ELECTRIC MOMENT OF THE NUCLEUS

E. L. Feinberg, Moscow

1. Introduction

Every new experimental discovery of properties of the atomic nucleus arouses in us a quite understandable interest. It is not difficult to imagine what significance for this theory is possessed by the new nuclear characteristic discovered and already rather thoroughly investigated during the last two years—the electric quadrupole moment of the nucleus.

It is known that the influence of the nucleus on the spectrum of the atom is determined by definite values of its charge, mechanical and magnetic moments, and mass. The charge \(Ze\), the most essential characteristic of the nucleus, determines the number of electrons in the shell and the positions of the separate terms. The so-called fine structure of these terms, caused by the different orientations of the electron spins relative to the mechanical moment of the whole shell, does not provide any new characteristics for the study of the nucleus. However, the resolving power of spectral instruments has made it possible to detect the complex structure of individual lines, split into a series of components, the distances between which have to be counted in hundredths and thousandths of an angstrom, or, in wave numbers, in thousandths of reciprocal centimeters. It is not difficult to see that here one must work almost at the limit of resolving power, which in the best instruments is of the order of \(10^6\). Indeed, in the visible region precisely at \(\Delta \lambda \sim 10^{-2}\ \text{Å}\) and \(\lambda \sim 6 \cdot 10^3\ \text{Å}\)

\[ \frac{\lambda}{\Delta \lambda} \sim 6 \cdot 10^5. \]

Some of these structures turned out to be ordinary multiplets with only very small splitting; however, the others, whose properties could not be explained by the usual considerations in such cases and to which the term “super-

“fine structure,” had an entirely different origin and made it possible to determine the mechanical (spin) and magnetic moments of the nucleus. Detailed information pertaining to this can be found in the book by S. E. Frisch1; however, we must reconstruct several basic propositions.

The energy of interaction of magnetic moments depends on their mutual orientation. Therefore the energy of a term depends on the different orientation of the nuclear spin \(I\) (and, consequently, of the nuclear magnetic moment \(\mu\), parallel or antiparallel to it) relative to the mechanical moment of the electron shell \(J\) (and of the magnetic moment of the shell \(\mu_e gJ\) parallel to it, where \(\mu_e\) is the Bohr magneton and \(g\) is the Landé factor). Since, for a given electronic configuration, several such orientations are possible in general, this naturally leads to a splitting of the energy level into sublevels. The nucleus, through its magnetic moment, affects the magnitude of the splitting. As for the number of components, it is determined by the number of possible relative positions of the vectors \(I\) and \(J\), which is equal to \(2I+1\) if \(I<J\), and \(2J+1\) if \(I>J\). Therefore, when \(J>I\), the number of components immediately makes it possible to judge the value of \(I\). However, in the analysis of terms with large \(J\), we rarely detect the splitting because of its smallness.

In such a case \(I\) has to be determined from the ratio of the distances between the components or from their intensities, which, of course, is less visual and not so convenient. These ratios do not depend on \(\mu\).

We see that analysis of the hyperfine structure makes it possible to determine both the spin (mechanical moment) of the nucleus \(I\) and its magnetic moment \(\mu\). There remains the fourth characteristic mentioned by us—the mass of the nucleus. The addition of one neutron to a nucleus gives an isotope whose electron shell remains exactly the same. However, the increase in mass nevertheless has an effect, above all in that the distribution of kinetic energy between the nucleus and the electron shell changes. This is the same effect that causes the change in the Rydberg constant in the transition from hydrogen to ionized helium. However, there too it explains (and rather well) the “isotopic shift” of lines in light elements. The observed isotopic shift in heavy elements must be understood in some other way. The proposed explanation reduces the entire effect to a change in the dimensions of the nucleus: the addition of a neutron pushes apart the boundaries of the nucleus, and, naturally, in the region newly occupied by it, where the Coulomb field of the nucleus previously prevailed, the potential will now be different, since the electric charge inside the nucleus will, of course, also be redistributed. In the unaffected region the field remains as before. This explanation can be accepted only as a qualitative one. The most serious objection to it is that such an effect should have been especially noticeable in light elements (thus, the transition

from the isotope ${}^{1}_{1}\mathrm{H}$ to the isotope ${}^{1}_{2}\mathrm{H}$ doubles the volume of the nucleus), whereas for them, as we have seen, taking this effect into account would be superfluous.

In one way or another, the value of the mass of the nucleus manifests itself in the isotopic displacement of lines or of entire systems of lines, whose direction the proposed considerations explain.

2. Experimental Part

The four characteristics enumerated exhausted the description of the properties of the nucleus up to 1935. The regularities observed by that time for the intensities and intervals of the hyperfine structure were explained quite satisfactorily by selecting suitable values of $I$ and $\mu$, which proved entirely reasonable. True, such a satisfactory situation did not arise at once, and above all as a result of overcoming experimental difficulties. These difficulties are quite understandable. First of all, the picture may be completely distorted by self-reversal of lines. Thus, for example, in the simplest case, when passing through a layer of unexcited atoms, a line may be strongly absorbed and, understandably, chiefly in its central part. As a result, observation will reveal two lines instead of one. Another hindrance is thermal motion, which gives a Doppler broadening of lines, often exceeding the distance between the components of the hyperfine structure. Thus even obtaining such a comparatively simple characteristic as the number of components is no simple matter.

Only in 1930, thanks to the use of a cooled discharge tube with a hollow cathode, onto which a thin layer of the metal under investigation is deposited by cathode sputtering, was it possible to obtain reliable photographs of the hyperfine structure. As a result, the determination of the magnetic and mechanical moments of the nuclei of metals became secure from the experimental side. However, as late as September 1933, when at the nuclear conference in Leningrad the question arose whether one could expect that the spin of radioactive elements would be measured in the near future, the answer given was negative[^2]. The contemporary method required no less than 1 g of the substance under investigation. But already a year later[^3] it proved possible to measure the mechanical moment of protactinium, the investigator having only 5, and then a further 3 mg of the preparation. This became possible thanks to an improvement of the apparatus, which now made it possible to study the hyperfine structure of the rarest elements.

This apparatus[^4a] (Fig. 1) consists of two hollow metal cylinders $A$ and $K$, serving as anode and cathode. One bottom of the anode $S$ is transparent; observation is made through it, while there is no other bottom. Opposite it is the opening of a small aluminum cylinder $C$, closed at the opposite end. This aluminum cylinder is tightly inserted into $K$, and only thin through channels $L$ connect the left part of the apparatus with the right. Both cylinders

are surrounded by a cooling jacket (not shown). Through the inlet opening \(M_1\) and the outlet \(M_2\), continuous circulation of the noble gas is maintained. The substance under investigation is placed in the recess of the cylinder \(C\); it may also be taken in the form of compounds. When the discharge passes, the ions of the noble gas filling the vessel, striking the specimen, knock atoms of the substance under investigation out of it and excite them.

Fig. 1

Owing to the sharp-current arrangement described, these atoms remain all the time in the cavity of cylinder \(C\) and are not washed out by the gas stream. They can only diffuse weakly into it. Therefore it is sufficient to have a very small quantity of substance in order to carry out observations for many hours under steady conditions. Another convenience is that the element under investigation may be taken in the form of any compounds, and non-metals can also be studied. Thanks to the continuous flushing of the vessel with noble gas, all impurities—even the weakest traces of foreign elements—are removed.

3. The essence of the observed effect

The use of this apparatus enabled Schuler and Schmidt in 1935 to record the hyperfine structure of the rare-earth element europium \({}_{63}\mathrm{Eu}^{46}\) for the line \(\lambda = 5831\) Å. The photograph shows that in fact there are two structures, belonging to the two isotopes of europium: \({}_{63}^{151}\mathrm{Eu}\) and \({}_{63}^{153}\mathrm{Eu}\). It is not difficult to distinguish the isotopic shift from the hyperfine splitting. As Pauli noted as early as 1924, in a magnetic field the Zeeman effect of two lines (or of two systems) must occur independently if these are two lines of two different atoms (isotopic shift). The Zeeman splittings of the components of the hyperfine structure, however, are in a definite mutual relation.

The choice of the element \({}_{63}\mathrm{Eu}\) was very fortunate because, in this element, the unfilled inner shell gives the entire electronic configuration a large spin moment and, consequently, a large \(J\) (\(^{7}/_{2}\) and larger). Since the number of lines (6) is smaller than

if \(2J+1\), then we may conclude that it is determined by the value \(I\), equal to \(5/2\). Thus \(I\) is determined exactly, and we can test how the ratio of the intervals justifies the predictions of the theory. The theory gives a very simple rule. The term energy is proportional to \(\mu g\,\mu_e \cdot \cos(I,J)\), where the expression \(\cos(I,J)\) must be replaced by its quantum-mechanical expression

\[ \frac{F(F+1)-I(I+1)-J(J+1)}{2IJ}. \]

Here \(F\) is the total mechanical moment of the system nucleus + electron shell. Subtracting from one another such expressions for \(F\), differing from one another by unity, we obtain the distances between the components of the hyperfine structure. It is easy to see that the successive distances are related as \(F : (F+1) : (F+2) : \ldots\). This simple interval rule, or, as it is called, the cosine law, serves to determine \(I\) in those cases when \(J<I\) and \(I\) cannot be determined from the number of components. Studying the intervals between the components of the Eu structure, one may notice that they do not obey the interval rule. The difference, although small, clearly lies outside the limits of experimental error. It turned out that the deviation is systematic and moreover proportional to \(\cos^2(I,J)\). If the sublevel energy is written in the form

\[ E=E_0+a_1\cos(I,J)+a_2\cos^2(I,J), \]

where \(E_0\) is the energy of the center of gravity of the structure, \(a_1\cos(I,J)\) is the energy of interaction of the magnetic moments of the nucleus and the shell, and the last term is the observed perturbation, then the coefficient \(a_2\) is a quantity of the order of \(10\%\) of \(a_1\).

How exactly the perturbation follows the law \(\cos^2(I,J)\) is clearly seen from measurements of the structure of another element, \({}^{175}_{71}\mathrm{Cp}\) (the same as \(\mathrm{Lu}\))\({}^{4в}\). In Fig. 2 the splitting of the term \({}^{3}D_3\) is shown: on the right, the measured splitting; on the left, that calculated from the cosine law under the condition that the lower and upper components coincide with the observed ones. The discrepancy \(\Delta\) (in \(10^{-3}\ \mathrm{cm}^{-1}\)) is incomparably greater than the possible experimental error (\(\sim 1\text{–}2\cdot10^{-3}\ \mathrm{cm}^{-1}\)). On the other hand, if the perturbation

\[ E_2=a_2\cos^2(I,J) \]

is plotted as a function of \(E_0+E_1\), then the curve of Fig. 3 is obtained, where the circles denote the measured quantities, and the solid line is drawn according to the formula \(E_2=a_2\cos^2(I,J)\)

Fig. 2

with an appropriate choice of the constant multiplier \(a_2\). It is difficult to demand a more complete agreement.

However, deviations from the interval rule could have been noticed earlier as well. They were, perhaps, only not so obvious. It always seemed natural to suppose that they are a consequence of the perturbation by neighboring terms, i.e., a consequence of the fact that, when the interaction of the magnetic moments of the nucleus and the shell is superposed, the wave function of the first approximation will contain, as components, the wave functions of many states.

Fig. 3.

Indeed, if before the interaction is superposed the system was in the state \(\psi_0\) with energy \(E_0\), while other states \(\psi_1, \psi_2 \ldots\) with correspondingly other energies \(E_1, E_2 \ldots\) were possible, then any new function \(\psi\) can be expanded in a series in the functions \(\psi_i\), since mathematically they form a complete system

\[ \psi = \sum_i C_i \psi_i . \]

Perturbation theory shows that when the perturbation \(V\) [in the present case proportional to \(\mu \cos (I,J)\)] is small, all coefficients \(C_i\), except \(C_0\), are small and equal to \(\dfrac{V_{0i}}{E_0 - E_i}\), where

\[ V_{0i} = \int \psi_0^* V \psi_i\, d\tau, \]

i.e., the function \(\psi\) is essentially reduced to the unperturbed \(\psi_0\), but other states are also admixed in some amount.

Accordingly, the energy of the perturbed system changes not simply by the mean value of the perturbation energy

\[ V_{00} = \int \psi_0^* V \psi_0\, d\tau, \]

but small terms are also added,

\[ \frac{|V_{0i}|^2}{E_0-E_i} \]

from all \(i\). Of course, these additions are small if \(E_0-E_i\) is large, i.e. the remaining terms lie far away; however, it is essential that they are precisely proportional to \(\cos^2(I,J)\), since

\[ |V_{0i}|^2 = \int \psi_0^* b\mu \cos(I,J)\psi_i\,d\tau \cdot \int \psi_0 b^*\mu \cos(I,J)\psi_i^*\,d\tau \]

(\(b\) is the proportionality factor). Thus the observed perturbations may be explained as a trivial admixture of neighboring terms. What is new is that, using Eu as an example, one can prove the untenability of such an explanation in the present case.

The quantity \(|V_{0i}|^2\) is proportional not only to \(\cos^2(I,J)\), but also to \(\mu^2\). However, for the two isotopes \(^{151}\mathrm{Eu}\) and \(^{153}\mathrm{Eu}\), for which, as we have seen, the total splitting of the fine structure is not the same (Fig. 3), the mechanical moments are equal (the number of lines is the same). Consequently, their \(\mu\)’s are simply different. Since the distances between components are in the ratio \(F:(F+1):\ldots\), and the proportionality factor for the two elements differs only by virtue of the difference in \(\mu\), it is not difficult to see that the total splittings are in the ratio

\[ \frac{\sum \delta\nu^{151}}{\sum \delta\nu^{153}} = \frac{\mu^{151}H(0)[F+1)+(F+2)+\ldots]}{\mu^{153}H(0)[(F+1)+(F+2)+\ldots]} = \frac{\mu^{151}}{\mu^{153}} \]

[here \(H(0)\) is the magnetic field at the center of the atom produced by the electrons of the shell]. But from analysis of the spectrum it follows that

\[ \frac{\sum \delta\nu^{151}}{\sum \delta\nu^{153}}=2.2. \]

Consequently, the anomalous perturbation found in \(^{151}\mathrm{Eu}\) should be \((2.2)^2\sim 5\) times greater than in \(^{153}\mathrm{Eu}\). Measurement of the constants \(a_2\) for both isotopes, however, shows that in the best case they are equal, and may even be \(^{151}a_2<{}^{153}a_2\).

Thus it may be regarded as established that a new perturbation has been found, proportional to \(\cos^2(I,J)\) and inexplicable from the point of view of the usual considerations. The supposition immediately arose that it is caused by an asymmetry in the distribution of electric charge in the nucleus, by its deviation from spherical symmetry,

4. The Influence of the Asymmetry of the Nuclear Charge on Terms

It is known that the field of any aggregate of charges situated inside some region can, outside this region, be represented as the sum of the fields of a point charge, a dipole, a quadrupole, an octupole, etc. This is a consequence of the mathematical fact that, for a point \(P\) at a distance \(R\) \((x,y,z)\) from some point \(O\) (let it be the origin of coordinates), near which arbitrary charges \(e_i\) are grouped [let their coordinates be \(r_i(\xi_i,\eta_i,\zeta_i)\)], the potential of these charges

\[ \varphi=\sum_i \frac{e_i}{\sqrt{(x-\xi_i)^2+(y-\eta_i)^2+(z-\zeta_i)^2}} =\sum_i \frac{e_i}{R_i} \]

can be expanded in a triple Taylor series in powers of \(\xi_i,\eta_i,\zeta_i\)

\[ \varphi=\sum_i \frac{e_i}{R_i} =\sum_i e_i\left\{ \xi_i\left(\frac{\partial \frac{1}{R_i}}{\partial \xi_i}\right)_{\xi_i=\eta_i=\zeta_i=0} +\right. \]

\[ \left. +\eta_i\left(\frac{\partial \frac{1}{R_i}}{\partial \eta_i}\right)_{\xi_i=\eta_i=\zeta_i=0} +\zeta_i\left(\frac{\partial \frac{1}{R_i}}{\partial \zeta_i}\right)_{\xi_i=\eta_i=\zeta_i=0} \right\}+ \]

\[ +\frac{1}{2}\sum_i e_i\left\{ \xi_i^2\left(\frac{\partial^2 \frac{1}{R_i}}{\partial \xi_i^2}\right)_0 +\ldots+ \zeta_i^2\left(\frac{\partial^2 \frac{1}{R_i}}{\partial \zeta_i^2}\right)_0+ \right. \]

\[ \left. +\xi_i\eta_i\left(\frac{\partial^2 \frac{1}{R_i}}{\partial \xi_i\partial \eta_i}\right)_0 +\ldots \right\} +\ldots =\varphi_0+\varphi_1+\varphi_2+\ldots \]

The first term here is the Coulomb potential, the second the dipole potential, the third the quadrupole potential, etc. They decrease with \(R\), respectively, as

\[ \frac{1}{R},\quad \frac{1}{R^2},\quad \frac{1}{R^3},\ldots . \]

In the case of a continuous distribution of charge we must replace summation over \(i\) by integration. In doing so, \(e_i\) is replaced by \(\rho\,dv\)—the density of protons in the nucleus, which is simply expressed through the wave function of the nucleus.

The interaction of an element of nuclear charge with an element of charge of the electron shell must be integrated, on the one hand, over the entire volume of the nucleus, as a result of which we obtain the action of the field of the whole nucleus on an element of shell charge, and, on the other hand, over the distribution of electrons in the shell. The Coulomb term then gives the usual value of the center of gravity of the term, while the dipole term vanishes (if only because the mean value of any

from the coordinates, by virtue of the axial symmetry of the nucleus, is zero). There remains the quadrupole field of the nucleus, which is of interest to us. Suppose that the vector \(I\), determining the symmetry axis of the nucleus, is directed along the \(z\)-axis. Then the mean values of the mixed terms (\(\xi\eta,\eta\xi\), etc.) are equal to zero. There remains

\[ \varphi_2=\int \rho\,d\xi d\eta d\zeta\,\frac{3}{2R^5} \left\{x^2\xi^2+y^2\eta^2+z^2\zeta^2-\frac{1}{3}R^2r^2\right\}. \]

Taking into account that

\[ \overline{\xi^2}=\overline{\eta^2}=\frac{r^2-\overline{\zeta^2}}{2}, \]

and

\[ x^2+y^2=R^2-z^2, \]

we obtain

\[ \frac{3}{2R^5}\{\ \}= \frac{1}{4R^5}(3\overline{\zeta^2}-r^2)(3z^2-R^2) \]

or, if \(\vartheta\) is the angle between \(R\) and the \(z\)-axis, and \(\theta\) is the angle between \(r\) and the \(z\)-axis,

\[ \varphi_2=\frac{1}{4R^3} \left[\int \rho r^2(3\cos^2\theta-1)\,dv\right] (3\cos^2\vartheta-1). \]

This expression has the usual form of the quadrupole potential

\[ \varphi_2=\frac{eq\cdot P_2(\cos\vartheta)}{R^3}, \]

where \(eq\) is the quadrupole moment, and \(P_2(\cos\vartheta)=3\cos^2\vartheta-1\) is the second Legendre polynomial in \(\cos\vartheta\). As was to be expected, in the case of a spherically symmetric charge distribution in the nucleus,

\[ \overline{\cos^2\theta}=\frac{1}{3},\qquad q=0. \]

To obtain the shift of the term \(\Delta E\), it is necessary further to multiply the expression written above by

\[ -e\,|\psi(x_1,y_1,z_1,\ldots,x_n,y_n,z_n)|^2\,dx_1\ldots dz_n, \]

where \(\psi(x_1,\ldots,z_n)\) is the wave function describing the behavior of all \(n\) electrons of the shell, and to integrate over all their coordinates. It is quite evident that in the case of a spherically symmetric shell, since \(\overline{\cos^2\vartheta}=\frac{1}{3}\), \(\Delta E=0\), even if the quadrupole moment of the nucleus differs from zero. This exactly corresponds to the experimental result, according to which the \(S\)-level does not experience anomalous perturbation. If \(J\) is parallel to \(I\) (and hence ...

read, and the \(z\)-axis; \(m_J=J\)), then integration over the shell gives a certain factor at

\[ \overline{\left(-\frac{eP_2(\cos\vartheta)}{R^3}\right)}_{m_J=J} \]

where the bar denotes multiplication by \(|\psi|^2\) and integration over the whole volume.

If the vectors \(I\) and \(J\) are not parallel, but, say, are rotated in the plane \(zox\) by an angle \(\alpha\) relative to one another, then one may proceed as follows. Let \(I\) still coincide with the \(z\)-axis. For integration over \(R\) we introduce a new coordinate system by means of the relations

\[ \begin{aligned} x&=x'\cos\alpha-z'\sin\alpha,\\ y&=y',\\ z&=x'\sin\alpha+z'\cos\alpha, \end{aligned} \]

the axis \(oz'\) of which coincides with the axis of symmetry of the shell \(J\). Substituting \(x,y\), and \(z\) into the expression \(\varphi_2\), we obtain

\[ \frac{3}{2R^5}\{\ \}= \frac{r^2}{8R^3}(3\cos^2\vartheta'-1)(3\cos^2\theta-1)(3\cos^2\alpha-1), \]

where \(\vartheta'\) is the polar angle in the new coordinate system. But now integration over the electron shell gives for \(3\cos^2\vartheta'-1\) the same value

\[ \overline{\left(-\frac{eP_2(\cos\vartheta)}{R^3}\right)}_{m_J=J} \]

as was previously obtained for \(3\cos^2\theta-1\). Consequently, rotation of \(J\) through an angle \(\alpha\) relative to \(I\) leads to multiplication by the factor

\[ \frac{1}{2}(3\cos^2\alpha-1). \tag{\(\lambda\)} \]

Of fundamental importance is the circumstance that here \(\cos\alpha\) enters squared.

Of course, such a classical treatment, which gives a visual picture of the possible nature of the observed effect, must be replaced by an exact quantum-mechanical consideration. This consideration was carried out in 1931 by Kramers\(^5\) for the general case of a perturbation of any nature depending on the mutual orientation of two “vectors” \(L\) and \(S\). He showed that if a system, without taking this perturbation into account, possesses a spatial degeneracy of two types (for example, spin and orbital degeneracy of the electron shell in Russell–Saunders coupling, or orbital degeneracy of the shell and the nucleus), then inclusion of the perturbation, different for different combinations of the unperturbed states (classically

we would say: depending on the mutual orientation of the vectors), leads to a multiplet splitting which does not shift the center of gravity of the term and which may be represented as a sum of shifts of the components of the term, quite analogous to the classical expansion of such a perturbation in spherical functions. The coefficients of the various terms of this sum can, of course, be determined only by taking into account the concrete character of the interaction; but the relative arrangement of the components follows from the most general symmetry considerations. The first term of this expansion gives the “quantum-mechanical law of cosines” and formally may be obtained from the classical expansion in spherical functions by replacing \(\cos (L,S)\) by

\[ \frac{R(R+1)-L(L+1)-S(S+1)}{2LS}, \]

where \(R=L+S\); the second term finds an important application in the problem of interest to us. It gives (if one sets \(R=F\), \(L=I\), \(S=J\))

\[ \Delta E = A\, \frac{\dfrac{3}{8}C(C+1)-I(I+1)J(J+1)} {I(2I-1)J(2J-1)}, \]

where

\[ C=F(F+1)-I(I+1)-J(J+1). \]

In agreement with the correspondence principle, for large \(I\) and \(J\), when the units may be neglected and \(C\) goes over into

\[ F^{2}-I^{2}-J^{2}=-\frac{\cos(IJ)}{2IJ}, \]

this formula takes the form

\[ \Delta E \to A\,\frac{1}{8}\,[3\cos^{2}(IJ)-1]. \]

Hence there follows, so to speak, the “quantum-mechanical law of the square of the cosine.”

Therefore, if one assumes that the observed perturbation is due to the asymmetry of the charge of the nucleus, its electric quadrupole moment, which we are entitled to conclude since the dependence on the angle \((I,J)=\alpha\) has the same form \((\lambda)\), then to the multiplier \(A\) one should assign the value

\[ eq\left(\overline{\frac{-eP_{2}(\cos\vartheta)}{R^{3}}}\right)_{m_J=J}, \]

and finally the expression for \(\Delta E\) takes the form\({}^{6}\)

\[ \Delta E = eq\left\{ \overline{ -\frac{e}{R^{3}}\,(3\cos^{2}\vartheta-1) } \right\}_{m_J=J} \times \]

\[ \times \frac{ \dfrac{3}{8}C(C+1)-\dfrac{1}{2}I(I+1)J(J+1) }{ IJ(2I-1)(2J-1) }, \]

where

\[ C=F(F+1)-I(I+1)-J(J+1), \]

and

\[ q=\overline{r^{2}\bigl(3\cos^{2}\theta-1\bigr)_{m_I=I}}\;{}^{*}). \]

If one singles out the term proportional to the quantum-mechanical square \(\cos(IJ)\), i.e. \(C(C+1)\), then, indeed, the total energy of the term will have the form

\[ E=E_0+a_1C+a_2C(C+1), \]

where \(a_1\) includes the magnetic interaction of the nucleus and the electron shell. In experiment we measure \(a_2\). Consequently, one can find the electric moment of the nucleus \(q\) from the formula

\[ q=-\frac{a_2\cdot 8IJ(2I-1)(2J-1)} {3e^2\left(\dfrac{3\cos^{2}\vartheta-1}{R^3}\right)_{m_J=J}}. \]

Here it is evident that, for comparison with experiment, it is necessary to know not only \(I\) and \(J\), but also the wave function of the shell, in order to carry out the integration. This is the essential difficulty. It turns out, however, that the inaccuracy in the knowledge of the complete function does not have a significant effect. Casimir, in treating the first data on Eu, proceeded from the assumption of Russell–Saunders coupling. In doing so, in order to determine certain coefficients he used part of the experimental data. It turned out that the remaining figures then fit the theoretical ones very accurately. Schuler, however, carries out calculations under the assumption both of Russell–Saunders coupling and of \((j,j)\) coupling. For \({}^{201}_{80}\mathrm{Hg}\) the following values are obtained for the nuclear moments:

\[ \left\{ \begin{array}{llll} \text{from the term }{}^{3}P_{1} & \text{R.–S.} & q=0.54\cdot10^{-24}\ \mathrm{cm}^{2} & (j,j)\quad 0.69\cdot10^{-24}\ \mathrm{cm}^{2},\\ \text{” ” }{}^{3}P_{2} & & q=0.44\cdot10^{-24}\ \mathrm{cm}^{2} & \quad 0.51\cdot10^{-24}\ \mathrm{cm}^{2}. \end{array} \right. \]

As we see, the fluctuations are small, and the final value accepted by the author, \(q=0.5\cdot10^{-24}\ \mathrm{cm}^{2}\), cannot but be regarded as quite plausible.

The correctness of the physical interpretation of the effect, i.e. the explanation of the observed perturbations proportional to \(\cos^{2}(I,J)\) by the influence of the asymmetry of the nuclear charge, is apparent not only from the fact that the dependence on the angle \((I,J)\) proves to be correct and that the \(S\)-level is indeed not perturbed, but above all from the fact that—

\({}^{*})\) The bar above, here and above, denotes multiplication by \(\rho\) and integration over the volume (this time of the nucleus).

calculation of \(q\) from different terms gives one and the same value. Thus \(q\) is a characteristic not of the electron shell, but of the nucleus. Particularly convincing in this respect is the example of bismuth. For it the following values of \(q^{4}\) are obtained:

\[ \begin{aligned} \text{neutral }{}^{209}_{83}\mathrm{Bi}\ \text{term }6p^{3}\,{}^{2}D_{3/2} &\quad q=-0.41\cdot 10^{-24},\\ \text{term }(6p_{3/2}7s)_{2} &\quad q=-0.36,\\ \text{singly ionized Bi}\quad (6p_{1/2}7s)_{2} &\quad q=-0.43,\\ (6p_{3/2}7s)_{1} &\quad q=-0.39,\\ \text{doubly ionized Bi}\quad 7p^{2}P_{3/2} &\quad q=-0.44 . \end{aligned} \]

The value inferred from this, \(q=-0.4\cdot 10^{-24}\ \mathrm{cm}^{2}\), is very convincing.

At present we have the following values of the nuclear electric moments (Table 1).

Among the important results confirming the above, one may include the data of Stanley Smith and Kowi8, who investigated the hyperfine structure of 25 terms of Tl II and did not observe any deviation from the sum rule, which is also understandable, since the mechanical moment of the nuclei \({}^{203}\mathrm{Tl}\) and \({}^{205}\mathrm{Tl}\) is equal to \(\frac{1}{2}\). Conversely, in Bi, Ester Minz9 found deviations from the Landé rule following the law \(\cos^{2}\), in agreement with the foregoing, but did not measure \(q\) accurately. Deviations from the interval rule for the hyperfine structure were also observed by Ebbe Rasmussen10 for Co \((I=7/2)\). It is significant that the data for In, obtained by the American physicists Bacher and Tambulian, coincide with Schuler’s data.

5. Analysis of the Results

In order, on the basis of the measured values of \(q\), to proceed to conclusions about the distribution of the electric charge, i.e. of the protons, in the nucleus, it is necessary to recall the meaning of \(q\)

\[ q=\overline{r^{2}(3\cos^{2}\theta-1)}_{m_I=I}. \]

First of all, concerning the sign. If the charge were spherically symmetric, \(\overline{\cos^{2}\theta}=\frac{1}{3}\), and \(q\) would be zero. With a flattened charge, i.e. if the concentration of charge in the plane perpendicular to the axis of rotation is greater than along the axis \(I\), the angles \(\theta\) close to \(\frac{\pi}{2}\) have greater weight, when \(\cos^{2}\theta\) is small, and consequently \(q\) has a negative sign. Conversely, a positive sign indicates an elongation of the charge along the axis.

Of course, this does not mean that the whole nucleus is so deformed. The arrangement of the neutrons in no way affects the electric

TABLE 1

field. As is evident from the table, for the overwhelming majority of elements the distribution of protons is flattened, and in fact quite considerably.

If we turn to a quantitative interpretation of the quantities \(q\), then here we obtain a number of very important results.

As is well known, for a long time in nuclear theory there persisted the same methods of interpretation that we apply to the electron shell of the atom. It was believed that nuclear particles could be represented as arranged in completed groups or layers, outside which there remain particles that have not fitted into them. At the same time (and this is the most essential point) it was assumed that these additional particles could be regarded as moving in the field of forces of the filled shell, which they disturb only slightly. Under such an interpretation, for example, the addition of one particle to a nucleus consisting of filled layers and therefore having no mechanical moment should set the additional particle in motion around the core of the same type as the motion of a valence electron in a monovalent atom. This point of view was refuted by Bohr (1936), who pointed out that the peculiarities of the nuclear state lead to essential differences, by virtue of which the interaction of an additional particle with the core substantially affects the whole nucleus, and one cannot speak of the preservation of any core whatever. The peculiarities of nuclear interaction as compared with Coulomb electrostatic interaction consist above all in the fact that this interaction is short-range. In contrast to the Coulomb interaction, which decreases inversely proportionally to the distance between the particles, the interaction of a proton and a neutron, or of two neutrons, is very large while these particles are at a small distance (of the order of \(10^{-13}\,\mathrm{cm}\)) from one another, but rapidly disappears when the distance exceeds this “radius of force.” Therefore, if in an atomic shell each electron interacts simultaneously with all the other electrons and the nucleus and, naturally, is little affected by the influence of one more added electron, then a nuclear particle in each of its positions interacts only with a small number of its neighbors. The addition of one new particle substantially affects its state. We see that the measure here may be the ratio of the radius of action of the forces \(a\) to the mean interparticle distance \(R\). For the electrons of an atom \(\frac{R}{a} \ll 1\), for nuclear particles \(\frac{R}{a} \sim 1\). In this respect the nucleus has a strong resemblance to a drop of liquid. In such a drop, van der Waals forces act between the molecules, decreasing very rapidly with distance, and here too \(\frac{R}{a} \sim 1\).

If from this point of view one analyzes the values of \(q\), the following is found. In \({}^{153}_{63}\mathrm{Eu}\), \(q\) reaches \(3.2\cdot 10^{-24}\). This nucleus may be regarded as formed by adding a proton to the preceding nucleus \({}^{152}_{62}\mathrm{Sm}\), which, as is usually the case for nuclei with an even number of protons and an even number of neutrons, gives no fine structure (\(I=0\)). Therefore it may be represented as a completed core. Then, according to the old interpretation, the addition of a proton should give its motion around the core on

on average distances which in the best case (\(\theta=0\)) have the order

\[ \sqrt{\frac{3.2}{2}\cdot 10^{-12}} \simeq 1.3\cdot 10^{-12}\ \text{cm}. \]

Meanwhile, the radii even of much heavier nuclei (\({}_{82}\mathrm{Pb}\)) are approximately equal to \(7\cdot 10^{-13}\) cm. Therefore such representations are entirely unsuitable. It should be assumed that the entire core of the nucleus participates in producing the electric moment. Unfortunately, precisely this result cannot serve as a decisive proof that the added proton causes a strong deformation of the whole core. The fact is that, although \({}^{152}_{62}\mathrm{Sm}\) apparently does not give hyperfine splitting and therefore we cannot directly measure its electric moment, as early as 1934 Schuler, measuring the isotopic displacement of \({}^{150}\mathrm{Sm}\) and \({}^{152}\mathrm{Sm}\), found an anomaly which he explained by the fact that, in the transition from \({}^{150}\mathrm{Sm}\) to \({}^{152}\mathrm{Sm}\), a substantial rearrangement of the nucleus takes place. Therefore it is possible that already \({}^{152}\mathrm{Sm}\) has a large electric moment, which cannot be detected by the ordinary method because \(I=0\), and that the addition of a proton, giving \({}^{153}\mathrm{Eu}\), does not cause a very large increase of it. However, it remains a fact that the addition of two neutrons to \({}^{151}_{63}\mathrm{Eu}\) manifests itself not in their moving along some orbits without deforming the core, but in the fact that within the core the protons are still more elongated along the axis and increase the electric moment of the nucleus twofold (from \(1.5\cdot 10^{-24}\) to \(3.2\cdot 10^{-24}\)). No less clearly, \({}^{175}_{71}\mathrm{Cp}\) shows that the electric moment is not determined by the “orbital” position of any one proton.

However, the remaining elements give \(q\)’s which can be explained by superposing the distribution of the added particle, which does not disturb the symmetric core. The mean distances obtained under such an interpretation are quite reasonable.

Thus, representing the nucleus in the form of a drop of proton-neutron liquid, one may imagine that a newly added particle “polarizes” this liquid. If a neutron is added, it will move outside, but to preserve equilibrium the protons previously present in the nucleus will shift toward the axis, which will increase \(q\). From this it is possible to understand why the addition of two neutrons to \({}^{151}\mathrm{Eu}\) increases \(q\) twofold. On the other hand, if the special stability of the \({}_{82}\mathrm{Pb}\) nucleus is interpreted as an indication of its symmetry, then one can understand the negative \(q\), i.e. the oblateness of the electric charge in \({}_{83}\mathrm{Bi}\), which differs from it only by the addition of one proton.

We thus see that the use of the values of the electric quadrupole moment of the nucleus may help in constructing a picture of the structure of the nucleus. It is annoying, of course, that at present we are able to measure charge asymmetry only for nuclei giving hyperfine structure. It is possible that, when the nature of the isotopic displacement becomes clearer, the study of the influence on it of the electric moment (of the type that may be assumed for Sm) will remove this limitation. However, be that as it may, the asymmetry of the electri-

...of the electric charge is a correction which, to a considerable extent, reduces to the correction that the Coulomb interaction gives to the basic nuclear interaction determining the general picture of the structure of the nucleus. Therefore a more subtle analysis of the nature of the electric moment will become possible only when at least this basic interaction and the basic structure of the nucleus have become sufficiently clear to us. For the present, this nuclear characteristic can serve only for very preliminary considerations of the type set forth above.

REFERENCES

1. S. E. Frisch, Atomic Nuclei and Spectra.

2. The Atomic Nucleus, Collection of Reports of the First All-Union Conference on Questions of the Atomic Nucleus, GTTI, 1934.

3. Zeeman’s Festschrift, Collection dedicated to the jubilee of Zeeman, The Hague, 1936.

4. Schüller and collaborators: a) Z. Physik, 93, 611, 1935; b) Z. Physik, 94, 457, 1935; c) Z. Physik, 95, 265, 1936; d) Z. Physik, 98, 239, 430, 1936; e) Z. Physik, 100, 113, 1936; f) Z. Physik, 102, 708, 1936; g) Z. Physik, 104, 468, 1936; h) Z. Physik, 105, 168, 1937; i) Z. Physik, 103, 443, 1936.

5. Kramers, Proc. of the Section of Sciences K. Akademie van Wetenschappen, Amsterdam, 34, 965, 1931.

6. Casimir, Physica, 2, 719, 1935.

7. Bacher & Tamboutian, Phys. Rev., 50, 1096, 1936.

8. Stanley Smith and J. Convey, Canad. Journ. Res. (A), 14, 129, 1936; cited according to Phys. Ber., 18, 66, 1937.

9. Estermann and U. Mintz, Journ. Franklin Inst., 222, 613, 1936.

10. Ebbe Rasmussen, Z. Physik, 102, 229, 1936.

1. S. E. Frisch. ↩