THEORY OF THE VAVILOV–CHERENKOV EFFECT

B. M. Bolotovskii

CONTENTS

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

I. The Vavilov–Cherenkov Effect in an Isotropic Medium

I.1. Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
I.2. Dielectric constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
I.3. Qualitative discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
I.4. Field of a moving point charged particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
I.5. Energy losses of a charged particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
I.6. Vavilov–Cherenkov radiation in a medium without dispersion . . . . . . . . . . . . . . . . . . . . 212
I.7. Longitudinal and transverse fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
I.8. Contribution of the losses caused by the Vavilov–Cherenkov effect to the total energy losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
I.9. Radiation field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
I.10. Duration of the radiation flash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
I.11. Radiation of a conductor carrying a current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
I.12. Reversal of the Vavilov–Cherenkov effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
I.13. Interference of Vavilov–Cherenkov radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
I.14. Quantum theory of the Vavilov–Cherenkov effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

II. The Vavilov–Cherenkov Effect in Crystals

II.1. Constitutive equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
II.2. Equations for the potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
II.3. Qualitative discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
II.4. Polarization waves of Vavilov–Cherenkov radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
II.5. Field of a moving point charged particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
II.6. Vavilov–Cherenkov effect in a uniaxial crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
a) The charge moves parallel to the optical axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
b) The charge moves perpendicular to the optical axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
II.7. Phase and group velocities of Vavilov–Cherenkov waves in a crystal . . . . . . . . . . . . . 235
II.8. The Vavilov–Cherenkov effect in an isotropic optically active medium . . . . . . . . . . . . 236
II.9. The Vavilov–Cherenkov effect in a gyrotropic crystal . . . . . . . . . . . . . . . . . . . . . . . . . . 237
II.10. The Vavilov–Cherenkov effect in an electron plasma placed in a magnetic field . . . . 239

Cited literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

INTRODUCTION

In 1934, papers by P. A. Cherenkov³⁴ and S. I. Vavilov¹⁸⁹ appeared, devoted to the visible glow of pure liquids under the action of γ-radiation. S. I. Vavilov pointed out that this glow could not be luminescence and suggested that the radiation is caused by the deceleration of electrons knocked out by γ-rays. Three years later, I. E. Tamm and I. M. Frank¹⁸¹ gave a theory of this phenomenon, showing that the source of the glow is electrons moving with a velocity exceeding the phase velocity of light in the medium.

Since the appearance of the work of I. E. Tamm and I. M. Frank, 20 years have passed. During this time the Cherenkov effect, or the Vavilov—Cherenkov effect, has been the subject of many theoretical, experimental, and applied investigations. Among the applied works, we should first of all note the use of Vavilov—Cherenkov radiation for the registration of fast charged particles (the technique of Vavilov—Cherenkov counters) and for the generation of microwaves. Vavilov—Cherenkov counters possess a whole series of advantages in comparison with other known detectors of charged particles (short resolving time, directionality of the radiation, etc.). How widespread such instruments as Vavilov—Cherenkov counters now are can be seen from the fact that the discovery of the antiproton was connected with their use.

The generation of radio waves by means of the Vavilov—Cherenkov effect is still a subject of laboratory investigation. The generation of radio waves as a result of the Vavilov—Cherenkov effect may, in principle, also occur in the ionized atmosphere surrounding the Sun and stars.

Many other possible applications of the Vavilov—Cherenkov effect are also being studied (a standard light source of low intensity, mechanisms for accelerating charged particles based on the reversal of the Vavilov—Cherenkov effect \(^{91}\), etc.).

Ten years ago a review by I. M. Frank on the Vavilov—Cherenkov effect \(^{62}\) was published in UFN. This review has retained its significance up to the present time. However, the numerous investigations that have appeared since then are almost not covered in the review literature. Jelley’s review, published in UFN \(^{106}\), gives a sufficiently complete picture of the existing experimental applications of the Vavilov—Cherenkov effect, but in this review little space is devoted to the exposition of the theory underlying the applications.

The present review sets itself the task of acquainting the reader with the contemporary state of the theory of the Vavilov—Cherenkov effect. The review is divided into two parts—the Vavilov—Cherenkov effect in an unbounded medium and the Vavilov—Cherenkov effect in the presence of boundaries. The author has not attempted to cover all aspects of the phenomenon under discussion. The Vavilov—Cherenkov effect in the atmosphere \(^{7,8,11,23,67,68,85,103,104,160,202}\), in ferromagnets \(^{94,95,97,100,101,153,168,169}\), the influence of multiple scattering on the angular width of Vavilov—Cherenkov radiation \(^{51,144}\), and certain other questions are not considered at all. The duration of the flash in the Vavilov—Cherenkov effect \(^{65,183}\), the reversal of the Vavilov—Cherenkov effect \(^{183,191}\), and certain other questions are discussed very briefly. All these questions may be studied in greater detail from the literature appended to the review*).

The bibliography contains only works directly connected with the Vavilov—Cherenkov effect. A few references to auxiliary literature are given in the text.

Before reading the second chapter it is desirable to become acquainted with the fundamentals of crystal optics, at least to the extent of the corresponding chapter of Sommerfeld’s book Optics \(^{178}\).

I. THE VAVILOV—CHERENKOV EFFECT IN AN ISOTROPIC MEDIUM

1.1. Maxwell’s equations.

Let an unbounded isotropic medium with dielectric constant \(\varepsilon\) and magnetic permeability \(\mu\) be traversed (the operator character of the quantities \(\varepsilon\) and \(\mu\) will be considered in more detail below)

*) The author is grateful to J. V. Jelley, who sent a new list of works on the Vavilov—Cherenkov effect carried out abroad.

moves uniformly and rectilinearly. We shall denote the velocity of the particle in the coordinate system in which the dielectric is at rest by \(\mathbf v\), and the charge of the particle by \(q\). The problem is to determine the fields—the electric field \(\mathbf E\) and the magnetic field \(\mathbf H\)—which arise in the medium during the motion of the charged particle. Let us write Maxwell’s equations:

\[ \left. \begin{aligned} \operatorname{rot}\mathbf E &= -\frac{1}{c}\frac{\partial \mathbf B}{\partial t},\\ \operatorname{rot}\mathbf H &= \frac{1}{c}\frac{\partial \mathbf D}{\partial t}+\frac{4\pi}{c}\mathbf j,\\ \operatorname{div}\mathbf B &= 0,\qquad &\mathbf D&=\varepsilon\mathbf E,\\ \operatorname{div}\mathbf D &= 4\pi\rho,\qquad &\mathbf B&=\mu\mathbf H. \end{aligned} \right\} \tag{1.1} \]

In these equations \(\rho\) and \(\mathbf j\) denote the charge density and the current density due to the moving point charge. If at the time \(t\) the particle is at the point \(\mathbf x=\mathbf v t\), then \(\rho(\mathbf x,t)\) and \(\mathbf j(\mathbf x,t)\) are expressed in terms of the delta function of the argument \(\mathbf x-\mathbf v t\):

\[ \left. \begin{aligned} \rho &= q\,\delta(\mathbf x-\mathbf v t),\\ \mathbf j &= q\mathbf v\,\delta(\mathbf x-\mathbf v t). \end{aligned} \right\} \tag{1.2} \]

It is convenient to pass, as is always done, from the fields \(\mathbf E\) and \(\mathbf H\) to the potentials \(\mathbf A\) and \(\varphi\), by means of which \(\mathbf E\) and \(\mathbf H\) are defined as follows:

\[ \left. \begin{aligned} \mathbf E&=-\operatorname{grad}\varphi-\frac{1}{c}\frac{\partial\mathbf A}{\partial t},\\ \mathbf H&=\frac{1}{\mu}\operatorname{rot}\mathbf A. \end{aligned} \right\} \tag{1.3} \]

Then from the equations for \(\mathbf E\) and \(\mathbf H\) one can pass to equations for the vector potential \(\mathbf A\) and the scalar potential \(\varphi\):

\[ \left. \begin{aligned} \Delta\mathbf A-\frac{\varepsilon\mu}{c^2}\frac{\partial^2\mathbf A}{\partial t^2} &=-\frac{4\pi\mu}{c}\mathbf j +\operatorname{grad}\left(\frac{\varepsilon\mu}{c}\frac{\partial\varphi}{\partial t} +\operatorname{div}\mathbf A\right),\\ \varepsilon\left(\Delta\varphi+\frac{1}{c}\frac{\partial}{\partial t}\operatorname{div}\mathbf A\right) &=-4\pi\rho. \end{aligned} \right\} \tag{1.4} \]

As is known, equations (1.4) do not determine \(\mathbf A\) and \(\varphi\) uniquely; therefore one may impose on \(\mathbf A\) and \(\varphi\) some additional condition, choosing it so as to simplify the system (1.4). If the condition

\[ \operatorname{div}\mathbf A+\frac{\varepsilon\mu}{c}\frac{\partial\varphi}{\partial t}=0, \tag{1.5} \]

is imposed on the potentials, the system of equations (1.4) takes the following symmetric form:

\[ \left. \begin{aligned} \frac{1}{\mu}\left(\Delta\mathbf A-\frac{\varepsilon\mu}{c^2}\frac{\partial^2\mathbf A}{\partial t^2}\right) &=-\frac{4\pi}{c}q\mathbf v\,\delta(\mathbf x-\mathbf v t),\\ \varepsilon\left(\Delta\varphi-\frac{\varepsilon\mu}{c^2}\frac{\partial^2\varphi}{\partial t^2}\right) &=-4\pi q\,\delta(\mathbf x-\mathbf v t). \end{aligned} \right\} \tag{1.6} \]

These equations for the potentials of the field created by a point particle are most often the starting point for the consideration of various problems connected both with Vavilov–Cherenkov radiation and, in general, with the passage of charged particles through matter.

However, in some cases (examples will be discussed below) it turns out to be more convenient to impose on the potentials, instead of the supplementary condition (1.5), the condition

\[ \operatorname{div}\mathbf A=0. \tag{1.7} \]

When this condition is satisfied, the system of equations (1.4) for \(\mathbf A\) and \(\varphi\) takes the form

\[ \left. \begin{aligned} \varepsilon \Delta \varphi &=-4\pi \hat{\rho}(\mathbf x-\mathbf v t),\\ \frac{1}{\mu}\left(\Delta \mathbf A-\frac{\varepsilon\mu}{c^2}\frac{\partial^2 \mathbf A}{\partial t^2}\right) &=-\frac{4\pi}{c}\,q\mathbf v\,\hat{\delta}(\mathbf x-\mathbf v t) +\frac{\varepsilon}{c}\operatorname{grad}\frac{\partial\varphi}{\partial t}. \end{aligned} \right\} \tag{1.8} \]

Of course, both systems of equations, (1.6) and (1.8), give the same expressions for the fields \(\mathbf E\) and \(\mathbf H\), and from this point of view it is immaterial which of them is solved. Therefore, if it is necessary to determine only \(\mathbf E\) and \(\mathbf H\), it is more convenient to use the system of equations (1.6), since in this case

\[ \mathbf A=\varepsilon\mu \frac{\mathbf v}{c}\,\varphi \tag{1.9} \]

and the matter reduces to the solution of only one equation of the system (1.6). However, it is often useful to know not the complete electromagnetic field, but only its transverse part, \(\mathbf E^{\mathrm{tr}}, \mathbf H\), satisfying the relation \(\operatorname{div}\mathbf E^{\mathrm{tr}}=0\). This part of the field describes the radiation of a charge passing through a dielectric. When it is necessary to determine the transverse field, it is more convenient to use the system of equations (1.8), since the vector potential \(\mathbf A\), satisfying the system of equations (1.8), immediately gives the transverse field

\[ \left. \begin{aligned} \mathbf E^{\mathrm{tr}}&=-\frac{1}{c}\frac{\partial \mathbf A}{\partial t},\\ \mathbf H&=\frac{1}{\mu}\operatorname{rot}\mathbf A. \end{aligned} \right\} \tag{1.10} \]

In what follows we shall use both the one and the other gauge of the potentials.

1.2. Dielectric constant. Let us now dwell on the meaning of the quantities \(\varepsilon\) and \(\mu\)—the dielectric constant and magnetic permeability entering into the field equations. If the medium in which the particle moves has no dispersion, i.e., waves of all frequencies propagate in it in the same way, then \(\varepsilon\) and \(\mu\) may simply be regarded as numbers. In this case

\[ \left. \begin{aligned} \mathbf D&=\varepsilon \mathbf E,\\ \mathbf B&=\mu \mathbf H, \end{aligned} \right\} \tag{1.11} \]

where \(\mathbf D\) and \(\mathbf B\) are the electric and magnetic inductions. If, however, the medium is dispersive, then \(\varepsilon\) and \(\mu\) have different values depending on the frequency of the electromagnetic wave propagating in the medium. Therefore, in order to write the relation, say, between \(\mathbf D\) and \(\mathbf E\), both of these quantities must be expanded in Fourier integrals with respect to time:

\[ \begin{aligned} \mathbf E(\mathbf x,t)&=\int \mathbf E_\omega(\mathbf x)e^{i\omega t}\,d\omega;\\ \mathbf D(\mathbf x,t)&=\int \mathbf D_\omega(\mathbf x)e^{i\omega t}\,d\omega. \end{aligned} \tag{1.12} \]

The relation between \(\mathbf D_\omega(\mathbf x)\) and \(\mathbf E_\omega(\mathbf x)\) turns out to be simple:

\[ \mathbf D_\omega=\varepsilon(\omega)\mathbf E_\omega, \]

where \(\varepsilon(\omega)\) is the dielectric constant for the frequency \(\omega\). In a dispersive medium the expression \(\varepsilon\mathbf E(\mathbf x,t)\) has the meaning of an integral over all frequencies:

\[ \varepsilon\mathbf E=\int \varepsilon(\omega)\mathbf E_\omega e^{i\omega t}\,d\omega . \tag{I.13} \]

By the formulas of operator calculus, \(\varepsilon(\omega)\) may be taken outside the integral sign, while at the same time making the formal replacement of \(\omega\) by \(-i\,\dfrac{\partial}{\partial t}\). Then from (I.13) we obtain

\[ \varepsilon=\varepsilon\left(-i\,\frac{\partial}{\partial t}\right). \tag{I.14} \]

Thus, the dielectric constant \(\varepsilon\) in the Fourier representation is a function of frequency, while in the coordinate representation it is an operator depending on differentiation with respect to time. Everything that has been said also applies to \(\mu(\omega)\).

I.3. Qualitative discussion. In this section we shall consider certain qualitative aspects of the phenomenon of the passage of a particle through a medium. For this purpose it is convenient to use the equations for the field potentials in the form (I.6).

If a charge moving uniformly in an unbounded homogeneous medium radiates, then the radiation field must be stationary with respect to the charge, i.e. it must move with the velocity of the charge. Let us see under what conditions the system of equations (I.6) for the field potentials can have as a solution a plane electromagnetic wave

\[ e^{i\mathbf k(\mathbf x-\mathbf v t)}, \tag{I.15} \]

for which the surfaces of constant phase are planes perpendicular to the wave vector \(\mathbf k\). These planes move in space with velocity \(\mathbf v\), equal to the velocity of the moving charge (Fig. 1).

Fig. 1.

Let us note that at all points, except the point \(\mathbf X=\mathbf v t\), where the moving charge is located, the system of equations (I.6) is homogeneous:

\[ \left. \begin{aligned} \frac{1}{\mu}\left(\Delta\mathbf A-\frac{\varepsilon\mu}{c^2}\frac{\partial^2\mathbf A}{\partial t^2}\right)&=0,\\ \varepsilon\left(\Delta\varphi-\frac{\varepsilon\mu}{c^2}\frac{\partial^2\varphi}{\partial t^2}\right)&=0. \end{aligned} \right\} \tag{I.16} \]

Substituting the plane wave (I.15) into the system (I.16), we find:

\[ \left. \begin{aligned} \frac{1}{\mu(\mathbf k\mathbf v)} \left[k^2-\varepsilon(\mathbf k\mathbf v)\mu(\mathbf k\mathbf v)\frac{(\mathbf k\mathbf v)^2}{c^2}\right]&=0,\\ \varepsilon(\mathbf k\mathbf v) \left[k^2-\varepsilon(\mathbf k\mathbf v)\mu(\mathbf k\mathbf v)\frac{(\mathbf k\mathbf v)^2}{c^2}\right]&=0. \end{aligned} \right\} \tag{I.17} \]

In these equations \(\varepsilon\) and \(\mu\) depend on the frequency \(\mathbf k\mathbf v\) of the plane wave (I.15). From equations (I.17) it follows that, for a given velocity \(\mathbf v\), two types of waves (I.15) can propagate in the medium, with two different relations between \(\mathbf k\) and \(\mathbf v\):

\[ k^2=\frac{\varepsilon\mu}{c^2}(\mathbf k\mathbf v)^2. \tag{I.18a} \]

II

\[ \varepsilon(\mathbf{k}\mathbf{v})=0. \tag{I.18б} \]

Electromagnetic waves satisfying condition (I.18a) are called Vavilov—Cherenkov waves. Condition (I.18a) can be given a simple form by using the fact that \(\mathbf{k}\mathbf{v}=kv\cos\vartheta\) (Fig. 1):

\[ \left\{ \begin{aligned} \cos^2\vartheta&=\frac{1}{\varepsilon_\mu\beta^2},\\ \cos\vartheta&=\pm\frac{1}{\sqrt{\varepsilon_\mu}\,\beta} =\pm\frac{v_\phi}{v}, \end{aligned} \right. \tag{I.19} \]

where \(v_\phi=\dfrac{c}{\sqrt{\varepsilon_\mu}}=\dfrac{c}{n}\) is the phase velocity of the electromagnetic wave in the medium. Of the two signs in (I.19), the plus sign should be chosen, since the wave vector of the propagating wave must have a positive component in the direction of the velocity.

Thus, the projection of the wave vector onto the particle velocity, or, what is the same, the quantity \(\cos\vartheta\), is determined. It remains to determine the component of the wave vector perpendicular to the velocity \(\mathbf{v}\). For the square of this component, \(k_r^2\), the following value is obtained:

\[ k_r^2=k^2\sin^2\vartheta. \]

Taking formulas (I.18a) and (I.19) into account gives

\[ k_r^2=\frac{(\mathbf{k}\mathbf{v})^2}{v^2}\left(\varepsilon_\mu\beta^2-1\right). \]

Hence \(k_r\) is determined up to a sign. The sign should be chosen so that the flux of electromagnetic energy is directed away from the particle path (see Section II.7).

Fig. 2.

Condition (I.19) determines a real Cherenkov wave only in the case when

\[ \frac{1}{n\beta}=\frac{v_\phi}{v}<1, \tag{I.20} \]

i.e., when the phase velocity of the electromagnetic wave is less than the velocity of the charged particle.

Condition (I.19) can be obtained from simple physical considerations, as was done by I. M. Frank. Suppose that at each point of its path an electron emits spherical waves propagating with velocity \(v_\phi=\dfrac{c}{n}\) (Fig. 2). The interference of these waves will lead to the formation of a wave propagating at an angle \(\vartheta\) to the direction of motion of the charge, with

\[ \cos\vartheta=\frac{1}{n\beta}. \]

This is precisely condition (I.19).

The phase velocity of Vavilov—Cherenkov waves is

\[ v_\phi=\frac{\omega}{k}=\frac{\mathbf{k}\mathbf{v}}{k}=v\cos\vartheta=\frac{c}{n}, \tag{I.21} \]

i.e., the same as for all transverse electromagnetic waves.

As follows from (I.20), the phase velocity of Vavilov—Cherenkov waves is always less than the velocity of the charged particle. The group velocity of the waves

Vavilov—Cherenkov

\[ v_g=\frac{\partial \omega}{\partial k}=\frac{\mathbf{k}}{k}\,\frac{2cn}{2n^2+\omega\frac{dn^2}{d\omega}}. \tag{I.22} \]

In the region of normal dispersion \(\left(\dfrac{dn^2}{d\omega}>0\right)\), the group velocity of Vavilov—Cherenkov waves is always less than the phase velocity.

Condition (I.18b) also describes waves that can be emitted by a charge moving in a medium uniformly and rectilinearly. These waves differ in many respects from Vavilov—Cherenkov waves. The condition for Vavilov—Cherenkov radiation (I.20) may be satisfied in a certain region of continuous variation of the frequency \(\omega=\mathbf{k}\mathbf{v}\). This means that Vavilov—Cherenkov radiation in an unbounded medium has a continuous spectrum. By contrast, condition (I.18b) is satisfied, generally speaking, for discrete fixed values \(\omega_\alpha=(\mathbf{k}\mathbf{v})_\alpha\), where \(\alpha\) is the number of the root of equation (I.18b).

The phase velocity of waves satisfying condition (I.18b) is determined by the equality

\[ v_\phi=\frac{\omega_\alpha}{k}=\frac{c}{\sqrt{\varepsilon(\omega_\alpha)\mu(\omega_\alpha)}}=\infty . \tag{I.23} \]

The group velocity of these waves is equal to zero. This means that the energy expended by the moving charge on exciting such waves in some volume element of the medium is not transported by the excited waves. Unlike Vavilov—Cherenkov waves, the energy of the oscillations is, as it were, “stored” in the medium, and the resulting local oscillations are analogous to the longitudinal oscillations of an electron plasma with frequency

\[ \omega_0=\sqrt{\frac{4\pi ne^2}{m}} \tag{I.24} \]

(\(n\) is the number of electrons per unit volume, and \(e\) and \(m\) are the charge and mass of the electron).

Waves satisfying condition (I.18b) are called polarization waves. Generally speaking, these waves may also be damped, since condition (I.18b) imposes restrictions only on the component of the wave vector parallel to the velocity.

Thus, a charged particle moving uniformly and rectilinearly in a substance can excite waves of two types: Vavilov—Cherenkov waves and polarization waves. Polarization waves are sometimes also called Borovsky waves. Below we shall show that Vavilov—Cherenkov waves are transverse, whereas Borovsky waves are longitudinal.*)

1.4. Field of a moving point charged particle. We now turn to finding the field of a moving charged particle. Below we shall proceed from the system of equations (I.6) for the potentials of the electromagnetic field. A solution will also be given of the system of equations (I.8), which immediately gives the separation of the fields into longitudinal and transverse ones.

In view of relation (I.9), it suffices to solve one of the two equations of system (I.6), say the first equation.

) See also V. L. Ginzburg, Theory of the Propagation of Radio Waves in the Ionosphere*, Gostekhizdat, 1948.

Represent \(\delta(\mathbf{x}-\mathbf{v}t)\) in the form

\[ \delta(\mathbf{x}-\mathbf{v}t)=\frac{1}{(2\pi)^3}\int e^{i\mathbf{k}(\mathbf{x}-\mathbf{v}t)}\,d\mathbf{k}. \tag{I.25} \]

Solving the first equation of the system (I.6) by “division” by the d’Alembert operator immediately gives an expression for the vector potential \(\mathbf{A}\):

\[ \mathbf{A}=\frac{q}{2\pi^2}\frac{\mathbf{v}}{c}\int \frac{\mu(\mathbf{k}\mathbf{v})\,e^{i\mathbf{k}(\mathbf{x}-\mathbf{v}t)}}{ k^2-\dfrac{\varepsilon\mu}{c^2}(\mathbf{k}\mathbf{v})^2}\,d\mathbf{k}. \tag{I.26} \]

If \(\varepsilon(\mathbf{k}\mathbf{v})\) and \(\mu(\mathbf{k}\mathbf{v})\) are complex quantities, i.e. if the medium absorbs electromagnetic waves, then the evaluation of this integral is a single-valued operation. If, however, the medium is transparent to electromagnetic waves (which corresponds to real functions \(\varepsilon\) and \(\mu\)), integration in (I.26) becomes ambiguous, since in this case the integrand contains poles on the path of integration, corresponding to the fulfillment of one of the two conditions (I.18). For the integral (I.26) poles occur when condition (I.18a) is fulfilled; for the integral

\[ \varphi=\frac{q}{2\pi^2}\int \frac{e^{i\mathbf{k}(\mathbf{x}-\mathbf{v}t)}\,d\mathbf{k}}{ \varepsilon(\mathbf{k}\mathbf{v})\left[k^2-\dfrac{\varepsilon\mu}{c^2}(\mathbf{k}\mathbf{v})^2\right]} \tag{I.27} \]

—when conditions (I.18a) and (I.18b) are fulfilled.

If one restricts oneself to retarded potentials, the integration in formulas (I.26) and (I.27) can be made single-valued; for this it is necessary to take all integrals in the sense of the principal value (P.V.) and, in the vicinity of the poles, to replace the singular expressions by the following\(^{96,26,27}\)*:

\[ \begin{aligned} \frac{1}{k^2-\dfrac{\varepsilon\mu}{c^2}(\mathbf{k}\mathbf{v})^2} &= \frac{P.V.}{k^2-\dfrac{\varepsilon\mu}{c^2}(\mathbf{k}\mathbf{v})^2} -i\pi\,\frac{\mathbf{k}\mathbf{v}}{|\mathbf{k}\mathbf{v}|}\, \delta\!\left\{k^2-\frac{\varepsilon\mu}{c^2}(\mathbf{k}\mathbf{v})^2\right\}, \\[6pt] \frac{1}{\varepsilon(\mathbf{k}\mathbf{v})} &= \frac{P.V.}{\varepsilon(\mathbf{k}\mathbf{v})} -i\pi\,\frac{\mathbf{k}\mathbf{v}}{|\mathbf{k}\mathbf{v}|}\, \delta\{\varepsilon(\mathbf{k}\mathbf{v})\}. \end{aligned} \tag{I.28} \]

For what follows it is convenient to introduce a cylindrical coordinate system \(r,\varphi,z\), with the \(z\)-axis coinciding with the line of motion of the charge. At the same time we introduce the notation \(\mathbf{k}\mathbf{v}=\omega\). Integrating expression (I.26) for \(\mathbf{A}\) over the components of the wave vector \(\mathbf{k}\) perpendicular to the particle velocity \(\mathbf{v}\), and taking into account the first relation (I.28), we obtain\(^{181}\)

\[ A_z=\int_{-\infty}^{\infty} e^{\,i\frac{\omega}{v}(z-vt)}\,a(\omega,r)\,d\omega, \tag{I.29} \]

where

\[ a(\omega,r)= \begin{cases} \dfrac{q}{\pi c}\,\mu(\omega)\, K_0\!\left(\dfrac{|\omega|}{v}\sqrt{1-\varepsilon\mu\beta^2}\,r\right), & \text{for } \omega>0,\\ \text{complex conjugate}, & \text{for } \omega<0. \end{cases} \tag{I.30} \]

The function \(K_0\) is a cylindrical function of imaginary argument (a Macdonald function). This function has a logarithmic singularity at zero:

\[ K_0(x)\approx \ln\frac{1}{x}\qquad (|x|\ll 1). \tag{I.31} \]

* It is assumed that \(\varepsilon(\mathbf{k}\mathbf{v})\) has zeros of no higher than first order.

and decreases exponentially for large values of the argument:

\[ K_0(x) \simeq \sqrt{\frac{\pi}{2x}}\, e^{-x}, \qquad |x| \gg 1 . \tag{1.32} \]

In an analogous manner the expression for the scalar potential \(\varphi\) is written as:

\[ \varphi = \int_{-\infty}^{+\infty} e^{i \frac{\omega}{v}(z-vt)} \Phi(\omega,r)\, d\omega , \tag{1.33} \]

where

\[ \Phi(\omega,r)= \begin{cases} \dfrac{q}{\pi v}\,\dfrac{1}{\varepsilon(\omega)}\, K_0\!\left(-\dfrac{|\omega|}{v}\sqrt{1-\varepsilon\mu\beta^2}\, r\right), & \text{for } \omega>0,\\[1.2ex] \text{complex conjugate,} & \text{for } \omega<0. \end{cases} \tag{1.34} \]

In the case of a transparent medium the expressions for \(a(\omega,r)\) and \(\Phi(\omega,r)\) are real if \(\varepsilon\mu\beta^2<1\), i.e. in that frequency interval in which the Vavilov—Cherenkov radiation condition is not satisfied. In this case \(\mathbf A\) and \(\varphi\) are expanded in harmonics that decay exponentially as \(r\to\infty\). A different picture occurs if the Vavilov—Cherenkov radiation condition is satisfied. In this case the field components corresponding to the frequency \(\omega\) do not decay exponentially. Putting

\[ \sqrt{1-\varepsilon\mu\beta^2} = -i\sqrt{\varepsilon\mu\beta^2-1} \qquad (\varepsilon\mu\beta^2>1) \tag{1.35} \]

(the sign before the root corresponds to the choice of outgoing waves) and using formula (1.32), we find that for \(\varepsilon\mu\beta^2>1\), i.e. when the Vavilov—Cherenkov radiation condition is fulfilled, the harmonics \(a(\omega,r)\) and \(\Phi(\omega,r)\) at \(r\to\infty\) describe a conical wave

\[ a(\omega,r),\quad \Phi(\omega,r) \simeq -\frac{i}{\sqrt{k_r r}}\, e^{i\frac{\omega}{v}(z-vt)+ik_r r-i\frac{\pi}{4}}, \tag{1.36} \]

where

\[ k_r=\frac{|\omega|}{v}\sqrt{\varepsilon\mu\beta^2-1} = \frac{|\omega|}{v}\operatorname{tg}\vartheta , \tag{1.37} \]

and \(\vartheta\) is the angle made by the direction of propagation of the Vavilov—Cherenkov wave with the velocity of the particle (see (1.19)). For \(\omega<0\) the right-hand side of equality (1.36) goes over into the complex conjugate expression.

Thus, if \(\varepsilon\mu\beta^2>1\), the field consists of waves going off to infinity at an angle \(\vartheta\) to the \(z\)-axis, along which the charge moves.

Starting from expressions (1.29) and (1.33) for the potentials \(\mathbf A\) and \(\varphi\), it is not difficult to obtain formulas for the fields \(\mathbf E\) and \(\mathbf H\). Only the components \(E_z\), \(E_r\), and \(H_\varphi\) turn out to be different from zero:

\[ \left. \begin{aligned} E_z&=-\frac{i}{c}\int e^{i\frac{\omega}{v}(z-vt)} \frac{1-\varepsilon\mu\beta^2}{\varepsilon\mu\beta^2}\, a(\omega,r)\,\omega\, d\omega,\\[1.2ex] E_r&=-\int e^{i\frac{\omega}{v}(z-vt)} \frac{1}{\varepsilon\mu\beta}\, \frac{\partial a(\omega,r)}{\partial r}\, d\omega,\\[1.2ex] H_\varphi&=-\int e^{i\frac{\omega}{v}(z-vt)} \frac{\partial a(\omega,r)}{\partial r}\, d\omega, \end{aligned} \right\} \tag{1.38} \]

where \(a(\omega,r)\) is determined by equality (1.30); the integrals over \(\omega\) are taken in the limits from \(+\infty\) to \(-\infty\).

The formulas (I.38) completely determine the field of a charge moving with a prescribed velocity in an unbounded medium.

I.5. Energy losses of a charged particle. Let us find expressions for the energy loss of a charged particle moving in a medium. This can be done in several ways. The first of them (Tamm and Frank \(^{181}\)) is as follows.

Let us surround the path of the particle by a cylindrical surface of radius \(b\), whose axis coincides with the particle path. Obviously, the flux of the Poynting vector through this surface characterizes the energy loss of the particle per unit time. Dividing this quantity by the particle velocity \(v\), we find the energy loss per unit path length.

In order that the expression obtained give the total energy loss of the particle, the radius of the cylindrical surface \(b\) must be made to tend to zero. This, however, cannot be done, since at small distances from the moving particle the classical electrodynamics of the medium is inapplicable. In what follows, unless otherwise stated, \(b\) will mean the minimum value for which the classical electrodynamics of the medium is still valid. The quantity \(b\), according to Schönberg’s estimate \(^{93}\), is equal to \(\dfrac{c}{\omega_0}\), where \(\omega_0\) is the plasma frequency of the medium, defined by (I.24). The energy loss of the particle in collisions with impact parameter smaller than \(b\) is determined by the quantum Bethe–Bloch formulas.

For the energy loss of the particle, defined as the flux of the Poynting vector through a cylindrical surface of radius \(b\), the following expression is obtained (Tamm and Frank \(^{181,184}\), Fermi \(^{59}\)):

\[ \left(\frac{dW}{dz}\right)_b = \frac{2q^2 b}{\pi v^2}\operatorname{Re} \int_0^\infty \mu(\omega) \left( \frac{1}{\varepsilon\mu} -\beta^2 \right) k_r^*(\omega)\, K_1\!\left(k_r^* b\right) K_0\!\left(k_r b\right) i\omega\,d\omega, \tag{I.39} \]

where

\[ k_r^2=\frac{\omega^2}{v^2}\left(1-\varepsilon\mu\beta^2\right), \tag{I.40} \]

and the sign of \(k_r\) should be chosen so that retarded potentials are obtained\(^*\). If \(\varepsilon\) and \(\mu\) are complex functions of frequency, it is necessary that \(\operatorname{Re} k_r>0\). If \(\varepsilon\) and \(\mu\) are real and \(\varepsilon\mu\beta^2>1\), then \(k_r=-i|k_r|\).

Expression (I.39) is the starting point in the analysis of the energy loss of a particle in matter. To obtain a numerical value of the energy loss, it is necessary to specify the analytic dependence of \(\varepsilon\) and \(\mu\) on frequency.

In the work of Tamm and Frank \(^{181}\), the energy losses in a transparent medium due to Vavilov–Cherenkov radiation were obtained in the following way. Let the cylinder radius \(b\) tend to infinity. Formula (I.39) will then give the loss due to radiation going off to infinity, i.e. due to Vavilov–Cherenkov radiation. Using the asymptotic expressions for \(K_0\) and \(K_1\) (see (I.32); the same formula also holds for \(K_1\)) and taking into account that it is necessary to determine only the real part of the integral (I.39), we obtain

\[ \left(\frac{dW}{dz}\right)_{b\to\infty} = -\frac{q^2}{c^2} \int_{\substack{\varepsilon\mu\beta^2>1\\ \omega>0}} \left( 1-\frac{1}{\varepsilon\mu\beta^2} \right) \mu(\omega)\,\omega\,d\omega, \tag{I.41} \]

Integration is carried out only over the frequency range for which the condition for Vavilov–Cherenkov radiation, written under the integral sign, is satisfied.

\[ \text{\(^*\) Formula (I.39) differs from the formula in the cited works by taking into account the quantity \(\mu\).} \]

Let us note here that in most works only the dielectric properties of the medium were taken into account and it was assumed that \(\mu=1\).

Formula (I.41) gives only part of the particle’s energy losses. Fermi\(^{59}\), starting from (I.39), obtained an expression for the total energy losses of the particle. In doing so, Fermi first considered an absorbing medium and in the final result let the damping tend to zero. The same result can also be obtained without introducing damping, by means of the replacement (I.28). In this case integration of formula (I.39) gives

\[ -\left(\frac{dW}{dz}\right)_b = \frac{q^2}{v^2}\sum_s \frac{\omega_s}{|\varepsilon'(\omega_s)|}\, k_s b\, K_1(k_s b)\,K_0(k_s b) + \frac{q^2}{c^2} \int_{\varepsilon\mu\beta^2>1} \left(1-\frac{1}{\varepsilon\mu\beta^2}\right)\mu\omega\,d\omega, \tag{I.42} \]

where

\[ k_s = k_r(\omega_s)=\frac{\omega_s}{v} \tag{I.43} \]

and \(\omega_s\) are the roots of equation (I.186). Here we have used the formula

\[ \delta\{\varepsilon(\omega)\} = \sum_s \frac{1}{|\varepsilon'(\omega_s)|}\, \delta(\omega-\omega_s). \]

Formula (I.42) shows that a particle moving through a medium emits Vavilov—Cherenkov waves, as well as waves of those frequencies for which the dielectric constant of the medium becomes zero.

The first term in (I.42) gives the so-called polarization losses. If for \(\varepsilon(\omega)\) one takes the expression

\[ \varepsilon(\omega)=1+\frac{\omega_0^2}{\omega_s^2-\omega^2}, \tag{I.44} \]

then the formula for the polarization losses takes the form\(^{59}\)

\[ \left(\frac{dW}{dz}\right)_{\mathrm{polar}} = -\frac{q^2\omega_0^2}{v^2}\, \frac{\Omega b}{v}\, K_0\!\left(\frac{\Omega b}{v}\right) K_1\!\left(\frac{\Omega b}{v}\right), \tag{I.45} \]

where \(\Omega=\sqrt{\omega_0^2+\omega_s^2}\) is the frequency at which \(\varepsilon(\omega)\) (I.44) becomes zero.

If \(\dfrac{\Omega b}{v}\ll 1\), formula (I.45) assumes the simple form

\[ -\left(\frac{dW}{dz}\right)_{\mathrm{polar}} = \frac{q^2\omega_0^2}{v^2}\ln\frac{v}{\Omega b}. \tag{I.46} \]

For \(\dfrac{\Omega b}{v}\gg 1\) (small velocities or large values of the cylinder radius \(b\)), formula (I.45) gives an exponentially small quantity. However, at very small particle velocities the whole approach becomes inapplicable (see, for example, Fermi, Nuclear Physics, Moscow, IL, 1951).

The losses due to Vavilov—Cherenkov radiation are determined by the second term of formula (I.42). If \(\mu=1\), and \(\varepsilon(\omega)\) is determined by formula (I.44),

the integral in the second term is equal to

\[ -\left(\frac{dW}{dz}\right)_{\mathrm{v.-ch.}}= \begin{cases} \dfrac{q^2\omega_0^2}{v^2}\left[-\beta^2-\ln(1-\beta^2)\right], & \text{for } \beta<\dfrac{1}{\sqrt{\varepsilon(0)}},\\[1.2em] \dfrac{q^2\omega_0^2}{v^2}\left[-\dfrac{1-\beta^2}{\varepsilon(0)-1}+\ln\dfrac{\varepsilon(0)}{\varepsilon(0)-1}\right], & \text{for } \beta>\dfrac{1}{\sqrt{\varepsilon(0)}}. \end{cases} \tag{I.47} \]

Figure 3 shows the graph of \(\varepsilon(\omega)\) (I.44). A straight line has been drawn on the graph, whose distance from the abscissa axis is equal to \(\dfrac{1}{\beta^2}\). The frequency region of Vavilov–Cherenkov radiation, corresponding to \(\varepsilon(\omega)>\dfrac{1}{\beta^2}\), is shaded. Polarization losses occur only at one frequency, at which the curve \(\varepsilon(\omega)\) intersects the abscissa axis.

Formula (I.42) gives that part of the particle’s energy losses which is described by the classical electrodynamics of a medium. To obtain the total energy losses, one must add to (I.42) the energy losses in collisions with impact parameter smaller than \(b\), calculated by quantum formulas.

Fig. 3.

The energy losses of a particle can also be determined by another method\({}^{29,121}\), by calculating the value of the electric field at the point where the particle is located. Obviously, the field brakes the particle and

\[ \frac{dW}{dx}=qE_z\bigg|_{\substack{z\to vt\\ r\to 0}}. \tag{I.48} \]

This formula, where \(E_z\) is taken from (I.38), gives for the energy losses the same results as the calculation of the flux of the Poynting vector *).

Let us note one more method for determining the energy losses of a particle. It consists in calculating the increase of the energy of the electromagnetic field in the medium in which the particle is moving. Obviously, the energy losses of the particle per unit time are determined by the expression

\[ \frac{dW}{dt}=-\frac{1}{8\pi}\frac{d}{dt}\int_V(\varepsilon E^2+\mu H^2)\,dV. \]

This method is especially convenient when using the Hamiltonian method\({}^{71,3,119}\).

I.6. Vavilov–Cherenkov radiation in a medium without dispersion. The whole discussion is substantially simplified if, from the very beginning, one neglects the dispersion of the medium\({}^{183}\). We shall assume that the dielectric constant \(\varepsilon\) does not depend on frequency (for simplicity we put \(\mu=1\)).

*) In formula (I.48), the limiting transition \(z\to vt,\ r\to 0\) should, generally speaking, be carried out after integration over all harmonics in the expansion for \(E_z\).

Then integration of formula (1.26) for $\mathbf A$ gives:

\[ \mathbf A=\frac{q\mathbf v}{c\sqrt{(z-vt)^2+r^2(1-\varepsilon\beta^2)}} , \qquad \text{for } \varepsilon\beta^2<1, \]

and, when the condition for Vavilov—Cherenkov radiation is satisfied, i.e. for $\varepsilon\beta^2>1$,

\[ \mathbf A= \begin{cases} \dfrac{2q\mathbf v}{c\sqrt{(z-vt)^2-r^2(\varepsilon\beta^2-1)}} & \text{for } z<vt,\quad vt-z>r\sqrt{\varepsilon\beta^2-1},\\[1.2ex] 0\quad \text{(in the rest of space).} \end{cases} \]

It follows from the last expression that for $\varepsilon\beta^2>1$ the field is discontinuous. The surface of discontinuity is described by the equation

\[ z-vt+r\sqrt{\varepsilon\beta^2-1}=0. \]

Discontinuous conical waves propagate along the $z$ axis with the velocity of the charge. The normal to these waves makes with the $z$ axis the angle

\[ \vartheta=\arccos \frac{1}{\sqrt{\varepsilon}\,\beta}. \]

Discontinuous waves of this type are well known in ballistics. If the velocity of a projectile is greater than the speed of sound in air, the projectile excites a narrow conical acoustic wave stationary with respect to the projectile (Mach wave). The very rapid increase in the resistance experienced by a projectile when its velocity becomes greater than the speed of sound is the result of the formation of these waves. The phenomena here turn out to be much more complicated than the corresponding optical phenomenon (boundary conditions on the surface of the projectile, nonlinearity of the equations of aerodynamics, etc.).

Analogous phenomena also occur in other areas of physics (radio engineering$^{157}$, meson physics$^{24,98}$). What is common to them is that the velocity of the source of the field exceeds the velocity of propagation of the field.

As early as 1904 A. Sommerfeld$^{176}$ calculated the resultant $F$ of the electromagnetic forces of interaction of all volume elements of a rigid spherical electron. In particular, he investigated the case of uniform motion of an electron in vacuum and found that $F=0$ for $v<c$ and

\[ F=-\frac{9q^2}{4\pi d^2}\left(1-\frac{1}{\beta^3}\right) \]

for $v>c$ ($d$ is the diameter of the electron). A very similar expression is obtained if the integral of the Vavilov—Cherenkov losses in a nondispersive medium is cut off at wavelengths of order $d$, i.e. at frequencies of order $\omega=\dfrac{c}{\sqrt{\varepsilon}\,d}$, and then $\varepsilon=1$ is put. Sommerfeld’s paper was written before the theory of relativity was developed. Replacing in Sommerfeld’s formulas the speed of light in empty space by the speed of light in the medium leads to the theory of Cherenkov radiation in a nondispersive medium.

The energy losses to radiation in a nondispersive medium are infinite:

\[ \frac{dW}{dz}=-\frac{q^2}{c^2}\int_0^\infty \left(1-\frac{1}{\varepsilon\beta^2}\right)\omega\,d\omega . \]

In reality, the case \(\varepsilon=\mathrm{const}\) is an abstraction, and the loss integral in a real medium is always cut off at some limiting frequency.

I.7. Longitudinal and transverse fields. Let us give another solution of the equations for the potentials (I.8) under the gauge (I.7)\(^{26,171}\). In this case the potential of the longitudinal field is written in the form

\[ \varphi=\frac{q}{\pi v}\int e^{\,i\frac{\omega}{v}(z-vt)} K_0\!\left(\frac{|\omega|}{v}\,r\right)\frac{d\omega}{\varepsilon(\omega)}, \tag{I.49} \]

and the vector potential of the transverse field in the form

\[ \mathbf A=\int e^{\,i\frac{\omega}{v}(z-vt)}\,\mathbf a(\omega,r)\,d\omega, \tag{I.50} \]

where

\[ a_r(\omega,r)= \begin{cases} \dfrac{iqc}{\varepsilon\pi v^2} \left[ K_1\!\left(\dfrac{|\omega|}{v}\,r\right) -\sqrt{1-\varepsilon\mu\beta^2}\, K_1\!\left(\dfrac{|\omega|}{v}\sqrt{1-\varepsilon\mu\beta^2}\,r\right) \right], & \text{for } \omega>0,\\[1.2em] \text{complex conjugate}, & \text{for } \omega<0; \end{cases} \tag{I.51} \]

\[ a_z(\omega,r)= \begin{cases} \dfrac{qc}{\varepsilon\pi v^2} \left[ K_0\!\left(\dfrac{|\omega|}{v}\,r\right) -(1-\varepsilon\mu\beta^2) K_0\!\left(\dfrac{|\omega|}{v}\sqrt{1-\varepsilon\mu\beta^2}\,r\right) \right], & \text{for } \omega>0,\\[1.2em] \text{complex conjugate}, & \text{for } \omega<0. \end{cases} \tag{I.52} \]

These potentials give, of course, the same expressions for the complete fields \(\mathbf E\) and \(\mathbf H\) as do the potentials (I.29) and (I.33). However, the separation of the fields into longitudinal and transverse is now carried out at once. It turns out that polarization losses are described by the longitudinal field:

\[ \left(\frac{dW}{dx}\right)_{\mathrm{polar}} =qE_z^{\mathrm{long}} =-q\,\frac{\partial\varphi}{\partial z}\bigg|_{\substack{z\to vt\\ r\to 0}}, \]

and Vavilov–Cherenkov losses by the transverse field

\[ \left(\frac{dW}{dx}\right)_{\mathrm{V.-Ch.}} =qE_z^{\mathrm{tr}} =-q\,\frac{1}{c}\frac{\partial A_z}{\partial t}\bigg|_{\substack{z\to vt\\ r\to 0}}. \]

I.8. Contribution of losses due to the Vavilov–Cherenkov effect to the total energy losses. In modern experimental techniques for detecting high-energy particles, either the ionization processes caused by the particles or the generation by them of Vavilov–Cherenkov radiation are used. It is therefore very important to clarify what fraction of the total energy losses of a particle is due to ionization and what fraction to Vavilov–Cherenkov radiation.

The formulas (I.45) and (I.47) for polarization losses and Vavilov–Cherenkov losses were obtained by Fermi\(^{59}\) under rather restrictive assumptions about the dielectric constant of the medium. (The expression (I.44) for \(\varepsilon(\omega)\).) Subsequently Fermi’s calculations were refined by Halpern and Hall, Wick, and others,* who used more general expressions for

\[ \text{* All the bibliography relating to this paragraph may be found by the reader in the abstract collection}^{57} \text{ and in the review by B. T. Price}^{158}. \]

of the dielectric constant, taking into account the presence of many absorption bands in the dielectric. In Wick’s works, for example, the loss integral (I.39) was evaluated under the assumption

\[ \varepsilon(\omega)=1+\omega_0^2\sum_k \frac{f_k}{\omega_k^2-\omega^2-i\gamma_k\omega}, \]

where \(f_k\) are oscillator strengths, and \(\gamma_k\) are damping coefficients.

All subsequent calculations confirmed the qualitative conclusions obtained by Fermi, in particular the conclusion that, owing to the polarization of the medium, the energy losses of a particle described by classical electrodynamics tend to a certain limiting value as the particle energy increases. (The proposition that screening caused by polarization of the medium can reduce ionization losses was first put forward by Swann.)

Several years before experiments were carried out to measure the energy losses of fast charged particles, the question of the relative magnitude of Vavilov—Cherenkov losses was investigated in the works of N. Bohr \(^{29}\) and M. Schönberg \(^{165}\). Starting from the approximation of a transparent medium, these authors showed that the increase of losses in the relativistic region is due mainly to Vavilov—Cherenkov radiation, whereas polarization losses increase only weakly with energy. From their studies it followed that the ionization losses of a fast particle should not increase in the relativistic region. This conclusion, however, was refuted by a whole series of experimental works devoted to determining the ionization losses of fast particles. The experiments showed that there is an increase in ionization losses, much more intense than followed from the theoretical analysis. Hence it followed that the relative magnitude of the energy losses due to Vavilov—Cherenkov radiation had been greatly overestimated by theory.

The possible causes of the discrepancy between theory and experiment, as well as modifications of the theory making it possible to bring it into agreement with experience, were discussed in a number of works.

M. Schönberg \(^{93}\) was one of the first to give a possible explanation of the fact that formula (I.41) for the energy losses due to Vavilov—Cherenkov radiation in a transparent medium*) gives overestimated results. Schönberg’s considerations may be illustrated by the following simple reasoning \(^{26}\). From formula (I.41) it is seen that the energy losses due to Vavilov—Cherenkov radiation turn out to be greatest in those spectral regions where \(\varepsilon(\omega)\) is large. In a transparent medium the dielectric constant is large in the region of frequencies close to an absorption line. This is seen from the simple formula (I.44) for the dielectric constant. At frequencies \(\omega\) close to \(\omega_s\), \(\varepsilon(\omega)\) assumes large values. Consequently, the greatest contribution to Vavilov—Cherenkov radiation is made by frequencies close to \(\omega_s\). But the field of Vavilov—Cherenkov radiation is determined by the condition (I.18a),

\[ k^2=\varepsilon(\omega)\frac{\omega^2}{c^2}, \]

which relates the wavelength of the radiation

\[ \lambda=\frac{2\pi}{k} \]

to its frequency. For \(\omega\) close to \(\omega_s\), \(\varepsilon(\omega)\) assumes large values, which, according to condition (I.18a), corresponds to short-wavelength radiation. It is precisely such waves that make the greatest contribution to formula (I.41). But such waves cannot be correctly described by classical electrodynamics. Classical electrodynamics of a medium assumes averaging of the field over a certain volume, and the wavelength

*) Here and below it is assumed that \(\mu=1\).

radiation in the medium therefore cannot be less than a certain value. For this minimum wavelength, below which the wavelength of Vavilov—Cherenkov radiation cannot be, according to Schonberg’s estimates, one obtains the value

\[ R=\frac{c}{\omega_0}\quad \left(\omega_0=\sqrt{\frac{4\pi n e^2}{m}}\right), \tag{I.53} \]

where \(c\) is the speed of light, \(e\) and \(m\) are the charge and mass of the electron, and \(n\) is the number of electrons per unit volume of the medium. The quantity \(\omega_0\) is identical with the frequency \(\omega_0\) entering expression (I.44). According to other estimates \(^{26}\), the limiting wavelength of Vavilov—Cherenkov radiation in the case when \(\varepsilon(\omega)\) is described by expression (I.44) is equal to

\[ R=\frac{c}{\sqrt{\omega_0^2+\omega_s^2}}, \tag{I.54} \]

where the quantity \(\omega_s\) is the frequency at which \(\varepsilon(\omega)\) (I.44) becomes infinite. To bring the theory into agreement with experiment it is necessary to exclude from consideration waves with length smaller than \(R\) (I.53). Discarding wavelengths smaller than \(R\) is equivalent to excluding from consideration collision parameters smaller than \(R\). In work \(^{93}\) this exclusion was carried out as follows. The path of a particle in the medium was surrounded by a cylindrical channel of radius \(R\). The energy losses of the particle in collisions with impact parameter smaller than \(R\) were determined from the Bethe—Bloch quantum formulae, while the energy losses due to Bohr radiation and Vavilov—Cherenkov radiation were determined with the aid of classical electrodynamics, the channel being regarded as empty in the classical calculation. When a particle moves along the axis of an empty channel in a dielectric, the spectrum of Cherenkov radiation is cut off on the short-wavelength side at wavelengths of the order of magnitude of the channel radius, while for longer wavelengths it practically does not change \(^{81}\).

The calculations of M. Huybrechts and M. Schonberg \(^{93}\) gave a decrease in the contribution of Vavilov—Cherenkov losses to the total losses and, in the authors’ opinion, are in satisfactory agreement with experiment.

It should be noted, however, that the results obtained in \(^{93}\) depend on the manner in which the exclusion of small collision parameters was carried out (the estimate of the channel radius, boundary conditions, etc.). This circumstance permits, in our opinion, one to speak only of qualitative agreement with experiment.

Another way that makes it possible to bring the theory into agreement with experiment consists in abandoning the concept of a transparent medium and in taking absorption into account. Let us illustrate this by a simple qualitative consideration. Taking absorption into account for the case (I.44) leads to the following expression for the dielectric constant:

\[ \varepsilon(\omega)=1+\frac{\omega_0^2}{\omega_s^2-\omega^2-i\gamma\omega}. \tag{I.55} \]

When damping is taken into account, \(\varepsilon(\omega)\) no longer becomes infinite, and values of \(\omega\) close to \(\omega_s\) do not make a large contribution to Vavilov—Cherenkov radiation. True, the definition of Vavilov—Cherenkov radiation in a medium with absorption is to some extent conventional, since Vavilov—Cherenkov radiation is absorbed in the medium, as is Bohr radiation, and does not escape to infinity. If one defines Vavilov—Cherenkov losses as the reaction on the particle from the field created by it

of the transverse field, the expression for the losses has the form

\[ -\frac{dW}{dz}=\frac{q}{c}\int a_z(\omega,r)\,i\omega\,d\omega\bigg|_{r\to 0}, \]

where \(a_z(\omega,r)\) is the Fourier component of the vector potential of the transverse field (I.52). For weak damping \((\operatorname{Re}\varepsilon \gg \operatorname{Im}\varepsilon)\) this gives\({}^{30}\)

\[ -\frac{dW}{dz} = \frac{q^2}{c^3} \int_{\operatorname{Re}\varepsilon\cdot\beta^2>1} \left(1-\frac{\operatorname{Re}\varepsilon}{|\varepsilon|^3}\right)\omega\,d\omega, \tag{I.56} \]

where \(\operatorname{Re}\) denotes the real part. The integral is taken over frequencies for which \(\operatorname{Re}\varepsilon\cdot\beta^2>1\).

Calculations of the energy losses of a charged particle in an absorbing medium have been carried out by many authors\({}^{*}\). In connection with recent experiments these works were refined in the papers of Sternheimer, Budini, and others. Taking account of absorption in the medium leads qualitatively to the same results as abandoning the classical treatment of collisions with small parameters, i.e., to a cutoff of the Vavilov–Cherenkov radiation spectrum near absorption bands and to a decrease of the losses due to Vavilov–Cherenkov radiation. Taking absorption into account also brings the theory into agreement with experiment. The two approaches considered do not exclude one another, but complement one another. If the damping in the medium is very small, then the cutoff of the Vavilov–Cherenkov radiation spectrum is explained mainly by M. Schönberg’s considerations. For media with appreciable damping, the cutoff of the spectrum is given simply by taking damping into account.

I.9. Polarizations. In what follows, when considering interference of Vavilov–Cherenkov radiation, it will be convenient to consider not the entire field, but only that part of it which describes radiation. To find the radiation field, one must make the substitution (I.28) in the integrals (I.26) and (I.27). The radiation field will be described by the terms under the integral that contain delta functions. The remaining terms give a field symmetric with respect to the plane \(z=vt\). The symmetric field does not describe the particle’s energy losses and cannot be associated with radiation.

For the radiation field one obtains the following result\({}^{163,126}\):

\[ E_{z\,\mathrm{V.-Ch.}} = \frac{q}{c^2} \int_{\varepsilon\beta^2>1,\ \omega>0} \left(\frac{1}{\varepsilon\beta^2}-1\right) J_0\!\left(\frac{\omega}{v}\sqrt{\varepsilon\beta^2-1}\,r\right) \cos\frac{\omega}{v}(z-vt)\,\omega\,d\omega, \tag{I.57a} \]

\[ E_{r\,\mathrm{V.-Ch.}} = -\frac{q}{v^2} \int_{\varepsilon\beta^2>1,\ \omega>0} \frac{\sqrt{\varepsilon\beta^2-1}}{\varepsilon} J_1\!\left(\frac{\omega}{v}\sqrt{\varepsilon\beta^2-1}\,r\right) \sin\frac{\omega}{v}(z-vt)\,\omega\,d\omega, \tag{I.57б} \]

\[ H_{\varphi\,\mathrm{V.-Ch.}} = -\frac{q}{cv} \int_{\varepsilon\beta^2>1,\ \omega>0} \sqrt{\varepsilon\beta^2-1}\, J_1\!\left(\frac{\omega}{v}\sqrt{\varepsilon\beta^2-1}\,r\right) \sin\frac{\omega}{v}(z-vt)\,\omega\,d\omega, \tag{I.57в} \]

where \(J_0(x)\), \(J_1(x)\) are Bessel functions.

In all the integrals \(\varepsilon\) is a function of \(\omega\). Note that \(qE_{z\,\mathrm{V.-Ch.}}(z=vt,r=0)\) gives the braking force caused by the reaction of Vavilov–Cherenkov radiation on the particle.

I.10. Duration of the radiation flash. Let us consider qualitatively how the Vavilov–Cherenkov radiation field changes at large distances from the particle. As was already said, the field at large

\({}^{*}\) See the note on p. 214.

at distances from the charge is a sum of waves of the form (I.36). These waves will undergo interference reinforcement if their phases are close, i.e., if, in some frequency interval,

\[ \frac{\omega}{v}(z-vt)+k_r r-\frac{\pi}{4}=\mathrm{const} \]

or

\[ z-vt+vr\,\frac{dk_r}{d\omega}=0 . \tag{I.58} \]

If this condition is not satisfied, then waves with close frequencies have different phases and the waves mutually cancel. Let us denote

\[ v\,\frac{dk_r}{d\omega}=g . \tag{I.59} \]

Suppose that in the Vavilov—Cherenkov frequency interval the quantity \(g\) lies within the limits from \(g_{\min}\) to \(g_{\max}\). Then, for a given instant of time \(t\), the radiation field will be different from zero in the space between the cones \(^{183}\)

\[ z+g_{\min}r=vt \quad \text{and} \quad z+g_{\max}r=vt, \tag{I.60} \]

since at points of space outside these cones equation (I.58) has no roots. Thus, at a specified point \(r,z\) the Vavilov—Cherenkov radiation field will be different from zero only during a finite time interval:

\[ \Delta t=\frac{g_{\max}-g_{\min}}{v}\,r . \tag{I.61} \]

The duration of the pulse in a dispersive medium increases with increasing distance \(r\) from the \(z\)-axis because of the spreading of the wave packet of the radiation pulse. The question of the duration of the Cherenkov-radiation flash was analyzed in detail by I. M. Frank \(^{65}\).

I.11. Radiation of a conductor with current. Above we spoke about the Vavilov—Cherenkov radiation of a point charged particle. Obviously, the source of Vavilov—Cherenkov radiation may be any system of charges or currents capable of creating an electric or magnetic field. It is only necessary that such a system move in the medium with a speed exceeding the phase velocity of light in some frequency interval. Then, in this same frequency interval, Vavilov—Cherenkov radiation will be emitted*.

The Vavilov—Cherenkov radiation of systems of electric charges has been considered many times \(^{64,26,50,82}\), and we shall discuss some results in § I.13. Here we shall briefly present some results of A. I. Morozov \(^{148}\), who considered the Vavilov—Cherenkov radiation produced by a conductor with current, moving in a medium with specified \(\varepsilon\) and \(\mu\). The velocities of conductors with current in machines of various kinds are very small compared with the phase velocity of light in the media surrounding them. Therefore it may seem that the problem under consideration is of no practical interest. However, already at present, in accelerators and in certain kinds of discharges, there exist current-carrying beams of particles which move as a whole with high velocities, approaching or exceeding the phase velocity of light in many substances. Investigation of the radiation of such beams is of interest, in particular, for the problem of generating radio waves by bunches of charged particles, and so—

* In order for Vavilov—Cherenkov radiation to be emitted, it is also necessary that a certain condition be fulfilled, imposed on the polarization of the Vavilov—Cherenkov wave. This condition will be discussed in Section II.

THEORY OF THE VAVILOV—CHERENKOV EFFECT

also in connection with the various methods of particle acceleration recently proposed, based on the interaction of beams of charged particles with an electron plasma and with one another.

Let us imagine an infinitely thin rectilinear conductor with current. For simplicity we shall consider the conductor neutral. We denote the current in the conductor by \(\mathbf j\). Let this conductor move as a whole with velocity \(\mathbf v\), perpendicular to \(\mathbf j\). We choose a coordinate system in which \(\mathbf j\) is parallel to the \(x\)-axis, and \(\mathbf v\) to the \(y\)-axis (Fig. 4), and in which the conductor lies in the plane \(xy\). In this coordinate system the field is described only by the component of the vector potential \(A_x\):

\[ A_x=\frac{j}{c}\int \frac{ e^{\,i\frac{\omega}{v}(z-vt)-\frac{\omega}{v}\sqrt{1-\varepsilon_\mu\beta^2}\,|z|} }{ \sqrt{1-\varepsilon_\mu\beta^2} } \,\frac{\mu(\omega)\,d\omega}{\omega}. \tag{I.62} \]

All the remaining components of \(\mathbf A\) and the scalar potential \(\varphi\) may be taken equal to zero.

As is seen from (I.62), the field of a moving rectilinear conductor with current is a sum of plane waves of the form

\[ e^{\,i\frac{\omega}{v}(z-vt)-\frac{\omega}{v}\sqrt{1-\varepsilon_\mu\beta^2}\,|z|}. \]

If the condition for Vavilov—Cherenkov radiation is not fulfilled, these waves decay with distance \(|z|\). If, however, \(\varepsilon\mu\beta^2>1\), the field is resolved into nondecaying waves, i.e. radiation takes place. The energy losses to Vavilov—Cherenkov radiation per unit length of the conductor per unit path can be found by determining the \(y\)-component of the force acting on the conductor from the field it creates, or by calculating the flux of the Poynting vector. The force acting on a unit length of the conductor is determined by the vector product of \(\mathbf j\) and \(\mathbf H\):

\[ \frac{dW}{dy}=F_y=\frac{1}{c}[\mathbf j\mathbf H]_y =-\frac{2j^2}{c^2v}\operatorname{Re}\int_0^\infty \frac{\mu(\omega)\,d\omega}{\sqrt{\varepsilon_\mu\beta^2-1}}. \tag{I.63} \]

Fig. 4.

This formula is valid for any \(\varepsilon(\omega)\) and \(\mu(\omega)\) satisfying the conditions

\[ \varepsilon(\omega)=\varepsilon^*(-\omega), \qquad \mu(\omega)=\mu^*(-\omega). \]

In the case of real \(\varepsilon\) and \(\mu\), the region of integration reduces to the region where \(\varepsilon\mu\beta^2>1\).

As is seen from formula (I.63), the radiation spectrum of a moving conductor with current increases as \(v\to c/\sqrt{\varepsilon\mu}\). If, instead of a neutral conductor with current, one takes a uniformly charged thread moving also with a superlight velocity, the radiation spectrum becomes proportional to \(\sqrt{\varepsilon\mu\beta^2-1}\), i.e. decreases as \(v\to c/\sqrt{\varepsilon\mu}\).

I. 12. Reversal of the Vavilov—Cherenkov effect

Let us consider Vavilov—Cherenkov radiation from the point of view of an observer moving with the electron. This was first done by I. E. Tamm \(^{183}\).

In the system where the electron is at rest, the medium moves with velocity \(-\mathbf v\). The fields in the rest system of the electron are determined from the fields in the rest system of the medium by means of Lorentz transformations. Calculations show \(^{183}\) that the force acting on the resting charge from the field created by it in the moving medium is determined by the same expression (I.41) as the force acting on the electron in the rest system of the medium. This means that the energy losses per unit length (in the rest system of the electron one should speak of the gain of energy by the charge) are equal in both coordinate systems. The phenomenon of the entrainment of a resting charge by a moving medium may be regarded as the inverse of the Vavilov—Cherenkov effect.

Not only does a moving medium entrain a charge, but a moving charge also entrains the medium, transferring to it part of the momentum lost \(^{87,203}\). However, the latter effect cannot be measured because of the enormous mass of the medium in comparison with the mass of the charged particle. It is difficult to verify the phenomenon of entrainment of a charge by a medium for another reason: we cannot obtain macroscopic volumes of matter moving with the necessary velocities. However, as was first pointed out by V. I. Veksler \(^{191}\), sufficiently dense beams of fast electrons can be used as the moving medium; a resting charge placed in the beam will experience an accelerating force from the beam of fast electrons. On this basis rests one of the variants of the coherent acceleration method proposed by V. I. Veksler.

I.13. Interference of Vavilov—Cherenkov radiation. In this paragraph we shall consider the interference of Vavilov—Cherenkov radiation. The results presented here were obtained mainly in the works of I. M. Frank \(^{60,63,64}\) (see also \(^{26,50}\)).

Let two point charges of equal magnitude \(q\) move in a medium with equal velocities \(\mathbf v\). The charges move along one line at a distance \(l\) one behind the other. Determining the energy losses of such a system to Vavilov—Cherenkov radiation by calculating the flux of the Poynting vector gives

\[ -\left(\frac{dW}{dz}\right)_{\text{V.-Ch.}} = \frac{2q^2}{c^2} \int_{\varepsilon\beta^2>1} \left(1-\frac{1}{\varepsilon\beta^2}\right) \left(1+\cos\frac{\omega l}{v}\right)\omega\,d\omega . \tag{I.64} \]

It is seen that the radiation spectrum of the system under consideration, consisting of two charges, differs substantially from the radiation spectrum of a single charge. These spectra differ by the factor

\[ 2\left(1+\cos\frac{\omega l}{v}\right), \tag{I.65} \]

which, at frequencies satisfying the condition

\[ l=\frac{2\pi v}{\omega}\,n \qquad (n=0,\,1,\,2,\ldots), \tag{I.66} \]

leads to a fourfold increase in the radiation intensity, while at frequencies for which

\[ l=\frac{\pi v}{\omega}(2n+1), \tag{I.67} \]

it vanishes.

The component along the \(z\)-axis of the wave vector of the Vavilov—Cherenkov wave is equal to \(\omega/v\), which corresponds to the wavelength \(\lambda_z=2\pi v/\omega\). Therefore it follows from conditions (I.66) and (I.67) that, if the wavelength fits an integer number of times between the charges, the intensity of radiation of such a wave is maximal. If the wavelength fits a half-integer number of times between the moving charges, the radiation intensity is zero.

From this simple example it is seen that the spectrum of Vavilov—Cherenkov radiation from systems of several charges is determined not only by the medium, but also by the mutual arrangement of the charges in the system. If the system of charges has no symmetry with respect to the line of motion, then, in addition to the braking force, there will also appear a force deflecting the system from rectilinear motion. All these phenomena are essential in investigating the possibilities of generating radio waves by beams of charged particles.

If, in the example considered, the charges of the two particles are taken to be different in sign \((q\) and \(-q)\), the sign before \(\cos \dfrac{\omega l}{v}\) under the integral in (I.64) changes:

\[ -\left(\frac{dW}{dz}\right)_{\mathrm{V.-Ch.}} = \frac{2q^{2}}{c^{2}} \int_{\varepsilon\beta^{2}>1} \left(1-\frac{1}{\varepsilon\beta^{2}}\right) \left(1-\cos\frac{\omega l}{v}\right)\omega\,d\omega . \tag{I.68} \]

This formula gives the losses to Vavilov—Cherenkov radiation of an electric dipole with moment \(p=lq\), oriented along the direction of the velocity. If the dimensions of the dipole \(l\) are small in comparison with \(\dfrac{v}{\omega}=\dfrac{\lambda_z}{2\pi}\), the factor \(1-\cos\dfrac{\omega l}{v}\) under the integral may be expanded in a series in \(\dfrac{\omega l}{v}\). Restricting oneself to the first nonzero term of the expansion gives the expression for the energy losses of a point dipole oriented along the direction of the velocity\(^{60}\),

\[ -\left(\frac{dW}{dz}\right)_{\mathrm{V.-Ch.}} = \frac{p^{2}}{c^{2}v^{2}} \int_{\varepsilon\beta^{2}>1} \left(1-\frac{1}{\varepsilon\beta^{2}}\right)\omega^{3}\,d\omega . \tag{I.69} \]

The expression for the energy losses of a point electric dipole of arbitrary orientation has the form

\[ -\frac{dW_p}{dz} = \frac{1}{q}(\mathbf{p}\nabla)^2 E_z \left| \begin{array}{c} z\to vt,\\ r\to 0 \end{array} \right., \tag{I.70} \]

where \(E_z\) is determined by formula (I.57a), \(\mathbf{p}\) is the vector of the dipole moment, and \(\nabla\) is the gradient operator. The calculation gives

\[ -\left(\frac{dW_p}{dz}\right)_{\mathrm{V.-Ch.}} = \frac{1}{c^{2}v^{2}} \int_{\varepsilon\beta^{2}>1} \left[ p_z^2+\frac{1}{2}p_r^2(\varepsilon\beta^{2}-1) \right] \left(1-\frac{1}{\varepsilon\beta^{2}}\right)\omega^{3}\,d\omega . \tag{I.71} \]

It should be emphasized that the dipole moment \(\mathbf{p}\) is measured in the rest frame of the medium.

I. M. Frank also obtained a formula for the energy losses of an arbitrarily oriented point magnetic dipole*):

\[ -\left(\frac{dW_{\mu}}{dz}\right)_{\mathrm{V.-Ch.}} = \frac{1}{c^{2}v^{2}} \int_{\varepsilon\beta^{2}>1} \left[ \mu_z^2+\frac{1}{2}\mu_r^2(\varepsilon\beta^{2}-1) \right] \left(1-\frac{1}{\varepsilon\beta^{2}}\right)\varepsilon\omega^{3}\,d\omega . \tag{I.72} \]

This formula is obtained from consideration of the interference of the radiation of two point magnetic charges.

Here, too, the magnetic moment \(\boldsymbol{\mu}\) is measured in the rest frame of the medium. The proper moments \(\mathbf{p}^0\) and \(\boldsymbol{\mu}^0\), measured in the frame where the particle

*) We note that, in order to obtain expressions (I.71) and (I.72) from the electrodynamics of continuous media with magnetic and electric polarization, one must start from the equations for moving media\(^{84}\), with the same \(\varepsilon\) taken everywhere.

are at rest, are related to \(\mathbf p\) and \(\boldsymbol\mu\) by the relations

\[ \left. \begin{aligned} \mathbf p&=\mathbf p^0-\left(1-\sqrt{1-\beta^2}\right) \frac{(\mathbf p^0\mathbf v)\mathbf v}{v^2} +\frac{1}{c}\,[\mathbf v,\boldsymbol\mu^0],\\ \boldsymbol\mu&=\boldsymbol\mu^0-\left(1-\sqrt{1-\beta^2}\right) \frac{(\boldsymbol\mu^0\mathbf v)\mathbf v}{v^2} -\frac{1}{c}\,[\mathbf v,\mathbf p^0]. \end{aligned} \right\} \tag{I.73} \]

It is interesting to note that the calculation of the energy losses due to Vavilov–Cherenkov radiation of a closed current loop moving in a medium with superluminal velocity leads to the same result (I.72) (see I.11). Here \(\boldsymbol\mu\) should be understood as the magnetic moment possessed by such a current loop.

A force also acts on a moving dipole, deflecting it from a rectilinear path. The origin of this force may be explained by the following example[^64]. Consider a dipole moving with superluminal velocity and oriented as shown in Fig. 5. The vertices of the radiation cones produced by each of the charges are displaced relative to one another, and, consequently, the phases of the waves corresponding to the two charges will be different. We shall regard the distance \(l\) between the charges as small in comparison with any wavelengths that can be radiated. In this case, if the phases of the waves from the two charges turn out to be the same, the waves will completely extinguish one another; as the phase difference increases, the resultant amplitude will increase. Therefore, in the case shown in Fig. 5, the intensity of the radiation directed upward will be greater than that of the radiation directed downward. Consequently, the momentum carried away by the radiation will give rise to a recoil force tending to deflect the dipole downward. The magnitude of this force for a point electric dipole is determined by the expression

Fig. 5.

\[ F_r=-\frac{1}{q}(\mathbf p\nabla)^2 \left(\mathbf E_r+\frac{1}{c}[\mathbf v,\mathbf H]_r\right) \bigg|_{\substack{r=0\\ z=vt}} = -\frac{p_rp_z}{c^4} \int_{\varepsilon\beta^2>1} \left(1-\frac{1}{\varepsilon\beta^2}\right) \varepsilon\omega^3\,d\omega, \tag{I.74} \]

where \(\mathbf E\) and \(\mathbf H\) are determined by the equalities (I.57a), (I.57b), and (I.57c). The deflecting force acting on a magnetic dipole is obtained from (I.74) by adding under the integral one more factor, \(\varepsilon\).

In an analogous manner one may consider the energy losses of any rigidly specified configuration of charges (multipoles[^64], extended charges of various shapes[^26,^50]).

For various applications (generation of microwaves by beams of charged particles, coherent acceleration) it is important to know whether the motion of a given system of charges—the source of radiation—will be stable. A complete treatment of the stability problem has not yet been carried out. Qualitatively, the stability of the motion may be judged from the change of the field inside a bunch of charged particles[^26]. Such a consideration leads to the conclusion that the motion of bunches whose dimensions are small compared with the radiated wavelength is unstable.

The radiation of a uniformly moving point charge can also be represented as the result of interference[^63]. From the first equation

of system (I.6) it is not difficult to find the equation for the Fourier component \(\mathbf{A}_\omega\) of the vector potential:

\[ \Delta \mathbf{A}_\omega+\frac{\varepsilon(\omega)\omega^2}{c^2}\mathbf{A}_\omega = \frac{q}{2\pi}\frac{\mathbf{v}}{v}e^{-i\frac{\omega}{v}z}\delta(x)\delta(y) = \frac{4\pi}{c}\mathbf{j}_\omega \tag{I.75} \]

(for simplicity we have put \(\mu=1\)).

The current \(\mathbf{j}_\omega\) may be regarded as the current of a certain polarization \(\mathbf{P}_\omega\), distributed along the line of motion of the charge. Formally this means that we introduce an auxiliary vector \(\mathbf{P}_\omega\), related to the current \(\mathbf{j}_\omega\) by the formula ordinarily defining the polarization current:

\[ \mathbf{j}_\omega=i\omega \mathbf{P}_\omega . \tag{I.76} \]

Comparison with (I.75) gives

\[ \mathbf{P}_\omega = -\frac{iq}{2\pi\omega}\frac{\mathbf{v}}{v} e^{-i\frac{\omega}{v}z}\delta(x)\delta(y), \tag{I.77} \]

and the equation for \(\mathbf{A}_\omega\) takes the form

\[ \Delta \mathbf{A}_\omega+\frac{\varepsilon(\omega)\omega^2}{c^2}\mathbf{A}_\omega = -\frac{4\pi i\omega}{c}\mathbf{P}_\omega . \tag{I.78} \]

It follows from the preceding consideration that the frequency component \(\omega\) of the field of a uniformly moving charge can be exactly imitated by the field of a continuous set of stationary harmonic oscillators, situated along the electron trajectory and oriented along its velocity. The field of the charge is obtained as a result of superposing the fields of all such oscillators situated over a length \(l\), equal to the path length of the electron in the medium. Carrying out this summation, we obtain\(^{18}\) that the amplitude of the field component of frequency \(\omega=\dfrac{2\pi c}{n\lambda}\) at an observation point at a distance \(R\) from the middle of the segment, much greater than the length of the segment \(l\) (Fig. 6), is equal to

Fig. 6.

\[ \mathbf{A}_\omega = \frac{q}{\pi}\frac{\mathbf{v}}{c} \frac{e^{-i\omega nR/c}}{R}\, \frac{ \sin\left\{\dfrac{\pi l}{\beta n\lambda}\left(1-\beta n\cos\vartheta\right)\right\} }{ \dfrac{2\pi c}{n\lambda}\left(1-\beta n\cos\vartheta\right) }. \tag{I.79} \]

For \(l\ll\lambda\) the field is identical with the field of a point dipole oriented along the velocity of the charge. Thus, for path lengths small in comparison with \(\lambda\), Vavilov–Cherenkov radiation has the same angular distribution as ordinary dipole radiation. For path lengths \(l\) comparable with \(\lambda\) or larger, a characteristic directionality of the radiation appears. For \(l\gg\lambda\) the radiation is emitted at an angle \(\vartheta\) determined by condition (I.19).

I.14. Quantum theory of the Vavilov—Cherenkov effect

Although the classical theory of the Vavilov—Cherenkov effect is well confirmed by experiment, it is of interest to explain this phenomenon by means of the laws of conservation of energy and momentum in quantum physics. This was first done by V. L. Ginzburg \(^{73,74}\) (see also \(^{174,175,100,101,46,186}\), etc.).

Let an electron, moving through a medium with velocity \(\mathbf v\), emit a quantum of light with energy \(\hbar \omega\) in a direction making an angle \(\vartheta\) with \(\mathbf v\). After the emission, the energy and momentum of the electron change. Denote the velocity of the electron after emission of the quantum by \(\mathbf v_1\). The momentum of a free electron is written in the form

\[ \mathbf p=\frac{m\mathbf v}{\sqrt{1-\dfrac{v^2}{c^2}}}. \tag{I.80} \]

The presence of the medium does not change the expression for the electron momentum, since the electron wavelength

\[ \Lambda=\frac{h}{p} \]

for a fast particle is much smaller than the interatomic distances in the medium.

However, the presence of the medium substantially affects the magnitude of the photon momentum

\[ \boldsymbol{\pi}=\hbar \mathbf k=\frac{\mathbf k}{k}\,\frac{2\pi\hbar}{\lambda}, \]

since the wavelengths of light that are important for the Vavilov—Cherenkov effect greatly exceed the interatomic distances in the medium. As the wavelength of light one should take not the wavelength in vacuum \(\left(\lambda=\dfrac{2\pi c}{\omega}\right)\), but that in the medium:

\[ \lambda=\frac{2\pi c}{\omega n} \qquad (n=\sqrt{\varepsilon}). \tag{I.81} \]

This leads to the following expression for the photon momentum in the medium *):

\[ \boldsymbol{\pi}=\frac{\mathbf k}{k}\,\frac{\hbar\omega n}{c}. \tag{I.82} \]

The laws of conservation of energy and momentum in the emission of a quantum by an electron are now written in the form

\[ \left. \begin{aligned} \mathbf p-\mathbf p_1&=\boldsymbol{\pi},\\ \sqrt{p^2+m^2}-\sqrt{p_1^2+m^2}&=\hbar\omega . \end{aligned} \right\} \tag{I.83} \]

From these two equations one can determine \(\cos\vartheta\) (Fig. 7):

\[ \cos\vartheta=\frac{1}{n\beta} \left(1+\frac{1}{2}\,\beta\,\frac{\Lambda}{\lambda}\,\frac{n^2-1}{n}\right), \tag{I.84} \]

where

\[ \Lambda=\frac{h}{mv}\sqrt{1-\beta^2} \]

is the Compton wavelength of the electron before emission—

*) In quantizing the electromagnetic field in a dielectric on the basis of Abraham’s energy-momentum tensor, for the photon momentum one obtains the expression

\[ \boldsymbol{\pi}=\frac{\mathbf k}{k}\,\frac{\hbar\omega}{cn}, \]

which differs from (I.82) by a factor of \(n^2\). However, in this case, upon radiation \(^{87,203}\), the momentum of the charged particle is transferred not only to the quantum but also to the medium; moreover, the momentum transferred to the dielectric is equal to

\[ \boldsymbol{\pi}_d=\frac{\mathbf k}{k}\,(n^2-1)\frac{\hbar\omega}{cn}. \]

The total transferred momentum \(\boldsymbol{\pi}+\boldsymbol{\pi}_d\) is exactly equal to expression (I.82). Therefore all further calculations coincide.

tion of the photon. The second term in the brackets is proportional to \(h\). For \(h \to 0\) we obtain the classical radiation condition.

As is seen from (I.84), taking into account the recoil of the electron in the emission of Vavilov—Cherenkov radiation introduces corrections to the classical consideration of order \(\Lambda/\lambda\)—the ratio of the electron Compton wavelength to the photon wavelength.

Quantum calculations of the Vavilov—Cherenkov effect have been carried out for particles of spin \(\frac{1}{2}^{73,174,100,101,186,150}\), \(1^{77}\), \(\frac{3}{2}^{75,76}\), and \(2^{165}\). The general procedure of calculation is as follows. The interaction between the particle and the electromagnetic field, quantized with allowance for the medium (i.e., with allowance for the equality \(k = h\omega n/c\)), is written down. Then the matrix element of this interaction corresponding to the emission of an electromagnetic quantum is computed. Below are given the results of these calculations for particles of various spins.

Particle of spin \(\frac{1}{2}^{175}\):

\[ -\left(\frac{dW}{dz}\right)_{\text{V.-Ch.}} = \frac{q^2}{c^2}\int \left[ 1-\frac{1}{n^2\beta^2} -\frac{h\omega}{pc\beta}\left(1-\frac{1}{n^2}\right) + \frac{n^2h^2\omega^2}{4p^2c^2}\left(1-\frac{1}{n^4}\right) \right]\omega\,d\omega. \tag{I.85} \]

Fig. 7.

Integration is carried out over the frequency region where the inequality \(\cos\vartheta < 1\) is satisfied (see (I.84)). In the extreme relativistic case \((p\to\infty)\) this formula goes over into the classical Frank and Tamm formula.

Particle of spin \(1\). For such a particle the following results have been obtained for the extreme relativistic case \(^{77}\):

\[ -\left(\frac{dW}{dz}\right)_{\text{V.-Ch.}} = \frac{q^2}{c^3}\int\left(1-\frac{1}{n^2}\right)\omega\,d\omega + \frac{q^2h^2}{4m^2c^5}\int\left(1-\frac{1}{n^2}\right)^2 n^2\omega^3\,d\omega. \tag{I.86} \]

The first term is due to transitions in which the polarization of the particle (the vector meson) remains unchanged. The second term is connected with the transition of the meson from an initial state with longitudinal polarization to a final state with transverse polarization. It is proportional to the square of the magnetic moment \(\mu \sim \frac{qh}{mc}\) and in form coincides with the radiation of a magnetic dipole perpendicular to the velocity. The difference from (I.72) consists in the replacement of \(n^2\) by \(n^4\), which may be caused by the assumptions made in deriving (I.72).

Particle of spin \(\frac{3}{2}\). In Ref. \(^{75}\) results are given for the extreme relativistic case. Let in the initial state the projection of the particle spin on the direction of its momentum be equal to \(\frac{3}{2}h\). Then

\[ -\left(\frac{dW}{dz}\right)_{\text{V.-Ch.}} = \frac{q^2}{c^2}\int\left(1-\frac{1}{n^2}\right)\omega\,d\omega + \frac{q^2h^2}{3m^2c^5}\int\left(1-\frac{1}{n^2}\right)^2 n^2\omega^3\,d\omega. \tag{I.87} \]

The first term is obtained for a transition without change of the spin projection (transition \(+\frac{3}{2}\to+\frac{3}{2}\)); the second—for a transition \((+\frac{3}{2}\to+\frac{1}{2})\). Other transitions in the limiting case under consideration do not occur. The second term can also be regarded as radiation of a magnetic moment.

Particle of spin \(2\). Cherenkov radiation of a particle with spin 2 was considered by M. Ya. Shirobokov \(^{165}\). Without giving here the rather cumbersome results obtained by him, we note that in the expression for the Vavilov—Cherenkov radiation of a particle of spin 2 there are terms describing

both radiation without change of polarization and quadrupole and octupole radiation (a particle with spin 2 has no magnetic dipole moment). Interference terms are also present.

Thus, the quantum theory of the Vavilov—Cherenkov effect makes it possible not only to take into account the recoil experienced by a charge during radiation, but also to consider the radiation of a particle possessing spin. However, the medium is then treated classically. It is assumed that the radiation occurs in a continuous medium whose optical properties can be described by the refractive index or, for nonmagnetic media (which are the ones under discussion), by the dielectric constant. Of interest is such a treatment of Vavilov—Cherenkov radiation in which the refracting medium is regarded not as continuous, but as consisting of many atoms interacting with the electromagnetic field and with the passing particle. The works of S. M. Nimtann \(^{150}\), D. A. Tidman \(^{187,188}\), and U. Fano \(^{58}\) are devoted to such a treatment. In the last work the microscopic theory of the Vavilov—Cherenkov effect is considered most fully and consistently. The author obtains the dispersion equation for the electromagnetic field in the medium by means of the method of collective variables, which gives a transition from the individual variables characterizing each particle of the aggregate of atoms to collective variables characterizing the entire aggregate (“medium”) as a whole \(^{*}\). As collective variables one may choose, for example, the particle density, the electric dipole moment per unit volume, etc. As a result, a macroscopic description of the medium is obtained. In such a scheme, the Vavilov—Cherenkov effect and polarization losses are explained by the interaction of the passing particle with elementary excitations.

II. THE VAVILOV—CHERENKOV EFFECT IN CRYSTALS

II.1. Material equations.

The electrodynamics of crystalline media differs from the electrodynamics of isotropic media in that the dielectric constant \(\varepsilon\) and the magnetic permeability \(\mu\) in crystals are not scalar, but tensor functions of frequency. Maxwell’s equations in a crystal can formally be written in the same way as in an isotropic medium. However, the relation

\[ \mathbf{D}=\varepsilon \mathbf{E} \tag{II.1} \]

in a crystal should be understood as the set of three equalities

\[ \begin{aligned} D_x &= \varepsilon_{11}E_x+\varepsilon_{12}E_y+\varepsilon_{13}E_z,\\ D_y &= \varepsilon_{21}E_x+\varepsilon_{22}E_y+\varepsilon_{23}E_z,\\ D_z &= \varepsilon_{31}E_x+\varepsilon_{32}E_y+\varepsilon_{33}E_z, \end{aligned} \tag{II.2} \]

where \(\varepsilon_{11}, \varepsilon_{12}\), etc., are components of the dielectric-constant tensor. Consequently, in a crystal the vector \(\mathbf{D}\) is not directed along \(\mathbf{E}\) and makes some angle with \(\mathbf{E}\).

Below we shall not take magnetic anisotropy into account and shall assume \(\mu=1\). The case of magnetic anisotropy was analyzed in the work of V. E. Pafomov \(^{153}\). In addition, we choose as coordinate axes the principal axes of the polarization ellipsoid in the crystal. In the chosen system, nonzero turn out—

\(^*\) See, for example, D. Bohm and D. Pines, Phys. Rev. 82, 625 (1951); 85, 338 (1952); 92, 509 (1953).

only the diagonal components of the dielectric-constant tensor, \(\varepsilon_{11}\), \(\varepsilon_{22}\), and \(\varepsilon_{33}\), are nonzero, and therefore relation (I.2) between \(\mathbf D\) and \(\mathbf E\) is substantially simplified, taking the form

\[ D_\alpha=\varepsilon_\alpha E_\alpha, \tag{II.3} \]

where the indices \(\alpha=1,2,3\) correspond to the axes \(x,y,z\).

In passing to the system of principal axes it should be borne in mind that the positions of the principal axes do not remain unchanged in space. Namely, in crystals of the triclinic and monoclinic systems the position of the principal axes of the polarization ellipsoid depends on the frequency. This phenomenon is called dispersion of the axes. Below we shall not take account of axis dispersion, keeping in mind that even in the case of triclinic or monoclinic symmetry the dispersion of the axes in the spectral region of Vavilov—Cherenkov radiation is expressed comparatively weakly.

II.2. Equations for the potentials. Maxwell’s equations in a crystal, as has already been said, have the form (I.1), where, however, under the simplifications we have made, \(\varepsilon\) should be regarded as a diagonal tensor. Following \({}^{71}\), let us pass from the equations for the fields to equations for the potentials \(\mathbf A\) and \(\varphi\), by means of which the fields \(\mathbf E\) and \(\mathbf H\) are expressed in the familiar way (see (I.3)). The equations for the potentials have the form:

\[ \left. \begin{aligned} \Delta \mathbf A-\frac{\varepsilon}{c^2}\frac{\partial^2\mathbf A}{\partial t^2} -\operatorname{grad}\operatorname{div}\mathbf A &= -\frac{4\pi}{c}\mathbf j+\frac{\varepsilon}{c}\frac{\partial}{\partial t}\operatorname{grad}\varphi, \\ \varepsilon_x\frac{\partial^2\varphi}{\partial x^2} +\varepsilon_y\frac{\partial^2\varphi}{\partial y^2} +\varepsilon_z\frac{\partial^2\varphi}{\partial z^2} +\frac{1}{c}\frac{\partial}{\partial t}\operatorname{div}\varepsilon\mathbf A &= -4\pi\rho . \end{aligned} \right\} \tag{II.4} \]

These equations are formally very similar to the equations for \(\mathbf A\) and \(\varphi\) in an isotropic medium (I.4). However, it should be borne in mind that in a crystal, for example, the vector

\[ \frac{\varepsilon}{c^2}\frac{\partial^2\mathbf A}{\partial t^2} \]

has the components

\[ \frac{\varepsilon_x}{c^2}\frac{\partial^2 A_x}{\partial t^2},\quad \frac{\varepsilon_y}{c^2}\frac{\partial^2 A_y}{\partial t^2},\quad \frac{\varepsilon_z}{c^2}\frac{\partial^2 A_z}{\partial t^2}. \]

The system of equations (II.4) can be simplified by imposing on the potentials a suitably chosen supplementary condition. In the electrodynamics of an anisotropic medium one may use a generalization of condition (I.7), putting \({}^{71}\)

\[ \operatorname{div}\varepsilon\mathbf A=0. \tag{II.5} \]

This condition leads to a simplification of the second equation of the system (II.4), which now takes the form

\[ (\nabla\varepsilon\nabla)\varphi=-4\pi\rho, \]

where

\[ (\nabla\varepsilon\nabla)= \varepsilon_x\frac{\partial^2}{\partial x^2} +\varepsilon_y\frac{\partial^2}{\partial y^2} +\varepsilon_z\frac{\partial^2}{\partial z^2}. \tag{II.6} \]

II.3. Qualitative consideration. Thus, if the supplementary condition (II.5) is fulfilled, the system of equations for the potentials of the electromagnetic field in a crystal consists of equation (II.6) for \(\varphi\) and equation (II.4) for \(\mathbf A\). Before carrying out the solution of these equations, let us give a qualitative analysis similar to that which was carried out in I.1 for the case of an isotropic medium. Let us consider the system of equations for \(\varphi\) and \(\mathbf A\) in a crystal in the absence of currents and charges. In this case the equations take the form

\[ \left. \begin{aligned} (\nabla\varepsilon\nabla)\varphi&=0,\\ \Delta\mathbf A-\frac{\varepsilon}{c^2}\frac{\partial^2\mathbf A}{\partial t^2} -\operatorname{grad}\operatorname{div}\mathbf A -\frac{\varepsilon}{c}\frac{\partial}{\partial t}\operatorname{grad}\varphi&=0. \end{aligned} \right\} \tag{II.7} \]

The equations for the potentials have this form in the system of principal axes. In this system the dielectric constant \(\varepsilon\) is a diagonal operator

\[ \varepsilon = \begin{pmatrix} \varepsilon_x & 0 & 0\\ 0 & \varepsilon_y & 0\\ 0 & 0 & \varepsilon_z \end{pmatrix}. \tag{II.8} \]

Let us consider under what conditions equations (II.7) can have as their solution a plane electromagnetic wave (I.15). Substitution of (I.15) into the system of equations (II.7), after simple transformations, gives the following two conditions:

\[ (\mathbf{k}\varepsilon\mathbf{k}) = k^2\left[s_x^2\varepsilon_x(\mathbf{k}\mathbf{v}) + s_y^2\varepsilon_y(\mathbf{k}\mathbf{v}) + s_z^2\varepsilon_z(\mathbf{k}\mathbf{v})\right] =0, \tag{II.9} \]

\[ \frac{s_x^2}{u^2-u_x^2} + \frac{s_y^2}{u^2-u_y^2} + \frac{s_z^2}{u^2-u_z^2} =0, \tag{II.10} \]

where \(u=\frac{\mathbf{k}\mathbf{v}}{k}\), \(v\cos\vartheta\), \(\mathbf{s}=\frac{\mathbf{k}}{k}\) is the unit vector in the direction of \(\mathbf{k}\),

\[ u_i=\frac{c}{\sqrt{\varepsilon_i}}; \]

equation (II.9) is evidently a generalization of equation (I.18б) to an anisotropic medium. Waves satisfying condition (II.9) may be emitted by a charge moving in the medium, and in emitting such waves the charged particle loses energy. These are polarization waves in the crystal.

Let us now consider condition (II.10), imposed on waves that can propagate in the medium with a given velocity. This condition is identical with the equation determining the refractive index \(n\) in the crystal in the direction \(\mathbf{s}\):

\[ \frac{s_x^2}{\dfrac{c^2}{n^2}-u_x^2} + \frac{s_y^2}{\dfrac{c^2}{n^2}-u_y^2} + \frac{s_z^2}{\dfrac{c^2}{n^2}-u_y^2} =0. \tag{II.11} \]

Comparing (II.10) and (II.11), we obtain the condition

\[ u^2=v^2\cos^2\vartheta = \frac{c^2}{n^2(\mathbf{s},\mathbf{kv})}, \tag{II.12} \]

where the refractive index \(n\) depends on the direction of the normal to the wave front and on the frequency of the wave \(\mathbf{kv}\).

As is known, in a crystal the refractive index \(n\) in a given direction has, generally speaking, two values: \(n_1(\mathbf{s},\mathbf{kv})\) and \(n_2(\mathbf{s},\mathbf{kv})\). Correspondingly, equation (II.12) splits into two:

\[ \left. \begin{aligned} \cos\vartheta_1 &= \frac{1}{n_1(\mathbf{s},\mathbf{kv})\beta},\\ \cos\vartheta_2 &= \frac{1}{n_2(\mathbf{s},\mathbf{kv})\beta}. \end{aligned} \right\} \tag{II.13} \]

The set of equations (II.13) is a generalization of condition (I.18a) for Vavilov–Cherenkov radiation to the case of an anisotropic medium. In an anisotropic medium, Vavilov–Cherenkov radiation forms, generally speaking, two complex conical surfaces. The generators of these surfaces are determined by the two equations (II.13). A real Vavilov–

—Cherenkov waves can exist only when the conditions

\[ n_1 \beta > 1 \quad \text{and (or)} \quad n_2 \beta > 1 . \tag{II.14} \]

are satisfied.

Therefore, for certain values of the velocity \(\mathbf v\), there may exist only one conical surface (if only one of the conditions (II.14) is satisfied), or none at all.

The phase velocity of Vavilov—Cherenkov waves in a crystal is determined by the equalities

\[ u_{1,2}=\frac{c}{n_{1,2}} . \tag{II.15} \]

From (II.14) it is clear that, for a Vavilov—Cherenkov wave,

\[ u_{1,2}<v, \]

i.e., also in the case of an anisotropic medium a moving charged particle excites only those waves whose phase velocity is less than the velocity of the particle.

Let us carry out the geometrical construction determining the possible angles between the velocity of the charge in the crystal \(\mathbf v\) and the normal \(\mathbf s\) to the front of a Vavilov—Cherenkov wave. To do this, in the system of principal axes let us lay off, in each direction from the origin of coordinates, segments equal to the phase velocities of light \(u_{1,2}\) (the velocities along the normal) in the given direction. We thus obtain the so-called surface of normals. Now lay off from the origin the vector of the charge velocity \(\mathbf v\). Figure 8 shows one of the sections of the surface of normals by a plane containing the vector \(\mathbf v\). In the section two closed curves are obtained, corresponding to two possible velocities in each direction. The given normal \(\mathbf u\) determines a Vavilov—Cherenkov wave if the straight line drawn through the ends of the vectors \(\mathbf u\) and \(\mathbf v\) is perpendicular to the vector \(\mathbf u\). It is easy to see that in this case one of the conditions (II.14) is satisfied.

Fig. 8.

II.4. Polarization of the Vavilov—Cherenkov wave. However, the condition (II.14) is not the only condition determining Vavilov—Cherenkov radiation in a crystal. The polarization of the wave plays an important role. If a charge emits an electromagnetic wave with electric vector \(\mathbf E e^{i(\mathbf k-\mathbf v t)}\), the inequality \(q(\mathbf E\mathbf v)<0\) must be satisfied, where \(q\) is the magnitude and \(\mathbf v\) the velocity of the moving charge. This is explained by the fact that, in emitting an electromagnetic wave, the charge must perform work against the field, and the amount of work performed per unit path is proportional to \(q(\mathbf E\mathbf v)\). Radiation of a wave for which the product \(q(\mathbf E\mathbf v)\) is zero or positive is impossible, since it is not accompanied by work of the charge against the forces of the field. Therefore, for example, in a negative uniaxial crystal (where the velocity of ordinary rays is less than the velocity of extraordinary rays), such as Iceland spar, when the charge moves along the optical axis (the optical axis in a crystal is the direction in which both phase velocities are equal), Vavilov—Cherenkov radiation consists only of a cone of extraordinary waves, although the phase velocity of ordinary waves in this case

obviously satisfies condition (II.14). This is explained by the fact that in a uniaxial crystal the electric vector of the ordinary wave is always perpendicular to the optical axis (i.e. \(\mathbf E\mathbf v=0\), since the vector \(\mathbf v\) in our example is parallel to the optical axis).

Of course, the absence or presence of energy losses to radiation follows unambiguously from the form of the solutions of the system (II.4), which we shall give below. Nevertheless, it is useful to bear in mind the simple considerations set forth above, which are valid for each harmonic of the solution.

Thus, in a crystal Vavilov—Cherenkov radiation is determined by two conditions: condition (II.14) and the condition \(q(\mathbf E\mathbf v)<0\), where \(\mathbf E\) is the electric vector of the emitted wave. In an isotropic medium the second condition can always be fulfilled for waves satisfying condition (I.20).

II.5. Field of a moving point charged particle

We now give expressions for the potentials \(\mathbf A\) and \(\varphi\) of the field of a charge moving in a crystal with velocity \(\mathbf v\):

\[ \left. \begin{aligned} \varphi &= \frac{q}{2\pi^2}\int \frac{e^{i\mathbf k(\mathbf x-\mathbf v t)}\,d\mathbf k}{(\mathbf k\varepsilon\mathbf k)},\\[6pt] \mathbf A &= -\,\frac{q^2}{2\pi^2 c}\int \Lambda^{-1}\left[ \mathbf s-\mathbf k\,\frac{(\mathbf k\varepsilon\Lambda^{-1}\mathbf s)} {(\mathbf k\varepsilon\Lambda^{-1}\mathbf k)} \right]e^{i\mathbf k(\mathbf x-\mathbf v t)}\,d\mathbf k, \end{aligned} \right\} \tag{II.16} \]

where

\[ \mathbf s=\mathbf v-\varepsilon\mathbf k\,\frac{\mathbf k\mathbf v}{(\mathbf k\varepsilon\mathbf k)}, \tag{II.17} \]

\[ \Lambda=\varepsilon\,\frac{(\mathbf k\mathbf v)^2}{c^2}-k^2. \tag{II.18} \]

Let us recall that in the system of principal axes of the crystal, which we chose,* the quantities \(\Lambda\) and \(\varepsilon\) are diagonal operators. Therefore, for example,

\[ \left. \begin{aligned} (\mathbf k\varepsilon\mathbf k) &= \varepsilon_x k_x^2+\varepsilon_y k_y^2+\varepsilon_z k_z^2,\\[4pt] \varepsilon\mathbf k &= (\varepsilon_x k_x,\ \varepsilon_y k_y,\ \varepsilon_z k_z),\\[4pt] (\mathbf k\varepsilon\Lambda^{-1}\mathbf k) &= \frac{\varepsilon_x k_x^2}{\Lambda_x} +\frac{\varepsilon_y k_y^2}{\Lambda_y} +\frac{\varepsilon_z k_z^2}{\Lambda_z}. \end{aligned} \right\} \tag{II.19} \]

Formulas (II.16) completely determine the field in the crystal produced by the moving charge. The potentials \(\mathbf A\) and \(\varphi\) can also be determined by Hamilton’s method \(^{71}\).

II.6. The Vavilov—Cherenkov effect in a uniaxial crystal

Let us now consider the main features of Vavilov—Cherenkov radiation in a crystal. This question was studied in a paper by V. L. Ginzburg \(^{71}\) and in a number of subsequent works—by A. A. Kolomenskii \(^{118}\), M. I. Kaganov \(^{109-113}\), V. E. Pafomov \(^{152,154}\), I. V. Polyubarinov \(^{159}\), Tanaka \(^{185}\), and others \(^{170,172}\).

For simplicity we shall restrict ourselves to the case of a uniaxial crystal and to the motion of a point charge parallel and perpendicular to its optical axis. Let us direct the \(z\)-axis of the system of principal axes along the optical axis of the crystal. Obviously, in the chosen coordinate system \(\varepsilon_x=\varepsilon_y=\varepsilon_r,\ \Lambda_x=\Lambda_y=\Lambda_r,\)

*) Below, in considering the Vavilov—Cherenkov effect in a gyrotropic crystal, a generalization of formulas (II.16) will be given that is free of this restriction.

where

\[ \Lambda_{\alpha}=\varepsilon_{\alpha}\frac{(k v)^2}{c^2}-k^2. \]

Below we shall not consider polarization losses, determined by the dispersion equation (II.9), and shall consider only the energy losses due to Vavilov—Cherenkov radiation. Vavilov—Cherenkov waves satisfy condition (II.10), or, what is the same, the condition \((\mathbf{k}\varepsilon\Lambda^{-1}\mathbf{k})=0\). This equation has, generally speaking, two different roots, which in a uniaxial crystal corresponds to the emission of ordinary and extraordinary waves. However, in addition to condition (II.10), the Vavilov—Cherenkov wave must also satisfy certain conditions imposed on the polarization (see above).

Let us now proceed directly to the consideration of Cherenkov radiation of a charge in a uniaxial crystal in two principal cases.

a) The charge moves parallel to the optical axis

\[ \text{(along the } z \text{ axis, Fig. 9)} \]

In this case ordinary waves are not emitted, since the vector of the electric field strength of an ordinary wave in a uniaxial crystal is perpendicular to the optical axis and, consequently, to the velocity of the charge. On the contrary, extraordinary waves are polarized in such a way that there is always a nonzero projection \(E_z\) of the electric vector onto the direction of the charge velocity.

Thus, when a charge moves along the optical axis of a uniaxial crystal, only a cone of extraordinary waves can be emitted. From symmetry considerations it is clear that this cone is circular, with a uniform distribution of intensity over the generators. The resulting radiation pattern qualitatively coincides with the case of an isotropic body.

Fig. 9.

However, the angle which the normal to the Vavilov—Cherenkov wave makes with the charge velocity, as well as the radiation intensity in the case under consideration, turn out to be different. Let us determine, for example, from the dispersion equation (II.10) the angle which the normal to the front of the Vavilov—Cherenkov wave makes with the line of motion of the charge. Eliminating the denominators and remembering that in our case \(u_x=u_y=u_r\), we obtain

\[ (u^2-u_r^2)\left[(s_x^2+s_y^2)(u^2-u_z^2)+s_z^2(u^2-u_r^2)\right]=0. \tag{II.20} \]

This equation has two roots. The first of them, \(u^2=u_r^2\), or \(\cos^2\vartheta=\dfrac{1}{\varepsilon_r\beta^2}\), corresponds to the emission of ordinary waves. But even in the case \(\varepsilon_r\beta^2>1\), ordinary waves are not emitted by a charge moving along the optical axis, since they are polarized perpendicular to the axis. The second root gives \(\cos^2\vartheta_e\) for the extraordinary wave:

\[ \cos^2\vartheta_e = \frac{1}{\varepsilon_z\beta^2} - \frac{1}{ 1+\dfrac{1}{\varepsilon_z\beta^2}-\dfrac{1}{\varepsilon_r\beta^2} }. \tag{II.21} \]

The quantities \(\varepsilon_z\) and \(\varepsilon_r\) are functions of the frequency \(\omega\).

Condition (II.21) can be written somewhat differently:

\[ \cos^2 \vartheta_e = \frac{1}{1+\dfrac{\varepsilon_z(\omega)}{\varepsilon_r(\omega)} \left[\varepsilon_r(\omega)\beta^2-1\right]} . \tag{II.21a} \]

It follows at once from this that radiation of extraordinary Vavilov—Cherenkov waves takes place only in the spectral region where the inequality

\[ \frac{\varepsilon_z}{\varepsilon_r}\left(\varepsilon_r\beta^2-1\right)>0 \tag{II.22} \]

is satisfied, since only in this case is \(\cos^2\vartheta_e<1\).

A more detailed investigation shows that the radial component of the wave vector of the Vavilov—Cherenkov wave is determined by the equality

\[ k_r^2=k_x^2+k_y^2= \frac{\omega^2}{v^2}\frac{\varepsilon_z}{\varepsilon_r} \left(\varepsilon_r\beta^2-1\right). \tag{II.23} \]

Therefore, in the presence of radiation condition (II.22) must be satisfied, since otherwise the radial component of the wave vector \(k_r\) becomes imaginary and the field is exponentially damped with increasing distance from the \(z\)-axis.

The magnitude of the energy losses of the charge to radiation of extraordinary Vavilov—Cherenkov waves can be determined by calculating the braking force acting on the charge due to the field of these waves. Obviously,

\[ \left(\frac{dW}{dz}\right)_{\text{V.-Ch.}} = q\left.\frac{(\mathbf{E}\mathbf{v})}{c}\right|_{\substack{\mathbf{x}=\mathbf{v}t\\ k_r^2>0}} = -\left.\frac{q}{cv}\left(\mathbf{v}\frac{\partial \mathbf{A}}{\partial t}\right)\right|_{\substack{\mathbf{x}=\mathbf{v}t\\ k_r^2>0}} . \tag{II.24} \]

Calculations give\(^{71,118,152}\)*)

\[ \left(\frac{dW}{dx}\right)_{\text{V.-Ch.}} = -\frac{q^2}{c^2} \int_{\frac{\varepsilon_z}{\varepsilon_r}(\varepsilon_r\beta^2-1)>0} \left|1-\frac{1}{\varepsilon_r\beta^2}\right|\omega\,d\omega , \tag{II.25} \]

where the inequality under the integral sign gives the region of integration.

Formula (II.25) has one interesting feature. Inequality (II.22) in a transparent medium can also be satisfied for \(\varepsilon_r\to 0\), \(\varepsilon_z<0\). But then the integrand, which gives the spectral intensity of Vavilov—Cherenkov radiation, becomes infinite, and the loss integral diverges. The problem of determining the losses in this case was addressed in the work of A. G. Sitenko and M. I. Kaganov\(^{170}\). Their considerations constitute a generalization to an anisotropic medium of the arguments\(^{93,26}\) presented in Section 1.8. The point is that in inequality (II.22), which gives the spectral region of integration, the quantity

\[ \frac{v^2}{\omega^2}k_r^2, \]

stands on the left, where \(k_r\) is the radial component of the wave vector of the Vavilov—Cherenkov wave (see formula (II.23)). As \(\varepsilon_r\to 0\), the quantity \(k_r\) becomes infinite. This corresponds to the emission of infinitely short Vavilov—Cherenkov waves. But such a process cannot be correctly described by the classical electrodynamics of a medium. Therefore, the spectral region of Vavilov—Cherenkov radiation for motion of a charge along the optical axis of a uniaxial crystal is more correctly determined by the double inequality

\[ k_{r,\max}^2> \frac{\omega^2}{v^2} \frac{\varepsilon_z(\omega)}{\varepsilon_r(n)} \left[\varepsilon_r(\omega)\beta^2-1\right]>0, \tag{II.26} \]

\[ \text{*) In work }^{71}\text{ this formula and the subsequent formula (II.32) contain misprints.} \]

where the quantity \(k_{r,\max}\) is the maximum value of the wave number for which macroscopic electrodynamics is still valid. Obviously, if inequality (II.26) is satisfied, \(\varepsilon_r\) cannot vanish, and the loss integral becomes finite.

b) The charge moves perpendicular to the optical axis

(along the \(x\)-axis, Fig. 10)

In this case both ordinary and extraordinary waves may be emitted; consequently, two Vavilov–Cherenkov radiation cones may simultaneously exist. The aperture of the cone of ordinary waves is obtained by setting equal to zero the first factor in equation (II.20):

\[ \cos \vartheta_0=\frac{1}{\sqrt{\varepsilon_r}\beta} =\frac{1}{n_0\beta}, \tag{II.27} \]

where \(n_0\) is the refractive index for ordinary rays.

The cone of ordinary waves is thus circular. However, as we shall see, the radiation intensity is not the same along its different generators.

Setting equal to zero the second factor of (II.20) gives the cone of normals of the extraordinary waves. Put

\[ \begin{aligned} s_x&=\cos\vartheta,\\ s_y&=\sin\vartheta\cos\varphi,\\ s_z&=\sin\vartheta\sin\varphi \end{aligned} \qquad \tag{II.28} \]

(\(\varphi\) is the angle between the \(y\)-axis and the projection of the wave vector onto the \(yz\)-plane). Then for the cone of extraordinary waves we obtain

\[ \cos^2\vartheta_e= \frac{\varepsilon_r\cos^2\varphi+\varepsilon_z\sin^2\varphi} {(\varepsilon_z-\varepsilon_r)\sin^2\varphi+\varepsilon_r\varepsilon_z\beta^2}. \tag{II.29} \]

Fig. 10.

Consequently, the cone of extraordinary waves is not circular, and the aperture of the cone depends not only on frequency, but also on the angle \(\varphi\). The condition for emission is, as usual, the requirement that the quantity (II.29) not exceed unity. This condition may be formulated differently by introducing the projection \(k_{yz}\) of the wave vector onto the \(yz\)-plane and requiring satisfaction of the inequality

\[ k_{yz}^2=k_y^2+k_z^2 =\frac{\omega^2}{v^2}\, \frac{\varepsilon_r(\varepsilon_z\beta^2-1)} {\varepsilon_r\cos^2\varphi+\varepsilon_z\sin^2\varphi} >0. \tag{II.30} \]

If this inequality is not satisfied, the field decreases exponentially with distance from the \(x\)-axis (the line of motion of the charge), i.e. there is no radiation.

The energy losses due to Vavilov–Cherenkov radiation are determined by means of formulas (II.16) and (II.24). We obtain \(^{71,152,172}\)

\[ \frac{dW_0}{dx} = -\frac{q^2}{2\pi c^2} \int \left( 1-\frac{1}{\varepsilon_r\beta^2} \right) \frac{\cos^2\varphi\, d\varphi\, \omega\, d\omega} {\cos^2\varphi+\frac{1}{\varepsilon_r\beta^2}\sin^2\varphi}, \tag{II.31} \]

\[ \frac{dW_e}{dx} = -\frac{q^2}{2\pi^2 c^2} \int \left( 1-\frac{1}{\varepsilon_z\beta^2} \right) \frac{\sin^2\varphi\, d\varphi\, \varepsilon_z(\omega)\,\omega\, d\omega} {(\varepsilon_z\sin^2\varphi+\varepsilon_r\cos^2\varphi) (\varepsilon_r\beta^2\cos^2\varphi+\sin^2\varphi)}. \tag{II.32} \]

where the indices \(o\) and \(e\) denote energy losses due to the emission of ordinary and extraordinary waves, respectively. The region of integration over \(\omega\) in the integral (II.31) is determined by the inequality \(\varepsilon_r\beta^2>1\), and in the integral (II.32) by the inequality (II.30). It is interesting to note that in the plane \(xz\left(\varphi=\dfrac{\pi}{2}\right)\) ordinary waves are not emitted, while in the plane \(xy(\varphi=0)\) extraordinary waves are not emitted. This is explained by the fact that the electric-intensity vectors of these waves are perpendicular to the velocity of the charge.

The spectral distribution of Vavilov—Cherenkov radiation when the charge moves perpendicular to the optical axis can be found by carrying out in (II.31) and (II.32) the integration over \(\varphi\). Suppose that \(\varepsilon_r\) and \(\varepsilon_z\) are constant positive quantities, independent of the frequency and satisfying the radiation conditions for both ordinary and extraordinary waves. Integration over \(\varphi\) for this case gives

\[ \left. \begin{aligned} \frac{dW_o}{dx}&=-\frac{q^2}{c^2}\int\left(1-\frac{1}{\sqrt{\varepsilon_r}\beta}\right)\omega\,d\omega,\\ \frac{dW_e}{dx}&=-\frac{q^2}{c^2}\int\left(1-\frac{1}{\sqrt{\varepsilon_z}\beta}\right)\frac{\omega\,d\omega}{\sqrt{\varepsilon_r}\beta}. \end{aligned} \right\} \tag{II.33} \]

The total intensity of radiation at frequency \(\omega\) is expressed by the formula

\[ I_e(\omega)+I_o(\omega)=\left(1-\frac{1}{\sqrt{\varepsilon_r\varepsilon_z\beta^2}}\right)\omega. \tag{II.34} \]

For \(\varepsilon_r=\varepsilon_z\), from (II.34) we obtain the radiation intensity in an isotropic medium.

As is seen from this formula, the total radiation intensity depends on the product \(\varepsilon_r\varepsilon_z\). Thus, when the charge moves perpendicular to the optical axis, additional possibilities arise in the sense of increasing the total radiation intensity. From comparison of formula (II.34) with formula (II.25), which gives the total radiation intensity when the charge moves along the optical axis, it is seen that for \(\varepsilon_z>\varepsilon_r\) the charge loses more energy if it moves perpendicular to the optical axis, whereas if \(\varepsilon_z<\varepsilon_r\), the charge loses more energy if it moves along the axis. This circumstance may prove very significant for the manufacture of Vavilov—Cherenkov counters from crystalline materials.

The case of radiation of a charge in a uniaxial crystal when moving at an arbitrary angle to the optical axis was investigated in the work of K. Tanaka \(^{185}\) and earlier by V. E. Pafomov \(^{152}\). Owing to the asymmetry of the cone of Vavilov—Cherenkov radiation, a charge moving in a crystal is acted upon not only by a braking force, but also by a force deflecting the charge from rectilinear motion. This force does not depend on the sign of the charge. In the crystal there are directions such that, when moving along them, the charge experiences no deflecting force, i.e. directions of stable motion. If the velocity of the charge makes some angle with such a direction, then the reaction force of the Vavilov—Cherenkov radiation tends to return the electron to the direction of stable motion.

Adjacent to the Vavilov—Cherenkov effect in a uniaxial crystal is the so-called parametric Vavilov—Cherenkov effect, considered by N. A. Khizhnyak and Ya. B. Fainberg \(^{199}\). Consider a medium consisting of alternating plane layers of two isotropic dielectrics: a layer of thickness \(l_1\) made of dielectric \(\varepsilon_1,\mu_1\), adjoining it a layer of thickness \(l_2\) made of dielectric \(\varepsilon_2,\mu_2\), and then the layers alternate periodically. For electromagnetic waves whose wavelength is sufficiently large in comparison with \(l_1+l_2\),

the inhomogeneous isotropic medium under consideration behaves like a uniaxial crystal with optical axis perpendicular to the plane layers, and*)

\[ \varepsilon = \begin{pmatrix} \varepsilon_r & 0 & 0\\ 0 & \varepsilon_r & 0\\ 0 & 0 & \varepsilon_z \end{pmatrix}, \qquad \mu = \begin{pmatrix} \mu_r & 0 & 0\\ 0 & \mu_r & 0\\ 0 & 0 & \mu_z \end{pmatrix}, \]

where

\[ \varepsilon_r=\frac{l_1\varepsilon_1+l_2\varepsilon_2}{l_1+l_2}, \qquad \mu_r=\frac{l_1\mu_1+l_2\mu_2}{l_1+l_2}, \]

\[ \varepsilon_z=\frac{\varepsilon_1\varepsilon_2(l_1+l_2)}{l_1\varepsilon_2+l_2\varepsilon_1}, \qquad \mu_z=\frac{\mu_1\mu_2(l_1+l_2)}{l_1\mu_2+l_2\mu_1}. \]

A charge moving through such a finely layered medium can emit Vavilov—Cherenkov waves.

The radiation of a charge in a biaxial crystal gives a more complicated picture, but the qualitative character of the radiation does not change. For a biaxial crystal, in contrast to a uniaxial one, the dispersion of the axes is important. If the direction of motion of the electron coincides with the optical axis for the frequency \(\omega\), then for other frequencies it can no longer be assumed that the electron moves along the optical axis.

II.7. Phase and group velocities of Vavilov—Cherenkov waves in a crystal. The conditions for Vavilov—Cherenkov radiation in a crystal (II.11)—(II.13) determine the angle formed by the normal \(\mathbf{k}\) to the Vavilov—Cherenkov wave and the velocity of the particle \(\mathbf{v}\). As is known, in a crystal the normal to the wave does not coincide with the direction of the energy flux, which is determined by the Poynting vector

\[ \mathbf{S}=\frac{c}{4\pi}[\mathbf{E}\mathbf{H}]. \]

The vector \(\mathbf{S}\) determines the direction of propagation of the ray in the crystal. For an arbitrary linear nonabsorbing crystal, the direction of the vector \(\mathbf{S}\) coincides with the direction of the group velocity of the wave:

\[ \mathbf{w}=\frac{\partial \omega}{\partial \mathbf{k}}, \]

and, thus, the group velocity of the Vavilov—Cherenkov wave is directed differently from the phase velocity. Whereas the phase velocity of the Vavilov—Cherenkov wave always makes an acute angle with the velocity of the particle, its group velocity in certain frequency intervals may be directed at an obtuse angle to the velocity of the particle \({}^{154}\).

The fact that in a crystal the group velocity of the Vavilov—Cherenkov wave does not coincide with the phase velocity leads to a number of interesting consequences, discussed by V. E. Pafomov \({}^{154}\). One of these consequences we have just indicated: the ray may make an obtuse angle with the velocity of the charge. The second feature concerns the choice of solutions of the field equations in the crystal.

Following \({}^{154}\), let us consider the motion of a charge along the optical axis of a uniaxial crystal. In this case the square of the radial component of the wave vector of the Vavilov—Cherenkov wave is determined by equality (II.23). From this relation \(k_r\) is determined up to a sign. Retarded potentials correspond to a positive projection of the wave vector on the radius, and advanced potentials to a negative one. Usually, as a

*) See, for example, S. M. Rytov, ZhETF 29, 605 (1955). The field averaged over the period of the structure is meant.

of the solutions, one takes the retarded potentials. However, this can be done only when the energy flux is thereby directed away from the radiating particle, i.e., when the projections of the wave vector \(\mathbf{k}\) and the group velocity \(\mathbf{w}\) onto the radius have the same signs. If, however, \(k_r\) and \(w_r\) have different signs, then the energy from the moving charge is carried away by the advanced potentials, which must then be taken as the solution.

II. 8. The Vavilov—Cherenkov effect in an isotropic optically active medium

Above we considered the Vavilov—Cherenkov effect in bodies that do not possess optical activity (do not exhibit rotation of the plane of polarization). Let us now consider certain features of the Vavilov—Cherenkov effect in media that possess optical activity. In what follows we shall also call such media gyrotropic.

We shall begin with the consideration of isotropic gyrotropic media, i.e., such media for which the optical activity is the same in all directions. A classic example of such media is an aqueous solution of sugar.

In an isotropic gyrotropic medium Maxwell’s equations are written in the form (I.1), as in an ordinary isotropic medium. The difference lies only in the material equations. Suppose that all field vectors in an isotropic gyrotropic medium are proportional to the factor \(e^{i\mathbf{k}(\mathbf{x}-\mathbf{v}t)}\). Then the material equations may be written in the form

\[ \left. \begin{aligned} \mathbf{D} &= \varepsilon \mathbf{E} + \frac{i\gamma}{k}[\mathbf{k}, \mathbf{E}],\\ \mathbf{B} &= \mathbf{H} - i\frac{(\mathbf{k}\mathbf{v})}{kc}\gamma \mathbf{E}, \end{aligned} \right\} \tag{II.35} \]

where \(\gamma\) is the parameter determining the rotation of the plane of polarization. (See, for example, M. Born, Optics, Kharkov—Kiev, 1937.)

In formula (II.35) the material equations are defined for plane electromagnetic waves and therefore for arbitrary fields, since arbitrary fields can be expanded into plane waves.

Substitution of the wave \(e^{i\mathbf{k}(\mathbf{x}-\mathbf{v}t)}\) into the system of Maxwell equations for an isotropic optically active medium gives the following relation between \(\mathbf{k}\) and \(\mathbf{k}\mathbf{v}\):

\[ \left(\frac{k^2c^2}{(\mathbf{k}\mathbf{v})^2}-\varepsilon+2\gamma\right) \left(\frac{k^2c^2}{(\mathbf{k}\mathbf{v})^2}-\varepsilon-2\gamma\right)=0. \tag{II.36} \]

This equation gives two values of \(\dfrac{\mathbf{k}\mathbf{v}}{kc}=\beta\cos\vartheta\), i.e., two values for the angle \(\vartheta\) between \(\mathbf{k}\) and \(\mathbf{v}\):

\[ \cos^2\vartheta_{1,2}=\frac{1}{\beta^2(\varepsilon \pm 2\gamma)}. \tag{II.37} \]

To these values of \(\cos\vartheta\) there correspond two plane electromagnetic waves, circularly polarized in opposite directions. The condition for Vavilov—Cherenkov radiation, consequently, is the fulfillment of the inequalities

\[ \beta^2(\varepsilon+2\gamma)>1 \quad \text{and (or)} \quad \beta^2(\varepsilon-2\gamma)>1. \tag{II.38} \]

Depending on the speed of the charge, in an isotropic optically active medium either one wave of frequency \(\omega\), circularly polarized, may be emitted, or two waves, circularly polarized with opposite directions of rotation, or none at all.

Equation (II.36) formally coincides with the equation for the refractive index in an isotropic optically active medium, if in it one pro-

make the obvious replacement \(\dfrac{kc}{(kv)} = n\). Consequently, in the case under consideration the refractive index has two values:

\[ n_{1,2}^2=\varepsilon \pm 2\gamma . \tag{II.39} \]

The energy losses of a charge to Vavilov—Cherenkov radiation in an isotropic optically active medium are expressed by the formula

\[ \frac{dW}{dx} = -\frac{1}{2}\frac{q^2}{c^2} \left\{ \int_{n_1^2\beta^2>1} \left(1-\frac{1}{n_1^2\beta^2}\right)\omega\,d\omega + \int_{n_2^2\beta^2>1} \left(1-\frac{1}{n_2^2\beta^2}\right)\omega\,d\omega \right\}, \tag{II.40} \]

where the inequalities determine the spectral regions of integration*). If the gyration parameter \(\gamma\) tends to zero, formula (II.40) goes over into (I.41), where one must put \(\mu=1\), i.e., it gives the energy losses to Vavilov—Cherenkov radiation in an isotropic medium. The gyration parameter \(\gamma\), in comparison with \(\varepsilon\), has the order of magnitude \(\dfrac{a}{\lambda}\), where \(a\) is the size of the molecules of the medium, and \(\lambda\) is the wavelength of the radiation. Therefore the energy losses to Vavilov—Cherenkov radiation in an isotropic optically active medium differ little from the losses in an isotropic medium without optical activity.

II.9. The Vavilov—Cherenkov effect in a gyrotropic crystal. Let us now consider the Vavilov—Cherenkov effect in gyrotropic crystals. In gyrotropic crystals the tensor of the dielectric constant \(\varepsilon'_{ik}\) can be written in the form

\[ \varepsilon'_{ik}=\varepsilon_{ik}+i\gamma_{ik}, \tag{II.41} \]

where \(\varepsilon_{ik}\) is the symmetric tensor of the dielectric constant, and \(\gamma_{ik}\) is the antisymmetric gyration tensor. The components of the tensors \(\varepsilon\) and \(\gamma\) are real functions of the frequency; moreover, \(\varepsilon_{ik}\) are even functions of \(\omega\), while \(\gamma_{ik}\) are odd functions. Maxwell’s equations in a gyrotropic crystal have the form (I.1), with the difference that instead of \(\varepsilon\) one must substitute \(\varepsilon'\) (II.41) and put \(\mu=1\).

The field of a charge moving uniformly and rectilinearly in a gyrotropic crystal is described by the potentials (II.16), if in (II.16) \(\varepsilon\) is replaced by \(\varepsilon'\). In this case formulas (II.17) and (II.18) also remain valid, while the relations (II.19) are replaced by the following:

\[ \left. \begin{aligned} ( \mathbf{k}\varepsilon' \mathbf{k})&=(\mathbf{k}\varepsilon \mathbf{k}) =\sum_{\alpha,\beta} k_\alpha k_\beta \varepsilon_{\alpha\beta},\\ \varepsilon'\mathbf{k} &= \left( \sum_\alpha \varepsilon'_{1\alpha}k_\alpha,\, \sum_\alpha \varepsilon'_{2\alpha}k_\alpha,\, \sum_\alpha \varepsilon'_{3\alpha}k_\alpha \right),\\ (\mathbf{k}\varepsilon'\Lambda^{-1}\mathbf{s}) &= \sum_{\alpha,\beta}k_\alpha s_\beta(\varepsilon'\Lambda^{-1})_{\alpha\beta}. \end{aligned} \right\} \tag{II.42} \]

It is not difficult to show that Vavilov—Cherenkov waves in a gyrotropic medium must satisfy the condition

\[ (\mathbf{k}'\Lambda^{-1}\mathbf{k})=0. \tag{II.43} \]

Since, according to (II.18),

\[ \Lambda=\varepsilon'\frac{(kv)^2}{c^2}-k^2 = k^2\left(\frac{\varepsilon'}{n^2}-1\right) \tag{II.44} \]

*) The polarization losses in this case, as in an inactive medium, are determined by the zeros of \(\varepsilon\).

(we took into account that \(\frac{kc}{k\nu}=n\)), then equation (II.43) coincides with the equation for determining the refractive index in a gyrotropic crystal. In a gyrotropic crystal the refractive index has two values. Therefore, just as in the case of a crystal without optical activity, the angle which the normal to the Vavilov—Cherenkov wave makes with the particle velocity is determined by the equalities*)

\[ \cos^2 \vartheta_{1,2}=\frac{1}{n_{1,2}^2\beta^2}. \tag{II.45} \]

The difference from the case of a nonactive crystal consists in the fact that \(n_1\) and \(n_2\) now depend on the components of the gyration tensor.

In addition to Vavilov—Cherenkov losses, a charge moving in a gyrotropic crystal may also experience polarization losses—at those frequencies at which all components of the tensors \(\varepsilon\) and \(\gamma\) simultaneously vanish. If \(\omega_\alpha\) is such a frequency value, then in a neighborhood of \(\omega_\alpha\) the expansion is valid

\[ \varepsilon'_{ik}=\left(\varepsilon^\alpha_{ik}+i\gamma^\alpha_{ik}\right)(\omega-\omega_\alpha)+\text{terms of higher order in }(\omega-\omega_\alpha). \]

Vavilov—Cherenkov radiation in a gyrotropic crystal was considered by A. A. Kolomenskii in 1951 by means of the Hamilton method, developed by him for the case of a gyrotropic medium \({}^{119}\). The quantum theory of the Vavilov—Cherenkov effect in a gyrotropic medium was given by I. V. Polubarinov \({}^{159}\) in 1953. In the work of A. G. Sitenko and A. A. Kolomenskii \({}^{172}\), which appeared in 1956, Vavilov—Cherenkov radiation for a charge moving in a uniaxial gyrotropic crystal was considered in detail. The tensor \(\varepsilon'\) in this case has the form

\[ \varepsilon'= \begin{pmatrix} \varepsilon_r & -i\gamma & 0\\ i\gamma & \varepsilon_r & 0\\ 0 & 0 & \varepsilon_z \end{pmatrix}. \tag{II.46} \]

In work \({}^{172}\) expressions were obtained for the energy losses of a charge moving in a uniaxial gyrotropic crystal parallel and perpendicular to the optical axis.

For the case of motion of the charge parallel to the optical axis, the losses to Vavilov—Cherenkov radiation are expressed by the formula

\[ -\frac{dW}{dz} = \frac{q^2}{v^2} \int_{n_1^2\beta^2>1} \frac{(1-\varepsilon_r\beta^2)(n_1^2-\varepsilon_r)-\beta^2\gamma} {\varepsilon_r(n_2^2-n_1^2)} \,\omega\,d\omega + \frac{q^2}{v^2} \int_{n_2^2\beta^2>1} \frac{(1-\varepsilon_r\beta^2)(n_2^2-\varepsilon_r)-\beta^2\gamma} {\varepsilon_r(n_1^2-n_2^2)} \,\omega\,d\omega, \tag{II.47} \]

where \(n_{1,2}\) are the values of the refractive index

\[ n_{1,2}(\vartheta)= \]

\[ = \frac{ (\varepsilon_r-\varepsilon_z)\sin^2\vartheta+\varepsilon_r\varepsilon_z(1+\cos^2\vartheta) \pm \sqrt{\left(\varepsilon_r^2-\gamma^2-\varepsilon_r\varepsilon_z\right)\sin^4\vartheta+4\gamma^2\varepsilon_z\cos^2\vartheta} } {2(\varepsilon_r\sin^2\vartheta+\varepsilon_z\cos^2\vartheta)} \tag{II.48} \]

at the angles \(\vartheta\) determined by the radiation condition (II.45). The expressions for the losses in the case of motion perpendicular to the optical axis prove to be

*) For the real existence of the Vavilov—Cherenkov wave it is also necessary that the corresponding conditions imposed on the polarization of the wave be fulfilled (see above).

rather cumbersome, but, as in the case of an isotropic optically active medium, the parameter \(\gamma\) is of order \(\dfrac{a}{\lambda}\) in comparison with \(\varepsilon\), and the difference from the case of a gyrotropic crystal is apparently small.

II.10. The Vavilov—Cherenkov Effect in an Electron Plasma Placed in a Magnetic Field. Optical activity may be produced artificially—by imposing a magnetic field on the medium. For the theory of the Vavilov—Cherenkov effect, the most interesting case of such a medium, in which the imposition of a magnetic field produces optical activity, is an electron plasma. If no magnetic field is imposed on an electron plasma, the Vavilov—Cherenkov effect is altogether impossible in it, since the dielectric constant of an electron plasma

\[ \varepsilon(\omega)=1-\frac{\omega_0^2}{\omega^2} \qquad \left(\omega_0^2=\frac{4\pi n e^2}{m}\right) \tag{II.49} \]

is less than unity. V. I. Veksler pointed out that the Vavilov—Cherenkov effect in an electron plasma may become possible if the plasma is placed in a magnetic field. This phenomenon was considered by A. A. Kolomenskii \(^{116,117,120}\).

An electron plasma placed in a homogeneous magnetic field behaves as a uniaxial gyrotropic crystal with optical axis parallel to the imposed field. The tensor of the dielectric constant of the plasma in a magnetic field has the form (II.46), where

\[ \varepsilon_r=\frac{\omega^2-\omega_0^2-\omega_H^2}{\omega^2-\omega_H^2}, \qquad \varepsilon_z=1-\frac{\omega_0^2}{\omega^2}, \qquad \gamma=\frac{\omega_0^2\omega_H}{\omega(\omega^2-\omega_H^2)}, \qquad \left(\omega_H=\frac{eH}{mc}\right). \tag{II.50} \]

Determination of the refractive index \(n\) from equation (II.43) shows that Vavilov—Cherenkov radiation is possible in two frequency regions:

\[ \text{a) } \omega<\omega_0 \qquad \text{and} \qquad \text{b) } \omega_0<\omega<\sqrt{\omega_0^2+\omega_H^2}. \tag{II.51} \]

In the relativistic case (\(\beta=1\)) only waves occupying the spectral region (b) are emitted. It is interesting to note that in a plasma placed in a magnetic field, Vavilov—Cherenkov radiation also occurs at small charge velocities (\(\beta\ll 1\)).

In work \(^{120}\) the energy losses to Vavilov—Cherenkov radiation by a charge moving in a plasma with a magnetic field along the optical axis (parallel to the imposed field) were determined.

We shall give the result obtained for the extremely relativistic case (\(\beta=1\)). The radiation losses are expressed by the formula

\[ -\frac{dW}{dx} = \frac{q^2\omega_0^2}{2c^2} \int \frac{\omega\,d\omega} {\left(\omega_H-\sqrt{\omega^2-\omega_0^2}\right)\sqrt{\omega^2-\omega_0^2}}, \tag{II.52} \]

where the integration is carried out over the region (II.516). The logarithmic divergence of the integral (II.52) at the upper limit (for \(\omega=\sqrt{\omega_0^2+\omega_H^2}\)) is explained by the fact that the limiting frequency corresponds to zero wavelength. Therefore, at the upper limit one cannot use the results of the macroscopic theory (see Section I.8 and (II.6a)). The integral (II.52) must be cut off at some limiting frequency

\[ \omega_{\max}<\sqrt{\omega_0^2+\omega_H^2}, \]

corresponding to the minimum wavelength admissible in the macroscopic theory.

There are no polarization losses in a plasma with a magnetic field, since there is no frequency at which \(\varepsilon_r\), \(\varepsilon_z\), and \(\gamma\) (II.50) simultaneously vanish.

An electron plasma with a magnetic field is encountered under a great variety of natural conditions, for example in the case of the terrestrial ionosphere, which is in the Earth’s magnetic field, or in the case of the ionized atmosphere of the Sun and stars. Streams of charged cosmic particles of various velocities pass through these regions of space. Therefore the generation of Vavilov–Cherenkov radiation may prove to be a fairly widespread phenomenon.

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8. D. D. Ivanenko, A. A. Sokolov. Classical field theory. 2nd ed., M.–L., 1951, § 27. Exposition of the classical theory of the Cherenkov effect.
9. D. D. Ivanenko, E. N. Tsytovich, the same question as in 94, 95, JETP 28, 291 (1955).
10. V. L. Indenbom, Nonbremsstrahlung production of mesons, JETP 21, 737 (1951).
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12. J. M. Jauch, K. M. Watson, Quantum theory of the Cherenkov effect, Phys. Rev. 74, 1485 (1948).
13. J. M. Jauch, K. M. Watson, the same question, Phys. Rev. 75, 1249 (1949).
14. J. V. Jelley, Registration of charged particles of cosmic radiation by the Cherenkov effect in distilled water, Proc. Phys. Soc. A64, 82 (1951).
15. J. V. Jelley, W. Galbraith, Light flashes of the night sky, Phil. Mag. 44, 619 (1953).
16. J. V. Jelley, W. Galbraith, Second part of work 68, J. Atm. Terr. Phys. 6, 304 (1955).
17. J. V. Jelley, Review article, Atomics 4, 81 (1953).
18. J. V. Jelley, Review article, Progr. Nucl. Phys. (Frisch) 3, 84 (1953). Translation into Russian, see UFN 58, 231 (1956).
19. J. V. Jelley, Calculation of a Cherenkov counter for the proton synchrotron of the University of Birmingham, AERE Report, No. NP/R 1770.
20. M. H. L. Jester, γ-ray spectrometer, University of California, Radiation Laboratory Report, N UCRL 2990.
21. M. I. Kaganov, Motion of a charged particle in an anisotropic dielectric with axial symmetry, JETP 23, 507 (1953).
22. M. I. Kaganov, Radiation of a charge moving uniformly along a circle in an anisotropic dielectric. Transactions of the Physics Division of the Physics-Mathematics Faculty of the A. M. Gorky Kharkov State Institute 4, p. 67 (1953).
23. M. I. Kaganov, Beam in a cylindrical waveguide loaded with an anisotropic dielectric, JETP 23, 517 (1953).
24. M. I. Kaganov, Beam in a cylindrical waveguide loaded with an anisotropic dielectric (kinetic theory), JETP 23, 523 (1953).
25. M. I. Kaganov, Interaction of charged particles with slow waves in an anisotropic dielectric. Dissertation. V. I. Lenin Kharkov State University, Kharkov, 1954.
26. A. Kantz, R. Hofstadter, Cherenkov counters for high-energy particles, Nucl. 12 (3), 36 (1954).
27. N. Klepikov, On the question of transition radiation, Vestnik MGU, No. 8 (1952).
28. A. A. Kolomensky, Radiation of an electron moving uniformly in an electron plasma placed in a constant magnetic field. Report of FIAN, 1951.
29. A. A. Kolomensky, Passage of a charged particle through a dielectric or an electron plasma placed in a magnetic field. Report of FIAN, 1952.
30. A. A. Kolomensky, Cherenkov radiation and polarization losses in a uniaxial crystal, DAN 86, 1097 (1952).
31. A. A. Kolomensky, Cherenkov effect in a gyrotropic medium, JETP 24, 167 (1953).
32. A. A. Kolomensky, Radiation of an electron moving uniformly in an electron plasma placed in a magnetic field, DAN 106, 982 (1956).

1. L. D. Landau, Editor’s appendix to N. Bohr’s book (see 28).

2. L. D. Landau and E. M. Lifshitz, Macroscopic Electrodynamics, Gostekhizdat, M.—L., 1957.

3. H. Lashinsky, Cherenkov radiation from extended electron beams moving near a medium with a complex refractive index, Columbia University Radiation Laboratory, N.Y., 1953.

4. M. A. Lampret, Interaction of an electromagnetic wave with a superluminal electron beam, Phys. Rev. 102, 299 (1956).

5. J. D. Lawson, Relation between Cherenkov and bremsstrahlung radiation, Phil. Mag. 45, 748 (1954).

6. M. L. Levin, Interaction of two parallel-flying Cherenkov electrons, ZhETF 20, 381 (1950).

7. V. M. Lenchenko, Energy losses of fast charged particles passing through matter with constant velocity. Dissertation. V. I. Lenin Central Asian State University, Tashkent, 1955.

8. Yin-yuan Li, Angular width of the cone of Cherenkov radiation, Phys. Rev. 80, 104 (1950).

9. Yin-yuan Li, Corrections to the preceding work, Phys. Rev. 82, 281 (1951).

10. S. J. Lindenbaum, A. Persner, Description of a Cherenkov counter, Rev. Sc. Instr. 25, 285 (1954).

11. J. G. Linhart, Cherenkov radiation in the motion of a charge parallel to the boundary of a dielectric, J. Appl. Phys. (USA) 26, 527 (1955).

12. J. G. Linhart, Review article, Research 8 (10), 402 (1955).

13. J. Linsley, Registration of multiply charged primary cosmic particles by means of Cherenkov counters, Phys. Rev. 93, 899 (1954).

14. J. Linsley, the same question, Phys. Rev. 97, 1292 (1955).

15. J. Linsley, N. Horwitz, Description of Cherenkov counters with thin lucite radiators, Rev. Sc. Instr. 26, 557 (1955).

136–138. L. Mallet, Spectral studies of the luminescence of water and other media under the action of γ-radiation, Compt. Rend. Acad. Sci. (Paris) 183, 274 (1926); 187, 222 (1929); 188, 445 (1929).

1. M. Mando, Cherenkov counters without focusing, Nuovo Cim. 12, 5 (1954).

2. J. Marshall, Cherenkov counters for fast electrons, Phys. Rev. 81, 275 (1951).

3. J. Marshall, Registration of particles by Cherenkov radiation, Phys. Rev. 86, 685 (1952).

4. J. Marshall, Review of Cherenkov counters, Ann. Rev. Nucl. Sci. 4, 141 (1954).

5. L. Marshall, Cherenkov radiation and the spectrum of solar radio emission. Preprint. The Enrico Fermi Institute for Nuclear Studies, The University of Chicago, 1956. (Astron. Journ. 124, 601–604, 1956; see also 213.)

6. R. L. Mather, Determination of proton energy by means of a Cherenkov detector, Phys. Rev. 84, 181 (1951).

7. J. W. Mather, E. A. Martinelli, Production of neutral mesons in hydrogen by protons with energy 340 MeV; application of Cherenkov counters, Phys. Rev. 92, 780 (1953).

8. J. W. Mather, E. A. Martinelli, Description of a Cherenkov counter, University of California Radiation Laboratory Report, No. 1646.

9. E. Maurer, H. Kolz, Review article, Zs. Angew. Phys. 2, 223 (1950).

10. A. I. Morozov, Cherenkov effect for a current-carrying conductor, Vestnik MGU, Phys.-Math. Series, No. 1, 72, 1957.

11. C. Muzikar, Cherenkov effect in a waveguide filled with a dielectric, Czech. Journ. Phys. 5 (1), 9 (1955).

12. S. M. Neamtan, Quantum theory of the Cherenkov effect, Phys. Rev. 92, 1362 (1953).

13. K. W. Ogilvie, Application of a Cherenkov counter in measurements of the proton component of cosmic rays, Canad. Journ. Phys. 33 (9), 555 (1955).

14. V. E. Pafomov, Cherenkov effect in an anisotropic medium. Diploma thesis, Moscow State University, Faculty of Physics, 1952.

15. V. E. Pafomov, Cherenkov radiation in anisotropic ferrites, ZhETF 30, 761 (1956).

16. V. E. Pafomov, On the propagation of Cherenkov rays in crystals, ZhETF 32, 360 (1957).

17. V. E. Pafomov, Radiation of a charge moving parallel to the plane interface of two dielectrics, ZhETF 32, 10 (1957).

18. V. E. Pafomov, Radiation of a charge in passing through a plane plate, ZhETF (in press).

19. J. R. Pierce, Interaction of moving charges with wave circuits, J. App. Phys. (USA) 26 (5), 627 (1955).

1. V. T. Price, Ionization produced by relativistic particles. This review contains a complete bibliography on the question of the contribution of Cherenkov losses to the total energy losses of particles. Rep. Progr. Phys. 18, 52 (1955).

2. I. V. Polubarinov, Quantum theory of the Cherenkov effect in anisotropic media. Diploma thesis. Moscow State University, Faculty of Physics, 1953.

3. A. P. Polikarov, On superluminal radiation of cosmic particles in the Earth’s atmosphere. Reports of the Bulgarian Academy of Sciences 7, No. 2, 29 (1954).

4. J. A. Rich, R. E. Slovacek, F. J. Studier, Cherenkov radiation from Co\(^{60}\) in water, J. Opt. Soc. Amer. 43 (9), 750 (1953).

5. S. Robin, Brief review, J. Phys. et Radium (Paris) 11, January (1950).

6. M. Schönberg, Energy losses in collisions of particles moving in a medium, Bull. centre phys. nuc. université libre de Bruxelles 20, 1 (1950); Nuovo Cim. 8 (3), 159 (1951).

7. L. I. Schiff, Quantum theory of the Cherenkov effect. Quantum Mechanics. McGraw Hill, N.-Y., 1949.

8. M. Ya. Shirobokov, Cherenkov effect for a particle with spin 2, ZhETF 19, 481 (1949).

9. Gyorgy Schmidt, Cherenkov effect in waveguides with diaphragms and other bounded volumes, Magyar Fizikai Folyoirat IV, 453 (1956).

10. A. G. Sitenko, Application of the method of normal oscillations in the classical theory of radiation. Dissertation. Problems connected with the passage of a charge through a medium are considered, in particular, radiation of a charge uniformly rotating along a circle in an isotropic medium, Cherenkov radiation in a waveguide filled with a dielectric, etc., Kharkov State University named after A. M. Gorky, 1952.

11. A. G. Sitenko, On the passage of a charged particle through a magnet, DAN 98, 377 (1954).

12. A. G. Sitenko, Cherenkov effect in a ferrodielectric, ZhETF 23, 2000 (1953).

13. A. G. Sitenko, M. I. Kaganov, On energy losses to Cherenkov radiation in a crystal, DAN 100, 681 (1955).

14. A. G. Sitenko, On the passage of a charged particle through a dielectric possessing losses. Scientific Notes of Kharkov University 64. Proceedings of the Physics Division of the Faculty of Physics and Mathematics 6, Kharkov (1955).

15. A. G. Sitenko, A. A. Kolomenskii, Cherenkov effect in a uniaxial gyrotropic crystal, ZhETF 30, 511 (1956).

16. K. Skarsvag, A. L. Ruby, Cherenkov radiation in a Norwegian reactor with heavy water, J E N E R Report, No. 22.

17. A. Sokolov, Quantum theory of the Cherenkov effect, DAN 28, 415 (1940).

18. A. Sokolov, D. Ivanenko, Quantum Field Theory, Gostekhizdat, Moscow–Leningrad, 1952. Paragraph devoted to the quantum theory of Cherenkov radiation.

19. A. Sommerfeld, After a superluminal electron in vacuum, Proc. Amst. Acad. 26 (1904).

20. A. Sommerfeld, the same question, Gött. Nachr., 1905, p. 201.

21. A. Sommerfeld, Optics, Moscow, IL, 1953.

22. R. B. Sutton, T. H. Fields, J. G. Fox, J. A. Kane, W. E. Mott, R. A. Smallwood, Application of Cherenkov counters in experiments on proton scattering by protons, Phys. Rev. 97, 783 (1955).

23. M. S. Svirskii, On the question of the absorption and emission of photons. Physics of Metals and Metallography, vol. II, issue 3, 397, 1956.

24. I. E. Tamm, I. M. Frank, Coherent radiation of a fast electron in a medium, DAN 14, 107 (1937).

25. I. E. Tamm, I. M. Frank, P. A. Cherenkov, Luminescence of pure liquids under the action of fast electrons, Izv. AN SSSR, OMEN, p. 30 (1938).

26. I. E. Tamm, Radiation of uniformly moving electrons, J. Phys. USSR 1, 439 (1939).

27. I. E. Tamm, I. M. Frank, Radiation of an electron during uniform motion in a refracting medium. Proceedings of the P. N. Lebedev Physical Institute of the USSR Academy of Sciences 2, No. 4 (1947).

28. K. Tanaka, Cherenkov effect in a uniaxial crystal, Phys. Rev. 93, 459 (1954).

29. T. Taniuti, Classical and quantum theory of the Cherenkov effect, Prog. Theor. Phys. 6, 207 (1951).

30. D. A. Tidman, Microtheory of the Cherenkov effect, Nuovo Cim. 3, 503 (1956).

31. D. A. Tidman, the same topic, Nucl. Phys. 2, 289 (1956).

32. S. I. Vavilov, On possible causes of the blue γ-luminescence of liquids, DAN 2, 457 (1934).

1. S. I. Vavilov, The Microstructure of Light, Publishing House of the Academy of Sciences of the USSR, Moscow, 1952.

2. V. I. Veksler, Coherent method of accelerating charged particles, Geneva Conference. CERN, Symposium I, 80 (1956). June 1956; also Atomic Energy II, No. 5, 427 (1957).

3. P. B. Weisz, B. L. Anderson, Attempt to detect Cherenkov radiation from cosmic-ray particles in water, Phys. Rev. 72, 431 (1947).

4. J. Winckler, Cherenkov counter for measuring the “albedo” of cosmic rays, Phys. Rev. 85, 1054 (1952).

5. J. R. Winckler, K. Anderson, Improved Cherenkov counter for the same measurements, Rev. Sci. Instr. 23, 765 (1952).

6. J. R. Winckler, K. Anderson, Measurements by means of Cherenkov counters of geomagnetic and other effects in cosmic rays, Phys. Rev. 93, 596 (1954).

7. J. R. Winckler, E. N. Mitchell, K. A. Anderson, L. Peterson, Measurements of Cherenkov radiation from positive and negative \(\pi\)-mesons, Phys. Rev. 98, 1411 (1955).

197*. H. Wyckoff, J. E. Henderson, Experimental verification of the formula

\[ \cos \vartheta = \frac{1}{n\beta}, \]

Phys. Rev. 64, 1 (1943).

198. J. R. Winckler, Cherenkov radiation of cosmic-ray particles, Phys. Rev. 87*, 241 (1952).

1. N. A. Khizhnyak, Ya. B. Fainberg, Parametric Cherenkov effect, ZhETF (in press).

2. R. J. Hanson, D. C. Moore, Gas Cherenkov counter with variable threshold, Nuovo Cim. 4, 1558 (1956).

3. W. R. Webber, Measurement of the flux of \(\alpha\)-particles and of Li, Be, B nuclei in primary cosmic rays by means of a Cherenkov detector, Nuovo Cim. 4, 1285 (1956).

4. N. M. Nesterova and A. E. Chudakov, On the observation of Cherenkov radiation accompanying extensive atmospheric showers of cosmic rays, ZhETF 28, 384 (1955).

5. G. Marx, G. Györgyi, Energy–momentum tensor of the electromagnetic field in a medium (see also [^7]), Annalen der Physik 16, 241 (1955).

6. J. Marshall, Cherenkov counters, CERN Symposium, 1956, Vol. I, p. 63.

7. J. M. Cassels, Measurements with Cherenkov counters, ibid., p. 74.

8. R. Hofstadter, Cherenkov counters, ibid., p. 75.

9. J. V. Jelley, On focusing Cherenkov counters, ibid., p. 76.

10. O. Chamberlain, C. Wiegand, Cherenkov velocity selector, ibid., p. 82.

11. I. Filosofo, T. Yamagata, Large Cherenkov counter for recording high-energy photons and electrons, ibid., p. 85.

12. A. I. Akhiezer, Ya. B. Fainberg, G. Ya. Lyubarskii, Cherenkov radiation and the stability of beams in waveguides with slow waves (linear accelerators), ibid., p. 220.

13. On measuring the velocity of particles by the method of interference of Vavilov–Cherenkov radiation, PTE, 1956, No. 3, p. 44.

14. N. L. Balasz, Cherenkov radiation of a neutral particle with a magnetic moment, Phys. Rev. 101, 1220 (1956).

15. U. E. Kruse, L. Marshall, J. R. Platt, Synchrotron radiation in the radio spectrum of the Sun. An addendum to the paper contains a correction to article 143.

16. W. B. Jones, H. R. Kratz, J. Rouvina, Cherenkov counter for total absorption, Rev. Sci. Instr. 28, 167 (1957).

17. G. M. Garibyan, On the theory of transition radiation, ZhETF (in press).

18. V. A. Shakhbazyan, On the question of Cherenkov radiation in an absorbing medium, Izv. AN Arm. SSR 9, No. 5, p. 90 (1956).