Nuclear Moments

B. T. Feld*)

I. Introduction

In recent years a great deal of information has been accumulated concerning the static properties of nuclei. To determine nuclear spins, magnetic moments, and electric quadrupole moments, various methods have been used: optical spectroscopy, molecular and atomic beams, nuclear induction, and microwave spectroscopy. At first stable nuclei in the unexcited state were investigated; then, as experimental techniques improved, the investigations were extended to radioactive nuclei \(^{1—5}\). A review of the present state of experimental technique has been given by Ramsey \(^{6}\).

A characteristic feature of the methods mentioned above is the high accuracy of the measurement of nuclear moments \(^{7—19}\). It is known that, in the investigation of the hydrogen isotopes, the discrepancy between the experimental data and the theoretical values served as a basis for revising the theory and for the further development of quantum electrodynamics; at present the theory correctly interprets the regularities of the atomic spectra of hydrogen.

In the case of complex atoms it is not possible to establish a definite connection between the hyperfine structure of spectra and nuclear moments, owing to the difficulties in obtaining exact expressions for the atomic wave functions. Whereas magnetic moments can be determined directly by the molecular-beam method or by nuclear induction, the accurate determination of electric quadrupole moments is still connected with measurements of the hyperfine structure of atomic spectra. Recent work by Sternheimer \(^{26}\) on the electric shielding of nuclei due to the polarization of inner electron shells, and by Koster \(^{27}\) on the influence of different electron configurations on the hyperfine structure of atomic spectra, has led to a better understanding of the difficulties inherent in the accurate measurement of nuclear quadrupole moments and has made it possible in a number of cases to give more accurate estimates. The theory of molecular hyperfine structure has again attracted

*) Annual Review of Nuclear Science, 1953, vol. 2, pp. 239–260.

attention of researchers ^{28—30}, as has the theory of magnetic resonance in liquids and solids ^{28, 31, 34}; these investigations are important not only from the point of view of a more accurate estimate of the magnitude of nuclear moments, but also for a better understanding of the liquid and solid states of matter.

The ground states of stable nuclei now constitute an ever smaller fraction of the nuclear states for which nuclear moments are known to us. True, most of these data refer to nuclear spins. The success of the independent-particle model of nuclei in the form in which it was proposed by Mayer ^{35} and by Haxel, Jensen, and Suess ^{36} in predicting the spins and moments of low nuclear levels gave a considerable impetus to the development of nuclear spectroscopy and initiated a revision of the systematics of nuclear magnetic and electric moments. The aim of the present article is a brief review of the current state of the question concerning methods for determining nuclear moments (including spins) and the connection of the observed quantities with possible models of the nucleus.

II. METHODS FOR MEASURING THE MOMENTS OF UNSTABLE NUCLEI

The usual methods of measuring nuclear moments are applicable to stable nuclei in the ground state, long-lived radioactive nuclei, and certain isomers, provided only that the latter can be obtained in sufficient quantity ^{1—5}. To determine the properties (parity, spin, moments) of all other nuclei, other methods are necessary. Such methods exist, and at present significant progress has been achieved in their development and application. As a result, a considerable amount of data has been obtained concerning nuclear energy levels, especially for light nuclei ^{37—39}.

The properties of short-lived nuclei are determined by studying their radiations. In particular, under favorable circumstances it is possible to determine the angular momentum and parity of the emitted particles. In this case one can determine the spin and parity of the nucleus before emission, if the spin and parity of the nucleus in the final state are known. Thus, ultimately, in order to determine the spin and parity of a nucleus in an excited state it is necessary to know the properties of the nucleus in the stable state. The latter can be determined by ordinary methods. The change of the parity and spin of a nucleus is most directly connected with the angular distribution of the nuclear radiation (with respect to some selected direction). Some conclusions can be drawn on the basis of the lifetime of the nucleus and the interaction of the radiation with atomic electrons.

1. ANGULAR DISTRIBUTION OF NUCLEAR DECAY PRODUCTS

Methods for determining the properties of excited nuclei from experiments on “angular correlations” were recently reviewed by Deutsch \(^{40}\). In our review we shall consider (more or less arbitrarily) three basic methods.

a) Emission of Polarized Nuclei

Polarization of nuclei is equivalent to establishing a nonuniform distribution of nuclear moments in a substance in which, ordinarily, all directions of the nuclear magnetic moments are equally probable. A number of experiments in this direction have been proposed and discussed by various authors \(^{41\text{–}46}\). Some experiments make use of the actual orientation of nuclear spins in an external magnetic field or in a previously oriented atomic field (“dipolarization”); others are based on interactions proportional to the square of the magnetic quantum number \(m^2\), analogous to the nuclear electric quadrupole interaction (“quadrupolarization”). In order to achieve even a small degree of polarization, all of them require a very low temperature, \(<1^\circ\mathrm{K}\), and some, in addition, strong magnetic fields.

If there is a system of polarized radioactive nuclei, then the intensity of the radiation in the general case depends on the angle between the direction of emission and the direction of polarization. The angular distribution of the radiation is a function of the total angular momentum, spin, and parity of the emitted particles (and, of course, of the degree of polarization). The theory of the angular distribution was developed by Spiers \(^{47,48}\) and others \(^{49,50}\).

The experimental realization of appreciable nuclear polarization is an extremely difficult problem. Nevertheless, certain successes were achieved by groups of workers at Oxford \(^{51,52}\) and Leiden \(^{53,54}\), who observed deviations from isotropy in the \(\gamma\)-radiation of \(\mathrm{Co}^{60}\), oriented by the method of Bleaney \(^{45}\). Starting from the decay scheme and the spin value \((I=5)\) proposed by Deutsch and Scharff-Goldhaber \(^{55}\), it is possible, on the basis of these experiments, to assign to the magnetic moment of \(\mathrm{Co}^{60}\) the value \(|\mu|\simeq 3\) nuclear magnetons.

b) Angular Distribution in Nuclear Reactions

In nuclear reactions the direction of the particles incident on the nucleus constitutes a natural axis relative to which the directions of the particles emerging from the nucleus are distributed. In order that this distribution deviate from spherical symmetry, it is necessary that the incident and outgoing particles have total angular momentum greater than \(\hbar/2\). A simple and well-known example is the photodisintegration of the deuteron \(^{56}\). In the case when

when the energy of the \(\gamma\)-quanta does not exceed the reaction threshold too greatly, two processes dominate. One of them—the photoelectric disintegration—corresponds to the absorption of an electric-dipole \(\gamma\)-quantum \((l=1)\), with the subsequent transformation of the basic \({}^{3}S_{1}\)-state of the deuteron into a free neutron and proton in \({}^{3}P\)-states. (Selection rules for electric-dipole absorption: \(\Delta L=\pm 1,\ \Delta S=0\).) The angular distribution (in the coordinates of the center of inertia) of the emitted protons and neutrons is described by the function \(\sin^{2}\theta\), where \(\theta\) is the angle measured from the direction of the \(\gamma\)-quantum. This distribution also follows from classical considerations, since the electromagnetic radiation acts on the instantaneous dipole moment of the deuteron through the electric vector, which is perpendicular to the direction of propagation of the \(\gamma\)-quanta.

The second process consists in dipole magnetic absorption, which corresponds to a “turning over” of one of the nuclear spins relative to the other \((\Delta L=0;\ \Delta S=\pm 1)\), and, consequently, to the transformation of the basic \({}^{3}S_{1}\)-state into a \({}^{1}S_{0}\)-state. The angular distribution in this reaction is spherically symmetric in the coordinates of the center of inertia. Such simple reactions, in which no intermediate nucleus is formed, are observed for the most part among the light elements. The angular distribution in these reactions is used for identifying the spins and parities of many levels in light nuclei.

However, for the majority of nuclear reactions it is necessary to take into account the formation of an intermediate nucleus, in which the spacing between energy levels is comparable with the spread in energy of the bombarding particles (especially in reactions involving heavy nuclei). Here we may distinguish two limiting cases. First, we may consider the case when the energy levels of the intermediate nucleus are so closely spaced that a continuum is formed, with a statistical distribution of the spins and parities of the levels. In this case the angular distribution of the reaction products is the sum of all possible processes of absorption and decay. If, nevertheless, the final state is discrete, then from the angular distribution of the reaction products one can determine the properties of this state, as was shown by Hauser and collaborators\(^{58}\) in the case of inelastic scattering of neutrons by medium and heavy nuclei.

The second case, interpreted more simply, consists in the fact that the intermediate nucleus has one (resonance) level. In this case, knowledge of the properties of the initial and final states and measurement of the angular distribution of the reaction products make it possible to determine the spin and parity of the intermediate nucleus at the level under consideration\(^{59,60}\). It is not very difficult in this case, by means of the determination alone of the maximum total cross section for the reaction of formation of the intermediate nucleus, to find the spin

of the latter, since[^58]

\[ \sigma_{\max}=\frac{4\pi\lambda^2(2J+1)}{(2s+1)(2I+1)}, \tag{1} \]

where \(J\) is the spin of the intermediate nucleus, \(s\) is the spin of the bombarding particle, \(I\) is the spin of the target nucleus, and \(\lambda\) is the wavelength of the bombarding particle divided by \(2\pi\).

Of special interest for the purposes of measuring nuclear spins, among other nuclear reactions, are \((d,p)\)- and \((d,n)\)-reactions (stripping reactions). In these reactions an intermediate nucleus is not formed; rather, one of the nucleons of the deuteron is “captured” directly into a level of the final nucleus. The final nucleus in this case often remains in the unexcited state or in a state close to it, while the uncaptured particle carries away the reaction energy. Thus the protons or neutrons appearing as a result of the reaction have a line spectrum of energies. Each group of particles corresponding to a definite energy has an angular distribution with one or several sharply pronounced maxima or minima in the direction of the incident beam.

Butler[^61] developed the theory of such reactions, on the basis of which, from the angular distribution of the emitted particle, one can determine the orbital angular momentum added to the nucleus upon capture of the other particle. If the spin and parity of the target nucleus are known, then the parities and the limits of possible spin values of the final nucleus can be determined. The applicability of this theory has been verified for a large number of target nuclei and a wide range of deuteron energies.[^62–^67]

b) Angular correlation between successive nuclear radiations

Another method for determining the orientation of a nucleus (the value or distribution of the azimuthal quantum number \(m\)) is possible if the nucleus undergoes a series of successive transformations. In this case there is a correlation between the directions of successive radiations, depending on the angular momenta and parities of the nuclei arising in the process of the transformations, and also on the nature of the emitted particles.

The theory of angular correlation was first developed by Hamilton for the case of successive emission of two \(\gamma\)-quanta. It was extended by many authors[^69–^77] to other cases, including: \(\beta\)-\(\gamma\),[^74–^77] internal conversion–\(\gamma\),[^70,^73] \(\alpha\)-\(\gamma\), etc.[^72,^74,^77] A simple physical picture of an effect of this type was given by Moon[^78] for the case of successive emissions of electric-dipole \(\gamma\)-quanta in nuclear transformations corresponding to a change of spin \(0—1—0\). Suppose that the first \(\gamma\)-quantum is emitted in the direction of the \(z\)-axis. The emission of the \(\gamma\)-quantum imparts to the nucleus an angular moment—

of motion equal to unity and directed in a plane perpendicular to the \(z\)-axis, for example in the direction of the \(x\)-axis. The second \(\gamma\)-quantum must carry away this unit of angular momentum, i.e. the direction of emission of the second \(\gamma\)-quantum must lie in the \(yz\)-plane, and all angles in this plane must be equally probable. Since the choice of the \(x\)-axis as the direction of the angular momentum in the intermediate state was completely arbitrary, the angular correlation can be obtained by rotating the angular distribution of the second \(\gamma\)-quantum (any direction in the \(yz\)-plane) about the \(z\)-axis. This leads to an angular correlation that is most probable along the \(\pm z\) direction and least probable in the \(xy\)-plane, i.e. to an angular correlation described by the formula \(1+\cos^2\theta\), where \(\theta\) is the angle between the directions of the two \(\gamma\)-quanta. The general case of nuclear transitions corresponding to nuclear spins \(J \to J' \to J''\) and to multipole radiation unfortunately does not reduce to such simple physical pictures. In this case the calculation is carried out by relatively simple means of quantum mechanics, if only it is assumed that all azimuthal quantum numbers are taken with respect to the direction of emission of the first quantum (Hamilton’s theorem)\(^{68,79,80}\).

In the most frequently encountered cases the expression for the angular correlation has the following form:

\[ W(\theta)=\sum_{i=0}^{l} a_i \cos^{2i}\theta=\sum_k b_k P_k(\cos\theta) \tag{2} \]

where \(l\) is equal to the least of the values \(L, L', J'\) (\(L\) and \(L'\) are the orbital angular momenta of the successive radiations, \(J'\) is the spin of the intermediate state). There are, however, cases in which expression (2) also contains odd powers of \(\cos\theta\), corresponding to the presence of interference effects arising when the given level decays by two or more paths\(^{74,75}\).

After the first successful measurements of \(\gamma\)-\(\gamma\) angular correlation, carried out by Brady and Deutsch\(^{82}\), this method was widely applied to many radioactive nuclei. Most of the results are explained by theory, but in some cases it proves inapplicable. Many of these anomalies can be understood as the result of partial or complete destruction of the angular correlation in the intermediate state owing to reorientation (“loss of memory”). Such reorientation occurs, for example, if the lifetime of the intermediate state is large in comparison with the period of precession of the nucleus in the magnetic field of the atom, molecule, or crystal*)\(^{69,83—85}\). Some authors proposed using this effect to measure the magnetic moment of the nucleus\(^{83,86,87}\)

*) The precession period of the nucleus is measured by a quantity inverse to the difference of the frequencies of hyperfine-structure lines.

in the intermediate state, observing the change (partial destruction) of the angular correlation when an external magnetic field is applied. Such an experiment was successfully carried out by Appleby and co-workers^88; for the magnetic moment of the excited nuclear level of Cd^111 with energy 247 kev \((I = 5/2)\) they obtained the value \(\mu = -0.85 \pm 0.22\) nuclear magnetons.

In the case of angular correlation involving \(\gamma\)-radiation, the parity of the levels cannot be determined unambiguously, owing to the fact that in the emission of an electric- or magnetic-multipole \(\gamma\)-quantum the same change in angular momentum occurs for opposite changes of parity.

To remove this ambiguity it is necessary to carry out some additional investigation of the \(\gamma\)-radiation. One of the methods is to measure the plane of polarization of one of the \(\gamma\)-radiations with respect to the direction of emission of the second^89,90. This method was applied by Metzger and Deutsch^91.

Of the other possible measurements of angular correlation we shall mention only \(\alpha\)-\(\gamma\) correlation. This type of correlation, if it is observed, can be interpreted most simply, since the \(\alpha\)-particle has no intrinsic angular momentum. Consequently, nuclear transformations between states \(J \to J'\), including as the first stage the emission of an \(\alpha\)-particle, end with the emission of a \(\gamma\)-quantum with \(L = |J - J'|\), \(\Delta m = 0\), and with a change of parity \((-1)^L\). This type of correlation was used for the identification of the Li^7 nuclear level corresponding to an excitation of 482 kev and arising in the reaction B^10 \((n,\alpha)\), for determining the spins and parities of nuclei in the resonance reaction F^19 \((p)\) Ne^20\(*\) \((\alpha)\) O^16\(*\) \((\gamma)\) O^16 ^95, and for the study of the ThX levels, which are a product of the natural \(\alpha\)-decay of RdTh^96,97.

2. β-SPECTROSCOPY

The investigation of the properties of \(\beta\)-spectra has given much information concerning nuclear energy levels. If some nuclear radiation is accompanied by \(\beta\)-radiation or follows it, then the angular correlation between them can be studied, as was indicated above. The theory of \(\beta\)-\(\gamma\) angular correlation was developed by Falkoff and Uhlenbeck^74 and used by a number of investigators. A review by Stevenson and Deutsch^98 is devoted to this question. Another type of correlation occurs between \(\beta\)-radiation and simultaneously emitted neutrinos. Since the neutrino cannot be directly observed, one has to resort to observing the correlation between the \(\beta\)-radiation and the recoil nucleus^101,102. Finally, recently an angular correlation has been observed between \(\beta\)-radiation and the accompanying continuous electromagnetic radiation^103.

The use of the characteristics of \(\beta\)-spectra for the investigation of the properties of nuclear energy levels is complicated by the circumstance that,

B. T. FELD

that the character of the interaction between nucleons and the electron-neutrino field has not yet been established, if indeed it can be established unambiguously at all[^104]. Nevertheless, despite the uncertainty of the interaction—and, consequently, of the selection rules—it still appears possible to relate the characteristics of the β-spectrum to the change in angular momentum and parity. There are two such methods.

a) Relative half-life periods

or \(ft\)-values

The original Fermi theory[^105] and its subsequent modifications[^104] predict a relation between the half-life period—\(t\), the maximum energy of the β-particles—\(W_0\), and the changes in angular momentum and parity occurring in the process of radioactive transformation. This relation is usually expressed through the value \(ft\) (relative half-life periods) and has the form

\[ ft=\frac{2\pi^3(2I_i+1)\ln 2}{g^2\sum |(kM|i)|^2}, \]

where \(t\) is the half-life period; \(I_i\) is the spin of the parent nucleus; \(g\) is Fermi’s constant, which is a measure of the strength of the interaction; \((k|M|i)\) is the matrix element for the β-transition, which must be summed over all spin projections; \(f=f(Z,W_0)\) is a factor determined by the theory[^104,^106].

The selection rules enter the theory through the matrix element, which can take a series of decreasing values depending on the change in angular momentum and parity. These values turn out to be more or less discrete, so that the corresponding β-transformations fall into groups. The group corresponding to the smallest value of \(ft\) (the largest matrix element) is called “allowed”; the next, larger value of \(ft\), gives the group “forbidden in first order,” then the group “forbidden in second order,” and so on. The selection rules are determined by the type of interaction and fall into two groups: the Fermi[^105] and Teller[^107] selection rules. Both groups of transition rules are compared in Table I[^104].

However, the interpretation of \(ft\)-values in reality has only limited applicability. There is a group of allowed transitions with \(ft<10^4\) (for the most part these are transitions between “mirror” nuclei, which are often called “superallowed”). Transitions with \(ft>10^4\) do not split so clearly into different groups; nevertheless, they can still be classified as allowed, weakly forbidden, forbidden in first order, and so on. In this case the overlapping of one group with another appears natural, since one may expect that the matrix elements will fluctu-

be transferred from one nucleus to another. In accordance with the latest observations (for example, \( \mathrm{He}^6 \to \mathrm{Li}^6\), \(\Delta J = 1\), without a change of parity,

Table I

Selection rules for \(\beta\)-decays

allowed transition with \(ft \cong 10^2—10^3\)) the Teller selection rules are the more widespread.

b) Shapes of \(\beta\)-spectra

The shape of a \(\beta\)-spectrum is revealed most clearly by plotting a certain function of the momenta and intensities of the \(\beta\)-particles as a function of energy. This plot is known as the “Curie plot” or “Fermi plot” \(^{105,109}\). On such a plot the allowed spectra are represented in the form of straight lines intersecting the \(x\)-axis at the point corresponding to the energy \(W_0\). One may expect that the curves corresponding to forbidden spectra will deviate noticeably from straight lines, and that the form of the deviation characterizes the nature of the interaction \(^{104}\).

Up to 1949, all reliably measured β-spectra were represented by linear Kurie plots, with the exception of RaE*)\(^{110,111}\). However, following the discovery by Linger and Price\(^{112}\) of the β-spectrum of Y\(^{91}\) of a “forbidden” form (deviating from linear), the number of such spectra has increased and continues to grow. Most spectra of “forbidden” form obey interactions that lead to Teller’s selection rules. A summary of achievements in this field is given in the excellent review by Wu\(^{113**)}\).

3. PROPERTIES OF ISOMERIC TRANSITIONS

If the spin of a nucleus in an excited state differs from the spin of the same nucleus in a state with a lower degree of excitation by a relatively large amount (for example, \(>3\)), and if the excitation energy is relatively small (for example, 500 kev), then the lifetime of the excited nucleus may be fairly large (for example, \(>10^{-1}\) sec.). Such states***) are called “isomeric states,” or simply isomers. In particular, isomeric nuclei pass into stable ones by emitting γ-quanta. In such isomeric transitions, a considerable part of the γ-radiation may undergo “internal conversion”; in this case an electron of definite energy is ejected from the \(K\)-, \(L\)-, or \(M\)-shell of the atom; the ejection of the electron is accompanied by characteristic X-radiation. The lifetime of an isomer and the probability of “internal conversion” of γ-quanta are functions (for a given energy) of the changes in the angular momentum and parity of the nucleus. These changes determine the multipolarity of the emitted γ-quanta, and the corresponding nuclear transitions are usually denoted as indicated in Table II.

Recently it has become known (see, for example,\(^{114,115}\)) that a diagram on which the logarithm of the lifetime is plotted as a function of the logarithm of the decay energy has a tendency to break up into a certain number of straight lines. Transformations corresponding to the different straight lines are accompanied by different transitions, for example \(E2\), \(M2\), \(E3\), etc., and the lifetime increases with increasing order of multipolarity (for a given transition energy).

Another approach consists in classifying isomers according to the observed “conversion coefficients.”\(^{114—115}\) The drawback of this method lies not only in the great experimental difficulties, but also in the inaccuracy of the theory. In some cases both

) Partial deviations from a straight line in the region of low energies may be explained by the finite thickness of the source.
) See UFN, vol. XLIV, 558 (1951).
**) Of course, the classification of isomeric states depends on the observer’s possibilities for measuring the delay time of transformations.

method of classifying isomeric transitions lead to contradictory results.

Nevertheless, it may be asserted that, thanks to the work of Goldhaber and Sunyar\(^{116}\), the identification of nuclear isomers is on the right track. With the aid of the theory of nuclear shells (see below) they established an empirical relation between the lifetime and the energy of isomeric transitions. This relation may be stated as follows: for transitions with \(|\Delta J| > 3\), the lifetime of an isomer is determined mainly by the change of spin, and not by changes in the order of multipolarity or of parity. The lifetimes in magnetic transitions are in agreement with the conclusions of Weiskopf’s theory\(^{117}\). The lifetimes in electric transitions are somewhat longer than the theory requires, with the exception of \(E2\) transitions, for which the lifetime is sufficiently small and often coincides with \(M1\).

Table II

Classification of isomeric transitions

These same authors established empirical curves for the conversion ratios from the \(K\) and \(L\) shells, which are more accurate than the theoretical curves\(^{118,119}\). These curves are of great assistance in identifying isomeric transitions.

III. THE INDEPENDENT-PARTICLE MODEL AND NUCLEAR MOMENTS

1. THE SHELL STRUCTURE OF THE NUCLEUS

The independent-particle model, which is by no means a new idea, has in recent years been revived anew. In accordance with this model (sometimes called the “quasi-atomic” model of the nucleus), each nucleon may be regarded as moving in a central field which is the average interaction of the given nucleon with the other nucleons of the nucleus. The nucleons occupy different energy levels in accordance with the Pauli principle.

(according to which two or more protons or neutrons cannot be in one and the same state). Thus, one should expect that the ground state of a nucleus has a shell structure which, in the case of completely filled shells, leads to especially stable nuclei (analogously to the noble gases among atoms). Although the revival of the independent-particle model was associated with the discovery of the phenomenon of filled shells, as a result of which the nucleon numbers 20, 28, 50, 82, and 126 were assigned the name “magic numbers,” these numbers could not be interpreted satisfactorily until Mayer[^35] and Haxel, Jensen, and Suess[^36] introduced the hypothesis of a strong spin-orbit coupling of the individual nucleons. The figure shows, schematically and not to scale, the ordering of nuclear energy levels under various assumptions concerning the form of the nuclear potentials.

The best approximation to the experimental data is given by a “rectangular well with rounded edges” with spin-orbit coupling increasing with increasing orbital angular momentum. It must be pointed out that up to the present there are still no satisfactory theoretical foundations either for the independent-particle model itself or for the assumption concerning strong spin-orbit coupling.[^123],[^125] Nevertheless, the experimental evidence in favor of this model is now so numerous, and its significance in the study of the properties of low-lying nuclear levels so great, that there is no doubt as to the empirical suitability of the model. In fact, the independent-particle model manifests itself even in the region of relatively high-lying energy levels. This fact is difficult to understand and explain.

The shell model of the nucleus not only successfully explains the “magic numbers,” but can predict the angular momenta and parities of almost all nuclei in the ground state and in states corresponding to small excitations. For this purpose it is necessary to supplement the model by certain rules concerning the coupling of more than one particle in a shell that is not completely filled. In essence, these rules assert that an even number of nucleons is coupled so that \(J=0\) and the parity is positive, whereas an odd number of nucleons, each of which has angular momentum \(j\), is coupled so that \(J=j\), and the parity corresponds to the parity of the individual nucleons. These rules can be derived from the assumption of a spin-independent \(\delta\)-shaped interaction law between nucleons,[^35] but they are also valid in the case of a spin-independent interaction of finite radius, provided only that the latter is not too large. Nevertheless, there are several exceptions, all of which follow the auxiliary rule according to which an odd number of nucleons with angular momentum \(j\) can sometimes give a state with \(J=j-1\) and a parity corresponding to the parity of the individual nucleon. Prob-

NUCLEAR MOMENTS

…the coupling of an odd number of neutrons with an odd number of protons (in odd–odd nuclei) is somewhat more complicated and will be considered below.

Order of the energy levels in the independent-particle model under various assumptions concerning the shape of the nuclear potential, including the effect of spin–orbit coupling. The energy scale is not observed. Nor should the order of levels within a given shell be taken too seriously.

The order in which levels are filled within a shell often differs from the scheme shown in the figure. First, several tri-

The evident circumstance is that the exact order of the levels depends on the form and magnitude of the nuclear potential, which may fluctuate from element to element. The second consists in the difference between the filling of neutron and proton shells.

More important is the effect of the energy gain when two nucleons are located on one and the same subshell; this effect increases rapidly with increasing orbital angular momentum. As a result of this effect, states with high angular momentum are preferentially filled by pairs of nucleons, so that the ground states of odd-odd nuclei have the lowest values of \(j\).

The effect of the energy gain in paired filling explains the fact that there are no odd-odd nuclei in the ground state with \(j>9/2\), despite the fact that the shells 50–82 and 82–126 contain subshells with \(h_{11/2}\) and \(i_{13/2}\), respectively. The latter states are encountered, however, in weakly excited nuclei belonging to certain regions of the Mendeleev periodic table.

The energetic closeness of levels with a large difference in \(j\)-values indicates the presence of “isomeric islands” in the periodic table at those places where the number of protons or neutrons is odd and almost reaches a “magic number,”\(^{128,129}\) for example in the nearly filled shell 28–50, where the levels with angular momenta \(p_{1/2}\) and \(g_{9/2}\) are adjacent. The properties of such isomers have apparently been studied quite well at present. In the case of even-even nuclei, experiment has shown that the first excited level is almost always even and has \(J=2\).

The independent-particle model has proved useful in the classification of \(\beta\)-transitions.\(^{130-132}\) It is necessary, however, to make additional assumptions concerning the transformations of odd-odd nuclei, relating to the mechanism of coupling of the odd nucleons;\(^{131}\) this question will be considered in the following section.

The applicability of the independent-particle model with strong spin-orbit coupling to light nuclei is open to question. This problem has recently attracted the attention of investigators, especially in application to the study of the properties of “mirror” nuclei.\(^{57,133-137}\)

2. NUCLEAR MOMENTS

After Schmidt discovered\(^{138}\) that measurements of the magnetic dipole moments of nuclei containing an odd number of protons or neutrons are in good agreement with the values predicted on the basis of the one-particle model, many considerations were developed concerning agreements and deviations

from the so-called “Schmidt limits”*) \(^{139}\). The further accumulation of data \(^{140,141}\) not only confirmed Schmidt’s observations, but also clarified the nature of the deviations observed. Thus, for example, the moments of all nuclei with an odd number of nucleons (with the exception of \(\mathrm{H}^3\) and \(\mathrm{He}^3\)) lie within the “Schmidt limits” along two curves which are approximately parallel to the Schmidt curves \(^{140}\). If it is assumed that the state of the odd nucleon is determined by the nearest Schmidt curve, then the state so determined is, for the most part, in complete agreement with the predictions of the independent-particle model with spin–orbit coupling. If this model is taken seriously, then it is necessary to explain the observed deviations of nuclear moments from the Schmidt curves. It is possible, however, not to accept the independent-particle model unreservedly. Thus, for example, following Davidson \(^{142}\), one may suppose that the wave function of the ground state of a nucleus is a mixture of the wave functions of individual particles and the wave function of many particles corresponding to the same value of \(J\), but to an \(L\) differing by unity**). The magnetic moment of such a state will lie between the corresponding Schmidt limit and the opposite Margenau–Wigner limit \(^{139}\); it can be brought into agreement with the measured value by an appropriate choice of the percentage ratio of the separate wave functions. This scheme, as well as some others \(^{142—144}\), may prove useful in predicting some as yet unmeasured moments.

*) The Schmidt limits are the moments of a single neutron or proton with spin \(s=\frac{1}{2}\), orbital quantum number \(l\), and total angular momentum \(j\). They are equal to:

(a) for a proton

\[ \mu=\mu_p+j-\frac{1}{2} \quad \text{for } j=l+\frac{1}{2}; \]

\[ \mu=\frac{\left(j+\frac{3}{2}-\mu_p\right)j}{j+1} \quad \text{for } j=l-\frac{1}{2}; \]

(b) for a neutron

\[ \mu=\mu_n \quad \text{for } j=l+\frac{1}{2}; \]

\[ \mu=-\frac{\mu_n j}{j+1} \quad \text{for } j=l-\frac{1}{2}. \]

Here it is assumed that the gyromagnetic ratio for the orbital motion of the proton and neutron is equal to 1 and 0, respectively. The magnetic moments of the free proton and neutron are equal to:

\[ \mu_p=2.793, \]

\[ \mu_n=-1.913 \]

nuclear magnetons.

**) Parity in this case remains a quantum number, since the parity of the wave function of several particles does not necessarily correspond to the value of \(L\).

In the following sections we shall consider two hypotheses that have recently been proposed to explain deviations of the observed values of nuclear moments from the values predicted on the basis of the strict independent-particle model. Both hypotheses (from our point of view) have the merit that they are based on the shell model and use simple physical ideas (more or less easily calculable).

a) The asymmetric-core hypothesis

A survey of nuclear electric quadrupole moments leads to two striking conclusions:

1) There is a strong correlation, both in sign and in magnitude, between nuclear quadrupole moments and nuclear shell structure \(^{145,146}\).

2) Many quadrupole moments are too large (by an order of magnitude) in comparison with the value that could be produced by a single nucleon. (Nuclei with an odd number of neutrons also possess an electric quadrupole moment.)

These facts lead us to the assumption that there exists some other mechanism, in addition to the independent-particle model, which produces an asymmetric charge distribution in the core of the nucleus and thereby leads to the appearance of a quadrupole moment.

This mechanism was proposed by Rainwater \(^{147}\) and used by other investigators \(^{148-153}\). Essential for this mechanism is the polarization of the nuclear core by an asymmetrically situated odd particle, which turns out to be energetically more favorable than spherical symmetry of the core. The existence of an asymmetric nuclear core was at one time considered not only in connection with the large values of quadrupole moments, but also in connection with a number of anomalies observed in the isotope shifts of spectra \(^{154,155}\). A review by Kopfermann \(^{156}\) is devoted to this question. Convincing arguments in favor of the asymmetric-core hypothesis follow from consideration of the moments of isotope pairs differing from one another by two neutrons and having the same spin (for example, \(\mathrm{Cl}^{35,37}\), \(\mathrm{Cu}^{63,65}\), \(\mathrm{Ga}^{69,71}\), \(\mathrm{Br}^{79,81}\), \(\mathrm{Eu}^{151,153}\)). If one assumes the existence of a linear dependence between the magnetic dipole and electric quadrupole moments for each pair and extrapolates the quadrupole moment to zero (absence of core asymmetry), then the value of the magnetic moment, to within the experimental errors, coincides with the Schmidt limits calculated on the basis of the shell model.

A. Bohr \(^{151}\) used the conclusions following from the concept of an asymmetric core to calculate nuclear spins and magnetic moments. He considered the motion of a nucleon in an approximately constant field produced by an asymmetric core

with rotational and vibrational degrees of freedom, following the analogy with the structure of a molecule constructed in the form of a symmetric top. Under certain reasonable assumptions concerning the coupling between the nucleon and the core of the nucleus (at least for one excess nucleon or one “hole” in a filled shell), for the nuclear spin one obtains \(J=j\). The moment depends on the type of coupling assumed. In general Bohr obtained better agreement with experiment than follows from the Schmidt curves*).

b) “Quenching” of nuclear moments in nuclear matter

Another explanation of the deviations from the Schmidt limits was given by Bloch \(^{157}\), Deshalit \(^{158}\), and Miyazawa \(^{159}\). In their opinion, the magnetic moment of a nucleus in the ground state can be calculated on the basis of a strict independent-particle model, with, however, the assumption that the anomalous magnetic moments of the nucleons are partially quenched in nuclear matter. Then the magnetic moments are determined by the Schmidt limits, but with the following values:

\[ 1<\mu_p<\mu_p^f, \]

\[ 0>\mu_n>\mu_n^f, \]

where \(\mu_p^f\) and \(\mu_n^f\) are the moments of free nucleons**).

In the language of meson theory, the anomalous magnetic moments of nucleons are a consequence of the virtual emission and absorption of mesons. On the basis of such a model Miyazawa \(^{159}\) showed that the anomalous moments must be, at least partially, quenched in nuclear matter, since the Pauli principle forbids such processes of meson emission that lead to an increase in the number of recoil nucleons with momenta corresponding to states already occupied by other nucleons. An estimate of this effect cannot be given exactly because of the shortcomings inherent in modern meson theory (which cannot even explain the anomalous magnitude of the magnetic moments of free nucleons). Consequently, the results of Miyazawa’s calculations, which indicate a noticeable “quenching” of the anomalous moments, should be regarded only as a qualitative indication of the existence of this effect.

Villars and Weisskopf \(^{160}\) used a phenomenological approximation to this problem, in which they considered the principal

*) Bohr’s model is a specific method of adding the wave functions of many particles to the wave function of the ground state of the nucleus; a method which, in a much more general form, was used by Davidson \(^{143}\). However, from these two models different consequences follow concerning nuclear moments and nuclear asymmetry.

**) See the footnote on p. 615.

properties of the interaction of nucleons, proceeding from experiments with small energies. Their estimate of the “switching-off” effect is closely connected with the estimate of the “exchange magnetic moment” in nuclei\(^{161—166}\). They arrived at a very small “switching-off” effect, which is insufficient to explain the deviations from the Schmidt limits*).

It seems probable that the “switching-off” effect exists, but possibly to a considerably smaller degree than is required for explaining the observed moments. It is also clear that this conclusion does not exclude from consideration the asymmetric-core model; electric quadrupole moments and other effects still require explanation. At present it seems probable that a combination of the two models may provide an explanation of all the anomalies in the behavior of nuclear moments.

c) Anomaly of hyperfine structure

From the foregoing one may conclude that, if certain changes are introduced into the independent-particle model, it will be possible to substantiate the observed deviations of nuclear magnetic moments from the Schmidt limits. In order to choose between the various models on the basis of magnetic effects, it is necessary to observe certain other phenomena that depend on the details of the chosen model. Such a phenomenon, as Kopfermann\(^{167}\) and Bitter\(^{168}\) pointed out, is the anomaly of hyperfine structure, which depends on the spatial distribution of magnetic moments in the nucleus. The theory of the hyperfine-structure anomaly was developed by Bohr and Weisskopf\(^{169}\) and by Bohr\(^{152}\). Its applications to the existing experimental data were considered by Eisinger, Bederson, and Feld\(^{170}\).

The anomaly of hyperfine structure consists in the fact that the splitting of hyperfine-structure lines \(\Delta \nu\) differs, generally speaking, from the value that follows from the representation of the nucleus as a point magnetic dipole, i.e. \(\Delta \nu = \Delta \nu_{\mathrm{point}}(1+\varepsilon)\). Since \(\Delta \nu_{\mathrm{point}}\) depends on the wave function of the electrons, in the general case it is impossible to determine \(\varepsilon\) by measuring \(\Delta \nu\) for only one isotope. Since \(\Delta \nu_{\mathrm{point}} \sim (2J+1)g\), where \(J\) is the nuclear spin and \(g=\mu/J\), then for two isotopes of one element (under the assumption that the electron wave functions of both isotopes are identical)

\[ \frac{\Delta \nu_1}{\Delta \nu_2} = \frac{(2J_1+1)g_1}{(2J_2+1)g_2} (1+\Delta), \]

where \(\Delta \simeq \varepsilon_1-\varepsilon_2\) is a quantity characterizing the anomaly of hyperfine

*) It is necessary to note that effects of this type, which lead to “switching-off,” may also change the orbital gyromagnetic ratio \(g\), i.e. make the proton \(g\) not equal to unity and give the neutron a \(g\)-factor different from zero. However, these effects must be much smaller than the effect of spin “switching-off.”

structure, which can be obtained from independent measurements of the ratios \(\dfrac{\Delta\nu_1}{\Delta\nu_2}\) and \(\dfrac{g_1}{g_2}\).

The difference in the value of \(\varepsilon\) for two isotopes arises as a result of the difference in the spin and orbital components of the magnetic moments. The value \(\varepsilon\) can be calculated for various models \(^{152,169,170}\). The results of the calculations are compared with the experimental data in Table III \(^{170}\). We note that the small anomaly in the isotopes

Table III

Comparison of the observed and calculated values of the anomaly of hyperfine structure

Model 1. “Switched-off” moment with \(\Delta\mu_e = 0\); \(\Delta\mu'_s\)—from experimental data.
Model 2. The model of Feynberg and Davidson \(^{142}\).
Model 3. Bohr’s model with an asymmetric core \(^{152}\).

\(\mathrm{Li}^{6,7}\) (\(\sim 0,01\%\)) is probably due to the electronic and nuclear structure of these light nuclei, while the anomaly in the isotopes \(\mathrm{Tl}^{203,205}\) (\(0,01—0,02\%\)) is due to the difference in nuclear radii \(^{173,174}\).

From the data of Table III it is impossible to choose between the various models. The model of an asymmetric core agrees better with experiment for the potassium isotopes and worse for the rubidium isotopes. Somewhat surprising is the agreement of the results calculated according to models 1 and 2, which, however, may also be accidental for these nuclei. The general conclusion is that the data on the anomaly of hyperfine structure give strong confirmation of the validity of those nuclear models which, in their essential features, are based on the model of independent particles with spin-orbit coupling.

d) Some remarks concerning odd-odd nuclei

The coupling of two or more nucleons in one and the same shell was considered in the preceding section, where rules were also given concerning the order of succession of deep nuclear levels.

in the model with \(j\)-\(j\) coupling. These rules, apparently, are well confirmed at least for nuclei with \(A > 50^{126, 127, 175}\). The problem of coupling odd neutrons and odd protons continues to be a subject of study.

Nordheim\(^{131}\), on the basis of empirical data, mainly relating to \(\beta\)-decay schemes, formulated the following rules:

(1) The individual configurations of neutrons and protons in odd-odd nuclei are the same as in nuclei with odd mass number \(A\), with an equal number of nucleons in the odd group of particles.

(2) If the odd groups of neutrons and protons belong to different Schmidt groups, then their resultant spins are subtracted.

(3) If the odd groups of neutrons and protons belong to one and the same Schmidt group, then their spins are coupled so as to give a value greater than the minimum possible one.

Kurath\(^{176}\), basing himself on the law of a \(\delta\)-shaped, spin-independent interaction between odd nucleons, obtained these rules for the nucleons \(1p\), \(1d\), and \(1f\), namely: a) for neutron and proton \((j_1)'(j_2)'\), \(j = |j_1 - j_2|\), and b) for neutron and proton \((j_1)'(j_2)^{-1}\), \(J = j_1 + j_2 - 1\). For an interaction with finite radius, in rule a) the value \(J = j_1 + j_2\) becomes more probable, while rule b) apparently remains unchanged.

The known data concerning spins and magnetic moments (in nuclear magnetons) of odd-odd nuclei are collected in Table IV. There are also given the configurations of neutrons and protons expected from the shell model (columns 5 and 6), the values of nuclear spins calculated by Nordheim (column 7) and Kurath (column 8), and the magnetic moments (column 9), calculated on the basis of the strict independent-particle model with \(j\)-\(j\) coupling and under the assumption that the spins are equal to the measured values (column 3), while the proton and neutron configurations correspond to the data given in this same table (columns 5 and 6). It is necessary to note that, owing to the “switching-off” effect, the nuclear moments tend to vanish for odd protons and neutrons, especially if they are in analogous states\(^{177}\); therefore the deviations from the calculated values should be somewhat smaller for nuclei with odd \(A\).

On the whole, the agreement of the data is not too bad, with the exception of \(\mathrm{Na}^{24}\). (Incidentally, it is interesting to note that if, contrary to our shell model, one assigns to the proton the state \(d_{3/2}\), and to the neutron \(d_{5/2}\), then for \(\mathrm{Na}^{24}\) we obtain \(\mu = -1.79\).) Nevertheless, we must say that it is still premature to speak of the success of the \(j\)-\(j\)-coupling model in predicting the spins of odd-odd nuclei.

Spins and magnetic moments of odd–odd nuclei
Table IV

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