Electron Scattering and the Structure of Nuclei*

R. Hofstadter

Contents

		Page
I.	Introduction	693
II.	Theory of scattering	695
III.	Various phenomena in scattering	703
IV.	Experimental part	714
V.	Results	723
VI.	Neutrons	744
VII.	Applicability of electrodynamics	746
VIII.	Comparison with other measurements of nuclear sizes	748
IX.	Summary	753
X.	Conclusions	756

I. Introduction

Only a few years ago, the principal information about the geometrical features of nuclear structure was obtained from comparisons of the energies of mirror nuclei, from capture cross sections (and total cross sections) for fast neutrons, from binding energies entering into the semiempirical Weizsäcker formula, and, in the case of the heaviest elements, from energies and lifetimes for $\alpha$-decay. All these methods lead to the same values of nuclear radii for a charged sphere, which was always regarded as a suitable model of the nucleus. The results obtained are expressed by the well-known formula for the radius of a homogeneous sphere

\[ R = r_0 A^{\frac{1}{3}} \cdot 10^{-13}\ \text{cm}. \tag{1} \]

In what follows we shall measure all distances in units equal to $10^{-13}\ \text{cm}$ (calling this unit a fermi). This formula gives, for example, that the boundary of the nuclear sphere in the case of gold is at a distance of $8.45$ fermis from the center of the nucleus, if for the quantity $r_0$ one adopts the well-agreed value close to $1.45$ fermis. Such a model leads to the same mass density for all nuclei, namely:

\[ \rho_M = \frac{A}{\left(\frac{4}{3}\right)\pi R^3} \frac{\text{nucleons}}{(\text{fermi})^3} = \frac{1}{\left(\frac{4}{3}\right)\pi r_0^3} \frac{\text{nucleons}}{(\text{fermi})^3} = 0.080\,\frac{\text{nucleons}}{(\text{fermi})^3} \tag{2} \]

and to a variable charge density in nuclei:

\[ \rho_C = \frac{Ze}{\left(\frac{4}{3}\right)\pi R^3} = e\,\frac{Z}{A}\, \frac{1}{\left(\frac{4}{3}\right)\pi r_0^3} = \frac{Z}{A}\cdot 0.080\, \frac{\text{proton charges}}{(\text{fermi})^3}. \tag{3} \]

* R. Hofstadter, Electron Scattering and Nuclear Structure, Phys. Rev. 28, 214 (1956).

These formulas, presented in Fig. 1, a and b, give the relative sizes and shapes of some nuclei.

Work carried out during the last several years by Lyman et al.,[^1] Hofstadter et al.,[^2–^4] Pidd et al.,[^5] and Fitch and Rainwater[^6] on the scattering of electrons and μ-mesoatoms has shown that the radii of heavy elements determined by these methods are approximately 20% smaller than the radii following from formula (1) with \(r_0 = 1.45\) fermis. The results obtained with μ-mesons indicate that the radii of light elements also have smaller radii, satisfying the same formula (1) with \(r_0 = 1.20\) fermis.

At the same time, Cooper and Henley[^7] showed that the data on light mirror nuclei can be explained if smaller values of the radii are likewise assumed. Subsequently this conclusion met with certain objections. The smaller nuclear radii obtained by the indicated methods are called “electromagnetic” radii.

Fig. 1. a) Density of nuclear matter for a homogeneous sphere (formula (2)). b) Charge density in nuclei for a homogeneous spherical model (formula (3)).

In the absence of a better name, the radii following from formula (1) may be called “nucleon” radii, since these larger values of the radii have been obtained from experiments on the interaction of nucleons with nuclei.

A great deal of work on the study of electron scattering was carried out at Stanford University during 1953–1956. These investigations have been a source of information on the charge density in nuclei from the proton to uranium. At present it seems timely to provide a review of the known data. Such a review will make it possible to bring together in one place the scattered material, so that researchers dealing with more general methods of determining nuclear radii may have the radii of “electron scattering” and the charge density in nuclei for comparison with their own data. The purpose of this review is the corresponding consolidation of the available data, including some unpublished work. This review cannot be regarded as a final summary of data on the sizes and shapes of nuclei obtained by the method of electron scattering. Planned work on electron scattering at high energies will undoubtedly change some of the known conclusions and lead to new, unexpected results. However, the emergence of significant contradictions seems unlikely, and this makes the review advisable.

The review contains no chronological or comprehensive consideration of work performed by other laboratories in the field of electron scattering, but the author hopes that the review offered here will serve as an incentive for those working in this and related areas of physics to collect data on nuclear radii and perhaps even to encourage new experiments.

The recently published review of data on the distribution of nuclear charge by Ford and Hill[^9] is one of the steps in this direction.

II. THEORY OF SCATTERING

a). Scattering by a point charge

At the basis of all phenomena of elastic scattering of charged particles lies the famous Rutherford formula. This formula

\[ \sigma(\vartheta)=\frac{z^2 Z^2 e^4}{16 E^2}\frac{1}{\sin^4 \frac{1}{2}\vartheta} \tag{4} \]

gives the dependence of the differential cross section \(\sigma(\vartheta)\) for the scattering of a moving point charge \((ze)\) with kinetic energy \(E\), by an immobile point charged center, such, for example, as a heavy nucleus \((Ze)\), which is the source of a strong electric field; \(\vartheta\) is the polar scattering angle. It was shown that Rutherford’s formula (4), obtained by means of classical mechanics, remains valid also in quantum mechanics \({}^{10}\).

The Rutherford scattering law describes the scattering of \(\alpha\)-particles and protons of medium energies. Therefore formula (4) is not relativistic and takes into account neither the spins of the colliding particles nor their possible identity.

The relativistic scattering of Dirac particles by a point nucleus, for example electrons, was considered by Mott in a well-known paper \({}^{11}\). In this case it is assumed that the scattered particle (electron) has spin (and a Dirac magnetic moment), whereas the scattering center (nucleus) has neither spin nor magnetic moment. Mott obtained an expression for the cross section of elastic scattering in the form of a series and also gave an approximate formula suitable for nuclei satisfying the inequality

\[ \frac{Z}{137}=Z\frac{e^2}{\hbar c}\ll 1. \tag{5} \]

This approximation was called Mott scattering and is described by the formula

\[ \sigma_M(\vartheta)= \left(\frac{Ze^2}{2mc^2}\right)^2 \left(\frac{1-\beta^2}{\beta^4}\right) \frac{1}{\sin^4 \frac{1}{2}\vartheta} \left(1-\beta^2\sin^2 \frac{1}{2}\vartheta\right), \tag{6} \]

where

\[ \beta=\frac{v}{c};\quad z=1; \tag{7} \]

\(v\) and \(c\) are the velocities of the scattered particle and of light, respectively; \(m\) is the electron rest mass. Formula (6) is written for the center-of-mass system. Under the experimental conditions described in our review, \(\beta\) is always very close to 1, and correspondingly in formula (6) \(\beta^4\) may be taken equal to 1, so that, with a high degree of accuracy,

\[ 1-\beta^2\sin^2\frac{\vartheta}{2}=\cos^2\frac{\vartheta}{2}. \tag{8} \]

The total energy of the electron is equal to

\[ E=\frac{mc^2}{(1-\beta^2)^{\frac{1}{2}}}, \tag{9} \]

whence

\[ 1-\beta^2=\left(\frac{mc^2}{E}\right)^2 . \tag{10} \]

Making these substitutions in formula (6), we obtain the relativistic Mott formula*) for the elastic scattering of electrons with spin by a point nucleus without spin and with charge \(Ze\),

\[ \sigma_M(\vartheta)=\left(\frac{Ze^2}{2E}\right)^2 \frac{\cos^2 \frac{1}{2}\vartheta}{\sin^4 \frac{1}{2}\vartheta}. \tag{11} \]

This formula indeed has a very simple form.

Equation (11) is satisfied very accurately if condition (5) is fulfilled. However, for heavier nuclei, when \(Z\) is large, it has been shown that formula (11), as was to be expected, leads to errors. Attempts to improve formula (11) were made by many investigators; we shall not describe the results of these calculations. McKinley and Feshbach \(^{13}\) summarized these results and themselves made improved corrections to formula (11) for nuclei with large values of \(Z\). Their work was confirmed by Dalitz \(^{15}\), who used the second Born approximation.

The authors mentioned \(^{13,15}\) also considered the case when \(Z/137\) is small, but condition (5) is not fulfilled, while the other conditions for which formula (11) is valid are preserved. The expression obtained has the form

\[ \sigma_F(\vartheta)=\left(\frac{Ze^2}{2E}\right)^2 \frac{\cos^2 \frac{1}{2}\vartheta}{\sin^4 \frac{1}{2}\vartheta} \left[ 1+\frac{\pi Z}{137} \frac{\left(\sin \frac{1}{2}\vartheta\right) \left(1-\sin \frac{1}{2}\vartheta\right)} {\cos^2 \frac{1}{2}\vartheta} \right]. \tag{12} \]

At small angles formulas (11) and (12) are equivalent, and even at large angles the discrepancy is small. The relative error arising from the use of (12) instead of (11) lies between 3 and 7% for Si and Zn at \(\vartheta=90^\circ\), respectively, and between 17 and 35% for Si and Zn at \(135^\circ\). Equation (12) cannot be applied to the case of heavy nuclei. It is impossible to give a simple formula, analogous to (12), suitable for all values of \(Z\). Numerical values of the scattering for large \(Z\) were given in the work of Feshbach \(^{14}\). This author also obtained the corresponding results for positron scattering. Bartlett and Watson \(^{16}\) performed exact numerical calculations for the heavy mercury nucleus (\(Z=80\)). It should be noted that the angular distribution given by (11) and (12) does not depend on the energy of the scattered particle. It should, however, be borne in mind that these expressions refer to scattering by an infinitely heavy center and are applicable only in the coordinate system of the center of mass. For real nuclei with ordinary mass values, the center of mass moves forward with considerable velocity when the energy of the scattered electrons exceeds 100 MeV. This leads to the angular distribution at high energies becoming even more strongly elongated forward. We shall return to this question later (see Section III, e).

Exact calculations of elastic scattering by a point charge in the case of copper and gold were carried out for high-energy electrons by Ienen,

*) The changes necessary for the transition to the laboratory coordinate system are given in formula (36).

Ravenhall and Wilson^17,18 by the phase-shift method. The authors compared their results with the corresponding data obtained on the basis of the first Born approximation (Fig. 1 in Ref. 17).

b) Scattering by a nucleus of finite size

The first allowance for the influence of the finiteness of the nucleus on electron scattering was apparently made by Guth^19. Later, and independently, analogous ideas were developed by Rose^20.

Elton^21, Felbakh^22, and Acheson^23, using more exact methods, considered problems connected with finite sizes in connection with experiments on electron scattering at low energies (up to 20 MeV), whereas Parson^24 dealt with an energy of 100 MeV for Pb. Somewhat later Smith^25, using the first Born approximation, examined this problem in detail and subsequently compiled the principal results in a review^26.

Smith’s results are sufficiently accurate only for light nuclei (small \(Z\)); however, some of his results apply both to elastic and to inelastic scattering. Schiff^27 performed analogous calculations, based on the first Born approximation, for high energies and obtained other interesting results in his work. Since the first Born approximation may be applied without danger to light nuclei, and since it gives the correct qualitative estimate of effects connected with the finite sizes of nuclei, we shall devote the next several paragraphs to this question.

The first Born approximation. Rose^20, Smith^22 and others showed that, analogously to formula (11) for a point charge, the formula for elastic scattering by a nucleus of finite size has the form

\[ \sigma_S(\vartheta)= \left(\frac{Ze^2}{2E}\right)^2 \frac{\cos^2 \frac{1}{2}\vartheta}{\sin^4 \frac{1}{2}\vartheta} \left| \int_{\substack{\text{nuclear}\\ \text{volume}}} \rho(\mathbf r)e^{i\mathbf q\mathbf r}\,d\tau \right|^2 , \tag{13} \]

where \(\rho(\mathbf r)\) is the charge density in the nucleus as a function of the radius-vector drawn from the center of the nucleus, and \(\hbar \mathbf q\) is the vector of the momentum transferred. The numerical value \(q\) for elastic scattering is equal to

\[ q=\frac{2E}{\hbar c}\sin \frac{1}{2}\vartheta =\frac{2}{\lambda}\sin \frac{1}{2}\vartheta , \tag{14} \]

as is seen from Fig. 2, where \(|\mathbf p_1|=|\mathbf p_0|\). \(\mathbf p_0\) and \(\mathbf p_1\) are the momenta before and after scattering, respectively (the primary and secondary momenta). \(\lambda\) in formula (14) is the de Broglie wavelength of the scattered high-energy electron,

\[ \lambda=\frac{\hbar}{p_0}, \tag{15} \]

and \(qr\) in (13) is thus a dimensionless phase factor.

An additional assumption is the absence of recoil of the nucleus; this is equivalent to saying that Fig. 2 refers to the center-of-mass system.

It can be shown^25 that the integral in (13) is simplified, so that

\[ \sigma= \left(\frac{Ze^2}{2E}\right)^2 \frac{\cos^2 \frac{1}{2}\vartheta}{\sin^4 \frac{1}{2}\vartheta} \left[ \int_0^\infty \rho(r)\frac{\sin qr}{qr}\,4\pi r^2\,dr \right]^2 . \tag{16} \]

The quantity standing in square brackets is the coefficient in the expression for the scattering cross section by a point charge (see (11)). By ana-

…in analogy with the notation used in the study of diffraction of electrons or X-rays, the quantity

\[ F=\frac{4\pi}{q}\int_0^\infty \rho(r)\sin(qr)\,r\,dr \tag{17} \]

has been called the “form factor” or “structure coefficient,” determined by the finite density of the nuclear charge distribution. Indeed, the analogy proves to be very close[^28] if the electron cloud around the atom is replaced by the proton cloud in the nucleus. If the charge density in (16) is normalized to unity, the form factor \(F\) is a dimensionless quantity.

Fig. 2. Momentum transferred \(q\) in electron scattering. For elastic scattering in the center-of-mass system \(|p_1|=|p_0|\).

In applying the first Born approximation, the central idea is the following: in order to obtain the actual scattering by a finite nucleus, it is sufficient simply to multiply the scattering cross section for a point charge by the square of the form factor obtained for the adopted model of the nucleus. These calculations can be carried out in a direct and usually quite simple way, since the problem reduces to the evaluation of a single integral (17). For light nuclei this is sufficient. Unfortunately, for medium and heavy nuclei such a method is not satisfactory. As is well known, the first Born approximation is equivalent to a treatment in which the incident and diffracted waves are regarded as plane waves. In reality, the waves are distorted by the strong electromagnetic field of the nucleus, so that they cannot be regarded as plane waves. Apparently, this is equivalent to the assertion that the first Born approximation describes single scattering in a force field, whereas the true scattering depends on multiple scattering in the same field.

In some cases the application of the Born method to elastic scattering provides a convenient way of analyzing electron scattering by light nuclei and also retains its value in the qualitative consideration of scattering by heavy nuclei. We shall return later to the question of the accuracy of the first Born approximation.

Using (17), one can obtain results for several models of the nucleus. To present the data in the most compact form, Table I gives a series of form factors for various distributions of nuclear charge density*). In this table \(a\) corresponds to the mean-square radius, “weighted” according to the charge,

\[ a=\int_0^\infty r^2 4\pi r^2\rho\,dr =4\pi\int_0^\infty \rho r^4\,dr, \tag{18} \]

where \(\int 4\pi r^2\rho\,dr\) is normalized to unity.

The ratio

\[ \frac{r}{a}=y \]

gives the distance from the center of the nucleus in units of the mean-square radius. The quantity \(x\) in the table is equal to \(qa\).

If the quantity \(qa\), where \(a\) is the mean-square radius, is small, all form factors can be represented by the simple expression

\[ F=1-(q^2a^2/6)+\cdots \tag{19} \]

*) This convenient form of the table was given by Chambers.

Table I

In this table \(\rho(r)\) is the charge-density function, \(a\) is the root of the mean-square radius for the given charge distribution, \(F(qa)\) is the form factor; \(x=qa\).

Model No.	Name of model	Expression for the charge density \(4\pi a^{3}\rho(r);\ y=r/a\)	\(F(qa);\ x=qa\)
I	Point	\(\delta\)-function	\(1\)
II	Uniform	\(\begin{cases}\dfrac{9}{5}\left(\dfrac{3}{5}\right)^{1/2}\ \text{for } y\leq \left(\dfrac{5}{3}\right)^{1/2} \\[4pt] 0\ \text{for } y\geq \left(\dfrac{5}{3}\right)^{1/2}\end{cases}\)	\(5\left(\dfrac{5}{3}\right)^{1/2}x^{-3}\left[\sin\left(\dfrac{5}{3}\right)^{1/2}x-\left(\dfrac{5}{3}\right)^{1/2}x\cos\left(\dfrac{5}{3}\right)^{1/2}x\right]\)
III	Gaussian	\(3\left(\dfrac{6}{\pi}\right)^{1/2}\exp\left(-\dfrac{3}{2}y^{2}\right)\)	\(\exp\left(-\dfrac{x^{2}}{6}\right)\)
IV	Exponential	\(12\sqrt{3}\exp\left(-(12)^{1/2}y\right)\)	\(\left(1+\dfrac{x^{2}}{12}\right)^{-2}\)
V	Shell	\(\delta(y-1)\)	\(x^{-1}\sin x\)
VI	“Shifted” exponential	\(\dfrac{200}{3}y\exp\left(-(20)^{1/2}y\right)\)	\(\left(1-\dfrac{x^{2}}{60}\right)\left(1+\dfrac{x^{2}}{20}\right)^{-3}\)
VII	—	\(\dfrac{75}{2}(30)^{-1/2}y^{2}\exp\left(-(30)^{1/2}y\right)\)	\(\left(1-\dfrac{x^{2}}{30}\right)\left(1+\dfrac{x^{2}}{30}\right)^{-4}\)
VIII	Yukawa I	\(\sqrt{2}\,y^{-2}\exp(-\sqrt{2}y)\)	\(\sqrt{2}\,x^{-1}\operatorname{tg}^{-1}\left(\dfrac{x}{\sqrt{2}}\right)\)
IX	Yukawa II	\(6y^{-1}\exp(-\sqrt{6}y)\)	\(\left(1+\dfrac{x^{2}}{6}\right)^{-1}\)
X	“Shifted”—Gaussian	\(\dfrac{50}{3}\left(\dfrac{5}{2\pi}\right)^{1/2}y^{2}\exp\left(-\dfrac{5}{2}y^{2}\right)\)	\(\left(1-\dfrac{x^{2}}{15}\right)\exp\left(-\dfrac{x^{2}}{10}\right)\)
XI	Generalized shell model	\(\dfrac{8}{\sqrt{\pi}}\dfrac{k^{3}}{(2+3a)}(1+ak^{2}y^{2})\times \exp(-k^{2}y^{2})\) where \(k=\left[\dfrac{3(2+5a)}{2(2+3a)}\right]^{1/2}\)	\(\left[1-\dfrac{ax^{2}}{2k^{2}(2+3a)}\right]\times \exp\left(-\dfrac{x^{2}}{4k^{2}}\right)\)
XII	Modified exponential	\(\dfrac{27}{\sqrt{2}}\left[1+(18)^{1/2}y\right]\exp\times \left[-(18)^{1/2}y\right]\)	\(\left(1+\dfrac{x^{2}}{18}\right)^{-3}\)

At high energies this approximation is inapplicable, since terms of higher orders must be taken into account.

Almost all the charge distributions used in the nucleus are included in Table I, or can, with more or less good approximation, be described by the models of this table. Apparently none of the distributions listed in Table I is a sufficiently good approximation for models with a repulsive core. The squares of some form factors are shown in Figs. 3 and 4.

Fig. 3. Square of the form factor for typical charge distributions.

Fig. 4. Square of the form factor for several commonly used charge distributions at small values of \(qa\). The model number corresponds to the number in Table I for the charge density.

The usual method presently employed by authors using the Born method consists in their attempting to fit the experimental data to one of the simple models. The search for a suitable model is quickly restricted to one or, perhaps, two of those given in Table I. After this, a selection of parameters is made that best satisfy the experimental data.

One may also reverse this process and calculate the charge distribution from the experimental form factor. This was done by Ravenhall \(^{30}\) in the analysis of exact data on \(\mathrm{C}^{12}\). Such a method is applicable when very accurate experimental data are available. In the author’s opinion, the accuracy presently available does not in most cases permit use of this method, although, apparently, it will not be long before the time when application of this method becomes possible.

Inverting (17), we obtain

\[ \rho(r)=\frac{1}{2\pi^{2}r}\int_{0}^{\infty} F(q)\sin(qr)\,q\,dq . \tag{20} \]

Schiff \(^{27}\) gave a method for processing experimental data, consisting in comparing the form factor \(F\) with certain experimental quantities which must be in agreement with the value of \(F\) when the correct model is chosen.

c) Analysis of phase shifts in electron scattering

In the works of Yennie, Ravenhall, and Wilson \(^{17,18}\), Brenner, Brown, and Alston \(^{31}\), and Elizabeth Baranger \(^{32}\), it was convincingly shown that for most models of medium and heavy nuclei the exact value of the elastic-scattering cross section differs greatly from that predicted by the first Born approximation. There are two basic types of such discrepancies. Both are shown in Fig. 5, taken from the work of Yennie et al. \(^{17}\). This figure refers to a uniform charge distribution for gold at an energy of about \(150\ \mathrm{MeV}\) and to an analogous distribution for copper at an energy of about \(225\ \mathrm{MeV}\). First of all, the Born approximation transforms the form factor into zero three times, whereas the exact calculations give, instead of zero, a minimum, and in some cases only points of inflection. Secondly, the dimensions given by the Born approximation are, generally speaking, larger than those given by exact calculations. This can be understood by noting that the de Broglie wavelength of the primary electron in the field of the electric forces of the nucleus must be smaller than that of a free electron. This follows directly from the fact that the effective kinetic energy will be greater in the field than outside the field because of the potential well. The Born approximation does not take this into account. Since electrons “measure” all lengths in units of \(\lambda\), nuclei will appear, in the Born approximation, where \(\lambda\) is not changed by the action of the nuclear field, larger than in the case of exact calculations. The same can be expressed differently by saying that diffraction properties are connected with the given values of \(qR\), where \(R\) is the parameter characterizing the radius. Since \(qR \sim R/\lambda\), a smaller value of \(R\) corresponds to the given diffraction properties at smaller \(\lambda\) than in free space. An argument of this kind was advanced by Yennie et al. \(^{17}\), but later it turned out that Gaussian and exponential charge distributions do not give direct confirmation of the expected properties. Apparently, this latter circumstance is explained by the fact that Gaussian and exponential charge distributions give a monotonic angular distribution, thereby not revealing diffraction properties, on the existence of which the arguments given above are based. This question is of particular importance for understanding the physical properties of the scattering phenomenon; one may hope that it will be clarified.

Fig. 5. Analysis of phase shifts according to Yennie et al. \(^{17,18}\) for a homogeneous spherical model in the case of gold and copper. Curves are shown corresponding to a point charge, and also to the results of the Born approximation. The experimental data refer to an energy of about \(150\ \mathrm{MeV}\) for gold and \(225\ \mathrm{MeV}\) for copper.

In any case, it has been shown that the first Born approximation cannot be used for heavy elements. From Fig. 5 it is seen that the Born approximation gives much better results for copper than for gold, although even in the case of copper the agreement is not very good. It is worse-

altogether in the region of diffraction minima, which correspond to three zero points of the Born approximation. For nuclei with \(Z\) less than 10, the Born approximation proves satisfactory, with the exception of regions close to the zeros.

One of the reasons for the unsuitability of the first Born approximation was clarified by Yennie et al.\(^{18}\) These authors found that, for gold, the scattering amplitude (generally speaking, a complex number) differs greatly from the scattering amplitude given by the Born approximation. In the Born approximation the scattering amplitude is real and may be positive or negative and zero at a diffraction minimum. The polar diagram (Fig. 6), taken from the work of these authors, demonstrates the typical behavior of the common logarithm of the modulus of the scattering amplitude as the angle \(\vartheta\) is varied for a model of type II (Table I, p. 699).

Since the Born approximation cannot be used for medium and heavy elements, and since there are as yet no other simple approximate methods, at present it is necessary to use exact phase-shift methods. At least up to the present time

Fig. 6. Polar diagram of the dependence of \(\lg\) of the modulus of the scattering amplitude on the angle \(\vartheta\) for a homogeneous model for gold, copper, and aluminum. The data for aluminum agree best with the experiment, indicating the suitability of the Born approximation for light nuclei.

Fig. 7. Fermi model. \(C\) is the distance from the center to the point where the density is equal to one half; \(T\) is the thickness of the surface layer (the distance over which the density changes from 90% to 10% of the maximum value).

no other ways of comparing experimental data with theory had been proposed, apart from choosing a definite model and calculating the angular distribution. If a discrepancy with experiment is found, the model is changed and new calculations are carried out. Successive attempts make it possible to settle on one model or on a number of suitable models. This method was described in the work of Yennie et al.,\(^{18}\) in which the smoothed Fermi model was introduced. This model has the form of a Fermi function\(^{33}\) (see (21)), and its form is shown in Fig. 7. As will be seen below, such a model proves to be very close to the actual shape of moderately heavy and the heaviest nuclei

\[ \rho(r)=\frac{\rho_1}{\exp\left(\frac{r-c}{z_1}\right)+1}. \tag{21} \]

Comparison of experimental data with results of the type shown in Fig. 8 is useful for choosing the model that best satisfies the experimental data. In the upper part of Fig. 8 three charge distributions are shown, and in the lower part—the three corresponding theoretical angular distributions for gold at 125 MeV, obtained by the phase-shift method. A uniformly charged sphere (rectangle) gives the most strongly pronounced diffraction properties. The smoothest curve is given, as one would expect, by the most smoothed angular distribution. Brown and Elton ^34 carried out calculations with similar models and arrived at analogous conclusions. Hill, Freeman, and Ford ^35 performed a similar analysis, using somewhat different models, and also obtained close results. Simplified models were considered by Glassgold ^36.

It is necessary to make a few brief remarks on the calculations of Yennie et al. ^18 and Brenner et al. ^31. These authors used the Dirac equation, applying it to a spherically symmetric static charge distribution. Quadrupole interactions were not considered *), and other dynamic effects, as well as possible dispersion corrections or correlations, were not taken into account. Dispersion corrections were considered by Schiff ^38, who showed that they are small. The assumptions introduced into the theory—that the Dirac equation is applicable to scattering, that the charge distribution may be regarded as static, that electron–nucleon nonelectromagnetic forces are absent, that Coulomb’s law is valid at small distances, etc.—can be checked by the agreement of the theory with experimental scattering data at various energies and by the consistency of these assumptions with data obtained in other areas of nuclear physics. Up to now there have been no visible reasons to doubt the suitability of the simple hypotheses chosen, with the possible exception of the proton case (see Sec. VII).

Fig. 8. Angular distribution for three nuclear models shown above. The uniform model gives the most noticeable diffraction phenomena.

Correlations between protons in the nucleus have recently been considered by Lewis ^39, who investigated nonpotential scattering and examined some details of the second Born approximation ^40.

III. VARIOUS PHENOMENA IN SCATTERING

Scattering can be divided into two large regions: elastic and inelastic scattering. In elastic scattering the kinetic energy of the two colliding particles in the center-of-mass system remains constant. In other words, one may say that neither excitation of the colliding particles nor formation of new ones occurs. In our case, when one of the colliding particles is an electron, it is sufficient to make sure that

*) This will be discussed in Sec. V, D.

the nucleus remains in the ground state before and after the collision, even if it acquires kinetic energy in the laboratory system. In inelastic scattering, phenomena of various types are observed; they will be named and considered below.

In reality, since in the electric field surrounding nuclei the electron emits a large number of soft quanta, it should be said that there is no truly elastic scattering. However, if a 300-MeV electron is scattered by a heavy nucleus and successively emits three quanta with energies 0.1, 1.0, and 3.0 eV, then the energy of the scattered electron differs so little from its initial energy that the scattering may be called elastic. In other words, if the detecting apparatus cannot distinguish an electron with energy 300 MeV from an electron whose energy is less than 300 MeV by \(1\cdot 10^{8}\) part of this value, then the collision is practically elastic. We shall use the term elastic scattering precisely in this sense. A correction for radiation may be introduced, if necessary, when comparing experimental data with the theory of elastic scattering.

a) Nuclear recoil

Before discussing the various types of inelastic scattering, let us consider scattering that appears inelastic in the laboratory frame of reference but is elastic when considered in the center-of-mass system. We mean simply the change in the energy of the scattered electron that occurs because of the transfer of energy and momentum to the scattering nucleus. For example, an electron with initial energy 400 MeV, after being scattered through \(60^\circ\) by a proton at rest, will have an energy of 326 MeV. The remaining energy (74 MeV) will be converted into the kinetic energy of the scattering proton, which will recoil at an angle determined by the laws of conservation of energy and momentum. The collision is, of course, relativistic.

The relativistic kinematics of the collision is analogous to the kinematics of the Compton effect, since at the high energies used in scattering experiments \((E \gg mc^{2})\) the electron energy

\[ E=(c^{2}p^{2}+m^{2}c^{4})^{1/2} \tag{22} \]

is, to a high degree of accuracy, determined by the approximate formula

\[ E=cp, \tag{23} \]

identical with the expression for the energy of x-ray quanta. In place of the electron with which the collision occurs in the case of the Compton effect, we substitute the mass of the scattering nucleus, which leads to the expression

\[ E_{n}=\frac{E^{2}}{mc^{2}}\frac{1-\cos \vartheta}{1+\left(\frac{E}{Mc^{2}}\right)(1-\cos \vartheta)}, \tag{24} \]

where \(E_{n}\) is the energy of the scattering nucleus, and \(M\) is its rest mass. This formula gives a simple and exact way of calculating the energy of the scattered electron, which is equal to

\[ E'=E-E_{n}. \tag{25} \]

The accuracy of this formula increases with increasing energy: at 20 MeV the error is less than 1%, and at 200 MeV it is less than 0.1%. In those cases where it is desirable to calculate exact values of the energy losses, one may

to make use of the kinematic equations of collision given in the literature, for example in Janossy’s book Cosmic Rays \(^{41}\).

An interesting conclusion from (24) follows when \(\vartheta\) is equal to \(\pi\). In this case we obtain the usual result of the Compton effect

\[ \bar E_n = E \frac{2\alpha}{2+2\alpha}, \tag{26} \]

where

\[ \alpha = \frac{E}{Mc^2} \tag{27} \]

and

\[ E' = E \frac{1}{1+2\alpha}. \tag{28} \]

If \(2\alpha \gg 1\), i.e., if the energy of the scattered electron is considerably greater than \(Mc^2\), then

\[ E' \to \frac{Mc^2}{2} \tag{29} \]

and the energy of the electron after scattering asymptotically approaches half the rest energy of the scattering nucleus. In a collision of an electron with a proton, the limiting energy of an electron scattered backward is therefore equal to 469 MeV. Thus, even at very high electron energies, for example at 10 Bev and perhaps at higher energies, if the cross sections are not too small, important scattering experiments are possible in which relatively uncomplicated apparatus is used, provided that only back scattering is studied.

Fig. 9. Energy of electrons scattered by a proton as a function of the angle in the laboratory system. The energy of the incident electron is close to 187 MeV. The solid curve was calculated from (24).

Fig. 10. Maxima of elastic scattering by protons of electrons with an initial energy of 187 MeV at angles of 60, 100, and 130°. The shifts in the position of the maximum occur as a consequence of proton recoil.

To demonstrate the application of (24) to a real scattering problem, Fig. 9 shows the experimentally determined kinetic energies of electrons scattered through various angles from a beam of 187-MeV electrons incident on a gaseous hydrogen target. The theoretical dependence determined by (24) is drawn as a solid line. The points correspond to experimental data. Small deviations at large and small angles arise because of energy losses experienced by electrons crossing the chamber wall at a small angle to it. The plotted energy values were measured from several series of data obtained with a magnetic spectrometer, from elastic-scattering maxima similar to those shown in Fig. 10. The initial energy of the electrons, obtained from the data

Fig. 9, where the average results of several series of measurements are given, differs within 1%.

The recoil energies according to (24) vary inversely with the mass number of the target nuclei, and therefore it becomes possible to distinguish the maxima of elastic scattering arising from two elements in a compound target, or even from two isotopes of one element. In studying the scattering of electrons by protons, polyethylene is a convenient substance, since the proton maximum of elastic scattering is located far from the carbon maximum and the problem of eliminating the background is greatly simplified. This method has a number of advantages, but when using it one must be sure that inelastic scattering by the heavier nucleus of the target does not fall at the energy of the maximum of elastic scattering by the proton of the element under investigation.

b) Inelastic scattering

If the scattering is elastic, the nucleus is observed in the ground state before and after scattering. In inelastic scattering the nucleus changes its state as a result of the scattering: the passage of the electron has caused a transition from the ground state of the nucleus to some excited state or to a level of the continuous spectrum. We shall consider the following possibilities.

1. Excitation of nuclear levels. The incident electron can transfer the nucleus into a definite excited state. Then the electron moves away from the nucleus with an energy reduced by the amount spent on exciting the nucleus. Fig. 11 shows such a phenomenon for carbon at an initial electron energy of 187 MeV and at a scattering angle of 80°. The maximum corresponding to elastic scattering occurs at 185.1 MeV. The small shift toward lower energies is explained by the recoil of the carbon nuclei and by direct losses of energy for ionization of the target. On the left, near 180.7 MeV, lies the maximum of inelastic scattering, approximately half as intense as the elastic maximum. This scattering arises from the excitation of the 4.43 MeV level of C\(^{12}\). In Fig. 11 smaller maxima are also noticeable, arising from scattering at the 7.65 MeV and 9.61 MeV levels and at higher levels. This type of scattering curve is characteristic of inelastic scattering, observed in many elements upon excitation of nuclear levels. Similar effects were found for beryllium\(^{3,44}\) and were also observed in lithium\(^{45}\), magnesium, silicon, sulfur, calcium, strontium\(^{46}\), etc.

Fig. 11. Maximum of elastic scattering from carbon near 185 MeV and maxima of inelastic scattering from excited states of carbon. The maximum near 180.7 MeV is associated with the 4.43 MeV level.

Inelastic scattering of the type described is of great interest, since it makes it possible to raise the nuclei under study to higher levels, and moreover to levels that cannot be obtained in most other methods. From these data one can also find the values of transition matrix elements and the nuclear angular momenta and parities of states. Qualitatively speaking, one may say that in such inelastic scattering the static characteristics of the nucleus in its ground state are manifested. From the experimental point of view, when attempting to separate the maximum of elastic scattering from the other maxima arising from inelastic scattering of the type described, certain difficulties arise. Below, the questions of inelastic scattering and its interpretation will be considered in greater detail.

2. Electron disintegration (distribution of nucleon momenta in the nucleus). The second type of inelastic scattering occurs when the incident electron causes the emission of a proton or neutron from the nucleus. This process may be called “electron disintegration.” It was discovered in deuterium ^48, and later was also observed in helium ^49 and other elements. On the curve \(BCDE\) in Fig. 12 there is shown the continuous spectrum of inelastic scattering arising owing to such a process in helium; the maximum \(A\) in this figure corresponds to elastic scattering of \(400\) MeV (more precisely, \(395\) MeV) electrons by \(\alpha\)-particles when observed at a scattering angle of \(45^\circ\). The figure also shows the maximum arising in the elastic scattering of electrons by free protons in hydrogen. Comparison of the two maxima reveals a difference in the recoil energies of the proton and the \(\alpha\)-particle. In scattering accompanied by emission of a nucleon, the energy of the scattered electron is reduced at least by the binding energy of the given nucleon in the nucleus. In the case of an \(\alpha\)-particle, emission of a neutron or proton requires approximately \(20\) MeV, and in Fig. 12 it is seen that the continuous spectrum of inelastic scattering approaches the abscissa axis on the high-energy side at a point separated from the maximum of elastic scattering by \(\alpha\)-particles by approximately this amount.

Fig. 12. Electron disintegration of the \(\alpha\)-particle at \(400\) MeV and \(45^\circ\). The elastic maximum is denoted by \(A\). The continuous spectrum of inelastic scattering \(BCDE\) is associated with the distribution of momenta of nucleons in the \(\alpha\)-particle. \(G\) indicates the formation of negatively charged \(\pi\)-mesons (see Sec. IV, in the text).

The part of the continuous spectrum formed by low energies arises as a result of the emission of a proton or neutron with an energy greater than that which corresponds to the position of the maximum for a free proton in Fig. 12. The emitted proton or neutron can also acquire an energy smaller than the recoil energy of a free neutron if this nucleon before emission has a velocity component directed opposite to the velocity of the scattered electron. If the compon—

the component of the nucleon velocity is parallel to the electron velocity, the recoil energy will be greater and the scattered electron will thus have a smaller energy. If, finally, the motion of the nucleon is perpendicular to the electron trajectory, the energy of the scattered electron in the first approximation will coincide with the energy of an electron scattered by a free proton, apart from the correction mentioned above for the energy required to remove the nucleon from the nucleus, i.e., for the binding energy. This explains the appearance of a maximum in the continuous spectrum of elastic scattering at an energy approximately 20 MeV lower than the energy corresponding to the maximum for a free proton in Fig. 12. A spectrum with analogous properties is shown in Fig. 13 for a scattering angle of 60°; in this case

Fig. 13. Elastic and inelastic scattering of electrons by alpha particles at 400 MeV and 60°. The area labeled “mesons” refers to negative π-mesons formed in the target and possessing the same momentum as the electrons that underwent scattering. For comparison, the scattering maximum from free protons is also shown.

the maximum of elastic scattering for α-particles is considerably smaller in comparison with the continuous spectrum of inelastic scattering. The same phenomenon is observed in the case of inelastic scattering on the deuteron, where the binding energy is only 2.23 MeV. The study of this inelastic distribution may therefore prove to be a source of information on the distribution of nucleon momenta in the nucleus.

3. Nuclear disintegration. In addition to processes analogous to those just considered, the simultaneous emission of more than one particle is also possible. One may expect the emission of such fragments as deuterons and α-particles. It is possible that even the fission of some heavy nuclei may occur. Such nuclear disintegrations undoubtedly arise and contribute to continuous spectra such as those shown in Fig. 12. Experimental separation of splittings in which one or several nucleons arise is difficult; it can be carried out in experiments using the coincidence method, or in special experiments searching for nuclear fragments.

4. Meson processes. The scattered electron can cause the emission by a nucleus or a nucleon of charged or neutral mesons of various types. It must be taken into account that, above a threshold close to 140 MeV, the electron energy can be converted into the energy of formation of such particles. In this case the energy of scattered electrons participating in such processes will belong to the low-energy region of the continuous spectrum of inelastic scattering. The π-mesons that arise may be

detected by the same apparatus that registers the electrons \({}^{40,50}\). Indeed, the “tail” of the spectrum in the region of small energies, marked in Fig. 13 by the inscription, owes its origin to \(\pi\)-mesons having the same momenta as the electrons recorded at the given spectrometer setting. The electrons themselves that form the mesons are not thereby separated from other scattered electrons falling into the region of the continuous spectrum owing to the motion of the nucleons, but there is no doubt that such electrons, forming mesons, are present in the spectrum.

1. Radiation. Both elastic and inelastic scattering of electrons is accompanied by the emission of photons. Radiation emitted predominantly forward is the well-known bremsstrahlung X-radiation, also observed in betatrons and synchrotrons. Bremsstrahlung may be detected at any angle and may be caused by an electron scattered through almost any angle. Therefore any elastic or inelastic maximum has, on the low-energy side, a “tail” falling off in the characteristic manner for radiative phenomena, i.e. inversely proportional to the energy of the emitted radiation. At large energy losses of the scattered electron (\(>10\) MeV), single X-ray quanta are emitted. At small energy losses, say less than \(1\) eV, the main part of the radiation is characterized by a Poisson distribution of the number of emitted photons. Often a large single energy loss may be accompanied by a shower of photons of very small energy. Radiative losses are apparently well described by the Bethe–Heitler theory; nevertheless it is of interest to investigate in detail the validity of a spectrum of the type \(1/E - E_0\) at large angles. This has not yet been done. For processes occurring on a finite nucleus, Biehl and Berkhof \({}^{51}\) formulated a theory based on the Born approximation.

The preceding remarks concerned the emission of real quanta. It is well known that in the process of scattering there also occurs virtual emission and absorption of radiation. Calculations that include consideration of this process, along with real radiation, were carried out by Schwinger \({}^{52}\), and later by Suura \({}^{53}\). The latter author made the applicability of Schwinger’s calculations more general. For many practical purposes, corrections compensating the real and virtual radiation processes for photon energies less than or equal to \(\Delta E\) must be introduced into the experimental scattering data. These corrections are given by Schwinger in the form

\[ I = I_0 e^{-\delta_r}, \tag{30} \]

where \(I\) is the measured scattering intensity, and \(I_0\) is the value compared with the theory that assumes only elastic scattering; \(\delta_r\) is the Schwinger radiative correction, which is described sufficiently well by the expression

\[ \delta_r=\frac{4\alpha}{\pi} \left\{ k\left[ \ln\left(\frac{2E}{mc^2}\sin\frac{1}{2}\vartheta\right)-\frac{1}{2} \right] +\frac{17}{12} \right\}, \tag{31} \]

where

\[ k=\ln\frac{E}{\Delta E}-\frac{13}{12}, \tag{32} \]

and \(E\) is the energy of the electron that has undergone scattering, \(\alpha\) is the fine-structure constant, and \(m\) is the rest mass of the electron. The practical quantity \(\Delta E\) reaches the value of the smallest energy that

can be distinguished from the main elastic maximum. It is usually close to the half-width of this maximum. The Schwinger correction is not very sensitive to the practical value of \(\Delta E\) and, moreover, depends very little on the angle. Therefore, when considering angular distributions it can usually be neglected; however, in those cases where sufficient accuracy is required, it must be taken into account. In investigations in which absolute cross sections are essential, this correction must be introduced. No detailed experimental study of the Schwinger correction has been made. A typical value of the Schwinger correction varies from \(14\%\) at \(40^\circ\) to \(17\%\) at \(135^\circ\), if \(E/\Delta E\) is close to 100.

The emission of real photons, i.e., bremsstrahlung of electrons emerging from targets of finite thickness, must also be taken into account. This leads to a correction of the type

\[ I_1 = I e^{\delta_B}, \tag{33} \]

where \(\delta_B\) is given by the expression\({}^{54}\)

\[ \delta_B = \frac{t}{\ln 2}\ln\frac{E}{\Delta E} \tag{34} \]

and where \(t\) is the average thickness of the target traversed by the electron (along the direction of the primary or scattered beam), and \(\Delta E\) is the full width of the elastic maximum at half height. \(I_1\) represents the intensity with the correction taken into account.

c) Magnetic scattering

For most nuclei, elastic scattering is completely determined by the electric charge of the nucleus, i.e., by the electric field of force surrounding the nucleus. However, because of the finite size of the nucleus, elastic scattering at large angles and high energies may turn out to be reduced by several orders of magnitude in comparison with the scattering expected for a point nucleus and Coulomb interaction. It is necessary to determine whether any elastic scattering remains after scattering by the charge falls below the experimentally distinguishable scattering. Another question connected with this is the following: will the neutron scatter high-energy electrons, and do the nuclei scatter electrons? To both of these questions an affirmative answer should be given: in experiments carried out at Stanford\({}^{42,55}\), magnetic scattering was indeed observed.

Proton. Elastic scattering of high-energy electrons by the magnetic moment of the proton was predicted by Rosenbluth\({}^{56}\) in 1950. He indicated the presence of a contribution to elastic scattering both from the Dirac component and from the Pauli component of the proton magnetic moment. The Pauli moment is usually denoted as the “anomalous” part of the magnetic moment of the proton. Rosenbluth’s results may be represented in the following form: for a proton with point charge and point magnetic moment the differential cross section \(\sigma_p\) is equal to\({}^{*}\):

\[ \sigma_p(\vartheta)=\sigma_{NS}\left\{1+\frac{q^2}{4M^2}\left[2(1+\mu)^2\tg^2\frac{1}{2}\vartheta+\mu^2\right]\right\}, \tag{35} \]

\[ \overline{\phantom{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx}} \]

\({}^{*}\) In accordance with usual practice, we write Rosenbluth’s formula in units \(\hbar=c=1\). In these units the nuclear magneton \(\mu\) consists of 1.79 nuclear magnetons (the Pauli moment) and 1.00 magneton (the Dirac moment). \(1.00+1.79=2.79\) is the total magnetic moment of the proton in nuclear magnetons.

where

\[ \sigma_{NS}=\frac{e^4}{4E^2}\frac{\cos^2\frac{1}{2}\vartheta}{\sin^4\frac{1}{2}\vartheta}\, \frac{1}{1+\left(\frac{2E}{M}\right)\sin^2\frac{1}{2}\vartheta} \tag{36} \]

and

\[ q=\frac{2}{\lambda}\, \frac{\sin\frac{1}{2}\vartheta} {\left[1+\left(\frac{2E}{M}\right)\sin^2\frac{1}{2}\vartheta\right]^{1/2}} . \tag{37} \]

Formula (36) is identical with (11) for \(Z=1\) (the proton), if in (11) the corresponding transformations are carried out for the transition from the center-of-mass system to the laboratory system. The subscripts \(NS\) indicate that (36) gives the scattering cross section for a proton without magnetic moment or spin (\(NS=\) no spin). The corresponding “\(q\)” in (14), the new value of \(q\) (\(\hbar=1\)) in formula (37), denotes the momentum transferred in the laboratory system and contains a new numerical factor connected with the transition to the new coordinate system. (We note that (37) follows directly from (24), if one bears in mind that \(1-\cos\vartheta=2\sin^2\frac{1}{2}\vartheta\) and puts \(\hbar q=p_n\), where \(E_n=p_n^2/2M\). Formula (37), however, is not connected with this approximation.)

Finally, from (35) it follows that for a point proton which scatters like a Mott proton, the scattering formula must be multiplied by a coefficient \(S_p\), which takes magnetic scattering into account:

\[ S_p=1+S=1+\frac{q^2}{4M^2}\left[2(1+\mu)^2 \operatorname{tg}^2\frac{1}{2}\vartheta+\mu^2\right]. \tag{38} \]

\(S_p\) depends on the energy, since \(q\) also depends on the energy \(\left[\lambda\propto\left(\frac{1}{E}\right)\right]\). \(S_p\) also depends on the scattering angle. The coefficient \(S_p\) is considerably greater than 1 at high energies (\(q\) large) and large angles \(\left(\operatorname{tg}\frac{1}{2}\vartheta\ \text{large}\right)\). Under these conditions the quantity \(S_p\) determines the scattering, and the main part of the scattering occurs because of the term \(S_p\), which contains the term \(\operatorname{tg}^2\frac{1}{2}\vartheta\). Together with (36), the coefficient \(S_p\) makes the behavior of the cross section at large angles flatter, owing to the fact that scattering by a point magnetic moment is appreciably more isotropic than Mott scattering by a pure charge.

If the proton possesses neither a point charge nor a point magnetic moment (as is to be expected from meson theory), then a form factor must be introduced into the consideration, the presence of which will lead to a reduction in the effective values of the charge and magnetic moment. Rosenbluth \(^{56}\) carried out similar calculations, using meson theory with weak coupling. Since, however, there is still no satisfactory meson theory, it is preferable to use phenomenological form factors to estimate the effects caused by the finite dimensions of the proton. The “dimensions” of the proton and its “shape” are ascribed to a virtual cloud of mesons, both charged and neutral, which may be emitted and reabsorbed by the proton. The phenomenological form factors \(F_1\) and \(F_2\) were introduced by Yennie, Levy, and Ravenhall \(^{57,42}\) in accordance with Rosenbluth’s scheme and the formalism given by Foldy \(^{58}\). \(F_1\) is introduced to take account of the pulsation of the extended charge and the pulsation

of the Pauli moment. \(F_2\) is an independent quantity taking into account the pulsation of the Pauli moment. Formula (39) shows how \(S_\rho\) changes from the introduction of \(F_1\) and \(F_2\):

\[ \sigma(\vartheta)=\sigma_{NS}\left\{F_1^2+\frac{q^2}{4M^2}\left[2(F_1+\mu F_2)^2 \operatorname{tg}^2 \frac{1}{2}\vartheta+\mu^2F_2^2\right]\right\}. \tag{39} \]

\(F_1\) and \(F_2\) are independent functions of the transferred momentum \(q\).

Neutron. In the case of the neutron, whose charge is zero, the “naive” approximation is \(F_1=0\). This is correct in the static limit \(q\to 0\). However, when the energy of the scattered electron increases and the wavelength correspondingly decreases, the electron passing through the neutron cloud becomes sensitive to the positive and negative charge (effective charge) of the clouds and may experience a deflection caused by these charges. The expected effect will be small if the dimensions of the positive and negative meson clouds of the neutron are small. Thus, \(F_1\) for the neutron reaches the value \(-\frac{1}{6}q^2 r^2\), as follows from (19), if the first static term is set equal to zero. The quantity \(F_2\) for the neutron has a clearer meaning, since the neutron has a static magnetic moment equal to \(-1.91\) nuclear magnetons. Consequently,

\[ F_{2n}=1-\frac{q^2 r_{2n}^{\,2}}{6}+\cdots, \tag{40} \]

if \(q r_{2n}\) is small.

A critical review of the ideas discussed above was made by Yennie et al. \(^{57}\). Let us also note here that the invalidity of electrodynamics, say, the inapplicability of Coulomb’s law at small distances, gives the same effect as finite dimensions. We shall return to this below (Section VII).

Deuteron. In the case of magnetic scattering by the deuteron, the expected cross section should be smaller than for the proton; since the static magnetic moment of the deuteron is \(\mu_D=0.858\) nuclear magneton, whereas for the proton it is \(2.79\) nuclear magnetons. Since the cross section is proportional to the square of the magnetic moment, the magnetic scattering of the deuteron will be approximately equal to \(1/9\) of the scattering of the proton. Thus, in electron scattering by the deuteron one should expect almost pure scattering by charge. This assumption follows from the results of Foldy \(^{59}\), who showed that the actual elastic scattering by the deuteron (neglecting very small quadrupole terms) is equal to:

\[ \sigma_D(\vartheta)=\sigma_{NS}\left\{1+\frac{2}{3}\frac{q^2}{4M^2}\left[2\mu_D^2 \operatorname{tg}^2 \frac{1}{2}\vartheta+\mu_D^2\right]\right\}F_D^2, \tag{41} \]

where \(\sigma_{NS}\) is defined as before, by (36) (of course, in equations (31) and (37) the deuteron mass is used instead of the proton mass \(M\)), \(\mu_D\) is the static magnetic moment of the deuteron, and \(F_D\) is the form factor obtained from the deuteron charge density, determined by the deuteron wave function in its ground state. The second term in brackets is the magnetic term, and its form confirms the remark made above about the smallness of magnetic scattering. The spin of the deuteron is equal to 1, and this explains the difference in the coefficients multiplying the \(q^2\) term in (35) and (41).

Of course, if the magnetic moment \(\mu_D\) is the sum of the extended moments of the neutron and proton, then a form factor arises by which \(\mu_D\) must be multiplied. However, the question of how this must be fulfilled—

...this modification (41) is complicated. It was briefly considered by Jancus\(^ {59}\) and in more detail by Ienti et al.\(^ {57}\) In any case, it is clear that scattering by magnetic moments in coherent elastic scattering from the deuteron will be smaller (by approximately \(1/9\)) than the corresponding scattering from the proton. Further details will be considered in Sections Vb and VI.

For the case of inelastic scattering accompanied by breakup of the deuteron, Jancus also showed that at large momentum transfers

\[ \sigma_D^{in}(\vartheta)=\sigma_{NS}\left\{1-F_D^2+ +\frac{q^2}{4M^2}\left[2\left(\mu_p^2+\mu_n^2-3F_D^2\right)\operatorname{tg}^2\frac{1}{2}\vartheta+\mu_p^2+\mu_n^2-3F_D^2\right]\right\} \tag{42} \]

or, if \(F_D\) is small,

\[ \sigma_D^{in}(\vartheta)=\sigma_{NS}\left\{1+\frac{q^2}{4M^2}\left[2\left(\mu_p^2+\mu_n^2\right)\operatorname{tg}^2\frac{1}{2}\vartheta+\mu_p^2+\mu_n^2\right]\right\}. \tag{43} \]

In these formulas \(\sigma\) is the inelastic-scattering cross section, \(\mu_p\) and \(\mu_D\) are the magnetic moments of the proton and deuteron, respectively, and \(F_D\) is the form factor of elastic scattering by the deuteron. At large momentum transfers \(F_D \simeq 0\). We shall return to (43) in Section VI, when we consider experiments with protons and neutrons.

It is obvious that other nuclei possessing spin, under the corresponding conditions, also exhibit elastic magnetic scattering. One may expect, however, that magnetic scattering will be important only for light nuclei with spin different from zero, for example \(\mathrm{Li}^7\). In medium or heavy nuclei the magnetic moment and spin are determined by only a few unpaired particles among the many particles forming the charge. Magnetic effects are localized near the surface and are associated with a form factor that apparently decreases even more rapidly than the form factor for scattering by charge.

d) Nonspherical shape of nuclei

Up to now, in considering effects connected with the finite dimensions of the nucleus, we have discussed only radial variations of the charge density. In reality, however, it follows from the spectroscopic literature on quadrupole moments of nuclei and from other data that the shape of nuclei lying between nuclei with closed shells (nuclei with magic numbers) is distorted and is not spherical. It is highly probable that the shape of nuclei close to nuclei with magic numbers differs only slightly from spherical. Data obtained in recent years in experiments on Coulomb excitation by alpha particles and protons give further grounds for believing that many nuclei in their ground state have an ellipsoidal shape. The Bohr–Mottelson model\(^ {60}\) explains such a shape from the point of view of the collective motion of nucleons bound together and forming a traveling bulge moving near the surface of the nucleus. Such motions correspond to the lower energy levels, called “rotational” by analogy with the rotational levels of a molecule. Whatever the actual shape of such nuclei—for example, nuclei of the rare earths, Ta, W, U, etc.—there is no doubt that there is something specific about this shape. The features of such nuclei are also revealed in the study of electron scattering. In this case the diffraction properties turn out to be considerably more smeared than in the case of spherical nuclei such as \(\mathrm{Pb}^{208}\) and \(\mathrm{Au}^{197}\).

In interpreting experimental data on electron scattering, it is also necessary to consider the influence of the ellipsoidal shape of the nucleus and, correspondingly, to average the effect of the nonspherical shape of the nucleus on scattering. Such averaging is equivalent to a rotation of the nuclear surface; it leads to an increase of the apparent nuclear surface in comparison with the actual one. This effect, however, does not cause a large change in the scattering, mainly because the surface layer is itself rather large. In any case, an increase of the surface is insufficient to explain the smoothed character of the scattering. From the fact that lower levels exist, it follows that scattering by these levels must also be included in the consideration: in electron scattering, transitions to these levels take place. A one-way transition from the ground state to excited rotational levels leads to inelastic scattering. However, because of the small energies of the transitions from the ground state to the excited rotational levels, such inelastic scattering remains unobservable at the resolutions presently achieved in scattering experiments. To distinguish rotational scattering from static scattering, a resolving power of about 1 in 2000 is required.

A theoretical treatment of the influence of the quadrupole shape of the nucleus on scattering was given by Schiff^47, and also by Downes, Ravenhall, and Yennie^61 for nuclei near tantalum. Further discussion of these works will be given after consideration of the experimental data.

IV. EXPERIMENTAL PART

In the experiments carried out at Stanford, two independent spectrometric installations were used. They will be considered separately below.

a) Installation for 190 MeV (the “intermediate” station)

The spectrometer for medium energies (up to 190 MeV) and the associated apparatus are surrounded by a closed vault, which begins approximately at the middle of the linear accelerator and extends for 12 m parallel to the accelerator. Figure 14 shows the general arrangement of the main parts of the installation. After passing through the brass collimator, groups of accelerated electrons, almost homogeneous in energy and passed through a slit in the uranium or brass collimator \(S\), are deflected and dispersed by the “deflecting magnet \(C\)” shown in Fig. 14^3,63. In this way a relatively monochromatic electron beam is selected; continuing on its path, it enters the field of the focusing magnet \(R\). The magnet \(R\) turns the beam back to its original direction and again focuses it at a point located at a distance of about 2.7 m from the edge of magnet \(R\). The basic idea

![Figure 14 diagram]

Fig. 14. General view (section) of the installation in the “intermediate station” and of the accelerator. The experiments here were carried out with the energy limited to 190 MeV because of the design of the spectrometer.

SCATTERING OF ELECTRONS AND THE STRUCTURE OF NUCLEI

The purpose of such double deflection, shown in Fig. 14, is to obtain an electron beam free from $\gamma$-radiation. The second magnet deflects the electrons away from the direction in which the beam of intense bremsstrahlung, produced in the target $S$, strikes the concrete shielding.

The apparatus shown in Fig. 14 produces vertical focusing of the beam onto the target. The wedge on the exit surface of magnet $R$ is intended to change the focal distance, which makes it possible to regulate the position of the spot accurately. A slight curvature of the wedge surface also helps to reduce the horizontal dimensions of the spot. The entire trajectory of the electrons from the accelerator gun to the scattering target is in high vacuum. The size of the spot on the target depends on the size of the exit collimator and is approximately $1$ mm in height and $3$ mm in width with a collimator of diameter $1.5$ mm, and $3\times 15$ mm with an $8$-mm collimator. In order to make the spot approximately circular and to obtain maximum intensity for a given diameter, rectangular collimators were often used. The largest spot sizes used were $3$ mm in height and $9$ mm in width.

Observation of the spot was carried out by means of a mirror and a telescope focused on a fluorescent plate of CsBr(Tl) crystal, $6.25\ \mathrm{cm}^2$ in area and $1$ mm thick. The fluorescence caused by the beam on this plate was very clearly visible, and it could be observed at a distance of $24$ m even at very low beam intensity—of the order of $10^6$ electrons per pulse (60 pulses per second). CsBr(Tl) is highly resistant to the action of the electron beam, apparently because a large part of the released energy leaves the crystal in the form of light instead of remaining in the crystal, heating it or causing a change in structure.

The beam of the “intermediate station,” focused on the target, contains at $188$ MeV from $2$ to $3\cdot 10^9$ electrons per pulse in an energy band of about $2$ MeV. Since the accelerator gives 60 pulses per second, this is equivalent to an average current of several hundredths of a microampere, for the beam apertures used. (At the “final station” (see below) more intense beams are obtained.) This is a very powerful beam, making it possible to measure small scattering cross sections.

Unfortunately, the ratio of the duration of the electron pulse to the dead time for the beam is not entirely favorable: the duration of the beam is $0.6\ \mu\mathrm{sec}$ per pulse, and the entire count must be completed in this short interval of time. The usual coincidence technique is useless here. At the same time there is a large background of gamma rays and fast neutrons, which makes the operation of scintillation counters (anthracene, NaJ(Tl)) extremely difficult and causes pulse pile-up. There are, however, effective ways of overcoming these difficulties: 1) a Cherenkov counter may be used as the detector, 2) the scattered electrons may be subjected to magnetic deflection and subsequent focusing onto a detector placed where it can be surrounded by shielding, 3) magnetic analysis greatly reduces the fraction of the radiation coming from the target that can be superposed on the phenomena under investigation. All this can be accomplished with the aid of a magnetic analyzing spectrometer, which selects electrons of definite momenta and thereby separates out phenomena of the desired type, for example, purely elastic scattering. Other phenomena can also be isolated, and in this way electrons of any energy can be counted with a small background. With the experimental arrangement described, for a beam of maximum intensity, with the magnet position corresponding to the maximum background, with closed

of the spectrometer entrance slit and the target in the operating position the greatest background is about 1 pulse in two minutes. Such a low background is achieved with the spectrometer, detector, and shielding described below. A photograph of the apparatus is shown in Fig. 15, where it may be noted that, while the scattering plane is horizontal, the magnetic spectrometer is arranged vertically. The electrons which have undergone scattering leave the scattering chamber (Fig. 14) in all directions. Some of them fall within the angular aperture of the entrance slit of the magnetic spectrometer. The entrance slit is made in lead and can be opened to a distance of about 2.5 cm in the horizontal scattering plane or completely closed. In a typical case it is used with a width of about 1.25 cm. The vertical dimensions of the slit can also be varied; usually they are close to 2.5 cm. Control of the entrance slit, which is mounted on the entrance surface of the spectrometer vacuum chamber, is carried out at a distance. The electrons pass through a thin window (75–150 μ of aluminum) of the magnetic chamber, located just behind the slit, and then enter the region of the magnetic field of the spectrometer, where they are analyzed by momentum.

Fig. 15. Semicircular spectrometer for 190 MeV, mounted on the gun carriage, at left. The upper platform carries the lead and paraffin surrounding the Cherenkov counter. At the bottom the brass scattering chamber with thin windows is visible. In front are visible the monitors—ionization chambers.

The spectrometer belongs to the type of double-focusing spectrometers proposed by Siegbahn and Svartholm \(^{64}\) and improved by Snyder et al. \(^{65}\). This type of spectrometer has the following essential features: the magnetic field is inhomogeneous and decreases as \(r^{-1/2}\), where \(r\) is the length of the radius vector drawn from the center of curvature of the field to the orbit. The mean radius of curvature is 40 cm, and the width of the pole surface is 15 cm; in this case the pole extends from a radius of 32.5 cm to a radius of 47.5 cm. The pole surfaces form an arc of \(180^\circ\), and the electrons are thus deflected through this angle. A small stray field extends beyond the entrance (and exit) slit, but the deflection in these regions is very small. The distance between the poles is 5 cm for the central trajectory, and the inclination of the poles is such that

\[ \frac{dy}{y} = - \frac{dr}{2r}. \tag{44} \]

Such an inclination provides the required dependence of the magnetic field on the radius. The pole surfaces thus have a linearly increasing narrowing. Projections are made at the inner and outer ends of the poles in order to prevent the field from falling too rapidly to zero.

The magnet itself weighs 2.5 tons and rests on four supports located on the carriage of an obsolete 40-millimeter twin antiaircraft mount.

design. It was obtained from the U.S. Navy, with the assistance of the Office of Naval Research. As can be seen from Fig. 15, a platform is placed on the magnet, serving as the base for the heavy shielding surrounding the Cherenkov counter. The shielding consists mainly of lead and paraffin and weighs about 2 tons. Inside the shielding is a small Cherenkov counter made of lucite in the shape of a truncated cone.

After the electrons are deflected by 180° in the magnet and analyzed by momentum, they leave the vacuum chamber through a thin window (0.15 mm aluminum) and pass through a horizontal aperture into a slot 2.5 cm thick, made in a uranium block. The vertical slot is made in a lead block and is usually fixed at a width of 1.25 cm. The horizontal opening of the slot determines the energy limits of the electrons passing through the spectrometer into the Cherenkov counter, while the vertical gap determines the effective width of the target. The dispersion of the instrument is 1.6% per 2.5 cm of horizontal opening of the slot.

The Cherenkov counter is made of well-polished lucite, is 10 cm long and has a diameter of 3.75 cm at the exit end, which is coupled to a Dumont 6292 photomultiplier. The conical shape of the counter allows light that has undergone internal reflection to reach the photomultiplier. The Cherenkov counter itself is enclosed in a lead casing. This entire system is placed in a brass tube connected to the head of the photomultiplier cathode follower and thus forms a single optical-electronic unit. The counter is fixed in a definite position behind the exit slot, while the shielding, platform, and magnet, rigidly connected to one another, rotate together with the zenith mounting. The angular position of the apparatus is controlled remotely; it is measured by means of a combination of selsyn indicators of high and low speed. Fixing the angular position with an accuracy better than 0.1° presents no difficulty.

The target is placed in vacuum, in the scattering chamber shown in Figs. 14 and 15. The target frame, designed for 6 positions, is made in the form of a vertical ladder and makes it possible to install the required target and, being remotely controlled, to change targets at will during the course of measurements. The angular position of the targets is also monitored remotely.

If it is necessary to work with gas, for example with hydrogen or helium, the gas-target chamber shown in Fig. 16 is placed inside the cone-shaped vessel described in the next paragraph.

Labels in Fig. 16:
For removal of gas; For admission of gas; Circular seal; Stainless-steel bottom, 0.23 mm; Cylinder of stainless steel, 0.42 mm thick.

Fig. 16. Typical gas-target chamber used at pressures up to 130 atm.

As stated earlier, the scattering foil or gas target is placed in an evacuated brass scattering chamber 50 cm in diameter. The geometry of the scattering is shown schematically in Fig. 17. In order that there be as little extraneous scattering material as possible in the path of the scattered electrons, the walls of the scattering chamber are made of Mylar film only 0.15 mm thick. This is achieved by fastening the walls made of this film to the thick walls of the chamber by means of a rubber

gaskets. The walls made of Mylar film extend from \(-150^\circ\) to \(-15^\circ\) and from \(+15^\circ\) to \(+150^\circ\), and \(3.75\ \mathrm{cm}\) above and below the scattering plane, which corresponds to a height of \(7.5\ \mathrm{cm}\) free of extraneous scatterers. Between \(-15^\circ\) and \(+15^\circ\) there are two brass supports, intended to counteract the force of atmospheric pressure compressing the top and bottom of the chamber. The front region between \(\sim -14^\circ\) and \(+14^\circ\) is occupied by an aluminum window \(0.15\ \mathrm{mm}\) thick. Mylar-film windows were also used in this part of the chamber, but usually they weakened and broke down after several hundred hours of irradiation by the beam. Aluminum windows last indefinitely.

Fig. 17. Diagram of the scattering geometry used when employing the gas-target chamber.

The scattering chamber, provided with the windows described, is made in the form of a conical vessel and can simply be removed from the base. It has an upper door through which the foils can be changed without removing the conical vessel from the base. The bottom of the scattering chamber contains a large number of well-insulated electrical feedthroughs. There are also devices for moving the monitoring and counting apparatus inside the conical vessel, in vacuum, by means of a large circular drive controlled remotely. At present a monitor operating on the principle of secondary electron emission is mounted on the circular drive; it can be placed either behind the scattering target or in front of it. Usually it is located behind the target. The monitor plates are sufficiently large (\(4.3\ \mathrm{cm}\) in diameter) for the entire beam to fall on them, even after it has been broadened by multiple scattering in the target. To measure the total number of electrons passing through the foil and the monitor, an ordinary electronic integrator is used, operating on the principle of charge accumulation. The advantage of the secondary-electron-emission monitor is its linearity and the absence of saturation. Unfortunately, its practical use is limited to beam intensities greater than \(10^6\) electrons per pulse. For weaker beams a monitor in the form of an ionization chamber is used, located outside the scattering chamber in the path of the beam that has undergone multiple scattering in the target. The output monitor is shown in Fig. 17.

The counting apparatus is simple and consists of an Elmore amplifier, model 501, whose pulses are fed to a counter with “gates,” constructed on the model of Narud’s instrument. The duration of the “gates” can be varied and is usually from 10 to 12 \(\mu\mathrm{sec}\). The pulse height, expressed as a function of the discriminating voltage on the counter with “gates,” shows a good plateau and repeatability in each series of measurements. To obtain curves characterizing the plateau, a twenty-channel pulse-height discriminator is very convenient. In the energy region in which the Cherenkov counter operates (\(84\text{–}190\ \mathrm{MeV}\)), no dependence of the pulse height on

energy, and thus in this region of energies the efficiency is constant.

The current in the magnet is maintained by means of a feedback amplifier with an accuracy better than \(0.1\%\). In this amplifier the input voltage is taken from a resistance connected in series with the magnet winding. The magnet winding is made of a hollow copper bus of square cross section, with a side equal to \(1.25\ \text{cm}\), cooled by water. The winding can carry a current of \(800\) amperes; the maximum magnetic field for the central trajectory is \(16\,500\) gauss. Such a field corresponds to approximately \(192\) MeV. The focusing at this maximum energy is not as good as at lower energies (\(150\) MeV), but is still sufficient. The energy calibration of the magnet was carried out: a) by the known energy of the incident electrons, when the magnet was placed in the forward direction; b) with a known energy of the incident electrons, by the recoil energy of hydrogen nuclei, determined by (24); c) by the known energies of inelastic scattering on excited levels of carbon; d) with the aid of a fluxmeter with a rotating coil; and, finally, e) with the aid of an instrument registering magnetic proton resonance. These methods give some discrepancy in the results, but up to now the experiments have not required a calibration with an accuracy exceeding the existing discrepancy (\(\sim 1\%\)). Measurement of the field by the method of magnetic induction is at present being carried out continuously.

In practice, to obtain data corresponding to a definite angle, it is necessary 1) to set the current in the magnet, 2) with the aid of the Cherenkov counter to count the number of electrons for a given value of the charge accumulated on the capacitor of the calibration monitor, 3) to find the ratio of these two quantities. The points corresponding to these quantities are plotted as a function of the current in the magnet, measured from the readings of a potentiometer giving the potential drop across the resistance in the magnet circuit. In Fig. 11 typical data obtained in this way are shown. From the elastic maximum on this graph one can see that the full width at half maximum is \(0.8\) MeV, i.e. about \(0.4\%\). Usually, with a corresponding loss in counting rate, a full width of \(0.2\%\) can be obtained.

Absolute counting can be performed approximately if an absolute calibration of the monitor is made\(^{66}\) and the effective solid angle is calculated according to Jaddu\(^{67}\). Exact absolute counting is at present impossible, but a semi-absolute standardization can be carried out by measuring an unknown scattering and simultaneously comparing these results with the intensity of electron scattering by protons. The effective cross section for scattering by protons can be taken from theory, corrected in accordance with the results of McAllister and Hofstadter\(^{42}\). At present, however, steps are being taken toward the direct measurement of absolute cross sections.

b) Spectrometer at 550 MeV (“end” station)

The dimensions of the large spectrometer (\(550\) MeV) considerably exceed the dimensions of the \(180\) MeV installation. On the other hand, the main units of the large installation are analogous to those considered by us above in Sec. IVa). Here it is necessary to consider only those parts of the installation where there is an essential difference or where new devices are used, such as, for example, the spectrometer itself.

Figure 18^68 shows the general arrangement of the equipment used in the study of electron scattering. Further details on the “end station” and on the accelerator may be found in Ref. 62. Figure 18 gives a diagram of the parts of the spectrometer, the platform with the target, the monitor, the detector, etc. Figure 19 presents a photograph.

Fig. 18. Experimental area of the 550-MeV spectrometer.

Here it will be appropriate to consider some details of the large magnet. A diagram of the magnet and the vacuum chamber is given in Fig. 20. The spectrometer, like the smaller spectrometer, is a modification of the Ziegban–Swartholm spectrometer with double focusing through 180°. This instrument, which weighs about 30 tons, is not an enlarged model of the 40-centimeter spectrometer. The maximum aperture used is close to 0.001 of the full solid angle. The radius of curvature of the central orbit is 90 cm, and the interpole gap for this orbit is 7.5 cm. The poles have a linear slope, determined by (44), and each edge of the poles is provided with a projection. The width of the pole surface is 37.5 cm. The dispersion of this spectrometer is 0.30% per 1 cm.

Fig. 19. Photograph of the 550-MeV spectrometer, surrounded by a protective labyrinth. The electron beam falls on the target located beneath the platform, passing through the vacuum tube visible in the foreground.

For the central trajectory the maximum value of the field is equal to 20,000 gauss, but because of saturation of the pole edges and other places in the gap—

magnet is rarely used at such a high value of the field. The cross-sectional area of the surface between the poles is \(3.5\times 5\ \text{cm}^2\), but it is not used completely because of the presence of a thick-walled bronze vacuum chamber. This chamber reduces the free internal dimensions to \(35\times 5\ \text{cm}\).

Figure 20 shows three radial channels passing through the outer yoke, each 10 cm in diameter. Similar channels, but of smaller dimensions, are also present in the vacuum chamber. They are used for inserting radial probes during magnetic measurements. The field measured in

Fig. 20. Sketch of the 550-MeV, 90-cm spectrometer and vacuum chamber.

the channels at \(30\) and \(120^\circ\) proves to be \(2\%\) less than the field in the \(90^\circ\) channel at the center of the magnet. The magnetization curve of the spectrometer, shown in Fig. 21, indicates that the field is proportional to the current up to \(14\,000\) gauss (i.e., up to 400 MeV). The region in which the field falls off as the square root of \(r\) lies between radii of \(83.6\) and \(96.2\ \text{cm}\), and at higher fields it contracts in such a way that for an energy of 550 MeV its width is only \(5\ \text{cm}\). Therefore, at large fields the vertical aperture is reduced by means of the entrance slit, in order to prevent electrons from penetrating into the saturation region in the gap. The stray field was measured and proved to be negligibly small for all energies. The fourth channel, shown in Fig. 20, allows bremsstrahlung from the target to leave the vacuum chamber through a thin window, while all electrons and positrons are deflected by the magnetic field. Further details of the construction are given in \({}^{68}\). The small continu—

Fig. 21. Magnetization curve of the 550-MeV spectrometer. Potentiometer readings are proportional to the current in the magnet windings.

The abundance of pulses makes it necessary to use a massive ten-ton shield to protect the Cherenkov counter from background radiation. This shield is supported by a platform located on the magnet itself, high above the field level. The magnet, platform, and shield can be moved radially on two rails, shown in Fig. 18. The magnet rests on the carriage of a paired five-inch antiaircraft gun, kindly provided by the U.S. Navy. Remote control of the angular refraction of the gun is carried out with an accuracy of 0.05° by means of standard gun selsyn indicators.

The Cherenkov counter is analogous to that described above, but its dimensions have been increased. The entrance diameter of the counter is 6.9 cm, the exit diameter 9.4 cm, and the length 12.5 cm. The truncated lucite cone is coupled to a five-inch Dumont photomultiplier. The baffle arrangement for this detector repeats the corresponding device of the small spectrometer.

The usual dimensions of the beam spot on the target are 9.3 mm in width and 3–6 mm in height. Under certain conditions they can be substantially reduced. This setup uses a gas target chamber, the schematic of which was given in Fig. 16. The length of the gas target has been increased with such calculation that, in studying scattering at small angles, the influence of the end windows would be less than in the small chamber. For targets made of foils and plates in the large spectrometer, a ten-position holder is used. The branch of the vacuum chamber of the spectrometer comes right up to the target itself, so that between the scatterer and the thin-walled window of the entrance part of the spectrometer there are only a few centimeters of air. (Some details are visible from Fig. 22.) Nevertheless, the instrument detects electrons scattered in such an air layer, and soon all experiments at the “terminal station” will be performed in vacuum, just as is done at the “intermediate station.”

Fig. 22. Details of the monitor, target holder, and entrance of the magnetic spectrometer.

Finally, Fig. 18 shows a large Faraday cylinder, recently installed and used for absolute cross-section measurements.

c) Properties of the Cherenkov counter

In the study of neutrons\(^ {50}\) (see below, Section VI), when scattering targets CH\(_2\) and CD\(_2\) were used, peculiarities were noted that occurred at large energies and large angles. Fig. 23 gives typical data pertaining to 120° and 550 MeV. Attention should be drawn to the usual sharp maximum of scattering by free protons at potentiometer division 127. In addition to this maximum, very large maxima are observed in the energy spectrum of CH\(_2\) and CD\(_2\), the center of which falls at division 90. Further, near the maximum for the free proton on the CD\(_2\) curve there is a small bulge. This

a flare of intensity is the object of study, but it is almost completely masked by the large maxima noted above. Investigations, which will not be considered here, show that the large maxima are associated with negative \(\pi\)-mesons formed in the target and having the same momenta as the scattered electrons under investigation. On careful study it turns out that the left part of the maxima corresponds exactly to the threshold velocity of \(\pi\)-mesons required for the excitation of a light pulse in a lucite (refractive index 1.50) Cherenkov counter. If the lucite is replaced by liquid \(\mathrm{C_8F_{16}O}\) (index 1.276), the \(\pi\)-meson maximum almost completely disappears and the deuteron flare of intensity can be observed well. Thus, if it is necessary to select electrons, one must use a Cherenkov counter with as small a refractive index as possible. In this case, of course, the index must be greater than 1, otherwise there will be no light flashes at all. A gas Cherenkov counter is very suitable for this purpose.

Fig. 23. Maximum of scattering on free protons, the incoherent deuteron maximum, and maxima of negative \(\pi\)-mesons. The first two maxima are connected with electron scattering, the remaining three (\(\mathrm{CH_2, CO_2, C}\)) with negative \(\pi\)-mesons having the same momenta as the electrons of inelastic scattering in this momentum interval. See also Sec. VI.

In Fig. 13 the ledge near 175 MeV corresponds to the background of \(\pi\)-mesons, observed in this case even for the lower energy of the primary electrons, equal to 400 MeV.

V. RESULTS

The results obtained with the aid of the two installations described above will be considered below in order of increasing atomic numbers.

Remark. The term “mean square radius,” often used in the text for the “equivalent homogeneous model,” corresponds to the quantity \(r_0\) defined by (52).

a) Proton

Electron scattering by the proton has been investigated for energies from 100 MeV to 550 MeV. McAllister and Hofstadter\({}^{42}\) studied scattering in gaseous hydrogen at a high pressure of 128 atm for energies 100, 188, 210, and 236 MeV. The very first experimental data showed a deviation from Rosenbluth’s\({}^{56}\) calculation according to (35) for scattering by a point charge with a magnetic moment. The data for 188 MeV are shown in Fig. 24. The upper curve \(c\) is the Rosenbluth curve for a point charge and a point magnetic moment having the full anomalous value \(1+\mu=2.79\) nuclear magnetons. The lower curve \(a\) corresponds to the absence of a magnetic moment and thus represents the Mott curve (formula (36)) for the laboratory coordinate system.

The large distance between curves a) and c) gives the theoretical value of the contribution of magnetic point scattering. Curve b) gives the Rosenbluth cross section for the case in which the magnetic moment of the proton is the purely Dirac magnetic moment, i.e., equal to one nuclear magneton. The experimental points shown in Fig. 24 lie between the Dirac curve and the curve for point charge and moment. This means that the magnetic moment of the proton is not point-like and that the proton must be assigned a certain form factor.

Fig. 24. Scattering of electrons by protons at an energy of 188 MeV. The experimental points lie below the Rosenbluth curve for point charges and magnetic moment, which is a consequence of finite sizes.

Without further investigations it is impossible to indicate a priori which of the two factors, the Dirac factor \(F_1\) or the Pauli factor \(F_2\) (see (39)), or both of them, is responsible for the small magnitude of the backscattering in Fig. 24. It follows from (39), however, that \(F_1\) and \(F_2\) give different angular distributions. One can show, for example, that a point charge \((F_1=1)\) and an extended Pauli moment \((F_2<1)\) lead to a form factor that is close to 1 at small transferred momentum \((q<1)\) and begins to fall rapidly at large values of the transferred momentum. On the other hand, an extended charge \((F_1<1)\) and a point Pauli moment \((F_2=1)\) give less scattering for small values of \(q\) and almost the full Rosenbluth point-scattering value at large transferred momenta. This is another way of making the qualitative statement that the charge is responsible for scattering through small angles at low energies, while the Pauli magnetic moment is responsible for almost all scattering through large angles at high energies. It should be remembered that even if \(F_2=0\), \(F_1\) contributes to the magnetic scattering caused by the Dirac magnetic moment. However, the effect of \(F_2\) is greater. An extended charge \((F_1<1)\) and an extended moment \((F_2<1)\) weaken the scattering both at small and at large values of \(q\). Thus, in principle, by studying scattering over a large range of energies and angles, one can separate the contributions to the scattering from \(F_1\) and \(F_2\).

The results for protons, obtained at energies of 100, 188, 210, and 236 MeV, were analyzed under the assumptions: 1) point charge \((F_1=1)\), extended moment \((F<1)\), 2) point moment \((F_2=1)\), extended charge \((F_2<1)\), and 3) extended charge and moment \((F_1<1,\; F_2<1)\).

Let us note that at low energies (\(q\) small) only the mean-square radius can affect the form factor, as follows from (19). Consequently, the determination of \(F_1\) and \(F_2\) at energies below 200 MeV gives the radius \(r_l\) for the Dirac cloud and the radius \(r_m\) for the mesonic Pauli cloud, but not the shape of each of these distributions as a function of radius. The analysis further gives
\[ r_e=r_m=0.74 \pm 0.24\ \text{fermi}, \]
where \(r_e\) and \(r_m\) are the radii associated with the Dirac and Pauli parts of the distribution of the proton charge density and magnetic moment. These values of \(r_e\) and \(r_m\) agree well with the experimental data at all energies. The solid curve in Fig. 24 represents the theoretical dependence for \(r_e=r_m=0.70\) fermi, which corresponds to the above-mentioned

assumption (3) and is a characteristic example of agreement between theory and experimental data. The accuracy of the experimental data is not sufficiently great to assert with certainty that assumptions 1) and 2) can be excluded.

Experiments of this type were continued by Chambers and Hofstadter^68 in the energy range from 200 to 500 MeV in the laboratory system. Under these conditions the experiment was carried out with polyethylene \((\mathrm{CH}_2)\). At 400 MeV, short control experiments with gaseous hydrogen were also performed. Figure 25 shows the peak corresponding to elastic scattering of 400-MeV electrons by protons in polyethylene through an angle of \(60^\circ\). Owing to the recoil of the proton, an electron at an angle of \(60^\circ\) carries away an energy equal to 326 MeV.

Fig. 25. Elastic scattering of 400-MeV electrons by protons in polyethylene through an angle of \(60^\circ\) in the laboratory system.

The quantity measured in this experiment is the area under the proton peak; it is proportional to the differential cross section*) for scattering through the given angle in the laboratory system. To obtain this area one must subtract the background produced by scattering in carbon, and extend the falling branch of the peak on the low-energy side, as shown in Fig. 25 by the dashed line \(AC\). If the areas under the proton peak are normalized, according to the monitor readings, to one scattered electron, it becomes possible to compare cross sections for different angles. A typical curve is shown in Fig. 26; the experimental points for 400 MeV are given with the corresponding error limits. The solid curve drawn above the experimental curve corresponds to Rosenbluth’s theoretical calculations (see (35)) for a point charge and a point magnetic moment. This is indicated by the notation \(r_e=0\) and \(r_m=0\) for the root-mean-square value of the charge radius (Dirac) and the radius of the moment (Pauli), respectively.

At the higher energies used in these experiments \((200, 300, 400, 500, 550\ \mathrm{MeV})\), powers of \(qa\) higher than the second begin to play a role in (19), and the form of the charge and magnetic-moment distributions becomes essential. Furthermore, one and the same distribution can satisfy the experimental data for all energies if the particular model and the interpretation given by (39) are correct. Without entering into the details of this problem, we give in Fig. 26 a curve corresponding to one particular model and satisfying the experimental data. In this case the model represents an exponential distribution both for the charge density and for the magnetic-moment density. The root-mean-square radius of each distribution is taken

*) In all these experiments a slit of constant width is used. Therefore, into the values of the areas one must introduce the correction well known in beta spectroscopy, which takes into account the dispersion of the apparatus \(\left(\text{constant } \frac{dp}{p}\right)\). This correction has been introduced in the usual way into all the cross sections.

equal to 0.80 fermi. For this model, from (39) and from the values of \(F_1\) and \(F_2\) obtained from column 4 of Table I, a theoretical curve can be constructed. It is shown by the solid line passing through the experimental points in Fig. 26. To obtain the best agreement, all the experimental points may be shifted upward or downward. No other attempts to improve the agreement were made. The indicated procedure is necessary because the absolute values of the cross sections are unknown. The ratio of the experimental values (normalized for best agreement with the theoretical curve at small angles) to the theoretical values for a point charge and moment gives the value \(F^2\) (the form factor)\(^2\). At all energies from 200 to 500 MeV the procedure of best agreement with the experimental data was carried out. The results obtained are shown in Fig. 27. The ordinates of this graph are the value \(F^2\), determined by the method indicated above; the abscissae are the square of the momentum transfer, \(10^{-26}\ \text{cm}^2\). The data for all energies and angles are well satisfied by the particular model considered.

Fig. 26. Typical angular distribution for elastic scattering of 400-MeV electrons by protons. The solid curve passing through the experimental points gives the theoretical distribution for protons in an exponential model with mean-square radius \(0.80\cdot 10^{-13}\ \text{cm}\).

Fig. 27. Square of the form factor as a function of \(q^2\). \(q^2\) is given in units of \(10^{-26}\ \text{cm}^2\). The solid curve is calculated for the exponential model with mean-square radius \(0.80\cdot 10^{-13}\ \text{cm}\).

The mutual consistency of the data for all energies and angles justifies the use of (39) and the phenomenological introduction of the form factors \(F_1\) and \(F_2\). However, good agreement with the experimental data is not a characteristic feature of this model in particular. The Gaussian model with \(r_e=r_m=0.72\) fermi also satisfies the experimental data well at all angles and energies. In this way many other models were studied, among them models II–X from Table I. For all these models it was assumed that \(r_e=r_m\) and that the shapes of the charge and magnetic-moment clouds coincide. The models giving the best

agreement with experiment are given in Table II. All other models do not satisfy the experimental data well enough for it to be worthwhile to consider them.

Table II

This table gives a summary of proton models and the corresponding values of the root-mean-square radius giving the best agreement with experiment. It is assumed that the Dirac and Pauli clouds have identical radii.

Model number	Form	R.m.s. radius giving the best agreement \((r_0=r_m)\), in fermis
III	\(\exp(-r^2)\)	\(0.72\pm0.05\)
IV	\(e^{-r}\)	\(0.80\pm0.05\)
VI	\(re^{-r}\)	\(0.78\pm0.05\)
VII	\(r^2e^{-r}\)	\(0.75\pm0.05\)
Mean (best agreement)	\(\ldots\)	\(0.77\pm0.10\)

Some models are shown in Fig. 28. Along the ordinate axis in this graph is plotted \(4\pi r^2\rho\). This quantity is proportional to the magnitude of the charge in a spherical shell of radius \(r\). The Gaussian, exponential, and quasi-exponential models all satisfy the experimental data equally well. Any model whose graph lies in Fig. 28 in the region occupied by these three models likewise gives the “best” approximation to the charge distribution in the proton. The distribution of the density of the magnetic moment has the same properties. The Yukawa model shown in the graph does not satisfy the experimental data. The same is true of the model of a uniformly charged sphere.

Fig. 28. The experimental data are satisfied equally well by the Gaussian, exponential, and “intermediate” exponential models. The Yukawa model does not satisfy the experimental data. Along the ordinate axis is plotted \(4\pi r^2\rho\).

All the models considered above are based on the assumption that the charge and meson clouds of Dirac and Pauli have identical radii and the same form. If one assumes that the forms of the clouds are different and \(F_1\) differs from \(F_2\), then the number of possible models increases enormously. Considerable effort was expended in attempts to find pairs of different radii and distributions agreeing with experiment. Many possibilities could be excluded. One example, shown in Fig. 29, demonstrates the typical behavior of a model with a magnetic cloud of small extent. It is impossible to choose a reasonable value of \(F_2\) corresponding to a small radius.

Without going into details, one may summarize the study of the models by saying that, if different radii are chosen for the Dirac and Pauli clouds, then the independent limits of the radii for each cloud lie approximately between 0.6 and 1.5 fermi. Increasing the accuracy of the experimental data may narrow these limits and make it possible to choose among the models indicated in Fig. 28. At present the experiment gives the least accuracy for determining the dimensions of the region near zero radius, extending to 0.3 fermi. Among all the models considered, the quasiexponential model with \(r_e = r_m = 0.78\) fermi best satisfies the experimental data. This does not mean that \(\rho = 0\) at \(r = 0\), for, as was already noted, the accuracy at \(r = 0\) is very small.

Fig. 29. Example of a model that does not satisfy the experimental data. This model has an insufficiently extended distribution of magnetic moment.

The interpretation of these experiments with protons will be continued in Sections VII and VIII. It is interesting to note, however, that the “Dirac radius” of the proton \((r_e)\) has the same dimensions as the “Pauli radius” \((r_m)\) and is very large. Indeed, it is three times greater than the Compton wavelength for the nucleon. We again draw attention to the fact that phenomena associated with finite dimensions can be explained with equal success by the assumption of point particles and by violations of the laws of electrodynamics.

b) Deuteron

Elastic scattering by the deuteron at high energies (192 MeV) was studied by McIntyre and Hofstadter \(^{70}\), and recently in considerably greater detail for 198 MeV and 400 MeV by McIntyre \(^{71}\). Scattering by the deuteron is of great interest, since this is the only stable system consisting of only two nucleons. For the problem of nuclear forces the deuteron occupies a position corresponding to that of the hydrogen atom in atomic physics. Because of this simplicity, the wave function of the ground state of the deuteron can be calculated for many possible nuclear potentials acting between the neutron and the proton. Among them one should mention the rectangular well, the potentials of Hulthén, Blatt–Kalckar, Gartenhaus, and the repulsive core potential. It is well known that all these potentials lead to identical results if one specifies the correct value of the deuteron binding energy \(\varepsilon = -2.226\) MeV and the triplet scattering length. All that can be extracted from nuclear experiments is the effective radius, a quantity essentially independent of the form of the deuteron potential at the scattering energies with which one has to deal. One may hope that, in studying the scattering of electrons by the deuteron, it will be possible to obtain new and independent data on the neutron–proton potential.

From the deuteron wave function \(\psi(r_{12})\), where \(r_{12}\) is the distance between nucleons, and from the adopted value of the effective radius, equal to 1.70 fermi, one can calculate, by the formula \(\rho=e|\psi|^2\), the charge density in the deuteron. For this potential, and hence for the specified charge density \(\rho\), one can calculate the deuteron form factor \(F_D\) (see (41)). The method of electron scattering gives a value of the form factor that is experimental and independent of theory. Thus the predictions of the theory can be compared with the results of electron-scattering studies. Such a comparison is made in Fig. 30 for three different potentials.

Fig. 30. Experimental data of McIntyre for 400 MeV. The square of the form factor is plotted along the ordinate axis. The angular form factors obtained from well-known nuclear potentials do not satisfy the experimental data.

The experiment makes it possible to normalize the deuteron data with respect to hydrogen. For this purpose the gas target is filled alternately with deuterium and hydrogen. This circumstance is very important, since experiments with protons give data that have decisive significance for calibration.

It follows from the graph that the observed scattering agrees with none of the theoretical curves, and it seems that in order to obtain agreement it is necessary to introduce an additional form factor. The poor agreement with experiment cannot be considered unexpected (Fig. 30) if the charge density in the deuteron \(e|\psi|^2\) is calculated in such a way as if the neutron and proton were points. As was indicated in Section IIIb, the introduction of this form factor requires great caution and is connected with consideration of the meson clouds surrounding the neutron and proton, as well as the Dirac “cores” of these particles.

Agreement with experiment in the graphs of Fig. 30 can, generally speaking, be achieved by increasing the dimensions of the effective region. For this it must be increased from 1.70 to at least 2.20 fermi, which apparently lies beyond the possible errors (\(\pm 0.03\) fermi) for the effective radius. On the other hand, McIntyre obtained excellent agreement with experiment by using, for the proton, a model of a finite Gaussian distribution with rms radius \(r_e=0.80\) fermi with a repulsive potential at the center according to Jankus\(^{59}\). In Fig. 31 a comparison is made with experimental data for three proton radii. The agreement of this analysis with the analysis of the proton data is remarkable. Analogous agreement is obtained with other potentials, for example with a Hulthén-type potential, if a somewhat larger value of the proton radius is adopted.

We have already pointed out that the introduction of a nucleon form factor, due to the finite dimensions of the neutron and proton, requires caution from the point of view of meson theory. One might expect a simple reduc-

of replacing the deuteron form factor \(F_D\) by the form factor of a proton of finite size. In fact, however, the finite dimensions of the neutron also have an effect. Since it is known that the specific interaction between the electron and the neutron is very small, the apparent size of the neutron may be assumed to be very small (see Sec. VI and \(^{90}\)). In that case only the finite size of the proton will be significant. In reality the situation is more complicated; it was investigated in the work of Yennie et al.\(^{57}\). These authors showed that in the deuteron the negative meson cloud of the neutron will neutralize the positive meson cloud of the proton, as a result of which only the action of the core in the proton and neutron remains. One might have expected the dimensions of the core to be very small, but to explain experiments with deuterons it seems necessary to introduce appreciable dimensions for the nucleons. This points to very large dimensions of the nucleon core (\(0.7\) fermi). Inelastic scattering by the deuteron was considered in Sec. IIIv, and we shall return to this question when considering the size of the neutron (Sec. VI).

Fig. 31. The introduction of a finite proton core makes it possible to satisfy experimental data with the aid of conditional form factors (McIntyre).

c) Alpha particle

Scattering by alpha particles (gaseous helium) was studied by McAllister and Hofstadter\(^{42}\) at \(188\) MeV and by Blankenbecler and Hofstadter\(^{49}\) at \(400\) MeV. Both experiments are in good agreement with one another and indicate the existence of very appreciable finite-size effects of the alpha particle. Since both the spin and the magnetic moment of the alpha particle are equal to zero, elastic scattering is caused by the action of the charge alone. However, inelastic scattering at large angles depends substantially on the magnetic moments of the nucleons.

Figure 32 gives the experimental data for helium at a pressure of about \(96\) atm. The gas-target chamber was filled alternately with helium and hydrogen at each given value of the angle. The scattering intensity was measured under the same experimental conditions. This made it possible to normalize the data for helium to scattering by the proton and to compare these data with the Mott curve (36), calculated for helium and denoted in Fig. 32 as the “theoretical Mott curve.” From Fig. 33 it is seen that, at angles greater than \(70^\circ\), elastic scattering is more than a hundred times smaller than the scattering expected for a point alpha particle. This graph presents the dependence of the square of the form factor on the scattering angle in the laboratory system, obtained from the data of Fig. 32. The points with the indicated error limits represent the experimental data, while the three solid curves correspond to three theoretical form factors for possible models of alpha particles with the indicated on the graph

...quantities of the radius. Of these three curves, evidently, the curve for the Gaussian model with a root-mean-square radius equal to 1.61 fermi gives the best agreement with the experimental data. This value is in good agreement with that obtained earlier.^42 Here no attempt was made to introduce finite nucleon sizes; however, in a final comparison of the data with the value of the form factor calculated from the nuclear theory of the alpha particle, it is necessary to take into account the effect of the finite dimensions of the nucleons.

Blankenbecler and Hofstadter^49 carried out a preliminary investigation, in need of repetition, of the continuous spectrum of inelastic scattering in helium for large values of \(q\): 400 MeV, \(60^\circ\). The results, shown in Fig. 13, were briefly discussed in Sections III6₂ and III6₄. Incoherent scattering from protons and neutrons in the alpha particle gives a cross section considerably exceeding the coherent scattering, to which there corresponds an elastic maximum at 373 MeV. As far as the author knows, at present there is no clear theory that would give the momentum distribution in the alpha particle with which the continuous spectrum of inelastic scattering in Fig. 13 could be compared.

Fig. 32. Experimental angular distribution of 400-MeV electrons scattered in helium. At the top is shown the curve corresponding to a point charge.

Fig. 33. Square of the alpha-particle form factor for three possible models. The Gaussian model with root-mean-square radius \(1.61 \cdot 10^{-12}\ \mathrm{cm}\) agrees best with the experimental data, indicated by points.

The size of the alpha particle can be compared with the radius of 1.61 fermi obtained from the phenomenological analysis of the data in Fig. 33, carried out with the help of the “best” Gaussian model. To make such a comparison, Dalitz and Ravenhall^72 calculated the root-mean-square radius for Clark’s wave function,^73 who used a variational method to obtain the correct value of the binding energy of the alpha particle. The radius obtained amounts to only \(2/3\) of the required value. Po-

apparently, this discrepancy is connected with the fact that in Clark’s calculations only two \(D\)-states are used.

It should be noted that in Fig. 13 at 352 MeV there is a clear maximum and, in the immediate vicinity of the continuum spectrum of inelastic scattering on the side of higher energies, there may be other irregularities. In view of the weakness of the evidence obtained, it is difficult to say to what extent this is an indication of an excited state of the alpha particle. To resolve this question, the investigation must be repeated.

г) Lithium and beryllium

Separated protons \( \mathrm{Li}^6 \) and \( \mathrm{Li}^7 \) were investigated by Strauch \(^{74}\), who found that both nuclei have, within a few percent, coincident root-mean-square radii. The charge densities in both isotopes are described rather well by model XII of Table I. Strauch \(^{75}\) found that the elastic-scattering experimental data are best satisfied by model XII with root-mean-square radius \(\langle a\rangle(\mathrm{Li}^6)=2.78\) fermi and \(\langle a\rangle(\mathrm{Li}^7)=2.71\) fermi, with an accuracy of \(\pm 2\%\). The ratio \(a(\mathrm{Li}^6)/a(\mathrm{Li}^7)\), which can be measured considerably more accurately than the radii themselves, is \(1.026 \pm 0.008\). The influence of the magnetic moment of \(\mathrm{Li}^7\) was calculated by a formula similar to Rosenbluth’s formula (35), and was taken into account in obtaining the above values of the nuclear radii. It is interesting that \(\mathrm{Li}^7\) has smaller dimensions than \(\mathrm{Li}^6\). This can be explained by the fact that the \(\mathrm{Li}^6\) nucleus behaves in many respects as if it had a deuteron outside its closed shell. The value \(r_0\) for \(\mathrm{Li}^6\) is 1.98 fermi, and for \(\mathrm{Li}^7\) 1.83 fermi for the equivalent uniform model (see (1)).

In the case of \(\mathrm{Be}^9\), earlier investigations showed that inelastic scattering by nuclear levels is very large. Elastic and inelastic scattering was again studied very intensively by Strauch \(^{75}\), who found good agreement with the old data. He analyzed the data for beryllium using model XII of Table I, which represents a modification of the exponential charge distribution, and obtained for the root-mean-square radius the value \(3.04 \pm 0.07\) fermi. This corresponds to the value 1.89 fermi in the equivalent uniform model.

Calculations according to the shell model for nuclei with a \(p\)-shell were carried out by Ferrell and Visscher \(^{76}\), who found that the root-mean-square radius of \(\mathrm{Li}^6\) is 2.8 fermi. This is in good agreement with the value 2.78 fermi obtained by Strauch. The experimental value for \(\mathrm{Li}^7\) (2.71 fermi) is higher than the theoretical value \(2.3 \pm 0.2\) fermi, and the experimental radius of \(\mathrm{Be}^9\) (3.04 fermi) is also greater than the theoretical value \(2.3 \pm 0.2\) fermi.

д) Carbon

\(\mathrm{C}^{12}\) is a relatively simple nucleus, which should therefore be carefully investigated. It was studied by Fregeau and Hofstadter \(^{43}\), and then in considerably greater detail by Fregeau \(^{77}\). In Section IIIв1 we gave typical results obtained for this scattering angle for carbon nuclei. From these results followed not only the existence of an elastic maximum, but also scattering from various levels of \(\mathrm{C}^{12}\). A summary of the data obtained up to the present time for an energy of 187 MeV showed—

shown in Fig. 34. The graph shows the behavior of the elastic maximum, whose magnitude changes by approximately \(2 \times 10^6\) times when the scattering angle is varied from 35 to \(138^\circ\). Fig. 34 also indicates the angular dependence of the scattering cross section to the excited levels of \(\mathrm{C}^{12}\): 4.43, 7.55, and 9.61 MeV. At large angles, scattering from each excited nuclear level begins to exceed elastic scattering. The angular distributions for scattering from the 4.43- and 9.61-MeV levels have a similar character and are less steep than the angular distributions for the 7.65-MeV level or for elastic scattering. Since the 4.43-MeV level corresponds to the \(0^+—2^+\) transition, and the 7.65-MeV level to the \(0^+—0^+\) transition, the difference in the angular distributions may be connected with radial oscillations in \(0—0\) transitions. On the basis of the fact that the 9.61-MeV level gives an angular distribution analogous to the 4.43-MeV level, the 9.61-MeV level may be assigned the transition \(0^+—2^+\).

Elastic scattering makes it possible to determine the radial charge density in the ground state of \(\mathrm{C}^{12}\). It proves possible to compare directly in one experiment scattering by carbon with proton scattering and thus carry out an “absolute” determination of the experimental form factor. In the first paper[^43], comparison of the “absolute” form factor with the form factor obtained for three selected models, i.e. for Gaussian, uniform, and exponential distributions, showed that the curve best satisfying the data lies between a Gaussian distribution with a root-mean-square radius of 2.47 fermi and a uniform distribution with a root-mean-square radius of 2.20 fermi. This gives a “best” value of the rms radius equal to 2.40 fermi. A considerably better determination was recently obtained by Fregeau[^77]; it proved to be in excellent agreement with this conclusion.

Fig. 34. Fregeau’s data on the dependence of the elastic and inelastic scattering cross section of 187-MeV electrons on the angle in the center-of-mass system.

Fregeau’s results are given in Fig. 35, where the dependence of \(F^2\) on the angle is shown for three models proposed by Ravenhall[^78] and for a recently proposed model of Morpurgo[^79], based on the oscillating-shell model. (See also model XI, Table I.) For this model the charge density \(\rho\) is the same for \(jj\) and \(LS\) coupling and is expressed by the formula

\[ \rho = \rho_0 \left(1+\alpha \frac{r^2}{a_0^2}\right)\exp\left[-\left(\frac{r^2}{a_0^2}\right)\right], \tag{45} \]

where

\[ \alpha = 4/3 \tag{46} \]

for the shell model, and \(a_0\) is a parameter proportional to the mean …

Fig. 35. Square of the form factor for \(C^{12}\). The figure shows the theoretical curves for model XI of Table I (corresponding to (45)). The parameter \(\lambda\) is a normalization coefficient, which should be equal to unity if theory and experiment coincide exactly. The value \(\lambda = 0.988\) for \(\alpha = 4/3\) is quite satisfactory.

Fig. 36. Charge distribution in model XI for three values of \(\alpha\). All three charge distributions satisfy the experimental data equally well. \(\alpha = 4/3\) has a certain advantage from the theoretical point of view.

Fig. 37. Charge distributions of Fig. 36, multiplied by \(4\pi r^2\). It should be noted that the distributions obtained are very close to one another.

quadratic value of the radius. To obtain the “best” agreement one should vary \(\alpha\), as is shown in Fig. 35, from which it follows that the value \(\alpha=4/3\) gives the best agreement with experiment. In Figs. 36 and 37 are given, respectively, the charge distributions \(\rho\), obtained from (45) for three values of \(\alpha\), and the quantity \(4\pi r^2\rho\) for the same values. The solid curves correspond to \(\alpha=4/3\). With the existing accuracy of the measurements there is no possibility of determining precisely the behavior of \(\rho\) near \(r=0\), but the charge density at large values of the radius is determined very well. Fregeau also investigated other models, and it turned out that all suitable models give curves \(4\pi\rho^2 r\), close to the curves of Fig. 37. For the model giving the “best” agreement, the mean-square radius is equal to \(2.40\pm 0.05\) fermi. The value \(r_0\) for the equivalent uniform model (see (1)) is equal to \(1.36\) fermi.

The properties of the inelastic-scattering curves were studied by Ravenhall and Morpurgo. In the case of the \(4.43\) MeV level both authors found that the \(LS\)-coupling scheme gives better agreement with experiment than the \(JJ\)-coupling scheme. Morpurgo was also able to explain the energy behavior of the ratio of the inelastic and elastic scattering cross sections. He also gave a satisfactory explanation of the constancy of the inelastic-scattering cross section (\(4.43\) MeV) at \(90^\circ\) in the energy region \(80\)—\(187\) MeV. Neither the \(LS\)-coupling scheme nor the \(JJ\)-coupling scheme gives exact quantitative agreement with the observed ratio of the cross sections of inelastic (\(4.43\) MeV) and elastic scattering for all the energies investigated. The experimental values are, on the average, larger than the theoretical ones by a factor of two, as was shown earlier by Ravenhall. In view of the preliminary character of the oscillating-shell model, this discrepancy does not seem serious. The angular distribution for (\(4.43\) MeV) inelastic scattering is well described by the theory.

It should be noted that the Born approximation introduces an error in the consideration of elastic and inelastic scattering. We pointed this out earlier (Sections IIb and IIc). For the case of elastic scattering, the reduction coefficient \((\gamma)\) of the mean-square radius, giving the ratio of the exact value to the value obtained from the Born approximation, was estimated by Ravenhall:

\[ \gamma=\frac{r_{\text{exact}}}{r_{\text{Born}}} =\frac{1}{1+\left(\dfrac{3Za}{2kR}\right)} \tag{47} \]

for a charge uniformly distributed in a sphere of radius \(R\). Here \(\alpha\) is the fine-structure constant, and \(k\) is the wave number of the scattered electron. In the case of carbon the exact value of the mean-square radius is thus \(2.37\pm 0.05\) fermi instead of \(2.40\pm 0.05\). The corrected \(r_0\) in (1) is equal to \(1.34\) fermi in the case of carbon \(C^{12}\).

For carbon nuclei a considerable amount of theoretical work has been carried out. The estimate of the rms radius made by Ferrell and Visscher \(^{76}\) for the ground state of \(C^{12}\) (\(a=2.3\pm 0.2\) fermi) is in good agreement with the experimental value \(2.37\) fermi. In addition to calculations of scattering at the \(4.43\) MeV level \(^{78,79}\), scattering at the \(7.65^{81}\) and \(9.61^{82}\) MeV levels was calculated. Schiff \(^{81}\) found that both the alpha-particle model of \(C^{12}\) and the elastic-liquid model give too large values of the matrix elements for the transition from the \(7.65\) MeV level to the ground state. He also used the independent-particle model and the \(JJ\)-coupling scheme to study two-nucleon transitions between the \(p_{3/2}\) and \(p_{1/2}\) shells. The transition matrix elements calculated in this case turn out to be approximately six times smaller. Schiff came to the conclusion

on the necessity of an intermediate-type model, i.e., a model of a more collective character than the model of independent particles (only with pair interactions), and less collective than the alpha-particle model of an elastic liquid. Glassgold and Galonsky \(^{83}\) showed that the alpha-particle model of the \(C^{12}\) nucleus agrees with the value of the charge-distribution radius in the ground state. However, this model, which proved successful for \(O^{16}\), in the case of \(C^{12}\) predicts states of \(5.54\) MeV, which in fact were not observed.

e) Magnesium, silicon, sulfur, argon, and strontium

Helm \(^{46}\) studied the scattering of electrons by even-even nuclei \({}_{12}\mathrm{Mg}^{24}\), \({}_{14}\mathrm{Si}^{28}\), \({}_{16}\mathrm{S}^{32}\), \({}_{18}\mathrm{A}^{40}\), and \({}_{38}\mathrm{Sr}^{88}\), having in mind, in addition to elastic scattering, also the investigation of \(0\)—\(2\) transitions between the ground and first excited states. The first excited levels are sufficiently far removed from the ground level, so that inelastic scattering can be separated from elastic scattering.

Helm’s experimental data for elastic scattering are shown in Fig. 38; the Freese and Hofstadter curve for \(C^{12}\) is also given there. These curves, whose ordinates are equal to the square of the form factor, reveal typical differential properties.

Fig. 38. Square of the form factor for elastic scattering of 187-MeV electrons by even-even nuclei. The diffraction stages occur at the same value of the abscissa. It follows from this that the radial parameter is proportional to \(A^{1/3}\). The graph is taken from Helm’s work \(^{46}\).

Fig. 39. Data on inelastic scattering for even-even nuclei. Along the ordinate axis is plotted the square of the form factor for inelastic processes.

Helm’s data for inelastic scattering, as well as some data for carbon, are shown in Fig. 39. Here the actual cross section is again divided by the cross section for a point charge in order to obtain an effective value (the square of the form factor) \(^{2}\) for inelastic scattering.

Helm introduced new charge distributions of interest, for example,

\[ \rho(r)=\int \rho_0(r)\rho_1(r-r')\,d^3r', \tag{48} \]

where \(\rho_0(r)\) is a uniform charge distribution for distances up to \(r=R\), i.e.

\[ \rho_0(r)= \begin{cases} \dfrac{3}{4\pi R^3}, & r\leq R,\\[4pt] 0, & r>R \end{cases} \tag{49} \]

and

\[ \rho_1(r)=\frac{1}{(2\pi g^2)^{3/2}} \exp\left[\left(-\frac{r^2}{2g^2}\right)\right]. \tag{50} \]

Helm called this particular model the “Gaussian uniform,” or \(gU\)-distribution. Such a formulation has the advantage that the resulting form factor is the product of two individual form factors

\[ F(q)=F_0(q)F_1(q), \tag{51} \]

where \(F_0\) and \(F_1\) are determined, as usual, from (17). Helm also used the \(uU\)-model, i.e. a uniform-uniform distribution. This method gives satisfactory results only within the range of applicability of the Born approximation.

The results obtained by applying the \(gU\)-model to the experimental data are given in Table III. Corrections have been introduced into the values obtained for the radii for errors arising from the use of the Born approximation (formula (47)). For comparison of the results with other experiments, one may calculate the root-mean-square radius for the \(gU\)-distribution and equate it to the root-mean-square radius of the “equivalent” uniform charge distribution given by (1). In this way one can obtain the equivalent \(r_0\) in (1). This quantity is given in the \(r_0\) column of Table III. The value of \(r_0\) obtained in this way has the form

\[ r_0=\left(\frac{5}{3}\right)^{1/2}aA^{-1/3}, \tag{52} \]

Table III

\(gU\)-distribution. This table includes Helm’s data for the radii of even-even nuclei and the data of Frege and Hofstadter for \(C^{12}\). The accuracy of the figures quoted is \(2\text{–}3\%\). Length in fermi.

Element	\(r_0\)	\(r_1=cA^{-1/3}\)	\(t\)
\(C^{12}\)	1.35	0.95	2.2
\(Mg^{24}\)	1.33	0.99	2.6
\(Si^{28}\)	1.29	0.97	2.8
\(S^{32}\)	1.30	1.03	2.6
\(Ca^{40}\)	1.28	1.08	2.4
\(Sr^{88}\)	1.20	1.08	2.3

where \(a\) is the root-mean-square radius of any charge distribution. The columns \(r_1\) and \(t\) refer to the parameters given by Hahn et al.\(^{33}\), which are the “half-density” distance \(c\) and the thickness of the surface layer \(t\) of the charge distribution, defined in that work (see also Sec. Vd). The quantity \(r_1\) is defined as

\[ r_1=cA^{-1/3}, \tag{53} \]

and \(t\) is equal to the thickness of the surface layer over which the density changes from 90 to 10% of its maximum value. The data obtained by Helm are given in Table III and will be discussed below in the section on nuclear radii. Helm’s \(uU\)-results are analogous to the \(gU\)-results.

The most interesting result on inelastic scattering is shown in Fig. 40. From the graphs presented there it is seen that, if the value of the experimental cross section is divided by the cross section for a point charge and normalized jointly with respect to the maximum value of the form factor, then several “universal” curves can be obtained in this way. This leads to the supposition that each electric multipole transition has its own universal curve for each value \(J = 0, 2, 3,\) etc. Generally speaking, this need not be a general rule, but in the present case such a rule holds. As a result,\(^{84}\) one can assign to \(\mathrm{S}^{32}\) \((2.25\ \text{MeV})\) \(J=2^+\) and to \(\mathrm{Ca}^{40}\) \((3.73\ \text{MeV})\) \(J=3\). Helm\(^{46}\) also explained the relative intensities of inelastic transitions from the point of view of Ravenhall’s theory, based on the ideas of Schiff,\(^{47}\) who gave the values of the radiative widths of levels.

Fig. 40. Dependence of the square of the form factor in inelastic scattering on the variable proportional to \(A^{1/3}\sin \dfrac{\vartheta}{2}\). After normalization of the maxima, Helm\(^{46}\) obtained “universal” curves for transitions of different multipolarity.

g) Moderately heavy and heavy elements

Hahn, Ravenhall, and Hofstadter\(^{33}\) investigated the elements Ca, V, Co, Sb, Hf, Ta, W, Au, Bi, Th, and U at an energy of 183 MeV and In, Au, and Bi at 153 MeV. Special attention was given to silver in connection with previously obtained data on this element. The first results with \(\mathrm{Au}^{197}\) and \(\mathrm{Pb}^{208}\) showed,\(^{4,85}\) that experiments at these energies make it possible to determine two basic parameters describing the charge distribution. These two parameters, denoted \(c\) and \(t\), were considered above. They are shown in Fig. 7 for a special model called the Fermi model. Generally speaking, these quantities relate to the parameter characterizing the radius and to the thickness of the surface layer. The aim of the investigation by Hahn et al. was to study changes of these two parameters in the region of nuclei from Ca to Bi. In doing so, only spherical nuclei were investigated. It is also desirable to know within what limits the values of both parameters can be varied so that they still give good agreement with the experimental data for the chosen model. Some nonspherical nuclei (Hf, Ta, W, Th, and U) were also experimentally studied, but without a detailed interpretation of the results.

The main experimental data are presented in Fig. 41, where, in order to show the diffraction phenomena better, the ordinates are divided by the Mott cross section. Since the Born approximation gives very poor results for many of the elements investigated, in this case the phase-shift method of Yennie et al.\(^{17,18}\) was used. Nevertheless, the angular positions of the diffraction minima in Fig. 41 occur at approximately the same values of \(A^{1/3}\sin \dfrac{\vartheta}{2}\). According to the Born approximation, this fact indicates that some parameter related to the radius varies approximately as \(A^{1/3}\). Such a quantity, as we see, is the parameter \(c\). Some of the nuclei were

selected for study because they are located near magic numbers, such as, for example, Ca, In, Sb, Au, Bi and, apparently, are spherical. Other nuclei were taken because they are represented by no more than one isotope (Ca, V, Co, In, Ta, Au, Bi). The main results are given below.

1.

Au was studied in considerable detail. The experimental results are presented in Fig. 42, for two energies, 153 and 183 MeV. The diffraction—

Fig. 41. Experimental data of Hahn et al.^33, showing that the radial parameter for different charge distributions varies according to the law \(A^{1/3}\). Diffraction phenomena are well observed and increase as the atomic number decreases.

Fig. 42. Comparison of experimental data for gold at 153 and 183 MeV with the theoretical curves of Hahn et al.^33, obtained by the method of phase shifts. The upper part of the graph refers to the determination of the best statistical agreement.

—like falls are especially clearly visible if the curves are examined along their length. The ordinates of these curves are not divided by the Mott coefficient, unlike in Fig. 41. The graph in the upper part of the figure refers to the parameters used by Hahn et al. to find the best agreement with experiment. The parameters \(s\) and \(r_0\) are related to \(t\) and \(c\). \(r_0\) is the coefficient in (1) for an equivalent uniform sphere.

In order to satisfy the experimental data, three trial models*) were chosen.^87 They had the following form:

Fermi:

\[ \rho(r)=\frac{\rho_1}{\exp\left(\frac{r-c}{z_1}\right)+1}; \tag{54} \]

modified Gaussian:

\[ \rho(r)=\frac{\rho_2}{\exp\left(\frac{r^2-c^2}{z_2^2}\right)+1}; \tag{55} \]

*) Here the terminology of Hahn et al.^33 is used.

trapezoidal:

\[ \begin{aligned} \rho(r)&=\rho_3; \quad 0<r<c-z_3,\\ \rho(r)&=\rho_3\,\frac{c+z_3-r}{2z_3}; \quad c-z_3<r<c+z_3,\\ \rho(r)&=0; \quad r>c+z_3. \end{aligned} \tag{56} \]

The parameters of these distributions are determined by the equations themselves. It is convenient to introduce the parameter \(c\), suitable for the various models:

\[ c=\frac{1}{\rho(0)}\int_0^\infty \rho(r)\,dr . \tag{57} \]

For models with a symmetrical surface, \(c\) is the distance from the center of the nucleus to the radius at which the charge density \(\rho\) falls to one-half of its value at the center. For the Fermi model the surface parameter \(t\), which gives the distance between the 90% and 10% values of \(\rho\), is equal to \(4.40z_1\). For the modified Gaussian model \(t=2.20z_2^2/c\), and for the trapezoidal model \(t=1.60z_3\).

The charge density at the center was varied in accordance with the model

\[ \rho(r)= \frac{\rho_k\left(1+\dfrac{wr^2}{c^2}\right)} {\exp\left(\dfrac{r-c}{z_a}\right)+1} \tag{58} \]

in order to detect the influence of a small and a large density at the center. In all cases, the method of least squares was used to find the model giving the smallest errors. Figures 43a and 43b show attempts to satisfy the experimental points for Au from Fig. 42. The best agreement was obtained for \(w=0.64\). However, from Fig. 43b it is seen that the Fermi model \(w=0\), expressed in \(4\pi r^2\rho\), differs so little from the curve giving the best agreement that the complication arising from the introduction of the third parameter \(w\) is not necessary at the present accuracy of the experiment.

Fig. 43. Various models of the charge density in gold, which lead to theoretical scattering curves very close to the curves giving the “best agreement” in Fig. 42. The distribution of the “best agreement” in Fig. 43a has a small dip toward the center, but the difference between the distribution with the existing dip (\(w=0.64\)) and the Fermi model (\(w=0.0\)) lies within the probable errors of measurement. This becomes clear from consideration of Fig. 43b. The curves shown here give \(4\pi r^2\rho\), i.e., the magnitude of the charge in a unit shell. Near the center the charge is very small, and the magnitudes of the charge in a shell for \(w=0.64\) and \(w=0.0\) are very close.

Further, Fig. 44 shows three models giving the “best agreement” within their type. At the existing accuracy of the measurements the experiment is not able to distinguish these three possibilities. It should be noted,

that the three charge-density curves shown intersect one another approximately at the same points, on the decline near the surface. From these results it follows that at the present time only two parameters can be determined, namely \(c\) and \(t\), or certain parameters close to these two. These two parameters represent general properties of all models satisfying the experimental data.

Fig. 44. Three models that satisfy the experimental data for gold equally well.

Fig. 45. Theoretical curves that satisfy the experimental data rather poorly.

One and the same model (the Fermi model), with the same numerical values of the parameters, satisfies the data for Au both at 153 and at 183 MeV, as is shown by the solid curves in Fig. 42. On the other hand, in Fig. 45 certain differences are visible. The two theoretical curves \(a\) and \(b\), exhibiting the difference, correspond to the Fermi model with parameters somewhat different from the parameters giving the best agreement. Both \(a\) and \(b\) refer to 183 MeV. The difference between models \(a\) and \(b\) is approximately the same as that between the models in Fig. 44.

2.

Data for other elements were compared with theory only by means of the Fermi model. The accuracy of the corresponding experiments is not as great as in the case of gold, and the different models at present cannot be resolved. Figure 46 shows both the experimental data for indium and the corresponding theoretical curves. The agreement proves to be very good.

Fig. 46. Theoretical and experimental curves for \(\mathrm{In}^{115}\) at two energies.

Investigations with the Fermi model show that the parameter \(c\) determines chiefly the angular position of the diffraction dips, whereas the parameter \(t\) is connected with the depth of the dips. Thus, the properties of the curves shown in Fig. 41 can be correlated with variations of \(c\) as \(A^{1/3}\). This conclusion is confirmed below.

3.

Data for various nuclei are summarized in Fig. 47. All these results are based on the Fermi model. Table IV gives the corresponding numerical data. From Table IV and Fig. 47 there follow several interesting conclusions:

Table IV

Results of the analysis of nuclei from the standpoint of the Fermi smoothed uniform distribution. All lengths are in fermis. Charge density is in \(10^{19}\) coulombs/cm\(^3\). Accuracy of these data: radial parameters \(\pm 2\%\); surface-layer thickness parameters \(\pm 10\%\). For light elements the error is probably larger. In the case of gold the accuracy is higher. \(R\) is the radius of the uniform charge distribution having the same value of the root-mean-square radius as the Fermi distribution.

Nucleus	\(c\)	\(t\)	\(R\)	\(c/A^{1/3}=r_t\)	\(R/A^{1/3}=r_0\)
\({}_{20}\mathrm{Ca}^{40}\)	3.64	2.5	4.54	1.06	1.32
\({}_{23}\mathrm{V}^{51}\)	3.98	2.2	4.63	1.07	1.25
\({}_{27}\mathrm{Co}^{59}\)	4.09	2.5	4.94	1.05	1.27
\({}_{49}\mathrm{In}^{115}\)	5.24	2.3	5.80	1.08	1.19
\({}_{51}\mathrm{Sb}^{122}\)	5.32	2.5	5.97	1.07	1.20
\({}_{79}\mathrm{Au}^{197}\)	6.38	2.32	6.87	1.096	1.180
\({}_{83}\mathrm{Bi}^{209}\)	6.47	2.7	7.13	1.09	1.20

a) The thickness of the surface layer for all the nuclei investigated is constant and close to \(t \simeq 2.4\) fermis. b) The parameter \(c\) varies as \(1.08\,A^{1/3}\), as was assumed in Fig. 41. Therefore the flat part of the curves moves away from the center as the atomic number increases. In the case of very light nuclei (\(Z \ll 6\)) it disappears. The quantity \(r_0\) (formula (1)) varies from \(\sim 1.19\) for heavy elements to 1.32 for Ca. This shift of \(r_0\) continues toward lighter elements, as can be seen from Table III (1.33 for \(\mathrm{Mg}^{24}\) and 1.35 for \(\mathrm{C}^{12}\)). The difference between Helm’s value 1.28 for \(\mathrm{Ca}^{40}\) (Table III) and the value of Hahn et al. is explained by the use of different models.

Fig. 47. Fermi models for various nuclei. As \(Z\) decreases, the mean charge density at the center increases.

For light nuclei, in which the central flat part of the curves is not shifted and for which, as, for example, in the case of carbon, a modified Gaussian distribution best satisfies the experimental data (Fregeau), one may expect a decrease

values of \(t\). In very light elements the dimensions of the entire nucleus do not yet exceed the dimensions of the surface layer \(\sim 2.4\) fermi.

Thus, the assumption of the constancy of \(r_0\) in (1) does not withstand comparison with the results presented. These models do not give a constant value of \(r_0\). The trends in the variation of \(r_0\) are as follows: small \(r_0\) for large \(A\), large \(r_0\) for small \(A\). In order for this conclusion to be extended to all nuclei, further investigations with new nuclei are necessary. It is possible that local changes of \(r_0\), which are exceptions to this rule, will then be discovered.

4.

For nonspherical nuclei I an and others obtained the results shown in Fig. 48. These nuclei display more smoothed, less sharply expressed diffraction properties than, for example, gold in Fig. 42. This has received a qualitative explanation as an indication of a quadrupole (ellipsoidal) distortion of the shape of nuclei.

From other experiments it is well known that the nuclei in Fig. 48 have long-lived levels arising under Coulomb excitation, which indicates large intrinsic quadrupole moments. This compels one to assume large deviations from spherical symmetry, which are connected with the collective motion of the outer nucleons \(^{60}\).

Downs and Downs with co-workers \(^{88}\) carried out detailed calculations using an improved Born approximation in order to determine the influence of quadrupole scattering on the filling of the diffraction minima observed in scattering by nonspherical nuclei.

Quadrupole effects give three contributions to the scattering: a) Ellipsoidal distortions of the shape must be averaged over all directions, which leads to a nuclear density \(\rho_S\) having a greater effective thickness of the surface layer than nuclei without shape distortion. Elastic scattering with quadrupole smoothing of this type is shown on the curve \(\sigma_S\) (Fig. 49) according to Downs. b) The second type of elastic scattering corresponds to a change in the orientation of the nuclear spin axis, i.e., a “spin flip.” For a nonspherical nucleus a transition of this type is possible.

Fig. 48. Experimental data for nuclei possessing considerable nonsphericity. In comparison with gold the diffraction phenomena are smoothed.

Such scattering, averaged over all orientations of the nuclei, is an independent addition to the elastic scattering. a) For oriented nuclei interference effects may be observed. c) Inelastic scattering, corresponding to transitions from the ground state to lower excited states, gives a contribution to the scattering usually called elastic. This contribution is so close to the elastic scattering (\(\sim 1\) part in 2000) that at present such scattering may practically be regarded as elastic.

As a result, b) and c), the contribution to the scattering (Fig. 49), denoted by \(\sigma_T\), is added to the elastic scattering \(\sigma_s\), forming the curve of total scattering \(\sigma_s + \sigma_T\). The latter curve can be compared with experiment. The curve \(\sigma_T\) thus corresponds to quadrupole scattering. The curve \(\sigma_T^{(2)}\) is an approximation obtained from the Born approximation \(\sigma_T^{(1)}\) for a pure quadrupole and from higher-order approximations, which do not vanish at the zeros of the Born quadrupole scattering. The curve \(\sigma_T^{(2)}\) was obtained in this way, and was not calculated, with the exception of the two points indicated by circles.

Fig. 49. Theoretical curves of Downs et al.\(^{88}\) for tantalum nuclei, calculated under the assumptions considered in the text concerning quadrupole scattering.

Fig. 50. Comparison of the calculations of Downs et al.\(^{88}\) with the experiments of Hahn and Hofstadter\(^{89}\) for tantalum.

Figure 50 shows the results of analogous calculations for tantalum, in which three values of the intrinsic quadrupole moment \(Q_0\) were used, corresponding to three states with an increasing degree of distortion of the shape of the nucleus. Figure 50 gives the experimental points of Hahn and Hofstadter\(^{89}\). It is evident that the experimental data are satisfied by an intermediate value of \(Q_0\) between 7 and 14 barns. However, Downs does not consider these calculations sufficiently reliable for determining the actual value of the quadrupole moment. The probable cause of the smoothing of the curves in Fig. 48 is the quadrupole contributions to the scattering which we have considered.

VI. NEUTRONS

Determination of the internal structure of the neutron, or at least the obtaining of such data about this particle as would correspond to the level of our knowledge of the structure of the proton, is of great interest. The study of the interaction between an electron and a neutron, pro-

ELECTRON SCATTERING AND THE STRUCTURE OF NUCLEI

carried out by Fermi, Rabi, Yuzom, and their collaborators\(^{90}\), indicated an unexpectedly small effective radius for the charge distribution. These experiments, however, gave no data on the size or shape of the neutron magnetic moment. One may expect that the method of electron scattering will be able to make a substantial contribution to this problem.

The information available at present can be obtained on the basis of the following considerations. If, in the inelastic scattering of an electron by the deuteron (or by Be\(^{9}\), in which a weakly bound neutron is located at the periphery of the nucleus), a large momentum transfer occurs, the neutron and proton may be regarded as essentially free particles, since in the deuteron they are weakly bound. Further, at large angles and high energies the scattering of electrons is almost completely determined by the magnetic moment of the nucleon (Rosenbluth). Accordingly, since the magnetic moment of the neutron is equal to \(-1.91\), and the magnetic moment of the proton to \(2.79\) nuclear magnetons, the scattering of electrons by the proton is \([(2.79/1.91)^2 \approx 2]\) approximately twice as large as the scattering by the neutron. Here it is assumed that the particles are points, or that the clouds of their magnetic moments have the same dimensions. However, if the neutron has dimensions smaller than the proton, magnetic scattering by the neutron may attain or even exceed magnetic scattering by the proton. If the dimensions are the same, scattering by the neutron will be equal to approximately one half of the scattering by the proton. Comparing the inelastic scattering of electrons by deuterium with the scattering of electrons by free protons in hydrogen, one can, from the deuterium–proton difference, obtain the neutron scattering cross section and thereby its dimensions.

These arguments are based on a detailed theory developed by Jankus\(^{59}\). Indeed, if \(F_D\) is small, (43) can be written in the following form:

\[ \sigma_D^{in}(\vartheta)=\sigma_{NS}\left[1+\frac{q^2}{4M^2}\left(2\mu_p^2 \operatorname{tg}^2 \frac{1}{2}\vartheta+\mu_p^2\right)\right]+ \]

\[ +\sigma_{NS}\cdot \frac{q^2}{4M^2}\left(2\mu_n^2 \operatorname{tg}^2 \frac{1}{2}\vartheta+\mu_n^2\right), \tag{59} \]

\[ \sigma_D^{in}(\vartheta)=\sigma_p+\sigma_n . \tag{60} \]

Formula (60) expresses the qualitative assertions made above in a more exact way. Accordingly, the deuterium–hydrogen difference method makes it possible to find the value \(\sigma_n\). Knowing \(\sigma_n\) and calculating the form factor, and knowing \(\sigma_p\) and the proton form factor, one can obtain information on the dimensions of the neutron.

In reality, such experiments are complicated by the smearing of the continuous spectrum of inelastic scattering caused by the motion of the neutrons and protons in the deuteron. Figure 51 shows the approximate form of the momentum distribution in the deuteron, obtained by Blankenbecler, Hofstadter, and Yearian\(^{50}\) in recent preliminary experiments. The proton maximum has a sharp form and a falloff on the side of low energies, caused by bremsstrahlung. Elementary scattering from a proton and neutron moving in the deuteron reveals the same falloff due to bremsstrahlung, but this radiative cross section lies in the region of the continuous spectrum of inelastic scattering. In order to carry out the subtraction, a correct estimate of the corresponding corrections is necessary.

At present only preliminary data are known. The first assumption following from these data is that scattering by the magnetic moment of the neutron is apparently smaller than scat—

scattering by the proton through an angle of \(135^\circ\) of electrons with energy \(500\ \mathrm{MeV}\), but the accuracy of this conclusion is low. If this result is maintained as the accuracy of the experiment increases, it will mean that the cloud of magnetic moment of the neutron has the same, or possibly somewhat smaller, dimensions than that of the proton.

Fig. 51. Inelastic scattering of electrons by deuterons through an angle of \(135^\circ\) at an incident-electron energy of \(500\ \mathrm{MeV}\). The upper curve gives the scattering in deuterated polyethylene \((CD_2)\), and the points in triangles give the corresponding contribution from scattering in pure carbon. The difference of the data for \(CD_2\) and \(C\) is shown by the lower dashed curve. These data have been corrected for the dispersion of the spectrometer. The dashed curve is a preliminary indication of the momentum distribution in the deuteron and corresponds to \(\sigma_D^{\mathrm{inel}}(135^\circ)=\sigma_p+\sigma_n\) in (59). The remainder, after subtracting from the area under the dashed maximum the \(\sigma_p\) for a free proton, corresponds to \(\sigma_n\).

Recent experiments indicate more definitely that the magnetic cloud of the neutron is not as small as follows from the static experiments.

Recently Chambers and Hofstadter \(^{91}\) obtained preliminary data on \(Be^9\), from which it follows that this method is also applicable to beryllium and, possibly, to certain other nuclei having isotopes differing by one neutron.

VII. APPLICABILITY OF ELECTRODYNAMICS

As was pointed out, the difference observed in electron-scattering experiments between the actual scattering cross section and the cross section expected for a point charge can be wholly or partly attributed to a violation, at small distances, of the Coulomb law of electric interaction of charges. In other words, the deviations that have been explained by the influence of the finite dimensions of the nucleus may be connected with a violation of the laws of electrodynamics, while the particles themselves may be pointlike. At present there are too many independent indications of the opposite kind for one to hope that such a point of view will prove correct for nuclei. It is still possible, however, to suppose that such an origin may be responsible for the “differences” between electromagnetic and nucleon dimensions; but this too seems improbable, since as experiments and their interpretation improve these differences are steadily decreasing.

In the case of the proton and neutron we face a different situation, characterized by the fact that modern electron-scattering experiments do not make it possible to determine the dimensions of these particles. Meson theories predict that the meson clouds around nucleons have dimensions lying somewhere between the Compton wavelengths for the nucleon and the \(\pi\)-meson, i.e., between \(0.2\) and \(1.4\) fermi, respectively, and probably closer to the smaller value. Although the methods of meson theory are fruitful in the sense of yielding qualitative results, quantitatively they are inadequate, and at present it is difficult to acquire confidence by relying on the predictions of any meson theory. Therefore, consistently with the known facts, one may ascribe the radial dimensions measured by Mac-

Alister, Chambers, and Hofstadter and equal to 0.77 fermi for the proton, to a breakdown of electrodynamics and suppose that in reality meson clouds are very small in comparison with 0.77 fermi. As for experiments on the scattering of electrons by the proton, both explanations are suitable here, and in fact both finite-size effects and the inadequacy of electrodynamics may simultaneously play a role here. The final explanation of the results of scattering experiments must, however, be the same in both cases. There are even some suspicions that finite size and the inadequacy of electrodynamics are two aspects of one and the same phenomenon.

A simple example showing how close these two possibilities are is the following. To explain the experiments on scattering by the cloud of proton charge one must assign finite dimensions and a definite model, for example a Gaussian one. The function of the Coulomb potential then remains finite at a radius equal to zero, whereas for a point proton the potential at zero goes to minus infinity. The algebraic difference of these two potential functions may be regarded as a law of deviation from the Coulomb interaction at small distances. The new potential, taking this deviation into account, corresponds to a “corrected Coulomb law.” Thus we have a special model indicating to what extent the laws of electrodynamics are violated at small distances, i.e. a new law of forces, valid for all ordinary distances, but with new properties at very small distances. An analogous situation obtains for the density of the proton’s magnetic charge. One and the same violation of the laws of electrodynamics describes both cases simultaneously.

There are some possible ways of establishing a distinction between finite dimensions and the inadequacy of electrodynamics; one such method is the scattering of electrons by electrons at sufficiently high energies \((\sim 20\,\text{Bev})\), when the de Broglie wavelength in the center-of-mass system becomes comparable with the small distances within which one may expect a violation of Coulomb’s law. Another method, more accessible with present experimental technique, consists in determining the dimensions of the neutron, which was discussed in Section VI. If the dimensions of the neutron turn out to be identical with the dimensions of the proton, this will indicate that electrodynamics is violated at certain small distances, and all objects having smaller dimensions, including the neutron and the proton, appear to have the same dimensions. On the other hand, if the dimensions (or the shape) of the neutron differ from the dimensions of the proton, then it seems reasonable to assert that the dimensions correspond, at least in part, to a real structural effect and that the laws of electrodynamics probably still remain valid even at small distances. A third method consists in observing whether the radii of very light nuclei, measured, for example, by the method of mirror nuclei, coincide with radii measured by electron scattering. If both methods of measurement give concordant results, this is an argument in favor of the validity of Coulomb’s law. Unfortunately, both methods of determining radii have an accuracy one order of magnitude worse than that needed to clarify the situation. Other methods, making only weak demands on nuclear theory, may also prove suitable for such investigations. It is also possible that experiments on bremsstrahlung or on the production of pairs at large angles, apparently in hydrogen or other nuclei, may help in solving this unresolved problem. It is possible that the study of mesoatoms will help here.

In connection with this, it is appropriate to consider the question of the size of the electron. One may think that many facts relating to the size of the nucleon also have a bearing on the radius of the electron. At present there is no possibility of satisfactorily resolving this question, for if a finite electron does make it possible to explain the phenomenon, then Dirac’s theory, the basis of calculations with electrons, cannot in this case be applied without reservations. In other words, at the present state of our knowledge, the point electron and Dirac’s theory are in agreement with one another. In general, owing to the recoil acquired by the electron in the virtual emission and absorption of photons from the radiation field, it must have dimensions and is not, strictly speaking, a point. But this size is, in essence, equivalent to the action of Schwinger’s radiative corrections. These effects are small and are an order of magnitude smaller than the proton radius. Thus, at present the question of the size of the electron remains unanswered. To obtain a clear answer to the question of the size of the electron, coordinated studies of electron scattering and other measurement methods are necessary.

VIII. COMPARISON WITH OTHER MEASUREMENTS OF NUCLEAR SIZES

There are many possibilities for measuring the size of nuclei and the distribution of charge in them. A recent review by Ford and Hill \(^{9}\) gives a good summary of the available methods. In addition to electron scattering, the following possible methods have also been used:

I. Methods sensitive to charge

a) Coulomb effects in mirror nuclei.
b) Mesoatoms and scattering of \(\mu\)-mesons.
c) Fine structure of X-ray spectra.
d) Isotopic shifts.
e) Hyperfine structure of hydrogen.

II. Methods sensitive to the range of action of nuclear forces

f) Experiments on neutron scattering at intermediate energies (14—25 MeV).
g) Neutron scattering at high energies (90 MeV).
h) Neutron scattering at ultrahigh energies (1.4 BeV).
i) Experiments on proton scattering (20 MeV — 340 MeV).
j) Experiments on \(\alpha\)-particle scattering (13—42 MeV).

Combinations of methods I and II

k) Weizsäcker’s semiempirical formula for the binding energy.
l) Alpha decay.

Probably there exist many other methods for which the dimensions of the nucleus are important and which accordingly make it possible to measure these dimensions \(^{92}\).

The purpose of this review is not a detailed discussion of the results obtained by the various methods listed above. This has been excellently done in the review by Ford and Hill. It is necessary, however, to point out that the present situation does not permit an exact comparison of the data from the various methods, chiefly because of the paucity both of experimental data and of their proper interpretation. Such a situation can be understood if one takes into account our insufficient knowledge of the laws of nuclear forces. Nevertheless, even now it is desirable to make a brief comparison of the data for several cases in which the data themselves and their interpretation are satisfactory. This will be done below.

a) On the basis of a nuclear model that includes exchange effects for pairs of mirror nuclei, Jankowicz calculated a quantity defined by him as the ratio of the Coulomb and meson radii for \(O^{17}\) and \(N^{15}\). These ratios are equal to 1.18 for \(O^{17}\) and 1.08 for \(N^{15}\). The experimentally found ratio for these radii is 1.27 and 1.17, under the assumption that the meson radius of light nuclei is described by (1) with \(r_0 = 1.20\) fermi. This reveals a discrepancy between the meson radius and the radius of mirror nuclei. On the other hand, if the true values of the meson radii lie near \(r_0 = 1.30\) fermi, instead of 1.20 fermi, the discrepancy is removed. It is interesting that electron scattering for nuclei in this region gives values lying close to \(r_0 = 1.32\).

Carlson and Talmi \({}^{93}\) also calculated the influence of parity effects on the Coulomb energy and used these effects to determine nuclear radii. The values of \(r_0\) obtained by these authors are given in Table V. For nuclei with \(A < 11\) and \(A > 28\) these calculations are less reliable than for nuclei with \(A\) between 11 and 28. The table demonstrates the smooth decrease of the radius between \(C^{13}\) and \(Al^{27}\) from the value \(r_0 = 1.34\) to 1.20 fermi. The results for \(A > 28\) show a new increase of \(r_0\) to values close to 1.32 fermi. If these results are not illusory, it follows from them that local changes of \(r_0\) may be expected in other places as well, and that one can hardly expect a smooth variation of \(r_0\) along the entire periodic system.

Table V

Data of Carlson and Talmi \({}^{93}\) for \(r_0\) in the case of mirror nuclei are included in this table

Nucleus	\(r_0\)	Nucleus	\(r_0\)
\(Li^{7}\)	1.489	\(F^{19}\)	1.259
\(Be^{9}\)	1.543	\(Ne^{21}\)	1.248
\(B^{11}\)	1.283	\(Na^{23}\)	1.217
\(C^{13}\)	1.340	\(Mg^{25}\)	1.230
\(N^{15}\)	1.305	\(Al^{27}\)	1.197
\(O^{17}\)	1.262

These data show that generalizations about nuclear radii cannot yet be made from only a few cases. Moreover, as follows from work on electron scattering, a single parameter is not enough to describe the charge distribution, and a simple mean-square radius is insufficient for this purpose. Thus, when comparing results obtained by different methods, one should bear in mind the frequently expressed consideration that different methods may measure different quantities representing radii. This, generally speaking, is especially important when the charge or electromagnetic radius is compared with the nucleon radius obtained by a method sensitive to the range of nuclear forces.

b) The results on \(\mu\)-mesoatoms appear to be in excellent agreement with the results on electron scattering by heavy nuclei. In the case of lead the agreement is within 1%. For lighter nuclei some discrepancies with the results of electron scattering may arise; however, up to now the conclusions obtained in the study of mesoatoms have had less accuracy in this region than in the case of heavy nuclei. There are also discrepancies between the mesoatomic data and the data on mirror nuclei \({}^{8}\). On the other hand, as was said above, the radii of mirror nuclei are in agreement with the radii from electron scattering.

The \(\mu\)-mesoatomic data (\(2P - 1S\) transitions) at present give only one parameter, which for light nuclei is the mean-square radius. For heavy nuclei the measured parameter is not exactly the mean-square radius and depends somewhat on the density distribution. Higher transitions also do not

sensitive, albeit weakly, to the charge distribution. The method of μ-mesoatoms is thus a very promising method that may help clarify the problem of nuclear radii.

c) The fine structure of the splitting of x-ray lines in the \(L\)-series was studied in detail in recent experiments by Schaklett and DuMond \(^{95,96}\), which, however, have not yet received a definite interpretation. The theoretical interpretation of these data presently available gives radii larger than the radii obtained by any other methods. The corresponding theory, however, presents considerable difficulties, and the interpretation of these data is apparently still in its early stage.

d) Isotopic shifts apparently provide a good way of determining the degree of concentration of nuclear matter, but at present this method has not added much that is new to the question of nuclear dimensions \(^{97,98}\).

e) Precise measurements of the hyperfine structure of hydrogen and a very precise determination of the fine-structure constant make it possible to determine the extent of the charge and of the magnetic moment of the proton. Calculations of the upper limit of the mean radius (but not the root-mean-square radius) were carried out by Møllerling et al. \(^{99}\) on the basis of an analysis of the hyperfine structure. The value obtained, \(R_m < 2.5\hbar/MC = 0.5\cdot 10^{-13}\) cm, also lies within the range of values given by electron scattering. Further study of this important source of data on nuclear dimensions is desirable.

f) Experiments with neutrons of 14–25 MeV are divided into two types: 1) capture (or absorption) cross section; 2) scattering cross section.

1) Experiments to determine the absorption cross section were carried out rather long ago by Sherr \(^{100}\) and Amaldi et al. \(^{101}\). The experimental cross section is described by the asymptotic formula

\[ \sigma_T = 2\pi R^2, \tag{61} \]

from which the radius is found. Generally speaking, here \(R\) denotes the radius at which the action of nuclear forces begins, and includes the “radius” of the neutron. The validity of the interpretation expressed by (61) is not entirely obvious, and in the region of neutron energies 14–25 MeV, for medium-heavy nuclei, it is very likely that the factor 2 in (61) should be replaced by 2.5. Substitution of this factor into (1) leads to the radius \(r_0 \simeq 1.30\) fermi, whereas processing the data \(^{122}\) of Sherr and Amaldi gives \(r_0\) close to 1.40 fermi. Sherr expressed his results in the form

\[ R = b + r'_0 A^{1/3}, \tag{62} \]

where \(b = 1.7\) fermi and \(r'_0 = 1.22\) fermi. But this is equivalent to \(r_0 = 1.37\)–1.40 fermi, within the experimental errors. The absorption cross section thus indicates that the values of \(r_0\) are close to \(\simeq 1.3\), if equality (61) is corrected.

2) The neutron scattering cross section at 14 MeV was measured by Kuhn \(^{103}\) and interpreted by Kaller, Fernbach, and Sherman \(^{104}\) using the approximation of the optical model of Fernbach, Serber, and Taylor \(^{105}\). Kaller et al. used a two-term potential function equivalent to a smoothed well. The potential contains an imaginary part responsible for absorption. The radius of the real part of the well was found to be \(1.22 A^{1/3} + 0.74\) fermi, which is equivalent to \(1.4 A^{1/3}\) for medium nuclei. Elliot \(^{106}\) also carried out experiments with 14 MeV neutrons for a number of elements and interpreted his results with the aid of a simple potential-

of the model

\[ V=-V_0(1+\xi),\quad \text{for } r<R_0, \tag{63} \]

where \(V_0=42\) MeV, \(\xi=0.15\), and

\[ R_0=1.32A^{1/3}\ \text{fermi}. \tag{64} \]

This radius is the interaction radius and includes both the “radius” of the neutron and the radius of the nuclear forces. Such an analysis therefore gives small values of the radii, lying not far from the electromagnetic radii (if the radius of the nuclear forces is taken into account).

g) Fernbach et al. \({}^{105}\) developed an optical model of the nucleus, according to which the nucleus is partially transparent at high energies and is regarded as a homogeneous medium characterized by a complex index of refraction. Without giving details here, let us note that the authors interpreted the careful experiments of Cook et al. \({}^{107}\) on the scattering of 90 MeV neutrons by a large number of elements and showed that all the data considered can be satisfactorily fitted by a spherical model of the nucleus with radius \(R\), where

\[ R=1.37A^{1/3}. \tag{65} \]

This is very close to the dependence \(R_0=1.3r^{1/3}\) obtained by Elliott \({}^{106}\), although the two models differ somewhat from each other. Nevertheless, if the radius of the nuclear forces is taken to be close to 1.0 fermi, the obtained values of the radius will closely coincide with the electromagnetic radii.

h) Recently, with the aid of the optical model, important experiments by Kerr et al. \({}^{108}\) on the absorption of 1.4 BeV neutrons by various elements have been interpreted. This led to a value for the radius of the sphere

\[ R=1.28A^{1/3}\ \text{fermi}. \]

Thus these experiments also indicate small radii. These same experiments were analyzed by Williams \({}^{109}\), who showed that the experimental data are in agreement with the charge distribution obtained from electron scattering and with small electromagnetic dimensions.

i) Experiments on proton scattering in the region of 20 MeV were performed by Cohen and Neidigh \({}^{110}\), by Dayton \({}^{111}\), and by others \({}^{112}\). The scattering cross section reveals excellent diffraction minima and maxima, which opens up splendid possibilities for studying many questions of nuclear structure by this method. If one uses a potential in the form of a rectangular well, it proves impossible to fit the obtained data by means of the optical model. However, Woods and Saxon \({}^{114}\) “rounded off” the edge of the well by using the potential

\[ V(r)=\frac{V+iW}{1+\exp\left(\frac{r-r_2}{a_1}\right)}, \tag{66} \]

where \(r_2\) is a parameter characterizing the nuclear dimensions, while \(a_1\) determines the diffuse character of the surface, in other words, the thickness of the “surface” layer. This formula is analogous to the Fermi model used by Hahn et al. \({}^{33}\) and by Hofstadter et al. \({}^{18}\). The Coulomb part of the interaction with the incident proton is ascribed to a uniformly charged sphere of radius \(r_0\), which is consistent with the electromagnetic values. For platinum, the parameters giving good agreement with the experimental curve are \(V=38\) MeV, \(W=9\) MeV, \(r_2=8.24\) fermi, and \(a_1=0.49\) fermi. Recalling that for the Fermi model \(t=4.40a_1\), we obtain \(t=2.16\) fermi, in good agreement with the value obtained by Hahn et al. \({}^{33}\) in the study of electron scattering. The model of Woods and Saxon

insufficiently well satisfies experiment at small \(Z\) (for example, Ni), but the agreement is improved if spin-orbit interaction is taken into account. However, the radius \(r_2\) nevertheless turns out to be larger than the electromagnetic radius. This is perhaps a reflection of the fact that the radius of action of nuclear forces appears as the effective “radius” of a nucleon in those cases when the radius is measured by a method sensitive to nuclear forces.

Hahn and Riddell \(^{116}\) noted earlier that, in order to obtain agreement with experimental data \(^{117}\) on the scattering of high-energy protons (340 MeV), a smooth falloff of the density at the edge of the nuclear sphere is necessary (the optical model). They also found that \(r_0 = 1.25\) fermi gives better agreement with experiment than the old, larger value of the radius.

k) Elastic scattering (13–42 MeV) of alpha particles by heavy nuclei was studied by Farwell and Wegner \(^{118}\). An interpretation of the observed abrupt energy drop was given by Blair \(^{119}\). The theory of these experiments is complicated, and the obtained estimates of radii should apparently be regarded as upper limits. Taking into account the radius of the \(\alpha\)-particle, they obtain the estimate \(r_0 = 1.5\) fermi. It is unclear how realistic this estimate is.

Fig. 52. A proton interacting with a gold nucleus, with the old value of the gold radius \(8.45 \times 10^{-13}\) cm. Owing to the finite size of the nucleon, already at this distance some interaction takes place. \(a\) corresponds to the radius of nuclear forces. It should be noted that the charge density in the proton is large in comparison with that in the gold nucleus.

l) In Weizsäcker’s semiempirical formula for nuclear binding energies there is an electrostatic term for a uniformly charged sphere, proportional to

\[ E_{\text{el}} = -\frac{3}{5}\,\frac{(Ze)^2}{r_0 A^{1/3}}, \tag{67} \]

and thus, in principle, from the packing-factor curve a value of \(r_0\) can be found. Unfortunately, estimates of \(r_0\) from 1.2 to 1.5 fermi are equally consistent with the available data. If a nonuniformly charged model is introduced, the problem becomes more complicated, but if the form of the model is known, a new value of \(r_0\) can be found.

m) We shall not consider the numerous works in which nuclear radii were determined from natural and artificial alpha radioactivity. This problem has a long history, and its study gave the first evidence for the existence of the quantum-mechanical tunneling effect. It is meaningful, however, to draw attention to recent works that introduced certain changes into the old theories. Tolhoek and Brussaard \(^{120}\) used a nuclear model with a surface layer of finite thickness, similar to the model of Hahn et al. \(^{33}\). This theory gives the value \(r_0 = 1.13\) fermi, in contrast to the old determinations, which in most cases gave \(r_0\) from 1.4 to 1.5 \(^{102}\). At present it is still unknown how good the Tolhoek and Brussaard model may be in practice. Finally, Fig. 52 reminds us that, when speaking of nuclear radii, one should take into account the finite dimensions of the nucleon. This graph gives the charge density in a gold nucleus

(in proton charge per cubic centimeter) as a function of distance from the center. The right-hand part of the plot shows the charge density in the final Gaussian model for the proton (root-mean-square radius 0.70 fermi) at a distance of 8.45 fermi between the centers of the proton and the gold nucleus. This distance corresponds to the conventional radius of gold \(r_0 = 1.45\) according to (1). In the graph, \(a\) denotes the approximate value of the radius of nuclear forces. It should be noted that there is a noticeable overlap both of the charges and of the regions where nuclear forces act. Therefore, the existence of a strong interaction at such a distance is not unexpected. This graph, in contrast to Fig. 1, clearly indicates that the cloud of proton charge has a high average density in comparison with attractive nuclei similar to the gold nucleus.

In conclusion to this section, we note that many of the new and some of the old determinations of radii are consistent with small values of the electromagnetic radii. Some other determinations stubbornly give large and differing values. Until the nature of nuclear forces has been clarified, a reasonable comparison of many of these “radii” is impossible. It is to be hoped that the constant references to the quantity \(r_0\) in this chapter will not cause one to attach excessively great importance to it. In reality, to characterize the size and shape of a nucleus it is necessary to have more than one single parameter (such as \(r_0\)).

IX. RESULTS

A listing of the conclusions from this review would require much space. The material presented can be divided into new data, still in need of verification, and older data that have been confirmed better. Some of the old results do not require repetition. Instead of repeating new and old conclusions, it makes sense to present a large part of the phenomena in the form of simple graphs and tables. Despite the incompleteness of such a method, it may help the emergence of new experimental and theoretical ideas. The remarks given below explain the graphs and tables.

In Fig. 53, a large part of the various phenomena observed in electron-scattering experiments is presented. These graphs, which are schematic in character, demonstrate phenomena in extended nuclei similar to carbon, at relatively large scattering angles and moderately high energies, for example \(70^\circ\) and \(400\) MeV. The relative scales in Fig. 53 are not exact.

The primary electron energy is denoted by \(E_0\). \(A\) is the maximum of elastic scattering, and the inelastic maxima arising from scattering by nuclear levels are indicated by the letters \(B, C, D\). The left-hand falloff of the elastic maximum is due to bremsstrahlung. Let us note that, owing to recoil phenomena, the elastic and inelastic maxima are shifted to energies smaller than \(E_0\). The continuous spectrum of inelastic scattering at lower energies is denoted by \(F\). Individual nucleons produce incoherent scattering of electrons, and all these individual cross sections, similar to that shown for the proton \(P\), add together, forming the large maximum \(F\). This broad maximum is located near the maximum of scattering by a free proton \(H\), shown for comparison, but lies below this maximum because of the binding energy of protons and neutrons in the nucleus. A noticeable part of the scattering in this incoherent maximum is connected with magnetic spin-flip processes.

Table VI

This table gives the radial parameters of nuclei indicated in the first column and the corresponding charge (and magnetic-moment) distributions. The determination of all quantities entering the table is given in the text, except for the parameters of Hill’s model (used only for \({}_{82}\mathrm{Pb}^{208}\)). All distances are given in \(10^{-13}\) cm (fermi). The accuracy of determining the parameters of the surface-layer thickness is close to \(\pm 10\%\) and may be somewhat worse in the case of light nuclei. The accuracy of determining the radial parameters is closer to \(\pm 2\%\), except perhaps for the case of Ta. For gold the accuracy is better than \(\pm 2\%\). \(\rho U\) is the charge density (in units of the proton charge per cubic fermi) for the equivalent uniform model. It may be compared with Fig. 1, б. The results for lithium and beryllium should be regarded as preliminary.

Nucleus	Type of charge distribution (see Table I)	Mean sq. radius	Radius of equivalent uniform model \((R)\)	\(r_0=\dfrac{R}{A^{1/3}}\)	Thickness of surface layer	Half-density radius	\(r_1=\dfrac{c}{A^{1/3}}\)	\(\rho U\)	\(A^{1/3}\)	Note	Reference number
\({}_{1}\mathrm{H}^{1}\)	III, IV, V, VII; the magnetic moment has the same distribution	\(0.77 \pm 0.10\)	1.00	1.00	…	…	…	0.239	1.00	The charge distributions in column 2 are equivalent to one another. The rms radius gives the mean value for all models. The magnetic distribution coincides with the charge distribution. The fact that \(R=1.00\) (column 4) is a chance coincidence	42, 55, 68
\({}_{1}\mathrm{D}^{2}\)	Charge distribution calculated from the deuteron wave function with the potentials given by Hulthén et al.	1.96	2.53	2.01	…	…	…	0.0147	1.26	…	71
\({}_{2}\mathrm{He}^{4}\)	III	1.61	2.08	1.31	…	…	…	0.053	1.59	…	42, 49
\({}_{3}\mathrm{Li}^{6}\)	XII	2.78	3.59	1.98	…	…	…	0.0153	1.82	…	75
\({}_{3}\mathrm{Li}^{7}\)	XII	2.71	3.50	1.83	…	…	…	0.0167	1.19	…	75
\({}_{4}\mathrm{Be}^{9}\)	XII	3.04	3.92	1.89	…	…	…	0.0157	2.08	…	75
\({}_{6}\mathrm{C}^{12}\)	XI	2.37	3.04	1.33	\(\sim 2.0\)	\(\sim 2.3\)	1.00	0.051	2.29	\(a=4/3\)	77

Continuation of Table VI

Nucleus	Type of charge distribution (see Table I)	Mean square radius	Radius of the equivalent uniformly charged sphere \((R)\)	\(r_0=\dfrac{R}{A^{1/3}}\)	Surface-thickness half-width	Density half-radius	\(r_1=\dfrac{c}{A^{1/3}}\)	\(\rho U\)	\(A^{1/3}\)	Note	Reference number
\({}_{12}\mathrm{Mg}^{24}\)	\(gU\)	2.98	3.84	1.33	2.6	2.85	0.99	0.051	2.88	…	46
\({}_{14}\mathrm{Si}^{28}\)	\(gU\)	3.04	3.92	1.29	2.8	2.95	0.97	0.056	3.04	…	46
\({}_{16}\mathrm{S}^{32}\)	\(gU\)	3.19	4.12	1.30	2.6	3.28	1.03	0.055	3.18	…	46
\({}_{20}\mathrm{Ca}^{40}\)	Fermi	3.52	4.54	1.32	2.5	3.64	1.06	0.052	3.42	…	33
\({}_{23}\mathrm{V}^{51}\)	Fermi	3.59	4.63	1.25	2.2	3.98	1.07	0.055	3.71	…	33
\({}_{27}\mathrm{Co}^{59}\)	Fermi	3.83	4.94	1.27	2.5	4.09	1.05	0.0662	3.89	…	33
\({}_{49}\mathrm{In}^{115}\)	Fermi	4.50	5.80	1.19	2.3	5.24	1.08	0.0605	4.87	…	33
\({}_{51}\mathrm{Sb}^{122}\)	Fermi	4.63	5.97	1.20	2.5	5.32	1.07	0.0572	4.96	…	33
\({}_{73}\mathrm{Ta}^{181}\)	Fermi plus quadrupole	5.50	\(\sim 7.10\)	\(\sim 1.25\)	\(\sim 2.8\)	\(\sim 6.45\)	\(\sim 1.14\)	0.0491	5.65	In connection with quadrupole effects, the radial distances have the meaning of “effective” radii	61, 88
\({}_{79}\mathrm{Au}^{197}\)	Fermi	5.32	6.87	1.180	2.32	6.38	1.096	0.0581	5.82	…	33
\({}_{82}\mathrm{Pb}^{208}\)	Hill et al. (reference 9) \(n=10,\ s=0\)	\(\sim 5.42\)	\(\sim 7.0\)	1.18	\(\sim 2.3\)	\(\sim 6.5\)	\(\sim 1.09\)	0.057	5.93	The Hill et al. model is analogous to the Fermi model	9
\({}_{83}\mathrm{Bi}^{209}\)	Fermi	5.52	7.13	1.20	2.7	6.47	1.09	0.054	5.935	…	33

In this respect the neutron and the proton are almost equivalent. The study of the question of how the individual maxima of protons and neutrons add up, forming a continuous spectrum for different scattering angles, is of great interest. If experiments are performed with light nuclei, such as, for example, D, H\(^3\), He\(^3\), He\(^4\), Li\(^6\), Li\(^7\), Be\(^9\), etc., and one proceeds to heavier nuclei, it is possible to study the interaction of the incident particles with the surrounding particles. These investigations should provide information on the distribution of momenta in the nucleus.

At still lower energies \(\pi\)-mesons are produced, and the region of the spectrum denoted in Fig. 53 by \(M\) and drawn as a dashed line corresponds to electrons scattered in processes of \(\pi\)-meson production.

Fig. 53. This graph gives an idea of the various phenomena observed in electron scattering under conditions of large momentum transfer. \(A\) corresponds to the maximum of elastic scattering, while \(B\), \(C\), and \(D\) refer to inelastic scattering from individual levels. \(H\) refers to the maximum from scattering by free protons, observed in scattering in air. \(P\) corresponds to the maximum from incoherent scattering by individual protons (or neutrons) in the nucleus. In comparison with the maximum for the free proton \(H\), this maximum is broadened by the motion of the intra-nuclear nucleons. \(F\) gives the sum of such maxima over all \(A\) nucleons of the nucleus. \(M\) corresponds to electrons produced in the scattering of \(\pi\)-mesons. Note that, because of the recoil of the nucleus, all electrons lie on the side of lower energies relative to \(E_0\). In this graph neither the vertical nor the horizontal axes are drawn to scale.

We see that in Fig. 53 most of the processes considered in our review have been indicated.

In Fig. 53 it was impossible to present the angular distributions corresponding to each of the processes shown in the diagram. This would require a graph in a space of several dimensions. Instead, we refer the reader to Fig. 38, Fig. 41, and Table VI.

The analysis of these data is given in Table VI. In those cases where this is possible, the table gives the values of the parameters characterizing the charge distribution. In other cases the charge distribution is described by the root-mean-square radius and the value \(r_0\). The most important features of Table VI are the approximate constancy of the thickness of the surface layer and the decrease of the flat region as one goes to lighter elements. The lightest nuclei and the proton cannot be described in a simple way. In general, the mean charge density at the center of nuclei increases as the atomic number decreases, reaching in the table the charge density maximum for the proton. The quantity \(U\rho\) is given for the equivalent uniform model.

It is of interest to consider the statistical approximation of Jensen and Luttinger \(^{121}\), who applied the Thomas–Fermi model to the nucleus and calculated the mean square of the angular-momentum moments. They showed that a surface-layer thickness of 1.8 fermi gives agreement with the predictions of the shell model. This result also follows from the data presented in Table V. Further work with this model may prove successful. Also desirable are more detailed theoretical approximations. Several such attempts have been made by various authors \(^{122-129}\). It may be hoped that some of these models will prove successful.

X. CONCLUSIONS

In this brief section we note that the method of electron scattering has opened broad possibilities for investigating problems of the sizes and shapes of nuclei and of internal nuclear dynamics. It should be borne in mind, however, that at present we still do not have complete information on these problems; the experiments performed are only a “scratching of the surface.” It is necessary to study a large number of nuclei, to increase the accuracy of the measurements, and to obtain absolute values of the cross sections. Improving the separation between elastic and inelastic scattering is a serious and still unsolved problem. It is necessary to have good electron counters that would distinguish high-energy electrons from other particles with the same momentum. These remarks may be summarized as follows: more extensive and better data are needed. Time is required so that the data obtained from electron scattering can be compared with and checked by data from other methods. Only in this way can the experiments and the conclusions drawn from them be tested. It is desirable to check the validity of electrodynamics by several independent methods, but this apparently requires apparatus that does not yet exist. Time is also needed for improving experimental technique. Further development of theoretical ideas is necessary in order to obtain a clear picture with respect to many details. In addition to the development of the theory of elementary particles, methods for analyzing many-body problems must be available. In this connection one should hope for the development of a successful quantitative meson theory. It may be hoped that, in the course of the next several years, some of these problems will be solved.

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