Radiation of Electrons Moving in a Medium at Superluminal Velocity

I. M. Frank

§ 1. Basic properties. § 2. Condition for the occurrence of radiation. § 3. Electron field. § 4. Energy of radiation. § 5. Comparison with theory. § 6. Conclusion.

§ 1. Basic Properties

In 1934, in the study of the luminescence of uranium salts, P. A. Cherenkov discovered and subsequently investigated the glow of liquids arising under the action of the $\gamma$-rays of radioactive preparations$^{1,2*}$).

As is well known, luminescence phenomena are extremely widespread in nature. In most liquids, for example, a glow is observed under the action of ultraviolet light, caused by negligible impurities contained in them$^3$. Many liquids also glow under the action of X-rays$^4$. A glow under the action of $\gamma$-rays had likewise been observed more than once before Cherenkov. For example, Malle$^5$ even obtained photographs of its spectrum.

In all probability, it was precisely the prevalence of the ordinary types of luminescence that was one of the reasons why, before Cherenkov, no special attention had been paid to the glow arising under the action of $\gamma$-rays.

Another reason that hindered its investigation was the low intensity of the glow. When even a strong radium preparation ($\sim 100$ mg) is brought near a liquid, the glow can be seen only after the eye has adapted to the dark. Convenient methods of photometry for such intensities did not previously exist.

*) A complete summary of P. A. Cherenkov’s experimental results is contained in his doctoral dissertation$^2$. Below, in addition, references are also given to the corresponding work used in the dissertation.

The results of Cherenkov’s experimental investigations and their theoretical consideration (S. I. Vavilov, I. E. Tamm, and I. M. Frank) showed that in this case, too, there occurs a new optical phenomenon, which subsequently received in the literature the name of Cherenkov radiation.

Already in his first paper,^1 devoted to this question, Cherenkov used the photometric method developed shortly before by Brumberg and Vavilov,^6 the so-called quenching method, based on the constancy of the threshold of visual excitation of the eye and making it possible to carry out measurements of extremely weak intensities of visible light. As a result of Cherenkov’s first investigations, the following was established:^1,2,7

The luminescence is universal. In any transparent liquid, when it is irradiated with $\gamma$ rays, a weak visible glow is observed. The same, as was later found, also occurs for solids; there, however, it is often masked by ordinary luminescence.

The first photometric data showed that, for the 19 liquids investigated, of the most diverse chemical composition and different density, the intensity of the glow is practically the same. The difference in glow intensities under measurements in identical conditions proved to lie within limits of not more than $\pm 15\%$, i.e., not beyond the limits of accuracy of the measurement.

The glow proved to be partially polarized, with the electric vector of the oscillations oriented along the direction of the beam of $\gamma$ rays.

The most thorough purification of the liquids from possible impurities did not reduce the brightness of the glow, and therefore the hypothesis that the glow is caused by small fluorescent impurities had to be rejected. To clarify the nature of the phenomenon, Cherenkov carried out a series of experiments customarily used for this purpose in S. I. Vavilov’s laboratory in the investigation of fluorescence of solutions of luminous dyes. These experiments consist in determining the brightness and polarization of the glow under various conditions. Thus, for example, raising the temperature increases the mobility of the molecules and lowers the degree of polarization of the glow, which depends on the orientation of the molecules. In addition, a change in temperature usually also affects the quantum yield of fluorescence. The quantum yield may also be arbitrarily reduced through collisions of the second kind, by admixing certain quenching substances to the liquid. Theory makes it possible, by means of such measurements, to determine the principal characteristic of emitters—the lifetime in the excited state*).

In the case of the glow discovered by Cherenkov, it turned out that there is no influence either of temperature or of quenching agents.

*) See, for example, the review report by S. I. Vavilov.^8

This means that the lifetime of the excited state of the emitters here, even if it is nonzero, is in any case several orders of magnitude smaller than for ordinary kinds of luminescence. This gave Vavilov \(^{9}\) grounds to assert that “the observed effect cannot in general be any kind of luminescence, for which a finite duration of excitation is a necessary characteristic feature.”

As is known, the mechanism of absorption of \(\gamma\)-rays consists in the gamma photon’s transferring its energy wholly or partially to electrons in the photoelectric or Compton effect. Thus, the transfer of energy from gamma rays to the medium occurs by means of electrons which, moving in the medium, are gradually slowed down. It was therefore natural to assume that the origin of Cherenkov radiation also occurs through the agency of electrons.

Since, however, the radiation, as has already been said, cannot be caused by fluorescence of the atoms and molecules of the medium under the action of electrons, S. I. Vavilov \(^{9}\) assumed that the emitters are the electrons themselves. The only mechanism of such radiation that was known at that time led to identifying the phenomenon with bremsstrahlung, and this was done. This hypothesis naturally explained almost all the properties of the radiation known at that time—the absence of a lifetime of the excited state, the universality of the radiation, and also its polarization. Indeed, fast electrons are ejected in the direction of the gamma rays, and the electric vector of the emitted waves will be directed along the direction of the velocity, if during the braking of the electron the direction of the electron’s motion does not change very strongly.

From the qualitative point of view, only the fact of the approximate constancy of the brightness of the radiation of different liquids remained unclear. The absorption coefficient of gamma rays, as is known, is proportional to the density of the substance. The range of the electrons which arise in this process is inversely proportional to the density. Therefore the constancy of the brightness means that the radiation is proportional to the number of electrons and to the path that they traverse. This contradicts the hypothesis of ordinary bremsstrahlung, for which the energy radiated over the full path of a particle in a substance with a given atomic number is determined only by the number of electrons generated and does not depend on the density of the substance. At the same time, however, bremsstrahlung should depend strongly on the atomic number of the substance in which the motion occurs. A considerable difficulty also arose in explaining the absolute brightness of the radiation. It proved to be too large in comparison with that expected for bremsstrahlung.

For further investigations, Vavilov’s assertion was very essential: that the phenomenon discovered by P. A. Cherenkov is not ordinary luminescence and that the electrons themselves emit;

P. A. Cherenkov’s measurements confirmed the correctness of this point of view.

Experiments performed with sources of β-rays²˒⁷˒¹⁰ (thin-walled ampoules filled with radium emanation) showed that electrons do in fact excite in liquids a luminescence identical in its properties with that produced by gamma rays. For a direct proof that, under irradiation by γ-rays as well, the luminescence is likewise caused by electrons, experiments were carried out on the influence of a magnetic field on the luminescence²˒¹¹. If the luminescence is caused by electrons, then the polarization of the luminescence must be determined by the direction of motion of the electrons. A magnetic field acts on the electrons, curving their trajectories. Therefore the application of a magnetic field should in this case cause a change in the polarization of the luminescence. If, however, the radiation is caused not by electrons but directly by gamma rays, then there should be no influence of the magnetic field. Thus this experiment appeared to be decisive in choosing between gamma rays and electrons as the exciters of the luminescence. The influence of the magnetic field proved to be very considerable, and the question was thereby resolved in favor of electrons. Unexpected, however, was the fact that this influence manifested itself chiefly in a change in the observed brightness of the luminescence.

As a result of these and of additional experiments set up by Cherenkov, it was shown that the radiation has a clearly expressed directionality²˒⁷˒¹²˒¹³. It turned out that the maximum intensity is observed at a quite definite acute angle to the direction of motion of the electron. The application of a magnetic field bends the electron’s trajectory and thereby changes the angular distribution of the luminescence. Therefore the intensity of the luminescence, measured at a specified angle of observation, changes when the magnetic field is switched on.

Fig. 1. Diagram of the apparatus for observing the angular distribution of the luminescence produced by fast electrons.

The directionality of the radiation is clearly illustrated by the following experiment performed by Cherenkov.

A liquid was poured into a thin-walled cylindrical glass vessel \(A\) (Fig. 1). At its side a source of γ-rays was placed, for example an ampoule with radium (the direction of the γ-rays is shown by the arrow). Then a transparent liquid poured into a transparent vessel appears dark if one looks at it from the same side on which the radium is placed, and luminous if one looks from the opposite side. In order to be able to obtain the distribution of the luminescence intensity for all angles at once, Cherenkov surrounded the vessel with a conical mirror (Fig. 1). When observed from above, in various parts of the mirror one sees

emission traveling at various angles to the direction of the \(\gamma\)-rays. The photographs obtained in this way are presented in Fig. 2. The photographs are arranged so that the direction of the \(\gamma\)-ray beam corresponds to the direction from bottom to top.

From Fig. 2 it is seen that only part of the conical mirror is luminous; the radiation proves to be strongly directed and concentrated inside a cone forming an acute angle with the direction of the \(\gamma\)-rays. A very characteristic feature of the radiation is that the maximum of the radiation does not coincide with the direction of the \(\gamma\)-rays (see Fig. 2), but forms an acute angle with it.

Fig. 2. Photograph of the angular distribution of the glow in cyclohexane (\(\mu=1.4367\)).

Photograph 2, \(a\), was obtained by irradiating cyclohexane (refractive index \(n=1.437\)) with gamma rays from a radium preparation, and photograph 2, \(b\), by exciting the same liquid with rays from radiothorium. The results of photometric measurements of these and analogous photographs for various liquids are presented in Fig. 3. It is known that the \(\gamma\)-rays of a radium preparation produce Compton electrons of, on the average, lower energy than the \(\gamma\)-rays of radiothorium (the product of the decay of radiothorium, ThC″, emits in this preparation). As is seen from Figs. 2 and 3, an increase in the energy of the Compton electrons shifts the maximum of the radiation toward larger angles. From Fig. 3 it is also seen that this angle for the maximum of the radiation increases with increasing refractive index of the liquid.

The last experimental fact that should be mentioned before a detailed consideration of the theory of the phenomenon is the spectrum of the radiation. As was found already by Mallet \(^{5}\) and confirmed by Cherenkov, it proved to be continuous and bounded on the short-wavelength side only by the absorption edge of the liquid itself or by the absorption edge of the optics used \(^{2,7}\).

§ 2. CONDITION FOR THE OCCURRENCE OF RADIATION

The phenomenon investigated by P. A. Cherenkov, both qualitatively and, apparently, quantitatively, can be explained with the aid of the concepts of classical electrodynamics. The immediate result of these concepts is the assertion that even a charge moving uniformly in a homogeneous medium must radiate

light, if its velocity of motion exceeds the phase velocity of light in this medium.

For an electron moving in water, such a critical velocity is already reached at an energy of approximately 250 KeV. It is known that a considerable fraction of the electrons emitted by radioactive substances or produced by gamma rays possess large energies. The expected properties of such radiation are identical with those observed by Cherenkov^14.

H₂O
n = 1.3371
\(\theta_{\text{theor}} = 33^\circ 30'\)
\(\theta_{\mathrm{Ra}} = 28^\circ\)

C₂H₂
n = 1.4367
\(\theta_{\text{theor}} = 38^\circ\)
\(\theta_{\mathrm{Ra}} = 34^\circ 30'\)

C₆H₆
n = 1.5133
\(\theta_{\text{theor}} = 41^\circ 30'\)
\(\theta_{\mathrm{Ra}} = 38^\circ 30'\)

C₁₁H₂O₂
n = 1.58043
\(\theta_{\text{theor}} = 44^\circ\)
\(\theta_{\mathrm{Ra}} = 41^\circ\)

\(\theta\)

○—○—○ Excitation of luminescence by Compton electrons from γ-rays of Ra \((\beta_{\mathrm{eff}} = 0.847)\)

●—●—● Excitation of luminescence by Compton electrons from γ-rays of ThC″ \((\beta_{\mathrm{eff}} = 0.896)\)

Fig. 3. Angular distribution of the intensity of luminescence.

As it turned out subsequently, the question of the field of an electron at superlight velocities had been discussed in the literature very long ago. S. I. Vavilov pointed to a statement made as early as 1901 by Lord Kelvin^15, that a moving atom must radiate during uniform motion if its velocity exceeds the velocity of propagation of waves. In this case the emitted wave must propagate in a direction satisfying a condition analogous to the Mach condition for sound waves. As will be seen from what follows, this statement is correct if the words “moving atom” are replaced by the words “moving charge.” Later (1904, 1905), and independently of Kelvin, the same question was considered by Sommerfeld^16.

Sommerfeld calculated the force acting on an electron in the case where the velocity of the electron in vacuum exceeds the velocity of light, and came to the conclusion that it must be braked by its own field. As Tamm subsequently showed^17, the force calculated by Sommerfeld is precisely equivalent to radiation damping according to the theory of Tamm and Frank^14, if one assumes that it propagates over

all frequencies down to wavelengths of the order of the classical radius of the electron, as should also occur when an electron moves in vacuum with superluminal velocity.

Subsequently it became clear that a velocity of motion exceeding the speed of light in empty space is unattainable. The possibility of motion in a medium, where such an excess of speed is possible, since the phase velocity is \(n\) times (where \(n\) is the refractive index) less than the speed of light in empty space, remained unnoticed. Therefore Sommerfeld’s theory, as well as Kelvin’s statement, was forgotten, and A. F. Ioffe drew our attention to it in connection with the theory of the Cherenkov effect \(^{14}\).

The condition under which an electron moving uniformly in a homogeneous medium can emit light is easily obtained both from wave and from quantum considerations.

Let the electron move uniformly along the \(z\)-axis with velocity \(v\). We shall assume that the motion of the electron began infinitely long ago. Then, in the coordinate system associated with the electron, the electron will appear to be at rest, and moreover for an indefinitely long time; consequently, in this coordinate system its field cannot be a function of time. This means that in the rest coordinate system the whole field pattern must move together with the electron with velocity \(v\). Therefore the field observed at some point \(x, y, z\) at the moment \(t\) must be identical to the field observed at the moment \(t+t_1\), but at the point \(x, y, z+vt_1\), since at the instants of observation these points are situated identically with respect to the electron. From the coincidence of the fields it follows that the same will be obtained also for any corresponding components of these fields.

Suppose that the electron radiates, i.e. that the electron field contains waves of frequency \(\omega\) diverging from the \(z\)-axis at some angle \(\theta_\omega\). Such waves are described by functions

\[ A(r)e^{i\omega\left(t-\frac{z\cos\theta_\omega+r\sin\theta_\omega}{c/n(\omega)}\right)}, \tag{2,1} \]

where \(n(\omega)\) is the refractive index for the frequency \(\omega\), and, consequently, \(c/n(\omega)\) is the phase velocity for this frequency, while \(r\) is the distance of the observation point from the \(z\)-axis. Since the motion is directed along the \(z\)-axis, the radiation must be symmetric with respect to this axis, that is, (2,1) can be represented as a function only of \(r\) and \(z\).

From what was said above about the components of the electron field it follows that, under a simultaneous change of \(t\) by an amount \(t_1\) and of the coordinate \(z\) by an amount \(vt_1\), both the amplitude and the phase of the wave must remain unchanged.

Since the amplitude, by definition, must not be a function of time, the condition of constancy of the amplitude will be satisfied,

if we assume that it does not depend on \(z\), which has already been done in (2,1). From the constancy of the phase under such a transformation we have

\[ i\omega\left(t_1-\frac{v t_1 \cos\theta_\omega}{c/n(\omega)}\right)=0, \tag{2,2} \]

whence, after cancellation by \(i\omega t_1\), we obtain

\[ \cos\theta_\omega=\frac{c}{v n(\omega)}; \tag{2,3} \]

obviously, this condition can be fulfilled and, consequently, a divergent wave can exist only in the case when

\[ \frac{c}{v n(\omega)}<1, \]

i.e. when the electron velocity \(v\) is greater than the phase velocity \(c/n(\omega)\) for the wave under consideration.

Condition (2,3), as is not difficult to see, is analogous to Mach’s condition for sound waves and can be obtained by simple qualitative considerations. Indeed, it follows from (2,3) that the ray component of the electron velocity, \(v\cos\theta_\omega\), for the angle \(\theta_\omega\), is equal to the wave velocity \(c/n(\omega)\). Thus, the wave produced by an electromagnetic disturbance at some point \(A\) on the electron trajectory will approach an observer situated in the direction \(\theta_\omega\) with the same velocity as the electron. Therefore it will arrive at the observer simultaneously with the waves produced by the electron at subsequent points of its path. It follows from this that the waves traveling in this, and only in this, direction will add together.

V. L. Ginzburg\(^{18}\) considered the question of the radiation of an electron moving uniformly in a crystalline medium. The refractive index in such media depends not only on the frequency, but also on the polarization and on the direction of propagation of the wave. Condition (2,3) is preserved in this case as well, if \(n(\omega)\) is understood as the refractive index corresponding to a wave of the given direction and polarization. Owing to the peculiarities of the behavior of the refractive index in crystals, the cross section of the cone for the directions \(\theta_\omega\) in this case may be not circular but complicated; moreover, two cones are possible, determined by the ordinary and extraordinary rays.

Simple quantum considerations also lead to a condition for the occurrence of radiation analogous to (2,3). This question was examined in detail by V. L. Ginzburg\(^{19}\). Here we shall restrict ourselves only to considering the simplest case, when the energy of the emitted photon is small in comparison with the energy of the electron, as is in fact the case in the Cherenkov effect. Let there be a freely moving electron with total energy \(W\) and momentum \(p\), where, as is known,

\[ W=\frac{mc^2}{\sqrt{1-v^2/c^2}},\qquad p=\frac{mv}{\sqrt{1-v^2/c^2}}. \tag{2,4} \]

From (2.4) it follows that if the energy of the electron has changed by a small amount \(\Delta W\), then the relation

\[ \Delta W = v\Delta p, \tag{2.5} \]

must be fulfilled, where \(\Delta p\) is the change in the absolute value of the momentum. Suppose that the change in the energy and momentum of the electron has occurred as a result of radiation. Then

\[ \Delta W = h\nu, \tag{2.6} \]

where \(h\nu\) is the emitted photon.

The momentum of the electron in absolute value will decrease by \(\Delta p\) and will become equal to

\[ p_1 = p - \Delta p, \]

while the direction of the momentum will change by an angle \(\psi\).

The photon emitted in the medium should be assigned a momentum equal to \(\dfrac{n}{c}h\nu\).\(^{19}\) Then the law of conservation of momentum gives

\[ \begin{aligned} \frac{n}{c}h\nu \cos\theta &= p - (p-\Delta p)\cos\psi,\\ \frac{n}{c}h\nu \sin\theta &= (p-\Delta p)\sin\psi, \end{aligned} \tag{2.7} \]

where \(\theta\) is the angle of emission of the photon with respect to the initial direction of motion of the electron. If the absolute value of the momentum of the emitted photon is small in comparison with \(p\), then the angle \(\psi\) is small, and consequently one may take \(\cos\psi = 1\). Then we obtain

\[ \frac{n}{c}h\nu\cos\theta = \Delta p. \tag{2.8} \]

Substituting expressions (2.6) and (2.8) into (2.5), we have

\[ h\nu = \frac{nv}{c}h\nu\cos\theta, \tag{2.9} \]

or

\[ \cos\theta = \frac{c}{nv}. \tag{2.10} \]

Thus, we again obtain condition (2.3).

If the condition for radiation is derived without the simplifications made above, then for the angle of radiation one obtains\(^{19}\)

\[ \cos\theta = \frac{c}{nv} \left( 1+\frac{h\nu}{2mc^2}(n^2-1)\sqrt{1-v^2/c^2} \right). \tag{2.11} \]

Thus, in the exact quantum expression there appears a correction term which, however, as is not difficult to see, is extremely small. Indeed, apart from coefficients of order unity, it contains the factor \(h\nu/mc^2=\lambda_0/\lambda\), where \(\lambda_0\) is the Compton wavelength \((\lambda_0=0.024\text{ Å})\).

Since radiation is possible only in the frequency range where \(n(\omega)>1\), since it is necessary that \(vn(\omega)\) be greater than \(c\), \(\lambda\) does not go beyond the ultraviolet part of the spectrum. Therefore \(\lambda_0/\lambda\) is always negligibly small.

Condition (2,11) may also be written in the following simple form:

\[ \cos\theta=\frac{c}{nv}+(\Lambda/\lambda)(n^2-1)/2n^2, \tag{2,12} \]

where \(\Lambda\) is the de Broglie wavelength for the moving electron*).

The difference between the quantum condition for radiation and the classical one is not accidental. It takes into account the recoil that the electron receives upon radiation. In the classical treatment the electron velocity is regarded as given and as not changing during radiation. The quantum treatment proceeds from the concept of a free electron. Here \(v\) is the initial velocity of the electron, which is greater than the final one. It is not difficult to convince oneself that (2,3) numerically coincides with (2,11) if by \(v\) one understands a velocity equal to some intermediate one between the initial and final velocities of the electron.

The difference between the quantum treatment and the classical one also consists in the following \(^{21}\). For \(v<c/n\), the conservation laws equally forbid both the emission of a single photon and the simultaneous emission of a group of photons. However, at superluminal velocity such a process of emission of several photons is possible, and condition (2,3) is not obligatory for them. If one leaves aside the possibility of emitting several photons at once, then from formula (2,3), or (2,10), in agreement with the experimental data it follows that Cherenkov radiation must be directed at an acute angle \(\theta_\omega\) to the direction of motion of the electron, and this angle increases with increasing electron velocity and refractive index of the medium (see Fig. 3).

§ 3. THE FIELD OF THE ELECTRON

The problem of the field of an electron moving in a medium is conveniently reduced to the consideration of the field of stationary oscillators located on its trajectory \(^{22}\). In this case the question of radiation is reduced to the consideration of the interference of waves emitted by such oscillating dipoles.

We shall, as before, assume that the electron moves along the \(z\)-axis, and that its coordinates at time \(t\) are

\[ z=vt,\quad x=y=0. \tag{3,1} \]

The charge density \(\rho\) and the current density \(\mathbf{j}\) can in this case be taken—

*) Several years after Ginzburg’s work, the same condition for the radiation of an electron in a medium was again published in a paper by Cox \(^{20}\). Cox’s formula (2,12) is identical to Ginzburg’s formula (2,11).

to be equal to

\[ \mathbf{j}=ev\boldsymbol{\delta}(z-vt)\delta(x)\delta(y), \tag{3,2} \]

\[ \rho=e\delta(z-vt)\delta(x)\delta(y), \tag{3,3} \]

where the symbol \(\delta\) denotes the Dirac delta-function, which, as is known, is equal to zero for any argument not equal to zero, and, when integrated from minus infinity to plus infinity, gives unity. Therefore \(\mathbf{j}\) and \(\rho\) in equations (3,2) and (3,3) are equal to zero everywhere except at the point at which the electron is located at the moment \(t\), and integration over volume gives the current \(ev\) and the charge \(e\), which corresponds to the conditions of the problem.

Let us expand \(\mathbf{j}\) and \(\rho\) as functions of \(t\) in Fourier integrals. According to the well-known theorem,

\[ F(t)=\frac{1}{\pi}\int_{0}^{\infty} d\omega \int_{-\infty}^{\infty} F(n)\cos \omega(n-t)\,dn \tag{3,4} \]

and, consequently, for example, for the component \(\omega\) of the current density we obtain

\[ \mathbf{j}_{\omega}(z,t)=\frac{ev}{\pi}\delta(x)\delta(y)\int_{-\infty}^{+\infty}\delta(z-vn)\cos\omega(n-t)\,dn. \tag{3,5} \]

Making a change of variable: putting \(\xi=z-vn\) and taking into account the known property of the \(\delta\)-function, according to which

\[ \int_{-\infty}^{+\infty}\delta(\xi)f(\xi)\,d\xi=f \]

we obtain

\[ \mathbf{j}_{\omega}(z,t)=\frac{e}{\pi}\mathbf{z}_{1}\cos\{\omega(z/v-t)\}\delta(x)\delta(y), \tag{3,6} \]

where \(\mathbf{z}_{1}\) is a unit vector directed along the \(z\)-axis.

In an analogous manner, for the component \(\omega\) of the charge density we have

\[ \rho_{\omega}(z,t)=\frac{e}{\pi v}\cos\{\omega(z/v-t)\}\delta(x)\delta(y). \tag{3,7} \]

Such currents and charges \(\mathbf{j}_{\omega}\) and \(\rho_{\omega}\), oscillating at each point with frequency \(\omega\), may be represented as the result of the existence at the same points of periodically varying dipole moments. Indeed, between the density of the dipole moment \(\mathbf{P}\) (i.e. the magnitude of the dipole electric moment contained in an element of volume, divided by the magnitude of this volume), the charge density, and the current density there exist the relations

\[ \dot{\mathbf{P}}_{\omega}=\mathbf{j}_{\omega}, \]

\[ \operatorname{div}\mathbf{P}_{\omega}=-\rho_{\omega}. \tag{3,8} \]

It is known that the specification of $\mathbf{j}_{\omega}$ and $\rho_{\omega}$ uniquely determines all components of the electric and magnetic fields of frequency $\omega$. Therefore, if one can find such fictitious $\mathbf{P}_{\omega}$ for which $\mathbf{j}_{\omega}$ and $\rho_{\omega}$, found from (3,8), coincide with the true distribution of currents and charges, then, consequently, the field specified with the aid of $\mathbf{P}_{\omega}$ must be identical with the desired one.

In our case this requirement is easily satisfied if it is assumed that there exists a spectrum of electric dipoles $\mathbf{P}_{\omega}$ of various frequencies $\omega$, directed along the $z$ axis and equal to

\[ \mathbf{P}_{\omega}=\frac{e}{\pi\omega} Z_1 \sin \{\omega(t-z/v)\}\delta(x)\delta(y). \tag{3,9} \]

Indeed, substitution of this value of $\mathbf{P}_{\omega}$ into (3,8) leads to quantities $\mathbf{j}_{\omega}$ and $\rho_{\omega}$ equal to (3,6) and (3,7), as is readily verified if one takes into account that in the present case

\[ \operatorname{div}\mathbf{P}=\frac{dP}{dz}. \]

Thus, the field of a moving electron can be replaced by an aggregate of stationary harmonic oscillators distributed along the $z$ axis, the spectrum of which extends from $\omega=0$ to $\omega=\infty$.

Fig. 4.

To determine the field at the observation point it is necessary to sum the fields of the individual dipoles, i.e. one has to integrate over the volume in which they are located. As is seen from (3,9), all the dipoles may be regarded as concentrated on the $z$ axis; moreover, integration over $x$ and $y$ always gives a factor equal to unity. The dipole moment of the oscillators located on the element $dz$ of the $z$ axis should therefore be taken equal to

\[ \mathbf{P}_{\omega}\,dz=\frac{e}{\pi\omega} Z_1 \sin \{\omega(t-z/v)\}\,dz . \tag{3,10} \]

Such a reduction of the problem of the field of a moving electron to the field of an aggregate of stationary oscillators makes it possible not only to determine the conditions for the occurrence of radiation, but also to calculate its energy in a simple way.

Let us consider, for example, the field of oscillators located on a bounded segment of the $z$ axis, for instance, in the interval from $-z_0$ to $+z_0$. This is equivalent to an electron which began its motion at the point

RADIATION OF ELECTRONS MOVING IN MATTER

\(z=-z_0\) and, having traversed with uniform velocity the distance \(2z_0\), stopped at the point \(+z_0\). We shall determine the field of the electron at the point of observation \(A\), distant from the origin of coordinates by a distance \(R_0 \gg 2z_0\) (see Fig. 4).

The radiation field observed at the point \(A\) is obtained as the result of interference of the waves emitted by the individual dipoles located on the segment of the \(z\)-axis under consideration. The Hertz vector at the point of observation is equal to

\[ \Pi_\omega=\int_{-z_0}^{+z_0}\frac{\mathbf{P}_\omega(z,t')}{R}\,dz, \tag{3,11} \]

where \(\mathbf{P}_\omega\) has the value (3,10), and \(t'\) is the time with retardation taken into account. With the choice \(R_0 \gg z_0\), the quantity \(R\) in the denominator of (3,11) may be regarded as constant and equal to \(R_0\), while

\[ t'=t-\frac{R_0-z\cos\theta}{c/n} \tag{3,12} \]

(see Fig. 4), where \(n\) is the refractive index of the medium, i.e. \(c/n\) is the velocity of light in the medium, and \(\theta\) is the angle formed by the direction \(R_0\) and the \(z\)-axis. Substituting \(t'\) from (3,12) into (3,10), we obtain

\[ \mathbf{P}_\omega(z,t')=\frac{e}{\pi\omega}z_1 \sin\left\{\omega\left[t-\frac{R_0 n}{c} -\frac{z}{v}\left(1-\frac{vn}{c}\cos\theta\right)\right]\right\}, \tag{3,13} \]

i.e.

\[ \Pi_\omega=\frac{ez_1}{\pi\omega R_0} \int_{-z_0}^{+z_0} \sin\left\{\omega\left[t-\frac{R_0 n}{c} -\frac{z}{v}\left(1-\frac{vn}{c}\cos\theta\right)\right]\right\}\,dz, \tag{3,14} \]

whence

\[ \Pi_\omega= \frac{2ev}{\pi\omega^2R_0} \frac{ \sin\left\{\omega\frac{z_0}{v}\left(1-\frac{vn}{c}\cos\theta\right)\right\} }{ \left(1-\frac{vn}{c}\cos\theta\right) } \sin\left[\omega\left(t-\frac{R_0 n}{c}\right)\right]. \tag{3,15} \]

If the velocity of the electron is greater than the phase velocity of light, then one can find a value \(\theta=\theta_\omega\) for which the coefficient of \(z\) in formula (3,14) vanishes. From formula (3,14) it is clear that this will occur if \(\theta\) satisfies condition (2,3), i.e.

\[ \cos\theta_\omega=\frac{c}{vn(\omega)}. \tag{3,13} \]

In this case all the waves arrive at the point of observation in the same phase, i.e. a summation of waves occurs. The magnitude \(\Pi_\omega\) (see (3,14) and (3,15)) will then increase proportionally to \(z_0\), and, consequently, fulfillment of condition (2,3) is not only a necessary but also a sufficient condition for the occurrence of radiation. For any other \(\theta\) the phase of the waves depends on \(z\), and the more so the larger the ...

\(\theta\) differs from \(\theta_\omega\). Therefore, when \(\theta\) is changed in either direction from \(\theta_\omega\), for a given \(z_0\) the quantity \(\Pi_\omega\) decreases and first becomes zero at such a \(\theta\) for which the phase \(\mathbf{P}_\omega\) in (3.13), as \(z\) changes from 0 to \(z_0\), changes by \(\pi\) (see (3.15)). In this case the field of the oscillators situated in the interval from \(-z_0\) to 0 completely extinguishes the field of the oscillators situated from 0 to \(+z_0\).

From consideration of (3.15) it is not difficult to see that the angular interval corresponding to this maximum of \(\Pi_\omega\) narrows the more strongly, the larger the value of \(z_0\). Hence it follows that the greater the electron range, the more directed its radiation is.

As for the polarization of the radiation, it is determined by the direction of all the dipoles along the \(z\) axis. The electric vector of the waves must therefore lie in the plane containing the ray and the \(z\) axis.

From consideration of (3.15) it is also clear that for any \(\theta\) and \(v\) not satisfying (2.3), \(\Pi_\omega\), generally speaking, is different from zero; i.e. the field of a part of the oscillators can remain uncompensated. When \(z_0\) is varied, the quantity \(\Pi_\omega\) in (3.15) will then change periodically from zero to some constant value. This occurs because a bounded electron trajectory is being considered. The electron begins to move at the point \(-z_0\) and stops at the point \(+z_0\), and therefore radiation corresponding to the acceleration of the electron at these points must exist. The dependence of the quantity \(\Pi_\omega\) on \(z_0\) is the result of interference of the fields corresponding to the emission and stopping of the electron.

From equation (3.15) the angular distribution is obtained directly; it also makes it possible to find the radiation energy*). From the formulas given above it is seen that, in agreement with the experimental data, the radiation of an electron of given velocity is determined by its range and does not depend either on the atomic number of the material of the medium or on its other atomic properties. (The influence of the refractive index on the magnitude of the energy will be considered below.) Thus the difficulty in explaining Cherenkov radiation, which had already been pointed out in connection with Vavilov’s hypothesis (§ 2), is removed. In addition, and also in agreement with experiment, the glow must be polarized, with the electric vector lying in a plane passing through the direction of motion of the electron.

§ 4. RADIATION ENERGY

In the preceding section the radiation field of an electron moving in a medium with a velocity exceeding the phase velocity of light was considered. When these results are compared with experiment, the question arises of the legitimacy of the idealization of the medium adopted there, under which

*) Such a method of consideration was used in Frank’s work \(^{22}\) in solving an analogous problem. Radiation for a bounded electron trajectory. See § 7 of the work of I. E. Tamm \(^{17}\).

real substance is replaced by a continuous homogeneous medium, characterized only by its macroscopic characteristic—the refractive index.

An electron moving in a substance interacts with the individual atoms and molecules located near its trajectory, and expends energy on their ionization and excitation. If one abandons the notions of the atomic structure of matter, then this interaction cannot be explained correctly. Indeed, the equations of motion of an electron in a homogeneous transparent medium at a velocity less than the phase velocity of light, analogous to the equations of motion of an electron in vacuum, show that the electron in this case does not radiate and experiences no braking forces. If, however, one assumes that the motion takes place in a homogeneous but absorbing medium, then the energy losses turn out to be infinitely large, as is not difficult to see on the basis of the results of the preceding paragraph. The field of the electron, as was shown there, may be represented as an aggregate of the fields of point oscillators. It is known, however, that the flow of energy through a sphere surrounding a point harmonic oscillator placed in an absorbing medium tends to infinity if the radius of the sphere is shrunk to zero*). The same, obviously, also holds for an aggregate of oscillators. In problems of radio engineering this feature leads to the necessity of taking account of the real dimensions of the radiator. Here, however, it evidently requires abandoning the representation of the medium as homogeneous, at least at small distances from the electron trajectory.

Until recently, the magnitude of the energy losses of an electron moving in a substance was calculated as the result of the action of the electron on individual atoms or on the electrons of atoms, these interactions being regarded as completely independent of one another.

Such a method of determining the atomic energy losses (in our literature they are usually called ionization losses, which is unfortunate, since in fact, besides ionization, they also include energy losses due to excitation of atoms and radiation) does not always give the correct result. Indeed, such a method incorrectly takes into account the energy losses due to Cherenkov radiation, which, if one starts from the results of the preceding paragraph, arise in the interaction of the electron with a homogeneous medium, i.e. as a result of the coherent action of many atoms.

It is further known that an essential role in the atomic energy losses is played by the interaction of the electron with atoms that are removed from the electron by distances many times exceeding the mean distance between atoms. It is obvious that in this case, besides the action of the electron field on the atom, one must take into account the interaction

*) See, for example, A. A. Petrovsky, Scientific Foundations of Wireless Telegraphy, St. Petersburg, 1913. I express my gratitude to Acad. N. D. Papaleksi for his remarks on this question.

polarized by it among themselves*). In the former theory of atomic losses this, as has already been indicated, was not taken into account. A corresponding correction of the theory was proposed by Fermi²³. It is obvious that the polarization of the medium is taken into account in the simplest way precisely in a macroscopic treatment. Fermi, using Maxwell’s equations, calculated the energy going in a homogeneous absorbing medium through the surface of a cylinder of prescribed radius surrounding the electron trajectory. The medium is thus divided into two regions: inside the cylinder and outside it. Outside the cylinder the medium is treated as homogeneous, and the energy going into it (distant interactions) is calculated with the aid of the macroscopic theory. As for the energy expended in the inner region (near interactions), they must then be calculated by the former method on the basis of atomic concepts. Fermi’s theory gives, for an electron velocity less than the velocity of light in the medium for small frequencies (i.e. for

\[ v < \frac{c}{\sqrt{\varepsilon}}, \]

where \(\varepsilon\) is the dielectric constant), a result that differs hardly at all from the result of the former theory. At high velocities, however, the energy losses prove to be smaller than had previously been supposed.

It follows from the very formulation of Fermi’s problem that in his theory Cherenkov radiation must enter into the general balance of energy losses. As Fermi himself indicates, his theory is analogous to the theory of the Cherenkov effect. It is a generalization of this theory²¹ to the case of a medium with absorption (a complex refractive index).

In what follows we shall consider only that part of the energy losses of the moving electron which is connected with the observed Cherenkov luminescence.

It is obvious that, in order to observe the radiation, it is necessary that the medium be transparent in some range of frequencies. If one restricts consideration only to these frequencies, the problem is substantially simplified. First of all, since absorption is absent, the corresponding quantity of the energy flux going through the surface of the cylinder surrounding the electron trajectory cannot depend on its radius. Therefore, in contrast to an absorbing medium, the prescription of the cylinder radius \(r\) is inessential, and the result will not change if one formally lets \(r\) tend to zero.

This means that all the energy given up by the electron to a transparent medium as a result of distant interactions consists entirely of radiation.

* The possibility that such a polarization effect might influence the magnitude of the energy losses was first pointed out by Swann (W. F. G. Swann, J. Frank. Inst., 226, 598 (1938)).

Next one should take into account the remark made by L. I. Mandelstam*), the validity of which can be justified by simple qualitative considerations. L. I. Mandelstam pointed out that if one imagines an electron moving along the axis of an empty channel drilled in a substance, and if the radius of the channel is much smaller than the wavelength of the emitted light, then the existence of the channel should not affect the intensity of the Cherenkov radiation.

Quantitatively this assertion is illustrated by Fig. 5, which shows the theoretical dependence of the intensity of Cherenkov radiation as a function of the channel radius^24. The radius of the channel is expressed in fractions of the wavelength and is plotted along the abscissa. The ordinates give the relative intensity of radiation of an electron with energy \(1 MeV\) (curve 2) and of an electron of infinitely large energy, \(v/c = 1\) (curve 1), for a refractive index of the medium \(n = 1.5\). The initial ordinate in both cases corresponds to radiation in a continuous medium \((r = 0)\). It is evident from the figure that, for a channel radius of one hundredth of the wavelength \((r = 10^{-7}\ \mathrm{cm}\) for visible light), the radiation is practically insensitive to the absence of substance inside the cylinder. Therefore the fact that near the trajectory of the electron, at distances from it of the order of atomic distances \((\sim 10^{-8}\ \mathrm{cm}\) in a dense substance), the medium cannot be regarded as homogeneously filled with matter should not affect the intensity of Cherenkov radiation.

Fig. 5.

In summary, one may therefore assert that if one is interested only in that part of the light radiation which emerges from the substance, then in the treatment one may replace the real substance by a homogeneous transparent medium with a specified refractive index. In this case all types of energy loss of the electron, except for light radiation, will be excluded from consideration. The energy of the radiation may then be calculated either as the energy flux through the surface of a cylinder^14,17 **), or as the work done in the motion of the electron against the field forces. The true energy losses of the electron in this theory are significant only insofar as they determine the change in the velocity of the elec-

*) L. I. Mandelstam — speech at the defense of the doctoral dissertation of P. A. Cherenkov, 1940.

**) A somewhat different method for determining the radiation energy was developed in the works of V. L. Ginzburg^19.

tron along its trajectory, which in this theory should be regarded as given.

Below is the calculation of the amount of work performed by an electron moving in a homogeneous transparent medium, which can easily be carried out with the aid of the formulas of the preceding paragraph.

Consider the field of the electron at the point \(z=z_1,\ x=y=0\), lying on the electron’s trajectory, at the moment when the electron is at it, i.e. at the moment \(t_1=z_1/v\). The Hertz vector at the observation point is determined from (3,11). The only difference is that the trajectory of the electron is regarded as unbounded, i.e. the integration with respect to \(z\) extends from \(-\infty\) to \(+\infty\),

\[ \Pi_\omega=\int_{-\infty}^{+\infty}\frac{\mathbf P_\omega(zt')}{R}\,dz; \tag{4,1} \]

since the observation point lies on the \(z\)-axis, it follows that

\[ R=z_1-z \qquad z\le z_1 \]

and

\[ R=z-z_1 \qquad z\ge z_1 . \tag{4,2} \]

As for the retardation time, it is equal to

\[ t'=t-\frac{Rn}{c}. \]

Substituting for \(\mathbf P_\omega\) its value from (3,10), we have

\[ \Pi_\omega=\frac{ez_1}{\pi\omega} \int_{-\infty}^{+\infty} \frac{\sin\left\{\omega\left(t-\frac{z}{v}-\frac{Rn}{c}\right)\right\}}{R}\,dz . \tag{4,3} \]

We make a change of variables, taking \(R\) as the variable of integration. Then the integral (4,3) splits into two integrals, since according to (4,2) the change of variable must be made separately for \(z\le z_1\) and for \(z\ge z_1\).

As a result we obtain

\[ \Pi_\omega=\frac{ez_1}{\pi\omega} \left[ \int_0^\infty \frac{\sin\left\{\omega\left[t-\frac{z_1}{v}+\frac{R}{v}\left(1-\frac{vn}{c}\right)\right]\right\}}{R}\,dR + \int_0^\infty \frac{\sin\left\{\omega\left[t-\frac{z_1}{v}-\frac{R}{v}\left(1+\frac{vn}{c}\right)\right]\right\}}{R}\,dR \right]. \tag{4,4} \]

Let us determine the magnitude of the force acting on the electron and directed along the \(z\)-axis, which determines the work performed in the displacement of the electron. This force is

\[ F=eE_z, \tag{4,5} \]

where \(E_z\) is the component of the electric-field intensity directed along the \(z\)-axis, and \(e\) is the charge of the electron.

The Hertz vector is related to the component \(\omega\) of the electric-field intensity by the relation

\[ \mathbf{E}_{\omega}=\frac{1}{n^2}\operatorname{grad}\operatorname{div}\Pi_{\omega} -\frac{1}{c^2}\ddot{\Pi}_{\omega}. \]

The vector \(\Pi_{\omega}\) in our case is directed along the \(z\)-axis and, consequently,

\[ \operatorname{div}\Pi_{\omega}=\frac{\partial \Pi_{\omega}}{\partial z_1} \]

and the \(z\)-component of

\[ [\operatorname{grad}\operatorname{div}\Pi_{\omega}]_z =\frac{\partial^2 \Pi_{\omega}}{\partial z_1^2}, \]

i.e.

\[ E_{z\omega}=\frac{1}{n^2}\frac{\partial^2\Pi_{\omega}}{\partial z_1^2} -\frac{1}{c^2}\frac{\partial^2\Pi_{\omega}}{\partial t^2}. \tag{4,6} \]

Hence, differentiating \(\Pi_{\omega}\) in (4,4) with respect to \(z_1\) and \(t\) and substituting in (4,6), we obtain

\[ E_{z\omega}=+\frac{\omega^2}{c^2} \left(1-\frac{c^2}{v^2 n^2}\right)\Pi_{\omega}. \tag{4,7} \]

We are further interested in the field intensity at the point \(z_1\) only at the moment when the electron is located there, i.e. when

\[ t_1=\frac{z_1}{v}. \]

Taking this into account for the component of the force \(F_{\omega}=eE_{z\omega}\), associated with the action of the field of frequency \(\omega\), from (4,4) and (4,7) we obtain

\[ F_{\omega}=\frac{e^2\omega}{\pi c^2} \left(1-\frac{c^2}{v^2 n^2}\right) \left[ \int_0^\infty \frac{\sin\left\{\dfrac{\omega R}{v}\left(1-\dfrac{vn}{c}\right)\right\}}{R}\,dR - \int_0^\infty \frac{\sin\left\{\dfrac{\omega R}{v}\left(1+\dfrac{vn}{c}\right)\right\}}{R}\,dR \right]. \tag{4,8} \]

It is known that

\[ \int_0^\infty \frac{\sin kR}{R}\,dR= \begin{cases} +\pi/2, & \text{if } k>0,\\ -\pi/2, & \text{if } k<0. \end{cases} \tag{4,9} \]

The factor multiplying \(R\) in the second integral (4,8) is always positive, and, consequently, this integral is equal to \(+\pi/2\). As for the first integral, two cases are possible here. If \(v<c/n(\omega)\), then \(\left(1-\dfrac{vn(\omega)}{c}\right)>0\), and the first integral is also equal to \(\pi/2\), and therefore their difference is zero.

Thus,

\[ F_\omega=0,\quad \text{if } v<\frac{c}{n\omega}. \tag{4,10} \]

Consequently, if the velocity of the electron is less than the phase velocity of light for the frequency \(\omega\), then, in the motion of the electron, no work is performed due to the field component of this frequency. Thus, in accordance with condition (2,3), the frequency \(\omega\) is absent from the radiation.

If, however, \(v>\dfrac{c}{n(\omega)}\), then \(\left(1-\dfrac{vn(\omega)}{c}\right)<0\), and consequently the first integral is equal to \(-\pi/2\). In this case we have:

\[ F_\omega=-\frac{e^2\omega}{c^2}\left(1-\frac{c^2}{v^2 n^2(\omega)}\right), \qquad v>\frac{c}{n(\omega)}. \tag{4,11} \]

The minus sign means that \(E_\omega\) is directed opposite to the motion of the electron, i.e. that the moving electron expends energy in the field of this frequency. The frequency \(\omega\) is emitted, and the work performed is determined by formula (4,11). Since

\[ F_\omega=eE_{\omega z} \quad \text{and} \quad F=eE_z=e\int_0^\infty E_{\omega z}\,d\omega, \]

in order to obtain the total force acting on the electron, one must integrate (4,11) over all frequencies for which \(F_\omega\) is different from zero. The work is equal to the product of the force by the path, and therefore \(F\), in absolute value, is equal to the work \(W\) performed by the electron per unit path, or, what is the same thing, to the energy radiated over \(1\ \mathrm{cm}\) of path. Thus we have

\[ W=\frac{e^2}{c^2}\int \omega\left(1-\frac{c^2}{v^2 n_\omega^2}\right)\,d\omega. \qquad \frac{vn(\omega)}{c}>1. \tag{4,12} \]

Equation (4,12) gives the spectral distribution and the absolute magnitude of the energy radiated along a path of \(1\ \mathrm{cm}\), and is the fundamental formula of the theory.

In comparison with experiment it is often necessary to know the quantum yield, i.e. the number of emitted photons. It is obtained, obviously, by dividing (4,12) by the magnitude of the photon energy, i.e. by \(\hbar\omega\).

Usually, in experiment, the radiation energy is determined in a comparatively narrow frequency interval corresponding to some portion of visible or ultraviolet light, for which the refractive index \(n(\omega)\) in formula (4,12) may be regarded as approximately constant. In addition, it must be taken into account that the velocity of the electrons decreases during motion in the medium. Therefore, in order to determine the total number of emitted photons, one must perform the integration (4,12) over the path \(l\), regarding \(v\) as a function of \(l\). Hence, for the total number of photons,

emitted by an electron with initial velocity \(\vartheta_0\) in the frequency interval from \(\omega_1\) to \(\omega_2\), we obtain

\[ N_{\omega_1\omega_2} = \frac{e^2}{hc^2}(\omega_1-\omega_2) \int_0^L \left(1-\frac{c^2}{\vartheta^2 n^2}\right)\,dl . \tag{4,13} \]

Here \(L\) is the path of the electron which it traverses during the time while its velocity decreases from \(\vartheta_0\) to the critical velocity \(\vartheta_{\mathrm{cr}}=\dfrac{c}{n}\), at which the radiation process ceases.

In Fig. 6 is shown the value, calculated by P. A. Cherenkov,\(^2\) of the energy and the number of photons emitted in a narrow spectral interval from \(\lambda=536\,m\mu\) to \(\lambda=556\,m\mu\), corresponding to his experimental conditions, for various liquids and electron energies.

In integrating (4,13), the dependence of the electron velocity on the path traversed was determined on the assumption that, for a fast electron, the loss of energy per unit path is a constant quantity (for unit density of the substance, it was taken to be \(1750\,KeV/cm\)).

Cherenkov radiation, if its emission is excited by the radiations of radioactive preparations, as has already been indicated, appears in the form of an extremely weak glow. This occurs because the number of emitting electrons in this case is comparatively small, and also because their range in the substance is small. The number of \(\beta\)-particles emitted by 1 g of radium corresponds to a current of only about \(10^{-8}\) ampere. As for the range, only for a few electrons, when moving in water, does it exceed \(1\) cm.

Fig. 6. Dependence of the energy emitted by an electron in the interval from \(\lambda=536\,m\mu\) to \(\lambda=556\,m\mu\), on its initial energy: a) ethyl iodide, b) water, c) carbon disulfide, d) benzene.

The probability of emission of photons, determined by (4,13), is, essentially speaking, very large. Thus, for example, it is known that an excited atom emits a photon in a time of \(10^{-8}\)—\(10^{-9}\) seconds. If one imagines the atom moving with the same velocity as a \(\beta\)-particle (i.e. of the order of \(c\)), then the emission of a photon would occur over a path of tens or even hundreds of centimeters. At the same time, a fast electron, as is seen from Fig. 6, must emit dozens of photons of visible light along its path, not exceeding one centimeter.

A quantum treatment of the radiation of an electron at superluminal velocity was carried out in the work of Ginzburg,\(^{19}\) which has already been mention-

was in connection with the condition for the occurrence of radiation*). Quantum theory, if one leaves aside the question of the role of the electron’s magnetic moment, gives for the energy radiated per unit path the value

\[ W=\frac{e^2}{c^2}\int \omega\left\{1-\frac{c^2}{n^2v^2}\left[1+\frac{n^4}{4}\left(\frac{h\nu}{mc^2}\right)^2+n^2\left(\frac{h\nu}{mc^2}\right)\right]\right\}\,d\omega . \tag{4,14} \]

It is not difficult to verify that this formula, obtained from Schrödinger’s theory, differs from the classical formula (4,12) by correction terms of the same character as the difference between the classically and quantum-mechanically determined angle \(\theta_\omega\) [formulas (2,3) and (2,11) in § 2]. Just as there, the physical meaning of these corrections consists in taking account of the recoil which the electron receives upon emission\({}^{19}\). The quantum formula coincides exactly with the classical one if it is expressed in terms of the angle \(\theta_\omega\). Indeed, taking (2,3) into account, formula (4,12) may be written as

\[ W=\frac{e^2}{c^2}\int_{\omega n/c>1}\omega\sin^2\theta_\omega\,d\omega . \tag{4,15} \]

Formula (4,13) is reduced to the same form, since in the nonrelativistic approximation (\(v/c\ll 1\) and \(n\gg 1\)) expression (2,11) for \(\theta_\omega\) has the form\({}^{19}\)

\[ \cos\theta_\omega=\frac{2c}{nv}\left(1+\frac{h\nu}{2mc^2}n^2\right). \tag{4,16} \]

The difference between the quantum and classical values of \(\theta_\omega\), as already indicated, is negligible, and, consequently, the same is true for the energy values.

§ 5. COMPARISONS WITH THEORY

The most comprehensive comparison of the observed properties of the radiation with theory was carried out by P. A. Cherenkov\({}^{2}\). In a number of respects the theory was also verified in works by American physicists carried out in recent years.

The main attention in these works was directed to the study of the most characteristic feature of the radiation, namely, to the verification of relation (2,3), which determines the directionality

\[ \cos\theta=\frac{c}{vn}. \tag{5,1} \]

The first results in this direction were obtained by P. A. Cherenkov\({}^{2,7,12,13}\) as a result of measurements of photographs with a conical mirror (see Figs. 1 and 2), presented in Fig. 3. As is evident from this figure, the position of the intensity maximum depends also on

*) A quantum treatment of the question is also given in Sokolov’s work\({}^{31}\).

liquid and on the gamma-ray source used. The values of \(\theta\) found from these data for the maximum brightness of the radiation are given in Table 1^2.

Table 1

Liquid	Chemical formula of the molecule	\(n\)	Excitation by Ra \(\gamma\)-rays: \(\theta\) exp.	Excitation by Ra \(\gamma\)-rays: \(\theta\) calc. \((\beta=0.847)\)	Excitation by ThC″ \(\gamma\)-rays: \(\theta\) exp.	Excitation by ThC″ \(\gamma\)-rays: \(\theta\) calc. \((\beta=0.896)\)
Water	\(\mathrm{H_2O}\)	1.34	\(28^\circ\)	\(28^\circ\)	\(33^\circ 30'\)	\(33^\circ 30'\)
Cyclohexane	\(\mathrm{C_6H_{12}}\)	1.44	\(34^\circ 30'\)	\(34^\circ 40'\)	\(38^\circ\)	\(39^\circ\)
Benzene	\(\mathrm{C_6H_6}\)	1.51	\(38^\circ 30'\)	\(38^\circ 40'\)	\(41^\circ 30'\)	\(41^\circ 35'\)
Ethyl cinnamate	\(\mathrm{C_{11}H_{12}O_2}\)	1.58	\(41^\circ 0'\)	\(41^\circ 50'\)	\(44^\circ 0'\)	\(45^\circ 10'\)

If one assumes that relation (5.1) is correct, then in the case of water, from the experimental value of \(\theta\) for the maximum of the radiation and the value \(n=1.34\), we obtain the effective value \(\beta=v/c\).

In the case of radium rays this gives \(\beta=0.847\) \((E \sim 450\ \mathrm{KeV})\), and in the case of the rays of the radiothorium preparation \(\beta=0.896\) \((E \simeq 640\ \mathrm{KeV})\). Thus, in agreement with the existing data on the \(\gamma\)-ray spectra of these preparations, the effective energy in the case of radium proves to be smaller than in the case of radiothorium ThC″. The comparatively small magnitude of this effective energy should not seem surprising, since the effective velocity of the electrons is obtained as a result of averaging over the entire path of the electron, beginning with its initial energy and down to the energy at which it ceases to radiate. For the same reason, the rather broadened maximum for the angular distribution is natural.

If one uses the effective value of \(\beta\) found for water, one can calculate the expected angles \(\theta\) for the maximum intensity in other liquids. As is seen from Table 1, the agreement between the calculated \(\theta\) and the experimental \(\theta\) is quite good.

Thus, if from these data the dependence of the angle \(\theta\) on the electron energy is verified only qualitatively, then for the dependence on \(n\), at least in a first approximation, quantitative agreement with the theory is obtained. For an exact verification of relation (5.1), it was necessary to use a directed monochromatic beam of electrons incident on a sufficiently thin layer of substance, in which the change in the velocity of the electrons would not make itself felt appreciably. Such experiments were carried out by Collins and Reiling^25 (1938), and then by Wyckoff and Henderson^26 (1943).

In both works the Cherenkov method with a conical mirror was used (Fig. 1). At the center of the mirror, however, there was placed not a liquid but a very thin plate, for example of mica, which was bombarded by a beam of electrons of definite energy, accelerated in a high-...

in a volt electron tube. From the photographs obtained by these authors it is evident (Fig. 7) that the directionality of the radiation under such conditions indeed becomes very great. Collins and Reiling determined the angle \(\theta\) in thin plates of mica, glass, and cellophane for electrons with energy \(1.9\,MeV\). To avoid total internal reflection of the radiation, the plates were inclined to the electron beam at an angle of \(45^\circ\), so that the observed radiation emerged at a small angle to the normal to the surface of the plate. The results obtained are given in Table 2.

Table 2

	Thickness in \(cm\)	\(n\)	\(\theta\) observed	\(\theta\) calculated
Mica . . . . . . .	0.002	1.59	\(53^\circ 30\)	\(52^\circ 10\)
Glass . . . . . . .	0.006	1.47	\(45^\circ 15\)	\(46^\circ 30\)
Cellophane . . . . . .	0.002	1.54	\(50^\circ 0\)	\(49^\circ 22\)

Since in these experiments both the velocity of the electron and the refractive index of the substances were known, comparison of the experimental values of \(\theta\) with those obtained from (5,1) could be carried out without any additional assumptions. The discrepancy between the measured and calculated angles everywhere proved to lie within the limits of experimental error.

Fig. 7. Typical radiation pattern in mica.

Let us note in passing that, as was to be expected, the intensity of the glow excited by a powerful beam of artificially accelerated electrons proves to be very considerable. Despite the small thickness of the plates, at a current of \(1\,\mu A\), exposures of only 10 seconds are sufficient to obtain photographs.

Wyckoff and Henderson \(^{26}\) measured \(\theta\) in mica for electron energies from \(240\,KeV\) to \(814\,KeV\). Such comparatively small energies were chosen because at these energies the electron velocity still depends substantially on the energy and, consequently, relation (5,1) can be checked over a wide range of angles. Fig. 7 gives the photographs obtained by them for energies of \(800\,KeV\) (left) and \(258\,KeV\) (right). In the photographs the reflection in the co-

...in a spherical mirror the glow arising in a mica plate inclined to the electron beam at an angle of \(20^\circ\). As can be seen from the photographs, in both cases only a sector of small angular width appears illuminated. At the lower voltage the angular distribution is somewhat more diffuse than at \(800\,KeV\), which is probably explained by electron scattering, which is considerably stronger at low electron energy.

The vertical line in both photographs represents the image in the mirror of the exit slit of the beam, so that the angular distance of the luminous spot from it gives a clear idea of the magnitude \(\theta\) and of its variation with the electron energy.

The positions of the intensity maximum, obtained by these authors from measurements of the photographs, are shown in Fig. 8.

Along the axis of abscissas is plotted the electron energy, and along the axis of ordinates—the angle \(\theta\). The dots and circles represent the experimental values obtained in two series of measurements, while the solid curve is the theoretical dependence (5,1). The deviations of the experimental points from the curve everywhere lie within the limits of error. In the authors’ opinion, real, perhaps, is only a slight discrepancy at small angles, on which, however, they do not insist.

Fig. 8. Angle of distribution of radiation in mica as a function of electron energy.

Let us note that, for a sufficiently thin target, discrepancies of precisely this kind are to be expected on the basis of theory. Indeed, for a luminous layer of finite thickness the radiation should not be absolutely directional. From consideration of the analogous problem in which, instead of restricting the layer of dense matter, the electron range in this matter is restricted (see § 3), one can find that the field amplitude depends on the angle in the following way \(^{17,27}\):

\[ A=A_0 \sin \theta\, \frac{ \sin\left\{\frac{\pi c}{v n}\frac{l}{\lambda}\left(1-\frac{v n}{c}\cos\theta\right)\right\} }{ \left(1-\frac{v n}{c}\cos\theta\right) }, \tag{5,2} \]

where \(l/\lambda\) is the ratio of the electron range to the wavelength. Obviously, only for \(l/\lambda \gg 1\) is condition (5,1) obtained for the radiation maximum.

Conversely, for \(l/\lambda \ll 1\) the angular distribution of the radiation has a dipole character. This result is evident, since the radiation is weak...

is in the present case from the fields of coherent dipoles located on a segment smaller than the wavelength. The maximum intensity will then be observed at an angle \(\pi/2\). Finally, in the intermediate case, when \(l\) is comparable with \(\lambda\), the factor \(\sin b\), especially for small \(b\), will lead to the angle for the radiation maximum being overestimated in comparison with (5.1). In the work of Wyckoff and Henderson one of the targets had thickness \(l = 0.00005\) inch \((1.3 \cdot 10^{-4}\ \text{cm})\), i.e. \(l/\lambda \sim 2\), and consequently the circumstance indicated had to be taken into account.

Comparing the results of Cherenkov, Collins-Riding, and Wyckoff-Henderson, it is necessary to conclude that relation (5.1) is justified not only qualitatively but also quantitatively in the interval of electron energies from \(250\ KeV\) to \(2\ MeV\) and for the interval of refractive indices from \(n = 1.34\) to \(n = 1.58\).

Considerably less complete data are available on the magnitude of the radiation energy. Quantitative results were obtained by Cherenkov\({}^{2,28}\), who placed inside a photometric sphere with white walls a glass vessel with liquid, and in it a thin-walled silvered ampoule with a radioactive substance (radium emanation). Electrons emerging from the ampoule entered the liquid and caused luminescence in it. The presence of the photometric sphere eliminated the uncertainty in the angular distribution of the radiation, which is very difficult to take into account. The luminescence is very weak, and therefore obtaining accurate quantitative data requires great experimental skill. In comparison with theory, difficulties also arise. Formula (4.12) gives the energy radiated per unit path for an electron of given velocity. In reality the \(\beta\)-particles of a radioactive preparation have a complex spectrum of velocities. The range in the liquid depends on the initial velocity, and the velocity of the electron decreases along its path according to a rather complicated law. All these circumstances can be taken into account only approximately. Cherenkov’s measurements nevertheless proved to be in good agreement with the theoretically expected magnitude. For the energy radiated in the wavelength interval from \(\lambda = 536\ m\mu\) to \(\lambda = 556\ m\mu\), Cherenkov obtained a value equal to \(4.1 \cdot 10^{-4}\ \text{erg/sec}\) per millicurie of radium emanation\({}^{*}\). The calculated value of the energy, taking all corrections into account, is \(3.51 \cdot 10^{-4}\ \text{erg/sec}\). It differs from the measured value by \(17\%\), which is considerably less than the possible errors.

An estimate of the yield was also made in the work of Collins and Riding\({}^{25}\); however, they did not fully use the considerable experimental possibilities that were at their disposal. Instead of exciting the luminescence by a monochromatic beam of electrons in a thin layer, i.e. eliminating the uncertainty in ranges and velocities, they excited it in a thick layer of water. The radiation

\({}^{*}\) One millicurie of radium emanation is the amount of this gas in equilibrium with 1 milligram of radium.

was measured at a certain angle, and, consequently, the result depended on the angular distribution of the radiation. Therefore the accuracy of the measurements proved to be very low. The energy value measured by them was approximately 2.5 times smaller than the calculated one, and the authors regard this discrepancy as lying within the limits of error.

Cherenkov investigated, by the same method, the relative magnitude of the radiation energy for various liquids. The conditions under which these measurements were carried out differ substantially from the measurements discussed in § 1 and which showed that the brightness does not depend on the nature of the liquid. There the source of the electrons was γ-rays, and from the laws of their absorption it follows that the number of electrons generated per unit volume had to be proportional to the density of the substance. At the same time, the range of each of the electrons decreased with increasing density, and as a result the glow proved to be insensitive to its change. Here the electrons were emitted by the preparation itself; their number was constant, and therefore this density effect had to occur.

According to (4,12), the radiation energy also depends on the refractive index. In the experiments with gamma rays it was not the energy that was measured, but the brightness of the glow, which, as is known, changes by a factor of \(n^2\) in passing from a dense medium into air. In addition, the observation was made at a specified angle, so that the difference in the angular distribution in different substances could have played some role. It happened by chance that these factors mutually compensated one another, and the glow proved insensitive to the change in \(n\). In measurements with a photometric sphere, however, the dependence on \(n\) should have appeared. The results obtained are given in Table 3, in which the radiation energy of water is taken as unity. The law of electron slowing was taken everywhere to be the same, and the specific energy losses per unit path proportional to the density (i.e., the range, and consequently also the radiation energy, inversely proportional to the density).

Table 3

Liquid	Chemical formula of the substance	Density	\(n\)	Radiation energy, exper.	Radiation energy, by formula (4,13)
Water	\(\mathrm{H_2O}\)	1.0	1.334	1.0	1.0
Benzene	\(\mathrm{C_6H_6}\)	0.879	1.505	1.93	1.92
Cyclohexane	\(\mathrm{C_6H_{12}}\)	0.787	1.433	1.61	1.77
Carbon disulfide	\(\mathrm{CS_2}\)	1.266	1.637	1.68	1.70
Isobutyl alcohol	\(\mathrm{C_4H_{10}O}\)	0.805	1.398	1.40	1.57
Carbon tetrachloride	\(\mathrm{CCl_4}\)	1.61	1.468	1.03	1.0

From this table it is clearly seen that there is a dependence both on \(n\) and on the density, and the results everywhere agree with the calculation within the limits of error.

Summarizing, one may assert that the radiation energy of fast electrons in any case agrees, in order of magnitude, with that theoretically expected. In contrast to braking radiation and in agreement with formula (4.12), the radiation energy is proportional to the path traversed by the electron in the liquid, and does not depend on the atomic number of the substance. The dependence on the refractive index also agrees with the theory.

Essentially, the dependence of the radiation energy on the velocity of the electron has remained uninvestigated, and the existence of the boundary at

\[ \frac{vn}{c}=1 \]

has not been established. There is only a series of experiments, carried out by Cherenkov\(^{1,7}\) with filtration of \(\gamma\)-rays, which show that the radiation, at least for the most part, is caused by fast electrons. Experimental proof of the existence of the boundary would be of interest, all the more so since some liquids, as has already been indicated, also glow under the action of X-rays\(^{4}\), although the nature of this glow is apparently quite different.

Formula (4.12) also gives the distribution of energy in the radiation spectrum. If \(n=\mathrm{const.}\), then the energy must be proportional to the frequency or, on the wavelength scale, inversely proportional to the cube of the wavelength.

Fig. 9. Distribution of energy in the spectrum of the luminescence caused by fast electrons.

As has already been mentioned, as early as Mallet discovered that the radiation spectrum is continuous and extends into the far ultraviolet\(^*\). Authors using an artificial electron beam likewise confirm that the spectrum is continuous\(^{25,26}\). The great intensity of the glow allowed them to speak also of the color of the radiation, which proved to be bluish-white\(^{26}\), in agreement with theoretical expectation. However, only Cherenkov made an attempt to obtain quantitative data on the distribution of energy in the radiation spectrum\(^{2,30}\). Photometry was successfully carried out in a comparatively narrow interval of wavelengths of the visible spectrum, from \(440\,m\mu\) to \(600\,m\mu\). In view of the weakness of the glow observed in the monochromator, photometry was performed by the method of quenching and therefore could be extended only to the sec-

\(^*\) An attempt to observe the ultraviolet part of the spectrum by the photon-counter method was made by Zavadovskaya\(^{2,29}\).

of wavelengths lying in the spectral region of sensitivity of the eye in twilight vision. The results are presented in Fig. 9. Along the abscissa is plotted \(1/\lambda^3\) (\(\lambda\) in Å), and along the ordinate the relative radiation energy. As is seen from the figure, the observed dependence fits well on a straight line passing through the origin. Thus, in this point as well there is agreement with theory.

The polarization of the glow was measured by Cherenkov \(^{2,1,7}\) only in experiments with \(\gamma\)-rays, in which the angular distribution of the electrons was evidently very diffuse. In spite of this, the degree of polarization nevertheless proved to be fairly considerable (25%). The sign of the polarization, as was already indicated, was found to be correct. The fact of complete polarization, predicted by theory, could have been established only in measurements in the presence of a parallel beam of electrons and a thin layer of substance, which, however, was done by no one.

The nature of the electronic emitters can also be elucidated in the study of the interference properties of the radiation. The distinction between dipoles, quadrupoles, etc., can be established by this method \(^{32}\). In the case where the emitter is oriented in a definite way, the same data can also be obtained from an investigation of the angular distribution of the intensity. With regard to the interference properties of Cherenkov radiation, only preliminary data were obtained by M. N. Alentsev; these do not contradict the idea that in this case chains of coherent dipoles radiate \(^{27}\). The interference properties, however, were not fully elucidated. One may suppose, for example, that by making an electron pass successively through two thin layers of substance, we shall obtain two luminous volumes whose radiation will be mutually coherent, i.e. capable of interfering \(^{27}\).

The angular distribution was already discussed above. In essence, only the position of the intensity maximum was investigated. As for the angular distribution itself, it would make sense to determine it in extremely thin layers of substance, in which it should lose its sharp directionality.

Finally, the last characteristic property of the emitters—the presence or absence in them of an excited state of finite duration—was, as already indicated (§ 1), clarified in Cherenkov’s first experiments and served as one of the starting points for the interpretation of the phenomenon.

§ 6. CONCLUSION

The experimental facts set forth above show that the radiation discovered by Cherenkov must indeed be regarded as the radiation of an electron moving with a velocity exceeding the phase velocity of light. Both the classical and the quantum treatments of the phenomenon, based on an idealization of the substance in which the phenomenon occurs, сафронова?

motion, by replacing it with a continuous homogeneous medium characterized only by its refractive index, gives results in agreement. These results, in all respects qualitatively, and in those that have been investigated in detail also quantitatively, are confirmed by experiment. Thus, the existing theory gives, in all its main points, a correct description of the phenomenon.

Cherenkov radiation is the only phenomenon of the optics of velocities exceeding the speed of light that has been investigated at the present time. This phenomenon shows that the peculiarities of the propagation of light in a medium can lead to fundamentally new results from the point of view of radiation theory. In this connection it is of interest to clarify how the macroscopic properties of a medium, i.e. the difference between the velocity of light in it and the velocity of light in vacuum, dispersion, and absorption, affect the properties of various radiators placed inside the medium. This influence proves especially significant for moving sources of light, but for sources of light at rest it must also be taken into account. Thus, for example, the radiation of an oscillator placed in a medium proves to be proportional to \(n\), if it is due to the presence in it of an electric dipole moment, and proportional to \(n^{3}\) for a magnetic oscillator. Hence it follows that, for example, for fluorescent molecules placed in different media, even in the absence of quenching, the rate of damping (a quantity inverse to the lifetime \(\tau\)) depends on the medium. For electric dipoles it is proportional to \(n\), and for magnetic ones to \(n^{3}\). The existing accuracy of measuring \(\tau\), apparently, may make it possible to detect this dependence.

When a source of light moves in a medium, what is essential, in contrast to vacuum, is not only its velocity and its change, but also the velocity of light in the medium and its change. Therefore, for example, the transition of an electron from one medium into another is to some extent equivalent to a change in its velocity: a uniformly moving electron must then radiate light\({}^{33}\). This radiation has the greatest intensity when the electron passes from a transparent medium into a medium that strongly absorbs light, or, conversely, for example, from vacuum into a metal or from a metal into vacuum. From the point of view of the generation of frequencies in the optical part of the spectrum, this case is equivalent to the complete stopping of an electron or to its acceleration from zero to full velocity. The intensity of such “transition radiation” is not large, but, apparently, it can readily be observed with an artificially produced beam of electrons of even comparatively small velocity. In the work of Ginzburg and Frank\({}^{33}\), from this point of view, the experiments of Kohn\({}^{34}\) are discussed, in which a glow was observed at the anticathodes of electron tubes, although, apparently, in these experiments the glow was caused by the presence of residual gas in the tube. In this connection it is interesting to note that Collins and Reiling\({}^{25}\) discuss the question of whether the glow investigated by Kohn is not identical with Cherenkov radiation. They reproduced these experiments of Kohn, making pass

RADIATION OF ELECTRONS MOVING IN MATTER

a beam of electrons through metal foils placed in air. The authors indeed observed a bluish-white glow at the surface of foils (of thickness \(0.001\ \mathrm{cm}\)) of aluminum, copper, silver, and platinum. They quite justifiably assert that this cannot be Cherenkov light, for the reason that, in particular, for silver the refractive index is less than unity. In all probability, these authors observed precisely “transition radiation,” provided only that it, as in Cohn’s experiments, was not masked by the glow of the gas.

Something analogous, in the sense of the existence of a transition from one medium to another, also occurs for Cherenkov radiation. Indeed, observation of the glow is usually carried out in air, while the generation takes place in a dense substance*). It is not difficult to see that Cherenkov radiation can emerge from the medium into the air only in the case when the electron moves at an acute angle to the surface separating the medium and the air, i.e. when it approaches it.

If, however, the electron moves parallel to the interfaces of the substance, and moreover in such a way that the radiation cannot emerge from it (motion in a cylindrical column of matter along its axis), then the energy losses to radiation in general become equal to zero\({}^{24}\). Thus, the presence of a transition of radiation from one medium to another may have fundamental significance.

Radiation entirely analogous to Cherenkov radiation should also arise when a particle possessing an electric or magnetic dipole moment moves with superluminal velocity. This circumstance was first pointed out by Ginzburg\({}^{19}\) in connection with the quantum theory of the Cherenkov effect. The presence of a magnetic moment in the electron leads, in the classical treatment, to additional radiation, the intensity of which is negligibly small in comparison with the radiation of the charge. In the general relativistic case, for the energy radiated per unit path, one obtains: for electric and magnetic dipoles oriented in the direction of the velocity\({}^{22}\)

\[ W=\frac{p^{2}(1-v^{2}/c^{2})}{v^{2}c^{2}}\int \omega^{3}\left(1-\frac{c^{2}}{v^{2}n^{2}\omega}\right)\,d\omega \tag{6,1} \]

and

\[ W=\frac{m^{2}(1-v^{2}/c^{2})}{v^{2}c^{2}}\int n^{2}(\omega)\omega^{3}\left(1-\frac{c^{2}}{v^{2}n^{2}(\omega)}\right)\,d\omega, \tag{6,2} \]

(where \(p\) and \(m\) are the electric and magnetic moments of the dipoles at rest) and, for dipoles oriented perpendicular to the direction of motion, respectively\({}^{22}\)

\[ W=\frac{p^{2}}{2v^{2}c^{2}}\int n^{2}(\omega)\omega^{3}\left(1-\frac{c^{2}}{v^{2}n^{2}(\omega)}\right)\,d\omega . \tag{6,3} \]

* The exception is, perhaps, only the glow arising in the liquid that fills the human eye. This glow is clearly visible if, in darkness, a preparation emitting gamma rays is brought up to the eye.

and

\[ W=\frac{m^2}{2v^2c^2}\int n^2(\omega)\omega^3\left\{2\left(1-\frac{1}{n^2(\omega)}\right)^2-\right. \]
\[ \left.-\left(1-\frac{v^2}{c^2n^2(\omega)}\right)\left(1-\frac{c^2}{v^2n^2(\omega)}\right)\right\}\,d\omega . \tag{6,4} \]

Let us compare, for example, the radiation of a longitudinal magnetic dipole (6,2) with the radiation of an electron (4,12). The ratio of the energy quantities is equal to

\[ \frac{m'\omega^2n^2}{e^2v^2} = \frac{4\pi^2m'^2}{\lambda^2e^2}\frac{c^2}{v^2}. \]

Taking the magnetic moment \(m'=m\sqrt{1-v^2/c^2}\) to be of the order of magnitude of the Bohr magneton, \(m'\sim 10^{-20}\), and putting \(\lambda\sim 10^{-5}\), \(e\sim 10^{-10}\), and \(v/c\sim 1\), we obtain for this ratio a value \(\sim 10^{-9}\). Thus, the radiation associated with the magnetic moment of the electron must be negligibly small in comparison with ordinary Cherenkov radiation.

It is clear from the formulas that the radiation for longitudinally oriented dipoles tends to zero as \(v\to c\), which is explained by the Lorentz contraction of the dipole moment. The radiation of a transverse dipole as \(v\to c\), like the radiation of a charge, tends to a constant value.

It is hardly possible to detect experimentally the radiation of a particle arising as a result of the presence of a magnetic moment. However, theoretically, its presence or absence is of fundamental significance, since it characterizes the magnetic properties of the particle. Ginzburg \(^{19,35}\) used, for this purpose, the calculation of such radiation in the extreme relativistic case for particles with spin zero, one half, one, and three halves. It turned out that particles with spin zero and one half do not give magnetic radiation, whereas for spin 1 and \(3/2\) radiation is obtained corresponding to the radiation of a transverse dipole (formula (6,4)). This result is explained by the fact that a particle with spin zero has no magnetic moment at all. A particle with spin one half possesses an active magnetic moment only if it is directed longitudinally—along the velocity or opposite to it. As a result of Lorentz contraction, in the extreme relativistic case this magnetic moment becomes equal to zero. Finally, for spin values 1 and \(3/2\), there will be radiation in the case where there is a component of the magnetic moment perpendicular to the direction of the velocity.

A different picture for the radiation is obtained in the case when a particle moves that has no constant component of the moment—for example, a magnetic or electric oscillator. Such a moving oscillator radiates at any velocity, and the observed frequency obeys the Doppler condition. The presence of a medium here too leads to very peculiar changes. At velocities smaller than the speed of light in the medium, the frequency emitted at an angle \(\theta\) to the direction of mot—

radiation, obeys the condition^22

\[ \omega_{\theta}=\frac{\omega_0\sqrt{1-v^2/c^2}}{1-\dfrac{v n(\omega_{\theta})}{c}\cos\theta}, \tag{6,5} \]

where \(\omega_0\) is the proper frequency of the moving oscillator. Equation (6,5) differs from the usual Doppler condition only in that the velocity of light in vacuum has been replaced in it by the velocity of light in the medium for the Doppler frequency \(c/n(\omega_{\theta})\). However, since \(n(\omega_{\theta})\), in the presence of dispersion, is a complicated function of \(\omega\), equation (6,5) may, for given \(v\), \(\theta\), and \(\omega_0\), have not one but several solutions. This means that the Doppler frequency may split into several components. Such a “complex” Doppler effect can in principle be observed if the Doppler frequency is close to the frequency of an absorption line of the gas in which the motion takes place.

L. I. Mandelstam*) pointed out that the possibility of the occurrence of the “complex” Doppler effect is determined by the magnitude of the group velocity of light in the medium. Indeed, it is not difficult to show that the condition derived earlier for the occurrence of the complex Doppler effect^22 proves to be equivalent to the requirement that the radial component of the velocity of motion be greater than the group velocity of light for one of the Doppler frequencies.

It has already been indicated above that the radiation energy of an oscillator depends on the refractive index. At relativistic velocities in a medium it also turns out to depend on the orientation of the dipole moment^22.

A special place, as in the case of the electron, is occupied by the case of velocities greater than the phase velocity of light. In contrast to the ordinary Cherenkov effect, for an oscillator there is no radiation satisfying the condition (2,3), i.e.

\[ \frac{v n(\omega_{\theta})}{c}\cos\theta=1. \]

Indeed, if this condition were fulfilled, then \(\omega_{\theta}\) in (6,5) would become infinite, which contradicts the requirement \(n(\omega_{\theta})>1\).

However, at superluminal velocities new Doppler frequencies may arise, which obey the condition^22

\[ \omega_{\theta}=\frac{\omega_0}{\dfrac{v n(\omega_{\theta})}{c}\cos\theta-1}. \tag{6,6} \]

These frequencies must necessarily give a complex Doppler effect. In addition, frequencies satisfying (6,5) must be observed simultaneously with them. A quantum treatment reveals the physical nature of these superluminal Doppler frequencies^24. In contrast to

*) L. I. Mandelstam—Lectures (unpublished).

ordinary radiation, they correspond to the case when the system emitting light spontaneously passes from the normal state to an excited one and at the same time emits a photon. The energy is then drawn from the kinetic energy of motion. This feature points to an analogy of the phenomenon with the Cherenkov effect. An atom moving with superluminal velocity must excite itself, emitting light in the process and losing kinetic energy. The excited atom will give radiation satisfying (6.5), in which it passes into the normal state. Then excitation will occur again, and so on. The process will be repeated many times, and the radiation will not cease until the velocity of the atom exceeds the velocity of light in the medium. Consequently, an atom moving with superluminal velocity will brake itself, giving up energy to radiation.

All that has been said about close interactions for the electron applies, of course, also to the atom. Therefore an atom in flight, owing to close interactions, will become ionized, and its electron shell will often be subjected to irregular perturbations. The question therefore arises as to what extent the phenomena considered above are at least in principle observable.

In this connection it should be noted that, just as is the case for the electron, the result of the consideration will not change if one assumes that the atom moves not in a continuous medium, but inside a channel of small diameter, compared with the wavelength of the emitted light[^24] (or, what is the same, near the surface of the substance). In this case the close interactions will be eliminated, and if it were possible to attain the necessary velocities, the phenomenon could be observed.

From what has been said it is evident that radiation in refracting media indeed possesses a number of essential features. Generalizing the interpretation of the phenomenon discovered by Cherenkov, one may assert that any system capable of interacting with radiation must radiate at the expense of its kinetic energy if its velocity exceeds the phase velocity of light.

LITERATURE

1. P. A. Cherenkov, DAN, 2, 451, 1934.
2. P. A. Cherenkov, Transactions of the Physics Institute of the USSR Academy of Sciences, vol. 2, No. 4, 1944.
3. L. A. Tumerman and S. I. Wawilow, Zeits. f. Phys., 54, 270, 1929.
4. A. K. Timiryazev (unpublished); H. S. Newcomer, Journ. Amer. Chem. Soc., 42, 1997, 1920.
5. M. L. Mallet, C. R., 183, 274, 1926; 187, 222, 1928; 188, 445, 1929.
6. E. M. Brumberg and S. I. Vavilov, Izv. AN OMEN, ser. VII, 919, 1933; DAN, 3, 405, 1934.
7. P. A. Cherenkov, Izv. AN OMEN, physical series, 455, 1937.
8. S. I. Vavilov, Izvestiya AN, physical series, 9, 283, 1945.
9. S. I. Vavilov, DAN, 2, 457, 1934.
10. P. A. Cherenkov, DAN, 14, 99, 1937.
11. P. A. Cherenkov, DAN, 3 (12), 413, 1936.

1. P. A. Cerenkov, Phys. Rev., 52, 378, 1937.
2. P. A. Cherenkov, DAN, 21, 323, 1938.
3. I. E. Tamm and I. M. Frank, DAN, 14, 107, 1937.
4. Lord Kelvin, Nineteenth Century, Clouds over Dynamical Theory of Heat and Light. Phil. Mag. Jouly, 1901.
5. A. Sommerfeld, Götting. Nachricht 99, 368, 1904; 205, 1905.
6. Ig Tamm, Journ of Phys., USSR, 1, 439, 1939.
7. V. L. Ginzburg, ZhETF, 10, 608, 1940.
8. V. L. Ginzburg, ZhETF, 10, 589, 1940; DAN, 24, 131, 1939.
9. R. T. Cox, Phys. Rev., 66, 106, 1944.
10. I. E. Tamm and I. M. Frank, Proceedings of the Physical Institute, vol. 2, No. 4, 1944.
11. I. M. Frank, Journ of Phys., USSR, 7, 49, 1943.
12. Enrico Fermi, Phys. Rev., 57, 485, 1940.
13. V. L. Ginzburg and I. M. Frank, The Doppler Effect at Superluminal Velocity (in press).
14. G. B. Collins and V. G. Reiling, Phys. Rev., 54, 499, 1938.
15. H. O. Wyckoff and J. E. Henderson, Phys. Rev., 64, 1, 1943.
16. I. M. Frank, DAN, 42, 341, 1944.
17. P. A. Cherenkov, DAN, 21, 117, 1928.
18. E. K. Zavadovskaya, Proceedings of the Siberian Physical-Technical Institute, IV, issue 3, 98, 1936; Phys. Zeit. d. Sowjetunion, 13, 244, 1938.
19. P. A. Cherenkov, DAN, XX, 653, 1938.
20. A. Sokolov, DAN, 28, 415, 1940.
21. S. I. Vavilov, DAN, 17, 459, 1937; O. Halpern and F. W. Doermann, Phys. Rev., 52, 937, 1937.
22. V. L. Ginzburg and I. M. Frank, ZhETF, 16, 15, 1946.
23. W. M. Cohn, Zeits. f. Phys., 73, 662, 1931.
24. V. L. Ginsburg, Journ. of Phys., USSR, 5, 47, 1941, 7, 115, 1943.