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
1.7. Neutrinos - ghosts between particles

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 to proton, electron and electron (anti)neutrino. The neutron is slightly heavier than the proton, the difference in mass is about 2.5 times the mass of the electron (this difference comes from the difference in the mass of "u" and "d" quarks and the difference in binding energy). However, the intermediate W boson, which mediates quark conversion, is about 80-times more massive than a proton or neutron, so there is a high potential barrier. So, despite the small energy surplus, the neutron does not immediately decay into a proton, but it takes 13 minutes on average to tunneling through this barrier.
  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. However, the earlier opinion of exchanging virtual pions is now abandoned...
 The daughter nucleus after conversion of beta^-
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.

A more detailed explanation of the origin, properties, meaning and detection of neutrinos is quite comprehensive, which is why we included it in a separate chapter :

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" (antiparticles, antiatoms, antimatter, "antiuniverses" are discussed in §1.5., section "Antiparticles - antiatoms - antimatter - antiworlds").
Note: Really to the end of the universe ?
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. And finally, 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⁺ positrons 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^- (about the short episode just before the annihilation, there is a small passage "Positronium" below), and since they are "antagonistic" antiparticles, they are mutually destroyed ("eat each other"): 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.
  Intuitively - simplified, approximately - we can imagine the emergence of gamma radiation during e⁺ e^- annihilation from the point of view of electromagnetism: Electrons and positrons interact by their charges by electromagnetic force. When these two identical opposite charges disappear at once during annihilation, a disturbance in the electromagnetic field is created, which spreads from the place of annihilation as an electromagnetic wave. The rest mass of the electron and the positron is also of electromagnetic origin, so when they suddenly disappear, the resulting electromagnetic wave also carries away their rest mass of 2x511 keV. The detailed interaction mechanism (the law of conservation of momentum - in the center of gravity system, the momentum is zero after complete braking) then leads to the emission of gamma radiation in opposite directions.
  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 :

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, resulting from the conversion of the parent ⁹⁹Mo.

After elution, the ^99mTc activity in the generator drops to almost zero, then rises and reaches a (local) maximum 23 hours after the previous elution, after which a radioactive equilibrium occurs and the ^99mTc activity decreases exponentially with a half-life of 67 hours of ⁹⁹Mo. After 23 hours from the last elution, the elution yield of ^99mTc is the highest; the generator can of course be eluted as needed in a shorter time, but with a lower yield of ^99mTc.
  These elution cycles can be repeated many times until the activity of the parent radionuclide falls below the applicable limit; for a Mo-Tc generator with an initial activity of approx. 10-40 GBq it's about 7-15 days. The relatively long half-life of the parent radionuclide allows for long-term use of the generator, and the short half-life of the resulting daughter radionuclide ensures a low radiation exposure to the patient.
Note: In the past, were also rarely used generators of the extraction type (by passing the methyl ethyl ketone through an aqueous solution of ⁹⁹Mo, extracting pertechnetate ⁹⁹Tc and separating it from the aqueous phase from the parent molybdenum) and sublimation type (using the difference between the volatility of molybdenum oxide and the resulting technetium oxide). Due to their excessive complexity and operational difficulty, they are no longer used, they have been pushed out by elution generators.
  ^99mTc is the most important radionuclide in nuclear medicine (see Chapter 4 "Radionuclide scintigraphy", §4.8 "Radionuclides and radiopharmaceuticals for scintigraphy", where Fig.4.8.1 shows the principle of operation and technical design of the ⁹⁹Mo- ^99mTc generator). Pure gamma emitter with a short half-life of 6 hours allows, without the risk of significantly increased radiation exposure, to apply to patients a very high activity of ^99mTc (of the order of hundreds of MBq) needed to obtain quality images in SPECT or dynamic scintigraphy. The radiation exposure is relatively low because there is no cospuscular radiation that would be absorbed into the tissue and give up its energy; radiation g then, due to its penetration, usually flies out of the organism, only a small part is absorbed.
  Gaseous radioactive krypton ^81mKr is obtained from the generator ⁸¹Rb (T1/2 = 4.85 hours) (EC) ® ^81mKr (T1/2 = 13s) - §1.4, passage " ^81mKr - scintigraphy of pulmonary ventilation". The air stream, guided through a tube through a container containing a layer of parent radionuclide ⁸¹Rb, carries away the released daughter ^81mKr, which the patient inhales, and a scintillation camera uses external detection of g radiation showing the distribution of this ^81mKr in the pulmonary alveoli - ventilatory scintigraphy of the lungs (§4.9.5 "Scintigraphy lung").
  In terms of production and properties of artificial radionuclides, generators are described in §1.4 "Radionuclides", section "Production of artificial radionuclides", section "Radionuclide generators".

Terminological agreement: g and X radiation
The high-energy region of electromagnetic radiation consists of X radiation (X-rays) and gamma radiation. In terms of spectral classification, g -radiation lies energetically behind the X-ray region, towards higher energies and short wavelengths. Both of these radiations have the same physical nature (photon radiation) and largely similar properties, they may differ in the way they originate. In connection with the above-described gamma radioactivity, the following terminological agreement was made on the distribution of g and X radiation according to their creation and origin :
l Radiation gama is called photon radiation generated in atomic nuclei (during deexcitation of excited nuclear levels) - even in case if it has a low energy of several keV.
l Radiation X is called radiation caused by jumps of electrons between internal levels in the atomic shell and braking radiation by electrons - even if it has a relatively higher energy of tens and hundreds of keV.
  Thus, we can have, for example, radiation g with an energy of only 14.4 keV (one of the energies emitted by the radionuclide ⁵⁷Co) and at the same time X-radiation with an energy of 75 kV (line K_a characteristic X-rays of lead), or braking X-rays with a continuous spectrum reaching energies of 120keV and higher (depending on the X-ray tube voltage). However, photon radiation with very high energies (of the order of MeV) is usually called gamma radiation, regardless of the way it is generated.

Internal conversion of gamma and X-rays ; conversion and Auger electrons
If the nucleus is part of an atom (which is almost always the case in our conditions), not all dexcitation in the nucleus may radiate as photons of radiation g. Competitive processes may occur, preventing the emission of part of the photons of g radiation during deexcitation of excited nuclear levels - especially the process of so-called internal electron conversion of gamma radiation (abbreviated only internal gamma conversion ), abbreviation IC (Internal Conversion). The energy of nuclear deexcitation does not radiate, but is transferred to an electron in the envelope, which then flies out as a so-called conversion electron.

Fig.1.2.7. Schematic representation of the internal conversion of radiation g to form conversion electrons and characteristic X-rays and the internal conversion of X-rays to form Auger electrons. The consequence of the internal conversion for the gamma radiation spectrum is shown on the right - the presence of the characteristic X-ray peak [the detailed structure of the K_a,b characteristic X-ray peaks, measured by a semiconductor Ge(Li) detector is shown in the box].

In the left part of Fig.1.2.7 all relevant processes are schematically shown. In particular, the blue arrow shows the basic case of "undisturbed" radiation of the photon g from the excited nucleus. The process of internal conversion can be simply imagined so that the gamma photon emitted during deexcitation of the excited nuclear level can "collide" with the envelope electron of its own atom, which takes all its energy (a photo-effect occurs), the gamma photon disappears and is replaced by an electron flies out, released due to the energy received from the bond in the atom (thicker red arrow). This phenomenon is commonly observed, it is called the internal conversion of gamma radiation (formerly also called the "internal photo effect") and the corresponding electrons are called conversion electrons.
Note: This older explanation of the internal conversion mechanism should now only be considered heuristic; the gamma radiation photon does not actually fly out of the nucleus at all (it is only virtual), but the deexcitation energy of the excited level of the nucleus is transmitted by the electromagnetic interaction to the nearest envelope electron directly and it then flies out of the atom with the kinetic energy given by the difference between the deexcitation energy of the nucleus and the binding energy of the electron in the envelope. Physical emission of the gamma photon is not necessary here, because the wave function of the envelope electrons partially penetrates the nucleus (there is a non-zero probability of the electron occurring in the nucleus region) and the interaction can occur immediately. Thus, despite its name, internal conversion is not a "two-stage" process (in which a gamma photon is first emitted and then absorbed by an electron that ejected), but a direct "one-stage" electromagnetic process.
Internal conversion is favored in cases where direct gamma photon radiation deexcitation is prohibited due to "spin mismatch" (discussed above in the section "Nuclear Metastability and Isomerism").
  The process of internal conversion most often occurs with electrons on the K shell (in the 1s state), which is closest to the nucleus, and the wave function of whose electrons most likely penetrates to the nucleus. However, it also occurs with less intensity on the L, M, ... shells (electrons in the 2s, 3s, ... state). If the energy of a given transition is not sufficient for the internal conversion on the K-shell, it cannot occur and the internal conversion takes place only from higher atomic orbits L, M .... An example is ²⁴¹Am with an excited state energy of 59.6 keV, which not enough for the binding energy of 118 keV on the K shell of the daughter neptunium, so the internal conversion occurs mainly on the L shell and the peaks of the characteristic X-ray of neptunium in the range of about 12-20keV are clearly visible in the spectrum- see spectrum in the "Americium" section §1.4. The process of internal conversion is more efficient when the energy difference of the levels of the excited nucleus is relatively small (comparable to the binding energies of the envelope electrons) and when the excited level does not have a large spin.
  The kinetic energy of the conversion electrons is given by the difference between the energy of the respective transition of the excited state of the nucleus and the binding energy of the envelope electron - see below "Discrete spectrum of conversion electrons".
  The probability of internal conversion is different for different atoms and excited nuclear levels and is characterized by the so-called coefficient of internal conversion, which indicates the ratio of the mean number of conversion electrons to the mean number of quanta g emitted during a given nuclear transition.
  The probability of internal conversion depends somewhat on the chemical composition of the radioactive preparation by a similar mechanism to that described above for "Electron capture" radioactivity.
Characteristic X-rays
As a result of internal conversion, a space, vacancy, is released on the inner shell (usually K) of the atomic shell after the conversion electron, so that the electron immediately jumps from a higher level (most often L) with a quantum of characteristic X-rays emission (lighter blue arrow in Fig.1.2.7). Hence the characteristic X-ray peaks, which often appear in the gamma-ray spectra (eg the 33keV peak in the radionuclide spectrum ¹³⁷Cs with energy g 662keV, Fig.1.2.7 right). There is a whole cascade of deexcitation in the atomic shell, in which several photons of X-rays are emitted, the spectrum of which has a fine structure with lines K_a,b, L_a,b, ...
  Intense characteristic X-rays are emitted especially in radionuclides converting with electron capture, where, however, the free space in the electron envelope is not caused by the emission of the conversion electron, but by the absorption of the envelope electron by the nucleus (see above "Electron capture", Fig.1.2.4 on the right).
Auger electrons
The internal conversion may undergo also X-rays emitted during electron jumps between the internal levels of the atom (eg after electron capture, photoeffect, internal conversion g). The conversion electrons emitted in this way are then called Auger's electrons (Pierre Auger first observed them in the nebula chamber in 1923) - a thinner red arrow in the left part of Fig.1.2.7. In one radioactive transformation, several Auger electrons can be emitted, corresponding to a cascade of deexcitation in the atomic shell and a fine structure (K_a,b , L_a,b , ...), mostly low-energy electrons. Intensive sources of Auger electrons are mainly radionuclides decaying by electron capture (such as ¹²⁵I); they are also emitted by some g- radionuclides (including metastable ones, eg ^99mTc) - due to the internal conversion of the nuclear g- transition and the subsequent series of deexcitation in the atomic envelope.
  A special case of the Auger effect is the filling of a free vacancy by the jump of an electron from a higher subsurface of the same electron shell (eg the transition L_I ® L_III). This process is called the Coster-Kronig transition in spectrometry (D.Coster and R.L.Kronig observed it in the X-ray spectrum in 1935) and is accompanied by the emission of photons or Auger (Coster-Kronig) electrons of very low energy.
Discrete spectrum of conversion electrons 
Conversion and Auger electrons have essentially the same properties as b^-, but differ significantly in the shape of their energy spectrum. The spectrum of radiation b is continuous (as explained in detail above - "Radioactivity beta"), while the spectrum of conversion and Auger detectors is discrete. The energy of these electrons is given by the difference between the primary energy of the transition (deexcitation) and the binding energy of the electron on the respective shell of the atomic shell; both of these energies have fixed quantum values. For mixed b+g radionuclides, the conversion electrons appear as discrete peaks on a continuous b-spectrum curve. These discrete conversion electron peaks (of known energies) are preferably used for the calibration of radiation spectrometers b, in particular magnetic spectrometers.
  Another difference compared to beta^- radioactivity is that during the internal conversion, of course, there is no emission of neutrinos, only electrons fly out (and subsequent characteristic X-rays + Auger electrons).

Mixed radioactivity - branched transformations
In addition to the basic types of radioactivity (alpha, beta^-,+, EC, gamma), some radioactive nuclei have so-called branched radioactive transformations of one parent nucleus by two different types of radioactivity - with certain probabilities - into two different daughter cores. The ratio of the frequency of each alternative mode transitions to the total number is called branching ratio. A branched-converting radionuclide essentially has two conversion constants and two partial half-lives. If the imaginary experiment were able to block various alternative types of transformation, we get for each line 1 and 2 different value for the decay constant l₁, l₂ and a half-life (T_1/2)₁ and (T_1/2)₂. The total value of l_tot of the conversion constant by both branches will then be given by the sum of the partial values of l_tot = l₁ + l₂ and the resulting half-life of this radionuclide can be considered as effective or total half-life both branches: (T_1/2)_tot = (T_1/2)₁ . (T_1/2)₂ /[(T_1/2)₁ + (T_1/2)₂] . The hypothetical partial half-lives (T_1/2)₁ and (T_1/2)₂ are inversely proportional to the relative proportion (branching ratio [%]) of the individual branches.
  The most common type of branched radioactive transformation is [beta^-] + [electron capture (EC) or beta⁺]. Of the natural radioisotopes, it occurs in potassium-40, which (with a half-life of T1/2 = 1.27.10⁹ years) is converted by beta^- radioactivity to argon ⁴⁰Ar (89%) and by electron capture to calcium ⁴⁰Ca (11%) - §1.4, passage "⁴⁰ K". It occurs in a number of artificial radionuclides, the most important of which are, for example, ¹⁵² Eu , ¹⁸⁶ Re , ¹⁹² Ir and several others.
  Branched [gamma (IT)] + [beta] transformation is observed for several radionuclides conversion. This is the case when a certain excited state of the nucleus is metastable (described above in the passage "Nuclear isomerism and metastability") with a long half-life. Then, instead of deexcitation of gamma photon emissions (or internal conversion), there may be a certain probability of an alternative beta-radioactive transformation of this excited nucleus to another neighboring nucleus. An extreme example is europium ^152mEu (see the passage "Europium" in §1.3), on the contrary, it also occurs to a small extent with the well-known technetium ^99m Tc . The preparation of lutetium-177 by the branching IT-beta transformation is facing the reaction ¹⁷⁶Lu (n, g )¹⁷⁷Lu, when a metastable contaminant ^177mLu is formed, which in 22% dexcites by successive isomeric transitions to the ground state ¹⁷⁷Lu and in 88% is converted independently by b^- -radioactivity to a relatively high excited state 1315keV of the daughter ¹⁷⁷Hf - see ¹⁷⁷ Lu .
  Furthermore, it is a branched transformation [alpha] + [beta^-], occurring in heavier nuclei. For natural radionuclides we are occurring in the decay series of uranium and thorium (§1.4, section "Decay series"), wherein the bismuth isotopes ^{211 212 213}Bi with both converted a radioactive isotopes of thallium on and radioactivity b to isotopes of polonium. Furthermore, actinium ²²⁷Ac is converted by alpha radioactivity to ²²³Fr and beta by conversion to ²²⁷Th. This leads to branching of the decay chains.
  Combined [alpha] + [spontaneous fission] transformations occur in heavy uranium and transuranium radionuclides. For uranium and lighter transuranium, the proportion of spontaneous fission is very small - the branching ratio of spontaneous fission is approximately: ²³⁵U (7.10^-9 %), ²³⁸U (5.5.10^-5 %), ²⁴¹Am (0.36%). However, for heavier transurans, the proportion of spontaneous cleavage is not negligible, sometimes even decisive. E.g. californium ²⁵²Cf (T1/2 = 2.56 y.) is 97% converted by alpha-radioactivity to ²⁴⁸Cm and in 3% spontaneous fission occurs into two lighter nuclei from the middle of the periodic table and 2-3 neutrons. Californium ²⁵⁴Cf with an effective half-life of 60.5 days is even converted in only 0.3% by alpha-radioactivity to ²⁵⁰ Cm and in the whole 99.7% spontaneous fission occurs ..!...

"Exotic" types of radioactivity
In addition to the standard types of radioactivity (a, b, g - described above), there are rarely occurs also less common, "exotic" types of radioactivity (and some other types are only presumed and not yet proven). So let's make a brief mention of some less common (and sometimes debatable) types of radioactivity :

Spontaneous fission of heavy nuclei
Some heavy nuclei in the uranium and transuranic regions, in addition to radioactivity a show another specific but less common type of radioactive transformation: spontaneous fission of heavy parent nucleus A into two lighter nuclei F1 and F2 *) (located roughly in the middle Mendeleev's periodic tables of elements), sometimes called fragments or fission products : ^NA_Z ® ^N1F1_Z1 + ^N2F2_Z2 + (2-3)n^o + Q(energy). In addition to these resulting nuclei, neutrons are also emitted during fission, in number usually 2-3. During the fission of heavy nuclei into lighter nuclei, considerable energy is released (derived from the difference of binding energies, see §1.3), which is carried away in the form of kinetic energy by the resulting nuclei and neutrons.
*) In addition to the usual binary fission, there is also a relatively rare type - the so-called ternary fission (0.2-0.3% of cases), in which the heavy nucleus splits into three fragments. Two of these fragments are medium-heavy nuclei from the middle of the periodic table, the third can also be a very light nucleus - helium ⁴He, tritium ³H, ⁵He is also observed (which decomposes to ⁶Li with a half-life of about 0.8 s). Interestingly, in spontaneous fission, a 25% higher proportion of ternary fission is observed than in fission induced by neutron absorption (this is probably related to the supply of some additional energy during neutron absorption by the nucleus).
  The cause of spontaneous fission of heavy nuclei lies in their structure and in the properties of nuclear forces. In §1.1, section "Atomic nuclei", it was mentioned that nuclear forces are of short range and exhibit a saturation state. Heavy nuclei therefore have a lower binding energy per nucleon than lighter nuclei from the center of the Mendeleev table of elements. The configuration of two lighter cores is therefore more energetically favorable than the configuration of one heavy core (see Fig.1.3.3 in §1.3, part "Nuclear energy. Fission and fusion of atomic nuclei"). The actual fission process is hampered potential barrier strong interaction holding the core together: in delaying one of the core portion from each other, it is necessary initially to overcome the large attractive force - increases potential energy that only after reaching the height of the barrier (after sufficient distance, where short-range nuclear forces begin to decline sharply) begins to decline. In order for fission to occur, it is necessary to overcome this energy that holds the nucleus together. The fission process is already energy efficient for cores with N> 80. However, the energy gain in the division of these lighter nuclei is very small and the height of the potential barrier is too high, so that fission does not occur. Only in very heavy nuclei do relatively weak nuclear forces (saturation state), so that the height of the "potential wall" by which the individual parts of the nucleus are held together is low *). Therefore, there is a certain probability that the fragments pass through a potential fission wall due to the "tunnel effect" (similar to radioactivity a) and fly apart with high energy.
*) For the heaviest transurans, the height of the potential wall for fission is already practically zero, so it is probably no longer possible to form transuranic elements above a certain Z- limit, which would not disintegrate immediately by spontaneous fission. This is also one of the reasons why a kind of "island of stability", expected by some experts for superheavy nuclei in the area Z » 120-150, probably doesn't exist.
  Spontaneous fission occurs, for example, with uranium ^235,238U (demonstrated in 1940 by G.N.Flerov and K.A.Petržak). It is a process that proceeds with scant intensity and compared with other kinds of radioactivity (in the case of heavy nuclei mainly a) is practically negligible *). However, efficient fission of heavy nuclei occurs upon neutron absorption, as will be shown in the following §1.3; in this case, however, this is not radioactivity, but a nuclear reaction. A small amount of neutrons, originating from spontaneous fission, can then initiate a chain fission reaction when the so-called critical amount of fissile material is reached (§1.3, part "Fission of atomic nuclei").
*) Exceptions are some heavy transurans (described in detail in §1.3, part "Transurans"), eg californium ^252,254Cf, for which spontaneous fission is a common process competing with alpha radioactivity. In practice, the isotope californium ²⁵²Cf is mainly used as a source of neutrons (see §1.4 "Radionuclides", passage "Transurans") in some laboratory applications.

Radioactivity "higher" than a-helium - "cluster" radioactivity
Radioactivity alpha may be regarded as "split" parent nucleus to a helium nucleus ( a particles) and the remainder of the core: ^NA_Z ® ⁴He₂ + ^N-4B_Z-2. From this point of view, radioactivity a and a spontaneous fission can be understood as two observed extreme cases of one and the same nuclear process. The natural question arises as to whether (non-fissile) radioactive transformation is possible with the emission of particles heavier than particles a, for example, carbon, nitrogen nuclei or the like..?. (Or less massive nucleon particles? - see next paragraph). This type of radioactivity was first demonstrated in Oxford in 1984 for the radium isotope ²²³ Ra , where a small amount (1:10 billion a) of decays was observed between the usual a -radioactivity with the emission of carbon isotope nuclei ¹⁴C: ²²³Ra ® ²⁰⁹Pb + ¹⁴C (+12MeV). Later, a similar type of decay was registered in other radium isotopes (^{222,224, 226} Ra ), as well as other heavy nuclei. Furthermore, eg for isotopes ²³¹Pa, ²³⁰Th, ²³³U, ²³⁷Np, ²⁴¹ Am,... in addition to the dominant a radioactivity, transformations with the emission of other heavier nuclei - neon ²⁴Ne, magnesium ²⁸Mg, silicon ^32,34Si are  rare occur.
  Since this is a nuclear transformation with the emission of nuclear fragments - clusters or groupings of nucleons (clusters), this rare type of nuclear transformation is called cluster radioactivity (cluster decay) or heavy ions radioactivity. It is a kind of "transient type" between alpha-radioactivity and spontaneous nuclear fission. Its mechanism is somewhat similar to that of alpha radioactivity (see above "Radioactivity alpha", Fig.1.2.2 right) - a quantum tunneling process in which a stronger bound group (cluster) of nucleons within the nucleus overcomes and penetrates the potential barrier of attractive short-range nuclear forces. The probability of this tunneling is very small. The energy of emitted clusters is substantially higher than in alpha radiation - ranges from 20MeV (at ¹⁴C conversion) to 95MeV (at ³⁴Si conversion).

Proton and neutron radioactivity ?
In the vast majority of nuclei, strong interactions between nucleons keep the nuclei in a relatively stable state - their radioactive transformation can occur by mutual transmutation of neutrons and protons (beta ^{-, +} radioactivity), deexcitation of excited levels (isomeric transition - g emission ), or by radiation a. We do not observe spontaneous emission of individual nucleons or light nuclei with the exception of alpha particles and rare cluster radioactivity. However, in nuclei with a high excess of protons or neutrons, the balance is so disturbed that the emission of individual nucleons or whole light nuclei is energetically possible and advantageous.
  Upon irradiation of some nuclei with protons and accelerated heavier ions, it was observed that in addition to the standard b⁺ radionuclides also form nuclei with a large excess of protons, which emit protons when they decay. This phenomenon has been observed, for example, in ¹⁷Ne, ²⁵Si or ⁴¹Ti. The proton can be emitted either directly from an unstable nucleus with excited nucleons (tunneling similar to the decay of a), or "in two stages" - following the previous decay of b⁺; a case of emission of 2 protons was also recorded. This process, in which the neutron deficient nucleus ^NA_Z of proton emissions is converted to the nucleus ^N-1B_Z-1, was called proton radioactivity or proton decay.
Note: The proton radioactivity presented here has nothing to do with the hypothesis of decay or "radioactivity" of the proton itself, discussed in §1.5 !
  The question is whether the emission of protons from short-lived excited neutron deficient nuclei can be considered as a separate type of radioactivity. The specific characteristics of this process (short half-life and very small relative proportions) suggest that in many cases it is rather a kind of retarded concomitant phenomenon in nuclear reactions *). In some cases, however, the lifetime (longer than the mere characteristic transit time of a nucleon through the nucleus) suggests that there is some resemblance to radioactivity. However, only proton radioactivity can be considered as true a one-step process, in which a proton is emitted from the ground state of the nucleus (or from the isomeric state). Proton radioactivity was demonstrated in the isomeric state of the cobalt core ⁵³ Co. Proton radioactivity of nuclei in the ground state was demonstrated in the 1980s in ¹⁵¹Lu: ¹⁵¹Lu ® ¹⁵⁰Yb + p (+1,2MeV), then in ¹⁴⁷Tm, as well as in lighter nuclei of isotopes cesium ¹¹³Cs and iodine ¹⁰⁹I with a high deficiency of neutrons. Such highly neutron-deficient nuclei are prepared by special experimental methods of bombarding suitable targets with accelerated heavy ions, with electromagnetic separation of the required radionuclides. This process is not of practical significance, the theoretical significance lies in the illustration of the variety and diversity of phenomena occurring in nuclear reactions and in strongly unbalanced nuclei.
*) A similar phenomenon, the emission of delayed neutrons, is encountered in the fission of heavy nuclei, where in addition to the "primary" neutrons emitted immediately during the actual fission, other neutrons with a relatively large time delay are emitted from the fragments. However, this process is two-step: b occurs first- decay of a chip with a longer half-life, after which only a rapid neutron emission from the excited daughter nucleus occurs at a time comparable to the kinematic nuclear time. Therefore, "neutron radioactivity" is not mentioned. We can also think about a free analogy with the so-called resonances in high-energy interactions of elementary particles, discussed in §1.5 "Elementary particles".

Another "exotic" types of radioactivity ?? - muon, pion, hyperon radioactivity ??
Studies of the properties of heavy nuclei far from the region of stability point to the possibilities of other, even more unusual types of radioactivity. It could be pion radioactivity, in which the heavy nucleus A would split into two medium-heavy fragments B and C (or several fragments) and simultaneously emit a p- meson :
^NA_Z ® p^±,0 + ^N1B_Z1 + ^N2C_Z2, whereas N = N1+N2, Z = Z1+Z2+Zp .
The muon radioactivity is discussed analogously : ^NA_Z ® m^± + n_m + ^N1B_Z1 + ^N2C_Z2 ; N = N1+N2, Z = Z1+Z2+Zm .
Or even hyperon radioactivity with lambda hyperon emission: ^NA_Z ® L⁰ + ^N1B_Z1 + ^N2C_Z2 ; N = N1+N2, Z = Z1+Z2 ,
or with hyperon emission S : ^NA_Z ® S^±,0 + ^N1B_Z1 + ^N2C_Z2 ; N = N1+N2, Z = Z1+Z2+ZS .
Possibly, emission of nuclei with bound hyperons - so-called hypernuclei or hyperfragments C_L, eg: ^NA_Z ® ^N1B_Z1 + ^N2CL_Z2 , followed by rapid meson or nucleon decay of the hypernucleus...

Stability and instability of atomic nuclei
The time stability or instability of an atomic nucleus is given by a complex interplay of strong, electromagnetic and weak interactions, determining particle processes and energy ratios in the nucleus, related to the number of protons and neutrons, their ratio and arrangement. Nuclei in which the energy level of a given configuration of protons and neutrons is minimal are stable over time. However, nuclei that are not in an energetically stable state tend to go into a more energetically stable state by a certain recombination of proton and neutron states - by transforming protons and neutrons, emitting particles or emitting quanta of "excess" energy; there are radioactive transformations of nuclei. In order for such a nucleus transformation to take place, two basic conditions must be met :
1. Existence of a nucleus with a lower energy level - usually "adjacent" or close to its proton and nucleon number (except for spontaneous decay of heavy nuclei "Fission of atomic nuclei").
2. A physical mechanism who performs the transformation an energy-richer nucleus into a lower-energy nucleus.
  The energy levels of proton and neutron configurations in nuclei are determined primarily by strong ones nuclear interaction, with minor corrections from the electromagnetic interaction (proton and neutron levels are occupied independently; it was discussed in §1.1, section "Nuclear structure", passage "Strong nuclear interaction"). The most common physical mechanism of nuclear transformations is the weak interaction, which through the transmutation of quarks within nucleons can mutually transform protons and neutrons - beta^-,+ radioactivity. However, this only happens when the energy condition is met: the new configuration of protons and neutrons must have less energy than the original configuration. For heavy uranium and transuranic nuclei, another mechanism of transformation is short-range strong nuclear interactions, causing relatively weak binding of heavy nuclei, and conversely strong binding of helium ⁴He₂ nuclei, which can "tunnel" from the peripheral parts of heavy nuclei as alpha particles (again when the energy condition is met). The short range of strong nuclear interaction is also responsible for the spontaneous fission of heavy nuclei. In gamma isomeric transitions, the mechanism is the electromagnetic interaction, thanks to which the energy difference of the excited nucleus radiates as a high-energy quantum - a photon of electromagnetic radiation.
  According to the shell model of nuclei due to the exclusion principle, a proton or neutron occurs in the same energy state only 1 time (or 2 times, with opposite spins). Another proton or neutron can only occupy a higher energy state. It is therefore energetically advantageous to fill the same energy state with a proton and then a neutron. From this it can be expected that nuclei with the same number of protons and neutrons will preferentially occur in nature. For lighter cores (N<»30) this rule is actually confirmed. However, electrostatic repulsion increases with large cores. Therefore, for heavier nuclei to ensure stability, neutrons are beginning to predominate, exhibiting only attractive nuclear forces and not electrical repulsion.
  In the above passages of this §1.2 "Radioactivity", we have discussed in detail the mechanisms by which radioactive transformations of various types of nuclei occur. Let us now look globally at the set of all known atomic nuclei (nuclides) in terms of the laws of their stability, or, conversely, instability - the "tendency" to spontaneous radioactive transformations.
  For a detailed and clear capture of the properties of nuclei, it is very useful to draw them in the so-called nuclide map. It is a rectangular network of small squares in which the nuclei are arranged in rows according to the increasing number of protons p (= Z) and in columns according to the increasing number of neutrons n (= N-Z). Each nuclide thus corresponds to one small square with coordinates [p, n], in which we can then write the name and selected characteristics of the nuclide *).
*) Such a table (map) of nuclides is ideologically somewhat similar to Mendeleev's table of chemical elements. In addition to the basic difference between the atomic nucleus and the envelope, there are two differences in notation: 1. Mendeleev's table is periodic, the table of nuclides is sequential. 2. Mendeleev's table is traditionally arranged according to the system of writing text in lines below each other (from left to right + from top to bottom), while our table of nuclides is arranged by the system of plotting quantities in Cartesian coordinates (from left to right + from bottom to top).
Note: Sometimes a special system of hexagonal arrangement is used nuclide tables ("trilinear arrangement"), in which, in addition, the hexagons representing the nuclei are color-coded depending on the half-life of the nucleus.

Fig.1.2.8. To clearly illustrate the properties of nuclei, their drawing on the nuclide map is used.
Left: A simple two-dimensional map of nuclides. Right: Three-dimensional map including binding energies.
The three-dimensional model was made within the diploma thesis under the guidance of Doc.Dr.J.Kleczek, DrSc.

A reduced schematic representation of the nuclide map is in the left part of Fig.1.2.8. Details of the characteristics of the nuclei are not marked here (they would not fit into a small picture), only stable nuclei are marked with black squares, empty (white) squares correspond to unstable, radioactive nuclides. From this diagram we see mainly three facts :
¨ 1. All existing nuclei form in the map of nuclides a band going to the right and up in an approximately diagonal direction, which gradually expands - with increasing number of protons increases the diversity of existing nuclei (there are more different isotopes).
¨ 2. Stable cores form in this general belt a kind of "stability line". For lighter nuclei (N<»30) with stable nuclei centered around the line n = p , i.e. having approximately the same number of protons and neutrons, are symmetric. For heavy stable nuclei gradually begins to dominate the number of neutrons over protons. It due to the fact that with increasing Z (= p) increases the repulsive Coulomb force of protons, which is to reduce the total energy of the core is advantageous to compensate by the presence of more neutrons. The ratio of the number of protons and neutrons decides on the stability and instability of nuclei through weak interactions, causing radioactivity beta (see note below "Weak interactions influence on the stability of the cores").
¨ 3rd Stability line ends with bismuth, ²⁰⁹Bi₈₃, which is the last, most heavy (almost) stable nucleus *). All the heavier nuclei (uranium and transuranic region) are already radioactive (usually radioactivity a, possibly also spontaneous fission).
*) Natural bismuth consists isotope ²⁰⁹Bi, which until recently was considered stable. In 2003 it was found that this isotope is slightly radioactive - with an extremely long half- 1.9.10¹⁹ years subject to alpha-radioactivity. From a practical point of view, however, natural bismuth appears to be non-radioactive (300 kg of bismuth would have an activity of 1Bq). Lead of ²⁰⁸ Pb is now considered to be the heaviest truly stable nuclide.
Influence of weak interactions on the stability of nuclei
If there was only a strong interaction (and electromagnetic) in the microworld, there could be "mononucleon" nuclei composed only of protons or only neutrons (mononeutron nuclei would not have an electron shell). Nuclear "monsters" composed of thousands of neutrons could also form. However, we do not observe anything like this in nature, there are no stable nuclei from either the two protons themselves or the two neutrons; even the neutron itself is unstable. Namely, there is another kind of force in nature - a weak interaction, which ruthlessly transforms beta (- or +) radioactivity each nucleus, in which a certain ratio between the number of protons and neutrons is disturbed (if the energy condition in the potential well given by a strong nuclear interaction is met). The mechanisms of these processes have been discussed above in the "Beta Radioactivity" section.

When interpreting the mechanisms of individual types of radioactivity, we have seen that the stability or instability of nuclei is determined by two interrelated factors: the number of protons and neutrons and the binding energy of the nucleus. An even more illustrative picture of the properties of nuclei is thus obtained by plotting the mean binding energy of a nucleus per nucleon for each nucleus in the third dimension (perpendicular to the drawing) of the nuclide map. Since each nucleon in the nucleus is bound in a kind of potential well, we carry this binding energy downwards. This creates a three-dimensional table of nuclides, having the shape of a "valley of nuclides" extending diagonally through a coordinate network (Fig.1.2.8 on the right).
  The most stable nuclei lie at the "bottom" of this valley of radionuclides - the binding energy of their nucleons is the largest. On the "left slope" of the nuclide valley there are nuclei rich in protons (sometimes referred to as the "proton slope"), which decay by b⁺ -radioactivity, on the "right slope" there are nuclei with a predominance of neutrons ("neutron slope"), decaying with b^- -radioactivity. These facts are clearly visible in cross sections of the valley of nuclides, where there are nuclei with the same number of nucleons (isobars) - the left part of Fig.1.2.9. The most stable cores are at the bottom of the valley. Nuclei with an excess of protons (left) or with an excess of neutrons (right) have excess energy *), which it tries to get rid of b^- or b⁺ decay. Therefore, the nuclei on the proton and neutron "slope" are b- radioactive.
*) This 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. This is analogous to nuclei with an excess of neutrons, where b^- decay converts a neutron into a proton, which occupies a free proton level with lower energy.

Fig.1.2.9. The "valley of nuclides" profiles in the three-dimensional table give a clear idea of the energy balance of the atomic nucleus bond.
Left: Example of a cross section through a valley of nuclides. Right: Longitudinal section through the valley of nuclides at the deepest point (corresponding to the relatively most stable cores).

The longitudinal section through the valley of nuclides in the deepest places is also interesting - Fig.1.2.9 on the right. This longitudinal section gives a curve of the dependence of the binding energy (per nucleon) on the nucleon number for the relatively most stable nuclei, ie nuclei occurring in nature. From this curve, we see that the most strongly bound nuclei have elements of the iron group that are the most stable and form the deepest "bottom" of the nuclide valley, while the light and heavy nuclei have smaller binding energies. It is clear that "falling to the bottom of the valley of radionuclides" in the longitudinal direction is possible in two directions: By merging light nuclei into heavier ones (nuclear fusion, thermonuclear reactions), or decay (a) or fission of heavy nuclei into lighter nuclei *); in both cases there is a release of nuclear energy. We will deal with this curve in detail in the following §3.3, part "Nuclear energy", in connection with nuclear energetics, where we will redraw it in the inverted form (Fig.1.3.2).
*) Nuclear fusion cannot take place spontaneously, but only at high kinetic energies of merging nuclei at high temperatures (inside stars or in a thermonuclear reactor, see §3.3, section "Fusion of atomic nuclei"). Radioactivity a it occurs spontaneously in heavy nuclei, whereby these nuclei "fall" along the valley of nuclides towards lighter and more stable nuclei. This process is often combined with b- radioactivity in the transverse direction of the proton or neutron slope. The cleavage of heavy nuclei can occur spontaneously, but usually only with negligible intensity (with the exception of heavy transurans) . Efficient fission of heavy nuclei occurs during neutron absorption, these processes will be discussed in the following §1.3 "Nuclear reactions", section "Fission of atomic nuclei".
  The nuclide valley of a three-dimensional table of nuclei thus gives us a clear energetic idea about why some nuclei are stable and others decay radioactively. Just as a stone released down a mountain slope rolls into a valley due to gravity, so nucleons in nuclei on energetic "slopes" try to get as low as possible to the bottom of the valley, where they are "pulled" by nuclear forces. Whether they "succeed" and nuclear transformation occurs, is determined by the energy balance in relation to the mechanism of radioactivity b, a, event. nuclear fusion or fission.
  Unstable nuclei lying "low on the slope" of the nuclide valley reach a stable configuration with only one radioactive transition (b^{+, -}, a). However, nuclei with a large excess of protons or neutrons, as well as heavy nuclei in the uranium and transuranic regions, lying high on the "slope" of nuclides, must undergo a whole cascade of individual radioactive transformations before resting on the "valley floor" in a stable configuration. This creates radioactive decay series (see §1.4 "Radionuclides", part "Decay series", Fig.1.4.1).

1.1. Atoms and nuclei		1.3. Nuclear reactions and nuclear energy

Vojtech Ullmann