INELASTIC DIFFRACTION PROCESSES AT HIGH ENERGIES

E. L. Feinberg

I. GENERAL CONSIDERATIONS

1. Introduction

In the course of the last two or three years there has appeared a series of more than twenty theoretical works1–16 in which certain distinctive features of collision processes of particles at high energies were discovered. All these works are united by one common trait. It turns out that at high energies, when the wavelengths of the particles participating in collisions become extremely small, nevertheless in a number of cases it is precisely the wave properties of the particles that begin to play a special role.

Thus, for example, we are accustomed to think that the crystalline structure of a body can affect the behavior of an electron incident on the body only when the electron wavelength is comparable with the lattice constant, i.e. if the electron has an energy of the order of \(10^2\)–\(10^4\) eV. It turns out, however, that if we are interested in the bremsstrahlung of an electron on nuclei, then at energies of the order of \(10^8\) eV (for the emission of soft photons—even earlier) and higher, i.e. when the lattice constant exceeds by \(10^4\) and more times the wavelengths of the electron and of the emitted quantum, the structure of the crystal becomes significant2. Moreover, even if the body is not crystalline, then in this case as well, when the energies are sufficiently high, bremsstrahlung, as it turns out, cannot be regarded as occurring on a single atom. Because of the wave properties of light, the influence of the other atoms of the medium proves to be fundamentally important. Thus, for example, independently of the structure of the body, for an electron with energy of the order of \(10^{12}\)–\(10^{13}\) eV (when the wavelength is \(10^8\)–\(10^9\) times smaller than the mean distance between atoms) in a dense medium of the type of lead, bremsstrahlung should fall off sharply3.

Another example concerns the collision of fast deuterons, having an energy of the order of 100 MeV, with nuclei. It has long been known that in this energy region there exists a characteristic stripping reaction

(stripping), which consists in the fact that the deuteron, grazing the edge of the nucleus, leaves in it one of its two nucleons, while the other nucleon flies past. The usual theory \(^{17,18}\) proceeds from the fact that nucleons can be regarded as points or balls, since their wavelength*) is tens of times smaller than the dimensions of nuclei. However, as it turns out \(^{11,12,13}\), diffraction of the deuteron and of nucleons by the nucleus (especially if the nucleus is opaque), which is a consequence of their wave properties, must give effects (deuteron stripping) of exactly the same order as the usually taken into account stripping.

The range of similar phenomena being studied is at present ever expanding, and the methods of treatment are being improved. Experimental possibilities already now permit the empirical study of some of these phenomena. It therefore seems timely to give a brief survey of what has been done (taking the opportunity to note that a number of the questions touched upon here were clarified as a result of joint discussions with I. Ya. Pomeranchuk).

Thus, here we shall be concerned with processes occurring in collisions of particles in the region of high energies, mainly above \(10^8\) eV (in a number of cases the peculiarities of the phenomenon appear only at energies of the order of \(10^{12}\) eV and higher), although one should expect that phenomena of this type may be encountered at any energies. We have in mind inelastic collisions; electromagnetic bremsstrahlung, production of \(\pi\)-meson pairs by \(\gamma\)-quanta, stripping of fast deuterons, etc. The processes considered are united by the fact that they are all connected with the wave properties of particles, often simply with processes of diffraction of particles by an object playing the role of a “third body.” This object may be a “macroscopic” accumulation of other particles (bremsstrahlung of an electron in a medium, in particular in a crystal lattice; electromagnetic radiation in the diffraction of a \(\pi\)-meson by a nucleus, etc.).

2. Region of space essential in the course of the process

First of all it is necessary to explain in what way wave properties, in particular diffraction phenomena, can manifest themselves at high energies.

The physical argumentation below will constantly rely now on wave, now on corpuscular characteristics of the process. It is therefore necessary to examine in more detail the limits of applicability of the corpuscular approach.

*) A nucleon with energy \(E\) MeV has a wavelength (divided by \(2\pi\))

\[ \lambda \approx \frac{1}{\sqrt{E}}\cdot 4.4\cdot 10^{-13}\ \text{cm}. \]

It is obvious that it is by no means always necessary to describe a particle by means of a wave distributed over all space. Some region of space, in a wave treatment, plays the decisive role. Only within the limits of this region is it impermissible to neglect wave properties.

Let us consider, for example, the bremsstrahlung of a fast electron on a nucleus. Such a process is basically classical in nature: in Coulomb scattering by a nucleus the electron acquires a momentum \(q_{\perp}^{(e)}\) perpendicular to its initial direction. If the acceleration is sufficiently large, for example, if \(q_{\perp}^{(e)}\) is sufficiently large, namely if \(q_{\perp}^{(e)} \gg mc\) (\(m\) is the electron mass, \(c\) the speed of light), so that in the coordinate system in which the electron was initially at rest it acquires a velocity of order \(c\), then electromagnetic radiation arises, distributed rather isotropically in the electron system. In the system of the nucleus (i.e. in the laboratory system) it is confined within an angle of order the ratio of \(q_{\perp}^{(e)}\) to the initial momentum of the electron \(\mathbf{p}_0\):

\[ \theta \sim \frac{q_{\perp}^{(e)}}{p_0}. \tag{1} \]

On the other hand, in a wave calculation in the Born approximation the effect appears in the second order of perturbation theory, and the matrix element factorizes into the product of two matrix elements. The first of them describes the emission of a photon by a free electron, the second—the scattering of the electron by the atom. Let us consider the second factor, which is proportional to the integral \(^{19,20}\)

\[ \int e^{\frac{i}{\hbar}\mathbf{q}\mathbf{r}} V(\mathbf{r})\,d\mathbf{r}, \tag{2} \]

where

\[ \mathbf{q}=\mathbf{p}_0-\mathbf{p}-\mathbf{k}=\mathbf{q}^{(e)}-\mathbf{k} \qquad \left(\mathbf{q}^{(e)}=\mathbf{p}_0-\mathbf{p}\right) \tag{3} \]

\(\mathbf{q}\) is the momentum transferred to the nucleus, \(\mathbf{p}\) is the final momentum of the electron, \(\mathbf{k}\) is the momentum of the emitted quantum, and \(V(\mathbf{r})\) is the field of the nucleus and of its screening atomic electrons (in general, the field of a “third body”). The behavior of the integrand determines the dimensions of that region of space (i.e. those values of \(\mathbf{r}\)) which contributes to the integral and plays an essential role in the process. Here, first of all, the factor \(\exp\left(\frac{i}{\hbar}\mathbf{q}\mathbf{r}\right)\) is important. We can decompose it into the factors \(\exp\left(\frac{i}{\hbar}q_{\perp}r_{\perp}\right)\) and \(\exp\left(\frac{i}{\hbar}q_{\parallel}r_{\parallel}\right)\), where \(q_{\perp}\) and \(q_{\parallel}\), as well as \(r_{\perp}\) and \(r_{\parallel}\), are the components of the vectors perpendicular and parallel to the initial momentum \(\mathbf{p}_0\).

The following circumstance is essential for what follows. If we divide the space of the atom into cylinders elongated along the motion of the electron and having transverse dimensions

if \(\Delta r_{\perp\,\mathrm{eff}}\sim \dfrac{\hbar}{q_\perp}\), then the radiations originating from different cylinders will interfere only weakly. Therefore they may be considered independently. One may even take as the initial state of the electron not a plane wave, but a packet with transverse dimensions of order \(\Delta r_{\perp\,\mathrm{eff}}\). Then, summing the intensity of the emitted radiation over all impact parameters, i.e. over all possible distances of such a packet from the nucleus, we obtain the usual result of the Born approximation (see the Appendix). The agreement of the results obtained by the two methods is proof of the independence of the radiations emitted by the various cylinders. Naturally, in this case it is inadmissible to take the impact parameter smaller than the dimensions of the packet. It is known\({}^{19}\) that the most essential values of \(q_\perp\) lie in the interval \(0<q<mc\). Consequently, the smallest possible size of the packet and, correspondingly, the smallest value of the impact parameter will be \(\dfrac{\hbar}{mc}\).

It is evident that in the longitudinal direction as well one may single out a volume with dimensions

\[ \Delta r_{\parallel\,\mathrm{eff}}\sim \frac{\hbar}{q_\parallel}. \tag{4} \]

Radiations emitted by different volumes will not interfere. Conversely, the radiations of different parts of this volume are coherent with one another.

Thus, we can single out a region of space in which, for a given impact parameter, a corpuscular interpretation is impossible and, consequently, localization of the particles is impossible. Throughout this region the interaction of the particles cannot be divided into stages. In it, in this sense, the interaction occurs “simultaneously.” This region has dimensions of order (4) in the longitudinal direction and of order

\[ \Delta r_{\perp\,\mathrm{eff}}\sim \frac{\hbar}{q_\perp} \tag{5} \]

in the transverse direction. It may be called the essentially wave region, or simply the essential, effective region for the course of the given process.

We now pass to the central point.

Whereas the transverse dimensions of the effective region are determined by the requirement that the state of the particle be sufficiently strongly disturbed (for example, that its transverse velocity become close to \(c\), and, consequently, that \(q_{\perp}^{(e)}\) become of order \(mc\)), the longitudinal dimensions are specified practically only by the law of conservation of energy. Thus, for example, if \(\mathbf{p}_0\), \(\mathbf{p}\), and \(\mathbf{k}\) are directed identically

(zero emission angle), then at relativistic energies^19

\[ q_{\parallel}=p_0-p-k= \]

\[ =\frac{1}{c}\sqrt{E_0^2-m^2c^4} -\frac{1}{c}\sqrt{(E_0-ck)^2-m^2c^4}-k \simeq \frac{m^2c^4}{2E_0(E_0-ck)}\,k. \tag{6} \]

It is easy to see that if the scattering angle is not zero but is of the order of the mean angle, \(\theta\sim \dfrac{mc}{p_0}\ll 1\), then in this case as well \(q_{\parallel}\) has the order indicated by relation (6). But this quantity can be very small if \(E_0\) and \(E_0-ck\) are sufficiently large. Therefore \(\Delta r_{\parallel\,\mathrm{eff}}\) may exceed not only \(\dfrac{\hbar}{mc}\), but also the wavelengths of all particles participating in the process and the dimensions of the atom, and may even, for \(E_0\sim E_0-ck\sim ck\sim 10^{11}mc^2\), acquire macroscopic dimensions \(\left(\dfrac{\hbar}{q_{\parallel}}\sim 1\ \mathrm{cm}\right)\). Thus, at relativistic energies the effective region is determined by the difference of the momenta of the participating particles. It is extremely elongated along the direction of motion. In particular, if \(q_{\parallel}\) becomes so small that at the distance \(\dfrac{\hbar}{q_{\parallel}}\) (along the axis of motion) not one atom but several fit, then the “third-body” field \(V(\mathbf r)\) in (2) must be understood as the sum of the fields of all these atoms \(\sum_i V(\mathbf r-\mathbf r_i)\), where \(\mathbf r_i\) characterizes the position of the \(i\)-th atom. All of them will act coherently, and the cross section will be proportional to the square of their number.

It is known that diffraction of X-rays can be obtained on an optical diffraction grating, at sufficiently small scattering angles, when the difference of the projections of the wave vectors onto the plane of the grating is very small (much less than the reciprocal of the grating constant). Similarly here, an electron that has emitted a photon and therefore has momentum \(\mathbf p_0-\mathbf k\), when scattered by a nucleus (atom) into a state with momentum \(\mathbf p\), can give a small projection of their difference.

Let us note that for the very same reason all the protons of one nucleus are, in the usual treatment, regarded as acting coherently, so that bremsstrahlung on a single nucleus is proportional to \(Z^2\). In fact, this is a consequence of the fact that all the protons of the nucleus are located at distances that are small both in comparison with \(\dfrac{\hbar}{q_{\parallel}}\) and in comparison with \(\dfrac{\hbar}{q_{\perp}}\) (although the wavelengths of the particles participating in the process may be smaller than the nuclear dimensions; for example, in the case of Pb this already occurs at an energy of the order of \(20\ \mathrm{MeV}\)).

Since the Coulomb forces deflecting the electron extend over large distances, up to the boundary of the atom, radiation can arise for wave packets for which the effective cylindri-

regions passing at the most varied distances from the nucleus, up to a distance equal to the radius of the atom. It is only necessary that they should (at least partially) include the region of action of the Coulomb field. In what follows (see §§ 4, 5, etc.) we shall consider analogous radiation processes in which the deflecting force field is short-range, nuclear. In that case it is necessary that the effective region pass through the nucleus or, at least, include its edge.

We have analyzed in detail the dimensions of the effective region using bremsstrahlung as an example. However, the relations found have a very general significance. Indeed, we have been speaking of a process in which the momentum given to the system (in our example, electron + photon) is equal to \(\mathbf q\), so that the momentum of this entire system is specified, at least, with the same accuracy. According to the uncertainty relation, we conclude that the process takes place in a region that cannot be localized more accurately than \(\hbar/q_\perp\) and \(\hbar/q_\parallel\) in the corresponding directions. Consequently, in the general case as well, the “essential region” in which the particles (electron, etc.) “simultaneously” experience the action of a third body is determined by relations (4) and (5).

From these general considerations it is clear that the effective region could also be determined from consideration of the factor in the matrix element that gives the radiation of a photon by a free electron, and not of the factor describing scattering by the field of the atom (2).

There is one more point requiring clarification. It was said above that radiation by an electron in scattering by an atom will occur effectively if \(q_{\perp}^{(e)} \sim mc\), in particular if the velocity acquired by it (in the system in which it was initially at rest) is of order \(c\). However, radiation is determined by acceleration, not by velocity. Therefore it is necessary that the momentum \(q_\perp\) be acquired nonadiabatically in a sufficiently short time \(\tau_0\). In the present case this time is the interaction time, i.e., the time of flight through the effective region, which has length \(\hbar/q_\parallel\). According to (4) and (6), in the rest frame of the atom it is equal to

\[ \tau \sim \frac{\hbar}{q_\parallel c} \sim \frac{2E_0(E_0-ck)\hbar}{m^2c^4\cdot ck}. \tag{7} \]

In the system in which the electron was initially at rest, it is smaller by a factor \(E_0/mc^2\). Therefore, in the principal region of photon energies,

\[ E_0 \sim E_0 - ck \sim ck, \]

we obtain

\[ \tau_0 \sim \frac{\hbar}{mc^2}. \tag{8} \]

Thus the time required to acquire a relativistic velocity also has a typical relativistic magnitude; the process is indeed nonadiabatic. Therefore radiation occurs.

3. Diffraction by a Nucleus

Using bremsstrahlung as an example, we have seen that, as the energy of the participating particles increases, the region of space essential for the process grows without bound. It turns out (see below) that, quite generally, in an elementary act, because the angles of emission of all particles decrease as the energy increases, the longitudinal component of the momentum \(\mathbf q\) transferred to the body with which the collision occurs (lattice, nucleus) becomes small and continues to decrease with increasing energy. Ultimately, at sufficiently high energy, \(q_{\parallel}/\hbar\) becomes very small in comparison with the inverse characteristic dimensions of the “third body”—with the inverse interatomic distance \(a\), or with the inverse nuclear radius \(R\). In other words, at sufficiently high energy the conditions begin to be fulfilled

\[ q_{\parallel} a \ll \hbar \tag{9} \]

or

\[ q_{\parallel} R \ll \hbar . \tag{10} \]

This means that the region of space effective for the entire elementary act becomes very large in comparison with \(a\) or \(R\), however small the wavelength may be.

This very important circumstance affects the course of various phenomena in different ways and is used in somewhat different ways in constructing their theory.

On the one hand, if the effective region of space in which the process takes place is so large, then, in addition to the normally considered “third body,” other particles of the medium also enter this region; the process in the medium will not proceed as it does at an isolated center. This proves especially significant for purely electromagnetic processes: bremsstrahlung of an electron in the Coulomb field of a nucleus, pair production, and so forth (see also below IV, 13). The role of this effect increases as the energy grows, when more and more particles of the medium are included in the essential region.

On the other hand, sometimes, as the energy increases, the number of particles of the medium falling into the essential region no longer increases; instead, all those already included are regarded collectively as a single “third body.” For example, the nucleus as a whole is regarded as one particle. In this case it is important that its scattering action on the other particles participating in the process must be known over a large effective region, i.e. mainly at large distances from it. The process is “external” with respect to the nucleus. Here the scattering action can be described phenomenologically, for example, by the formulas of the theory of diffraction of waves by a sphere, or by known data on the behavior of scattered particles far from the nucleus. This circumstance proved very fruitful in the study of

nuclear processes (emission of photons in the interaction of \(\pi\)-mesons with nuclei, etc.).

Let us consider diffraction by a nucleus in more detail.

It is well known that particles absorbed by nuclei consequently undergo diffraction scattering by them. If the wavelength of the particles is small in comparison with the nuclear radius, then the scattering may be treated by the classical theory of wave diffraction in the Kirchhoff approximation. The simplest method can be illustrated by the example of scattering by an absolutely absorbing nucleus. For nucleons the nucleus is black at energies of the order of \(30\)—\(100\) MeV or above several GeV\(^{22}\), and for \(\pi\)-mesons at energies above \(1\)—\(2\) GeV\(^{23}\).

Let the radius of the nucleus be \(R\). Draw through the nucleus a plane perpendicular to the momentum \(\mathbf p_0\) of the incident particle, and take the direction of the momentum as the \(z\)-axis. We may assume that in this plane, outside the nucleus, the wave function of the particle is unperturbed (the amplitude is equal to unity), while over the entire shadow surface of the nucleus and inside the nucleus it is equal to zero. Expanding in a Fourier integral with respect to \(x\) and \(y\) the function

\[ \psi(x,y,0)= \left\{ \begin{array}{ll} 0, & \sqrt{x^2+y^2}<R\\ 1, & \sqrt{x^2+y^2}>R \end{array} \right\} \equiv 1- \left\{ \begin{array}{ll} 1, & \sqrt{x^2+y^2}<R\\ 0, & \sqrt{x^2+y^2}>R \end{array} \right\} = \]

\[ =1-\frac{R}{2\pi\hbar}\iint \frac{dq_xdq_y}{q_\perp}\, J_1\!\left(\frac{q_\perp R}{\hbar}\right) e^{\frac{i}{\hbar}(q_xx+q_yy)} \tag{11} \]

(where \(q_\perp=\sqrt{q_x^2+q_y^2}\) is denoted here, since, as will be clear from what follows, \(q_\perp\) has the same meaning as the quantity denoted in the same way earlier), we obtain, for \(z\ne0\), the scattered wave from the second term if under the integral we add
\(\exp\left\{\dfrac{i}{\hbar}\sqrt{p_0^2-q_x^2-q_y^2}\,z\right\}\):

\[ \psi_{\mathrm{sc}}= \]

\[ =-\frac{R}{2\pi\hbar}\iint \frac{dq_xdq_y}{q_\perp}\, J_1\!\left(\frac{q_\perp R}{\hbar}\right) e^{\frac{i}{\hbar}\left(q_xx+q_yy+\sqrt{p_0^2-q_x^2-q_y^2}\,z\right)} . \tag{11a} \]

From this one can find the flux in the direction making an angle \(\vartheta\) with \(\mathbf p_0\) (\(q_\perp=p_0\sin\vartheta\sim p_0\vartheta\)), and then find the differential cross section for diffraction scattering. With this method it is especially clearly seen that diffraction scattering is the consequence of any processes which remove the particle from the initial beam, cutting a cylindrical “hole” in the primary flux, with subsequent motion of the flux “contracting” as a result of diffraction. In particular, “absorption” must also include inelastic scattering, changing

INELASTIC DIFFRACTION PROCESSES

also the energy of the particle. The nuclear radius \(R\) reflects the total cross section of the nucleus with respect to all such processes.

Carrying out the calculations, it is easy to find that the fraction of particles scattered through an angle \(\vartheta\) is equal to\({}^{21}\)

\[ \frac{d\sigma(\vartheta)}{\pi R^2} = \frac{2}{\sin\vartheta} J_1^2\!\left(\frac{p_0 R}{\hbar}\sin\vartheta\right)d\vartheta \tag{12} \]

(\(J_1\) is the Bessel function of the first order).

The principal diffraction maxima lie at \(R p_0 \sin\vartheta \sim \hbar\), i.e. at \(q_\perp \sim p_0\vartheta \sim \dfrac{\hbar}{R}\). For large nuclei these \(q_\perp\) are substantially smaller than \(\mu c\), where \(\mu\) is the mass of the \(\pi\)-meson \(\left(\dfrac{\hbar}{R}=A^{-1/3}\mu c\right)\). If we are interested in large \(q_\perp\), then \(J_1^2\!\left(\dfrac{p_0 R}{\hbar}\vartheta\right)\) may be replaced by its asymptotic value

\[ \frac{2\hbar}{\pi p_0 R\vartheta} \sin^2\!\left(\frac{1}{\hbar}p_0 R\vartheta-\frac{\pi}{4}\right) \]

and, for estimates, the square of the sine of the rapidly oscillating argument may be replaced by its mean value \(\dfrac{1}{2}\). In this case the fraction of particles scattered with transverse momentum \(q_\perp\) is equal to

\[ \frac{d\sigma(q_\perp)}{\pi R^2} = \frac{2\hbar}{\pi R}\, \frac{dq_\perp}{q_\perp^2}. \tag{12a} \]

When, upon scattering, a particle acquires a transverse momentum \(q_\perp=p_0\vartheta\), the nucleus receives the same momentum in the opposite direction in the given act of scattering. It is important that it receives it as a whole, and if the mass of the nucleus is sufficiently large, the recoil energy will be small, and no processes of excitation or splitting can occur.

Integrating over \(\vartheta\) exceeding some specified angle, it is easy to obtain the probability that particles acquire a sufficiently large momentum. It has already been said above that large \(q_\perp\) correspond to small sizes of the essential region. If these become smaller than the mutual distance of the particles in the nucleus, then, perhaps, it will no longer be possible to regard the interaction as occurring with many nuclear particles at once. In general, in the presence of an absolutely black sphere, the field of a scalar wave, initially characterized by momentum \(\mathbf p_0\) and normalized to the volume \(\Omega\), may be written in the Kirchhoff approximation as (cf. \({}^{24}\))

\[ \psi_{\mathbf p_0} = \frac{1}{\Omega^{3/2}} \left( e^{\frac{i}{\hbar}\mathbf p_0\mathbf r} + \frac{i p_0}{2\pi\hbar} \int_{\substack{r'>R\\ \mathbf r'\perp \mathbf p_0}} \frac{d\mathbf r'}{|\mathbf r-\mathbf r'|} e^{\frac{i}{\hbar}p_0|\mathbf r-\mathbf r'|} \right), \tag{13} \]

where the integration is carried out over a large cross-section of the sphere (the two-dimensional vector \(\mathbf r'\)), and the cross-section should be chosen perpendicular to the vector \(\mathbf p_0\). For \(\rho_0 R \gg \hbar\), the second term in the brackets (the integral) makes a substantial contribution only for those points \(\mathbf r\) which lie in the “geometrical shadow” of the sphere and near it. Outside the shadow (and outside the diffraction transition region from the shadow to the illuminated region) the field remains practically unperturbed.

The function \(\psi_{\mathbf p_0}\) may be regarded as an exact (in the sense that it is obtained not by perturbation theory) solution of the Schrödinger equation in the presence of an absorbing sphere, satisfying the boundary condition—the presence of an incoming wave.

In a number of cases it is necessary to know the solution of the equation in the presence of an absorbing sphere when the boundary condition requires that at infinity there be an outgoing wave. Such a solution describes a particle produced in the nucleus or near it. It can be obtained, for example, from the following considerations.

The function (13) gives a solution of the Schrödinger equation in the presence of absorption in the sphere, which can be represented by a complex interaction potential \(U(r)\) ([21], § 18):

\[ \left(-\frac{\hbar^2}{2M}\nabla^2+U(r)\right)\psi_{\mathbf p_0} =E\psi_{\mathbf p_0}, \tag{14} \]

where the index \(\mathbf p_0\) means that at infinity the momentum of the particle is equal to \(\mathbf p_0\). In the case where the nucleus is the source, the potential \(U\) must be replaced by the complex conjugate \(U^*\). Consequently, the solution “with an outgoing wave” \(\psi^{(-)}\) of the Schrödinger equation may be the function \(\psi^*_{\mathbf p_0}\) complex-conjugate to \(\psi_{\mathbf p_0}\). However, in such a function the term corresponding to the first term in (13) will have the character \(e^{-\frac{i}{\hbar}\mathbf p_0\mathbf r}\) and therefore at infinity will describe a wave going out with momentum \(-\mathbf p_0\). Therefore, as \(\psi^{(-)}\) it is better to take the complex conjugate of another particular solution of equation (14), namely the solution giving diffraction scattering of a wave incident with initial momentum \(-\mathbf p_0\). Then \(\psi^{(-)}_{\mathbf p_0}=\psi^*_{-\mathbf p_0}\) will not only be a solution of the Schrödinger equation in the presence of an absorbing body, but at infinity will behave as a plane wave with momentum \(\mathbf p_0\). Thus, a particle of momentum \(\mathbf p_0\) produced in the nucleus and undergoing scattering on this same nucleus, if it is regarded as absolutely absorbing, is described, within the range of applicability of the Kirchhoff approximation, by the function

\[ \psi^{(-)}_{\mathbf p_0} = \frac{1}{\Omega^{3/2}} \left( e^{\frac{i}{\hbar}\mathbf p_0\mathbf r} - \frac{i p_0}{2\pi\hbar} \int_{\substack{\mathbf r'\perp \mathbf p_0\\ r'<R}} \frac{d\mathbf r'}{|\mathbf r-\mathbf r'|} e^{-\frac{i}{\hbar}p_0|\mathbf r-\mathbf r'|} \right). \tag{15} \]

As is known, in general, in order to obtain the function of the emerging particle, one must take the function of the particle incident from outside and scattered in the given field, replace in it the initial momentum by the opposite one, and pass to the complex conjugate\(^{25}\).

Analogous formulas for spinor waves (the Dirac equation) were obtained in \(^{5}\). The consideration is easily generalized to the case of a partially transparent sphere.

The cross sections of many of the processes considered below can be estimated simply by selecting from the total diffraction cross section the cases with sufficiently large \(q_\perp\).

In the first works on the circle of questions considered by us, when it was necessary to take into account the influence of the nucleus, this influence was treated precisely by the theory of diffraction on a sphere. But recently one more essential step was taken\(^{14}\). The point is that in reality we always need to know only the scattering action of the nucleus on the particles participating in the process at large distances from the nucleus (only for this reason is the Kirchhoff approximation suitable). Often this action is known from independent experiments on scattering by the given center or from some calculations. All cross sections in such cases are expressed not through the radius of an absolutely absorbing sphere or its absorption and refraction indices, but through the amplitudes of scattering of the particles by the given center. However, the formulas of diffraction on a sphere are convenient in that with their help one can obtain simple estimates of the cross sections of processes. Thus, for example, if the given process can arise only under the condition that the recoil is sufficiently large, i.e. \(q_\perp\) exceeds some \(q_{\perp\min}\), then according to (12a) the cross section for such an effect is equal to

\[ \sigma=\int_{q_{\perp\min}}^{\infty} d\sigma(q_\perp)=2R\,\frac{\hbar}{q_{\perp\min}} . \tag{16} \]

Here there arises the characteristic proportionality to the first power of \(R\). If, as is the case for many processes, \(q_{\perp\min}\sim \mu c\), then \(\sigma\sim R\dfrac{\hbar}{\mu c}\).

It is appropriate to examine one more question touched upon above: how large \(q_\perp\) (for \(q_{\parallel}R\ll\hbar\)) may be allowed when using the formulas of diffraction on a sphere. If \(q_\perp>\mu c\), then the essential region has a width small in comparison with the distances between nucleons in the nucleus, \(r_0=\dfrac{\hbar}{\mu c}\). In particular, if this region crosses the blurred edge of the nucleus, then diffraction scattering may not occur. The point is that here the properties of the medium (in the direction perpendicular to \(p_0\)) change little over the extent \(q_\perp\). It is known, for example, that light does not undergo reflection from a blurred boundary of a body if

the refractive index changes little over the normal projection of a wavelength (in the approximation of geometrical optics there is no reflection at all\(^{26}\)). Likewise here, where the role of the wavelength is played by \(\dfrac{\hbar}{q_\perp}\), when

\[ \dfrac{\hbar}{q_\perp} \ll \dfrac{\hbar}{\mu c}=r_0 \]

(where \(r_0\) is at the same time the width of the transition region of the nucleus), the cross section of the effects must fall off sharply (in wave-mechanical calculations this is manifested in the fact that the rapidly oscillating factor \(\exp\left(\dfrac{i}{\hbar}q_\perp r_\perp\right)\) sharply reduces the value of the matrix elements).

Conversely, if \(q_\perp \ll \dfrac{\hbar}{r_0}\), then the edge may be regarded as sharp. Here the situation is the same, for example, as in the diffraction of visible light by the edge of the Moon’s disk. Observing on the Earth light diffracted through a sufficiently small angle \(\vartheta\), we may neglect the irregularities of the lunar surface and regard the Moon as a disk with a sharp edge, if the momentum imparted to the Moon in the perpendicular direction

\[ q_\perp = |\mathbf{k}-\mathbf{k}_0| = 2k_0 \sin \dfrac{\vartheta}{2} \simeq k_0 \vartheta \]

\[ \left(\dfrac{k_0}{\hbar}=\dfrac{2\pi}{\lambda}\text{ is the wave number of the light wave}\right) \]

is small in comparison with the irregularities of the Moon’s surface.

Fig. 1.

Let us note that blurring of the edge in the longitudinal direction plays no role if \(q_{\parallel}R \ll \hbar\): in this respect the edge of the nucleus is always sharp.

In spite of these considerations, for heavy nuclei the situation is somewhat more favorable: the permissible \(q_\perp\) may appreciably exceed \(\mu c\). This is connected with the fact that, for a point particle passing through the nucleus at a distance \(r_\perp\) from the center of the nucleus, in the region \(r_\perp \sim R\) the total effective absorption decreases with increasing \(r_\perp\) more sharply than the density of nuclear particles decreases with increasing radial distance \(r\) (Fig. 1). In fact, the total absorption, for example, in motion along the line \(AB\), may be large, although this line passes through a region of low density of nuclear matter. More definite estimates are difficult, since \(\pi\)-mesons and nucleons can hardly be considered pointlike, although, apparently, the radius of a nucleon is nevertheless substantially smaller than \(\dfrac{\hbar}{\mu c}\). We shall assume that in the case of heavy nuclei one may use the theory of diffraction by a sphere for \(q_\perp\) several times exceeding \(\mu c\).

Let us now explain how one can get rid\(^{14,15}\) of representing the nuclear field in the form of an absolutely black sphere. Functions of the form (13) and (15), when computing matrix elements, need be known mainly for \(|\mathbf r-\mathbf r'|\gg R\), and moreover for points \(\mathbf r\) lying in a direction making a small angle \(\vartheta\) with the vector \(\mathbf p_0\) (because of the condition \(p_0R\gg\hbar\), the shadow cone is very narrow). Therefore one may put in the exponent under the integral

\[ p_0|\mathbf r-\mathbf r'|\simeq p_0r-p_0\frac{\mathbf r\mathbf r'}{r} = p_0r-\mathbf p'_0\mathbf r', \]

where \(\mathbf p'_0\) is the vector directed from the nucleus to the observation point \(\mathbf r\), \(p'_0=p_0\). Neglecting \(\mathbf r'\) in comparison with \(\mathbf r\) in the denominator of the integrand, we have, instead of (13),

\[ \psi_{\mathbf p_0}\simeq \frac{1}{\Omega^{3/2}} \left( e^{\frac{i}{\hbar}\mathbf p_0\mathbf r} + \frac{e^{\frac{i}{\hbar}p_0r}}{r} \frac{i}{2\pi} \iint e^{-\frac{i}{\hbar}\mathbf p'_0\mathbf r'} \,d\!\left(\frac{\mathbf p_0}{\hbar}\mathbf r'\right) \right) = \]

\[ = \frac{1}{\Omega^{3/2}} \left( e^{\frac{i}{\hbar}\mathbf p_0\mathbf r} + \frac{e^{\frac{i}{\hbar}p_0r}}{r}A(\vartheta) \right), \]

\[ A(\vartheta)= \frac{i}{2\pi} \iint e^{-\frac{i}{\hbar}\mathbf p'_0\mathbf r'} \,d\!\left(\frac{\mathbf p_0}{\hbar}\mathbf r'\right) \tag{13a} \]

(obviously, in fact, \(A(\vartheta)\) depends only on the angle between \(\mathbf p_0\) and \(\mathbf p'_0\)). This expression is a special case of the general asymptotic formula for the scattering of a particle of momentum \(\mathbf p_0\) by a force center. Consequently, performing all calculations with the aid of the more general formula of the form (13a), we can then substitute for \(A(\vartheta)\) the scattering amplitude for one or another field, if it has been found by some independent method.

Taking account of the nonsphericity of the nucleus somewhat changes the angular distribution of the scattered particles\(^{32}\). However, for the estimates that interest us, this change is apparently insignificant.

4. General characterization of the effects studied

Before presenting the quantitative results in detail, let us enumerate the separate effects studied and find the criterion for their occurrence.

Consider, for example, processes of a purely electromagnetic character. We have already found (see (6)) that for the bremsstrahlung of an electron having energy \(E_0\), mass \(m\), and momentum \(p_0\), when a quantum of energy \(ck\) is emitted, in the relativistic region,

\[ q_{\parallel}\simeq \frac{m^3c^4}{2E_0(E_0-ck)}\,k. \tag{17} \]

Here attention should be paid, at least, to the following effects.

1) If \(E_0\) becomes so large (or \(k\) so small) that condition (9) is satisfied, then the bremsstrahlung of a particle moving along a crystallographic axis of the crystal will occur under the combined (coherent) action of all \(N_{\mathrm{eff}}\) nuclei located on the segment \(\dfrac{\hbar}{q_{\parallel}}\):

\[ N_{\mathrm{eff}} \sim \frac{\hbar}{q_{\parallel} a}. \tag{17a} \]

This means that the effective charge of the scattering center will be not \(Ze\), but \(N_{\mathrm{eff}}Ze\); the cross section of the effect will be proportional to \(N_{\mathrm{eff}}^{2}\), and a maximum of the bremsstrahlung will appear, having an interference origin\(^2\). It turns out that this effect should be very substantial for \(E_0 > 50\) MeV.

It should be noted that in the emission of quanta of extremely high energy this effect does not occur. Namely, if \(E_0 - ck\) is small, then \(q_{\parallel}\) is large and may become greater than \(\dfrac{\hbar}{a}\). Then the collision again occurs with a separate atom. Similar considerations, with the corresponding modifications, are also valid for the other effects listed below.

2) In motion both in an amorphous and in a crystalline body, if \(E_0\) is sufficiently large or \(k\) sufficiently small, the electron may undergo, over the path \(\dfrac{\hbar}{q_{\parallel}}\), such strong multiple Coulomb scattering that it will go outside the angle \(\dfrac{mc^2}{E_0}\) and the radiation process will be disturbed. Therefore, as the energy increases, bremsstrahlung in a medium, starting from some electron energy, must weaken\(^3\). It turns out that in dense matter a sharp drop of the radiation should occur at \(E_0 \sim 10^{11}—10^{12}\) eV.

3) Both of these effects should, under practically the same conditions, also occur for the production of electron pairs by a quantum, because the interaction of the components of the pair with the “third body” has the same character\({}^{2,3}\).

4) For soft bremsstrahlung quanta, the difference of the refractive index of the medium \(\sqrt{\varepsilon}\) from unity should manifest itself: over such a large path \(\Delta r_{\parallel \mathrm{eff}}\), the additional phase shift of the photon

\[ \left(\sqrt{\varepsilon}-1\right)\frac{k \Delta r_{\parallel \mathrm{eff}}}{\hbar} \tag{18} \]

may become sufficient for the phase relations over the path \(\dfrac{\hbar}{q_{\parallel}}\) to become completely upset and for the radiation process to be disturbed\(^6\). As a result, the bremsstrahlung spectrum of very fast

electrons, as the investigation shows, also changes its character. In dense media this occurs, roughly speaking, for \(ck \lesssim 10^{-4}E_0\).

Let us consider, on the other hand, nuclear interactions, where, as was indicated, the circumstance that the process is external with respect to the nucleus is especially important.

5) In the interaction of a nuclear-active particle with a nucleus as a whole, this particle may emit a photon if it has an electric charge, or a \(\pi\)-meson (in particular, if this particle is a nucleon)\(^1\).

Thus, if a charged \(\pi\)-meson with energy \(E_0\) and mass \(\mu\), incident on a nucleus, emits a photon of energy \(ck\), then at zero angle of emission (neglecting the recoil of the nucleus)

\[ q_{\parallel}=\frac{1}{c}\sqrt{E_0^2-\mu^2c^4} -\frac{1}{c}\sqrt{(E_0-ck)^2-\mu^2c^4}-k \simeq \]

\[ \simeq \frac{1}{2}\frac{\mu^2c^4}{E_0(E_0-ck)}\,k. \tag{19} \]

When \(E_0\) becomes sufficiently large, so that condition (10) is satisfied, the essential region of space will become much larger than the dimensions of the nucleus. The nucleus will act as a whole, and the generation of quanta will occur, mainly, far outside the nucleus.

This process can also be understood in another way. The possibility of absorption of a \(\pi\)-meson in the nucleus leads to the fact that, along with absorption, diffraction of the \(\pi\)-meson by the nucleus must also occur. In this diffraction the meson receives a recoil, and for such large scattering angles \(\vartheta\), for which the momentum \(q_{\perp}(\simeq p_0\vartheta)\) transferred to the meson is sufficiently large, emission of photons will occur.

In the region of high energies this will be diffraction by a black sphere, and when the wavelength of the \(\pi\)-meson is sufficiently small, one can use the Kirchhoff approximation of diffraction theory. The total radiation has an energy of the order of \(\dfrac{e^2}{\hbar c}E_0\) and a spectrum in the main frequency region of the type \(\dfrac{dk}{k}\).

6) In a similar manner one may phenomenologically consider the process of formation of \(\pi\)-meson pairs by a photon\(^8,{}^9\). Its cross section, as it turns out, does not depend on the energy of the quantum and is of the order

\[ \frac{1}{12\pi}\frac{e^2}{\hbar c} \]

of the geometrical cross section of the nucleus. At the same time, along with simple pair production, when the nucleus as a whole experiences only a weak recoil, a very peculiar process is possible: one of the mesons formed, or both of them, may be absorbed in the same nucleus, which as a result will produce a star. Thus arises a new mechanism of formation of photostars, in particular, those accompanied by the emission of one fast \(\pi\)-meson\(^ {14,\,15}\).

7) A charged \(\pi\)-meson or another electrically charged and at the same time nuclear-interacting particle, diffracting on a nucleus, can acquire a sufficient momentum \(q_\perp\) to cause, by its electric field, the formation of an electron–positron pair\({}^{16}\).

8) If a nucleon of high energy \(E_0\) (mass \(M\)) interacts with a nucleus, it can emit a \(\pi\)-meson (energy \(E_\pi\)). At small angles of emission

\[ q_\parallel = \frac{1}{c}\sqrt{E_0^2-M^2c^4} - \frac{1}{c}\sqrt{(E_0-E_\pi)^2-M^2c^4} - \frac{1}{c}\sqrt{E_\pi^2-\mu^2c^4} \simeq \frac{M^2c^3E_\pi}{2E_0(E_0-E_\pi)} + \frac{\mu^2c^3}{2E_\pi}. \tag{20} \]

If condition (10) is satisfied, then the possibility arises of external generation of \(\pi\)-mesons\({}^{4}\). In this case it is necessary phenomenologically to take into account both the scattering of the \(\pi\)-mesons and the scattering of the nucleon. This process can occur only for \(E_\parallel \gg A^{1/3}Mc^2\), where \(A\) is the number of nucleons in the nucleus.

9) When fast nuclei, for example deuterons, fall on a nucleus, their diffraction occurs. Consequently, they acquire transverse momentum, and therefore external (diffraction) breakup of the deuteron can occur quite independently of the action of the electric field of the nucleus\({}^{11,12,13}\). Thus, for a nonrelativistic deuteron with kinetic energy \(E_0\) (binding energy \(\varepsilon_D\)), taking the energies of the emitted neutron and proton to be close (which is confirmed by the subsequent calculations), we obtain, again neglecting the recoil of the nucleus,

\[ q_\parallel \simeq \sqrt{4ME_0} - 2\sqrt{\,2M\,\frac{E_0-\varepsilon_D}{2}\,} \simeq \frac{\varepsilon_D}{\sqrt{ME_0}}\,M. \tag{21} \]

Condition (10) can be satisfied already at an energy exceeding several MeV. In the case of a completely opaque nucleus, the cross sections for this process and the energy and angular distributions turn out, in general, to be the same as for the stripping reaction.

In all processes (5)—(9), in which the nucleus participates as a whole, a distinctive feature is that the entire nucleus at once receives the momentum transferred to it. Taking

\[ q=\sqrt{q_\perp^2+q_\parallel^2}\simeq q_\perp \]

not very large, namely \(q_\perp \ll \mu c\) (see above, § 3), we find that the energy received by a nucleus of atomic weight \(A\),

\[ \frac{q^2}{2MA} \ll \mu c^2\,\frac{\mu}{2MA} \sim \frac{12\,\text{MeV}}{A} \tag{22} \]

is very small, and the nucleus, as a rule, experiences only a small recoil, but is not destroyed.

10) Yet another application of the same principle may be found in considering head-on collisions of fast nucleons with nuclei, in which the nucleus is destroyed and new particles are produced. Often in this case, in the energy region \(10^9—10^{12}\) eV, attempts are made to construct a cascade process inside the nucleus or, at any rate, to consider successive collisions with individual nuclear nucleons. It turns out, however, that such a treatment of successive collisions is not always valid. Sometimes one must assume that the incident particle interacts at once with all the nucleons of the nucleus lying in its path, i.e. with the whole “tunnel”\({}^{10}\).

II. ELECTROMAGNETIC PROCESSES IN NUCLEAR COLLISIONS

5. Emission of photons in diffraction and capture of mesons

Let us first of all consider the electromagnetic processes accompanying the collision of fast charged \(\pi\)-mesons with nuclei. The momentum of the \(\pi\)-meson is assumed to be so large that the wavelength of the meson is much smaller than the nuclear radius \(R\).

The nuclear interaction at present does not lend itself to theoretical calculation. However, the process of electromagnetic radiation accompanying the nuclear interaction can be studied in detail thanks to the features discussed above.

The general character of the process is determined by the fact that, in scattering or in absorption in a nucleus, the \(\pi\)-meson must radiate the electromagnetic field carried along with it and because, at least in the region of soft quanta, the photon spectrum has the character \(\frac{dk}{k}\), while the probability of radiation differs from the cross section of the basic process by a factor \(\frac{e^2}{\hbar c}=\frac{1}{137}\) (and by factors depending logarithmically on the energy).

Here two interactions play a role: the interaction of the \(\pi\)-meson with the electromagnetic field and its nuclear interaction with the nucleus (we shall not take into account the comparatively weak electromagnetic interaction with the nucleus). The possibility of a complete and sufficiently rigorous calculation is connected with the fact that the first can be treated by perturbation theory, whereas the second (and this constitutes the essence of the method) can be treated exactly, not by perturbation theory, since the wave function of the \(\pi\)-meson, taking into account the influence of the nucleus, is known in the essential region (far from the nucleus). Namely, if one assumes that the nucleus is an absolutely black sphere with respect to the \(\pi\)-meson (which, as follows from experiment, is already valid for \(E_0 > 1 \div 2 \cdot 10^9\) eV\({}^{23}\)), then the pseudoscalar wave function of the \(\pi\)-meson (13) is the superposition of the initial plane wave and the wave diffracted by the sphere.

Here there are plane waves with momentum different from \(\mathbf{p}_0\), which expresses the acceleration acquired by the \(\pi\)-meson in diffraction scattering by the nucleus. This acceleration will be sufficient for the \(\pi\)-meson to emit electromagnetic radiation if, in the system where the \(\pi\)-meson was initially at rest, it acquires a sufficiently large momentum in a sufficiently short time. Just as in the analogous case for the electron (7), (8), one can verify that it is necessary that only the single condition \(q_\perp > \mu c\) be fulfilled. Selecting, in the Fourier expansion \(\psi_{\mathbf{p}_0}\), scattering events with \(q_\perp > \mu c\), one can estimate the cross section of the process. According to (16) one obtains \(\sigma \sim R \dfrac{\hbar}{\mu c}\). In such an estimate, inadmissibly large values of \(q_\perp\), which are taken into account here, cannot affect the order of magnitude of the cross section.

A more complete theory can be obtained either by applying perturbation theory to the transition, under the action of radiation, from state (13) to state (15) (a plane wave plus a converging one; integration over the cross section perpendicular to \(\mathbf{p}'\)), or¹ by writing the equation for \(\psi\) in the field of a quantized electromagnetic wave with vector potential \(\mathbf{A}\):

\[ \left\{\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\nabla^2+\left(\frac{\mu c}{\hbar}\right)^2\right\}\psi = \frac{2e}{ic}\,\mathbf{A}\nabla\psi, \tag{23} \]

replacing \(\psi\) by

\[ \psi=\psi_{\mathbf{p}}e^{-\frac{i}{\hbar}E_p t} + \sum_{\mathbf{p}'} e^{-\frac{i}{\hbar}E_{\mathbf{p}'}t}\psi_{\mathbf{p}'} \tag{24} \]

and using the Green function (in the presence of a black sphere) of the resulting equation for \(\psi_{\mathbf{p}'}\):

\[ G(\mathbf{r},\mathbf{r}') = \frac{e^{\frac{i}{\hbar}p'|\mathbf{r}-\mathbf{r}'|}}{4\pi|\mathbf{r}-\mathbf{r}'|} - \frac{p'}{2\pi i\hbar} \iint \frac{e^{\frac{i}{\hbar}p'|\mathbf{r}-\mathbf{s}|}}{|\mathbf{r}-\mathbf{s}|} \frac{e^{\frac{i}{\hbar}p'|\mathbf{r}'-\mathbf{s}|}}{4\pi|\mathbf{r}'-\mathbf{s}|} \,dS \tag{25} \]

(the integral is over the cross section of the nucleus perpendicular to the vector \(\mathbf{r}-\mathbf{r}'\)). As a result one obtains an expression for the wave function of a meson emerging with momentum \(\mathbf{p}\) and accompanied by a photon of momentum \(\mathbf{k}\). In calculating the cross section the question arises whether the \(\pi\)-meson may be regarded as pointlike. For a point \(\pi\)-meson, in diffraction scattering by a nucleus of radius \(R\), the cross section for emission of a quantum \(\mathbf{k}\) contains, as a factor, a certain function of the dimensionless parameter \(\mathfrak{R}\left(\dfrac{\mu R}{\hbar}\right)\), which for \(\mu cR \gg \hbar\) turns into

\[ \frac{0.56}{R}\frac{\hbar}{\mu c}. \]

If \(c\mu R \gg \hbar\) \((A^{1/3}\gg 1)\), then

\[ \sigma^{d}(k)\simeq 2.3\,\frac{e^2}{\hbar c}\, \frac{E_0-ck}{E_0}\, \frac{1}{k}\, \frac{R\hbar}{\mu c} \sim 2.3\,\frac{e^2}{\hbar c}\, \frac{E_0-ck}{E_0}\, \frac{A^{1/3}}{k} \left(\frac{\hbar}{\mu c}\right)^2 . \tag{26} \]

The integral of the product of the emitted energy and the cross section is equal to

\[ c \int k \sigma^{d}(k)\, dk \simeq 2E_0 R^2 \frac{e^2}{\hbar c}\,\mathfrak{R}\!\left(\frac{\mu c R}{\hbar}\right) \longrightarrow 1.12\, \frac{R\hbar}{\mu c}\,\frac{e^2}{\hbar c}\,\mathfrak{R} E_0 \tag{27} \]

(the last expression is written for a heavy nucleus, \(A^{1/3} \gg 1\)).

Emission of \(\gamma\)-quanta may arise not only in diffraction scattering of \(\pi\)-mesons, but also when they are captured by a nucleus. It may be called stopping radiation. This is radiation accompanying the flux of mesons into the nucleus. It is proportional to the flux, and therefore its cross section is proportional to \(R^2\) (and not to \(R\), as in the case when we select part of the scattering events).

This radiation has much in common with diffraction radiation. In particular, for it too the region of photon generation is extremely elongated, since its dimensions are determined by the same formulas (4) and (19), i.e. it too is external with respect to the nucleus. In the present case the cross section can be calculated, since the complete solution \(\psi\) has been found. Integrated over frequencies \(k > k_{\min}\) and over emission angles of the quantum \(\vartheta < \vartheta_{\max}\), the cross section proves to be equal to

\[ \sigma^{(c)} = \frac{e^2 R^2}{\hbar^2 c^2} \left( \ln \frac{\vartheta_{\max}^{2} E_0^{2}}{137\mu^2 c^4} \right) \left( \ln \frac{E_0}{c k_{\min}} - 1 \right). \tag{28} \]

Taking into account considerations of relativistic invariance concerning the possible form of the \(\pi\)-meson form factor, one can arrive at definite conclusions about its influence on photon radiation. Namely, this form factor, as one may conclude, must have the form

\[ F = F\left\{ \frac{kE_0}{2\mu^2 c^3} \left( \vartheta^2 + \frac{\mu^2 c^4}{E_0^2} \right) \right\}. \tag{29} \]

One may therefore express the hope that a detailed experimental study of such electromagnetic radiation will make it possible in the future to approach the determination of such an important quantity as the form factor of the \(\pi\)-meson. The product of the total energy radiated in capture and the cross section turns out, in order of magnitude, to be equal to \(\dfrac{R^2}{137}E_0\).

The radiation cross section at small angles is especially large. Thus, for

\[ \vartheta = \frac{\mu c^2}{E_0} \]

the differential radiation cross section in capture is equal to

\[ \frac{e^2}{4\pi \hbar c}\, R^2 \left( \frac{E_0}{\mu c^3} \right)^2 \frac{E_0 - ck}{E_0}\, \frac{1}{k} \tag{30} \]

and for \(E_0 \sim 40 \mu c^2\) is equal to \(R^2 \dfrac{E_0 - ck}{E_0}\dfrac{1}{k}\), which has the order of magnitude of nuclear cross sections.

An analogous radiation process was also considered for a particle with spin \(\dfrac{1}{2}\).

If, instead of the black-nucleus model, one uses the scattering amplitude of the \(\pi\)-meson, \(A(\vartheta)\), in the given force field, then, naturally, integration over the angles cannot be carried out until the specific form of \(A(\vartheta)\) is substituted. Such a calculation was performed, in particular, for a semitransparent nucleus \({}^{14}\).

6. Formation of \(\pi\)-Meson Pairs by \(\gamma\)-Quanta on a Nucleus

The process considered in § 5 has characteristics whose order of magnitude can be predicted in advance. However, its study made it possible not only to obtain exact formulas, but also to develop an apparatus applicable to a number of other processes. Thus, in \({}^{8,9}\) the process of formation of pairs of \(\pi\)-mesons by a \(\gamma\)-quantum on a nucleus was considered; moreover, the action of the nucleus as a black body with respect to mesons is taken into account by the fact that the state of the produced meson is described by the function (15). Here, therefore, the interaction of the meson with the nucleus is taken into account sufficiently accurately, and not by perturbation theory (under the assumption, which is not of fundamental character, that the nucleus is absolutely black).

The expression for the effective cross section for pair production has a complicated form. However, in the case of heavy nuclei, whose radius is large compared with the Compton wavelength of the \(\pi\)-meson, the formulas are substantially simplified. The differential cross section for production of a pair with momenta \(\mathbf p_1, \mathbf p_2\) by a quantum with momentum \(\mathbf k\) is equal to

\[ d\sigma = \frac{e^2}{2\pi^2\hbar c} R^2 \frac{J_1^2(\varkappa R/\hbar)\,E_1(ck-E_1)\,E_1 b^2\,db\,d\varkappa} {c^3\varkappa^3 k^3\,(b^2+\mu^2 c^2)^2}. \tag{31} \]

Here

\[ \mathbf p_1 = \frac{\mathbf k}{k} \left(p_1-\frac{k_1^2}{2p_1}\right) +\mathbf k_1; \qquad \mathbf p_2 = \frac{\mathbf k}{k} \left(p_2-\frac{k_2^2}{2p_2}\right) +\mathbf k_2; \]

\[ \varkappa = k_1+k_2; \qquad \mathbf b = \frac{1}{2}(\mathbf k_1-\mathbf k_2); \qquad E_1 = c\sqrt{p_1^2+\mu^2 c^2}, \]

where \(\mathbf k_1, \mathbf k_2\) are two-dimensional vectors perpendicular to the direction of the quantum; \(k_1^2 \ll k^2,\ k_2^2 \ll k^2\). In formula (31) the \(\pi\)-mesons are regarded as pointlike. Otherwise, in this formula there appears a factor having the meaning of the form factor of the process under consideration. In the presence of a strong interaction between \(\pi^+\)- and \(\pi^-\)-mesons, in (31) also

a factor appears that takes this interaction into account. If (31) is integrated over all \(\varkappa\) and over \(b < b_{\max}\), then the following energy distribution of the pairs is obtained:

\[ d\sigma=\frac{e^2R^2}{2\hbar c}\, \frac{E_1(ck-E_1)\,dE_1}{c^3k^3} \left[ \ln \frac{\mu^2c^2+b_{\max}^2}{\mu^2c^2 e} + \frac{\mu^2c^2}{\mu^2c^2+b_{\max}^2} \right]. \tag{32a} \]

The integral cross section is equal to

\[ \sigma=\frac{e^2R^2}{12\hbar c} \left[ \ln \frac{\mu^2c^2+b_{\max}^2}{\mu^2c^2 e} + \frac{\mu^2c^2}{\mu^2c^2+b_{\max}^2} \right]. \tag{32б} \]

For large \(b_{\max}\) the “oddness” of \(\pi\)-mesons is essential. From formulas (31)—(32б) the following conclusions may be drawn:

a) The cross section for the production of \(\pi^+\), \(\pi^-\)-pairs not accompanied by excitation of the nucleus, at very high energies, does not depend on the energy of the quantum. It is proportional to \(R^2\sim A^{2/3}\), where \(A\) is the number of nucleons in the nucleus. Its order of magnitude is

\[ \frac{e^2R^2}{10\hbar c}. \]

b) The sum of the transverse momenta of the \(\pi^+\)- and \(\pi^-\)-mesons \((\varkappa=\mathbf{k}_1+\mathbf{k}_2)\) is of order \(\hbar/R\). The distribution in \(\varkappa\) is given by formula (31).

c) The effective values \(\mathbf{b}=\frac{1}{2}(\mathbf{k}_2-\mathbf{k}_1)\) are connected with the “sizes” of the \(\pi\)-particles and the properties of their interaction. It should be noted that for the transformation \(\gamma\to 2\pi^0\) the process under consideration is forbidden. This is connected with the law of charge parity: the \(\gamma\)-quantum is charge-odd, the \(\pi\)-meson charge-even (the nucleus performs only black-body functions and therefore must be regarded as charge-even). Consequently, at high energies the cross section of the process \(\gamma\to 2\pi^0\) is small in comparison with the cross section of the process \(\gamma\to\pi^+ + \pi^-\).

Alongside the case considered above of the production of a pair of \(\pi\)-mesons by a \(\gamma\)-quantum, when both \(\pi\)-mesons go off to infinity, there is also possible such a process of absorption of a \(\gamma\)-quantum in which, at a large distance from the nucleus, a virtual pair of \(\pi\)-particles is formed, of which one goes off to infinity, while the other is absorbed by the nucleus and a star (nuclear explosion) is produced. In addition, a process is possible in which both \(\pi\)-mesons are absorbed by the nucleus. The theory of these processes\(^{14}\) reveals a number of their peculiar features which, it may be hoped, make it possible to distinguish such cases among other processes of photodisintegration of nuclei.

The differential cross section of the process in which a quantum of energy \(ck\) gives one \(\pi\)-meson with energy \(E_\pi\) turns out to be equal to\(^{14}\)

\[ d\sigma=\frac{3\pi e^2}{16\hbar c}\, \frac{E_\pi(ck-E_\pi)}{c^3k^3}\, R\,\frac{\hbar}{\mu c}\,dE_\pi, \tag{33} \]

so that the total cross section is approximately equal to

\[ \sigma \simeq \frac{\pi}{32}\,\frac{e^2}{\hbar c}\,R\,\frac{\hbar}{\mu c} \tag{33a} \]

(here and throughout it is assumed that \(R \gg \frac{\hbar}{\mu c}\), and the meson form factor is not taken into account).

For the process in which both mesons are absorbed by the nucleus, neglecting the form factor, the total cross section for the production of a photostar proves to be equal to

\[ \sigma=\frac{e^2 R^2}{4\hbar c}\ln\frac{k}{\mu c}. \tag{33b} \]

Thus this cross section increases logarithmically with energy. Such a mechanism of photostar production may become significant at high energies. The result in this case too can be expressed in terms of the scattering amplitudes of \(\pi\)-mesons\(^{14}\).

7. Formation of electron–positron pairs in the diffraction and capture of a \(\pi\)-meson

This process\(^{16}\) is a process of higher order by a factor of \(\frac{e^2}{\hbar c}\) in comparison with the process of emission of a \(\gamma\)-quantum by a \(\pi\)-meson (§ 5). Since the electromagnetic interaction is taken into account in the calculation by perturbation theory, the calculations are carried out according to a similar scheme. It is again necessary to separate the process connected with diffraction (its cross section, as in (27), is essentially proportional to the first power of \(R\)) from the process connected with absorption (the cross section, as in (28), is proportional to \(R^2\)). Namely,

\[ \sigma^{(d)} \simeq \frac{2.6}{3\pi}\left(\frac{e^2}{\hbar c}\right)^2 \frac{R\hbar}{\mu c} \left(\ln\frac{E_0}{mc^2}\right)^2, \tag{34} \]

\[ \sigma^{(c)} \simeq \left(\frac{e^2}{\hbar c}\right)^2 \frac{2R^2}{3\pi} \left(\ln\frac{E_0}{mc^2}\right)^2 \left(\ln\frac{E_0}{\mu c^2}\right). \tag{34a} \]

A characteristic feature here is the weak dependence of (34a) on the mass \(\mu\) of the incident particle.

III. PURELY ELECTROMAGNETIC PROCESSES

8. Bremsstrahlung of electrons in a crystal

The possibility of the simultaneous influence of different atoms of a crystal on bremsstrahlung was first noted by Williams\(^{27}\). However, he confined himself to an estimate which led him (as is now clear) to the incorrect conclusion that bremsstrahlung at very high energies, because of the ordering of the positions of the atoms, must

always drop. The criterion for the appearance of the effect, given without derivation, likewise was not confirmed. Subsequently[^28] the influence of the crystal was considered in greater detail; however, in that work too the results were not quantitative in character, and the conclusions were not confirmed by the subsequent investigation under discussion here.

In the work of Ter-Mikaelian[^2] there was studied consistently, within the framework of ordinary perturbation theory (in practice, for the calculations, as in[^28], the Weizsäcker–Williams method was used, but the same formulae are also obtained by considering the system in which the lattice is at rest), the bremsstrahlung of a relativistic electron incident in a given direction with respect to the crystallographic axes (for simplicity a cubic lattice was considered). It turned out to be very important that the thermal vibrations of the lattice sites were taken into account.

To calculate this process, as follows from (2), in the ordinary calculations it is simply necessary, instead of \(V(\mathbf r)\), to substitute the total potential of all the nuclei of the crystal \(\sum_i V(\mathbf r-\mathbf r_i)\), where \(\mathbf r_i\) is the position of the \(i\)-th nucleus. The matrix element decomposes into a sum of elements, and, just as in the analogous case of X-ray scattering, the cross section for the process in which the lattice receives the momentum \(\mathbf q\) will differ from the bremsstrahlung cross section on an isolated atom by the interference factor:

\[ \left|\sum_i e^{\frac{i}{\hbar}\mathbf q \mathbf r_i}\right|^2, \]

where the sum is taken over the instantaneous positions of the nuclei \(\mathbf r_i\). Averaging over the thermal vibrations leads to the replacement of this factor, as usual in the theory of X-ray scattering, by

\[ N_1N_2N_3(1-e^{-2\mathfrak M})+e^{-2\mathfrak M}\left|\sum_i e^{\frac{i}{\hbar}\mathbf q\mathbf r_{i0}}\right|^2 . \]

Here \(N_1, N_2, N_3\) are the numbers of sites fitting in the directions of the different axes, \(2\mathfrak M=\frac{1}{\hbar^2}q^2\overline{u^2}\), where \(\overline{u^2}\) is the mean square of the thermal displacement of a site, and \(\mathbf r_{i0}\) are the equilibrium positions of the sites. Correspondingly, the cross section splits into an “amorphous” part, for which the cross section is proportional to the total number of atoms of the crystal \(N_1N_2N_3\), and an “interference radiation.” The cross section of the amorphous radiation is less than the sum of the radiations on the same number of isolated atoms by an amount which depends on the temperature and, for different elements, is of the order of \(10\)—\(20\%\). With increasing temperature the correction decreases. This radiation does not depend on the angle of incidence of the electron.

More complicated and more interesting is the interference part. Since the perpendicular component of the momentum transmitted to a nucleus, \(q_\perp\), is usually large \((q_\perp \gg mc)\), nuclei situated in one plane perpendicular to the motion do not give an interference effect (the “essential region” always lies in the field of only one of these nuclei). Nuclei situated along the motion, however, do give this effect. Refinement of the estimates that can be obtained from formulas (9) and (16) leads to the following value of the energy determining the appearance of the effect:

\[ \frac{E_0}{mc} \gg \frac{am}{4\pi \hbar c}\frac{\varepsilon}{1-\varepsilon} \sim a_{\text{\AA}} \frac{137}{2\pi}\frac{\varepsilon}{1-\varepsilon}, \tag{35} \]

where \(a_{\text{\AA}}=10^8 a\) is the lattice constant in angstroms, \(a\) is the lattice constant in cm, \(\varepsilon=\dfrac{ck}{E_0}\) is the fraction of the electron energy carried away by the photon. For \(a_{\text{\AA}}\sim 3\), \(\varepsilon\sim \dfrac12\), this gives \(E_0 \gg 50\) MeV. At higher energies (or with respect to the emission of softer quanta than \(\varepsilon\sim \dfrac12\)) the cross section grows in proportion to the square of the effective number of atoms \(N_{\mathrm{eff}}\) lying along the direction of motion.

The conditions for the appearance of interference maxima were investigated; it was shown how the transition to a crystal of arbitrary thickness is made, and the limits of applicability of the formulas obtained were given. The dependence on the direction of incidence of the electron was studied.\(^{2}\) Thus, the interference part of the bremsstrahlung cross section of a beam of electrons with angular width \(\theta\), incident along the axis of a crystal, per nucleus is equal to

\[ \frac{1}{N_1N_2N_3}\,\sigma_{\mathrm{int}}(\varepsilon)\,d\varepsilon = \sigma_{\mathrm{B.-G.}}(\varepsilon)\,d\varepsilon\,\frac{\pi\rho}{a\theta} \tag{36} \]

under the condition that

\[ \frac{2\pi}{a}\sqrt{\overline{u^2}} = \theta_{\max} > \theta > \theta_{\min} = \frac{amc}{2\hbar}\frac{\varepsilon}{1-\varepsilon}\frac{mc^2}{E_0}. \]

Here \(\sigma_{\mathrm{B.-G.}}\) is the usual Bethe–Heitler cross section, \(\rho=Z^{-1/3}\dfrac{\hbar^2}{me^2}\) is the screening radius of the atom. The interference radiation falls off for \(\theta<\theta_{\min}\). For tungsten, \(\theta_{\max}=2.4^\circ\) at absolute zero, \(\theta_{\max}=4.8^\circ\) at the Debye temperature. Since \(\dfrac{\pi\rho}{a}\sim \dfrac{2\pi}{a_{\text{\AA}}}\dfrac{1}{Z^{1/3}}\sim 1\), then for sufficiently small \(\theta\), for example for \(\theta\sim\theta_{\max}\), the radiation intensity may many times exceed the radiation intensity on an isolated atom.

It is essential that the integral over all angles strictly gives the same intensity as the radiation on an isolated atom. Thus, when a beam of electrons passes through a sufficiently thic—

solid polycrystal, the difference from the usual formulas disappears. The action of the structure of the medium for a single crystal consists, when the angle of incidence of the electron is varied, in the appearance of sharp and narrow maxima, compensated by a slight lowering of the general background outside the maxima*).

9. The influence of multiple scattering on the bremsstrahlung of electrons

In the case of an amorphous medium, where there are no interference phenomena, at sufficiently high energy another effect begins to show itself—the influence of multiple Coulomb scattering of the electron by the atoms of the medium (in the case of a crystal it is also present, but in the presence of the interference effect it plays a subordinate role\(^2\)). The existence of this effect was pointed out by Landau and Pomeranchuk\(^3\). They also gave estimates and approximate limiting formulas in the case of the classical theory of bremsstrahlung (which is quite rigorous for the emission of soft quanta \(ck \ll E_0\) and sufficient for estimating the entire effect). A more exact quantitative theory in the classical case\(^7\), and especially the quantum theory, proved to be complicated. Its results, in particular, confirm the approximate estimates obtained by means of the classical consideration.

The essence of the effect in question can be understood as follows. If, during the time of motion of the particle through the essential region of length

\[ \frac{\hbar}{q_{\parallel}} \sim \frac{2E_0(E_0-ck)\hbar}{m^2c^4k} \]

multiple Coulomb scattering in the given medium takes the electron outside the limits of the angle

\[ \vartheta \sim \frac{mc^2}{E_0}, \]

then the entire radiation process is disturbed.

Over this length the mean square of the scattering angle is equal to

\[ \overline{\vartheta_s^2} = \left(\frac{E_s}{E_0}\right)^2 \frac{r_{\parallel \mathrm{eff}}}{L} \sim \frac{2E_s^2\hbar}{m^2c^4kL}\, \frac{E_0-ck}{E_0}, \]

where \(E_s \simeq 21\) MeV, \(L\) is the shower unit of length (in cm) in the given medium. If

\[ \vartheta_s^2 > \left(\frac{mc^2}{E_0}\right)^2, \]

i.e., if

\[ \frac{E_0}{mc^2} \gtrsim \frac{1}{60} \sqrt{ \frac{E_0}{E_0-ck}\, \frac{kL}{\hbar} }, \tag{37} \]

then the radiation intensity will decrease. Even for hard

) Note added in proof.
Recently F. Dyson and H. Überall, Phys. Rev. 99*, 604 (1955), having become acquainted with work\(^3\), but not knowing of work\(^2\), also came to the conclusion that, because of the large size of the effective region, bremsstrahlung of electrons in a crystal must possess interference features. In their note only some estimates are given.

quanta, putting for the estimate \(ck \sim \dfrac{1}{2}E_0\), we obtain the condition

\[ \frac{E_0}{mc^2} > \frac{E_{\text{scatt}}}{mc^3} = \frac{1}{2}\frac{Lmc}{\hbar}\left(\frac{mc^2}{E_s}\right)^2 \simeq \frac{1}{7200}\frac{Lmc}{\hbar}, \tag{38} \]

where \(E_{\text{scatt}}\) denotes the characteristic energy determining the appearance of the effect (in papers 3, 6, and 7 it is denoted by \(E_0\)). For lead \(L \sim 5\cdot 10^{-1}\ \text{cm}\), and for air under normal conditions \(L \sim 3\cdot 10^4\ \text{cm}\). Accordingly, for \(E_0 \gtrsim 2\cdot 10^6 mc^2\) in lead and for \(E_0 \gtrsim 10^{11}mc^2\) in air, the bremsstrahlung cross section should decrease in comparison with its usually accepted value. For softer quanta the effect sets in at still lower electron energies. At a given energy \(E_0\), it affects the emission of photons with energy

\[ ck < 3600\left(\frac{E_0}{mc^2}\right)^3 \frac{\hbar c}{L} \tag{39} \]

(in particular, in lead for \(ck < 4\cdot 10^{-7}\dfrac{E_0}{mc^2}E_0\)). Thus, for example, an electron with energy \(E_0 \sim 5\cdot 10^{10}\ \text{eV}\) in lead should produce very few photons with energy less than \(0.01E_0 \sim 5\cdot 10^8\ \text{eV}\).

With a more consistent estimate in the region of validity of the classical treatment (\(ck \ll E_0\)), it turns out that, in contrast to the usual Bethe–Heitler formula, for the intensity of radiation of frequency \(\omega = \dfrac{ck}{\hbar}\)

\[ dI_{\text{B.-H.}}= \frac{e^3}{3\pi}\left(\frac{E_s}{mc^2}\right)^2\frac{d\omega}{L} = \frac{16}{3}ne^2(Zr_0)^2\ln(191 Z^{-1/3})\,d\omega \tag{40} \]

(\(n\) is the number of atoms of the medium per unit volume, \(Z\) is the atomic number of the element, \(r_0\) is the electron radius), the radiation intensity is described by the formula\(^3\)

\[ dI=\frac{e^2}{2}\frac{E_s}{E_0}\, \frac{\sqrt{\omega\,d\omega}}{\sqrt{6\pi Lc}} \left(E_s=\sqrt{4\pi\cdot137}\,mc^2\simeq 21\ \text{MeV}\right), \tag{41} \]

valid for

\[ \omega < \frac{c}{6L}\left(\frac{E_0E_s}{m^2c^4}\right)^2; \qquad E_0 > \frac{m^2c^4}{E_s}\sqrt{\frac{6\omega L}{c}}. \]

Thus, the bremsstrahlung cross section at high energies decreases as \(\dfrac{1}{E_0}\), while the photon spectrum has the character

\[ \frac{dI}{\omega}\sim \frac{d\omega}{\sqrt{\omega}}. \]

At sufficiently high energies, electrons and positrons in a medium acquire the properties of a penetrating component. As has already been said, the same is true for pair production. Consequently, photons also become penetrating.

To obtain this formula, one considers the classical expression for the energy emitted within an element of solid angle \(d\Omega\) at frequency \(\omega\) by an electron moving along a definite

trajectory:

\[ dI=\frac{e^2}{4\pi^2c^3}\,\omega^2 d\omega d\Omega \left|\int_{-\infty}^{+\infty}[\mathbf{v}\mathbf{n}]\,e^{i\left(\frac{\mathbf{k}\mathbf{r}}{\hbar}-\omega t\right)}\,dt\right|^2 . \tag{42} \]

where \(\mathbf{r}=\mathbf{r}(t)\) and \(\mathbf{v}=\mathbf{v}(t)\) are the radius vector and velocity of the electron as functions of time \(t\), \(\mathbf{n}\) is the unit vector in the direction of radiation, and \(c\mathbf{k}=\hbar\omega\mathbf{n}\). The classical approximation to the usual Bethe–Heitler formula is obtained if one assumes that \(\mathbf{r}=\mathbf{v}t\), with the velocity, in a certain region of size \(a\), changing from one constant value \(\mathbf{v}_1\) to another value \(\mathbf{v}_2\) (the formula will then be valid for radiation of wavelengths \(\lambda \gg a\)). In particular, for \(\mathbf{v}_2=0\) one obtains “stopping radiation.” However, in a medium, owing to multiple scattering, \(\mathbf{v}\) changes continuously near the values \(\mathbf{v}_1\) and \(\mathbf{v}_2\), and it may be assumed with sufficient accuracy that only the components of the radius vector and velocity perpendicular to \(\mathbf{v}_1,\mathbf{v}_2\) fluctuate. Their square increases linearly with time. Thus, for example, for radiation strictly along the direction of the initial motion \((\mathbf{k}\mathbf{v}_1=kv_1)\), in the phase for \(-\infty<t<0\), there will on the average be not

\[ \frac{k r}{\hbar}-\omega t=\omega t\left(\frac{v_1}{c}-1\right), \]

but an expression of the type

\[ \omega t\left[\frac{v_1}{c}\left(1-\frac{1}{2}\vartheta^2(t)\right)-1\right], \]

where \(\vartheta(t)\) is the angle of multiple scattering, which is assumed small

\[ \left(\cos\vartheta\simeq 1-\frac{\vartheta^2}{2}\right). \]

For \(kv_1\vartheta \gtrless 2kc\left(1-\frac{v_1}{c}\right)\) an essential additional term appears in the phase, and because of the rapid oscillation of the integrand the expression \(dI\) (see (42)) decreases. To obtain the true radiation, formula (42) must be averaged over all values of the perpendicular components of the velocity, or, equivalently, over \(\vartheta\). Initially, instead of averaging \(dI\), the phase was averaged; this also leads to the estimate expressed by formula (41)\(^3\).

Subsequently, Migdal\(^7\), with the aid of an elegant method, carried out rigorous calculations valid for \(E\gg mc^2\). A quantum theory of the phenomenon was also given. Namely, in order to average over scattering acts (in the classical case) the quantity \(dI\), defined by formula (42), this expression was transformed to such a form that it could be expressed through the Fourier components (which appear owing to the exponential factor in (42)) of the probability \(W_{\mathbf{k}}\) of a given value of the velocity. In the space of the angular variable, as it turns out, \(W_{\mathbf{k}}\) satisfies an equation of the Fokker–Planck type. It was possible to solve it. As a result it was found that the radiation intensity per unit path and in the frequency interval \(d\omega\) is described by the formula*)

\[ dI'=\{dI'\}_0\Phi(s)=\frac{4e^2}{3\pi c}\left(1+\frac{v}{c}\right)Q\Phi(s), \tag{43} \]

*) Here the misprints that crept into \({}^7\) have been corrected.

where

\[ s=\frac{1-\dfrac{v}{c}}{4}\sqrt{\frac{\omega}{Q}}, \qquad Q=2\pi n(Zr_0)^2\ln(191Z^{-1/3}). \]

\(\{dI\}_0\) is the expression for the radiated energy without allowing for multiple scattering (the classical approximation to the Bethe—Heitler formula). \(\Phi(s)\) is a tabulated function:

\[ \Phi(s)=3s\int_0^\infty e^{-sx}\frac{\cos sx+\sin sx}{\operatorname{ch}^3 \dfrac{x}{2}}\,dx+ \]

\[ {}+24s^2\int_0^\infty e^{-sx}\frac{\sin sx}{\operatorname{sh}x}\,dx-6\pi s^2 . \tag{43a} \]

For \(s\to\infty\) (scattering negligibly small), \(\Phi(s)\to 1-\dfrac{48}{7s^4}\), and formula (40) is obtained; for \(s\to 0\), \(\Phi(s)\to 6s\), and the result differs from the approximate formula (41) by a factor of order unity. From the complete formula it has been found that in lead, for photons with energy \(ck=\dfrac{1}{2}E_0\), the deviation from the Bethe—Heitler formula reaches 50% if \(E_0=3\cdot 10^{12}\) eV. The quantum treatment, necessary for obtaining an exact formula in the region \(ck\sim E_0\), is much more complicated. When scattering is taken into account, the particle producing the bremsstrahlung is no longer described by a wave function, and one has to consider the density matrix. For it a quantum kinetic equation is constructed, which is subsequently solved. The result is expressed by the following formulas. The intensity of bremsstrahlung of frequency \(\omega\) per unit path is equal to

\[ dI'=\frac{8e^2Q}{3\pi c(\mathbf p_0+\mathbf p)^2} \left\{ \frac{p_0^2-p^2}{2}\Phi(s)+ \frac{(\mathbf p_0-\mathbf p)^3}{4}G(s) \right\}\,d\omega, \tag{44} \]

\[ G(s)=48s^2\left(\frac{\pi}{4}-\int_0^\infty e^{-2st}\frac{\sin 2st}{\operatorname{sh}t}\,dt\right) \]

(\(\mathbf p_0\) and \(\mathbf p\) are the momenta of the electron before and after radiation). For \(\omega\to 0\) this formula goes over into the formula of the classical approximation (43).

The interest of this investigation lies not only in obtaining the formulas given above, which provide a very complete and closed theory of the process, in particular containing the quantum theory of the effect, but also in the development of a special method of the quantum kinetic equation, which may find application in other problems as well.

10. Polarization of the Medium

As was shown by Ter-Mikaelian\(^6\), the consideration of the influence of multiple scattering on bremsstrahlung must be supplemented by allowance for the polarization of the medium. Namely, it is necessary to take into account the [[unclear: word beginning “ot-”]]

INELASTIC DIFFRACTION PROCESSES

the difference of the photon propagation velocity from unity. For this it is necessary in formula (42) to add the factor \(\sqrt{\varepsilon}\) and, instead of \(ck=\hbar\omega\), to put

\[ ck=\sqrt{\varepsilon}\hbar\omega, \tag{45} \]

where the dielectric constant of the medium may be taken equal to

\[ \varepsilon=1-\frac{4\pi nZe^2}{m\omega^2}. \tag{46} \]

The effect, as is seen from this, is essential only for the softest quanta, so that one may restrict oneself to the classical approximation. Performing, after this replacement, again the averaging of the phase of the subintegral expression in (42) over scattering events, we see that in the phase, besides the term of the type \(\omega t\left(1-\frac12\vartheta^2(t)\right)\), caused by scattering, there appears the term \(\omega t(\sqrt{\varepsilon}-1)\simeq -\frac{2\pi nZe^2}{m\omega}\). If it turns out to be larger than the term caused by scattering (low frequencies), then the polarization of the medium plays a greater role than scattering. In general the results are as follows:

a) As long as the electron energy does not exceed a certain critical value, i.e.

\[ E_0<E_{\mathrm{crit}}=\left(\frac{mc^2}{E_s}\right)^2\frac{6L}{c}\sqrt{\frac{4\pi nZe^2}{m}}\,mc^2 \tag{47} \]

(for dense media of the middle and end of the periodic system \(E_{\mathrm{crit}}\sim 10^4mc^2\sim 10^{10}\ \mathrm{ev}\)), the influence of multiple scattering has no time at all to manifest itself, since the polarization of the medium is substantial. In this case, in the region

\[ \omega>\omega_p^{(1)}\simeq \sqrt{\frac{4\pi nZe^2}{m}\frac{E_0}{mc^2}} \tag{48} \]

there is valid the formula obtained when the influence of the medium is neglected—the classical analogue of the Bethe–Heitler formula, coinciding with it for \(ck\ll E_0\):

\[ dI=\frac{e^2}{3\pi}\frac{d\omega}{L}\left(\frac{E_s}{mc^2}\right)^2 . \tag{48′} \]

In the region

\[ \sqrt{\frac{4\pi nZe^2}{m}}\ll \omega \ll \sqrt{\frac{4\pi nZe^2}{m}\frac{E_0}{mc^2}}=\omega_p^{(1)}. \]

(For \(\omega\lesssim \sqrt{\frac{4\pi nZe^2}{m}}\) the approximate formula (46), strictly speaking, is inapplicable) we have:

\[ dI=\frac{m}{12\pi^2LnZ}\left(\frac{E_s}{E_0}\right)^2\omega^2d\omega. \tag{49} \]

b) When the electron energy increases and becomes greater than the critical one \((E_0>E_{\mathrm{crit}})\), there appears a frequency region in which

a decisive role is played by multiple scattering. Namely, for

\[ \left(\frac{4\pi nZe^2}{m}\right)^{2/3} \left(\frac{6E_0^2L}{E_s^2c}\right)^{1/3} \equiv \omega_p^{(2)} \ll \omega \ll \frac{E_0}{\hbar}\frac{E_0}{mc^2} \left(\frac{E_s}{mc^2}\right)^2 \frac{\hbar}{Lmc} \equiv \omega_s \tag{50} \]

formula (41) is valid. For \(\omega \ll \omega_p^{(2)}\) formula (49) remains valid, and for \(\omega \gg \omega_s\), formula (40). For \(E_0 \gg E_{\mathrm{scatt}}=mc^2\frac{Lmc}{\hbar}\left(\frac{mc^2}{E_s}\right)^3\), this region, indicated by relation (50), covers almost the entire high-frequency region. It is necessary, however, to emphasize that in the emission of quanta of extremely large energy the usual Bethe—Heitler formulas for bremsstrahlung are always valid. This is connected with the circumstance already noted in § 3—namely, with the fact that

Fig. 2.

for very small \(E_0-ck\) the value of \(q_{\parallel}\) is large and the effective region is small, so that sufficiently large scattering does not have time to occur. Indeed, according to (37), near the upper boundary of the spectrum, i.e., if \(E_0-ck \ll E_0\), so that \(ck\sim E_0\), multiple scattering becomes significant only under the condition

\[ E_0-ck \gg \frac{mc^2}{3600}\sqrt{\frac{Lmc}{\hbar}} = \frac{1}{2}\frac{Lmc}{\hbar} \left(\frac{mc^2}{E_s}\right)^2mc^2 \equiv E_{\mathrm{scatt}}, \tag{51} \]

where \(E_{\mathrm{scatt}}\) is the quantity introduced above that characterizes the effect. In lead \(L \sim 0.5\ \mathrm{cm}\), \(E_{\mathrm{scatt}} \sim 2\cdot 10^{12}\ \mathrm{eV}\). Thus, for the region of quantum energies \(E_0 - E_{\mathrm{scatt}} \ll ck \ll E_0\), emission occurs as on an isolated atom. In general the picture can be represented schematically by the graph in Fig. 2, where, however, the scale is not maintained: in reality \(\hbar\omega_p \ll E_0\), etc.

An interesting feature of the influence of the medium is, in particular, that the infrared catastrophe is eliminated. Consequently, when checking experimentally the radiative corrections to various effects calculated by theory, the action of the medium must be taken into account, since these corrections are usually strongly influenced by the infrared catastrophe.

IV. NUCLEAR PROCESSES

11. Emission of Mesons by Nucleons in Diffraction Scattering

The indicated process\(^4\) differs from the other processes considered in that it cannot be reliably calculated quantitatively. This is hindered by the absence of a consistent meson theory, and by the inapplicability in this case of perturbation theory, which can give only an unreliable indication of the order of magnitude of the cross section.

Condition (10), upon substituting (20), gives the energy threshold for the occurrence of the effect. If the meson carries away a small fraction of the energy, of order \(\dfrac{\mu}{M}E_0\), then it must be

\[ E_0 \gg \frac{c\mu R}{\hbar}\,Mc^2 \sim A^{1/3}Mc^2 . \tag{52} \]

If the meson energy is of the order of the initial energy of the nucleon, then it must be

\[ E_0 \gg \frac{M}{\mu}\,A^{1/3}Mc^2, \tag{52'} \]

i.e., the threshold increases substantially. For the generation of several mesons in one act, the threshold increases proportionally to their number. A distinguishing feature of such a generation process would be that the mesons would fly (in the center-of-mass system) only forward, i.e., the so-called “second cone” would be absent.

To estimate the magnitude of the cross section, it should be taken into account that we can regard diffraction as occurring on the entire nucleus as a whole only in the case when the perpendicular component of the transferred momentum is not very large, namely, if its reciprocal does not greatly exceed the mutual distance of particles in the nucleus. This latter may be taken equal to \(\dfrac{\hbar}{\mu c}\). Consequently, it must be \(q_\perp \lesssim \mu c\). In this case the velocity acquired by the nucleon (in its rest system)

small: \(v \sim \dfrac{\mu}{M} c\). Therefore the emission of mesons cannot be very strong. Indeed, application (according to the scheme used in § 4) of perturbation theory under the assumption of a pseudoscalar coupling of pseudoscalar mesons gives, for the emission of one meson,

\[ \sigma \sim \frac{g^2}{\hbar c}\left(\frac{\hbar}{\mu c}\right)^2 A^{1/3}\left(\frac{\mu}{M}\right)^2 \sim \frac{g^2}{\hbar c}\left(\frac{\mu}{M}\right)^2 \frac{\sigma_0}{A^{1/3}}, \tag{53} \]

where \(g\) is the coupling constant, and \(\sigma_0\) is the geometrical cross section of the nucleus. Even for \(\dfrac{g^2}{\hbar c} \sim 10 \div 20\), the cross section is appreciably smaller than the geometrical one. This effect, both as a consequence of the condition (51)—(52) for its occurrence and according to formula (53), must be relatively more noticeable for light nuclei.

Since meson interactions are strong, one should suppose that a similar effect of generation of a \(\pi\)-meson by a \(\pi\)-meson diffracted on a nucleus is possible. In such a case the smallness of the mass of the scattered meson plays a favorable role.

12. Diffraction Disintegration of the Deuteron

The calculation of the diffraction disintegration of the deuteron can be carried out quite fully, despite the fact that the form of the interaction forces between nucleons is unknown. It is necessary to know only the wave function of the deuteron. Since it is reliably known only at proton—neutron distances exceeding the range of action of the forces \(\dfrac{\hbar}{\mu c}\), only that part of the disintegration cross section corresponding to the transfer of not very large momenta \(q\) can be calculated quite reliably. The investigation was carried out for deuterons with energies in the interval of the order of \(30\)—\(170\) MeV, when the energy of each of the incident nucleons is such that the target nucleus may be regarded as opaque. Having written the function of a deuteron diffracted as a whole on a black sphere representing the target nucleus (here one must take into account the finite dimensions of the deuteron, and therefore the function of the internal motion in the deuteron \(\varphi(|\mathbf r_p-\mathbf r_n|)\) enters), one may, in order to obtain an estimate, simply select those scattering events in which the momentum transferred to the deuteron is sufficient for disintegration of the deuteron (say, is of order \(\mu c\)). Hence, according to formula (12a), for an estimate of the cross section we obtain \(\sigma \sim RR_d\) (\(R\) is the radius of the nucleus, \(R_d\) the radius of the deuteron). Thus, the cross section has the order of the stripping cross section and is smaller than the geometrical cross section of the target nucleus. Therefore, taking \(R_d \ll R + R_d\) (this assumption is not of fundamental significance), one may apply perturbation theory. One can verify that the cross section for disintegration with emission of a proton and a neutron having momenta \(p_p\) and \(p_n\) is equal to

\[ d\sigma(p_p,p_n)=2\pi\left|\langle \psi_{p_p}^{(-)}\psi_{p_n}^{(-)}|U|\psi_d\rangle\right|^2\delta(E_i-E_f), \tag{54} \]

INELASTIC DIFFRACTION PROCESSES

where \(\psi_{p_p}^{(-)}\psi_{p_n}^{(-)}\) are the functions of the produced proton and neutron of the form (15), \(\psi_d\) is the wave function of the deuteron diffracted as a whole (therefore, a function of the form (13)), and \(U\) is the proton–neutron interaction operator. If one restricts oneself to the part of the processes corresponding to momentum transfer \(q \ll \mu c\), then the nuclear edge may obviously be regarded as sharp, the deuteron function may be replaced by
\(\varphi \sim \dfrac{1}{r}\exp\{-\alpha|r_p-r_n|\}\), \(\alpha=\sqrt{M\varepsilon_D}\), and \(U\) may be treated as a delta-like function of the distance. The latter simplification is possible because the relative velocity of the emitted nucleons is small—of the order of their velocity inside the deuteron.

Thus, falling apart from a weak jolt, the deuteron gives a proton and a neutron with the same distribution in angles and energies as in stripping, since in both cases these distributions are specified by the velocity distribution in the deuteron. However, in the present case, in contrast to stripping, both the neutron and the proton are emitted in each event. The cross section of the process, if one restricts oneself to the cases \(q_\perp<q_{\perp\max}\ll\mu c\), is found to be

\[ \sigma \simeq \frac{3}{2}\,RR_d^2\,\frac{\mu c}{\hbar}\,\frac{q_{\perp\max}}{\mu c}, \tag{55} \]

where \(q_{\perp\max}\) is the maximum of the values of the transferred momentum allowed in the calculation. For \(q_{\perp\max}\sim\mu c\) we obtain \(\sigma\sim RR_d\) (as is clear from formula (12a), this is what should have been obtained, since to break up the deuteron it is necessary to impart to it a momentum \(q_\perp\sim \dfrac{\hbar}{R_d}\sim \dfrac{\mu c}{2}\)).

Experimental verification of the indicated effect requires coincidence experiments. It must be noted, however, that experiments observing the number of protons in the stripping reaction have always given excessively large values. In particular, recently, without taking into account the possibility of the indicated process, a group of American physicists\(^{29}\) obtained from their data an astonishingly large value of the nuclear radius: \(R=r_0A^{1/3}\) with \(r_0=1.6\div1.7\cdot10^{-13}\) cm. It is possible that in reality the effect under discussion manifested itself here, and the excess number of protons was incorrectly ascribed to the stripping effect.

Let us note that such a diffraction breakup has a purely kinematic character and is determined by transverse momenta (this is seen, in particular, from the fact that the cross section does not depend on the deuteron energy). Therefore it must proceed in the same form also at relativistic deuteron energies (when the target nucleus again becomes opaque).

As for energies smaller than those considered above, diffraction processes of course play a role here as well; however, they cannot be separated from all the other processes of interaction of the deuteron with the nucleus. They are automatically taken into account in the quantum theory of deuteron reactions.

13. Collision of a Nucleon with a Nucleus

It was already noted above (§ 4) that, because of the increase in the dimensions of the essential region, it may turn out to be impermissible to regard the collision, accompanied by meson generation, of a fast nucleon or meson that has entered the nucleus with the nucleons of the nucleus as a process of successive collisions.^10 In more detailed estimates the situation is complicated by the fact that the nucleon—nucleon energy transferred in the collision may not be considered small, and the target nucleon cannot be considered at rest.

Indeed, an incident nucleon of energy \(E_0\), generating, for example, a \(\pi\)-meson with energy \(E_\pi\) and remaining with energy \(E\), at the same time transfers a certain energy \(W=E_0-E-E_\pi\) to the recoil nucleon, i.e. the momentum received by the incident nucleon and by the meson field, if the emission angle is zero, is equal to

\[ q_{\parallel} = \frac{1}{c}\sqrt{E_0^2-M^2c^4} - \frac{1}{c}\sqrt{E^2-M^2c^4} - \]

\[ -\frac{1}{c}\sqrt{E_\pi^2-\mu^2c^4} \simeq \frac{E_0-E-E_\pi}{c} + \frac{M^2c^3}{2E_0EE_\pi} \left[ E_\pi(E_0-E)+\frac{\mu^2}{M^2}E_0E \right]. \]

For \(E_0\sim E\sim E_\pi\), and also for \(E_\pi\ll E_0\), we have

\[ q_{\parallel}\sim \frac{W}{c}+\frac{Mc^2}{2E_0}\cdot Mc . \]

Thus, if the recoil energy transferred to the nucleon does not exceed \(\mu c^2\sim 140\) MeV, then for \(E_0>\dfrac{M}{\mu}Mc^2\) we have \(\dfrac{\hbar}{q_{\parallel}}>\dfrac{\hbar}{\mu c}=r_\mu\), i.e. it cannot be assumed that the collision occurs with an individual nucleon of the nucleus. At large emission angles, \(\vartheta\sim\sqrt{\dfrac{Mc^2}{E}}\), which corresponds to isotropy in the center-of-mass system, the picture of individual collisions is preserved, whereas at angles of order \(\dfrac{M}{E_0}\) it is inadmissible (of course, if \(W<\mu c^2\)). At a very small recoil energy and small emission angles, the collision takes place at once with the whole tube. Apparently, there do indeed exist experimental indications^30 testifying against ideas of successive collisions inside the nucleus not only at high energies, where the Fermi–Landau theory is valid, but also at energies of the order of \(10^{10}\)–\(10^{11}\) eV. This can be reconciled with data showing that cosmic-ray nucleons, colliding with the nuclei of atoms in air, transfer to recoil nucleons (on average, along the path of a nucleon in the nucleus, three particles are encountered) an energy of the order of 400 MeV, i.e. about 130 MeV per recoil nucleon.^31 If these data are confirmed, then it will indeed be necessary to consider that the picture of successive collisions in the nucleus is not always applicable.

It is easy to see that the final formulas for processes involving nuclear forces (Chs. II, IV) contain certain estimates (the thickness of the transition region, etc.), because of which the numerical coefficients may prove to be inaccurate.

CONCLUSION

The ten different effects analyzed above not only illustrate the peculiarities of diffraction processes in the interaction of particles, but also show the variety of possible phenomena in which diffraction and, in general, typically wave-like features of particle behavior may manifest themselves. The processes studied so far have, as a rule, involved high energies. However, diffraction splitting of the deuteron already occurs at energies that cannot be regarded as large. In any case, here the particles are by no means relativistic. It is not impossible that in the future similar examples will be found in the region of still lower energies.

The role of the wave properties of particles, revealed in the study of all these processes, shows once again how cautiously one must use classical models in which localization of particles is assumed, even if the wavelengths of the particles are extremely small. A closely related example here is the question of the interaction of particles of ultrahigh energy, leading to multiple production of particles[^33].

It was natural to attempt[^34],[^35] to consider collisions with a large impact parameter by methods just as classical as those used for central collisions. Indeed, for large impact parameters of two nucleons, the interaction cannot, during the time of collision, spread over the entire region occupied by the nucleons[^36]. Therefore one might have thought that separate “overlapping” parts of the meson fields of different nucleons, regarded as nuclear matter distributed in space, could independently enter into local interaction with one another.

However, it turns out[^36],[^37] that such a treatment is inadmissible, being too “classical.” The quantum properties of the meson field completely change the result. This conclusion follows both from the consideration of a concrete quantum scheme for the structure of the meson field[^36], and, in a more general form, from the uncertainty relation for time and energy[^37],[^38] (in much the same way as the dimensions of the region essential for bremsstrahlung are obtained in § 2 both from a concrete scheme of the process and from the uncertainty relation for coordinate and momentum).

The processes considered in the present review have so far not found experimental confirmation. In some cases this is due to the fact that such phenomena have not been looked for,

since their possibility was established only quite recently. In other cases the necessary energies are still unattainable (although they may become attainable in the near future). In some cases, however, it is not so easy to distinguish these phenomena from ordinary ones. It appears, nevertheless, that they deserve attention.

Addendum

1. Let bremsstrahlung of an electron on a bare nucleus take place. If the electron is described by a plane wave, then the cross section of the process in which the nucleus receives momentum \(\mathbf{q}\) is proportional to the square of the matrix element[^19]:

\[ |M_c|^2 \sim \left|\int e^{\frac{i}{\hbar}\mathbf{q}\mathbf{r}}\frac{Ze^3}{r}\,d\mathbf{r}\right|^2 = \left|\frac{4\pi Ze^3\hbar^2}{q^2}\right|^2 \sim \frac{1}{q^4}. \tag{A.1} \]

Moreover, the cross section contains factors of the type \((E_0-p_0\cos\theta)^{-2}\), etc.

If the incident electron is described by a packet with transverse dimensions \(\Delta r_\perp\sim \dfrac{\hbar}{q_\perp}\) and longitudinal size of order \(\Delta r_\parallel\sim \dfrac{\hbar}{q_\parallel}\) (as we know, \(q_\parallel\ll q_\perp\)), passing at a distance \(\bar r_\perp\) from the nucleus, then, when the transition matrix element is calculated, the uncertainties of the momenta \(\mathbf{p}_0\), \(\mathbf{p}\), and so on will change the factors of the type \((E_0-p_0\cos\theta)^{-1}\) comparatively little, but will change the integral (A.1), which must be taken over \(r_\perp\) within the limits of a small area of size \((\Delta r_\perp)^2\), and over \(r_\parallel\equiv z\) within the limits from \(-\dfrac{\hbar}{q_\parallel}\) to \(+\dfrac{\hbar}{q_\parallel}\):

\[ |M'_c|^2 = \left| \int_{(\Delta r_\perp)^2} e^{\frac{i}{\hbar}\mathbf{q}_\perp\mathbf{r}_\perp}\,d\mathbf{r}_\perp \int_{-\hbar/q_\parallel}^{+\hbar/q_\parallel} \frac{Ze^3 e^{\frac{i}{\hbar}q_\parallel z}}{\sqrt{r_\perp^2+z^2}}\,dz \right|^2 . \tag{A.2} \]

Taking \(\Delta r_\perp=\dfrac{\hbar}{q_\perp}\ll \bar r_\perp\), we may approximately replace the exponential factors by unity, and \(r_\perp\) by \(\bar r_\perp\), so that the integral over \(r_\perp\) simply gives \((\Delta r_\perp)^2\). Therefore

\[ |M'_c(\bar r_\perp)|^2 \sim \left|\frac{4\pi Ze^3\hbar^2}{q_\perp^2}\right|^2 \left\{ \ln \frac{ \sqrt{\dfrac{\bar r_\perp^{\,2}q_\parallel^2}{\hbar^2}+1}+1 }{ \bar r_\perp q_\parallel/\hbar } \right\}^2 . \tag{A.3} \]

Consequently, with the packet width chosen here, we obtain the correct dependence on \(q_\perp\simeq q\). The normalization volume \(V\left(=\dfrac{\hbar^3}{q_\parallel q_\perp^2}\right)\), as always, drops out of the result, since

its dimensions are large in comparison with the wavelength. The probability of the process, therefore, depends only very weakly on the impact parameter \(\bar r_\perp\) and on \(q_\parallel\). Averaging over all possible parameters from \(\bar r_{\perp\min}=\Delta r_\perp\) to \(\bar r_\perp=\dfrac{\hbar}{q_\parallel}\), instead of the square of the logarithm we obtain a numerical factor very close to unity. For large parameters \(\left(\bar r_\perp>\dfrac{\hbar}{q_\parallel}\right)\) the logarithmic factor, and consequently also the cross section, decreases rapidly.

1. In the case of a nucleus screened by an electron shell, if \(\dfrac{\hbar}{q_\parallel}\) is greater than the radius of the shell (the principal case), then, first, in integration over \(z\) the limits will be not \(\pm \dfrac{\hbar}{q_\parallel}\), but \(\pm b\), where \(b=a_0 Z^{-1/3}\) is the radius of the electron shell; second, in averaging over impact parameters one must take \(\Delta r_\perp<\bar r_\perp<b\), and consequently the integration over \(z\) again leads only to a factor close to unity; third, in order that the packet enter inside the atom, it is necessary that \(\Delta r_\perp<b\), i.e., \(q_\perp\) must be not less than \(\dfrac{\hbar}{b}\). Upon integrating the probability of the process over the scattering angles this will lead to the usual logarithmic factor.

2. If along the line of the initial motion of the electron there is a chain of atoms, then the potential \(\dfrac{Ze^3}{r}\) should be replaced by \(\sum_i \dfrac{Ze^3}{|\mathbf r-\mathbf r_i|}\), where \(|\mathbf r-\mathbf r_i|\) is the distance to the \(i\)-th nucleus. When \(\dfrac{\hbar}{q_\parallel}\) exceeds the distance between nuclei, the integral over \(z\) splits into a sum of identical integrals, the number of which \(\nu\) is equal to the number of atoms on the segment \(\dfrac{\hbar}{q_\parallel}\), on which the oscillating factor can be replaced by its effective value, \(\nu\sim\dfrac{\hbar}{q_\parallel a}\), where \(a\) is the distance between atoms. Thus, integration over \(z\) gives a factor that grows rapidly as \(q_\parallel\) decreases. This leads to the interference effects considered in work \(^{2}\).

3. If interference effects are absent (an amorphous medium), then the matrix element (A. 1) is not changed in the presence of the medium. It is then necessary to take into account the weaker influence of the medium on the second factor in the matrix element of the second approximation of perturbation theory, which describes the emission of a photon by the electron and gives in the cross section factors of the type \((E_0-p_0\cos\vartheta)^{-2}\), etc. Here the wave properties of light manifest themselves. The influence of the medium consists in multiple scattering of the electron, which disrupts the process of bremsstrahlung radiation \(^{3}\).

References

1. L. D. Landau and I. Ya. Pomeranchuk, JETP 24, 505 (1953).
2. M. L. Ter-Mikaelyan, JETP 25, 289, 296 (1953).
3. L. D. Landau and I. Ya. Pomeranchuk, DAN 92, 535, 735 (1953).
4. I. Ya. Pomeranchuk and E. L. Feinberg, DAN 93, 439 (1953).
5. A. I. Akhiezer, DAN 94, 651 (1954); A. I. Akhiezer and I. Ya. Pomeranchuk, DAN 94, 821 (1954).
6. M. L. Ter-Mikaelyan, DAN 94, 1033 (1954).
7. A. B. Migdal, DAN 96, 49 (1954); 105, 77 (1955).
8. I. Ya. Pomeranchuk, DAN 96, 265 (1954).
9. I. Ya. Pomeranchuk, DAN 96, 481 (1954).
10. E. L. Feinberg, JETP 28, 242 (1955).
11. E. L. Feinberg, JETP 29, 115 (1955).
12. A. I. Aliev and E. L. Feinberg, JETP 30, 115 (1956).
13. A. I. Akhiezer and A. G. Sitenko, Scientific Notes of Kharkov State University 64, 9 (1955).
14. Yu. A. Vdovin, Some Processes of Interaction of High-Energy Quanta with π-Mesons and Nucleons. Author’s abstract of dissertation, MEPhI Press, Moscow, 1955.
15. Yu. A. Vdovin, DAN 105, 947 (1955).
16. E. Ya. Rabinovich, diploma thesis, MEPhI, 1953.
17. R. Serber, Phys. Rev. 72, 1114 (1947).
18. A. I. Akhiezer and I. Ya. Pomeranchuk, Some Problems in the Theory of the Nucleus, Gostekhizdat, Moscow, 1950, § 14.
19. V. Heitler, The Quantum Theory of Radiation, Gostekhizdat, 1940, § 17.
20. H. Bethe, W. Heitler, Proc. Roy. Soc. 146, 83 (1934).
21. A. I. Akhiezer and I. Ya. Pomeranchuk, Some Problems in the Theory of the Nucleus, Gostekhizdat, Moscow, 1950; see §§ 18, 20.
22. V. I. Goldanskii, A. I. Lyubimov, B. V. Medvedev, UFN 48, 531 (1952).
23. Cosmic-Ray Physics, ed. J. Wilson, IL, 1954.
24. M. Born, Optics, ONTI, Kharkov, 1937, §§ 46, 47.
25. A. Sommerfeld, Atombau u. Spektrallinien, 1939, Ch. VI, § 4 and Ch. VII, § 2.
26. See, e.g., Ya. L. Alpert, V. L. Ginzburg, E. L. Feinberg, Propagation of Radio Waves, Gostekhizdat, Moscow, 1953, § 65.
27. E. J. Williams, Kgl. Danske Vid. Selsk. Mat.-fys. Med. 13, 276 (1935).
28. B. Feretti, Nuovo Cimento 7, 118 (1950).
29. G. P. Millburn, W. Birnbaum, W. E. Crandall, L. Schecter, Phys. Rev. 95, 1268 (1954); W. Heckrote, Phys. Rev. 95, 1279 (1954);
30. I. L. Rozental and D. S. Chernavskii, UFN 52, 185 (1954); 55, 121 (1955).
31. S. N. Vernov, N. L. Grigorov, G. T. Zatsepin, A. E. Chudakov, Izv. Acad. Sci. USSR, Phys. Ser. 19, No. 4 (1955).
32. S. I. Drozdov, JETP 28, 734, 736 (1955).
33. E. Fermi, Progr. Theor. Phys. 5, 570 (1950).
34. W. Heisenberg, Zeits. f. Phys. 133, 65 (1952).
35. H. J. Bhabha, Proc. Roy. Soc. 219, 293 (1953).
36. E. L. Feinberg and D. S. Chernavskii, DAN 81, 795 (1951).
37. E. L. Feinberg and D. S. Chernavskii, DAN 91, 511 (1953).
38. L. D. Landau, Izv. Acad. Sci. USSR, Phys. Ser. 17, 51 (1953).