Alpha Decay

L. L. Goldin, L. K. Peker, G. I. Novikova

1. Introduction

Several specific features of $\alpha$-decay are striking even upon a cursory examination of the experimental data. First of all, attention is drawn to the fact that the energies of the $\alpha$-particles of all radioactive substances lie within a narrow energy interval around 6 MeV. Only in a few cases are $\alpha$-rays emitted with energies below 4.5 MeV or above 8.5 MeV, whereas the number of known $\alpha$-active nuclei approaches two hundred.

Rutherford’s experiment$^{R27}$, in which uranium was bombarded with $\alpha$-particles from ThC′, played an enormous role in understanding the process of $\alpha$-decay. Uranium emits $\alpha$-particles with an energy of only 4 MeV, whereas the energy of the $\alpha$-particles from ThC′ is 8.8 MeV. It was natural to expect that the $\alpha$-particles from ThC′ would penetrate into uranium nuclei. Ordinary Rutherford scattering of $\alpha$-particles in the Coulomb field of nuclei should, at large scattering angles, have given way to scattering caused by nuclear forces. However, the scattering of $\alpha$-particles revealed no deviations from the usual law of Coulomb scattering.

As Gamow$^{G28a}$ and Gurney and Condon$^{G28b}$ first showed, the noted properties of $\alpha$-rays find a natural explanation if one assumes that $\alpha$-decay occurs by means of the tunnel effect, that is, by the penetration of $\alpha$-particles through the high potential barrier created by the Coulomb repulsive forces between the nucleus and the $\alpha$-particle. With this interpretation of $\alpha$-decay, the absence of anomalies in the scattering of $\alpha$-particles from ThC′ on uranium is explained by the fact that, despite the low energy of the emitted $\alpha$-particles, the full height of the Coulomb barrier of the uranium nucleus is substantially greater than the energy of the $\alpha$-particles from ThC′. The narrow range of energies in which $\alpha$-decay is observed is explained by the extremely sharp dependence of the barrier penetrability on the energy of the $\alpha$-particles; a change in the decay energy by 10% leads to an increase in the lifetime by more than 3 orders of magnitude. At energies

below \(3.5\text{--}4\) MeV, the period of \(\alpha\)-decay increases so much that radioactivity cannot be detected.

Attention is drawn to the sharp separation of the regions of \(\alpha\)-active and \(\alpha\)-stable nuclei. Nuclei with charge greater than 82 (lead), with rare exceptions, prove to be \(\alpha\)-active (in some cases the \(\alpha\)-activity is masked by more strongly expressed \(\beta\)-activity). Nuclei with charge less than 82 prove, on the contrary, to be stable with respect to \(\alpha\)-decay. In this region only very rare nuclei are \(\alpha\)-radioactive, possessing a particularly large neutron deficiency. Such nuclei are sharply unstable with respect to \(K\)-capture and are not observed in nature. The only exception is \( \mathrm{Sm}^{147} \), present in natural samarium in an amount of 15% and having a half-life of the order of \(10^{11}\) years.

The sharp separation of the region of \(\alpha\)-active nuclei from the region of \(\alpha\)-stable nuclei is connected with the shell structure of nuclei. Alpha decay becomes energetically possible already in the region of atomic weights around 180. However, in this region it cannot yet be detected experimentally because the \(\alpha\)-decay energy is small and the lifetimes are correspondingly enormous. Alpha decay in this case can, of course, become observable if the difference in binding energies of the corresponding nuclei is for some reason anomalously large. Such a case occurs for some nuclei in the rare-earth region (the nucleus \( \mathrm{Sm}^{147} \) has already been noted above), where the sharp increase in binding energy associated with the neutron shell \(N = 82\) has a strong effect. The shell \(Z = 82\) raises the decay energy in the region \(Z > 83\) and lowers it for \(Z \leq 82\) so much that the region of \(\alpha\)-active nuclei becomes sharply separated.

When considering \(\alpha\)-active nuclei, one should bear in mind that pure \(\alpha\)-emitters in the region \(Z > 82\) are analogues of stable nuclei in the region \(Z < 82\). As is known, the masses of nuclei possessing a given \(A = N + Z\) lie on a parabola, in the lower part of which are located \(\beta\)-stable nuclei, and on the branches—\(\beta\)-active nuclei. The same holds for nuclei with \(Z > 82\). Only in this case there are no stable nuclei at all; in the lower part of the parabolas are located \(\beta\)-stable nuclei exhibiting pure \(\alpha\)-activity, and on the branches—\(\beta\)-unstable nuclei emitting, in addition to \(\alpha\), also \(\beta\)-rays. In those cases where the \(\beta\)-instability is especially pronounced, the \(\alpha\)-activity may turn out to be practically unobservable if the lifetime associated with it is sufficiently long.

Alpha-active nuclei emit, as a rule, not one but several monochromatic groups of \(\alpha\)-particles. The most intense group is usually associated with the transition from the ground state of the parent nucleus to the ground state of the daughter nucleus. In addition, there may be transitions from the ground state of the parent nucleus to excited levels of the daughter nucleus (the fine structure of the \(\alpha\)-spectrum) and transitions from excited levels of the parent nucleus (long-range \(\alpha\)-particles).

The ratio between the intensity of long-range alpha particles and of the $\alpha$-particles of the main group depends on the ratio between the probabilities of $\alpha$-decay and $\gamma$-emission. Since the lifetimes with respect to $\gamma$-emission are, generally speaking, incomparably shorter than the times associated with $\alpha$-emission, the parent nuclei, as a rule, have already managed to de-excite before $\alpha$-decay, and the decay proceeds from the ground state.

The ratio between the lifetimes for $\alpha$- and for $\gamma$-decay depends sharply on the excitation energy of the nucleus, since the probability of $\alpha$-decay increases with energy incomparably faster than the probability of $\gamma$-emission. At strong excitations, $\alpha$-decay from excited levels may become measurable and is observed in a number of cases (ThC′, RaC′). It is interesting to note that both nuclei presently known to emit long-range particles are Po isotopes with atomic number 84. The daughter nuclei are lead isotopes, i.e. they have a closed proton shell. Po$^{212}$ (ThC′) has, in addition, 128 neutrons, so that the daughter nucleus Pb$^{208}$ also has a closed neutron shell. This leads to very large energies of the $\alpha$-particles (8.776 MeV and 7.680 MeV) and to short lifetimes ($3 \cdot 10^{-7}$ and $1.6 \cdot 10^{-4}$ sec) for $\alpha$-decay. The long-range $\alpha$-particles have still higher energy (10.536; 10.417 and 9.489 MeV—for ThC′) and correspondingly still shorter $\alpha$-decay times. However, even in the two cases indicated, the intensity of the long-range $\alpha$-rays does not exceed 0.02% of the intensity of the main lines.

As for transitions from the ground state of the parent nucleus to excited levels of the daughter nucleus, such transitions are the rule rather than the exception. The richness of the $\alpha$-particle spectrum is determined by the density of levels of the daughter nucleus. Since the intensity of lines falls rapidly with energy, levels with low excitation energy are important. Such levels are found in strongly elongated nuclei—they are often associated with the rotation of the nucleus. Near closed shells the nuclei are almost spherical, and fine structure is absent. With increasing distance from $Z = 82$, $N = 126$, the nuclei acquire a noticeable elongation (the ratio of semiaxes is of order 1.3), and the spectrum contains many (up to 10) lines.

In considering experimental data, one should distinguish between the energy of the $\alpha$-particles and the energy of $\alpha$-decay. The energy of $\alpha$-decay is the sum of the kinetic energy of the $\alpha$-particle and that of the daughter nucleus. The large difference in the masses of the $\alpha$-particle and the nucleus leads to the fact that only a small part of the energy falls to the share of the nucleus. As is easy to see,

$$ E_{\text{decay}} = E_{\alpha}\frac{A}{A - 4}, \tag{1.1} $$

where $E_{\text{decay}}$ is the decay energy, $E_{\alpha}$ is the energy of the $\alpha$-particle, and $A$ is the atomic weight of the parent nucleus.

The energy of α-particles is usually known with such high accuracy that the correction

\[ \frac{A}{A-4} \]

turns out to be significant. This correction must also be introduced when calculating the spacing between the levels of the daughter nucleus from the difference of α-particle energies.

Table 1

Energies of α-particles of reference substances

No.	Substance	Energy of α-particles, MeV
1	Bi$^{212}$ (ThC)	$6.0861 \pm 0.0024$
2	Bi$^{214}$ (RaC)	$5.5051 \pm 0.0022$
3	Po$^{210}$	$5.3006 \pm 0.0026$
4	Po$^{214}$ (RaC′)	$7.6804 \pm 0.0009$
5	Po$^{216}$ (ThA)	$6.7746 \pm 0.0013$
6	Po$^{218}$ (RaA)	$5.9982 \pm 0.0008$
7	Rn$^{220}$ (Tn)	$6.2823 \pm 0.0013$
8	Ra$^{224}$ (ThX)	$5.6814 \pm 0.0011$

The comparatively long lifetime of α-active nuclei leads to a small intrinsic line width, which even for long-range α-particles does not exceed fractions of a millielectronvolt. At the full energy of α-particles, measured in megaelectronvolts, such widths are completely inaccessible to observation, so that the lines appear quite monochromatic.

In precision measurements of α-particle energy, the method of magnetic analysis is used most often. Absolute measurements of energy are made rarely. Usually the energy of the α-particles under study is compared with the energy of α-particles emitted by reference substances. We shall give the latest data for the energy of α-particles of these substances B$^{54a}$.

2. EXPERIMENTAL TECHNIQUE, α—γ CORRELATIONS

At present, measurement of the energy of α-particles is carried out almost exclusively by two principal methods: the method of magnetic analysis and the method of the pulsed ionization chamber.

The simplest is analysis by means of an ionization chamber (see, for example, G$^{51a}$).

To measure the energy of α-particles, one uses the pulse produced by the arrival at the chamber electrode of electrons knocked out of atoms as a result of collisions with α-particles. The more prolonged ion pulse—the result of the arrival of ions at the second electrode—is usually not used, since the ions move too slowly. The magnitude of the electron pulse depends not only on the number of electrons, but also on the place of their formation.

Indeed, suppose that one ion pair has been formed in the chamber. When the electron and the ion reach the chamber electrodes, a quantity of electricity equal to one electron charge will have passed through it. Part of this quantity of electricity will pass during the motion of the electron, and the remaining part during the motion of the ion. The ratio of the parts is determined, roughly speaking, by the paths traversed by them.

electron and ion (more precisely, by the potential difference between the point of formation of the ion pair and the corresponding electrodes of the chamber). Under these conditions the magnitude of the electron pulse is not proportional to the energy of the $\alpha$-particle and cannot serve for its measurement.

In order to achieve proportionality between the magnitude of the electron pulse and the energy of the $\alpha$-particle, a grid is introduced into the chamber. The formation of ions takes place in the space between the source and the grid, which is not used for measurements. Electrons accelerated in the source—grid field pass through the grid into the space grid—collecting electrode, which already serves for current measurement. Since all electrons traverse this space from beginning to end, the current in it is strictly proportional to the number of electrons and, consequently, to the energy of the $\alpha$-particle that produced them.

The chambers are connected to pulse-height analyzers. Ionization chambers make it possible to measure the energies of $\alpha$-particles with a resolution of the order of 50 kev. Since the distances between lines in $\alpha$-spectra often prove to be less than 50 kev, the chambers are of little use for analyzing $\alpha$-spectra. They play, however, a very important role in measurements with small amounts of substance, since almost half of the $\alpha$-particles emitted by the source are used for the measurements.

Recent improvements E55 have made it possible to bring the resolving power of chambers to 25 kev, which is a substantial step forward.

For several decades the principal method for studying $\alpha$-spectra has been magnetic spectrometers. They may be of different types. At present three large magnetic spectrometers are in operation. The semicircular spectrometer of Rosenblum’s group (see, for example, C51b) uses a large permanent magnet at Bellevue (France). In it the usual Danysz focusing of particles through an angle of $180^\circ$ takes place. The source and collector are in the magnetic field. The position of focusing of the $\alpha$-particles depends on their energy. The spectrometer of the group of Azaro and Perlman A52e (America) contains a sector electromagnet and in geometry resembles a mass spectrometer. The source and collector are outside the magnetic field, which substantially facilitates handling of the spectrometer but worsens the accuracy of measuring the energy of $\alpha$-particles. The Soviet $\alpha$-spectrometer G55, used by the authors of the present article, has double focusing in direction (focusing angle $254^\circ$), as proposed by Svartholm and Siegbahn S47a. The source and collector are placed inside a large electromagnet (mean diameter—1 m). The last two magnetic spectrometers have a resolution of 7.5 kev at a luminosity of the order of $10^{-4}$.

The sources of $\alpha$-spectrometers are thin films of $\alpha$-active substances, obtained recently most often by evaporating chlorides or oxides in vacuum. Electrolysis is used more rarely. Other methods are practically unsuitable because of the nonuniformity

and the great thickness of the resulting layers, which noticeably broaden the peaks of the $\alpha$-spectra.

The collectors of $\alpha$-particles are most often photographic plates, which have the advantage over counters that they make it possible to study at once an entire region of the $\alpha$-spectrum. The plates are then processed under a microscope by counting the number of tracks.

The classical method of determining the energy of $\alpha$-particles from their range in air is now almost never used.

An important method for studying $\alpha$-rays is the method of processing photographic plates impregnated with an $\alpha$-active substance. Plates sensitive not only to $\alpha$-particles but also to slow electrons are most often chosen. The energy of the $\alpha$-particles is estimated from the length of the tracks. These measurements are not very accurate, but in some cases they yield valuable information about the nature of $\alpha$-emitters. For example, the presence of internal-conversion electrons emerging from the same point as the $\alpha$-particles serves as evidence that not all the $\alpha$-radiation goes to the ground level of the daughter nucleus.

Recently the method of $\alpha$–$\gamma$ coincidences has undergone considerable development. The energy of the $\alpha$-particles is measured (with an accuracy of about 5%) by a scintillation counter. Another counter registers the $\gamma$-rays emitted simultaneously with the $\alpha$-particles. The method described makes it possible to observe weak lines against the background of a substantially stronger line corresponding to transition to the ground state. In this way, for example, $\alpha$-particles of $\mathrm{Po}^{210}$ going to the first excited level of $\mathrm{Pb}^{206}$ (800 kev) were detected. The intensity of this line is only $10^{-5}$ of the intensity of the main group of $\alpha$-particles.

A new and interesting method for studying $\alpha$-rays is the study of angular correlations between $\alpha$-particles and $\gamma$-rays. From the angular correlation it is possible to determine the angular momentum carried away by the $\alpha$-particle in the case where, after the decay, the daughter nucleus remains in an excited state. Unfortunately, a number of serious difficulties are encountered here. The lifetimes of the excited states prove to be long enough for a noticeable fraction of the nuclei to have time to reorient under the action of the magnetic forces of the electron shell or the electric forces of the crystal lattice. As a result, the angular correlations turn out to be less sharply expressed than is predicted by theory. The low efficiency of the coincidence method makes it necessary to use large solid angles, which substantially smears out the pattern of the angular distribution. Despite all the shortcomings, the method has made it possible in several cases to show that the first excited levels of even-even nuclei have spin 2.

Recently, the discrepancy between the theoretical and observed value of the angular correlation $(\alpha-\gamma)$ has begun to be used to determine the magnitude of the quadrupole moment of nuclei. Since the other methods are still also very imperfect, the results obtained in this way are of some interest.

3. CLASSICAL THEORY OF α-DECAY

The theory of α-decay is distinguished by an unusual fate. The first successes in this field were achieved by Gamow \(^{G28a}\) and by Gurney and Condon \(^{G28b}\) at the dawn of quantum mechanics. The theory thus constructed in 1928 has been repeatedly revised and improved. At the same time, however, no very serious progress has been achieved, and although nonrigorous arguments have long since been replaced by scrupulous calculations, the quantitative agreement of the theory with experiment has, in recent times, perhaps even somewhat worsened. The new ideas advanced by A. Bohr (the theory of the nonspherical nucleus) \(^{B53b,B54f}\) have made it possible to advance substantially in the theory of the fine structure of α-spectra, but so far have given little for the general theory of α-decay.

Already the first works \(^{G28a,G28b}\) explained the dependence of the lifetimes of α-active nuclei on the energy of α-decay and on the parameters of the nuclei by the tunneling effect—the passage of α-particles under the Coulomb barrier. Theoretical calculations up to the present have been carried out only for a spherically symmetric barrier, since before the work of A. Bohr due attention was not paid to the shape of heavy nuclei. A noticeable nonsphericity of α-active nuclei, characterized by a ratio of the semiaxes of the ellipsoid of about 1.3, must strongly affect the period of α-decay, since even small changes in the shape of the barrier are essential for the tunneling effect. Undoubtedly, α-decay is substantially facilitated in the region located near the major semiaxis of the ellipsoid. The importance of taking account of the nuclear shape is becoming ever more obvious, so that the appearance of theoretical work in this field should be expected.

The second difficulty of the theory—the calculation of the so-called pre-exponential factor—also still remains far from resolution. The theory usually considers the behavior of a system consisting of two bodies—of a ready-made α-particle and a daughter nucleus—since it is impossible to give an exact answer to the question of how an α-particle is formed in the nucleus and what times are associated with its formation. However, as will be seen from what follows, this question may be, if not solved, then at least circumvented.

The indicated difficulties lead, in particular, to the fact that at present the theory can say nothing about the dependence of the half-life on the angular momentum carried away by the α-particle, although a considerable number of works have been devoted to this question \(^{O49b,Z33,P47,H53b}\), etc. Some progress in this respect was achieved by L. D. Landau (see § 9).

As is known (see, for example, \(^{L49}\), equality (50.9)), the penetrability of a barrier is described by the formula

\[ D=e^{-S(E,l)}, \]

\[ S(E,l)=\frac{2}{\hbar}\left|\int_a^b p\,dx\right|, \tag{3.1} \]

where \(D\) is the permeability of the barrier, \(p=\sqrt{2m(E-U)}\) is the momentum of the particle, and \(a,b\) are the “turning points,” i.e., the points at which the potential energy of the particle is equal to the total energy and at which a particle obeying classical mechanics would have changed the sign of its velocity. The decay probability is obtained by multiplying the barrier penetrability by a factor determining the probability of decay in the absence of the barrier. Thus, even the crudest considerations lead, for the probability of \(\alpha\)-decay, to the formula

Fig. 1. Potential for an \(\alpha\)-particle.

\[ \lambda = f e^{-S(E,l)} \tag{3.2} \]

Refinement of formula (3.2) requires additional considerations concerning the forces acting on the \(\alpha\)-particle and its state inside the nucleus.

It is usually assumed that the potential energy of the \(\alpha\)-particle is equal to the Coulomb energy outside the nucleus and to a constant inside it. For a spherical nucleus, therefore (Fig. 1):

\[ \left. \begin{aligned} U &= U_0; && r<R;\\ U &= \frac{2Ze^2}{r}+\frac{\hbar^2}{2M}\frac{l(l+1)}{r^2}; && r>R \end{aligned} \right\} \tag{3.3} \]

(\(Z\) is the charge of the daughter nucleus). The term \(\frac{\hbar^2}{2M}\frac{l(l+1)}{r^2}\) describes the “centrifugal barrier” that appears in the Schrödinger equation for the radial wave function of a particle with angular momentum \(l\) (see, for example, equation (32.12) of [49]). This term is a small correction to the first term, which characterizes the Coulomb energy. If, as is usually done, one introduces the quantity \(\sigma\), which determines the relative magnitude of the centrifugal energy, then for heavy nuclei

\[ \sigma=\frac{\hbar^2}{2M}\frac{l(l+1)}{R^2}:\frac{2Ze^2}{R}\simeq 0.002l(l+1). \tag{3.4} \]

In (3.2) one can single out a factor depending on the centrifugal energy. It turns out to be of order 1 (0.7 for \(l=1\); 0.37 for \(l=2\)). We do not write it out here, because equation (3.2) does not take into account circumstances that affect much more substantially the dependence of the \(\alpha\)-decay probability on the difference between the angular momenta of the parent and daughter nuclei. In studying the dependence of the \(\alpha\)-decay probability on the angular momentum carried off by the \(\alpha\)-particle, one cannot ignore the nonsphericity of nuclei, which proves to be substantially more important than the centrifugal barrier. In comparing theory with experiment one should also keep in mind that specifying the spin of the parent and daughter nuclei does not, in general, determine the angular momentum carried away by the \(\alpha\)-particle. This angular momentum may take

$2l+1$ values ($I$ is the smaller of the spins of the parent and daughter nuclei) and is determined uniquely only in the case when the spin of the parent or daughter nucleus is zero. The parity of the wave function of the $\alpha$-particle is specified by the parity of the state of the parent and daughter nucleus. At the same time, for the $\alpha$-particle the parity of the wave function is determined by the parity of the angular momentum $l$ carried away by it. Therefore, of the $2I+1$ values of $l$, either only even $l$’s or only odd ones can occur. Thus, the observed $\alpha$-decay constant determines the probability of a process proceeding through several channels.

Let us dwell on the physical meaning of the radius $R$. The radius $R$ should not be identified, as is often done, with the nuclear radius. It is an effective “channel radius,” which, roughly speaking, includes the nuclear radius, the radius of the $\alpha$-particle, and the range of the nuclear forces that hold the $\alpha$-particle in the nucleus. The separation of $R$ into its component parts cannot be carried out in an unambiguous manner. The matter is still more complicated for nonspherical nuclei, when it is not known which particular dimension should be called the “radius.” An increase of transparency in the region of the “tip” of an elongated nucleus in any case leads to an increase of the effective radius, calculated from $\alpha$-decay, in comparison with the mean one. In particular, there is nothing strange in the fact that the radii $R$ do not coincide with the radii determined from experiments on the scattering of fast electrons or from the study of $\mu$-mesoatoms. Following generally accepted terminology, we shall nevertheless use the name “nuclear radius” for $R$, in the hope that the remarks made above will protect the reader from misunderstandings.

Let us now turn to the pre-exponential factor in (3.2), written for the time being in the form $f$. This factor must take into account the probability of formation of the $\alpha$-particle and the speed of its motion in the nucleus. The theory does not make it possible to calculate it in any satisfactory way; however, there are a number of methods for estimating its magnitude.

Most authors do not even attempt to calculate the probability of formation of the $\alpha$-particle. Gamow$^{G49b}$, Segrè$^{Z33,H53b}$, and Preston$^{P47}$ confine themselves to the simple solution of the problem of the motion of a ready-made $\alpha$-particle in the field of the daughter nucleus. Gamow solves the problem by means of an approximate quasiclassical treatment, while Segrè and Preston try to solve the Schrödinger equation for the radial part of the wave function. In doing so, the external (Coulomb field) and internal (constant potential) solutions of the Schrödinger equation are matched.

The $\alpha$-decay constant according to Gamow has the form

\[ \lambda=\frac{8\pi\hbar}{mR^2}\,e^{-\frac{4e^2}{\hbar}\frac{Z}{V}(2\alpha_0-\sin 2\alpha_0)}, \tag{3.5} \]

\[ \cos^2\alpha_0=\frac{RE}{2Ze^2}. \tag{3.6} \]

In formula (3.5), \(m\) is the mass of the \(\alpha\)-particle (strictly speaking, the reduced mass of the system \(\alpha\)-particle + daughter nucleus), \(R\) is the radius of the nucleus, more precisely the channel radius, \(Z\) is the charge of the daughter nucleus, \(V\) is the velocity of the \(\alpha\)-particle after leaving the nucleus (more precisely, the velocity of the \(\alpha\)-particle in the system of the daughter nucleus), and \(E\) is the energy of \(\alpha\)-decay.

Let us compare formulas (3.2) and (3.5). The exponent in (3.5) is obtained by direct integration in (3.2) with the aid of (3.3) (without taking account of the centrifugal term).

The pre-exponential factor was obtained from the consideration that the wave function of the \(\alpha\)-particle first vanishes at the boundary of the nucleus (radius \(R\)). (Alpha particles obey Bose statistics, so that there is no Pauli exclusion principle for them. Therefore, in an unexcited daughter nucleus they must all be on the lowest level. Thus, the radial wave function must have no nodes.) This condition is not exact; the wave function inside the nucleus can be determined more rigorously by imposing the condition of smooth matching at the boundary.

In this way Zeksel \(^{Z33}\) and Preston \(^{P47}\) obtained:

\[ \lambda=\frac{2v}{R}\frac{\mu^{2}\operatorname{tg}\alpha_{0}}{\mu^{2}+\operatorname{tg}^{2}\alpha_{0}} e^{-\frac{4e^{2}Z}{\hbar v}(2\alpha_{0}-\sin 2\alpha_{0})}, \tag{3.7} \]

\[ \mu=-\operatorname{tg}\alpha_{0}\operatorname{tg}(\mu kR), \tag{3.8} \]

\[ k=\frac{1}{\hbar}\sqrt{2mE}. \tag{3.9} \]

Here \(\alpha_{0}\) is defined by equality (3.6). The quantity \(\mu\) entering (3.7) is determined from equation (3.8) and in calculations may be regarded as a simple parameter. The physical meaning of \(\mu\) can be understood from the formula relating it to the potential energy \(U_{0}\) of the \(\alpha\)-particle inside the nucleus:

\[ \mu=\sqrt{1-\frac{U_{0}}{E}}. \tag{3.10} \]

Formula (3.10) makes it possible to find the value of \(U_{0}\).

Formula (3.7) is, of course, more rigorous than (3.5); however, calculations by it require solving the transcendental equation (3.8). Since in deriving (3.5) and (3.7), as already stated above, very crude assumptions were made about the shape of the nucleus (the nucleus was regarded as spherical) and of the potential barrier, and the question of the formation of the \(\alpha\)-particle was not considered, the advantage of one formula over the other is not of essential importance, and it is usually preferred to use the simpler formula (3.5).

Landau \(^{L37}\), Blatt and Weisskopf \(^{B54}\), and Devaney \(^{D53c}\) estimate the pre-exponential factor from the density of nuclear levels. In the notation adopted above, the formula of Blatt and Weisskopf

has the form

\[ \lambda=\frac{2D}{\pi\hbar}\, \frac{\mu^{2}\tan\alpha_{0}}{\mu^{2}+\tan^{2}\alpha_{0}}\, e^{-\frac{4e^{2}Z}{\hbar v}(2\alpha_{0}-\sin 2\alpha_{0})}. \tag{3.11} \]

This equation replaces (3.7) in the system (3.7), (3.8), (3.9). \(D\) is the mean spacing between levels with the same parity and angular momentum, while \(\lambda\) is the probability of emission of \(\alpha\)-particles with a given energy and momentum. Although formula (3.11) appears to be substantially more satisfactory than the preceding ones, it still cannot be compared directly with experiment. The lower levels that manifest themselves in the study of the fine structure of \(\alpha\)-spectra, as is now well known, are for the most part rotational in nature and differ in angular momentum both from the ground level and from one another. The spin and parity of the other levels most often have not been established. In any case, there are no grounds for thinking that they coincide with the spin and parity of the ground state. Thus, for the time being there are no experimental indications of how \(D\) changes from nucleus to nucleus. In any case, it is plainly inadmissible to suppose that \(D\) can be calculated from the spacing between the ground state and the first excited level, as Blatt and Weisskopf \(^{\mathrm{B54}}\) and Deveney \(^{\mathrm{D53c}}\) attempt to do.

From what has been said, the reason is clear why these authors obtain values of nuclear radii that vary strongly from nucleus to nucleus.

As for the numerical values of the pre-exponential factor, in formula (3.5) it is, in order of magnitude, equal to \(5\cdot10^{21}\), in formula (3.7) to \(2\cdot10^{21}\), and in formula (3.11) to \(2\cdot10^{20}\). We also note that Bethe calculated the pre-exponential factor on the assumption that the \(\alpha\)-width of the level (without taking the barrier factor into account) is equal to the neutron width, which is roughly incorrect.

In calculations it is usually assumed that only the Coulomb field of the nucleus acts on the \(\alpha\)-particle, and the field of the atomic electrons is neglected. This is more or less correct, since the most essential part of the barrier is located in the immediate vicinity of the nucleus. The influence of the atomic electrons was taken into account in several works \(^{\mathrm{A51a,\ B52b,\ D53c}}\). It leads to the result that, instead of the emission energy of the \(\alpha\)-particle in formulas (3.5), (3.7), and (3.10), there should stand the energy \(E_t\), determined by formula (3.12) \(^{\mathrm{D53c,\ B52d}}\),

\[ E_t=E+73\,(Z_{\mathrm{dau}})^{4/3}\,\mathrm{eV}+65\,Z^{1/3}\,\mathrm{eV}. \tag{3.12} \]

The second term in (3.12) takes into account the lowering of the barrier, and the third the rearrangement of the shell occurring in the transition from the parent nucleus to the daughter nucleus.

These corrections are small and, in the present state of the theory, inessential.

As is clear from the preceding, the theory is not sufficiently accurate to allow one reasonably to predict half-life periods. However, useful results concerning the radii of heavy nuclei can be obtained from it. Taking the quantity \(\dfrac{RE}{2Ze^2}\) (in 3.6) to be small (which is suitable only for very rough calculations), instead of (3.5) one may obtain the simple formula

\[ \ln \lambda = \ln \frac{8\pi\hbar}{mR^2} - \frac{4\pi Ze^2}{\hbar v} + \frac{8e\sqrt{m}}{\hbar}\sqrt{ZR}. \tag{3.13} \]

In the region of \(\alpha\)-active nuclei the quantities \(R\) and \(Z\) vary little.

Then

\[ \lg \lambda = C - \frac{D}{\sqrt{E}}. \tag{3.14} \]

This approximate relation, well known from experiment, is called the Geiger—Nuttall law.

Formulas (3.5), (3.7), (3.11), (3.13) are very sensitive to the nuclear radius \(R\), and, conversely, the radius \(R\), calculated from these formulas, depends little on errors in the other quantities. For example, an error in the pre-exponential factor by a factor of 10 changes the nuclear radius by only 4%. Therefore theoretical formulas are used chiefly for calculating nuclear radii. When, however, it is necessary to find approximate values of the decay constant, it is preferable to use the empirical relations, which will be discussed below.

Let us examine formula (3.13). First of all, note that \(R\) and \(Z\) vary little from nucleus to nucleus.

The factor \(\dfrac{4\pi Ze^2}{\hbar v}\) for \(\alpha\)-active nuclei is, in order of magnitude, equal to 160. A change in the energy of the \(\alpha\)-particles by a factor of two will change its value by a factor of \(\sqrt{2}\), i.e. approximately by 50. Thus \(\lambda\) changes by a factor of \(10^{22}\). This is why the energies of \(\alpha\)-particles are confined to a narrow interval \((4 \div 9)\) MeV.

4. ALPHA DECAY TO THE GROUND STATE OF THE DAUGHTER NUCLEUS

A necessary condition for \(\alpha\)-decay is the requirement that the mass of the parent nucleus be greater than the sum of the masses of the daughter nucleus and the \(\alpha\)-particle. Only in this case is the decay energetically possible. Let us make the corresponding estimates.

In the first approximation the binding energy of nuclei may be found with the aid of the semiempirical Weizsäcker formula, which separates the binding energy into parts associated with nuclear forces, with electrostatic forces, and with the forces of surface tension, and takes into account—

contributing a general tendency toward equalization of the numbers of protons and neutrons.

Following Blatt and Weisskopf \(^{54}\), we write the binding energy in the form

\[ E=-U_v A+4U_\tau \frac{T_\tau^2}{A}+4U_c Z(Z-1)A^{-1/3}+U_s A^{2/3}, \tag{4.1} \]

where \(E\) is the binding energy of the nucleus, \(A\) is the atomic weight, \(Z\) is the nuclear charge, \(T_\tau=\frac{1}{2}(N-Z)\) is the neutron excess, \(U_v=14\) MeV, \(U_c=0.146\) MeV, \(U_\tau=18.1\) MeV, \(U_s=13.1\) MeV. Formula (4.1) makes no claim to accuracy—it does not even take into account the presence of shells.

From formula (4.1) it is easy to find that the energy of \(\alpha\)-decay is equal to

\[ \begin{aligned} E_\alpha&=E(Z,A)-E(Z-2,A-4)+B_\alpha=\\ &=-28-72.5\,\frac{(N-Z)^2}{A^2}+2.34\,\frac{Z}{A^{1/3}} -0.78\,\frac{Z^2}{A^{4/3}}+35\,\frac{1}{A^{1/3}}. \end{aligned} \tag{4.2} \]

In formula (4.2) the \(\alpha\)-decay energy is expressed in MeV, \(B_\alpha=28\) MeV is the binding energy of the \(\alpha\)-particle.

Formula (4.2) makes it possible to estimate correctly the \(\alpha\)-decay energies in the region \(A\sim225\). For \(235<A<240\) it gives underestimated values for the energy (by approximately \(1\) MeV); for \(A<225\), overestimated values. For \(A<225\) the deviations gradually increase with decreasing atomic number, i.e., as one approaches the shells \(Z=82\), \(N=126\), reaching \(4.5\) MeV in \(^{50b}\). Thus, this semiempirical formula cannot be used for quantitative estimates. It does, however, make it possible to explain correctly the main regularities of the \(\alpha\)-decay energy.

Substitution of numbers into formula (4.2) shows that, beginning with \(A=190\), nuclei are unstable with respect to \(\alpha\)-decay. As was noted above (§ 1), this instability still remains insignificant for a long time because of the enormous lifetimes, so that in fact only nuclei heavier than lead exhibit measurable \(\alpha\)-activity.

Let us consider the dependence of the energy of \(\alpha\)-particles on \(N\), i.e., the quantity \(\left(\dfrac{\partial E_\alpha}{\partial A}\right)_Z\):

\[ \left(\frac{\partial E_\alpha}{\partial A}\right)_Z =-11.7\,\frac{1}{A^{4/3}} -145\,\frac{A-2Z}{A^2}\left[1-\frac{A-2Z}{A}\right] -0.78\,\frac{Z}{A^{4/3}}\left(1-\frac{4}{3}\frac{Z}{A}\right). \tag{4.3} \]

This quantity is less than zero for all \(\alpha\)-emitters. Thus, for each element the energy of \(\alpha\)-particles must increase

with decreasing atomic weight, which is also observed experimentally. The circumstance that, in elements that do not possess measurable $\alpha$-activity (mercury, gold), $\alpha$-activity nevertheless appears in isotopes with a considerable deficiency of neutrons is also connected with the rule noted.

The regularity that has been established is well confirmed experimentally. In Fig. 2 the curves of the dependence of the $\alpha$-decay energy on atomic weight for different elements are shown. The solid lines are drawn through all isotopes of the given element, and the dashed lines through even-even isotopes.

Let us first of all pay attention to the middle part of the graph, between mass numbers 215 and 240. The curves for each element reveal a regular decrease with increasing mass number (number of neutrons), which agrees well with the theory. At the same time the curves drawn only through even-even nuclei prove to be substantially smoother, which is also natural, since in even-even nuclei, whose ground states are characterized by one and the same spin (equal to zero) and the same parity (positive), the individual peculiarities of the nuclei have a smaller effect.

If for the moment one disregards the break in the curves in the region $A = 211$, it is easy to detect the same regular course in the region $A < 209$. Here again there is a decrease of the $\alpha$-decay energy with increasing mass number at constant $Z$.

The jump in the region $A = 211$ is explained by the filling of the shell of 126 neutrons. The formation of a closed shell is energetically advantageous. The appearance of extra (above 126) neutrons causes a sharp increase of the potential energy. A comparatively large potential energy is also possessed by a shell that is not quite closed. This leads to the fact that the decay of nuclei having 128 neutrons is the most energetically favorable. Indeed, the greatest decay energy is possessed by $\mathrm{Bi}^{211}$, $\mathrm{Po}^{212}$ and $\mathrm{At}^{213}$ with 128 neutrons. $\mathrm{Em}^{214}$ and $\mathrm{Fr}^{215}$, which also have 128 neutrons and should possess large decay energies, are unknown.

Let us note that the sharp decrease of the $\alpha$-decay energy on passing through the shell leads to the fact that $\mathrm{Bi}^{209}$ and $\mathrm{Bi}^{208}$, having 126 and 125 neutrons, are practically stable, while the unstable $\mathrm{Bi}^{207}$—$\mathrm{Bi}^{204}$ do not exhibit noticeable $\alpha$-activity.

The curves of Fig. 2 make it possible to predict the decay energy of unknown isotopes. Naturally, the predictions prove to be especially reliable for even-even isotopes.

In a completely analogous way behave the lighter $\alpha$-emitters situated in the vicinity of the shell of 82 neutrons. Here one should expect the greatest $\alpha$-decay energy for nuclei with 84 neutrons. The lightest of the known $\alpha$-active isotopes of gadolinium, ${}_{64}\mathrm{Gd}^{148}$ (84 neutrons), indeed has the greatest

Fig. 2. Dependence of the energy of \(\alpha\)-decay on atomic weight (even–even isotopes are connected by dashed lines).

energy of $\alpha$-decay—$3.27$ MeV. The $\alpha$-decay energy of $\mathrm{Gd}^{149}$ and $\mathrm{Gd}^{150}$ is, respectively, $3.1$ and $2.8$ MeV.

The alpha-active nucleus ${}_{62}\mathrm{Sm}^{147}$ contains 85 neutrons (decay energy $2.24$ MeV) and has a lifetime of $1.5\cdot 10^{11}$ years. The greatest $\alpha$-decay energy should be possessed by $\mathrm{Sm}^{146}$, with 84 neutrons. It is interesting to note that this isotope still remains unknown, although samarium isotopes with atomic weights 143, 144, 145, 147, 148, 149, 150, 151, etc., are known. It is possible that the large $\alpha$-decay energy and, correspondingly, the short lifetime prevented the discovery of this isotope.

The only known alpha-active europium isotope, ${}_{63}\mathrm{Eu}^{147}$, has 84 neutrons.

Let us now turn to the dependence of the $\alpha$-decay energy on the atomic number

\[ \left(\frac{\partial E_\alpha}{\partial Z}\right)_N = 145\,\frac{N-Z}{A^2}\left(1+\frac{N-Z}{A}\right) +\frac{2.34}{A^{1/3}}\left(1-\frac{1}{3}\frac{Z}{A}\right) -\]
\[ {}-11.7\,\frac{1}{A^{4/3}} -1.56\,\frac{Z}{A^{4/3}}\left(1-\frac{2}{3}\frac{Z}{A}\right). \tag{4.4} \]

As is easy to see, this quantity is always positive, so that the $\alpha$-decay energy must increase with increasing nuclear charge. The dependence of the $\alpha$-decay energy on the nuclear charge is shown in Fig. 3.

Fig. 3. Dependence of the $\alpha$-decay energy on the atomic number (points with the same number of neutrons are connected).

It is evident from the figure that the decay energy does indeed increase with increasing atomic number. This increase is accompanied by substantially larger fluctuations than occur for the curves of the dependence of the $\alpha$-decay energy on the number of neutrons (see Fig. 2).

Thus, the individual properties of nuclei manifest themselves more sharply with a change in the number of protons than with a change in the number of neutrons. This is possibly connected with the fact that the number of neutrons in heavy nuclei substantially exceeds the number of protons.

5. ALPHA DECAY TO THE GROUND STATE OF THE DAUGHTER NUCLEUS

(CONTINUATION)

In this section we consider transitions from the ground state of the parent nucleus to the ground state of the daughter nucleus. The fine structure of \(\alpha\)-rays will be discussed below.

As was noted in § 3, \(\alpha\)-decay makes it possible to calculate nuclear radii. It was also pointed out there that the quantities obtained from these formulas include both the radius of the nucleus and the radius of the \(\alpha\)-particle, as well as the range of action of the nuclear forces acting between the \(\alpha\)-particle and the nucleus. Thus the nuclear radii calculated from \(\alpha\)-decay are certainly too large.

The crudeness of the theory leads to the fact that the chief value is represented not by the absolute values of the radii, but by their change from nucleus to nucleus. In this case the choice of formula, whether somewhat more rigorous or somewhat less rigorous, does not play a major role. We shall use formula (3.5), since it leads to comparatively simple calculations.

Since the magnitude of the pre-exponential factor has a weak influence on the other parameters, we shall take, for \(R\) in this factor, the value \(9 \cdot 10^{-13}\ \mathrm{cm}\) and shall use the formula

\[ B + \ln T = 0.548 \frac{A - 4}{A}\frac{Z - 2}{\sqrt{E_\alpha}}(2\alpha_0 - \sin 2\alpha_0). \tag{5.1} \]

\[ R = 2.88\,\frac{Z - 2}{E_\alpha}\,\frac{A - 4}{A}\cos^2\alpha_0. \tag{5.2} \]

In formulas (5.1) and (5.2) the following notation is used: \(Z\) is the charge of the active nucleus, \(A\) is its atomic weight, \(E_\alpha\) is the energy of the \(\alpha\)-particles, expressed in MeV, \(T\) is the half-life, and \(R\) is the nuclear radius in \(10^{-13}\ \mathrm{cm}\). The constant \(B\) has the following values:

\[ B = \begin{cases} 21.85, & \text{if } T \text{ is expressed in seconds,}\\ 23.63, & \text{if } T \text{ is expressed in minutes,}\\ 25.40, & \text{if } T \text{ is expressed in hours,}\\ 26.78, & \text{if } T \text{ is expressed in days,}\\ 29.35, & \text{if } T \text{ is expressed in years.} \end{cases} \]

In the calculations, the partial half-life corresponding to the emission of \(\alpha\)-particles with energy \(E_\alpha\) was substituted into formula (5.1).

In the presence of fine structure, the calculation was usually carried out for the most intense group of $\alpha$-particles. The results of the calculation are presented in the form of the graph in Fig. 4, in which nuclei with the same number $Z$ are connected.

Fig. 4. Radii of $\alpha$-active nuclei (isotopes of the same element are connected).

The data of Fig. 4 lead to two principal conclusions. Despite the enormous difference in the decay constant, reaching $10^{25}$ times, the radii calculated from formulas (5.1), (5.2) lie within the limits $7 \div 10 \cdot 10^{-13}\ \mathrm{cm}$, which indicates the reasonableness of the theory. At the same time, the nonuniform variation of the radius from nucleus to nucleus indicates its crudeness.

Fig. 5. Radii of even-even $\alpha$-active nuclei.

If one restricts oneself only to even-even nuclei, the nuclear radii change substantially more smoothly (Fig. 5). Still more reasonable results$^{53e}$ are obtained if, in the graph for even-even nuclei,

combine nuclei having the same isotopic number \(N-Z\) (Fig. 6). Let us note that the masses of even-even nuclei possessing the same isotopic number, depending on whether \(N-Z\) is divisible by 4 or not, are described by the formula \(A=4K\) or \(A=4K+2\).

The radii of nuclei are usually described by the formula

\[ R=r_0 A^{1/3}, \]

which indicates the constancy of the density of nuclear matter. The radii of \(\alpha\)-active nuclei lead to the value \(r_0=1.45\cdot 10^{-13}\) cm. This value is larger than the radii of nuclei obtained from electron scattering or from data on \(\mu\)-mesoatoms. There one obtains values of the order \(r_0=1.2\cdot 10^{-13}\) cm. As has already been mentioned, this difference is quite natural.

Fig. 6. Radii of even-even \(\alpha\)-active nuclei (points belonging to isotopes with equal \(N-Z\) are connected).

Let us note, finally, that the nuclear radii calculated from the theory of \(\alpha\)-decay do not obey the law \(R=r_0 A^{1/3}\), even if one disregards fluctuations of \(R\) from nucleus to nucleus, which are connected with the nonsphericity of nuclei. Therefore one should not, as is often done, study the variation of \(r_0\) calculated by the formula

\[ r_0=\frac{R}{A^{1/3}}, \]

since this has no physical meaning.

The study of nuclear radii by means of \(\alpha\)-decay is of little use for predicting the properties of new isotopes. More useful is the empirical dependence of the decay energy on the mass number \(^{50a}\), shown in Fig. 2. The comparatively smooth course of the curves makes it possible to predict the energy of \(\alpha\)-decay of unknown isotopes with an accuracy of the order of \(0.1\) MeV.

Let us note that, in addition to the sharp break in the curves in the region \(A=211\), connected with the shell of 126 neutrons, a disturbance of the monotonic course of the berkelium and californium curves is observed in the region \(A=250\). These breaks are attributed to the subshell \(N=152\) \(^{654a}\). In the indicated region, however, too few nuclei are as yet known to make definite assertions.

In some cases the energy of $\alpha$-decay can be determined not only by interpolation or extrapolation of the curves in Fig. 2, but also by direct calculation with the aid of $\alpha$—$\beta$ cycles. As an example we give the calculation of the $\alpha$-decay energy of $\mathrm{Am}^{242}$, carried out by Perlman, Ghiorso, and Seaborg $^{\mathrm{P}50\mathrm{a}}$. The course of the calculation is shown in Fig. 7 and requires no explanation.

Fig. 7. Calculation of the decay energy of $\mathrm{Am}^{242}$ with the aid of “$\alpha$—$\beta$ cycles.”

The dependence of the lifetime of even-even $\alpha$-active nuclei on the energy released in $\alpha$-decay can be represented in the form of the curves in Fig. 8, joining nuclei with equal $Z$. As is seen from Fig. 8, the points lie on a system of smooth, almost parallel curves. Let us note that not all isotopes of Po and Em are plotted on the graph. Isotopes with a number of neutrons less than 130—132 are not shown on it, since they do not lie on the curves. There is, of course, nothing surprising in this: the transition through the filled shell $N = 126$ cannot fail to affect the lifetimes of the nuclei.

Fig. 8. Dependence of lifetime on $\alpha$-decay energy for even-even nuclei.

The curves of Fig. 8 can be represented by formula (3.14)

\[ \lg \lambda = C - \frac{D}{\sqrt{E}}, \tag{5.3} \]

where $\lambda$ is the decay probability (per second), and $E$ is the energy of the $\alpha$-particles.

Alpha Decay

in MeV. The quantities \(C\) and \(D\) are not entirely constant. In B55 the values for \(C\) and \(D\) given in Table II are recommended.

Table II

\(Z\)	\(C\)	\(D\)	\(Z\)	\(C\)	\(D\)
84	50,15	128,8	92	52,55	143,1
86	50,94	132,7	94	53,35	147,4
88	51,51	136,2	96	53,97	151,3
90	51,94	139,4	98	54,40	154,7

For characterizing \(\alpha\)-transitions it is convenient to use the decay coefficient \(F\), defined by the formula

\[ F=\frac{\lambda_{\text{obs}}}{\lambda_{\text{calc}}}. \tag{5.4} \]

The determination of \(\lambda_{\text{calc}}\) is carried out with the aid of formula (5.3) or graphs of the type shown in Fig. 8. The values of \(F\) for even-even nuclei are small and usually lie between 0,8 and 1,5. For odd nuclei the decay coefficients behave irregularly, varying from nucleus to nucleus within the range from 1 to several tens of thousands. This question will be considered in more detail in § 7.

Table III gives the values of the decay coefficients for even-even nuclei with \(Z \geq 88\).

Table III

Decay coefficients \(F\) for even-even nuclei

\(\alpha\)-active nucleus	\(F\)	\(\alpha\)-active nucleus	\(F\)	\(\alpha\)-active nucleus	\(F\)	\(\alpha\)-active nucleus	\(F\)
\(\mathrm{Ra}^{220}\)	0,6	\(\mathrm{Th}^{232}\)	1,6	\(\mathrm{Pu}^{232}\)	0,1	\(\mathrm{Cm}^{242}\)	0,6
\(\mathrm{Ra}^{222}\)	1,3	\(\mathrm{U}^{228}\)	0,8	\(\mathrm{Pu}^{234}\)	0,6	\(\mathrm{Cm}^{244}\)	0,8
\(\mathrm{Ra}^{224}\)	1,0	\(\mathrm{U}^{230}\)	1,3	\(\mathrm{Pu}^{236}\)	1,0	\(\mathrm{Cf}^{244}\)	0,8
\(\mathrm{Ra}^{226}\)	0,8	\(\mathrm{U}^{232}\)	0,6	\(\mathrm{Pu}^{238}\)	0,6	\(\mathrm{Cf}^{246}\)	0,8
\(\mathrm{Th}^{224}\)	1,3	\(\mathrm{U}^{234}\)	0,8	\(\mathrm{Pu}^{240}\)	0,8	\(\mathrm{Cf}^{248}\)	1,3
\(\mathrm{Th}^{226}\)	1,3	\(\mathrm{U}^{236}\)	1,0	\(\mathrm{Pu}^{242}\)	0,8	\(\mathrm{Cf}^{250}\)	0,6
\(\mathrm{Th}^{228}\)	0,8	\(\mathrm{U}^{238}\)	1,0	\(\mathrm{Cm}^{238}\)	1,6	\(\mathrm{Cf}^{252}\)	1,3
\(\mathrm{Th}^{230}\)	0,6			\(\mathrm{Cm}^{240}\)	1,0

The decay of nuclei with \(Z<88\) shows significant deviations from formula (5.3). These deviations are observed not only for odd nuclei, but also for even nuclei.

For illustration, in Figs. 9a–9d graphs are given of the variation of \(\dfrac{1}{F}\) for \(\mathrm{Bi}(Z=83)\), \(\mathrm{Po}(Z=84)\), \(\mathrm{At}(Z=85)\), and \(\mathrm{Em}(Z=86)\), taken from work \(^{54}\).

Fig. 9a. Dependence of \(\dfrac{1}{F}=\dfrac{\lambda_{\mathrm{calc}}}{\lambda_{\mathrm{obs}}}\) on the number of neutrons for \(Z=83\) (Bi).

Fig. 9b. Dependence of \(\dfrac{1}{F}=\dfrac{\lambda_{\mathrm{calc}}}{\lambda_{\mathrm{obs}}}\) on the number of neutrons for \(Z=84\) (Po).

As is seen from the figures, the graph of \(\dfrac{1}{F}\) is represented in the form of two curves, one of which passes through nuclei with even \(A\), and the oth—

... through nuclei with odd \(A\). The curves have a clearly pronounced maximum at \(N=126\) neutrons. Comparing the figures with one another, one may also note the tendency of \(\frac{1}{F}\) toward a maximum at \(Z=82\).

This increase of \(\frac{1}{F}\) is undoubtedly connected with the fact that, as the state with filled proton and neutron shells is approached, the deformation of the nuclei decreases and, consequently, so do the effective radius of \(\alpha\)-decay and the probability of emission of the \(\alpha\)-particle.

Let us make one more remark. In existing theories of \(\alpha\)-decay, the \(\alpha\)-particle emitted from the nucleus must pass through the potential barrier corresponding to the daughter nucleus. Therefore—

Fig. 9b. Dependence of
\[ \frac{1}{F}=\frac{\lambda_{\mathrm{calc}}}{\lambda_{\mathrm{obs}}} \]
on the number of neutrons for \(Z=85\) (At).

Fig. 9c. Dependence of
\[ \frac{1}{F}=\frac{\lambda_{\mathrm{calc}}}{\lambda_{\mathrm{obs}}} \]
on the number of neutrons for \(Z=86\) (Rn).

— the value of the radius of the daughter nucleus must enter into the \(\alpha\)-decay radius. Thus, according to existing theories, the \(\alpha\)-active nuclei with \(Z=84\) and \(N=128\) should have the largest value of \(\frac{1}{F}\).

However, as we have seen, this expectation of the theory is not justified (the maxima of \(\frac{1}{F}\) are observed at \(Z=82;\ N=126\)).

This fact may be interpreted as indicating that in \(\alpha\)-decay the most important role is played by the properties (including the radius) of the parent, and not the daughter, nucleus \(^{\mathrm{C}54}\).

6. NONSPHERICITY OF NUCLEI AND THE ROTATIONAL STRUCTURE OF EXCITED LEVELS

The presence of large electric quadrupole moments in nuclei most clearly illustrates the deviation of nuclear shapes from spherical form.

In Fig. 10, taken from \(^{\mathrm{B}54f}\), the quadrupole moments of nuclei with atomic weights between 140 and 200 are shown. On the same graph, the magnitude of the quadrupole moment that would have to be produced by a single nucleon with the same angular momentum as the entire nucleus is indicated by hatching. The quadrupole moments of nuclei with \(A > 200\) are known only in a few cases and therefore are not marked in the figure.

Fig. 10. Quadrupole moments of heavy nuclei.

From the graph in Fig. 10 two main conclusions follow:

1) the shape of the nucleus differs little from spherical in the region of filled shells and differs strongly from it in regions far removed from filled shells;

2) deviations of nuclear shapes from spherical form are, generally speaking, large and cannot be explained by the motion of one or several nucleons around a spherically symmetric “core” formed by the remaining nucleons. On the contrary, one must assume that many nucleons participate in producing the asymmetry. Bohr and Mottelson \(^{\mathrm{B}53b}\) explain the deformation of nuclei by the strong interaction of the outer nucleons with the remaining part of the nucleus, leading to an appreciable deformation of the whole nucleus.

Thus, in regions far from filled shells, nuclei should, in the first approximation, be regarded as ellipsoidal. At present there is no reason to consider this ellipsoid triaxial; therefore, in what follows, we shall assume that nuclei have the shape of an ellipsoid of revolution. In the region \(140 < A < 200\) almost all nuclei are elongated.

In the region \(A > 200\), as already indicated, there are almost no experimental data, so that it cannot be stated with confidence whether the nuclei are elongated or flattened. It is possible that both kinds of nuclei are represented in appreciable numbers. Since for nuclei far from \(N = 126\) an elongated nuclear shape appears more

...natural; in what follows, for simplicity of exposition, we shall assume this, although the main results do not depend on whether the nucleus is elongated or flattened.

The total angular momentum of the nucleus must be conserved. The angular momentum, generally speaking, is not directed along the axis of the nucleus, so that the nuclear axis rotates about the angular-momentum axis. Thus, in measuring quadrupole moments, the asymmetry of the nucleus is not manifested in full measure: the “intrinsic” quadrupole moments—moments in a coordinate system rigidly connected with the nucleus—are still larger than the observed ones.

The nonspherical shape of nuclei leads to the appearance of excited levels of a new type—levels corresponding to rotation of the nucleus about an axis perpendicular to the symmetry axis. Of course, this motion should not be imagined as a simple classical rotation of a solid ellipsoid. The nucleus least of all resembles a rigid body.

Despite the specific character of nuclear rotation, the formula describing the energy associated with rotation coincides exactly with the formula for the energy of a rotator. As is known, the kinetic energy of a rotating body is described by the formula \(E=\frac{M^{2}}{2J}\), where \(M\) is the total angular momentum, and \(J\) is the moment of inertia.

Fig. 11. Angular momentum of a nucleus with odd \(A\).
\(I\) is the total angular momentum.
\(K\) is the “intrinsic” angular momentum, \(R\) is the “rotational” angular momentum.

The total angular momentum of the nucleus, generally speaking, consists of two parts: the part \(R\), associated with the rotation of the nucleus (the collective motion of the nucleons), and the part \(K\), associated with the motion of individual nucleons (the “intrinsic” angular momentum). By symmetry, \(K\) is directed along the nuclear axis.

The vector model of Fig. 11 illustrates the composition of the moments \(R\) and \(K\) into the resultant moment \(I\). As follows from the figure, \(R^{2}=I^{2}-K^{2}\). In the formula for the kinetic energy of the rotator one should substitute \(R^{2}\), since it is precisely this quantity that characterizes the rotation of the nucleus. Replacing \(I^{2}\) by \(I(I+1)\), as is always done in quantum mechanics, we find:

\[ E_{\mathrm{rot}}=\frac{\hbar^{2}}{2J}\{I(I+1)-I_{0}(I_{0}+1)\}. \tag{6.1} \]

The smallest angular momentum \(I_{0}\) for a given rotational band is, naturally, equal to \(K\).

In even-even nuclei the ground state has spin equal to zero. Then formula (6.1) takes the simpler form:

\[ E_{\mathrm{rot}}=\frac{\hbar^{2}}{2J}I(I+1). \tag{6.2} \]

The moment of inertia of the nucleus entering formulas (6.1), (6.2) is connected with its elongation. For small deviations from a spherical form,

\[ J=k\left(\frac{\Delta R}{R}\right)^2, \tag{6.3} \]

where \(k\) is a proportionality coefficient, \(\Delta R\) is the difference in the lengths of the semiaxes of the nucleus, and \(R\) is the mean radius of the nucleus.

For spherical nuclei \(\Delta R=0\); the effective moment of inertia vanishes, and the energy of the first rotational level becomes infinite. Therefore a spherical nucleus (unlike a rigid body) cannot rotate. Conversely, the greater the nonsphericity of the nucleus, the smaller the spacing between levels of rotational character, and the more sharply they are expressed.

The intrinsic angular momentum of the nucleus is made up of the angular momenta of the individual nucleons. Let \(\Omega\) denote the projection of the angular momentum of a nucleon on the symmetry axis. \(\Omega\) takes half-integer positive and negative values. Owing to symmetry, a change in the sign of \(\Omega\) does not change the energy. Therefore the nucleon states are doubly degenerate. A pair of nucleons occupying both degenerate states makes no contribution to the angular momentum. Accordingly, the ground states of even-even nuclei have a moment equal to zero, whereas the ground states of odd nuclei have a moment \(I_0\) equal to the moment \(\Omega\) of the last (unpaired) nucleon.

Finally, let us note that upon rotational excitation of the nucleus the parity of the state is conserved. The selection rules for \(I\) have different forms for even and odd nuclei. For even nuclei with \(I_0=0\), the quantity \(I\) may take the values

\[ I=0,2,4,6\ \text{etc.} \tag{6.4} \]

For odd nuclei,

\[ I=I_0,\ I_0+1,\ I_0+2\ \text{etc.} \tag{6.5} \]

Let us now consider the place occupied by rotational levels in the general system of nuclear levels. The very existence of rotational levels is possible only if the moment of inertia does not depend, or depends only weakly, on the excitation of the nucleus. This condition is equivalent to the following requirement: the spacing between the levels of the rotational structure must be substantially smaller than the spacing between levels of nonrotational character. The spacing between rotational levels, as we have seen, increases as one approaches closed shells. Therefore, near shells the levels have a nonrotational character. The lower levels of heavy nuclei far from shells, on the contrary, have a clearly expressed rotational character.

The requirements formulated above must be refined somewhat. As Landau pointed out, rotational levels can appear in those cases where their energy is less than the energy of the nearest nonrotational level of the same parity with an angular momentum differing by one unit from the angular momentum of the ground state of the rotational system.

Let us also note that formula (6.1) cannot be used in the case of spin equal to one half. In the latter case large corrections associated with degeneracy have to be introduced into the formula; we do not give them here.

Corrections containing \([I(I+1)]^2\) may be introduced into formulas (6.1) and (6.2). They should cause a shift of the levels that is the larger the higher the levels are situated. The available experimental data, however, are not sufficiently accurate for the theory to be tested. For this reason we do not give these correction terms either.

7. ROTATIONAL LEVELS AND THE FINE STRUCTURE OF \(\alpha\)-RAYS

As was explained in § 3, the probability of \(\alpha\)-decay depends sharply on the energy carried away by the \(\alpha\)-particle. Therefore the probability of \(\alpha\)-decay to excited levels of the daughter nucleus is small. An appreciably intense \(\alpha\)-decay is possible only to the very lowest excited levels of nuclei, lying no more than 200–300 keV above the ground state.

At large deformations, however, several rotational levels fit within this range, and in the case of odd nuclei sometimes also several levels of nonrotational nature.

The question of how to determine experimentally whether a given level is a rotational satellite of the ground state is by no means simple. The spin of rotational levels can be found from rules (6.4) and (6.5), the application of which requires only knowledge of the spin of the ground state and the number of the excited level. Thus, in formulas (6.1) and (6.2), which determine the energy of the rotational levels, only one unknown parameter \(J\) remains, and the ratio of the energies contains no unknown quantity. By comparing the experimentally found ratio of the level energies with the calculated one, it is possible to determine whether the levels under consideration belong to one rotational band. As is easy to see, for such a test it is necessary to know at least three levels belonging to one rotational band. In the case where the spin of the ground state is unknown, the matter is still further complicated.

An important method for identifying rotational levels is the determination of their spin and parity from the multipolarity of \(\gamma\)-transitions. In some cases it is possible to measure the lifetime of the excited levels. The identification of rotational levels can then be carried out quite convincingly, since the formulas for

the lifetimes of single-particle transitions and transitions between rotational levels differ by hundreds of times (transitions between rotational levels occur substantially faster).

Let us take, as an example, the system of excited levels of \(U^{236}\), arising in the \(\alpha\)-decay of \(Pu^{240}\). The alpha particles emitted by \(Pu^{240}\) have energies \(5.159\) MeV (75.5%), \(5.115\) MeV (24.5%), and \(5.004\) MeV (0.085%) [56]. The decay scheme is shown in Fig. 12.

Fig. 12. Alpha decay of \(Pu^{240}\) and the scheme of excited levels of \(U^{236}\).

The ground state of the nucleus \(U^{236}\), as in all even-even nuclei, is even and has spin 0. The first excited level has spin 2 and positive parity, and the second excited level has spin 4 and positive parity. The sequence of angular momenta and the constancy of parity exactly follow rule (6.4) for even-even nuclei. The energies of the second and first excited levels should theoretically, according to (6.2), be related as

\[ \frac{E_2}{E_1}=\frac{I_2(I_2+1)}{I_1(I_1+1)}=\frac{4\cdot 5}{2\cdot 3}=3.33. \]

The measured ratio of the energies is

\[ \frac{158}{45}=3.5. \]

Thus, the rotational nature of the first excited levels of \(U^{236}\) is beyond doubt.

The study of the rotational structure of the lower levels of even-even nuclei by means of \(\alpha\)-decay is made difficult by the fact that the intensity of the transition to the second excited level is anomalously small. Thus, in the case of \(Pu^{240}\) considered here, only \(\sim 0.1\%\) of the total number of alpha particles go to the second excited level of \(U^{236}\). Alpha transitions to the third excited rotational level of even-even nuclei have so far never been directly observed at all. (Indirect data from the \(\gamma\)-radiation accompanying \(\alpha\)-decay are available in some cases.)

At present a number of even-even nuclei are known for which the first two excited levels definitely have a rotational nature. Let us list such even-even nuclei:

\[ U^{230},\quad U^{232},\quad U^{234},\quad Pu^{238},\quad Pu^{240},\quad Cm^{242}. \]

For all these nuclei, decays are known to the ground state and to two excited levels with spins, respectively, 2 and 4.

At present several odd nuclei are known whose \(\alpha\)-decay leads to excited rotational levels of the daughter ...

nucleus. Of greatest interest is the $\alpha$-decay of $\mathrm{Am}^{241}$, which has by now been studied in detail. The scheme of excited levels of the daughter nucleus $\mathrm{Np}^{237}$, given in [55], is shown in Fig. 13.

The developed system of rotational levels begins with the second excited level of $\mathrm{Np}^{237}$, with energy $59.8$ keV. The $\alpha$-decay to this level has the maximum intensity $(85\%)$. To the rotational system associated with this level belong the levels with excitation energies $103.2$; $157.2$; $224$ and $305$ keV. (The level $305$ keV, not yet confirmed by measurements of the $\gamma$-spectrum, is shown by a dotted line.) These levels, according to rule (6.5), should be assigned spins increasing monotonically by one.

Fig. 13. Alpha decay of $\mathrm{Am}^{241}$ and the level scheme of $\mathrm{Np}^{237}$ (the level whose existence has not been sufficiently reliably confirmed is shown by a dotted line).

Let us compare the level energies with formula (6.1). The formula contains two unknown parameters $J$ and $I_0$. The presence of 5 levels constituting one rotational band makes it possible to choose these constants and to test the theory. Analysis of the level energies shows that the principal level of the structure—the level with energy $59.8$ keV—should be assigned spin $5/2$. The energies of the remaining levels (relative to the level $59.8$ keV), according to (6.1), should be in the ratio

$$ \left(\frac{9}{2}\times\frac{7}{2} - \frac{7}{2}\times\frac{5}{2}\right): \left(\frac{11}{2}\times\frac{9}{2} - \frac{7}{2}\times\frac{5}{2}\right): $$

$$ :\left(\frac{13}{2}\times\frac{11}{2} - \frac{7}{2}\times\frac{5}{2}\right): $$

$$ :\left(\frac{15}{2}\times\frac{13}{2} - \frac{7}{2}\times\frac{5}{2}\right) = 1:2.28:3.86:5.72. $$

The experimentally measured ratio is $1:2.25; 3.74:5.62$. In addition to the transitions described, in $\mathrm{Am}^{241}$ there is also observed a weak decay to the ground and to the first excited level of $\mathrm{Np}^{237}$. These transitions are comparatively weak $(0.39\%$ and $0.24\%)$.

The ground level of $\mathrm{Np}^{237}$ has the same spin $(5/2)$ as the ground level of the rotational system, but opposite parity. It is possible that the first excited level is a rotational satellite of the ground state [54c].

Thus, \( \mathrm{Np}^{237} \) is an example of a nucleus having several (two) rotational systems of levels, independent of one another. Finally, let us note that \( \mathrm{Np}^{237} \) has two more levels (268 keV and 433 keV) which do not belong to these rotational bands and are not observed in the \(\alpha\)-decay of \( \mathrm{Np}^{241} \). They are known from the decay of \( \mathrm{U}^{237} \), which after a \(\beta\)-transition \(5^5\) goes to \( \mathrm{Np}^{237} \).

It is natural to pose the question why different levels of the nucleus find themselves in such an unequal position with respect to \(\alpha\)-decay, as occurs in the case of \( \mathrm{Np}^{237} \). Why is the principal rotational system of levels excited by strong groups of \(\alpha\)-particles, while \(\alpha\)-decay to the ground and first excited levels almost does not occur? Why do two levels, well known from \(\beta\)-decay, not show up at all in \(\alpha\)-decay? It is clear that the answers to these questions must proceed from an analysis of the structure of the nucleus in different excited states. Our knowledge, however, is at such a level that there is no possibility of solving the problem quantitatively by calculating the corresponding matrix elements. Qualitative considerations will be set forth in the next section.

Fig. 14. Energy of the first excited level of even-even nuclei as a function of the number of neutrons in the nucleus.

Fig. 15. Ratio of the energies of the 2nd and 3rd to the energy of the 1st excited level as a function of the number of neutrons.

Let us now consider how the distance between the ground and first excited levels changes depending on the number of neutrons in the nucleus. In Fig. 14, borrowed by us from \(P^{54a}\), the corresponding data are given. As also follows from the theory (see § 6), the energy of the first excited level increases as one approaches the closed shell of 126 neutrons and decreases upon moving away from it. Apparently, somewhere in the region of 144–150 neutrons a minimum of this distance is reached.

As we have already noted, the energies of the excited rotational levels must be in a strictly definite ratio. Thus, the energy of the second level must be 3.33 times greater than the energy of the first level, and the energy of the third level—7 times greater. The available experimental data are presented in Fig. 15. Since the data on \(\alpha\)-decay for the second levels are incomplete, and for the third levels are entirely absent, they were supplemented by data obtained from an analysis of \(\gamma\)-radiation. As follows from the figure, at a sufficient distance from the shell \(N=126\) the ratios of the energies of the excited levels come very close to the theoretical ones. At \(N<138\) systematic deviations are observed in the ratio

\[ \frac{E_2}{E_1} \]

(the energy of the second to the energy of the first excited level) toward smaller values. For the doubly magic nucleus \(\mathrm{Pb}^{208}\) this ratio is equal to 1, which possibly indicates the vibrational nature of the levels. Let us note that even among even-even nuclei not all levels have a rotational nature. Thus, in the \(\alpha\)-decay of \(\mathrm{Th}^{228}\) the level \((1-)\) of the daughter nucleus \(\mathrm{Ra}^{224}\) appears (Fig. 16). This level is located between the first rotational level \(2+\) and the second rotational level \(4+\). The level \((1-)\) also appears in other even-even nuclei (\(\mathrm{Th}^{226}\), \(\mathrm{Ra}^{222}\)).

Fig. 16. Scheme of \(\alpha\)-decay of \(\mathrm{Th}^{228}\) and levels of \(\mathrm{Ra}^{224}\).

At present it is difficult to say what is responsible for the appearance of the level \(1-\). It is not known whether this level is a single-particle one, situated for some reason anomalously low, or whether it is connected with collective motion—the appearance of such levels is possible when the symmetry of the nucleus is broken (pear-shaped deformation).

8. INTENSITY OF LINES IN \(\alpha\)-SPECTRA. FAVORED AND UNFAVORED TRANSITIONS

As has already been noted, in odd nuclei the total half-lives of \(\alpha\)-decay and the partial half-lives of transitions to particular excited levels can undergo appreciable fluctuations. These fluctuations are conveniently characterized quantitatively by calculating the value of the decay coefficient \(F\) (see § 5), equal to the ratio of the observed decay constant to that calculated from formula (5.3). It is assumed here that formula (5.3) can be applied

for individual lines of the $\alpha$ spectrum (in the following paragraph it will be shown that this is probably not so).

Calculating $F$ for various $\alpha$ lines of odd nuclei, it is not difficult to see that in almost every $\alpha$ spectrum there are lines for which $F$ is of the order of $1^{55}$. We shall call such transitions favored. For some of these transitions it has been possible to show that the spin of the daughter nucleus in the corresponding state is equal to the spin of the parent nucleus, so that the $\alpha$ particles do not carry away angular momentum. It is possible that this is the case in all instances. In Table IV, for each nucleus, that transition of the $\alpha$ spectrum has been selected which has the largest $F$.

Table IV

Favored transitions in odd nuclei

Element	Atomic weight	$\alpha$-particle energy ($Mev$)	$F$	Element	Atomic weight	$\alpha$-particle energy ($Mev$)	$F$
$_{88}$Ra	219	8,0	0,3	$_{93}$Np	231	6,28	25,1
$_{88}$Ra	221	6,71	0,3	$_{93}$Np	233	5,53	2,0
$_{88}$Ra	223	5,596	0,2	$_{93}$Np	235	5,06	0,5
				$_{93}$Np	237	4,77	0,4
$_{89}$Ac	223	6,64	0,3	$_{94}$Pu	235	5,85	0,4
$_{89}$Ac	225	5,80	0,3	$_{94}$Pu	239	5,150	0,3
$_{89}$Ac	227	4,942	0,2	$_{94}$Pu	241	4,893	1,3
$_{90}$Th	233	7,55	0,5	$_{95}$Am	237	6,01	0,1
$_{90}$Th	225	6,57	0,4	$_{95}$Am	239	5,75	0,2
$_{90}$Th	227	5,704	0,2	$_{95}$Am	241	5,476	0,6
$_{90}$Th	229	4,85	0,5	$_{95}$Am	243	5,267	0,8
$_{91}$Pa	227	6,46	0,6	$_{96}$Cm	241	5,89	0,1
$_{91}$Pa	231	4,722	0,4	$_{96}$Cm	243	5,777	0,5
				$_{96}$Cm	245	5,34	0,3
$_{92}$U	227	6,8	2,0	$_{97}$Bk	243	6,20	0,1
$_{92}$U	229	6,42	0,3	$_{97}$Bk	245	5,90	0,3
$_{92}$U	233	4,816	0,6	$_{98}$Cf	249	5,82	0,3
$_{92}$U	235	4,20	0,3

The principal favored transition is usually followed by an entire system of transitions to rotational levels.

Let us consider, as an example, the $\alpha$ decay of $\mathrm{U}^{233}$. The most complete scheme of levels of the daughter nucleus $\mathrm{Th}^{229}$, given in $\Gamma^{56}$, is shown in Fig. 17. The transition to the ground level of $\mathrm{Th}^{229}$ is favored. An appreciable number of $\alpha$ particles go to the excited levels of $\mathrm{Th}^{229}$. All five excited levels have a clearly pronounced rotational character. Two spin numbers are assigned to these levels in Fig. 17.

The first of these characterizes the magnitude of the nuclear spin, and the second—the projection of it on the nuclear axis. All rotational levels of the given band have the same projection of the spin on the symmetry axis of the nucleus.

In \( \mathrm{Np}^{237} \), following the level at \(59.8\ \text{keV}\) (Fig. 13), whose decay is favored, there comes a whole series of levels belonging to one rotational band.

Of interest is the decay of \( \mathrm{Pa}^{231} \), shown in Fig. 18. Here the transition to the comparatively high-lying 6th level is favored. It may be that, as a result of this, a number of transitions to lower levels have comparable probability. Apparently, the change in the energy of the \(\alpha\)-particles is compensated by a change in the degree of forbiddenness. In discussing the theory of \(\alpha\)-decay (§ 3), among the factors influencing the magnitude of the pre-exponential factor, we noted the probability of formation of the \(\alpha\)-particle in the nucleus. This probability, of course, is especially large in the case when \(\alpha\)-decay does not cause a large

Fig. 17. Scheme of the \(\alpha\)-decay of \( \mathrm{U}^{233} \) and of the levels of \( \mathrm{Th}^{229} \).

Fig. 18. Scheme of the \(\alpha\)-decay of \( \mathrm{Pa}^{231} \) and of the levels of \( \mathrm{Ac}^{227} \).

rearrangement of the nucleus. Therefore it should be expected that transitions to those levels of the daughter nucleus whose structure is close to the structure of the parent nucleus will be favored.

Apparently, formation of the \(\alpha\)-particle occurs most readily from a pair of nucleons occupying in the nucleus both degenerate states,

—as has already been noted, the states of nucleons in nuclei having an axis of symmetry are doubly expressed. If this is so, it is not difficult to understand the rule noted at the beginning of this paragraph: favored transitions do not change the angular momentum of the nucleus, or, more precisely, do not change the projection of the angular momentum on the nuclear axis, i.e., the state of the odd nucleon.

Let us return to the consideration of the $\alpha$-decay of $\mathrm{Am}^{241}$ (Fig. 13). The transition to the 59.8-keV level of $\mathrm{Np}^{237}$ is undoubtedly favored. The spin of this level, $5/2$, is equal to the spin of the $\mathrm{Am}^{241}$ nucleus in its ground state. True, the $\mathrm{Np}^{237}$ nucleus in its ground state also has spin $5/2$, but the parity of this state is opposite to the parity of the 59.8-keV level. Evidently, the parity of the $\mathrm{Am}^{241}$ ground state coincides with the parity of the 59.8-keV level, and not with the parity of the ground level.

Of course, what has been set forth above should not be understood to mean that the retardation of $\alpha$-decay is caused precisely by the angular momentum carried away by the $\alpha$-particle. In any case, a change of spin by 1 or 2 does not introduce into $\alpha$-decay factors noticeably different from unity. This assertion can readily be checked experimentally from the probability of $\alpha$-decay to the first excited level of even-even nuclei. This decay occurs from the level 0 of the parent nucleus to the level $2+$ of the daughter, so that the $\alpha$-particle carries away an angular momentum equal to two.

Table V presents experimental data characterizing the transition to the first excited level of even-even nuclei—

Table V

Alpha decay to the first excited level
$(2+)$ of even-even nuclei

Element (parent)	Atomic weight	Level energy (keV)	Decay coeff. $F$	Element (parent)	Atomic weight	Level energy (keV)	Decay coeff. $F$
$_{84}\mathrm{Po}$	206	163	0.3	$_{92}\mathrm{U}$	234	52	0.8
$_{84}\mathrm{Po}$	210	800	0.6	$_{92}\mathrm{U}$	236	50	0.9
				$_{92}\mathrm{U}$	238	45	0.67
$_{88}\mathrm{Ra}$	224	240	0.8	$_{94}\mathrm{Pu}$	234	45	0.3
$_{88}\mathrm{Ra}$	226	188	1.1	$_{94}\mathrm{Pu}$	236	47	0.45
				$_{94}\mathrm{Pu}$	238	43	0.6
$_{90}\mathrm{Th}$	226	110	0.8	$_{94}\mathrm{Pu}$	240	44	0.6
$_{90}\mathrm{Th}$	228	84	1.1	$_{94}\mathrm{Pu}$	242	45	0.45
$_{90}\mathrm{Th}$	230	68	1.0	$_{96}\mathrm{Cm}$	242	44	0.6
$_{90}\mathrm{Th}$	232	65	1.2	$_{96}\mathrm{Cm}$	244	43	0.55
$_{92}\mathrm{U}$	230	70	0.9	$_{98}\mathrm{Cf}$	246	42	0.4
$_{92}\mathrm{U}$	232	58	0.9

Since small changes of the moment, as we have seen, do not have a substantial effect on the decay probability, it remains to suppose that the matter lies in a change of the internal structure of the nucleus, so that favored transitions are those that do not require a rearrangement of the nucleus. Nuclei with the same structure also have the same angular momenta. This is why, in favored transitions, the angular momentum of the nucleus does not change. Similarly to what takes place for even nuclei, the transition to the first rotational satellite of a given level of an odd nucleus is favored if the transition to the ground level of the rotational structure is favored.

The application of formula (5.3) to decay to the second excited level \((4+)\) of even-even nuclei leads to serious difficulties (see § 9).

If one starts from (5.3), large hindrance coefficients \(\frac{1}{F}\) must be assigned to transitions to the \(4+\) level; the graph in Fig. 19 was constructed for these coefficients. As is seen from the graph, the quantity \(\frac{1}{F}\), approximately equal to ten for the Th isotopes, rises sharply with increasing atomic number \(Z\), reaching, for \(\mathrm{Cm}^{242}\), a value equal to several hundreds. Beyond \(\mathrm{Cm}^{242}\), \(\frac{1}{F}\) falls again.

In the figure: “data obtained from \(\alpha\)-spectra”; “data obtained from \(\gamma\)-spectra”; axes \(\frac{1}{F}\) and \(Z\).

Fig. 19. Hindrances \(\frac{1}{F}\) of the partial decay time of two excited levels of even-even nuclei.

Rasmussen attempted a qualitative explanation of this phenomenon. The wave functions of an \(\alpha\)-particle with moment \(l\) contain a Legendre polynomial of degree \(l\). The Coulomb barrier surrounding the nucleus is most transparent at the elongation of the nucleus (at the “nose”) and least transparent in the middle part of the ellipsoid. If it should turn out that the formation of \(\alpha\)-particles at the “nose” is for some reason hindered, then the principal role in \(\alpha\)-radiation would begin to be played by an intermediate region of the nuclear surface, where the nodes of the Legendre polynomials are located. If, for example, the \(\alpha\)-active region lies in the region of the nodes of the 4th Legendre polynomial, the emission of \(\alpha\)-particles with moment 4 will be very improbable.

With such an explanation, the probability of decay to the \(4+\) level depends essentially on the ratio of the semiaxes of the nucleus. The graph shown in Fig. 19, with its clearly expressed maximum, qualitatively agrees…

confirms this conclusion. The value of such arguments is, of course, small until, on the basis of the considerations expressed, calculations have been performed that can be compared with experiment. Another approach to explaining this phenomenon was proposed by L. D. Landau (see § 9).

Bohr and Fröman \({}^{555}\) undertook an interesting attempt to calculate the intensities of \(\alpha\)-transitions in odd nuclei. As was already noted above, in even-even nuclei the probability of transition to the first excited level corresponds to formula (5.3) for favored transitions, while the probability of transitions to the second excited level turns out to be greatly reduced, but varies from nucleus to nucleus in a regular way.

Transitions in odd nuclei are more complicated than transitions in even nuclei already because the spins of the initial and final states in them do not determine in a unique manner the angular momentum carried away by the \(\alpha\)-particle. In fact, the \(\alpha\)-particle may carry away any angular momentum from \((I_{\rm final}-I_{\rm initial})\) to \((I_{\rm final}+I_{\rm initial})\) (half of the angular momenta are forbidden by parity). Nevertheless, the calculation of transition probabilities can be carried out with the aid of Clebsch—Gordan coefficients, if the probabilities of emission of \(\alpha\)-particles with different angular momenta are taken as empirical coefficients from the \(\alpha\)-decay of even nuclei.

Bohr and Fröman \({}^{555}\) calculated the ratio of the intensities of successive \(\alpha\)-lines of the rotational structure for \(\mathrm{Am}^{241}\) and \(\mathrm{U}^{233}\). For \(\mathrm{Am}^{241}\) they found: \(100:14:2.2:0.02\). Experiment gives \(100:15:1.9:0.018\). For \(\mathrm{U}^{233}\) their calculations led to the ratio \(100:13:1.8:0.2\), whereas from experiment for the corresponding lines there follows the ratio \(100:18:1.9:0.08\). Thus the calculation describes well the intensities of several first rotational levels of the daughter nuclei. (See also the end of § 9.)

9. INTENSITY OF \(\alpha\)-TRANSITIONS TO ROTATIONAL LEVELS

(CONTINUATION)

New considerations on the intensity of \(\alpha\)-decay to rotational levels were recently put forward by L. D. Landau \({}^{556}\), who proposed simple formulas for calculating the intensities of \(\alpha\)-lines.

For simplicity we shall consider, for the inverse process of \(\alpha\)-decay, the absorption of \(\alpha\)-particles by the daughter nucleus. The probability of decay and the capture cross section, according to the principle of detailed balance, are proportional.

Let a flux of \(\alpha\)-particles be incident on an elongated daughter nucleus. Since the transparency of the potential barrier surrounding the nucleus in the region of the “nose” has a sharply expressed maximum, the principal fraction of the \(\alpha\)-particles will penetrate the nucleus precisely through the “nose.” The transparency of the barrier also depends on the direction of motion of the \(\alpha\)-particle. It is easiest to reach the nucleus by moving “across the barrier” in the direction of the electric field. Thus, for the principal fraction of the \(\alpha\)-particles penetrating into the nucleus, the projection of the angular momentum on the major axis of the ellipsoid is equal to zero.

The probability of capture of an \(\alpha\)-particle with angular momentum \(l\) and energy \(E\) is proportional to the transparency of the barrier and to the squared modulus of the wave function near the “tip” of the nucleus, i.e. at \(r=a,\ \vartheta=0\), so that

\[ W=C\left|Y_{l,0}(a,0)\right|^{2}e^{-S(E,l)} =C(2l+1)e^{-S(E,l)} . \tag{9.1} \]

(We note that for a perfectly spherical nucleus, when the considerations stated above are inapplicable, the factor \(2l+1\) also appears before the exponential, although for another reason: it takes into account the statistical weight of the state with angular momentum \(l\). For strongly nonspherical nuclei the statistical weight is equal to unity, since of all \(\alpha\)-particles with angular momentum \(l\) only \(\alpha\)-particles with zero projection of the angular momentum on the axis of symmetry are absorbed.) In formula (9.1), \(C\) is a normalization factor independent of \(E\) and \(l\). Equality (9.1) is, of course, approximate. In a more exact treatment one would have to assume that the capture probability is equal to the integral of the product of the squared modulus of the wave function, the barrier penetrability, and the sticking probability, which, generally speaking, depends not only on the energy but also on the angle \(\vartheta\). As regards the sticking coefficient, it is difficult to say anything except that it is apparently close to unity. There is no reason to expect a strong dependence of it on the energy or on the angle. For simplicity we shall regard the sticking coefficient as a constant and include it in \(C\).

The difference in energy of the various groups of \(\alpha\)-particles is small. We therefore expand \(S(E,l)\) in a series:

\[ S(E,l)=S(E_0,0)+\alpha_1\Delta E+\alpha_2 l(l+1)+\cdots . \tag{9.2} \]

(In the expansion (9.2) it is taken into account that the energy depends not directly on \(l\), but on the quantity \(l(l+1)\).)

Since for rotational levels \(\Delta E\) is determined by formula (6.3), we find:

\[ S(E,l)-S(E_0,0)=\alpha_1\left[I(I+1)-I_0(I_0+1)\right]+\alpha_2 l(l+1). \tag{9.3} \]

Let us first consider even-even nuclei. The daughter nucleus, being in the ground or an excited state, after absorption of the \(\alpha\)-particle passes into the ground state of the parent nucleus, which has zero angular momentum. Therefore, for even-even nuclei

\[ \begin{gathered} I_0=0;\\ I=l. \end{gathered} \tag{9.4} \]

Thus, for rotational levels of even-even nuclei,

\[ S(E,l)-S(E_0,0)=\alpha l(l+1). \tag{9.5} \]

Substituting (9.5) and (9.4) into (9.1), we find

\[ W_l=W_0(2l+1)e^{-\alpha l(l+1)} , \tag{9.6} \]

This formula contains the only parameter subject to determination from experiment, \(\alpha\) (the theory makes it possible to calculate \(\alpha\) from the geometrical dimensions of the ellipsoid).

Let us return to Table V, where the observed intensities of \(\alpha\)-decay to the first excited level are compared with formula (5.3), or, equivalently, (3.5) and (3.6). As has already been explained, the latter formulas describe well the decay to the level \(2+\), and very poorly to the level \(4+\). Since, however, these formulas do not contain the obligatory factor \((2l+1)\), equal to five for the level \(2+\), their agreement with experiment argues rather against these formulas than in their favor.

Let us apply formula (9.6) to the \(\alpha\)-decay of \(\mathrm{Pu}^{240}\) (intensity of the transition to the ground state \(76\%\), to the level \(2+\)—\(24\%\), to the level \(4+\)—\(0.07\%\)). If in formula (9.6) we take \(\alpha=0.46\), then the intensity of the transition to the level \(2+\) should be equal to

\(5e^{-6\cdot 0.46}=0.32\)

(the observed value \(\dfrac{24\%}{76\%}=0.32\)), and the intensity of the transition to the level \(4+\) should be equal to

\(9e^{-20\cdot 0.46}=9\cdot 10^{-4}\),

which also agrees well with experiment.

Thus formula (9.6) describes the intensities of the rotational levels of even-even nuclei, without leading to the need to introduce large delays \(\dfrac{1}{F}\).

It is interesting to compare the theory with experiment for the following levels of the rotational structure (for example, for the level \(6+\)). Formula (9.6) predicts an extremely small probability of decay to this level. The levels \(6+\) have in fact never been directly observed in \(\alpha\)-decay. Their presence in some cases (\(\mathrm{U}^{234}\), \(\mathrm{Pu}^{238}\)) can be inferred from weak \(\gamma\)-radiation of the corresponding energy. Since there are no data in the literature on the intensity of \(\alpha\)-decay to this level, a comparison of theory with experiment for the level \(6+\) cannot be made (we note that for large angular momenta the next terms in the expansion \(S(E,l)\) should begin to play a role, so that the approximate theory may lose applicability).

In the case of odd nuclei the situation is more complicated.* The transition from the level \(I_0\) of the parent nucleus to the level \(I\) of the daughter may be accompanied by the emission of \(\alpha\)-particles with different angular momenta, so that the decay probability according to Ter-Martirosyan is determined by the formula

\[ W_I^{I_0} = C \sum_{l=|I-I_0|}^{I+I_0} \left| C_{I0,l0}^{I_0 I_0} \right|^2 (2l+1) e^{-\alpha_1\{I(I+1)-I_0(I_0+1)\}-\alpha_2 l(l+1)} . \tag{9.7} \]

* L. D. Landau believes that formula (9.6) is also applicable to odd nuclei.

In equality (9.7), besides the normalization constant \(C\), constant for all levels of the rotational band, there enter the squares of the Clebsch—Gordan coefficients \(C^{I'I_0'}_{I I_0,\,lm}\).

The Clebsch—Gordan coefficients appear in the expansion of the state with a given angular momentum of the daughter nucleus (spin \(= I\), projection of the spin on the \(z\)-axis equal to \(I_0\)) and with a given angular momentum of the \(\alpha\)-particle \((l,m)\) in terms of the states of the parent nucleus (spin \(I'\), spin projection \(I_0'\)).

In \(\alpha\)-decay to levels of a single rotational band, the projection of the angular momentum on the nuclear axis does not change, so that \(I_0' = I_0,\ m = 0\).

For reference we give a table of some Clebsch—Gordan coefficients.

Table VI

Squares of some Clebsch—Gordan coefficients
\[ I_0 = 3/2 \]

	\(I=3/2\)	\(I=3/2\)	\(I=5/2\)	\(I=5/2\)	\(I=7/2\)	\(I=7/2\)	\(I=9/2\)	\(I=9/2\)	\(I=11/2\)	\(I=11/2\)
	\(l=0\)	\(l=2\)	\(l=2\)	\(l=4\)	\(l=2\)	\(l=4\)	\(l=4\)	\(l=6\)	\(l=4\)	\(l=6\)
	1	\(1/5\)	\(12/35\)	\(2/63\)	\(1/7\)	\(1/7\)	\(2/11\)	\(4/143\)	\(7/99\)	\(15/143\)

\[ I_0 = 5/2 \]

	\(I=5/2\)	\(I=5/2\)	\(I=5/2\)	\(I=7/2\)	\(I=7/2\)	\(I=7/2\)	\(I=9/2\)	\(I=9/2\)	\(I=9/2\)	\(I=11/2\)	\(I=11/2\)	\(I=11/2\)	\(I=13/2\)	\(I=13/2\)	\(I=13/2\)
	\(l=0\)	\(l=2\)	\(l=4\)	\(l=2\)	\(l=4\)	\(l=6\)	\(l=2\)	\(l=4\)	\(l=6\)	\(l=4\)	\(l=6\)	\(l=8\)	\(l=4\)	\(l=6\)	\(l=8\)
	1	\(5/14\)	\(1/42\)	\(5/14\)	\(30/231\)	\(1/286\)	\(1/10\)	\(5/22\)	\(5/143\)	\(70/429\)	\(16/143\)	\(1/221\)	\(6/143\)	\(45/286\)	\(15/442\)

Formula (9.7) contains two parameters \(\alpha_1\) and \(\alpha_2\), to be determined from experiment. Fortunately, this can often be done without much arbitrariness, since transitions in odd nuclei are fairly rich in rotational levels.

In connection with formula (9.7) it should be noted that the attempt of Bohr and Fröman (see § 8) to calculate the intensities of \(\alpha\)-lines on the basis of data for even nuclei is apparently not satisfactory, since, as already mentioned, in the description of even-even nuclei one parameter enters instead of the two in odd nuclei.

Let us consider the \(\alpha\)-decay of \(\mathrm{Am}^{241}\). As experiment shows, the intensities of the lines of the rotational structure are in the ratio
\[ 1:0.15:0.019:1.8\cdot10^{-4}:2.5\cdot10^{-5}. \]

If one takes \(a_1=0.050,\ a_2=0.364\), then the calculated ratio of intensities is \(1:0.12:0.021:2.1\cdot10^{-4}:2.9\cdot10^{-5}\), in satisfactory agreement with experiment.

Such agreement is not always observed. The intensities of the successive lines of the \(\alpha\)-spectrum of \(U^{233}\) are in the ratio

\(1:0.18:0.019:8\cdot10^{-4}:5\cdot10^{-4}:3.5\cdot10^{-4}\). The rapid and regular change in intensity of the first four lines is replaced, for the following lines, by a very weak change. It is not difficult to describe the intensities of the first lines with the aid of any of the proposed theoretical formulas. The explanation of the intensities of the last lines, however, encounters considerable difficulties.

In discussing the formulas given above, one should bear in mind the basic assumption made in deriving them, namely, that all \(\alpha\)-radiation originates from a small region of the nucleus near the “nose.” This assumption, while plausible, of course requires further verification.

10. CONCLUSION

In summing up, it should be noted that the theory of \(\alpha\)-decay is in an unsatisfactory state. In essence, nothing has been done except to calculate the transparency of the barrier for a spherical nucleus, which is entirely insufficient, since \(\alpha\)-active nuclei are clearly nonspherical. Nothing can be said about the probability of formation of \(\alpha\)-particles in nuclei.

The theory of \(\alpha\)-spectra has at present achieved unquestionable successes, having found that a noticeable part of the lower excited levels of nuclei has a rotational nature. At the same time, many regularities (especially in odd nuclei) remain unexplained. There are no convincing calculations explaining the forbidden character of transitions to the level \(4+\) in even-even nuclei. The question of the intensities of various lines of \(\alpha\)-spectra is especially obscure, although the first attempts to clarify this question are already being made.

In the study of \(\alpha\)-decay, a number of important empirical regularities have been noted. These regularities make it possible to predict reliably the \(\alpha\)-decay energies of unknown isotopes and (somewhat less well) their lifetimes.

Recently the amount of experimental data on \(\alpha\)-decay has noticeably increased; nevertheless, the available data are clearly insufficient. To construct a theory it is necessary to obtain information on \(\alpha\)-decay to the level \(6+\) in even-even nuclei. More intensive study of long-range \(\alpha\)-particles is needed. Data are needed on angular correlations of \(\alpha\)- and \(\gamma\)-rays. It is important to have more complete information on the \(\alpha\)-spectra of odd elements.

TABLE

OF α-RADIOACTIVE ISOTOPES

(data as of April 1, 1956)

Isotope	Mode of decay	Half-life	Energy of α-particle (in MeV)	Energy of γ-radiation (in keV)	Data on α–γ and γ–γ coincidences	Decay scheme	Basic characteristics of the nucleus
\({}_{83}\mathrm{Bi}^{<198}\)	\(\alpha\) T48	1.7 min N50a	6.2 — ion. chamber N50a
\(\mathrm{Bi}^{198}\)	\(\alpha\) — \(5\cdot10^{-2}\%\) EC — 99% N50a	7 min N50a	5.83 — ion. chamber N50a
\(\mathrm{Bi}^{199}\)	\(\alpha\) — \(10^{-2}\%\) EC — 99% N50a	\(\sim 25\) min N50a	5.47 — ion. chamber N50a
\(\mathrm{Bi}^{201}\)	\(\alpha\) — \(3\cdot10^{-3}\%\) EC — 99% N50a	62 min N50a	5.15 — ion. chamber N50a
\(\mathrm{Bi}^{203}\)	\(\alpha \sim 10^{-5}\%\) D52a EC N50a	12 h N50a	4.85 — range in photoemuls. D52a
\(\mathrm{Bi}^{209}\)	\(\alpha\) F51	\(2\cdot10^{17}\) yr R52b	2.9 — range in photoemuls. R52b				\(I = 9/2\) M50c
\(\mathrm{Bi}^{210}\) (RaE)	\(\beta^{-}\) — 99% \(\alpha\) — \(5\cdot10^{-5}\%\) B47	4.88 d L53a	\(\dfrac{\gamma(E_\gamma>90)}{\beta^{-}} =\) \(= 0.00084\) B53c				\(I = 0?\) F53a \(I = 1\) S54c

Nuclide	Decay	Half-life	α-particle data	γ-rays / electrons	Other data		Decay scheme	Spin
Bi\(^{210}\)	\(\alpha\), \(\beta^-\) or EC (\(\sim 0.3\%\)) H53a	\(2.6\cdot 10^6\) years H53c	4.94 — ion. chamber L54 No long-range \(\alpha\) — photoemuls. L54	No \(\gamma\), no \(e^-\) scint. counter L54				\(I \ge 4\) L54
Bi\(^{211}\) (AcC)	\(\alpha\) — 99.68% \(\beta^-\) — 0.32% C31	2.15 min S54f	6.621 (82.6%) 6.274 (17.4%) spectrum V52b	\(\sim 350\) \((e_k/\gamma=0.18)\) \((K/L=5.5)\) T52a F52a	\((6.274\alpha)\ (350\gamma)\ \vartheta\) G53b		text<br>Bi^211<br> ↙ α 16% ↘ β^-<br>Bi^207 ───────→<br> ↘ θ ↗ θ 84%<br> ↘ ↗<br> C31<br>	\(I=\frac{1}{2};\) \(I=\frac{3}{2}\) F54e
Bi\(^{212}\) (ThC)	\(\alpha\) — 35.4% \(\beta^-\) — 64.6% M53d	60.5 min C31	6.083 (27.2%) 6.047 (69.9%) 5.765 (1.7%) 5.622 (0.15%) 5.603 (1.1%) 5.481 (0.016%) spectrum R51a	40 (strong); 144; 164; 288; 328; 432; 452; 472 spectrum S46 40 (\(\sim 4\%\)) \((e/\gamma \ge 14)\) K47 See also S52a, M54g, S37 120; 435 W52a	\((6.04\alpha)\ (40\gamma)\ \vartheta\) H53d			\(I=1\) H53h
Bi\(^{213}\)	\(\alpha\) — 2% \(\beta^-\) — 98% H50b E47	46 min E47	5.86 — ion. chamber E47
Bi\(^{214}\) (RaC)	\(\alpha\) — 0.04% \(\beta^-\) — 99% CK1	19.7 min C31	5.52 (37%) 5.47 (46%) 5.33 (17%) spectrum C48	62.5 191 spectrum, coinc. C51a See also D53b, M52d, P53

Isotope	Mode of decay	Half-life	Energy of α-particles (in MeV)	Energy of γ-radiation (in keV)	Data on α—γ and γ—γ coincidences	Decay scheme	Main characteristics of the nucleus
$_{84}\mathrm{Po}^{197?}$		4 min R54b	6.040 — spectr. R54b
$\mathrm{Po}^{198?}$		6 min R54b	5.935 — spectr. R54b
$\mathrm{Po}^{199?}$		11 min R54b	5.846 — spectr. R54b
$\mathrm{Po}^{200}$	α, EC K51	11 min K51 8 min R54	5.770 — spectr. R54b
$\mathrm{Po}^{201}$	α, EC K51	18 min K51	5.671 — spectr. R54b
$\mathrm{Po}^{202}$	α, EC K51	52 min K51 55 min R54	5.575 — spectr. R54b
$\mathrm{Po}^{204}$	α — 1% EC — 99% K51	3.8 h K51	5.370 — spectr. R54b
$\mathrm{Po}^{205}$	α — 0.074% EC — 99%	1.5 h K51	5.2 — ion. cham. K51

$\mathrm{Po}^{206}$	$\alpha \sim 10\%$ EC $\sim 90\%$ T47a	9 days T47a	5.218 (96%) 5.064 (4%) spectr. R53f	800 — abs. T47a
$\mathrm{Po}^{207}$	$\alpha \sim 10^{-2}\%$ EC 99% T47a	5.7 hours T47a	5.10 — ion. chamber K51	1300—abs. T47a
$\mathrm{Po}^{208}$	$\alpha$ T47a	2.93 years T50b	5.109 (100%) 4.784 (0.1%) R54b	No $\gamma$ T47a
$\mathrm{Po}^{209}$	$\alpha > 90\%$ EC $< 10\%$ P50a	$\sim 100$ years H53a	4.877 — spectr. H53a	100 (0.07%) 200 (0.2%) 550 (0.5%) 870 (1%) scint. spectr. H53a 270 ($\sim 0.75\%$) 570 ($\sim 0.75\%$) 865 ($\sim 0.75\%$) D55a	(270 $\gamma$) (570 $\gamma$) D55a		diagram: levels labeled $0$, $570$, $640$, $865$; transitions $0.777$, $0.225$, $0.865$; $\mathrm{Po}^{209}$; D55a
$\mathrm{Po}^{210}$ (RaF)	$\alpha$ C31	138.4005 days E54	5.2984 — spectr. H38 5.3006 $\pm$ 0.0026 B54a	800 ($e/\gamma \sim 0.03$) ($K/L \sim 3.7$) spectr. A51b 800 ($1.8 \cdot 10^{-3}\%$) ($e/\gamma \sim 0.07$) No 80 $\gamma$ G51c 770 ($\sim 10^{-3}\%$) 84 ($\sim 10^{-3}\%$) Z48	(4.5 $\alpha$) (800 $\gamma$) $β$ 800 $\gamma$ — E2 B52a		diagram: levels labeled $(2+)800$, $(0+)0$; transition $\sim 10^{-3}\%$; $\sim 100\,d$; $\mathrm{Po}^{210}$; G51c, B52a

Isotope	Mode of decay	Half-life	Energy of α-particles (in MeV)	Energy of γ-radiation (in keV)	Data from α—γ and γ—γ coincidences	Decay scheme	Main nuclear characteristics
Po²¹¹ (AcC′)	α	0.52 sec S54f	7.434 — spectr. L34 (7.43) (99%) 6.895 (0.50%) 6.509 (0.53%) No 6.34 (<0.02%) spectr. H55a	562; 88) coincid. sc. M54b
Po²¹¹	α	25 sec J54	8.70 (7.0%) 7.85 (2.5%) 7.14 (90.5%) ion. cham. J54	550; 1060 scint. J54	(560 γ) (1060 γ) J54	![decay scheme: visible diagram with levels labeled 0, 570, 870, 1634, 2230; transitions including 570, 1060, and 1360; states labeled Po²¹¹ with half-lives 25 sec and 0.32 sec; daughter level J54]
Po²¹² (ThC′)	α C31	2.9·10⁻⁷ sec H53e	8.777 (100%) 9.885 (0.004%) 10.416 (0.002%) 10.4885 (0.017%) spectr. R51a; B54a
Po²¹³	α E47	4.2·10⁻⁶ sec J43	8.336 — ion. cham. E47

Nuclide	Decay	Half-life	Energy	Radiation / levels	Coincidences / γ-rays			Notes
Po\(^{214}\) (RaC′)	\(\alpha\) C31	\(1.58 \cdot 10^{-4}\) sec B53h	7.683 — spectrum. S51 9.069 (0.002%) 8.280–10.509 (11 lines) spectrum. L34; B54a	850; 1770; 2200; 2400 R55a	(1120 γ) (1761γ) δ (608γ) (1235γ) δ (608γ) (2090γ) δ ID53b (607 γ) (770 γ) (607 γ) (930 γ) (607 γ) (1120 γ) (607 γ) (1240 γ) (607 γ) (1380 γ) (607 γ) (1520 γ) (607 γ) (1850 γ) R55a
Po\(^{213}\) (AcA)	\(\alpha\) — 99% \(\beta^{-}\) — \(5 \cdot 10^{-4}\)% A50	\(1.83 \cdot 10^{-3}\) sec W42	7.365 — spectrum. L34 7.383 — spectrum. B54a					\(I = {}^{5}/_{2}\) M54i
Po\(^{216}\) (ThA)	\(\alpha\) [C31] \(\beta\) — stable. H53a	0.158 sec W42	6.7746 — spectrum. B54a
Po\(^{217}\)	\(\alpha\) H53a	\(\sim 10\) sec M56	6.54 M56
Po\(^{218}\) (RaA)	\(\alpha\) — 99% \(\beta^{-}\) — 0.03% K43	3.05 min C31	5.996 — spectrum. B53d 5.9982 B54a
\(_{85}\)At\(^{<202}\)	\(\alpha\), EC B51	43 sec B51	6.50 — ion. cham. B51

Isotope	Mode of decay	Half-life	Energy of α-particles (in MeV)	Energy of γ-radiation (in keV)	Data from α—γ and γ—γ coincidences	Decay scheme	Main nuclear characteristics
$\mathrm{At}^{<203}$	$\alpha$, EC B51	1.7 min B51	6.35 — ion. cham. B51
$\mathrm{At}^{203}$	$\alpha$, EC B51	7 min B51	6.10 — ion. cham. B51
$\mathrm{At}^{205}$	$\alpha$, EC B51	26 min B54e	5.90 — ion. cham. B51; B54e
$\mathrm{At}^{207}$	$\alpha \sim 10\%$ EC $\sim 90\%$ B51	2.0 h B51 B54e	5.75 — ion. cham. B51; B54e
$\mathrm{At}^{208}$	$\alpha \sim 0.5\%$ EC $\sim 99\%$ H50c	1.7 h H50c	5.65 — ion. cham. H50c
$\mathrm{At}^{209}$	$\alpha \sim 5\%$ EC $\sim 95\%$ B51	5.5 h B51	5.65 — ion. cham. B51	See M54b
$\mathrm{At}^{210}$	$\alpha$ — 0.17% EC — 99% H53a	8.3 h K49b	5.519 (32%) 5.437 (31%) 5.355 (37%) spectrum H55a	See M54b	See M54b		$I=4-$; $5-$ or $6+$ M54b
$\mathrm{At}^{211}$	$\alpha$ — 40.9% EC — 59.1% N51	7.5 h K49b	5.862 — spectrum H55a	0.671 scint. spectrum M54b	$(671\gamma)(\alpha, Kx)$ M54b		$I=4-$; $5-$ or $6+$ M54b

$\mathrm{At}^{212}$	$\alpha$ H53a	0.22 sec W54
$\mathrm{At}^{213}$	$\alpha$ H53a		9.2 — range in photoemulsion H53a
$\mathrm{At}^{214}$	$\alpha$ M49	$\sim 2 \cdot 10^{-6}$ sec M49	8.78 — ion. cham. M49; M51
$\mathrm{At}^{215}$	$\alpha$ K44	$\sim 10^{-4}$ sec G48	8.00 — ion. cham. G48; M51
$\mathrm{At}^{216}$	$\alpha$ K43	$\sim 3 \cdot 10^{-4}$ sec M49; M51	7.79 — ion. cham. G48; M51
$\mathrm{At}^{217}$	$\alpha$ E47; H47	0.021 sec E47 0.018 sec H47	7.00 — ion. cham. H47
$\mathrm{At}^{218}$	$\alpha$ — 99% $\beta^{-}$ — 0.1% W48	1.5–2.0 sec W48	6.7 — ion. cham. W48
$\mathrm{At}^{219}$	$\alpha \sim 97\%$ $\beta^{-} \sim 3\%$ H53a	$\sim 0.9$ min H53a	6.27 — ion. cham. H53a
${}_{86}\mathrm{Em}^{[[unclear: mass number]]}$	$\alpha$ M55a	3 min M55a	6.27–6.30 ion. cham. M55a
$\mathrm{Em}^{206?}$	$\alpha$ M55a	6.2 min M55a	6.22 — ion. cham. M55a

Isotope	Decay mode	Half-life	α-particle energy (in MeV)	γ-radiation energy (in keV)	Data on α—γ and γ—γ coincidences	Decay scheme	Principal nuclear characteristics
Em207	α B54e	11 min M55a B54e	6.12 — ion. cham. M55a 6.09 — ion. cham. B54e
Em208	α ∼ 20% EC ∼ 80% H53a M55b	23 min M55b 21 min M55a	6.141 (100%) spectr. M55b
Em209	α ∼ 17% EC ∼ 83% M55b	30 min M55b	6.037 (100%) spectr. M55b
Em210	α — 4% EC — 96% M55b	2.7 hours M52a M55b	6.037 (100%) spectr. M55b
Em211	α — 25% EC — 75% M52a EC/α = 2.8 M55b	16 hours M52a M55b	5.847 (33.5%) 5.779 (64.5%) 5.613 (2%) spectr. M55b; P54a	70; 150; 400; 600 M53a 70; 169; 234 M55b
Em212	α H50c	23 min H50c M55b	6.262 — spectr. M53a; M55b
Em215	α M52b	∼ 10−6 sec M52b	8.6 — ion. cham. M52b

Column 1	Column 2	Column 3	Column 4	Column 5	Column 6	Column 7	Column 8	Column 9
$\mathrm{Em}^{216}$	$\alpha$ M49	$\sim 10^{-4}$ sec M51	8.01 — ion. chamber H53a
$\mathrm{Em}^{217}$	$\alpha$ M51	$\sim 10^{-3}$ sec M51	7.74 — ion. chamber M51
$\mathrm{Em}^{218}$	$\alpha$ S48	0.019 sec S48	7.127 (100%) spectrum. 6.53 (weak) $\alpha-\gamma$ coincid. P54a	609 — spectrum. S54b
$\mathrm{Em}^{219}$ (An)	$\alpha$ C31	3.92 sec C31	6.807 (69%) 6.542 (15%) 6.418 (12%) 6.197 (4%) R36; B54a	67; 124; 198 (strong) 267; 321; 392; 589 (weak) conversion spectrum. S37 123; 270; 590 cryst. spectrum. F40			Level scheme shown: levels labeled 0, 67, 270, 397, 622; transitions labeled 67, 270, 397, 622; top labeled $\mathrm{Em}^{219}$; reference R36.
$\mathrm{Em}^{220}$ (Th)	$\alpha$ C31	54.5 sec C31	6.278 — spectrum. B53d $a_0 = 6.2823$ B54a 6.282 (100%) 5.747 ($\sim 0.3\%$) spectrum. P54a
$\mathrm{Em}^{221}$	$\alpha \sim 20\%$ $\beta^- \sim 80\%$ M53a	25 min M53a; H53a	6.0 M56

Isotope	Mode of decay	Half-life	Energy of $\alpha$-particles (in MeV)	Energy of $\gamma$-radiation (in keV)	Data from $\alpha$—$\gamma$ and $\gamma$—$\gamma$ coincidences	Decay scheme	Main characteristics of the nucleus
Em$^{222}$ (Rn)	$\alpha$ C31 $\beta$ — stable H53a	3.825 days C31 T51c	5.482 — spectr. B53d 5.4861 B54a
$_{87}$Fr$^{212}$	$\alpha$ — 44% EC — 56% M55b H50c	19.3 min H50c M55b	6.409 (37%) 6.387 (39%) 6.339 (24%) spectr. P54a; M55b
Fr$^{217}$	$\alpha$ H53a		8.3 range in photoemulsion H53a
Fr$^{218}$	$\alpha$ M51	$5\cdot 10^{-3}$ sec M51	7.85 — ion. cham. M51
Fr$^{219}$	$\alpha$ G48	0.02 sec M51	7.30 — ion. cham. M51
Fr$^{220}$	$\alpha$ G48	27.5 sec M51	6.69 — ion. cham. M51
Fr$^{221}$	$\alpha$ E47 H47	5 min E47 4.8 min H50b	6.30 ($\sim$75%) 6.05 ($\sim$25%) ion. cham. H50b	220—conv. el. spectr. conv. spectr. H53a

Fr²²²	α 0.01—0.1% β⁻ — 99% H53a	14.8 min H53a
Fr²²³ (AcK)	β⁻ α — 4·10⁻³ % H53a	21 min P39		49.8; 80; 215; 310 none <40 H54d
₈₈Ra²¹³	α M52a	2.7 min M55b	6.90 — ion. cham. M53a, M55b
Ra²¹⁹	α M52b	∼10⁻³ sec M52b	8.0 — ion. cham. M52b
Ra²²⁰	α M51	3·10⁻² sec M51	7.43 — ion. cham. H53a
Ra²²¹	α M51	30 sec M51	6.71 — ion. cham. M51
Ra²²²	α S48	38 sec S48	6.554 (100%) spectr. 6.23 (weak) α — coincid. P54a	330 — spectr. S54b

Isotope	Mode of decay	Half-life	Energy of α-particles (in MeV)	Energy of γ-radiation (in keV)	Data on α—γ and γ—γ coincidences	Decay scheme	Principal characteristics of the nucleus
Ra²²³ (AcX)	α C31 β — stable. H53a	11.2 days C31 11.68 days H54b	5.860 (sl.) 5.730 (9%) 5.704 (53%) 5.596 (24%) 5.528 (9%) 5.487 (2%) 5.419 (3%) spectrum. reference in H53a 5.750 (11%) 5.719 (53%) 5.607 (25%) 5.540 (9%) 5.433 (2%) spectrum. H53a	144; 155; 180; 270; 340 cryst. spectr. F40 26; 64; 81; 99; 116; 154; 164; 180; 232; 268; 280; 322; 348; 444 conv. spectr. S37
Ra²²⁴ (ThX)	α C31 β — stable. H53a	3.64 days C31	5.681 (95%) 5.448 (4.6%) 5.194 (0.4%) spectrum. R49a α₀ = 5.6814 B54a α₀ = 5.679 B53d	241 — E2 ($e_K/\gamma = 0.13$) ($e_{L_3}/\gamma = 0.08$) R52a R54c	(5.45α) (241γ) δ M54f	![decay scheme for Ra224 showing levels 0, 241, 935, transitions labeled 0.4%, 4.6%, 95%, and daughter Rn220/R52a]
Ra²²⁵	β E47 α — 10⁻⁴ M56

$\mathrm{Ra}^{226}$	$\alpha$ C31 $\beta$ — stable H53a	1590 years C31 1622 years H53a	$\alpha_0 = 4.779$ B54a $\alpha_0$ (94.3%) $\alpha_{188}$ (5.7%) No $\alpha$ in the region 3.6–4.4 Mev (with accuracy 0.02%) spectrum A52c	188; $(e_K/\gamma = 0.15)$ 660 R54d 186 — E2 $(e/\gamma = 0.9)$ V52c F54e	(4.61α) (186γ) $\vartheta$ M54f $(\alpha)$ (188 γ) $\vartheta$ $(\alpha)$ (663 γ) $\vartheta$ R54b	Diagram: $\mathrm{Ra}^{226}$ level scheme; levels marked 0 and 188; branches marked 5.7% and 94.3%; H53a
${}_{89}\mathrm{Ac}^{221}$	$\alpha$ H53a		7.6 — range in photoemuls. H53a
$\mathrm{Ac}^{222}$	$\alpha$ M51	5.5 sec M52b	6.96 — ion. cham. M51
$\mathrm{Ac}^{223}$	$\alpha$ — 99% EC — 1% M51	2.2 min M51	6.64 — ion. cham. M51
$\mathrm{Ac}^{225}$	$\alpha$ E47	10 days E47 H50b	5.80 — ion. cham. H50b
$\mathrm{Ac}^{227}$	$\alpha$ — 1.2% $\beta^{-}$ — 99% P39	22.0 years H50d	4.942 — spectrum H53a 4.95 (85%) 4.95–0.350 (15%) range in air G47	37 (0.2%) 300? L50 37 (0.22%) 300? (0.2%) R53g			$I = {}^{3}/_{2}$ T51b
${}_{90}\mathrm{Th}^{223}$	$\alpha$ M52b	~0.1 sec M52b	7.55 — ion. cham. M52b

Isotope	Mode of decay	Half-life	Energy of α-particles (in MeV)	Energy of γ-radiation (in keV)	Data from α—γ and γ—γ coincidences	Decay scheme	Main characteristics of the nucleus
Th²²⁴	α M51	∼1 sec M51	7.13 — ion. cham. H53a
Th²²⁵	α ∼ 90% EC ∼ 10% M51	8.0 min M51	6.57 — ion. cham. M51
Th²²⁶	α S48	30.9 min S48	6.336 (77%) 6.228 (21%) 6.100 (1.8%) 6.037 (0.6%) spectrum. P54a	109; ∼130; ∼190; 240 spectrum. S54b	(6.100α) (240γ) δ S54b	[[figure: decay scheme with levels labeled 0, 109, 240, 300 and branches to Th²²⁶; S54b]]
Th²²⁷ (RaAc)	α [C31] β — stable. H53a	18.9 days C31 18.17 days H54b	6.030 (19%) 6.001 (5%) 5.972 (21%) 5.952 (13%) 5.922 (∼2%) 5.860 (4%) 5.796 (2%) 5.749 (17%) 5.728 (∼1%) 5.704 (15%) 5.651 (∼2%) spectrum. reference in H53a	50; 57; 80; 101; 113; 129; 208; 240; 258 cryst. spectr. F40 49.8; 87; 140; 235 H54d 29.95; 31.62; 50.13; 61.57; 100.4; 113.3; 173.1; 205.0; 234.6; 236.1;		Decay scheme see in F55b

				256.4; 286.3; 304.8; 312.8; 334.7 conversion spectrum F54d; F55b
Th\(^{228}\) (RaTh)	\(\alpha\) C31 \(\beta\)—stable H53a	1.90 years C31	5.421 (71%) 5.3385 (28%) 5.208 (0.4%) 5.173 (0.2%) spectrum A53b	89 — E2 \((e/\gamma \sim 16)\) 137 — E1 \((e/\gamma \ll 1)\) 169 — E2 \((e/\gamma \sim 1.2)\) 212 — E1 \((e/\gamma \ll 1)\) scint. spectrum A53b 84.4 132.3 — E1 167 214 — E1 prop. counter N54a	\((\alpha)\) \((83\ \gamma)\) ? No \((\gamma)(\gamma)\) No \((\alpha)(86.8\ \gamma)\) [B53e] \((\alpha)\) \((30\ e^-\) conv.) \((\alpha)\) \((e^-\) conv. from \(84\ \gamma)\) J53	Level-scheme diagram: Th\(^{228}\) \((4+)\) 253 \((1-)\) 217 \((2+)\) 84 \((0+)\) 0 transitions labeled 0.084, 0.132, 0.217 S54b
Th\(^{229}\)	\(\alpha\) E47 \(\beta\)—stable H53a	\(\sim 10^4\) years E47 7 340 years H50b	5.02 (\(\sim 10\%\)) 4.94 (\(\sim 20\%\)) 4.85 (\(\sim 70\%\)) ion. chamber H50b				\(I = 5/2\) M54i

Isotope	Mode of decay	Half-life	Energy of α-particle (in MeV)	Energy of γ-radiation (in keV)	Data from α—γ and γ—γ coincidences	Decay scheme	Principal characteristics of the nucleus
Th\(^{230}\) (Io)	α C31 β — stable H53a	\(8.2\cdot 10^4\) years C31 \(8.0\cdot 10^4\) years H53a	(4.685) (76.3%) 4.619 (23.4%) 4.546 (0.07%) 4.474 (0.20%) 4.439 (0.07%) 4.363 (0.08%) 4.293 (0.07%) 4.209 (0.06%) spectrum. R48; R54a	68 (1.05%) 142 (0.14%) 255 (0.05%) coinc. count R53b 67.76 — E2 141 M1 or M2 257 small mult. 203 large. » spectrum. conv. R54f 184; 250 spectrum. S54b 210—4— 68—2— F54f	(140 γ) (68 γ) No (68 γ) (Lx) (250 γ) (68 γ) (250 γ) (140 γ) coinc. count R53b (α) (68 γ) θ (α) (142 γ) θ ion. cham. and scint. count F54f (α) (68 γ) θ T53a (4.47α) (e− conv.) (4.61α) (e− conv.) No (4.44α)(e− conv.) (4.68α) (e− conv.) ion. cham. V53b

$\mathrm{Th}^{232}$	$\alpha$ C31 $\beta$—stable H53a	$1{,}39\cdot 10^{10}$ years K38 Spont. fiss. H53a $1{,}4\cdot 10^{18}$ years H53a Spont. fiss. $4\cdot 10^{17}$ years P47	3,994 (76%) spectrum 3,93 (24%) from $\gamma$ rad. P54a 4,00 (75%) 3,95 (25%) P54b		($\sim 20\%\alpha$) (75 yr) A52d ($\sim 24\%\alpha$) ($\sim 55$ yr) D52b
$_{91}\mathrm{Pa}^{225}$	$\alpha$ H53a	2,0 sec H53a
$\mathrm{Pa}^{226}$	$\alpha$ M51	1,8 min M51	6,81 — ion. chamber M51
$\mathrm{Pa}^{227}$	$\alpha \sim 85\%$ EC $\sim 15\%$ M51	38,3 min M51	6,46 — ion. chamber M51
$\mathrm{Pa}^{228}$	$\alpha \sim 2\%$ EC $\sim 98\%$ M51	22 hours M51	6,09 (75%) 5,85 (25%) ion. chamber M51
$\mathrm{Pa}^{229}$	$\alpha \sim 1\%$ EC $\sim 99\%$ M51	1,5 days H53a	5,69 — ion. chamber M51
$\mathrm{Pa}^{230}$	$\alpha \sim 0{,}003\%$ M51 EC $\sim 92\%$ $\beta^- \sim 8\%$ H53a	17,0 days C48		940 H53a

Isotope	Mode of decay	Half-life	Energy of α-particles (in MeV)	Energy of γ-radiation (in keV)	Data from α—γ and γ—γ coincidences	Decay scheme	Principal nuclear characteristics
Pa²³¹	α β — stable. H53a	3.43·10⁴ years W49 3.2·10⁴ years G30	5.0490 (12%) 5.0205 (23%) 5.0060 (28%) 4.9740 (1.5%) 4.9420 (24%) 4.8476 (1.5%) 4.7270 (10%) 4.7040 (0.8%) 4.6710 (1.3%) spectrum. G55 5.046 (8.7%) 5.018 (23%) 5.001 (28%) 4.938 (27%) 4.843 (1.5%) 4.724 (10%) 4.667 (1.4%) 4.627 (0.3%) spectrum. H55c	95; 294; 323 M28 44; 56; 66 T53b T52b 27 — E1 R53g M54a 357 — M1 331 — M2 301 — M1 383—E1 or M2 259 — M1 198 — E2 102 — E2 82—M1 or M2 64 — E2 57 — E2 38 — E2 spectrum. conv. F53b F54e		![decay scheme: Pa-231 to Ac-227 levels; labels include 365, 332, 325, 205, 109, 76.5, 44.4, 29, 0; transition labels 96, 126, 180, 213, 332, 365; references G55]	I = ³⁄₂ M50c
₉₂U²²⁷	α M52b	1.3 min M52b	6.8 — ion. chamber M52b
U²²⁸	α ∼ 80% EC ∼ 20% M52b	9.3 min M51	6.67 — ion. chamber H53a

U²²⁹	α ∼ 20% EC ∼ 80% M52b	58 min M51	6.42 — ion. chamber M51
U²³⁰	α C48	20.8 days C48	5.888 (68%) 5.819 (31%) 5.662 (0.8%) spectrum P54a	70; 160; 230 S54b			Decay scheme shown: U²³⁰; levels 0, 40, 20, 230; labels (0+), (2+), (4+), (1−); transitions marked 1%; 31%; 68%; S54b
U²³¹	α—5.5·10⁻³% EC — 99% H53a	4.3 days H53a	5.45 — ion. chamber H53a	51; 64; 76 conversion spectrum H53a
U²³²	α H53a	74 years S54i	5.318 (68%) 5.261 (32%) 5.134 (0.3%) spectrum A55a P54a	60 (32%) 130 (0.57%) 270 (0.0096%) 330 (0.070%) spectrum S54h 57.9 (0.21%) E2 131 (0.075%) 268 (0.004%) E2 326 (0.004%) spectrum A55a	(30% α) (60 γ) photoemuls. D52b		Decay scheme shown: U²³²; levels 0, 60, 190, 330; labels (0+), (2+), (4+), (6−); transitions marked 0.3%; 32%; 68%; A55a, S54h
U²³³	α S47c	1.62·10⁵ years 1.63·10⁵ years 1.2·10⁵ years H53a	4.816 (83.5%) 4.773 (14.9%) 4.717 (1.6%) 4.656 (0.07%) 4.582 (0.04%) 4.489 (0.03%) spectrum G56	42.8 (0.05%) 56.1 (0.01%) prop. count. W52b 40; 80 (0.8%; e/γ ∼ 8) 310 (0.1%; e/γ ∼ 3) H53a	(α) (40 γ) photoemuls. B52b (α) (40 γ) (α) (90 γ) (α) (360 γ) H53a		Decay scheme shown: U²³³; levels 0, 42.8, 100, 164, 237, 300; spin labels (5/2), (7/2), (9/2), (7/2), (3/2), (5/2); transitions marked 0.03%, 0.04%, 0.07%, 1.6%, 14.9%, 83.5%; G56	I = 5/2 S55

Isotope	Mode of decay	Half-life	Energy of α-particles (in MeV)	Energy of γ-radiation (in keV)	Data from α—γ and γ—γ coincidences	Decay scheme	Principal characteristics of the nucleus
U²³⁴	α C31 β — stable H53a	2.52·10⁵ years K52 2.475·10⁵ years F52c	4.763 ion. cham. H53a (4.763) (74%) 4.707 (23%) 4.593 (3%) ion. cham. in coincidence with prop. counter V53a 4.768 (72%) 4.717 (28%) spectr. G55	53:93:118 = = 1:0.2:0.4 coinc. spectr. H53a	(α) (55 γ) photoemuls. T55b	Levels: 0, 52, 173 → U²³⁴; transitions marked 52, 22?, 14?, 31?; H53a
U²³⁵	α C31 β — stable H53a	7.13·10⁸ years F52c Sp. div. 1.9·10¹⁷ years S52b	4.58 (10%) 4.47? (~3%) 4.40 (83%) 4.20 (4%) ion. cham. G51a	94:143:184: :289:386 = = 0.9:0.2: :0.1:0.05 coinc. spectr. H53a		Levels: 0, 114, 183, 366 → U²³⁵; transitions marked 94?, 69?, 297?, 102?	I = ¹/₂ S55
U²³⁶	α G51b	2.39·10⁷ years F52c 2.46·10⁷ years J51	4.499 — ion. cham. J51		(27% α) (50 γ) photoemuls. D52b	Levels: 0, ~50 → U²³⁶; transitions marked ~27?, ~73?; H53a
U²³⁸	α C31 β — stable	4.49·10⁹ years K49a	4.18 — ion. cham. A47		(22% α) (45 γ) photoemuls.

Nuclide	Decay	Half-life	Radiation energy / intensity	Gamma rays / notes	Conversion / photoemulsion	Scheme / figure	Spin / notes
${}_{93}\mathrm{Np}^{231}$	$\alpha$ EC? M50b	H53a	$4.51\cdot10^9$ years N39 Sp. del. $1.3\cdot10^{16}$ years P47 $\sim 50$ min M50b	$4.182$ (77%) ion. cam. $4.135$ (23%) by en. $\gamma$ P54a $6.28$ ion. cam. M50b		D52b $(24_{\sigma}/\alpha)$ $(\sim 50\gamma)$ photoemuls. A52b	(level scheme shown) $U^{235}$ $0.23\%$; $\sim71\%$ $\sim 50$ H53a
$\mathrm{Np}^{233}$	$\alpha \sim 10^{-3}\%$ EC 99% M50b	35 min M50b	$5.53$ ion. cam. M50b
$\mathrm{Np}^{235}$	$\alpha \sim 5\cdot10^{-3}\%$ EC $(L/K>9)$ J52	410 days J52	$5.06$ ion. cam. J52	No $\gamma$ H54h
$\mathrm{Np}^{237}$	$\alpha$ $\beta$ — stable H53a	$2.2\cdot10^6$ years H53a	$4.872$ (3.1%) $4.816$ (3.5%) $4.787$ (53%) $4.767$ (29%) $4.713$ (1.7%) $4.674$ (3.3%) $4.644$ (6.0%) $4.589$ (0.5%) $4.52$ (0.02%) ion. cam. M55c $4.866$ (2.5%) $4.803$ (3.4%) $4.781$ (54%) $4.762$ (29.5%) $4.702$ (2.3%) $4.644$ (8.3%) spectr. K56	87; E54 20; 29; 56.8; 86.9; 145; 175; 200 proport. and sc. count. M55c	(80/α) (soft $\gamma$) photoemuls. D52b		$I=5/2$ B54c

Isotope	Decay mode	Half-life	Energy of α-particles (in MeV)	Energy of γ-radiation (in keV)	Data from α—γ and γ—γ coincidences	Decay scheme	Principal nuclear characteristics
\({}_{94}\mathrm{Pu}^{232}\)	\(\alpha \geq 2\%\) EC \(\leq 98\%\) H53a	36 min H53a	6.58 ion. chamber H53a
\(\mathrm{Pu}^{234}\)	\(\alpha \sim 4\%\) EC \(\sim 96\%\)	9.0 hr 8.5 hr H53a	6.19 ion. chamber H53a 6.19 (86%) ion. chamber 6.14 (14%) by γ energy P54a
\(\mathrm{Pu}^{235}\)	\(\alpha \sim 0.002\%\) EC 99% H53a	26 min H53a	5.85 ion. chamber H53a
\(\mathrm{Pu}^{236}\)	\(\alpha\) \(\beta\)—stable H53a	2.7 yr H53a Sp. fiss. \(3.5 \cdot 10^{9}\) yr G52a	5.75 ion. chamber run in air H53a		(20% α) (45 γ) photoemuls. D52b	\(\mathrm{Pu}^{236}\) Levels: 45, 0 Branches marked: 20%, 80% H53a
\(\mathrm{Pu}^{238}\)	\(\alpha\) \(\beta\)—stable H53a	89.6 yr 92 yr H53a Sp. fiss. \(3.8 \cdot 10^{10}\) yr H53a	5.495 (72%) 5.452 (28%) 5.352 (0.09%) spectr. A54b 5.491 (69%) 5.450 (31%) Γ55	17 (13) 43.8 (0.038%) E2 99 (0.0080%) E2 150 (0.0010%) E2 coinc. spectr. A54b 13 (6%) 18 (11%) 42 (0.4%)	(α) (44 γ) δ M54f	\(\mathrm{Pu}^{238}\) Levels: 2.85; (4+) 145; (2+) 44; (0+) 0 Branches marked: 0.09%; 28%; 72% A54b

Nuclide	Decay mode	Half-life	Radiation energies and intensities	Ratios / notes	Method	Level scheme / figure	Spin and parity / references
\(\mathrm{Pu}^{239}\)	\(\alpha\) K46	\(24.3\cdot10^3\) years C49 \(24.4\cdot10^3\) years F54c	5.150 (69%) 5.137 (20%) 5.100 (11%) spectrum A52a 5.147 (72.5%) 5.134 (16.8%) 5.0963 (10.7%) spectrum F55	\(\sim170\) R51b 39:53,1:100: :124:384 \(=\) \(=0.4:1.4:1.1:\) :0.5:0.3 conversion spectrum, coincidence spectrum F52b 52 \((7\cdot10^{-5}\) per \(\alpha)\) 38.5 \((2\cdot10^{-5}\) per \(\alpha)\) proportional counter W52b 49.6 F52b	\((\alpha)\) (50 \(\gamma\)) \((\alpha)\) (35 \(\gamma\)) photoemulsion D52b	Level scheme: \(\mathrm{Pu}^{239}\); levels 57, 13, 0; branches 1%, 20%, 80%; A52a	\(I=1/2\) B54b B54c B54d
\(\mathrm{Pu}^{240}\)	\(\alpha\) \(\beta\)—stable H53a	\(6.3\cdot10^3\) years F54c \(6.24\cdot10^3\) years W51	5.162 (76%) 5.118 (24%) 5.014 (0.1%) spectrum P54a 5.159 (75.5%) 5.115 (24.5%) 5.004 (0.085%) spectrum F56			Level scheme: \(\mathrm{Pu}^{240}\); levels \((4+)\) 158, \((2+)\) 45, \((0+)\) 0; branches 0.005%, 23.5%, 76.5%; F56
\(\mathrm{Pu}^{241}\)	\(\alpha\)—\(10^{-3}\)% \(\beta\)—99% T50a	14 years T50a 13 years M53b	4.893 (75%) 4.848 (25%) spectrum P54a	100:145 \(=5:1\) scint. counter F52b			\(I=5/2\) B54c
\(\mathrm{Pu}^{242}\)	\(\alpha\) T50a \(\beta\)—stable H53a	\(\sim5\cdot10^5\) years T50a	4.898 (80%) 4.854 (20%) spectrum P51a

Isotope	Mode of decay	Half-life	Energy of α-particles (MeV)	Energy of γ-radiation (keV)	Data from α—γ and γ—γ coincidences	Decay scheme	Main nuclear characteristics
$_{95}\mathrm{Am}^{237}$	α 0.005% EC 99% H53a	$\sim 1.3$ h H53a	6.01 ion. cham. H53a
$\mathrm{Am}^{239}$	α 0.003% EC 99% H53a	12 h H53a	5.75 ion. cham. H53a	48 (50%) A55c 300 (10%) H53a
$\mathrm{Am}^{241}$	α β—stable H53a	470 years H52a	5.535 (0.34%) 5.503 (0.21%) 5.476 (84.2%) 5.433 (13.6%) 5.379 (1.4%) spectr. A54a 5.5408 (0.39%) 5.5082 (0.24%) 5.4820 (85%) 5.439 (12.8%) 5.386 (1.66%) 5.321 (0.015%) 5.241 (0.002%) T55	26.38; E1 33.22; M1 + E2 (?) 43.43; E2 + M1 59.62; E1 → M2 101; 165; 208; 268; 330; 370; 433; 69? 124? 193? spectr. conv. B55 26.38 $\left(\dfrac{e^-+\gamma}{\alpha}=0.22\right)$ 33.24 ($\sim 0.1$) 43.4 (0.12) 56.0 (0.005) 59.62 (0.77) 99.5 (0.0018) 128 (?) ($6\cdot 10^{-5}$) spectr. J55	(α) (59.7 γ) (α) (26.3 γ) B52c (α) (60 γ) ∂ R55b (43 γ) (59 γ) (43 γ) (26 γ) (43γ + 26γ) (33γ) T55b	![[unclear: decay-scheme diagram with levels labeled Am²⁴¹ and energies 0, 33.1, 59.8, 102.2, 157.2, 224, 268, 303, 433; spin labels including $I+5/2$, $I+3/2$, $I+1/2$, $I-1/2$, $I-3/2$, $I-5/2$; transition labels including 6.03%, 0.24%, 0.24%, 0.10%, 0.63%, and references r55]]	$I=5/2$ F53c

Am$^{242}$	$\alpha$; $\beta^{-}$; EC H53a $\alpha/\beta^{-}=0.01$ S50	$\sim 100$ years S50		See also M54h D55b T55b 38; 53 conv. spectr. H53a			$I \leq 2$ $I \ne 3$ G52b
Am$^{243}$	$\alpha$ S50	$8.8\cdot10^{3}$ years D53a $7.6\cdot10^{3}$ years A54a	5.340 (0.17%) 5.309 (0.16%) 5.267 (87.1%) 5.224 (11.5%) 5.169 (1.1%) spectr. S54a 5.267 (84%) 5.225 (13%) 5.171 (3%) spectr. A54a	75 (80%) E1 ($e/\gamma < 0.25$) A54a S54a	(80% $\alpha$) (75 $\gamma$) A54a S54a	Level scheme diagram: Am$^{243}$; levels marked $\gamma(3/2+)$, $\gamma(7/2+)$, $\gamma(5/2+)$, $(1/2-)$ with energies 173, 116, 74, 32, 0; transitions labeled 11%, 15%, 0.1%, 0.2%, 87%, 1.0%; S54a	$I=5/2$ K54b
$_{96}$Cm$^{238}$	$\alpha > 10\%$ EC $< 90\%$ H53a	2.5 hours H53a	6.50 — ion. cham. H53a
Cm$^{240}$	$\alpha$ No EC H53a	26.8 days H53a sp. fiss. $7.9\cdot10^{6}$ years G52a	6.25 — ion. cham. H52b

Isotope	Mode of decay	Half-life	Energy of α-particles (in MeV)	Energy of γ-radiation (in keV)	Data from α—γ and γ—γ coincidences	Decay scheme	Main nuclear characteristics
Cm\(^{241}\)	α ∼ 0.2% EC 99% H53a	35 days H53a	5.90 — ion. chamber H53a 5.95 — ion. chamber P54a
Cm\(^{242}\)	α R50a	162.5 days Sp. fission \(7.2 \cdot 10^6\) years H50a 163 days H54f	6.110 (73.7%) 6.066 (26.3%) 5.965 (0.035%) spectr. A53a	44 (\(e/\gamma = 620\)); 100 (\(e/\gamma = 5\)); 157 coinc. spectr. A53a	(23% α) (45 γ) photoemuls. D52b
Cm\(^{243}\)	α R50a	∼ 100 years T50a	5.985 (6%) 5.777 (78%) 5.732 (13%) 5.679 (3%) spectr. A53a P54a	226; 277 A53a	(5.777 α) (226 γ) (5.777 α) (277 γ) A53a

\(\mathrm{Cm}^{244}\)	\(\alpha\) R50a	19.2 years S54d 18.4 years F54b spont. fiss. \(1.4 \cdot 10^{7}\) years G52a	5.798 (75%) 5.755 (25%) spectrum A53a			diagram labeled \(\mathrm{Cm}^{244}\), 0, 43, 25%, 75%, A53a
\(\mathrm{Cm}^{245}\)	\(\alpha\) H51	\(>500\) years H51 \(2 \cdot 10^{4}\) years H54e	5.6 — ion. cham. H51 5.36 no weak ones ion. cham. H54e
\(\mathrm{Cm}^{246}\)	\(\alpha\) F54b	\(4 \cdot 10^{3}\) years F54b
\({}_{97}\mathrm{Bk}^{243}\)	\(\alpha \sim 0.1\%\) EC 99% T50a	4.6 hours G54c T50a	6.72 (30%) 6.55 (53%) 6.20 (17,%) ion. cham. G54c, T50a
\(\mathrm{Bk}^{245}\)	\(\alpha \sim 0.1\%\) EC 99% H51	4.95 days H51	6.33 (18%) 6.15 (48%) 5.90 (34%) ion. cham. H51

Isotope	Decay mode	Half-life	Alpha-particle energy (in MeV)	Gamma-radiation energy (in keV)	Data on \(\alpha\)—\(\gamma\) and \(\gamma\)—\(\gamma\) coincidences	Decay scheme	Basic nuclear characteristics
\(\mathrm{Bk}^{249}\)	\(\alpha?\ \beta^{-}\) \((\beta^{-}/\alpha \approx 10^{5})\) M54	1 yr. D54 290 days M54 Sp. fiss. \(>2\cdot10^{9}\) years M54	5.4 — ion. cham. D54 M54
\({}_{98}\mathrm{Cf}^{244}\)	\(\alpha;\ \mathrm{EC}?\) T50c	45 min T50c G54c	7.15 — ion. cham. T50c; T50e; G54c
\(\mathrm{Cf}^{246}\)	\(\alpha\) G51d	35.7 hr H51 H55c Sp. fiss. \(\sim 2000\) years H53a	6.753 (78%) 6.711 (22%) (6.608) \((>0.08\%)\) spectr. H55d	42; 145 spectr. H55d		Decay scheme diagram shown with levels \(145\), \(42\), \(0\); alpha branches labeled approximately \(7.006\), \(22\%\), \(78\%\); reference H55d.
\(\mathrm{Cf}^{248}\)	\(\alpha\) G54c	225 days G54c 250 days Sp. fiss. \(7\cdot10^{3}\) years H54c	6.26 — ion. cham. G54c

Cf$^{249}$	$\alpha$ T54	400 years G54b 470 years spont. fiss. $>5\cdot10^6$ years M54 550 years spont. fiss. $>10^6$ years D54	6,00 (10%) 5,82 (90%) G54b
Cf$^{250}$	$\alpha$ G54b D54	12 years spont. fiss. $5\cdot10^3$ years G54b 9,4 years spont. fiss. $>10^4$ years D54 10,0 years spont. fiss. $1,5\cdot10^4$ years M54e	6,033 (90%) 5,99 (10%) ion. chamber M54
Cf$^{252}$	$\alpha$ G54b D54	2 years spont. fiss. 100 years G54b 2,1 years spont. fiss. 60 years D54 2,2 years spont. fiss. 66 years M54e	6,117 (90%) 6,08 (10%) ion. chamber M54e

Isotope	Mode of decay	Half-life	Energy of $\alpha$-particles (in MeV)	Energy of $\gamma$-radiation (in keV)	Data from $\alpha$—$\gamma$ and $\gamma$—$\gamma$ coincidences	Decay scheme	Main characteristics of the nucleus
${}_{99}\mathrm{E}^{247}$	$\alpha$; EC? G54c	7.3 min G54c	7.35 G54c
$\mathrm{E}^{253}$	$\alpha$ S54g	20 days S54g C54c $\sim 30$ days T54 Sp. fission $>10^5$ years F54a	6.64 S54g 6.63 C54
$\mathrm{E}^{254}$	$\alpha$	$>2$ years H55b	6.44 H55b
$\mathrm{E}^{255}$	$\alpha$	30 days C54c
${}_{100}\mathrm{Fm}^{250}$	$\alpha$	30 min A54c	7.7 A54c

\( \mathrm{Fm}^{254} \)	\(\alpha\)	3 h H54a 3.3 h Sp. fis. 240 d F54a	7.22 C54c 7.17 F54a	42 \((2\cdot 10^{-2}\%)\) 94 \((4.4\cdot 10^{-2}\%)\) spectr. A55b			[[decay-scheme diagram with levels labeled \(0\), \(42\), \(136\), \( \mathrm{Fm}^{254}\), \( \mathrm{A55b}\), and transitions labeled \(0.03\%\), \(42\), \(94\)]]
\( \mathrm{Fm}^{255} \)	\(\alpha\)	15 h C54c	7.1 C54c

The table contains all \(\alpha\)-radioactive isotopes known as of April 1, 1956, from \({}_{83}\mathrm{Bi}\) to \({}_{100}\mathrm{Fm}\) (several \(\alpha\)-active isotopes lying in the rare-earth region are not included in the table).

The following designations have been adopted:

\[ \begin{aligned} \mathrm{EC} &\text{— electron capture,}\\ e_K^c &\text{— conversion electrons on the }K\text{-shell,}\\ e_L^c &\text{— conversion electrons on the }L\text{-shell,} \end{aligned} \]

Sp. fis. — spontaneous fission,
spectr. — spectrometer,
scint. counter — scintillation counter,
conv.-electron spectr. — conversion-electron spectrometer,
ion. chamber — ionization chamber,
cryst. spectr. — crystal spectrometer,
absorpt. — radiation energy determined by absorption.

In the column “\(\gamma\)-radiation energy” are entered the \(\gamma\)-rays associated with \(\alpha\)-radiation; \(\gamma\)-rays associated with \(\beta\)-radiation, as a rule, were not entered in the table, although references to the relevant literature are given in the tables.

In the column “Data on \(\alpha\)—\(\gamma\) and \(\gamma\)—\(\gamma\) coincidences,” the letter \(\theta\) marks experiments on angular correlations between \(\alpha\)- and \(\gamma\)-rays.

References to Hollander, Perlman, and Seaborg’s Table of Isotopes (H53a) are given in our tables in those cases where they refer to data not published in print readily accessible to Soviet readers.

References Cited

a) Literature published in Soviet editions

B48. Bethe H. A., Nuclear Physics, Part II. Gostekhizdat, Moscow–Leningrad (1948).

B54. Blatt and Weisskopf, Theoretical Nuclear Physics. Foreign Literature Publishing House, Moscow (1954).

B55. Baranov S. A., Shlyagin K. N., Collection of Reports at the Session of the Academy of Sciences of the USSR, 1955, p. 251.

G55. Goldin L. L., Tretyakov E. F., Novikova G. I., Collection of Reports at the Session of the Academy of Sciences of the USSR, 1955, p. 226.

G56. Goldin L. L., Novikova G. I., Tretyakov E. F., Report at the 6th Conference on Nuclear Spectroscopy, 1956.

K56. Kondratyev L. N., Novikova G. I., Vorob’ev A. M., Goldin L. L., Report at the 6th Conference on Nuclear Spectroscopy, 1956.

L49. Landau L., Lifshitz E., Quantum Mechanics, Gostekhizdat (1949).

L56. Landau L. D., Report at the 6th Conference on Nuclear Spectroscopy, 1956.

N55. Nosov V. G., Reports of the Academy of Sciences of the USSR 103, 65 (1955).

P47. Perfilov N. A., ZhETF 17, 476 (1947).

S54. Sliv L. A., Perker L. K., Reports of the Academy of Sciences of the USSR 99, 727 (1954).

b) Literature published in foreign editions

A

A47. Alder F., Huber P., Metzger F., Helv. Phys. Acta 20, 234 (1947).

A50. Avignon P., J. Phys. Rad. 11, 521 (1950).

A51a. Ambrosino G., Piattier H., C. R. 232, 400 (1951).

A51b. Alburger D. E., Friedlander G., Phys. Rev. 81, 523 (1951).

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ALPHA DECAY

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V

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W

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Z

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