when a particle crosses a potential barrier higher than the energy of the particle, is called a tunneling phenomenon - a particle that does not have sufficient kinetic energy (and therefore cannot "fly" over the potential barrier) can still penetrate the barrier with a certain probability, as if a hidden "tunnel" had been drilled in it. The particle, on the other side of the barrier, seemed to "tunnel". The tunneling phenomenon has a probabilistic character. It occurs either in a large number of incident particles, some of which manage to "tunnel", or the individual particle must be able to perform a series of "failed attempts" before it can "release" from binding in the nucleus
(eg alpha-radioactivity) or in the material (eg thermoemission).
  Probability w of quantum tunnel passage of a particle with kinetic energy E
kin through a potential barrier of height Vo (> Ekin) and width d is approximately equal to
» exp ( - 2d. Ö [2m. (Vo - Ekin) / h 2 ] ) .
This probability decreases exponentially with the width d of the potential barrier.
  The tunneling phenomenon, which is typically a quantum-mechanical effect associated with the wave properties of particles, is significantly applied in many phenomena of the microworld - in atoms and atomic nuclei (eg in
alpha radioactivity, nuclear reactions - especially in thermonuclear fusion), in electrical phenomena in conductors and semiconductors. In an electric field, electrons can be emitted from metals (thermoemission, photoemission) even if the kinetic energy of the electrons is lower than the corresponding output work; a tunnel scanning microscope is based on the quantum tunnel emission of electrons from the surface of conductive substances. Thanks to the tunnel effect, many processes at the microscopic level can take place even at significantly lower energies than would be necessary according to classical physics.

Schrödinger's cat
It is a somewhat morbid *) and absurd thought experiment, which the Austrian physicist E.Schrödinger composed in 1935 to show one paradoxical property of the superposition of states in quantum physics: the mere fact that a stochastic phenomenon is observed influences its result! Schrödinger thus sought to point out the absurdity of this interpretation, which was later more or less accepted in quantum physics ...
*) Unfortunately, the virtual morbidity of this experiment was replaced in a few years by real morbidity committed by his national compatriots, who used the same hydrogen cyanide to murder thousands of innocent people in concentration camps..!.. However, Schrödinger himself was a staunch opponent of german nazism.
The following objects are placed in a hermetically sealed opaque box : 
Live cat; 2. Ampoule with poisonous gas - hydrogen cyanide
(in the first version it was a rifle aimed at a cat) ; 3. A sample of radioactive material containing one radionuclide atom with a half-life of 1 hour; 4. A radiation detector electronically coupled to a mechanism capable of breaking the ampoule.
When the radioactive atom decays, the detection device registers it, the electrical device inside the box breaks the ampoule of poison, and the cat dies.
  The decay of a radioactive atom is a stochastic quantum phenomenon - we cannot predict the time when the nucleus will decay, only the appropriate probability; after one hour, there is about a 50% chance that the nuclide has decayed. According to the notions of quantum mechanics, a radionuclide that is not observed is in a superposition of a decayed and an undecayed nucleus, as if it were in both states at the same time. So the whole follow-up system in the box should be in a superposition of states: [decayed radionuclide - dead cat] and [undeayded nuclide - live cat]. However, if we open the box, we will of course see only one of these states.
  The paradox is that in interaction with a suitable quantum system (radionuclide), a cat can get to a state where it seems to be alive and dead at the same time. According to quantum mechanics, the system in the box is described by a wave function that contains a combination of these two possible mutually exclusive states - that is, at every point in time, the cat is alive and dead at the same time. Only when we open the lid of the box, external observation decides whether the cat really lives or not. The imaginary experiment points to the imperfection (incompleteness) of quantum mechanics in the complex understanding of nature, in the transition between the micro- and macroworld.
  This imaginary experiment was popular especially in the period of building concepts of quantum mechanics, when it played a certain heuristic role. From today's point of view, he is no longer very convincing... Rather than the contradiction of physical reality, it expresses the paradox of its formal description.
Note: A possible bizarre solution to this paradox is sometimes considered from the point of view of the hypothesis of an infinite number of parallel worlds in which all possibilities are realized (it is discussed in §5.7 "Anthropic principle and existence of multiple universes", passage "Concept of multiple universes"). In one universe the cat is dead, in the "neighboring" universe the experiment survived ...

Quantum entaglement and teleportation. Quantum computers.
An interesting and for classical physics completely surprising consequence of the fundamental nonlocality of the quantum description of particles by means of wave functions is a phenomenon called quantum entanglement. It consists in the fact that two particles, whose quantum state is "intertwined" originally by a common wave function, remain in a sense still connected by a kind of "invisible bond", even at any distance. If the state of one of the entangled particles changes, the state of the other particle also changes, "immediately" - a kind of "teleportation" of information occurs, according to some opinions "at superluminal speed"..?..
(will be discussed below) .
  As mentioned above,
the evolution of a quantum system is described by a wave function. It is a (thought.) wave propagating through space, with the object ("particle") "occurring", in a non-local sense, everywhere at the head of this wave. When an object interacts with another quantum object or measuring instrument, a "wave function collapse" occurs, and the object is temporarily localized and can be described in particle form. The collapse of the wave function takes place non - locally - the wave function, according to this interpretation, suddenly disappears from the whole space..?.. It is only a theoretical-hypothetical idea....
  According to the usual so-called Copenhagen interpretation of quantum mechanics, the investigated system consists of the quantized objects themselves and of classical measuring instruments or observers. The collapse of the wave function locates the information that the observer obtained by measuring.....
  If we have a pair of spatially separated quantum subsystems that form part of a single system, these subsystems are bound to each other through a common original wave function. The measurement (interaction) of one subsystem thus bound forces the other bound subsystem to immediately go to the corresponding (complementary) state, regardless of the spatiotemporal distance. This phenomenon is referred to as EPR-nonlocality (Einstein-Podolsky-Rosen) or EPR-paradox .
  A.Einstein and his collaborators B.Podolský and N.Rosen formulated the thought experiment outlined below to show the internal contradiction and incompleteness of quantum physics. It seems paradoxical that without the presence of exchangeable (mediating) particles or fields, it is possible to immediately influence a particle that is, for example, at the opposite end of the universe - a kind of "haunting effect from a distance"! According to the special theory of relativity, it can be expected that only places between which the space-time connection is limited by the speed of light can be in causal contact. However, quantum mechanics, due to its nonlocality of wave functions, can in a sense temporarily violate this causal requirement of relativistic physics. Now, quantum entanglement is no longer considered paradoxical. By performing measurements on one particle, no mass or energy is transferred to the other particle. And as for information, both observers must use "classical" communication using a signal with sub-light speed to confront the measured results; this ensures the STR causality of both measurements
(see also the commentary to Fig.1.1.3 below).
  Mutual nonlocal interconnection or "intertwining" of quantum states is referred to in English as entanglement. It is a quantum correlated state of a system of two or more particles, in which the state of one particle cannot be measured without influencing the other
(and therefore it makes no sense to talk about the states of individual particles). Both particles have a common non-local wave function.
  This can be illustrated by the example of a particle initially at rest, with zero momentum, which breaks up into two identical particles flying apart, each with a spin of 1/2. The law of conservation of the total momentum implies that if we measure a spin projection 1/2 to a certain axis for one particle, the other particle must have a spin projection to the same axis -1/2 (and vice versa). Therefore, if we measure the spin on one particle, immediately we learn (we judge) the spin value of the second particle, no matter how far. As if this information was spreading immediately, contrary to the special theory of relativity *)! From a quantum-mechanical non-local point of view, when measured on one particle, the common wave function "collapses" in the whole space
(in imaginary idea?), which is reflected in the state of the other particle. However, the actual verification and reconstruction of the measured quantum state of the second particle is only possible with a classical communication channel with sublight speed!
*) Therein lies the oft-stated misconception about the superluminal speed of quantum teleportation. If we know that both particles are quantum linked and have oppositely oriented spin, measuring the spin of one particle naturally implies the opposite spin of the other particle, without the complexity of its measurement and teleportation of information. And if we don't know it, it is generally necessary to transmit information about the state of particles with a communication signal at sublight or light speed.
  Another typical example of "entangled" particles is a pair of photons generated simultaneously in a quantum process. It can be a pair of photons emitted from an atom after its excitation, the polarization of which can be correlated (in the simplest case to the opposite, perpendicular to each other). If we measure a single photon polarization e.g. in the direction of the horizontal X axis, the second photon polarization is in the direction Y. This may occur with radiation gamma produced during annihilation particle and antiparticle (see §1.2 and 1.5). Quantum entangled photons are also formed in nonlinear optical crystals by the impact of coherent monochromatic radiation from a laser, where some incident photons "split" into two bound photons with lower energy, the polarizations of which are complementary to each other.
Quantum teleportation
The term "teleportation"
(Greek tele = distance , Latin portare = carry, transfer ; ie transfer, relocation in distant) often found in science fiction, generally refers to the process by which a given object (even a person in science fiction) disappears in its original place and appears in another place (in science fiction, for example, at the other end of the universe, immediately or at superlight speed). In a more sophisticated embodiment, it is an indirect relocation : the object is disassembled and analyzed in one place, the obtained complete information about its construction is transferred to another remote location, where then an exact copy is created (reconstructed) using this information.of the original object. This copy is not created from the original matter, but from particles of the same kind (eg atoms) in a new place, which are assembled into the same structure as the original object - it is not a physical transfer of objects - their matter (substance), but information transfer. And it definitely doesn't go through infinite or superlight speed!
  Quantum interconnected (entangled) particles can in principle be used for the so-called quantum teleportation of information about the state of another particle that interacts with one of them. In 1993, Ch.Bennet proposed the following indirect method (Fig.1.1.3) :

Fig.1.1.3. Simplified arrangement principle for quantum teleportation.

Let us have an O1 observer (sender's laboratory) and an O2 observer (recipient's laboratory). At the O1 observer, we create a pair of entangled particles A and B so that the A particle remains at the O1 and the B particle is sent to the O2 observer. The O1 observer then realizesthe interaction of particle A with the third particle C carrying the information (state) to be teleported; measures the resulting states of particles A and C after interaction. The original state of particle C is deleted, but thanks to entanglement, this information appears (in coded form) on the distant particle B, whose state B´ is measured by the observer O2. In order for the O2 observer correctly determined the original state of particle C which was in the place O1 , the observer O1 must connect with the observer O2 using a classical (non-quantum, causal) communication channel (eg electromagnetic signal) and tell him what result of the states of particle A and C after measured the interaction. The O2 observer then confronts the result of his measurement of the state of particle B with the data communicated by the observer O1 (using them to decode his result by linear transformations of the type of rotations in the vector base...), while the final result is to determine (reconstruct) the original state of particle C in place O1 - this corresponds to the teleportation of this information.
  Thus, both a nonlocal "EPR" channel of entangled particles and a normal (causal) communication channel are required to perform quantum teleportation. This is the only way to decode teleported information. It is this necessity of classical communication that effectively makes it impossible to send information at super-light speeds. Thus, quantum teleportation does not violate the principles of causality of the special theory of relativity - EPR is no longer a paradox...

  The process of quantum teleportation in its current understanding can only work within elementary particles and cannot be used for teleportation of macroscopic objects. There is no known way in which a set of quantum-bound states could interact with a macroscopic object in a targeted manner. In addition, this method transmits only the value of one observable quantity, not the complete quantum state.
  Quantum teleportation of the polarization state of UV photons was first performed experimentally in 1997 in the laboratory of quantum optics and spectrometry in Innsbruck. Later, quantum teleportation was performed on excited states of calcium ions
40Ca+ (in the same laboratory in Innsbruck) and beryllium 9Be+ (at the National Institute of Standards and Technology in the USA). In 2017, we managed to teleport photons between the ground station and the Chinese satellite Micius to a distance of 1400 kilometers.
Quantum electronics, quantum computers
An important quantum property of particles, electrons, protons and whole atoms is spin - the intrinsic rotational momentum of a particle. Particles with non-zero spin can act as magnetic dipoles that respond to an external magnetic field. The spin of an electron is oriented in two opposite directions
(spin projection) , which are referred to as upward-pointing states |á> and down |â>. When such a particle passes through a suitably configured (inhomogeneous) magnetic field, the particles with spin |á> deflects to one side, while particles with spin |â> tilts to the opposite side. They therefore fall into different places of electronic detectors. On this principle, the so-called spintronics develops - electronics, which, in addition to the charge of electrons, also uses the orientation of their spin. Digital application of the principles of this quantum electronics leads to quantum computers.
  On these remarkable quantum properties have therefore recently "rushed" some experts in the field of computer and cybernetics, who have reformulated them into their digital terminology and, together with physicists, have begun to work on the possibilities practical applications in this area. In current
(already classical) digital and computer technology, the basic unit of information is "bit" - an electronic signal whose state takes two values (it is digital), which are expressed as "0" or "1". It is usually realized by two normalized well-distinguishable values of voltage in the gate circuit. Groups of combinations of these bits ("bytes", groups of 8 bits) then express the codes and numerical values of all data in the binary system.
  In quantum informatics, the so-called quantum bit or qubit
(quantum bit) is introduced as the basic unit as a quantum bit version - digital units of information. While the classical bit is always in either the |0> or |1> state, the qubit also carries arbitrary values between 0 and 1 during the process - it also includes all superpositions of probabilities of these states. In the wave function, information about all superposition coefficients is carried in parallel.
Qubit state |q> is written as: |q> = A . |0> + B . |1>, where A and B are complex probability coefficients of states |0> and |1>, for which A
2 + B2 = 1 holds.
  The complex superposition state "between 0 and 1" is implicitly contained only in the free state, without any interaction. The specific explicit value |0> or |1> is acquired by qubit only at the moment of measurement (interaction). During the interaction
(when we "look" at it, detect it, decode it), the state of the qubit "flips" - in quantum terminology, "collapses" - to one side or the other, whichever is most likely at that moment.
  Quantum computers mainly use three specific phenomena from the quantum microworld: quantum superposition of states, quantum interconnectedness and interference (constructive or destructive) of quantum states. The quantum computer is based on three basic principles :
1. Technical realization of qubits ; 2. Quantum entanglement of qubits ; 3. Detection and decoding of quantum states (logical operations with qubits) .
1 :
Suitable two-level quantum-mechanical microsystems can be used for technical realization of qubits
(photons, electrons, atoms). So far, four methods for realizing qubits have been tested :
Utilization of photon polarization ;
¨ Use the spin of particles, especially electrons ;
Use excited atoms,  especially hydrogen atoms ;
¨ Use of superconducting conductors arranged in so-called Josephson junctions (they are described in §2.5, section "Microcalorimetric detectors", passage "SQUID") .
  Several other special quantum phenomena could be used (.....), which have not yet been realized.
2 :
entaglement of qubits can be performed by mutual interaction of particles, atoms or ions. Interconnected quantum states are often realized by means of a laser pulse. If we managed quantum interconnected N qubits, the number of superposition coefficients is 2
N. Operations with these interconnected qubits are parallel (it takes place with all superposition coefficients), which gives the potential for high performance of electronic storage, transmission and analysis of information.
3 :
For the recognition and decoding of quantum states of qubits and for logical operations with them, use methods depending on the type of qubits used. For polarized photons, these are optoelectronic methods. The measurement of the plane of polarization can be performed by placing a polarization filter in the path of the photon, through which only photons with a certain plane of polarization (state |1>) pass, while photons polarized perpendicular to the plane of the filter do not pass (state |0>). Photons polarized in other planes will behave as qubits in the superimposed state - according to the angle of rotation of the polarization, the amplitude will change the probability that the photon will pass or not pass through the polarization filter. Magnetoelectronic methods are used for spin orientations.
Perspectives of quantum computers ?
The application of the laws of quantum properties in informatics and computer technology promises attractive possibilities for quantum computers that some computational tasks might be able to solve many times faster than conventional computers. Often mentioned, but in fact marginal, is the possibility of perfect quantum cryptography
(protection of transmitted data using quantum keys).
  The problem is to build quantum computers with a large enough number of qubits. Large sets of particles no longer behave according to the laws of quantum physics and begin to follow the laws of classical mechanics and electrodynamics. Possibilities of creating so-called modular quantum systems are being explored : construction of many small quantum processors and their interconnection by a small number of nodal quibits, which would not disturb their quantum properties
(the modules themselves remain relatively isolated from each other). Quantum chips are developed in "high-tech" microelectronics laboratories, in which pairs of entangled ions (in ion traps) or entangled photons (in optical crystals or micro ring resonators) are generated and analyzed. These could become essential elements of truly usable quantum computers.
  A big problem is the prevention of quantum decoherence of qubits, for which it is necessary to completely isolate the system from the disturbing influences of the environment, including thermal oscillations. Therefore, the operating temperature is a significant technical obstacle to the practical use of the quantum computers being developed so far. For the correct function of qubit circuits, it is necessary to cool them to a very low temperature close to absolute zero (of the order of milliKelvins), which requires a complex cryogenic technique.
  So far, quantum computers are in the stage of experimental verification and improvement of basic physical and technical principles. Manipulating qubits is incomparably more difficult than bits in electronic computers. There will be a long and difficult journey to create truly usable and powerful quantum computers ..!..
  After initial enthusiasm, many computer experts are now somewhat skeptical of quantum computers. They will definitely not be "self-saving"! They will not be universal, so they can no replace classic digital electronic computers (we will probably never have quantum PCs at home...). They will be suitable only for some special areas (eg searching for information in large unsorted data files, factorization decomposition of numbers into coefficients, fast Fourier transform, ...), where they can significantly speed up classical algorithms. Due to the quantum-stochastic nature of qubit states, quantum computers use probabilistic algorithms (individual processes are repeated a million times), with an effort for fast repairability and convergence. Whole pure quantum (100-%) computers they are not feasible - and it probably wouldn't even make sense. Rather, they will be quantum coprocessors for large specialized computing systems.

Atomic structure of matter
The question of the structure and composition of matter is one of the most basic and important questions that people ask nature - along with questions about the origin, size and structure of the universe, or questions of the origin of life. In earlier times, when humans did not have the means to gain a deeper insight into the microscopic dimensions of the interior of matter, it was by no means easy to make any credible claims about the invisible microstructure of matter. Scholars therefore resorted to various assumptions and hypotheses, formed by analogy with what was seen with the naked eye.
   In this context, a fundamental question arose: does matter have a continuous or granular structure? In other words: matter is infinitely divisible to smaller and smaller particles, or in this division do we finally come across the smallest, further indivisible isolated particles?
Intricate historical development of the concept of atoms
The Greek ancient philosopher Demokritos
(5th century BC, partly followed the views of his teacher Leukippos of Miletus) was a supporter of this second possibility of the smallest indivisible particles, arguing that if matter were indefinitely divisible, it would not remain in the end, nothing that would carry the properties of the substance. Therefore, each substance must be composed of indivisible particles which carry the properties of that substance. He called these smallest indivisible particles "atomos" - Greek. "indivisible".
Note: It would be a complete misunderstanding to consider Demokritos as the discoverer of atoms or the creator of atomic theory! Demokritos knew nothing about real atoms, his opinion was just one of many speculative hypotheses, mutually equivalent at the level of knowledge at the time; other scholars at the time rightly considered matter to be infinitely divisible ...
  By the way, the above-mentioned philosophical argument about the bearers of the properties of matter would no longer stand. We know that an increase in quantity, or a combination of more quantities, can create a new quality. The properties of the system are created only by a combination of the properties of its subcomponents. And it's the same with matter. A specific substance may not have any elementary carrier of its properties - these properties are created only by a specific "construction" of a substance from particles, which themselves have completely different properties ...
   The idea of atoms then fell into oblivion for a long time. It was not until the turn of the 17th and 18th centuries, when feudalism and the church lost their absolute power, that the former alchemists - charlatan and often fraudulent "goldsmiths" in the service of the rich and powerful - gradually was replaced serious scientists who no longer wanted a recipe for making gold or various elixirs, but they tried to penetrate into the true essence of the construction of matter. It was then that the idea of the basic building blocks of matter (Descartes, Hook) began to appear again in connection with the study of the behavior of substances, especially gases (the dependence of pressure on the volume of gas). From the 18th century, with the gradual liberation from earlier alchemical superstitions and prejudices, chemistry emerged as an independent discipline. Through a series of experiments, R.Boyl and A.L.Lavoisier came to terms chemical element as a substance that can no longer be broken down into two or more other substances. During chemical experiments, important laws of chemical processes were determined :
- The law of conservation of mass and energy of substances entering the reaction and resulting reaction products in a closed system (M.V.Lomonosov 1748, A.L.Lavoisier 1774) ;
The law of constant merging ratios (J.L.Proust and J.Dalton 1799), the law of multiple merging ratios (J.Dalton 1802) and the law of constant volume ratios (Gay-Lussac 1805), observed in reactions in the gaseous state.
  These laws became the experimental basis for clarifying the question of the internal structure of the elements. J.Dalton gave a natural explanation of all these important laws in 1808 in his atomic hypothesis, according to which each element is composed of a large number of mutually identical atoms, indivisible particles characterized by a certain characteristic mass and other properties. The atoms of the same element are the same, the atoms of different elements differ in mass and other "chemical" properties. These atoms are the basic indivisible building block of matter that participates in chemical reactions - the merging of elements consists in the joining of two or more atoms. In this concept, the law of conservation of mass in chemical reactions is an outward manifestation of the indestructibility (and also the "uncreatability") of atoms. The new bound whole, created by merging an integral number of atoms, was called a molecule
(lat. Moles = mass); the name comes from A.Avogadra from 1811, who also discovered the first relationships between molecular, weight and volume amount of substances.
Note: We now know that the law of conservation of mass and merging ratios differ slightly from ideal values. This is in connection with the relationship between equivalence of mass and energy E = mc 2given by the binding energy of the reaction, the mass defect of atoms and nuclei, the difference between the mass of a proton and a neutron. These aspects will be discussed below where appropriate.
  Electrolysis played an important role in understanding the composition of substances and in discovering a number of elements: the decomposition of substances (mostly their aqueous solutions) by the action of an electric current from Volt electric cells (batteries). First it was the electrolysis of water into hydrogen and oxygen, then the electrolysis of various alkalis, acids and salts into hydrogen, oxygen, sodium, potassium, calcium, copper, zinc, chromium and other elements. An interesting question arose: "If electricity can decompose substances into elements, could it even combine elements into more complex substances?". She already anticipated later knowledge of electrical origin of all chemical reactions (see "Atomic interactions" below).
Periodic Table of the Elements
In 1869, D.I.Mendelev systematically dealt with the chemical properties of various elements. He found that the chemical properties of elements periodically depend on their relative atomic mass (atomic weight). He proposed to arrange the elements in a table, in order of increasing atomic weight into horizontal rows (forming periods) so that elements of similar properties get under each other. The definitive explanation of this Mendeleev's periodic table of elements was made possible by the development of the physics of atoms - see below "
Bohr's quantum model of the atom", passage "Occupancy and configuration of energy levels of atoms" and "Interaction of atoms - Chemical merging atoms - molecules".
  A similar story, explaining the essence of the observed regularity and repetitive structure and properties at different scales is repeated in other areas of science :
- In particle physics - §1.5 passage "Unitary symmetry and particle multiplets" and part "Standard model - unified understanding of elementary particles", passage "Preon hypothesis".
- In astrophysics of stars - §4.1"The role of gravity in the formation and evolution of stars"- The Hertzprung-Russell diagram, in the book "Gravity, Black Holes and the Physics of Spacetime".
  At the turn of the 19th and 20th centuries, when a sufficient amount of experimental data from the field of chemistry and physics was collected, is was realized that pure elements are composed of "indivisible" basic particles - atoms (which bear their properties), which can combine - merge - into molecules in compounds. Next experiments in the early 20th century showed that even the atom is not an indivisible (unstructured) elementary particle, but has its own complex electro-mechanical structure. In terms of construction materials, atoms are not the last, smallest and most fundamental particles of the substance, but only one of the important hierarchical units of substance structure.

The structure of atoms
Although physics and chemistry during the 19th century increasingly convincingly showed that substances consist of atoms and molecules, the nature and structure of the atoms themselves practically nothing was known until the end of the 19th century. Experiments with electrolysis carried out by M.Faraday in 1836 have already shown that chemical compounds have a lot in common with electrical phenomena. The first significant penetration into the structure of the atom was the discovery of the electron, made in 1895 by J.J.Thomson in the study of electric discharges in gases *) and the discovery that all atoms contain electrons.

Electric discharges
Electric shocks in the air, known as spark jump among bodies sufficiently electrified by static electricity, have been known for a long time (for the development of knowledge about electrical phenomena, see also §1.1 "Historical development of knowledge about nature, space, gravity", passage "Electricity and magnetism" in the book "Gravity, black holes and space-time physics"). In 1743, M.Lomonosov suggested that lightning and aurora borealis are manifestations of electric discharges in the air (he was right about that lightning; aurora borealis is more about the interaction of high-energy particles from the Sun with the upper layers of the Earth's atmosphere). In the last decades of the 19th century a number of researchers have studied high-voltage electric discharges in dilute gases (as early as 1838, M. Faraday observed a strange fluorescent arc between the cathode and the anode, connected to an electrical voltage in a tube with dilute air). Glass tubes or flasks with sealed electrodes were used for this - discharge lamps, filled with air or other gases, diluted by means of a vacuum pump to a pressure of about 10-3 atmospheric (approx. 100 Pa). The most famous were the so-called Geissler tubes, used since 1850. These differently shaped lamps, beautifully lit in different colors (according to the type of gas charge), were very attractive; they evolved into "neon" tubes. We now know that the light manifestations of an electric discharge are caused by ionization and excitation of gas atoms, caused by the impact of electrons (accelerated in an electric field), followed by deexcitation with the emission of photons.

Discharge lamps - gas-filled tubes with electrodes to which a high voltage >100 V is applied Cathode ray tubes - very dilute gas-filled tubes with electrodes (discharge lamps - Crookes tubes), to which a very high voltage > 1-10 kV is applied

   Later, electric discharges were studied in even more dilute gases at a pressure of about 10 -6 atm., when the visible discharge ceases. In 1859-76, J.Plücker, J.Hittorf, and E.Goldstein observed a faint fluorescence of the flask opposite the cathode: as if some radiation came out of the cathode, hence the cathode radiation. In 1880, W.Crookes designed a special glass flask with sealed electrodes, the so-called Crookes cathode ray tube, into which various objects, screens, and minerals were inserted between the cathode and the anode. At voltages of about 1000V and higher, Crookes found that with sufficient dilution of the gas, invisible so-called cathode rays emanate from the direction of the negative electrode, causing the bulb to fluoresce in places opposite the cathode. The objects and screens inserted between the cathode and the anode cast sharp shadows in this luminescence, some of the minerals exposed to the cathode rays fluorescent. Vacuum tubes, screens, and X-rays tubes have evolved from Crookes cathode ray tubes, however, the diluted gas was replaced by a vacuum and the cold cathode by replaced by heated cathode, which thermoemission supplies the necessary electrons ("cathode rays").

1895: J.J.Thomson - discovery of electrons => first model of atom

In 1895, J.J.Thomson studied the deflection of these cathode rays in electric and magnetic fields and found that cathode rays are made up of very light negatively charged particles, whose charge corresponded to an elementary electric charge (roughly determined from Faraday's laws of electrolysis and later specified by Millikan's experiments). In this way, he discovered the first elementary particles of the microworld - electrons - and revealed the corpuscular nature of cathode rays, which are formed by a stream of fast-flying electrons. We now know that these electrons came from gas atoms ionized by the impact of other electrons accelerated in an electric field. Name electron (Greek electron = amber; static electricity was observed on amber objects in ancient Greece) comes from G.J.Stoney, who in 1891 dealt with Faraday's laws of electrolysis in connection with Dalton's atomic concept and concluded that the electric charges needed to exclude individual types of atoms are integral multiples of a certain small basic, elementary charge, representing a kind of "atoms" of electricity (electricity was until then considered a continuous "fluid"). Further experiments with cathode ray tubes led to the discovery of X-rays (§3.2 "X-rays - X-ray diagnostics").  
Note: The electrical phenomena in discharge lamps and cathode ray tubes were dealt with by a number of researchers in the given period, either independently or in mutual connection or cooperation. It is also probable that researchers "sunken patriots" who did not penetrate the general consciousness also came to new discoveries of knowledge in their remote laboratories. Therefore, arguing about the particular primacy of individual specific researchers can be problematic - and it is basically useless... It is important that by joint research they significantly contributed to revealing the laws of electricity and the microworld (cf. also the passage "Significant scientific discoveries - chance or method?" in §1.0).
  Electrons have (conventionally) negative electric charge and according to the first experiments were more than 1000 times lighter than electrically neutral atoms; we now know that electrons are 1837 times lighter than a hydrogen atom. Thus, each atom must contain a sufficient amount of positively charged mass to balance the negative charge of its electrons, and this positively charged component represents almost the entire mass of the atom. Based on these findings, J.J.Thomson proposed in 1898 the idea that atoms are a miniature homogeneous sphere of positively charged matter, into which electrons are nested - Fig.1.1.4 on the left. This Thomson model of the atom was also called the "pudding model", due to its resemblance to the English pudding with baked raisins.

Fig.1.1.4. To develop ideas about the structure of atoms.
Left: Thomson's "pudding" model of the atom. Middle: Rutheford experimental arrangement of scattering
a -particles by metal foil. Right: Difference of dispersion a particles by atoms for the case of Thomson model and model of atom with nucleus.

A more detailed experimental investigation of the structure of atoms was undertaken in 1909-11 by E.Rutheford, who, together with his collaborators H.Geiger and E.Marsden, performed important experiments with alpha particle scattering (a maximum energy of 7.7 MeV emitted by the natural radionuclide 226Ra and its decay products, especially polonium) during their passage through a thin gold foil (thickness about 3.10-4 mm, which corresponds to about 104 atomic layers) - Fig.1.1.4 in the middle; alpha particles after passage and scattering of the foil are labeled as a'. These particles were observed visually by Rutheford and co-workers according to the flashes in the scintillation layer (zinc sulfide) with which the flask surrounding the irradiated foil was coated from the inside.
Note : Scattering experiments (mostly with high-energy electrons and protons on accelerators) are generally the most important method of investigating the structure of the microworld and the properties of particle interactions - see §1.5 "Elementary particles".
  According to Thomson's model of the atom, it was expected that heavy and fast alpha particles would easily "pierce" the thin gold foil - they would pass through the foil either directly or with only a small scattering (Fig.1.1.4 at the top right); the uniformly sparse distribution of charge and mass inside the "pudding" atom causes only weak electrical forces when passing heavy alpha particles. The passed alpha particles should then leave their light traces only on a small area on the back of the flask, in a direct direction from the emitter.
  However, the experiment showed, that in addition of this, a number of particles
a' scattered by a large angle, some were even reflected in the opposite direction - Fig.1.1.4 in the middle *). In order for heavy alpha particles (more than 7,000 times heavier than an electron) moving at high speed (almost 2.107 m/s) to disperse in this way, large forces had to act on them inside the atoms, which would not be possible with the Thomson model with a relatively light, sparsely dispersed positive mass in which light electrons are embedded. Although most alpha particles easily penetrated by the fringes of atoms, some of them had to bounce off "something" small, heavy, and positively charged inside the atom.
*) Most flashes, as expected, appeared on the back of the flask in a straight line from the emitter, which corresponded to the passage of alpha particles through "gaps" between atoms, far from the nuclei. However, the particles passing through the inner part of the gold atoms showed considerable angles of deflection.
  To clarify these experimental results, Rutheford abandoned Thomson's model and proposed an image of an atom composed of a very small nucleus (less than ten thousandths of the diameter of the whole atom *), in which the positive charge and almost the entire mass of the atom are concentrated, and from electrons located at a certain (relatively large) distance from the nucleus. Namely nearby of this extremely small, heavy, and the positively charged nucleus, around which, according to the Coulomb law, very high electric field intensities acts, the alpha-particles that fly just around the nucleus are effectively scattered (Fig.1.1.4 bottom right). If the alpha particle flew relatively far from the nucleus, it did not disperse almost at all. The closer the path
a- particle approached the nucleus, the more it dissipated (due to the greater electrical repulsive forces).
*) Later measurements showed that the nucleus is even 100,000 times smaller than an atom (!) and clarified the structure of the atomic nucleus (described below in the "Atomic Nucleus" section).
  However, the electrons in this Rutheford model of the atom cannot be at rest, because the electrostatic force would draw them to the nucleus and the atom would collapse - they must move, orbit the nucleus *) along paths where the electric attractive force is balanced by centrifugal force, analogous to this is the case with planets in the solar system - a planetary model.
*) In quantum mechanics, an alternative explanation of the structure of atoms is sometimes given : electrons cannot fall on the nucleus due to quantum-mechanical uncertainty relations (discussed above in the "" section) and the fermion character of electrons. Electrons cannot acquire a smaller distance, or a lower energy level in the electric field of the nucleus than the lowest basic one; if we tried to "push" them even closer to the nucleus, they would "defend" themselves with an intense repulsive force - electrons as if their wave nature "did not fit " into such a small space - as if they suffered from a "claustrophobic effect". The so-called Fermi pressure of degenerate electrons arises here, which counteracts the electrical attraction of the nucleus. Electrons in the atomic shell according to Pauli's exclusion principle "pair" into pairs with opposite spin in areas of space called orbitals and the quantum-mechanical oscillations resulting from corpuscular-wave dualism prevent them from occupying smaller volumes. In our interpretation, however, for better comprehensibility and continuity between classical and quantum physics, we will stick to the usual "planetary" explanation - the orbit of electrons around the nucleus.
  Only in the quantum model of the atom will we use corpuscular-wave dualism to explain the quantization of electron orbits (Fig.1.1.6) and use the "claustrophobic effect" of quantum uncertainty relations to explain the repulsive forces in the nucleus at subnuclear scales
(see below "Strong nuclear interactions").
All nature is almost absolute emptiness ! 
Our whole nature is only emptiness, "polluted" by an almost negligible amount of matter. Everything around us is made up of only a small amount of real - "solid", concentrated - matter. This somewhat paradoxical statement clearly follows from the knowledge of the structure of atoms. An atom is not some solid mass ball, but it consists of a very dense nucleus of only
10-13 cm and an almost empty electron package. The nucleus, bearing more than 99.9% by weight of the atom, is about 100,000 smaller than the whole atom. This can be compared to a large sports stadium (representing an atom), in the center of which is a small children's ball representing the core; a few electrons would circle on the area of the stadium and the rest would be completely empty. As can be seen from Fig.1.1.4 on the right, the energy particle, in collision with the atom, in most cases flies through it as if through empty space; only if it accidentally hits the nucleus will there be a reflection or interaction.
  Thus, an atom is actually an empty space, "polluted" by several protons, neutrons and electrons. About 99.98% of each atom is an empty vacuum. Even our body, which is built of these atoms, is mostly formed by emptiness: the whole "real" mass of our body could theoretically be compressed into a ball with a diameter of about 1
mm, the rest would be emptiness. The same emptiness forms and all objects around us ...
Matter is mainly the field  
Thus, if atoms are predominantly empty space, at the mechanistic point of view there is a paradoxical question: "How is it possible that the most individual material objects do not penetrate with each other? Why do we see clear boundaries fixed objects?"- why don't we go trough the wall, or why don't we "penetrate the wood" while sitting and fall through the chair due to the force of gravity? The answer to this paradoxical question of why the material normally does not pass, is the electromagnetic field - a force evoked by charged protons and electrons in atoms. If the bodies get close to each other, their atoms (due to the deformation of the electronic configuration) begin to electrically repel each other and under normal circumstances there can be no interpenetration of atoms. When sitting on a chair, we are actually hovering ("levitating") over the upper layer of atoms of the chair's material, on the "pillow" of the electric field. The "penetration" of atoms occurs only at higher forces and energies, as discussed below in the section "
Interaction of atoms".
  It is similar with weight. The sum of the rest masses of the basic building blocks - quarks and electrons - of our body would represent only about 1% of the mass of our body
(§1.5, passage "Quark structure of hadrons"). The predominant part of the mass of our body (and of each material object) is formed by the kinetic energy of the building particles and the energies of the fields, according to the relation E = mc2.

Gnoseolological note :
Trajectories and orbits of particles <--> quantum states ?
Our understanding of nature is largely based on experience, expressed in classical mechanics
(and event. the theory of relativity).
That is why, even when transitioning to quantum physics, we use the idea of particle movement along certain trajectories and electron orbits along (quantized) orbits in the atom for better understanding.
However, according to the opinion of current quantum physics, these trajectories and orbits of particles do not exist, particles only occupy certain quantum states..?..
We will reflect on this dual epistemological view at a number of places in our discussion of atoms, particles and the physics of the microworld.

Planetary atomic model
E.Rutheford therefore based on the above-mentioned scattering experiment with alpha
particles during their passage of thin metal foils, formed the first realistic model of the atom - now commonly known planetary model according to which the atom consists of a positively charged core, around which orbit the negatively charged electrons (Fig.1.1.5 - where, however, the planetary model is already draw in an improved version of Bohr's model). The attractive electric force acting under Coulomb's law between the negative electrons and the positive nucleus is balanced by the centrifugal force created by the circular circulation of the electrons.
  For the motion of an electron with charge -e and mass me in the electric Coulomb field of a nucleus with charge +Z.e (Z is an atomic number, now called a proton number- see below "Atomic nucleus"), according to 2. Newton's law of force and Coulomb's law of electrostatics, the equation of motion apply
     me.d2r/dt2   =  F  =   -(1/4p.eo).(Z.e2/r2).ro   ,
where r is the position vector from the nucleus to the electron location, r is the instantaneous distance of the electron from the nucleus, r
o is a unit radius-vector pointing from the nucleus to the electron. The nucleus is considered to be motionless and infinitely heavy compared to the mass of the electron me . This equation of motion expresses the motion of an electron in the central field of the nucleus along Keppler orbits (generally an ellipse, hyperbola, parabola), similar to the motion of planets in the central gravitational field (for a detailed mathematical analysis in §1.2 "Newton's law of gravitation" in book "Gravitation, black holes and space - time physics"). In the simplest case of a circular orbit path of radius r we get a simple equation of motion
     me.v2/r  =  (1/4p.eo).Z.e2/r2  ,      
indicating the orbital velocity v the electron depending on the radius of orbit r. This basic equation of the planetary model of the atom can also be easily obtained as a condition of the balance of the centrifugal force m
e .v2/r, acting on the electron in a circular motion, and the attractive electric force (1/4p.eo).Z.e2/r2 of nucleus from the Coulomb law.
  However, the original planetary model had the drawback of being in conflict with classical electrodynamics: According to Maxwell's equations of electrodynamics, every electric charge moving with acceleration, emits electromagnetic waves. So every electron orbiting the nucleus (circular motion is uneven - the direction of the velocity vector changes - centripetal acceleration) should generate a periodically changing electromagnetic field, which would be manifested by the emission of electromagnetic waves carrying away the kinetic energy of the orbiting electron - see §1.5 "Electromagnetic field. Maxwell's equations.", Larmor formula (1.61'), monograph "Gravity, black holes and space-time physics". An electron braked in this way would orbit in a spiral and fall closer and closer to the nucleus, the intensity and frequency (equal to the frequency of the circular motion of the electron) of the radiation would increase, until the electron finally hit the nucleus *). Such an "electric collapse" the planetary atom would run very fast, in about 10-10 seconds for the hydrogen atom.
*) By substituting into the mentioned Larmor radiation formula -(dE/dt) = (2/3).(1/4p.eo).q2a2/c3 of the charge of the electron q = e and the acceleration of its circular motion a = v2/r = (1/4p.eo)Z.e2/me.r2 we get for the time change of the radius of circulation r (decreasing r , descending along a spiral) differential relation dr/dt = -(4/3).(1/4p.eo).(Z.e4/me2c3r2). By integrating its inverse form from r(t = 0) = rat - original radius of the atom rat » 10-10 m in the initial time t = 0, to r (t = tcol ) = rnuc - impact on the nucleus of radius rnuc » 10-14 m, we get for the collapse time tcol the value tcol = (4p2.eo2me2c3/Z.e4).(rat3 - rnuc3) » 10-10/Z [sec].
  Fortunately, we do not observe anything like that - atoms exist here and are stable! In addition, atoms with electrons in different orbits would emit different frequencies of electromagnetic radiation continuously, in contrast to experimentally observed discrete spectra of atomic radiation composed of individual spectral lines of precisely given wavelengths
(frequencies and energies) characteristic of different atoms (see below "Radiation of atoms") .

Fig.1.1.5. Schematic ilustration of the planetary structure of an atom in which negative electrons orbit a positively charged nucleus.
According to Bohr's model, electrons orbit the nucleus only along quantized discrete orbits on which they do not radiate. When an electron jumps from a higher to a lower path, the corresponding energy difference radiates as a quantum (photon) of electromagnetic radiation.
Is it an atom ?
Looking at this picture, almost every educated person says that "it is an atom". This answer is only partially correct, because in reality it is only a model of the atom. If we could reduce ourselves in the sci-fi concept to the size of one picometer and penetrate the atom, we would not see any "balls" - electrons orbiting another "ball" - nuclei. At most we could see only fluctuating and rippling fields, with different densities distributed around the orbitals.
  However, this drawing is quite apt (cf. the passage "
Ball Model" in §1.5 "Elementary particles "), can clearly display and understand most important processes in atoms - excitation and deexcitation, emission of photons of characteristic radiation, chemical fusion of atoms, processes of interaction of atoms with ionizing radiation, internal conversion of gamma and X radiation with emission of conversion and Auger electrons, other accompanying phenomena in radioactivity... That's why we will use it often.

Bohr's quantum model of the atom
The mentioned serious shortcomings of the planetary model of the atom was repaired in 1913 by the Danish physicist Niels Bohr, who based on experimental knowledge and in the spirit of the ideas of the emerging quantum mechanics supplemented the original planetary model of the atom with three important postulates :

The planetary model, supplemented by these three postulates, represents the famous Bohr model of the atom (Fig.1.1.5), which successfully explains the most important quantum properties of atomic structure, including discrete line spectra of radiation emitted by atoms (see below). Bohr's model has retained its validity to this day (with the relevant generalizations mentioned below).

The Atom and the Planetary System: Similarities and Differences
After discovering that the atom is a system of positively charged nuclei and negatively charged electrons bound by an electric force, the well-researched solar system bound by gravity became the inspiration for clarifying the structure of this atomic system. There is an obvious analogy on three points :

Based on these analogies, Rutheford's planetary model of the atom was created. However, there are also fundamental differences between the planetary system and the atom :

  These differences have forced Bohr's above-mentioned modification of the planetary model of the atom. Nevertheless, the planetary concept of the atom is still used in some illustrative qualitative considerations...
  One of the main differences between the classical electromechanical and quantum understanding of atoms is the mechanism of radiation of atoms. Radiation from atoms is not emitted continuously, but after quantum, and the frequency of the radiation f not given by the frequency of a periodic electron circulation, but the energy difference E of stationary orbits of electrons, combined with the relationship E = h.f between the energy of electromagnetic quantum (photons) and the frequency f corresponding electromagnetic waves (cf. above-mentioned "
Corpuscular-wave dualism").
Atoms are very empty !
If we compare the typical size of an atom 10-8 cm - the orbits of electrons - and the size of a dense nucleus 10-13 cm, we see how extremely empty the atom is: as much as 99.9999999999999% of the volume of the atom is empty space (vacuum)...

Why don't atoms glow at rest? - wave mechanism of quantization
The mechanism of quantization in Bohr's model of the atom can be most clearly understood by the idea of the corpuscular-wave behavior of the electron as it moves in orbit around the atomic nucleus. We will first consider the simplest case - the hydrogen atom.
  From a corpuscular point of view, an electron of mass m
e and charge -e, orbiting a proton of charge +e along a circular path of radius r with velocity v , is acted upon by a centrifugal force FC = me.v2/r and Coulomb attractive electrostatic force FE= (1/4p.eo).e2/r2. The condition of equilibrium (stability) of the path is then FC=FE, e.g. me.v2/r = (1/4p.eo).e2/r2, from which the relations for the radius of the path and the orbital velocity of the electron follow

  e 2                             e
r = ----------------  ,    v = -----------------  .
pe o m e v 2            Ö (4 pe o m e r)                  

However, if these relations are fulfilled, the electron could orbit according to classical ideas at any distance r from the center of the atom.
  From a wave point of view, a circulating electron can be considered as a wave whose Broglie wavelength is
l = h/me.v . In order for such a "electron wave" to orbit continuously along a path of radius r , an integral number of wavelengs l must be "deposites" on this path, ie either one complete Broglie electron wave l , or 2 wavelengths per circumference 2pr, 3l/circumference, 4l/circumference, etc. - Fig.1.1.6, in above. Only then do all the electron waves stack and follow each other smoothly along the entire circumference of the path. If less than a few wavelengths occur along the path (Fig.1.1.6 in below part), the wave continuity is broken and the path is not stable, there will be discontinuity and disturbing interference, which is formed into a quantum of electromagnetic radiation - a photon is emitted, which carries the appropriate amount energy and the electron goes to the nearest stable orbit with an integer number of Broglie wavelengths.

Fig.1.1.6. Above: An electron orbits a nucleus along a stable orbit indefinitely and without radiation, if its orbit contains an integer number n of Broglie wavelengths of the electron. Bottom: With a non-integer number of wavelengths, the "wave continuity" is broken and the orbit is unstable - a photon is emitted and the electron goes to a stable orbit with an integer number of wavelengths.

A circular path of radius r has a circumference of 2p r, so the condition for the stability of the path is
p r n = n. l , n = 1,2,3,4, ..... ,
where r
n denotes the radius of the path which contains n wavelengths l = h/me.v. Substituting for the orbital velocity from the planetary model v = e/Ö(4p.eo.mer) we get that only those electron paths are stable, whose radius is given by the relation

h2 eo            
n = n2 ------------ , n = 1,2,3, ..... .
p me e2               

For n = 1 we get the lowest (basic, unexcited) orbit of an electron in a hydrogen atom with radius r1 = 0.529.10-8 cm. This value is called the Bohr radius and is also considered as one of the model values of the electron radius re (see the discussion in §1.5, passage "Size of elementary particles ...").
   The integer n is called the main quantum number and determines not only the order of the "allowed" quantum path, but also the energy of the electron in a given quantum path: The total energy E of an electron in orbit is given by the sum of its kinetic energy E
k = (1/2) me,v2 and the potential energy v = e/Ö(4p.eo. mer) in the Coulomb electric field of the nucleus (we choose the zero potential point at infinity; the sign "-" means that the force acting on the electron is attractive). Thus E = Ek + Ep = me.v2/2 - e2/(4p.eor), which after substituting v = e/Ö(4p.eomer) gives E = e2/(8p.eor). For the allowed paths of the orbital radius rn they then discrete values of energy En are obtained :

                  m e e 4     1                           
n  =   - ----------. ---- , n = 1,2,3, .....,
8e o h 2    n 2                    

which are referred to as energy levels or shells. These levels are all negative (related to the fact that we have chosen the potential of the electrostatic field to zero at infinity), which means that the kinetic energy of the electron in the quantum path is not enough to free the electron from the attractive force of the nucleus and escape from the atom. The absolute value of electron energy |En| indicates the work (energy) that we would have to supply the electron to transfer it from a given quantum path n to infinity, ie to free it from the attraction of the nucleus, ie to release it from the atom.
   If an electron orbits on the lowest quantum path n = 1, we say that it is in basic (unawakened) state. The transition to at higher quantum path is possible only by supplying energy - excitation of the atom, which may occur either photon absorption or action of Coulomb electric forces when passing charged particle impact or another atom (e.g at higher temperatures). During the transition from this higher energy level n to the lower energy level n-1 , ie during deexcitation , the energy difference is emitted in the form of a quantum (photon) of electromagnetic waves of energy E = E
n-1-En and wavelength ... ..
If the energy supplied to the electron is higher than the binding
|En|, the electron is released from the field of the nucleus and flies out - ionization of the atom occurs.

An improved Bohr model ; quantum numbers
The original Bohr model applied to a hydrogen atom and considered only circular orbits of electrons. With the improvement of experimental spectrometric methods, it has been shown that the spectral lines of atoms are not simple, but double and multiple - the spectra show a fine structure. To explain this fine structure, Bohr's followers, especially A.Somerfeld, supplemented and perfected Bohr's original model of the atom.
   In addition to circular orbits, elliptic orbits of electrons with a longer (major) half-axis given by the principal quantum number n have been proposed, while the minor (shorter) half-axis is characterized by a second quantum number l , which can take discrete values from 0
£ l £ n-1. This quantum number 1 , formerly referred to as the minor quantum number, is now called the orbital quantum number and determines the magnitude of the angular momentum M1 of the electron in a given orbit. Quantum mechanical analysis gives a quantum value for the angular momentum :
l  = (h / 2p) . Ö[ l (l-1) ] , l = 0, 1, 2, ......, n-1 .
The duplication and fine structure of the spectral lines can then be explained by the transitions between energy levels with different quantum numbers n to different sub-levelss differing in the value of l , on which the total energy E depends only little.
   An electron orbiting at a velocity v along a circular path of radius r represents, from an electrical point of view, a miniature current loop flowing through an electric current I = e.v/2
p r (the v/2p.r indicates how many times an electron with charge e has passed through a given path point per unit time). This current loop generates a magnetic field and its magnetic moment is me = pr2.I = r.e.v/2 = (e/2me).me.r.v = (e/2me).M, where M is the orbital angular momentum of the electron. Since the angular momentum M is quantized (Ml = l.h/2p, l = 0,1,2, ......, n-1), the orbital magnetic moment of the electron me on a given quantum path is
m e  = m l . e.h / 2m e  = m l . m B , m l = 0, ± 1, ± 2, ....., ± l ,
where m
l is the magnetic quantum number and the constant mB is called the Bohr magneton - represents the smallest, elementary quantum of magnetic moment.
   In addition to the orbital magnetic moment caused by the movement of an electron in orbit, the electron also has its own so-called spin magnetic moment and its own "rotational" angular momentum - spin.

These properties are often simply explained by the rotation of an electron around its own axis - a rotating electron would have its rotational angular momentum and corresponding magnetic moment. However, this explanation is not entirely consistent, as the "circumferential speed" of the electron would have to significantly exceed the speed of light (contrary to the special theory of relativity) and it would not be possible to explain what force compensates for the enormous centrifugal force and holds the electron together. Spin must be considered as a purely quantum property of a particle, for which we do not have an exact classical model.

   For own angular momentum, i.e. the spin of the electron, then is hold that its projection to the axis of rotation can take only two values: either -
1/2 h , or +1/2 h; the spin magnetic moment of the electron is then given by the Bohr magneton: ± mB. For the momentum of the electron Ms and the spin magnetic moment ms the following applies: Ms = s . h , ms = -(e/me).Ms = ±mB, where s = 1/2 or -1/2. The number s is called the spin number, and in the electron can take values ±1 / 2 (in §1.5 "elementary particles" encounter with particles, e.g. mesons p , for which three values of the spin number are possible: -1, 0, +1).
   The interaction between the magnetic fields excited by the spin and orbital angular momentum of electrons, the so-called spin-orbital interaction, leads to the splitting of energy levels of electrons in atoms into nearby "subsurfaces", which is reflected in the spectra of radiation from atoms by splitting spectral lines into fine structure.
E.g. for hydrogen, the lowest energy level of the electron n = 1 is split into two subsets with the consensus and dissent spin of the electron and proton. The transition between these two states corresponds to the absorption or radiation of electromagnetic radiation with a wavelength of 21 cm. The emission and absorption of this atomic hydrogen radiation is very important in radio astronomical observations of outer universe.

a - the constant of the fine structure
For the construction of atoms (as well as atomic nuclei), the force with which particles interact with electromagnetic fields is of particular importance. In general, this force is expressed by Coulomb's law of electrostatics and Lorentz force acting on a charge moving in a magnetic field. In quantum physics, where the electrical charge is quantized in multiples of the elementary electron charge e , performs an interesting ratio that represents electrical, quantum and relativistic properties of electromagnetic interaction of charged particles in a vacuum: it is called fine structure constant *)
a = e 2 /2 e o h c = 0.0072973525376 = 1 / 137.0359996868 ,
where e is the elementary charge of the electron,
h is Planck's constant (reduced), c is the speed of light, e is the electric permittivity of the vacuum. The constant of a fine structure is a dimensionless quantity, its numerical value does not depend on the choice of units.
*) The name comes from the fact that this constant appears in the relations for the splitting of spectral lines of radiation of atoms into a fine structure due to the so-called spin-orbital interaction.between spin and orbital angular momentum of electrons, resp. between the magnetic fields excited by them. This constant was first used in 1916 by one of the pioneers of atomistics, A. Sommerfeld, in the study of the fine structure of electron levels in an atom. He extracted this dimensionless value from earlier Rydberg constants expressing the wavelengths of spectral lines at electron jumps between levels in an atom and interpreted it as a measure of relativistic deviation of spectral lines about Bohr's model (ratio of velocity v1 electron in the first orbit of Bohr's hydrogen atom in vacuum a = v1/c).
   This important physical constant characterizes the strength of the electromagnetic interaction, acting as coupling constant in quantum electrodynamics. It co-determines the properties of atoms, molecules and substances composed of them, as well as the properties of atomic nuclei, including nuclear reactions. Possible variability of basic natural constants during the evolution of the universe is sometimes discussed , and the fine structure constant could be a suitable tool for sensitive spectrometric analysis of radiation from outer space (see also the passage "
Origin of natural constants" in §5.5 "Microphysics and cosmology" monography "Gravity, black holes and space - time physics").

Gnoseolological note :
Trajectories and orbits of particles <--> quantum states ?
Our understanding of nature is largely based on experience, expressed in classical mechanics
(and event. the theory of relativity).
That is why, even when transitioning to quantum physics, we use the idea of particle movement along certain trajectories and electron orbits along (quantized) orbits in the atom for better understanding.
However, according to the opinion of current quantum physics, these trajectories and orbits of particles do not exist, particles only occupy certain quantum states..?..
We will reflect on this dual epistemological view at a number of places in our discussion of atoms, particles and the physics of the microworld.

Occupancy and configuration of electron levels of atoms
Let us now move from a hydrogen atom to more complex atoms with more electrons. Imagine that we have created a nucleus with Z protons and we place it in a space containing free electrons. By electric forces, this nucleus will attract electrons, which will gradually occupy the individual "allowed" quantum orbits around the nucleus until an electron shell formed by Z electrons is formed and the atom becomes electrically neutral.
Electron orbits, shells, levels, orbitals 
The electron orbits are quantized, so from an energetic point of view it would be most advantageous, if all electrons occupied the lowest energy level with the main quantum number n = 1. However, such "crowding" of electrons to one level does not take place. Conversely, according to the so-called Pauli exclusion principle *), only one electron can be in the same quantum state, so if the lowest energy levels are occupied, other electrons must occupy ever higher and higher levels.
*) This exclusion principle was derived by the Swiss physicist W.Pauli in 1925 on the basis of a series of experimental studies of the distribution of electrons in atoms. Later, this exclusion principle was theoretically justified as a consequence of the quantum-statistical behavior of particles with antisymmetric wave functions (relative to particle transposition) - the so-called fermions, which include also electrons (see §1.5 "Elementary particles").
   Using Pauli's exclusion principle, we can determine how many electrons can simultaneously orbit in the paths (subshells) corresponding to the main quantum number n : there are possible n-1 values of the orbital quantum number l , with for each l there are 2.1 +1 different values of magnetic quantum number m
l and two possible values of spin magnetic number ms (+1/2, -1/2). Thus, each subshell can contain a maximum of 2.(2.1 + 1) electrons and each shell with the main quantum number n a maximum of n-1 of these subshells, ie a maximum total of
l = 0 Sn-1 2. (2.l - 1) = l = 0 S n-1 4.l + 2.n = 4. (n-1) .n / 2 + 2n = 2 n 2
electrons. Said set of electrons forms the n-th shell (sphere, level) of the atom. These energy levels, corresponding to the discrete values of the principal quantum number n , are denoted by letters (in the direction from the inside of the nucleus): K, L, M, N, O, P. The number of electrons that can orbit at a given level is therefore not arbitrary, but is limited by the maximum number 2n
2 :
K (n = 1): max. 2 electrons, L (n = 2): max. 8 electrons, M (n = 3): max. 18 electrons,
N (n = 4): max. 32 electrons, O(n = 5): max. 50 electrons, P (n = 6): max. 72 electrons.
Electrons occupy paths gradually, starting with the K shell.
   This system of occupying electron shells and subshells in atoms, together with the analysis of the binding electric force of electrons, makes it possible to understand the most important laws of the chemical behavior of elements. If we sort the chemical elements in order by atomic number, elements with similar chemical and physical properties are repeated at regular intervals. This empirically determined periodic law was formulated by D.I.Mendelev in 1869 in his periodic table of elements, which was supplemented by his followers and specified in its current form
(see below the section "Interaction of atoms", passage "Periodic chemical properties of atoms").
   An analysis of the electron configuration mainly shows, that the electrons in a fully occupied shell, termed closed shell, are strongly bound, because the positive charge of the nucleus significantly exceeds the negative charge of the internal electrons causing electrically "shield". Distribution of the effective charge in of an atom containing only closed shells is perfectly symmetrical, the atom has no dipole moment, does not attract other electrons and its own electrons are strongly bound, such atoms do not enter chemical bonds, are chemically inert - this manifests itself in helium
2He, neon 10Ne, argon 18Ar, krypton 36Kr, xenon 54Xe, radon 86Rn *).
*) The fully occupied sphere K is for helium and the fully occupied sphere K and L for neon. However, in the case of heavier inert gases Ar, Kr, Xe, Rn, the filling of the outer shell is only 8 electrons, which is related to the dependence of the binding energy on the orbital quantum number, as a result of which the filling of some subshells can become energetically disadvantageous.
   In contrast, atoms with one electron in the outer shell easily lose this electron because it is weakly bound: it is relatively far from the nucleus, whose charge is shielded by internal electrons to an effective value of only +e - this explains the high chemical reactivity of alkali metals (and also hydrogen) with valence +1. On the contrary, atoms that lack one electron in the outer shell for closure try to obtain this electron by the attractive force of an incompletely shielded nuclear charge, which explains, for example, the increased reactivity of halogens. For chemical reactions, see the section "
Interaction of atoms" below, passage "Chemical fusion of atoms - molecules".

Due to the wave-stochastic laws of quantum physics, electrons in the atomic shell do not orbit along a predetermined path (precise orbit), but it is possible to determine only the area in which the electron is located with a certain probability. The region in which the electron is most likely to occur (> 95%) is called the orbital. The electrons in the shell are grouped into finer spatial configurations, orbitals, depending on the minor (orbital) quantum number l , which determines the type of orbital and the value of the magnetic moment of the electron (discussed above). In a given shell, according to the increasing minor quantum number, the orbitals are denoted sequentially by the letters s, p, d, f. On the basic shell n = 1 there is only one orbital 1s, on the shell n = 2 there are orbitals 2s and 2p, ..., on the shell n = 4 can be orbitals 4s, 4p, 4d, 4f respectively. Due to the quantum exclusion principle, electrons gradually occupy the s- orbital shell K, followed by s- and p- orbitals of the shell L, s- , p- and d- orbitals of the shell M, etc ... Each orbital is filled with one and then the other electron with opposite spins. The way orbital is occupied is sometimes called Hund's rule. Orbitals can be differently oriented in space, according to the magnetic quantum numbers m (takes values -l, ..., 0, ... + l). These orbitals of the same type, which differ only in spatial orientation, have the same energy - they are energetically degenerate.
   When an atom is placed in an external magnetic field, the originally uniform energy of the electrons within the orbital spreads to several different (though close) energy levels, depending on the different orientations of the orbitals. The motion of each electron inside the shell creates a magnetic moment
(the vector of which depends on the spatial arrangement of the orbitals), to which a force acts in the external magnetic field. This interaction leads to a fine splitting of the energy levels of the electrons. A very fine distribution of energy levels inside the orbitals also occurs due to the opposite spins (+1/2, -1/2) of electrons  (Stern-Gerlach experiment).
Designation of atoms of elements 
In connection with the structure of atomic nuclei
(see section "Atomic nuclei" below), atoms are characterized by two basic parameters :
- Proton number Z (formerly called atomic number A), indicating the number of protons in the nucleus - and for a neutral (non-ionized) atom, also the number of electrons in the envelope. This also gives the position of the element in Mendeleev's periodic table of elements.
- The nucleon number N (formerly called the mass number) indicates the total number of nucleons in the nucleus of an atom - the sum of protons and neutrons. Characterizes weight of atom (since the nucleons in the nucleus represent more than 99.99% by weight of the atom).
   A commonly used way to write these numbers in certain element X is via the upper and lower index:
NXZ. E.g. hydrogen 1H1, nitrogen 14N7, sodium 23Na11,... It will be discussed in more detail below in the "Atomic Nuclei" section.

Excitations and spectra of atomic radiation
According to Bohr's model, electromagnetic radiation is generated in the electron shells of atoms when electrons pass from higher levels to lower ones. Since the energy levels of atoms are quantized, photons of radiation with very specific energies are emitted from the shell of the atom - the spectral distribution of energies and wavelengths is not continuous, but discrete.
Excitation, deexcitation and ionization of atoms
Excitation of atoms
In order for an electron to transition from a higher to a lower level, the atom must first be supplied with energy leading to its excitation *) - to transition the electron to a higher energy level. This energy can be supplied either by a Coulomb electromagnetic interaction of an incoming charged particle (electron, proton, collision with another atom), or by photon radiation.
*) We are considering "already finished" atoms here, not a situation where atoms are just emerging - the formation of atoms is, of course, also accompanied by quantum excitations and radiation.
   Under normal circumstances, electron levels in atoms with the principal quantum number n are mostly excited to values of n+1, n+2 or n+few values of the principal quantum number. However, with the help of strong pulses of the electric field, electrons can also reach highly excited states with values of n~100 and more. Such atoms with highly excited electrons are sometimes called Rydberg atoms
(after J.R.Rydberg, who measured the spectra of excited atoms, especially hydrogen, at the beginning of the 20th century). The diameter of such atoms already reaches macroscopic dimensions of the order of micrometers. Using sophisticated methods with a combination of laser and microwave radiation, it was possible to prepare excited atoms with dimensions of almost one millimeter!

Strong electrical impulses or radiation can cause electrons in atoms to reach highly excited states - Rydberg atoms are created.

An important property of Rydberg atoms is their higher sensitivity to the electric field gradient. This makes it possible to "operatively" manipulate atoms in demanding atomic and particle experiments - see e.g. §1.5, passage "Artificial production of antimatter".
   Excited atoms themselves are usually very unstable. They deexcite either one-time to the ground state with the emission of visible or UV light, or jumps between higher levels with the emission of microwave radiation. The Rydberg valence electrons are excited into high orbitals considerably distant from the nucleus. A somewhat unexpected consequence of the high excitation of electrons to large values of the principal quantum number is the extended lifetime of the atom. This is due to the fact that an electron can only absorb or emit a photon with an appropriate energy equal to the difference between the levels (in the case of Rydberg electrons, these are energies in the deep infrared region). The lifetime of a Rydberg atom is proportional to n4.
   Highly excited Rydberg atoms can bind to each other by sharing electrons when they approach and form a regular arrangement into hexagonal planar clusters, the so-called Rydberg mass. During the formation of this cluster, valence highly excited electrons are delocalized and trapped in the potential wells created between the atomic nuclei. This confinement prevents electrons from leaving the cluster and causes the Rydberg matter to have a long lifetime. Another property is the very weak interaction of this matter with electromagnetic radiation - its "darkness". These properties make Rydberg matter a certain candidate for dark matter in the universe, or for some part of it (§5.6, section "Future development of the universe. Dark matter" in the book "Gravity, black holes ...).

Deexcitation of atoms
To dexcitation of the atom and emission of radiation then has mostly occurs spontaneously
(according to some concepts of quantum field theory, spontaneous deexcitations and emission of radiation is initiated by constant quantum fluctuations of vacuum). Electrons in atoms can only go over between existing discrete energy levels. Even here, however, there are certain limitations caused by the law of conservation of angular momentum. The photon carries the released angular momentum difference between the respective levels through its spin, which is equal to 1. The transition between levels whose angular momentum differs, for example, by 1/2, is therefore not possible by photon deexcitation - we say that this transition is "forbidden". If an electron occupies such a higher energy level, it cannot spontaneously go into a lower (basic) energy state - it remains "trapped" at a higher level for a longer time: such a state is called metastable or isomeric (atoms that are stuck in a metastable state differ from the original atoms only in the energy state). Deexcitation can be induced either Coulombically (non-radiation) by interaction with surrounding atoms or particles, or through a higher level by radiation.
   A similar mechanism of "allowed" and "forbidden" transitions - gamma-deexcitation - can be found even in the atomic nucleus in §1.2, part "
Gamma radiation", passage "Nuclear isomerism and metastability".
Metastable states of atoms are used in some radiation applications :
- Quantum light generators - LASERs, by irradiation with light, they "pump" electrons to metastable levels and also, by means of light, trigger a mass return transition, avalanche deexcitation, leading to an intense flash of light.
- Thermoluminescent dosimeters, in the sensitive substance of which electrons are excited to a metastable state due to exposure to ionizing radiation, and during later evaluation, deexcitation is induced by heating
(§2.2 "Photographic detection of ionizing radiation", passage "Thermoluminescent dosimeters").
   The very process of deexcitation of the excited energy level and formation of the emitted photon is very fast, but not immediate. According to the laws of quantum electrodynamics, deexcitation process of the electron in the atomic packing is about 10
-16 seconds.
Ionization of atoms 
If the atom, or any of its electron, energy supplied is higher than the binding energy of an electron at a certain energy level, this electron is released from the atom and wiil fly out - occurs ionization of the atom, from which it becomes an ion
(Greek ion = pilgrim, going). When an electron is ejected, the neutral atom becomes a positively charged particle, a cation. If there are enough electrons in the environment, the electron will be recaptured - the electron will recombine with the ion, creating a neutral atom and photon radiation of the electron's binding energy. Similarly, a "extra" electron (in an electric discharge, by another atom) may be passed to a neutral atom; the result is a predominant negative electric charge, an anion (negative ion) is formed.
   Ionization occurs by the impact of fast-flying particles - ionizing radiation - into atoms, by electric discharges, by the collision of fast-moving atoms in a substance heated to a high temperature
(several thousand degrees), by dissolving salts in water, by mechanical friction of substances ("static electricity"). Whenever an atom or molecule loses or gains an electron and the number of electrons no longer corresponds to the number of protons in the nucleus, an ion is formed - the atom or molecule carries a positive or negative electric charge. Atoms and molecules that have one or more free or missing electrons have an increased tendency to chemical reactions(see below "Interaction of atoms") due to electrical interactions with other atoms. At this reactions, atoms and molecules can formed, that othewise are already electrically neutral (they have the same number of protons and circulating electrons, so they are not ions), but they have unpaired electrons in their orbits: they are called radicals because they are chemically very reactive, with each other and with others surrounding atoms and molecules (their essential significance for radiation effects on matter and living tissue is discussed in more detail in Chapter 5 "Biological effects ionizing radiation").
Emitted radiation  
The energy of the emitted photons is given by the energy difference between the levels of electrons in the atom. The energy distribution of electron levels is completely characteristic of the atoms of a given element, so by measuring the spectrum emitted by a certain substance, we can determine the element whose atoms are located there - this forms the content of the atomic spectrometry.
   Spectral distribution of wavelengths, resp. the frequency or energy of the photons of the electromagnetic radiation emitted by the substances can, in extreme cases, have diametrically different shapes :

In terms of the positional relationship between the primary energy source, radiating atoms and the spectrometer, we encounter two types of spectra :

Interaction of atoms
The atomic structure of matter makes it possible to explain naturally and from a uniform physical point of view a number of important phenomena at the atomic and subatomic level, from which all properties and manifestations of matter are derived - chemical reactions and molecular properties, structure and properties of solids, liquids and gases, all thermal phenomena (kinetic theory of heat), electrical, magnetic and optical properties of substances.

Chemical fusion of atoms -> molecules
Each atom binds a number of electrons in its electron shell exactly equal to the number of protons, so that the atom is electrically neutral. However, this electrical neutrality of the atoms is fully manifested only at greater distances, where the field of the positively charged nucleus is perfectly "shielded" by the negative electrons in the envelope. In the close vicinity of the atom, however, we can encounter residual manifestations of electric forces *), caused by vector folding of electric field intensities from protons in the nucleus and from electrons located in different places of the electron configuration of the envelope. When twoo atoms are closely approached, these electric forces (initially repulsive) can lead to such a rearrangement of the configuration of electrons on the outer shells (e.g., to the sharing or transfer of electrons) that electric attractive forces can arise that permanently bind the atoms together to form a molecule - Fig.1.1.7. We say that there has been a chemical fusion of atoms. The combination of rearranged electron orbitals of individual atoms creates common molecular orbitals. From the energetic point of view, chemical bonding results in such a rearrangement of electrons (electron density) in the outer valence layers of nearby atoms, which has a lower energy than isolated atoms and is therefore more stable.
*) Interestingly, although electric forces have a long (unlimited) range, their "residual manifestation" - the "chemical" forces between atoms - are short-range. In the vector composition of electric forces from protons in the nucleus and electrons in the envelope, these forces are canceled at greater distances, but at short distances a non-zero "residue" remains. A similar mechanism is encountered in the atomic nucleus in short-range nuclear forces between nucleons, which are a residual manifestation of long-range strong interactions between quarks - see below "The structure of the nucleus", part "Strong nuclear interactions".
   Due to the energy released during the chemical bonding of atoms, molecules are formed in an energetically excited state. Deexcitation occurs either by emitting infrared radiation or by direct electromagnetic interaction with surrounding atoms and molecules. Radiation deexcitations is applied in reactions in thin gaseous medium, while in the dense medium of liquids and solids is dominant direct dexcitation with the participation of surrouding atoms and molecules. In both cases, the energy released during the chemical fusion is eventually transferred to the surronding atoms and molecules of the substance in the form of kinetic energy of motion - the substance is heated, the heat of reaction is generated (we mean here exothermic reactions, see below).
  When atoms approach each other, they are initially electrically repelled (eponymous charged electrons in envelopes). That is, so that the atoms can get close enough to each other - such that their orbitals blend and a chemical bond can form - a certain electrical repulsive barrier must be overcome. Appropriate activation energy must be supplied to the atoms. This is done by the kinetic energy of the thermal motion of atoms - a certain minimum temperature is required to carry out chemical reactions in reaction mixture. At low temperatures, chemical reactions do not take place *). At high temperatures, chemical reactions take place faster, but the mean kinetic energy of atoms and molecules can exceed the binding energy of atoms in molecules - during collisions, the molecules break down, the chemical compound again decomposes.
*) Another possibility of stimulating chemical reactions is irradiation with ionizing radiation. In the irradiated substance, electrons are released from the atoms and positive ions are formed. The resulting electric forces allow the merging of atoms without the need to impart kinetic energy to overcome repulsive forces. Radiation stimulation of chemical reactions plays an important role in the cold gas-dust clouds in space (see "Cosmic radiation"). However, already "finished" molecules are decomposed by ionization radiation - occurs radiolysis of compounds.
Yet unexplored possibility of chemical reactions at low temperatures is the mutual intersection of the wave functions of atoms trought a tunneling phenomenon ...

Energy and the kinetics of chemical reactions
As mentioned above, to effect a merger of two atoms it is necessary to supply them with certain activation kinetic energy Q
A. On the contrary, during the actual merger, the binding energy of the atoms in the molecule QR is released. From the point of view of energy balance, their difference Q = QR -QA - reaction energy is important. According to the sign of the reaction energy, chemical reactions are divided into two groups :
Endothermic (endoenergetic) reactions Q <0 ,
wherein the binding energy of the atoms in the molecule is less than the kinetic energy of the interacting atoms, "consumed" to overcome the repulsive electrical forces. Endothermic reactions cannot be maintained spontaneously, the activation energy must be supplied continuously from the outside; the rate of such reactions is then given by the "supply" of this energy. An example is the formation of carbon disulphide in the passage of sulfur vapors through a hot coal: C + 2 S
® CS2.
Exothermic (exoenergic) reactions Q> 0 , where there is a "release" and gain of energy, which is drawn from the binding energy of atoms in molecules. For exothermic reactions, there are several possibilities for their kinetics. Time course - kinetics exothermic reactions - decisively depends on the concentration of interacting atoms in the reaction mixture, pressure, temperature, the presence of other types of atoms or molecules.
   With a sufficiently high concentration of reacting atoms, a situation may arise where the released reaction energy during the merging of two atoms is efficiently transferred by electromagnetic interaction to the surrounding atoms. These atoms thus gain kinetic energy, causing them to merge immediately, releasing more energy - which is passed on and causes other atoms to merge. Upon delivery of the initial (initiating) activation energy, a chain chemical reaction is formed *). If the reaction mixture contains a large number of atoms in a sufficiently high concentration, this chain reaction has the character of an explosion: its velocity increases exponentially, in a small moment (of the order of ms) practically all atoms in the reaction mixture combine. Suddenly released heat of reaction heats the mixture to a high temperature (of the order of thousands of degrees), which causes a rapid expansion - the explosion of the reaction mixture. A well-known example is the ignition of a mixture of hydrogen and oxygen, a small spark of locally elevated temperature is sufficient. If the concentration of one of the components is lower, or the individual components are fed to the reaction space gradually, an equilibrium chain reaction having the character of a continuous combustion can be established.
*) The nuclear chain reaction has a similar kinetics, but a different mechanism fission of heavy nuclei of uranium or plutonium by neutrons - see §1.3, section "Fission of atomic nuclei".
   At low concentrations of reacting atoms, the chain reaction does not occur. When atoms are combined into a molecule, binding energy is released, which is emitted in the form of infrared photons. However, these photons fly away, the probability of their absorption by other distant atoms in a sparse environment is negligible. For chemical reactions to take place in a sparse environment, the activation energy must be supplied externally continuously (the situation is similar to endothermic reactions).  

Fig. 1.1.7. Symbolic representation of the mechanism of joining atoms and their electrical bonds in molecules.
Left: Covalent bond of two atoms caused by electron sharing. Right: An ionic bond of atoms caused by the handover of an electron from one atom to another.

Types of chemical bonding
When two atoms approach each other, there are basically three diferent possibilities of their interaction :

In addition to the above-mentioned purely covalent and purely ionic bonds, many molecules undergo a mixed type of bond in which the atoms share electrons unequally.
Metal bond 
In addition to the covalent and ionic bond between two atoms, there is another type of bond in which the electrons of the outer valence layer are not shared by two nuclei or atoms, but by a large number of atoms. This applies to metals. Metal atoms are characterized by a small number of electrons in the outer shell, usually one or two electrons. These external valence electrons from a larger number of nearby atoms can then form a single continuous cloud - the so-called electron gas, in an array of regularly spaced nuclei surrounded by electrons of the inner layers. A metal crystal is a kind of huge "molecule", made up of regularly distributed cations, between which binding electrons move freely. The electrostatic attractive forces between these cations and the electrons in the cloud then form a bond called a metal.

From a physical point of view, chemical bonds are described using several parameters, four of which are mentioned here :

In a similar way as atoms, molecules or atoms with molecules can also react together . A more detailed analysis of the mechanisms of atomic bonding belongs to the field of physical chemistry. The merging of specific types of atoms and the properties of the formed molecules (reactions of their further merging or decomposition) then form the main content of chemistry.
Periodic chemical properties of atoms
In the pre-scientific period, alchemists studied substances. However, they had no idea about atoms and their nuclei, but they also did not recognize elements and compounds. They judged the substances according to their external manifestations and a few simple "cooking" reactions that they were able to carry out. In the 18th century, when earlier errors of alchemy were gradually abandoned and distinguished by a number of attempts chemical elements and compounds, classical chemistry arose, as a science of the combination of elements, the properties of compounds, their other mutual reactions of compounding and decomposition. The most important finding was the discovery of the periodicity of the properties of elements according to their relative atomic weight (we now know that the atomic or proton number Z decides): if elements are sorted sequentially according to their atomic number, their chemical properties repeat after a certain sequence of elements (see also above part "Atomic structure of matter", passage "Periodic table of elements"). The systematic culmination of classical chemistry was the creation of a periodic table of elements (Mendeleev compiled its first version in 1869), in which the elements are arranged according to the ascending atomic number into horizontal rows (forming periods) so that the elements of similar properties get under each other (into columns). Mendeleev left a few blanks in the table and made the bold hypothesis that new elements would be discovered later to fill these gaps; it really came true.
   In the first half of the 20th century it was revealed that the periodicity of elements is based on the quantum behavior of electrons orbiting the atomic nucleus, in the laws of occupancy of individual orbitals (explained above in the section "Planetal and Bohr model of the atom", passage "Occupancy and configuration of electron levels").
The first period has only one type of orbital called "s", which can be occupied by one (for hydrogen) or two (for helium) electrons. The atoms of the second and third periods have, in addition to one orbital "s", three orbitals of type "p". Each of these 4 orbitals can again be filled with one or two electrons, with a total possible number of 8 electrons, creating a period of eight elements. The fourth and fifth periods have, in addition to the orbitals "s" and "p", a third type "d", which adds another 10 places for electrons - the length of the period is thus extended to 18 elements. The last two rows of the table contain heavy atoms with four orbitals of types "s", "p", "d" and "f" and have a period of length 18 + 14 = 32 elements. The last element with Z = 118 has all orbitals "s, p, d, f" filled with electrons.- see §1.3, section "Transurans"), a completely new table row will have to be created for them. From element 121, a new, as yet unknown type of orbital "g" would be added, which would extend the periodicity (number of columns) up to 50 elements. From the point of view of the chemical properties of atoms, however, this has only a debatable significance, superheavy nuclei immediately disintegrate and possibly their volatile atoms have different properties than those corresponding to the periodic table :
Violation of periodicity ?  
The exact periodicity of the physico-chemical properties applies only to lighter elements. The principle of similar behavior of elements in the same column of the periodic table may be violated for heavy atoms due to relativistic effects in an electronic package. With a high number of protons, the electric charge of the nucleus is high, which also leads to a high velocity of electrons on the internal orbitals. In heavy atoms, the internal electrons reach orbital velocities that partially approach the speed of light (they become "relativistic"), so the effects of the special theory of relativity are beginning to apply here. Due to relativistic contraction, the size (shrinkage) of the internal orbitals decreases. Reducing the radius of the inner orbitals results in an increase in the electrical "shielding" of the positive charge of the nucleus by these electrons, so that more distant electrons (no longer relativistic) are attracted to the nucleus by less force. External orbitals, especially valence ones, are less bound to heavy atoms than would correspond to a conventional non-relativistic quantum model of an atom. And also the energy distant between the outer levels is smaller. This is reflected in the optical properties of the elements and in some specific chemical reactions. Relativistic quantum mechanical effects cause atoms of very heavy elements in the region of transurams to behave chemically differently than we would assume based on their location in the columns of Mendeleev's periodic table
(will be discussed in §1.3, at the end of the section "Transurans", in the section "Chemical properties of transurans") .

Bonds of atoms and molecules in solids and liquids
In addition to the above-mentioned radiation phenomena and chemical fusion processes, electrical forces, given by electronic configurations of atomic shells, are also responsible for tight clustering of large numbers of atoms and molecules into solids and liquids as well as their properties - flexibility, strength, compressibility, electrical, magnetic and optical properties, thermal properties.
   The solids contain primarily the ionic and covalent bond analogs mentioned above in connection with the chemical fusion of atoms into molecules. In addition, significantly weaker so-called van der Waals forces are applied in liquids (and partly also in amorphous solids).
Van der Waals forces
All atoms and molecules (including atoms of inert gases of helium, argon, xenon, etc.) show a weak short-range mutual attraction, which is caused by the so-called van der Waals forces *). The basis of van der Waals forces are the attractive forces between the electrical dipole moments of atoms or molecules. For polar molecules that have a permanent electric dipole moment (such as an H
2O molecule, where the end of a molecule with an oxygen atom has a higher electron concentration and is more negative than the opposite part of a molecule with hydrogen atoms), the molecules orient each other with their ends of opposite polarity, creating an attractive electrical force.
*) Based on phenomenological considerations in 1873, J.D. van der Waals introduced these attractive "cohesive" forces (forces holding together) between molecules into his well-known equation of state of imperfect (ie real) gas, generalizing the equation of state for perfect gases in order to be able to explain gas condensation.
   However, the polar molecule can also attract molecules that do not normally have a permanent dipole moment: the electric field of the polar molecule when approached causes such a redistribution of charge in the second molecule that induces a dipole electric moment in the same direction as the polar molecule's moment - the result is an attractive force. A more detailed electrical analysis shows that the magnitude of this force FW ~ a.d2/r7 is proportional to the square of the dipole moment d and inversely proportional to the 7th power of the distance r ; a is a constant indicating the polarizability of the molecule.
   However, even for nonpolar molecules and for closed shell atoms where the electron distribution is symmetric on average and the mean dipole moment d is zero, the instantaneous dipole moment shows quantum fluctuations in magnitude and direction. Although the mean value of the dipole moment
< d > is zero, the mean value of the square of the dipole moment < d2 > it is not zero, but has a small finite value - this creates an effective attractive force between the two fluctuating electric dipole moments, which is proportional to ~ <d2>/ r7.
   Van der Waals forces are much weaker than the forces of ionic and covalent bonds. In addition, the high power of their indirect distance dependence, r
-7, causes short-range forces that only apply when molecules or atoms are close together (doubling the distance between two molecules will reduce the attractive force acting between them by more than 120 -times).
   Van der Waals forces cause the condensation of gases into liquids and the solidification of liquids into solids even when the mechanism of ionic or covalent bonding does not apply (eg to inert atoms with closed shells). These forces are also the basis for other properties of substances, such as viscosity, surface tension, adhesion, friction.
According to the state, we divide substances into three well known basic groups :  

Thermal motions of atoms and molecules
Atoms and molecules that make up substances are never at rest with each other, but perform constant movements. According to the kinetic theory of heat, the movements of atoms and molecules in substances are the cause and essence of all thermal phenomena. In solids, atoms and molecules exhibit oscillating motion in the crystal lattice. In gases and liquids, a disordered motion of elastically colliding *) atoms and molecules takes place (can be observed as the known Brownian motion).
*) At sufficiently high temperatures, however, these collisions of atoms and molecules are no longer elastic, the excitation of atoms and molecules occurs with subsequent deexcitation accompanied by radiation. At even higher temperatures, atoms are ionized and molecules decompose, a plasma state is formed, sometimes called the 4th state (see below the passage "Plasma - 4th state of matter") .
   Mechanical impacts of gas atoms and molecules on the walls of the vessel cause reaction forces that cause gas pressure.
The instantaneous velocity of the individual condensing gas molecules varies and changes irregularly over time, both in size and direction. In statistical mechanics, the so-called Maxwell-Boltzmann's law is derived by the statistical distribution of kinetic energies of moving molecules in a (ideal) gas ......... In a gas heated to the (absolute) temperature T, the mean kinetic energy <ek> per molecule is proportional to the temperature according to the relation: <ek> = (3/2) .kT, where k is the so-called Boltzmann constant, whose numerical value is k = 1.380x10-23 Joule/Kelvin. This constant is a kind of "conversion factor" between the energy measure of the temperature of a substance and a phenomenologically established temperature scale in degrees Kelvin (°K; the relationship between the absolute Kelvin scale and the "water" Celsius scale is T [°K] = 273 + t [°C] ).
Since the kinetic energy
ek of a molecule of mass m is related to its velocity v the known relation ek = (1/2) m.v2, for the velocity of molecules (so - called mean square velocity <vkv> - is the square root of the mean value of the square of the velocity of molecules) followed the relation: <vkv> = Ö<v2> = Ö(3kT/m). For ordinary gases at temperatures common in the Earth's air, these speeds are in the order of hundreds of meters per second. E.g. for hydrogen at 0 °C (= 273 °K) is <vkv> » 1300 m/s.
Mechanical impacts of atoms and molecules on the walls of the vessel cause reaction forces that cause gas pressure . Pressure P is expressed as the force acting per unit area, this force being given by the rate of change in time of the momentum of the incident particles. The momentum p = m.v a molecule of mass m is related to its kinetic energy by the relation
ek = p2/m. With each elastic impact on the wall, the molecule changes its momentum to the opposite, ie the total change in its momentum is Dp = 2.p. The momentums of the particles are oriented chaotically in all three directions in space, so that the number of particles hitting the wall is on average only 1/3 of their total number. The number of incident particles is further given by their number no in volume unit. After taking into account all these circumstances, the pressure is given by the relation: P = (1/3)<v2>, resp. P = (1/3).r .<v2>, where r is the gas density. The pressure of the gas on the walls of the vessel is thus directly proportional to the density of the gas and the mean value of the square of the velocity of its molecules.
............ equation of state ....
Heat propagation
Heat, ie disordered or oscillating motions of atoms and molecules, spreads in substances from one place to another in three basic ways :

The dependence between the absorbed amount of heat (energy) DQ and the temperature increase DT of a heated body of mass m is important for the thermal properties of substances. This dependence generally has a complex nonlinear course, but if there are no phase transitions (changes of state) and we do not move in a large temperature range (within the limit DT ® 0), this dependence is approximately linear: DQ = m.C. DT. Coefficient C in this dependence is called the specific or measure heat given substance.
................... ..............
   A more detailed study of thermal energy and thermal properties of substances forms the content of a special area of physics - thermals and thermodynamics.

Electromagnetic and optical properties of substances
The atomic and molecular structure of matter makes it possible to explain naturally and from a uniform physical point of view the interactions of electric and magnetic fields, electromagnetic waves (and especially light), with substances. All electrical and magnetic phenomena originate from the basic building blocks of the atom - electrons, as carriers of the elementary negative electric charge, and protons carrying a positive elementary charge. And forces acting - interactions - electric and magnetic fields with atoms and molecules of the material environment cause all the peculiarities and differences of electromagnetic phenomena in comparison with these phenomena in vacuum. It will be shown below that from a macroscopic point of view, the electromagnetic field in a material environment (especially dielectric) can be described by essentially the same Maxwell's equations as in vacuum, in which the values of electrical permittivity
eo and magnetic permeability mo of vacuum are replaced by respective coefficients e and m for the substance.

Electrical phenomena in the material environment
Based on the properties of atoms and molecules, it is possible to explain electrostatic phenomena, including the very "formation" of an electric charge. Interactions of atoms and molecules in substances (in the simplest case by mechanical friction of two bodies) can release a certain number of external electrons from atoms. If a large number of these electrons accumulate on one of the interacting bodies, this body with an excess of electrons has a negative electric charge, while in the other body with an excess of protons a positive electric charge is applied. Such electrically charged bodies with charges Q
1 and Q2, placed in a vacuum at a distance r, will exert a force on each other according to the known Coulomb's law F = k. Q1.Q2/r2, where k is a coefficient expressed in a system of SI units using the so-called vacuum permittivity eo : k = 1/4peo .
   If electrically charged bodies are placed in a material environment, in addition to their mutual Coulomb interaction, their electrical interactions with atoms and molecules of matter will also occur. The basic nature of this interaction will depend primarily on whether or not the substance contains freely moving electric charge carriers.
  Solid state physics describes the electrical properties of these substances using the so-called band theory, according to which electrons in matter are combined into energy bands, separated from each other by unoccupied bands of "forbidden" energies. Discrete energy states of electrons orbiting individual atoms in solids in orbits propagate into energy bands due to interaction with other atoms in the solid, but there are certain gaps between these bands - the so-called bands of forbidden energies, which electrons cannot acquire. The energetically highest occupied band is the valence band , followed by the forbidden band and above it lies the so-called conduction band of electrons, which already behave as free. If the forbidden band is wide, the conduction band is completely unoccupied in the equilibrium (basic) state, all electrons are bound and the substance is electrically non-conductive. Otherwise, electrons jumping into the conductivity band cause the electrical conductivity of the substance.
   From this electrical point of view, substances are divided into two extreme groups :

- substances that contain freely moving electric charges (or their carriers). The electric field, with its force effects, sets the carriers of the electric charge in motion - an electric current is generated, which lasts until the rearranged electric charges disrupt the electric field; the charges equalize. According to the nature of moving electric charge carriers, electrical conductivity is divided into two types :
-> Electron conductivity caused by freely moving electrons. The conductivity band is so close that it overlaps with the valence band and the outer electrons pass freely into the conduction band. It occurs mainly in metals, where part of the outer electrons is not bound in atoms in the crystal lattice, but is freely dispersed and forms a so-called electron gas. Metals are therefore very good conductors of electricity and also heat. The large amount of weakly bound electrons in the metal conduction band allows relatively easy release of electrons from their surface. Heating the metal (to a temperature above about 400 °C) by increasing the kinetic energy of the electrons causes the thermoemission of the electrons. Similarly, the impact of electromagnetic radiation, light and harder radiation leads to the photoemission of electrons - a photoelectric effect (it was analyzed in more detail above in the section "Corpuscular-wave dualism", passage "Photoelectric effect").
-> Ionic conductivity caused by the movement of positively or negatively charged ions - atoms with missing or excess electrons in the envelope. This type of conductivity occurs in solutions with dissociated molecules - so-called electrolytes, or in ionized gases (electric discharges).
  The movement of electric charges in conductors is not completely free, the carriers of electric charge collide with atoms and molecules in matter, thus transferring to them part of their electrically obtained kinetic energy. Electric current generates heat, conductors put resistance to electric current (expressed in Ohms). The only exception is the phenomenon of so-called superconductivity, when electrons
(connected in so-called Cooper pairs forming Bose-Einstein condensate) move completely freely in the conductor and the electrical resistance drops to zero (§1.5, passage "Fermions as bosons; Superconductivity").
-> Semiconductivity. A special group of substances are semiconductors, substances with a narrow band gap, where electrons jumping from the valence band to the conduction band (by thermal motion or photoexcitation) become negative conductivity carriers and empty spaces in the valence band - so-called holes - effectively appear as positive conductivity carriers. By incorporating suitable impurities of elements that provide conductivity electrons (donors) or accept electrons from valence band bonds (acceptors) into semiconductor materials, an increase in their conductivity and a predominance of free negative ("n") or positive ("p") carriers can be achieved. Very important electrical phenomena occur at the interface adjacent semiconductors of type "n" and "p" - rectifying "diode" effect at the n-p interface, amplifying "transistor" effect at the p-n-p or n-p-n interface, as well as optoelectric phenomena. The most important semiconductor materials are germanium and silicon.
  In addition to wide use in electronics (transistors, diodes, LEDs, integrated circuits, computer processors, memory chips, optoelectric display chips, ...), germanium and silicon are also used in semiconductor detectors of ionizing radiation (§2.5 "Semiconductor detectors"). For some purposes, a combination of tellurium, cadmium and zinc is also used here - CZT detectors.
2. Non-conductors (insulators, dielectrics)
- substances in which no freely moving electric charges are present
(the conduction band is separated from the valence band by a wide band of forbidden energies, so that electrons from atoms do not get into it). Here the electric charge of the inserted bodies can persist, the non - conductive substance is able to separate (isolate) charges of various sizes and signs. Atoms and molecules remain generally electrically neutral, but the force of the electric field leads to a certain rearrangement of the charge distribution in atoms and molecules - the so-called dielectric polarization (Fig.1.1.8 on the right). Originally, the spatially symmetric charge distribution in the time average *) is slightly deformed due to electric forces - the positive charge is effectively shifted in the direction of the field, the negative charge in the opposite direction. The effect of so-called sliding charges arises.
*) This applies to atoms and so-called non-polar molecules with a symmetric spatial distribution of positive and negative charges. In addition, there are polar molecules, in which the atoms are bound by ionic bonds, with an asymmetric charge distribution forming a miniature electric dipole. However, the orientation of these molecular electric dipoles in the substance is completely disordered due to thermal movements, so that their electrical effects are canceled outwards (Fig.1.1.8 in the middle). However, the external electric field exerts a force on the individual dipoles and partially orients them in the direction of the field - there is an orientational polarization of the dielectric (Fig.1.1.8 on the right). In addition, the force of the field will somewhat increases the dipole moment of the polar molecules thus oriented.

Fig.1.1.8. Polarization of dielectric atoms and molecules and the formation of sliding charges.
Left: The electric field between two electrodes of charge +Q and -Q has an intensity E
o in vacuum. Middle: In the absence of an external electric field, atoms and non-polar molecules have a symmetrical charge distribution on average, and polar molecules have random chaotic orientations of their dipole moments. Right: The action of an external electric field deforms the originally symmetrical charge distribution in atoms and non-polar molecules - they become electric dipoles; for polar molecules, dipole moments are oriented. In both cases, the dipole moments are oriented opposite to the electric intensity vector Eo external field - dielectric polarization effectively reduces the intensity of the applied field from the maximum vacuum value Eo to the value E .

The result of electrical interaction with atoms and molecules (non-polar and polar) of the dielectric is the formation of electrical dipoles oriented in the field direction. The electric field of the electric dipoles d induced in this way adds to the original acting field Eo - and since it is in the opposite direction, it effectively reduces the value of the electric field intensity, reduces the electric force to the value E < Eo . For not very strong electric fields, the polarization P is directly proportional to the intensity of the electric field: P = k . E , where the coefficient k is called dielectric succeptibility (polarizability) of the dielectric. Coulomb's law still applies to the force action of electric charges in a substance, but in the proportionality constant, instead of the permittivity of the vacuum eo, performs the permittivity of the substance e, also called the dielectric constant, e : e = eo + k = er . eo, where er = 1 + k is the so-called relative permittivity of substance. The relative permittivity of substances is always greater than 1, for non-polar and dilute substances only slightly (for air only 1.006), for polar substances it can be quite high (for water er = 81).
Magnetic phenomena in the material environment
Magnetic phenomena are a manifestation of the interactions of moving electric charges. The moving charges, creating a current I in the length element dl , generate a magnetic field of intensity B *) at a distance r according to the Biot-Savart-Laplace law: d B = k. I. [dl
´ ro] / r2, where ro is the unit direction vector from the measured point to the current element and k is the proportionality constant expressed in the system of SI units using the so-called vacuum permeability mo : k = mo/4p. The magnetic field then shows force effects on each electric charge q moving at a speed v: F = q. [B ´ v]; this so-called Lorentz force acts perpendicular to the direction of movement of the charge.
*) For historical reasons, the quantity B is called not intensity but magnetic induction .
   If we insert a substance into a magnetic field, the atoms and molecules of the substance will interact with the magnetic field, leading to the magnetization of the substance. This is because the electrons moving in the atomic shells generate their elementary electric currents ("current loops"), which excite their elementary magnetic fields expressed by the so-called magnetic moment m = I. S , defined as the product of the current I and the area S of the current loop. In atoms, elementary current loops and magnetic moments are caused by two types of electron motion: 1. The cycle of an electron along its path - trajectory or orbital magnetic moment; 2.Due to electron spin - spin magnetic moment. The resulting magnetic moment of an atom is the vector sum of the moments of all its electrons. During this vector addition, three significant cases can occur :
a) All moments are compensated for each other, the resulting moment is zero. In such atoms, when inserted into a magnetic field, the electron paths are deformed so that additional magnetic moments are induced, the field of which is directed
(in connection with the so-called Lenc rule of the opposite effect) against the direction of the external field. Thus, the field weakens, such substances are called diamagnetic.
Only spin moments are compensated. In the external magnetic field then occurs twisting the magnetic moments of the individual atoms in a direction consistent with the external field, thereby amplifying the resulting magnetic field. Such substances are called paramagnetic. However, the tendency of magnetic moments is counteracted by the thermal motion of atoms, which in turn puts atoms into a state of chaotic disorder - according to the so-called Curie's law, the magnetic amplification effect (so-called magnetic susceptibility) is inversely proportional to absolute temperature.
c) Atoms have uncompensated spin moments (this occurs with atoms that do not have a fully occupied electron level). In this case, some substances may occur in certain small areas spontaneous orientation of all magnetic moments in one direction - the so-called magnetic domains is created, which is magnetized to a saturated state (the size of these domains is about 10
-6 -10-2 cm). Under normal circumstances, these domains in the substance are randomly distributed and oriented, so that their magnetization is canceled. However, when an external magnetic field is inserted, these domains are easily oriented so that the vector of their magnetization is directed in the direction of the field - there is a total magnetization of the substance, which significantly amplifies the applied magnetic field. Such substances with significant magnetic properties are called ferromagnetic (according to iron, which is the oldest known substance of this kind). Ferromagnetic properties disappear at higher temperatures, when the domains of spontaneous magnetization decay and the substance acquires paramagnetic properties (the relevant boundary temperature, characteristic for a given substance, is called the Curie temperature ) .
   The excitation of the magnetic field in the medium can again be expressed using the Biot-Savart-Laplace law, but in the proportionality constant instead of the vacuum permeability
mo, performs the magnetic permeability of the substance m = mr . mo, where mr = m/mo is the so-called relative permeability of the substance, indicating the "amplifying" or "attenuating" effect of the substance on the magnetic field (here is mean the "permeability, transmisivity" for the magnetic field).
For diamagnetic substances is
mr <1, the paramagnetic agent is mr > 1; in both of these cases, however, the value of mr is very close to 1. For ferromagnetic substances, mr reaches high values of the order of 103 -105 (here, however, it is not a constant, but a variable whose value depends on the intensity of the magnetic field; for strong fields, the state of magnetization saturation is reached, the hysteresis effect also manifests itself).
The magnetic force action of minerals - permanent magnets - known since antiquity - has long been separated from electrical phenomena in the understanding of science. The mentioned theory of magnetic moments of atoms and molecules shows that even in permanent and natural magnets the origin of the magnetic field lies in the interactions of moving charges. These are so-called magnetically hard ferromagnetic substances (mostly containing iron), which retain a certain remanent magnetization even without an external magnetic field.

Electromagnetic waves in matter
Propagation of electromagnetic waves in matter at the classical level given by Maxwell's equations (§1.5 "Electromagnetic field. Maxwell's equations" books "Gravity, black holes and spacetime physics") , in which instead of vacuum values of electrical permittivity eo and magnetic permeability mo are the respective coefficients e and m for the given substance (their origin was discussed above in the passages "Electrical and magnetic phenomena in the material environment"). If the medium contains free charge carriers, the non-zero current density j will appear on the right side of Maxwell's equations, which in the simplest (linear) case is given by Ohm's law - this expresses a direct ratio between the specific conductivity of material s and the current density j flowing through the material after application of electric field E : j = s.E. Using the ohmic resistivity rohm = 1/s, the current density can be expressed equivalently as: j = E/rohm.
   These Maxwell's equations in the material environment have a wave solution
2 E / x2 + 2 E / y2 + 2 E / z2 = em .¶2 E / t2 + sm .¶E / t ,
which differs from the normal vacuum wave equation in that it also has a term with the first derivative according to time
s.m .¶E/t, which describes losses - damping - absorption of waves in a given material due to excitation of currents (sometimes called the "telegraph equation", because the attenuation of the signal in the telegraph line behaves in an analogous way). The resulting attenuation causes that if an electromagnetic wave of circular frequency w = 2p f with an input intensity I0 falls into the substance, with increasing depth d its intensity I will decrease according to the exponential law
      I (d) = I
0 . e - [ Ö ( w.s.m / 2)]. d
with absorption coefficient
Ö(w.s.m/2), increasing with wave frequency and specific conductivity of the material. Sometimes the value of the effective depth of penetration of the electromagnetic wave into the substance de = Ö(2/ws.m) = Ö(rohm /p.F.m) is introduced, at which the amplitude of the electromagnet. waves drop to 1/e. With good dielectrics with low conductivity (0, rohm®¥), electromagnetic waves pass almost without attenuation to great depths. On the contrary, in metals with very good conductivity for free electrons (s®¥, rohm®0) electromagnetic waves almost do not penetrate *), they are reflected from their surface (in electronics this manifests itself as the so-called "skin effect" for conductors through which high-frequency alternating current flows - it flows only at the surface of the conductors). The space surrounded by a sufficiently dense wire mesh therefore functions as a so-called Faraday cage, shielded against external electromagnets. waves.
*) Eg. for copper having a conductivity s = 5.8x107 S/m, or electrical resistivity rohm = 1.68x 10-8 W.m, the electromagnetic wave of frequency 1MHz penetrates to a depth of about 65 micrometers, at a frequency of 300MHz the penetration depth de is only 3.8 micrometers.
  These simple laws, resulting from the classical electrodynamics of the continuum, apply accurately enough to the "usual" electromag. waves of longer wavelengths
(much greater than interatomic distances) and not too high frequencies (max. up to "optical" frequencies of about 1014 Hz, in rare cases for some optically transparent dielectrics up to 1015 Hz). Due to the very high frequencies and short wavelengths, the atoms do not "manage" to react so quickly, the response of the substance is no longer synchronous. Atoms begin to oscillate in the crystal lattice and in molecules (where their vibrational and rotational modes can be excited), for higher energies the electrons in the atoms also oscillate (excitations and ionizations can occur). Quantum laws of discrete energy levels are beginning to be applied, electrons rising from the valence band to the conduction band are appearing. Radiation absorption consists in the exchange of energy of penetrating photons with the environment, in which part of the energy is converted into the kinetic energy of the atoms of matter - heat. The passage or absorption of radiation is spectrally selective, significantly depending on the wavelength (frequency). And for very high frequencies - high photon energies - the classical optical laws disappear, discrete quantum interactions occur (see §1.6, passage "Gamma radiation interactions"). Depth dependence of radiation absorption still retains exponential character, but absorption coefficients are no longer related to continuum electrodynamics (independent of permeability and specific conductivity of material), but are given by effective cross sections of radiation interaction with matter atoms (very complex functional dependences for different materials and energies radiation). For electromagnetic radiation X and gamma, the absorption is discussed in §1.6, section "Absorption of radiation in substances", Fig.1.6.5.
  The ability of a material environment to attenuate the radiation that passes through it is called opacity
(lat. opacitas = shading, shadow). It expresses the degree of "opacity" of a substance, quantitatively expressed by the ratio of the intensity of incident radiation and radiation transmitted through the substance. For longer wavelengths, the opacity is caused by the above-mentioned ohmic losses, for shortwave radiation, the absorption of electromagnetic radiation is the result of electrons in atoms - excitation and ionization, or absorption and scattering by free electrons.
Note :  Some very good dielectric (capable of forming a transparent crystal) when they are heterogeneous and multi-crystalline, can be opaque due to multiple refractions, reflections and scatterings between individual crystals.
  other ("foreign") atoms or molecules in the crystal lattice form in the regular lattice the local "centers" of different binding energies of the atoms, which can oscillate into other energy modes by the electromagnetic wave and thus affect the optical properties of the substance. This usually leads to increased absorption of radiation of certain wavelengths ...
Optical properties of substances
As mentioned above in the section "
Electromagnetic fields and radiation", light is an electromagnetic wave of short wavelength (approx. 360-750nm). The optical phenomena of refraction and reflection of light are described at the macroscopic level by simple laws of geometric optics. At the microscopic level, however, these simple laws are the result of much more complicated interactions of electromagnetic waves with atoms and molecules of matter. As an electromagnetic wave passes through a material, the electrons in atoms and molecules are subjected to electric and magnetic forces, under the influence of which they move. The reaction to the electrical component of the wave is an oscillating motion of electrons in the material, the magnetic field causes a circular motion. These movements cause periodic polarization of the atoms and molecules of matter, which affects the properties of the wave and its propagation. The higher the effective polarization induced by the wave (depends on the coefficients e, m and on the difference of the wave frequency from the natural frequency of oscillations of atoms and molecules in the substance - will be discussed below), the slower the electromagnetic waves propagate in a given optical environment.
   It should be noted that the dimensions of the atoms of matter are significantly smaller (about 4 orders of magnitude) than the wavelength of visible light. Such an electromagnetic wave therefore does not "see" individual atoms and molecules, but interacts with the "collective" response of millions of atoms or molecules. From a macroscopic point of view, therefore, the response of a material to such "long" electromagnetic waves can be described by two standard parameters known from the science of electricity and magnetism :
- electrical permittivity
e characterizing the polarization response to an electric field;
- magnetic permeability
m, which expresses the reaction of orbiting electrons (forming elementary "current loops") to a magnetic field.
   The electromagnetic wave will then be a wave solution of Maxwell's equations, in which instead of vacuum values of electrical permittivity
eo and magnetic permeability mo the respective coefficients e and m for a given substance will appear ("Electromagnetic field and radiation"). In a dielectric medium transparent to electromagnetic waves of the appropriate wavelength, the velocity c´= 1/Öem of the propagation of this wave will be less than c = 1/Öeomo in vacuum *). From Huygens' law of waves, it follows that at the interface of two optical materials with different velocities c1 and c2 the propagation of waves will change the direction of propagation - the refraction of light according to Snell's law sin a/sin b = c1/c2 = n, where the refractive index is given by the permittivity and permeability n = Öem. The law of reflection follows from the same Huygens law from an environment into which electromagnetic waves cannot penetrate (which are mainly materials with free-moving electrons, such as metals, or some optical interfaces).
*) Clearly, we can imagine this slowing down of light in such a way that individual photons are repeatedly absorbed by atoms or molecules in matter and then emitted again. This causes them to "delay" in time, which appears macroscopically to slow down. However, in the intervals between the radiation by one atom and the absorption by an adjacent atom, they move at a basic velocity c in a vacuum!
However, it is only an auxiliary idea, the interaction here does not occur at the level of individual photons with atoms, but collectively with many thousands of atoms. An even coarser example: A classic express train (not a special express train) and a passenger train, if pulled by the same type of locomotive, move at the same speed between the stations. As a result, the passenger train moves more slowly due to time delays at many stops...
  In material optical environments, the speed of light is slightly lower than in a vacuum and depends somewhat on the wavelength, ie the frequency of light - the so-called dispersion *). E.g. in water the speed of light for red light is (rounded) 226 000 km/s, for violet 223 000 km/s; it is even slower in crystals and glass. Of all natural materials, diamond has the highest refractive index (n = 2.42), in which the speed of light is only 123,881 km/s - this leads to significant optical effects of refraction and reflection of light in diamond crystals, which is the source of its aesthetic popularity as jewelry.

*) The dispersion phenomenon is caused by the frequency dependence of the polarization of the dielectrics in the variable electromagnetic field of the passing wave. Charged particles (negative electrons and positive nuclei), which are part of atoms and molecules, are held around their equilibrium positions by elastic (quasi-elastic) electric forces. In the field of these forces, each atom or molecule has a certain own frequency of oscillations f
o. Due to the incident electromagnetic wave, the charged particles in the molecules and atoms perform forced oscillations with a frequency equal to the frequency of the incident wave f . If this frequency is far from the frequencies fo of the natural oscillations of atoms or molecules, the resulting effective polarization is small and light passes through the medium at a little reduced speed; at the same time absorption and dispersion are small. If these frequencies are close, partial resonance occurs and the speed of light differs significantly from the vacuum value of c , or the refractive index differs significantly from one. For f <fo the refractive index will increase with frequency and will be quite high in the vicinity of fo , for f> fo the refractive index will decrease with frequency ("anomalous" dispersion). Significant resonant absorption occurs for frequencies f close to fo, the material is almost opaque to light of this wavelength. In the visible light range, most materials show a "normal" dispersion, in which the refractive index increases with frequency. For other wavelengths (beyond the region of the resonant frequency) we can also encounter anomalous dispersion.
  When the wavelength of the electromagnetic wave is shortened, ie with the growth of the energy of the photons, the individual interaction with the individual atoms and molecules of the substance begins - the laws of geometric optics gradually disappear. For the area of softer X-rays, the effects of diffraction on the crystal lattice of the substance are applied, for harder X-rays and g-rays, any optical phenomena of reflection and refraction longer do not show, this ionizing radiation interacts hard with individual atoms through photo- effect, Compton scattering and the formation of electron-positron pair (see §1.6 "Ionizing radiation" passage "Interaction of gamma rays" Fig.1.6.3 ).

Electro-mechanical, electro-thermal, electro-chemical, electro-optical phenomena
Mutual electromagnetic interactions of atoms and molecules and their interactions with external electric and magnetic fields are the cause of many other related phenomena on the border of electricity and mechanics, thermals, chemistry, optics, biophysics. We can name for example :
¨ Piezoelectric phenomenon - mechanical deformations of some crystals (eg quartz) cause the opposite charges on the walls of these crystals. Conversely, if we apply to the opposite walls of the crystal electrodes with opposite charges, the crystal is slightly deformed in this direction (electrostriction). A similar electrical effect occurs when heating crystals - a pyroelectric phenomenon.
¨ Magnetostriction- change in length dimensions and volume caused by magnetization of ferromagnetic substances.
¨ Thermoelectric phenomenon - the formation of electrical voltage or current when heating materials to different temperatures. Conversely, the formation of thermal gradients in the passage of electric current. The cause of these phenomena is thermal movement and diffusion of free carriers of electric charge. These include the Thomson effect in a conductor with a temperature gradient, or the Seebeck and Peltier effects at the interface of two conductors with different Fermi levels, where a contact potential arises.
¨ Photoelectric effect- electron emission or change in the electrical properties of the substance during light irradiation. When electromagnetic waves hit a substance, it interacts with atoms and electrons in the valence or conduction band. Upon absorption of this energy by a weakly bound electron in the conduction band, its photoemission can occur - an external photoelectric effect. If the radiant energy is absorbed by an electron in the valence band, it can jump into the conduction band - an internal photoelectric effect, which creates free carriers of electric charge and the conductivity of the material occurs (or increases).
¨ Electroluminescence- emission of photons of light by the effect of the passage of an electric current. Photons of light are created when electrons jump from a higher energy level of the conduction band to a lower level of the valence band (the electron recombinates with the hole), or through the level of a suitable admixture. In the so-called LED diodes, this phenomenon occurs in the area of the p-n transition.
¨ Electrochemical phenomena - change in the chemical composition of compounds and chemical reactions caused by the passage of an electric current. It is mainly electrolysis - the excretion of substances on the electrodes when an electric current passes through a solution of dissociated compounds (electrolyte).
¨ Electric discharges in gases- passage of electric current through ionized gas. The formation of free carriers of electric charge - electrons and ions, or ionization, occurs either by heating to a high temperature, or by the absorption of electromagnetic or corpuscular radiation of sufficient quantum energy. Ionization can also be caused and maintained by electrons and ions accelerated by the electric field between the electrodes during own discharge.

Plasma - 4th state of matter
At high temperatures, in an electric discharge or by the action of ionizing radiation, electrons are ejected from the gas atoms and the atoms themselves become positive ions. Such a partially or fully ionized gas is called plasma
(Greek plasma = ductile material; the electric discharge copies the shape of the tube and its shape is easily influenced by electric and magnetic fields). Plasma is sometimes referred to as the 4th state of matter (1st solid, 2nd liquid, 3rd gas, 4th plasma). In order to distinguish this ionized substance from other situations with electrically charged particles, we require two additional properties in the physical definition of plasma :
- Electrical neutrality on a macroscopic scale (on average the same number of electrons and positive ions) - we do not consider charged particle beams to be plasma;
- Collective behavior caused by a long-range interaction of sufficiently close charged particles - it is not a very dilute or weakly ionized gas by plasma.
  Thus, the general physical definition of plasma is: " Plasma is a set of particles with free charge carriers that is globally neutral and exhibits collective behavior ". This definition also includes exotic forms of the substance, such as quark-gluon plasma
(§1.5, passage " Quark-gluon plasma -"5th state of matter" ") .
   Plasma has significant electrical properties: it is electrically conductive, it reacts to a magnetic field, it can generate electric and magnetic fields on its own, complex electro- and magneto-dynamic processes take place in it. It is these phenomena that are very important in astrophysical processes in hot ionized gases in space.
  In ordinary terrestrial nature, plasma occurs relatively rarely in atmospheric discharges, lightning. From a global perspective, however, plasma is a very important form of matter - most of the observed substance in the universe is in the plasma state. Plasma is of great importance for achieving thermonuclear fusion -
§1.3, part "Fusion of atomic nuclei".

Atomic nucleus
Let us now look deep into the interior of the atom - directly into the atomic nucleus itself. Before we deal with the structure of the atomic nucleus, it is worth noting its size compared to the size of the atom. "Diameter" of an atom is of the order of
»10-8 centimeters (thus far below the resolution of the optical microscope - atom is much smaller than the wavelength of light; even of electron microscopy atoms are not directly observable). However, the core is even 100,000 times smaller! - its "diameter" is only »10-13 cm. At the same time, almost the entire mass (more than 99.9%) of the atom is concentrated in the nucleus. The density with which matter is "crammed" in the atomic nucleus is therefore unimaginably high - r » 1014 g/cm3 !
   It is not easy to imagine such a huge density: if, for example, a box of matches were filled with nuclear matter, it would weigh about a billion tons (!) - it would break through the table, soil and rock and fall into the center of the Earth. Apart from atomic nuclei, we do not encounter such a high density anywhere in the surrounding nature. However, wonderful bodies called neutron stars have been discovered in space. They are stars at the end of their lives with depleted nuclear "fuel", gravitationally collapsed to the size of only tens of kilometers, they are composed of neutrons with a density of also »1014 g/cm3 .They rotate quickly and when the charged particles interact with a strong magnetic field, electromagnetic radiation is created, which "sweeps" the surrounding space as the star rotates, similar to the light of a rotating beacon - we observe them as pulsars. Details can be found in Chapter 4 "Black Holes", §4.2 of the book "Gravity, Black Holes and the Physics of Spacetime".
   It follows from the very fact of such small dimensions and fantastic densities in the atomic nucleus - even without knowledge of the specific structure of the nucleus, that great forces will act in the atomic nuclei and there will be "high energy" in the game (this will be discussed in §1.3, part "Nuclear energy").
Note: At the same time, it is quite clear from these facts, that alchemists trying to transmute elements (eg to turn lead into gold) had no smallest chance of success! By the methods at their disposal (grinding, hammering, annealing, burning, chemical fusion) they only "scraped" atoms along their uppermost (valence) shells. If they wanted to change the element, they would have to penetrate a hundred thousand times deeper into the interior of the atom, change the nucleus, and only then would they achieve transmutation. Of course, they did not have the resources, energy or knowledge to do so. Now, in principle, nuclear physics can do it by methods of "bombarding" nuclei with elementary particles accelerated to high energies (or neutrons in reactors) - however, only a tiny amount of transmuted elements can be prepared in this way.

Atomic nucleus structure
The existence of a positively charged, very small and dense atomic nucleus was convincingly proved by the above-mentioned scattering experiments of E.Rutheford et al. from 1911
(Fig.1.1.4), but nothing could be deduced from these experiments about the nature and structure of the atomic nucleus. The discovery of a proton, a positively charged heavy particle, also made by Rutheford in tracking alpha particle traces in the Wilson's clode chamber, played a key role for revealing the structure of atomic nuclei (a similar key importance as discovery of eletron for in uncovering the structure of atoms).
Upon impact of the particles a in the nitrogen nuclei, a reaction of 4a2 + 14N7 ® 17O8 + 1p1 occurred. Two traces emanated from the collision site, one corresponding to the oxygen nucleus, the other to a positive particle identical to the hydrogen nucleus - this particle was called a proton. A proton as an elementary particle is denoted "p", or alternatively, according to chemical terminology, "H" or 1H1 as a hydrogen nucleus. The reality of its positive elementary charge is sometimes characterized by the index "+", ie p+. Further measurements were used to gradually determine the properties and physical characteristics of the proton, see §1.5 "Elementary particles".
   The idea was immediately offered that the nuclei of atoms were composed of protons. It was also supported by the remarkable regularity in the masses of the atoms - that the masses of all the atoms are almost exactly integer multiples of the mass of the hydrogen atom. However, the model of a nucleus composed of protons alone encountered two problems :
First, it was the electrical Coulomb repulsion of uniformly charged protons, which would be extremely strong at such short distances, and at the time no other forces were known to counter it and maintain the stability of the nucleus. Furthermore, the masses of all atoms except hydrogen were about half that actually observed.
Therefore, models in which protons and electrons combine in the nucleus were temporarily proposed: the nucleus of an element with atomic number Z it would consist of 2.Z-protons (ie twice the protons) and Z-electrons, whose negative charge would compensate for the excess positive charge. The proton-electron model gave approximately the correct mass values for light nuclei, but not for heavy nuclei. Radioactivity
b , in which electrons are emitted from nuclei, seemingly supported this "nuclear electron" model. However, other properties of the cores were no longer in line with this model (eg the magnetic moment of the cores would be significantly higher).
   The missing article to clarify the structure of the atomic nucleus was supplemented by the discovery of the neutron, made by J.Chadwick in 1932 during experiments with bombardment of beryllium nuclei by particles
a. It turned out that these neutrons, particles about as heavy as protons but without an electric charge, are probably the mysterious missing component that is together with the protons in the atomic nuclei. At the same time, the composition of proton and neutron nuclei naturally explained the existence of isotopes: isotopes of one element contain the same number of protons (therefore they have the same chemical behavior), but different numbers of neutrons, so they differ only in mass.
   The Wilson cloud chamber was after colliding particles
a the core Be observed only one foot, which belonged carbon nucleus C . When Chadwick performed a detailed analysis of a and C particle tracesfrom the point of view of the laws of conservation of energy and momentum, he came to the conclusion that during the collision, in addition to the carbon core, another relatively heavy and energetic particle must be formed, which does not carry an electric charge and therefore does not create an ionization trace in the nebula. Thus, a reaction of 4a2 + 9Be4 ® 12C6 + 1n0 occurs; the newly discovered neutral particle (slightly more than a proton's mass) was called a neutron, labeled "n". The electrical neutrality of a neutron is sometimes characterized by the index zero "o", ie no. Other experiments and physical properties of the neutron were gradually determined by further experiments, see again §1.5 "Elementary particles".
  Thus, it was found that atomic nuclei consist of two types of heavy particles (nucleons): protons and neutrons, while these protons and neutrons are held in the nucleus by a new, hitherto unknown, type of force - the so-called nuclear forces (see below).

Fig.1.1.9. Schematic representation of the structure of the atomic nucleus. The right part shows the energy levels of the nucleus, the excited nucleus and its deexcitation by gamma photon emission.

In Fig.1.1.9, the atomic nucleus is imaginarily "magnified" a total of about 1014 times and its structure is schematically shown here. The nucleus consists of particles of two kinds collectively called nucleons: positively charged protons p+ and neutrons no without electric charge. The number of protons in the nucleus, called the proton number Z, unambiguously determines the configuration of electrons on the individual shells of the atomic shell (each nucleus "picks up" so many electrons to be electrically neutral) and thus the chemical nature of the atom - proton number Z is also a serial number in Mendeleev's periodic table of chemical elements. Number Z is therefore sometimes called an atomic number. The total number of nucleons, called nucleon number N, determines the mass of an atomic nucleus in multiples of the mass of a proton or neutron; mass number N is also sometimes called the mass number and is denoted A. Nuclei with the same number of protons, which have different numbers of neutrons, are called isotopes - the chemical properties of the respective atoms are the same, differing only in mass (see the section "Physical and chemical properties of isotopes" below). We denote nuclei by letters of chemical designation according to Mendeleev's table of elements (here we generally denote X), while we add a nucleon number as a superscript and a proton number as a subscript: NXZ - eg hydrogen 1H1, helium 4He2, carbon 12C6, uranium 238U92. Since the proton number is uniquely determined by the name of the element in Mendeleev's table, the subscript is often omitted (eg instead of 12C 6, only 12C is abbreviated writed).
   The sizes (diameters) of atomic nuclei
(in view of strong interaction) range from about 1.6 fm (ie 1.6.10-13 cm) for a hydrogen atom - diameter of 1 proton, up to about 15 fm (ie 1.5.10-12 cm) for the heaviest atoms from the uranium region and nearby transuranium (for particle size in the microworld see also §1.5, passage "Size, dimensions and shape of particles?").
Physical and chemical properties of isotopes
The different number of neutrons in the nuclei of isotopes naturally affects their physical and, in part, to a much lesser extent, their chemical properties. The physical properties of isotopes can be divided into nuclear and atomic. Different nuclear properties of different isotopes of the element lie in three aspects :

Different courses (type, cross section) of interactions and nuclear reactions when bombarded nuclei of different isotopes of the particles, or when they collide (this is discussed in detail in §1.3, "Nuclear reactions and nuclear energy") .
- Stability or instability - possible radioactivity of nuclei, depends on the number of neutrons, due to the number of protons (§1.2 "Radioactivity", especially part "Stability and instability of nuclei"). It is often sufficient if there is only one neutron more or less in the nucleus and the relevant isotope is already radioactive (the properties of radioactive isotopes are studied in detail in §1.4 "Radionuclides").
- Furthermore, there are different values of the magnetic moment of nuclei, depending on the number of paired and unpaired protons and neutrons - this may be important in the analytical method nuclear magnetic resonance (see §3.4, section "Nuclear magnetic resonance").
   A somewhat different atomic properties of different isotopes are due to differences in atomic weight, given by different number of neutrons in the nuclei of the same element. This is more pronounced especially for light elements with a low proton number. Hydrogen atom 2H1 - deuterium D, is 2 times heavier than ordinary hydrogen 1H1, and its oxygen compound, "heavy water" D2O, has a density about 10% higher than ordinary "light" H2O. The freezing point of heavy water is ( instead of 0 °C for ordinary water) 3.8 °C, boiling point 101.4 °C.
   The chemical properties of atoms - the ways in which they are bound and reacted with other atoms - are determined by the configuration of the electrons in the atomic shells, which depends on the number of protons in the nucleus, not on the number of neutrons. Thus, different isotopes of the same element have the same chemical properties. This is very important in nuclear chemistry and in the applications of radionuclides in laboratory methods
(§3.5 "Radioisotope tracking methods"), biology and medicine (especially in nuclear medicine - Chapter 4 "Radioisotope scintigraphy"). The only way in which the chemistry of different isotopes of the same element may differ somewhat is the rate of chemical reactions. A larger number of neutrons in the nucleus means that the atoms of such a higher isotope are heavier and thus move slightly slower in the reaction mixture than lighter isotopes. Therefore, chemical reactions with heavier isotopes will proceed somewhat more slowly under otherwise identical conditions - the kinetic isotope effect. This is most pronounced in deuterium, where it even leads to the biological toxicity of this isotope of otherwise biogenic hydrogen. Replacing ordinary hydrogen with its heavier isotope deuterium significantly slows down the rate of biochemical reactions - it acts as a "brake" on many life processes in cells. This has negative effects especially in higher organisms (where a higher deuterium content, above about 30%, can even cause death).

Strong nuclear interactions
If we look at this model of the nucleus in terms of the laws of electricity, a fundamental objection or question arises immediately: How is it possible that the nucleus holds together? According to Coulomb's law, charged protons of the same sign will repel, so that they would immediately "scatter" into the surrounding space, no nuclei and atoms (except hydrogen) could exist. In reality, however, (fortunately) nothing like this happens, the cores usually hold us nicely "together". Therefore, in addition to the electric repulsive forces, there must be other forces that are attractive and stronger than the electric ones - these forces then overcome the electric repulsive forces and keep the core together. They are called strong nuclear interactions; their nature will be briefly discussed below. They are about 100 times stronger than electric forces, but they have one specific peculiarity - they have a short range. They work effectively only up to a distance r
» 10-13 cm, while for larger distances they are already negligibly weak - with distance r they decrease rapidly exponentially. The potential of these forces is often modeled by the so-called Yukawa potential
            U (r) = g. e
- d . r / r ,
where g is a constant expressing the strength of the interaction and the parameter d = 1,6.10
-13 cm characterizes the range of nuclear forces.
Note: The characteristic length of 10-13 cm, important in nuclear physics, is sometimes called 1 Fermi; it is also 1 femtometer [fm].
  Each nucleon can therefore interact directly only with a limited number of adjacent nucleons - nuclear forces show saturation. This is the main reason for the reduced stability of heavy nuclei, as will be shown below (§1.2, §1.3). Since electrostatic (Coulomb) proton repulsion is long-range and acts significantly throughout the nucleus, there is a limit to the ability of strong nucleon interactions to prevent the decay of large nuclei. At this boundary is the core of bismuth 209B83, which is the heaviest stable core *); all heavier nuclei with Z> 83 and N> 209 are already spontaneously transformed into lighter nuclei (especially by radioactivity a) - see §1.2 "Radioactivity".
*) Until recently, bismuth-209 was indeed considered the most heawy stable nuclide. In 2003, however, its weak radioactive transformation by alpha decay with a very long half-life of 2.10
19 years to 205Tl was demonstrated in the Orsay nuclear laboratories. With such a long half-life, the radioactive conversion is practically unobservable and the 209Bi isotope appears to be stable. Lead of 209Pb is now considered to be the heaviest truly stable isotope.
  Nuclear forces do not depend on the type of nucleons, they are charge independent. Thus, strong nuclear interactions act both between protons and protons, and between protons and neutrons or between neutrons and each other - protons and neutrons belong to a group of particles called hadrons (see below §1.5 on elementary particles). A more detailed analysis showed that nuclear forces are spin-dependent (the interaction between nucleons depends on the angle between the spin and the junction of both particles) - the interaction between two nucleons with parallel spins is somewhat different from the interaction of nucleons with antiparallel spins.
Influence of weak interactions on the structure of nuclei
If there were only strong
(and electromagnetic) interactions in the microworld, there could also be "mononucleon" nuclei composed only of protons or only neutrons (mono-neutron nuclei would not have an electron shell). Nuclear "monsters" composed of thousands of neutrons could also form. However, we do not observe anything like this in nature, there are no stable nuclei from either the two protons themselves or the two neutrons; even the neutron itself is unstable. In nature, namely, also another kind of force is at work - a weak interaction, which mercilessly transforms by beta (- or +) radioactivity any nucleus in which a certain ratio between the number of protons and neutrons is violated. The mechanisms of these processes are discussed in §1.2, section "Radioactivity beta".

The nature of strong interactions between nucleons
According to an older concept proposed by H.Yukawa in 1935, nuclear forces are caused by the exchange of
p- mesons between nucleons. Although this idea seemed to explain quite well some of the then known properties of nuclear forces, further research has shown that the real cause of nuclear forces (and strong interactions between hadrons in general) is to be found at a deeper level - in the internal structure of protons, neutrons, p-mezons and other hadrons. According to today's concept, the primary cause of "strong interactions" between hadrons is gluons mediated interactions between quarks inside the hadrons. The observed "strong" interactions between hadrons, and thus nuclear forces, are a kind of "residual manifestation" of these primary interactions between quarks. Simply put, we can imagine that gluons partially "seep" from the inside into the immediate vicinity of protons or neutrons and cause attractive nuclear forces there.
   It is noteworthy that the inherent strong interactions between quarks are expected to have a long range, while the observed short range of the resulting interactions between hadrons (and thus nuclear forces) is due to the "residual manifestation" mechanism of these forces (for further discussion see §1.5 "Elementary Particles") section "
Interaction of elementary particles", "Quark structure of hadrons" and "Four types of interactions").
: It is interesting that similar mechanisms are seen in interactions and chemical compounding of atoms: short range "chemica" forces between the atoms of the residual speech dlouhodosahových electric forces from protons and electrons in the envelope, which are added together in vector: at greater distances it is canceled, at short distances it remains a non-zero "residue" - see above "
Interaction of atoms".   
   In the commonly used name "strong nuclear interaction" the word "nuclear" is inserted, because we mostly deal with the properties of the atomic nucleus, in which these interactions apply most significantly. In particle physics, the name "strong interactions" sufficed, as these are fundamental forces acting generally between interacting hadrons - as a mentioned consequence of a strong interaction between quarks forming hadrons. Nuclear forces are only a special manifestation of these strong interactions.

Fig.1.1.10. Graphical ilustration of nuclear force potentials for neutron and proton as a function of distance. The right part of the figure shows the discrete (quantum) energy levels of nucleons in the potential well of the nucleus.

Figure 1.1.10 graphically shows how the potentials of the forces acting between the nucleus and the nucleon depend on the distance. In an imaginary experiment, imagine that with the nucleon we are slowly approaching toward to an atomic nucleus. For a neutron no (left) without an electric charge, only the field of strong interaction acts, so at greater distances the force is negligible and at distances of the order of 10-13 cm an attractive force acts, which binds the neutron to the nucleus. For a positively charged proton p+, an electric repulsive force will act at greater distances according to Coulomb's law (blue curve in Fig.1.1.10), and only when we overcome it (we say that we have overcome the Coulomb barrier) and the proton approaches the nucleus at a distance close to 10-13 cm (1 fm), an attractive nuclear force (red curve) begins to act, overcoming the electrical repulsive force, and "ties" the proton to the nucleus. In both cases, the resultant force curve is marked in green in the figure. At even smaller distances (tenths of fm) inside the core, the attractive force is in both cases effectively replaced by a repulsive force *), preventing complete shrinkage of the core; its origin is related to the quantum principle of uncertainty and to the exclusion principle of fermions.
*) Nuclear interactions at subnuclear distances At short distances of the order of tenths of fm, nuclear interactions have a repulsive character 
. This "repulsion" of nucleons as they approach each other at a distance of <<1 fm is not a specific property of a nuclear strong interaction (which is a residual manifestation of a strong interaction between quarks within nucleons) or "incompressibility" of nucleons, but is only an "effective force" that it is a consequence of quantum relations of uncertainty and fermion character of nucleons (Pauli exclusion principle). Nucleons cannot reach a smaller distance, or a lower energy level in the field of nuclear forces than the lowest basic one; if we tried to "push" them even closer together, they would "defend" themselves with an intense repulsive force - nucleons as if with their wave nature did not "fit" into such a small space...
This effect leads to the so-called Fermi presure of degenerate fermion gas in the final stages of stellar evolution, which can stop the gravitational collapse and balance the massive gravitational forces (§4.2 "Final Stages of Stellar Evolution. Gravitational Collapse" in the monograph "Gravity, Black Holes, and the Space Physics"). However, only if the mass of the star is not too large - in that case strong gravity, according to the general theory of relativity, occurs to create an event horizon that "overpows" even this quantum counterpressure and a complete catastrophic gravitational collapse wins and forms a black hole.
  The course of the interaction potential of nucleons for distances of tenths of fm is largely speculative, is not implemented in nucleus and cannot be verified experimentally. We cannot "take two protons in hand", push them close to each other, and measure the forces by which they are attached or repelled. This can only be done by collision experiments at high kinetic energies. At medium energies (units up to tens of MeV), the dependence according to Fig.1.1.10 is measured in scattering experiments, but only with a part of the repellent component. For a larger interaction "approach" of nucleons, it is necessary to use larger collision energies of hundreds of MeV. Here, however, a new phenomenon manifests itself from about 300 MeV: the production of
p- mesons (pions). We are already encountering the fact that short-range nuclear forces are a residual manifestation of a long-range strong interaction between quarks within nucleons, mediated by gluons. In an effort to get as close as possible to the nucleons, we no longer get any real interaction potential, because the nucleons "melt" into the quark-gluon plasma, cease to exist "individually" and we observe a number of secondary particles reflecting the properties of the new state of hadron matter (§1.5, "Quark structure of hadrons"). Nucleons at higher energies are not incompressible, but change into new states and particles.

  Experimental measurements (with scattering of high-energy electrons) show that the approximate relation R = d . N1/3 applies to the radius of the nuclei, where N is the nucleon number of the nucleus and the parameter d has the value d = 1.3·10-13 cm - this is the range of strong interaction. It follows from this relationship that the volume of the nucleus is directly proportional to the nucleon number N and thus each nucleon occupies approximately the same volume in the nucleus. Thus, the nucleus can be considered as a set of nucleons with an approximately constant density of nuclear mass.
  Neutrons without an electric charge show only attractive nuclear forces - they help to "stabilize" the nucleus. For each nucleus there is a certain ratio of protons and neutrons, for which the nucleus is the most stable (this ratio is close to 1:1 for light nuclei, for heavy nuclei it is up to about 1:1.5 in favor of neutrons). If we add or remove some neutrons to a nucleus with a stable configuration of protons and neutrons, usually such a nucleus will no longer be stable, but will spontaneously decay (or transform) - it will be radioactive
(the relevant mechanisms will be analyzed in §1.2 "Radioactivity").
Excited states of the nucleus 
Nucleons are located in the nucleus and move in the field of nuclear forces (strong interactions), in which they can have different (binding) energy - they move in a kind of "potential pit". According to the laws of quantum physics, nucleons cannot have continuously variable energy in this field, but only certain quantized energy values. Thus, similar to the electrons in the atomic shell, the nucleons are located at discrete energy levels in the nucleus (Fig.1.1.9 and 1.1.10, both on the right). Proton and neutron levels differ somewhat due to electrical interaction and are occupied independently, with Pauli's exclusion principle limiting the occupancy of each such level by a maximum of two protons and two neutrons with opposite spins. The lowest energy level of the nucleus corresponds to the ground state, but the nucleus can (by supplying energy - excitation ) get to a higher energy state - to the so-called excited energy levels - as if the nucleus were "inflated", the nucleons are "farther apart" (Fig.1.1.9), occupy higher levels. The energetically excited nucleus usually "collapses" very quickly - the levels are deexcited, while the corresponding energy difference is emitted in the form of a photon of electromagnetic radiation - radiation
g (see §1.2, section "Gamma radiation").
   In addition to the ground state, atomic nuclei
(except hydrogen) have a number of excited states (energy levels), only some of which are involved in radioactive transformations. The other excited states arise during the bombardment of nuclei by energetic particles from accelerators.
Metastable levels and nuclear isomerism
The lifetime of excited nuclear levels is usually very short (
»10-15-10-6 sec.), but there are situations where the lifetime of the excited level is in the order of seconds, minutes and even several hours, days and years ! - such levels are called metastable and we speak of the isomeric state of the nucleus. Such a nuclear isomer is often considered to be a separate nuclide and is denoted by the superscript "m" at the nucleon number e.g. 99mTc. This phenomenon occurs when there is an energy level near the ground state of the nucleus, which differs significantly from the ground state by its angular momentum - spin (at least 3h , ie DI ³ 3), see the core shell model below. Then the radiation g emitted at the transition from such a level to the ground state must have a higher multipolarity (E3, M3 or higher) - transitions between such levels are unlikely, they are "forbidden", so the corresponding lifetimes can take on large values. Isomerism and metastable states do not occur in light nuclei (where there are no excited levels with DI ³ 3), but only for nuclei with a nucleon number from 40; details are explained by the shell model of nucleus. An important example is the metastable technetium 99mTc, which deexcites with a half-life of 6 hours, see the following §1.2 "Radioactivity", section "Gamma radiation".
  However, some nuclear isomers have such different quantum properties from the ground state (especially the spin value) that there is no transition to the ground state of
g- radiation photon emissions , but to the radioactive conversion of beta- , beta+ or electron capture, to another neighboring nucleus (§ 1.2, section "Nuclear isomerism and metastability").

Binding energy of atomic nuclei
As already mentioned, protons and neutrons are bound in the nucleus by an attractive strong interaction. Associated with this binding force of nucleons is a certain potential energy called the binding energy of the nucleon or the whole nucleus. The total binding energy of the nucleus E
v means the energy required to completely decompose the nucleus into individual free nucleons *). At the same time, this binding energy is equal to the energy that would be released when the nucleus was formed from individual nucleons.
*) For the time being, we are leaving aside the mechanisms by which such a distribution or composition of nuclei can be carried out; we will deal with this in §1.3 "
Nuclear reactions and nuclear energy".
  According to the relativistic concept of mass-energy equivalence, due to the binding energy, the total mass of the nucleus m(Z, N) is always somewhat less than the sum of the masses of its free protons Z.m
p and neutrons (N-Z).mn. The difference between the resulting rest mass of the nucleus and the total rest mass of the free nucleons of which the nucleus is composed,
D m  =  Z.m p + (N-Z) .m n - m (Z, N)  ,
is called the weight loss or defect. According to Einstein's equivalence relationship between mass and energy, the weight loss is related to the total binding energy of the nucleus by the relation E
v º DE = Dm. c2. If we divide the total binding energy of the nucleus Ev by the number of nucleons N, we get the average binding energy per one nucleon` Ev = Ev/N.
   Weight loss is expressed either in grams or in atomic units of mass (1/12 of the mass of a carbon atom
12C), binding energy is usually expressed in megaelectronvolts [MeV] in nuclear physics. E.g. the helium nucleus 4He2 has a mass loss Dm = 0.5061.10-25 g, the binding energy Ev º DE = 28 MeV, the binding energy per nucleon is 7 MeV.
Sometimes the rate of weight loss is expressed by the so-called cogestion coefficient d = (Dm/m), sometimes multiplied by 10000.
   The binding energy per nucleon
(or compression coefficient) with a proton number initially increases rapidly, the largest is for nuclei around iron, then again slightly decreases - see Fig.1.3.3 in §1.3 "Nuclear reactions and nuclear energy". This dependence is also crucial for the possibilities of obtaining nuclear energy. Fig.1.3.3 will be mentioned here for clarity :

Fig.1.3.3. Dependence of the mean binding energy Ev per one nucleon on the nucleon number of the nucleus. In the initial part of the graph, the scale on the horizontal axis is slightly stretched to better see the differences in binding energy for the lightest nuclei. The right part of the figure concerns two ways of releasing bound nuclear energy, which will be discussed in detail in §1.3 "Nuclear reactions and nuclear energy".

Binding energy and stability of atomic nuclei
As with other bound systems, the binding energy of atomic nuclei can be expected to be closely related to their stability. It is related to the "external" stability in the supply of energy to the nucleus from the outside
(usually by scattering some particles bombarding the nucleus), as well as to the "internal" stability or instability caused by internal mechanisms in nucleons and their bonds. In the following §1.2 "Radioactivity" we will get acquainted with the processes and mechanisms by which nuclei can transition from higher energy configurations to lower energy configurations during radioactive transformations. In order for every nuclear process (and every physical process in general) to take place, two basic conditions must be met: the energy balance and the existence of the appropriate mechanism by which the process takes place. The stability of light and medium-heavy nuclei is determined by the ratio of the number of protons and neutrons. According to Fig.1.1.10, the quantum energy levels of protons and neutrons in the field of nuclear forces are given, which are occupied independently. If at these levels a situation arises where the energy of a different configuration of protons and neutrons is energetically lower enough, "will work" a mechanism of weak interaction, which is able to mutually convert protons and neutrons - there is a radioactive transformation of beta (beta- or beta+ depending on whether there is an excess of neutrons or protons). For heavy nuclei in the uranium and transuranic regions, alpha radioactivity occurs due to the inability of a strong interaction, due to the short range, to hold such a large number of nucleons together. In §1.2 "Radioactivity" we will analyze all these processes in terms of mechanisms and energy balance on a 3-dimensional table of nuclides, mapping the binding energies of nucleons in nuclei - part "Stability and instability of atomic nuclei", Fig.1.2.8 and 1.2.9 in §1.2.

Atomic nucleus models
Due to their size of
»10-13 cm and quantum character, atomic nuclei are completely beyond the scope of any direct observation. To understand the various processes with atomic nuclei, it is necessary to get some at least approximate ideas about nuclei and their internal arrangement. Atomic nucleus models are certain schematic representations, fictitious constructions, and analogies that explain, with greater or lesser success, certain properties or processes in light and heavy atomic nuclei. There are several models, each of which usually explains only some of the specific nuclear processes for which it was created (with the exception of the shell model, which is more general). Here we will only briefly mention some of the more commonly used models :

Diversity of atomic nuclei
Currently, more than about 2,600 species of different nuclei are known, differing in the number of protons or neutrons. Of these, the stable nuclei are 270, the other nuclei are radioactive. There are 340 nuclides in terrestrial nature - 270 stable and 70 radioactive. Let's list some important specific cores of elements. The simplest element is hydrogen
1H1 (hydrogenium), whose nucleus consists of only a single proton p+, around which a single electron e- orbits. The addition of one neutron no forms the nucleus of heavy hydrogen 2H1 - deuterium. The heaviest isotope of hydrogen is tritium 3H1, containing a proton and 2 neutrons; however, two neutrons per proton are "a little too much", the equilibrium configuration is broken, and tritium 3H1 is already decaying radioactively (decaying b- with a half-life of 12.6 years to helium 3). Another important light nucleus is helium 4He2 containing two protons and two neutrons (there is also a small amount of 3He).
   Other important nuclei include carbon
12C6, nitrogen 14N7, oxygen 16O8, sodium 13Na11, sulfur 33S1 , ....., iron 56Fe26, ...., gold 197Au79 etc. The heavier the nucleus, the more different isotopes it has, some of which are stable, but most is radioactive. The last stable cores are lead 208Pb82 and bismuth 209Bi83; all heavier nuclei are already radioactive - we gradually get into the area of uranium nuclei (235,238U92 and other isotopes) and transuranic nuclei (plutonium, americium, californium, einsteinium, fermium, mendelejevium ...). The heaviest known cores (such as 258Lw103 and higher) are already disintegrating so quickly after their artificial production, that it is difficult to prove their existence at all. The preparation of heavy transurans is briefly mentioned in §1.3 "Nuclear reactions", part "Transurans". The properties of the three important transurans (plutonium, americium, californium) are given in §1.4, section "The most important radionuclides".

Stability of atomic nuclei
The temporal stability or instability of atomic nuclei is due to the complex interplay of strong, electromagnetic and weak interactions between nucleons (and even within nucleons). In principle, the nuclei are held together in a stable configuration by the predominance of the strong attractive nuclear interaction of nucleons over the weaker electrical repulsive force between protons. For too large nuclei, a strong nuclear interaction, due to its short range, is not enough to bind the nucleus strong enough, which can lead to the emission of nucleons (alpha radioactivity), or even to the fission of the nucleus. Within the nucleons themselves, there are strong and weak interactions between quarks; these weak interactions can lead to transmutations of quarks within nucleons and thus to mutual transformation between protons and neutrons - this results in the instability of the nucleus, in its transformation into another nucleus (beta radioactivity). The mechanisms of different types of radioactivity will be dealt with in §1.2 "
Strong, electromagnetic and weak interactions determine the energy conditions in the nucleus according to the number of protons and neutrons, their mutual ratio and arrangement. From the energetic point of view, we will analyze the causes of nuclear instability in §1.2, section "
Stability and instability of atomic nuclei".

The origin of atomic nuclei and the origin of elements - we are the descendants of stars!
Cosmic alchemy
D.I.Mendeleev and his followers systematized the individual elements known in nature into the periodic table. Chemists have studied in great detail the properties of all these elements and their compounds, which are the cause of the variety and diversity of the world. But let's ask ourselves a curious question: Where did the individual elements come from? How did their atomic nuclei form?
   They werw constructed, metaphorically speaking, by God "with his hands" already at the creation of the world - ie all the elements were created already at the creation of the universe? Or did they originate during the further evolution of the universe? Contemporary astrophysics and cosmology clearly lean towards the second possibility - it has developed a fascinating "scenario" of the chemical evolution of the universe - cosmic nucleogenesis.
   According to the standard cosmological model, the universe was born before about 13 to 15 billion years ago in a very hot and dense state - when called "Big Bang".
Within the classical general theory of relativity, the actual act of the origin of the universe (the big bang) has the character of a point so-called singularity with zero volume, infinite curvature of spacetime, infinite energy density. According to the quantum theory of gravity, however, spacetime in the microscales of the so-called Planck-Wheeler lengths »10-33 cm show such large quantum fluctuations in geometry (metrics) that even the topology of spacetime fluctuates - spacetime has a "foamy"constantly spontaneously fluctuating microstructure. According to the concepts of quantum cosmology, the universe was born of quantum space-time foam; what's more - together with our Universe, more universes could have been born in this way ! (cf. "Anthropic Principle or Cosmic God ").
   Individual phases of the evolution of the universe after the "big bang", accompanied by rapid expansion and cooling universe, are divided into 4 significant stages differing in the dominant physical interactions and processes that took place at that time
(described in detail in §5.4 "Standard Cosmological Model. The Big Bang." of the book "Gravity, Black Holes and the Space Physics") :

  1. Hadron era
    - lasted only ten thousandths of a second, the mass-energy density was higher than the nuclear density of 10
    14 g/cm3, the temperature exceeded 1012 °K. The majority of matter was formed by a mixture of constantly evolving heavy particles - protons, neutrons, mesons, hyperons and their antiparticles, the dominant role was played by a strong interaction between these hadrons.
  2. Lepton era
    - lasted about 10 seconds. When the temperature drops below 10
    12 °K, nucleons with antinucleons annihilated (except for a small excess of nucleons due to baryon asymmetry, which led to the formation of all later matter in space), electrons, positrons, photons, neutrinos (with antineutrins) took over the energetically dominant role. At the beginning of the lepton era, the remaining protons and neutrons were constantly transformed at high temperatures by the following reactions: n + e+ ß® p + n, n + n ß® p + e-. Due to the slightly higher mass, neutrons converted to protons faster than protons to neutrons, so the number of neutrons decreased. The complete disappearance of neutrons was prevented by the fact that due to a decrease in the temperature of space at time t »10 s to about 3.1012 °K, protons and neutrons could begin to merge into stable nuclei of helium 4He2 (via deuterium and tritium). In about 100 seconds from the beginning of the expansion, about 25% of helium was formed, the other 75% remained in the form of hydrogen (ie free protons). This was the initial nucleosynthesis in the end, because due to the rapid decrease in density, nuclei heavier than helium could form only in negligibly small amounts (the chain of two-particle interactions pa, na, aa does not continue also because there are no stable nuclei with 5 and 8 nucleons). This primordial nucleosynthesis is analyzed in more detail in the section "Lepton era. Initial nucleosynthesis" §5.4 of the monograph "Gravity, black holes and space-time physics".
    Further nucleosynthesis could continue for up to 100 million years, after the formation of the first stars with sufficient density inside them to combine helium into carbon, etc., as outlined below.
    Note: The original opinion of the founder of the theory of the hot early universe, G. Gamov, that all elements of Mendeleev's periodic table were "cooked" in the earliest universe proved to be wrong . During the Big Bang (in the Lepton era) only the lightest elements hydrogen and helium were formed, the other heavier elements were (nuclear) synthesized only in the stars.
  3. The era of radiation
    - after annihilation of most positrons with electrons, electromagnetic radiation, which is initially so high-energy that it does not allow the formation of atoms , takes over the government in space for about 300,000 years - electrons are immediately ejected by radiation. .....
  4. The era of matter
    - when the temperature drops below about 3000 °K, the energy of the photons decreases to such an extent that they are no longer able to ionize hydrogen and helium atoms, so that electrons can bond to protons and
    a-particles indisturbedly - hydrogen gas and helium are formed, which is already for existing electromagnetic radiation transparent. Thus, radiation is separated from matter, matter becomes the main factor in the evolution of the universe. The era of matter lasts until now, the temperature of electromagnetic radiation has dropped from the then 3000 °K to today's 2.7 °K - we observe them as microwave relic radiation .

  To recap, at the end of the Lepton era, all the matter in the universe consisted of only hydrogen (75%) and helium (25%). This situation lasted for about 300,000 years throughout the radiation era, when the universe expanded and the temperature dropped. When the temperature dropped below about 3000 °K, the hydrogen and helium atoms already retained their electrons - hydrogen gas and helium were formed, and the era of matter that lasts so far has begun. The gas envelope was very thin, but had an inhomogeneous structure. Local densities began to shrink under their own gravity, creating nuclei of galaxy clusters and galaxies. There, the gravitational shrinkage and densification of the gases continued, increasing the pressure and temperature (adiabatic compression).
  When the temperature inside some of the thickening clouds reached about 10
7 degrees, the kinetic energy of the nuclei began to overcome the repulsive electrical forces between the positively charged nuclei - thermonuclear reactions ignited. The thermonuclear reaction is the fusion of atomic nuclei at high temperatures, with lighter nuclei forming heavier nuclei (§1.3). High temperature is needed for positively charged nuclei to overcome electrical (Coulomb) repulsive forces with their kinetic energy and to approach each other at a distance of »10-13 cm, where due to attractive strong interactions, both nuclei can merge and combine to release a considerably large binding energy. This released energy is then the source of the star's light and heat, and further shrinkage stops - the gravitational forces are balanced by the pressure of the radiation and the thermal motion of the ionized gas due to the released nuclear energy.
Thermonuclear reactions and nucleosynthesis in stars 
(more detailed analysis is in §4.1 "
Gravity and evolution of stars", part "Evolution of stars" - "Thermonuclear reactions inside stars" monograph "Gravity, black holes and space-time physics").
   For most of the star's life is thermonuclear combining hydrogen to helium
  which is the most energy efficient. After consuming hydrogen inside the star for some time, gravity prevails, the star continues to shrink, and the pressure and temperature rise so much that the helium nuclei begin to coalesce into carbon (4He + 4He ® 8Be + g , 8Be + 4He ® 12C + g, reactions 3a(=4He 12C+g). After depletion of helium, further shrinkage of the star's interior occurs, and at ever-increasing temperatures, other thermonuclear reactions accompanied by carbon combustion take place ( eg 12C + a ® 16O + g, 16O + a ® 20Ne + g, 20Ne + a ® 24Mg + g, 12C + 12C ® 24Mg, etc . .. ) and at higher temperatures also oxygen (16O + 16O ® 24Si+ a, resp. ® 31P + p, resp. ® 32S + g). The nuclei of silicon and other elements in the hot thermonuclear plasma capture neutrons, protons and a-particles, creating other heavier elements. In addition to carbon, many similar nuclear reactions produce oxygen, nitrogen, ..., magnesium, ..., ... silicon, ... calcium, ... chromium, ... and finally iron.
Note: In order for a star to be able to synthesize heavier elements, it must have enough mass for gravity to cause sufficiently high pressures and temperatures inside it. Small stars can only make helium from hydrogen, more massive stars like our Sun will form nuclei down to magnesium, and much larger stars will have a whole sequence of thermonuclear reactions.

  How do the thermonuclearly created heavier elements get away from stars ?
If no other processes took place in the stars apart from thermonuclear nucleosynthesis, all the heavier elements "cooked" in this way would remain forever trapped in the interior of the stars by strong gravity and would not contribute in any way to the chemical evolution of the universe, not even life could arise. Fortunately, there are two processes that release the synthesized heavier elements from the gravitational grip of the stars and enrich the surrounding interstellar space with them :
--> Thermoemission of gases from the upper layers of the "atmosphere" of stars - the stellar wind (passage "Stellar wind"), which continuously carries a small amount of the star's gas, with an admixture of synthesized heavy elements, into the surrounding space.
--> A supernova explosion that ejects a substantial amount of the star's material, including a large amount of thermonuclearly synthesized elements, into the surrounding space. And during the proper explosion, it will create many other even heavier elements (see the section "Supernova explosion. Neutron star. Pulsars." in the §4.2).
   For iron nuclei, the sequence of thermonuclear reactions ends because the elements around iron have the highest binding energy, so the nuclear synthesis of heavier elements is no longer an exothermic reaction (energy must be supplied). All nuclear reactions releasing energy cease, the active life of the star ends - the final phase of stellar evolution occurs.
   What happens next? It depends on the remaining mass of the star. If this mass is not higher than about 1.25 masses of the Sun, the star (compressed by gravity from the original several hundred thousand kilometers to an average of several thousand kilometers and a density of the order of thousands of kilograms per cm
3) remains in equilibrium, when gravitational forces are balanced by the so-called Fermi pressure of degenerate electron gas in a fully ionized substance. A star in this state is called a white dwarf (until it glows hot with the remaining heat; then it becomes a black dwarf ); such a star is not very important for the chemical evolution of the universe - the heavier elements synthesized during its evolution remain gravitationally "trapped" inside the white dwarf and will not enter the surrounding universe.
   If the star has a residual mass greater than about 1.25 times the mass of the Sun (the so-called Chandrasekhar limit), the pressure of the electron gas is no longer able to balance such enormous gravitational forces, gravity will win and shrinkage will continue. Electrons are "pushed" into the nuclei and absorbed by them (there is a massive electron capture); they combine there with protons to form neutrons and flying neutrinos: e
- + p+ ® no + n (inverse b-decay). As a result, the electron content in the star decreases and their Fermi pressure decreases, making the star's substance easier to compress - further shrinking and absorption of electrons. The process continues at an avalanche-increasing rate: in a fraction of a second, the star will shrink violently, almost all of which protons and electrons merge into neutrons (atomic nuclei dissolve and cease to exist). At this stage, equilibrium can (but does not have to!) re-occur - a neutron star is formed, which has a diameter of only a few tens of kilometers and is composed of a neutron "substance" with a density of »1014 g/cm3 of the same order as the density in atomic nuclei.
The gigantic gravitational forces are balanced by the Fermi pressure of the degenerate neutron "gas". Fast-rotating neutron stars are observed in space as so-called pulsars - they emit a cone of directed electromagnetic radiation as they rotate, which, like a beacon, we observe as very regular rapid flashes of radiation.

   During the implosion leading to the formation of a neutron star, a large amount of energy is suddenly released, which is partially emitted by neutrinos and electromagnetic radiation (not only infrared and visible light, but mainly hard X-rays and gamma radiation), while the outer layers of the star expand rapidly into space and form then a glowing nebula: the formation of a neutron star is accompanied by a supernova explosion, which emits a huge amount of energy and the outer layers of the star are "scattered" into the surrounding space.
If the burned star has a residual mass more than twice the mass of the Sun, the gravitational forces are already so great that they can overcome the Fermi forces between neutrons, the catastrophic gravitational collapse does not stop at the neutron star stage and continues until the star, acording to the general theory of relativity, falls below its gravitational radius, crosses the horizon and a black hole is formed (details in Chapter 4 "Black Holes" of the book "Gravity, Black Holes and the Physics of Spacetime").

A supernova explosion is essential for the chemical evolution of the universe in two ways :

  Stars such as our Sun and the entire solar system formed from clouds of gases "remelted and boiled" by earlier generations of stars *); these clouds have already been enriched with heavy elements.
*) The stars of the first generation, which formed in the period of about 300-700 thousand years after the big bang from hydrogen and helium (other elements were not yet in space at the time), probably had quite large masses of about 100-300¤. According to the laws of stellar evolution, therefore, they evolved very rapidly- after about 3-5 million years, they exploded like supernovae and introduced heavier elements into the interstellar matter, which were formed in them by thermonuclear fusion. The next generation of stars, which formed from this substance enriched with heavier elements, no longer reached such masses, and their lifetimes were hundreds of millions of years to several billion years. Our Sun probably formed as a 3rd generation star made of material enriched after the explosion of 2nd generation stars (and previously 1st generation).

  Stars can be described as a kind of "alchemical cauldrons" of the universe, in which heavier elements are formed from lighter elements by thermonuclear synthesis. Alchemists, who often looked at the stars (also engaged in astrology ) at night with religious reverence, called on them and begged for help, had no idea that these stars had been doing (and still do) on a huge scale what they did for billions of years before, they tried unsuccessfully on a small scale - transmutation - to transform elements. So with a bit of exaggeration, we can proudly say that “we are all descendants of the stars!"- every atom of carbon, oxygen, nitrogen, sulfur, etc. in our body formed long ago billions of years ago in the "fiery nuclear furnace" of the interior of an old star, now long extinct. All elements on Earth, except hydrogen, which is primordial and helium, come from the "dust" of stars burned out long before the creation of our solar system. We are the "ashes" - a kind of 'recycled waste' thermonuclear fusion of ancient stars ...
  An exception are light elements deuterium, lithium, beryllium and boron are not the direct product of thermonuclear reactions in stars (on the contrary, in these reactions they are "burned"), but they were formed by the primordial nucleosynthesia and interaction of cosmic rays with other nuclei, especially carbon, nitrogen and oxygen, which are fragmented into lighter nuclei by high-energy cosmic rays.
Fusion of neutronon stars
Another way of creating heavier elements in space occurs with the close orbit of two neutron stars and their merging - fusion, "collision". In this process, a large amount of neutron matter is ejected, which immediately "nucleonizes" to form atomic nuclei (§4.8, passage "Collisions and fusions of neutron stars"). This creates a large number of cores, with a relatively higher proportion of heavy elements. Due to the huge number of neutrons, the r-process of rapid repeated neutron capture by lighter nuclei takes place intensively, during which very heavy nuclei are also effectively formed - from the area around iridium, platinum, gold, to a group of uranium.

The evolution of stars from the point of view of relativistic astrophysics is described in more detail in the book "Gravity, Black Holes and the Physics of Spacetime", §4.1 "The role of gravity in the formation and evolution of stars"; cosmic nucleosynthesis from the point of view of nuclear (astro) physics is outlined in the work "Cosmic Alchemy", a synthetic view of the evolution of the universe in the work "Anthropic Principle or Cosmic God".

Nuclear astrophysics ® atomic astrochemistry
According to the laws of nuclear astrophysics, light atomic nuclei were formed at the beginning of the universe by primordial cosmological nucleosynthesis, heavier nuclei by thermonuclear synthesis inside stars. These nuclei are originally "bare", without electron shells - gamma radiation and sharp collisions at high temperatures will not allow the formation of a permanent electron shell, electrons are immediately ejected from the atomic shell, complete ionization of atoms. No chemical reactions and compound formation can occur here. In the ejected clouds, these nuclei enter cold interstellar space, where the nuclei capture their free electrons with their electrical attraction, filling them with electron orbits to form complete atoms of elements. Chemical reactions may already occur between them.
   The probability of collision and merging of two or more atoms in the sparse gaseous state of cold interstellar clouds is very small. However, there are two important mechanisms of chemical reactions in space :
¨ "Cold" astrochemistry
For the formation of molecules from atoms in space, solid dust particles condensed in an ejected nebula are very important. There, the atoms are close to each other and can exchange electrons - chemical reactions and the synthesis of molecules from atoms in interstellar space take place on grains of dust. They can also be stimulated by radiation from surrounding stars and cosmic radiation. By interacting with radiation, neutral atoms become ions, which, thanks to attractive electrical forces, are able to carry out reactions and bonds to molecules even at very low temperatures (at which normal chemical reactions do not take place).
¨ "Hot" astrochemistry
Gas envelopes
can function as "space chemical laboratories" around some stars, especially around red giants rich in carbon and oxygen. There are large differences in temperature and pressure in the individual areas of the envelope and there is intense radiation. The kinetic energy of the thermal motion of atoms overcomes the repulsive electric forces, and the atoms can approach by sharing the valence electrons and merging them into molecules. Temperatures are higher in the interior and compounds of silicon, magnesium, aluminum, sodium, etc. may be formed. In the lower temperature, compounds with longer carbon chains may be formed.
Intense chemical reactions then occur in protoplanetary disks and the planets formed around them around stars, where there is sufficient density and often favorable temperature.
  Using radio astronomy spectrometry, a large number of molecules not only inorganic (water, carbon dioxide, ammonia, ...) but also more than 100 different types of "organic" molecules composed of hydrogen, carbon, oxygen, nitrogen were discovered in interstellar clouds. Some are composed of more than 10 atoms, in addition to methane, there are also polycyclic aromatic hydrocarbons, aldehydes, alcohols and the like.
  One stellar giant (or several of these stars) on the inside of one of the spiral arms of the Milky Way, which exploded as a supernova about 7 billion years ago, was important to our Earth and solar system - from the cloud ejected by it, enriched with heavier and biogenic elements, the germinal nebula for the Sun and our entire solar system condensed. We don't know where the remnant of this previous star is, it ended up like a black hole...

Occurence of elements in nature
Cosmic nucleosynthesis outlined above - primordial cosmological and stellar - led to the current average representation of individual elements in space according to Fig.1.1.12 above. By far the most abundant elements in the universe are hydrogen and helium. In principle, it can be said that the element is more abundant in the universe, the smaller the proton (atomic) number, so the fewer protons it contains in the nucleus, the simpler it is - the easier it is to form in nuclear reactions. Exceptions are the light elements lithium (Li), beryllium (Be) and boron (B), the significantly lower occurrence of which is due to the fact that they "burn" to helium inside the stars before the main conversion of hydrogen to helium takes place. The opposite exception is a group of very stable elements (with high binding energy of nuclei, so it is easier to "survive" the final stages of stellar evolution) around iron (Fe), whose content is increased. Very slight occurrence of elements, which do not have stable isotopes - technetium (Tc), promethium (Pm) and actinides such as polonium (Po) to palladium (Pa), is due to their radioactivity with a not too long half-life; these elements can be formed in trace amounts by neutron capture. Thorium (Th) and uranium (U) are also unstable (radioactive), but with very long half-lives (of order 108 -1010 years), so after its formation in supernovae, it is sufficient to persist for a long time in interstellar clouds, stars and planets.
  The regular "oscillations" in the representation between adjacent elements, which can be seen in the graph (especially in the areas between Z = 8-20, 30-40, 45-60 and 62-75), are related to the slightly higher binding energy of nuclei with an even proton number than nuclei with an odd number of protons. These even nuclei are therefore somewhat more stable - they are easier to form in nuclear reactions and are "more resistant" to destruction during the turbulent final stages of stellar evolution. Therefore, they occur a little more abundantly compared to their "odd" neighbors.
The chemical evolution of the universe is still ongoing, so the current representation of the elements will change in the distant future; there will be mainly a decrease in light elements, which will merge into heavier elements. See also §5.6 "
The future of the universe. The arrow of time. Hidden matter." the mentioned monograph "Gravity, black holes ...".

Fig.1.1.12. Relative representation of elements in nature depending on their proton (atomic) number Z, related to hydrogen Z = 1.
Above: The current average representation of elements in universe. Bottom: Occurrence of elements on Earth (in the Earth's crust) and terrestrial planets.
Due to the large range of values, the relative representation of the elements (relative to hydrogen Z = 1) on the vertical axis is plotted on a logarithmic scale; however, this can optically distort a large difference in the representation of hydrogen and helium compared to heavier elements, especially in the upper graph.

Representation of elements in nature - selection mechanisms
The basic representation of individual elements in global "cosmic" nature is therefore given primarily by primordial cosmological nucleosynthesis (see §5.4 "Standard cosmological model. The Big Bang. Shaping the structure of the universe.", Passage "Primary nucleosynthesis") - 98 % of light elements of hydrogen and helium (with trace amounts of lithium) and only 2% of heavier elements formed by nucleosynthesis in previous generations of stars (is described in §4.1 "The role of gravity in the formation and evolution of stars" passage "Thermonuclear reaction inside star "), which in the final stages of their lives ejected these stars into interstellar space (§4.2 "Final stages of stellar evolution. Gravitational collapse , part "Supernova explosion, neutron stars, pulsars"). Such is the approximate representation of elements in existing stars, interstellar matter, nebulae, gas-dust clouds.
However, in more detail, in different places in the Universe and at different times, the chemical composition of matter can differ significantly, the chemical composition is differentiated - selection mechanisms are applied "favoring" some elements and suppressing others :
Time selection factor 
is given by the degree of stability or instability (radioactivity) of the elements. When a supernova explodes, they are ejected and basically all isotopes of all elements are formed. Soon after this grandiose event, we could find not only stable elements in the vicinity of the supernova, but also a number of radioactive elements. In terms of astronomical distances and time scales, however, only stable nuclei will be preserved for further development, and of radioactive nuclei, only those whose half-life of radioactive decay is sufficiently long, greater than about 10
8 years. Unstable nuclei with a shorter half-life will decay billions of years after a supernova explosion (transform into other, stable nuclei), so we will no longer find them in matter.
Gravitational selection factor 
causes spatial selection of lighter and heavier elements - gravitational density differentiation, especially in planetary systems around stars. Heavier elements (such as iron and nickel) descend towards the center and are concentrated in the cores of the planets, while lighter substances (such as silicates) float to the surface - there is a planetary differentiation of density. Gravity in co-production with radiation pressure and its thermal effects acts as a "mass separator", separating light elements and molecules from heavier ones in protoplanetary disks around emerging stars (§4.1 "
The role of gravity in star formation and evolution" of the mentioned monograph, part "Planets"). In the inner parts of planetary systems (such as our solar system), therefore, smaller planets with a higher content of heavier elements - terrestrial planets - are formed, while in more distant regions, large planets composed mainly of light gases - gas giants. The relative representation of elements on Earth and other terrestrial planets is therefore diametrically different from the average representation of elements in space - see. Fig.1.1.12 below. The main difference is the significantly higher proportion of heavier elements (due to hydrogen) and the practical absence of helium (see below "Helium - an element of the sun god").
Chemical selection factor 
related to the different reactivity of the elements and to the properties of the resulting compounds. It is mainly due to the difference between dense and refractory compounds of silicon and many metals, compared to volatile compounds of hydrogen, carbon and other elements. As well as the inert properties of helium and other "rare" gases.
Rare and exotic elements in nature
As mentioned above, during thermonuclear reactions inside stars and then during a supernova explosion, the nuclei of virtually all elements of Mendeleev's table are formed, including heavy nuclei up to transurans. They are thereby different isotopes of these elements, including radioactive. For further development, however, only stable nuclei will be preserved, and of the radioactive ones, those whose half-life of radioactive decay is sufficiently long, longer than about 10
8 years.
Extinct radionuclides
Unstable nuclei with a shorter half-life have already decayed (transformed into other, stable nuclei) billions of years after the explosion of our "parent" supernova . These extinct radionuclides - "burnt out" or "extinct" - no longer occur in our terrestrial nature
(or occur very rarely if they are continuously formed by natural processes such as cosmic rays or decay chains of long-lived radionuclides - §1.4 "Radionuclides"). Their earlier existence can be deduced from the analysis of the representation of their stable decay products (daughter nuclides). An example is iodine 129I, which decays to a stable xenon 129Xe with a half-life of 15.7.106 years; it was found in increased concentrations in iodine samples in meteorites. Furthermore, aluminum 26Al, which decomposes into magnesium 26Mg; or iron 60Fe .......
  So-called primary radionuclides (such as
40K, 232Th, 235,238U) have been preserved from radioactive nuclei, although their amount is lower than at the beginning - see §1.4 "Radionuclides". No transurans have been preserved, as well as radioactive isotopes of other elements with half-lives of less than about 108 years. In the following §1.2 "Radioactivity" will be discussed in detail the laws of radioactive transformations. Virtually all light and medium-heavy elements up to bismuth (ie with a proton number less than 84) have their stable isotopes, represented in nature.
  A notable exception is technetium Tc
43, which does not have a stable isotope (the most stable is 98Tc with a half-life of 4.2 million years, ...); about 30 isotopes of technetium are known. Therefore, it practically does not occur in terrestrial nature, and its place in Mendeleev's periodic table remained empty for a long time. Artificial technetium was first found in 1937 by chemists C.Perrier and E.Segrém in a sample of molybdenum, which had previously been irradiated by accelerated deuterium nuclei by nuclear physicist T.Lawrence ( 96Mo+2H®97Tc+n, resp. 98Mo+2H®97Tc+2n). Later (in 1962) a trace amount of technetium was found in uranium ore (approx. 1 mg Tc per 1 kg U), where it is formed as one of the fission products during spontaneous fission 235U. A relatively large amount of technetium is formed in nuclear reactors during the fission of uranium - in fuel cells about 27 mg of Tc is produced for every gram of 235U split. Due to the long half-life, technetium is one of the difficult components of nuclear waste. It is also interesting that this exotic and practically unknown element, thanks to its metastable isotope 99mTc, which is a pure g-emitter, has become a very important radionuclide, on which most methods of so-called radiosotope scintigraphy in nuclear medicine are based - see chap. 4 "Scintigraphy".
  Another such element from the center of Mendeleev's periodic table, which does not have a stable isotope and therefore occurs only in a small proportion, is promethium (Pm). And of course all the elements heavier than bismuth - actinides such as polonium, radium, radon, transurans. Thorium and uranium also do not have stable isotopes, but thorium-232 and uranium-238,235 are commonly found in nature due to their very long half-lives (as mentioned above).

Helium - an element of the sun god
The terrestrial story of the second most abundant element in the universe - helium
4He2 - is also interesting. Helium is so rare in terrestrial nature, that it has not been known for a long time and was surprisingly first discovered not on Earth, but on the Sun ! This happened in 1868, when the French astronomer Pierre Janssen examined the spectrum of solar radiation in detail and noticed that in addition to the spectral lines of hydrogen, carbon, oxygen and other known elements, there are spectral lines of a hitherto unknown "solar" element, which was called helium (Helios = ancient Greek sun god ). Only later was helium found on Earth, first in uranium ores (in 1895 W.Ramsay, P.T.Cleve and N.A.Langley), then in the natural gas from which it is now mined.
  Why is helium, which is abundandly widespread in space in general, so rare on Earth? This is because helium is too light and an inert gas that does not combine with anything (
valence electrons He completely fill the valence orbital 1s and thus prevent chemical reaction with other elements). Earth's gravity will not hold it, at Earth's temperature helium rises in the atmosphere and escapes into space from the upper layers of the atmosphere. Hydrogen gas behaves in the same way, but due to its high reactivity, it has combined with oxygen into heavier water molecules (which the Earth's gravity will sustain) and has been preserved in large quantities on Earth.
Note: Only large material planets (such as Jupiter) will retain more helium in the atmosphere due to stronger gravity.
  Thus, helium remained on Earth only in closed underground spaces, from where it could not escape into the atmosphere. All this helium on Earth probably comes from radioactive
a-decay natural radioactive substances, uranium and thorium - the a particles itself is the nucleus of helium, see below §1.2 "Radioactivity", part of "Radioactivity Alpha". It is estimated that about 3,000 tons of helium are produced in the Earth's interior per year. Most of the helium thus formed remains absorbed in the crystal lattice inside the rocks, some of which is released in the gas phase into the cavities in the Earth's crust. These closed underground spaces are also a reservoir of natural gas, from which helium is isolated by fractional distillation and liquefaction (in natural gas helium is present in a concentration of up to 7%). The most common use of liquid helium is as a cooling medium, as it has the lowest boiling point of all substances 4.22 °K = - 268.9 °C. Boiling liquid helium can achieve very low temperatures at which many conductors show superconductivity (§1.5 "Elementary particles", passage "Fermions as bosons; Superconductivity"). Liquid helium is therefore used for cooling superconducting electromagnets (§1.5, passage "Electromagnets in accelerators") in nuclear magnetic resonance, accelerators, tokamaks (§1.3 "Nuclear reactions", part "Tokamak") .

1.0. Physics - fundamental
natural science
  1.2. Radioactivity

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Vojtech Ullmann