SPATIAL CHARGE DISTRIBUTION AND “ELECTRIC RADIUS” OF HEAVY NUCLEI

G. F. Drukarev

1. INTRODUCTION

At the present time, experimental data and theoretical calculations have accumulated that make it possible to refine the values of the “electric radii” of heavy nuclei and to clarify certain details of the spatial distribution of the nuclear charge.

To investigate the charge distribution one may use those phenomena in which the electromagnetic interaction of particles plays the principal role and in which the influence of the finite size of the nucleus is manifested sufficiently clearly.

This group of phenomena includes:

1) scattering of electrons with energy considerably exceeding the electron rest energy,
2) the Coulomb energy of heavy nuclei,
3) fine-structure effects of X-ray terms,
4) the isotope shift in atomic spectra,
5) the spectrum of X-rays emitted by negative μ-mesons in transitions between stationary states in heavy mesoatoms.

In experiments on electron scattering, two energy regions are distinguished: low energies, when the de Broglie wavelength is greater than the dimensions of the nucleus, and high energies, when the wavelength is smaller than or of the order of the dimensions of the nucleus. For the theoretical interpretation of an experiment, some analytic expression is chosen for the charge density \(\rho\), usually spherically symmetric, depending on one or several parameters, and the parameters are selected for which the best agreement between theoretical calculations and experiments is achieved. The theoretical interpretation also includes an estimate of the role of such factors as the electric quadrupole and magnetic dipole moments of the nucleus, radiative effects, etc.

Data on the scattering of low-energy electrons make it possible to determine one parameter characterizing the nuclear radius. Data

…on the scattering of high-energy electrons at the present time make it possible to determine two parameters: the nuclear radius and the thickness of the surface layer.

The Coulomb energy of heavy nuclei depends comparatively weakly on the form of \(\rho(r)\). Therefore, data on the Coulomb energy make it possible to determine only the nuclear radius.

Effects in atomic spectra, as it turns out, at present cannot be used to determine the nuclear radius. In the fine structure of X-ray terms, radiative effects play a large role, for which no accurate estimates are yet available for heavy elements. In the isotope shift, an important role is played by the compressibility of the nucleus, for which there are likewise no reliable estimates.

The study of the X-ray spectra of mesoatoms can in principle give information both about the nuclear radius and about the form of \(\rho\). However, from experiment there are as yet known only the energy of a single transition \(2P - 1S\) and the comparatively inaccurate value of the doublet splitting. From these data one can obtain information only about the nuclear radius.

Below are presented the results of some investigations of recent years on the effects listed.

2. THEORY OF ELECTRON SCATTERING BY THE NUCLEUS

Let us first give some information from the theory of scattering of electrons by a point charge \(Ze\). We shall use a system of units in which \(\hbar = c = 1\).

Represent the differential effective cross section in the form

\[ d\sigma = \sigma(\theta)\,d\Omega . \tag{2.1} \]

For \(\sigma(\theta)\) in the Born approximation one can obtain a simple analytic expression.

In the energy region of interest to us, \(E \gg m\), the cross section \(\sigma\) has the form

\[ \sigma_0 = \frac{Z^2 e^4 \cos^2 \dfrac{\theta}{2}} {4E^2 \sin^4 \dfrac{\theta}{2}}; \tag{2.2} \]

\(\sigma_0\) has its smallest value at \(\theta = \pi\) and increases monotonically as \(\theta\) decreases.

The Born approximation is valid only for small \(Z\). For large \(Z\) it leads to a large error. A specific feature of the extremely relativistic case under consideration is that if the Born approximation proves inapplicable at some energy, then it will not be applicable at all higher energies either.

When the scattering of fast electrons is treated exactly, \(\sigma\) is determined in the form (see, for example, \({}^{3}\))

\[ \sigma(\theta)=\left(1+\operatorname{tg}^{2}\frac{\theta}{2}\right)|f(\theta)|^{2}\quad (E\gg m), \tag{2.3} \]

where \(f(\theta)\) is expressed by the series

\[ f=\frac{1}{2ip}\sum_{j=\frac12}^{\infty}\left(j+\frac12\right)\left(e^{2i\delta_j}-1\right)\times \]

\[ \times\left(P_{j+\frac12}(\cos\theta)+P_{j-\frac12}(\cos\theta)\right); \tag{2.4} \]

\(p\) is the electron momentum, \(\delta_j\) is the phase, which is determined from the solution of the Dirac equation. In a spherically symmetric field \(V(r)\), the Dirac equation reduces to a system of two equations for the radial functions \(f_j\) and \(g_j\).

In the limiting case \(E\gg m\) these equations have the form

\[ \left. \begin{aligned} \frac{dg_j}{dr}+\frac{\varkappa}{r}g_j-(E-V)f_j&=0,\\ \frac{df_j}{dr}-\frac{\varkappa}{r}f_j+(E-V)g_j&=0, \end{aligned} \right\} \tag{2.5} \]

where

\[ \varkappa=\mp\left(j+\frac12\right)= \begin{cases} -(1+l) & \text{for } j=l+\frac12,\\ l & \text{for } j=l-\frac12. \end{cases} \tag{2.6} \]

In the case when scattering occurs by a point charge,

\[ V=-\frac{Ze^{2}}{r}. \]

The asymptotic expressions for \(g_j\) and \(f_j\) have the form

\[ \left. \begin{aligned} g_j&\sim \cos\left(pr+Ze^{2}\ln 2pr-\frac{l+1}{2}\pi+\delta_j^{(0)}\right),\\ f_j&\sim \sin\left(pr+Ze^{2}\ln 2pr-\frac{l+1}{2}\pi+\delta_j^{(0)}\right). \end{aligned} \right\} \tag{2.7} \]

For the phase \(\delta_j^{(0)}\) in this case there exists an analytic expression. Thus, the determination of \(\sigma(\theta)\) reduces to the summation of (2.4).

Let us note that direct summation of the series (2.4) is complicated by certain specific features of the phases of the Coulomb field. In the work of Yennie, Ravenhall, and Wilson\({}^{3}\), a method was proposed

computing \(f(\theta)\), which makes it possible to avoid these difficulties. At present there exist tables (Fishbach \(^{4}\)) and approximate formulas that make it possible to represent the tabular data in analytic form (Parzen and Weinright \(^{5}\)).

Let us determine what changes should be expected in the scattering pattern by an extended nucleus as compared with scattering by a point charge. Obviously, the greatest changes will be in the angular distribution of the scattered electrons in the region of large angles. This follows from the fact that the region of large angles corresponds to small impact distances, at which the difference between an extended nucleus and a point one is manifested. A general idea of the nature of the change in the angular distribution can be formed by using the analogy between the scattering of particles and the diffraction of waves. If, for example, diffraction of light by a drop of liquid takes place, the diffracted wave may be represented as the result of interference of waves emanating from various portions of the drop. In the case when the dimensions of the drop are comparable with the wavelength, the intensity of the diffracted wave changes very sharply when the direction changes. In certain directions, waves emanating from different portions of the drop will be weakened as a result of interference. In these directions, consequently, the intensity of the diffracted wave will be reduced.

Fig. 1. Angular distribution for \(Z=79\) and energy \(150\) MeV. \(1\)—point charge, \(2\)—uniformly charged sphere (exact calculation), \(3\)—uniformly charged sphere (Born approximation).

An analogous picture may also be expected in the scattering of electrons in the case when the electron wavelength is comparable with the dimensions of the nucleus: in the angular distribution of the scattered electrons, minima will be observed at definite angles (Fig. 1). In the case when the electron wavelength is large in comparison with the dimensions of the nucleus, the minima will practically not be noticeable.

In addition, one may expect an overall decrease in the number of scattered electrons in a given solid angle as compared with the case of scattering by a point charge, since the magnitude of the potential in the central part of the smeared-out charge is smaller than the magnitude of the potential near a point charge (Fig. 1).

Mathematically, the effect of the smearing of the charge is expressed in a modification of the expression \(\sigma(\vartheta)\). In the Born approximation this modification reduces to multiplying \(\sigma_0(\vartheta)\), defined by (2.1), by a certain factor

\[ \sigma=\sigma_0\cdot |F^2|, \tag{2.8} \]

where \(F\) is the nuclear form factor. It is related to the charge density \(\rho\) by the relation

\[ F=\frac{\int e^{i\mathbf{q}\mathbf{r}} \rho\,dr}{\int \rho\,dr}. \tag{2.9} \]

Here \(\mathbf{q}\) is the momentum transferred to the nucleus in scattering, with

\[ q=2p\sin\frac{\vartheta}{2}. \tag{2.10} \]

At low energies, in the case of a spherically symmetric \(\rho\), one may, with sufficient accuracy, take

\[ F=1-\frac{1}{6}q^2\langle r^2\rangle, \tag{2.11} \]

where

\[ \langle r^2\rangle=\frac{\int r^2\rho\,dr}{\int \rho\,dr}. \tag{2.12} \]

In this case the scattering pattern is determined by only one parameter, \(\langle r^2\rangle\).

At high energies the form factor is no longer determined by a single parameter, but depends essentially on the form of \(\rho\). For a uniform distribution \(\rho=\mathrm{const}\) for \(r\leq R\), the quantity \(\sigma(\vartheta)\) has sharp diffraction minima at definite values of \(qR\). If, however, \(\rho\) is represented by a function with a maximum at the center of the nucleus and a sharp falloff toward the edges, then the diffraction minima are smoothed out and become weakly pronounced.

In the exact theory there is no analytic expression for the form factor. By analogy with (2.8), one may introduce the quantity \(|F|^2\) as the ratio of the scattering cross section by an extended charge to the scattering cross section by a point charge. However, \(|F|^2\) must then be determined by numerical integration of the Dirac equation.

The problem reduces to the integration of (2.5), with \(V(r)\) for \(r>R\) given by the expression

\[ V=-\frac{Ze^2}{r}, \]

and for \(r \leq R\) it is related to \(\rho\) by Poisson’s equation

\[ \Delta V = -4\pi e \rho(r). \tag{2.13} \]

As a result of the fact that the field is not Coulombic in the region \(r \leq R\), an additional phase shift appears in comparison with the phase of the Coulomb field, so that

\[ \delta_j = \delta_j^{(0)} + \delta_j'. \tag{2.14} \]

The quantity \(\delta_j'\) is found by matching the solutions of equations (2.5) at the boundary of the nucleus for \(r = R\).

For electrons of low energies, of all the \(\delta_j'\) practically only \(\delta'_{1/2}\) plays a role. The additional phase shift \(\delta'_{1/2}\) at low energies is determined, as can be shown (see, for example, \(^{22}\)), by the quantity \(\langle r^2\rangle\). Thus the picture of electron scattering in the exact theory, just as in the Born approximation, is determined by only one parameter, \(\langle r^2\rangle\).

At high energies several phases \(\delta_j'\) play a role. The large number of numerical calculations carried out in recent years \(^{3,6,7}\) makes it possible to establish certain characteristic features of the angular distribution \(\sigma(\theta)\). Thus, for the case of a uniform charge distribution, or an almost uniform one with a rounded edge, \(\sigma(\theta)\) has diffraction minima at certain definite values of \(qR\). In this respect there is agreement with the conclusions made on the basis of the Born approximation. However, there is also an essential difference. The depth of the diffraction minima in scattering by heavy nuclei, according to exact calculations, is considerably smaller than according to the Born approximation (see Fig. 1). It is precisely here that the Born approximation requires the greatest correction. Since the depth of the minima depends on the character of the charge distribution inside the nucleus, the Born approximation cannot be used for interpreting data on the scattering of fast electrons by heavy nuclei.

Let us now dwell on those causes which produce a difference between the elastic scattering of electrons by a spherically symmetric charge considered above and the scattering of electrons by nuclei observed in experiment. These include the presence in the nucleus of an electric quadrupole and a magnetic dipole moment. In addition, owing to the finite energy resolution, what is in fact measured is the cross section not only of elastic scattering but also of “almost elastic” scattering, in which the electrons lose energy within the limits of the resolving power of the instruments. Consequently, when comparing theory with experiment it is necessary to take into account excitation of the nucleus by electrons within the limits of the resolving power and, in addition, radiative effects. The relative role of the effects listed has been estimated in the Born approximation by a number of authors.

The influence of the quadrupole moment was estimated by Schiff\(^8\). He comes to the conclusion that the square of the form factor averaged over all orientations of the nucleus, \((|F|^2)_{\mathrm{av}}\), even for such a strongly deformed nucleus as \(\mathrm{Ta}^{181}\), does not change appreciably near the diffraction minima.

The results of some as yet unpublished estimates concerning the influence of the quadrupole moment are also given in the review by Ford and Hill\(^1\).

The correction associated with the presence of a magnetic moment plays a role, at the energies under consideration, only for light nuclei. For heavy nuclei the influence of the magnetic moment may be neglected.

Inelastic scattering of electrons accompanied by excitation of the nucleus was investigated in the Born approximation by Smith\(^9\) for several simple models of the nucleus. For heavy nuclei his calculations have only illustrative significance. An indirect conclusion that in some cases the effect of excitation of the nucleus by electrons is relatively small can be drawn on the basis of experimental data.

Thus, for example, in one of Hofstadter’s works with collaborators\(^ {15}\), scattering by \(\mathrm{Pb}^{208}\) and \(\mathrm{Au}^{197}\) nuclei was compared. The first excited level in the \(\mathrm{Pb}^{208}\) nucleus has an energy of \(2.6\ \mathrm{MeV}\); at the resolving power of the analyzer used in the experiment, it was possible to isolate the elastic scattering. In the \(\mathrm{Au}^{197}\) nucleus the energy of the first level is less than \(100\ \mathrm{keV}\), and the analyzer could not separate elastically scattered electrons from those which were scattered inelastically, exciting the nucleus. However, the angular-distribution curves for \(\mathrm{Au}^{197}\) and \(\mathrm{Pb}^{208}\) differ very little. From this the authors concluded that the influence of inelastic processes is insignificant. There are, however, cases in which the processes of nuclear excitation in electron scattering play a comparatively large role. These are nuclei in which there exist rotational levels associated with the collective motion of nucleons. As the latest experiments of Hofstadter\(^ {16}\), as well as the as yet unpublished theoretical estimates mentioned in work\(^1\), show, in scattering by Ta, W, Th, and U nuclei the angular distribution is strongly smoothed and the diffraction structure is almost unnoticeable.

The radiation correction in the energy region under consideration and at the high resolving power achieved in the experiment is insignificant. It is almost the same at all angles and changes the magnitude of \(\sigma(\theta)\) by only a few percent.

There are still a number of effects whose influence under the conditions considered is negligibly small. These include, for example, dispersion scattering of electrons by the nucleus, which, strictly speaking, must be taken into account together with the potential scattering (the effect of dispersion scattering in the Born approximation was estimated by Schiff\(^ {10}\)); \(\beta\)-interaction, and others.

3. EXPERIMENTS ON THE SCATTERING OF FAST ELECTRONS AND THEIR THEORETICAL INTERPRETATION

The scattering of electrons of low energies was investigated in the experimental works of Lyman, Hanson, and Scott12 (energy 15.7 MeV, scattering in Cu, Ag, and Au), Hammer, Raka, and Pidd13 (energies 33 and 43 MeV, scattering in Sn and W), and also in the work of Hofstadter, Fechter, and McIntyre14 (energy 25 MeV, scattering in Au), devoted mainly to electrons of high energy. An interpretation of the results of the first work was given by Fishbach and Bitter11. The analysis reduces to determining an additional phase shift \(\delta'_{1/2}\) and selecting the corresponding \(\langle r^2\rangle\). The calculations are carried out under the assumption of a uniformly charged nucleus with radius \(R\). From the definition of the quantity \(\langle r^2\rangle\) there follows the relation

\[ \langle r^2\rangle=\frac{3}{5}R^2. \tag{3.1} \]

\(R\) may be called the equivalent radius. This is the radius of such a uniformly distributed charge which gives the same value of \(\langle r^2\rangle\) as the true distribution.

Table I gives the values of the equivalent radii obtained from the analysis of experiments12 and 13.

Table I

*) Unpublished data, cited in 1.

Let us now consider the works on the scattering of electrons of high energy, published by Hofstadter and collaborators in 195314, 195415, and 195616.

In the 1953 work, the scattering of electrons with energies 125 MeV and 150 MeV was investigated. The energy resolution was 1.5%. The angular distribution (in arbitrary units) was measured for scattering by Ta, Au, and Pb nuclei.

A theoretical interpretation of these experiments was given on the basis of the Born approximation by the authors, and also by Schiff. From the absence of sharp diffraction minima, expected in the Born approximation for a uniform or nearly uniform charge distribution, the conclusion was drawn that the charge density is strongly concentrated toward the center, approximately according to an exponential law.

However, after an exact calculation was made the basis of the interpretation, it turned out that this conclusion was incorrect.

In the 1954 work the authors introduced several improvements into the apparatus and improved the energy resolution. The angular distribution was measured at several energies for Au\(^{197}\) and Pb\(^{208}\). The experiments were performed for Au\(^{197}\) at 84, 126, 154, and 183 MeV. A typical curve \(\sigma(\theta)\) is shown in Fig. 2. Traces of diffraction structure are visible on the curve. The arrows indicate points whose positions correspond to smoothed diffraction minima. The positions of the first smoothed minima on all curves approximately satisfy the relation

\[ E \sin \frac{\theta}{2} \sim 57\ (E-\text{in MeV}). \]

For the second smoothed minima,

\[ E \sin \frac{\theta}{2} \sim 106 . \]

Analysis of the new experimental data was carried out on the basis of exact calculations by Yennie et al.\(^3\). Cross sections were calculated for the following forms of the density distribution: exponential

\[ \rho=\rho_0 e^{-\frac{r}{a}}; \]

Gaussian

\[ \rho=\rho_0 e^{-\left(\frac{r}{b}\right)^2}; \]

uniform distribution \(\rho=\mathrm{const};\ r<R\); Fermi

\[ \rho=\rho_0 \frac{1}{1+e^{K(r-B)}}, \]

so named because of its analogy with the Fermi distribution, and several others. The calculations were performed for an energy of 125 MeV for the Au\(^{197}\) nucleus at certain values of the parameters \(a, b, K, B\). With the aid of several interpolations, a recalculation was made to other energies and other parameter values. Comparison with experiment definitely indicates that the exponential and Gaussian distributions are unsuitable. The closest agreement between calculation and experiment is obtained for an almost uniform charge distribution with a rounded edge of the type shown in Fig. 3. The rounded edge corresponds to a surface layer in which the charge density falls to zero.

As calculations show, in the energy region under consideration the exact shape of the edge (i.e., the law of decrease of the density in the surface layer) turns out to be insignificant: only the thickness of the surface layer is important.

In the work of Ravenhall and Yennie\(^{6}\), a uniform charge distribution with a rounded edge was specially considered. The authors introduce two parameters characterizing such a distribution, namely the radius

\[ c=\frac{1}{\rho(0)}\int_{0}^{\infty}\rho(r)\,dr \tag{3.2} \]

and the thickness of the surface layer \(s\)

\[ s^{2}=-\frac{4}{\rho(0)}\int_{0}^{\infty}(r-c)^{2}\frac{d\rho}{dr}\,dr. \tag{3.3} \]

For a distribution of the Fermi type

\[ \frac{\rho}{\rho_{0}}=\frac{1}{1+e^{K(r-B)}} \tag{3.4} \]

the quantity \(c\) is the distance from the center of the nucleus to the middle of the surface layer, and \(s^{2}\) is proportional to the mean-square thickness of the surface layer. Up to terms \(\left(\frac{s}{c}\right)^{4}\), the mean-square radius \(\langle r^{2}\rangle\) is expressed in terms of the parameters \(s\) and \(c\) as follows:

\[ \langle r^{2}\rangle=\frac{3}{5}c^{2} \left[ \frac{1+\frac{5}{2}\left(\frac{s}{c}\right)^{2}} {1+\frac{3}{4}\left(\frac{s}{c}\right)^{2}} \right]. \tag{3.5} \]

If \(s/c\ll 1\), then approximately

\[ \langle r^{2}\rangle=\frac{3}{5}c^{2}. \tag{3.6} \]

In this case \(c\) coincides with the equivalent radius (see (3.1)).

It turns out that in the energy region under consideration the scattering cross section is practically completely determined by the parameters \(c\) and \(s\). The authors verified this for distributions with different forms of rounding for the gold nucleus. Up to angles \(\sim 105^\circ\), the angular distributions for different forms with the same \(s\) and \(c\) differ by no more than a few percent. The best agreement with experiment was obtained for the parameter values

\[ s=1.65\cdot 10^{-13}\ \text{cm};\qquad c=6.63\cdot 10^{-13}\ \text{cm}. \]

The corresponding value of the equivalent radius is

\[ R=1.20\cdot 10^{-13}A^{1/3}. \]

In the work of Hahn et al.\(^{16}\), angular distributions in scattering by the nuclei Ca, V, Co, In, Sb, Au, Bi were subjected to analysis.

at 153 and 183 MeV. The analysis was carried out on the basis of the Fermi charge distribution (3.4). For the gold nucleus, some other distributions were also considered, having the form of an almost uniform distribution with a rounded edge:

a) modified Gaussian

\[ \rho=\frac{\rho_0}{1+e^{\frac{(r-C)^2}{z_2}}}, \tag{3.7} \]

b) trapezoidal

\[ \left. \begin{aligned} \rho&=\rho_3; &&0<r<C-z_3,\\ \rho&=\rho_3\cdot\frac{C+z_3-r}{2z_3}; &&C-z_3<r<C+z_3,\\ \rho&=0; &&r>C+z_3, \end{aligned} \right\} \tag{3.8} \]

where \(\rho_2,\ \rho_3,\ C,\ z_2\) and \(z_3\) are parameters to be adjusted. The best agreement of the calculated angular distribution with the experimental one for the nucleus \(\mathrm{Au}^{197}\) at 183 MeV is attained for the forms of the charge distribution shown in Fig. 3. As is seen from the figure, the distributions are close to one another. In Fig. 4 the distributions for 153 and 183 MeV are shown for the optimum parameters of the Fermi distribution, and the experimental points are plotted.

Fig. 3. Charge distribution in the nucleus \({}_{79}\mathrm{Au}^{197}\), corresponding to the optimum parameters. Ordinates—charge density in units of \(10^{19}\) coulomb/cm\(^3\). Abscissae—distance from the center in units of \(10^{-13}\) cm.
1—Fermi distribution, 2—modified Gaussian, 3—trapezoidal.

Fig. 4. Angular distribution in scattering by the nucleus \({}_{79}\mathrm{Au}^{197}\). Solid curves—calculation for the optimum parameters; points—averages from five measurements.

The results of the analysis for Au and other nuclei are presented in the form of a table of optimal parameters. In so doing, to characterize the thickness of the surface layer the authors use, instead of \(s\), another parameter \(t\). This parameter is the distance over which \(\rho\) falls from \(0.9\rho(0)\) to \(0.1\rho(0)\).

The optimal values of the parameters \(c\), \(t\), and also of the equivalent radius \(R\), are given in Table II. The error in the radial parameters is estimated at \(\pm 2\%\), and in the layer thickness at \(\pm 10\%\).

Table II

4. COULOMB ENERGY OF HEAVY NUCLEI

The Coulomb energy of heavy nuclei can be determined from an analysis of nuclear masses based on the semiempirical formula for nuclear energy. This formula can be written in the form

\[ E=-a_1A+a_2A^{2/3}+a_3\frac{Z^2}{A^{1/3}}+a_4\frac{(N-Z)^2}{4A}. \tag{4.1} \]

The third term on the right-hand side of (4.1) is the Coulomb energy. The constant \(a_3\) depends on the form of the charge distribution in the nucleus \(\rho\) and on the radius of the nucleus. For a uniformly charged nucleus the relation is

\[ a_3=\frac{3e^2}{5R}. \tag{4.2} \]

The question of determining the coefficients \(a_i\) that give the best approximation of (4.1) to the experimental data has been the subject of many works. The most recent works pertaining to the region of heavy nuclei of interest to us were carried out by Green and Engler and by Green[^18].

In the work of Green and Engler the mass defect \(\Delta=M-A\) is represented as a function

\[ \Delta=\Delta_m+J\cdot(D-D_m)^2, \tag{4.3} \]

where \(D=N-Z\) is the neutron excess, \(\Delta_m\), \(D_m\), and \(J\) are certain

functions of the mass number, and the problem is posed of determining the optimal \(\Delta_m\), \(J\), and \(D_m\) for which the best agreement of (4.3) with experimental data is achieved.

As a first, and quite good, approximation, suitable for \(A>10\), the functions used are

\[ \left. \begin{aligned} \Delta_m^{(r)} &= \frac{(A-100)^2}{100}-64 \text{ thousand mass units},\\ J^{(r)} &= \frac{25}{A},\\ D_m^{(r)} &= \frac{0.4A^2}{A+200}, \end{aligned} \right\} \tag{4.4} \]

then corrections are fitted to them and the optimal \(\Delta_m^{(0)}\), \(D_m^{(0)}\), \(J^{(0)}\) are found.

On the other hand, it turns out to be possible to express \(\Delta_m\), \(J\), and \(D_m\) in terms of the coefficients \(a_1\), \(a_2\), \(a_3\), and \(a_4\) in (4.1). To do this, from (4.1) the mass defect is determined, which the authors denote by \(\Delta^w\), representing it in the form

\[ \Delta^w = \frac{1}{2}(A+D)\Delta_n + \frac{1}{2}(A-D)\Delta_H - a_1A + a_2A^{2/3} + \]

\[ + a_3\frac{(A-D)^2}{4A^{1/3}} + a_4\frac{D^2}{4A}. \tag{4.5} \]

where \(\Delta_n\) and \(\Delta_H\) are the mass defects of the proton and neutron, and the value \(D\) is found that satisfies the condition

\[ \left(\frac{\partial \Delta^w}{\partial D}\right)_{A=\mathrm{const}}=0. \tag{4.6} \]

One obtains:

\[ \left. \begin{aligned} \Delta_m^w &= -\left[ a_1-\frac{(3\Delta_n+\Delta_H)}{4} \right]A + \frac{(a_4+\Delta_n-\Delta_H)}{4}D_m + a_2A^{2/3}, \\ D_m^w &= A\left[ \frac{a_3}{a_4}A^{2/3} - \frac{\Delta_n-\Delta_H}{a_4} \right] \left[ 1+\frac{a_3}{a_4}A^{2/3} \right], \\ J^w &= \frac{a_4}{4A} \left( 1+\frac{a_3}{a_4}A^{2/3} \right). \end{aligned} \right\} \tag{4.7} \]

The procedure for determining \(a_1\), \(a_2\), \(a_3\), \(a_4\) reduces to selecting the values of these coefficients so that \(\Delta_m^w\), \(D_m^w\), and \(J^w\) from (4.7) are as close as possible to the optimal \(\Delta_m^{(0)}\), \(D_m^{(0)}\), and \(J^{(0)}\).

In Green’s work\(^{18}\), the method of least squares was used to select the quantities \(a_i\), and the best statistical values of \(a_i\) were found. Assuming the nuclear charge to be uniformly distributed, Green, using formula (4.2), determined that \(R=1.216\cdot A^{1/3}\cdot 10^{-13}\,\mathrm{cm}\). The probable error is estimated at \(1\%\).

5. EFFECTS IN ATOMIC SPECTRA

The effects in atomic spectra that are connected with the finite dimensions of the nucleus include changes in the magnitude of the doublet splitting of x-ray terms in comparison with the value for a point nucleus, and the isotopic shift. The first effect was considered in the works of Schawlow and Townes[^19][^20]. In the doublet \(2P_{1/2}—2P_{3/2}\), the influence of the smearing of the nuclear charge is reflected mainly in the level \(2P_{1/2}\). Therefore the problem reduces to considering the shift \(\Delta E\) of the energy of the \(2P_{1/2}\) level in the field of a smeared charge as compared with the point-field case.

Fig. 5. \(\dfrac{\Delta E}{h\Delta\nu}\) as a function of \(\left\langle r^{1.48}\right\rangle\) for \(Z=92\). The abscissa is \(\left\langle\left(\dfrac{r}{r_0}\right)^{1.48}\right\rangle\), where \(r_0=1.5\cdot 10^{-13}(238)^{1/3}\,\text{cm}\).

Perturbation theory makes it possible to obtain easily a simple expression for \(\Delta E\) (see, for example,[^22]):

\[ \Delta E=\frac{4\pi Ze^2C}{2\sigma(2\sigma+1)}\left\langle r^{2\sigma}\right\rangle . \tag{5.1} \]

Here \(C\) is the normalization coefficient in the wave function of the \(2P_{1/2}\) state,

\[ \sigma=\left[1-\left(\frac{Z}{137}\right)^2\right]^{1/2}. \tag{5.2} \]

For \(Z=92\) we have \(2\sigma\simeq 1.48\). According to (5.1), \(\Delta E\) is determined by a single nuclear parameter \(\left\langle r^{2\sigma}\right\rangle\).

However, for heavy nuclei perturbation theory is poorly applicable. A sufficiently accurate determination of \(\Delta E\) in heavy elements requires rather cumbersome calculations. The method for calculating \(\Delta E\) and the results for some \(Z\) are given in [^20].

In order to estimate the error introduced by using perturbation theory, Schawlow and Townes constructed a plot of the dependence of \(\dfrac{\Delta E}{h\Delta\nu}\) on \(\left\langle\left(\dfrac{r}{r_0}\right)^{1.48}\right\rangle\) for \(Z=92\), where \(h\Delta\nu\) is the magnitude of the doublet splitting, \(r_0=1.5\cdot10^{-13}A^{1/3}\) (in this case \(A=238\)) (Fig. 5). The dashed line shows the linear dependence following from (5.1) according to perturbation theory. The points are values obtained from numerical calculation.

As a result of their calculations, Schawlow and Townes come to the conclusion that agreement between theory and experiment, under the assumption of a uniform charge distribution, is achieved for

\[ R=(2.1\pm0.2)\cdot10^{-13}A^{1/3}. \]

This value of the radius is excessively large and indicates that,

besides the effect of smearing of the charge, other circumstances also play a role.

In searching for these circumstances, Shoupp and Townes drew attention to the fact that the magnitude \(\Delta E\) depends very strongly on \(Z\). (The calculated values of \(\Delta E\) in the region \(60 \leq Z \leq 95\) fit the empirical formula

\[ \frac{\Delta E}{h\Delta\nu}=De^{b(Z-60)}, \]

where \(b\) and \(D\) are numerical coefficients.) The same characteristic feature—a strong dependence on \(Z\)—is possessed by the effects of vacuum polarization and radiative effects. Calculation of the shift \(\Delta E\) in heavy elements under the action of these effects is a very difficult problem. At present, for large \(Z\), only the shift under the action of vacuum polarization has been calculated, in the work of Wichmann and Kroll \(^{21}\). The magnitude of the shift found by them is of approximately the same order as the shift due to smearing of the charge. But the sign of the polarization shift turns out to be opposite to that which would be needed in order to bring the value of the nuclear radius into agreement with the data of other experiments. The question of the magnitude of the shift under the action of other quantum-electrodynamical effects remains open. It is important to note, however, that the radiative shift should be opposite in sign to the polarization shift.

Let us turn to the isotopic shift. Consideration of this effect in connection with the possibility of determining the nuclear radius was carried out by Hill and Ford \(^{22}\) (see also the review by Striganov and Dontsov \(^{2}\)).

They note that the corrections to the perturbation-theory formula (5.1), which should be made on the basis of more exact calculations, depend almost neither on the shape nor on the radius of the nucleus. Therefore the authors believe that, to good accuracy, one may put

\[ \Delta E=\varphi(Z)\langle r^{2\sigma}\rangle, \tag{5.3} \]

where \(\varphi\) is a function only of \(Z\), independent of the nuclear radius, and (5.3) is applicable both to \(P\)- and to \(S\)-levels. Starting from this formula, they obtain the changes \(\delta\Delta E\) in comparing two isotopes:

\[ \delta\Delta E=\varphi(Z)\,\delta\langle r^{2\sigma}\rangle. \tag{5.4} \]

If \(\rho\) has the form

\[ \rho=\rho_{0}f\left(\frac{r}{r_{0}}\right) \tag{5.5} \]

and \(f\) is the same for all nuclei at a given \(Z\), then

\[ \frac{\delta\Delta E}{\Delta E}=2\sigma\frac{\delta r_{0}}{r_{0}}. \tag{5.6} \]

If, further, it is assumed that \(r_{0}\sim A^{1/3}\), then \(\frac{\delta r_{0}}{r_{0}}=\frac{\delta A}{3A}\) and, consequently,

\[ \delta\Delta E=\left(\frac{2\sigma}{3A}\right)\varphi(Z)\langle r^{2\sigma}\rangle. \tag{5.7} \]

However, as the authors note in connection with this conclusion, the assumption of the sameness of \(f\left(\dfrac{r}{r_0}\right)\) for isotopes is, strictly speaking, not valid because of the effect of nuclear deformation, while the assumption \(r_0 \sim A^{1/3}\) is not exact because of nuclear compressibility.

The compressibility effect cannot at present be determined with sufficient accuracy, and therefore data on the isotope shift cannot be used for a more or less precise determination of the nuclear radius. Rough estimates of compressibility \(^{22}\) lead to the value of \(R\) for a rectangular distribution equal to

\[ 1.6 \cdot A^{1/3} \cdot 10^{-13}\ \text{cm}. \]

A calculation neglecting compressibility for a uniformly charged nucleus gives \(R = 0.92 \cdot A^{1/3} 10^{-13}\ \text{cm}\).

6. X-RAY SPECTRUM OF \(\mu\)-MESONIC ATOMS

A slow \(\mu\)-meson, as is known, can be captured into one of the “orbits” near the nucleus and form a \(\mu\)-mesonic atom for some time. Capture into a \(K\)-“orbit” by means of a series of radiative and nonradiative transitions (Auger effect) occurs, according to estimates by Fermi and Teller \(^{23}\), in \(10^{-13}\)–\(10^{-14}\) sec, whereas the decay time of the \(\mu\)-meson is \(\sim 21 \cdot 10^{-8}\) sec, and the time of nuclear capture for large \(Z \simeq 80\) is \(\sim 7 \cdot 10^{-8}\) sec.

Wheeler \(^{24}\) showed that the transitions of the \(\mu\)-meson during capture that occur between states with large quantum numbers are mainly nonradiative, while transitions belonging to states with small quantum numbers occur with the emission of x rays.

Since for the \(\mu\)-meson the radius of the “orbit” is approximately 200 times smaller than the radius of the corresponding electronic orbit, the extent of the nuclear charge must have a large influence on the spectrum of x rays emitted by \(\mu\)-mesons in heavy elements.

Among the experimental investigations of x rays emitted by \(\mu\)-mesons, the most accurate and systematic are the experiments of Fitch and Rainwater \(^{25}\), in which the energy of the x rays was determined with an accuracy of 1%.

Table III gives the energies of the \(2P - 1S\) transition in different substances. There, for comparison, are also given theoretically calculated transition energies for a point nucleus.

The theoretical interpretation of experiments on \(\mu\)-mesonic x rays is given in the works of Fitch and Rainwater \(^{25}\), Cooper and Henley \(^{26}\), Wheeler \(^{27}\), Hill and Ford \(^{28}\).

As estimates show, at large \(Z\) the main effect causing the displacement of the \(\mu\)-meson energy levels relative to

values for a point nucleus is the smearing of the charge. The problem is thus reduced to solving the Dirac equation for a \(\mu\)-meson in the field of a nucleus with a smeared charge.

Table III

Fitch and Rainwater determined, on the basis of the uniform-distribution model, the radii of certain nuclei for which the calculated value of the \(2P - 1S\) transition energy agrees with the measured value. The results are given in Table IV.

Table IV

Detailed calculations for the nucleus \(\mathrm{Pb}^{208}\) were carried out by Hill and Ford. They set themselves the task of calculating, for various laws of nuclear-charge distribution \(\rho\), the energy levels \(1S\), \(2S\), \(2P\), and \(3D\), and of choosing the parameters in the analytic expressions for \(\rho\) so that the difference of the levels \(2P_{1/2} - 1S\) corresponded to the experimentally found value \(6.02\) MeV. Representing the density \(\rho\) in the form

\[ \rho = \rho_0 f(x), \tag{6.1} \]

where \(x = \dfrac{r}{r_0}\), and \(r_0\) is a parameter characterizing the extent of the charge, Hill and Ford considered many families \(f(x)\), whose form varied from exponential to rectangular.

The Dirac equation was integrated numerically for the levels \(1S\), \(2S\), \(2P_{1/2}\), \(2P_{3/2}\). For the levels \(3D_{3/2}\) and \(3D_{5/2}\), the shift (relative to the positions for a point nucleus) was determined by perturbation theory. In addition, in their calculations Hill and Ford took into account corrections arising from the following effects.

1) Vacuum polarization. For light nuclei this effect makes a much larger contribution to the level shift than does the finite size of the nucleus. For heavy nuclei, on the contrary, this is a small correction. An estimate of the polarization correction for several levels in the field of the Pb nucleus was given by Hill and Ford.

2) Other radiative effects are estimated by Hill and Ford by analogy with the calculation of the radiative shift for hydrogen.

3) Discreteness of the nuclear charge. This effect was considered by Cooper and Henley. To estimate it they calculated the interaction of the $\mu$-meson with a proton lattice in which the protons are arranged symmetrically in spherical layers. The number of protons in a layer and the distance between layers are chosen so that the change of the potential along the radius gives no shift of the energy levels as compared with their positions for a uniform distribution of continuous charge. According to Cooper and Henley’s estimate, for the Pb nucleus the level shift caused by charge discreteness in such a model is no more than $0.1\%$.

4) Polarization of the nucleus by the $\mu$-meson. This effect was also considered by Cooper and Henley. Starting from a uniformly charged nucleus model, they estimated the addition to the energy of the $\mu$-meson in the $1S$ state in the second approximation of perturbation theory. Without going into the details of the calculations, we give the result. The shift of the $1S$ level due to polarization of the lead nucleus is, according to the maximum estimate: for $\mathrm{Pb}^{207}$ — $3.7\%$ of the energy of the $2P-1S$ transition, for $\mathrm{Pb}^{208}$ — $2.7\%$.

5) Inaccuracy in determining the mass of the $\mu$-meson. The corresponding correction was estimated by Hill and Ford, as well as by Cooper and Henley. The latter indicate that increasing the mass of the $\mu$-meson by $5\%$ increases the calculated energy of the $2P-1S$ transition in lead by $0.2\%$.

A summary of the corrections, with an indication of the upper limit of possible errors for the $2P-1S$ transition, is given in Table V.

Table V

As can be seen from Table V, the observed value of the transition energy \(2P-1S\), equal to \(6.02\) MeV, corresponds to the value \(5.95 \pm 0.7\) MeV, associated specifically with the finite-size effect. In addition to these corrections, there also exists a number of others which were not taken into account in the calculations of Hill and Ford either because they are altogether too small, or because they are absent owing to the peculiarities of the \(Pb^{208}\) nucleus. The former include: screening by atomic electrons; the non-electromagnetic interaction of the \(\mu\)-meson with the nucleus, etc. The latter include the correction associated with the quadrupole moment of the nucleus.

Let us now consider some results of the calculations of Hill and Ford. Figure 6 gives the dependence of the energies on the nuclear radius for a uniform distribution. As was to be expected, the energy of the \(1S\) level depends most strongly on the nuclear radius.

Figure 7 shows the dependence on the nuclear radius (for a uniform distribution) of the transition energies \(2P_{1/2}-1S\), \(2P_{3/2}-1S\), \(2S-2P_{1/2}\), and \(2S-2P_{3/2}\), and of the doublet splitting \(2P_{3/2}-2P_{1/2}\).

A transition energy \(2P_{3/2}-1S\) equal to \(6\) MeV corresponds, for a uniformly charged nucleus, to

\[ R = 1.17\, A^{1/3}\cdot 10^{-13}\ \text{cm}. \]

It is important to note that, at a fixed transition energy \(2P_{3/2}-1S\), the energies of other transitions change quite noticeably depending on changes in the shape \(f(x)\).

As one example, the family \(f(x)\) of the form

\[ f_n(x)=\sum_{k=0}^{n}\frac{x^k}{k!}e^{-x}. \tag{6.2} \]

was considered. When \(n\) is varied from \(0\) to \(\infty\), the family \(f_n\) changes from exponential to rectangular. By choosing the parameters \(r_0\) and \(\rho\) so that the transition energy \(2P_{3/2}-1S\) is fixed, and varying \(n\), one can study the dependence of the energies of other transitions on the form \(f\).

Figure 8 shows the change in the transition energies \(2S-2P_{1/2}\), \(2S-2P_{3/2}\), and of the doublet splitting \(2P_{3/2}-2P_{1/2}\) at an unchanged energy \(2P_{3/2}-1S\). The energy \(2P_{3/2}-1S\) changes by \(15\%\) when \(n\) is varied from \(0\) to \(\infty\).

Consequently, on the basis of data on the energies of other transitions, besides \(2P_{3/2}-1S\), one can draw conclusions not only about the magnitude of the nuclear radius, but also about the form of the charge distribution. However, such data are not yet available at present.

In conclusion, in Table VI on p. 411 we shall give a summary of the values of the radius of heavy nuclei in the region of Pb, determined from various sources.

Fig. 6. Dependence of the energy levels of the \(\mu\)-meson in the field of a uniformly charged Pb nucleus on the nuclear radius. Ordinates—energy in MeV. Abscissae—radius in units of
\[ \lambda = 1.87 \cdot 10^{-13}\ \text{cm}. \]

Fig. 7. Dependence of the transition energies \(2P—1S\), \(2P—2S\), and of the doublet splitting on the radius of a uniformly charged nucleus. Ordinates—energy in MeV. Abscissae—radius in units of
\[ \lambda = 1.87 \cdot 10^{-13}\ \text{cm}. \]
\(D\)—doublet splitting \(2P_{3/2}—2P_{1/2}\), increased by a factor of 10.

Fig. 8. Dependence of the energies of the transitions \(2S—2P_{1/2}\), \(2S—2P_{3/2}\), and of the doublet splitting \(2P_{3/2}—2P_{1/2}\) on the form of the charge density \(f(x)\), at fixed transition energy \(2P_{1/2}—1S\), \(6.00\) MeV. Ordinates—energies in MeV. Left scale—for the transitions \(2S—1P\). Right scale—for the doublet splitting. Abscissae—the parameter \(n\). \(n=\infty\) corresponds to a rectangular distribution, \(n=1\) to an exponential one.

Table VI

CITED LITERATURE

I. REVIEWS

1. D. Hill and K. Ford, Ann. Rev. of Nuclear Sci. 3, 25 (1955).
2. Striganov and Dontsov, UFN 55, 315 (1955).

II. ELECTRON SCATTERING

a) Theory and calculations

1. D. Jennie, D. Ravenhall, R. Wilson, Phys. Rev. 95 291 (1954).
   (There are also references there to some earlier works not mentioned here.)
2. H. Feshbach, Phys. Rev. 88, 295 (1953).
3. G. Parzen, T. Wainwright, Phys. Rev. 96, 188 (1954).
4. D. Ravenhall, D. Jennie, Phys. Rev. 95, 239 (1954).
5. G. Brown, L. Elton, Phil. Mag. 46, 164 (1955).
6. L. Schiff, Phys. Rev. 92, 987 (1953).
7. I. Smith, Phys. Rev. 95, 271 (1954).
8. L. Schiff, Phys. Rev. 98, 756 (1955).
9. F. Bitter, H. Feshbach, Phys. Rev. 92, 837 (1953).

b) Experiment

1. Lyman, Hanson and Scott, Phys. Rev. 84, 626 (1951).
2. Hammer, Raka, Pidd, Phys. Rev. 90, 341 (1953).
3. R. Hofstadter, H. Fechter, I. McIntyre, Phys. Rev. 92, 978 (1953).
4. R. Hofstadter, B. Hahn, A. Knudsen, I. McIntyre, Phys. Rev. 95, 512 (1954)
5. B. Hahn, D. Ravenhall, R. Hofstadter, Phys. Rev. 101, 1131 (1956).

III. COULOMB ENERGY OF HEAVY NUCLEI

1. A. Green, N. Engler, Phys. Rev. 91, 40 (1953).
2. A. Green, Phys. Rev. 95, 1006 (1954).

IV. EFFECTS IN ATOMIC SPECTRA

1. A. Schawlow, G. Townes, Science 115, 284 (1952).
2. A. Schawlow, G. Townes, Phys. Rev. 100, 1273 (1955).
3. Wichmann, Kroll, Phys. Rev. 101, 843 (1956).
4. D. Hill, K. Ford, Phys. Rev. 94, 1630 (1954).

V. μ-MESON X-RAYS

1. E. Fermi, E. Teller, Phys. Rev. 72, 399 (1947).
2. J. Wheeler, Rev. Mod. Phys. 21, 133 (1949).
3. V. Fitch, I. Rainwater, Phys. Rev. 92, 789 (1953).
4. L. Cooper, E. Henley, Phys. Rev. 92, 801 (1953).
5. Wheeler, Phys. Rev. 92, 812 (1953).
6. D. Hill, K. Ford, Phys. Rev. 94, 1630 (1954).