DIFFRACTION SCATTERING OF FAST NEUTRONS AND CHARGED PARTICLES

A. I. Akhiezer and I. Ya. Pomeranchuk

INTRODUCTION

It is known that if obstacles—opaque bodies—lie in the path of propagation of light, then deviations from the laws of geometrical optics may be observed, the so-called diffraction phenomena, characterized by the absence of a sharp boundary between the regions of light and shadow. Diffraction phenomena are directly connected with the wave nature of light and appear the more strongly, the smaller the dimensions of the opaque bodies in comparison with the wavelength of light. Diffraction phenomena are observed not only in the case of the propagation of light; for example, the diffraction of electrons and of other particles is known, in which the wave properties of these particles are manifested.

A very important and interesting example of the manifestation of diffraction phenomena is provided by the scattering of sufficiently fast particles by atomic nuclei. In order to visualize more clearly the picture of this phenomenon, let us first consider the scattering of neutrons—particles that have no electric charge. The scattering of such neutral particles by nuclei under certain conditions must be analogous to the diffraction of light by an obstacle having the shape and dimensions of the nucleus. Diffraction phenomena in the region of light are observed if, in the path of propagation of the light, there is an opaque, i.e. light-absorbing, screen. If the scattering of neutrons by nuclei is considered, then, in a certain energy region, nuclei likewise behave with respect to the neutrons incident upon them as opaque, absorbing screens. For this the neutron energy must not be too small and must not fall within the region of the resonance energies of the nucleus, since in this region the role of each individual nuclear level is significant and therefore a phenomenological treatment of the nucleus as a certain absorbing body, analogous to an opaque optical screen, is impossible. On the other hand, at very large

at such energies the nucleus becomes transparent for particles incident upon it, and therefore in this case the analogy with the propagation of light near an opaque screen also disappears. However, there exists a broad range of energies in which the mean free path of the incident particle in nuclear matter is small in comparison with the dimensions of the nucleus. Under such conditions the nucleus is opaque for particles incident upon it.

This corresponds to energies of the incident particle less than approximately \(30\text{--}40\) MeV for light nuclei and \(200\) MeV for heavy nuclei\(^1\). If the energy of the particle is below this limit, but sufficiently large that the role of individual nuclear levels is not manifested, then the nucleus may be regarded purely phenomenologically as a certain body, opaque to particles, which absorbs them. If, moreover, the particles are electrically neutral, then we must expect a complete analogy between the scattering (elastic) of particles by a nucleus and the diffraction of light by an opaque body having the shape and dimensions of the nucleus. Theoretical calculations and experiments confirm this proposition.

It is possible, however, to go further and not necessarily restrict oneself to the case of neutral particles. For neutral particles especially simple relations hold, but diffraction phenomena must also be observed in the case of scattering of sufficiently fast charged particles. The diffraction pattern in this case is much more complicated than in the case of light, and it goes over into the ordinary pattern if the charge of the particle is equal to zero.

In the present article we shall set forth in detail the theory of diffraction phenomena occurring in the scattering of fast particles by absorbing nuclei, considering both neutral and charged particles.

1. PROBABILITY OF STICKING

Let us determine under what conditions a nucleus may be considered absolutely absorbing (i.e. opaque) with respect to neutrons incident upon it. To answer this question it is necessary to consider the general properties of collisions between neutrons and nuclei.

It turns out to be possible to describe a number of properties of nuclear reactions occurring with heavy nuclei without a detailed theory of nuclear forces. This is connected with the possibility of applying, in the study of the properties of heavy nuclei containing a large number of strongly interacting particles, the methods of statistical physics\(^2,\,3,\,4,\,5\).

In order to clarify the features of the course of nuclear reactions, let us compare atomic and nuclear collisions. In the well-studied case when the interaction energy of an incident particle (say, an electron) with the individual electrons of an atom is small in comparison with the energy of the particle itself, the transfer of energy from the particle to an atomic electron is a rare phenomenon. The most...

it is probable that particles will pass through the atom without loss of energy or, in other words, that the particles will be elastically scattered.

A small magnitude of the interaction is associated with a small magnitude of the acceleration of the incident particle. Hence it is clear that the time during which the particle is in the region of the atom is, in order of magnitude, equal to the quotient of some length, of the order of the dimensions of the atom, divided by the velocity of the particle.

Different relations obtain in nuclear collisions, which occur with the participation of particles whose energy is considerably less than the binding energy of atomic nuclei. An essential circumstance is that the interaction between the colliding particle and the nucleus is very large. The energy of this interaction is of the same order of magnitude as the energy of interaction between the individual particles constituting the nucleus. The strong interaction between the colliding particle and the nucleus leads to the result that, very soon after contact with the nucleus, the incident particle loses a considerable part of its energy, which is transferred to the other particles composing the initial nucleus.

This distribution of the energy, initially concentrated on the incident particle, among all the available particles is such that none of them, generally speaking, will possess sufficient energy immediately to leave the system that has been formed—the initial nucleus plus the incident particle. Only after a long time has elapsed, when, owing to fluctuations, some particle acquires sufficient energy to overcome the attractive forces acting on it from the other particles, will it be able to leave the nucleus.

We thus arrive at an important conclusion: the colliding particle and the nucleus may be regarded as a single quantum-mechanical system which exists, without disintegrating, for a long time, if the latter is measured on the characteristic nuclear scale. This means that the lifetime of the system is considerably greater than the time during which the particle colliding with the nucleus can traverse a distance of the order of the nuclear dimensions without being delayed in it. If the velocity of the particle is taken to be of the order of \(4 \cdot 10^9\) cm/sec and the dimensions of the nucleus are taken to be of the order of \(10 \cdot 10^{-13}\) cm, then the “characteristic nuclear time” will be of the order of \(3 \cdot 10^{-22}\) sec.

During the lifetime of the system under consideration its properties do not differ from those of ordinary nuclei in highly excited states. It is therefore customary to call the system consisting of the nucleus plus the incident particle an “intermediate nucleus.”

We said above that only after a long time has elapsed can the excitation energy of the compound nucleus accidentally become concentrated on some particle, which can therefore leave the compound nucleus. In this case it is by no means necessary that the particle emitted from the compound nucleus be of the same kind as the original one.

particle that led to the formation of this nucleus. On the contrary, generally speaking, it is unlikely that the nature of both particles should be the same, since there are many possible modes of decay of the compound nucleus. It is even less likely that, with the same nature of the incident and emitted particles, the internal state of the nucleus would remain unchanged; it is more probable that the nucleus remaining after the emission of the particle will be in an excited state.

If the emitted particle is of the same kind as the incident one, and the internal states of the initial nucleus and of the nucleus remaining after the decay of the compound nucleus coincide, then we are dealing with elastic scattering of particles. One may say that elastic scattering is a comparatively rare event in nuclear collisions involving fast particles, since the decay of the compound nucleus may, generally speaking, occur in various ways, in which the emitted particle differs in its nature from the incident particle, while the remaining nucleus is in an excited state. Thus, in a nuclear collision one must distinguish two stages: the formation of a quasistationary, long-lived compound nucleus and the decay of the latter. In other words, a nuclear collision proceeds according to the scheme: initial nucleus \(+\) incident particle \(\to\) intermediate nucleus \(\to\) residual nucleus \(+\) emitted particle.

Since the binding between the particles in the compound nucleus is extremely strong and this nucleus lives relatively “long,” the second stage of the decay of the compound nucleus must be regarded as a separate event, independent of the first stage—the formation of the compound nucleus, i.e. the “entangling” of the incident particle in nuclear matter. The final result of a nuclear collision is determined by competition among the various possible processes of decay of the compound nucleus compatible with the general conservation laws.

In order for the concept of a compound nucleus as a quasistationary system to be meaningful, it is evidently necessary to assume a sufficiently large total number of particles among which the energy, initially associated with the incident particle, is distributed. As for the energy of the incident particle, it must not be too large, since in the region of high energies the nucleus becomes “transparent” to the particle. The concept of a compound nucleus may be used if the mean free path of the incident particle in nuclear matter is small in comparison with the dimensions of the nucleus. For light nuclei this corresponds to energies of the incident particle less than approximately \(50\ \mathrm{MeV}\), and for heavy nuclei—to energies less than \(250\ \mathrm{MeV}\). Here we shall consider the case of sufficiently small energies of the incident particle, when it is legitimate to use the concept of a compound nucleus.

Like ordinary stable nuclei, the compound nucleus is characterized by the spectrum of its energy levels. Since the compound nucleus is only a quasistationary, and not a stable—

system, then these levels possess a certain width. The width of a level is related to the lifetime of the system situated at this level.

According to the uncertainty principle, the width of a level \(\gamma\) is, in order of magnitude, equal to the quotient obtained by dividing the quantum constant \(\hbar\) by the lifetime \(\tau\):

\[ \gamma \simeq \frac{\hbar}{\tau}. \tag{1,1} \]

The probability \(W\) per unit time for the transition of the system from the state corresponding to the level under consideration into any other possible state is equal to

\[ W=\frac{1}{\tau}. \tag{1,2} \]

If the decay of a compound nucleus situated in some state can occur by various paths, then \(W\) may be represented in the form

\[ W=\sum_i W_i, \tag{1,3} \]

where \(W_i\) is the probability of decay of the \(i\)-th type. Correspondingly, one may speak of the partial width \(\gamma_i\) of the level under consideration, corresponding to decay of the \(i\)-th type, understanding by \(\gamma_i\) the probability \(W_i\) multiplied by \(\hbar\). Obviously, the total width of the level is equal to the sum of its partial widths:

\[ \gamma=\sum_i \gamma_i. \tag{1,4} \]

One may, for example, speak of the neutron width \(\gamma_n\), the radiative width \(\gamma_\gamma\), the widths \(\gamma_\alpha\) and \(\gamma_p\) with respect to the emission of an \(\alpha\)-particle and a proton, etc., understanding by these quantities the probabilities of emission from the compound nucleus of a neutron, a \(\gamma\)-quantum, an \(\alpha\)-particle, a proton, etc., measured on the energy scale.

Since the lifetime of a compound nucleus is large when measured in characteristic nuclear units, the widths of the levels of the compound nucleus turn out to be small in comparison with nuclear binding energies. It does not follow from this, however, that the widths of the levels are necessarily small in comparison with the spacings between neighboring levels. In principle two cases are possible: when the widths of the levels are small in comparison with the spacings between neighboring levels, and when the widths are of the same order as the spacings between levels, or even greater than them. The first case occurs in the region of excitation energies of compound nuclei that arise when a nucleus is irradiated by slow particles (for example, slow neutrons). The second case takes place for levels of compound nuclei arising when nuclei are bombarded by fast particles. In this case the spacing between neighboring levels is usually smaller than their widths.

Only in the first case can one speak of separate discrete levels; in the second case we are dealing essentially with a continuous spectrum. A very important feature of the first case is the sharply expressed dependence of the probability or effective cross section for formation of the compound nucleus on the energy of the incident particle. This dependence has a resonance character: for certain values of the energy of the incident particle the probability of formation of the compound nucleus, and consequently also the cross section for some process of nuclear disintegration, become especially large. Resonance phenomena of this kind are in fact observed, for example, in the interaction of slow neutrons with nuclei.

In the second case (overlapping levels) there is no sharply expressed resonance dependence of the cross section on the energy. The magnitude of the cross section in this case is due to the action of a large number of levels. The observed cross section is the result of averaging the cross sections associated with each level over a large number of levels. The study of this case is simplified by the fact that, at large excitations, statistical methods may be used.

If the excitation energy of the compound nucleus is concentrated on some particle, then the latter leaves the compound nucleus. The following types of transformation of the compound nucleus are possible: emission of a neutron or of a charged particle (for example, a proton or an $\alpha$-particle), emission of a $\gamma$-quantum and, finally, fission of the nucleus.

Of the processes listed above, fission occurs only in the case of the heaviest nuclei. As for the emission of charged particles, the probability of emission of a charged particle, because of barrier effects, is in general less than that of a neutral one.

We know that each possibility of decay is associated with a certain broadening of the level of the system. Therefore, to each decay process one may assign a partial level width. Let us consider the partial widths corresponding to neutron emission and to emission of a $\gamma$-quantum, which are usually called the neutron and radiation widths.

The neutron width $\gamma_n$ depends essentially on the excitation energy of the compound nucleus. If the latter is less than the binding energy of the neutron in the nucleus, then $\gamma_n$ is equal to zero. Below we shall see that if the excitation energy is close to the binding energy of the neutron, so that the kinetic energy of the neutron emitted from the nucleus is small, then $\gamma_n$ is proportional to the velocity of the emitted neutron.

With increasing excitation energy the neutron width increases strongly. This is promoted, above all, by the increase in the velocity of the neutron, and also by the circumstance that, at large excitation energies of the compound nucleus, the nucleus remaining after emission of the neutron may itself be in an excited state. Therefore the number of possibilities associated with emission of a neutron-

DIFFRACTION SCATTERING OF FAST NEUTRONS

significantly increases, which also leads to a strong increase of \(\gamma_n\).

The radiative width \(\gamma_r\) is different from zero for arbitrarily small excitations of the nucleus. For excitation energies not exceeding the binding energy of the neutron in the nucleus, the radiative width coincides with the total width of the level, since the neutron width then vanishes. At sufficiently large excitation energies the radiative width is considerably smaller than the neutron width. In practically all cases in which one has to deal with nuclei of medium atomic weight (\(\sim 100\)), \(\gamma_r\) does not exceed, in order of magnitude, \(0.1\ \mathrm{eV}\), and is rather less than this value. The neutron width for such nuclei, at energies of the emitted neutrons of the order of \(10\text{--}100\ \mathrm{KeV}\), already exceeds the radiative width.

Let us determine how the partial widths may be related to other quantities characterizing the state of the nucleus. For this purpose we shall use the so-called principle of detailed balance, which establishes a relation between the probabilities of direct and inverse transitions. In the present case the problem is to establish the relation between the width of a level and the formation cross section of a compound nucleus.

Let us imagine a large vessel of volume \(\Omega\), containing nuclei of the most diverse kinds. Nuclear reactions may occur between the nuclei. We shall consider a state of statistical equilibrium, in which the number of disintegrations per unit time of nuclei of species \(C\), proceeding according to the scheme \(C \to A + a\) (\(A, a\) being the reaction products), is equal per unit time to the number of recombinations of the type \(A + a \to C\).

Denote the probabilities of the disintegration and recombination processes, referred to unit time, respectively by \(W^C_{Aa}\) and \(W^{Aa}_C\). The condition of equality between the number of recombinations of the type \(A + a \to C\) and the number of disintegrations of the type \(C \to A + a\) may be written in the form

\[ g_C W^C_{Aa} = g_A g_a W^{Aa}_C, \tag{1,5} \]

where \(g_A\) is the statistical weight of the state in which the nucleus \(A\) is found.

Let \(E_A, E_C,\ldots\) denote the values of the internal energy of the nuclei \(A, C,\ldots\), understanding by internal energy the difference between the total energy of the nucleus at rest and the rest energy of its constituent particles. Let, further, \(\mathbf{p}_A, \mathbf{p}_C\) denote the momenta of the nuclei \(A, C,\ldots\). We shall assume that the momentum of the nucleus \(A\) lies in the interval between \(\mathbf{p}_A\) and \(\mathbf{p}_A + d\mathbf{p}_A\).

If \(E_A\) belongs to a discrete spectrum, then by the statistical weight \(d\) we must understand the product \((2i+1)\) (where \(i\) is the angular momentum of the nucleus \(A\) in the state with energy \(E_A\)) by

\[ \frac{\Omega\, d\mathbf{p}_A}{(2\pi\hbar)^3}. \]

If \(E_A\) belongs to a continuous spectrum and the internal energy of the nucleus \(A\) lies in the interval \((E_A, E_A + dE_A)\), then the statistical

the weight \(g_A\) is equal to

\[ g_A=\rho(E_A)\,dE_A\,\frac{\Omega\,dp_A}{(2\pi\hbar)^3}, \]

where \(\rho(E_A)\,dE_A\) is the number of levels of nucleus \(A\) in the energy interval \((E_A,E_A+dE_A)\), each level being counted as many times as its degeneracy.

By \(C\) we mean the compound nucleus, whose energy lies in the continuous spectrum; the state of nucleus \(A\), however, we consider discrete. By \(a\) we mean the particle leaving nucleus \(C\). Thus,

\[ g_C=\rho(E_C)\,dE_C\,\frac{\Omega\,dp_C}{(2\pi\hbar)^3}; \qquad g_A=(2i+1)\,\frac{\Omega\,dp_A}{(2\pi\hbar)^3}; \tag{1,6} \]

\[ g_a=(2s+1)\,\frac{\Omega\,dp_a}{(2\pi\hbar)^3}, \]

where \(s\) is the spin of particle \(a\), and \(i\) is the angular momentum of nucleus \(A\).

Substituting (1,6) into (1,5), we obtain:

\[ W^C_{Aa\rho}(E_C)\,dE_C\,dp_C = (2i+1)(2s+1)\,W^{Aa}_C\,dp_a\,dp_C\,\frac{\Omega}{(2\pi\hbar)^3}. \tag{1,7} \]

Let us introduce the momentum vector of the relative motion of particles \(A\) and \(a\), which we shall denote by \(\mathbf p\),

\[ \mathbf p= \frac{M_aM_A}{M_a+M_A}(\mathbf v_a-\mathbf v_A) = \frac{M_A\mathbf p_a-M_a\mathbf p_A}{M_a+M_A} \]

(here \(M_A, M_a\) are the masses of particles \(A\) and \(a\), and \(\mathbf v_A, \mathbf v_a\) are their velocities). It is easy to verify that the equality

\[ dp_A\,dp_a=dp_C\,dp \]

holds. (This equality means that the Jacobian of the transition from the variables \(\mathbf p_A,\mathbf p_a\) to the variables \(\mathbf p_C=\mathbf p_A+\mathbf p_a\) and \(\mathbf p\) is equal to unity.) Using this equality, we rewrite (1,7) in the form

\[ W^C_{Aa}\,dE_C = (2i+1)(2s+1)\, \frac{1}{\rho(E_C)}\,W^{Aa}_C\,dp\, \frac{\Omega}{(2\pi\hbar)^3}. \tag{1,8} \]

We shall now use the law of conservation of energy, according to which

\[ E_C=E_A+E_a+E, \]

where \(E_a\) is the binding energy of particle \(a\), and \(E\) is the energy of the relative motion of particles \(A\) and \(a\):

\[ E=\frac{p^2}{2M},\qquad M=\frac{M_aM_A}{M_a+M_A}. \]

It follows from the written equality that \(dE_C=dE\). Replacing \(dp\) by \(4\pi p^2\,\dfrac{dp}{dE}\,dE_C\), we obtain from (1,8) the relation

\[ W^C_{Aa} = W^{Aa}_C\, \frac{(2i+1)(2s+1)}{\rho(E_C)}\, \frac{4\pi M p\Omega}{(2\pi\hbar)^3} = \]

\[ = W^{Aa}_C\, \frac{(2i+1)(2s+1)}{(2j+1)}\, D_j\, \frac{4\pi M p\Omega}{(2\pi\hbar)^3}, \tag{1,9} \]

where \(j\) is the angular momentum of the compound nucleus, and \(D_j\) is the mean spacing between degenerate levels of the compound nucleus whose angular momentum is equal to \(j\). We assume here that the degeneracy multiplicity of a level of the compound nucleus with excitation energy \(E_C\) is equal to \(2j+1\).

Multiplying \(W^{Aa}_C\) by \(\hbar\), we shall find the partial width for decay of the type \(C \to A+a\), averaged over all levels of the compound nucleus whose energy lies near \(E_C=E_A+E_a+E\). Let us denote this averaged width by \(\Gamma^{C}_{Aa}\). Let us also introduce the cross section for formation of the compound nucleus as a result of the collision of particles \(A\) and \(a\), equal to

\[ \sigma^{Aa}_C=\frac{W^{Aa}_C}{(v/\Omega)}, \]

where \(v\) is the velocity of the relative motion of particles \(A\) and \(a\). We shall finally rewrite formula (1,9) in the form

\[ \Gamma^{C}_{Aa}= \frac{(2j+1)(2s+1)}{2j+1}\, D_j\, \frac{\sigma^{Aa}_C}{2\pi^2 \lambda^2}, \tag{1,10} \]

where \(\lambda\) is the wavelength of particle \(a\) divided by \(2\pi\).

Thus, using statistical considerations, one can relate the partial width of a level to the probability of the inverse process, i.e. to the cross section for formation of the compound nucleus. At the same time it should once again be emphasized that the width in question is averaged over a large number of levels lying near the excitation energy \(E_C\).

Multiplying \(\sigma^{Aa}_C\) by \(\Gamma^{C}_{Bb}/\Gamma^C\), where \(\Gamma^C\) is the total width of the level of the compound nucleus, and \(\Gamma^{C}_{Bb}\) is the partial width with respect to decay of \(C\) into nucleus \(B\) and particle \(b\), and summing the resulting expression over all values of the angular momentum of the compound nucleus \(j\) compatible with the law of conservation of angular momentum, we find the averaged effective cross section for the process \(A+a \to B+b\):

\[ \sigma^{Aa}_{Bb}= \frac{2\pi^2\lambda^2}{(2i+1)(2s+1)} \sum_j (2j+1)\, \frac{\Gamma^{C}_{Aa}\Gamma^{C}_{Bb}}{\Gamma^C D_j}. \tag{1,11} \]

Let us now proceed to the determination of the cross section for formation of the compound nucleus \(\sigma^{Aa}_C\). Let us first recall that the collision parameter of the particles \(A\) and \(a\), i.e. the distance between the initial direction of motion of \(a\) and the center of the nucleus \(A\), is, in order of magnitude, \(r_l=l\lambda\), where \(\lambda\) is the wavelength of the particle far from the nucleus divided by \(2\pi\), and \(l\hbar\) is the angular momentum of particle \(a\) relative to \(A\). We shall consider the case of fast incident particles, when \(\lambda \ll R\).

Let us denote by \(\sigma_l\) the total effective cross section for formation of the compound nucleus as a result of collision of nucleus \(A\) with a particle \(a\) possessing angular momentum \(l\), irrespective of what value the angular momentum of the compound nucleus will then have.

If \(\alpha_{lj}\) is the probability that the compound nucleus formed will have angular momentum \(j\), then

\[ \sigma_C^{Aa}=\sigma_j^{Al}=\alpha_{lj}\sigma_l . \tag{1,12} \]

To determine \(\sigma_l\), let us find the number of particles \(a\) with angular momentum \(l\) incident per unit time on the nucleus \(A\). Since it is assumed that \(\lambda \ll R\), we may use a quasi-classical treatment.

Far from the nucleus, particles with angular momentum \(l\hbar\) evidently pass through the area of a ring situated perpendicular to the direction of motion of the particles, the inner and outer radii of the ring being respectively \(r_l=l\lambda\) and \(r_{l+1}=(l+1)\lambda\). Let the flux of particles, i.e. their number incident per unit time per unit area, be equal to unity. Since the area of the ring is \((2l+1)\pi\lambda^2\), the required number of particles incident on \(A\) per unit time and possessing angular momentum \(l\) is \((2l+1)\pi\lambda^2\). This is evidently the maximum possible value of \(\sigma_l\), corresponding to the case in which all incident particles are absorbed by the nucleus. In reality only some fraction of them is absorbed; we shall denote this fraction by \(\zeta_l\).

Thus, the cross section for formation of the compound nucleus may be represented in the form

\[ \sigma_l=(2l+1)\pi\lambda^2\zeta_l \quad (0\leq \zeta_l\leq 1). \tag{1,13} \]

The quantity \(\zeta_l\) is called the probability of adhesion of the particle to the nucleus, or, in other words, the probability of formation of the compound nucleus.

Let us now determine \(\alpha_{lj}\), i.e. the probability that, as a result of the fusion of the nucleus \(A\) and the particle \(a\), a compound nucleus with angular momentum \(j\) arises. We shall assume that the adhesion probability \(\zeta_l\) does not depend on \(j\). In other words, we suppose that the probability of the appearance of some value \(j\) is determined only by the statistical weight \(j\), i.e. by the number of different possible orientations of the angular-momentum vector having the value \(j\). The number of these orientations is \(2j+1\). Therefore we set

\[ \alpha_{lj}=g_{lis,j}\, \frac{2j+1}{(2s+1)(2l+1)(2i+1)}, \tag{1,14} \]

where \(g_{lis,j}\) is the number of ways in which the given value \(j\) can be obtained by vectorially adding \(l\), \(i\), \(s\).

If at least one of the numbers \(l\), \(i\), \(s\) is equal to zero, then \(g=1\). If \(s=1/2\), then for the maximum and minimum values of \(j\), equal respectively to \(l+i+\frac12\) and \(|l-i|-\frac12\), the quantity \(g=1\); for all intermediate values of \(j\), the quantity \(g=2\), since such values can be obtained in two ways. It is easy to verify that the quantity \(\alpha_{lj}\) defined in the indicated manner is a normalized probability, i.e. \(\sum_j \alpha_{lj}=1\), with the summation—

...tion is distributed over all possible values of \(j\) compatible with the conservation law for angular momentum.

Using formulas (1.13) and (1.14), let us represent the formation cross section of the compound nucleus \(\sigma^{Al}_{j}\) in the form

\[ \sigma^{Al}_{j}=\pi \lambda^{2} g_{lis,j}\frac{2j+1}{(2s+1)(2i+1)}\zeta_l . \tag{1.15} \]

Substituting this expression into formula (1.10), we find the averaged width \(\Gamma^{j}_{Al}=\Gamma^{C}_{Aa}\):

\[ \Gamma^{j}_{Al}=g_{lis,j}D_j\frac{\zeta_l}{2\pi}. \tag{1.16} \]

This formula relates the probability (i.e. the partial width) of decay of the compound nucleus \(C\) (as a result of which a particle \(a\) with angular momentum \(l\) and a nucleus \(A\) appear) to the probability \(\zeta_l\) for particle \(a\) to stick to nucleus \(A\), and to the mean distance between levels of the compound nucleus \(D_j\) (the levels in question are, of course, levels of \(C\) with excitation energy close to \(E_i=E_A+E_a+E\)). The quantity \(\zeta_l\) is determined by the energy of the incident particle \(E\) and its angular momentum \(l\).

From (1.16) an important conclusion follows: if the sticking probability \(\zeta_l\) is, in order of magnitude, equal to unity, then the partial width \(\Gamma^{j}_{Al}\) has the order of magnitude of the distance between levels of the compound nucleus \(D_j\), and conversely. In other words, in this case the levels of the compound nucleus overlap. It must be emphasized that (1.16) determines only the partial width for some given mode of disintegration of the compound nucleus, averaged over a large number of closely situated levels.

According to (1.16), the sticking probability for \(l=0\) is equal to

\[ \zeta_0=\frac{2\pi}{g_{0is,j}}\frac{\Gamma^{j}_{A0}}{D_j}. \tag{1.17} \]

The neutron width at small neutron energies is proportional to the square root of the neutron energy. Let us recall that \(\zeta_l\) with \(l\ne 0\) is very small at small energies, since in this case the smallest impact parameter will be larger than the nuclear radius. Using the experimental data\({}^{6}\) on \(\Gamma_n\), we have:

\[ \zeta_0\approx 10^{-3}\frac{2\pi}{g_{0is,j}}\frac{\sqrt{E}}{D} \tag{1.18} \]

(\(E\) and \(D\) in electron-volts). It follows from this relation that for \(E=1\) MeV, \(\zeta_0\) is of order unity if \(D\) is of order \(10\) eV for such excitations of the intermediate nucleus. This estimate shows strong absorption by the nucleus of neutrons with energies of order \(1\) MeV. (We are speaking of absorption leading to the formation of an intermediate nucleus, and not of radiative capture.)

Analogous relations also hold for charged particles (protons, deuterons, \(\alpha\)-particles) in the case when they have an energy exceeding the barrier height by a quantity of order \(1\) MeV.

However, the estimate (1.18) is not valid if \(E > 1\) MeV. In order to clarify the behavior of \(\xi_l\) at high energies, we shall next consider\({}^{7}\) a certain semiquantitative method for determining the capture probability.

We shall proceed from the fact that a free particle, upon entering a nucleus, is absorbed by the latter and loses its individuality. Therefore, when considering the formation of a compound nucleus from the phenomenological point of view, we must characterize nuclear matter by a large absorption of nuclear particles. To obtain a qualitative picture we shall make the simplest assumption, namely that the number of particles absorbed per unit time at a certain point of the nucleus is proportional to the density of particles at that point.

The continuity equation with absorption taken into account can be written in the form

\[ \frac{\partial \rho}{\partial t} + \operatorname{div}\mathbf{j} = -\frac{2}{\hbar}\sigma \rho, \tag{1.19} \]

where \(\mathbf{j}\) and \(\rho\) are, respectively, the current density and the particle density, while \(\sigma\) is a certain quantity characterizing the absorption, which we shall call the absorptive potential. This quantity depends on the coordinates, and also on the energy and other quantities characterizing the particle (spin, direction of motion, etc.). For qualitative orientation we shall assume that the dependence on spin and direction of motion is immaterial. As for the dependence on the particle energy, it can become noticeable only over a very large interval of energy variation, of the order of magnitude of the binding energy of particles in the nucleus (\(\sim 10\) MeV). Considering a small interval of energy variation, one may regard \(\sigma\) as independent of the particle energy.

Thus, we assume that \(\sigma\) depends only on the coordinates of the particle. It is easy to see that the presence of an absorptive potential in (1.19) can be taken into account if the Schrödinger equation for the incident particle is written in the form

\[ \Delta \Psi + \frac{2m}{\hbar^2}(E - V + i\sigma)\Psi = 0, \tag{1.20} \]

where \(E\) is the kinetic energy of the particle far from the nucleus, and \(V\) is its potential energy.

To prove the equivalence of equations (1.19) and (1.20), multiply (1.20) by \(\Psi^{*}\) (\(\Psi^{*}\) is the quantity complex-conjugate to \(\Psi\)) and subtract from the product obtained the expression complex-conjugate to it. Using the definitions of \(\rho\) and \(\mathbf{j}\):

\[ \rho = \Psi^{*}\Psi,\qquad \mathbf{j}=\frac{\hbar}{2im}\left(\Psi^{*}\nabla\Psi-\Psi\nabla\Psi^{*}\right), \]

we then obtain (1.19).

Thus, the absorptive potential appears as the imaginary part of the effective complex potential energy of the particle \(V - i\sigma\).

Inside the nucleus, \(\sigma\) is, in order of magnitude, equal to the energy of the nuclear interaction, which is about \(30\) MeV. Let us note that neither \(\sigma\) nor \(V\) undergoes a sharp jump at the boundary of the nucleus, since the nuclear forces by which \(V\) and \(\sigma\) are in fact determined have a finite range of action. It should also be borne in mind that the boundary of the nucleus cannot be regarded as sharp, owing to the presence of zero-point oscillations of the nuclear particles and to the finiteness of the range of action of the nuclear forces. Outside a narrow boundary layer, which we shall call the region of diffuseness of the nuclear boundary, the potential \(V-i\sigma\) may be neglected.

The theory set forth is semi-quantitative; nevertheless, as we shall see below, it correctly conveys the main features of the behavior of \(\zeta_l\) at large and small energies.

Let us write the potential energy of the particle in the form \(V=V_n+V_l\), where \(V_n\) is associated with the nuclear forces, and \(V_l\) with the Coulomb and centrifugal forces. We shall assume that \(V_n\), \(\sigma\), \(V_l\) depend only on \(r\)—the distance to the center of the nucleus. Introduce the radial function \(U=r\Psi\); the Schrödinger equation for it has the form

\[ \frac{d^2 U}{dr^2}=\Phi(r)U, \tag{1,21} \]

where

\[ \Phi(r)=\frac{2m}{\hbar^2}\left[-W(r)+V_n(r)-i\sigma(r)\right] \]

and

\[ W(r)=E-V_l(r). \]

Outside the nucleus, but near its surface, where \(V_n\) and \(\sigma\) still play an essential role, we shall regard \(W(r)\) as constant, equal to

\[ W=E-V_l(R), \]

where \(R\) is the radius of the nucleus.

Let us first consider the case of large energies of the incident particle, when the wavelength of the particle is considerably smaller than the radius of the nucleus. In this case one may use the method of the quasiclassical approximation. The condition for the applicability of this method is that the modulus of the change of \(\Phi\) over the length of a wavelength, equal to \(\frac{1}{\sqrt{\Phi}}\), must be small in comparison with the absolute value of the function \(\Phi\) itself.

Since the quantity \(W\) is assumed large, while \(V_n\) and \(\sigma\) do not undergo a sharp change at the boundary of the nucleus, it may be assumed that the conditions for applicability of the quasiclassical approximation method are fulfilled.

Using this method, we represent the solution of equation (1,21) in the form

\[ U=\operatorname{const.}\,\Phi^{-1/4}\exp\left(\int_0^r \Phi^{1/2}\,dr\right). \tag{1,22} \]

We must, in doing so, take \(\sqrt{\Phi}\) with such a sign that \(U\) decreases in the direction toward the center of the nucleus. Such a choice of sign corresponds to our basic assumption about absorption of the particle by the nucleus. Thus, the real part must be positive. For \(r<R\) the quantity \(\Phi\) lies in the third quadrant of the plane of the complex variable (the real and imaginary parts of \(\Phi\) are negative). Therefore \(\Phi^{1/2}\) must lie in the fourth quadrant of the complex plane.

Let us now imagine that \(r\) increases; then \(\sigma\) and \(V_n\) decrease and, finally, outside the nucleus become equal to zero. Since \(\Phi^{1/2}\) must remain in the fourth quadrant all the time, the imaginary part of \(\Phi^{1/2}\) must be negative also outside the nucleus. Therefore the solution (1,22) outside the nucleus has the form

\[ U=\mathrm{const.}\,\Phi^{-1/4}\exp\left\{-i\frac{\sqrt{2m}}{\hbar}\int \sqrt{E-V_l(r)}\,dr\right\}, \tag{1,22'} \]

i.e., it represents a purely incident wave. Since we have not obtained a reflected wave, this means that the probability of sticking \(\zeta_l\) is equal to unity.

Thus, for fast particles entering the sphere of action of the nucleus,

\[ \zeta_l=1. \]

The relation obtained has a simple physical meaning. Since we are considering the case of fast particles, i.e., of small wavelengths, classical mechanics is valid. But in that case it is quite clear that if a particle enters the sphere of action of the nucleus, i.e., if its collision parameter is smaller than the nuclear radius, then the particle is absorbed by the nucleus; in other words, \(\zeta_l=1\). Deviations from the relation \(\zeta_l=1\) are connected with the inaccuracy of the quasiclassical approximation which we have used.

Let us now turn to the consideration of the case of small energies of the incident particle, when its wavelength is large in comparison with the nuclear radius.

Suppose that the incident particle is a slow neutron whose orbital angular momentum is equal to zero. In this case \(V_l=0\). Outside the nucleus equation (1,21) has the form

\[ \frac{d^2U}{dr^2}+\frac{2m}{\hbar^2}EU=0, \]

whence

\[ U=\mathrm{const.}\,\sin(kr+\eta)=\mathrm{const.}\,(e^{ikr+i\eta}-e^{-ikr-i\eta}), \tag{1,23} \]

where

\[ k=\sqrt{\frac{2mE}{\hbar^2}} \]

and \(\eta\) is the complex scattering phase. The latter can be related to the value of the logarithmic derivative of \(U\) on the surface of the nucleus, i.e., at \(r=R\):

\[ \left(\frac{1}{U}\frac{dU}{dr}\right)_{r=R}=\chi. \tag{1,24} \]

Since we assume that the wavelength of the incident particle is considerably greater than the dimensions of the nucleus, the value of this derivative outside the nucleus does not undergo any substantial change over distances of order \(R\). Therefore the introduction of the quantity \(\chi\) is meaningful, despite the diffuseness of the nuclear boundary.

The quantity \(\chi\) should be regarded as complex, since owing to the possibility of absorption of particles the effective potential energy is complex. We shall write \(\chi\) in the form

\[ \chi = -\frac{1}{b} e^{-i\frac{\varphi}{2}}, \qquad \varphi > 0, \]

where \(b\), in order of magnitude, determines the width of the region of diffuseness of the nuclear boundary, and \(\varphi\) is a real phase depending on the complex potential \(V - i\sigma\). (The circumstance that \(\varphi > 0\) follows from the expression for the sticking coefficient \(\zeta_0\); see below (1,25).) In the first approximation one may assume that \(b\) and \(\varphi\) do not depend on the energy.

Since \(kR\) and the absolute value of \(\eta\) are small compared with unity (the latter condition means that the scattering cross section is small compared with \(\pi \lambda^2\)), we shall represent (1,23) in the form

\[ U = \operatorname{const.}(kr + \eta). \]

The boundary condition (1,24) leads to the result

\[ \eta = -kR + k b e^{i\varphi/2}. \]

The probability of sticking of particles with angular momentum \(l = 0\) is equal to [see below (3,8) \((\beta_0 = e^{2i\eta_0})\)]:

\[ \zeta_0 = 1 - |\beta_0|^2 = 4kb \sin \frac{\varphi}{2}, \tag{1,25} \]

where we have neglected terms of order \(|\eta|^2\).

Above we said that \(b\) and \(\varphi\) in the region of small energies may be regarded as independent of the neutron energy. Formula (1,25) then shows that, for slow neutrons whose wavelength is large compared with \(R\), the sticking probability is proportional to \(k\), i.e. to \(\sqrt{E}\). We note that \(\zeta_0\) becomes of order unity at \(E \simeq 1\) MeV, if \(b \simeq 10^{-13}\) cm. This result agrees with the conclusions that were obtained earlier (1,18). Thus, for neutrons whose energy is \(E \gg 1\) MeV, nuclei are “black.” (At very high energies nuclei become transparent.)

We have determined the dependence of the sticking probability on the energy of the incident particle in the limiting cases of high and low energies, when the wavelength of the particle is small or large compared with the nuclear radius. To obtain a general formula for \(\zeta_l\), valid at all energies, a detailed theory of nuclear forces is necessary.

2. OPTICAL THEORY OF DIFFRACTION SCATTERING OF FAST NEUTRONS

We shall begin by considering the diffraction scattering of fast neutrons, regarding the nucleus as an absolutely “black” body, absorbing all neutrons incident upon it.

We shall assume that the nucleus has the form of a sphere of radius \(R\) and that the neutron wavelength is considerably smaller than the dimensions of the nucleus. When this condition is fulfilled, the nucleus may be regarded as absolutely “black” for neutrons that fall within the sphere of action of the nucleus.

Let us estimate the neutron energy \(E_0\) at which its wavelength, divided by \(2\pi\) and denoted below by \(\lambda\), becomes equal to the radius of the nucleus. Taking the radius of the nucleus to be approximately

\[ R = 1.5 \cdot 10^{-13} A^{1/3}\,\text{cm}, \]

where \(A\) is the mass number, we obtain:

\[ E_0 = \frac{\hbar^2}{2MR^2} = \frac{8.2}{A^{2/3}}\,\text{MeV}. \]

If \(A=40\), then \(E_0 \simeq 0.8\) MeV; for \(A=100\), \(E_0 \simeq 0.5\) MeV; and for \(A=238\), \(E_0 \simeq 0.25\) MeV.

Thus, we see that if the neutron energy exceeds several million electron-volts, the condition

\[ \lambda \ll R \]

is satisfied.

On the other hand, it is well known that if the wavelength of light is small in comparison with the dimensions of an obstacle, then it is possible to obtain a general formula describing diffraction phenomena.

We shall obtain the diffraction pattern in the scattering of fast neutrons, first on the basis of this optical diffraction formula, and then we shall verify that the same pattern will occur if one proceeds from the rigorous quantum-mechanical theory of scattering. First of all, let us recall this formula and, using it, obtain the diffraction pattern for the case in which light is incident on an absolutely black absorbing sphere of radius \(R\).

The general formula determining diffraction has the following form\(^8\):

\[ u = \int_S \frac{k}{2\pi i r}\, u_0 e^{ikr}\, df_n . \tag{2,1} \]

Here \(u\) is the field sought at some point \(P\), \(u_0\) is the field at some point of the integration surface \(S\), \(df_n\) is the projection of the surface element \(df\) onto the direction of the ray \(\mathbf{n}\) going from the light source to the element \(df\), \(\mathbf{r}\) is the distance from the point \(P\) to the surface element \(df\), and \(\mathbf{k}\) is the wave vector. The integration surface, in general of arbitrary form, closes the aperture in an opaque screen and limi-

is determined by its edges. The values of the field \(u_0\) are assumed here to be the same as if there were no screens at all.

In what follows we shall be interested in the case when both the light source and the observation point \(P\) are at very large distances from the screens. In this case, which is known as Fraunhofer diffraction, the rays coming from the light source fall on the screens as a parallel beam; likewise, the rays going from the screens to the observation point are parallel. Therefore, in Fraunhofer diffraction we are dealing with a change in the direction of a beam of light that has undergone diffraction near the screens. The intensity of the light is in this case a function of the angle of deviation of the light from its original direction. This angle is called the angle of diffraction.

In the case of Fraunhofer diffraction all rays incident on the screen have the same direction, i.e. the same wave vector \(\mathbf{k}\). Therefore the field \(u_0\) on the surface of integration in formula (2.1) may be represented in the form

\[ u_0 = A e^{i k \rho}, \]

where \(A\) is a constant, \(\rho\) is the radius vector drawn from a certain point \(O\) to the surface element \(df\) (Fig. 1).

Fig. 1.

As is seen from this figure, the distance \(r\) from \(df\) to the observation point \(P\) may, for large \(r\), be approximately replaced by \(r = R_0 - \rho \mathbf{n}'\), where \(\mathbf{n}'\) is the unit vector in the direction of \(\mathbf{R}_0\), i.e. in the direction of the ray going from the screen to the observation point. Using this relation and noting that in the integrand the quantity \(1/r\) may be regarded as constant and equal to \(1/R_0\), we write the general formula (2.1) in the case of Fraunhofer diffraction in the form

\[ u = \mathrm{const.}\int\limits_S e^{i(\mathbf{k}-\mathbf{k}')\rho}\,df_n, \tag{2.2} \]

where

\[ \mathrm{const.}=\frac{A k e^{i k R_0}}{2\pi i R_0} \]

and \(\mathbf{k}' = k\mathbf{n}'\) is the wave vector of the diffracted ray.

From formula (2.2) there follows an important property of the so-called complementary screens, i.e. two such plane screens, of which

one has an aperture where the other is opaque. It consists in the fact that the diffraction patterns of complementary screens are identical. To be convinced of the validity of this assertion, it is enough to note that if, as the surface of integration in (2.2), one takes the entire infinite plane in which the aperture lies, then (2.2) reduces to \(\delta(\mathbf{k}-\mathbf{k}')\), where \(\delta(f)\) is the Dirac \(\delta\)-function. Therefore, for \(\mathbf{k}\ne\mathbf{k}'\) this integral vanishes and, consequently, for complementary screens with \(\mathbf{k}\ne\mathbf{k}'\) the values of the function \(u\) differ only in sign. For \(\mathbf{k}=\mathbf{k}'\) the above-formulated property of complementary screens, generally speaking, does not hold.

We are interested in diffraction by an absolutely black sphere of radius \(R\), on which a parallel beam of rays is incident. The diffraction pattern in this case, according to what has been said above, coincides with the diffraction pattern from a circular aperture of radius \(R\) in an opaque screen, perpendicular to the plane of which the light is incident. It is important to note that the diffraction patterns also coincide for \(\mathbf{k}=\mathbf{k}'\), since in this direction in both cases there is a maximum of intensity.

Thus, let us consider diffraction by a circular aperture of radius \(R\). Denote the projection of the vector \(\mathbf{k}'\) onto the plane of the aperture by \(x\) (the vector \(\mathbf{k}\), as was said above, is perpendicular to this plane).

Introduce cylindrical coordinates \(z, r, \varphi\) with an axis passing through the center of the aperture perpendicular to its plane. From symmetry considerations it is clear that the intensity of the diffracted rays will not depend on \(\varphi\); therefore it is sufficient to restrict ourselves to considering a ray lying in the plane \(\varphi=0\). Formula (2.2) leads to the following expression for the field of the diffracted wave:

\[ u=\mathrm{const.}\int_{0}^{R}\int_{0}^{2\pi} e^{-ixr\cos\varphi}\,r\,dr\,d\varphi = \]

\[ =\frac{Ak}{iR_0}e^{ikR_0}\int_{0}^{R} J_0(xr)\,r\,dr, \tag{2.3′} \]

where

\[ J_0(x)=\frac{1}{2\pi}\int_{0}^{2\pi} e^{-ix\cos\varphi}\,d\varphi \]

is the Bessel function of order zero.

Using the relation known from the theory of Bessel functions,

\[ \int_{0}^{R} J_0(xr)\,r\,dr=\frac{R J_1(xR)}{x}, \]

where \(J_1(x)\) is a Bessel function of the first kind, let us rewrite \((2,3')\) in the form

\[ u=-iAkR\,\frac{e^{ikR_0}}{R_0}\,\frac{J_1(\chi R)}{\chi}. \tag{2,3} \]

Multiplying \(|u|^2\) by \(R_0^2\,d\omega\), where \(d\omega\) is the element of solid angle in which the vector \(\mathbf{k}'\) lies, we find the intensity of light \(dI\) that has undergone diffraction in this direction:

\[ dI=R^2A^2k^2\left|\frac{J_1(\chi R)}{\chi}\right|^2\,d\omega. \]

Noting that \(A^2\pi R^2\) represents the total intensity of the light incident on the aperture, which we denote by \(I_0\), we represent \(dI\) in the form

\[ dI=I_0\,\frac{J_1^2(kR\sin\theta)}{\pi\sin^2\theta}\,d\omega, \tag{2,4} \]

where \(\theta\) is the diffraction angle, related to \(\chi\) by the relation \(\chi=k\sin\theta\).

The same formula determines the distribution of light in diffraction by an absolutely black absorbing sphere\(^{10}\). In this case \(I_0\) should be understood as the total intensity of the light incident on the cross-sectional area of the sphere. We see that in this case, just as in diffraction by a circular aperture, the intensity maximum lies at \(\chi=0\), i.e. in the direction of propagation of the incident beam of light. In other words, the intensity maximum lies at the center of the shadow obtained according to the laws of geometrical optics.

The diffractive elastic scattering of fast neutrons by nuclei must be determined by an analogous formula in the case when the nuclei completely absorb the neutrons incident on them. Here \(\mathbf{k}\) and \(\mathbf{k}'\) should be understood as the wave vectors of the incident and scattered neutrons, and \(I_0\) as the total intensity of the neutron beam incident on the cross-sectional area of the nucleus.

In the case of neutron scattering, it is more convenient to express formula \((2,4)\) in terms of effective cross sections. If one divides \(dI\) by the flux density of the incident neutrons, equal to \(I_0/\pi R^2\), then we obtain the differential scattering cross section \(\sigma(\theta)d\omega\), referred to the element of solid angle \(d\omega\). It is determined by the following formula:

\[ \sigma(\theta)d\omega=R^2\left|\frac{J_1(kR\sin\theta)}{\sin\theta}\right|^2\,d\omega. \tag{2,5'} \]

Let us note that, by the very meaning of optical diffraction theory, deviations from geometrical optics must be small; this means that the diffraction angle must be small. From the formula \((2,5')\) obtained by us for the scattering cross section, it is easy to conclude that the effective scattering angle is small and, in order of magnitude, is equal to

\(\lambda/R \ll 1\). Indeed, for large values of the argument the Bessel function \(J_1(x)\) is asymptotically equal to

\[ J_1(x) \approx \left(\frac{2}{\pi x}\right)^{1/2} \sin\left(x-\frac{\pi}{4}\right), \quad x \gg 1 . \tag{2,6} \]

Therefore, in the region of angles satisfying the condition

\[ \frac{1}{kR} \ll \sin\theta < 1, \]

the differential cross section \(\sigma(\theta)\) decreases inversely proportionally to \(\sin^3\theta\). Owing to the rapid decrease of the effective cross section at scattering angles exceeding \(\lambda/R\), the principal role in the integral effective cross section is played by angles of order \(\lambda/R\).

Taking this circumstance into account, we replace \(\sin\theta\) in \((2,5')\) by \(\theta\) and write \(\sigma(\theta)\,d\omega\) in the form

\[ \sigma(\theta)d\omega = R^2\left|\frac{J_1(kR\theta)}{\theta}\right|^2 d\omega . \tag{2,6'} \]

Fig. 2.

An idea of the character of the angular dependence of the cross section is given by Fig. 2, which shows the dependence of the function

\[ \left|\frac{2J_1(x)}{x}\right|^2 \]

on \(x\). This function has a sharp maximum at \(x=0\) and then, oscillating, tends to zero as \(x\) increases.

In the region of small scattering angles, when \(\theta \ll \lambda/R\), the Bessel function \(J_1(kR\theta)\) may be replaced by \(kR\theta/2\), and therefore

\[ \sigma(\theta)d\omega = \frac{k^2R^2}{4}\,d\omega, \quad \theta \ll \frac{\lambda}{R}. \tag{2,7} \]

Thus, for very small angles \(\theta\) the differential cross section tends to a constant limiting value.

If the angles \(\theta\) satisfy the condition

\[ \frac{\lambda}{R} \ll \theta \ll 1, \]

then the Bessel function may be replaced by the asymptotic expression (6), and \(\sigma(\theta)d\omega\) is equal to

\[ \sigma(\theta)d\omega = \frac{2}{\pi}R\lambda\, \frac{\sin^2\left(\frac{R}{\lambda}\theta-\frac{\pi}{4}\right)} {\theta^3}\,d\omega, \quad \frac{\lambda}{R} \ll \theta \ll 1 . \tag{2,8} \]

The effective cross section in this region of angles oscillates rapidly with “frequency” \(R/\lambda\), the amplitude of the oscillations decreasing inversely proportionally to the cube of the scattering angle. In view of this, as already noted—

noted above, large angles do not play a role; angles of the order of \(\lambda/R\) prove to be effective in the integral cross section.

Thus, we see that diffraction scattering of neutrons is characterized by a sharp asymmetry; the neutrons are scattered predominantly forward. The effective scattering angle is, in order of magnitude, equal to \(\lambda/R\). At large angles secondary maxima occur, distinguished, however, by an intensity considerably smaller than that of the principal maximum.

Let us determine the total effective scattering cross section \(\sigma\). Integrating (2,5) and extending the limits of integration in \(x=kR\theta\) from zero to infinity, we find:

\[ \sigma = 2\pi R^2 \int_0^\infty \frac{J_1^2(x)}{x}\,dx. \]

The last integral (see 11) is equal to \(\frac{1}{2}\). Therefore

\[ \sigma = \pi R^2,\qquad \lambda \ll R. \tag{2,9} \]

Thus, the integral cross section of elastic scattering of fast neutrons whose wavelength is considerably smaller than the radius of the nucleus is equal to the area of the transverse cross section of the latter.

3. QUANTUM-MECHANICAL THEORY OF SCATTERING AND ITS APPLICATION TO FAST NEUTRONS

We shall now show that formula (2,5), determining the cross section of diffraction scattering of neutral particles, can be obtained from the general quantum-mechanical theory of particle scattering. We shall present a derivation of this formula based on the general theory of scattering, with the intention of subsequently generalizing it to the case of charged particles, whose diffraction cannot be described by the elementary formula (2,2).

Let us first recall some general relations that hold in elastic scattering of particles.

Let a monochromatic beam of particles, described by the plane wave \(e^{ikz}\) (\(k\) being the magnitude of the wave vector of the particles), fall on a scatterer in the direction of the \(z\)-axis. Far from the scatterer the scattered wave has the form

\[ \frac{1}{r}\,e^{ikr} f(\theta), \]

where \(r\) is the distance to the scatterer and \(\theta\) is the scattering angle. The quantity \(f(\theta)\) has the dimension of length and is called the scattering amplitude. The scattering amplitude is directly connected with the differential cross section \(\sigma(\theta)d\omega\) of elastic scattering of particles into the solid angle \(d\omega\) \((d\omega = 2\pi\sin\theta\,d\theta)\):

\[ \sigma(\theta)d\omega = |f(\theta)|^2\,d\omega. \tag{3,1} \]

The scattering amplitude can be expressed in terms of the so-called “phases at infinity”\(^{12}\), which determine the asymptotic behavior of the radial wave functions of the particle for different values of the particle’s orbital angular momentum with respect to the scatterer. Since the method of calculating the scattering amplitude by means of the phases at infinity is extremely important, we shall dwell on it in somewhat greater detail.

Let us first note that the following expansion of the plane wave \(e^{ikz}\) in a series of Legendre polynomials\(^{13}\) holds:

\[ e^{ikz}=\sum_{l=0}^{\infty}(2l+1)i^l P_l(\cos\theta) f_l(kr), \tag{3,2} \]

where

\[ f_l(kr)=\left(\frac{\pi}{2kr}\right)^{1/2} J_{l+\frac12}(kr) \]

and \(J_{l+\frac12}\) is the Bessel function of order \(l+\frac12\).

Each separate term in the sum (3,2) is a wave function of a free particle with a definite value of the orbital angular momentum of the quantity of motion; namely, the term with index \(l\) corresponds to the angular momentum of the particle with respect to the scatterer equal to \(\hbar l\).

From the theory of Bessel functions it is known that, for large \(r\), the asymptotic formula

\[ f_l(kr)\sim (kr)^{-1}\sin\left(kr-\frac{l\pi}{2}\right). \]

is valid. Hence it is seen that the wave function of a free particle for large \(r\) behaves as

\[ \frac{i}{2kr}\sum_{l=0}^{\infty}(-1)^l(2l+1)\{e^{-ikr}-(-1)^l e^{ikr}\}P_l(\cos\theta). \tag{3,3} \]

Therefore the function

\[ \frac{i}{2kr}(-1)^l(2l+1)\{e^{-ikr}-(-1)^l e^{ikr}\} \tag{3,3'} \]

is the asymptotic expression for the radial wave function of a free particle with angular momentum \(\hbar l\). In this expression the first term represents an incident wave, and the second an outgoing wave with angular momentum \(l\).

If the particle undergoes deflection in a force field, i.e. is not free, then the asymptotic expression for its wave ...

for the function at large \(r\) will be determined, instead of (3.3), by the formula

\[ \Psi \sim \frac{i}{2kr}\sum_{l=0}^{\infty}(-1)^l(2l+1)\{e^{-ikr}-(-1)^l\beta_l e^{ikr}\}P_l(\cos\theta). \tag{3.4} \]

This function differs from the function describing free motion by other amplitudes of the outgoing waves: the amplitude of the outgoing wave with angular momentum \(l\) in formula (3.4) differs by the factor \(\beta_l\) from the amplitude of the outgoing wave with angular momentum \(l\) in the case of free motion. The quantities \(\beta_l\) differ from unity in the presence of scattering. If the scatterer does not absorb particles, then in modulus they are equal to unity, since the intensity of the outgoing wave is then equal to the intensity of the incident wave. In this important case of absence of absorption the quantity \(\beta_l\) may be represented in the form

\[ \beta_l=e^{2i\eta_l}, \]

where \(\eta_l\) is a real quantity determining the change, upon scattering, of the phase at infinity of the \(l\)-wave describing the motion of a particle with angular momentum \(l\hbar\). This quantity is called the phase at infinity. The asymptotic expression for the radial wave function of a non-free particle with angular momentum \(l\) has the form

\[ e^{i\eta_l}\frac{1}{kr}\sin\left(kr-\frac{l\pi}{2}+\eta_l\right). \]

Using formulas (3.3) and (3.4), it is easy to determine the scattering amplitude \(f(\theta)\). To do this one must subtract (3.3) from (3.4) and equate the result to \(\frac{1}{r}e^{ikr}f(\theta)\). We obtain, in this way, the following expression for \(f(\theta)\), relating the scattering amplitude to the quantities \(\beta_l\) introduced above:

\[ f(\theta)=-\frac{i\lambda}{2}\sum_{l=0}^{\infty}(2l+1)(\beta_l-1)P_l(\cos\theta), \tag{3.5} \]

where \(\lambda\) is the wavelength of the particle divided by \(2\pi\).*
For free motion, when \(\beta_l=1\), this quantity, as it should, vanishes.

Let us determine the integral cross section of elastic scattering. Using formula (3.1) and noting that

\[ \int |P_l(\cos\theta)|^2\,d\sigma=\frac{4\pi}{2l+1}, \tag{3.1'} \]

* In this formula the existence of spin of the particle is not taken into account. In what follows formula (3.5) will be applied to neutrons, whose interaction with nuclei in the energy region of interest to us may be regarded as independent of the orientation of the neutron spin, since absorption of a neutron by a nucleus is determined only by the orbital angular momentum of the neutron and does not depend on its spin (see formula (3.10)).

we obtain, integrating (3.1) over the solid angle, the following expression for the integral cross section:

\[ \sigma=\pi\lambda^{2}\sum_{l=0}^{\infty}(2l+1)|(\beta_l-1)|^{2}. \tag{3,6} \]

If there is no absorption of particles, then \(\beta_l=e^{2i\eta_l}\) (\(\eta_l\) is real) and

\[ \sigma=4\pi\lambda^{2}\sum_{l=0}^{\infty}(2l+1)\sin^{2}\eta_l. \tag{3,6'} \]

Let us note that formula (3,5) is more general than formula (3,6′), since it is also applicable in those cases when absorption of particles occurs. In this case the modulus of \(\beta_l\) is less than unity, since the intensity of the outgoing wave is less than the intensity of the incident wave.

Before analyzing formula (3,6) in the case of interest to us, namely the scattering of fast neutrons by nuclei, we shall derive here also the general formula for the cross section of absorption of particles by nuclei and relate this cross section to the quantities \(\beta_l\). For this purpose it is simplest to define the flux of particles through the surface of a sphere of large radius surrounding the nucleus. This flux, taken in the direction of the inward normal to the sphere, obviously determines the number of particles absorbed per unit time by the nucleus. The flux is determined by the well-known formula

\[ S=\frac{i\hbar}{2M}r^{2}\int\left(\Psi^{*}\frac{\partial\Psi}{\partial r}-\Psi\frac{\partial\Psi^{*}}{\partial r}\right)d o, \tag{3,7} \]

where \(r\) is the radius of the sphere, \(do\) is an element of solid angle, and \(M\) is the mass of the particle described by the wave function \(\Psi\). For \(\Psi\) we must, evidently, use the asymptotic expression (3,4), valid far from the nucleus. Noting that

\[ \frac{\partial\Psi}{\partial r} = -\frac{\Psi}{r} + \frac{1}{2r}\sum_{l=0}^{\infty}(2l+1)(-1)^l \left\{e^{-ikr}+(-1)^l\beta_l e^{ikr}\right\}P_l(\cos\theta), \]

and using the normalization condition for the spherical functions (3,1′), we obtain the following result:

\[ S=\frac{\pi\hbar}{Mk}\sum_{l=0}^{\infty}(2l+1)(1-|\beta_l|^{2}). \]

To determine the effective absorption cross section \(\sigma_{\text{abs}}\), one must divide \(S\) by the flux density of incident particles, equal to \(\frac{\hbar k}{M}\). As a result we obtain the following expression for the cross section

absorption of particles:

\[ \sigma_{\text{capt}}=\pi \lambda^2 \sum_{l=0}^{\infty}(2l+1)(1-|\beta_l|^2). \tag{3,8} \]

If \(|\beta_l|=1\), then, as was to be expected, \(\sigma_{\text{capt}}=0\).

Thus, we see that both the elastic-scattering cross section and the capture cross section are completely determined by the quantities \(\beta_l\). The quantity \(1-|\beta_l|^2\) entering (3,8) is the sticking coefficient \(\zeta_l\) (1,13).

Introducing \(\zeta_l\), we rewrite (3,8) in the form

\[ \sigma_{\text{capt}}=\pi \lambda^2 \sum_{l=0}^{\infty}(2l+1)\zeta_l. \tag{3,8'} \]

This formula, relating the capture cross section to the sticking coefficients, admits a simple physical interpretation, as follows.

Consider a parallel beam of fast particles incident on a nucleus. We shall assume that the wavelength of the particles is much smaller than the dimensions of the nucleus. In this case a quasiclassical treatment is applicable. We may therefore equate the expressions for the angular momentum of a particle as given by classical and quantum mechanics. In classical mechanics the angular momentum of a particle is equal to \(M v_\infty r_\infty\), where \(v_\infty\) is the velocity of a particle of mass \(M\) at an infinite distance from the nucleus, and \(r_\infty\) is the shortest distance at which the particle would have passed the nucleus, moving in a straight line. (This quantity, as indicated earlier, is called the impact parameter.) In quantum mechanics the angular momentum is equal to \(\sqrt{l(l+1)}\,\hbar\). Equating the two expressions, one can determine the impact parameter:

\[ r_\infty=\frac{\hbar}{M v_\infty}\sqrt{l(l+1)}. \]

Noting that \(\hbar/M v_\infty=\lambda\) and that in the quasiclassical approximation the quantum numbers are large, we obtain the following expression for the impact parameter:

\[ r_\infty=l\lambda. \tag{3,9} \]

Using (3,9), it is easy to see that, far from the nucleus, particles with angular momentum \(l\) pass through the area of a ring oriented perpendicular to the direction of motion of the particles, with the inner and outer radii of the ring equal respectively to \(l\lambda\) and \((l+1)\lambda\). Suppose that the density of the particle flux, i.e. the number of particles incident per unit time on a unit area, is equal to unity. Since the area of the ring is equal to \((2l+1)\pi\lambda^2\), the number of particles incident per unit time and having angular momentum \(l\) is equal to \((2l+1)\pi\lambda^2\).

Multiplying \((2l+1)\pi\lambda^2\) by \(\zeta_l\), we determine the fraction of these particles absorbed by the scatterer. Therefore the sum \(\sum_l \pi\lambda^2(2l+1)\zeta_l\) is the total cross section for particle capture. Let us note that, although our arguments, strictly speaking, are valid for \(l\gg 1\), the formula obtained, which coincides with \((3,8')\), is valid for all \(l\).

The quantities \(\beta_l\) and \(\zeta_l\) are complicated functions of the energy of the incident particle. Considerable simplifications occur in the case of fast particles, to which we shall now turn.

Let us begin again with fast neutrons, whose wavelength we shall assume to be small in comparison with the dimensions of the nucleus. At the same time we shall assume that the neutron energy is not sufficiently large for the nucleus to be transparent to neutrons. Under these conditions the nucleus behaves as an absolutely black body absorbing neutrons, provided only that they fall within the sphere of action of the nucleus; in other words, a neutron is absorbed by the nucleus if the collision parameter is smaller than the nuclear radius. In the case of interest to us, when \(\lambda\ll R\), the collision parameter for a neutron with orbital angular momentum \(\hbar l\) is, as indicated above, \(\lambda l\). We therefore come to the conclusion that the sticking coefficient is equal to unity if \(l\leq R/\lambda\), and is equal to zero if \(l>R/\lambda\), i.e.

\[ \zeta_l= \begin{cases} 1, & \text{if } l\leq R/\lambda,\\ 0, & \text{if } l>R/\lambda \end{cases} \qquad (\lambda\ll R). \tag{3,10} \]

It follows from this that \(\beta_l\) is equal to zero if \(l\leq R/\lambda\).

If \(l>R/\lambda\), then the neutron passes outside the nucleus and does not interact with it. Therefore, for angular momenta \(l>R/\lambda\), the quantity \(\beta_l\) is equal to unity. Thus,

\[ \beta_l= \begin{cases} 0, & \text{if } l\leq R/\lambda,\\ 1, & \text{if } l>R/\lambda. \end{cases} \tag{3,10'} \]

Before calculating the scattering amplitude with these values of \(\beta_l\), let us note that the sharp separation of the values of the angular momenta for which the sticking coefficient is equal to zero and to unity is, of course, approximate in character. This is connected, first, with the inaccuracy of the quasiclassical treatment that we used above in determining \(\beta_l\), and, second, with the lack of sharpness of the nuclear boundary, which arises owing to the motion of the nuclear particles located at the boundary of the nucleus, and the finite range of action of the nuclear forces. Therefore one may say that the introduction of a critical angular momentum \(l_0=[R]/\lambda\), separating the values of the angular momenta for which \(\zeta\) is equal to zero or unity, is meaningful with an accuracy of order unity.

We shall show that, nevertheless, we may use the values of \(\beta_l\) given above in determining the scattering cross section of fast neutrons,

for which \(\lambda \ll R\). The calculation of the scattering amplitude with these values of \(\beta_l\) will, evidently, give the correct result if changing \(l_0\) by unity does not lead to an appreciable change in the scattering amplitude. Writing \(f(\theta)\) in the form

\[ f(\theta)=-\frac{i\lambda}{2}\sum_{l=0}^{l_0}(2l+1)P_l(\cos\theta), \tag{3.11} \]

we arrive at the following condition for the applicability of our method of calculation:

\[ \left|P_{l_0}(\cos\theta)-P_{l_0\pm1}(\cos\theta)\right|\ll \left|P_{l_0}(\cos\theta)\right|. \tag{3.11'} \]

Since \(l_0\gg1\), for the Legendre polynomials one may use the asymptotic expression\(^7\)

\[ P_l(\cos\theta)\simeq \left(\frac{2}{\pi l\sin\theta}\right)^{1/2} \cos\left[\left(l+\frac{1}{2}\right)\theta-\frac{\pi}{4}\right]. \tag{3.12} \]

(This formula is valid if \(2(2l+3)\sin\theta\gg1\).) Substitution of (3.12) into (3.11) leads to the result that the scattering angle must be small compared with unity.

Thus, the use of the obtained quasiclassical values of \(\beta_l\) leads to the correct result in calculating \(f(\theta)\) in the case of small scattering angles.

We shall now show that, for \(\lambda\ll R\), small scattering angles play the principal role. For this purpose we use the equality\(^ {14}\)

\[ (2l+1)P_l(\cos\theta)=P'_{l+1}(\cos\theta)-P'_{l-1}(\cos\theta). \tag{3.13} \]

The scattering amplitude may be represented in the form

\[ f(\theta)=\frac{i\lambda}{2}\{P'_{l_0+1}(\cos\theta)+P'_{l_0}(\cos\theta)\}. \tag{3.14} \]

Substituting here, instead of \(P_{l_0}(\cos\theta)\), expression (3.12), we obtain:

\[ f(\theta)=i\lambda \left(\frac{2}{\pi}l_0\right)^{1/2} \frac{1}{\sin^{3/2}\theta} \sin\left[\left(l_0+\frac{1}{2}\right)\theta-\frac{\pi}{4}\right]. \tag{3.14'} \]

This expression shows that, apart from oscillations, the scattering cross section decreases with increasing angle inversely proportional to \(\sin^3\theta\). Hence it follows that small scattering angles play the principal role in elastic scattering of neutrons with \(\lambda\ll R\), for which our treatment is fully legitimate. Replacing \(\sin\theta\) by its argument, we obtain:

\[ f(\theta)=iR \left(\frac{2}{\pi l_0}\right)^{1/2} \frac{ \sin\left[\left(l_0+\frac{1}{2}\right)\theta-\frac{\pi}{4}\right] }{\theta^{3/2}}. \tag{3.15} \]

This formula is valid if \(2(2l_0+3)\theta\gg1\).

We note that for large values of the argument, the asymptotic expression is valid for the Bessel function \(J_1(z)\),

\[ J_1(z)=\left(\frac{2}{\pi z}\right)^{1/2}\sin\left(z-\frac{\pi}{4}\right). \tag{3,16′} \]

Comparison of (3,16′) with (3,15) shows that, for \(2(2l_0+3)\theta \gg 1\),

\[ f(\theta)=iR\,\frac{J_1(l_0\theta)}{\theta}. \tag{3,16} \]

This expression for the scattering amplitude, coinciding with that obtained earlier in the optical-diffraction treatment, has an essential advantage over (3,15), since (3,16) proves to be valid for arbitrarily small angles \(\theta\). Indeed, as \(\theta \to 0\), the exact formula (3,11) for \(f(\theta)\) gives:

\[ f(0)=\frac{i\lambda}{2}\sum_{l=0}^{l_0}(2l+1). \]

Since \(l_0 \gg 1\), this sum may be replaced by an integral,

\[ f(0)\simeq \frac{i\lambda}{2}\int_0^{l_0}2l\,dl =\frac{i}{2}\,\frac{R^2}{\lambda}. \]

As is easily seen, (3,16) also leads to the same result.

Squaring the modulus of \(f(\theta)\) and multiplying it by the element of solid angle, we find the differential scattering cross section

\[ \sigma(\theta)d\omega=|f(\theta)|^2d\omega =R^2\left|\frac{J_1\left(\dfrac{R\theta}{\lambda}\right)}{\theta}\right|^2d\omega . \tag{3,17} \]

This formula coincides exactly with formula (2,5), obtained from elementary optical considerations. We see that both the ordinary theory of diffraction and the exact theory of scattering, using the basic assumption that the nucleus is opaque to neutrons, lead to identical results, which, of course, was to be expected.

Integrating (3,17) over angles, we obtain the total cross section of elastic scattering of fast neutrons caused by their absorption:

\[ \sigma=\pi R^2. \tag{3,17′} \]

It is equal to the area of the transverse section of the nucleus.

It is important to note that the very fact of the presence of elastic scattering is a direct consequence of the absorption of neutrons. The removal of neutrons from the beam will be caused both by their absorption and by scattering. Obviously, the neutron absorption cross section in the case under consideration of small wavelengths is also equal to \(\pi R^2\), so that the total cross section of all processes is equal to \(2\pi R^2\). To verify this,

one should use formula (3.8) and substitute into it the values \(\zeta_l\) given by (3.10). We obtain the following expression for the neutron capture cross section:

\[ \sigma_{\text{capt}}=\pi \lambda^2 \sum_{l=0}^{[R/\lambda]} (2l+1)\simeq \pi R^2 . \]

Thus, the capture cross section, i.e. the inelastic scattering of fast neutrons, for \(\lambda \ll R\) is equal to the area of the transverse section of the nucleus (a “black” nucleus). The presence of absorption gives rise to additional elastic scattering (not connected with the formation of the compound nucleus and the re-emission of the neutron), occurring mainly at small angles, which may be called diffraction scattering, and whose total cross section, just as that of inelastic scattering, is equal to \(\pi R^2\). The total cross section for all processes, both elastic and inelastic scattering of fast neutrons, is equal to \(2\pi R^2\). This quantity determines the removal of neutrons from the beam.

4. DIFFRACTION SCATTERING OF FAST CHARGED PARTICLES

Let us proceed to the consideration of the scattering of fast charged particles that can be absorbed by nuclei[^15].

Here too, diffraction phenomena must occur, to some extent analogous to the diffraction of fast neutrons. The phenomenon, however, is complicated by the presence of charge on the particles. Because of this we cannot use the elementary optical theory to describe the scattering of interest to us, which may be characterized as the diffraction of charged rays around an absolutely black charged body. To determine the amplitude of this scattering, one must from the outset use the general formula of scattering theory (3.5). We must first of all establish what form the quantities \(\beta_l\) and \(\zeta_l\) now have. In contrast to the case of neutrons, for which \(\beta_l\) is equal to unity if the impact parameter exceeds the radius of the nucleus, in the scattering of charged particles \(\beta_l\) is always, for all arbitrarily large values of \(l\), different from unity. This is connected with Coulomb scattering, which exists for arbitrarily large values of the impact parameter. If scattering takes place at small angles, then the particle passes at a large distance from the nucleus and cannot be captured by it. Consequently, at very small angles the scattering must be Coulomb scattering. Noticeable deviations from purely Coulomb scattering must set in at angles exceeding some minimum angle. This conclusion is confirmed below by calculation.

We shall, just as in the case of fast neutrons, assume that a charged particle entering the sphere of action of the nucleus is absorbed by the latter. In considering neutrons, we assumed,

that absorption takes place in the case when the collision parameter does not exceed the radius of the nucleus. For charged particles the condition for absorption must be formulated somewhat differently.

In the case of fast particles of interest to us, whose wavelength is small in comparison with the dimensions of the nucleus, a quasi-classical treatment is valid. We may therefore say that, for absorption of the particle, it is in any case necessary that the shortest distance \(r_0\) between the nucleus and the particle, moving according to the laws of classical mechanics, not exceed the radius \(R\) of the nucleus:

\[ r_0 \leq R . \tag{4,1} \]

The quantity \(r_0\) is easily related to the collision parameter \(r_\infty\). Denoting the velocity of the particle at closest approach to the nucleus and at an infinite distance from the nucleus, respectively, by \(v_0\) and \(v_\infty\), we have, on the basis of the law of conservation of angular momentum,

\[ \frac{\dot r_0}{r_\infty}=\frac{v_\infty}{v_0}. \]

The ratio of the velocities is, evidently,

\[ \frac{v_\infty}{v_0}= \frac{1}{\sqrt{1-\dfrac{Zee'}{r_0E}}}, \]

where \(E\) is the energy of the incident particle, \(e'\) its charge, and \(Ze\) the charge of the nucleus. Therefore

\[ r_0= \frac{r_\infty}{\sqrt{1-\dfrac{Zee'}{r_0E}}}. \]

Inequality (4,1) takes the form

\[ r_\infty \leq R\sqrt{1-\frac{Zee'}{RE}} . \tag{4,2} \]

This is the condition which the collision parameter must satisfy in order that the particle can be absorbed by the nucleus.

Below we shall assume that, if condition (4,2) is fulfilled, then the particle is absorbed by the nucleus. If \(e'=0\), this condition becomes the condition for absorption of neutrons, which we used above. In the case of negatively charged particles the collision parameter may exceed the radius of the nucleus. In the case of positively charged particles the collision parameter must be smaller than the radius of the nucleus. In this latter case the quantity \(\dfrac{Zee'}{R}\) evidently represents the height of the barrier preventing the passage

particle into the nucleus. Denoting the height of the barrier by \(B\), we rewrite (4,2) in the form

\[ r_{\infty} \ll R \sqrt{1-\frac{B}{E}} . \]

As \(E\) increases, the radical tends to unity; therefore, for \(E \gg B\) the condition for absorption of charged particles coincides with the condition for absorption of neutrons.

In the quasiclassical approximation, which may be used in the case of fast particles, the collision parameter is, as is well known, \(\lambda_{\infty} l\), where \(\lambda_{\infty}\) is the wavelength of the particle at an infinite distance from the nucleus, and \(\hbar l\) is the angular momentum of the particle.

It follows from this that, for absorption of a particle, its angular momentum \(l\) must not exceed the value

\[ l_0=\frac{R}{\lambda_{\infty}}\sqrt{1-\frac{Zee'}{RE}}: \]

\[ l \ll l_0=\frac{R}{\lambda_{\infty}}\sqrt{1-\frac{Zee'}{RE}} . \tag{4,2′} \]

This condition replaces, in the case of charged particles, the condition \(l \ll \dfrac{R}{\lambda}\), which pertains to the case of neutrons. It is easy to see that the quantity \(\lambda_{\infty}\left(1-\dfrac{Zee'}{RE}\right)^{-1/2}\) is the wavelength of the particle at the surface of the nucleus. Denoting it by \(\lambda_0\), we can rewrite (4,2′) in the form

\[ l \ll \frac{R}{\lambda_0}. \]

Thus, we assume that a particle whose wavelength is considerably smaller than the radius of the nucleus is absorbed by the nucleus if the angular momentum of the particle does not exceed \(l_0\), i.e. the sticking coefficient \(\zeta_l\) is equal to unity if \(l \ll l_0\). Hence it follows that for \(l \ll l_0\), \(\beta_l=0\).

Just as in the case of neutrons, it should be borne in mind that at very high energies the nucleus becomes transparent to particles incident upon it. We shall assume that the energy of the particle lies below the limit at which the nucleus becomes transparent to particles.

Let us give the values of the quantity \(l_0\) (which may be called the critical value of the angular momentum of the particle) for various particles as a function of the energy \(E\). For heavy nuclei at the end of the periodic system the height of the potential barrier is approximately \(12\text{–}14\) MeV for protons and \(25\text{–}28\) MeV for \(\alpha\)-particles. For such nuclei and protons with energy of the order of \(15\) MeV the quantity \(l_0\) is \(\sim 4\), and for proton energies of the order of \(25\) MeV \(l_0 \approx 7\). For heavy nuclei and \(\alpha\)-particles with energy \(50\) MeV, \(l \approx 20\); for an \(\alpha\)-particle energy equal to \(25\) MeV, \(l_0 \approx 5\).

In the case of light nuclei \((A \simeq 100)\) and protons with energy \(25 \mathrm{MeV}\), \(l_0 \simeq 6\); for the same nuclei and \(\alpha\)-particles with energy \(50 \mathrm{MeV}\), \(l_0 \simeq 17\).

Table I

Values of \(l_0\) for protons, deuterons, and \(\alpha\)-particles

Particle	\(E\)	\(A=40\)	\(A=66\)	\(A=112\)	\(A=174\)	\(A=238\)
Proton	10	2.33	2.15	0	—	—
Proton	15	3.45	3.73	3.57	3.2	—
Proton	25	4.96	5.63	6.16	6.64	6.83
Proton	50	7.44	8.64	10.0	11.4	12.1
Proton	100	—	—	—	17.2	18.9
Proton	500	—	—	—	—	44.8
Deuteron	10	3.33	3.14	0	—	—
Deuteron	15	4.88	5.28	5.74	4.53	2.6
Deuteron	25	7.0	7.86	8.66	9.38	9.46
Deuteron	50	10.4	12.2	14.0	15.8	17.1
Deuteron	100	15.5	18.1	21.3	24.4	26.8
Deuteron	500	35.1	41.5	49.2	57.0	63.2
\(\alpha\)-particle	10	—	—	—	—	—
\(\alpha\)-particle	15	3.77	—	—	—	—
\(\alpha\)-particle	25	8.08	6.74	4.82	—	—
\(\alpha\)-particle	50	13.8	17.2	18.5	19.9	—
\(\alpha\)-particle	100	—	23.3	32.0	35.3	—
\(\alpha\)-particle	500	—	—	—	88.4	—

Table I gives the values

\[ l_0=\frac{1.5 \cdot 10^{-13} A^{1/3}}{\lambda} \sqrt{1-\frac{B}{E}} \qquad (B \text{ is the barrier height}) \]

for protons, deuterons, and \(\alpha\)-particles as a function of the nuclear mass \(A\) and the particle energy \(E\).

If the orbital angular momentum of the particle exceeds \(l_0\), then the sticking coefficient is equal to zero. Hence it follows that the coeffi-

the scattering coefficients \(\beta_l\) for \(l>l_0\) are equal to unity in absolute value. In the case of neutrons passing outside the nucleus, we simply considered \(\beta_l\) equal to unity. In the case of charged particles passing at an arbitrarily large distance from the nucleus, scattering caused by Coulomb forces always takes place; therefore \(\beta_l\) for \(l>l_0\) is different from unity.

We may assume that for \(l>l_0\) the phases at infinity will be the same as in the case of purely Coulomb scattering. In the latter case, as is known, the phase at infinity \(\eta_l\) is equal to\({}^{16}\)

\[ \eta_l=\arg \Gamma(l+1+i\alpha), \]

where \(\Gamma(x)\) is the gamma function, \(\alpha=Ze^2/\hbar v\), and \(v\) is the velocity of the particle at an infinite distance from the nucleus. Thus, for \(l>l_0\)

\[ \beta_l=e^{2i\eta_l}. \]

Now we can formulate the problem of determining the scattering amplitude of fast charged particles by an absorbing nucleus as follows: it is required to find the quantity \(f(\theta)\), determined by formula (3,5), in which the \(\beta_l\) are equal to:

\[ \beta_l= \begin{cases} 0, & \text{if } l\leq l_0,\\ e^{2i\eta_l}, & \text{if } l>l_0. \end{cases} \tag{4,3} \]

Before proceeding to the calculation of \(f(\theta)\), let us note, just as we did above in considering neutron scattering, that the sharp separation of the values of the orbital angular momenta for which the sticking coefficient is zero and unity is approximate in character. It may be said that the introduction of the critical angular momentum \(l_0\) has meaning with an accuracy up to quantities of order unity. Therefore the calculation of \(f(\theta)\) with the values of \(\beta_l\) given above will give the correct result if, in the sum written below (4,4), the role of the intermediate terms, i.e. the terms with values of \(l\) close to \(l_0\), is insignificant. In other words, our method of calculating \(f(\theta)\) will give the correct result if, when \(l_0\) is changed by a quantity of order unity, the scattering amplitude changes little.

To find out to what conditions this requirement leads, let us write the scattering amplitude with the values of \(\beta_l\) defined above:

\[ f(\theta)=\frac{1}{2}i\lambda\sum_{l=0}^{l_0}(2l+1)P_l(\cos\theta)- \]

\[ -\frac{1}{2}i\lambda\sum_{l=l_0}^{\infty}(2l+1)(e^{2i\eta_l}-1)P_l(\cos\theta). \tag{4,4} \]

A change of \(l_0\) by an amount of order unity will not affect the scattering amplitude if the conditions

\[ \left. \begin{aligned} &\left|P_{l_0}(\cos \theta)-P_{l_0\pm1}(\cos \theta)\right| \ll \left|P_{l_0}(\cos \theta)\right|,\\ &\left|\eta_{l_0\pm1}-\eta_{l_0}\right| \ll \left|\eta_{l_0}\right|. \end{aligned} \right\} \tag{4,5} \]

are satisfied. The first of these inequalities we have already considered above and have seen that it leads to the condition

\[ \theta \ll 1. \]

Let us now consider the second inequality. Using Stirling’s formula*), which determines \(\Gamma(z)\) for large \(|z|\), one can represent \(e^{2 i \eta_l}\) in the case of large values of \(l\) in the form

\[ e^{2i\eta_l}= \frac{\Gamma(1+l+i\alpha)}{\Gamma(1+l-i\alpha)} \simeq e^{2i\alpha\ln l},\qquad l\gg 1. \]

From this it is easy to conclude that the second of the inequalities (4,6) leads to the condition

\[ \alpha \ll l_0. \]

Thus, by introducing the quasiclassical quantity \(l_0\), which separates the values of \(l\) for which the sticking coefficient is equal to zero and to unity, we obtain the correct result for the scattering amplitude in the region of small scattering angles, provided only that the wavelength of the particle at the surface of the nucleus is small in comparison with its radius and provided that \(\alpha \ll l_0\).

As will be seen from what follows, for \(\bar\lambda \ll R\) the principal role is played by small scattering angles, for which our consideration is entirely legitimate.

In Table II are given the values of

\[ \alpha=\frac{Z}{137}\sqrt{\frac{M c^2}{2E}} \]

(\(M\) is the mass of the particle) for various particles as functions of \(Z\) and of the particle energy \(E\). Let us proceed to the study of formula (4,4), which determines the scattering amplitude.

Let us first note that the infinite series defining \(f(\theta)\) diverges, since the phase \(\eta_l\) does not tend to zero as \(l\to\infty\). This is connected with the property of the Coulomb interaction already noted earlier: scattering caused by Coulomb forces occurs at arbitrarily large distances between the particles. Nevertheless, the series (4,4) is summable; it may be regarded as the limit of a power series whose terms differ from the corresponding terms of (4,4) by factors \(\varepsilon^l\), where \(\varepsilon<1\). If \(\theta\ne0\), then there exists a limit of the sum of this series as \(\varepsilon\to1\); we regard the sum of the series (4,4) as equal to this limit.

*) This formula has the form \(^{17}\)

\[ \Gamma(z)\simeq \sqrt{2\pi}\, z^{z-\frac12} e^{-z}. \]

Table II

Values of \(\alpha\) for protons, deuterons, and \(\alpha\)-particles

Particle	\(E\)	\(Z=20\)	\(Z=50\)	\(Z=70\)	\(Z=80\)	\(Z=92\)
Proton	10	0.996	2.49	3.48	3.98	4.58
Proton	15	0.813	2.03	2.85	3.31	3.74
Proton	20	0.705	1.76	2.47	2.82	3.24
Proton	25	0.63	1.57	2.21	2.52	2.91
Proton	50	0.445	1.11	1.56	1.78	2.05
Proton	100	0.315	0.785	1.1	1.26	1.45
Proton	250	0.199	0.497	0.697	0.796	0.917
Proton	500	0.141	0.352	0.494	0.564	0.648
Deuteron	10	1.41	3.52	4.94	—	—
Deuteron	15	1.15	2.87	4.03	4.68	5.28
Deuteron	20	0.996	2.49	3.49	3.99	4.58
Deuteron	25	0.891	2.22	3.11	3.56	4.12
Deuteron	50	0.629	1.57	2.21	2.52	2.9
Deuteron	100	0.445	1.11	1.55	1.78	2.05
Deuteron	250	0.281	0.702	0.985	1.12	1.3
Deuteron	500	0.2	0.498	0.698	0.798	0.916
\(\alpha\)-particle	10	3.98	9.92	13.9	—	—
\(\alpha\)-particle	15	3.25	8.12	11.4	—	—
\(\alpha\)-particle	20	2.82	7.04	9.88	—	—
\(\alpha\)-particle	25	2.52	6.28	8.85	10.1	—
\(\alpha\)-particle	50	1.78	4.44	6.23	7.12	8.2
\(\alpha\)-particle	100	1.26	3.14	4.1	5.04	5.8
\(\alpha\)-particle	250	0.796	1.91	2.99	3.15	3.67
\(\alpha\)-particle	500	0.565	1.41	1.97	2.23	2.59

The infinite sum entering into (4,4) can be related to the amplitude of Coulomb scattering, which differs from the infinite sum (4,4) in that in it the summation is carried out not from \(l_0\), but from zero. Therefore \(f(\theta)\) can be written in the form

\[ f(\theta)=\frac{1}{2} i\lambda \sum_{l=0}^{l_0}(2l+1)e^{2i\eta_l}P_l(\cos\theta)+f_{\mathrm{Coul}}(\theta), \tag{4,6} \]

where \(f_{\text{Coul}}(\theta)\) is the amplitude of Coulomb scattering, equal, as is known, to

\[ f_{\text{Coul}}(\theta) = -\frac{Ze^{3}}{Mv^{2}}\, \frac{e^{-i\alpha\ln \sin \frac{\theta}{2}+2i\eta_{0}}}{1-\cos\theta} = -\alpha\lambda\, \frac{e^{-2i\alpha\ln \frac{\theta}{2}+2i\eta_{0}}}{1-\cos\theta}, \qquad e^{2i\eta_{0}}=\frac{\Gamma(1+i\alpha)}{\Gamma(1-i\alpha)}. \tag{4,6'} \]

Formula \((4,6)\) is inconvenient to use; we shall therefore transform somewhat the basic relation \((4,4)\).

For this purpose consider the series

\[ \sum_{l=0}^{\infty}(2l+1)P_l(\cos\theta). \]

This series diverges. However, it is summable in the sense indicated above, and its sum for \(\theta\ne 0\) is equal to zero. This is easily verified if one recalls the relation, known from the theory of Legendre polynomials,

\[ (1-2z\varepsilon+\varepsilon^2)^{-1/2} = \sum_{l=0}^{\infty}\varepsilon^l P_l(z), \]

from which it follows that

\[ \sum_{l=0}^{\infty}(2l+1)P_l(z)\varepsilon^l = \frac{1-\varepsilon^2}{(1-2z\varepsilon+\varepsilon^2)^{3/2}}. \]

If \(z\) is different from unity, then

\[ \lim_{\varepsilon\to 1} \sum_{l=0}^{\infty}(2l+1)\varepsilon^l P_l(\cos\theta) =0, \qquad \theta\ne 0. \]

Subtracting this series from \((4,4)\), we represent the amplitude \(f(\theta)\) in the form

\[ f(\theta) = -\frac{i\lambda}{2} \sum_{l=0}^{\infty}(2l+1)e^{2i\alpha\ln l}P_l(\cos\theta). \tag{4,7} \]

This series also does not converge in the ordinary sense; however, it too is summable. The divergence is physically connected with the exceptional role of small angles \(\theta\) in Coulomb scattering. From formula \((4,6')\) it is seen that, if the amplitude of Coulomb scattering is multiplied by \((1-\cos\theta)\), then we obtain a finite quantity at arbitrarily small angles. For this reason it proves expedient to transform formula \((4,7)\), after first multiplying it by \((1-\cos\theta)\) and using the relation, known from the theory of Legendre polynomials,

\[ (2l+1)xP_l(x)=(l+1)P_{l+1}(x)+lP_{l-1}(x). \]

We obtain the following expression for \((1-\cos\theta)f(\theta)\):

\[ \begin{aligned} (1-\cos\theta)f(\theta) &=-\frac{i\lambda}{2}\sum_{l_0}^{\infty} l^{2i\alpha} \left\{(2l+1)P_l(\cos\theta)-\right.\\ &\qquad\qquad\left.-(l+1)P_{l+1}(\cos\theta)-lP_{l-1}(\cos\theta)\right\}=\\ &=-\frac{i\lambda}{2}\left\{l_0(l_0-1)^{2i\alpha}P_{l_0}(\cos\theta) -l_0^{2i\alpha+1}P_{l_0-1}(\cos\theta)+\right.\\ &\qquad\left. +\sum_{l_0}^{\infty} \left[(2l+1)l^{2i\alpha}-l(l-1)^{2i\alpha}-(l+1)^{2i\alpha+1}\right] P_l(\cos\theta)\right\}. \end{aligned} \tag{4.8} \]

Since \(l_0\gg 1\), the expression in square brackets in the last formula, to an accuracy of \(1/l^2\), is equal to \(-(2i\alpha)^2l^{2i\alpha-1}\); therefore formula (4.8) may be rewritten in the form

\[ \begin{aligned} (1-\cos\theta)f(\theta) &=-\frac{i\lambda}{2}\left\{ l_0^{2i\alpha+1}\left[P_{l_0}(\cos\theta)-P_{l_0-1}(\cos\theta)\right]\right.\\ &\qquad\left. -2i\alpha l_0^{2i\alpha}P_{l_0}(\cos\theta) -(2i\alpha)^2\sum_{l_0}^{\infty}l^{2i\alpha-1}P_l(\cos\theta) \right\}. \end{aligned} \tag{4.8′} \]

Above we indicated that our method of determining the scattering amplitude is valid only for small scattering angles and large \(l_0\). If \(\theta\ll 1\) and \(l\gg 1\), then, as is known, the relation \(P_l(\cos\theta)\simeq J_0(l\theta)\) is valid, where \(J_0(x)\) is the Bessel function of order zero. Taking also into account the inequality \(\alpha\ll l_0\), from which it follows that the relative change of the factor \(l^{2i\alpha}\) in passing from \(l\) to \(l+1\) is much smaller than unity, we may replace the infinite sum entering (4.8′) by the integral

\[ \sum_{l_0}^{\infty} l^{2i\alpha-1}P_l(\cos\theta) \simeq \theta^{-2i\alpha}\int_{l_0\theta}^{\infty} z^{2i\alpha}\frac{J_0(z)}{z}\,dz . \]

In addition, the difference \(P_{l_0}(\cos\theta)-P_{l_0-1}(\cos\theta)\) may be replaced by \(\theta J_0'(l_0\theta)=-\theta J_1(l_0\theta)\). Therefore, finally, we obtain the following expression for \(f(\theta)\):

\[ f(\theta)=-i\lambda\left\{ l_0^{2i\alpha+1}\frac{J_1(l_0\theta)}{\theta} +\frac{1}{\theta^2}\left[ 2i\alpha l_0^{2i\alpha}J_0(l_0\theta) +(2i\alpha)^2\theta^{-2i\alpha} \int_{l_0\theta}^{\infty} z^{2i\alpha}\frac{J_0(z)}{z}\,dz \right]\right\}. \tag{4.9} \]

For \(\alpha=0\) this expression goes over into the expression already known to us for the scattering amplitude of neutral particles.

The integral entering here, which we shall denote by \(L_\alpha(x)\), where \(x=l_0\theta\), is not expressible in the general case in terms of known transcendental functions. It may be called the diffraction integral

of the problem under study. Below we shall use the following notations for its real and imaginary parts:

\[ C_\alpha(x)=\int_x^\infty \cos(2\alpha\ln z)\,\frac{J_0(z)}{z}\,dz, \]

\[ S_\alpha(x)=\int_x^\infty \sin(2\alpha\ln z)\,\frac{J_0(z)}{z}\,dz. \]

The form of these functions for the value \(\alpha=1\) is shown in Fig. 3.

Fig. 3.

Squaring the modulus \(f(\theta)\) and multiplying it by the element of solid angle \(d\sigma\), we obtain the differential scattering cross section, which-

can be represented in the form

\[ \sigma(\theta)\,d\omega = \frac{4\alpha^{2}\lambda^{3}}{\theta^{4}}\,N(l_{0}\theta,\alpha)\,d\omega, \tag{4,9'} \]

where \(\sigma_{R}(\theta)=\dfrac{4\alpha^{2}\lambda^{3}}{\theta^{4}}\) is the differential scattering cross section in a purely Coulomb field at small scattering angles\(^*\), and \(N(l_{0}\theta,\alpha)\) is a factor determining the deviation of the cross section from Rutherford’s formula. This factor is equal to

\[ \begin{aligned} N(x,\alpha)={}&J_{0}^{2}(x) +\left(\frac{x}{2\alpha}\right)^{2}J_{1}^{2}(x) +4\alpha^{2}\{C_{\alpha}^{2}(x)+S_{\alpha}^{2}(x)\} \\ &+2C_{\alpha}(x)\{\alpha\sin(2\alpha\ln x)J_{0}(x) -x\cos(2\alpha\ln x)J_{1}(x)\} \\ &-2S_{\alpha}(x)\{\alpha\cos(2\alpha\ln x)J_{0}(x) +x\sin(2\alpha\ln x)J_{1}(x)\}. \end{aligned} \]

The difference from the cross section in a purely Coulomb field is due to the possibility of absorption of particles by nuclei. In Fig. 4 the dependence of the factor \(N\) on \(l_{0}\theta\) and \(\alpha\) is shown.

5. INVESTIGATION OF THE DIFFRACTION INTEGRAL

The diffraction integral can be expressed through known functions only in some limiting cases, which we shall consider first of all.

It is easy to obtain the expansion of the diffraction integral in a series in increasing powers of \(x\). For this purpose note that\({}^{19}\)

\[ \int_{0}^{\infty} x^{m}J_{\nu}(x)\,dx = 2^{m}\, \frac{\Gamma\!\left(\dfrac{1}{2}+\dfrac{m}{2}\right)} {\Gamma\!\left(\dfrac{1}{2}-\dfrac{m}{2}\right)}. \]

Fig. 4.

\[ N(0.2;x) \]

\[ N(1.732;x) \]

\[ x=l_{0}\theta \]

\[ N(\alpha,x) \]

\(^*\) The amplitude of this scattering for \(\theta\ll 1\) is, according to\({}^{18}\),

\[ f_{\mathrm{Coul}}(\theta) = -\frac{2\lambda}{\theta^{2}}\, e^{-2i\alpha\ln\frac{\theta}{2}+2i\eta_{\alpha}}. \]

Setting in it \(m=2i\alpha-1\), we obtain:

\[ \int_0^\infty z^{2i\alpha}\frac{J_0(z)}{z}\,dz = \frac{2^{2i\alpha}}{2i\alpha}\, \frac{\Gamma(1+i\alpha)}{\Gamma(1-i\alpha)} = \frac{1}{2i\alpha}e^{2i\alpha\ln 2+2i\eta_0}, \]

where \(\eta_0\) is the phase at infinity in the Coulomb field for \(l=0\).

Using this formula and expanding \(J_0(z)\) in a power series in \(z\), we obtain:

\[ \begin{aligned} L_\alpha(x) &= \int_x^\infty z^{2i\alpha}\frac{J_0(z)}{z}\,dz = \frac{1}{2i\alpha}e^{2i\alpha\ln 2+2i\eta_0} \\ &\quad - \int_0^x z^{2i\alpha} \left\{ 1-\frac{z^2}{2^2}+\frac{z^4}{2^4(2!)^2}-\cdots \right\}\,dz \\ &= \frac{1}{2i\alpha}e^{2i\alpha\ln 2+2i\eta_0} - \frac{x^{2i\alpha}}{2i\alpha} + \frac{x^{2i\alpha+2}}{(2i\alpha+2)2^2} - \frac{x^{2i\alpha+4}}{(2i\alpha+4)2^4\cdot4} +\cdots . \end{aligned} \tag{5,1} \]

If \(x\ll 1\), then

\[ L_\alpha(x) = \frac{1}{2i\alpha}e^{2i\alpha\ln 2+2i\eta_0} - \frac{x^{2i\alpha}}{2i\alpha}. \tag{5,1'} \]

This expansion is valid for all values of \(\alpha\).

Let us now consider the limiting case of large \(x\). Replacing in this case, under the integral sign, the Bessel function by its asymptotic representation

\[ J_0(z)\simeq \left(\frac{2}{\pi z}\right)^{1/2} \sin\left(z+\frac{\pi}{4}\right),\qquad z\gg 1, \]

and integrating by parts, we obtain:

\[ \begin{aligned} L_\alpha(x) &\simeq \left(\frac{2}{\pi}\right)^{1/2} \int_x^\infty z^{2i\alpha-3/2} \sin\left(z+\frac{\pi}{4}\right)\,dz \\ &= \left(\frac{2}{\pi}\right)^{1/2} \left\{ x^{2i\alpha-3/2} \cos\left(x+\frac{\pi}{4}\right) - \frac{2i\alpha-3/2}{x}\, x^{2i\alpha-3/2} \sin\left(x+\frac{\pi}{4}\right) +\cdots \right\}. \end{aligned} \]

This expansion may be used if \(\alpha\ll x\).

Restricting ourselves to the first term and noting that the Bessel function \(J_1(x)\) for \(x\gg 1\) is determined by the formula

\[ J_1(x)\simeq -\left(\frac{2}{\pi x}\right)^{1/2} \cos\left(x+\frac{\pi}{4}\right), \]

we obtain the following asymptotic expression for \(L_\alpha(x)\), valid for large \(x\) and \(\alpha\) satisfying the condition \(\alpha\ll x\):

\[ L_\alpha(x)\simeq -x^{2i\alpha-1}J_1(x),\qquad x\gg 1,\ \alpha\ll x. \tag{5,2} \]

This formula is not valid for \(\alpha > x\). We shall therefore derive an asymptotic formula for \(L_\alpha(x)\), valid for large \(x\) and large \(\alpha\). (For \(x \ll 1\) and arbitrary \(\alpha\), \(L_\alpha(x)\) is determined by formula \((5,1')\).)

Using again the asymptotic expression for \(J_0(z)\), we represent \(L_\alpha(x)\) in the form \((x \gg 1)\)

\[ L_\alpha(x)=\frac{1}{2}\left(\frac{2}{\pi}\right)^{1/2} \int_x^\infty z^{-3/2} \left\{ e^{i\left(2\alpha\ln z-z+\frac{\pi}{4}\right)} + e^{i\left(2\alpha\ln z+z-\frac{\pi}{4}\right)} \right\}dz . \tag{5,3'} \]

We shall assume that \(2\alpha > x\). In this case, the method of “passage” may be applied to compute \(L_\alpha(x)\). Denote the exponents of the first and second terms in braces by \(b_-(z)\) and \(b_+(z)\). Notice that the derivative of \(b_-(z)\) vanishes at the point \(z_0=2\alpha\), which lies in the interval of integration; the zero of the derivative of \(b_+(z)\), however, does not lie in this interval. Since the principal role in the integral is played by the region of values of \(z\) close to the zero of the derivative of the exponent, where the exponential factor changes little and where, therefore, mutual cancellation of the different parts of the integral does not occur, it follows from what has been said above that the contribution to the integral from the second term will be considerably smaller than that from the first. We shall therefore neglect the second term and replace \(b_-(z)\) by the first two terms of the expansion of this function in powers of \(z-z_0\):

\[ b_-(z)=b_-(z_0)+\frac{(z-z_0)^2}{2}\,b_-''(z_0) =2\alpha\ln 2\alpha-2\alpha+\frac{\pi}{4} -\frac{(z-2\alpha)^2}{4\alpha}. \]

Taking the pre-exponential factor out from under the integral sign at the point \(z=2\alpha\), and extending the limits of integration from \(-\infty\) to \(+\infty\), we obtain:

\[ \begin{aligned} L_\alpha(x) &\simeq \frac{1}{2}\left(\frac{2}{\pi}\right)^{1/2}(2\alpha)^{-3/2} e^{i\left(2\alpha\ln 2\alpha-2\alpha+\frac{\pi}{4}\right)} \int_{-\infty}^{+\infty} e^{-i\frac{(z-z_0)^2}{4\alpha}}\,dz \\ &= \frac{1}{2\alpha}\,e^{i(2\alpha\ln 2\alpha-2\alpha)} . \end{aligned} \tag{5,3} \]

Let us now determine the conditions of applicability of this asymptotic expression. First of all it is necessary that the zero of \(b_-'(z)\) lie in the interval \((x,\infty)\), whence it follows that \(2\alpha\) must be greater than \(x\). In the integral entering into \((5,3)\), the width of the region of effective values of \(z-z_0\) is, in order of magnitude, \(\alpha^{1/2}\). For the applicability of the method it is necessary that this width be considerably smaller than \(z_0\), whence it follows that \(\alpha\) must be considerably greater than unity.

Finally, let us consider the question of the admissible values of \(x\). Since in the integral \((5,3)\) large values of \(x\), lying near \(z_0=2\alpha \gg 1\), are essential, we could use the asymptotic representation of \(J_0(z)\) without making the special assumption that \(x \gg 1\). Therefore the quantity \(x\) must only be less than \(2\alpha\).

otherwise it may be arbitrary. This argument is not applicable to small \(x\), since in this case a second essential range of values of \(z\), lying near zero, appears in the integral. This range, as follows from \((5,1')\), gives the same contribution to the integral as the range \(z \sim z_0\).

In the limiting cases considered, the diffraction integral is expressed in terms of elementary functions. In other ranges of the variables, numerical integration is necessary for determining \(L_\alpha(x)\); the results of this integration are presented in Fig. 3.

6. SOME LIMITING CASES OF DIFFRACTION SCATTERING OF CHARGED PARTICLES

Let us clarify the behavior of the amplitude and of the scattering cross section of fast charged particles in certain limiting cases. We shall begin with the case of small \(\alpha\).

The case of small \(\alpha\)

First consider the region of small scattering angles \(\theta\), satisfying the condition \(l_0\theta \ll 1\). For small values of the argument, the diffraction integral is determined by formula \((5,1')\). Using this formula and noting that for \(x \ll 1\), \(J_0(x) \simeq 1\), \(J_1(x) \simeq \dfrac{x}{2}\), we write \(f(\theta)\) in the form

\[ f(\theta)=\frac{\lambda}{2}l_0^2 e^{2i\alpha\ln l_0+i\frac{\pi}{2}} -\frac{2\alpha\lambda}{\theta^2}e^{-2i\alpha\ln\frac{\theta}{2}+2i\eta_0}, \tag{6,1'} \]

where \(l_0\theta \ll 1\).

The first term in this formula, independent of the angle \(\theta\), is the amplitude of diffraction scattering, while the second term, inversely proportional to \(\theta^2\), is the amplitude of Coulomb scattering. The differential scattering cross section for \(l_0\theta \ll 1\) is equal to

\[ \sigma(\theta)=\frac{4\alpha^2\lambda^2}{\theta^4} \left\{1+\frac{(l_0\theta)^4}{16\alpha^2} +\frac{(l_0\theta)^3}{2\alpha}\sin\left(2\alpha\ln\frac{l_0\theta}{2}-2\eta_0\right)\right\}. \tag{6,1} \]

The factor in braces, which determines the deviation from the Rutherford formula, may differ appreciably from unity. If

\[ (l_0\theta)^2 \ll 2\alpha,\quad \text{i.e.}\quad \theta \ll \frac{\sqrt{2\alpha}}{l_0}, \]

then this factor is practically equal to unity and the cross section is determined by the Rutherford formula

\[ \sigma(\theta)=\frac{4\alpha^2\lambda^2}{\theta^4},\quad \theta \ll \frac{\sqrt{2\alpha}}{l_0},\quad \alpha \ll 1. \tag{6,2} \]

If

\[ \frac{1}{l_0}\gg \theta \gg \frac{\sqrt{2\alpha}}{l_0}, \]

then the factor reduces to \(\dfrac{(l_0\theta)^4}{16\alpha^2}\), the cross section does not depend on the scattering angle and is equal to

\[ \sigma(\theta)=\frac{1}{4}\lambda^2 l_0^4,\quad \frac{1}{l_0}\gg \theta \gg \frac{\sqrt{2\alpha}}{l_0},\quad \alpha \ll 1. \tag{6,3} \]

This expression coincides with the scattering cross section of fast neutrons in the case when \(\theta \ll \dfrac{1}{l_0}\).

Let us now consider scattering angles exceeding \(\dfrac{1}{l_0}\). For small values of \(\alpha\), the diffraction integral is determined by formula (5,2). With increasing \(l_0\theta\) it decreases as \((l_0\theta)^{-3/2}\). In expression (4,9), which determines \(f(\theta)\), the principal role is played by the first term

\[ f(\theta)=i\lambda l_0^{2i\alpha}\frac{l_0}{\theta}J_1(l_0\theta),\quad 1\gg\theta\gg\frac{1}{l_0},\quad \alpha\ll 1. \tag{6,4} \]

We see that under these conditions the scattering amplitude of charged particles differs from the neutron scattering amplitude only by a phase. The scattering cross section for both is the same and is equal to

\[ \sigma(\theta)=\lambda^2 l_0^2\left|\frac{J_1(l_0\theta)}{\theta}\right|^2,\quad 1\gg\theta\gg\frac{1}{l_0},\quad \alpha\ll 1. \tag{6,4′} \]

Thus, in the limiting cases considered, the scattering cross section is determined either by Rutherford’s formula or by formula (2,6′), which describes the diffraction scattering of neutrons; moreover, one or the other case occurs depending on which scattering is greater—Coulomb or diffraction. The quantity \(\dfrac{\sqrt{2\alpha}}{l_0}\) separates regions of angles with different scattering laws.

Fig. 5.

In Fig. 5 the different regions of scattering are shown schematically.

Let us proceed to the investigation of the case of large \(\alpha\).

Case of large \(\alpha\)

We begin with the consideration of small scattering angles, \(\theta\ll\dfrac{1}{l_0}\). In this case the diffraction integral is determined by formula (5,1′). The scattering cross section is determined by formula (6,1), valid for arbitrary values of \(\alpha\). In particular, Coulomb scattering takes place

for angles \(\theta \ll \dfrac{\sqrt{2\alpha}}{l_0}\). However, as we shall now see, the region of applicability of Rutherford’s formula is extended considerably toward large angles, which need only satisfy the condition

\[ \theta \ll \frac{2\alpha}{l_0}. \]

In order to verify this, let us consider angles \(\theta \gg \dfrac{1}{l_0}\). If \(\alpha \gg l_0\theta \gg 1\), then the diffraction integral is determined by formula (5.3). The scattering amplitude takes the form

\[ f(\theta)=i\lambda\left\{ l_0^{2i\alpha}\frac{l_0}{\theta}J_1(l_0\theta) +\frac{2i\alpha}{\theta^2}l_0^{2i\alpha}J_0(l_0\theta) -\frac{2\alpha}{\theta^2}e^{-2i\alpha\left(\ln \frac{\theta}{2}-\ln\alpha+1\right)} \right\} \simeq -\frac{2i\alpha\lambda}{\theta^2} e^{-2i\alpha\left(\ln \frac{\theta}{2}-\ln\alpha+1\right)} . \tag{6.5} \]

(The last term is considerably larger than the first two.) Exactly the same formula gives the scattering amplitude in a Coulomb field in the case when \(\alpha \gg 1\). Indeed, the phase factor \(e^{2i\eta_0}\) for \(\alpha \gg 1\) can be represented in the form

\[ e^{2i\eta_0}=\frac{\Gamma(1-i\alpha)}{\Gamma(1+i\alpha)} \simeq i e^{-i\alpha+2i\alpha\ln\alpha}. \]

(We have used Stirling’s formula.) Therefore the amplitude \(f_{\text{Coul}}(\theta)\) for \(\alpha \gg 1\) is equal to

\[ f_{\text{Coul}}(\theta)= -\frac{2i\alpha\lambda}{\theta^2} e^{-2i\alpha\left(\ln \frac{\theta}{2}-\ln\alpha+1\right)}, \]

which coincides with (6.5).

Thus, in the range of angles

\[ \frac{1}{l_0}\ll \theta \ll \frac{2\alpha}{l_0} \]

the scattering is purely Coulomb scattering.

In the range of angles

\[ \frac{2\alpha}{l_0}\ll \theta \ll 1 \]

the diffraction integral is determined by formula (5.2). In expression (4.9) for the scattering amplitude the first term is considerably larger than the other two. Therefore

\[ f(\theta)=i\lambda l_0^{2i\alpha}\frac{l_0}{\theta}J_1(l_0\theta) \]

and

\[ \sigma(\theta)=\lambda^2 l_0^2\left|\frac{J_1(l_0\theta)}{\theta}\right|^2 \simeq \frac{2}{\pi}l_0\lambda^2 \frac{\sin^2\left(l_0\theta+\frac{\pi}{4}\right)}{\theta^3}, \qquad \frac{2\alpha}{l_0}\ll \theta \ll 1. \tag{6.6} \]

We have obtained an expression for the cross section of diffraction scattering of neutrons.

Thus, for \(\alpha \gg 1\), in the angular region \(\theta \ll \dfrac{2a}{l_0}\) the scattering is Coulomb scattering, while in the angular region \(\dfrac{2a}{l_0} \ll \theta \ll 1\) it is diffraction scattering. The boundary between the two angular regions with different scattering laws occurs at \(\theta \sim \dfrac{2a}{l_0}\), whereas in the case \(\alpha \ll 1\) such a boundary occurs at \(\theta \sim \dfrac{\sqrt{2\alpha}}{l_0}\) (see Fig. 5). We see, therefore, that in the case \(\alpha \gg 1\) the angular region in which the scattering cross section is determined by Rutherford’s formula is relatively wider than in the case of small \(\alpha\).

Let us note that in the case \(\alpha \ll 1\) the amplitudes of Coulomb and diffraction scattering coincide in order of magnitude at \(\theta \sim \dfrac{\sqrt{2\alpha}}{l_0}\). In the case \(\alpha \gg 1\) these amplitudes at \(\theta \sim \dfrac{2a}{l_0}\) do not coincide; on the contrary, the amplitude of Coulomb scattering exceeds the amplitude of diffraction scattering by a factor of \(\alpha^{1/2}\). Equality of the amplitudes occurs at \(\theta \sim \dfrac{4a^2}{l_0}\), and only at larger angles does the amplitude of diffraction scattering become greater than the amplitude of Coulomb scattering. Above, however, we have seen that the scattering acquires the features of diffraction already at angles \(\theta > \dfrac{2a}{l_0}\), and not at angles exceeding \(\dfrac{4a^2}{l_0}\), i.e., the scattering becomes diffraction scattering at angles much smaller than those at which the amplitudes of both kinds of scattering are compared. Hence there follows an important conclusion: near the angle \(\theta \sim \dfrac{2a}{l_0}\) there occurs a sharp decrease of the scattering cross section, by order of magnitude by a factor of \(\alpha\). The approximate course of the cross section for \(\alpha \gg 1\) has the form shown in Fig. 6.

Fig. 6.

The sharp decrease of the scattering cross section occurs because of the change in the width of the region of effective values of the variable in the diffraction integral when the scattering angle passes through the value \(\theta_0 = \dfrac{2a}{l_0}\).

It was already noted above that the width of the range of effective values of \(z\) in the diffraction integral in the case when \(z_0=2\alpha\gg l_0\theta\) is, in order of magnitude, \(\Delta z\sim \sqrt{\alpha}\) (\(\alpha\gg 1\)). If \(l_0\theta\gg 2\alpha\), then, as follows from the derivation of formula (5.2), \(\Delta z\sim 1\). Thus, when the scattering angle passes through the value \(\theta_0\sim \dfrac{2\alpha}{l_0}\), the width of the range of effective values of \(z\) decreases by a factor of \(\sqrt{\alpha}\). This leads to a decrease of the scattering amplitude by \(\sqrt{\alpha}\) and of the cross section by a factor of \(\alpha\). Since \(z=l\theta\), the width of the range of effective values of the orbital angular momenta \(l\), which play a role in the sum (4.8), is, in order of magnitude,

\[ \Delta l\sim \frac{\Delta z}{\theta_0}. \]

If \(\theta<\theta_0\), then

\[ \Delta l\sim \frac{\sqrt{\alpha}}{\theta_0}\sim \frac{l_0}{\sqrt{\alpha}}. \]

If \(\theta>\theta_0\), then

\[ \Delta l\sim \frac{1}{\theta_0}\sim \frac{l_0}{\alpha}. \]

It is easy to estimate the interval of angles \(\Delta\theta\) in which a sharp change of the cross section occurs. This interval is, in order of magnitude,

\[ \Delta\theta\sim \frac{\Delta z}{l_0}\sim \frac{\sqrt{\alpha}}{l_0}\sim \frac{\theta_0}{\sqrt{\alpha}}. \]

It is \(\sqrt{\alpha}\) times smaller than the limiting angle \(\theta_0\approx \dfrac{2\alpha}{l_0}\), separating the regions of Coulomb and diffraction scattering.

Summing up, we may say that the elastic scattering of fast charged particles capable of being absorbed by nuclei is, generally speaking, not described by the Rutherford formula.

If

\[ \alpha=\frac{Ze^2}{\hbar v}\ll 1, \]

then the scattering is determined by the Rutherford formula only at very small angles, small in comparison with

\[ \frac{\sqrt{2\alpha}}{l_0}\left(l_0=\frac{R}{\lambda}\sqrt{1-\frac{Zee'}{RE}}\right). \]

At angles larger than \(\dfrac{\sqrt{2\alpha}}{l_0}\), the cross section of elastic scattering of charged particles coincides with the cross section of elastic scattering of fast neutrons. This scattering, in which the chief role is played by angles \(\theta\ll 1\), may be called diffraction scattering, since it has the same character as the diffraction of light by an absolutely black sphere. The difference between neutrons and charged particles appears only in the phase of the scattering amplitude. In the range of angles from \(\dfrac{\sqrt{2\alpha}}{l_0}\) to \(\dfrac{1}{l_0}\) the effective cross section of this scattering does not depend on the scattering angle. At large angles characteristic diffraction oscillations appear.

If \(\alpha\gg 1\), then the Rutherford formula holds at angles smaller than \(\dfrac{2\alpha}{l_0}\). At larger angles the scattering again has the character of diffraction scattering. Near the angle \(\theta_0\sim \dfrac{2\alpha}{l_0}\) there occurs a sharp decrease of the scattering cross section, in order of magnitude by a factor of \(\alpha\).

Since the cross section of diffraction scattering is on the average inversely proportional to the cube of the scattering angle, and not to the fourth power, as is the case for scattering in a purely Coulomb field, it may be said that, owing to the presence of an absorbing nucleus, scattering through large angles becomes more probable. As a result, the mean scattering angle proves to be considerably larger than in the case of purely Coulomb scattering.

The scattering cross section in the limiting cases of small and large \(\alpha\) is determined by the following formulas:

If \(\alpha \ll 1\), then

\[ \sigma(\theta) \simeq \sigma_R(\theta) \quad \text{for } \theta \ll \frac{\sqrt{2\alpha}}{l_0}, \]

\[ \sigma(\theta) \simeq \sigma_R(\theta) \left| \frac{l_0\theta}{2\alpha} J_1(l_0\theta) \right|^2 \quad \text{for } \frac{\sqrt{2\alpha}}{l_0} \ll \theta \ll 1. \]

If \(\alpha \gg 1\), then

\[ \sigma(\theta) \simeq \sigma_R(\theta) \quad \text{for } \theta \ll \frac{2\alpha}{l_0}, \]

\[ \sigma(\theta) \simeq \sigma_R(\theta) \left| \frac{l_0\theta}{2\alpha} J_1(l_0\theta) \right|^2 \quad \text{for } \frac{2\alpha}{l_0} \ll \theta \ll 1, \]

where

\[ \sigma_R(\theta)=\frac{4\alpha^2\lambda^2}{\theta^4}. \]

The regularities of diffraction scattering of fast neutrons set forth above have been experimentally investigated by a number of authors \(^{20,21}\). In this, a qualitative agreement between theory and experimental data was found.

A fairly clear picture of neutron diffraction scattering was obtained in the work of Amaldi, Bocciarelli, Cacciapuoti, and Trabacchi \(^{21}\). They showed that the total cross section for the removal of neutrons from the beam is twice as large as the cross section for inelastic scattering (and absorption). In the case of lead they studied the angular distribution of elastically scattered neutrons and showed its agreement with the diffraction formula \((2,5')\), with the radius of the lead nucleus \(R\) equal to \(8.7\cdot 10^{-13}\) cm. This value agrees with other data. In this work \(^{21}\) neutrons with an energy of about 14 MeV were used. It is of interest to investigate the diffraction scattering of neutrons and charged particles at considerably higher energies, up to such values at which nuclei begin to become transparent.

CITED LITERATURE

1. R. Serber, Phys. Rev. 72, 1114 (1947).
2. N. Bohr, Uspekhi Fiz. Nauk XVI, 425 (1936).
3. L. Landau, ZhETF 7, 819 (1937).
4. Ya. Frenkel, Sow. Phys. 9, 533 (1936).
5. V. Weisskopf, Phys. Rev. 52, 295 (1937).
6. H. A. Bethe, Nuclear Physics, Part II, GTTI, p. 158 (1948).

1. H. Bethe, Phys. Rev. 57, 1125 (1940).
2. L. Landau and E. Lifshitz, Field Theory, 2nd ed., GTTI, p. 168 (1948).
3. V. I. Smirnov, A Course of Higher Mathematics, vol. 3, p. 670 (1939).
4. M. Born, Optics, ONTI, p. 216 (1937).
5. Whittaker and Watson, A Course of Modern Analysis, vol. II, GTTI (1934).
6. L. Landau and E. Lifshitz, Quantum Mechanics, part I, p. 453 (1948).
7. Mott and Massey, Theory of Atomic Collisions, p. 31 (1936).
8. V. I. Smirnov, A Course of Higher Mathematics, vol. 3, p. 650 (1939).
9. A. Akhiezer and I. Pomeranchuk, ZhETF 16, 396 (1946).
10. L. Landau and E. Lifshitz, Quantum Mechanics, part I, p. 477 (1948).
11. V. I. Smirnov, A Course of Higher Mathematics, vol. 3, p. 477 (1939).
12. L. Landau and E. Lifshitz, Quantum Mechanics, part I, p. 476 (1948).
13. G. N. Watson, Theory of Bessel Functions, 1945.
14. H. Aoki, Proc. Mat. Phys. Jap. 21, 232 (1939); S. Kikuchi, H. Aoki, T. Wakatuki, Proc. Mat. Phys. Jap. 21, 410 (1939); Phys. Rev. 55, 1264 (1939); T. Wakatuki, S. Kikuchi, Proc. Mat. Phys. Jap. 21, 650 (1939).
15. E. Amaldi, D. Bocciarelli, B. N. Cacciapuoti, G. C. Trabacchi, Nuovo Cimento 3, 203 (1946).

NOTE ADDED IN PROOF

In a recently published paper by Fernbach, Serber, and Taylor (Phys. Rev. 75, 1352, 1949), the diffraction scattering of neutrons by semitransparent nuclei is considered. Nuclei are semitransparent in the case when the mean free path of neutrons in them is comparable with the radius of the nucleus itself. This situation occurs for neutrons with energies of the order of 100 MeV or higher. Using a quasiclassical treatment and introducing the absorption and refraction coefficients of neutrons, equal to

\[ k=\frac{3A\sigma}{4\pi R^3}, \qquad n=\sqrt{1-\frac{V}{E}}, \]

where \(\sigma\) is the effective scattering cross section of neutrons by one nuclear particle, \(A\) is the atomic number, and \(V\) is the average potential energy, one can easily find the amplitude of diffraction scattering. If a particle has traversed a path \(s\) in the nucleus, then the amplitude of the \(\psi\)-function has decreased by the factor \(e^{-ks/2}\) and, moreover, has acquired the phase factor \(e^{i(n-1)ks}\). Therefore, immediately behind the nucleus the wave field has the form:

\[ r \gg R \quad \psi=\mathrm{const}=\psi_0, \qquad r \ll R \quad \psi=e^{-\frac{ks}{2}+i(n-1)ks}. \]

Expanding this function in a Fourier series, we find the amplitude of the waves scattered through the given angle \(\theta\):

\[ f(\theta)=k\int_0^R \left[1-e^{-\frac{ks}{2}+i(n-1)s}\right] J_0(k\rho\sin\theta)\,\rho\,d\rho, \]

\[ s=2\sqrt{R^2-\rho^2}. \]

If \(k\to\infty\), \(f(\theta)\) becomes the amplitude corresponding to absolutely black nuclei.

Comparison of the expression obtained for \(f(\theta)\) with experiment may make it possible to find \(V\).