EXPERIMENTAL FOUNDATIONS OF THE NUCLEAR SHELL MODEL

M. I. Korsunskii

1. INTRODUCTION

Up to the present time several simple models of the structure of the atomic nucleus have been proposed. The most important among them are the liquid-drop model and the model of nucleon shells. Each of these simple models satisfactorily explains particular nuclear phenomena, but none can explain the entire set of known properties of atomic nuclei.

The liquid-drop model of the nucleus explains rather well many phenomena connected with atomic nuclei, in particular the fission of atomic nuclei; the theory of this phenomenon on the basis of the liquid-drop model was first given by Ya. I. Frenkel’¹. Common to the liquid-drop model of the nucleus, as well as to a number of other models (for example, the continuous model), is the statistical method of treatment. Models in which the statistical method is applied give good results for excited nuclei, but, as has become clear recently, are inapplicable to the fundamental, weakly excited states of nuclei. This latter circumstance is manifested especially clearly in a number of important phenomena, such as isomerism, periodicity in properties, regularities in the magnitude of nuclear spin, and others. These phenomena are well explained by the nucleon-shell model.

In the present review we propose to illuminate the principal experimental data that testify in favor of the model of nucleon shells.

The conception of the nucleus as a system formed by nucleons moving as independent particles in a self-consistent field, which is formed by all the nuclear particles, arose and was developed as early as 1932–1934²–⁸, immediately after the appearance of D. D. Ivanenko’s hypothesis² that protons and neutrons are structural constituents of atomic nuclei. According to this model,

nuclear particles—protons and neutrons—since, having spin equal to one half, they cannot all be in one and the same quantum state. In heavy nuclei containing a large number of particles, protons and neutrons, like electrons in atoms, must be distributed among states characterized by various quantum numbers—$n$ (the principal quantum number), $l$ (the quantum number determining the angular momentum), $m$ (the quantum number determining the projection of the angular momentum), and $s$ (the spin number).

The distribution of particles over the various energy levels (the number of particles that can be found at a given energy level is determined by the number of allowed states for that level) will depend on the character of the forces acting on the nuclear particles. The aggregate of particles situated at one or several energy levels close to one another in magnitude is called (by analogy with electrons in the atom) a nuclear shell. The successive filling of nuclear shells by nucleons should lead to the formation of especially stable isotopes, just as the filling of electron shells leads to the formation of atoms of the noble gases. Of course, the analogy between nucleon shells in the nucleus and electron shells in the atom is a very remote one and, to a certain extent, purely formal, since nuclear forces by their nature differ from the forces acting on an electron in an atom.

As soon as it became clear that the nucleus is a system of strongly interacting particles, the development of the model of nucleon orbits, which had been the first proton-neutron nuclear model, came to a halt, since the idea of nucleon orbits seemed incompatible with the idea of the strong interaction of the particles forming the nucleus. Indeed, one can speak of any orbital motion of one particle among a group of other particles making up the nucleus (Fig. 1) only in the case where the mean free path of the particle with respect to energy exchange is sufficiently large and, in any case, exceeds the dimensions of the nucleus.

Fig. 1.

At first glance it seems that this condition cannot be fulfilled, for it is known that the effective cross section of $n$-$p$ scattering at an energy of the order of 20 MeV is approximately equal to $0.3 \cdot 10^{-24}\ \text{cm}^2$, which, at the existing density of particles in the nucleus, gives a value of the mean free path approximately equal to $1/8$ of the nuclear radius; that is, in other words, the interaction between nuclear particles is so strong that along a path, on the average equal to $1/8$ of the nuclear radius,

occur collisions between particles and, consequently, an exchange of energy between the colliding particles. The presence of such a strong interaction and of a rapid exchange of energy between particles, which underlies the liquid-drop model of the nucleus, seemed incompatible with the idea of nucleon shells. However, as was pointed out by Weisskopf^9 and Fermi^10,11, such a rapid exchange of energy will take place only for an external (extraneous) nucleon that brings a large excess of energy into the nucleus. Such a fast particle, having entered the nucleus, will in fact undergo frequent collisions (in accordance with the above-mentioned mean free path), as a result of which it will transfer its excess energy to particles that were previously in the nucleus (formation of a compound nucleus). The small mean free path ensures a rapid and complete exchange of energy between the nucleon that has entered the nucleus (with high energy) and the nucleons that were already in the nucleus. However, the exchange of energy between nucleons situated in a normal, unexcited nucleus will not occur so often. The mean free path with respect to energy exchange for a nucleon situated in the nucleus must be considerably greater than \(1/3\) of the nuclear radius. The point is that in an unexcited normal nucleus the nuclear particles for the most part occupy the lowest possible energy states; therefore a “collision” between them does not lead to an exchange of energy, since neither one of the interacting particles can transfer part of its energy to the other particle without passing into a state of lower energy, while all states of lower energy, according to the Pauli principle, prove to be occupied. Therefore most “collisions” between nucleons (normally situated in the nucleus) do not lead to a change in the state of motion, and the actual “mean free path” proves to be considerably larger than what follows from the data of the liquid-drop model of the nucleus and from experimental data on the interaction of neutrons with nuclei. If all the particles situated in the nucleus occupy levels with the lowest possible energy values, then there will be no collisions at all that lead to a change of energy.

The application of the Pauli principle to the nucleus indicates, on the one hand, the necessity of distributing nucleons among different states, and, on the other hand, the possibility of the existence of orbital motion in the nucleus, which creates, at least in principle, the admissibility of the nuclear shell model.

In the shell model the state of a nucleon in the nucleus is defined as the state of a particle situated in some averaged self-consistent field that is constant in time.

Such an approximation in the case of the electron shell, as is well known, is quite good; however, the possibility of applying it

M. I. KORSUNSKY

for a nuclear system raises many doubts. In the nucleus there is no such central body as the nucleus itself is in the electron shell of an atom, and which alone determines the greater part of the potential energy. On the other hand, the forces acting between nucleons are typical forces with a short range of action. Any given nucleon is acted upon by only a small number of other nucleons, and therefore the statistical character of the phenomenon in the case of the nucleus will be expressed much more weakly than in the case of the atom.

Until quite recently decisive importance was attached to these negative arguments. However, as it became clear that the consequences following from the simple model of nucleon shells satisfactorily explain an ever larger number of properties of atomic nuclei, the sharpness of these negative arguments was somewhat blunted.

Thus, for example, Fermi, pointing out that the model of nucleon orbits is not so very implausible, writes\(^ {10}\): “When one nucleon passes near neighboring nucleons, the potential wells may be situated so close to one another that, upon superposition, they form an approximately uniform potential.” Consequently, Fermi admits the possibility of an approximate representation by some averaged, time-independent field. To the extent that the model of nucleon orbits agrees with experimental data, it is justified.

Before proceeding to an exposition of the facts testifying in favor of the existence of nucleon orbits, we shall give here some consequences following from the idea of the motion of a nucleon in a time-independent potential field. First of all it is necessary to choose the potential of this self-consistent field. This potential may be chosen in the form of a rectangular potential well, i.e., one may set the potential \(U=0\) for \(r>R\) (where \(R\) is the radius of the nucleus) and \(U=-U_0\) for \(r \leq R\). One may also choose the potential inside the nucleus in the form of the potential of a harmonic oscillator. In what follows a harmonic three-dimensional oscillator is considered with terms shifted by the additive constant \(U_0\):

\[ U=-U_0+\frac{1}{2}cr^2=-U_0+\frac{1}{2}M\omega^2r^2, \tag{1} \]

where \(M\) is the mass of the particle moving in the field \(U\), \(\omega\) is the frequency of the “classical” oscillator, and \(r\) is the distance characterizing the position of the particle. The energy of a particle in such a field is determined\(^8\) by the expression*)

\[ E=-U_0+\hbar\omega\left(N+\frac{1}{2}\right), \tag{2} \]

*) The potential functions for the proton and the neutron, generally speaking, must be different, since the proton has charge, whereas the neutron does not.

where

\[ N=n_1+n_2+n_3+1, \tag{3} \]

and the integers \(n_1, n_2, n_3\) are quantum numbers associated with oscillations along the axes \(x, y\), and \(z\), respectively.

In the case \(N=1\) the wave function may be represented in the form

\[ \psi=e^{-\frac{1}{2}\rho^2}, \tag{4} \]

where

\[ \rho=\frac{M\omega r^2}{\hbar}, \tag{5} \]

and \(\hbar\) is Planck’s constant divided by \(2\pi\).

In such a state the angular quantum number \(l\) is equal to zero. This, according to spectroscopic notation, is the \(1s\) level.

For \(N=2\)

\[ \psi=e^{-\frac{1}{2}\rho^2}\rho\cos\theta . \tag{6} \]

In such a state the angular quantum number \(l\) is equal to one. This is, consequently, the \(2p\) level.

In spectroscopy it is customary to consider that the principal quantum number \(n\), corresponding to the lowest level with angular quantum number \(l\), is equal to \(l+1\). In a number of works concerning nuclear shells, the lowest level with a given \(l\) is assigned the quantum number 1, independently of the value of \(l\). This is inconvenient if one has in mind the analogy with atomic spectroscopy. In the notation adopted here, between the “energy quantum number” \(N\), the principal quantum number \(n\), and the angular quantum number \(l\) there will exist the relation

\[ N=2n-l-1. \tag{7} \]

In the case \(N=3\) the function \(\psi\) will have several values:

\[ \psi_1=e^{-\frac{1}{2}\rho^2}\left(\rho^2-\frac{3}{2}\right) \tag{8} \]

and

\[ \begin{aligned} \psi_2&=e^{-\frac{1}{2}\rho^2}\rho^2\left(\frac{3}{2}\cos^2\theta-\frac{1}{2}\right),\\ \psi_3&=e^{-\frac{1}{2}\rho^2}\rho^2\sin^2\theta\sin^2\varphi, \end{aligned} \tag{9} \]

i.e., for the case \(N=3\) two states are possible. One of them

(function \(\psi_1\)) has \(l=0\) and represents the level \(2s\), while the other state (characterized by the functions \(\psi_2\) and \(\psi_3\)) has \(l=2\) and corresponds to the level \(3d\).

For \(N=4\) one obtains states with the value of \(l\) equal to one (the level \(3p\)) and \(l=3\) (the level \(4f\)).

Thus, for the oscillator potential the levels are found to be highly degenerate. The second level \(s\) coincides with the first level \(d\), etc.

Let us note that levels with an even value of the number \(N\) correspond to odd values of the angular quantum number \(l\).

For a given energy (given \(N\)) the number of different quantum levels in the scheme \((nl)\), depending on whether \(N\) has an even or odd value, will be equal either to \(\frac{1}{2}N\) or to \(\frac{1}{2}(N+1)\). To these levels there correspond values of the number \(l\) equal to \(N-1, N-3, N-5\), etc., and principal quantum numbers \(n\) equal to \(N, N-1, N-2,\ldots\), respectively. The total statistical weight of the energy level \(N\), taking spin into account (a factor of 2 in the statistical weight), is equal to \(N(N+1)\). Thus the weights of the levels with \(N=1,2,3,4,5,6,7,\ldots\) are respectively \(2, 6, 12, 20, 30, 42, 56\). Consequently, the total number of quantum states for which \(N\) is less than or equal to \(N_0\) is equal to \(2, 8, 20, 40, 70, 112, 168\) (for values of \(N_0\) from one to seven).

Fig. 2. Energy levels in an oscillator potential, a potential well of infinite depth, and a well with a finite wall height.

Thus, if the self-consistent field for nuclear particles were expressed by an oscillator potential, then the filling of nuclear shells would have to occur in nuclei containing particles (protons or neutrons) in the amounts

\[ 2,\ 8,\ 20,\ 40,\ 70,\ 112,\ 168. \tag{10} \]

If, as the potential of the self-consistent field, one chooses a rectangular potential well with infinitely high[^5] or finite walls[^6], then in this case as well the sequence of levels proves to coincide with that which occurs for the oscillator potential. However, in this case there disappears that “accidental” degeneracy of levels with different \(l\), but identical values of \(N\), which occurs in the case of the oscillator po-

tial. The oscillator level \(N\) splits (in the case of a rectangular well) into levels with given \(l\) and \(n\) in such a way that the level with the largest \(l\) lies below all the others.

In Fig. 2 the arrangement of levels is compared for the oscillator potential, the infinite rectangular well, and the well with finite walls.

In this figure all the levels of the infinite well are represented that lie below the value \(100 \dfrac{\hbar^2}{MR^2}\) (where \(R\) is the radius of the well). The numerical values of the energies of these levels, expressed in units of

\[ \frac{\hbar^2}{MR^2}, \]

are given in Table I.

Table I

Energy levels of a potential well with infinitely high walls, in units of \(\dfrac{\hbar^2}{MR^2}\)

\(l\)	First level: level designation	First level: energy	Second level: level designation	Second level: energy	Third level: level designation	Third level: energy	Fourth level: level designation	Fourth level: energy
0	\(1s\)	4.93	\(2s\)	19.74	\(3s\)	44.42	\(4s\)	78.96
1	\(2p\)	10.12	\(3p\)	29.85	\(4p\)	59.45	\(5p\)	98.92
2	\(3d\)	16.61	\(4d\)	41.35	\(5d\)	75.96	\(6d\)	—
3	\(4f\)	24.40	\(5f\)	54.25	\(6f\)	93.83	\(7f\)	—
4	\(5g\)	33.51	\(6g\)	68.49	\(7g\)	—	\(8g\)	—
5	\(6h\)	43.76	\(7h\)	83.98	\(8h\)	—	\(9h\)	—
6	\(7i\)	55.27	\(8i\)	—	\(9i\)	—	\(10i\)	—
7	\(8j\)	67.98	\(9j\)	—	\(10j\)	—	\(11j\)	—

According to the scheme presented above, one should expect the following sequence of filling of quantum states. The first two neutrons and the first two protons are placed on the \(1s\) level and form the first shell. In the nucleus \({}^{4}_{2}\mathrm{He}\) (two protons and two neutrons) the first shell is filled. In nuclei with a large number of protons and neutrons, the next six particles are placed on the \(2p\) level. These particles form the second shell, and so on.

In Table II the energy levels of the infinite potential well are grouped into groups (shells), and there are given: the number of quantum states \((\eta_l)\) for each level, the number of states \(N_l\) with approximately the same energy. The data given in the last line \((\sum N_l)\) show the numbers of neutrons and protons for which (in the case of a potential well with infinite walls) filling of the group of shell levels should occur.

Table II

Filling of shells (neutrons or protons) in a potential well with infinitely high walls

Shell	1	2	3	4	5	6	7	8	9	10
	\(1s\)	\(2p\)	\(3d\quad 2s\)	\(4f\)	\(3p\)	\(5g\)	\(4d\quad 6h\quad 3s\)	\(5f\quad 7i\)	\(4p\)	\(8j\quad 6g\)
\(n_i\)	2	6	\(10\quad 2\)	14	6	18	\(10\quad 22\quad 2\)	\(14\quad 26\)	6	\(30\quad 18\)
\(N_i\)	2	6	12	14	6	18	34	40	6	48
\(\Sigma N_i\)	2	8	20	34	40	58	92	132	138	186

Depending on the degree of filling of the shell, we must expect in nuclei a relatively greater or lesser degree of stability. When a new shell begins, the binding energy of the newly added particles must be less than the binding energy of the preceding particles that have filled the shell. On the basis of such a scheme, one should expect that the 3rd, 9th, 21st, etc. proton or neutron should be bound considerably more weakly than the 2nd, 8th, or 20th.

2. On “Magic Numbers”

The very fact of the distribution of nucleons among shells would be reliably established, and the character of this distribution would be rigorously determined, if we could determine experimentally the energy levels of all nucleons in the nucleus, as has been done by means of X-rays for electrons in the atom. At present, however, we do not have such direct data at our disposal. Therefore the presence of nuclear shells must be judged from a number of more or less indirect data.

The first conclusions about the existence of nuclear shells were made on the basis of various regularities observed among the isotopes of stable nuclei. In recent years a number of works by M. A. Levitskaya\(^{12—16}\), A. P. Z’noiko\(^{17—19}\), and other investigators\(^{20—25}\) have been devoted to this question. It is known, for example, that all light elements from lithium to oxygen (atomic number 8) have only two stable isotopes, of which one (the lighter) corresponds to the equality \(N=Z\), while in the other (the heavier) the number of neutrons is greater by one than the number of protons. Oxygen also has such isotopes, but, in addition, oxygen has a third isotope with mass number 18. Beginning with oxygen, the regularity in the number of isotopes is different—elements with odd \(Z\) have only one sta-

ble isotope (F, Na, Al, P), while elements with even \(Z\) have three isotopes (O, Ne, Mg, S). The change that occurs at \(Z=8\) in the character of the isotopic composition of the elements may be a consequence of the fact that the particles have filled one of the nuclear shells. By tracing such changes in the character of the isotopic composition of the elements, one can reveal the presence of other nuclear shells. Unfortunately, however, the regularities in isotopic composition have a complicated and insufficiently distinct character. Instead of the isotopic composition of individual elements, one may use other properties of nuclei that vary periodically (or, more precisely, conditionally periodically) over the table of stable nuclei. Thus one may consider, as A. P. Znoiko did, the change in the specific charge with increasing number of particles in the nucleus, or the abundance of elements and their principal isotopes, as M. L. Chepelevetskii did\({}^{23}\). The presence of irregularities in the course of the specific charge \(\dfrac{Z}{A}\), as well as the presence of irregularities in the abundance of elements, undoubtedly indicates that nucleons in nuclei are distributed in definite groups. However, the data on the composition of such groups (shells) are not obtained with sufficient definiteness. More complete data on the composition of such groups can be obtained from consideration of the values of the binding energy and mass defects of nuclei, from regularities in the magnitude of the nuclear spin, and from data on the magnetic and quadrupole moments of nuclei. These data may at the same time give an idea of the character of nuclear levels (the quantum numbers \(j\), \(l\), \(s\)) and, consequently, serve as a basis for the systematics of nuclear levels. It turned out that from these, and also from certain other data, it follows that, apparently, the series of numbers:

\[ 2,\ 8,\ (14),\ 20,\ (28),\ 50,\ 82,\ 126 \tag{10a} \]

represents the numbers of nucleons filling successively arranged nuclear shells. These numbers are often called “magic” numbers in the foreign literature. It should be noted that the series of numbers (10a) does not coincide with the series of numbers (10), which corresponds (see Table II) to the filling of levels in the case of a rectangular potential well. This discrepancy will be discussed later; for now let us turn to those experimental facts which confirm the special stability of nuclei whose number of nucleons is expressed by table (10a).

3. BINDING ENERGY OF NUCLEONS

The existence of such especially stable nuclei could be judged, first of all, from the magnitude of the binding energy. Stable nuclei must possess the greatest binding energy. Nucleons attached to such nuclei will be bound to the nucleus more weakly than

those particles which form a filled group characterizing the given stable nucleus.

The stability of the nucleus \({}^{4}_{2}\mathrm{He}\), containing two protons and two neutrons, is well known. The energy released upon the addition of the next nucleon increases sharply in the series of nuclei \({}^{1}_{1}\mathrm{H}\), \({}^{2}_{1}\mathrm{H}\), \({}^{3}_{2}\mathrm{He}\), \({}^{4}_{2}\mathrm{He}\). The nuclei \({}^{5}_{2}\mathrm{He}\) and \({}^{5}_{3}\mathrm{Li}\), however, which are formed by adding to the \({}^{4}\mathrm{He}\) nucleus one neutron or one proton, respectively, are unstable. The binding energy of the fifth particle in these nuclei is not positive but negative, i.e. when a nucleon is added to the \({}^{4}\mathrm{He}\) nucleus, energy is not released but absorbed. The nucleus \({}^{5}_{2}\mathrm{He}\), for example, decays into \({}^{4}_{2}\mathrm{He}\) and a neutron with the release of energy.

It is difficult to use directly the data on binding energy to establish the special stability of nuclei with 8, 20, etc. particles, since on the curve expressing the dependence of the energy of attachment of one nucleon to a nucleus on the number of particles in the nucleus (Fig. 3), sharp maxima are clearly visible which correspond to nuclei with even numbers of protons and neutrons. The effect of the evenness of particles in light nuclei (and not only in light ones) obscures the change in the nucleon binding energy that is due to the fact of filling a nuclear shell.

Fig. 3. Dependence of the binding energy of the last nucleon on the number of nucleons among stable light nuclei.

In light nuclei it obscures the change in the nucleon binding energy that is due to the fact of filling a nuclear shell. From the curve in Fig. 3 it is evident that nuclei with an even number of particles—\({}^{8}_{4}\mathrm{Be}\), \({}^{12}_{6}\mathrm{C}\), \({}^{16}_{8}\mathrm{O}\), \({}^{20}_{10}\mathrm{Ne}\)—are especially stable with respect to the removal of one nucleon*). However, if we compare the value of the energy released upon the addition to these nuclei of one or

*) The nucleus \({}^{8}_{4}\mathrm{Be}\) is in fact unstable, but it is unstable with respect to fission of the nucleus—decay of the nucleus into two alpha particles; with respect to the removal of one nucleon, however, the nucleus \({}^{8}_{4}\mathrm{Be}\) must be very stable.

two new particles, then the special stability of the nucleus \({}^{16}_{8}\mathrm{O}\) will immediately appear.

Table III gives the value of the energy released upon the attachment to nuclei containing an even number of particles \(({}^{8}_{4}\mathrm{Be}, {}^{12}_{6}\mathrm{C}, {}^{16}_{8}\mathrm{O}, {}^{20}_{10}\mathrm{Ne})\) of one neutron (column 2) or one

Table III

Nucleus	Energy (in MeV) released by attachment to the nucleus: one neutron	Energy (in MeV) released by attachment to the nucleus: one proton	Energy (in MeV) released by attachment to the nucleus: one neutron and one proton	Energy (in MeV) released by attachment to the nucleus: two neutrons
\({}^{8}_{4}\mathrm{Be}\)	1.668	−0.182	8.25	8.475
\({}^{12}_{6}\mathrm{C}\)	4.959	1.943	12.493	13.122
\({}^{16}_{8}\mathrm{O}\)	4.142	0.605	9.729	12.182
\({}^{20}_{10}\mathrm{Ne}\)	6.759	3.626	13.506	17.162

proton (column 3), or a neutron and a proton (column 4), or two neutrons (column 5). (The data presented in Table III were calculated from the values of the masses of light nuclei given in the paper by B. S. Dzhelepov and L. N. Zyryanova\({}^{26}\).) As is seen from this table, the energy of attachment of one or two nucleons increases along this series of nuclei in the direction from \({}^{8}_{4}\mathrm{Be}\) to \({}^{20}_{10}\mathrm{Ne}\), i.e., with an increase in the number of particles in the nucleus the binding of the nucleon to the nucleus increases. Oxygen, however, falls out of this series. The energy of attachment of a neutron to oxygen proves to be smaller than in carbon and neon. The same is true with respect to the attachment to the nucleus of a proton, or of one neutron and one proton, or of two neutrons. It is clear that all these particles are bound to oxygen more weakly than to the other even nuclei located in the series \({}^{8}_{4}\mathrm{Be}\)—\({}^{20}_{10}\mathrm{Ne}\). This is what one should expect if, in the oxygen-16 nucleus, the filling of a nuclear shell took place.

As we have already indicated above, a direct comparison of data on the energies of attachment to a nucleus of the next nucleon is made difficult by the fact that other effects are superimposed on the effects caused by the filling of a nuclear shell, among which the parity effect is of dominant importance. At the same time, the very

energy effects associated with the filling of shells are not especially large. Therefore the jumps in binding energy that occur when shells are filled can be detected only as a result of very accurate measurements.

So long as the mass defects of medium and heavy nuclei were measured with insufficient accuracy \(\left(\dfrac{\Delta m}{m} > 10^{-5}\right)\), the course of the mass-defect curve appeared completely smooth, and there was no possibility of noticing any kinks on this curve. In recent years, a substantial step forward has been made in mass-spectrographic measurements. Two series of works have appeared in which the mass defects of medium and heavy nuclei have been measured with comparatively high accuracy*).

Both series of works revealed distinct kinks in the mass-defect curve. Nevertheless, conclusions from mass-spectrographic measurements must be approached with great caution, since the authors apparently tend to overestimate the accuracy of their measurements**), and the entire jump in binding energy at \(N\) and \(Z = 50\) and at \(N = 82\) does not exceed \(2.5\text{--}3\) MeV. In general, comparison with the energies of nuclear reactions shows that the data of Collins et al. possess considerably higher accuracy than Dacours’s data. However, they pertain only to two regions of the Mendeleev periodic system, namely to the region \(32 \leq A \leq 70\) and to the region \(102 \leq A \leq 136\). From these data it follows that after \(Z = 28\) and \(Z = 50\) small jumps are observed in the value of the packing fraction.

Figure 4 gives somewhat less reliable values of the packing fraction of various nuclei, calculated by Dacours and Preston\(^{30}\) on the basis of the values they measured for the masses of various isotopes. As is seen from this graph, in the dependence of the packing fraction on the value of the mass number of the nuclei, kinks are observed at positions corresponding to the isotope \({}^{90}_{40}\mathrm{Zr}\) (50 neutrons), the tin isotope \({}^{120}_{50}\mathrm{Sn}\) (50 protons), and the lead isotope \({}^{208}_{82}\mathrm{Pb}\) (82 protons and 126 neutrons).

The portion of the curve corresponding to mass-number values from 126 to 150 was investigated by Dacours\(^{27}\) later. Fig. 5 rep—

*) In the works of Dacours\(^{27\text{--}32}\) with collaborators, \(\dfrac{\Delta m}{m}\) is estimated to be approximately \(5 \cdot 10^{-6}\), which gives an error in the determination of the mass at \(m = 100\) of about \(0.5\) MeV. Another series of works was carried out by Nier, Collins, and others\(^{32a,32b}\). These authors succeeded in achieving still higher accuracy, equal, according to their estimate, to \(0.1\) MeV at \(A = 60\) and about \(0.2\) MeV at \(A = 120\).

**) That the authors in fact underestimate their errors is evident, for example, from the fact that according to Dacours the error in determining the mass of \(\mathrm{Cr}^{50}\) is 100 keV, and according to Collins et al. it is 70 keV, whereas the results of the two measurements differ from one another by 2 MeV.

produce the dependence of the magnitude of the binding energy, calculated per nucleon, on the value of the mass number. In the graph there is clearly

Fig. 4. Behavior of the packing coefficient with change in mass number.

revealed a kink in the course of this dependence, falling on the isotope \({}^{140}_{58}\mathrm{Ce}\), which contains 82 neutrons.

The presence, in the curves of Figs. 4 and 5, of kinks at the places corresponding to nuclei with a number of particles 50 and 82 indicates a change in the magnitude of the binding energy of nucleons when the number of particles (protons or neutrons) in the nucleus becomes greater than 50 or 82.

Fig. 5. Binding energy per nucleon in the region of medium mass numbers.

An analogous conclusion may also be drawn on the basis of experimental values of the neutron binding energy, determined by means of various nuclear reactions. Harvey\(^{33}\) systematized data relating to reactions of the type \((n,\gamma)\), \((d,p)\), and \((\gamma,n)\) for a number of isotopes*). Harvey’s data show an interesting dependence of the neutron binding energy on the number of particles in the nucleus. In order to exclude the influence on the binding energy of the parity factor, Harvey determines the dependence on

*) The study of neutron binding energies was also carried out by Fizer\(^{32a}\). More extensive material collected by this author confirms Harvey’s data.

numbers of particles in the nucleus are not the magnitudes of the neutron binding energy, but the differences between the actual neutron binding energy, measured experimentally, and the binding energy calculated from the semiempirical formula

\[ M(A,Z)=A-0.00081Z-0.00611A+0.014A^{2/3}+ \]
\[ +0.083\left(\frac{1}{2}A-Z\right)^2A^{-1}+0.000627Z^2A^{-1/3}+\delta, \tag{11} \]

where \(A\) is the mass number, \(Z\) is the nuclear charge, \(M(A,Z)\) is the mass of the nucleus of the given isotope, and \(\delta\) is equal to zero for odd \(A\), has a value equal to \(+\dfrac{0.036}{A^{3/4}}\) when \(A\) is even and \(Z\) is odd, and a value equal to \(-\dfrac{0.036}{A^{3/4}}\) when \(A\) is even and \(Z\) is even. From relation (11) it is easy to calculate the neutron binding energy. For this it is sufficient to determine the difference between the mass of the neutron and the quantity \(M(A+1,Z)-M(A,Z)\). The difference \(\Delta E\) between the experimental value of the binding energy of the \((N+1)\)-th neutron and its value calculated on the basis of the semiempirical formula (11) is shown in Fig. 6.

Fig. 6.

In Fig. 6, a gives the dependence of \(\Delta E\) on the number of neutrons in the nucleus—\((N+1)\)—for nuclei with the number of neutrons from 30 to 83, and in Fig. 6, b the same dependence is given for nuclei containing from 100 to 148 neutrons. As is seen from the curves obtained, sharp jumps are observed in the course of this dependence, falling at nuclei with neutron numbers 50, 82, and 126.

It should also be borne in mind that the effect of filling a nuclear shell is obscured not only by a change in the parity of the number of particles in the nucleus, but also by the circumstance that particles of two types are present in nuclei, and that as the total number of particles in the nucleus increases, both the number of protons and the number of neutrons change. Therefore, usually, along with the filling (as the number of particles in the nucleus increases) of some shell by particles of one

of a given type in the nucleus, the content of particles of the other type also increases. It may turn out that an increase in the nucleus of the number of particles of this other type will contribute to an increase in the binding energy, as a result of which the shell-filling effect, expressed in a decrease of the binding energies at the beginning of the filling of a new shell, may be obscured.

Apparently, the detection of a jump in the value of the binding energy of a nucleon, occurring at the beginning of the filling of a new shell, can be carried out more distinctly if one follows the change in binding energy as the nuclear shell is filled by the given particles, under the condition that the number of particles of the other type remains unchanged in the nucleus. Such a consideration can be made in the region of heavy nuclei, for which the existence of many radioactive isotopes and isotones (nuclei whose composition contains the same number of neutrons) has been established.

Kravtsov^34 compared the data for the binding energy of neutrons obtained in reactions of the type \((\gamma,n)\), \((n,\gamma)\), \((d,p)\), and \((d,t)\), occurring with heavy nuclei, with the values of the energies of \(\alpha\)- and \(\beta\)-decays of heavy nuclei and, on the basis of this comparison, calculated, from the value of the mass of the atom \({}^{208}_{82}\mathrm{Pb}\) known from mass-spectroscopic data, the mass values of atoms of most known heavy isotopes, and also the attachment energy to the nucleus of the last nucleon (proton and neutron). In Tables IV and V are given data borrowed from the mentioned article by Kravtsov, with some changes from a later work by Fizer^33a.

Table IV

Values of the binding energy of the last \((N+1)\)-th neutron (in MeV) for isotopes of heavy nuclei

\(Z\) / \(N\)	120	122	124	126	128	130
81	6.8	6.5	6.2	3.82	3.88	—
82	7.1	6.4	6.71	3.87	3.77	3.5
83	7.4	7.1	6.8	4.22	4.38	4.24
84	7.5	7.2	6.6	4.85	4.31	4.09
85	7.6	7.4	7.1	5.1	4.3	4.23
86	—	—	8.1	4.9	4.7	4.58
87	—	—	7.2	5.5	5.0	4.9

Table IV gives the values of the binding energy of the last \((N+1)\)-th neutron for nuclei with one and the same value of \(Z\),

but with different values of \(N\). Such nuclei are all placed in one horizontal row of this table. (To avoid the influence of parity, the table presents only data for the binding-energy value of the \((N+1)\)-st neutron for nuclei containing an even number of neutrons.) Nuclei with different \(Z\) are arranged in different rows.

As is seen from Table IV, the attachment energy of the 127th (and also of the 129th and 131st) neutron is considerably smaller than the attachment energy of the 121st, 123rd, and 125th neutrons. A jump in the binding energy on passing the neutron number through 126 is observed among all heavy nuclei with \(Z\) varying from 81 to 87. We note the considerable magnitude of this jump in binding energy; for all these values of \(Z\) it is equal to \(2\)—\(2.5\) MeV.

Table V gives the binding-energy values of the \((Z+1)\)-st proton for various nuclei containing different numbers of protons but the same number of neutrons (isotones), and again

Table V

Values of the binding energy of the last \((Z+1)\)-st proton (in MeV) for isotones (nuclei with an equal number of neutrons) of heavy nuclei

\(N \backslash Z\)	78	80	82	84
115	5.6	4.6	1.8	—
116	5.1	5.0	1.9	—
117	5.8	5.2	2.0	1.2
118	6.8	5.6	2.1	1.4
119	6.3	5.4	2.4	1.5
120	6.7	5.5	2.6	1.6
121	7.1	5.9	2.9	1.7
122	—	6.0	3.1	2.0
123	—	6.3	3.8	2.2
124	—	6.5	4.0	2.5
125	—	7.2	4.12	3.0
126	—	7.4	4.18	3.4
127	—	—	4.53	4.0

to exclude the influence of parity, Table V compares only nuclei with even values of \(Z\). As is seen from Table V, for all isotones with neutron numbers from 115 to 127, without exception, a jump is observed in the binding energy of the last \((Z+1)\)-st proton when the proton number is equal to 82. The magnitude of the jump is \(2.5\)—\(3\) MeV. If one takes into account that the error of the data given in Table V is \(0.5\) MeV, then the magnitude of this jump may be regarded as constant for all isotonic nuclei with neutron numbers from 115 to 126.

Comparing the binding energy of particles in various heavy nuclei, Kraetsov\(^ {95}\) came to the conclusion that separate proton and neutron shells do not exist in nuclei, but only common proton-neutron shells, and that a jump in binding energy is observed only in those nuclei in which shells are being filled simultaneously by both protons and neutrons. However, the data given in Table V indicate quite the opposite: namely, that the jump in binding energy when the proton number in nuclei passes through 82 is observed for any number of neutrons (within the range 115–126) in the nucleus, and that, consequently, the filling of proton shells takes place independently of the filling of neutron shells.

For light nuclei it is impossible to trace the presence of jumps in the binding energy in the way this was done in Tables IV and V, since comparatively few isotopes having the same number of particles of one type are known among light nuclei. However, quite definite indications can be obtained if one compares the mass defects of light nuclei with the same isotopic number. Of special interest are nuclei for which the isotopic number \(N - Z\) is equal to zero. In these nuclei the proton and neutron shells must close simultaneously; therefore the jump in binding energy should be manifested distinctly.

Figure 7 shows the dependence of the mass defect on the atomic number \(Z\) for nuclei with isotopic number equal to zero.

Fig. 7. Mass defect of even nuclei with isotopic number equal to zero.

Along the ordinate axis is plotted the quantity \(A - M\), i.e., the value of the mass defect taken with the minus sign. To eliminate the influence of the parity of the number of particles in constructing the curve of Fig. 7, we

we restricted ourselves only to nuclei with even \(Z\). As can be seen from Fig. 7, the dependence of the mass defect on the number of particles is not monotonic, but undergoes a number of jumps and sharp changes in the slope of the curve. The jumps occur for nuclei with the number of protons and neutrons equal to 8, 14, and 20.

The presence of such jumps and breaks in the course of the mass defect is also observed for nuclei with isotopic number \(N - Z = 1, 2, 3\). A comparison of the course of the mass defect for these nuclei (Fig. 8) with the course of the mass defect of nuclei with isotopic number equal to zero shows that breaks in the course of the mass defect occur when the number

Fig. 8. Mass defect of isotopes of even elements with isotopic number 1 (curve \(I\)), 2 (curve \(II\)), and 3 (curve \(III\)). The ordinate values indicated in the figure refer to curve \(I\). For curve \(II\), the zero value of \(A - M\) is shifted upward by two grid squares; for curve \(III\), by four grid squares.

of neutrons in the nucleus becomes equal to 8, 20, and 28, and when the number of protons in the nucleus is equal to 8, 14, 20, and 28. The absence on the curves of Fig. 8 of a jump at the number of neutrons equal to 14 would seem to confirm Weizmann’s\(^{31}\) idea about the difference in the filling of proton and neutron shells. However, the question of the order of filling of proton and neutron shells is complex, since a jump at \(N = 14\) is nevertheless observed, but only for nuclei with \(N < Z\). On the other hand, for such nuclei there is no jump in the transition from \(Z = 14\) to \(Z = 16\).

The dependence of the mass defect on the number of particles (protons) for nuclei with isotopic numbers 4, 5, and 6 is presented in Fig. 9. The jump in the course of the mass defect at \(N = 20\) and 28, and also at \(Z = 28\), is also clearly visible on these curves.

4. ABUNDANCE OF ISOTOPES

Data on the relative abundance of isotopes and on the isotopic numbers of various nuclei can also be used to illustrate the special properties of nuclei possessing a number of nucleons corresponding to the values of the “magic” numbers.

The special stability of the nucleus \({}^{40}_{20}\mathrm{Ca}\), containing 20 protons and 20 neutrons, is indicated by the fact that this isotope is the last in the series of nuclei with equal numbers of protons and neutrons. As is known, with increasing charge the ratio between the number of protons and neutrons in stable nuclei changes. Nuclei with equal numbers of protons and neutrons already prove not to be the most stable. Beginning with oxygen (atomic number 8), among stable nuclei with odd \(Z\) we do not encounter any for which there would be an equal number of protons and neutrons. It should be emphasized that the first nucleus with odd \(Z\) not containing isotopes for which \(Z = N\) is the nucleus with \(Z = 9\), in which the third proton shell begins to be filled. Among nuclei with even \(Z\), isotopes with \(Z = N\) are also found in elements with atomic number greater than 8. The last of such elements is \({}^{40}_{20}\mathrm{Ca}\). The next even element, titanium, already has no stable isotopes with equal numbers of protons and neutrons; the isotope \({}^{44}_{22}\mathrm{Ti}\) is not found among stable nuclei. The even element preceding calcium, argon, has the isotope \({}^{36}_{18}\mathrm{Ar}\), containing equal numbers of protons and neutrons; however, its relative abundance is very small. \({}^{36}_{18}\mathrm{Ar}\) is contained in argon in an amount of only \(0.31\%\)*). The relative

Fig. 9. Mass defect of nuclei with isotopic number 4 (curve I), 5 (curve II), and 6 (curve III). The value \(A - M\), indicated in the figure at the left, refers to curve I; at the right, to curve II. For curve III the origin of counting is shifted upward by two grid squares.

*) It is possible, however, that the small abundance of \(\mathrm{A}^{36}\) is a consequence of the fact that on the earth the greater part of argon arose from the decay of \(\mathrm{K}^{40}\), in which the isotope \(\mathrm{A}^{40}\) is obtained.

the abundance of $_{20}^{40}\mathrm{Ca}$ is very great: the isotope $_{20}^{40}\mathrm{Ca}$ is contained in calcium in an amount of 96.96%. The fact that $_{22}^{44}\mathrm{Ti}$ (22 protons and 22 neutrons) is absent in nature, $_{18}^{36}\mathrm{Ar}$ (18 protons and 18 neutrons) is stable but not very abundant, whereas $_{20}^{40}\mathrm{Ca}$ (20 protons and 20 neutrons) is stable and has a high abundance, testifies to the relatively great stability of the group consisting of 20 protons and 20 neutrons. It should also be added that the element calcium (20 protons) has a larger number of isotopes than its neighbors with even atomic number (in this region of the periodic table, elements with odd atomic numbers consist of only one isotope, and therefore we exclude them from consideration). Argon, for example (atomic number 18), has three stable isotopes; titanium (atomic number 22), five isotopes; chromium (atomic number 24), four isotopes; whereas calcium (atomic number 20) has six stable isotopes. Still more striking is the difference in the isotopic number for the extreme isotopes of calcium and of neighboring nuclei. For argon this difference is 4. For titanium it is also 4, while for calcium it is 8.

Nuclei with a number of neutrons equal to 20 occur among stable isotopes much more often than nuclei with 18 and 22 neutrons (there are no stable nuclei containing 19 and 21 neutrons). Thus, 5 stable nuclei are known that contain 20 neutrons ($_ {16}^{36}\mathrm{S}$, $_{17}^{37}\mathrm{Cl}$, $_{18}^{38}\mathrm{Ar}$, $_{19}^{39}\mathrm{K}$, $_{20}^{40}\mathrm{Ca}$), whereas only three different nuclei containing 18 neutrons are known ($_ {16}^{34}\mathrm{S}$, $_{17}^{35}\mathrm{Cl}$, $_{18}^{36}\mathrm{Ar}$). Stable nuclei with a number of neutrons equal to 22 ($_ {18}^{40}\mathrm{Ar}$, $_{20}^{42}\mathrm{Ca}$, $_{19}^{41}\mathrm{K}$) and 24 ($_ {20}^{44}\mathrm{Ca}$, $_{21}^{45}\mathrm{Sc}$, $_{22}^{46}\mathrm{Ti}$) are likewise known only three each.

Thus, nuclei containing 20 neutrons or 20 protons have relatively great stability.

In support of the assertion of the special stability of nuclei containing 50, 82, or 126 neutrons and 50 or 82 protons, a number of data may also be cited. Thus, the element with atomic number 50 (tin) has the largest number of stable isotopes of all known elements—ten. The neighboring elements (with even atomic number)—cadmium (atomic number 48) and tellurium (atomic number 52)—have eight stable isotopes each. The greatest difference between the mass numbers of the extreme isotopes is also found for tin—12; for the neighboring elements, cadmium and tellurium, it is 10. The difference between the mass numbers of the extreme isotopes is also 12 for Xe, whose heaviest isotope has 82 neutrons.

It is known, further, that for all even isotopes with atomic number $Z > 40$ the relative abundance of the most abundant isotope does not exceed 35%, and in the interval of $Z$ from 30 to 40 it does not exceed 57%.

Exceptions are:

1. The isotope \({}^{88}_{38}\mathrm{Sr}\) (containing 50 neutrons)—its abundance is 82%.

2. The isotope \({}^{138}_{56}\mathrm{Ba}\) (containing 82 neutrons)—its abundance is 71.66%.

3. The isotope \({}^{140}_{58}\mathrm{Ce}\) (containing 82 neutrons)—its abundance is 90%.

It is also known that the abundance of the lightest of the even isotopes is small and, as a rule, does not exceed 2%. There are, however, three elements for which the lightest isotope has a relatively large abundance, considerably exceeding 2%.

1. The isotope \({}^{90}_{40}\mathrm{Zr}\) (number of neutrons 50)—abundance 48%.

2. The isotope \({}^{92}_{42}\mathrm{Mo}\) (number of neutrons 50)—abundance 15.5%.

3. The isotope \({}^{142}_{60}\mathrm{Nd}\) (number of neutrons in the nucleus 82)—abundance 25.95%.

Figures 10 and 11 show diagrams indicating the distribution of stable isotopes with various numbers of neutrons, close to 52 and 82.

Fig. 10. Diagram of the abundance of stable isotopes among nuclei with numbers of neutrons close to 52.

close to 52 and 82. From these diagrams it is evident that there are significantly more isotopes with neutron numbers 50 and 82 than with other (neighboring) numbers. Seven isotopes with neutron number 82 are known; isotopes

with neutron number 50 there are known to be six, whereas the average number of isotopes for an even number of neutrons not equal to 82 or 50 is only 3–4.

Among even isotopes there is usually only one nucleus with an odd mass number, i.e. with an odd number of protons. Exceptions to this rule are the isotopes with neutron numbers 20, 50, and 82. There are two known such isotopes with neutron number 20

Visible entries in the diagram:
\(N^\circ=78\): \(\mathrm{Ce}^{136}\) \(<1\); \(\mathrm{Ba}^{134}\) 2.43; \(\mathrm{Cs}^{133}\) 100; \(\mathrm{Xe}^{132}\) 26.96; \(\mathrm{Te}^{130}\) 33.1.
\(N^\circ=79\): \(\mathrm{Ba}^{135}\) 6.59.
\(N^\circ=80\): \(\mathrm{Ce}^{138}\) \(<1\); \(\mathrm{Ba}^{136}\) 7.81; \(\mathrm{Xe}^{134}\) 10.54.
\(N^\circ=81\): \(\mathrm{Ba}^{137}\) 11.32.
\(N^\circ=82\): \(\mathrm{Sm}^{144}\) 3; \(\mathrm{Nd}^{142}\) 25.95; \(\mathrm{Pr}^{141}\) 100; \(\mathrm{Ce}^{140}\) 90; \(\mathrm{La}^{139}\) 100; \(\mathrm{Ba}^{138}\) 71.66; \(\mathrm{Xe}^{136}\) 8.95.
\(N^\circ=83\): \(\mathrm{Nd}^{143}\) 13.
\(N^\circ=84\): \(\mathrm{Nd}^{144}\) 22.6; \(\mathrm{Ce}^{142}\) 10.
\(N^\circ=85\): \(\mathrm{Sm}^{147}\) 17; \(\mathrm{Nd}^{145}\) 9.2.
\(N^\circ=86\): \(\mathrm{Sm}^{148}\) 14; \(\mathrm{Nd}^{146}\) 16.5.
\(N^\circ=87\): \(\mathrm{Sm}^{149}\) 15.
\(N^\circ=88\): \(\mathrm{Gd}^{152}\) 0.2; \(\mathrm{Eu}^{151}\) 49; \(\mathrm{Sm}^{150}\) 5; \(\mathrm{Nd}^{148}\) 6.8.
\(N^\circ=90\): \(\mathrm{Gd}^{154}\) 1.5; \(\mathrm{Eu}^{153}\) 50.9; \(\mathrm{Sm}^{152}\) 26; \(\mathrm{Nd}^{150}\) 5.95.

Fig. 11. Diagram of the abundance of stable isotopes among nuclei with neutron numbers close to 82.

(these are \({}^{39}_{19}\mathrm{K}\) and \({}^{37}_{17}\mathrm{Cl}\)), two isotopes with neutron number 50 (\({}^{89}_{39}\mathrm{Y}\) and \({}^{87}_{37}\mathrm{Rb}\))* and two isotopes with neutron number 82 (these are \({}^{139}_{57}\mathrm{La}\) and \({}^{141}_{59}\mathrm{Pr}\)). Among the isotopes with number 82, two (\({}^{144}_{62}\mathrm{Sm}\) and \({}^{142}_{60}\mathrm{Nd}\)) are the extreme light isotopes of the elements neodymium and samarium, while the other two isotopes, \({}^{138}_{56}\mathrm{Ba}\) and \({}^{136}_{54}\mathrm{Xe}\), are the extreme (heaviest) isotopes of xenon and barium. An analogous situation also obtains for iso-

* However, although \(\mathrm{Rb}^{87}\) occurs in nature, it is radioactive and is included in the table of stable isotopes only because of its enormous half-life. Therefore there exists only one \(\beta\)-stable nucleus with \(N=50\) and even \(A\).

EXPERIMENTAL FOUNDATIONS OF THE NUCLEAR SHELL MODEL

isotopes with a number of neutrons equal to 50. All these isotopes, as well as the isotopes with number 82, are extreme isotopes—the heaviest or the lightest—which also indicates the relatively great stability of nuclei with 50 and 82 neutrons.

Finally, it should be noted that nuclei containing neutrons in numbers 50, 82, and 126 possess a relatively greater abundance in comparison with nuclei containing another number of neutrons. This assertion is well illustrated by Fig. 12, which shows an approximate diagram of the relative abundance of various isotopes with an even mass number. Along the \(x\)-axis of this diagram is plotted the number of neutrons contained in the nucleus, along the \(y\)-axis the isotopic number of the given isotope, and along the \(z\)-axis the relative abundance of the given isotope \(\eta\).

Fig. 12. Percentage content of stable isotopes (along the \(z\)-axis). Along the \(x\)-axis is plotted the number of neutrons in the nucleus; along the \(y\)-axis, the isotopic number \((N-Z)\).

As can be seen from Fig. 12, the abundance of nuclei has a distinct maximum when the number of neutrons contained in the nucleus is equal to 50, 82, and 126 (the graph represents nuclei with a number of neutrons exceeding 38). It should be noted that the relatively large abundance of isotopes with particle numbers 50 and 82 was pointed out by Selinov\(^{20}\).

The relative stability of nuclei with proton number 82 and neutron number 126 is also evidenced by the following facts:

1) Lead (atomic number 82) is, in all radioactive families, the final product of radioactive decay.

2) In the series of radioactive emitters of \(\alpha\)-particles with a given \(Z\), the minimum energy of \(\alpha\)-particles is observed for those nuclei in which the number of neutrons in the nucleus is equal to 126. This means that such nuclei

more stable than nuclei with another number of neutrons. Conversely, the $\alpha$-particles with the greatest energy are emitted by radioactive elements with a number of neutrons equal to 128. This recalls the situation encountered in the filling of the electron shell of an atom. In a series of atoms in which one and the same electron shell is being filled, the greatest value of the electron energy occurs for that atom with which the filling of the shell begins, and the smallest electron energy for those atoms in which the filling of the shell is completed. In accordance with this, we may expect that an $\alpha$-particle (a pair of neutrons and a pair of protons) will have the greatest energy in those nuclei in which

Fig. 13. Energy of $\alpha$-decay (along the $z$ axis). Along the $x$ axis is plotted the atomic number $Z$, and along the $y$ axis the ratio $\dfrac{N}{Z}$.

a new shell begins to be filled. The existence of such a maximum in the energy of $\alpha$-particles is clearly seen in Fig. 13, which shows a three-dimensional diagram of the energy of $\alpha$-particles emitted in the radioactive decay of various nuclei. Along the $x$ axis of this diagram is plotted the value of $Z$, along the $y$ axis the ratio of the number of protons to the number of neutrons, and along the $z$ axis the energy of the $\alpha$-particle. In Fig. 13 a maximum at $Z=84$ is clearly visible, and it is seen that for $Z>100$ the appearance of a second maximum is indicated. The fact that the maximum energy of $\alpha$-particles falls on nuclei with $Z=84$ indicates that the construction of a new shell begins in the nucleus and that, consequently, in nuclei with $Z=82$ the proton shell is filled. At present, $\alpha$-radioactivity has been discovered in a whole series of nuclei with $N>82$. In this case the nuclei with $N=84$ possess the maximum energy of $\alpha$-decay[^35a].

It is also interesting to note that the cross section for the capture of neutrons is extremely small for nuclei containing 50, 82, or 126 neutrons. Griffith[^36], working with neutrons from

source Ra–γ–Be, found that yttrium 89 has a neutron-capture cross section 10–20 times smaller than those of the neighboring isotopes (the number of neutrons in yttrium 89 is 50). A minimum of the neutron-capture cross section also occurs for \({}^{139}_{57}\mathrm{La}\) and \({}^{141}_{59}\mathrm{Pr}\) (both have 82 neutrons). Meshcheryakov\({}^{37}\) found that, for neutrons obtained from the \((d,d)\) reaction, there is a deep minimum in the capture cross section for lanthanum 139, praseodymium 141, and barium 138 (the number of neutrons in all of them is 82), and also for bismuth, whose nucleus contains 126 neutrons.

A considerably more detailed investigation of the capture cross sections of fast neutrons was carried out by Hughes and collaborators\({}^{37a, 376}\). In this group of works it was shown that all nuclei with the number of neutrons equal to 50, 82, or 126 have anomalously small capture cross sections for neutrons with an effective energy of \(1\) MeV. The indicated isotopes have cross sections 10–50 times smaller than neighboring nuclei; moreover, for other nuclei with \(A > 80\) the capture cross section is never small.

One of the principal reasons for this phenomenon should be regarded as the decrease in the binding energy of a neutron absorbed by “magic” nuclei. The compound nucleus formed in this process has an excitation energy approximately \(1.5\) MeV lower than usual. As a result of the decrease in excitation energy, the level density of the compound nucleus drops noticeably, and therefore so does the capture cross section. It is possible, however, that the level density decreases somewhat more strongly than can be obtained from statistical theory.

5. QUADRUPOLE MOMENT OF NUCLEI AND “MAGIC” NUMBERS

Data on the magnitude of the quadrupole moment of nuclei can also be used to consider the question of the special stability of nuclei having protons or neutrons in an amount corresponding to one of the values of the “magic” series of numbers.

Indeed, nuclei with filled shells, both proton and neutron shells, must possess spherical symmetry. Of course, a nucleus in which only one of the shells—proton or neutron—is filled will not be completely symmetric; nevertheless it is natural to expect that in such nuclei the degree of asymmetry will be smaller than in nuclei with an unfilled shell.

Equality to zero of the quadrupole moment \(Q\) of a nucleus means that we have a symmetric nucleus. A positive value of \(Q\) means that the nucleus is elongated in the direction of the spin axis, while a negative value of \(Q\) means that the nucleus is flattened in the direction of the spin axis.

The quadrupole moment has now been measured for many nuclei; therefore one can determine how the value of \(Q\) changes as the number of particles in the atomic nucleus increases, and establish,

whether there are any peculiarities in the course of the dependence of the quantity \(Q\) on the mass number \(A\) for nuclei with a “magic” number of protons or neutrons. It turns out that such peculiarities do exist38,39.

Fig. 14. Dependence of the quadrupole moment on the atomic number.

The explanation is as follows38,39. Figures 14 and 15 illustrate this clearly. Figure 14 shows the course of the quadrupole moment \(Q\) as a function of \(Z\)—the number of protons in the nucleus. Figure 15 shows the course of the quadrupole moment \(Q\) as a function of the number of neutrons in the nucleus.

Fig. 15. Dependence of the quadrupole moment on the number of neutrons in the nucleus.

In these figures the experimental data relating to nuclei with an odd number of protons are marked by circles, while the data relating to nuclei with an odd number of neutrons are marked by circles with a dot inside. Circles with a cross inside mark the values of \(Q\) calculated on the basis of the established experimen-

...experimentally\(^{38}\) the relations between the quadrupole and magnetic moments of nuclei*).

In both Figs. 14 and 15 it is clearly seen that, for nuclei with numbers of protons 2, 8, 20, 50, 82 and for nuclei with numbers of neutrons 2, 8, 20, 50, 82, and 126, the quadrupole moment is either equal to zero or has a small value.

Let us add that on the curve in Fig. 14, showing the dependence of \(Q\) on \(Z\), there are also visible minima corresponding to values of \(N\) equal to 20, 50, 82, and 126.

Thus, on the basis of Figs. 14 and 15 we may conclude that, in nuclei containing protons and neutrons in the numbers 2, 8, 20, 50, 82, and 126, the degree of asymmetry is smaller than in nuclei with another number of particles. This is in agreement with the greater stability of such nuclei.

It should be noted that the quadrupole moments of nuclei having \(Z = Z_{\mathrm{cl}} + 1\), where \(Z_{\mathrm{cl}} = 28, 50, 82\), are always negative, as is seen from the examples of the nuclei \(\mathrm{Cu}^{63}\), \(\mathrm{Cu}^{65}\), \(\mathrm{Sb}^{121}\), \(\mathrm{Sb}^{123}\), \(\mathrm{Bi}^{209}\). Conversely, for nuclei with \(Z_+ = Z_{\mathrm{cl}} - 1\) the quadrupole moments (where they have been measured) are positive. Thus, for example, for the nuclei \(\mathrm{N}^{14}\), \(\mathrm{Al}^{27}\), \(\mathrm{In}^{113}\), and \(\mathrm{In}^{115}\) the quadrupole moments have a positive sign. Therefore one may put forward the supposition that in all cases, in going from a nucleus with \(Z_{\mathrm{cl}} - 1\) to a nucleus \(Z_{\mathrm{cl}} + 1\), the quadrupole moment changes sign. Apparently, the same effect is observed also in going from \(N_{\mathrm{cl}} - 1\) to \(N_{\mathrm{cl}} + 1\), but for these cases there are still too few data.

Thus, nuclei with a number of protons somewhat smaller than 8, 14, 28, 50, 82 have an elongated form, whereas nuclei in which the number of protons somewhat exceeds the indicated numbers are, apparently, flattened.

Consequently, one of the most important features of “magic” nuclei is that these nuclei have a number of nucleons of one or both types at which a change in the shape of the nucleus occurs.

* From experimental data\(^{38}\) for the quadrupole \(Q\) and magnetic \(\mu\) moments of nuclei it follows that, for nuclei with a positive value of \(Q\), the relation

\[ \frac{\mu_1}{\mu_2}=\frac{Q_1}{Q_2}=\mathrm{const}=c, \]

holds; for nuclei with a negative value of \(Q\),

\[ \frac{\mu_1}{\mu_2}=\frac{Q_2}{Q_1}=c. \]

On the basis of these experimentally found relations one can calculate the quadrupole moment of a nucleus from the known value of its magnetic moment.

6. ON THE SYSTEMATICS OF NUCLEAR LEVELS

The totality of all the facts cited above indicates that nuclear levels are filled by particles in amounts of 2, 8, 20 (28?), 50, 82, 126, whereas in a rectangular potential well the filling of levels should have occurred when the number of particles in the nucleus was equal to one of the numbers of the series

\[ 2,\ 8,\ 20,\ 34,\ 40,\ 58,\ 92,\ 138. \]

What, then, is the reason for the discrepancy between these two series of numbers, and can this discrepancy be explained within the framework of the simple model of nucleon orbits? It turns out that the discrepancy between these two series of numbers can be explained by a certain relative displacement of the levels of the rectangular well. Assuming that the cause of such a displacement of levels is the strong interaction between nuclear particles, which is not taken into account by the model of nucleon orbits, Nordheim\(^{40,41}\) suggested that as a result of such an interaction the levels corresponding to small values of the orbital angular momentum are shifted into the region of lower energies. Therefore the \(2s\) level turns out to lie below \(3p\), and the \(4d\) level below the \(5g\) level, etc. In consequence of this the levels \(3p\) and \(4d\) (second levels with orbital angular momentum 1 and 2) begin to be filled earlier than the \(5g\) level (the first level with orbital angular momentum equal to 4). In exactly the same way the \(5f\) level (the second level with orbital angular momentum equal to 3) begins to be filled earlier than the \(6h\) level (the first level with orbital angular momentum equal to 5). As a result of such a displacement the levels are grouped into shells differently than follows from the scheme of levels of a rectangular well. According to Nordheim the order of alternation of levels in nuclei proves to be as follows:

\[ \begin{aligned} &1s \text{ (first shell) — two particles,}\\ &2p \text{ (second shell) — 6 particles,}\\ &2s,\ 3d \text{ (third shell) — 12 particles,}\\ &4f,\ 3p,\ 4d \text{ (fourth shell) — 30 particles,}\\ &5g,\ 5f \text{ (fifth shell) — 82 particles} \end{aligned} \]

and so on. As levels close in energy, \(2s\) and \(3d\) are grouped into one shell. The levels \(4f\), \(3p\), and \(4d\) are likewise grouped into one shell.

Such a grouping of levels gives, for the number of particles completely filling the nuclear shells, values coinciding with the “magic numbers.” Indeed, in the first shell there will be two particles. In the second shell there will be 6 particles; consequently, filling of both shells (the second “magic” number) will occur in nuclei with eight particles. The third shell consists of particles filling the \(2s\) level (two particles) and the \(3d\) level (10 particles). All three

shells will be filled (the third “magic” number) in a nucleus containing 20 particles. The fourth shell turns out to be filled in a nucleus with a total number of particles equal to fifty (thirty particles are contained in one fourth shell). The fifth shell will be filled when the number of particles in the nucleus reaches 82.

Of course, the presence of a strong interaction between nuclear particles may substantially alter the conclusions following from the picture of particles moving in the self-consistent field of the nucleus, and may lead to a strong change in the arrangement of the levels. However, Nordheim’s conclusions are of a purely qualitative character, and the grouping of levels in a shell that he proposed (although it coincides with the “magic” numbers) is poorly justified and, apparently, incorrect. In particular, Nordheim’s grouping of levels contradicts the data on nuclear isomerism. With such a grouping of levels, isomerism among nuclei of the fourth and fifth groups should not occur, and in order to explain its existence Nordheim has to assume that in some groups of nuclei the order in which the levels are filled changes. A new shell begins to be filled before the preceding one has been filled.

Another possible cause of the relative displacement of nuclear levels (in comparison with the levels of a rectangular well) could be that the averaged field is not accurately represented by a rectangular well. Some change in the form of the potential well may lead to a noticeable displacement of the levels.

It has already been suggested^42–45 that in heavy nuclei, in the central part of the potential well, the potential may increase somewhat relative to the bottom of this well. This rise of the potential is due to the fact that the density of particles in the central part of heavy nuclei, owing to electrostatic repulsion, is somewhat lowered, which leads to an increase of the potential. Figure 16 shows the approximate course of the density in the nucleus and the corresponding potential function with a central rise. For such a form of the potential, the order of succession of the levels for a rectangular well will be disturbed. The levels will be shifted. The shift of the levels will be the greater, the higher the potential of the central part is raised relative to the bottom of the well. Different levels will undergo shifts of different magnitude. The \(s\) levels will be shifted more strongly than the \(p\) or \(d\) levels.

Fig. 16. \(A\)—density of particles in the nucleus and the potential function in light nuclei; \(B\)—density of particles in the nucleus and the potential function in heavy nuclei.

Figure 17 illustrates the displacement of levels of a rectangular potential well of radius \(R\), in which the bottom is raised by an amount \(D'\), expressed in units of

\[ \frac{\hbar^2}{MR^2}. \]

The rise of the potential is taken over the interval

\[ r=\frac{R}{2}. \]

The energy of the levels in Fig. 17 is also given in units of

\[ \frac{\hbar^2}{MR^2}. \]

As is seen from this figure, the presence of a rise in the central part of the potential well leads to a change in the order in which the levels occur. Consequently, the order of filling of the levels in this case will also differ from the order of filling of the levels in a simple rectangular well. For example, depending on the magnitude of \(D'\), the following orders of level filling may be obtained:

\[ 1s,\ 2p,\ 3d,\ 4f,\ 2s,\ 3p,\ 5g,\ 4d \]

or

\[ 1s,\ 2p,\ 3d,\ 4f,\ 5g,\ 4d,\ 3p,\ 2s,\ 6h, \]

or

\[ 1s,\ 2p,\ 3d,\ 4f,\ 5g,\ 6h,\ 4d,\ 3p,\ 2s. \]

Fig. 17. Displacement of energy levels with increasing rise \(D'\) in the central part of the potential well. The rise is taken over the interval

\[ r=\frac{R}{2}. \]

\(D'\) is given in units of

\[ \frac{\hbar^2}{MR^2}. \]

The numbers of filling of the levels in this case will be, respectively:

\[ 2,\ 8,\ 18,\ 32,\ 34,\ 40,\ 58,\ 68, \]

\[ 2,\ 8,\ 18,\ 32,\ 50,\ 60,\ 66,\ 68,\ 90, \]

\[ 2,\ 8,\ 18,\ 32,\ 50,\ 72,\ 82,\ 88,\ 90. \]

Thus, by selecting the height of the rise of the potential well, one can find conditions under which the experimentally observed “magic” numbers 50 and 82 would turn out to be level-filling numbers.

It should be noted, however, that the number 20 does not turn out to be “magic”; instead of it the number 18 should be “magic.” There are also no experimental indications that the number 32 is “magic.” The discrepancy of the small values of the “magic” numbers with what is obtained from the assumption of the existence of a rise in the central part of the potential well (18, 32 instead of 20) cannot, however, be regarded as a contradiction, since the rise of the potential can occur only in heavy nuclei, where there are many charged particles and the influence of repulsive forces will be relatively large.

In light nuclei, whose charge is still small, the decrease in the density of particles in the nucleus may be very small or even absent altogether, so that for such nuclei the order of alternation of the levels will be the same as in a simple rectangular well, i.e. the levels will be filled in the following sequence:

\[ 1s,\ 2p,\ 3d,\ 2s,\ 4f,\ldots, \]

which also gives, for the magic numbers in the region of light nuclei, the correct values 2, 8, 20. As the number of particles in the nucleus increases—in particular, as the number of protons increases—the character of the potential well begins to change, and the particle levels shift. The level \(2s\) undergoes an especially strong shift. For a certain number of particles in the nucleus, the level \(2s\) will be situated above the level \(4f\), and with a further increase in the number of particles in the nucleus the level \(2s\) will shift so strongly that it will coincide with the level \(5g\), or will even be situated above it.

Thus, owing to the change in the shape of the potential well that occurs as the number of charged particles in the nucleus increases, a displacement of the levels must take place, the order of their sequence must change, and in some nuclei (for a certain number of particles in the nucleus) the levels \(2s\) and \(5g\) will be situated close to one another, so that if the ground state of the last particle is the state \(2s\), then the nearest excited state to it will be the state corresponding to the level \(5g\). There may also be nuclei in which the level \(5g\) will prove to be the ground state, and the nearest level to it will be \(2s\). This crossing of levels with strongly different values of the orbital angular momenta is of great importance in explaining the phenomenon of nuclear isomerism.

A third possible reason for the discrepancy between the series of numbers (10) and (10a) may be the splitting of levels corresponding to one and the same value of the orbital quantum number \(l\). Such splitting may occur, for example, in the case of the existence of strong spin-orbit coupling. Since the spin-orbit coupling will be the stronger the larger the orbital angular momentum \(l\), the splitting of the levels (with a given \(l\)) will be more sharply expressed for larger \(l\). Mayer \(^{45—48}\) showed*) that the splitting of nuclear levels due to spin-orbit coupling occurs in such a way that levels with the larger total angular momentum \(j=l+\frac{1}{2}\) will have a binding energy greater than levels with angular momentum \(j=l-\frac{1}{2}\). The magnitude of this splitting will be proportional to \((2l+1)A^{-2/3}\), where \(A\) is the mass number of the given nucleus.

*) The shell-filling scheme obtained under the assumption of strong spin-orbit coupling was examined in greatest detail by Haxel et al. \(^{48a}\).

3 Usp. Fiz. Nauk, vol. LII, issue 1.

Levels with \(j=l+\dfrac{1}{2}\), as a consequence of the lowering of their energies, will tend to approach the levels corresponding to a smaller value of the quantum number of the oscillator potential; as a result, for large values of \(l\), the order of succession of levels characteristic of a rectangular potential well will be disturbed.

The number of particles located on each of the split levels is determined by the value of \(j\) and will be equal to \(2j+1\).

The assumption of the presence of spin-orbit coupling makes it possible to explain naturally and simply the observed values of the “magic” numbers. The first shell consists of only one level, \(1s_{1/2}\). In the second shell (for which \(l=1\)) there are two sublevels, \(2p_{3/2}\) \(\left(j=1+\dfrac{1}{2}\right)\) and \(2p_{1/2}\) \(\left(j=1-\dfrac{1}{2}\right)\). Since the splitting of the levels for \(l=1\) is still very small, these two sublevels are close in energy, and the particles located on these sublevels may be combined into one shell. In exactly the same way, the sublevels with \(l=2\), i.e., the sublevels \(3d_{5/2}\) \(\left(j=2+\dfrac{1}{2}\right)\) and \(3d_{3/2}\) \(\left(j=2-\dfrac{1}{2}\right)\), are not yet strongly split; that is, the particles located on them may also be combined into one shell. The third nuclear shell is formed from the particles of both \(d\) levels, and also from the particles located on the level \(2s_{1/2}\).

The splitting of the \(4f\) level is already more considerable, and it is possible that the sublevel \(4f_{7/2}\) (containing 8 particles) forms an independent shell. This shell should be filled when the number of particles is equal to 28. Some experimental data, which we have already mentioned, indicate the possibility of the existence of such a “magic” number as well.

The particles located on the sublevels \(4f_{5/2}\), \(3p_{3/2}\), \(3p_{1/2}\) form the next shell. To them are also added the particles of the sublevel \(5g_{9/2}\). The \(5g\) level undergoes such a strong splitting that the sublevel \(5g_{9/2}\) falls into one shell with the sublevels \(4f_{5/2}\), \(3p_{3/2}\), and \(3p_{1/2}\).

The next shell consists of the sublevels \(5g_{7/2}\), \(4d_{5/2}\), \(4d_{3/2}\) (\(4d\) are the second levels with orbital angular momentum 2), \(3s_{1/2}\) (the third level with \(l=0\)) and \(6h_{11/2}\).

The grouping of levels into shells proposed by Mayer on the basis of the concept of spin-orbit coupling is given in Table VI. As is evident from this table, the hypothesis of the existence of strong spin-orbit coupling gives the correct filling numbers of nuclear shells.

It is curious to note that the filling numbers of nuclear shells 50, 82, 126 can be obtained theoretically by applying the methods of quantum statistics. This question was the subject of

Table VI

Arrangement of levels in the case of spin-orbit coupling

Value of the oscillator-level quantum number $N$	Levels in a rectangular well	Terms with spin-orbit coupling taken into account	Number of particles filling the level	Number of particles filling the shell	Number of particles in the nucleus when the shell is filled
1	$1s$	$1s_{1/2}$	2	2	2
2	$2p$	$2p_{1/2}$	4	6	8
2	$2p$	$2p_{3/2}$	2	6	8
3	$3d$	$3d_{5/2}$	6	12	20
3	$3d$	$3d_{3/2}$	4	12	20
3	$2s$	$2s_{1/2}$	2	12	20
4	$4f$	$4f_{7/2}$	8	[[unclear: value appears as 8?]]	[[unclear: value appears as 28?]]
4	$4f$	$4f_{5/2}$	6	[[unclear: value appears as 8?]]	[[unclear: value appears as 28?]]
4	$3p$	$3p_{3/2}$	4	22	50
4	$3p$	$3p_{1/2}$	2	22	50
4		$5g_{9/2}$	10	22	50
5	$5g$	$5g_{7/2}$	8	32	82
5	$4d$	$4d_{5/2}$	6	32	82
5	$4d$	$4d_{3/2}$	4	32	82
5	$3s$	$3s_{1/2}$	2	32	82
5		$6h_{11/2}$	12	32	82
6	$6h$	$6h_{9/2}$	10	44	126
6	$5f$	$5f_{7/2}$	8	44	126
6	$5f$	$5f_{5/2}$	6	44	126
6	$4p$	$4p_{3/2}$	4	44	126
6	$4p$	$4p_{1/2}$	2	44	126
6		$7i_{13/2}$	14	44	126

works of Ivanenko and Rodichev^50, Ivanenko and Sokolov^51, Born and Yang^52.

Considering the nucleus as a completely degenerate system of particles, they obtained the distribution of particles over states with a given orbital angular momentum \(l\) in the form

\[ N(l)=K(l+1)^3, \tag{12} \]

where \(K\) is a function independent of \(l\) and determined only by the radial distribution of the particle density in the nucleus \(\rho(r)\), and \(N(l)\) is the number of particles in states with orbital angular momentum less than or equal to \(l\). Let us note that for the numbers \(N(l)\) the following relation will hold:

\[ [N(l+1)]^{1/3}-[N(l)]^{1/3}=2k^{1/3}=\mathrm{const}, \tag{13} \]

i.e., if \(N(l)\) is interpreted as the number of particles in the nucleus at which a given nuclear shell is filled (a “magic” number), then for heavy nuclei (for which alone the methods of statistics can be applied) we should expect that the difference between the cube roots of neighboring “magic” numbers will have a constant value. Table VII below, in which the differences of the cube roots of the “magic” numbers are presented, shows that relation (13), following from quantum statistics, is well satisfied for the “magic” numbers 28, 50, 82, 126.

Table VII

Number of particles in the filled shell (“magic” number) . .	28	50	82	126
Cube root of the “magic” number . . . . . .	3.04	3.69	4.35	5.01
Difference of the cube roots of the “magic” numbers . . . . .	0.65	0.66	0.66

It further turned out that if the density distribution in the nucleus \(\rho(r)\) is chosen in the form

\[ \rho(r)= \begin{cases} \rho_0, & \text{for } r<R_0,\\ \rho_0 e^{-\left(\dfrac{r-R_0}{a}\right)^2}, & \text{for } r>R_0, \end{cases} \tag{14} \]

i.e., if one assumes that inside the nucleus the particle density is constant, while at the boundary it decreases according to a Gaussian law, and determines the quantities \(\rho_0\), \(a\), \(R_0\) from the condition that the constant in relation (13) is equal to 0.66, then for the shell-filling numbers \(N(l)\) one can obtain correct values corresponding to the experimentally established

“magic” numbers. In this case the values of the quantities \(\rho_0\), \(a\), and \(R_0\) are obtained in the following form:

\[ \left. \begin{aligned} \rho_0&=\frac{1.04}{\frac{4}{3}\pi r_0^3},\\ a&=0.33r_0A^{1/3},\\ R_0&=0.67r_0A^{1/3}, \end{aligned} \right\} \tag{15} \]

where \(r_0=1.5\cdot 10^{-13}\). The quantity \(R_0\) proves to be about thirty percent smaller than the usually adopted value of the nuclear radius.

Table VIII gives a comparison of the quantities \(N(l)\), calculated under assumption (14), with the “magic” numbers:

Table VIII

\(l\) . . . . . . .	3	4	5	6
\(N(l)\) . . . . . . .	27.1	49.5	81.5	125.2
“Magic” numbers	28	50	82	126

As is seen from Table VIII, the agreement between the “magic” numbers and the calculated values of \(N(l)\) is good.

7. NUCLEAR SPIN

The value of the spin and magnetic moment has now been established for many stable isotopes, and also for some radioactive isotopes.

As a rule, the spins and magnetic moments of nuclei containing an even number of protons and neutrons are equal to zero. The spins and magnetic moments of nuclei with an odd mass number are always different from zero. Table IX gives the values of the spin and magnetic moment of various nuclei containing an odd number of particles \({}^{53}\).

The magnetic moment in Table IX is expressed in units

\[ \frac{e\hbar}{2M_pc} \]

(the nuclear magneton), where \(M_p\) is the proton mass, \(c\) is the speed of light, \(\hbar\) is Planck’s constant divided by \(2\pi\), and \(e\) is the electron charge.

The fact that the spins of many odd nuclei are not equal to one half undoubtedly testifies to the existence of some structure within nuclei. A comparison of the values of the spin and magnetic moment gives grounds for concluding that orbital motion exists inside nuclei. In the presence of orbital motion within

Table IX

Values of spins and magnetic moments of nuclei

Nuclear shell	Number of protons in the nucleus	Isotope	Spin	Magnetic moment	Number of neutrons in the nucleus	Isotope	Spin	Magnetic moment
I	1	${}^{3}_{1}\mathrm{H}$	$1/2$	2.97864	1	${}^{3}_{2}\mathrm{He}$	$1/2$	−2.12741
II	3	${}^{7}_{3}\mathrm{Li}$	$3/2$	3.2558	—	—	—	—
II	5	${}^{11}_{5}\mathrm{B}$	$3/2$	2.6885	5	${}^{9}_{4}\mathrm{Be}$	$3/2$	−1.177${}^{54\mathrm{a}}$
II	7	${}^{15}_{7}\mathrm{N}$	$1/2$	−0.2829	7	${}^{13}_{6}\mathrm{C}$	$1/2$	0.7022
III	9	${}^{19}_{9}\mathrm{F}$	$1/2$	2.6285	9	${}^{17}_{8}\mathrm{O}$	$5/2$	−1.89${}^{54\mathrm{a}}$
III	11	${}^{23}_{11}\mathrm{Na}$	$3/2$	2.2171	11	${}^{21}_{10}\mathrm{Ne}$	$3/2$	<0${}^{54\mathrm{a}}$
III	13	${}^{27}_{13}\mathrm{Al}$	$5/2$	3.6408	13	${}^{25}_{12}\mathrm{Mg}$	$5/2$	−0.8546${}^{55}$
III	15	${}^{31}_{15}\mathrm{P}$	$1/2$	1.13165	15	${}^{29}_{14}\mathrm{Si}$	$1/2$	−0.554${}^{54\mathrm{a}}$
III	17	${}^{35}_{17}\mathrm{Cl}$	$3/2$	0.8219	17	${}^{33}_{16}\mathrm{S}$	$3/2$	+0.643${}^{54\mathrm{a}}$
III	17	${}^{37}_{17}\mathrm{Cl}$	$3/2$	0.6841	19	${}^{35}_{16}\mathrm{S}$	$3/2$	—
III	19	${}^{39}_{19}\mathrm{K}$	$3/2$	0.391
III	19	${}^{41}_{19}\mathrm{K}$	$3/2$	0.215
IV	21	${}^{45}_{21}\mathrm{Sc}$	$7/2$	4.7556${}^{54\mathrm{a}}$	—	—
IV	23	${}^{51}_{23}\mathrm{V}$	$7/2$	5.1478	23	${}^{43}_{20}\mathrm{Ca}$	$7/2$	−1.315${}^{56\mathrm{a}}$
IV	25	${}^{55}_{25}\mathrm{Mn}$	$5/2$	3.4681
IV	27	${}^{57}_{27}\mathrm{Co}$	$7/2$	4.6
IV	27	${}^{59}_{27}\mathrm{Co}$	$7/2$	4.6484${}^{54\mathrm{b}}$
IV	29	${}^{63}_{29}\mathrm{Cu}$	$3/2$	2.2261	29	${}^{53}_{24}\mathrm{Cr}$	$3/2$	(−) 0.45${}^{54\mathrm{a}}$
IV	29	${}^{65}_{29}\mathrm{Cu}$	$3/2$	2.3845

Continuation of Table IX

Nuclear shell	Number of protons in the nucleus	Isotopes with an odd number of protons: isotope	Isotopes with an odd number of protons: spin	Isotopes with an odd number of protons: magnetic moment	Number of neutrons in the nucleus	Isotopes with an odd number of neutrons: isotope	Isotopes with an odd number of neutrons: spin	Isotopes with an odd number of neutrons: magnetic moment
IV	31	${}^{69}_{31}\mathrm{Ga}$	3/2	2,0267
IV	31	${}^{71}_{31}\mathrm{Ga}$	3/2	2,5614
IV	33	${}^{75}_{33}\mathrm{As}$	3/2	1,435054
IV	35	${}^{79}_{35}\mathrm{Br}$	3/2	2,1057
IV	35	${}^{81}_{35}\mathrm{Br}$	3/2	2,2696
IV	37	${}^{85}_{37}\mathrm{Rb}$	5/2	1,3532	37	${}^{67}_{30}\mathrm{Zn}$	5/2	0,8737857
IV	37	${}^{87}_{37}\mathrm{Rb}$	3/2	2,7501
IV	39	${}^{89}_{39}\mathrm{Y}$	1/2	−0,14	41	${}^{73}_{32}\mathrm{Ge}$	9/2	—
IV	41	${}^{93}_{41}\mathrm{Nb}$	9/2	6,1659	43	${}^{77}_{34}\mathrm{Se}$	1/2	0,5332658
IV	43	${}^{99}_{43}\mathrm{Tc}$	9/2	5,680555	45	${}^{79}_{34}\mathrm{Se}$	7/2	—
IV	47	${}^{107}_{47}\mathrm{Ag}$	1/2	−0,11154a	47	${}^{83}_{36}\mathrm{Kr}$	9/2	−0,9704
IV	47	${}^{109}_{47}\mathrm{Ag}$	1/2	−0,129
IV	49	${}^{113}_{49}\mathrm{In}$	9/2	5,486	49	${}^{87}_{38}\mathrm{Sr}$	9/2	−1,1
IV	49	${}^{115}_{49}\mathrm{In}$	9/2	5,500
V	51	${}^{121}_{51}\mathrm{Sb}$	5/2	3,35454a	51	${}^{91}_{40}\mathrm{Zr}$	5/2
V	51	${}^{123}_{51}\mathrm{Sb}$	7/2	2,54654a
V	53	${}^{127}_{53}\mathrm{I}$	5/2	2,8086	53	${}^{95}_{42}\mathrm{Mo}$	(5/2)	−0,91454a
V	53	${}^{129}_{53}\mathrm{I}$	7/2	2,61751a

Continuation of Table IX

Nuclear shell	Isotopes with an odd number of protons: number of protons in the nucleus	Isotopes with an odd number of protons: isotope	Isotopes with an odd number of protons: spin	Isotopes with an odd number of protons: magnetic moment	Isotopes with an odd number of neutrons: number of neutrons in the nucleus	Isotopes with an odd number of neutrons: isotope	Isotopes with an odd number of neutrons: spin	Isotopes with an odd number of neutrons: magnetic moment
V	55	${}^{131}_{55}\mathrm{Cs}$	$5/2$	$3,48^{55a}$	55	${}^{97}_{42}\mathrm{Mo}$	$(5/2)$	$-0,933^{55a}$
V	55	${}^{133}_{55}\mathrm{Cs}$	$7/2$	$2,5771$	55	${}^{99}_{44}\mathrm{Ru}$	$5/2^{565}$
V	55	${}^{135}_{55}\mathrm{Cs}$	$7/2$	$2,7271$	57	${}^{101}_{44}\mathrm{Ru}$	$5/2$
V	55	${}^{137}_{55}\mathrm{Cs}$	$7/2$	$2,8397$	59	${}^{105}_{46}\mathrm{Pd}$	$(5/2)$	$(-0,6)$
V	57	${}^{139}_{57}\mathrm{La}$	$7/2$	$2,7760$	63	${}^{111}_{48}\mathrm{Cd}$	$1/2$	$-0,5949$
V	59	${}^{141}_{59}\mathrm{Pr}$	$5/2$	$3,8^{556}$	63	${}^{113}_{48}\mathrm{Cd}$	$1/2$	$-0,62238$
V	63	${}^{151}_{63}\mathrm{Eu}$	$5/2$	$3,4$	65	${}^{115}_{50}\mathrm{Sn}$	$1/2$	$-0,91779$
V	63	${}^{153}_{63}\mathrm{Eu}$	$5/2$	$1,5$	67	${}^{117}_{50}\mathrm{Sn}$	$1/2$	$-0,99982$
V	65	${}^{151}_{65}\mathrm{Tb}$	$3/2$		69	${}^{119}_{50}\mathrm{Sn}$	$1/2$	$-1,0460$
V	67	${}^{165}_{67}\mathrm{Ho}$	$7/2$		71	${}^{123}_{52}\mathrm{Te}$	$1/2$	$-0,73188^{54a}$
V	69	${}^{169}_{69}\mathrm{Tm}$	$1/2$		73	${}^{125}_{52}\mathrm{Te}$	$1/2$	$-0,88235^{54a}$
V	71	${}^{175}_{71}\mathrm{Lu}$	$5/2$	$2,6$	75	${}^{129}_{54}\mathrm{Xe}$	$1/2$	$-0,7766^{54a}$
V	73	${}^{181}_{73}\mathrm{Ta}$	$7/2$	$2,1$	77	${}^{131}_{54}\mathrm{Xe}$	$3/2$	$+0,7$
V	75	${}^{185}_{75}\mathrm{Re}$	$5/2$	$3,143^{54a}$	79	${}^{135}_{56}\mathrm{Ba}$	$3/2$	$+0,8346$
V	75	${}^{187}_{75}\mathrm{Re}$	$5/2$	$3,175^{54a}$	81	${}^{137}_{56}\mathrm{Ba}$	$3/2$	$0,9351$
V	77	${}^{191}_{77}\mathrm{Ir}$	$3/2$
V	77	${}^{193}_{77}\mathrm{Ir}$	$3/2$
V	79	${}^{197}_{79}\mathrm{Au}$	$3/2$	$0,20$
V	81	${}^{203}_{81}\mathrm{Tl}$	$1/2$	$1,61166$
V	81	${}^{205}_{81}\mathrm{Tl}$	$1/2$	$1,6275$

Continuation of Table IX

Nuclear shell	Number of protons in nucleus	Isotope	Spin	Magnetic moment	Number of neutrons in nucleus	Isotope	Spin	Magnetic moment
VI	83	${}^{203}_{83}\mathrm{Bi}$	$9/2$	4.08	101	${}^{171}_{70}\mathrm{Yb}$	$1/2$	0.45
VI	89	${}^{227}_{89}\mathrm{Ac}$	$3/2^{54a}$		103	${}^{173}_{70}\mathrm{Yb}$	$5/2$	−0.65
VI	91	${}^{231}_{91}\mathrm{Pa}$	$3/2^{54a}$		105	${}^{177}_{72}\mathrm{Hf}$	$\left(\frac{1}{2},\frac{3}{2}\right)$
VI	93	${}^{237}_{93}\mathrm{Np}$	$5/2^{55b}$		107	${}^{179}_{72}\mathrm{Hf}$	$\frac{1}{2},\frac{3}{2}$
VI	95	${}^{241}\mathrm{Am}$			109	${}^{183}_{74}\mathrm{W}$	$1/2$
VI					113	${}^{189}_{76}\mathrm{Os}$	$1/2$
VI					117	${}^{195}_{78}\mathrm{Pt}$	$1/2$	0.6059
VI					119	${}^{199}_{80}\mathrm{Hg}$	$1/2$	0.50413
VI					121	${}^{201}_{80}\mathrm{Hg}$	$3/2$	−0.5590
VI					125	${}^{207}_{82}\mathrm{Pb}$	$1/2$	0.58950

Number of neutrons	Isotope	Spin	Magnetic moment
83	${}^{143}_{60}\mathrm{Nd}$	$7/2$	−1.0
85	${}^{145}_{60}\mathrm{Nd}$	$7/2$	−0.65
62	${}^{147}_{62}\mathrm{Sm}$	$7/2$	(−) 0.68
64	${}^{149}_{62}\mathrm{Sm}$	$7/2$	(−) 0.55

of the nucleus, the total angular momentum of the nucleus (“nuclear spin”) will be composed not only of the spins of the individual particles located in the nucleus, but also of their orbital moments.

From Table IX it is seen that among nuclei containing an even number of neutrons and an odd number of protons, there are those,

in which the value of the spin is equal to \(^{1}/_{2}\), while the magnetic moment is negative. Such are, for example, the nuclei \({}^{15}_{7}\mathrm{N}\), \({}^{89}_{39}\mathrm{Y}\), \({}^{107}_{47}\mathrm{Ag}\), \({}^{109}_{47}\mathrm{Ag}\).

If we assume that in the given case the spin of the nucleus owes its value \((^{1}/_{2})\) to the spin of one odd proton (and not to its orbital moment), then the magnetic moment of the nucleus would have to be the same as the magnetic moment of the proton. Meanwhile, in all the nuclei listed the magnetic moment differs in magnitude from the magnetic moment of the proton and, most important, has the opposite sign (is negative). Thus, the spin of these nuclei is determined not only by the spins of its constituent particles.

Further, among nuclei containing an even number of protons and an odd number of neutrons, we find some whose spin is also equal to \(^{1}/_{2}\), but whose magnetic moment has a positive value. Such are, for example, the nuclei \({}^{13}_{6}\mathrm{C}\), \({}^{171}_{70}\mathrm{Yb}\), \({}^{195}_{78}\mathrm{Pt}\), \({}^{199}_{80}\mathrm{Hg}\), \({}^{207}_{82}\mathrm{Pb}\). Meanwhile, if the spin of these nuclei were due to the spin of the odd neutron, then the magnetic moment of such nuclei would have to have not a positive, but a negative value, since the magnetic moment of the neutron is negative and has the value \(-1.91\).

The comparison given here of the magnetic moments and spins of the nuclei \({}^{15}_{7}\mathrm{N}\), \({}^{89}_{39}\mathrm{Y}\), \({}^{107}_{47}\mathrm{Ag}\), \({}^{109}_{47}\mathrm{Ag}\), \({}^{13}_{6}\mathrm{C}\), \({}^{171}_{70}\mathrm{Yb}\), \({}^{195}_{78}\mathrm{Pt}\), \({}^{199}_{80}\mathrm{Hg}\), \({}^{207}_{82}\mathrm{Pb}\) compels us to consider it established with certainty that the spin of these nuclei is due not only to the spins of its constituent particles, but also to the orbital moments of these particles. The existence of nuclei with a large value of the spin also confirms this proposition. Consequently, comparison of the spins and magnetic moments of nuclei gives us grounds to conclude that orbital motion undoubtedly exists in nuclei. Moreover, such a comparison, carried out for all nuclei, can establish the correctness of the systematics of nuclear levels, clarify the order of their filling, and group them into particular shells. To explain the values of the spins and magnetic moments of nuclei given in Table IX, Mayer advanced the following propositions as the basis for the systematics of nuclear levels:

1. Every even number of identical nuclear particles situated on a level with one and the same value of the total moment \(j\) is always coupled in the nucleus in such a way that the resultant spin and magnetic moment due to them are equal to zero.

2. Every odd number of identical nuclear particles situated in a state characterized by the value of the total moment \(j\) is coupled in the nucleus in such a way that the latter has spin equal to \(j\), and a magnetic moment the same as that possessed by one particle situated in this state. In other words, the spin and magnetic moment of nuclei are due to the orbital motion and the spin of only one unpaired particle (the one-nucleon model).

1. The energy of a nuclear particle moving inside a rectangular potential well with spin–orbit coupling depends on the parameters characterizing the motion (the quantum numbers \(n, m, l\)) in such a way that, as the number of particles in the nucleus increases, doublet levels change the order of their arrangement. This is expressed in the following:

a) For a given \(l\), levels with \(j=l+\dfrac{1}{2}\) will have an energy lower than levels with \(j=l-\dfrac{1}{2}\), and therefore will be filled earlier.

b) Pairs of spin levels*) in a given shell, originating from one level of the rectangular potential well, have a tendency to come closer together and even to change the order of their arrangement. Such are, for example, the pairs of levels \(d_{3/2}\) and \(s_{1/2}\), or \(f_{5/2}\) and \(p_{3/2}\), or \(g_{7/2}\) and \(d_{5/2}\). Such a rapprochement of levels and even their crossing (as the number of particles in the nucleus increases) may also occur with the pairs of levels \(p_{1/2}\) and \(g_{9/2}\), \(d_{3/2}\) and \(h_{11/2}\), \(p_{1/2}\) and \(i_{13/2}\). All these pairs of levels may cross, and their arrangement may prove to be different from that given in Table IX.

1. Here attention is drawn to the circumstance that the binding energy of a pair of identical particles on one level, calculated per one particle, is greater than the binding energy of a single particle. This increase in the particle binding energy (when the level is filled by pairs of particles) will be the greater within a given shell, the larger the total moment \(j\). This leads to the fact that, in the case of an odd number of particles, states with large \(j\) will be observed not as often as could have been expected on the basis of Table VI. For example, suppose that the energy of one particle on the level \(3s_{1/2}\) is slightly smaller than on the level \(6h_{11/2}\) (i.e., the level \(3s_{1/2}\) should have been filled earlier than the level \(6h_{11/2}\)). However, a pair of particles on the level \(6h_{11/2}\) may then have an energy smaller than a pair of particles filling the level \(3s_{1/2}\). If this occurs, then the process of filling the shell will proceed as follows: first the particle will occupy the level \(3s_{1/2}\). If a second particle, identical with the first, is added, then both will occupy the level \(6h_{11/2}\) (the level \(3s_{1/2}\) will become vacant). The third particle will again occupy the level \(3s_{1/2}\); the fourth particle, together with the third, will likewise end up on the level \(6h_{11/2}\), and not \(3s_{1/2}\) (the level \(3s_{1/2}\) will become vacant), etc. Although in the end the level \(6h_{11/2}\) will be filled, the nuclear spin (for an odd number of particles) will have the value one half, and not \(11/2\).

These four assumptions constitute the basis of the systematics of nuclear levels. They explain fairly well the experimentally observed values of nuclear spins.

*) Levels with different orientation of the spin with respect to the orbital moment are meant.

Let us turn again to Table IX. In this table all nuclei are divided into groups in accordance with the filling of nuclear shells by particles.

The first group includes nuclei in which the first shell is being filled. The first shell contains one level \(1s_{1/2}\). The shell is filled by two particles. Consequently, there will be only two nuclei with an odd number of particles in this shell—\({}^{2}_{1}\mathrm{H}\), with one proton, and \({}^{3}_{2}\mathrm{He}\), with one neutron. With an odd number of particles on the level \(1s_{1/2}\), the moment of the nucleus must be equal to \(1/2\). The spins of both nuclei \({}^{2}_{1}\mathrm{H}\) and \({}^{3}_{2}\mathrm{He}\) are indeed equal to one half.

The second group includes nuclei in which the second shell is being filled. The second shell contains two levels, \(2p_{3/2}\) and \(2p_{1/2}\). Consequently, among the nuclei of this group containing an odd number of particles, there must occur those whose spin is equal to \(3/2\) (the unpaired particle occupies the level \(2p_{3/2}\)), as well as those whose spin is equal to \(1/2\) (the unpaired particle occupies the level \(2p_{1/2}\)).

The second shell is filled by six particles, of which four must be on the level \(2p_{3/2}\), and two on the level \(2p_{1/2}\). Table IX gives five nuclei \({}^{7}_{3}\mathrm{Li}\), \({}^{11}_{5}\mathrm{B}\), \({}^{15}_{7}\mathrm{N}\), \({}^{9}_{4}\mathrm{Be}\), and \({}^{13}_{6}\mathrm{C}\), belonging to the second group. In three of them (\({}^{7}_{3}\mathrm{Li}\), \({}^{11}_{5}\mathrm{B}\), and \({}^{15}_{7}\mathrm{N}\)) the protons are contained in an odd number, and in the remaining ones (\({}^{9}_{4}\mathrm{Be}\) and \({}^{13}_{6}\mathrm{C}\)) there is an odd number of neutrons. All these nuclei have spin either \(3/2\) or \(1/2\), which is in agreement with the assumptions made above. Comparing the spin values of the nuclei \({}^{7}_{3}\mathrm{Li}\), \({}^{11}_{5}\mathrm{B}\), \({}^{15}_{7}\mathrm{N}\), \({}^{9}_{4}\mathrm{Be}\), and \({}^{13}_{6}\mathrm{C}\), one can also draw a conclusion about the order of filling of the levels of the second shell. Since \({}^{7}_{3}\mathrm{Li}\) (containing three protons, of which two are in the first shell and one in the second shell), \({}^{11}_{5}\mathrm{B}\) (having three protons in the second shell), and \({}^{9}_{4}\mathrm{Be}\) (having five neutrons, of which three are in the second shell) have spin equal to \(3/2\), while the nuclei \({}^{15}_{7}\mathrm{N}\) and \({}^{13}_{6}\mathrm{C}\), having five particles each in the second shell, have spin \(1/2\), it follows that in the second shell the level \(2p_{3/2}\) is filled earlier than the level \(2p_{1/2}\). The latter (the level \(2p_{1/2}\)) is filled after all four particles have filled the level \(2p_{3/2}\). This order of filling of levels corresponds to Mayer’s scheme, according to which the level \(l+\frac{1}{2}\) has a lower energy than the level \(l-\frac{1}{2}\).

The third group of nuclei includes those in which the third shell is being filled. The third shell contains the levels \(3d_{5/2}\), \(3d_{3/2}\), \(2s_{1/2}\). Consequently, among nuclei belonging to this group there must occur those in which the total moment will have the value \(5/2\) (these are nuclei in which the unpaired particle occupies the level \(3d_{5/2}\)), as well as those in which

the total moment will be \(^{3}/_{2}\) (level \(3d_{3/2}\)) and \(^{1}/_{2}\) (level \(2s_{1/2}\)). The third shell is filled with 12 particles, of which 6 fill the level \(3d_{5/2}\), 4 fill the level \(3d_{3/2}\), and 2 particles fill the level \(2s_{1/2}\).

Table IX contains 14 nuclei belonging to the third group, of which 8 contain an odd number of protons, and 6—an odd number of neutrons. The spins of these 14 nuclei have indeed proved to be equal either to \(^{5}/_{2}\), or to \(^{3}/_{2}\), or to \(^{1}/_{2}\). The value of the spin \(^{3}/_{2}\) occurs in those nuclei which in the third shell have either 9 protons (these are the nuclei \({}^{35}_{17}\mathrm{Cl}\), \({}^{37}_{17}\mathrm{Cl}\)), or 11 protons (the nuclei \({}^{39}_{19}\mathrm{K}\), \({}^{41}_{19}\mathrm{K}\)), or 9 neutrons (the nucleus \({}^{33}_{16}\mathrm{S}\)), or 11 neutrons (the nucleus \({}^{35}_{16}\mathrm{S}\)), i.e., in those nuclei in which the construction of the third shell is completed (altogether there are 12 particles in the third shell). Consequently, the level \(3d_{3/2}\) in this shell is filled last, after the levels \(3d_{5/2}\) and \(2s_{1/2}\). The level \(2s_{1/2}\) is filled before the level \(3d_{3/2}\), for the nuclei \({}^{31}_{15}\mathrm{P}\) and \({}^{29}_{14}\mathrm{Si}\), containing 7 particles each in the third shell, have spin \(^{1}/_{2}\). This means that in the first six nuclei of the third group the level \(3d_{5/2}\) is filled, and then the level \(2s_{1/2}\) is already being filled (nuclei with seven particles have spin \(^{1}/_{2}\)). The fact that the level \(3d_{5/2}\) is filled first is also confirmed by the fact that the nuclei \({}^{27}_{13}\mathrm{Al}\) and \({}^{25}_{12}\mathrm{Mg}\), having five particles each in the third shell, have spins \(^{5}/_{2}\). Thus, in the third shell the following order of filling of the levels occurs: \(3d_{5/2}\), \(2s_{1/2}\), and \(3d_{3/2}\). Such an order in the sequence of levels is consistent with the course of the mass defect of light nuclei, which indicates the presence of a filled group in nuclei containing 14 particles (protons). Such a group can consist precisely of particles situated on the level \(3d_{5/2}\). Indeed, recent measurements have shown that the spin of \(\mathrm{O}^{17}\) is equal to \(^{5}/_{2}\), and the magnetic moment is almost exactly equal to the magnetic moment of the neutron. This indicates that the odd neutron is located in the \(\mathrm{O}^{17}\) nucleus on the level \(3d_{5/2}\). Thus, the third shell begins to be filled from the level \(3d_{5/2}\). However, this order is then violated: for \({}^{19}_{9}\mathrm{F}\), containing one proton in the third shell, the spin is not equal to \(^{5}/_{2}\), but to \(^{1}/_{2}\). The magnitude of the magnetic moment of \({}^{19}_{9}\mathrm{F}\) indicates that this spin corresponds to the level \(s_{1/2}\). This means that for \(\mathrm{F}^{19}\) the level \(2s_{1/2}\) has the lowest energy; however, as the number of particles increases, a shift of the levels occurs and the level \(2s_{1/2}\) is located above the level \(3d_{5/2}\), as was also the case for \(\mathrm{O}^{17}\). (It is also possible that this is the result of the action of rule 4.) It might have been supposed that the answer to the question of at what number of particles in the nucleus the crossing of the levels \(2s_{1/2}\) and \(3d_{5/2}\) occurs could be obtained from the values of the spins of the nuclei \({}^{23}_{11}\mathrm{Na}\) and \({}^{21}_{10}\mathrm{Ne}\), containing three particles each in the third shell. It should have been expected that the spins of these nuclei would be either \(^{5}/_{2}\)

(if three particles occupy the \(3d_{5/2}\) level), or \(1/2\) (if two particles occupy the \(3d_{5/2}\) level, and one particle, in accordance with rule 4, occupies the \(2s_{1/2}\) level). However, it turned out that the spins of \({}^{23}_{11}\mathrm{Na}\) and \({}^{21}_{10}\mathrm{Ne}\) are not \(5/2\) and not \(1/2\), but \(3/2\). Moreover, the value of the magnetic moment of the nucleus \({}^{23}_{11}\mathrm{Na}\) shows that the state of the third particle is more likely the \(p_{3/2}\) state than the \(d_{3/2}\) state.

However, in the third shell, according to Mayer’s scheme, there should be no \(p_{3/2}\) level; thus the nucleus \({}^{23}_{11}\mathrm{Na}\) falls outside Mayer’s scheme.

In the fourth group are combined nuclei in which the fourth shell is being filled. Table IX gives 26 nuclei belonging to this group whose spins have been measured. Twenty of them are nuclei containing an odd number of protons. According to Mayer’s scheme, the fourth shell consists of the levels \(4f_{7/2}\), \(4f_{5/2}\), \(3p_{3/2}\), \(3p_{1/2}\), and \(5g_{9/2}\). Consequently, the spins of nuclei belonging to the fourth shell may have the values \(1/2\), \(3/2\), \(5/2\), \(7/2\), and \(9/2\). And indeed, among the fourth group we find all the indicated spin values.

What, then, is the order of filling of the levels of this shell? The first of the levels entering the fourth shell to be filled is the \(4f_{7/2}\) level. In fact, the nucleus \({}^{45}_{21}\mathrm{Sc}\), containing 21 protons, i.e. having one proton in the fourth shell, has spin \(7/2\); the nucleus \({}^{51}_{23}\mathrm{V}\), containing 23 protons, i.e. having 3 protons in the fourth shell, likewise has spin \(7/2\), and, finally, the nucleus \({}^{59}_{27}\mathrm{Co}\), containing 27 protons, i.e. having 7 protons in the fourth shell, likewise has spin \(7/2\). The spin of the radioactive nucleus \({}^{57}_{27}\mathrm{Co}\) has recently been measured and was also found to be \(7/2\). The nucleus \({}^{43}_{20}\mathrm{Ca}\), which has 23 neutrons, according to the latest measurements also has spin \(7/2\). The \(4f_{7/2}\) level is filled by eight particles. Consequently, \({}^{59}_{27}\mathrm{Co}\) must be the last among the nuclei of the fourth group possessing spin \(7/2\).

Thus, the fourth shell begins with the filling of the \(4f_{7/2}\) level (perhaps this level should be singled out as an independent shell, so strongly is it displaced relative to the \(4f_{5/2}\) level?). It should be noted that the nucleus \({}^{55}_{25}\mathrm{Mn}\) falls outside Mayer’s scheme, just as does the nucleus \({}^{23}_{11}\mathrm{Na}\). \({}^{55}_{25}\mathrm{Mn}\) has five protons belonging to the fourth shell. Its spin should be \(7/2\). However, \({}^{55}_{25}\mathrm{Mn}\) has spin not \(7/2\), but \(5/2\). The value of the magnetic moment of this nucleus shows that the odd proton in \({}^{55}_{25}\mathrm{Mn}\) more likely occupies the \(d_{5/2}\) level than the \(f_{5/2}\) level. But according to Mayer’s scheme there is no \(d_{5/2}\) level in the fourth shell.

The last among the levels of the fourth shell to be filled is the \(5g_{9/2}\) level. This follows from the fact that the nuclei \({}^{113}_{49}\mathrm{In}\), \({}^{115}_{49}\mathrm{In}\), and \({}^{87}_{38}\mathrm{Sr}\), containing 49 particles in their composition, i.e. nuclei which lack only one particle for the filling of the fourth shell,

have spin equal to \(9/2\). Before the \(5g_{9/2}\) level, the \(3p_{1/2}\) level is filled. Filling of the \(3p_{1/2}\) level begins in a nucleus containing 19 particles in the fourth shell, and a total of 39 particles. After the \(3p_{1/2}\) level, filling of the \(5g_{9/2}\) level begins. In \({}^{93}_{41}\mathrm{Nb}\), and \({}^{99}_{43}\mathrm{Tc}\), the unpaired particle occupies the \(5g_{9/2}\) level; however, in \({}^{103}_{45}\mathrm{Rh}\), \({}^{107}_{47}\mathrm{Ag}\), and \({}^{109}_{47}\mathrm{Ag}\) the spins again turn out to be equal to one-half. This means that the \(3p_{1/2}\) and \(5g_{9/2}\) levels cross.

The \(4f_{5/2}\) level is filled before the \(3p_{1/2}\) level and is filled predominantly by pairs of particles in accordance with rule 4.

Thus, the order of filling of the levels belonging to the fourth shell is as follows:

\[ 4f_{7/2},\quad 3p_{3/2},\quad 4f_{5/2},\quad 3p_{1/2},\quad 5g_{9/2}, \]

where the \(3p_{1/2}\) and \(5g_{9/2}\) levels may cross in the process of filling the fourth shell.

In the fifth group of nuclei, filling of the fifth shell takes place. This shell includes the levels \(5g_{7/2}\), \(4d_{5/2}\), \(4d_{3/2}\), \(3s_{1/2}\), and \(6h_{11/2}\). Consequently, in this group there may be nuclei with the following spin values: \(3/2\), \(5/2\), \(7/2\), and \(11/2\). Spin \(9/2\) among this group of nuclei should not occur. And indeed, among 37 different nuclei of this group for which the spin value has been established, we do not find a single nucleus with spin \(9/2\).

The first levels to be filled in this group are \(5g_{7/2}\) and \(4d_{5/2}\). These levels are evidently situated extremely close to each other, since two isotopes of antimony—\({}^{121}_{51}\mathrm{Sb}\) and \({}^{123}_{51}\mathrm{Sb}\)—and two isotopes of iodine—\({}^{127}_{53}\mathrm{I}\) and \({}^{129}_{53}\mathrm{I}\)—which possess different numbers of neutrons (the number of neutrons is even, and should not affect the magnitude of the spin), have different nuclear spins. Thus, \({}^{121}_{51}\mathrm{Sb}\) has spin \(5/2\), while the nucleus of the isotope \({}^{123}_{51}\mathrm{Sb}\) has spin \(7/2\). Similarly, \({}^{127}_{53}\mathrm{I}\) has spin \(5/2\), and \({}^{129}_{53}\mathrm{I}\) has spin \(7/2\). According to the latest data, the same picture obtains for Cs: the radioactive isotope \({}^{131}_{55}\mathrm{Cs}\) has spin \(5/2\), whereas the stable isotope \({}^{133}_{55}\mathrm{Cs}\) has spin \(7/2\).

Filling of these two (\(5g_{7/2}\) and \(4d_{5/2}\)) levels should occur with 14 particles in the fifth shell, i.e., it should have ended when the number of particles (of the given type) in the nucleus had reached 64. However, we find spin values \(5/2\) and \(7/2\) also in such nuclei as \({}^{175}_{71}\mathrm{Lu}\), \({}^{181}_{73}\mathrm{Ta}\), \({}^{185}_{75}\mathrm{Re}\), and \({}^{187}_{75}\mathrm{Re}\), containing respectively 71, 73, and 75 protons, of which 21, 23, and 25 protons are in the fifth shell. This means that in the fifth shell one of the levels, apparently \(6h_{11/2}\), is filled only by pairs of particles. That it is precisely the \(6h_{11/2}\) level that is filled only by pairs of particles is also indicated by the fact that nuclei with spin,

equal to \(11/2\), are not encountered among nuclei of the fifth group. The filling of the \(6h_{1/2}\) level therefore begins earlier than that of the \(5g_{7/2}\) level, and the \(4d_{5/2}\) level is filled completely.

The last levels to be filled in the fifth shell are \(3s_{1/2}\) and \(4d_{3/2}\). In nuclei containing an odd number of protons, the \(4d_{3/2}\) level is filled earlier than the \(3s_{1/2}\) level; while in nuclei containing an odd number of neutrons, the \(4d_{3/2}\) level is filled after the \(3s_{1/2}\) level.

As for nuclei of the sixth group, in which the sixth shell is being filled, nothing definite can be said about the order in which the levels are filled, since measurements of spin and magnetic moment have been made for only a few nuclei belonging to this group. At first it was assumed that, since the nucleus \({}^{209}_{83}\mathrm{Bi}\), which has one proton in the sixth shell, has a spin equal to \(9/2\), the first level to be filled in the sixth proton shell is \(6h_{9/2}\). However, later measurements showed that the spins of the nuclei \(\mathrm{Nd}^{143}\), \(\mathrm{Nd}^{145}\), \(\mathrm{Sm}^{147}\), and \(\mathrm{Sm}^{149}\) are equal to \(7/2\). Therefore the filling of the sixth shell evidently begins with the \(5f_{7/2}\) level. The last level to be filled in this shell is \(4p_{1/2}\), as is seen from consideration of the data on the nucleus \(\mathrm{Pb}^{207}\).

Consideration of the spin values of various nuclei indisputably testifies in favor of the idea that orbital motion exists in nuclei. The systematics of nuclear levels proposed by Mayer agrees well with the experimental values of nuclear spins. Some exceptions \(\left({}^{23}_{11}\mathrm{Na},\ {}^{55}_{25}\mathrm{Mn}\right)\) may be explained by the assumption that sometimes the lowest levels turn out to be those whose spin and magnetic moment are determined by the orbital motion not of one particle (as Mayer assumed), but of three (and more) particles. The spin of the nucleus \(\mathrm{Se}^{79}\), which has 5 particles in the \(5g_{9/2}\) level, is also equal to \(7/2\), and not \(9/2\), as would be expected on the basis of the one-particle model.

8. MAGNETIC MOMENTS OF NUCLEI

As we have already noted, the spin of a nucleus is determined both by the spins of the particles entering the nucleus and by their orbital motion. According to Mayer’s hypothesis, the spin of odd nuclei \(I\) is equal to \(j\), the value of the total angular momentum of the unpaired particle. In the case of strong spin-orbit coupling, \(j\) will be equal either to \(l+\frac{1}{2}\), or to \(l-\frac{1}{2}\). According to Schmidt’s rule[^39], the magnetic moment of a particle \(\mu_l\), having orbital angular momentum equal to \(l\), in the presence of spin-orbit coupling must be equal either to

\[ \mu_l = g_l l + g_s, \tag{16} \]

if \(I = l + \frac{1}{2}\), or

\[ \mu_l = g_l\frac{(l+1)(2l-1)}{2l+1} - g_s\frac{2l-1}{2l+1} = \frac{2l-1}{2l+1}\bigl(g_l(l+1)-g_s\bigr), \tag{17} \]

if \(I=l-\dfrac{1}{2}\). In expressions (16) and (17) the numbers \(g\) represent the ratios of the magnitude of the magnetic moment to the mechanical one (the symbol “\(l\)” refers to the orbital moment, and the symbol “\(s\)” to the intrinsic moment of the particle). Consequently, for protons \(g_l=1\), and \(g_s=2.79\). For neutrons \(g_l=0\) (the neutron has no charge; consequently, the magnetic moment corresponding to its orbital motion must be equal to zero), and \(g_s=-1.91\).

According to (16) and (17), the magnetic moments of nuclei must be a two-valued function of \(I\), different for nuclei with an odd number of protons and for nuclei with an odd number of neutrons. In Fig. 18, the solid lines express the dependence of \(\mu_l\) on \(I\) for nuclei with an odd number of protons

Fig. 18. \(a\)—magnetic moments of nuclei with an odd number of protons (\(N\) even), \(b\)—magnetic moments of nuclei with an odd number of neutrons (\(Z\) even).

(Fig. 18, \(a\)) and for nuclei with an odd number of neutrons (Fig. 18, \(b\)). On this graph the points indicate the values of the magnetic moments of various nuclei, taken from Table IX. As is seen from the figures, the experimental values of the magnetic moments of nuclei do not strictly follow the Schmidt rule: they do not lie on the curves corresponding to relations (16) and (17), but are situated between them, with one group of points lying closer to the lower curve, corresponding to \(j=l-\dfrac{1}{2}\), and the other group of points lying closer to the upper curve, corresponding to the value \(j=l+\dfrac{1}{2}\). Relations (16) and (17), therefore, do not determine the exact value of the magnetic moment, but rather are its asymptotic value.

Thus, the conclusions following from Mayer’s scheme with regard to magnetic moments do not agree very well with the experimental data. However, the circumstance that all nuclei can be divided into two groups, one of which has a magnetic moment more nearly corresponding to relation (16), and the other one more nearly to (17), shows that spin–orbit coupling in the nucleus does nevertheless exist.

The reason leading to the deviation of the magnitude of the nuclear magnetic moment from relations (16) and (17) has not yet been established. Attention has been drawn to the fact[^61][^62] that the magnetic moments of nuclei containing an odd number of protons lie between the curves corresponding to relations (16) and (17), and other similar curves calculated from relations (16) and (17) under the assumption \(g_s = 1\).

In Fig. 18, \(a\), these curves are shown by dashed lines. The fact that the magnetic moments of nuclei lie between the two Schmidt curves for \(g_s = 2.79\) and \(g_s = 1\) gave grounds for suggesting that the intrinsic magnetic moment of a particle (proton and neutron) in the nucleus is not conserved, but varies within the limits from \(g_s = 1\) to \(g_s = 2.79\).

For some nuclei, \({}^{127}_{53}\mathrm{I}\), \({}^{175}_{71}\mathrm{Lu}\), \({}^{55}_{25}\mathrm{Mn}\), \({}^{23}_{11}\mathrm{Na}\), and \({}^{75}_{33}\mathrm{As}\), the discrepancy with relations (16) and (17) is apparently due to the fact that the stable state of these nuclei is determined not by a single odd particle alone, but by the state of identical particles (in what follows we shall denote such states respectively by capital letters \(D\) (instead of \(d\)), \(F\) (instead of \(f\)), etc.).

It should be noted that for such light nuclei as \(\mathrm{C}^{13}\), \(\mathrm{N}^{15}\), and \(\mathrm{O}^{17}\), the value of the magnetic moment is very close to the Schmidt curve, i.e., to the value for the one-particle model.

The magnetic moments of other nuclei, such as \(\mathrm{Li}^{7}\) and \(\mathrm{Be}^{9}\), although they differ from the Schmidt values, can nevertheless be explained from the assumption of the coupling of the spins of several particles in one shell. It is therefore quite possible that the assumption of a change in the gyromagnetic ratio for protons and neutrons bound in the nucleus is not necessary.

9. \(\beta\)-SPECTRA AND NUCLEAR SHELLS

Knowledge of the nuclear spin and of the parity of the wave functions determining the ground state of the nucleus is of substantial importance in \(\beta\)-spectroscopy.

As follows from the theory of \(\beta\)-spectra, depending on the magnitude of the change of spin \(\Delta I\) and the change of parity, we obtain allowed or, to varying degrees, forbidden \(\beta\)-transitions.

The division of \(\beta\)-transitions into allowed and forbidden depends on the version of the theory of \(\beta\)-decay. The Teller selection rules (corres—

give the following possibilities:

1) $\Delta I=0,1$ — the parity does not change (allowed transition),

2) $\Delta I=0,1$ — the parity changes (singly forbidden transition),

3) $\Delta I=2$ — the parity changes (singly forbidden transition),

4) $\Delta I=2$ — the parity does not change and $\Delta I>2$ (doubly and multiply forbidden transitions).

All these cases of transitions must differ from one another in the magnitude of the function $\lg(ft)$.

In addition to the well-known selection rules with respect to spin and parity, some authors have proposed selection rules with respect to orbital angular momentum. These are the so-called $L$-forbidden transitions, which in the simplest case for decays allowed with respect to spin and parity are written as follows: $\Delta I=0,1;$ $\Delta l=2$, parity does not change.

The $L$-forbiddenness cannot be regarded as strict, since the orbital angular momentum in the nucleus is not conserved. Therefore, although, following Nordheim and others, we shall in what follows single out the group of $L$-forbidden transitions, it should be borne in mind that this group cannot be sharply separated from the ordinary allowed transitions.

Before the appearance of the theory of nuclear shells, the distribution of $\beta$-transitions among groups was an entirely empirical matter. There were no grounds for judging in which nuclei particular transitions should occur. The hypothesis of nuclear shells, in particular Mayer’s supposition that the state of a nucleus (its spin, parity) is determined by one unpaired particle moving in a rectangular potential well with spin-orbit coupling, gives a basis for establishing exactly what changes must take place in the parity of the functions and in the magnitude of the nuclear spin in a $\beta$-transition. In clarifying the question of the change of parity of the functions in a $\beta$-transition, we shall keep in mind that the functions corresponding to the states $s$, $d$, $g$, $i,\ldots$ are even, while the functions corresponding to the states $p$, $f$, $h,\ldots$ are odd.

Further (we shall confine ourselves here to considering $\beta$-transitions in nuclei with odd atomic weight), in a $\beta$-transition the type of the unpaired particle changes. For example, a nucleus with an odd number of neutrons in $\beta$-decay is transformed into a nucleus with an odd number of protons, and consequently, in a $\beta$-transformation the shell of the nucleus may change completely. Thus, the nucleus ${}^{105}_{45}\mathrm{Rh}$, containing (before decay) 60 neutrons and 45 protons, has a spin and parity corresponding to 45 particles, i.e. to the fourth group of nuclei (in which the fourth nuclear shell is being filled), while the nucleus ${}^{105}_{46}\mathrm{Pd}$ formed as a result of the transformation has 59 neutrons and 46 (an even number)

protons. The spin and parity of this nucleus are determined by 59 odd particles, and it belongs to the 5th group of nuclei (in which the fifth shell is filled).

The spin and parity of a nucleus depend precisely on the number of particles present in the given nucleus in an odd quantity. Table X indicates the normal state which the corresponding odd particle has in the unexcited nucleus, as well as the order of filling of the levels that follows from the experimental values of the nuclear spin.

The data given in Table X make it possible to establish the change in the spin and parity of the nucleus which (according to Mayer) may occur in a β-transition. We can compare this change in spin and parity with the experimentally established value of \(\lg(ft)\) for each β-decay, and consequently with the degree of forbiddenness of this β-transition. Such a comparison has been made here only for nuclei with odd mass number and is presented in Table XI. All the transitions listed in Table XI occur either to the ground state of the final nucleus or to a measurable excited state for which the parity and total angular momentum of the nucleus are known exactly. Transitions to excited states are not considered, since the parity and spin of these states are unknown.

The first of the two numbers appearing in the third column of the table denotes the number of particles present in the initial nucleus in an odd quantity; the second number in this column denotes the number of particles present in an odd quantity in the nucleus formed as a result of β-decay. The fourth column gives the state of the odd particle in the initial nucleus and in the product nucleus. The fifth column indicates the change in spin, and the sixth—the change in parity (“yes” means that the parity of the nucleus changes in the β-transition; “no” means that the parity is preserved). Finally, the seventh column gives the value of \(\lg(ft)\). The data for these quantities are taken from the paper by Mayer and Nordheim \(^{63}\).

As can be seen from the data of Table XI, all β-transitions can be divided into groups within which \(\lg(ft)\) varies very little. The change in spin and parity calculated according to Mayer’s scheme for each such group coincides, for the overwhelming number of β-transitions, with the change in parity and spin required by the Teller selection rules. Let us note here that among the numerous group of allowed transitions there are transitions both with \(\Delta I = 0\) and with \(\Delta I = 1\). There is no difference in the value of \(\lg(ft)\) for these cases (this fact may be regarded as a confirmation of the Teller selection rules, since according to the Fermi selection rules only transitions with \(\Delta I = 0\) are allowed).

It is noteworthy that the change in parity following from Mayer’s scheme agrees with the selection rules for all the β-transitions considered without exception; as for the change in spin, for

Table X

State of the odd particle in nuclei with an odd mass number

Nuclei with an odd number of protons	Nuclei with an odd number of protons	Nuclei with an odd number of protons	Nuclei with an odd number of protons	Nuclei with an odd number of neutrons	Nuclei with an odd number of neutrons	Nuclei with an odd number of neutrons	Nuclei with an odd number of neutrons
number of particles	state of the odd particle	spin	parity	number of particles	state of the odd particle	spin	parity
3 or 5	\(p_{3/2}\)	\(3/2\)	odd	3, 5
7	\(p_{1/2}\)	\(1/2\)	»	7
9	\(d_{5/2},\ S_{1/2}\)	\(5/2,\ 1/2\)	even	9	Same as for nuclei with an odd number of protons	Same as for nuclei with an odd number of protons	Same as for nuclei with an odd number of protons
11	\(D_{3/2}\ \bigl(d_{5/2}\bigr)\)	\(3/2,\ (5/2)\)	»	11	Same as for nuclei with an odd number of protons	Same as for nuclei with an odd number of protons	Same as for nuclei with an odd number of protons
13	\(d_{5/2}\)	\(5/2\)	»	13	Same as for nuclei with an odd number of protons	Same as for nuclei with an odd number of protons	Same as for nuclei with an odd number of protons
15	\(s_{1/2}\)	\(1/2\)	»	15	Same as for nuclei with an odd number of protons	Same as for nuclei with an odd number of protons	Same as for nuclei with an odd number of protons
17, 19	\(d_{3/2}\)	\(3/2\)	»	17, 19	Same as for nuclei with an odd number of protons	Same as for nuclei with an odd number of protons	Same as for nuclei with an odd number of protons
21, 23	\(f_{7/2}\)	\(7/2\)	odd	21, 23	Same as for nuclei with an odd number of protons	Same as for nuclei with an odd number of protons	Same as for nuclei with an odd number of protons
25	\(F_{5/2},\ (f_{7/2})\)	\(5/2,\ (7/2)\)	»	25	Same as for nuclei with an odd number of protons	Same as for nuclei with an odd number of protons	Same as for nuclei with an odd number of protons
27	\(f_{7/2}\)	\(7/2\)	»	27	Same as for nuclei with an odd number of protons	Same as for nuclei with an odd number of protons	Same as for nuclei with an odd number of protons
29, 31	\(p_{3/2}\)	\(3/2\)	»	29, 31	Same as for nuclei with an odd number of protons	Same as for nuclei with an odd number of protons	Same as for nuclei with an odd number of protons

Continuation of Table X

Nuclei with an odd number of protons: number of particles	Nuclei with an odd number of protons: state of the unpaired particle	Nuclei with an odd number of protons: spin	Nuclei with an odd number of protons: parity	Nuclei with an odd number of neutrons: number of particles	Nuclei with an odd number of neutrons: state of the unpaired particle	Nuclei with an odd number of neutrons: spin	Nuclei with an odd number of neutrons: parity
33, 35	$p_{3/2}, (f_{5/2})$	$3/2,\ (5/2)$	odd	33, 35	Same as for nuclei with an odd number of protons	Same as for nuclei with an odd number of protons	Same as for nuclei with an odd number of protons
37	$p_{3/2},\ f_{5/2}$	$3/2,\ 5/2$	same	37	Same as for nuclei with an odd number of protons	Same as for nuclei with an odd number of protons	Same as for nuclei with an odd number of protons
39	$p_{1/2}$	$1/2$	same	39	Same as for nuclei with an odd number of protons	Same as for nuclei with an odd number of protons	Same as for nuclei with an odd number of protons
41, 43	$g_{9/2}$	$9/2$	even	41, 43	$g_{9/2},\ p_{1/2}$	$9/2,\ 1/2$
45	$p_{1/2},\ g_{7/2}$	$1/2,\ 7/2$	odd	45	$p_{1/2},\ g_{7/2}$	$1/2,\ 7/2$
47	$p_{1/2}$	$1/2$	same	47	$g_{9/2},\ p_{1/2}$	$9/2,\ 1/2$
49	$g_{9/2}$	$9/2$	even	49	$g_{9/2}$	$9/2$
51, 53	$g_{7/2},\ d_{5/2}$	$7/2,\ 5/2$	even	51, 55	$d_{5/2},\ (d_{7/2})$	$5/2,\ (7/2)$	even
55, 57	$g_{7/2},\ d_{5/2}$	$7/2,\ 5/2$	same	57—61	$(d_{5/2};\ g_{7/2};\ s_{1/2})$	$(5/2;\ 7/2,\ 1/2)$	same
59	$d_{5/2}$	$5/2$	same	63—75	$s_{1/2},\ (d_{3/2};\ g_{7/2})$	$1/2,\ (3/2,\ 7/2)$	same
61	$(d_{5/2})$	$(5/2)$	same	77—81	$d_{3/2}$	$3/2$	same
63	$d_{5/2}$	$5/2$	same	83—99	$f_{7/2},\ h_{9/2}$	$7/2;\ 9/2$	odd
65	$d_{3/2}$	$3/2$	same	101	$p_{1/2}$	$1/2$	same

Continuation of Table X

| \multicolumn{4}{c}{Nuclei with an odd number of protons} | \multicolumn{4}{c}{Nuclei with an odd number of neutrons} |
|---:|:---:|:---:|:---:|---:|:---:|:---:|:---:|
| number of particles | state of the unpaired particle | spin | parity | number of particles | state of the unpaired particle | spin | parity |
| 67 | \(g_{7/2}\) | \(7/2\) | even | 103 | \(f_{5/2}\) | \(5/2\) | odd |
| 69 | \(s_{1/2},\ (d_{5/2})\) | \(1/2,\ (5/2)\) | ” | 105—107 | \(p_{1/2},\ p_{3/2},\ (f_{5/2},\ h_{9/2})\) | \(1/2,\ 3/2,\ (5/2,\ 9/2)\) | ” |
| 71, 73 | \(g_{7/2}\) | \(7/2\) | ” | 109—111 | \((p_{1/2},\ p_{3/2},\ h_{9/2})\) | \((1/2,\ 3/2,\ 9/2)\) | ” |
| 75 | \(d_{5/2}\) | \(5/2\) | ” | 113, 115 | \(p_{1/2},\ (p_{3/2})\) | \(1/2,\ (3/2)\) | ” |
| 77, 79 | \(d_{3/2}\) | \(3/2\) | ” | 117, 119 | \(p_{1/2}\) | \(1/2\) | ” |
| 81 | \(s_{1/2}\) | \(1/2\) | ” | 121 | \(p_{3/2}\) | \(3/2\) | ” |
| 83 | \(h_{9/2}\) | \(9/2\) | odd | 123 | \((p_{3/2},\ f_{5/2})\) | \((3/2,\ 5/2)\) | ” |
| | | | | 125 | \(p_{1/2}\) | \(1/2\) | ” |
| | | | | 127, 129 | \((g_{7/2},\ d_{5/2})\) | \((9/2,\ 5/2)\) | even |

Table XI

Nucleus symbol	Sign and energy of the particle	Number of particles in the initial and final nuclei	Initial and final state of the nucleus	Change of spin	Change of parity	\(\lg(ft)\)	Note
\(^{23}_{10}\mathrm{Ne}\)	\(-4,1\)	\(13—11\)	\(d_{5/2} — D_{3/2}\)	1	no	4,9	Allowed transitions \(\Delta l = 0,1\); parity does not change
\(^{25}_{11}\mathrm{Na}\)	\(-3,7\)	\(11—13\)	\(D_{3/2} — d_{5/2}\)	1	no	5,2	Allowed transitions \(\Delta l = 0,1\); parity does not change
\(^{35}_{16}\mathrm{S}\)	\(-0,17\)	\(19—17\)	\(d_{3/2} — d_{3/2}\)	0	no	5,0	Allowed transitions \(\Delta l = 0,1\); parity does not change
\(^{45}_{20}\mathrm{Ca}\)	\(-0,22\)	\(25—21\)	\(f_{7/2}(F_{5/2}) — f_{7/2}\)	\(0(1)\)	no	5,6	Allowed transitions \(\Delta l = 0,1\); parity does not change
\(^{43}_{21}\mathrm{Sc}\)	\(+1,13\)	\(21—23\)	\(f_{7/2} — f_{7/2}\)	0	no	4,8	Allowed transitions \(\Delta l = 0,1\); parity does not change
\(^{47}_{21}\mathrm{Sc}\)	\(-0,61\)	\(21—25\)	\(f_{7/2} — f_{7/2}(F_{5/2})\)	\(0(1)\)	no	5,6	Allowed transitions \(\Delta l = 0,1\); parity does not change
\(^{49}_{21}\mathrm{Sc}\)	\(-1,8\)	\(21—27\)	\(f_{7/2} — f_{7/2}\)	0	no	5,5	Allowed transitions \(\Delta l = 0,1\); parity does not change
\(^{45}_{22}\mathrm{Ti}\)	\(+1,2\)	\(23—21\)	\(f_{7/2} — f_{7/2}\)	0	no	4,7	Allowed transitions \(\Delta l = 0,1\); parity does not change
\(^{47}_{23}\mathrm{V}\)	\(+1,65\)	\(23—25\)	\(f_{7/2} — f_{7/2}(F_{5/2})\)	\(0(1)\)	no	4,7	Allowed transitions \(\Delta l = 0,1\); parity does not change
\(^{49}_{24}\mathrm{Cr}\)	\(+1,45\)	\(25—23\)	\(f_{7/2}(F_{5/2}) — f_{7/2}\)	\(0(1)\)	no	4,5	Allowed transitions \(\Delta l = 0,1\); parity does not change
\(^{51}_{25}\mathrm{Mn}\)	\(+2,0\)	\(25—27\)	\(F_{5/2} — f_{7/2}\)	1	no	5,1	Allowed transitions \(\Delta l = 0,1\); parity does not change
\(^{53}_{26}\mathrm{Fe}\)	\(+2,8\)	\(27—25\)	\(f_{7/2} — F_{5/2}\)	1	no	5,0	Allowed transitions \(\Delta l = 0,1\); parity does not change
\(^{61}_{27}\mathrm{Co}\)	\(-1,3\)	\(27—33\)	\(f_{7/2} — f_{5/2}\)	1	no	5,2	Allowed transitions \(\Delta l = 0,1\); parity does not change
\(^{63}_{30}\mathrm{Zn}\)	\(+2,36\)	\(33—29\)	\(p_{3/2} — p_{3/2}\)	0	no	5,4	Allowed transitions \(\Delta l = 0,1\); parity does not change
\(^{69}_{30}\mathrm{Zn}\)	\(-1,10\)	\(39—31\)	\(p_{1/2} — p_{3/2}\)	1	no	4,6	Allowed transitions \(\Delta l = 0,1\); parity does not change
\(^{71}_{30}\mathrm{Zn}\)	\(-2,1\)	\(41—31\)	\(p_{1/2} — p_{3/2}\)	1	no	4,5	Allowed transitions \(\Delta l = 0,1\); parity does not change
\(^{73}_{31}\mathrm{Ga}\)	\(-1,4\)	\(31—41\)	\(p_{3/2} — p_{1/2}\)	1	no	5,9	Allowed transitions \(\Delta l = 0,1\); parity does not change
\(^{75}_{32}\mathrm{Ge}\)	\(-1,1\)	\(43—33\)	\(p_{1/2} — p_{3/2}\)	1	no	5,0	Allowed transitions \(\Delta l = 0,1\); parity does not change
\(^{77}_{32}\mathrm{Ge}^{*}\)	\(-2,8\)	\(45—33\)	\(p_{1/2} — p_{3/2}\)	1	no	4,8	Allowed transitions \(\Delta l = 0,1\); parity does not change
\(^{71}_{33}\mathrm{As}\)	\(+0,6\)	\(33—39\)	\(p_{3/2} — p_{1/2}\)	1	no	5,1	Allowed transitions \(\Delta l = 0,1\); parity does not change
\(^{77}_{33}\mathrm{As}\)	\(-0,7\)	\(33—43\)	\(p_{3/2} — p_{1/2}\)	1	no	5,7	Allowed transitions \(\Delta l = 0,1\); parity does not change
\(^{73}_{34}\mathrm{Se}\)	\(+1,29\)	\(39—33\)	\(p_{1/2} — p_{3/2}\)	1	no	5,3	Allowed transitions \(\Delta l = 0,1\); parity does not change
\(^{81}_{34}\mathrm{Se}\)	\(-1,5\)	\(47—35\)	\(p_{1/2} — p_{3/2}\)	1	no	4,8	Allowed transitions \(\Delta l = 0,1\); parity does not change

EXPERIMENTAL FOUNDATIONS OF THE MODEL OF NUCLEAR SHELLS

Continuation of Table XI

Nuclear symbol	Spin and energy, MeV, of the particle	Number of odd particles in the ground and final nuclei	Initial and final state of the nucleus	Spin change	Parity change	$\lg(ft)$	Note
${}^{83}_{34}\mathrm{Se}^{*}$	$-3.4$	$49—35$	$p_{1/2} — p_{3/2}$	1	no	5.2
${}^{75}_{35}\mathrm{Br}$	$+1.6$	$35—41$	$p_{3/2} — p_{1/2}$	1	»	5.6
${}^{77}_{35}\mathrm{Br}$	$+0.36$	$35—43$	$p_{3/2} — p_{1/2}$	1	»	5.0
${}^{83}_{35}\mathrm{Br}$	$-1.05$	$35—47$	$p_{3/2} — p_{1/2}$	1	»	5.3
${}^{85}_{35}\mathrm{Br}$	$-2.5$	$35—49$	$p_{3/2} — p_{1/2}$	1	»	5.1
${}^{77}_{36}\mathrm{Kr}$	$+1.7$	$41—35$	$p_{1/2} — p_{3/2}$	1	»	5.4
${}^{89}_{40}\mathrm{Zr}^{*}$	$+1.07$	$49—39$	$p_{1/2} — p_{1/2}$	0	»	5.8	Allowed transitions $\Delta l = 0, 1$; parity does not change
${}^{91}_{42}\mathrm{Mo}$	$+3.7$	$49—41$	$g_{9/2} — g_{9/2}$	0	»	5.8	Allowed transitions $\Delta l = 0, 1$; parity does not change
${}^{105}_{45}\mathrm{Rh}$	$-0.57$	$45—59$	$g_{9/2} — g_{7/2}$	1	»	5.5	Allowed transitions $\Delta l = 0, 1$; parity does not change
${}^{107}_{48}\mathrm{Cd}$	$+0.32$	$59—47$	$g_{7/2} — G_{7/2}$	0	»	4.9	Allowed transitions $\Delta l = 0, 1$; parity does not change
${}^{121}_{50}\mathrm{Sn}$	$-0.38$	$71—51$	$d_{3/2} — d_{5/2}$	1	»	5.0	Allowed transitions $\Delta l = 0, 1$; parity does not change
${}^{127}_{52}\mathrm{Te}$	$-0.76$	$75—53$	$d_{3/2} — d_{5/2}$	1	»	5.6	Allowed transitions $\Delta l = 0, 1$; parity does not change
${}^{127}_{55}\mathrm{Cs}$	$+1.2$	$55—73$	$d_{5/2} — d_{3/2}$	1	»	4.7	Allowed transitions $\Delta l = 0, 1$; parity does not change
${}^{141}_{60}\mathrm{Nd}$	$+0.7$	$81—59$	$d_{3/2} — d_{5/2}$	1	»	5.2	Allowed transitions $\Delta l = 0, 1$; parity does not change
${}^{19}_{8}\mathrm{O}$	$-4.5$	$11—9$	$D_{3/2} — s_{1/2}$	1	»	5.5
${}^{31}_{14}\mathrm{Si}$	$-1.8$	$17—15$	$d_{3/2} — s_{1/2}$	1	»	5.9
${}^{63}_{28}\mathrm{Ni}$	$-0.05$	$35—29$	$f_{5/2} — p_{3/2}$	1	»	6.8
${}^{65}_{28}\mathrm{Ni}$	$-2.10$	$37—29$	$f_{5/2} — p_{3/2}$	1	»	6.6
${}^{61}_{29}\mathrm{Cu}$	$+1.22$	$29—33$	$p_{3/2} — f_{5/2}$	1	»	4.9	$l$-forbidden transitions; $\Delta l = 2$; $\Delta I = 1$; no parity change
${}^{67}_{29}\mathrm{Cu}$	$-0.65$	$29—37$	$p_{3/2} — f_{5/2}$	1	»	5.5	$l$-forbidden transitions; $\Delta l = 2$; $\Delta I = 1$; no parity change
${}^{65}_{30}\mathrm{Zn}$	$+0.32$	$35—29$	$f_{5/2} — p_{3/2}$	1	»	7.0	$l$-forbidden transitions; $\Delta l = 2$; $\Delta I = 1$; no parity change
${}^{69}_{32}\mathrm{Ge}$	$+1.0$	$37—31$	$f_{5/2} — p_{3/2}$	1	»	6.0	$l$-forbidden transitions; $\Delta l = 2$; $\Delta I = 1$; no parity change
${}^{109}_{46}\mathrm{Pd}$	$-1.0$	$63—47$	$d_{5/2} — G_{7/2}$	1	»	6.2	$l$-forbidden transitions; $\Delta l = 2$; $\Delta I = 1$; no parity change

Continuation of Table XI

Decay symbol	Particle energy and sign	Number of unpaired particles in the initial and final nuclei	Initial and final state of the nucleus	Change of spin	Change of parity	\(\lg(ft)\)	Notes
\({}^{87}_{35}\mathrm{Br}\)	\(-8.0\)	35—51	\(p_{3/2} — d_{5/2}\)	1	yes	7.3
\({}^{87}_{36}\mathrm{Kr}\)	\(-3.2\)	51—37	\(d_{5/2} — p_{3/2}\)	1	»	7.0
\({}^{89}_{37}\mathrm{Rb}\)	\(-3.8\)	37—51	\(p_{3/2} — d_{5/2}\)	1	»	6.6
\({}^{111}_{46}\mathrm{Pd}\)	\(-3.5\)	65—47	\(s_{1/2} — p_{1/2}\)	0	»	6.8
\({}^{111}_{47}\mathrm{Ag}\)	\(-1.0\)	47—63	\(p_{1/2} — s_{1/2}\)	0	»	7.2
\({}^{113}_{47}\mathrm{Ag}\)	\(-2.2\)	47—65	\(p_{1/2} — s_{1/2}\)	0	»	7.0
\({}^{115}_{47}\mathrm{Ag}\)	\(-3.00\)	47—67	\(p_{1/2} — s_{1/2}\)	0	»	6.4
\({}^{115}_{48}\mathrm{Cd}\)	\(-1.13\)	67—49	\(s_{1/2} — p_{1/2}\)	0	»	6.8
\({}^{117}_{48}\mathrm{Cd}\)	\(-1.50\)	69—49	\(s_{1/2} — p_{1/2}\)	0	»	6.1
\({}^{115}_{49}\mathrm{In}^{*}\)	\(-0.83\)	49—65	\(p_{1/2} — s_{1/2}\)	0	»	6.6
\({}^{117}_{49}\mathrm{In}\)	\(-1.73\)	49—67	\(p_{1/2} — s_{1/2}\)	0	»	6.2	Singly forbidden transitions; \(\Delta I = 0,1\); parity changes
\({}^{119}_{49}\mathrm{In}\)	\(-2.7\)	49—69	\(p_{1/2} — s_{1/2}\)	0	»	6.2
\({}^{137}_{54}\mathrm{Xe}\)	\(-4.0\)	83—55	\(f_{7/2} — g_{7/2}\)	0	»	6.3
\({}^{139}_{56}\mathrm{Ba}\)	\(-2.27\)	83—57	\(f_{7/2} — g_{7/2}\)	0	»	6.7
\({}^{141}_{57}\mathrm{La}\)	\(-2.9\)	57—83	\(g_{7/2} — f_{7/2}\)	0	»	7.6
\({}^{141}_{58}\mathrm{Ce}\)	\(-0.56\)	83—59	\(f_{7/2} — d_{5/2}\)	1	»	7.7
\({}^{143}_{59}\mathrm{Pr}\)	\(-0.93\)	59—83	\(d_{5/2} — f_{7/2}\)	1	»	7.6
\({}^{147}_{60}\mathrm{Nd}\)	\(-0.7\)	87—61	\(f_{7/2} — d_{5/2}\)	1	»	7.0
\({}^{147}_{61}\mathrm{Pm}\)	\(-0.23\)	61—85	\(d_{5/2} — f_{7/2}\)	1	»	7.6
\({}^{151}_{62}\mathrm{Sm}\)	\(-0.076\)	89—63	\(f_{7/2} — d_{5/2}\)	1	»	6.9
\({}^{165}_{66}\mathrm{Dy}\)	\(-1.28\)	99—67	\(f_{7/2} — g_{7/2}\)	0	»	6.1
\({}^{169}_{63}\mathrm{Er}\)	\(-0.33\)	101—69	\(p_{1/2} — s_{1/2}\)	0	»	6.1
\({}^{171}_{68}\mathrm{Er}\)	\(-1.49\)	103—69	\(f_{5/2} — d_{5/2}\)	0	»	7.0

Continuation of Table XI

Nuclide symbol	Energy-particle sign	Number of unpaired particles in the initial and final nuclei	Initial and final state of the nucleus	Change in spin	Change in parity	lg \((ft)\)	Note
\({}^{177}_{71}\mathrm{Lu}\)	\(-0.49\)	71—105	\(g_{7/2} — f_{5/2}\)	1	yes	6.8	Singly forbidden transitions; \(\Delta I = 0,1\); parity changes
\({}^{181}_{72}\mathrm{Hf}\)	\(-0.40\)	109—73	\(p_{1/2} — s_{1/2}\)	0	»	7.2	Singly forbidden transitions; \(\Delta I = 0,1\); parity changes
\({}^{185}_{74}\mathrm{W}\)	\(-0.43\)	111—75	\(p_{3/2} — d_{5/2}\)	1	»	7.5	Singly forbidden transitions; \(\Delta I = 0,1\); parity changes
\({}^{187}_{74}\mathrm{W}^{*}\)	\(-1.33\)	113—75	\(p_{3/2} — d_{5/2}\)	1	»	7.8	Singly forbidden transitions; \(\Delta I = 0,1\); parity changes
\({}^{199}_{78}\mathrm{Pt}\)	\(-1.8\)	121—79	\(p_{1/2} — d_{3/2}\)	1	»	6.3	Singly forbidden transitions; \(\Delta I = 0,1\); parity changes
\({}^{205}_{80}\mathrm{Hg}\)	\(-1.6\)	125—81	\(p_{1/2} — s_{1/2}\)	0	»	5.4	Singly forbidden transitions; \(\Delta I = 0,1\); parity changes
\({}^{209}_{82}\mathrm{Pb}\)	\(-0.68\)	127—83	\(g_{9/2} — h_{9/2}\)	0	»	5.6	Singly forbidden transitions; \(\Delta I = 0,1\); parity changes
\({}^{213}_{83}\mathrm{Bi}\)	\(-1.3\)	83—129	\(h_{9/2} — g_{9/2}\)	0	»	6.0	Singly forbidden transitions; \(\Delta I = 0,1\); parity changes
\({}^{37}_{16}\mathrm{S}\)	\(-4.3\)	21—17	\(f_{7/2} — d_{3/2}\)	2	»	7.1	Singly forbidden transitions; \(\Delta I = 2\); parity changes
\({}^{41}_{18}\mathrm{A}\)	\(-2.55\)	23—19	\(f_{7/2} — d_{3/2}\)	2	»	8.6	Singly forbidden transitions; \(\Delta I = 2\); parity changes
\({}^{85}_{36}\mathrm{Kr}\)	\(-0.74\)	49—37	\(g_{9/2} — f_{5/2}\)	2	»	9.2	Singly forbidden transitions; \(\Delta I = 2\); parity changes
\({}^{89}_{38}\mathrm{Sr}\)	\(-1.46\)	51—39	\(d_{5/2} — p_{1/2}\)	2	»	8.5	Singly forbidden transitions; \(\Delta I = 2\); parity changes
\({}^{91}_{38}\mathrm{Sr}\)	\(-3.2\)	53—39	\(d_{5/2} — p_{1/2}\)	2	»	8.0	Singly forbidden transitions; \(\Delta I = 2\); parity changes
\({}^{91}_{39}\mathrm{Y}\)	\(-1.56\)	39—51	\(p_{1/2} — d_{5/2}\)	2	»	8.7	Singly forbidden transitions; \(\Delta I = 2\); parity changes
\({}^{95}_{40}\mathrm{Zr}\)	\(-1.0\)	55—41	\(d_{5/2} — p_{1/2}\)	2	»	9.8	Singly forbidden transitions; \(\Delta I = 2\); parity changes
\({}^{95}_{43}\mathrm{Tc}^{*}\)	\(+0.4\)	43—53	\(p_{1/2} — d_{5/2}\)	2	»	8.3	Singly forbidden transitions; \(\Delta I = 2\); parity changes
\({}^{103}_{44}\mathrm{Ru}\)	\(-0.8\)	59—45	\(d_{5/2} — p_{1/2}\)	2	»	8.5	Singly forbidden transitions; \(\Delta I = 2\); parity changes
\({}^{123}_{50}\mathrm{Sn}^{*}\)	\(-1.42\)	73—51	\(h_{11/2} — g_{7/2}\)	2	»	9.1	Singly forbidden transitions; \(\Delta I = 2\); parity changes
\({}^{123}_{52}\mathrm{Sb}\)	\(-0.62\)	51—73	\(g_{7/2} — h_{11/2}\)	2	»	9.4	Singly forbidden transitions; \(\Delta I = 2\); parity changes
\({}^{137}_{55}\mathrm{Cs}\)	\(-0.53\)	55—85	\(g_{7/2} — h_{11/2}\)	2	»	9.6	Singly forbidden transitions; \(\Delta I = 2\); parity changes
\({}^{171}_{69}\mathrm{Tm}\)	\(-1.0\)	69—101	\(d_{5/2} — p_{1/2}\)	2	»	9.5	Singly forbidden transitions; \(\Delta I = 2\); parity changes

Continuation of Table XI

Nucleus symbol	Sign and energy of the particle	Number of particles in the initial and final nuclei	Initial and final state of the nucleus	Spin change	Parity change	\(\lg(ft)\)	Note
\({}^{87}_{37}\mathrm{Rb}\)	\(-0.13\)	37—49	\(p_{3/2} — g_{9/2}\)	3	yes	16.5	Doubly and more forbidden transitions
\({}^{99}_{43}\mathrm{Tc}\)	\(-0.30\)	43—55	\(g_{9/2} — d_{5/2}\)	2	no	13.0	Doubly and more forbidden transitions
\({}^{115}_{49}\mathrm{In}\)	\(-0.63\)	49—65	\(g_{9/2} — s_{1/2}\)	4	»	23.2	Doubly and more forbidden transitions
\({}^{129}_{53}\mathrm{I}\)	\(-0.12\)	53—75	\(g_{7/2} — s_{1/2}\)	3	»	13.5	Doubly and more forbidden transitions
\({}^{135}_{55}\mathrm{Cs}\)	\(-0.21\)	55—79	\(g_{7/2} — d_{5/2}\)	2	»	13.1	Doubly and more forbidden transitions
\({}^{137}_{55}\mathrm{Cs}\)	\(-1.19\)	55—81	\(g_{7/2} — d_{3/2}\)	2	»	12.2	Doubly and more forbidden transitions
\({}^{187}_{75}\mathrm{Re}\)	\(-0.043\)	75—111	\(d_{5/2} — h_{9/2}?\)	2?	yes?	17.7	Doubly and more forbidden transitions

for some nuclei there is no agreement with the data following from the single-particle model. These are the Na nuclei already mentioned above, with particle number 11, Mn with particle number 25, and some others, for which Table XI proposes states due to several particles (these states are denoted by the corresponding capital letter).

10. ISOMERISM OF ATOMIC NUCLEI

Perhaps the strongest argument in favor of the idea that particles in the nucleus have orbital motion is the fact of the existence of nuclear isomerism.

As has now been established\(^{64}\), isomers are nuclei in an excited state. The excitation energy of isomeric nuclei is usually small—several tens of keV, although isomers with a considerably larger excitation energy are also encountered—several hundreds of keV.

Isomeric nuclei exist in an excited state for a comparatively long time. The half-life (more precisely, the half-transition period) of many isomers is measured in hours, days, months, and even years. The longest period (among isomers known at present) has been established for the isomer \({}^{95}_{43}\mathrm{Tc}\). The time during which half of the excited \({}^{95}_{43}\mathrm{Tc}\) nuclei pass into the normal state is approximately 5 years.

Such a long residence of nuclei in an excited state is due to the fact that, at a comparatively small excitation energy—

there is a large difference between the ground and excited states of a nucleus in the magnitude of the total angular momentum. The experimentally observed lifetimes of nuclei in an excited isomeric state are well explained\(^{64—69}\) by the fact that the transition of a nucleus from the excited state to the normal one is connected with a change of the angular momentum by 3, 4, 5 units \((\hbar)\). Therefore, in the transition of a nucleus from the excited state to the normal one, multipole radiation is emitted. The degree of multipolarity of the radiation depends on how many units the nuclear spin changes by in the transition, while the character of the radiation (magnetic or electric multipole radiation) is determined by the change in the parity of the functions characterizing the state of the particle in the nucleus.

According to the selection rules, when the angular momentum of the nucleus changes by an amount \(\Delta I\), radiation of \(2^l\)-polarity must occur, where \(l=\Delta I\)*). The character of this radiation will depend on the change of parity. In the emission of radiation corresponding to an electric multipole of \(2^l\)-polarity, the parity of the function must change by \((-1)^l\), whereas in the emission of magnetic radiation of the same polarity the parity of the function must change by \((-1)^{l+1}\), i.e., in the same way as the parity of the functions changes in the emission of electric radiation of polarity \(2^{l+1}\). Table XII indicates what change of angular momentum and of parity of the functions must occur for the given radiation to arise.

Knowing the character and multipolarity of the radiation, one can determine the intensity of the radiation and, consequently, the duration of the nucleus’s stay in the excited state. If by \(T_\gamma\) we denote the time during which half of the excited nuclei de-excite by radiation, then, according to\(^{68}\), in the case of electric radiation,

\[ T_\gamma = 0.69\left[ \frac{2(l+1)}{l[1\cdot3\cdot5\ldots2(l+1)]^2}\, \omega\,\frac{e^2}{\hbar c}\left(\frac{\omega}{c}\rho\right)^{2l} \right]^{-1}, \tag{18} \]

where \(l\) characterizes the multipolarity of the radiation, \(e\) is the electron charge, \(c\) is the speed of light, \(\hbar\) is Planck’s constant divided by \(2\pi\), \(\rho\) is the radius of the nucleus, and \(\omega\) determines the energy of the quantum by the relation

\[ E=h\nu=\hbar\omega. \tag{19} \]

*) The radiation of a system of charges may be represented in the form of a series whose terms are electric dipole radiation, electric quadrupole radiation and magnetic dipole, electric octupole \((2^3)\) and magnetic quadrupole, electric \(2^4\)-pole and magnetic octupole, etc.

At a low excitation energy of the nucleus, the intensity of the radiation corresponding to a higher polarity decreases sharply with increasing polarity. Thus, if the transition is not forbidden, the overwhelming intensity should belong to electric dipole radiation. If electric dipole radiation is forbidden, the greatest intensity will belong to electric quadrupole or magnetic dipole radiation. If these transitions are also forbidden, octupole radiation will have the greatest intensity, etc.

Table XII

Selection rules for multipole radiation

Transition with emission of an electric multipole	Transition with emission of an electric multipole	Transition with emission of an electric multipole	Transition with emission of an electric multipole	Transition with emission of an electric multipole	Transition with emission of an electric multipole
Order of polarity	dipole	quadrupole	octupole	16-pole	32-pole
Minimum value \(l\)	1	2	3	4	5
Change of parity	yes	no	yes	no	yes
Transition with emission of a magnetic multipole	Transition with emission of a magnetic multipole	Transition with emission of a magnetic multipole	Transition with emission of a magnetic multipole	Transition with emission of a magnetic multipole	Transition with emission of a magnetic multipole
Order of polarity	dipole	quadrupole	octupole	16-pole	32-pole
Minimum value \(l\)	1	2	3	4	5
Change of parity	no	yes	no	yes	no

In the case of magnetic radiation \(T_\gamma\) is determined by the relation

\[ (T_\gamma)_{\text{magn}}=(T_\gamma)_{\text{electr}} \left[ \frac{\hbar}{mcp} \left(\mu l-\frac{l}{l+1}\right) \right]^{-2}, \tag{20} \]

where \(\mu\) is the magnetic moment of the proton, expressed in nuclear magnetons (\(\mu=2.79\)).

The study of the phenomenon of isomerism of atomic nuclei, begun by the work of Kurchatov, Mysovskii, and Rusinov\(^{77}\), showed that the existence of isomeric states is not a great rarity, but is observed in many nuclei. At the present time the existence of isomeric states has been established in 101 nuclei, i.e., approximately 10% of all known nuclei possess isomeric states.

Isomeric nuclei occur to the same extent both among stable nuclei and among radioactive ones. Naturally, the question then arises: why, if isomerism is a general property of nuclei, is it observed only in 10% of atomic nuclei and not in all? If, however, isomerism, as is actually the case, is limited to some particular circle of nuclei, then what are the peculiarities of those nuclei in which isomeric states can exist?

Perhaps it would even be more correct to pose another, almost opposite, question: why in atomic nuclei an excited

state closest to the ground state so often has an angular momentum differing from that of the ground state by several units? Perhaps it was precisely this circumstance that seemed most surprising in the phenomenon of nuclear isomerism. An explanation of this was provided by the nuclear shell model. Let us note first of all that isomerism is a property inherent only in quite definite groups of nuclei. Table XIII clearly illustrates this circumstance.

Table XIII

Number of odd particles (protons or neutrons) in odd-even nuclei	Number of stable and radioactive isotopes having the given number of protons or neutrons	Number of isomers among these nuclei
From 1 to 37	94	0
39—49	58	32
53—61	57	0
63—81	69	27
83—97	56	2

In this table a comparison is made between the number of known stable and radioactive isotopes and the number of isomers for various groups of nuclei containing an odd number of particles. The first such group includes all nuclei for which the odd number of particles lies in the interval from 1 to 37. In all, 94 such nuclei are known. Among these 94 nuclei there is not a single isomer.

The second group includes all nuclei in which the odd number of particles is in the interval from 39 to 49. Among the 58 nuclei possessing such a number of particles, 32, i.e. more than half, are isomers and, consequently, can remain for a long time in an excited state.

The third group, numbering 57 nuclei, includes those for which the odd number of particles lies in the interval between 53 and 61. Among these 57 nuclei there was not a single isomer. Conversely, in the fourth group of nuclei, containing an odd number of particles in the interval from 63 to 81, there again turned out to be many isomers. Of the 69 known nuclei of this group, 27 proved to be isomers. In the fifth group of nuclei, containing an odd number of particles in the interval from 83 to 97, isomers again occur rarely. Among the 56 known nuclei of this group only 2 isomers were found. The fact that the isomers are distributed among all nuclei as peculiar islands is also well illustrated by Fig. 19.

Thus, among nuclei for which the odd number of particles (protons or neutrons) lies in the intervals 1—37, 53—61, and 83—97,

isomeric states is not observed. Isomeric states among nuclei containing an odd number of particles are, for some reason, encountered when the number of odd particles in the nucleus lies in the intervals from 39 to 49 or from 63 to 81. Isomerism, therefore, is not a general property of nuclei.

To this it should also be added that, among the known isomers of odd-odd nuclei, i.e., nuclei that contain both an odd number of protons and an odd number of neutrons, the overwhelming majority (18 out of 27) have either neutrons or protons (or both) in numbers from 39 to 49 or from 63 to 81.

Fig. 19. “Islands of isomerism.” Number of isomeric nuclei as a function of the number of particles in odd nuclei. Solid lines—nuclear isomers with an odd number of neutrons. Dashed lines—nuclear isomers with an odd number of protons.

Naturally the question arises why precisely such nuclei (with an odd number of particles in the interval 39–49 or 63–81) can be found in an isomeric state. An answer to this question is provided by a model of the nucleus that allows for the existence of orbital motion in the nucleus.

The hypothesis of spin-orbit coupling in the nucleus and the assumption that nuclear spin is determined by the motion of one unpaired particle make it possible to explain the presence of these strange islands of isomerism among nuclei, as well as a number of features relating to the radiation emitted by isomeric nuclei.

Let us consider in detail the consequences that follow from the Mayer scheme for nuclear isomerism. As we have already indicated, isomers are excited nuclei whose moment differs substantially from the moment of the nucleus in its normal state. Consequently, in order for isomerism to occur,

it is necessary that, close to the ground state, there be a level differing greatly from the ground state in the magnitude of the angular momentum. It is evident, as follows from the definition of what constitutes a nuclear shell, that close levels must belong to one and the same nuclear shell. Thus, we arrive at the conclusion that isomerism can occur only in those nuclei in which shells are being filled that contain levels with strongly differing values of the angular momenta. Since such filling does not occur in all nuclei, but only in certain groups of nuclei, it is evident that isomerism cannot be a general property of nuclei, but must be observed only in certain groups of nuclei. This naturally and simply explains the existence of islands of isomerism. In which groups of nuclei, then, should isomerism be observed? It is evident that in the first and second groups of nuclei, i.e., in the groups of nuclei in which the first and second nuclear shells are being filled, there should be no isomers, since in the second group of nuclei the maximum change of spin (in the transition from one shell level to another) is equal to unity, while in the first shell the particles can in general have only one value of the spin.

In the third nuclear shell there are three levels: \(3d_{5/2}\), \(3d_{3/2}\), and \(2s_{1/2}\). The maximum difference in the angular momenta of the different levels in this group of nuclei can be equal to \(2\hbar\). This occurs in the case when the neighboring levels (the ground and excited ones) are \(3d_{5/2}\) and \(2s_{1/2}\). Such cases are quite possible, since the filling of these levels takes place in the following order: \(3d_{5/2}\), \(2s_{1/2}\), and \(3d_{3/2}\), i.e., the levels \(3d_{5/2}\) and \(2s_{1/2}\) are in fact neighboring levels. Since, however, the parity of the \(d\) and \(s\) levels is the same, then according to the selection rules (see Table XII), when a particle passes from the \(2s_{1/2}\) level to the \(3d_{5/2}\) level, electric quadrupole radiation must be emitted. Since in light nuclei, owing to the smallness of the nuclear radius, the distance between the levels will be relatively large, the quadrupole electric transition (with a large excitation energy) will occur very rapidly. Consequently, in the third group of nuclei as well (in which the third shell is being filled) there should be no isomers.

In the fourth nuclear shell there are the levels \(4f_{7/2}\), \(4f_{5/2}\), \(3p_{3/2}\), \(3p_{1/2}\), and \(5g_{9/2}\). The normal order of succession of the levels in a rectangular potential well is precisely as indicated here, but as a result of spin–orbit coupling, and also, possibly, as a result of a change in the shape of the potential well, as the number of particles in the nucleus increases the order of alternation of the levels in the fourth shell may change.

With the indicated arrangement of levels, isomerism can arise only when the filling of the \(5g_{9/2}\) and \(3p_{1/2}\) levels begins. Indeed, the difference of angular momenta between the pairs of neighboring levels \(4f_{7/2}\) and \(4f_{5/2}\), \(3p_{3/2}\) and \(4f_{5/2}\), \(3p_{3/2}\) and \(3p_{1/2}\), is equal to unity; therefore,

upon transition from one of these levels to a neighboring one, magnetic dipole radiation must arise (the parity of all these levels is the same—they are all odd), and the probability of dipole radiation (even magnetic) is large; therefore \(T_{\gamma}\) will be small; in other words, the nucleus in the excited state will exist only for a short time.

Only when all these levels are filled and the particles are located at the level \(3p_{1/2}\) can an isomeric state arise, in which the particle, instead of the normal level \(3p_{1/2}\), will be found at the nearest level to it, \(5g_{9/2}\). The difference in the total moment of these levels is four units. Further, since the parity of the levels \(3p_{1/2}\) and \(5g_{9/2}\) is different (the level \(5g_{9/2}\) is even, and \(3p_{1/2}\) is odd), then in the transition from the level \(5g_{9/2}\) to the level \(3p_{1/2}\) magnetic \(2^4\)-pole radiation must arise. The probability of such radiation is very small; therefore a nucleus in which the particle is found at the level \(5g_{9/2}\) instead of the normal \(3p_{1/2}\) will exist in the excited state for a very long time.

Thus, isomers may arise in those nuclei in which filling of the level \(3p_{1/2}\) begins. True, it is possible that the level \(5g_{9/2}\) will begin to be filled earlier than \(3p_{1/2}\), i.e. the level \(5g_{9/2}\) will be normal and \(3p_{1/2}\) excited; nevertheless, in this case too the excited state will be isomeric. Consequently, regardless of which of the levels \(3p_{1/2}\) or \(5g_{9/2}\) begins to be filled earlier, nuclei with an odd number of particles in which the levels \(4f_{7/2}\), \(4f_{5/2}\), and \(3p_{3/2}\) are filled may be isomers.

Since the occupation numbers of the levels \(4f_{7/2}\), \(4f_{5/2}\), and \(3p_{3/2}\) are respectively 8, 6, and 4, and the third shell is filled when the number of particles is equal to 20, all these levels must be filled when the number of particles in the nucleus reaches 38; consequently, filling of the levels \(3p_{1/2}\) or \(5g_{9/2}\) must begin in nuclei with a number of protons or neutrons equal to 39; consequently, isomerism in nuclei with an odd number of particles should be observed in nuclei containing 39 (or more) particles of that kind. And indeed, as we saw in Table XIII, nuclei with a number of particles less than 39 are not isomers. Nuclei with a number of particles 39 \(\left({}^{87}_{39}Y,\ {}^{89}_{39}Y,\ {}^{91}_{39}Y\ \text{and}\ {}^{69}_{30}Zn\right)\) possess isomeric states.

Further, as already mentioned, filling of the fourth shell occurs when the number of particles in the nucleus becomes equal to 50, i.e. the last odd nuclei in which both levels \(3p_{1/2}\) and \(5g_{9/2}\) are not yet filled will be nuclei with an odd number of particles 49. Thus, isomers may be encountered among nuclei containing an odd number of particles in the interval 39–49. But, as was already indicated in Table XIII, the majority of isomers are encountered precisely among such nuclei, and the majority of such nuclei are isomers.

Among nuclei of the fifth group isomers may also be encountered. Indeed, the levels \(5g_{7/2}\), \(2d_{5/2}\), \(2d_{3/2}\), enter the fifth shell,

$3s_{1/2}$ and $6h_{11/2}$. It is obvious that if neighboring levels in a nucleus turn out to be $4d_{5/2}$ and $6h_{11/2}$, or $3s_{1/2}$ and $6h_{11/2}$, then in those nuclei in which the filling of these levels begins, the excited states will be isomeric.

The order of alternation of the levels entering the fifth shell has not been established very precisely and, moreover, it is (as follows from the values of nuclear spins, see Table IX) different for nuclei with an odd number of protons and for nuclei with an odd number of neutrons. However, as follows from the values of nuclear spins, the first levels filled in the fifth shell are $5g_{7/2}$ and $4d_{5/2}$. The occupation numbers of these levels are 8 and 6, respectively. Thus, the filling of these levels should have ended when the nucleus contained 64 particles (50 particles fill the first four shells). Consequently, isomeric states could arise in nuclei with a number of particles equal to 65. Since the fifth shell is completely filled at 82 particles, isomeric nuclei may occur among odd nuclei with a number of odd particles in the range 65—81. As is seen from Table XIII, a considerable number of isomers belong to such a group of nuclei. Let us note that isomeric nuclei also occur for a number of particles in the nucleus equal to 63 (${}^{111}_{48}\mathrm{Cd}$, ${}^{113}_{50}\mathrm{Sn}$). Apparently this is a consequence of the fact that the $6h_{11/2}$ level begins to be filled earlier than the levels $5g_{7/2}$ and $4d_{5/2}$ are completely filled.

Thus, the single-particle model of the nucleus, with allowance for spin–orbit coupling, not only explains the existence of islands of isomerism, but also predicts with remarkable accuracy which particular nuclei should be isomeric.

Let us dwell further on some details:

1. As has already been indicated, the change in the arrangement of nuclear levels can be explained not only by the fact that spin–orbit coupling exists, but also by the fact that the form of the potential function differs from a rectangular well. In particular, Finberg^42 found that for a certain form of the potential well one can obtain shell occupation numbers 50 and 82. In addition, he found that in this case, upon filling the fourth nuclear shell, there should occur a crossing of levels capable of leading to the emergence of isomerism. Namely, upon filling the fourth shell, in some nuclei with a number of particles between 20 and 50 there should occur a crossing of the $3s$ and $5g$ levels. In this case the difference in angular momenta between the ground level and the excited level nearest to it can reach four units. It should be pointed out, however, that in the transition from the $5g$ level to $3s$ and conversely no change of parity will occur (both levels are even). Consequently, if Finberg’s point of view were correct, i.e., if the change in the arrangement of the levels were due not to the presence of spin–orbit coupling but to a change in the form of the potential well, then the radiation of isomers (belonging to the fourth nuclear shell)

should have been electric 24-pole radiation. According to the hypothesis of the presence in the nucleus of spin-orbit coupling, the isomeric transition is the transition between the levels \(5g_{7/2}\) and \(3p_{1/2}\), which is accompanied not by electric, but by magnetic 24-pole radiation. Thus, establishing the character of the isomeric transition (its multipolarity, change of parity) can decide what is the cause of the change in the arrangement of nuclear levels. A comparison of the excitation energy and half-life of the isomers, measurement of the conversion coefficient of this radiation, measurement of the ratio of conversion in the \(K\)- and \(L\)-shells of atoms show that in reality, among the isomers belonging to the fourth group of nuclei, de-excitation proceeds with the emission of 24-pole magnetic, and not electric, radiation. This circumstance speaks in favor of the assumption that the cause of the change in the grouping of nuclear levels is spin-orbit coupling.

1. Mayer’s hypothesis, therefore, correctly explains not only the fact of the existence of isomers among nuclei with particle number 39–49, but also the character of the isomeric radiation. Likewise, in general, this hypothesis also correctly describes the character of the radiation emitted by isomeric nuclei of the fifth group, i.e., those containing an odd number of particles in the interval from 63 to 81. As we have already noted, the isomers of this group of nuclei are formed as a result of the fact that the levels \(6h_{11/2}\), \(4d_{3/2}\), and \(3s_{1/2}\) turn out to be close to one another.

If the arrangement of levels in the nucleus is precisely as indicated here, then the ground state of the nucleus will be \(3s_{1/2}\) (spin \(1/2\)), while the isomeric state will be the state in which the odd particle occupies the level \(6h_{11/2}\). Although between the ground state \((3s_{1/2})\) and the excited state \((6h_{11/2})\) there is an intermediate level \((4d_{3/2})\), the nucleus will remain in the excited state for a very long time (provided only that the excitation energy is not too large), since the change of total angular momentum both in the transition \(6h_{11/2}—4d_{3/2}\) \((\Delta I = 4)\) and in the transition \(6h_{11/2}—3s_{1/2}\) \((\Delta I = 5)\) will be rather large. At the same time, the probability of transition from the level \(6h_{11/2}\) to \(4d_{3/2}\) may turn out to be considerably greater than the probability of transition from the level \(6h_{11/2}\) directly to the level \(3s_{1/2}\) (despite the fact that the energy difference \(W(6h_{11/2}) — W(4d_{3/2})\) is smaller than \(W(6h_{11/2}) — W(3s_{1/2})\)), since the change of spin in this transition is smaller than in the transition from \(6h_{11/2}\) to \(3s_{1/2}\). In this case the transition from the excited (isomeric) state to the ground state will take place in two stages. First there will occur a transition from the level \(6h_{11/2}\) to \(4d_{3/2}\), and then from the level \(4d_{3/2}\) to \(3s_{1/2}\). The character of this two-stage transition will be as follows: in the first stage, magnetic 24-pole radiation will be emitted (the levels \(6h_{11/2}\) and \(4d_{3/2}\) have different parity, and when the angular momentum changes by 4 units with a change of parity, magnetic 24-pole radiation is emitted); in the second stage—magnetic dipole radiation (the levels \(4d_{3/2}\) and \(3s_{1/2}\) have the same parity).

It is noteworthy that in many isomers belonging to the indicated group of nuclei (with an odd number of particles in the interval from 63 to 81), the existence of a two-cascade isomeric transition has been found. Such transitions have been detected both from the magnitude of the energy of conversion electrons and directly—by \((\gamma,\gamma)\)- and \((\gamma,e)\)-coincidences.

Table XIV gives data on radiation in the isomeric transition of nuclei with the number of particles in the interval 63–81 (the letters \(E\) or \(M\) in the fifth column denote, respectively, electric or magnetic transitions; the numeral characterizes the multipolarity of the radiation; for example, \(E3\) is an electric octupole transition, \(M4\) is a magnetic \(2^4\)-pole transition, etc.).

As is seen from the table, indeed, among the isomers of the fifth group of nuclei two-step isomeric transitions are often observed, corresponding to such an arrangement of levels—\(6h_{11/2}\), \(4d_{3/2}\), and \(3s_{1/2}\).

It is of interest to trace how the relative arrangement of these levels changes as the number of particles in the nucleus changes. In Fig. 20, a

Fig. 20. Distance between the levels \(6h_{11/2}\), \(4d_{3/2}\), and \(3s_{1/2}\) as a function of the number of neutrons in the nucleus: \(a\)—tellurium isotopes, \(b\)—xenon isotopes, \(c\)—barium isotopes.

is shown the change in the relative arrangement of the levels \(6h_{11/2}\), \(4d_{3/2}\), and \(3s_{1/2}\) in various tellurium isotopes as the number of neutrons increases from 69 to 81 (in this figure, as also in Figs. 19, b and 19, c, the energy value of the \(4d_{3/2}\) level is taken as zero). In Figs. 20, b and 20, c it is shown how the relative arrangement of these same levels changes in the Xe and Ba isotopes. In all three figures it is seen that the difference in energy of the levels \(6h_{11/2}\) and \(4d_{3/2}\) increases with an increase in the number of neutrons filling the shell,

Table XIV

Isomers of odd-odd nuclei with the number of neutrons from 63 to 81

Nucleus	Number of neutrons in the nucleus	Excitation energy of the isomeric state, in keV	Transition energy from the isomeric state to an intermediate state, in keV	Type of transition from the isomeric state to an intermediate state	Type of transition from an intermediate state to the ground state	Ground level	Isomeric level	Intermediate level
\({}^{111}_{48}\mathrm{Cd}\)	63	369	147	\(E3\)	\(E2\)	\(3s_{1/2}\)	\(6h_{11/2}\)	\(4d_{5/2}\)
\({}^{113}_{48}\mathrm{Cd}\)	65	Isomeric transition not established	Isomeric transition not established	Isomeric transition not established	Isomeric transition not established	\(3s_{1/2}\)	\(6h_{11/2}\)	—
\({}^{115}_{48}\mathrm{Cd}\)	67	Isomeric transition not established	Isomeric transition not established	Isomeric transition not established	Isomeric transition not established	—	—	—
\({}^{117}_{50}\mathrm{Sn}\)	67	317	159	\(M4+E5\)	\(M1\)	\(3s_{1/2}\)	\(6h_{11/2}\)	\(4d_{3/2}\)
\({}^{119}_{50}\mathrm{Sn}\)	69	89,7	65,3	\(M4\)	—	\(3s_{1/2}\)	\(6h_{11/2}\)	\(4d_{3/2}\)
\({}^{121}_{52}\mathrm{Te}\)	69	295	82	\(M4+E5\)	\(M1\)	\(3s_{1/2}\)	\(6h_{11/2}\)	\(4d_{3/2}\)
\({}^{121}_{50}\mathrm{Sn}\)	71	Isomeric transition not established	Isomeric transition not established	Isomeric transition not established	Isomeric transition not established	—	—	—
\({}^{123}_{52}\mathrm{Te}\)	71	247,5	88,5	\(M4+E5\)	\(M1+E2\)	\(3s_{1/2}\)	\(6h_{11/2}\)	\(4d_{3/2}\)
\({}^{125}_{52}\mathrm{Te}\)	73	145,1	109,7	\(M4\)	—	\(3s_{1/2}\)	\(6h_{11/2}\)	\(4d_{3/2}\)
\({}^{127}_{54}\mathrm{Xe}\)	73	300	175	\(E3?\)	—	\(3s_{1/2}\)	\(6h_{11/2}\)	\((4d_{3/2})\)
\({}^{127}_{52}\mathrm{Te}\)	75	88,5	—	\(M4\)	—	\(4d_{3/2}\)	\(6h_{11/2}\)	—
\({}^{129}_{54}\mathrm{Xe}\)	75	201	163	\(M4\)	—	\(3s_{1/2}\)	\(6h_{11/2}\)	\(4d_{3/2}\)
\({}^{129}_{52}\mathrm{Te}\)	77	196	—	\(M4\)	—	\(4d_{3/2}\)	\(6h_{11/2}\)	—
\({}^{131}_{54}\mathrm{Xe}\)	77	163	—	\(M4\)	—	\(4d_{3/2}\)	\(6h_{11/2}\)	\(3s_{1/2}\)
\({}^{133}_{56}\mathrm{Ba}\)	77	287,2	275,5	\(M4\)	—	\(3s_{1/2}\)	\(6h_{11/2}\)	\(4d_{3/2}\)
\({}^{131}_{52}\mathrm{Te}\)	79	183,2	—	\(M4\)	—	\(4d_{3/2}\)	\(6h_{11/2}\)	—
\({}^{133}_{54}\mathrm{Xe}\)	79	232	—	\(M4\)	—	\(4d_{3/2}\)	\(6h_{11/2}\)	—
\({}^{135}_{56}\mathrm{Ba}\)	79	300	—	\(M4\)	—	\(4d_{3/2}\)	\(6h_{11/2}\)	—
\({}^{135}_{54}\mathrm{Xe}\)	81	520	—	\(M4\)	—	\(4d_{3/2}\)	\(6h_{11/2}\)	—
\({}^{137}_{56}\mathrm{Ba}\)	81	669	—	\(M4\)	—	\(4d_{3/2}\)	\(6h_{11/2}\)	—

and at the end of the filling of the shell reaches a value of 500–600 kev. The \(3s_{1/2}\) level also shifts (with respect to the \(4d_{3/2}\) level) with an increase in the number of particles filling the shell. In Fig. 20 it is clearly seen how, with an increase in the number of particles, the \(4d_{3/2}\) and \(3s_{1/2}\) levels approach and cross. Thus, in \({}^{121}_{52}\mathrm{Te}\), \({}^{123}_{52}\mathrm{Te}\), and \({}^{125}_{52}\mathrm{Te}\) the \(4d_{3/2}\) level is situated between \(3s_{1/2}\) and \(6h_{11/2}\), whereas in \({}^{127}_{52}\mathrm{Te}\), \({}^{129}_{52}\mathrm{Te}\), \({}^{131}_{52}\mathrm{Te}\), and \({}^{133}_{52}\mathrm{Te}^{68}\) the \(3s_{1/2}\) level lies between \(6h_{11/2}\) and \(4d_{3/2}\). Exactly the same situation occurs also for the xenon isotopes. In the nuclei \({}^{127}_{54}\mathrm{Xe}\), \({}^{129}_{54}\mathrm{Xe}\) the \(4d_{3/2}\) level is situated between \(3s_{1/2}\) and \(6h_{11/2}\), while in \({}^{131}_{54}\mathrm{Xe}\), \({}^{133}_{54}\mathrm{Xe}\), and \({}^{135}_{54}\mathrm{Xe}\) it is situated between \(6h_{11/2}\) and \(4d_{3/2}\); in \({}^{133}_{56}\mathrm{Ba}\) the \(4d_{3/2}\) level lies between the \(6h_{11/2}\) and \(3s_{1/2}\) levels, and in the nuclei \({}^{135}_{56}\mathrm{Ba}\) and \({}^{137}_{56}\mathrm{Ba}\) it apparently lies between the \(4d_{3/2}\) and \(6h_{11/2}\) levels.

This displacement of the \(4d_{3/2}\) and \(3s_{1/2}\) levels explains why two-cascade isomeric transitions are not observed in all isomers of the fifth group. Two-cascade transitions are observed only in those nuclei in which the \(4d_{3/2}\) level lies between the \(6h_{11/2}\) and \(3s_{1/2}\) levels.

1. Consideration of isomeric states undoubtedly confirms the presence of the \(6h_{11/2}\) level in the fifth shell. If this level did not exist, isomerism among the nuclei of the fifth group could not be observed. In any case, the appearance of multipole \(2^4\)-magnetic radiation could not occur. Meanwhile such radiation is observed in almost all isomeric transitions of nuclei of the fifth group. Two-step transitions of the type \(6h_{11/2}\to 4d_{3/2}\to 3s_{1/2}\) are also observed. All this convinces one that the \(6h_{11/2}\) level in the fifth shell really exists. These data are important for confirming the correctness of the nuclear systematics, since the absence among the nuclei of the fifth group of such nuclei in which the spin is equal to \(11/2\) may cast doubt on the reality of the existence of the \(6h_{11/2}\) level in the fifth shell. The absence of such nuclei (and the simultaneous presence of the \(6h_{11/2}\) level in the fifth shell) means that the \(6h_{11/2}\) level is filled only by pairs of particles and that the energy of an unpaired particle on the \(6h_{11/2}\) level is considerably greater than on the \(4d_{1/2}\) and \(3s_{1/2}\) levels.

From the point of view of the shell model it was not possible to explain the properties of some transitions in the region \(39\leq Z\leq 49\) and \(39\leq N\leq 49\). In particular, the transitions in \({}^{99}_{43}\mathrm{Tc}\), \({}^{103}_{45}\mathrm{Rh}\), \({}^{107}_{47}\mathrm{A}\), \({}^{109}_{47}\mathrm{Ag}\), and some others, by the magnitude of the half-life and of the internal-conversion coefficient, were usually attributed to magnetic octupole transitions, whereas the shell theory with strong spin-orbit coupling required magnetic transitions \(2^4\) for these nuclei. The use of the formulas derived in work \(^{69}\) gave Goldhaber and Sunyar \(^{76}\) the possibility of showing that these transitions in reality are

by electric octupole transitions, which is also confirmed by investigation, using refined formulas, of the internal-conversion coefficients.

Goldhaber succeeded in explaining the reason for the appearance in these nuclei of octupole electric transitions instead of magnetic transitions[^24]. Indeed, if one assumes that, with 3, 5, or 7 identical particles present in the \(5g_{9/2}\) shell, all of them can contribute to the spin of the nucleus, then in this case the nuclear spin can take values smaller than \(9/2\) (an analogous picture occurs for \(\mathrm{Mn}^{55}\), where 3 particles in the \(4f\) state give spin \(5/2\)). In particular, the nuclear spin may then be equal to \(7/2\), and its parity will be positive, as is required in order for an electric octupole transition to the \(3p_{1/2}\) state to occur. Sometimes the \(g_{9/2}\) state with spin \(i=7/2\) may turn out to be the ground state, and in other cases an excited state. From the measurement data it was possible to predict that the spin of \({}^{79}_{34}\mathrm{Se}\) is \(7/2\), which was excellently confirmed experimentally.

11. INVESTIGATION OF EXCITED NUCLEAR LEVELS

Recently the investigation of excited nuclear levels has made great progress. The study of the conversion coefficients of radiation produced in the transition from an excited state to the normal state, determination of the ratio of the conversion coefficients in the \(K\)- and \(L\)-shells, study of the angular correlation between \(\gamma\)-quanta emitted in cascade transitions and between quanta and conversion electrons, and investigation of the influence of a magnetic field on the magnitude of the angular correlation—all this makes it possible, from the known value of the spin and parity of the ground state, to determine the values of these quantities in the excited state.

Knowledge of these quantities is of essential importance for the nuclear-shell model, for from the character of the excited levels (spin and parity) one can test the correctness of the systematics of nuclear levels. L. K. Pekar, L. A. Sliv, and A. V. Zolotavin[^70] processed the available experimental material for the middle group of elements \(20<Z<70\) and compiled schemes of the excited levels of these nuclei. As a result of the analysis carried out, they succeeded in establishing that all the schemes can be divided into three groups. The first type of schemes of excited levels is observed in those nuclei in which the number of particles corresponds to the beginning of filling of some shell. In such nuclei, among the nearest excited levels, characterized by a small excitation energy, there are usually other levels of the given shell, arranged in the order of their normal sequence. An example of such schemes may be the scheme of excited levels of \({}^{131}_{55}\mathrm{Cs}\) presented below, shown in Fig. 21. In this cesium nucleus the number of neutrons is even, while the number of protons—55—is odd. Of this number, five protons

is in the fifth shell, i.e. the fifth shell in \({}^{131}_{55}\mathrm{Cs}\) has only begun to be filled. The ground state of \({}^{131}_{55}\mathrm{Cs}\) is \(4d_{5/2}\). Among its excited levels we find the levels \(g_{7/2}\), \(d_{3/2}\), \(s_{1/2}\), i.e. other levels belonging to the fifth shell and arranged in the order in which the levels of this shell are filled.

The second type of scheme is encountered in nuclei in which the particles are in an almost filled shell. Weak excitation in these nuclei (with a small excitation energy, up to \(1\) MeV) occurs when a particle occupying one of the lower filled

Fig. 21. Scheme of excited levels in the nucleus \({}^{131}_{55}\mathrm{Cs}\).

Fig. 22. Scheme of excited levels in the nucleus \({}^{131}_{54}\mathrm{Xe}\).

levels of the given shell passes to an unfilled higher-lying level. Pekar, Sliv, and Zolotavin proposed calling such excitation of the nucleus “hole” excitation. With this character of excitation, among the excited levels we shall again find the same levels that form the given nuclear shell, but the order in which they appear among the excited levels will be the reverse of that obtained when the shell is being filled. An example of such a scheme is furnished by the level scheme of \({}^{131}_{54}\mathrm{Xe}\), Fig. 22. The isotope \({}^{131}_{54}\mathrm{Xe}\) contains 77 protons; consequently, the construction of the levels of the fifth shell is almost completed in it. The ground level is \(4d_{3/2}\). Among the excited levels we find the levels \(s_{1/2}\), \(h_{11/2}\), \(d_{5/2}\), \(g_{7/2}\). Their order of succession is the reverse of that which occurs when the shell is filled; the normal order of succession

levels in the fifth shell: \(4d_{5/2}\), \(5g_{7/2}\) (or \(5g_{7/2}\), \(4d_{5/2}\)), \(6h_{11/2}\), \(3s_{1/2}\), \(4d_{3/2}\). The third type consists of mixed schemes, in which the excited levels have both normal and hole character. These schemes are observed for nuclei whose shells are already considerably filled with particles, but are still far from being filled.

A remarkable conclusion following from the work of Pecker, Sliv, and Zolotavin is that the observed excited levels of nuclei correspond in character to those levels which should occur in the nucleus according to the model of nucleon shells.

It is interesting to note one more feature of low-energy excited levels. According to the nucleon-shell model, low-energy excited levels should belong to one and the same shell. As Pecker, Sliv, and Zolotavin have shown, this is indeed the case.

The arrangement of these levels within one shell is such that neighboring levels usually prove to be levels with the same parity, for example \(4f_{7/2}—4f_{5/2}\), \(4f_{5/2}—3p_{3/2}\), \(3p_{3/2}—3p_{1/2}\) (fourth shell), or \(5g_{7/2}—4d_{5/2}\), \(4d_{5/2}—4d_{3/2}\), \(4d_{3/2}—3s_{1/2}\) (fifth shell). If, however, within one shell neighboring levels turn out to be levels of different parity, for example \(3p_{1/2}\) and \(5g_{7/2}\) (fourth shell), or \(4d_{3/2}\) or \(3s_{1/2}\) and \(6h_{11/2}\) (fifth shell), then they differ greatly in the value of the total angular momentum. With such an arrangement of levels, electric dipole radiation should not be observed. And indeed, a characteristic feature of low-energy nuclear \(\gamma\)-radiation is the absence in it of lines corresponding to electric dipole transitions.

12. ANGULAR DISTRIBUTION OF PARTICLES IN \((d,p)\) AND \((d,n)\) REACTIONS AND NUCLEAR SHELLS

Recently Bete and Butler \(^{71}\) pointed out experiments that can serve as a test of the correctness of the one-particle model with spin-orbit coupling. These authors note that in most cases the \((d,p)\) and \((d,n)\) reactions proceed by means of the so-called stripping process. In such a process, as is known, the deuteron is split; one of its constituent particles penetrates into the nucleus, while the other moves away from the nucleus. In this case it turns out \(^{72}\) that the particle moving away from the nucleus will have an angular distribution depending on the value of the orbital angular momentum \(l\) which the particle penetrating into the nucleus brings into it. In the nucleon-shell model, \(l\) corresponds to the value of the orbital angular momentum of the state occupied by the particle that has entered the nucleus. If, for example, \(l=0\), then in the angular distribution there will be observed a sharply expressed

... at small angles (a maximum in the forward direction). If, along with \(l=0\), states with another value of \(l\) also turn out to be allowed, then other maxima will appear, at angles the larger the greater the value of \(l\). Figure 23 gives calculated curves showing the form of the angular distribution of a particle leaving the nucleus for the cases \(l=0\), \(l=1\), and \(l=2\). The scales of these curves are not matched. The values of the maxima of these curves

Fig. 23. Angular distribution (calculated) of particles liberated in the reactions \((d,p)\) and \((d,n)\), proceeding by the stripping process. The scales of these three curves are not matched.

actually differ from one another much more than in Fig. 23.

The curves shown in Fig. 23 differ strongly from one another not only in the magnitude of the maximum but, more substantially, in the position of the maximum. Therefore, having experimentally determined the angular distribution of the outgoing particle formed as a result of the stripping process and comparing it with the curves of Fig. 23, one can establish how the orbital angular momentum of the nucleus changes as a result of the penetration into it of the particle split off from the deuteron. Having determined \(l\), one can, from the parity and the magnitude of the spin of the initial nucleus, determine the spin and parity of the resulting nucleus. Thus, for example, the angular distribution of protons in the reaction \({}^{16}_{8}\mathrm{O}(d,p){}^{17}_{8}\mathrm{O}\), for the case when \({}^{17}_{8}\mathrm{O}\) is formed in the normal state, agrees fairly well with the curve of Fig. 22 corresponding to the value \(l=2\).

The state of \({}^{16}_{8}\mathrm{O}\) is even \((+)\), the spin is zero. Since the neutron that entered the \({}^{16}_{8}\mathrm{O}\) nucleus changed its orbital angular momentum by an amount \(l\) equal to 2, the normal state of the \({}^{17}_{8}\mathrm{O}\) nucleus must be even and have spin either \(5/2\) or \(3/2\). The value \(5/2\) would correspond to the \(3d_{5/2}\) level, from which, according to Mayer’s scheme, the filling of the third nuclear shell should begin. It is interesting to note that

Fig. 24. Angular distribution of protons in the reaction \({}^{31}\mathrm{P}(d,p){}^{32}\mathrm{P}\). The solid curves are calculated for \(l=0\), \(l=2\). The dashed curve represents the curve for the case \(l=0\) reduced tenfold.

the angular distribution of protons arising as a result of neutron capture to an excited level of \({}^{17}_{8}\mathrm{O}\) (excitation energy \(0.88\) MeV) differs\({}^{75}\) from the angular distribution of protons arising upon neutron capture to the ground level of \({}^{17}_{8}\mathrm{O}\). It corresponds to the curve of Fig. 22 with the value \(l=0\). Consequently, the excited state of \({}^{17}_{8}\mathrm{O}\) is even and has spin equal to one half. According to Mayer’s scheme, in the third shell the level \(2s_{1/2}\) should follow the level \(3d_{5/2}\). The spin of the nucleus in which the odd particle occupies this level must therefore be equal to \(1/2\). Thus,

the study of the angular distribution of particles formed in stripping reactions \((d,p)\) and \((d,n)\) makes it possible to compare the parities of nuclei and thus to verify conclusions following from the nuclear shell model. However, in some cases, as Bethe and Butler[^71] have indicated on the basis of experimental data on the angular distribution of particles formed in the stripping process, the conclusions of the nuclear shell model can also be directly tested with respect to the type of level occupied in the nucleus by the unpaired particle.

Let us suppose that we know the spin and parity both of the initial nucleus and of the nucleus formed as a result of the reaction, and let us suppose that

Fig. 25. Angular distribution of protons in the reaction \({}^{35}\mathrm{Cl}(d,p){}^{36}\mathrm{Cl}\).

the selection rules allow several different values of \(l\) for the particle that has entered the nucleus. If it is immaterial to the nucleus which of the possible (by the selection rule) values the particle acquires, then in the angular distribution of the outgoing particle there will dominate the maximum corresponding to the smallest of the possible (by the selection rule) values of \(l\), since the magnitude of the maximum in the angular distribution of the outgoing particle falls sharply with increasing \(l\). If, however, the value of \(l\) is not immaterial to the nucleus—for example, if, as the nuclear shell model indicates, the levels corresponding to the smaller possible values of \(l\) are already occupied—then in the angular distribution there should be no maximum corresponding to the value of \(l\) forbidden by the nuclear shell model.

For the most part, the values of \(l\) allowed by the nuclear shell model coincide with the smallest value of \(l\) permitted by the selection rules, but in some reactions such a coincidence is absent.

These reactions, in the opinion of Bethe and Butler, can serve as a test for the model of nuclear shells. In Table XV below such reactions are given. Some of these reactions, ${}^{31}_{15}\mathrm{P}(d,p){}^{32}_{15}\mathrm{P}$ and ${}^{35}_{17}\mathrm{Cl}(d,p){}^{36}_{17}\mathrm{Cl}$, were then subjected to special study$^{73,74}$. In Figs. 24 and 25 a comparison is given of the experimentally found angular distribution of protons in these reactions with that calculated theoretically for various values of $l$. As is seen from the figures, in both reactions the value $l=0$, allowed by the selection rules but contradicting the systematics of levels, in fact has no place.

Table XV

Reaction	Spin and parity of states: initial	Spin and parity of states: final	$l$ (from the systematics of nuclear levels)	Values of $l$ allowed by the selection rules
${}^{31}\mathrm{P}(d,p){}^{32}\mathrm{P}$	$1/2+$	$1+$	2	0 and 2
${}^{35}\mathrm{Cl}(d,p){}^{36}\mathrm{Cl}$	$3/2+$	$2+$	2	0, 2 and 4
${}^{37}\mathrm{Cl}(d,p){}^{38}\mathrm{Cl}$	$3/2+$	$2-$	3	1 and 3
${}^{41}\mathrm{K}(d,p){}^{42}\mathrm{K}$	$3/2+$	$2-$	3	1 and 3
${}^{45}\mathrm{Sc}(d,p){}^{46}\mathrm{Sc}$	$7/2-$	$4+$	3	1, 3, 5 and 7
${}^{51}\mathrm{V}(d,p){}^{52}\mathrm{V}$	$7/2-$	2 or $3+$	3	1, 3, 5 and 7

Thus, the data on the angular distribution of protons in $(d,p)$ reactions also confirm the correctness of Mayer’s nuclear systematics.

13. CONCLUSION

Thus, it has turned out that the single-particle model with strong spin-orbit coupling explains well a large number of diverse phenomena. It predicts the correct values of the “magic” numbers, gives a correct description of the magnitudes of the spins of nuclei, indicates the correct course of the change of the quadrupole and magnetic moments of nuclei. This model gives a correct interpretation of the division of $\beta$-transitions into allowed and forbidden ones and, quite naturally and simply, explains the existence of the enigmatic islands of isomerism. Moreover, this model correctly indicates among precisely which nuclei isomerism can be observed and in which...

there should be none of them in the nuclei. This model correctly predicts the character of the measured transitions—their multipolarity, parity, and also the presence of two-cascade transitions. The angular distribution of particles in the reactions \((d,p)\) and \((d,n)\) shows that the type of levels ascribed by the nuclear-shell model to nuclear particles is in agreement with the character of the observed angular distribution.

The shell model, as Pekar, Sliv, and Zolotavin have shown, for the most part correctly describes the types of excited levels nearest to the ground state.

Such agreement of a simple model with experimental data, and the successful explanation of so large a number of diverse phenomena, testify that the nuclear-shell model to some extent correctly describes the properties of atomic nuclei.

Of course, one should not expect that the simple scheme proposed by Mayer is exhaustive. There are data indicating that this scheme apparently must be developed. We have already pointed out that the spin and magnetic moments of some nuclei \(\left({}^{22}_{11}\mathrm{Na}, {}^{21}_{10}\mathrm{Ne}, {}^{55}_{25}\mathrm{Mn}\right)\) do not agree with Mayer’s scheme. Further, in studying isomers in the fourth group of nuclei, consisting of 39–49 particles, it was found \(^{76}\) that the moment of the excited state of some nuclei \(\left({}^{73}_{34}\mathrm{Se}, {}^{77}_{34}\mathrm{Se}, {}^{81}_{34}\mathrm{Se}, {}^{103}_{45}\mathrm{Rh}, {}^{105}_{45}\mathrm{Rh}, {}^{107}_{47}\mathrm{Ag}, {}^{109}_{47}\mathrm{Ag}\right)\) is equal to \({}^{7}/_{2}\), and not \({}^{9}/_{2}\), as it should have been if, in the excited nucleus, the particle occupied the level \(5g_{9/2}\). A state with spin \({}^{7}/_{2}\) has also been found in \({}^{83}_{36}\mathrm{Kr}\) and \({}^{85}_{37}\mathrm{Sr}\) (in these nuclei the level \({}^{7}/_{2}\) is situated between \(3p_{1/2}\) and \(5g_{9/2}\)). In the nuclei \({}^{79}_{34}\mathrm{Se}\) and \({}^{77}_{32}\mathrm{Ge}\), the level with moment \({}^{7}/_{2}\) apparently corresponds to the ground state. These data, as well as data for some other isomers, indicate that, in addition to levels corresponding to the one-particle model, there are also other levels in the nucleus. It is very probable that these levels, as we have already mentioned, result from the fact that in nuclei containing not one particle on a level, but three, five, etc., there may exist states that are due not to one particle, but to three (or more) particles located on the given level.

Although the parity of these levels (not included in Mayer’s scheme) agrees with the assumption that they are due to several particles, there are nevertheless as yet no reliable data that would make such an interpretation unambiguous.

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Note added in proof

D. D. Ivanenko and collaborators have drawn attention\(^{78—80}\) to the presence of irregularities in the course of the dependence of isotopic shift and nuclear radius on atomic number. The irregularities in the course of the curves occur for nuclei with a special “magic” number of particles.

Addendum to the list of cited literature

1. D. Ivanenko and N. Kolesnikov, DAN, 89, 253 (1953). Periodic regularities in isotopic shifts and nuclear radii.
2. D. Ivanenko and S. Larin, ZhETF, 24, 3 (1953). Nuclear volume and nuclear shells.
3. D. Ivanenko and N. Kolesnikov, DAN, 81, 771 (1951). N. Kolesnikov, ZhETF, 24, 246 (1953). Double β-decay and nuclear periodicities.
4. A. Sokolov and D. Ivanenko, Quantum Field Theory, Moscow, 1952.
5. L. V. Groshev and I. S. Shapiro, Spectroscopy of Atomic Nuclei, Moscow, 1951.