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ON THE SCATTERING OF FAST PARTICLES BY NUCLEI
L. Rosenfeld*)
One of the most direct and effective methods for investigating the distribution of mass and charge in a nucleus is the study of the scattering of sufficiently fast particles by nuclei. Below I shall set forth recent work carried out in this direction at the University of Manchester.
1. SCATTERING OF FAST ELECTRONS BY NUCLEI
The most complete and accurate experimental data at our disposal at present are the data concerning the elastic scattering of fast electrons; these data give direct information on the distribution of charge inside the nucleus. In order to clarify the connection between the observed angular distribution of scattered fast electrons and the distribution of charge in the nucleus that gives rise to such scattering, Stanford physicists proposed a method consisting in calculating, by numerical integration methods, the angular distribution of scattered particles corresponding to various assumed charge distributions, and in choosing the distribution that best agrees with the empirical data. It is possible, however, to obtain a better idea of this connection from an analytical treatment, which in particular indicates what information about the charge distribution inside the nucleus can in general be obtained from a given group of scattering results. The answer to this question was given in the work of Bodmer¹ and was further developed by Fowler².
Bodmer’s work is based on the observation that the contribution of a partial wave with azimuthal quantum number**) \(k\) to the cross section
*) Translated from the author’s manuscript.
**) We have \(k=-(l+1)\) for \(j=l+\dfrac{1}{2}\) and \(k=l\) for \(j=l-\dfrac{1}{2}\).
the scattering depends only on the magnitude of the ratio
\[ K_k(r)=\frac{f_k(r)}{g_k(r)} \tag{1} \]
of the “small” and “large” radial components of the Dirac wave function, taken at some distance \(r=r_1\), large compared with the radius of the charge distribution. Further, for a given spherical distribution of electric charge \(V(r)\) and for an initial energy \(E\), the function \(K_k(r)\) must satisfy a differential equation of the Riccati type:
\[ \frac{dK_k}{d\xi}=-\frac{2k}{\xi}K_k-(\varepsilon+1-U)K_k^2-(\varepsilon-1-U), \tag{2} \]
\[ \left( \xi=\frac{r}{\lambda_{\text{Compton}}};\qquad \varepsilon=\frac{E}{mc^2};\qquad U=\frac{V}{mc^2} \right) \]
with the appropriate boundary conditions. The solution of this equation, for not too large values of \(\varepsilon\), can be obtained by means of a rapidly convergent iterative process. In this way one can avoid solving the Dirac equation, which cannot be obtained in analytic form for an arbitrary function \(V(r)\), and obtain an explicit expression in terms of the potential \(V(r)\) and the charge-density distribution function \(\rho(r)\), suitable for discussion. Thus it was found that, for energies less than approximately \(50\) MeV (above this value the method becomes inapplicable), the contribution of each partial wave depends only on the corresponding “moment” of the charge distribution
\[ \int \rho(r)\, r^{2|k|}\, d v . \tag{3} \]
In particular, the \(S\)-wave \((k=-1)\), which gives the principal effect, determines only one parameter—the “quadratic moment” of the charge distribution. Instead of this parameter it is more convenient to introduce the radius \(R\) of a uniform spherical distribution that produces the same quadratic moment; such an equivalent radius of the nucleus is defined by the relation
\[ \frac{3}{5}ZeR^2=\int \rho r^2\, d v . \tag{4} \]
Experiments carried out at such comparatively low energies do not make it possible to obtain anything more concerning the charge-density distribution than this equivalent radius.
Bodmer’s method becomes unsuitable as soon as the radial component \(g_k(r)\) acquires zeros within the region of the charge distribution. This difficulty can be circumvented, as was shown
by Fowler^2, by the simple use of the well-known substitution
\[ K_k=\operatorname{tg}\varphi_k . \tag{5} \]
The equation for \(\varphi_k\), obtained from equation (2), can be solved just as simply as equation (2). In this case it turns out that the parameter of the charge distribution, determined by the partial wave of order \(k\), is now, instead of the moment (3), the expression
\[ \int\limits_{0}^{2|k|/\varepsilon} U(\xi)\left[ \frac{\xi}{\varepsilon^2\xi^2+(2|k|+1)^2} \right]^{2|k|} d\xi + \int\limits_{2|k|/\varepsilon}^{\xi_1} U(\xi)\,d\xi . \tag{6} \]
Expression (6), for large \(|k|\), goes over into the moment (3); however, for the most significant partial waves with \(|k|\leq 2\), the value of the parameter (6) is to some extent sensitive to the form of the charge distribution. According to Fowler’s estimates, the dependence of \(S\)-scattering on the form of the charge distribution will be, for example, most distinct for electron energies of the order of \(70\) MeV, when the relative changes of \(\varphi_1\) may reach \(30\%\).
In connection with the Stanford results, which concern the elastic scattering of electrons in the energy region \(150\)—\(200\) MeV, it should be pointed out that the values of the equivalent radius obtained up to now give a definite deviation from the usual formula \(R=r_0A^{1/3}\). This deviation consists in the fact that the factor \(r_0\) is not constant, but continuously decreases from values of the order of \(1.35\) sp*) for \(\mathrm{C}^{12}\) to the constant value \(1.2\) sp for elements with atomic weight \(A\gtrsim 88\). Such a tendency is in agreement with less direct data on the \(\beta\)-decay of light mirror nuclei^4, and its possible existence had already been indicated on the basis of the latest data of 1953 (see^1, p. 1050); at that time, however, the necessity was stressed of corroborating this point of view by scattering experiments. Now, apparently, we have obtained such confirmation. This effect is of special interest, since it indicates an incipient violation, in heavy nuclei, of the rigid saturation conditions of nuclear interactions.
It should now be noted that the scattering process is considered on the basis of the idea of a stationary charge distribution inside the nucleus. Obviously, the action of the stationary distribution represents only the averaged action of the true distribution of the protons constituting the nucleus. It is therefore necessary to clarify to what extent the distribution of charges
*) For convenience, the abbreviation sp has been introduced here for units of \(10^{-13}\) cm.
of individual protons affects the angular distribution of the scattered electrons. In order to consider this “granulation effect,” we represent the proper function of the scattering state in the form
\[ \psi=\psi_S+\psi_g, \tag{7} \]
where \(\psi_S\) represents the scattering state corresponding to the mean potential \(\langle V\rangle\), and \(\psi_g\) the corresponding contribution arising from the difference \(V-\langle V\rangle\) between the actual potential \(V\) and its mean value \(\langle V\rangle\). It is easy to see that \(\psi_g\) can be represented in the form
\[ \psi_g=\lim_{\varepsilon\to 0}(E-K_e-H_n-V+i\varepsilon)^{-1}(V-\langle V\rangle)\psi_S, \tag{8} \]
where \(K_e\) is the kinetic energy of the scattered electron and \(H_n\) is the Hamiltonian of the nucleus. Further, let \(\psi_k\) correspond to the state of the system in which the nucleus is in the ground state and the electron moves freely with momentum \(\hbar\mathbf{k}\). The differential cross section of elastic scattering for incident electrons with momentum \(\mathbf{k}_0\), acquiring momenta \(\mathbf{k}\), making an angle \(\theta\) with the initial momentum \(\mathbf{k}_0\), can be written in the form
\[ \sigma(\theta)=\frac{(2\pi)^2 k_0^2}{\hbar^2 c^2}|f(\theta)|^2,\qquad f(\theta)=\langle\psi_k|V|\psi\rangle . \tag{9} \]
It follows from (8) and (9) that a lower bound for the influence of the granulation effect on \(f(\theta)\) can be estimated with the aid of the expression
\[ f_g(\theta)\simeq \lim_{\varepsilon\to 0} \langle\psi_k|V(E-K_e-H_n+i\varepsilon)^{-1}(V-\langle V\rangle)|\psi_{k_0}\rangle . \tag{10} \]
A detailed discussion and estimate of this expression were carried out by Squires\(^{5}\). For electron energies of the order of those used at Stanford, it proved possible, on the one hand, to describe the state of the protons by the “impulse approximation,” and, on the other hand, to neglect their recoil energy; this leads to the fact that in relation (10) the term \(E-H_n\) should be replaced by the constant value of the energy of the incident electrons \(E_{in}\). Then relation (10) can be rewritten in a somewhat different form. Put
\[ \rho(\mathbf{r}_1,\mathbf{r}_1)=\int dv_3\ldots dv_A\, |\varphi(\mathbf{r}_1,\ldots,\mathbf{r}_A)|^2 . \tag{11} \]
This expression represents the spatial correlation function of the nucleus when all nucleons are in the ground state.
states \(\psi(\mathbf r_1,\ldots,\mathbf r_A)\); let, furthermore, the expression
\[ \rho(\mathbf r_1)=\int dv_2\rho(\mathbf r_1,\mathbf r_2) \tag{12} \]
represent the mean nuclear density. If we now denote by \(V_i\) the potential of the \(i\)-th proton and introduce, for brevity, the notation
\[ \langle ij\rangle \equiv \lim_{\varepsilon\to 0}\langle \mathbf k|V_i(E_{in}-K_e+i\varepsilon)^{-1}V_j|\mathbf k_0\rangle, \tag{13} \]
then we immediately find:
\[ \begin{aligned} f_g^{(0)} \simeq {}& Z\int dv_1\rho(\mathbf r_1)\langle 11\rangle - Z\int dv_1dv_2\rho(\mathbf r_1)\rho(\mathbf r_2)\langle 12\rangle +{}\\ &{}+Z(Z-1)\int dv_1dv_2\left[\rho(\mathbf r_1,\mathbf r_2)-\rho(\mathbf r_1)\rho(\mathbf r_2)\right]\langle 12\rangle . \end{aligned} \tag{14} \]
The first two terms in this formula correspond to repeated scattering by the same proton; the third term corresponds to the effect of the spatial correlation of protons. The granulation effect should be greater for heavy nuclei and at large scattering angles. Squires considered the case of Au, since for this case there are data on the scattering of electrons of energy \(183\) MeV. He estimated the contribution of the granulation effect to the differential cross section at \(\theta=90^\circ\); it turned out that this contribution amounts to about \(4\%\) of the measured value. It is possible that corrections for the magnetic field and radiation, as well as purely experimental possibilities for the accuracy of the measurements, are of approximately the same magnitude. Be that as it may, the granulation effect entails no greater influence than a change of only a few percent in the charge-density distribution obtained from electron-scattering data.
2. SCATTERING OF \(\mu\)-MESONS THROUGH LARGE ANGLES
The first experiments*) on the scattering of \(\mu\)-mesons revealed an anomalously large cross section for large scattering angles. This phenomenon was carefully studied in Manchester for a wide range of meson energies. A Wilson chamber was used for the investigation, into which plates of lead and iron were introduced \(^{6,7}\). According to their energy, the observed mesons could be divided only into separate groups with a broad spectral distribution; for each group the angular distribution was obtained
*) A review of these data is given in \(^{6}\). See also \(^{8}\).
from a very small number of observed events, especially for scattering at large angles; thus the uncertainty inherent in these measurements is already quite large simply on statistical grounds.
The theory with which the experimental data are compared is the theory of multiple scattering, which depends rather sharply on the assumed energy spectrum of the incident mesons\(^8\) and takes into account, in a rather indirect way, the charge distribution of the individual scattering nuclei. Nevertheless, the influence of the finite extension of the nucleus, at least for energies \(\gtrsim 1\) Bev corresponding to the energy of the highest group present in the experiment, is quite appreciable.\(^7\) For iron it proves very difficult to decide, on the basis of the experimental data, which theoretical curve should be preferred: the curve calculated for a nucleus of finite radius, or the curve for the limiting case of a point charge. The data for lead reveal a certain predominance of scattering through large angles; the differential cross section shows an angular dependence closer to the angular distribution corresponding to scattering by a point charge than to the distribution corresponding to a spatially distributed charge. But even in this case, according to the most recent critical analysis of the experimental data,\(^8\) the existence of such an “anomaly” can by no means be regarded as fully established.
The possible cause of anomalous scattering of this type has given rise to numerous conjectures. Although it now seems quite possible that any existing anomaly is considerably smaller than it was formerly thought to be, the theoretical discussion of the question should in any case conclude with an investigation of the electromagnetic interaction of \(\mu\)-mesons with nuclei. This question was discussed chiefly by Gatto\(^9\) and Fowler.\(^{10,11,12}\) It was established first of all that inelastic effects (including excitation of the “giant” dipole resonance, which was considered as a possible phenomenon) do not make any appreciable contribution to the scattering cross section at large angles. Moreover, the supposition that anomalous scattering might be due to an anomalous magnetic moment of the \(\mu\)-meson must be rejected, since\(^ {12}\) the existence of an appreciable anomalous magnetic moment of the \(\mu\)-meson is excluded by data on bremsstrahlung and pair production produced by \(\mu\)-mesons. On the other hand, it appears that the stars produced by \(\mu\)-mesons underground have a larger cross section than the cross section obtained from the assumption of a purely electromagnetic interaction;\(^{12}\) however, such estimates depend to a considerable extent on the assumed mechanism of energy exchange between the meson and the nucleus.
3. SCATTERING OF FAST NEUTRONS BY α-PARTICLES
The scattering of nucleons by nuclei is not amenable to as detailed a theoretical investigation as the scattering of electrons, since the laws of nuclear interaction are not known at present. Nevertheless, so global a description of the process as is furnished by the optical model of the nucleus gives a surprisingly good explanation of the general regularities of the phenomenon. According to this model, one should expect elastic scattering to depend primarily on the real part of the nuclear potential and, consequently, scattering gives information only about this quantity, which represents a certain average interaction between the nucleons constituting the nucleus and the incident nucleon*).
The case of neutron scattering by α-particles is of special significance, since it is sufficiently simple to permit a more detailed study of the interaction, if certain assumed laws of interaction between nucleons are adopted. For this reason the data relating to the scattering of slow neutrons were recently reconsidered from this point of view by Spie[^14]; he showed, on the one hand, that the observed phases of \(P\)- and \(S\)-scattering are, within the limits of experimental error, compatible with a definite attractive potential and, on the other hand, that the interaction potential between the neutron and the α-particle can, to a good approximation, be represented in the form of a certain central interaction between nucleons, including direct and exchange interaction forces, similar to the forces introduced by Serber. The data, however, are not sufficiently accurate to make it possible to say anything definite about the form of the neutron–α-particle potential.
In this respect the results of experiments on the scattering of fast neutrons indicate that the effective nuclear “radius” for this process has the form
\[ R = r_0 A^{1/3} + a, \tag{15} \]
assuming the existence of a “rim” of thickness \(a\), approximately equal to the range of the nuclear forces. The study of the scattering of fast neutrons by α-particles, carried out by Squire[^5], gave us some understanding of the reasons for the appearance of such a “fringing field.” Analyzing the contribution made to the scattering by various kinds of interactions between nucleons, Squire showed that it is precisely the Wigner-type component that extends beyond the nuclear surface and that it is chiefly responsible for the existence of the “fringing field.” He also established that for the most—
*) This point of view was thoroughly developed by Restell[^13], who proposed a general method for determining the magnitude of the nuclear potential from a suitable set of experimental scattering data.
for better agreement with the experimental data one should take a combination similar to the combination used by Serber (although a larger amount of experimental data is required in order to draw a final conclusion). In addition, he showed that the polarization of the $\alpha$-particle caused by the neutron cannot be neglected, but that the polarization energy by itself cannot give rise to the appearance of a peripheral field*).
References
- A. R. Bodmer, Proc. Phys. Soc. A66 (1953), 1041.
- G. N. Fowler, Proc. Phys. Soc. A68 (1955), 559.
- R. Hofstadter et al., Phys. Rev. 99 (1955), 1509; 101 (1956), 1131.
R. H. Helm, HEPL, report No. 40, Feb. 1956. - O. Kofoed-Hansen, Nuc. Phys. (in press).
- E. J. Squires, Manchester Ph. D. thesis, 1956.
- B. Leontic and A. W. Wolfendale, Phil. Mag. 44 (1953), 1101.
- I. B. McDiarmid, Phil. Mag. 45 (1954), 933.
- G. D. Rochester and A. W. Wolfendale, Phil. Mag. 45 (1954), 980.
- R. Gatto, Nuovo Cim. 10 (1953), 1559.
- G. N. Fowler, Nuc. Phys. 1 (1956), 119.
- G. N. Fowler, Nuc. Phys. 1 (1956), 125.
- G. N. Fowler and A. W. Wolfendale, Nuc. Phys. (in press).
- P. Rostall, Manchester Ph. D. thesis, 1955.
- E. von der Spuy, Nuc. Phys. (in press).
*) To estimate the direct influence of the interaction between nucleons, Squires used the Born approximation and then checked its applicability. The polarization energy was calculated in the “impulse approximation.”