Radioactivity, radionuclides, radiation

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

1.2. Radioactivity

Experience with the substance of which the surrounding world is composed shows that most substances are relatively stable over time and change only over longer time scales. However, these changes are mostly of a chemical nature, the atoms of the commonly occurring elements themselves are almost always stable. Practically until the end of the 19th century physicists and chemists firmly believed that atoms are immutable and eternal.
Discovery of radioactivity
At the turn of the 19th and 20th centuries however, phenomena have been discovered in which some substances emit invisible penetrating radiation. As early as 1896, during experiments with the luminescence of minerals and crystals, H.Becquerel observed that even without exposure to external light, some minerals (uranium compounds) emit special invisible radiation (Becquerel initially called them "uranium radiation"), which penetrates the light-tight envelope of photographic plates and causes them to blacken.
Henry Becquerel experimented with the luminescence of various minerals in Paris, including the uranium minerals he had from his father. He exposed the minerals to the sun's rays and judged their luminescence by blackening the photographic plates. One day, with the minerals ready for the exposition, the sky overcast and Becquerel, disappointed, placed the minerals in a drawer on the photo plates. After a few days, he called up the photographic plate (accidentally, perhaps to compare or check the quality of the plate, or whether the minerals have a chemical effect on the photographic plate..?..) and to his surprise he saw a black image of the minerals on the plate. No external light or any luminescence could cause this image, as the plate was still wrapped in black paper and the mineral was not in the sunlight, it was still in the darkness of the drawer. After further experiments, he come to an opiniont, hat a kind of invisible radiation emanated directly from the interior of some minerals, which penetrated the cover paper and exposed the photographic plates.
Note: A brief reflection on the extent to which the discovery of radioactivity was the result of chance or methodological procedure is given in §1.0 "Physics - fundamental natural science", passage "Significant scientific discoveries - chance or method?".

This phenomenon was then dealt with by Marie Sklodowska-Curie, her husband Pierre Curie and G.Bemont, who found other "glowing" elements, polonium and radium, in uranium ore. This phenomenon was called radioactivity (these substances "actively emitted radiation"). It was later found that radioactive elements change their chemical nature when emitting this radiation - radioactivity is accompanied by the transformation (transmutation) of the nuclei of atoms of one element to another element. In 1899 with the properties of radioactive ("uranium") radiation was dealt with by E.Rutheford, who found two different components in this radiation :
- soft component , which is absorbed by a sheet of paper and whose range in the air is only a few centimeters; he called it rays a.
- harder component , about 100 times more penetrating than a , which passes through a thin aluminum sheet; here he called radiation b .
Shortly afterwards, in 1900, P.Villard observed that radium emitted even more penetrating radiation, which is able to penetrate tens of centimeters of concrete; he called them radiation g. Later it turned out to be electromagnetic radiation with a very short wavelength, shorter than X-rays.
   Furthermore, at this time, Mr and Mrs Curie and A.Beckerel discovered that b- rays have a negative electric charge, the specific value of which is close to an electron (these results were then refined by W.Kaufmann in 1902) - it turned out that radiation b is a stream of electrons. In the years 1903-1908, E.Rutheford performed a number of experiments with the passage of radioactive radiation in the field of strong magnets. He found that the deviation of the rays a in the transverse magnetic field is significantly smaller and is directed to the opposite side than in the case of negatively charged radiation b. Finally, he showed that radiation a is a current of doubly ionized helium atoms, i.e.flow of helium nuclei. He found that the radiation g did not deflect in the magnetic field.
   In 1908, E.Rutheford and his co-workers discovered by spectroscopy that two new gases appeared in a closed tube with a sample of radium (RaCl ₂ chloride ) that had not been there before: one had spectral lines of helium, the other was then unknown and was called radio emanation, now radon. It turned out that radioactivity is a spontaneous decay of the atomic nucleus, in which the starting element changes to another - the element is transmutated. In 1919, the same Rutheford first achieved artificial transmutation of a non-radioactive nucleus: by bombarding nitrogen with particles a from a radioactive source (radium) it caused the reaction ¹⁴N₇ + ⁴He₂ ® ¹⁷O₈ + ¹H₁, which he first observed in a chamber filled with various gases, by means of a fluorescent screen recording the flying protons ¹H₁. The transformation of the nitrogen nucleus after the impact of the particle a was confirmed in 1923 by Rutheford's collaborator P.M.S.Blackett in a Wilson nebula chamber filled with nitrogen, where between a large number of "free" a particle trajectories in nitrogen cases where the trajectory of the particle a ended in "branching" into two new tracks: the long and narrow path of the proton and the wide short track belonging to the oxygen nucleus.
Artificial radioactivity
All these basic findings have been made on natural radioactivity, observed mainly in the heaviest elements. In 1934, F.Joliot-Curie and I.Joliot-Curie first created artificial radioactivity.
It happened when they irradiated aluminum with rays a. They observed that the aluminum thus irradiated, emits radiation even when the irradiation with the rays a was stopped, while the intensity of the radiation gradually decreases. By nuclear reaction with particles a, aluminum was converted to radioactive phosphorus ³⁰P, which was emitted by the positron e⁺ decays into silicon :
      a + ²⁷Al₁₃® ³⁰P₁₅ + n ; ³⁰P₁₅®(b⁺;2,5min)® ³⁰Si₁₄ + e⁺ .
Positron radiation was observed by J.-Curie in the ionization chamber. §1.3 "Nuclear reactions" is devoted to nuclear reactions.    
For some artificially created radioactive substances (eg already at the first ³⁰P) they then observed a new type of radioactivity b⁺, where instead of negative electrons, positively charged positrons are emitted. Gradually, a number of artificial radioisotopes were created, which showed all kinds of radioactivity - b^-,+, g, a. In 1940, G.N.Flerov and K.A.Petržak discovered that uranium, in addition to radioactivity a in a very small number of cases, also decays by spontaneous fission into pairs of medium-heavy nuclei, releasing neutrons; this spontaneous cleavage is characteristic of all transurans, where it is even more pronounced.
   As in other fields of physics, terminology, quantities, and units related to radioactivity and radiation have undergone a long and complex development that has left some illogicalities and ambiguities - to be specified below.
   We will not describe in more detail the next historical development of knowledge about radioactivity (and the heroic efforts of researchers) and we will enter directly into current physical knowledge about these phenomena.

General regularities of atomic nucleus transformations
Nuclear transformation means a change in the composition or energy state of a nucleus. In this chapter we will be interested in the spontaneous transition of nuclei to a more stable state with lower energy. First, we give a brief definition of radioactivity :

In this process, part of the binding energy of the nucleus is released in the form of kinetic energy of decay products (and often also in the form of electromagnetic radiation) - high-energy (ionizing) radiation is emitted. There is either a transformation of the nuclei of one element into the nuclei of another element - transmutation (for radioactivity a and b ), or energy deexcitation of the levels of the same nucleus (radioactivity g - isomeric transition). Radioactive transmutation is also called radioactive decay. In addition to the name radionuclides, the name radioisotopes is often used (these are certain specific isotopes of nuclei, showing radioactivity). Substances and articles containing radionuclides are referred to as radioactive emitters. The properties of specific radionuclides are detailed devoted to §1.4 "Radionuclides", the resulting radiation is then discussed in detail in §1.3 "Ionizing radiation".

Fig.1.2.1. Basic general scheme of radioactive transformation and exponential law of radioactive decay.

The most basic general scheme of radioactive transformation is shown in the left part of Fig.1.2.1. The nucleus A, called the parent, spontaneously (ie without external intervention - only due to internal forces and mechanisms acting in the nucleus *) transforms into a "slightly smaller" nucleus B called the daughter, while the particle C called radiation flies out. This particle radiation carries energy and the composition difference between nuclei A and B. The law of conservation of energy and baryon number (symbolically A = C + B for both of these characteristics) must be met. In order for radioactive transformation to take place, the mass-energy condition m(B) + m(C) < m(A) must be met according to the law of energy conservation, where m(A) is the mass of the parent nucleus A (analogously to the daughter nucleus B), m(C) is the rest mass of the emitted particle C. When nuclear transformation is released kinetic energy DE = [m(A)-m(B)-m(C)].c², which carries most of the emitted particles C, a small portion also resulting core B reflected due to the law of action and reaction. The energy difference between the basic states of the parent and daughter systems is called the energy of transformation Q.
*) Cause of radioactivity
The mechanisms of each type of radioactivity will be discussed in detail below. Here we can only give a brief summary of the causes of spontaneous nucleus transformation :
Radioactivity a is caused by quite understandable instability of too heavy nuclei (which strong short-range interaction ceases to be able to sustain), on whose "periphery" a very stable helium nucleus is formed, which can be emitted under certain circumstances by the contribution of a quantum tunneling phenomenon (Fig.1.2.2). Even simpler is the cause of radioactivity g - it is a mere "reorganization" of nucleons during deexcitation of an energetically excited nucleus, the energy difference of which is emitted by a photon g (this is analogous to the deexcitation of electrons in the atomic shell).
The cause of radioactivity b is more complex: it lies deep in the subnuclear region - in the quark structure of nucleons, where weak interaction can cause mutual transformations of quarks "u" and "d", and thus transformations of protons and neutrons (see below "Mechanism of decay b; weak interactions", Fig.1.2.5).
From an energy point of view, we will analyze the causes of radioactivity below in the section "Stability and instability of atomic nuclei". Here we can only state, that out of more than 1,800 known nuclides (natural and artificial), only 266 are stable, while others are radioactive - they spontaneously transforms themselves more or less rapidly into other nuclei (for more details see §1.4 "Radionuclides").
   Before we will describe the individual specific types of radioactivity, we will mention those aspects that all types of radioactivity have in common: these are mainly units of radioactivity and the exponential law of radioactive decay.

Units of radioactivity
As with any physical phenomenon that we want to quantify, it is necessary to determine the quantities and units of radioactivity in which we will measure its "strength", "intensity" or magnitude. The relevant quantity is called the activity (of emitter, preparation or set of nuclei in general) and is defined as the number of nuclei that is converted per unit time, or equivalent as the decrease in the number of nuclei (not yet converted) per unit time. The activity of a radionuclide is not a constant, but decreases with time as the original nuclei gradually disintegrate. The instantaneous value of A(t) activity at time t is therefore :
        A (t)  =  - d N (t) / d t  ,
where N(t) is the number of nuclei not yet converted at a given time t. The number of emitted particles per unit time, ie the intensity of radioactive radiation, is also proportional to this activity.
   Since radioactivity is a phenomenon in which the atomic nuclei of one element transform into the nuclei of another element over time, wherein the time is measuring in seconds, the natural unit of activity is 1 decay per 1 second. This unit was named 1 Becquerel in honor of the French pioneer in the field of radioactivity Henri Becquerel : 1 Bq = 1 decay / 1 second (on average *). And its decimal multiples: kilobecquerel (1kBq = 10³ Bq), megabecquerel (1MBq = 10⁶ Bq), gigabecquerel (1GBq = 10⁹ Bq). The greater the radioactivity of a given substance (sample) in Bq, the more nuclei per second are transformed by us and the more intense radiation the substance emits to its surroundings.
*) In the definition of a unit of radioactivity, the word "on average" (is given in brackets); under the influence of stochastic laws of quantum physics, radioactive decay does not take place evenly, but shows irregular statistical fluctuations. Therefore, we would measure a slightly different number of decayed nuclei every second - the results should be averaged, or measured for a longer time and the results normalized to 1 second.
Old units of activity - Ci, mCi, mCi 
At the beginning of the radioactivity research, one of the heroic acts was the separation of 1 gram of pure ²²⁶Ra radium from several tons of uranium ore by Marie-Sklodowska-Curie and her husband Pier Curie. Since it was the first known pure radioisotope, 1 gram of radium 226 was taken in their honor as a standard and the basic unit of radioactivity, which was called 1 Curie (1 Ci). The respective decimal fractions were then milicurie (1 mCi = 10^-3 Ci) and microcurie (1 mCi = 10^-6 Ci). Later, when many other radionuclides were discovered or prepared and the very nature of radioactivity was known (as shown in Fig.1.2.1), disadvantages of this randomly generated unit were seen. Conversion between the old and the current unit of activity is: 1Ci @ 37 GBq (i.e. 1 mCi @ 37 MBq and 1 mCi @ 37 kBq).
   In addition to the total activity of the emitter, in many cases it is necessary to know the specific (normalized) activity of the relevant preparation or sample. The specific activity is usually given as mass activity, which is the activity of the mass unit of the emitter - 1 kg, but in practice usually 1 gram: A_1g = A/M [Bq/g], where A is the total activity and M is the weight of the preparation. For liquids (or gaseous) preparations, volume activity is often used, ie activity per unit volume - liter, in practice mostly milliliters: A_1ml = A/V [Bq/ml], where A is the total activity and V is the volume of the preparation.
   The higher the activity of the emitter, the more intense it glows. The total energy output P[W] of the emitter of activity A [Bq] is: P = A.DE, where DE is the energy released during one decay (converted to Joules: 1eV = 1.6x10^-19 J), ie the difference between the energies of the parent and daughter nuclei, reduced by the resting energy of the emitted particle formed during the decay (as mentioned above). E.g. a emitter with an activity of 1GBq, formed by a radionuclide with energy DE = 1MeV/decay, will have an energy output of approx. 1.6x10^-4 W = 0.16 mW. Part of the energy performance of the radiator is converted into heat (kinetic energy scattered nuclei and the absorbed energy of radiation already at source), the residue carries ionizing radiation - constitutes the actual radiation power of radiator; in absorbing radiation in the substance, then the dose rate (see §5.1, "The effects of radiation on substance. Basic quantities dosimetry.").

The exponential law of radioactive decay
Radioactive transformation of nuclei is a stochastic quantum-mechanical phenomenon, so it is not possible to predict the time in which a particular nucleus will transform. Only the probability l with which the core of a given species transforms per unit time (1 second) can be determined. Let us have a radioactive substance (sample), in which at the initial time t = 0 there is a total of N_o of the same radioactive nuclei A, which will be gradually transformed into nuclei B (according to the diagram in Fig.1.2.1). We are interested in how fast the number of parent cores A will decrease (and thus also increase the number of daughter cores B) - in other words we want to determine the functional dependence of N(t) of the instantaneous number of N (remaining) parent nuclei on time t. The number of nuclei DN, that decay in a short time Dt, will be proportional to the current (instantaneous) number of nuclei N(t) and the probability factor l called the decay or transformation constant; in a sufficiently short time interval Dt, a fixed part of the present number of N radioactive nuclei is always converted. Thus, the original number of nuclei changes over time Dt by :
      D N(t)  =  - l . N (t). D t  .
If we go from finite differences to infinitesimal differentials (D-->d), we can write the differential equation after dividing by dt :
      dN(t) / dt  =  - l . N (t)  ,
whose integration gives the function N(t) = const. e^-l.t, where e @ 2,718 is the transcendent so-called Euler number - the basis of natural (Napier) logarithms. The boundary condition N(t=0) = N_o then gives the value of const = N_o for the integration constant, so we can write the resulting time law of radioactive decay :

The graph is a descending curve called an exponential (red curve in Fig.1.2.1 on the right). On the contrary, the number of nuclei of the daughter element B increases according to the exponential law N_B(t) = N_o. (1 - e^{-l .t}) - blue curve in Fig.1.2.1 on the right.
  An important quantity is the value of the time in which exactly half of the original number of nuclei is transformed: it is called the half-life of transformation (decay) and is denoted by T_1/2 . Thus N(T_1/2) = N_o/2, which after substituting into the above derived exponential law: N_o/2 = N_o.e^-l.T1/2 and logarithmization, gives l = ln2/T_1/2 @ 0,693/T_1/2. The exponential law of radioactive decay can therefore be written in the most commonly used form :

After the time t = T_1/2 has elapsed, exactly half of the nuclei are converted: N(T_1/2) = N_o/2. After the next half-time elapsing, the half of the half of nuclei number will remain, ie a quarter : N(2.T_1/2) = N_o/4. And so on to infinity, so only in the limit t®¥ will the limit be N(t®¥) = 0 and all nuclei will really transform *).
*) This is only theoretically. In fact, there is only a finite number of parent radioactive nuclei in each radioactive emitter or preparation. After a sufficient number of half-lives (tens or hundreds of T_1/2 ) have elapsed, in practice the last the parent nucleus will always eventually disintegrate (transform) and the sample will be non-radioactive.
  In addition to the decay constant l and the half-life T_1/2, the mean lifetime of the nucleus t = 1/l = T_1/2/ln2 is sometimes introduced, which is the time (t = t) for which the activity drops to 1/e @ 0.3679 of its original value. All three of these variables (half-life T_1/2, conversion constant l and mean lifetime t) indicate how fast a radionuclide transforms or decays.
The relationship between half-life and activity 
The number of atomic nuclei decaying per unit time (1s) is equal to -dn(t)/dt = l.n(t) = n(t). ln2/T_1/2; and this is by definition the activity in Bq (the value of T_1/2 must be given in seconds; for the number of nuclei we use the symbol small n so that it does not confuse with the nucleon number large N). Therefore, if we currently have n radioactive nuclei in the preparation at time t, its immediate activity will be
                         A [Bq] = n. l = n. ln2 / T_1/2 ;
it will therefore be directly proportional to the number of radioactive nuclei n, indirectly proportional to the half-life T_1/2 and will decrease with time according to the exponential law n(t) = n_o.e^-l.t = n_o.e^{-(ln2/T1/2).t} .
  If we have a radionuclide with nucleon number N, then the mass unit of 1 gram contains approximately n @ 1/(N.m_n) @ 6.10²³/N atoms (m_n is the mass of the nucleon; a binding energy that is less than 1% can be neglected here; the constant 6.10²³ is the known Avogadro number expressing the number of atoms in one gram molecule of the isotope). The specific activity A_{1g [Bq]} (mass activity) of 1 gram of pure radionuclide with nucleon number N and half-life T_{1/2 [sec.]} is thus given by the relation
                          A _1g @ (6.10 ²³ /N).ln2/T _1/2 @ 4.16.10 ²³ / (N.T _1/2) .
The radionuclide is thus the "stronger radioactive", the shorter its half-life and the smaller its nucleon number. For some known long-term radionuclides there is a specific activity :

The total activity A_[Bq] of the mass M_[g] of pure radionuclide with nucleon number N and half-life T_{1/2 [sec.]} is then determined by the relation
                            A = M.A_1g @ 4,16.10 ²³ .M / (N.T _1/2 ) .
In practice, pure carrier-free radionuclides are usually not present *), on the contrary, in most preparations they are very dilute, their concentration is small; the specific activity usually does not exceed a few GBq/gram, in some samples it does not even reach the level of 1Bq/g.
*) After all, pure carrier-free radionuclides with high activity would often melt or evaporate with the heat released - see "Thermal effects of radioactivity" below. E.g. 1 g of pure radioiodine ¹³¹I (T1/2 = 8 days) would have a colossal activity of 4,600,000 GBq, its energy output would be almost 750 W - by the heat released would evaporate immediately of it! Exceptions are radionuclides with a long half-life, which may well exist concentrated in larger quantities of many grams and kilograms - eg radium or uranium (for uranium 235 and plutonium, however, beware of critical mass and ignition of the chain fission reaction! - see the section "Nuclear fission" in §1.3 "Nuclear reactions").

Different decay rates and half-lives of radioactive decays
Each radionuclide has its very definite, specific and characteristic half-life *). The shorter the half-life T1/2, the faster the radionuclide transforms (decays). However, the half-life values are very different for different radionuclides. We know radionuclides with unusually long half-lives of the order of billions of years (these include some natural radionuclides such as ⁴⁰K with T1/2 = 1.3.10⁹ years or ²³⁸U with a half-life of 4.5.10⁹ years), hundreds of thousands of years, thousands of years (eg ¹⁴C radiocarbon with T1/2= 5730 years), hundreds and tens of years (eg ¹³⁷Cs with a half-life of 30 years), medium-long half-lives of years and tens or units of days (eg ⁵⁷Co with T1/2 = 270 days, radioiodine ¹³¹I with a half-life of 8 days), several hours or minutes (eg ^99mTc with T1/2 = 6 hours, ¹⁸F with T1/2 = 110min., ¹⁵O with a half-life of 2.2 minutes), even very "short-term" radionuclides with half-lives of the order of seconds or fractions of seconds (eg ^81mKr with T1/2 = 13sec., heavy transurans Z> 111 with half-lives sometimes of the order of milliseconds).
*) Rarely, however, are radionuclides that have two different half- lives in the same nucleus. This is because some nuclei can decay by two different mechanisms, each with a different probability and thus a different half-life. An example is the core ⁸⁰Br₃₅, which with a half-life of 17.6 min. disintegrates b^- decay to ⁸⁰Kr₃₆, and on the other hand with a half-life of 4.38 hours by b⁺ decay and electron capture transforms into a nucleus ⁸⁰Se₃₄. Another interesting example from the field of transurans is Californium ²⁵²Cf₉₈, which, on the one hand, decays with a half-life of 2.65 years a-decays to ²⁵⁰Bk (and then to the whole decay series), on the other hand, with a half-life of 85 years, it disintegrates by spontaneous fission into cores from the center of Mendeleev's table; while neutrons are emitted.
   These huge differences in half-lives are due to the different probabilities with which, according to quantum laws, the relevant processes take place inside the nuclei, which will eventually result in radioactive transformation. A number of factors in its construction determine whether a nucleus is stable or will disintegrate and at what rate (probability). Above all, these are configurations of energy levels of protons and neutrons in the field of nuclear forces, related to the relative number of protons and neutrons and the total number of nucleons. The higher the energy levels in the field of nuclear forces occupy protons and neutrons (an important circumstance here is that the proton and neutron levels occupy independently), the greater the likelihood of converting the nucleus to a lower energy configuration. Therefore, nuclei with a large excess of protons or neutrons, as well as nuclei with an enormously high total number of nucleons, usually decay considerably rapidly, ie with a short half-life, by radioactive decay by the mechanisms discussed in the next passages of this chapter.
   Enormous differences in the half-lives of radioactive nuclei raise the question of the limit values of half-lives that are still of physical significance, ie the question of what is the shortest and longest half-life values can in principle occur :

• What is the shortest half-life ?
  In §1.3 we will see that atomic nuclei (some stable, but mostly radioactive) are formed during nuclear reactions. Nuclear reactions involve a number of processes associated with the emission of various particles. Some emissions occur almost immediately after the interaction, others with a greater or lesser time delay - this is especially the case when a radioactive nucleus has formed. In the case of the above-mentioned very short-lived radionuclides, it may be debatable whether it is at all a nucleus with at least temporarily bound nucleons, and thus it makes sense to speak of its radioactive decay, or only of an immediately decaying configuration.
    Logical criterion for assessing this circumstances is to compare the life time of the "core" (resp. he delay time the emission of respective radiation) with the characteristic time during which a nucleon flying with energy several MeVs passes trough a typical nucleus diameter, i.e. distance »10^-13 cm. This so-called characteristic nuclear time is » 10^-21 sec. Only those nuclear configurations whose lifetime is significantly longer than this characteristic nuclear time, can be considered as separate nuclei and their subsequent decay can be considered as radioactivity. Otherwise, these are only kinematic effects in the nuclear reaction (eg, the ejection of a particle from the nucleus), which have no their origin in the real nucleus - the relevant radiation nuclear process is not referred to as radioactivity.
    However, from a practical point of view, even some short-term processes, which in a theoretical comparison have a lifetime somewhat longer than the characteristic nuclear time, cannot be analyzed experimentally and have essentially no real meaning. This problem is encountered in the research of strongly nonequilibrium nuclei, for example in attempts to form heavy transurans with Z> 115, which decay so fast that their volatile existence is very difficult to prove.
• What can be longest half-life of radioactive decay ?
  In the area of extremely long half-lives, we do not face the problem of proving the existence of nuclei - these undoubtedly exist and can be analyzed experimentally. The problem is to prove that they are not completely stable, but they decay apart over time, albeit extremely slowly.
    From a theoretical point of view, a logical criterion for assessing the stability or radioactivity could be a comparison of the mean lifetime of a given nucleus with the duration of the universe. According to current cosmological ideas, the universe has existed here for about 13¸15 billion years. If we have a total of N nuclei with a half-life of T1/2, then during the existence of the universe, of which about N.(exp)[-1n2.14.10⁹/T1/2] nuclei decayed.
  .......................
    An extreme case of the enormously long half-life is the radioactivity hypothesis of the simplest hydrogen nucleus, ie proton instability according to some grandunification theories (see §3.5) - the proton should decay with a half-life of about 10³³ years. If we compare it with the estimate of the total amount of approx. »10²⁷ protons in the universe, then during the existence of the universe may not have been able to disintegrate a single proton - protons can be boldly considered stable. However, in cosmological considerations (or rather speculations) about the global evolution of the universe in the distant future, even such extremely long half-lives can be significantly applied (there is "enough time") - see §5.6 "The Future of the Universe. The Arrow of Time." in the monograph "Gravity, black holes and space - time physics".
    From a practical point of view, it should be noted that the activity A of a preparation containing N nuclei with a half-life of T1/2 is A[Bq] = N. l = N.ln2 /T1/2, whereas the radioactivity is measured by detecting the emitted radiation (a, b, g), whose intensity is proportional to the activity A . For nuclides with an enormously long half-life (T1/2 > »10¹² years), the density (frequency) of the emitted radiation will be very low, significantly lower than the level of the natural radiation background, so that its experimental demonstration is very difficult. Therefore, in some nuclei with an extremely long half-life (mostly between 10¹⁴ and 10¹⁸ years) their radioactivity is often not even safely proven - eg ⁴⁸Ca, ^64,70Zn, ⁷⁶Ge, ⁸²Se, ^92,100Mo, ¹¹⁶Cd, ¹²⁴Sn, ¹²³Sb, ^136,142Ce, ^144,145,150Nd, ^147-149Sm, ¹⁵²Gd, ¹⁵⁶Dy, ¹⁶⁵Ho, ^{180,182,183,186}W, ¹⁹²Os, ^190,192,198Pt, ¹⁹⁶Hg, ²⁰⁹Bi..
    For stable nuclei, the half-life is set to infinity: T1/2 ® ¥ (and l® 0).

Radioactive transformation takes place in atomic nuclei, which are hidden deep inside atoms, whose electron shell effectively shields all chemical, mechanical and thermal influences, as well as the action of external fields. Radioactive decay is therefore independent of external normal physical and chemical influences and conditions (pressure, temperature, state, chemical form, external field, etc.) - there is no way to speed it up or slow it down. However, this statement is not entirely absolute. Below in the section "Independence of radioactive decay on external conditions", radiation or chemical influences, that may to some extent change the course of radioactive transformations in nuclei, will be discussed.
  Long-term stable rates or half-lives of some natural radionuclides are used to determine the ages of objects of organic origin and mineral rocks - it is described in more detail in §1.4, section "Radioisotope (radiometric) dating".

Mixtures of radionuclides, decay series, radioactive equilibrium
The (mono-) exponential law of decay derived from above applies only to a radioactive substance consisting of radioactive nuclei of the same species with a precise half-life value, whose daughter nuclei are already stable and do not decay further. However, if the preparation contains two or more species of radionuclides with different half- lives, the dependence of immediate activity on time will no longer be a (mono) exponential function, but will be a combination of two or more exponential dependences with different half-lives corresponding to represented radionuclides - it will be biexponential or multiexponential dependence in general - Fig.2.1.B (a).
  Specific situation in the time regularity of decay occurs when it is a radionuclide X, whose daughter nuclei are not stable, but continue to decay - it is generally a so-called decay series of "generically" related radionuclides - Fig.2.1.B (b, c). Disintegration dynamics here will depend on the ratio of half-lives of the parent radionuclide primary X and subsidiaries, further decaying radionuclides Y. If the half-life of the primary radionuclide is significantly longer than the half-lives of the daughter radionuclides, the rate of decay will be established at this longest (control) half-life - the decay series will be in radioactive equilibrium, at which the ratio of the activity of the parent and daughter radionuclides will be kept constant.
   By radioactive equilibrium we generally mean a situation where - despite radioactive decay - the relative number of nuclei of a certain radionuclide remains constant because these radioactive nuclei are constantly supplemented by some production mechanism, while the production rate of the daughter radionuclide is equal to its decay rate. This production mechanism can be :
l Nuclear reactions
This is the case in nature with cosmogenic radionuclides, which are constantly produced in the atmosphere by cosmic radiation (§1.6 "Ionizing radiation", part "Cosmic radiation", Fig.1.6.7). Also in the artificial production of radionuclides by a constant flow of neutrons in the reactor or protons in a cyclotron, a radioactive balance can be achieved between the production nuclear reaction and the decay of the resulting radionuclide (§1.4 "Radionuclides", section "Production of artificial radionuclides").
l Radioactive decay of the parent radionuclide having a longer half-life to daughter radionuclide with a shorter half-life.
   In terms of time can recognize two types of radioactive equilibrium of "generationally connected" radionuclides X and Y (as well as situation when the balance does not occur) :
¨ Long time equilibrium ,
also called secular equilibrium (lat. saecularis = centennial, permanent, long-lasting, secular). The already mentioned equilibrium concentration of cosmogenic radionuclides has this character. In the decay radioactive series, it occurs when the default parent radionuclide X has a much longer half-life than the daughter radionuclide Y; then, in terms of the time horizon of the half-life of the daughter radionuclide, we can consider its activity to be constant - Fig.2.1.B (c). In the long run, even the secular equilibrium is only approximate; overall, in fact, the "equilibrium" amount of the daughter radionuclide slowly decreases with the half-life of the parent radionuclide. Secular equilibrium occurs, for example, in the decay series of primary long-term radionuclides ²³²Th, ^235,238U (§1.4 "Radionuclides").
¨ Temporary equilibrium (passing, fleeting, unsteady),
also called transient equilibrium (lat. transitivus = transient, short-lived) is established when the half-life of the parent radionuclide X is only slightly longer than the daughter Y - Fig.2.1.B (b).
¨ No equilibrium
In a situation where the parent radionuclide X has a shorter half-life than the daughter Y, no radioactive equilibrium can be established at any time. If the parent radionuclide X was at the initial time t = 0 is prepared without the content of daughter radionuclides, then when its decay the amount of daughter radionuclides Y first increases, goes through a maximum and then decreases with the half-life of daughter Y.
Time dynamics of radioactive equilibrium
In the simplest case of subsequent conversion of two "generationally bound" radionuclides X(l_X)®Y(l_Y)®Z(stable), the rate of change of nuclei of parent radionuclide X and daughter radionuclide Y will be given by a system of two differential equations: dN_X/dt = - l_X.N_X , dN_Y/dt = l_X.N_X - l_Y.N_Y . By integrating it [under boundary conditions at t=0: N_X=N_X(0), N_Y=N_Y(0)] we obtain a biexponential time law for the daughter radionuclide Y :
            N_Y(t) = N_X(0).[l_X/(l_Y-l_X)] . (e^-lX.t - e^-lY.t ) + N_Y(0).e^-lY.t  ;
for the parent radionuclide X, of course, the classical monoexponential law N_X(t) = N_X(0) . e^-lX.t remains.

Fig.2.1.B. Time dynamics of radioactivity in a mixture of two radionuclides.
a) In a mixture of two independent radionuclides X , Y, each of them is converted according to its own half-life and the total activity of the preparation is given by the sum of both exponential functions.
b), c) In the decay series of two generically related radionuclides X --> Y, the decay dynamics depends on the ratio of the half-lives of the primary parent radionuclide X and the daughter, further decaying radionuclide Y; depending on this relation l_X and l_Y a transient or secular equilibrium of both radionuclides can then be established.
d) Specific radioactive dynamics of the radionuclide molybdenum-technetium generator during repeated elutions of the daughter ^99mTc, emerging from the conversion of the parent ⁹⁹Mo.

If l_X < l_Y, then after a sufficiently long time is exponential element e^-lY.t negligibly small compared with e^-lX.t. The law of time for the daughter radionuclide is then simplified to: N_Y(t) = N_X(0).[l_X/(l_Y-l_X)].e^-lX.t = N_X(t).l_X/(l_Y-l_X) - after a sufficiently long time, the equilibrium ratio between the daughter and the parent radionuclide is established, transient equilibrium (Fig. b), at which the activity of the daughter radionuclide decreases with the half-life of the parent radionuclide. The activity of the daughter radionuclide A_Y = l_Y.N_Y = N_X(t).l_Y/(l_Y-l_X) is maintained slightly higher than the instantaneous activity of the parent radionuclide (at l_X < l_Y the ratio l_X /(l_Y - l_X) is greater than 1), in the ratio A_Y/A_X = l_Y/(l_Y-l_X).
  If l_X << l_Y, then for short times compared to the half-life of the radionuclide X is l_X .t << 1 and the first exponential term e^{-l X .t} can be approximated by 1; after several half-lives of the daughter radionuclide Y (when the second exponential term is close to 0) an equilibrium amount of this radionuclide is then established N_Y = (l_Y/l_X).N_X - secular equilibrium (Fig. c). The equilibrium activity A_Y = l_Y.N_Y = l_X.N_X = A_X will be equal to the activity of the parent radionuclide X. In the long run, however, even this "equilibrium" amount decreases with the half-life of the parent radionuclide X.
   These regularities will be illustrated below in the case of the so-called radionuclide generator - part "Radioactivity gamma", passage "Pure gamma-radionuclides; technetium ^99m Tc; radionuclide generators". Specific types of generators for obtaining short-term radionuclides are described in §1.4, section "Radionuclide generators".
Note: The relationships given in this paragraph apply exactly provided that all disintegrations of the X ® Y ® Z chain take place in 100% of cases. If this is not the case, appropriate correction factors should be introduced and the resulting activity of the daughter isotopes will be lower.. This is the case with decay chains (b) -> (metastable g), if a certain percentage of decays do not take place at the excited metastable level, but at the ground state of the daughter core (an example is the mentioned Mo-Tc generator).
   For a decay series of a larger number of generically coupled radionuclides, their decay time dynamics is given by a system of a corresponding larger number of linear differential equations. E.g. for series 3 radionuclides X(l_X)®Y(l_Y)®Z(l_Z)®W(stable) it will be a system of three equations: dN_X/dt = - l_X.N_X, dN_Y/dt = l_X.N_X - l_Y.N_Y, dN_Z/dt = l_X.N_X - l_Y.N_Y - l_Z.N_Z . Under the initial conditions N_X(0)=N_Xo, N_Y(0)=N_Z(0)=0 we get the usual monoexponential decay law for the parent radionuclide X, for Y the above biexponential law, for the third member of the series Z it will be a combination of 3 exponential terms :
            N_Z(t) = N_Xo.l_X.l_Y . {e^-lX.t/[(l_Y-l_X)(l_Z-l_X)] - e^-lY.t/[(l_Y-l_X)(l_Z-l_Y)] - e^-lZ.t/[(l_Z-l_X)(l_Y-l_Z)]} .
For more complex decay series, additional exponential terms with corresponding exponents and coefficients are present. Particularly complex decay series with many secondary radionuclides occur in heavy uranium and transuranic nuclei; these decay series will be discussed in §1.4 "Radionuclides", Fig.1.4.1.

Some together phenomena accompanying radioactivity
Backscattering of nuclei
Due to the emission of particles and quants of radioactive radiation, due to the law of conservation of momentum - action and reaction - backs bounce of the daughter nucleus occurs, which takes over a certain small part of the kinetic energy of decay. This phenomenon of "hot" daughter nuclei is essentially reflected at each radioactive decay, but with most of the radioactivity alpha, as a particles capable have high weight and are usually emitted with high kinetic energy - with high momentum. For conventional alpha radionuclides with nucleon number N @ 220-240 that emit a particles capable of energy of about 4-7 MeV, the kinetic energy of the reflected daughter nuclei is around 100keV. The nuclei reflected in this way brake very quickly in the material on a path of approx. 500 nm, along which they cause ionization. The energy of chemical bonding of atoms in compounds is only about 50-100 eV.
  The back reflection of nuclei can thus lead to the release of atoms from the crystal lattice of minerals, or from chemical bonding in molecules *). The kinetic energy of the reflected nuclei will also have ionizing and thermal effects (see below). In common nuclear applications, core rebound is usually not physically manifested. However, for some accurate spectrometric measurements, it can have a significant effect, along with the thermal movements of atoms. Examples are Mössbauer spectrometry (see §1.6, section "Interaction of gamma and X-rays", passage "Mössbauer effect" and §3.4, section "Mössbauer spectroscopy"), or measuring the exact shape of spectrum b to determine the rest mass of neutrinos (see section below "Neutrinos", passage "Resting mass of neutrinos"). However, it can manifest itself significantly chemically :
*) (radio) Chemical effects of reflection 
When a radioactive atom is bound in a molecule of a compound, the reflection of nuclei can have a significant chemical effect. The kinetic energy of the reflection of nuclei, and thus of the respective atoms, is usually significantly higher than their chemical binding energy in the molecules of the compounds. Therefore, after radioactive transformation, the atom of the daughter nuclide is usually "ejected" - dissociated - from the chemical bond in the original molecule. This may have an adverse effect on the reduced radiochemical stability of radioactive preparations, especially radiopharmaceuticals (see §1.4, section "" In vivo generators "in nuclear medicine" or §3.6, part "Radioisotope therapy", passage "Physical and biological factors"). If alpha-transformation occurs in the organism, especially inside the cell, the ionizing effects of reflected nuclei may contribute to radiobiological effects (§3.6, passage" Beta and alpha radionuclides for therapy ") .
Thermal effects of radioactivity
A logical but little-known phenomenon in common applications accompanies all types of radioactivity: it is heat. During radioactive transformation, particles (quantum) of radiation fly out of the nucleus at high speed. According to the law of action and reaction, this will "throw" the nucleus (and in fact the whole atom) in the opposite direction - it will be given the kinetic energy of motion. Similarly, in the absorption of radiation, energy is transferred to the substance at the level of the kinetic energy of the atoms. And the kinetic energy of the motion of the atoms of matter is nothing but heat. With each further and further radioactive transformation, the atoms of the substance will thus oscillate to greater and greater kinetic energy - the radioactive substance will heat up. At low activities used mostly in practice, this phenomenon is unobservably weak, but strong emitters "warm up" quite clearly (eg in radiators for radiotherapy); the strongest emitters must even be cooled to prevent thermal damage (or even melting or evaporation) - this is especially the case for spent fuel cells from fission nuclear reactors (§.1.3, section "Nuclear reactors", section "Problem of cooling of nuclear reactors").
Note: The energy use of nuclear reactors in power plants is also based on similar thermal effects.
  The heat released by the radioactive decay of natural radionuclides, uranium ^235,238U, thorium ²³²Th and potassium ⁴⁰K, is probably an important source of geothermal energy, heating the Earth's interior - see §1.4 "Radionuclides", part "Natural radionuclides", passage "Geologic significance of natural radioactivity".
Electrical effects of radioactivity
Charged particles a or b (see below) carry away the electric charges from the radioactive emitter and according to the law of conservation of electric charge in the emitter substance, then predominated the opposite charge than the sign of the charge emitted particles. The actual radioactive emitter a or b is therefore gradually electrically charged. At low flows radiation, or if the emitter is least partially conductivelly electrically connected to the earth, this effect is negligible. However, electrically insulated radiator with intense flux of radiation b or a will gradually be positively or negatively charged to a high electrical potential of up to hundreds of kV (depending on the electrical capacity of the radiator body). This phenomenon is fully manifested only in a vacuum, because in the air the radiation causes ionization, the environment becomes partially electrically conductive and the charge is continuously removed from the radiator body.
  The electrical and thermal effects of radioactivity are used in small electrical sources - see §1.3, section "Radionuclide volta cells ("atomic" batteries )".

Independence of radioactive decay on external conditions
Radioactive decay is a spontaneous process - it is caused by the internal mechanisms of the atomic nucleus. It is independent of external normal physical and chemical influences and conditions (pressure, temperature, state, chemical form, external field, etc.), there is nothing to accelerate or slow down this disintegration. This is because the nucleus is hidden deep inside the atom, whose electron shell effectively shields all chemical, mechanical and thermal influences, as well as the action of external fields. However, this proven statement is not entirely absolute.
  Above all, it may not apply under extreme conditions. If we heat the radioactive substance to a very high temperature tens or hundreds of millions of degrees, the nuclei in an already completely ionized plasma will acquire such a high kinetic energy, that they will overcome the Coulomb repulsive force during collisions and nuclear reactions will occur, changing the speed and nature of radioactive decay (see §1.3 "Nuclear reactions"). Extremely strong electromagnetic or gravitational fields would have a similar effect (not yet available here on Earth, but they probably occur in universe around compact gravitationally collapsed objects, see eg Chapter 4 "Black Holes" of the book "Gravity, Black Holes and Space-Time Physics"). The Electron capture can be affected (slowed down or completely stopped) by heating the substance to about 10⁴ degrees, when the complete ionization of atoms, including the L and K shells, from which electron capture occurs; in addition, the rate of electron capture can be very slightly affected by chemical bonds (see "Specifics of electron capture" below). The rate of decay by electron capture can also be affected by the action of external pressure on radioactive atoms (higher pressure leads to an increase in electron density and thus to an increase in the probability of electron capture by the nucleus, but only at very high pressures). Complete ionization of atoms can significantly affect the half-life of beta ^- decay of some (isolated) radionuclides by the effect of so - called b-decay into bound electronic states. Gamma-radioactivity can be affected by the external environment through the effect of internal conversion of radiation g , which depends on the electronic configuration of the atomic shell. For fully ionized atoms, the internal conversion coefficient would become zero.
  The structure of nuclei, and thus radioactive decay, can be altered by irradiation with neutrons, fast protons or other particles *) causing nuclear reactions. Furthermore, the chemical bonding of the radionuclide may to some extent affect the electron capture and the process of internal conversion of gamma radiation (see below).
*) Even neutrinos can alter the course of beta radioactivity in nuclei: the capture of an (electron) neutrino induces the transmutation of a neutron to a proton, which is beta^- radioactivity, analogously the interaction of an antineutrinos with a proton induces a beta+ radioactive conversion. In practice, however, this phenomenon is not observed due to the very small effective cross section of the neutrino interaction (it could only be applied in a huge flow of neutrinos during a supernova explosion ).
  But these are all special or "exotic" situations. However, in almost all situations occurring in practice in the application of nuclear and radiation methods (Chapter 3), we can consider the course and half-life of radioactive decay to be completely independent of external conditions.

What happens to the atom during the radioactive transformation of the nucleus? Chemical properties of radioactive substances
Nuclei are in most circumstances part of atoms, for the structure of which they play a decisive role (the nucleus is the "boss" of the atom). The chemical properties of radioactive elements are, until the moment of radioactive decay, completely identical to the properties of their non-radioactive isotopes *). Thus, radioactive atoms (before their radioactive transformation) can enter the same compounds as the same non-radioactive atoms. This is used in a number of radiochemical technologies, including medical applications of radionuclides (see §3.5 "Radioisotope tracking methods", §3.6 "Radiotherapy", section "Radioisotope therapy" and §4.8 "Radionuclides and radiopharmaceuticals for scintigraphy").
*) Slight differences in kinetics (rate) during the chemical reactions caused by the different weight of isotope atoms (with different number of neutrons in the nucleus) are chemically usually insignificant. However, they can be used for isotope separation.
  However, during the radioactive transformation of the nucleus, changes in the atomic shell necessarily also occur *). When the radioactivity a the charge of the nucleus is reduced by 2 protons, which resultis in the release of two electrons from an atom envelope. For b ⁺ the charge of the nucleus decreases by one proton, so that one electron is released from the shell. During electron capture, the inner electron is absorbed by the nucleus, followed by a series of deexcitation jumps of electrons in the envelope, accompanied by the emission of characteristic X-rays and other soft radiation photons. Conversely, during the conversion of b^-, the charge of the nucleus increases by one proton, so that the resulting daughter atom takes up one electron from the surroundings. During this reconfiguration of the electron shell, following immediately after the radioactive transformation of the nucleus, the energy levels of the electrons are also shifted from the original levels of the parent atom to the new levels of the daughter atom, when emitting of soft electromagnetic radiation. The reconfiguration of the electron shell results in a change in the number of valence electrons and thus change in the chemical properties of the atom (its oxidation number). If the original parent atom was part of the compound, after the radioactive transformation, this chemical bond is usually broken and the daughter atom is eliminated from the compound - there will be dissociation, or establishes a new chemical bond.
*) With the exception of radioactivity g, in which the charge of the nucleus does not change, so there is no reconfiguration in the electron shell. However, there may be a so-called internal conversion of g radiation (see the "Gamma radioactivity" section below), which is accompanied by the emission and skipping of electrons in the envelope.
  The newly formed atoms just after the radioactive transformation of the original parent nucleus are called nascent atoms (lat.nascendi = birth). They have an initially deformed and excited electron shell, they have a non-zero electric charge - they are in the state of positive or negative ion, due to the above-mentioned back reflection of nuclei they have high kinetic energy ("hot atoms"). This leads to high chemical reactivity of nascent atoms after nuclear transformation.

Types of radioactivity
Radioactivity is divided and classified not according to the parent and daughter nuclei, but according to the type of radiation emitted (particle C in Fig.1.2.1). These types of radioactivity are indicated by the first three letters of the Greek alphabet - a, b, g. This terminology has a random, purely historical origin. Radioactive radiation was at that time labeled in the order in which it was discovered, without knowing its true physical nature. The only thing known at the time was that alpha radiation in the electric field deflects toward the negative electrode, beta radiation to the positive electrode, and gamma radiation is not bent by the electric or magnetic field. If at the time the "school-maniac pedants" they waited only a few years with the establishment of mandatory terminology "alpha-beta-gamma" in texbooks, now the radioactivity a would be called helium, b^- electron, b⁺ positron and g photon radioactivity. These would be far more apt names, expressing the true physical nature ...

Alpha radioactivity
The basic scheme of radioactivity a is shown in Fig.1.2.2. When this transformation is emitted nuclear particles a which is a helium nucleus ⁴He₂ - thus comprises two protons of p⁺ and 2 neutrons n^o :

Fig.1.2.2. Basic scheme of radioactivity a .

From the parent nucleus with N nucleons and Z protons, particles a carries out 2protons and 2neutrons, so the resulting daughter nucleus will have N-4 nucleons and Z-2 protons - in Mendeleev's periodic table of elements will be shifted 2 places to the left towards simpler elements. In order to radioactivity a may occur, must be satisfied by weight ~ energy condition m(Z-2, N-4) + m(a) < m(Z, N), where m(Z, N) is the mass of the nucleus with atomic number Z and the nucleon number N, m(a) º m(2,4) is the rest mass of the particle a.
  For too large nuclei, the strong nuclear interaction, due to its short range, is not enough to bind the nucleus strong enough against repulsive electrical forces between protons. Figuratively speaking, by alpha radioactivity heavy nuclei "get rid of excess nucleons" to make them lighter and more stable. These aspects are discussed in more detail below in the section "Stability and instability of atomic nuclei".
  Example of alpha radioactivity may be a remodeling of radium ²²⁶Ra₈₈ ® ²²²R₈₆ + ⁴He₂ (sa) at radon 222. Daughter nuclei after a -decay is often also radioactive (a or b), usually forming whole radioactive series (see §1.4 "Radionuclides", passage "Decay series"), until they reach a stable configuration. In addition, after the a-decay the product nucleus mostly in the excited state, so their dexcitation accompanied by gamma rays (see "Radioactivity gamma").
The spectrum of radiation a
Particles a carries the energy difference between the parent and daughter nuclei DE = [m(Z,N)-m(Z-2,N-4)].c², which is constant - all particles a during the transformations of a given type of nucleus, they have the same kinetic energies E_a = [m(Z,N)-m(Z-2,N-4)-m(a)].c² - radiation spectrum a is line, discrete. By spectrometric measurement of a number of a -radionuklides it was found that the shorter the half-life of a given radionuclide, the higher the energy of the emitted radiation a. This dependence between the half-life T_1/2 , resp. the decay constant l = (ln2)/T_1/2, and the energy E_a the radiation a gives the approximate Geiger Nutall relationship : ln l = A . ln E_a + B, where A and B are constants. These empirical dependence corresponds well with the mechanism of disintegration a the tunnel phenomenon of emission of particles a, mentioned below. For most a -radionuklides the energy emitted alpha particles in the range of about 4 - 8 MeV, reaching a rate of about 2-5% of the speed of light. The lowest alpha energy 1,830MeV is observed in the long-term ¹⁴⁴Nd (half-life 2.3x10¹⁵ years), the highest alpha energy 11,650MeV was measured at ^212mPo (half-life 45sec.).
  Kinetic energy E_a the emitted alpha particles are a small percentage lower than the total decay energy Q, because when the a- particle is emitted, the nucleus is reflected with low kinetic energy in the opposite direction due to the law of conservation of momentum. For a parent nucleus with mass number N , the resulting alpha-particle energy will be related to the decay energy by E_a @ Q. (N-4) / N.
The mechanism of disintegration a
In the right part of Fig.1.2.2 is intellectually shows the mechanism of radiate particles a. We can simply imagine that a heavy nucleus with more than 210 nucleons is already so large that the overall attractive field of strong interactions, due to its short range, no longer acts strong enough in the peripheral regions of the nucleus. It is not enough to sufficiently balance the mutual repulsion of protons, which has a long range (decreases with the square of the distance). This is used by some nucleons, which "cluster" so that 2 protons and 2 neutrons form a local stronger bound "cell", which then the so-called tunnel effect (described in §1.1, passage "Quantum tunnel phenomenon") overcomes the potential barrier of binding nuclear energy and flies out as particles a *). If a positively charged alpha particle (bound in the nucleus by an attractive strong short-range interaction) crosses the potential barrier, it is expelled from the nucleus by the repulsive electrostatic force in the region beyond the reach of nuclear forces, or rather ejected at high speed. By analyzing the permeability of this potential barrier through the tunnel effect, the mentioned Geiger-Nutall dependence between energy and half-life can be derived .
*) The alpha particle in the heavy nucleus is bound by a strong nuclear interaction with a potential wall height of about 25 MeV. However, the energy of the particles during decay is in the range of only about 4-9 MeV, depending on the particular radionuclide, so that the particle itself does not have enough energy to overcome this barrier. According to classical physics, emission could not occur. According to quantum physics, however, there is a small, but non-zero, probability that a particle can pass through a higher potential dam (§1.1, passage "Quantum tunneling").
Furthermore, the question may arise, why heavy nuclei are emitted just particles aº ⁴ He ₂ and perhaps no individual protons, neutrons, deuterons, or heavier than helium nuclei (e.g. carbon)? The cause is high binding energy particles a (28.3MeV). To escape from the nuclear field, a particle needs a certain kinetic energy. Individual nucleons usually do not have sufficient energy available, while the emission of a strongly bound particle a (whose mass is less than the mass of the nucleons of which it is composed) is more energetically advantageous. However, the emission of individual nucleons occurs in strongly excited nuclei, in some heavy nuclei radioactivity higher than alpha was also demonstrated - with the emission of 14-C even in heavier nuclear clusters (see below "Exotic types of radioactivity").
  For the assessment of the binding energies of nucleons in different nuclei, we recommend Fig.1.3.3 in §1.3., Section "Nuclear energy".
  Despite the fact that the letter a is the first in the Greek alphabet, alpha radioactivity is the least "important" of all types of radioactivity. This is for two reasons :
1. Radioactivity a occurs only in the heaviest nuclei from the very end of the Mendeleev table (N> 210) , mainly in the area of uranium and transurans (light nuclei simply "do not have the power" to radiate such a heavy particle as the nucleus helium a). Typical alpha emitters are, for example, radium ²²⁶Ra, plutonium ²³⁹Pu, americium ²⁴¹Am.
It is rare in some moderate radionuclides (see eg terbium ¹⁴⁹Tb ). The lightest element in which alpha radioactivity has been recorded is tellurium ^107-109Te₅₂. In connection with alpha radioactivity, beryllium is sometimes referred - its "exotic" isotope ⁸Be₄ *). With an extremely short half-life of 6.7.10^-17 s, it decays into two helium nuclei (particles a): ⁸Be₄®⁴He₂+⁴He₂. However, this is not about alpha radioactivity in the true sense of the word, but rather about fission highly unstable core into two equal parts.
*) Despite its high instability, beryllium ⁸Be is important as an intermediate in the thermonuclear synthesis of helium to carbon inside stars (see §4.1 "Gravity and evolution of stars", section "Evolution of stars", passage "Helium combustion" of the book "Gravity, black holes and the physics of spacetime").
2. Radiation a, thanks to its double positive charge, very effectively pulls electrons out of the shell of atoms when it enters a substance, thus rapidly losing energy and braking at about 0.1 mm in substances of water or tissue density (however, internally applied alpha-radionuclides are effective in radioisotope therapy - §3.6, section "Radioisotope therapy"). Alpha-emitters are used only sporadically in some detection instruments (e.g. gas density detectors, smoke detectors) or neutron generators (if we mix a-radiator with the target of a suitable material such as beryllium, particles a penetrate into the nuclei of the target material and by reaction (a,n) the neutrons are emitted from here), which are used in the laboratory, e.g. for neutron activation analysis.
Note: In terms of natural radioactivity, however, a-radioactivity is important. Most natural radionuclides on earth (except e.g. potassium ⁴⁰K) is formed of heavy elements - thorium ²³²Th and uranium ^{238, 235}U, which is a -radioactive. During the gradual decay of these radionuclides in decay series (§1.4, passage "Decay series"), many more nuclei are formed, while the particles a themselves after their braking add two electrons, thus creating neutral atoms of gaseous helium ⁴He₂. It is estimated that about 3,000 tons of helium are produced in the interior of the Earth in this way per year. It is assumed that practically all helium occurring on Earth was formed during the radioactive a decay of natural radionuclides thorium and uranium (the story of helium is briefly mentioned in the conclusion of §1.1 "Atoms and atomic nuclei", part "Nucleogenesis"). Most of the helium formed in this way remains absorbed in the crystal lattice inside the rocks, part of which is released in the gas phase into the cavities in the earth's crust, from where it is mined with natural gas.

Radioactivity beta
On the contrary, the most common and most important type of radioactivity is radioactivity b. There are three types of radioactivity b, which we will gradually discuss :

Radioactivity b ^-
The basic scheme of radioactivity b^- is in Fig.1.2.3. In this nuclear transformation, the particle b^- is emitted from the parent nucleus at high speed, which is nothing more than an ordinary electron e^- - the same as in the atomic shell.
Note: The names "radiation b", "particles b" come from a time when they were not yet known to be electrons.

Fig.1.2.3. Beta radioactivity.
Left: Basic scheme of radioactivity b^-. Middle: Continuous energy spectrum of radiation b. Right: Enlarged detail of the end of the spectrum for zero and non-zero rest masses of neutrinos.

If we remember the composition of the nucleus (§1.1), we immediately see an obvious paradox: how can (negative) electrons fly out of (positive) nuclei when there are only positive protons and uncharged neutrons, but no electrons?
Note: For some time, physicists believed that electrons b^- came from the atomic shell. However, it was found that in fully ionized beta-radioactive atoms depleted of electron shells, electrons b^- continue to radiate unchanged. Thus, electrons b^- really fly out of the atomic nucleus.
  When b^- radioactivity was found to occur in nuclei with an excess of neutrons, an explanation for the formation of b^- radiation was found^.: One of the "redundant" neutrons "wants" to become a rarer proton *) - it will do so by transforming: n^o ® p⁺ + e^- (+ n) ( let's not notice the particle n yet). The proton p⁺, as a legitimate nucleon, remains bound by a strong interaction in the nucleus, while the electron e^- flies out at high speed as radiation b^- (carries away the energy difference between nuclei A and B).
*) That "neutron wants to become a proton" is, of course, just a light allegorical expression; the actual disintegration mechanism b will be briefly discussed below in connection with weak nuclear interactions , Fig.1.2.5. However, a brief simplified explanation can be discussed here :
Cause of radioactivity b^- ; stability and instability of the neutron in the nucleus
The primary cause of radioactivity b^- lies in the conversion of neutrons to protons, electrons and antineutrinos - generally in the quark structure of nucleons, where weak interaction can cause mutual transformations of quarks "u" and "d", and thus transformations neutrons (see below "Mechanism of decay b ; weak interactions", Fig.1.2.5). After all, if a neutron is free, it is unstable - with a half-life of less than 15 minutes, it decays due to a weak interaction by the mentioned b -dissolution n^o ® p⁺ + e^- + n . In light of the described mechanism of b^- radioactivity, consisting in the decay of a neutron inside a nucleus, and the fact of instability of a (free) neutron, a paradoxical question may arise: How can stable nuclei containing neutrons exist at all? Why don't all the neutrons inside them decay into protons, electrons and neutrinos? The answer is that neutrons and protons in the nucleus cannot be seen as free - they are part of a higher whole, bound by a strong interaction. The stability of this whole, the nucleus, is then determined not so much by the stability of the isolated particles, but by the binding energy of the nucleons. If in a given bounded configuration a nucleus coposed of p protons and N neutrons has a lower total mass (~energy) than a core composed of the p +1 protons and n -1 neutrons, not to b ^--decay occur, neutrons will be "forced" to behave as stable. Otherwise, when a nucleus composed of p protons and n neutrons has a higher total energy ~ mass than a nucleus of p+1 protons and n -1 neutrons (at least by the rest mass of the electron, ie by 511keV), a situation arises where b^- decay may occur.
  Sometimes we can meet with the following argument :
If a neutron is strongly interacted in the nucleus together with protons, there is a so-called parity neutron-proton transformation: there is a continuous change of n ® p ® n, etc., mediated by the exchange of virtual p- mesons (which previously considered to mediate strong interactions between nucleons). The neutron thus bond in the nucleus, before it "manages" disintegrate by b -decay, changes (mutual exchange of p-meson with the closest proton) into a proton, and the newly born neutron "runs the decay time again from 0", again transforms into a proton etc. As a result, the short half-life of the neutron (13 sec.) by mutual conversion with protons is constantly "renewed" - neutron is stable in the core. Only when there is a "surplus" of neutrons in the nucleus, does there arise a certain probability that one of the neutrons will not "find" a suitable partner for meson exchange in time and will transfom by b-decay; this is then observed as the b ^-- radioactivity of such a nucleus.
  During the conversion of b^- the nucleon number does not change, but since one neutron has changed to a proton, the proton number increases by 1 - the daughter nucleus shifts by one place to the right in Mendeleev's periodic table. This remarkable fact that b^- decay produces a "more complex" nucleus than the original, has played a crucial role in the cosmic nucleosynthesis of elements heavier than iron inside stars and especially during supernova explosions (see "Cosmic Alchemy - We Are Descendants of Stars!" §1.1, §4.1 "Gravity and evolution of stars" of the book "Gravity, Black Holes and the Physics of Spacetime", or the syllabus "Cosmic Alchemy"). The heavy elements were created precisely due to the shift rule "to right" in beta^- radioactivity.
  Typical example of radioactivity beta is the b-conversion of tritium ³H₁ ® ³He₂ + e^- + n' to helium 3, or carbon ¹⁴C₆ ® ¹⁴N₇ + e^- + n' to nitrogen. Frequently used beta-emitters are cesium ¹³⁷ Cs , cobalt ⁶⁰ Co , iodine ¹³¹ I , iridium ¹⁹² Ir and others.
  In order for radioactivity b^- to occur, the mass-energy condition must be met m(Z+1, N) + m_e < m(Z, N), where m(Z, N) is the mass of the nucleus with proton number Z and nucleon number N, m_e is the rest mass of the electron.
The energy balance of nuclei can be well explained by the shell model of the atomic nucleus structure discussed in the previous §1.1. If there is an excess of neutrons over protons in the nucleus, neutrons will fill slightly higher energy levels than protons. The nucleus can then go into a lower energy state by converting the neutron by b^- decay to a proton, which goes to a free lower energy proton level.
  In terms of the mass of quarks "u" and "d", the stability and instability is discussed in §1.3, passage "Stability and instability of quarks, hadrons, nucleons".
Violation of parity conservation and asymmetry of the angular distribution of electrons b
For completeness, we mention one subtle property of radioactivity b - violation of the law of conservation of quantum parity number (parity is defined in §1.5 "Elementary particles and accelerators", section "Physical parameters of particles; quantum numbers"). This property leads to an asymmetry of the angular distribution of the flying electrons b with respect to the direction of the momentum vector (spin) of a given radioactive nucleus. This phenomenon was experimentally demonstrated on ⁶⁰Co cores. Relevant experiments and theoretical analysis are discussed in §1.5, passage "CPT symmetry of interactions".
  Violation of the conservation of parity and other symmetries have some theoretical importance in particle physics and interactions ("CPT symmetry of interactions") as well as in cosmology (§5.4 "Standard cosmological model. The Big Bang. Shaping the structure of the universe." in book "Gravity, Black Holes and Physics of spacetime"), but is insignificant in practice for the beta radioactivity itself. It manifests itself only in specially prepared experiments, at very low temperatures and strong magnetic fields. It is never occurs in nature or in physical practice, because the nuclei of radioactive elements are chaotically oriented and the radiation is on average isotropic in all directions.

Double decay b
A rare type of beta radioactivity has been reported in some nuclei - so-called double decay b. It consists in the simultaneous conversion of two neutrons in the nucleus into protons, emitting two negative electrons: 2n^o ® 2p⁺ + 2e^- + 2n´ and two electron antineutrinos. This type of decay rarely occurs in some nuclei with an even nucleon number (eg N = 48 or N = 116), in which there three isotopes occurs ^NA_Z, ^NB_Z+1, ^NC_Z+2, whose weights gradually decrease in a specific way (first by a smaller, then by a larger difference than 511keV). Overall, the mass ~ energy condition m(Z, N) > m(Z+2, N) + 2m_e must be met and at the same time the situation when normal beta decay cannot take place - either m(Z, N) < m (Z+1, N) + m_e , or some mechanism of its "disadvantage" is applied. In this case, it is possible to directly convert nucleus A to nucleus C : ^NA_Z ® ^NC_Z+2 + 2e^- + 2n´ by double beta decay. Denoted by (2n2b ), or (nnbb ) - two - neutrino double beta decay. It has been observed, for example, in nuclei ⁴⁸Ca₂₀, ¹¹⁶Cd₄₈, ¹³⁰Xe₅₄, half-lives are very long, about 10²⁰ years. Both double b^-b^- decay and double b⁺b⁺ decay are possible, as well as double electron capture.
  For some nuclei (such as ⁷⁶Ge) the possibility of neutrino-free double decay b is considered, in which the neutrino would not be emitted - (0nbb). We imagine the internal mechanism in two phases. In the first step, in the nucleus b -radioactivity, one neutron n₁ is converted into a proton: n₁ ® p₁ + e₁ + n, after which in the second step the resulting neutrino is absorbed by another neutron: n + n₂ ® p₂ + e₂. The emitted and absorbed neutrino is only virtual, the result is the transformation of (Z, N) ® (Z+n, N) + e₁ + e₂, in which two neutrons are converted into protons (the proton number increases by 2) and only two electrons are emitted. The neutrino-free double beta decay would violate the law of conservation of the lepton number (0®1+1). Experimental demonstration and analysis of this process could help to refine the determination of the mass of the neutrino (reduce the current upper limit of the mass) and would show that the neutrino is a so-called Majoran particle, ie the neutrino and the antineutrinos are identical.

b^--decay into bound electron states
Another small interest is shown in the comparison of the radioactivity of beta^- in "bare" nuclei (in fully ionized atoms) and nuclei surrounded by electrons in atoms. In classical beta-radioactivity, an electron is emitted from the nucleus, leaves the atom and flies into the environment. However, at low energies, there may be a situation where the emitted beta-electron does not leave the atom, but remains bound to one of the lower orbitals of the daughter atom. This cannot happen with neutral atoms whose internal electron orbits are already occupied. However, for fully ionized b^- radioactive atoms, emitted b-electrons can be bound to the low orbitals of the daughter atom (which are all free here). Full ionization of the atom thus can create an additional "channel" of beta decay to bound electron state (bound-state b^- decay).
  This phenomenon is observed in ¹⁶³Dy, which is a stable isotope under normal conditions. However, when fully ionized (¹⁶³Dy⁶⁶⁺), it undergoes a b- decay to a K and L shell of ¹⁶³Ho with a half-life of 47 days. Another case is ¹⁸⁷Re, which is in neutral atomic form b-radioactivity converts with a very long half-life of 42.10⁹ years to ¹⁶³Os. At full ionization, however, this half-life is only 33 years - the b -decay  channel to a bound electron state has reduced the half-life by a billion times! Under normal terrestrial conditions, the influence of beta-radioactivity by complete ionization of the substance is not commonly encountered, but it is assumed that it can be assert in astrophysical processes of cosmic nucleosynthesis - see §1.1, section "Cosmic alchemy".

Radiation spectrum b . Neutrinos
What about the energy spectrum of emited electrons of beta radiation? Electron b should carry away the energy difference DE = E_A - E_B between the parent A and daugher B core which is DE = [m(Z, N) - m(Z+1, N)].c². This difference is always constant, so all electrons b should fly with the same kinetic energy E_b = [m(Z, N) - m(Z+1, N) -m_e].c² - the spectrum should be line, as indicated by the red dashed line in Fig.1.2.3 in the middle. However, if we measure the actual spectrum of radiation b, we get a different result: the spectrum will be continuous *) from zero and with E_A-E_B energy will end - a thick black curve in Fig.1.2.3 in the middle. The vast majority of electrons b therefore fly out with energy much less than would correspond to the law of conservation of energy!
*) The continuous spectrum of radiation b was first measured in 1914 by J.Chadwick using bending in a magnetic field (the first prototype of a magnetic spectrometer) and detection by G.-M. tube.
  After some initial doubts about the validity of the law of conservation of energy for radioactivity b (these doubts were initially expressed also by N.Bohr) the following solution was proposed: in addition to the electron b, at the same time another (not yet observed) very light and electrically neutral particle n flies out of the nucleus, which carries away the appropriate kinetic energy, which it "shares" with the flying electron, in accordance with the law of conservation of energy. The Swiss physicist W.Pauli came up with the hypothesis of this particle in 1930. The Italian physicist Enrico Fermi (a pioneer in the research of radioactivity b and nuclear physics in general) likened this strange particle to a "small neutron" (tiny and neutral) - Italian neutrino - and this name and the designation "n" it has remained. Neutrino remained a hypothetical particle for more than 20 years, experimentally was demonstrated in the 50s, see below. The physical properties of neutrinos and the possibilities of their detection are discussed in more detail below in the section "Neutrinos".
  If the neutrino "steals" almost all the energy of decay, the electron b is "slipped out" from the nucleus with the low energy (case I. marked in the spectrum in Fig.1.2.3 in the middle). Conversely, if the electron b manages to capture most of the energy, it will fly out with high kinetic energy (case II. in the middle part of Fig.1.2.3). And case III. in the middle of the spectrum it occurs when an electron and a neutrino "share" the energy roughly equally. In most cases, the neutrino carries about 2/3 of the total energy and about 1/3 of the maximum energy will remain on the electron - this corresponds to the wide peak of the continuous spectrum of radiation b in Fig.1.2.3 in the middle.
  Beta radiation therefore has a continuous energy spectrum, it contains electrons with energies from zero to a certain maximum energy, which is characteristic for a given radionuclide. Typical values of the maximum energy of most beta emitters are tens of keV to MeV units (the lowest energy of 2.5 keV is observed for radionuclide ¹⁸⁷Re, the highest energy of 16.6MeV was measured at ¹²N).
The shape of the radiation spectrum b
The exact shape of the curve of the radiation spectrum b follows from the analysis of the energy distribution of the emitted electrons within the framework of Fermi's theory of weak interaction. It follows from this theory that the intensity N(p) of radiation b of momentum p and energy E_b is given by the relation N(p) = (E_bmax-E_b)².p².F(Z,p), where E_bmax is the maximum decay energy b and the constant F (expressing the correction for the Coulomb field of the nucleus) includes the relevant constants including the proton number Z. From this relation follows for the spectrum b the equation E_bmax-E_b = Ö[N(p)/p².F(Z,p)]. If we plot the function Ö[N(p)/p².F(Z,p)] on the vertical axis as a function of the energy E_b on the horizontal axis, we get a linear dependence called the Fermi-Kurie graph. It is a descending line that intersects the horizontal (energy) axis at a point indicating the maximum decay energy b. Fermi-Kurie graphs are sometimes used in accurate spectrometric analysis of radiation b. The shape of the end section of this spectrum (see Fig.1.2.3 on the right) - deviation from the linear dependence in the Fermi-Kurie graph, can also be used to determine the mass of neutrinos, see "Resting mass neutrino" below.
  To determine the mean energy` E<sub>b</sub> rays b from the shape of the radiation spectrum b can come out in the first approximation of the laws of the approximate` E_b ~ E_bmax/3. For higher energies, this value is relatively moves slightly to the right, so that more accurate empirical formula is: `E_b » (E_bmax/3).(1+¹/₄ÖE_bmax); (in [MeV]). Certain impact on the final energy has a Coulomb electric effect of flying particles b with a charge of the nucleus of protons Z, which leads to a further correction factor (1 - ¹/₅₀.ÖZ).
Inverse beta decay
The standard beta^- radioactivity discussed above, also called beta decay, consists in the conversion of the neutron n^o to the proton p⁺ with the emission of the electron e^- and (electron) neutrino n: n^o ® p⁺ + e^- + n. However, there is the opposite process, a nuclear reaction in which a proton can combine with an electron to form a neutron and emit a neutrino: p⁺ + e^- ® n^o + n - is called inverse beta decay (electron uptake by an proton is the opposite of the emission of an electron from the nucleus during beta^- -decay). It occurs either spontaneously in atoms during radioactive electron capture (described below in the section "Electron capture"), or is forced by external forces - either by the kinetic energy of accelerated electrons, or by very strong gravity. In universe, this process of neutronization takes place avalanche during the gravitational collapse of massive stars - the supernova explosion and the formation of a neutron star (it is discussed in detail in §4.2, passage "Supernova explosion. Neutron star. Pulsars." in book "Gravity, black holes and spacetime physics ").
Note.: For inverse beta decay is sometimes considered the interaction of neutrinos with protons to form neutrons and positrons: n'_e + p⁺ ® n^o + e⁺, or with neutrons to form a proton and electron: n_e + n^o ® p⁺ + e^-, which are used to detection of neutrinos in neutrino detectors (described below "Interactions and detection of neutrinos").

Neutrinos - "ghosts" between particles
Although elementary particles will be systematically discussed until later in §1.5 "Elementary particles", beta radioactivity is a good opportunity to mention in more detail here very interesting and remarkable particles of the microworld - neutrinos. We will do so somewhat more generally, not only in direct connection with radioactivity b *). However, the overall classification of neutrinos in the systematics of other elementary particles will be discussed in §1.5.
* ) The rationale for the existence of neutrinos and the origin of their name was discussed above in the section "Radioactivity b^-", passage "Radiation spectrum b .Neutrinos". Neutrinos, once formed as an additional not very convincing hypothesis trying to explain "something that was missing", have become real and very interesting particles not only for nuclear physics, but also for astrophysics and cosmology.
  Neutrinos are tiny particles (with a rest mass close to zero - see below passage "Rest mass of neutrinos"), which do not have an electric charge and do not show a strong nuclear interaction; they show only a weak nuclear interaction *), which is so weak and short-range, that neutrinos almost do not interact with the substance and fly freely (like "ghosts"; in addition, it shows the ability to "reincarnate" each other - oscillations between three different types, see below).
*) And, of course, universal gravitational action is assumed, which is a negligible force at the microscopic level and we are not interested in it yet. However, for astrophysics and cosmology, the gravitational action of neutrinos, of which there are huge quantities in the universe, can be of considerable importance, see the passage "Rest mass of neutrinos" below.
  Many billions of neutrinos pass through our bodies every second. It is estimated that here on Earth, every cm², including the surface of our body, flies about 50 billion neutrinos every second *) (coming mainly from thermonuclear reactions inside the Sun), but we do not have to worry about their harmful effects on our health, during one lifetime only one or two of these neutrinos will be trapped in our body. The vast majority of neutrinos are able to fly freely throughout our globe. It is estimated that the neutrino could completely capture up to a layer of lead 1000 light-years thick!
*) This huge number of neutrinos occurs here on Earth near the Sun and in space near every star, in which huge amounts of neutrinos are formed during thermonuclear reactions. In distant interstellar and intergalactic space, the density of neutrinos is estimated at about 300 neutrinos/m³, which are mainly relict neutrinos.
  Neutrinos are produced in huge numbers in a various processes in space - from the "big bang" lepton era, to thermonuclear reactions in stars, to supernova explosions (see below). Due to their stability and permeability, these particles are constantly accumulating in the world. So it can be judged that (along with photons) neutrinos are the most abundant particles in the universe - we are as if "immersed in the invisible sea of neutrinos", which was born at different times and in different parts of the universe. They are ubiquitous, but almost elusive particles..!..

Formation and types of neutrinos
Neutrinos - very light particles (at rest mass close to zero), without electric charge, with spin 1/2, moving at a speed close to the speed of light - are inseparable accompanying particles in all processes with elementary particles with weak interaction. Their emissions accompany the formation of electrons during the decay of pions and muons, the mutual conversion of protons and neutrons (especially during radioactivity b), as well as a number of processes in the collisions of elementary particles at high energies. Neutrinos, along with photons, are among the most abundant particles in the universe.
  According to its origin, specific place and mechanism of origin, we can divide neutrinos into five groups :

• Relict neutrinos
  from the earliest periods of the universe's evolution. In the so-called lepton era, less than 1 second after the Big Bang, when the temperature dropped below about 10¹⁰ °K, the neutrinos practically stopped interacting with other substances (electrons, neutrons, protons). Since then, huge amounts of "relict" neutrinos have moved freely through the expanding universe (now their density is estimated at about 300 neutrinos/cm³ in every bit of outer space, even here on Earth). As a result of the cosmological expansion of the universe, the energy of relic neutrinos has dropped to very low values, so we have no reaction available to detect them.
  The cosmological aspects of these processes are described in §5.4 "Standard Cosmological Model. The Big Bang." of the book "Gravity, Black Holes and the Physics of Spacetime".
• Stellar neutrinos
  In the interior of all stars, the thermonuclear synthesis of hydrogen nuclei to helium takes place (whether it is a direct p-p chain or CNO cycle), in the later stages of stellar evolution also the synthesis of heavier elements (see the passage "Cosmic alchemy - we are descendants of stars!" in §1.1, too §4.1 "Gravity and evolution of stars" of the book "Gravity, Black Holes and the Physics of Spacetime", or the syllabus "Cosmic Alchemy") . In the chain of these nuclear reactions, in addition to strong interactions, the weak interactions also participate - beta decays, in which neutrinos are formed with energies up to about 20MeV *). Thanks to its permeability, neutrinos easily escape from the interior of stars and spread to the surrounding space. A large amount of neutrinos is emitted from the interior of the Sun, which "floods" the entire solar system. In our Earth, the flux of these solar neutrinos is about 6.10¹⁰ neutrinos/second/cm², most of which fly freely across the globe.
  *) Most neutrinos are formed in the "start" proton-proton reaction p⁺ + p⁺® ²H + e⁺ + n_e, but their energy (<0.42MeV) is low for detection methods. Suitable for our detection methods are neutrinos formed in one of the sub-branches of the reaction, in which the boron nucleus changes by b⁺ decay to beryllium nucleus, positron and neutrino: ⁸B ® ⁶Be + e⁺ + n_e, where the neutrino can have maximum energy up to about 14MeV. This reaction is only about 0.01%, but the corresponding solar neutrinos are well detectable.
• Neutrinos from a supernova explosions
  A supernova explosion is accompanied (and actually caused) by the rapid absorption of almost all electrons by protons (p⁺ + e^- ® n^o + n - a inverse b-decay), with a sudden colossal amount of neutrinos is emitting (estimated at 10⁵⁷ neutrinos). There is also a massive neutron fusion of nuclei in the outer layers, accompanied by subsequent b^- decay, also with the emission of neutrinos.
  Several neutrinos of this origin were detected in 1987 after the explosion of the supernova SN 1978A in the Large Magellanic Cloud, see below. The processes during the supernova explosion are described in more detail in §4.2 "The Final Phases of Stellar Evolution. The Gravitational Collapse" of the book "Gravity, Black Holes and the Physics of Spacetime".
  High-energy neutrinos 
  In addition to low- and medium-energy neutrinos (of the order of MeV), high-energy neutrinos can also be formed during a supernova explosion. The rapidly expanding shell of the ionized mass in a supernova explosion can use the Fermi mechanism to accelerate protons to high energies. The interaction of these protons with the shell material produces (via pions and muons), among other things, high-energy neutrinos (1 ¸1000 TeV), which are then part of the primary cosmic rays.
• Neutrinos from secondary cosmic radiation (atmospheric neutrino)
  Interactions of hard cosmic radiation (primary) particles with nitrogen and oxygen nuclei in the upper atmosphere (at altitudes of about 20-30 km above the ground) create sprays of secondary cosmic radiation, of which neutrinos are a tertiary part. In primary interaction occurs there produce pions p^± (and also K mesons), which disintegrate rapidly on muon m^± and (muon's) neutrinos: p^- ® m^- + n'_m , p⁺ ® m⁺ + n_m. Muons continue to decay continuously into electrons, positrons and neutrinos (muon's and electron's): m^- ® e^- + n'_e+ n_m , m⁺ ® e⁺ + n_e+ n'_m . Neutrinos of this origin are referred to as atmospheric neutrinos. Cosmic radiation is discussed in more detail in §1.6, section "Cosmic radiation".
• Neutrinos from natural radioactivity , geoneutrinos
  In the Earth's crust there are radioactive decays of natural radionuclides, especially uranium, thorium and potassium 40 (see §1.4 "Radionuclides", part "Natural radionuclides", passage "Geological significance of natural radioactivity"), during which electron (anti) neutrinos are formed, which due to their origin are called geoneutrinos.
  Detection of geoneutrin , which from the inaccessible depths of the Earth's interior passes freely through thick layers of rock to the surface, could provide valuable information on radioactive elements inside the Earth. By analyzing the directions of arrival of neutrinos in detectors, located in different places, we can get information about the spatial distribution of radioactive elements. By spectrometric analysis of neutrino energy it is possible to find out in principle which radioactive nuclei emit them: neutrinos from the decay of potassium 40 and uranium 235 have a maximum energy of about 1.3MeV, from the decay series ²³²Th about 2.2MeV, from the series ²³⁸U the maximum energy of neutrinos is about 3.2MeV. After the improvement of the detection technique, it will be possible to use the detection of geoneutrinos for the geological search of uranium or thorium deposits deep underground.
    In the air, in water and in the bodies of plants and animals, there is also a beta- decay of cosmogenic radionuclides - radiocarbon ¹⁴C and tritium ³H, also with the emission of neutrinos.
• Neutrinos from nuclear reactors and accelerators (neutrinos of artificial origin)
  Here on Earth, nuclear reactors are intense sources of neutrinos in modern times. During the itself fission of uranium or plutonium nuclei, neutrinos are not formed, but they are subsequently formed during the radioactivity of the fission products. Heavy nuclei (such as ²³⁵U) have a higher neutron to proton ratio for their stability than stable medium-heavy nuclei. The medium-heavy nuclei that are formed during fission therefore have a significant excess of neutrons, which they "get rid of" with a few b^- radioactive transformations until stable nuclei are formed. This produces "beta" electrons and electron (anti) neutrinos, and gamma radiation during deexcitation of nuclear levels. After each fission, about 5-6 neutrinos are formed. During this subsequent b- radioactive decay of fission products with an excess of neutrons, a large amount of electron antineutrins (with an average energy of about 4MeV) is formed. They are sometimes called "reactor neutrinos". A powerful reactor produces about 10²⁰ neutrinos per second.
    To a lesser extent, neutrinos are formed as "by-products" of particle interactions in accelerators (there are energies of tens of MeV to hundreds of GeV). Accelerators then enable the targeted production of high-energy neutrino beams, especially muon ones :
     High-energy protons from the accelerator are allowed to collide with the nuclei of the target, creating a large number of mesons p (and many other particles - kaons, hyperons, ...) . Charged pions are passed through a vacuum decay tunnel, where they decay during flight (they have a half-life of only about 2.10^-8 sec.) to form muons and muon neutrinos: p^- ® m^- + n'_m , p⁺ ® m⁺ + n_m. These muon neutrinos, arising from high-energy pions, also have high energy and are closely collimated in the direction of the original pion beam (in the direction of the decay tunnel axis) - see " CNGS + OPERA Experiment " below.  
   The production of high-energy electron neutrino beams is more difficult. One possibility could be the acceleration of lighter b- radioactive nuclei with a short half-life (eg ^6,8He, ⁹Li, or the like) on a linear or circular accelerator, which would then be fed to a magnetic storage ring with longer linear sections. Accelerated nuclei would emit electron neutrinos during their b- conversion, which would (in the laboratory reference system) also be "accelerated" to high energies and fly out in beams collimated along said linear sections. It has not been implemented yet.

According their quantum physical nature, three kinds of neutrinos n are observed, according to the type of the associated charged lepton :

• Electron neutrinos
  Neutrinos, operative in beta radioactivity, are just one of three kinds of neutrinos - are neutrino electron n_e accompanying the emission of an electron and a positron in the transformation of beta. However, electron neutrinos (at least in their primary origin) are the predominant type of neutrinos - they are formed during thermonuclear reactions inside stars, during the explosion of supernovae, in natural and artificial radioactivity, in nuclear reactors.
• Muon and tauon neutrinos
  In processes with muons m and tauons t (see §1.5 "Elementary particles"), muon neutrinos n_m are emitted (they are an important part of "atmospheric" neutrinos formed in sprays of secondary cosmic radiation) and tauon neutrinos n_t. Most of their properties are very similar to electron neutrinos, they differ in the way of their interactions with elementary particles - instead of processes with electrons (such as b decays) they are accompanied by weak interactions with the participation of muons and tauons.

We distinguish these individual types of neutrinos according to which type of charged lepton they form from or which lepton is formed during their interaction, whether it is an electron, muon or tauon. In connection with the law of conservation of the lepton number, it is assumed that for each of the three types of neutrinos there is also a corresponding antineutrino n'.
Difference between neutrinos and antineutrinos - helicity of neutrinos. Goldhaber's experiment.
It is discussed whether the neutrinos are Dirac or Majoran particles - whether the antineutrins are different or identical to the neutrinos. The only way to distinguish a neutrino/antineutrino pair from each other is their helicity (rotation, spiral) - the orientation of the spin (internal angular momentum) of a particle with respect to the direction of its motion. Direct measurement of neutrino helicity would be very difficult (beyond the scope of current experimental techniques). The only viable option is the indirect determination of the helicity of the neutrino through the law of conservation of the momentum in the radioactive process electron capture EC (see "Electron capture" below). During electron capture, only neutrino is emitted from the nucleus, which we are essentially unable to detect. However, the daughter nucleus after EC is often in an excited state, during the deexcitation of which a gamma photon is emitted , which in its helicity carries certain information about the angular momentum balance. Under certain circumstances, this information can be decoded to determine the helicity of the flying neutrino.
  This indirect determination of neutrino helicity was achieved in 1958 by an American team led by M.Goldhaber (Brookhaven National Laboratory) by a very sophisticated experiment. They used to measure the helicity of 963keV gamma deexcitation photons emitted from the excited state ¹⁵²Sm generated by electron capture in the nuclear isomer of europium ^152mEu (§1.3, passage "Europium"). This nuclide was chosen because the start and end states have the same spin number 0^-. During electron capture, the neutrino and deexcitation photons are then emitted with opposite spin (due to the law of conservation of angular momentum). If we measure the polarization of the photon, we also find the helicity of the neutrino. The ^152mEu emitter was placed in a ferrite magnet, which filters (selects) the flying deexcitation quantum according to the polarization ...... These gamma photons then fall into the ring from the inactive ¹⁵²Sm, in which they can cause resonant absorption.with deexcitation, whose photons are registered by a scintillation detector... A relatively complex analysis of the results of measured impulses in confrontation with the balance of spins (angular momentums) gave the helicity of electron neutrinos the value Hn_e = -1 ± 0.3 - neutrinos are left-rotatory, while antineutrinos right-rotatory. This is consistent with the finding of parity violation in a weak interaction (see §1.5, passage "CPT symmetry of interactions").

Oscillations of neutrinos
In neutrinos we encounter a surprising and from the classical point of view incomprehensible phenomenon - that there are spontaneous transformations between individual types of neutrinos - the so-called oscillation of neutrinos. Neutrino during their flight while electron n_e, then turns into a muon and taun neutrino and then again on the electron etc. (but see note "physical reincarnation?" at the end of the segment). Metaphorically, it can be said with a bit of exaggeration that neutrinos behave like mythical "ghosts" in a sense: they can go through anything without hindrance (even walls...) and are able to "reincarnate" each into other ...  
  We imagine the phenomenon of neutrino oscillation as a consequence of the quantum-mechanical interference of three quantum states of a neutrino particle, which is an internal superposition of the "1", "2" and "3" neutrino states. These states are described by quantum waves of different wavelengths *), which, when propagated through space, alternately reach the same phases and different phases, which manifests itself as electron, m - or t - neutrinos.
*) In order for this mechanism of interference of different wavelengths to work, it is necessary that the individual quantum states have (in the spirit of corpuscular-wave dualism) different masses - so that the neutrino has a generally non-zero rest mass (at least some of them). In the Standard Model particle interactions, reactions of neutrinos with charged leptons (eg m ^- ® e ^- + n _m + n _e ') are called reactions of so-called weak charged currents and in quantum theory they are described by total Lagrangian or Hamiltonian , including both charged lepton components and neutrino, with coefficients given by the so-called Pontecor-Makai-Nakagawa-Sakat (PMNS) mixing matrix. This matrix, which describes the mixing of neutrino fields, is parameterized by three mixing angles q and three complex phases d . The corresponding Schrödinger equation then provides the overall wave solution as a coherent superposition of "wave clusters" corresponding to the individual eigenvalues of masses. The probability of finding one's own state of mass is then expressed by a periodic function of time , which is reflected in the movement of the neutrino by the periodic dependence of the neutrino species along the path - the oscillation of neutrinos in space and time. The magnitude and time course of the oscillations depends on the degree of "mixing" between the individual pure states "1", "2", "3", which are described by the so-called mixing angles q ₁₂ , q ₂₃ , q ₁₃ , resp. squares of the sinuses of these angles.
  In the simplest case of the motion of a relativistic neutrino with kinetic energy E _n , which is a coherent superposition of two mass components with the difference of quadrates of rest masses D m _n , the probability P finds a certain neutrino state (eg n _e ): P_n®ne » 1 - sin²2q.sin²[(Dm_n²/4hc.E_n).L], where L is the distance from the point of emission of neutrinos and q is the mixing angle parameterizing the PMNS matrix (which is two-dimensional here). For each value of energy E _n , then there are some so-called oscillatory length L_o at which L argument . D m _n ² /4 h C.E. _n in the sine function changed by P and the probability P _{n ®n e} again takes the value 1: L _a = 4 h C.E. _N / D m _n ² . The oscillation length is directly proportional to the neutrino energy E _n and indirectly proportional to the square of the mass difference Dm _n ^{2 of} both species (states) of neutrinos. For commonly used units of length and energy, the oscillation length can be expressed in a simplified form: L _{o [m]} » 2.5 . E _{n [MeV]} / D m _n ² _[eV 2 _] . The value of D m _n ² can be determined experimentally by measuring the dependence of the number of observed neutrinos on the energy E _n for a certain fixed distance L and the known spectrum of energies E _n neutrinos. KamLAND's experiments are based on n _e «n _m oscillationsvalue D m _n ² » 8.10 ^-5 eV ² , in Super-Kamiokande the value D m _n ² » 2.10 ^-3 eV ² is based, which are oscillations of all three types of neutrinos. For common solar and reactor neutrinos with MeV energies, the oscillation length is relatively short, in the order of tens of km. For "atmospheric" neutrinos (which are formed by interactions of high-energy cosmic rays in the Earth's atmosphere) with an energy of tens of GeV or higher, the typical oscillation length reaches hundreds to thousands of kilometers (to test the oscillations, neutrinos detected at great distances from the point of origin must be used; even neutrinos formed in the atmosphere on the opposite side of the globe, as mentioned below in the Super-Kamiokande experiment).
  Neutrino oscillation is a quantum-stochastic process whose probability after neutrino emission is initially low, but increases along its path length. At greater distances from the source emitting neutrinos of a certain species (usually electron), we will register a mixture of neutrinos of all three species, represented roughly homogeneously in a ratio of 1: 3 (it depends on the values of the above-mentioned mixing angles q). The influence of this phenomenon on the detection of individual types of neutrinos will be briefly mentioned below.
Note: Physical reincarnation?
The phenomenon of oscillations of neutrinos is difficult to understand from the classical point of view, it might seem that it violates the laws of conservation. However, from the point of view of quantum physics, neutrinos do not "physically" transform into each other. Just wave function neutrinos is a superposition of three states (it is parameterized aforementioned mixing angles). Depending on the distance from the point of origin will then show the various probabilities that the neutrino will interact as electron, the muon or tau. In a sense, we can imagine that even the velocity of the neutrino oscillates. In quantum physics these phenomena are not considered to be a violation of energy conservation or symmetry.

Interaction and detection of neutrinos
The detection of different types of radiation will be discussed in detail in Chapter 2 "Detection and spectrometry of ionizing radiation". However, the detection of neutrinos differs considerably in its technique and principles from the more or less "routine" detection of radiation a, b, g, often used in technical practice. These are (at least so far) rather unique and delicate experiments, trying to prove the very existence of neutrinos and some of their basic properties (in this sense, it is somewhat similar to experiments with the detection of gravitational waves). Therefore, the methods of neutrino detection will be mentioned here.
  Detecting neutrinos is very difficult - they do not show an electromagnetic or strong interaction, only a weak interaction (we will mention gravity below). The effective cross section of this weak interaction is very small, as it is mediated by exchanges of heavy intermediate bosons W and Z (according to the Standard Model of Particle Interactions, see §1.5 "Elementary Particles"); this large mass of exchangeable particles strongly suppresses the probability of interactions, in contrast to the electromagnetic processes mediated by the exchange of photons with zero rest mass. In principle, two types of processes can be used to detect neutrinos (+ the third process for the detection of high-energy neutrinos, mentioned at the end of this passage) :

• 1. Interaction of neutrinos with nucleons ( reverse beta decay process ; particle production)
  The electron neutrino n_e interacts (weakly interacts) in the atomic nucleus with the neutron n^o and causes a reaction there n_e + n^o ® p⁺ + e^-, or with a proton by the reaction n'_e + p⁺ ® n^o + e⁺. The effective cross section of this interaction is about 10^-47m².(E_n /1MeV)².
  At high neutrino energies, particle interactions with nucleons can occur to form electrons, muons, or tauons.

The first successful detection of neutrinos
using the process n_e+p⁺ ® n^o+e⁺ was achieved in 1956 by F.Reines and C.Cowan from the laboratories in Los Alamos, who used a powerful nuclear reactor in the Savanah River as a source of neutrinos with an antineutrinos flow of about 10¹³ n/cm²/s. A liquid scintillator (triethylbenzene) with admixture of camium in a large vessel with a volume of 1400 liters was used as a target and at the same time a detector. When a neutrino interacts with a proton contained in a scintillator, a neutron n^o (with a kinetic energy of several keV) and a positron e⁺ (with a kinetic energy of 0 ¸ 8MeV) are formed. Positron e⁺ is braked very rapidly in the liquid (for about 10^-10 s) and then annihilated with one of the electrons (e⁺ + e^- ® g + g ) to form two annihilation photons g, each of which has an energy of 511keV. The ionization energies of the positron together with the annihilation photons (total energy 1 ¸ 8MeV) cause a light flash in the scintillator registered by the photomultipliers. The neutron slows down with collisions with the nuclei (the process of deceleration and diffusion takes up to 30 msec) and is captured by the cadmium nucleus in the reaction n^o + ¹¹³Cd₄₈ ® ¹¹⁴Cd₄₈ + g; the excited isotope cadmium, when deexcited, emits 2-4 quantum gamma with a total energy of 9MeV within a few microseconds and returns to the ground state. These g- quants also produce a flash of light captured by the photomultipliers in the scintillator. The interaction of the neutrino with the proton ( n_e + p⁺ ® n^o + e⁺) is thus revealed by two consecutive electrical signals from the photomultipliers: 1. a pulse of amplitude corresponding to 1-8MeV coming from the positron registration; within about 25 msec, a 2nd pulse with an amplitude of 3-10MeV originating from neutron capture in the cadmium nucleus appears. During measurements lasting more than 100 hours, an average of 36 cases of said reaction (n_e + p⁺ ® n^o + e⁺) neutrinos per hour, which gave an effective cross section of the reaction of about 10^-43 cm².
Radiochemical detection of neutrinos
To measure the interaction with a neutron (n_e + p⁺ ® n^o + e⁺) it is necessary to choose a nucleus where the conversion of a neutron to a proton leads to a radioactive nucleus emitting radiation that can be easily detected. In practice, ³⁷Cl chlorine cores were first used for these measurements- a large tank filled with about 600 tons of tetrachlorethylene C₂C_l4 (otherwise a common chemical cleaner, which had to be specially cleaned) was placed at a depth of about 1.5 km underground in an abandoned gold mine Homestake in South Dakota, USA - to minimize radiation background from cosmic rays and terrestrial sources. The experiments took place under the leadership of R. Davis in the 1960s. Electron neutrinos induce a reverse b- decay reaction in chlorine nuclei : n + ³⁷Cl ® ³⁷Ar + e ^-; the threshold energy of neutrinos here is 814keV. The resulting argon ³⁷Ar decays back to ³⁷Cl with a half-life of 35 days by electron capture. At the same time, Auger electrons with an energy of about 2.6 keV are emitted from the atomic shell of the daughter chlorine (due to the characteristic X-radiation, which is 90% subject to internal conversion), which can be detected by a gas proportional detector. To do this, however, it is necessary to extract the few ³⁷Ar atoms formed from the entire volume of about 400,000 liters of tetrachlorethylene in the tank, which is an extremely difficult technical problem. In 1968, the results of the experiment were evaluated, neutrinos were successfully detected, but their amount represented only about 30% of the neutrinos expected from astrophysical models of thermonuclear reactions inside the Sun.
  Another material suitable for radiochemical detection of neutrinos is gallium ⁷¹Ga, in the nuclei of which electron neutrinos induce the reaction n + ⁷¹Ga ® ⁷¹Ge + e^- with the subsequent decay of germanium ⁷¹Ge by electron capture accompanied by the radiation of Auger electrons, similarly to the decay of ³⁷Ar. The advantage of gallium is a significantly lower threshold energy of 233 keV of detected neutrinos, the disadvantage is the higher price of gallium compared to chlorine. On this basis, the SAGE and GALEX experiments were successfully performed in the early 1990s, which successfully detected low-energy neutrinos, also in lower numbers than expected.
  In this way, neutrinos have been successfully detected both from laboratory sources and from space - especially from the Sun, where huge amounts of neutrinos are formed during thermonuclear reactions. However, one mystery occurred here permanently: all measurements gave approximately 3 times less value of the flow of electron neutrinos than expected from the analysis of thermonuclear reactions on the Sun. This mystery of solar neutrino deficiency has been solved until many years later, when improved neutrino detection methods demonstrated the effect of neutrino oscillation - the spontaneous interconversion of electron, muon and tauon neutrinos, with earlier methods being able to detect only electron neutrinos (1/3).
CNGS (CERN Neutrinos to Gran Sasso) + OPERA (Oscillation Project with Emulsion Tracking Apparatus)
This is an new interesting measuring system type transmitter ® receiver (detector) neutrinos, which targeted the investigation of neutrino oscillations, especially on taun neutrinos and refinement neutrino masses. It started operating in 2006. The neutrino transmitter is at CERN, the neutrino receiver (detector) is installed in an underground laboratory at a depth of 1400 m below the Gran Sasso mountainin in Italy. The neutrino transmitter is a large proton accelerator, the synchrotron SPS (Super Proton Synchrotron) at CERN, from which protons, accelerated to 400GeV energy, collide with the nuclei of the target to form a large number of mesons p (and a number of other particles - kaons, hyperons, ...). Charged pions are separated by a magnetic field and led to a 1.2 km long vacuum decay tunnel, where they decay during flight (they have a half-life of only about 2.10 ^-8 sec.) to form muons and muon neutrinos : p^- ® m^- + n'_m , p⁺ ® m⁺ + n_m . The average energy of the neutrinos formed is about 17GeV; to register tauon neutrinos using tauons, we need a very high energy of the original muon neutrinos, higher than the resting energy of the tauon (m_0t »1,2GeV/c²). Due to relativistic effects, the neutrino beam is relatively closely collimated approximately in the direction of the original proton and pion beam.
  The whole system [pion beam + decay tube] is directed to the 732 km distant underground laboratory LNGS (Laboratori Nazionali del Gran Sasso) in Italy, where neutrinos arrive through the ground (other particles are quickly absorbed in the shield at the end of the decay tunnel, other muons in rock underground). Here, an OPERA detection system is installed, consisting of 150,000 alternating layers ("sandwiches", bricks) of lead *) 1 mm thick, interspersed with thin sheets of photographic emulsion. Lead serves as a target for the neutrino interaction and a photographic emulsion for registration and visibility of reaction products.
*) To reduce the radiation background, the so-called "Roman lead" from ships carrying lead was used to build a water pipe in Rome and sunk sometime around the 60s BC. At the bottom of the sea, this lead lay intact for 2,000 years, so most of the natural radioactive isotopes decayed in it during that time.
  This "old" technology of nuclear photoemulsions was used because it is the only one that provides a very high spatial resolution »1 mm of registered particle trajectories, needed to capture short tauon trajectories of very short lifetime (of the order of picoseconds). However, the evaluation of photoemulsions is fully automated here (using technologies developed in Japan). Detection of particle trajectories using nuclear photoemulsions, including a technology called ECC (Emulsion Cloud Chamber), a kind of "emulsion fog chamber", is discussed in more detail in §2.2, section "Particle Detectors". The OPERA detection system is very massive, those 150,000 sandwiches together contain about 110,000 m² of film with photoemulsion and more than 100,000 m² lead sheets with a total weight of more than 1200 tonnes. This power is forced by two circumstances: 1. Very low effective cross section of neutrino-substance interactions; 2. The distance of 730km between the transmitter and the detector (corresponding to the flight time of neutrinos 2.43ms) is too short, so only a very small part (approx. 1-2%) of neutrinos can oscillate.
  During the flight from the transmitter to the detector, neutrinos oscillate, so in addition to the original muon ones, also electron and taunine neutrinos arrive here. In the reactions of muon neutrinos in lead, muons are formed, in the reactions of electron neutrinos electrons, in the reactions of tauon neutrinos tauons are formed (which we are mainly looking for here). The resulting tauons decay into pions in 64% of cases, electrons in 18% and muons in 17% (when emitting neutrinos escaping from the detection space). Based on the recording of traces of particles in the photoemulsion, the kinematics of the original interaction are reconstructed by microscopic scanning, which makes it possible to determine the position of the formation of the tauon and the place of its decay. There are also plastic scintillators in the system, the signal of which indicates the moment of interaction and makes it possible to determine in which photoemulsions a useful record can be expected; these are then evaluated. The system also includes electronic particle "trackers" and magnetic spectrometers. In May 2010, a taun neutrino, created by oscillations from an originally muon neutrino, was registered here for the first time.

A courious report - mystery: Superlight speed neutrinos?

In september 2011, a report appeared in the press that in the CNGS + OPERA experiment , the slightly superluminal velocity of muon neutrinos emitted from the accelerator at CERN and detected in the Gran Sasso underground laboratory was measured . Based on synchronization of send and receive times (using GPS) and statistical evaluation of the detection of about 16,000 muon neutrinos, CNGS experimenters measured that the neutrino traveled a distance of 732km about 60 nanoseconds before flying at the speed of light (the velocity of these neutrinos would be about 2 , 5 thousandths of a percent higher than the speed of light). And immediately bombastic speculations about the invalidity of the special theory of relativity were deduced from this ("Einstein was refuted", "Physics textbooks must be rewritten", etc.). All physicists hoped it was a "false alarm", that some systematic error will be found (probably in the synchronization of the time of sending and receiving). Otherwise it would be a big "trouble"! This can be compared to a situation where you build a house and just before completion a laboratory tells you to demolish it, because a certain hidden defect was found in the bricks used. At the same time, hundreds of houses were built from exactly the same bricks, which have been standing for many decades ..! .. The special theory of relativity (STR) is verified with high accuracy for all known phenomena, for all other particles. It is based on the existence of a maximum velocity of propagation of interactions , which is equal to the speed of propagation of electromagnetic waves (and therefore also light) in a vacuum. Particles with zero rest mass , which are mainly photons , move at this speed . And since neutrinos have been shown to have a non-zero rest mass (albeit indirectly), they should move slightly slower than light. If no error could be found, I would suggest exploring, for example, the following speculative possibility of explanation : Some supersymmetric unitary field theories contain tachyons - hypothetical particles moving only at superluminal speeds (see §1.5, section " Hypothetical and model particles ", passage " Tachyons ", or in more detail §1.6 "Four- dimensional spacetime and special theory of relativity ", passage " Tachyons " in the book "Gravity, black holes and physics of spacetime"). These particles do not formally violate the special theory of relativity      , but due to some of their "pathological" properties, they probably cannot exist as real. However, they can "exist" as virtual tachyons . There could then be a quantum "mixing" of wave functions of real neutrinos with a slight contribution of the wave functions of virtual tachyons. This mix of quantum states could then effectively cause slightly superluminal neutrino velocities as a quantum effect against the STR background. In quantum physics, it is common for real particle states to be affected by virtual particles without violating the laws of physics. But it would remain to explain There would certainly be more possibilities of "explanation", but all of them would probably also be "pulled by the hair" (eg to declare the measured rate of neutrinos as the new maximum rate of propagation of interactions; or the manifestation of hidden extra-dimensions of multidimensional microstructure of spacetime). .) . However, we are ahead of that. In any case, if a systematic error cannot be found, the effect must be verified by a completely independent experiment . Only then can we take it seriously and it will be justified to draw some far-reaching conclusions. So far, it is only about scientific-physical "folklore" ...

Note 1: After all, there is a contradiction with the astronomical observation of the supernova explosion in the Large Magellanic Cloud (a neighboring galaxy 170,000 light-years away) in 1987, from which a 12-neutrino spray was registered (see below " KamiokaNDE Neutrine Detector "), which arrived about 3 hours before the flash of light was astronomically registered. This time difference does not mean that perhaps neutrinos are slightly faster than light, but it is explained by the mechanism of the supernova explosion (below in the section " Resting mass of neutrinos ") and suggests that neutrinos move at the speed of light.with an accuracy of the order of billionths of a percent. If the neutrino moved at the speed as "measured" in the above experiment, a flash of light from this supernova would reach us a few years later (about 4 years) than the neutrino spray! So this is a strong argument against the superluminal speed of neutrinos. It should be noted, however, that these are mainly lower-energy electron (anti) neutrinos; however, we do not see any reason yet why muon neutrinos (which also have a rest mass perhaps higher than electrons) should behave differently ... Note 2: R. van Elburg recently investigated the effect of time dilation (according to a special theory of relativity) between the Earth's reference systems and the moving GPS satellite. The relativistic dilation of time caused by the movement of the clock on board the GPS relative to the Earth's reference system would lead to a time difference of 32ns in CERN and 32ns in Gran Sasso. That might explain the difference 60ns ..? ..

A technical error has been found ! - the "bubble burst" At the end of February 2012, the CNGS + OPERA experimenters themselves found two technical errors: incorrect connection of the optical cable connecting the computer to the signal from the GPS satellite and an error in the oscillator used to synchronize the external GPS signal with the OPERA experiment control clock. These problems could skew the measurements and cause a false result that the neutrino beam covered a distance of 732km from CERN to the Gran Sasso detector 60 nanoseconds before it corresponds to the speed of light ...

• 2. Elastic scattering of a neutrino on an electron : n + e ^- ® n ´ + e ^- ´
  (occurs for all types of neutrinos). The fast-flying neutrino n collides with the electron e^-, bouncing off it (usually at a large angle or even in the opposite direction) as a neutrino n´ with lower energy, transferring part of its energy to the electron. The reflected electron e^- mostly moves in the direction of the original (incident) neutrino n and can be detected.
  The effective cross section of this interaction is about 10^-48 m².(E_n/1MeV).

   If this electron e^-´ moves in an optic environment (eg water) at a speed higher than the speed of light in this environment, it emits Cherenkov radiation (mechanism of its origin and properties see §1.6, passage "Cherenkov radiation"), which can be detected by photomultipliers. Although this method of neutrino detection has a higher threshold energy (5MeV), its advantage is its applicability even for other types of neutrinos than electron's.

Detection and spectrometry of neutrinos using various types of their interactions.
Left: Some detection principles. Right: SuperkamiokaNDE photomultiplier system.

Kamioka NDE Neutrinous Detector
A detector of this type, called the Kamioka NDE (Kamioka Neutrino Detection Experiment *), was designed under the leadership M.Koshiba in 1982 in Japan - in the Kamioka tin mine (in the mountains ... called the "Japanese Alps") in at a depth of 820 m, a reservoir containing about 20,000 tons of high-purity water was built. Photons of Cherenkov radiation were registered by almost 1000 large photomultipliers; another 120 photomultipliers involved in anticoincidence surrounded this system in a geometry of 4p. The electronic system processing the pulses from the individual photomultipliers made it possible to locate the place of interaction of the neutrino, determine its energy and approximately the direction of arrival. So this detector was already a spectrometer, working in real time (in contrast to the additional detection of Auger electrons in earlier radiochemical detectors). In addition to refine the results of radiochemical detectors and measuring the high-energy part of the spectra of neutrons from the Sun was in this detector also achieved another major success: on 23 February 1987 was registered glimmer 12 neutrinos coming from the supernova SN 1987A in the Large Magellanic Cloud (a neighboring galaxy 170,000 light years away).
*) Note: The Kamioka NDE detector was originally designed to demonstrate proton decay (NDE stands for "Nucleon Decay Experiment") - according to some versions of the unitary theories of great unification (GUT = Grand Unification Theory, unite strong, weak and electromagnetic interactions) the proton should not be a stable particle, but would decay (eg p⁺ ® p^o + e⁺) with a half-life of T1/2 > 10^30-40 years. This half-life is so long that perhaps not a single proton in the universe may have decayed since its inception! Not surprisingly, the detection of proton decay was not successful. However, the lucky idea of using this device to detect neutrinos was very successful, which gave the acronym NDE a new meaning.
Super Kamioka NDE 
   As a continuation of the KamiokaNDE detector, an even larger Super KamiokaNDE detector was built in 1996, located in an old zinc mine 1700 m below the surface of Mount Iken Yama near the town of Kamioka. The cylindrical tank with a diameter of 34 m and a height of 36 m, on the inner walls of which are placed 11,146 large photomultipliers (photocathode diameter almost 50 cm), is filled with almost 50,000 tons of superpure water. Photomultipliers detect Cherenkov radiation of electrons or muons created by the collision of electron or muon neutrinos with protons and neutrons. Using the ratio of the amount of production of electrons and muons, the system is able to distinguish between electron and muon neutrinos. In 1998, oscillations of atmospheric neutrinos were demonstrated on this apparatus. Atmospheric neutrinos are constantly formed in the upper layers of the atmosphere during the decay of pions and later muons, created by the interaction of hard cosmic radiation with the atmosphere - see §1.6, section "Cosmic radiation", passage "Secondary cosmic radiation". Muon and electron neutrinos are formed (in the ratio 2n_m : 1n_e). Neutrinos of energy E_n coming "from below" pass through the whole Earth (distance L from the place of their origin) and have more time to undergo oscillations, in contrast to neutrinos coming "from above", which passed only a few kilometers of atmosphere and less than 2km of soil. This vertical anisotropy in the relative proportion of muon neutrinos was reliably measured by the detection system. A significant decrease in the detected frequency of muon neutrinos for L/E_n around 500km/GeV was observed and in addition a re-increase was measured - "revival" of the frequency of detection of muon neutrinos for lengths of the order L » 10³-10⁴ km/GeV, caused by oscillating behavior of neutrinos in distances greater than half the oscillation length L_o neutrinos for a given energy E_n (as discussed above in the section "Oscillations of neutrinos").
Neutrin SNO detector
Another significant improvement was implemented in the neutrino SNO spectrometer (Sudbury Neutrino Observatory) located at a depth of 2 km in a mine near Sudbury in Ontario, Canada. Here, inside a vessel with 7000 tonnes of "light" water (¹H₂O), another vessel with 1000 tonnes of heavy water (²H₂O, ie D₂O) is placed. More than 9,500 external photomultipliers monitors the flashes of Cherenkov radiation through light guide tubes. Neutrinos cause three types of reactions in the detector material (water and heavy water) :
a) Absorption of an electron neutrino by a neutron in the deuterium, in which the neutron changes into a proton and an energetic electron. Deuteron, which is a weakly bound nucleus, then decays into two protons and an electron: n_e + ²H ® p + p + e^-. A fast-flying electron causes Cherenkov radiation.
b) A fast-flying neutrino "collides" with a neutron or proton in the deuterium and, in elastic scattering, transfers some of the kinetic energy, causing the deuteron to decay into a proton and a neutron: n + ²H ® n_scattered + p + n^o. The released neutron is then absorbed by another deuteron, emitting a photon of radiation g. This photon g, during a photo effect or Compton scattering in matter, emits an electron that causes Cherenkov radiation.
c) Neutrino "collides" with an electron and, when elastically scattered, accelerates it to such an extent that it causes Cherenkov radiation.
  Process a) is only possible with the electron neutrino n_e, while processes b) and c) can induce by any neutrino. Processes a) and b) occur only in deuterium in heavy water, process c) occurs equally on electrons in light and heavy water. By analyzing these processes, it is possible to measure independently both the flux of electron neutrinos and the flux of all neutrinos (ie electron + muon + tauon) together. The measurement results showed that the lack of solar neutrinos in all previous experiments is due to neutron oscillations.
KamLAND - neutrino scintillation detector from nuclear reactors
Japanese KamLAND detector ( Kamioka Liquid Scintillator Neutrino Detector; originally there was KamiokaNDE, which was redesigned to KamLAND) consists of a spherical vessel with a diameter of 13 m, filled with a liquid scintillation agent detecting positrons formed by the capture of antineutrinos by a proton. Scintillation flashes are registered by a system of more than 18,000 photomultipliers distributed on the inner wall of the vessel. The scintillation vessel is surrounded by an external Cherenkov detector with 3,200 tons of water. Neutrino energy can be determined approximately from positron scintillation. The resulting neutrons are captured by hydrogen nuclei in the path up to about 10 cm, while photons of radiation g are emitted with an energy of about 2MeV, also causing scintillation detected by a photomultiplier system. The device is designed to detect antineutrins from surrounding nuclear reactors, while determining the energy spectrum and the proportion of electron antineutrins depending on the distance traveled. The detector enabled a more detailed study of neutrino oscillations, which confirmed and supplemented the results from both systems mentioned above.
DUNE (Deep Underground Neutrino Experiment)
is built to study the properties of neutrinos, especially in connection with the not yet sufficiently researched and understood phenomenon of neutrino oscillations. This international advanced project is based on the cooperation of two nuclear laboratories in particular :
-> Neutrinos will be "manufactured" at the Fermi National Accelerator Laboratory (Fermilab) in Batavia, Illinois, as part of the Long-Baseline Neutrinos Facility (LBNF) project. Here, the powerful proton accelerator PIP-II (800 MeV) will produce by proton impacts on the carbon target the intense neutron beams (muon (anti)neutrinos in the energy range 1-5 MeV), collimated northwest towards the Sanford Lab. These neutrinos in Fermilab will first pass through a smaller "monitoring" detector of transmitted (non-oscillating) neutrinos.
-> The resulting detection of these neutrinos will be performed at the Sanford Underground Research Facility (Sanford Lab) in Lead, South Dakota, 1,300 km from the source, using a large cryogenic underground detector in depth 1.5-kilometer (in the former Homestake gold mine) containing 70,000 tons of liquid argon cooled to -184 °C, with LArTPC time-projection chambers (these detectors are described in Chapter 2 "Radiation detection and spectrometry", §2.3, passage "Drift and time-projection proportional chambers"), with simultaneous detection of scintillation flashes in argon.
  Neutrinos will oscillate continuously during their 1,300 km underground journey, so all 3 types of neutrinos will fall into the remote detector. By measuring the representation of different types of resulting neutrinos registered in the remote detector, depending on the energy, it will allow a more detailed study of the dynamics of neutron oscillations. One of the goals is also to improve the current low success rate of tauon neutrino detection.
A large cryogenic underground detector in DUNE could also detect neutrinos from a supenova explosions. It is also planned to look for products of possible proton decay ...
  The DUNE experiment can be considered a significant improvement on the previous CNGS + OPERA neutrino experiment (described above "CNGS+OPERA") as well as the famous Kamioka - Super Kamioka NDE neutrino detector. Completion of the large underground detector, start-up of the neutrino beam and start of measurements in DUNE is planned for 2026-27.

• 3. Interaction of high-energy neutrinos with protons to form muons m .
  Charged muons m, then during high-speed movement in the clear optical medium emit Cherenkov radiation which can be detected by a photomultiplier.

Detection of neutrinos in glaciers
An interesting and somewhat curious option for the detection of fast neutrinos is the use of massive masses of natural ice in large glaciers located mainly in Antarctica. When a high-energy neutrino hits a proton (in one of the nuclei of a water-ice molecule), a high-energy muon m is formed, which leaves a bluish light cone of Cherenkov radiation along its path of ice movement. Its direction allows you to determine the direction of the path of the original neutrino. At kilometer depths inside the glacier at high pressures, the ice is highly transparent, compact and bubble-free, so flashes of muons can be detected at distances of tens to hundreds of meters. Geometric placement of the photomultipliers, together with the coincidence analysis of the detected pulses, enables the spatial reconstruction of the cone.
  When optical sensors - photomultipliers - are turned down into the Earth's interior, neutrinos coming from the opposite side, the northern hemisphere, passing through the globe are detected. Interfering muons from secondary cosmic rays are completely shielded from this direction. The system is able to detect not only common neutrinos with energies of several MeV, but also high-energy neutrinos with energies of the order of TeV and higher.
AMANDA
The first system of this kind is the AMANDA (Antarctic Mion And Neutrino Detector Array) built in 1996-2000 at the Amundsen-Scot Polar Station in Antarctica. It consists of more than 700 photomultipliers, housed in pressure-resistant glass spheres, embedded under Antarctic ice in 19 shafts over 2 km deep *). The photomultipliers are powered by electric cables, the detected pulses are led by light cables to the evaluation device. The achieved angular resolution for neutrinos from cosmic rays is around 1°.
*) The ice shafts are "drilled" with a stream of hot water at 80° C, photomultipliers are lowered into them, after which the shaft freezes again after a few hours. However, the photomultipliers frozen in ice will remain electrically connected to the evaluation center.
ICECUBE
A sequel is an even larger system for detecting neutrinos in the Antarctic glacier, called ICECUBE (Ice Cube), completed in December 2010. It consists of 5160 photomultipliers embedded in 86 shafts at various depths of 1450-2450 m under ice. These shafts are arranged horizontally in a hexagonal grid with a spacing of 125 meters. A chain of 60 photomultipliers with vertical distances of 17 meters *) was launched (when the ice melts) on the rope into each these shafts, which then froze in the ice. By detecting light flashes, a cube of 1x1x1 kilometer of ice is covered. All photomultipliers are equipped with digital microprocessors with fast data transfer to remote evaluation computers.
*) Note: In the middle of the field, 8 strings of photomultipliers are distributed more densely, with a horizontal distance of 70 meters and a vertical spacing of 7 meters (so-called "DeepCore"). In addition, a photodetection unit with two photomultipliers facing downwards is placed on the surface above each depth chain of the photomultipliers. This "IceTop" surface field serves as an anticoincidence and calibration detector for the IceCube.
  The IceCube detection system has registered several interesting cases of high-energy neutrinos during its operation. E.g. from September 2017, when a high-energy neutrino was detected, probably coming from the very active blazar TXS 0506+056. There is real hope of successful detection of neutrinos from a supernova ("Supernova explosion. Neutron star. Pulsars."), if it explodes in our Galaxy or in a neighboring one. Or maybe from the fusion of neutron stars ("Collisions and fusion of neutron stars").
Submarine detection of neutrinos
Cherenkov radiation, caused by the passage of muons, can also be detected in water at great depths in the sea (where sunlight no longer penetrates). The first prototype of an underwater neutrino detector was the DUMAND ( Deep Underwater Muon And Neutrino Detector) with 24 photomultipliers off the coast of the Big Island of Hawaii. The first functional neutrinos detector of this type is the BAJKAL with 192 photomultipliers, which successfully works at a depth of 1500 m below the surface of the Siberian lake Baikal.
  The submarine detection system neutrinos ANTARES (Astronomy with a Neutrino Telescope and Abyss environmental RESearch) *) was built in 2006-2007 in the Mediterranean Sea about 40 km from the French coastal city of Toulon. To a depth of 2400 m below sea level (where daylight no longer penetrates), 12 vertical supporting ropes, each 450 m long, carrying a total of 900 photomultipliers was gradually launched. The detection system covers an area of about 200x200m. Flashes of Cherenkov radiation from flying muons cause electrical impulses in the photomultipliers, which is detected in coincidence. This radiation from muons must be distinguished from the background caused by the bioluminescence of submarine organisms and from the Cherenkov radiation of a large number of electrons of energy around 1MeV, arising from the b- decay of radioactive potassium ⁴⁰K.
*) In addition to neutrino detection, the ANTARES experiment is also part of interdisciplinary underwater and oceanographic research, such as monitoring of the underwater environment, mainly bioluminescence; also contains seismographic sensors.
  The ANTARES experiment will continue in the coming years with the NEMO (NEutrino Mediterranean Observatory) projects about 80 km from the coast of Sicily and NESTOR (NEutrinos from Supernova and TeV Sources, Ocean Range) off the coast of Greece, which will detect neutrinos in volume of the order of 1 km³. This will cover a geographical area largely complementary to the aforementioned Antarctic IceCube detector.
Radio detection of neutrinos using Askaryan radiation
Askaryan radiation (see §1.6, passage "Askaryan radiation") is tested for neutrino detection, especially in Antarctica, where high-energy neutrinos pass through a layer of ice. The ANITA (....) antenna, located on a balloon above Antarctica, detects these radio pulses. It works in collaboration with photomultipliers detecting Cherenkov radiation in Antarctic ice in the IceCube system.
  The main task of large ice and submarine detectors is to search for high-energy neutrinos, which could have formed during stormy cosmic events, especially during the explosion of supernovae and during the very formation of the universe, during the Big Bang. Detection of such neutrinos could be a valuable source of information about processes that are otherwise unobservable. The localization of sources of high-energy neutrinos could elucidate the mechanism of proton acceleration and thus answer the question of the origin of high-energy cosmic radiation.

The importance of neutrino detection
In addition to basic (mostly purely theoretical) research into the properties of neutrinos and particle interactions with their participation, neutrino detection can also be important for the study of various processes here on Earth and in space. Neutrinos, due to their extreme penetration, are the only particles that are able to "bring out" information about nuclear and particle processes from the interior of massive, large or compact objects, from which no other radiation absolutely penetrates.
   Here on Earth, an increased number of detected neutrinos in certain places may indicate the presence of deposits of natural radioactive substances (even at great depths underground) - uranium and thorium, during the radioactive decay of which, in decay series, neutrinos (electron antineutrinos) are also formed.
   Detection of the flow of solar neutrinos makes it possible to test the instantaneous intensity of thermonuclear reactions inside the Sun (especially the proton-proton cycle). Even the high density and thickness of the plasma in the solar interior do not prevent neutrinos from leaving the region of their birth almost immediately and thus "bring out" relevant information (unlike photons, which for hundreds of thousands of years "penetrate" plasma, with gradual energy degradation, from interior to surface before they radiate; they can only carry information about the surface layers of the Sun).
   Neutrinos also provide important information about turbulent processes in outer space. They are above all supernova explosions in which a colossal amount of neutrinos (electron n_e) is emitted. Relict neutrinos, originating from the Lepton era, can provide important information about the dynamics of the earliest stages of the universe's evolution and the formation of its structure.
   However, the wider use of the possibilities provided by neutrinos is tied up to the improvement of neutrino detection techniques.

Rest mass of neutrinos
The original Fermi theory assumed that the rest mass of a neutrino was zero. However, in the early 1980s, extensive discussions developed about the rest mass of neutrinos: whether the neutrino has zero rest mass (and is therefore of a wave nature - as a quantum of radiation it propagates at the speed of light c), or a non-zero, albeit very small, rest mass m_0n ( and is therefore a particle moving slower than light). These discussions were fueled by the first successful attempts to detect neutrinos in the 1970s and 1980s, which showed that the flux of solar neutrinos is about 3 times lower than expected based on the analysis of thermonuclear and subsequent reactions inside the Sun. This the solar neutrino deficit was referred to as the "solar neutrino mystery", or even as the "neutrino scandal". In 1985, Miseyev, Smirnov and Wolfstein hypothesized that neutrinos "oscillate" between electron, muon, and tauon neutrino states during their flight, leading to them becoming alternately visible and invisible to detectors capable of detecting only electron neutrinos at the time. However, the mechanism of oscillations can only work if the neutrinos have a non-zero rest mass (at least two species-neutrino states); was discussed above in the section "Neutrino Oscillations". There are several ways to measure, or at least estimate, the rest mass of neutrinos.
   In principle, the mass of the neutrino m_0n could be determined on the basis of the law of conservation of energy in b-decay, if we knew the difference between the masses DM of the parent and daughter nuclei. By measuring the maximum energy of the flying electrons E_bmax, it is possible in principle to determine the rest mass of an electron neutrino on the basis of the law of conservation of energy - the heavier the neutrino, the less kinetic energy remains for the electron b; the rest mass of the neutrino is then m_0n = (DM.c² - E_bmax)/c². The dependence of the end part of the continuous beta spectrum on the rest mass of the neutrino is shown in Fig.1.2.3 on the right, which we will mention here again for clarity :

Fig.1.2.3. Beta radioactivity. Left: Basic scheme of radioactivity b^-. Middle: Continuous energy spectrum of radiation b. Right: Enlarged detail of the end of the spectrum for zero and non-zero rest masses of neutrinos.

However, when direct measuring the difference between the masses DM of the parent and daughter nuclei by mass spectrometry and the energy E_bmax by an electron spectrometer, the measurement errors are significantly larger than the required value m_0n.c² [eV]. Therefore, the linearization of the spectrum by means of the above-mentioned transformation - Fermi-Kurie graph (passage "Shape of the beta radiation spectrum") is used for the energy analysis of radiation b. If the rest mass of the neutrino is zero, the Fermi-Kurie graph will be linear up to the maximum value: the end section will also be linear and will intersect the energy axis at the point of maximum energy b. In the case of a non-zero rest mass of a neutrino, the beta electron will always be deprived of the energy necessary to produce this non-zero mass. In the initial sections of the spectrum, this is only slightly apparent and the linearity of the FK graph is retained here. However, in the final section of the FK-graph, in the case of a non-zero mass of the neutrino on the linear dependence, a small "bend" appears, the spectrum decreases faster and is "prematurely" terminated (reaches zero) at a somewhat lower energy E_bmax - m_0n.c², compare with Fig.1.2.3 on the right. Due to the transformation that must be used to linearize the spectrum b, when analyzing the shape of the Fermi-Kurie spectrum, we find the square of the mass of the neutrino m_0n².
  A suitable b-radionuclide for these measurements is tritium ³H. The measurements are very difficult because we are looking for effects much smaller than the "blur" of energy caused by the recoil of nuclei (according to the law of action and reaction at emission b, the nucleus is reflected in the opposite direction) and by thermal motion; measurements were performed at temperatures close to absolute zero and the investigated b -radioactive nuclei ³H was bound in a high molecular weight substance so that the recoil converted to the whole molecule was small. In some of the new planned experiments, a gaseous radioactive source of ³H will be used to minimize the energy losses of electrons in the spectrum.
   Initial measurement based on a detailed analysis of the shape of the end portion of the continuous spectrum of radiation b, giving initially a relatively high value of m_on » 40 eV, later however, the value decreased to 5 eV, which during the measuring error greater than ⁺5 eV admit even a zero value. Only recently have experiments leaned towards a non-zero rest mass of neutrinos - the so-called oscillation of neutrinos has been proven (as previously described experiments Super-Kamiokande, SNO, KamLAND) - spontaneous conversion between electron neutrino n_e , muon n_m and taun n_t , which can occur only when the nonzero rest mass. However, these neutrino oscillation measurements do not provide any absolute range of neutrino masses; they only say that at least two of the three neutrinos have a non-zero rest mass and for the most heavy of them give an estimate of the lower limit > 0.05 eV.
  The latest results of measuring the shape of the ³H tritium b spectrum on special electrostatic spectrometers with magnetic collimation (in the laboratories in Troick and Mainz) show the upper limit m_on < 2.3 eV. New planned experiments and possibly analysis of neutrino-free double beta decay should further reduce and refine this limit. One of these forthcoming experiments is KATRIN (Karlsruhe Tritium Neutrino Experiment), built in international collaboration in a special Tritium Laboratory in Karlsruhe. It will consist of a gaseous tritium source, one smaller "filtration" spectrometer and a giant main spectrometer of particles b (diameter 10m, length 23m). Another planned independent experiment is called MARE (Microcalorimeter Arrays for a Rhenium Experiment), which will measure the radiation b of the radionuclide ¹⁸⁷Re, which has the lowest energy of all only 2.5keV, but an extremely long half-life T1/2 = 4,3.10¹⁰ years; the specific activity and intensity of the radiation is therefore very low. Instead of the usual spectrometric methods, a large number of cryogenic microcalorimeters will be used (their principle is briefly outlined in §2.5 "Semiconductor detectors"), in which a slight increase in temperature caused by complete absorption of particle b in the sample Re will be electronically detected. These future experiments are expected to have a neutrino mass sensitivity of about 0.2 eV.
  Astronomical observations provide other possibilities for determining (or estimating) the rest mass of neutrinos. One of the possibilities, unfortunately very rare, is the observation of a supernova explosion: in the initial phase, it produces an intense flash of light and a massive outburst of neutrinos. If we could observe a flash of light (or a flash of harder photon radiation - X, g) and at the same time detect a "flash" of neutrinos from the same supernova, then we can determine from the time difference between the arrival of a photon and a neutrino flash, how much slower than light the neutrinos moved through space travel. From this, according to the laws of the special theory of relativity, it would be possible to determine the rest mass of neutrinos *)
*) Purely theoretically: a zero time difference would correspond to a zero rest mass; the greater the time difference, the greater the resting mass of neutrinos. Unfortunately, the reality is more complicated. During the initial phase of a supernova explosion - the formation of a neutron star - a huge amount of neutrinos is quickly emitted, which fly almost unobstructed and immediately fly into space. However, a flash of light (and electromagnetic radiation in general) forms for a long time and "laboriously penetrates" the dense mass and the gas envelope. Therefore, the neutrino from the supernova usually arrives at a distant observer a little earlier than a flash of light. The neutrino flash is fast - it lasts only a few tens of seconds, the light flash is delayed, it gradually increases and reaches its maximum only after a few hours. Measurements during the supernova explosion in the Large Magellanic Cloud in 1987 did not show (with regard to the mentioned facts) a difference in the speed of light and neutrinos and thus showed a very small rest mass of neutrinos.
   Indirect (and unfortunately model-dependent) estimates of the rest mass of neutrinos can be derived in cosmology from the measurement of relic radiation anisotropy and from the analysis of the mechanisms of formation of large-scale structures in the early stage of universe.

Important in the universe ?
An unbiased person may be surprised, "what are the physicists' worries about?" - if almost nothing like neutrino has a negligibly small or completely zero rest mass! - depends on the? However, interest in this issue was also stimulated by a completely opposite side of scientific research than the microworld - by relativistic astrophysics and cosmology - see Chapter 5 "Relativistic Cosmology" of the book "Gravity, Black Holes and the Physics of Spacetime". The standard cosmological model of the origin and evolution of the universe shows, that shortly after the Big Bang - in the so-called lepton era - such a huge amount of neutrinos was formed in the universe that if their rest mass was greater than about 5 eV, it would be able to "close the universe" by its gravitational action - the current expansion of the universe would stop (in the distant future), the universe would begin to shrink and end up in the "fiery furnace" of large crash. Otherwise, the universe would expand constantly (and the end state would be a kind of "thermal death" - stopping all the processes that release energy, dropping the temperature to absolute zero...); details in §5.6 "The Future of the Universe. The Arrow of Time.". Such a gnoseological situation is characteristic of contemporary fundamental physics, astrophysics and cosmology - that also the tiniest particles we know (almost "nothing" like neutrinos) can, with their ephemeral properties, "have or do not have a rest mass" decide the fate of the largest thing - the whole universe. One of the paradoxes of microworld physics is also that we need to have the largest instruments for research the smallest particles - see §1.5 "Elementary particles".
Note: Here we briefly touched on evolution of space in connection with neutrinos. However, the situation is much more complicated - neutrinos are only one of the "candidates" for dark matter in the universe, there are also a number of scenarios for the evolution of the universe, just as there may be "more universes" (see the work "Anthropic Principle or Cosmic God "). In any case, the now determined resting mass of neutrinos shows that neutrinos would probably not be enough to close the universe, or metagalaxy. However, there has been speculation that a suitable "candidate" for dark matter could be neutralines - boson superpartners to neutrinos, the existence of which is predicted by so-called supergravity theories (see §1.5 "Elementary Particles" and §B.6 "Unification of Fundamental Interactions. Supergravity. Superstrings" in book "Gravity, Black Holes...."). In addition, recent observations of the distant universe suggest that in addition to dark matter, the universe is filled with a kind of "dark energy" causing the accelerating expansion of the universe - see the conclusion of the already mentioned chapter "The Future of the Universe. Time arrow.".

Radioactivity b ⁺
Another type of beta radioactivity is radioactivity b⁺. Its basic scheme is in the left part of Fig.1.2.4 :

Fig.1.2.4. Basic scheme of b⁺ radioactivity and electron capture.

During this nuclear transformation, the nucleus emits a particle b⁺, which is an antiparticle to the electron e^- - positron e⁺. From the point of view of electric charge, the emission of b⁺ is not as paradoxical as b^-, but there are no positrons in the nucleus neither. Radioactivity b⁺ occurs in radionuclides in which protons predominate over neutrons (so-called neutron-deficient nuclei ). Thus, the mechanism of the formation of b⁺ can be allegorically explained as follows: Some of the "redundant" protons get "tired" of being a proton and want to become a rarer neutron - it will do so by transforming p⁺ ® n^o + e⁺ + n. The neutron n^o as a legitimate nucleon remains bound by a strong interaction in the nucleus, while the positron e⁺ flies out at high speed like a b⁺ particle. With radioactivity b⁺, the nucleon number does not change (as with b^-), but the proton number decreases by 1 - the daughter nucleus is shifted one place to the left in the Mendeleev periodic table.
An example of beta⁺ radioactivity is the positron conversion of carbon ¹¹C₆ ® ¹¹B₅ + e⁺ + n for boron, or fluorine ¹⁸F₉ ® ¹⁸O₈ + e⁺ + n for oxygen.  
  In order for b⁺ radioactivity to occur, the mass-energy condition m(Z-1, N) + 2m_e < m(Z, N) must be met, where m(Z, N) is the mass of the nucleus with proton number Z and nucleon the number N, m_e is the rest mass of the electron (which is identical with the rest mass of the positron). Thus, the difference between the weights of the mother and daughter nuclei must be greater than 1.022 MeV (=2.m_e).
  Here again, the energy balance can be well explained by the shell model of the atomic nucleus structure discussed in the previous §1.1. If there is an excess of protons over neutrons in the nucleus, protons will fill slightly higher energy levels than neutrons. The nucleus can then go into a lower energy state by converting the proton by a b⁺ -conversion to a neutron, which goes to a free lower energy neutron level .
Note: At first glance, it may seem paradoxical that a lighter particle - a proton, may spontaneously transform into a heavier one particle - neutron, in violation of the law of conservation of energy! Indeed, the free proton is never disintegrate by b⁺ -decay on neutron (according to the classical theory the proton is stable, although some hypotheses grandunific theories predict the possibility of instability and decay of protons, but a different mechanism in addition to the almost infinite life span greater than about 10³³ years). In nuclei, however, we cannot look at protons and neutrons as free; they are part of a higher whole (core) bound by a strong interaction. It is this bond that will provide the energy needed to convert the proton. The stability of the nucleus is then co-determined not only by the sum of the masses of nucleons, but also by the binding energy of nucleons. If in the coupled configuration, the core consisting of p protons and N neutrons, has a higher total energy ~ weight than the nuclei of p -1 protons and n +1 neutrons (at least by a rest mass of the electron, i.e. 511 keV), the situation arises, when b⁺ decay may occur. And it is completely in accordance with the law of conservation of energy.

Positron radiation b ⁺
Everything about the continuous energy spectrum and neutrino emission applies here analogously to b^- (only in the conversion of b^- antineutrino is emitted, while in the case of b⁺ neutrino). As for the further fate radiated by b⁺, it would be the same as in b^- only in a vacuum - the positron e⁺ is as stable a particle as e^-, so in a vacuum it would "fly off to the other end of the universe" *).
*) Note: Claim that the emitted particle in vacuum "flies at the end of the universe" is here meant rather theoretical and figuratively. Movement of particles in space is actually influenced by three factors :
1.. Gravity that its universal action bends the paths of all particles, including photons.
2. Magnetic field shaping paths of electrically charged particles.
3 . The interaction of charged particles with electromagnetic cosmic background radiation, wherein the particles reverse Compton scattering lose energy.
   These phenomena cause the actual range of particles even in a free space is not unlimited. Charged particles with lower energies remain "trapped" in the space of the galaxy, where they will be complicatedly to orbit in magnetic and gravitational fields. Although high-energy particles can escape the galaxy's magnetic field, they will be inhibited by inverted Compton scattering on the ubiquitous relic electromagnet. radiation. Neither particles nor photons of electromagnetic radiation are able to overcome the horizons of events arising according to the general theory of relativity in the gravitationally curved space-time of the universe ("Gravity, black holes and space-time physics").
  In matter, however, the fate of b⁺ is diametrically different (see Fig.1.6.1 in §1.6, section "Interaction of charged particles - directly ionizing radiation". As long as the positron has a high velocity, it pulls electrons out of the shell with its electric forces as it passes around the atoms, and thus ionizes, similarly to the beta^- electron. However, after sufficient braking (in water or tissue after about 1-4 mm), the positron e⁺ meets the electron e^- *), and since they are "antagonistic" antiparticles, they are mutually destroyed ("eaten"): they are annihilation e⁺ + e^- ® 2 g - transforms into two quantum of hard radiation g with energies of 511keV, which fly out of the annihilation site exactly in opposite directions (at an angle of 180^o in the center of gravity system). This fact is used in the scintigraphic method of positron emission tomography PET, as described in §4.3. Thus, if we have a sample of the radioactive substance b⁺, positrons with electrons annihilate already inside this sample, so that we do not register practically any positrons in its vicinity, but such a sample will be a source of intense hard radiation g with an energy of 511keV. And just as when a radiolabel labeled with b⁺ radionuclide is applied to the organism - each positron at a distance of about 1-3 mm from the place of its origin annihilates with an electron in the tissue and we can detect two quantums of g radiation in coincidence about 511 keV energy flying in opposite directions - this is the basis of PET scintigraphy (see the passage "PET cameras" in Chapter 4 "Radioisotope scintigraphy").
*) Positronium
Just before the actual annihilation, the electron e^- and the positron e⁺ can orbit for a around itself (they orbit the common center of gravity) - they form a special bound system (similar to a hydrogen atom) called positronium (Ps). The dimension of the "atom" of the positron is twice the hydrogen atom, the binding energy of the positron is 6.8 eV. Depending on the mutual orientation of the electron and positron spins, the positronium can be in either the singlet state ¹S₀ with oppositely oriented spins - the so-called parapositon p-Ps (1/4 cases), or in the triplet state ³S₁ with consistently oriented spins - the so-called orthopositonium o-Ps (3/4 cases).
  However, this system of positronium is unstable, the two particles approaching each other in a spiral under the emission of electromagnetic waves; in p-Ps in about 120ps they "fall" on each other and there is annihilation on two photons g, each with an energy of 511keV. In the case of o-Ps, annihilation to two photons is prohibited by quantum selection rules (related to the law of conservation of the spin momentum - each of the photons has spin 1), so o-Ps would decay in a vacuum with a relatively long lifetime of about 140ns by emissions of 3 photons with a continuous energy spectrum (the total energy of 1022keV is divided by the photons in a stochastic way). In the substance, however, the positron bound in o-Ps much earlier is enough to annihilate with some "foreign" electron from the environment, which has the opposite spin orientation - again, two photons g with energies of 511 kV are formed.
  The annihilation of a positron with an electron produces 2 gamma photons in the vast majority of cases, as mentioned above. Sometimes, however, there may arise even more , but with a very small probability (the likelihood that the e^-e⁺ - anihilation in 2+n photons is proportional a^-n, where a= 1/137 is the fine structure constant). If a positron interacts with an electron bound in an atomic shell, the extinction of such a pair may be accompanied by the emission of only a single photon, and some of the energy and momentum may be transferred to either the atomic nucleus or one of the other electrons; however, the probability of this process is very small and does not apply in practice.
  The lifetime of positrons in substances is in the order of hundreds of picoseconds. However, the exact value depends on local electron densities and configurations, which is used in the PLS (Positron Lifetime Spectroscopy) spectroscopic method . The investigated material is locally irradiated with a b⁺- g emitter (most often ²²Na), wherein the positron lifetime is determined by measuring the delayed coincidence between the detection of photon radiation g of irradiating radionuclides (from Na²² on, it is g 1274 keV) annihilation photons, and detecting of g 511 keV.
  The e⁺ positrons emitted by beta⁺ radioactivity are actually a kind of "visitors from the anti-world" - particles of antimatter. The properties of antiparticles, antimatter, antworlds or "antiuniverses" are discussed in §1.5 "Elementary particles and accelerators", passage "Antiparticles - antiatoms - antimatter - antworlds" (including the possibilities of "production" and the use of antimatter, with a somewhat light sci-fi story about the encounter of a "human " with an "anti-human"...).

The difference between the energy spectrum b ^- and b ⁺
The spectrum of beta radiation drawn above in Fig.1.2.3 is theoretical and ideological, it does not take into account the electric forces between electrons or positrons and the atomic nucleus that emits them. In fact, the continuous spectrum b^- and b⁺ differs somewhat in shape in the low energy region. In b^- there is a higher proportion of electrons of lower energies, because the electrons are slowed down by electric attractive forces in the field of a positively charged nucleus during the trip from the nucleus. Positrons b⁺, on the other hand, are accelerated by electrical repulsion after leaving the nucleus, gaining additional kinetic energy - there is a smaller share of low energies in the positron spectrum. Thus, in the region of very low spectral energies, the electric Coulomb interaction causes the enrichment of slow electron emissions and the decrease of the slow positron emission :
        
                        A typical shape of the continuous spectrum of beta ^- and beta ⁺ radiation

Electron capture
The last, and somewhat strange, type of beta radioactivity is electron capture. It is an alternative process to the decay of b⁺ in nuclei with an excess of protons. An "excess" proton can achieve its goal of "becoming a neutron" not only by converting p⁺ ® n^o + e⁺ + n to b⁺ radioactivity, but also in another way. Around the nucleus, electrons orbit in tha atom. A proton that "wants to change into a neutron" can "to reach out" into orbit, capture an electron *) and merge with it: p⁺ + e^- ® n^o + n - Fig.1.2.4 right. This process is called electron capture (EC - Electron Capture ); since the capture of an electron from the level of K , or L or M of the atomic shell can occur, this process is sometimes also called K-capture, or L or M capture. The most common is the K-capture of an electron in the singlet state s, because this orbital has a relatively significant overlap in the region of the nucleus. Electron capture is also sometimes called inverse beta decay (electron absorption is the opposite of the process of electron emission from the nucleus at beta^- - decay). The shift rules between the nucleon and proton numbers of the parent and daughter nuclei are the same as for b⁺, i.e. (Z, N) ® (Z-1, N).
*) It's just a humorous-allegorical formulation! According to the laws of quantum mechanics, part of the wave function of the orbiting electron extends to the region of the nucleus, so that the described process of electron capture can occur directly and immediately...
  During electron capture, no corpuscular radiation is emitted from the nucleus (apart from elusive neutrinos), the energy is divided into neutrino energy and electron binding energy when absorbed by a proton in the nucleus. So is it radioactivity at all in the sense of our initial definition and scheme in Fig.1.2.1? The answer is yes! First, there is transmutation of the nucleus. Second, the electron from the higher shell (L) immediately jumps to the vacant space after the electron on the shell K, emitting characteristic X-rays; each electron capture decaying radionuclide is a source of intense X-rays. And third, as we will see below, the daughter nucleus B is usually formed in an excited state and emits radiation g during deexcitation. Electron capture is often accompanied by the emission of so-called Auger electrons (see below in the section "Gamma radioactivity", passage "Internal conversion of radiation g"), arising from the internal conversion of characteristic X-rays.
  Electron capture is an alternative ("competitive") way to decay those neutron-deficient nuclei that have enough energy to emit a positron. Electron capture occurs primarily when the mass of the parent nucleus m(Z, N) is greater than the mass of the daughter nucleus m(Z-1, N), but by a smaller difference than the rest mass of the electron and positron 2m_e = 1.022 MeV. Positron emission b⁺ cannot occur in this case (not enough conversion energy is available), so electron capture is the only possible way of conversion. If the difference m(Z, N) - m(Z-1, N) is > 2m_e, it is possible to convert the parent nucleus (Z, N) to the nucleus (Z-1, N) by both emission b⁺ and electron capture (usually the conversion of b⁺ occurs preferentially).
  Electron capture converts eg 57-cobalt ⁵⁷Co₂₇ + e^- ® ⁵⁷ Fe ₂₆ + n to iron, or 125-iodine ¹²⁵I₅₃ + e ^- ® ¹²⁵Te₅₂ + n to tellurium.
Two peculiarities of electron capture :
¨ 1. Electron capture EC is the only type of radioactive transformation of the nucleus (in addition to the process of internal gamma conversion during deexcitation of excited levels - isomeric transition, is described below in the section "Internal conversion of g radiation"), in which electron atomic envelope also participates. The rate, probability, or half-life of a nucleus by electron capture can be slightly affected by chemical bonding of a given atom. The rate of conversion is slightly greater (i.e., the half-life is shorter) for the EC-radioactive isotope in elemental form than if it were part of a compound. In the chemical bond, the density of the orbital electrons of the EC-radioactive atom is partially shifted to the adjacent bound atom, which somewhat reduces the probability of electron capture EC by the nucleus. This is observable for light EC-radioactive atoms, whose valence electrons orbit relatively close to the nucleus (especially s-electrons, whose orbitals have a relatively significant overlap into the nucleus region, are involved). For heavier atoms, the chemical effect on the EC is negligible. Furthermore, the conversion rate by electron capture can be increased by applying external pressure to radioactive atoms or molecules - higher pressure leads to an increase in electron density and thus to an increase in the probability of electron capture by the nucleus.
  These effects have been demonstrated mainly in beryllium ⁷Be, as it is a small atom whose valence electrons are close to the nucleus. Various compounds or ⁷Be implants in different media (Au, Al ₂ O ₃ , LiO₂ , graphite, fullerene C₆₀) were investigated. A difference of 0.9 % was observed between the half-lives in metallic and dielectric media. When a high pressure of 270 kilobars was applied to ⁷BeO, it was found that the decay rate of ⁷Be increased by 0.59%, in ⁷Be(OH)₂, at the same pressure, the EC conversion rate of ⁷Be increased by 0.88 %. External influencing on the decay rate of electron capture have also been observed for ¹⁰⁹In and ¹¹⁰Sn isotopes implanted in Au and Al foils (into the respective crystal lattices).
  The course of electron capture also depends on the ionization of the respective atom. The radioactivity of isotopes that decay by pure EC can be slowed down (or theoretically even stopped ) if their atoms are fully ionized (cf. for interest with the opposite effect of acceleration of low-energy b^--conversions by complete ionization in the rare so-called b - decay into bound electronic states, described above). Under normal terrestrial conditions, we do not encounter such a situation, but it is assumed that it can be able to affect astrophysical processes of cosmic nucleosynthesis - see §1.1, section "Cosmic alchemy".
  During a supernova explosion , elements are synthesized by the rapid fusion of neutrons (the so-called r - process), launched into space in a fully ionized state. Therefore, those that are EC-radioactive cannot decay radioactively for a long time (until they meet and connect to electrons in colder regions of space). This could cause some anomalies in the distribution of elements. The holmium ¹⁶³Ho is considered, which under normal (atomic) conditions is converted to ¹⁶³Dy by electron capture. Fully ionized ¹⁶³Dy with the above-mentioned process b -decay to bound electron states, decays to ¹⁶³Ho.......
¨ 2. Since in EC there is no emission of corpuscular radiation, the energy difference Q parent and daughter nuclei is reflected in the excitation of the daughter nuclei (with one or more excited levels, followed by radiation g) and the rest of the energy is carried away by the neutrino. If we could measure the spectrum of these neutrinos, in the EC would be line (discrete), as opposed to continuous spectrum of neutrinos from b^-,+ decays.
Mixed radioactivity beta ^- , beta ⁺ , electron capture
The possibilities of individual types of radioactive transformations are mainly determined by the energy balance between the respective "neighboring" nuclei (discussed in more detail below in the final part of this chapter "Stability and instability of atomic nuclei"). If the mass-energy inequality m(Z, N)> m(Z+1, N) is satisfied for the given initial nucleus (Z, N), the b^- transformations will occur. Under the conditions m(Z, m)> m(Z-1, N) <m(Z-1, N) + 2m_e , the nucleus (Z, N) will decay by electron capture. If m(Z, N)> m(Z-1, N) + 2m_e , it is possible to convert the nucleus (Z, N) to the nucleus (Z+1, N) by both beta⁺ decay and electron capture. If both the inequality m(Z, N)> m(Z+1, N) and the inequality m(Z, m)> m (Z-1, N) + 2m_e are satisfied at the same time, then all three types of beta conversion can occur simultaneously. An example is the ⁶⁴Cu radioisotope, which decays from 39% of b^- emissions, from 19% of b⁺ emissions and 42% of cases by electron capture.

Mechanism of beta conversion. Weak interactions.
When interpreting radioactivity b, we have clarified the formation of radiation b^- and b⁺ by the mutual transformation of neutrons and protons in a situation where due to the increased number of neutrons or protons, higher energy levels are occupied. But what is the internal cause or mechanism of these transformations? The first theory of decay b was presented by E.Fermi in 1934, which introduced the so-called weak interaction acting between elementary particles.
  According to the laws of quantum physics, Fermi described the emitted electron and neutrino by wave functions depending on their momentum, expressing the probability of electron and neutrino emission in a certain range of momentum as the product of squares of these wave functions and a special expression containing the wave function integral of the original neutron and the resulting proton, multiplied by the constant g_F characterizing the strength of the interaction leading to electron and neutrino emission - newly introduced weak interactions. The radiation spectrum b corresponding to this Fermi expression has a continuous shape according to Fig.1.2.3 on the right in accordance with the experiment.
  The original Fermi theory was formulated in the spirit of quantum physics, but not quantum field theory - the weak interaction was conceived as "contact" or "point" - the decay products arise at the same point and moment as the neutron disappears (no intermediate particles here did not assume). However, in the spirit of quantum field theory, each interaction should be caused by the quantum of the field - the particle mediating the interaction. This concept was developed in 1967-8 by S.Weinberg, A.Salam and S.Glashow, who, due to the weak interaction, introduced three mediating particles W⁺, W^-, Z^o (with positive and negative charge and without charge) and by adding photons created a unified theory of weak and electromagnetic interactions - the so-called electroweak interaction. This concept was experimentally verified in 1983: intermediate bosons W⁺, W^-, Z^o with masses m_W = 82GeV and m_Z = 93GeV were discovered in interactions of upstream proton-antiproton beams (270GeV versus 270GeV) of the collider of the large proton synchrotron at CERN, whose modes of decay were in good agreement with the predictions of the Weinberg-Salam model.

Fig.1.2.5. Schematic representation of the mechanism of b^- -neutron decay (top) and b⁺ -proton transformation (bottom) within the standard model of elementary particles.

Now, within the standard model of elementary particles (see §1.5), the beta decay mechanism is explained by the quark transmutation scheme according to Fig.1.2.5. The neutron n^o (with zero electric charge) consists of quarks u - d - d; quark u has charge +2/3, quarks d charge -1/3. One of the quarks d is transformed into a quark u by the action of a field of (electro)weak interactions by mediating a virtual intermediate boson W^-, which carries away the charge -1. From the virtual boson W^- then the electron e^- and the antineutrino n´ are formed, which fly out in different directions. The result of the transformation is a proton p⁺ consisting of quarks u - u - d. The transitional stage between the initial and final state in the middle of the image lasts only a small moment (approx. 10^-27 sec.) and is not directly observable. Analogous diagram can be drawn for the conversion of b⁺ (bottom of the image), wherein the proton p⁺ due to transmutation quark u to at quark d for mediating intermediate boson W⁺ is converted to a neutron n^o, positron e⁺ and neutrino n. Feynman's diagram of the transmutation of a quark inside a proton during beta conversion is drawn in Fig.1.5.1.F in §1.5 "Elementary particles".
Complexity of beta radioactivity 
When we compare individual types of radioactivity according to the complexity of their mechanism, we can state :
  Alpha radioactivity is basically simple (Fig.1.2.2). In a nucleus with a large number of nucleons, where the nuclear forces of short-range at the peripheral parts are not strong enough, small, more strongly bound groups of two protons and two neutrons are formed, which can (with the help of tunneling) leave the nucleus and fly off as alpha particles - helium-4 nucleus.
  Gamma conversion  (Fig.1.2.6) is also simple. In the energetically excited state, the nucleus deexcites to the ground (or lower) state, while the energetic difference is emitted as a gamma radiation photon by electromagnetic interaction.
  However, beta radioactivity is very complex ! Its mechanism is hidden not only inside the nucleus, but even deeper - inside the nucleons themselves. It consists in the transmutation of quarks "u" or "d" through the intermedial meson W, which leads to the mutual transformation of protons and neutrons (Fig.1.2.5).

Electromagnetic radioactive transformations
These are radioactive processes of nuclear deexcitation caused by electromagnetic interaction. We know two types of these nuclear deexcitation: 1. Direct emission of gamma photons; 2. Internal conversion with emission of orbital electrons from an atom. Below we will discuss them mainly in connection with radioactivity.
Note: In addition to the ground state, atomic nuclei have a number of excited states (energy levels), only a part of which is applies in radioactive transformations. The other excited states arise only during the bombardment of nuclei by energetic particles from accelerators.

Gamma radioactivity
So far, in interpreting radioactivity, we have focused on our own mechanisms of radioactive transformation and on the properties of emitted corpuscular radiation (a, b^-, b⁺). Therefore, we have deliberately drawn the basic schemes of radioactive transformation in Figures 1.2.1 to 1.2.4 in a somewhat simplistic way so that we can focus on the nature of radioactive transformations. Now notice the behavior of the daughter nucleus immediately after the radioactive transformation. Fig.1.2.6 shows a complete scheme of radioactive decay, including the behavior of the daughter nucleus :

Fig.1.2.6. A typical diagram of radioactive decay of the parent core A to the excited daugher nucleus B* and its
subsequent deexcitation by photon radiation g into the resulting daughter nucleus B .

After such a large "event", as radioactive transformation means for the atomic nucleus, the resulting daughter nucleus seldom remains in an unexcited basic energy state. After changes in the number or type of nucleons in the nucleus, the nucleons may not immediately occur in the lowest energy states. The released energy results in the daughter nucleus B, after radioactive transformation, mostly being formed in the energetically excited state B*; we can imagine that the nucleus is "inflated" ¹) - nucleons are more distant from each other, occupy higher energy levels - cf. Fig.1.1.9. Such a "inflated" nucleus B* usually "collapses" very quickly ²), nucleons rearrange to a lower energy state - deexcitation occurs, in which the respective energy difference is emitted in the form of one or more quantum - photons - hard electromagnetic radiation - gamma radiation ³). The emission of gamma quantum stabilizes the energy conditions in the nucleus. The daughter core B then remains in the ground state.
¹ ) In the case of non-spherical nuclei, which can perform a rotational motion, in addition to the level excitation, we also encounter rotational excitation. Here, too, deexcitation is accompanied by the emission of the photon g.
² ) In some cases, the excited nuclear state can last for a very long time - see below "Metastability and nuclear isomerism".
³ ) If the core is part of an atom (which is almost always), alternative way deexcitation of excited nuclear levels may be the emissions of the envelope electron by the process of internal conversion - see below "Internal conversion gamma and X". For very heavy nuclei in field transurans highly excited states of these heavy unsymmetrical nuclei, besides conventional gamma-deexcitation, can succumb to spontaneous fission (§1.3, section "Fission of atomic nuclei" and "Transurans").
  We can therefore pronounce the following definition :

Gamma-ray spectrum
The energy levels of the atomic nucleus are quantized , so that all photons g emitted by a given type of deexcitation will have the same energies - the spectrum of g- rays is line *), discrete. If the B* nucleus has more excited levels, several groups of monoenergetic photons g will be emitted, so that the spectrum will be formed by several discrete lines - peaks in the measured spectrum (measurement of gamma radiation spectra is discussed in §2.4, section "Gamma ray scintillation spectrometry").
*) Width of gamma lines
Gamma-photons at radioactivity come from nuclear transitions between discrete levels with precisely given energies, so they are essentially monoenergetic, their ideal physical spectrum represents a sharp line on the energy E_g. Very small fluctuations in energy values are caused by quantum uncertainty relations and backscatter of nuclei during gamma-photon emission. Furthermore, since these photons practically never come from free nuclei, but are emitted from radioactive atoms contained in a substance (in a certain material), some photons interact with the substance before leaving the sample. It can also blur their energy somewhat. The thermal motion of the atoms in the sample causes by Doppler effect the small frequency - energy shifts according to different velocities of emitting atoms (nuclei), which is reflected in the Doppler extension of the gamma line. However, the magnitude of all these extensions is generally very small compared to the effects in the actual detection of radiation. The real, physical, gamma-ray spectrum can therefore be considered practically line - discrete, monoenergetic.
  Gamma-photons of annihilation radiation, created by annihilation of positrons with electrons e⁺ e^- ® 2 g, have an energy of 511 keV, are basically monoenergetic. However, electrons and positrons annihilate together at somewhat different residual braking rates, which the Doppler effect leads to frequency-energy shifts. This results in a Doppler broadening of the 511keV annihilation peak from positron radionuclide samples. The 511 keV annihilation peak is somewhat wider than other "nuclear" near-energy gamma peaks in the spectrum. Under normal laboratory conditions, this additional Doppler broadening of the annihilation peak is about 1.2 keV and is observable only on high energy resolution spectrometers (semiconductor Ge detector or magnetic spectrometer).
In our spectrometric measurements of a number of radionuclides (eg ¹⁸ F , ²² Na ., ⁶⁸ Ga , ¹³¹ I , ¹²⁴ I , ¹³⁷ Cs , ¹⁵² Eu , ...) on a semiconductor HPGe detector, we observed FWHM half-widths of about 1.4-1.5keV in "nuclear" gamma peaks of medium energies between 400-700keV, in annihilation the 511keV peaks of the FWHM half-width were approximately 2.5-2.7keV (Doppler broadeningwas observed).
  The energy of radiation g from radionuclides is usually in the range of about 5keV to 5MeV. The lowest energy of photon radiation from nuclear transitions is observed at ^110mAg(1.1 keV) and ^99mTc(2.17 keV; but the main peak here is 140keV); however, these low energies are almost completely subject to internal conversion (see "Internal conversion of gamma and X-rays" below) and their intensity is immeasurably weak. So far, the highest energy of 11.26 MeV gamma radiation from radioactivity was recorded for the short-lived radionuclide ²⁰Na. However, during interactions of high-energy particles, gamma radiation of much higher energies of the order GeV or TeV can also occur!
A typical example of the process according to Fig.1.2.6 may be the b- radioactivity of cobalt ⁶⁰Co. In the actual b- decay, the cobalt nucleus first emits an electron e^- (b- particle) and an electron antineutrino n_e', thereby converting to the nickel core ⁶⁰Ni in the excited state: ⁶⁰Co ® ⁶⁰Ni* + e^- + n_e'. This newly formed excited nucleus is then freed of excess energy by radiating a quantum of g : ⁶⁰Ni* ® ⁶⁰Ni + g. The daughter ⁶⁰Ni* has two excited levels, so that the emitted quantum g here has an energy of 1173 keV and 1332 keV (see §1.4, passage "Cobalt", Fig. "Co-60").
  The spectrometry of radiation g is discussed in §2.4, part "Scintillation spectrum of radionuclides". Gamma-spectrum (scintillation and semiconductor), a number of important radionuclides are, together with the description, displayed in §1.4, the "Most important radionuclides". Looking at the spectrum implies an interesting regularity: if radioactive transformation occurs to a greater number excited states of the daughter nucleus, then usually more likely to lower energy levels than higher excited states - gamma spectra are represented by higher intensity peaks of lower energies, while high-energy peaks are much weaker, are often visible until at high magnification and longer acquisition time.
  The diagram in Fig.1.2.6 shows two important facts for gamma radiation emitted by radionuclides :   
1. Radiation g is temporally following and concomitant after the emission of corpuscular radiation during its own nuclear transformation. This delay is usually quite slight (nanoseconds and less), but can sometimes be very significant (see below "Nuclear isomerism and metastability") ..!..
2. Most radionuclides are mixted emitters - either a+g or b+g. Only some emitters are pure a or pure b; radioactive conversion sometimes occurs directly to the ground state of the daughter nucleus (this is the case, for example, with tritium ³H or carbon ¹⁴C). However, pure g emitters do not exist in nature! Nevertheless, we can produce pure gamma emitters artificially. This is made possible by the remarkable property of some nuclear excited levels: their metastability (see below "Nuclear isomerism and metastability").
Angular (directional) correlations of gamma radiation
The quantum of ionizing radiation generated during radioactivity are generally emitted completely randomly and isotropically in all directions *). If, however, at one and the same decay events are emitted simultaneously from the core of two or more quanta, between the directions of their trip may occur the angular correlations. Let's show it in gamma radiation :
  In most cases, when emitting independent gamma photons from the nucleus, these photons fly completely randomly, isotropically in all directions (angles). However, if cascade deexcitation occurs with the formation of two gamma photons emitted promptly in succession, then after the emission of the first photon the nucleus is polarized - its excited intermediate state is oriented in a certain way, depending on the angular momentum carried by the emitted photon. The second (subsequent) gamma photon will then preferably be emitted in a certain direction, correlating with the emission angle of the first photon. The increased intensity of the second gamma radiation is then observed at a certain angle (eg 90°, 180°) with respect to the direction of the first radiation. The condition for the observation of these g-g angular correlations at normal laboratory temperature of the preparation is a sufficiently short lifetime of the intermediate excited level in comparison with the average frequency of thermal collisions of emitter atoms (to avoid angular momentum transmission and nuclear orientation). Isotopes in which angular correlations g-g occur are, for example, ⁶⁰ Co , ¹¹¹ In , ⁷⁵Se, ¹⁶⁹Yb, ⁸¹ Rb and many others. The perfect 100% angular correlation is in the annihilation g-radiation, which arises during the annihilation of electron-positron pairs to two 511 keV photons, flying in exactly opposite directions at an angle of 180°.
  Furthermore, in some radionuclides that are converted by electron capture to excited states of the daughter nucleus, angular g- X correlations occur between gamma radiation and characteristic X-radiation accompanying electron capture (e.g. at ¹²⁵ I , ¹⁹⁸Hg). An angular correlation b-g is also observed for combined beta-gamma emitters.
*) The random emission of radiation quanta isotropically in all directions is related to the fact that under normal conditions the atomic nuclei (their spins) are oriented completely randomly and change chaotically due to thermal movements. Only if the radioactive sample is placed in a strong magnetic field (several Tesla), are the magnetic moments and spins of the nuclei partially polarized and then even single-photon radiation will show an increased emission at a certain angle correlating direction of the magnetic field (magnetic induction vector).
Coincidence tomographic scintigraphy
These angular correlations were once experimented with in coincidence tomographic scintigraphy (§4.3 "Tomographic scintigraphy", passage "Technical development of tomographic scintigraphy"), but this has not been succesfull in practice. The perfect angular correlation is in the annihilation g- radiation, which arises during the annihilation of electron-positron pairs to two 511keV photons, flying in exactly opposite directions at an angle of 180° (in the center of gravity reference system) - see §1.5, section "Elementary particles and their properties , passage "Positrons"). This 100% angular correlation is widely used in gamma imaging by the method of coincidence positron emission tomography in nuclear medicine (§4.3, part "Positron emission tomography PET").

Rate of deexcitation and emission g. Nuclear isomerism and metastability.
The vast majority of nuclear excited states are very unstable and deexcite almost immediately (after the order of 10^-12 s.) by gamma photon radiation. In some cases, however, nuclear excited levels do not deexcite immediately (the cause is discussed below), but only after a certain longer average time has elapsed - we say that they are metastable. This phenomenon is also called nuclear isomerism - nuclei can exist in two isomeric states, called isomers. Metastable isomers of a certain isotope are denoted by the index "m" located after the mass (nucleon) number of the nucleus - eg ^99mTc, or Tc-99m. If a given isotope has more isomers, the indices "m1, m2, ..." are denoted, in order of higher energy of the respective excited states - e.g. ^{152 m2}Eu. Metastable isomers are generally produced after the radioactive decay of the daughter nuclei (as in the ^99mTc), but they may also exist separately without prior radioactive decay, they can be prepared in nuclear reactions.
Isomerism metastable levels were first observed at b-decay of thorium ²³⁴Th to ^234mPa, in the case of an artificial radionuclide, then in the case of bromine, ^80mBr produced by irradiation ⁷⁹Br with neutrons. More than 300 nuclear isomers are now known, some of which are widely used - above all ^99mTc.
Cause of metastability and isomerism :
The main mechanism of suppression of gamma-deexcitation of induced nuclear levels, to which metastable isomers owe their long half-lives, is the relative "prohibition" of gamma transition due to large difference in angular momentums (spins) between the respective nuclear levels - "spin mismatch". Photon emission from the excited state of the nucleus leads to a change in the spin of the nucleus. Because the photon has spin 1, transitions with DI =1are most probable and fastest with a lifetime of excited states of about 10^-16-10^-10 seconds. Transitions with DI = 2 are also very fast to lifetimes of the excited state in the range of 10^-11-10^-4 s. In both of these cases, therefore, the gamma-ray photons are emitted practically simultaneously with the previous transformation a or b (for the rate of deexcitation and emission g we therefore observe the same half-life as for the previous corpuscular transformation). Transitions between levels with DI> 2 are much less likely, relatively "forbidden". Spin changes greater than 2,3,4, ... are in principle possible, but each such unit of change causes a significant inhibition of the transition probability (by about 5 orders of magnitude). Unless there is another possibility of deexcitation *), it will be reflected in a longer lifetime of the excited level (10^-3 s , seconds, minutes, hours, years, occasionally even longer...). Deexcitation and photon emission are then governed by its own half-life, independent of the half-life of the previous transformation a or b. The nuclide in the excited nuclear state with a longer half-life of the gamma transition is called the nuclear isomer and is denoted by the superscript "m" at the nucleon number - e.g. ^99mTc , ^81mKr.
*) However, if one or more other energy levels lie between the excited and ground state of the formed nucleus, deexcitation occurs in such a way that fast transitions with DI = 1 or DI = 2 are preferentially applied and a metastable state does not arise.   Metastability of excited levels 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 (at least 3 h , ie DI ³ 3). 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, so that 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 in nuclei with a nucleon number from 24, from N> 40 they occur relatively often. The shell model of nucleus explains the details.
Transformation half - lives of nuclear isomers
 The deexcitation half-lives of metastable levels are very different for various nuclides, depending on the quantum nuclear configurations. They are usually very short, on the order of nanoseconds or microseconds. Isomers with half-lives of the order of seconds, minutes, hours and more are possible for practical use. The shortest-lived radionuclide that has practical use is the ^81mKr krypton isomer with a half-life of only 13 seconds (see §1.4, passage "Rubidium-Krypton").
  However, long-term nuclear isomers are particularly interesting from a physical point of view, even for their numerous applications. Undoubtedly the most important of these is technetium ^99mTc with a half-life of 6 hours, described below (see also references). The ^99mTc core has spin +1/2 and is converted to the basic state of ⁹⁹Tc with spin +9/2. An example of a truly long- lived isomer is hafnium ^187m2Hf with a high excitation energy of 2.45MeV and a half-life of 31 years, or holmium ^166m1Ho with a half-life of 1200 years.
  The most stable known nuclear isomer is tantalum ^180mTa (as a natural primordial isotope contained in natural tantalum in a concentration of 0.012%) with an extremely long half-life >10¹⁵ years! - it is therefore almost stable; while ¹⁸⁰Ta in the basic state is beta^- and EC radioactive with a half-life of only 8.1 hours. Remarkable stability of ^180mTa is caused by a large spin mismatch: the excited level of ^180mTa (energy 75keV) with spin -9 should be photon deexcited to the base ¹⁸⁰Ta with spin +1; this large spin difference causes inhibition of photon deexcitation by a factor of 10³⁵ relative to the usual "permissible" transition of 10^-12 s, leading to a half-life of 10²³ s, i.e. about 10¹⁵ years. Direct beta decay to ¹⁸⁰Hf or ¹⁸⁰W is also blocked here due to spin mismatch...
Note: The half-life of the g -deexcitation, which takes place by electromagnetic interaction, is independent of the half-life of the previous corpuscular radioactive transformation (alpha or beta), but depends only on the specific configuration of the excited level relative to the lower or basal level (see mechanisms above).
Other (non-photon) radioactivity of isomers
Metastable excited levels of isomers usually "decay" by conventional electromagnetic deexcitation to emit photon g. As with other excited states, an alternative method of deexcitation may be the emission of the envelope electron by the process of internal conversion - see "Internal conversion of gamma and X" below.
  However, some nuclear isomers have such different quantum properties from the ground state (especially the spin value) that there is no isomeric transition to the ground state of g- radiation photon emissions, but radioactive conversion of beta^- , beta⁺ or electron capture to another neighboring nucleus. An example is the metastable ^152mEu, which does not pass at all to the basic ¹⁵²Eu, but from 73% is converted to ¹⁵²Gd by b^- -radioactivity and by 27% by electron capture and b⁺ to ¹⁵²Sm (half-life is 9.3 hours).
Note: However another metastable isomer europium designated ^152m2Eu pass regularly with a half life of 96 minutes with isomeric transition to basic ¹⁵²Eu with photon emission g -radiation with energy 89.8keV ...
  Another example is metastable lutetium ^177mLu with a half-life of 160.4 days, which in 22% of cases passes by normal g -deexcitation to basic ¹⁷⁷Lu (which is then converted to ¹⁷⁷Hf by beta-radioactivity with a half-life of 6.68 days, but at the same time in 78% is converted directly to ¹⁷⁷Hf by direct beta-decay. In the latter case, the baseline of ¹⁷⁷Lu is "bypassed" and, in addition, beta-conversion from metastable lutetium goes to different (substantially higher) excited hafnium levels than with the usual beta-radioactivity of ¹⁷⁷Lu. There are more similar cases of branched g-b transformation of metastable states. After all, even in the known technetium ^99mTc, its metastable level of 140 keV, in addition to the usual isomeric g- transition to the ground state of ⁹⁹Tc, passes in a small percentage (0.003%) by beta-radioactivity directly to excited states of ruthenium ⁹⁹Ru (see §1.4, part "Molybdenum-Technetium", Fig.Tc99m) .
  For very heavy transuranic nuclei the highly excited metastable states of these heavy asymmetric nuclei in addition to normal gamma-deexcitation may undergo spontaneous fission (§1.3, section "Fission of atomic nuclei" and "Transurans"), mostly with a short half-life. These "fissile isomers" are denoted by the index "f" (instead of "m"), e.g. ^240fPu.
Monopole E0 transitions 0⁺ ® 0⁺
A special case of strongly forbidden deexcitation is the transition from an excited state with zero spin and positive parity to a ground state, which also has zero spin and positive parity - from an electrical point of view it is a "monopole" transition, which does not occur in electrodynamics. In such a case, the usual deexcitation does not take place with the emission of one quantum gamma (the law of conservation of angular momentum would be violated), but three alternative mechanisms may come into play :
¨ Internal conversion (discussed below in the section "Internal conversion") - electrical transfer of the transition energy to the electron in the atomic shell, which is emitted and carries away the difference between the transition energy E0 and the binding energy of the electron. It is the main process here in the case of low energies E0 excitation;
¨ Emissions of electron-positron pair (from Dirac's vacuum background) with oppositely oriented spins can occur at high excitation energies - if the E0 transition has an energy greater than twice the rest mass of the electron, i.e. > 1.022 MeV;
¨ Simultaneous emission of two gamma photons with oppositely oriented spins. It is a higher order process with a low probability of about 10^-4 -10^-3.
A practical example of this monopole E0 transition and its manifestation in the gamma radiation spectrum is given in §1.4, passage "Y-90". 

Pure gamma-radionuclides; technetium ^99mTc ; radionuclide generators
The most important example of the formation of a metastable nuclear isomer is the radionuclide molybdenum ⁹⁹Mo, which with a half-life T1/2 = 2.66 days b^- -radioactivity changes to ⁹⁹Tc - to its excited level of ⁹⁹Tc* with energy 140keV, which is metastable with a half-life of T1/2 = 6.02 hours. The excited nucleus in the metastable state is denoted by the index "m", so here the metastable technetium has the designation ^99mTc. If during this time separated daughter technetium from the parent molybdenum, we get a pure preparation of technetium, containing excited metastable nuclei ^99mTc, which during gradual deexcitation emit g radiation with an energy of 140keV - so we obtained a pure gamma emitter.
  Laboratory equipment for obtaining shorter-term radionuclides from the original parent radionuclides with a longer half-life are called generators. Molybdenum-technetium generator for the preparation of ^99mTc consists of a glass cylinder with a small "chromatographic" column, in which it is on a suitable sorbent (eg Al₂O₃ , SiO₃ , ZrO₂) in the insoluble chemical form of the oxide the longer-term parent isotope ⁹⁹Mo is absorbed. Radioactive conversion produces a short-lived radionuclide ^99mTc, which is released from the insoluble bond and rinsed with water (or saline) - so-called elution - elutes from the column a solution of sodium pertechnetate Na ^99mTcO₄ containing atoms of pure g- emitter ^99mTc; b -radioactive ⁹⁹Mo remains absorbed in the column.
  The continuous conversion of the parent ⁹⁹Mo leads again after the elution to the accumulation of ^99mTc in the generator, so that the elution can be repeated over time.
The principle and technical design of the ⁹⁹Mo/^99mTc generator is shown in Fig.4.8.1 in §4.8, which we present here again for clarity :

Fig.4.8.1. Elution ⁹⁹Mo - ^99mTc generator.
Left: Principle functional diagram of the elution generator. In the middle: Technical design of a sterile generator with an evacuated elution vial.
Right: Decomposition scheme of molybdenum ⁹⁹Mo to technetium ^99mTc, deexcitation to ⁹⁹Tc and slow transformations to stable ruthenium ⁹⁹Ru.

From above in the section "Exponential law of radioactive decay", passage "Mixtures of radionuclides, decay series, radioactive equilibrium" the general equation of subsequent decay of radionuclides A(l_A)®B(l_B)®C(stable) was derived. If we apply this equation (multiplied by the factor l_Tc to get the instantaneous activity in [Bq] from the instantaneous number of nuclei) to our case of the Mo-Tc generator ⁹⁹Mo(l_Mo=0,0105hod^-1)®^99mTc(l_Tc=0,1155hod^-1)®⁹⁹Tc("stable") and taking into account that 87% ⁹⁹Mo decays to a metastable excited level of ^99mTc, we get for the time dependence of the immediate activity of the required technetium ^99mTc relation: A_99m-Tc(t) = 0,957.A_Mo(t=0) . (e^-0,0105.t - e^-0,1155.t ), where A_Mo(t=0) is the activity of ⁹⁹Mo at time t = 0 of the previous elution, time t is in hours. The activity of ⁹⁹Mo changes with time T according to the decay law A_Mo(T) = A_Mo(0).e^-lMo^.T = A_Mo(0).e^-0,0105.T. Substituting this basic decomposition of molybdenum we obtain the final relation for the instantaneous activity of the eluted ^99mTc at time T from the delivery of the molybdenum generator and at time t from the last elution :
            A _99m-Tc (T, t) = 0.957.A _Mo (0) .e ^{- 0 , 0105.T} . ( e ^{- 0.0105.t} - e ^{- 0.1155.t} )   ,
where A_Mo(0) is an activity of ⁹⁹Mo at time T = 0 of the generator production, times t and T are in hours. To determine the actually eluted ^99mTc activity, we must also take into account the elution efficiency, which is usually approximately 75-85%. This time dynamics of ^99mTc activity during repeated elutions of the Mo-Tc generator is plotted in Fig.2.1.B (d), which we present here again for clarity :