CHERENKOV RADIATION*)

J. V. Jelley

CONTENTS

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 232
    a) Cherenkov’s discovery . . . . . . . . . . . . . . . . . . 232
    b) Qualitative explanation of the phenomenon . . . . . . . . . 233
    c) Survey of further work by Russian investigators . . . . . . 236
        1) Radiation spectrum . . . . . . . . . . . . . . . . . . 236
        2) Absolute intensity of the radiation . . . . . . . . . . 238
        3) Dependence of \(\theta\) on \(n\) and \(\beta\) . . . . . . . 239

II. Theoretical interpretation . . . . . . . . . . . . . . . . . . 240
    a) The theory of Frank and Tamm . . . . . . . . . . . . . . . 240
    b) Development of the elementary theory . . . . . . . . . . . 243
    c) Quantum treatment of the phenomenon and “magnetic” Cherenkov radiation . . . 248
    d) A charged particle moving near the surface of a dielectric; Cherenkov radiation as a possible source of microwaves . . . . . . . . . . . . . . . . . . . 250

III. Later experimental work . . . . . . . . . . . . . . . . . . . 251
    a) Experiments with artificially accelerated particles . . . . . 251
    b) Cherenkov radiation from cosmic-ray particles . . . . . . . . 254
    c) Radiation in aqueous solutions of radioactive isotopes . . . . 259
    d) Cherenkov radiation in the atmosphere . . . . . . . . . . . . 262

IV. Practical applications of Cherenkov detectors . . . . . . . . . 263
    a) General considerations . . . . . . . . . . . . . . . . . . 263
    b) Photomultiplier . . . . . . . . . . . . . . . . . . . . . 266
    c) Counters with focusing properties . . . . . . . . . . . . . 268
    d) Photographic instrument of high precision . . . . . . . . . 271
    e) Proton selector . . . . . . . . . . . . . . . . . . . . . 276
    f) Albedo of cosmic rays . . . . . . . . . . . . . . . . . . 279

V. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 281

) Progress in Nuclear Physics*, vol. 3, pp. 84—130. Translated by A. A. Ilyina.

J. V. JELLEY

I. INTRODUCTION

a) Cherenkov’s Discovery

In the early period of the history of radioactivity several observers noted that various substances, including solutions of mineral salts, emit a weak light under the influence of radiation from radioactive substances. It was established over time that this radiation was, in the main, fluorescence light and behaved in many respects in the same way as the radiation of various substances arising under the action of ultraviolet rays.

While studying the luminescence of solutions of uranium salts under the action of $\gamma$-rays, Cherenkov discovered in 1934 that a very weak radiation is visible even in the case of pure liquids[^1]. A detailed study of a large number of pure liquids led him to the conclusion that the nature of this new phenomenon was different from fluorescence. This effect, which soon came to be called the Cherenkov effect, was interpreted three years later by Frank and Tamm[^2] on the basis of classical electromagnetic theory.

Fig. 1. Cherenkov apparatus used in his first works[^1].

It is interesting to note that eight years earlier Mallet[^3] had obtained ultraviolet spectra emitted by water irradiated with $\gamma$-rays. Undoubtedly this phenomenon should be identified with the phenomenon discovered by Cherenkov, although Mallet’s work is rarely cited as the first observation of this effect.

In his first experiments Cherenkov used a very simple apparatus, the diagram of which is shown in Fig. 1. A small ampoule containing 104 mg of radium was placed in position $R_1$ in a wooden block $B$, in which a platinum crucible $A$ was also set. The liquid under investigation was located in $A$ above the radium source. By means of an optical system consisting of a collimator $L_1$, a prism $P$, and a telescope $L_2L_3$, it was possible to observe at $E$ the weak glow of the liquid near the source. The field of view was limited by a diaphragm $D$; other elements of the optical system were: a graduated wedge $W$ for measuring relative intensities, filters $F$ of various colors for rough spectral analysis, and a Nicol polarizing prism $N$ for studying the polarization of the radiation.

With an eye adapted to the dark, Cherenkov could measure the relative intensities of the light of 16 pure liquids, among which were: distilled water, paraffin, xylene, toluene, glycerin

and various alcohols. In these experiments he used as a criterion the moment of extinction of the light by an optical wedge. Cherenkov’s main conclusions amounted to the following: for all 16 liquids the interval of observed values of relative intensities was comparatively small (from 65 for isobutyl alcohol to 81 for carbon tetrachloride and 71 for distilled water). It was shown that the spectral distribution of the radiation differs little in passing from one liquid to another, and that the emitted wavelengths are concentrated in the blue and violet regions of the spectrum. In addition, it was established that there was no diminution of the intensity of the glow when silver nitrate, potassium iodide, or other compounds known as strong quenchers of fluorescence were added to the liquid. Changes of temperature from room temperature to \(100^\circ\) C did not affect the radiation intensity, whereas the strong influence of temperature that occurs in the case of fluorescence is well known. These temperature changes, of course, produced very large variations in the viscosity of some of the liquids investigated. In experiments with polarization a small effect was observed, indicating partial polarization, in which the electric vector lay in the same direction as that of the exciting \(\gamma\)-radiation. In the polarization experiments the source was placed in position \(R_2\) in Fig. 1 and was rotated about the axis passing through \(A\) and \(P\). If the relative intensities of the rays emerging from the Nicol prism with vertical and horizontal orientations of the plane of polarization are denoted by \(I_v\) and \(I_h\), then in Cherenkov’s experiments values close to 10 were obtained for the ratio \(\dfrac{I_v+I_h}{I_v-I_h}\), so that \(I_v/I_h=1.2\), i.e., the partial polarization was \(\sim 20\%\).

b) Qualitative explanation of the phenomenon

Although later we shall return once more to the discussion of Cherenkov’s subsequent works, even at this stage it is useful, anticipating the work of Frank and Tamm\(^2\), to consider qualitatively the physical nature and origin of Cherenkov radiation. If a fast charged particle moves in a dielectric medium with constant velocity, the electromagnetic impulse associated with it temporarily polarizes the medium near the particle’s trajectory. In this process the individual atoms follow the electromagnetic pulsation of the particle and thereby themselves become emitters of an electromagnetic wave. In the general case the waves emitted by them, coming from all parts of the trajectory, interfere in such a way that, at a point situated at some distance from the trajectory, the intensity of the resultant field proves to be zero.

However, if, as often happens, the velocity of the particle exceeds the phase velocity of light in this medium, the elementary waves,

emitted from all parts of the trajectory, can be in phase at a certain point of observation and give a resultant field there. From the Huygens construction shown in Fig. 2, it is seen that this radiation can be observed only at a certain angle \(\theta\) with respect to the path of the particle; this is the angle at which the elementary waves from the points \(P_1, P_2\), and \(P_3\), lying on the trajectory \(AB\), are coherent and form the plane wave front \(BC\). This coherence occurs when the particle traverses the path \(AB\) in the same time in which light traverses the path from \(A\) to \(C\).

Fig. 2. Huygens construction showing the formation of coherent radiation.

The condition of coherence, therefore, consists in the following: during the time interval \(\Delta \tau\), the particle must traverse the path from \(A\) to \(B\), while the wave front of the radiation passes from \(A\) to \(C\). If the velocity of the particle is \(\beta c\), where \(c\) is the velocity of light in vacuum, and if \(n\) is the refractive index of the medium, then \(AB=\beta c\Delta \tau\) and \(AC=\left(\frac{c}{n}\right)\Delta \tau\). Hence it follows that

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

This relation is fundamental, and, as we shall see, it was the one first confirmed by Cherenkov. From this relation it is seen that:

1) For a medium with a given refractive index \(n\), there exists a critical velocity \(\beta_{\min}=\frac{1}{n}\), below which there will be no radiation. At this critical velocity the radiation will be parallel to the direction of the particle.

2) For a relativistic particle, for which \(\beta=1\), there will be observed a maximum emission angle \(\theta_{\max}\), which is given by the expression

\[ \theta_{\max}=\arccos\left(\frac{1}{n}\right). \tag{2} \]

3) The radiation must be located chiefly in the visible (or near-visible) region of the spectrum, for which \(n\) is positive. Emission in the region of X-rays is impossible, since \(n\) will be less than unity and equation (1) will not be satisfied.

4) It can be seen that, within the limits of velocities determined by conditions (1) and (2), the emission angle increases with the velocity of the particle.

Since the probability of emission of light is the same for all directions of the vector \(AC\), which makes an angle \(\theta\) with the direction \(AB\), the light is emitted inside a certain cone with half-angle \(\theta\) at the vertex and with its axis coinciding with the direction \(AB\) (Fig. 3). Further, if, as often happens, the thickness of the layer of medium \(AB\) is small

in comparison with the distance at which the radiation is observed; then the light is concentrated on the conical surface, and the distribution of intensity over the angles \(\theta\) should approach a \(\delta\)-function.

From all that has been said, it should be clear that this phenomenon is to some extent analogous to the V-shaped shock wave observed in acoustics when a projectile flies through the air at a speed exceeding the speed of sound. A simpler case is the formation of a bow wave by a ship moving on water; in this case the speed of the ship is greater than the speed of waves on the surface of the water.

Fig. 3. Formation of the cone of Cherenkov radiation and the polarization vectors.

Returning to the optical phenomenon, one must clearly keep in mind that here there are two further conditions which must be satisfied in addition to those established by relation (1). First, the length \(l\) of the particle’s path in the medium must be large in comparison with the wavelength \(\lambda\) of the radiation; otherwise diffraction effects must predominate, and the light will propagate within the limits of the angle \(\theta_{\mathrm{diff}} \sim \lambda/l\), instead of appearing only at one angle given by relation (1). Second, the velocity of the particle must be constant during its passage through the medium, or, more precisely, the differences between the time intervals in which the particle traverses successive path segments \(\lambda\) must be small in comparison with the period \((1/\nu)\) of the emitted wave.

In connection with the second condition it is necessary to emphasize at this stage that Cherenkov radiation should not be identified with radiation emitted by an accelerating or decelerating charged particle, which corresponds to “bremsstrahlung,” such as X-rays with a continuous spectrum. We stress this especially here because not only did Vavilov\(^{4}\) later give this unconfirmed interpretation of Cherenkov’s initial experiments, but much later, even after the work of Frank and Tamm, the same incorrect explanation of this phenomenon was advanced by Collins and Reiling\(^{5}\).

There are several characteristic features of bremsstrahlung that distinguish it from Cherenkov radiation\(^{6}\).

First, radiative losses in the process of bremsstrahlung occur rather as a result of a few collisions involving the transfer of large amounts of energy, and not as a result of a large number of collisions involving the transfer of small amounts of energy. The greater part of the radiation emitted in this process has an energy quantum,

comparable with the energy of the particle. The total energy radiated in this process is, in general, considerably greater than the losses in the Cherenkov process, but the fraction of bremsstrahlung falling in the visible region is utterly negligible. Secondly, whereas the intensity of Cherenkov radiation depends only weakly on the physical properties of the medium and is connected only with the refractive index \(n\), bremsstrahlung depends substantially on the atomic number \(Z\) of the substance, being proportional to \(Z^2\).

Further, the intensity of Cherenkov radiation does not depend on the mass of the particle, whereas bremsstrahlung is inversely proportional to the square of the particle mass, i.e., if \(m\) and \(M\) are respectively the masses of the electron and the proton, then the intensities of bremsstrahlung are in the ratio \((M/m)^2\), which is equal to \(3.4\cdot 10^6\)¹. Collins and Reiling⁵ and Mather⁷ found that the intensity of Cherenkov radiation for electrons and protons is approximately the same. Moreover, at high values of the particle energy \(E_0\), bremsstrahlung is concentrated inside a cone with solid angle \(\Omega \sim mc^2/E_0\) relative to the particle path, i.e., at large particle energies the angle of the emitted radiation would be smaller, whereas for Cherenkov radiation the opposite should be observed. Finally, the process of bremsstrahlung entails large angles of deflection of the particle, whereas from what has been said it is clear that in the case of Cherenkov radiation only very small changes in the direction or velocity of the particle are permissible without disrupting the coherence that produces Cherenkov radiation.

c) Review of further work by Russian investigators

Following Vavilov’s supposition that this effect might be due to bremsstrahlung, Cherenkov in the course of the next two years carried out a series of experiments; part of this work was done before the communication of Frank and Tamm, and part somewhat later. The study of the phenomenon under the influence of a magnetic field was carried out by Cherenkov in 1936⁸ with a source of \(\gamma\)-rays and a cuvette with water. This experiment showed that the radiation can more readily be attributed to secondary electrons arising in the medium than to the \(\gamma\)-rays themselves. Then in 1937, almost at the same time as Frank and Tamm developed their theory, two more works appeared—

Fig. 4. Cherenkov’s first setup for photographic recording of the phenomenon: a—the instrument; b—appearance of the photographic image obtained with this instrument¹¹.

CHERENKOV RADIATION

...works. In the first of them\(^9\) it was reported that the phenomenon is observed with \(\beta\)-rays, and in the second\(^ {10}\) a simple experiment was described which at first showed the asymmetry of the light intensity with respect to the direction of the exciting radiation, as had been predicted by Frank and Tamm. With a source of \(\gamma\)-rays of insufficient parallelism and a thick layer of substance, only a rough check was possible of the relation

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

The first photographic observations of this effect appeared in 1937.\(^ {11}\) Figure 4, \(a\), shows the simple apparatus used in these experiments. A source of \(\gamma\)-rays, equivalent to 794 mg of radium, was placed on one side of a conical mirror \(M\), inside which there was a thin-walled glass vessel \(B\), filled with water or benzene. The light emitted through the walls of the vessel was reflected upward and focused by the objective \(L\) \((f:1.4)\) onto a photographic plate. With an exposure of 72 hours an image was obtained, of the general form shown in Fig. 4, \(b\), indicating an asymmetric distribution of the light, in which two regions are more intense than the others.

Fig. 5. Cherenkov’s experiment for determining the spectral distribution of the radiation\(^ {65}\): \(a\)—spectrometer; \(b\)—observed intensity distribution in comparison with a Hefner lamp; \(c\)—intensity function \(E(\lambda)\), plotted as a function of \(1/\lambda^3\).

In 1938 three more detailed experiments were carried out to check the theoretical predictions.

1) The spectrum of the radiation was investigated\(^ {65}\) with the aid of the apparatus shown in Fig. 5, \(a\). The \(\beta\)-rays from a radon source \(Rn\) (\(\sim 100\,mC\)) in a small glass ampoule irradiated benzene poured into a thin-walled glass vessel \(V\). The image of \(V\) was focused on the slit \(S\) of one of two monochromators \(M_1\) and \(M_2\) (placed one behind the other). The light was visually observed in \(E\)...

and, with the aid of a neutral wedge \(W\), the relative radiation intensities were determined (by the quenching-threshold method) in the wavelength interval \(4300\)—\(6000\) Å. A comparison was then made with the light spectrum of a standard Geffner lamp, which illuminated a magnesium-oxide screen placed in the same position as that initially occupied by the source \(Rn\).

The energy-distribution curves of these two sources are given in Fig. 5, б. From the known spectral distribution of the Geffner lamp and the two measured curves, the curve of the true distribution of the radiation arising in benzene, \((E\lambda)\), was obtained. In constructing the curve of \(E(\lambda)\) as a function of \(1/\lambda^3\) (see Fig. 5, в), excellent agreement between experiment and theory is seen (the \(1/\lambda^3\) law is derived in the following section).

2) The absolute radiation intensity\(^{12}\) was then measured using the apparatus shown in Fig. 6. A radon source (\(\sim 150\, mC\)) was used; it was sealed in an ampoule \(Rn\) and placed at the center of a small spherical vessel \(V\), filled with water. The vessel \(V\) was located close to the surface

Fig. 6. Apparatus used by Cherenkov for determining the absolute intensities of Cherenkov radiation\(^{12}\).

of the integrating spherical cavity \(C\) (the inner surface of which was coated with magnesium oxide). The cavity itself was made inside a lead block \(Pb\), which protected the observer from the intense \(\gamma\)-radiation of the source. The Cherenkov radiation arising in the water, under the assumption of a uniform distribution over the spherical surface \(C\) and emerging from the aperture \(D_1\), was observed visually at \(E\) through a constant-deviation spectrometer \(S\).

With a suitable choice of the slit width \(\delta\), a band was observed with a wavelength interval of \(5360\)—\(5560\) Å and center at \(\lambda \simeq 5460\) Å (the wavelength of the green line of the mercury spectrum). The wedge \(W\) was used

and in this case to determine the threshold extinction. A comparison was made with the intensity given by the mercury lamp \(L\), the light of which entered the integrating sphere through the aperture \(D_2\) and the prism \(P\). The absolute intensity of the light was measured with a thermopile placed inside \(C\) above the aperture \(D_2\); the liquid filter \(F\) absorbed everything except the green part of the spectrum. The thermopile was calibrated, in turn, against a Hefner lamp.

In these experiments the absolute intensity of the radiation was found to be \(4.1 \cdot 10^{-4}\) erg/sec/1 mC of radon. The intensity calculated from the theory of Frank and Tamm under the same conditions is \(3.5 \cdot 10^{-4}\) (in the same units), which may be regarded as remarkably good agreement with the experimental value. In applying the theory to this special problem, the change in the value of \(\beta\) along the path of the electrons and the energy distribution of the \(\beta\)-particles of the source were taken into account.

The radiation intensities in various liquids were also measured relative to the intensity in water; the results of these experiments are collected in Table 1 and compared with the calculated values.

Table 1

Liquid	Formula	\(n\)	Relative intensity, experim.	Relative intensity, calc.
Water	\(\mathrm{H_2O}\)	1.334	1.0	1.0
Benzene	\(\mathrm{C_6H_6}\)	1.505	1.93	1.92
Cyclohexane	\(\mathrm{C_6H_{12}}\)	1.433	1.61	1.77
Carbon disulfide	\(\mathrm{CS_2}\)	1.637	1.68	1.70
Isobutyl alcohol	\(\mathrm{C_4H_{10}O}\)	1.398	1.40	1.57
Carbon tetrachloride	\(\mathrm{CCl_4}\)	1.468	1.03	1.00

3) Dependence of \(\theta\) on \(n\) and \(\beta\). In concluding this series of experiments before the war, Cherenkov applied the photographic method already described \(^{11}\) to the study of the dependence of \(\theta\) on \(n\) and \(\beta\) \(^{13}\). In this work he used \(\gamma\)-rays both from radium \(^{14}\) and from ThC″ \(^{13}\), measuring the angles \(\theta\) from microphotometric records. Changes in \(\theta\) as a function of \(\beta\) were obtained by determining the energies of two sources of \(\gamma\)-rays and from the Klein—Nishina theory of Compton scattering \(^{6}\). Changes in \(\theta\) with \(n\) were studied using four liquids as examples: water, cyclohexane, benzene, and ethyl citrate.

In both cases good agreement with theory was obtained. Cherenkov published a review of his early works in 1937 \(^{14}\).

II. THEORETICAL INTERPRETATION

a) Frank and Tamm’s Theory

Here we shall briefly outline the theoretical interpretation of the phenomenon developed by Frank and Tamm\(^2\). These authors showed, on the basis of a purely classical theory, that “an electron moving in a medium emits light, even if it moves uniformly, in the case when its velocity is greater than the velocity of light in this medium.” Since Frank and Tamm applied their theory to the emission of radiation in the visible region (i.e., the part of the spectrum for which the wavelengths of the radiation were much larger than the dimensions of the radiating molecules of the medium), it was possible to use the methods well developed in the electromagnetic theory of Maxwell and Lorentz, and to treat the medium macroscopically.

We shall begin with the dynamical relation between the polarization \(\mathbf P\) and the field strength \(\mathbf E\), namely:

\[ \frac{\partial^2 \mathbf P}{dt^2}+\sum \omega_s^2 \mathbf P_s=\alpha \mathbf E, \tag{3} \]

where \(\omega_s\) are the frequencies of the molecular oscillators of the medium and \(\alpha\) is a constant for the given medium.

Expanding then all variable fields into Fourier integrals, in order to represent the components of the electromagnetic disturbance associated with the moving particle, i.e.

\[ \mathbf E=\int_{-\infty}^{+\infty}\mathbf E_\omega e^{i\omega t}\,d\omega,\qquad \mathbf P=\int_{-\infty}^{+\infty}\mathbf P_\omega e^{i\omega t}\,d\omega \quad \text{and so on,} \tag{4} \]

we obtain the relation between \(\mathbf P\) and \(\mathbf E\) as a function of the refractive index \(n\) of the medium, namely:

\[ 4\pi \mathbf P_\omega=(n^2-1)\mathbf E_\omega . \tag{5} \]

Neglecting the absorption of light in the medium, regarding \(n\) as real, and assuming that the conductivity of the medium (and its magnetic susceptibility) is zero, we obtain the following equations for calculating the strengths of the electric and magnetic fields \(\mathbf E\) and \(\mathbf H\) from the vector and scalar potentials \(\mathbf A\) and \(\varphi\):

\[ \begin{aligned} \mathbf H_\omega&=\operatorname{rot}\mathbf A_\omega,\\ \mathbf E_\omega&=-\nabla\varphi_\omega-\frac{1}{c}\frac{\partial \mathbf A_\omega}{\partial t}. \end{aligned} \tag{6} \]

The equations which determine these potentials are as follows:

\[ \nabla^2 \mathbf A_\omega+\frac{\omega^2 n^2}{c^2}\mathbf A_\omega =-\frac{4\pi}{c}\mathbf I_\omega . \tag{7} \]

and

\[ \operatorname{div}\mathbf A_\omega+\frac{i\omega}{c}n^2\varphi=0, \tag{8} \]

where \(I_\omega\) is the current density.

These Maxwell equations are then solved, and, after the boundary conditions have been introduced, only three components of the field vectors \(\mathbf E\) and \(\mathbf H\) remain which do not vanish in the case of the solution for \(\beta n>1\). The solution for \(\beta n<1\) is of little interest to us, since in this case the field of the electron decreases exponentially with distance from the axis and therefore no radiation takes place.

The three components in cylindrical coordinates \(z,\rho\), and \(\varphi\), in which the \(z\)-axis coincides with the trajectory of the particle, may be written in the following form:

\[ \left. \begin{aligned} H_\varphi&=-\frac{a}{\sqrt{\rho}}\int \sqrt{s}\,d\omega\cos\chi,\\ E_\rho&=-\frac{a}{c\sqrt{\rho}}\int \frac{\sqrt{\beta^2n^2-1}}{\beta^2n^2}\, \frac{\omega d\omega}{\sqrt{s}}\cos\chi,\\ E_z&=+\frac{a}{c\sqrt{\rho}}\int \left(1-\frac{1}{\beta^2n^2}\right) \frac{\omega d\omega}{\sqrt{s}}\cos\chi, \end{aligned} \right\} \tag{9} \]

where

\[ \left. \begin{aligned} a&=\frac{e}{c}\sqrt{\frac{2}{\pi}}, \qquad \chi=\omega\left(t-\frac{z\cos\theta+\rho\sin\theta}{c/n}\right)+\frac{\pi}{4}, \\[6pt] s^2&=\frac{\omega^2}{v^2}\left(\beta^2n^2-1\right). \end{aligned} \right\} \tag{10} \]

In these equations \(e\) and \(v\) denote, respectively, the charge and the velocity of the particle. The arrangement of the components of the vectors is shown in Fig. 3. In this elementary theory three assumptions have been made:

1) that the light is observed at a distance \(d\) large in comparison with the wavelength \(\lambda\) of the radiation;

2) that the velocity of the particle is essentially constant over distances comparable with \(\lambda\);

3) that the path length \(l\) is large in comparison with \(\lambda\).

We now wish to obtain an expression for the total energy liberated in this process per unit length of the electron’s path. Equations (9) can be integrated only over the region of positive \(\omega\), and we shall restrict ourselves only to the frequency intervals determined by the condition \(\beta n(\omega)>1\). The total energy \(W\) radiated by the electron through the surface of a cylinder of length \(l\) (the axis of the cylinder coincides with the path of the particle) is given by the expression

\[ W=2\pi\rho l\int_{-\infty}^{+\infty}\frac{c}{4\pi}[\mathbf{E}\mathbf{H}]_\rho\,dt. \tag{11} \]

Using the formula

\[ \int_{-\infty}^{+\infty} \cos(\omega t+\alpha)\cos(\omega' t+\beta)\,dt = \pi\delta(\omega-\omega') \]

one obtains the following expression for the radiated energy:

\[ W=\frac{e^{2}l}{c^{2}}\int_{\beta n>1}\omega\,d\omega\left(1-\frac{1}{\beta^{2}n^{2}}\right). \tag{12} \]

It is of interest to obtain at once a numerical value of the order of magnitude of the effect for comparison with other sources of energy loss of a charged particle passing through matter. Substituting in (12) the approximate value of \(n^{2}\), determined by the equations

\[ n^{2}(\omega)=1+\frac{B}{\omega_{0}^{2}-\omega^{2}}, \qquad n^{2}(0)=\varepsilon=1+\frac{B}{\omega_{0}^{2}}, \tag{13} \]

where \(\varepsilon\) is the dielectric constant, \(B\) is another constant, and \(\omega_{0}\) is a certain mean frequency of the molecules of the medium, and integrating from \(\omega=0\) to \(\omega=\omega_{0}\), we obtain the energy loss per unit path for a fast electron \((\beta\sim 1)\)

\[ \frac{dW}{dl} = \frac{e^{2}\omega_{0}^{2}}{2c^{2}}(\varepsilon-1)\lg\left(\frac{\varepsilon}{\varepsilon-1}\right). \tag{14} \]

Assuming \(\omega_{0}\sim 6\cdot 10^{15}\ \mathrm{sec}^{-1}\), we find that \(\frac{dW}{dl}\) is of the order of several kiloelectronvolts per centimeter.

Spectral distribution. From expression (12) one can derive the spectral distribution of the radiation. Usually it is assumed here that the refractive index is constant over the visible region of the spectrum. If this assumption is accepted, then one can obtain an expression for the amount of energy radiated per unit path length:

\[ \frac{dW}{dl} = \frac{e^{2}}{c^{2}} \left(1-\frac{1}{\beta^{2}n^{2}}\right) \int_{\beta n>1}\omega\,d\omega, \tag{15} \]

which, since the energy of a quantum is equal to \(h\nu\) and \(\omega=2\pi\nu\), can be represented in the form

\[ \frac{dN}{dl} = \frac{e^{2}}{\hbar c^{2}} \left(1-\frac{1}{\beta^{2}n^{2}}\right)d\omega \]

quanta per unit path between \(\omega\) and \(d\omega\),

\[ \tag{16} \]

where

\[ \hbar=h/2\pi. \]

If this is expressed as an energy distribution on the wavelength scale, then we obtain:

\[ \frac{dW}{dl}=4\pi^2 e^2 \sin^2\theta \left(\frac{1}{\lambda^3}\right)d\lambda . \tag{17} \]

From this expression it is evident that the light will be concentrated at the violet end of the spectrum.

b) Deepening of the elementary theory

Following the work of Frank and Tamm, a more complex and more detailed development of the same problem appeared in Tamm’s paper \(^{15}\). In this work a more rigorous proof of the basic relation (1) was found, and a more detailed discussion was given of the expressions for the intensity and the spectral distribution. Expressions for the field of the excited electron inside and outside the radiation cone are derived first for a nondispersive and then for a dispersive medium. In addition, the problem was solved from the point of view of an observer moving together with the particle, in order to obtain expressions for the forces acting on the electron in the reference system in which it is at rest. Finally, ionization effects slowing down the particle and other processes were investigated, i.e. the conditions under which the influence of the electron’s acceleration on its coherent radiation could be neglected were established. This condition of constancy of velocity can be expressed by the following relation:

\[ T\frac{dv}{dt}\ll \left(\frac{c}{n}\right), \tag{18} \]

where \(T\) is the period of the wave under consideration and \(dv/dt\) is the deceleration of the electron. In other words, this means that the change in \(\beta n\) during one period \(T\) must always be much less than unity. In practice this condition is always easily satisfied in the visible region, even for the principal source of losses, namely ionization losses, which amount to \(\sim 2\ \mathrm{MeV}/\mathrm{g}/\mathrm{cm}^2\).

In view of the small magnitude of the energy losses in the Cherenkov effect, Tamm pointed out in his article that the total intensity of the coherent visible radiation is completely insignificant in comparison with the ordinary process of bremsstrahlung. The possibility of experimental detection of the radiation is connected only with differences in the spectral distribution of the two types of radiation, and this difference is such that in the visible region the coherent radiation is in fact much more intense than the intensity of bremsstrahlung.

A particularly interesting conclusion follows from Tamm’s work if we put \(n=1\) and obtain an expression for the energy loss

of the electron when moving with constant velocity in vacuum. We may here briefly consider this hypothetical case for a point charge and a nondispersive medium for which \(n=\mathrm{const}\). The energy loss given by expression (12) is in the general case finite for a refractive index decreasing to unity and even below unity (X-rays), so that the integral thereby becomes finite; the radiation is cut off at the wavelength \(\lambda\) for which \(n=1\). If now \(n=\mathrm{const}\), the energy loss must be infinite. This divergence is eliminated if we take into account the finite dimensions of the electron, since we may assume that the radiation is limited to waves which are larger than the “diameter” \(d\) of the classical electron. Integrating (12) from \(\omega=0\) to \(\omega=c/nd\) (i.e., from \(\lambda=\infty\) to \(\lambda/2\pi=d\)), we obtain:

\[ \frac{dW}{dl}=\frac{e^2}{2n^2d^2}\left(1-\frac{1}{\beta^2 n^2}\right). \tag{19} \]

This result, obtained by Tamm, very much resembles the expression obtained fifty years ago by Sommerfeld \(^{16}\) in studying the paradox of the self-interaction of a moving electron in vacuum. Sommerfeld calculated the vector sum of the electromagnetic forces of interaction of all elements of a rigid spherical electron and found that the resultant force \(F\) vanishes (in the case of vacuum, for which \(n=1\)) only if \(v<c\), but is equal to

\[ F=\frac{9e^2}{4\pi d^2}\left(1-\frac{1}{\beta^2}\right), \tag{20} \]

if \(v>c\). From this he concluded that an electron can maintain its state of constant velocity greater than the speed of light only if some external force \(F\) acts, directed opposite to the force (20). He further concluded that the work of this external force must be converted into radiation, so that in our notation \(F=\dfrac{dW}{dl}\). It is easy to see that his equation (20) differs from Tamm’s equation only by a numerical factor. It may be noted, of course, that Sommerfeld’s work was carried out before the establishment of the special theory of relativity, when it was considered possible to examine particle velocities exceeding the speed of light \(c\).

It is also of interest to note that Klein and Sommerfeld \(^{17}\), somewhat later, in studying the wave resistance experienced by projectiles moving with velocities exceeding the speed of sound, obtained an equation analogous to the equation for the optical case.

Following the solution of the problem for an isotropic medium, Ginzburg \(^{18}\) extended the theory to the case of an electron moving with po-

...with constant velocity through the crystal. In this case it turns out that the radiation has a more complicated character; instead of the single circular cone of rays observed in an isotropic body, here in general two noncircular cones of rays must be observed, and the intensity of the radiation will not be the same on different generators of these conical surfaces. The polarization of the radiation also differs from the polarization in the case of an isotropic medium. Ginzburg calculated the form of the conical surface, the intensity, and the polarization of the radiation in a uniaxial crystal for two cases, when the electron moves along the optical axis and perpendicular to it. He concludes with a brief description of the phenomena that may be expected in the more complicated case of biaxial crystals.

Tanaka^19 solved a more general problem, in which the direction of motion of the electron is at an arbitrary angle with respect to the optical axis of a uniaxial crystal.

Frank^20 considered interference phenomena that may be expected in Cherenkov radiation and whose features are due to the large angle of divergence of the light waves.

Angular distribution of Cherenkov radiation. As will be shown somewhat later, there are experimental limits of resolution attainable in measuring the angles \(\theta\) (see (1)), and therefore the accuracy with which the velocities of charged particles can be found is limited. Apart from these practical limits of resolution, Li^21 pointed out that here there is a natural source of scatter of the values of \(\theta\), even for particles of the same energy and of infinite path length; this follows from the following: in a more detailed study of the effect one must assume that the particle velocity changes by small jumps upon emission of photons. Schiff^22 calculated the distribution \(I(\vartheta)\) for the spread in the direction of photon emission about the value \(I(\vartheta_0)\), where \(\vartheta_0\) is the angle usually denoted by \(\theta\) in formula (1); \(I(\vartheta)\) is obtained in the form

\[ I(\vartheta)=I(\vartheta_0)\frac{\sin^2 y}{y^2}, \tag{21} \]

where

\[ y=2\pi\frac{1}{\lambda}\left[\frac{1}{\beta n}-\cos\theta\right], \]

and \(\lambda\) and \(l\), as before, are the wavelength and the effective path length. It may be noted at once that for an infinite path length, i.e. \(l\to\infty\), \(I(\vartheta)\) tends to the \(\delta\)-function derived from the simple elementary theory. For finite \(l\), the half-width of the distribution \(\Delta\theta\left(\frac{1}{2}\right)\) about the maximum value \(\theta_0\) is determined by the condition \(\sin y=y/\sqrt{2}\), i.e.

\[ \frac{2\pi nl}{\lambda}\sin\theta_0\,\Delta\theta\left(\frac{1}{2}\right)\sim160^\circ . \tag{22} \]

Li indicated that in this analysis \(l\) should be understood not as the total path length in the medium, but as the path length along the particle trajectory between two successive photon emissions. From the Frank and Tamm theory it follows that the mean free path is \(l_{\mathrm{cp}}\simeq 10^{-3}\) cm for a medium with \(n_{\mathrm{cp}}=1.5\) (within the wavelength range \(2000\)—\(20\,000\) Å) and an electron beam with energy \(0.5\) MeV. In the visible region we therefore obtain:

\[ \Delta\theta\left(\frac{1}{2}\right)\sim \frac{1^\circ}{\sqrt{1-\frac{1}{\beta^2 n^2}}}\sim 2^\circ . \]

This angle is of the same order of magnitude as the angle obtained from Ginzburg’s calculations\(^{18}\) for the separation of two cones in the case of anisotropic media. Li suggested that this result may explain why neither Jelley and Henderson\(^{23}\) nor Gardiner and Henderson\(^{24}\) were able to observe a double cone in experiments with electrons passing through mica; he also pointed out that chromatic separation of photons can never be observed. From equation (1), neglecting the decrease of \(n\) with \(\lambda\), it can be seen that cones of blue light should occur outside cones of red light. After this, Møzer\(^{7}\) observed the half-width of the Cherenkov image to be smaller than the natural half-width calculated above, and also found chromatic dispersion. In a recent communication Li\(^{25}\) indicated that his first estimate of the natural spread was too high, since he had assumed zero coherence between the elementary waves coming from separate elements of the path \(l\). In fact, there is considerable partial coherence here between elementary waves coming from successive path elements, which leads to an effective value of \(l\) having an intermediate value between the mean free path \(\sim 10^{-3}\) cm and the physical path length through the medium, which in many cases is \(\sim 10^{-1}\) or greater.

There is another effect of the finite value of \(l\) which influences the radiation characteristic and which should also be mentioned here. In deriving (1) and (12), \(l\) was assumed infinite and, as has already been noted, no radiation is obtained below the velocity threshold \(v=c/n\). However, for finite \(l\), when \(v<c/n\), it is the maximum of the radiation intensity that disappears, not the radiation itself; its total intensity decreases continuously with the electron velocity beyond the threshold \(c/n\).

The influence of multiple scattering and diffraction on the finite width of the image in the Cherenkov effect was studied in detail in a recent communication by Dedrick\(^{26}\).

Following the mathematical treatment of the Cherenkov effect given by Frank and Tamm, other works also appeared: those of Schiff\(^{22}\), Fermi\(^{27}\), Beck\(^{28}\), and Taniuti\(^{29}\).

In Fermi’s work, Cherenkov radiation appeared as a fraction of the total energy loss of a particle (passing through a condensed medium), included among the energy losses calculated from the theory of ionization30. Work on the theory of ionization losses at relativistic energies and the related polarization effects was also carried out by Schönberg31.

Beck solves the problem by a more general method than Tamm, since he assumes that the electron moves with constant velocity in vacuum before it enters the dielectric. In this case, in addition to the usual Cherenkov radiation, an additional radiation term is obtained in the field equations, which gives the onset of the effect at the point where the particle enters the medium. This radiation is sometimes called transition radiation. For small particle velocities its intensity is insignificant, but at high velocities it can become observable under suitable conditions. The number of photons \(dN\) in the wavelength interval \(d\lambda\), associated with this transition radiation, is given by the expression

\[ dN \simeq \frac{4ne^2\beta^2}{\pi hc}\, \frac{\sin^3\theta\,d\theta}{[1-n^2\beta^2\cos^2\theta]^2}\, \frac{d\lambda}{\lambda}, \tag{23} \]

which shows that here there is a sharp maximum in the forward direction at the angle

\[ \theta=\left[\frac{6}{5}\left(1-\beta^2 n^2\right)\right]^{\frac{1}{2}} . \tag{24} \]

Integrating over all angles in order to obtain the total energy emitted in this process per unit frequency interval, we obtain:

\[ dW=h\nu\,dN= \frac{2e^2}{\pi c} \left[ \frac{1}{2}(1+\beta^2 n^2)\lg\left(\frac{1+n\beta}{1-n\beta}\right)-n\beta \right]h\,d\nu . \tag{25} \]

The absolute value of the energy yield near the critical velocity \(v=c/n\) is of the order of one photon in the visible region per 100 incident electrons. It seems that no observations of this phenomenon have been reported, which is not surprising in view of the extremely low intensity of this radiation. The most recent work on studying the relation of the Cherenkov effect to the energy losses of relativistic particles was done by Budini32,33, who predicted a logarithmic increase in the radiation yield at ultrarelativistic energies in dense media, analogous to the increase of ionization losses at such energies. He obtained an expression for the radiation losses similar to that derived by Frank and Tamm, but containing additional terms at high energies. Sternheimer34 also considered this effect in relation to ionization losses.

c) Quantum treatment of the phenomenon and “magnetic” Cherenkov radiation

In the classical theory of Frank and Tamm it was assumed that the velocity of the particle is constant during the emission of light; in other words, the back action of the emitted radiation on the particle was neglected. A quantum treatment of the phenomenon was given by Ginzburg^35. In this work the “recoil” experienced by the particle as a result of photon emission was taken into account. The first result obtained in this theory was an expression for the coherence conditions, which differed slightly from that originally obtained from the classical theory (1). This result is identical with that obtained later by Cox^36 from an elementary consideration of the conservation of momentum and energy.

Let \(u\) be the velocity of the particle (of rest mass \(m\)) in the medium before emission of the photon. We shall assume that in some part of its path a photon \(h\nu\) is emitted at an angle \(\theta\) with respect to the initial direction of the particle, and that this leads to an instantaneous loss of part of the energy, so that after this the particle moves with velocity \(v\) at an angle \(\varphi\) with respect to the initial direction.

The law of conservation of momentum leads to the following equations:

\[ mv\left(1-\frac{v^2}{c^2}\right)^{-\frac{1}{2}}\cos\varphi+\frac{h}{\lambda}\cos\theta = mu\left(1-\frac{u^2}{c^2}\right)^{-\frac{1}{2}} \tag{26} \]

and

\[ mv\left(1-\frac{v^2}{c^2}\right)^{-\frac{1}{2}}\sin\varphi-\frac{h}{\lambda}\sin\theta=0, \tag{27} \]

whereas the requirement of conservation of energy gives

\[ mc^2\left(1-\frac{u^2}{c^2}\right)^{-\frac{1}{2}} = mc^2\left(1-\frac{v^2}{c^2}\right)^{-\frac{1}{2}}+h\nu. \tag{28} \]

Eliminating \(\varphi\) and \(v\), and putting \(\nu=c/n\lambda\), where \(n\) is the refractive index of the medium, we obtain:

\[ \cos\theta=\frac{c}{nu}+ \frac{h\left(1-\frac{u^2}{c^2}\right)\left(n^2-1\right)} {2mun^2\lambda}, \tag{29} \]

which in turn may be written in the form

\[ \cos\theta=\frac{1}{\beta n}+ \left(\frac{\Lambda}{\lambda}\right)\frac{n^2-1}{2n^2}, \tag{30} \]

where \(\Lambda\) is the de Broglie wavelength of the particle,

\[ \lambda=\frac{h\sqrt{1-\beta^2}}{mu}, \]

and \(\lambda\) is the photon wavelength. It can be seen that expression (30) differs from the original Cherenkov relation (1) by the appearance of the second term. Since in practice \(\Lambda\) is always much smaller than \(\lambda\), the deviations from the classical formula are insignificant.

Further, Ginzburg calculated the energy loss by means of quantum theory and considered three cases: an electron without a magnetic moment, a particle having no electric charge but possessing a magnetic moment, and an electron with a magnetic moment. For a nonmagnetic electron the results coincide with the classical ones.

The results for a particle with zero charge and with a magnetic moment are of interest, since from time to time suggestions have been made that the Cherenkov effect in the case of a moving magnetic dipole could be used to detect fast neutral particles.

For “magnetic” radiation from an uncharged magnetic dipole with moment \(\mu\), moving with velocity \(v\), Ginzburg obtained the following expression for the energy loss:

\[ W_\mu=\frac{\mu^2 l}{v^2 c^2}\int \omega^3 d\omega\left(1-\frac{1}{\beta^2 n^2}\right)n^2 \tag{31} \]

under the condition that the dipole-moment vector \(\boldsymbol{\mu}\) is parallel to the direction of the particle. Comparing this with (12) for a charged particle, we find:

\[ \frac{W_\mu}{W}\sim\left(\frac{\mu\omega n}{ev}\right)^2 . \tag{32} \]

If we put \(e=4.8\cdot10^{-10}\) CGSE, \(v=2\cdot10^{10}\ \mathrm{cm/sec}\), \(n=1.6\), \(\mu=10^{-20}\ \mathrm{erg\cdot g^{-1}}\), and give \(\omega\) its maximum value \(10^{15}\ \mathrm{sec}^{-1}\), then we obtain \(W_\mu/W\sim 2.5\cdot10^{-12}\). If \(\boldsymbol{\mu}\) is perpendicular to \(\mathbf{v}\), then we obtain:

\[ \frac{W_\mu}{W}=\frac{2\mu\omega n}{ev}\sim 3\cdot10^{-6}. \tag{33} \]

In any case it is evident that there is no practical possibility of detecting the “magnetic” Cherenkov effect.

Ginzburg’s results show that for a nonrelativistic magnetic electron, described by the Pauli and Dirac equations, the relations following from quantum theory differ from those obtained by the classical theory, whereas in the extremely relativistic region the results coincide with what can be obtained for a classical nonmagnetic electron.

A more general treatment of the effect, based on quantum electrodynamics, was given by Jauch and Watson\(^{37,38}\), who investigated

two cases: when the medium is at rest with respect to the observer and the particle moves with constant velocity, and the inverse case, when the particle is at rest and the medium moves.

The results obtained by them both for a medium without dispersion and for a dispersive medium agree in their main features with the results of Cox and Frank and Tamm.

d) A charged particle moving near the surface of a dielectric; Cherenkov radiation as a possible source of microwaves

Ginzburg\(^{39,40}\) showed that Cherenkov radiation should be observed when a charged particle moves near the surface of a dielectric parallel to it. In this case the radiation should contain all wavelengths that are large in comparison with the distance between the trajectory of the particle and the surface. An obvious generalization of this may be the case in which a particle is directed into an evacuated channel of circular cross-section in a dielectric. For the resulting radiation the conditions \(d \leq 0.1\,\lambda/n\) must be fulfilled. In this way it would be possible to obtain Cherenkov radiation in the absence of losses due to ionization, but for practical reasons this would not be feasible in the region of optical wavelengths.

We have already seen that the spectrum of Cherenkov radiation is cut off in the region of X-rays as a result of anomalous dispersion; however, on the long-wavelength side there is no such limit, provided, of course, that the medium is free of absorption bands in this region. Ginzburg further pointed out that microwave Cherenkov radiation should be observed and that, possibly, it can provide sources of considerable power in the region of wavelengths that are usually difficult to obtain by other means.

Let us consider a band \(d\lambda\) with its center at \(\lambda \sim 1\), so that \(\omega = 2\cdot 10^{12}\) and \(d\omega = 2\cdot 10^{11}\); then, according to formula (12), the energy of the radiation emitted in this band, \(\Delta W\), should be equal to

\[ \Delta W=\frac{e^{2}l}{2c^{2}}\,\omega\,d\omega, \tag{34} \]

if we assume that \(\beta n\) is much greater than unity and that we may neglect dispersion. For a single electron at a path length of \(20\) cm and with the above-indicated values of \(\omega\) and \(d\omega\), we obtain the value \(10^{-15}\) erg for \(\Delta W\). Of course, this quite insignificant energy can be increased many times if a stream of electrons is used, and if this stream is concentrated, still greater success can be achieved. If the width of the beam is small in comparison with \(\lambda\), then the radiation from the individual electrons of the beam will be coherent, and \(e\) in formula (34) will in this case be replaced by \(\nu e\), where \(\nu\) is the number

electrons in the bunch. If \(N\) bunches pass through the system per second, then the radiated power \(\Delta U\) will be

\[ \Delta U=\frac{e^{2}l}{2c^{2}}\,v^{2}N\omega\,d\omega =8\cdot 10^{-22}\,lvl\omega\,d\omega\ \text{erg/sec}, \tag{35} \]

where \(I\) is the mean current strength, equal to \(evN\), expressed in amperes. If we take the admissible value for the current to be \(I=10^{-2}\) a and \(v=10^{9}\), then we obtain \(\Delta U=6\) kW. Ginzburg proposes that a medium with a very high dielectric constant \(\varepsilon\), of order 50, be used, so that \(n=7\); the electron energy \(V\) required in order to satisfy the condition \(\beta n \gg 1\) would then not be excessively large; for \(n=7\), \(V\sim 10\) kV. The spectrum should be continuous and should have a noticeable maximum at \(\lambda_{\max}\sim dn\). On the long-wavelength side of this maximum the spectrum should follow formula (17), whereas on the short-wavelength side the distance of the particle from the dielectric should limit the coherence of the radiation.

Abel \(^{41}\) investigated Ginzburg’s proposal in considerable detail. He considered the special case of a cylindrical waveguide partially loaded with a dielectric medium, and then examined the spectral distribution of Cherenkov radiation produced by an electric charge moving along the axis. The radiation is concentrated in different ways according to the natural frequencies of oscillation of the free waves in the waveguide. Abel then describes how a considerable fraction of the energy could be concentrated in one definite overtone and, finally, discusses the possibility of creating a microwave-region amplifier based on these principles.

There is no doubt that considerable technical difficulties will have to be overcome before this method of exciting microwaves becomes practically acceptable.

III. LATER EXPERIMENTAL WORKS

(a) Experiments with artificially accelerated particles

After the early works of the Russian investigators, a certain interval of time elapsed during which only slight progress was made in this direction.

Undoubtedly, the principal cause of this was the world war. In the resumption and expansion of work in the postwar period two factors played a role: the development of accelerators, which yielded an enormous number of monoenergetic particles, and the invention of photomultipliers, which were the first instruments sufficiently sensitive to register individual photons.

with sufficient effectiveness. In this section we shall give a brief survey and discussion of the experimental work determined by the factors indicated for this period.

Collins and Reiling^5 used electrons with energies of 2 MeV from an electrostatic generator to investigate this radiation. Their electron beam, which was more monochromatic in comparison with the β-ray sources previously used by Cherenkov, was made parallel and fell on thin films of various substances placed on the axis of a conical mirror in the apparatus shown in Fig. 7, a. With a beam whose current was about 10 μA, it was possible to obtain images of much greater intensity than those previously observed by Cherenkov. Owing to the small spread in energy values, the good collimation of the electrons, and the low energy losses in the thin films, these authors were able to obtain images with good angular resolution in \(\theta\), as is seen from Fig. 7, b. They investigated radiation from glass, mica, and cellophane, and also from water. Thin films of water were obtained by introducing it into a small aperture in a copper plate. The results of the experiments of Collins and Reiling reduce to the following.

Fig. 7. Experiments of Collins and Reiling^5: \(a\)—apparatus; \(b\)—positions at which the photographic image appeared; \(c\)—recorded photographic spectrum of Cherenkov radiation (2) in comparison with the spectrum of a tungsten lamp (1).

1) A dependence of the angle \(\theta\) on the refractive index \(n\) was found for electrons of the same energy and compared with relation (1). These results are given in Table II.

Table II

Medium	Thickness, cm	\(n\)	\(\theta_{\mathrm{obs}}\)	\(\theta_{\mathrm{calc}}\)
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	\(50^\circ 0'\)	\(49^\circ 22'\)

In these experiments the film, in turn, was mounted in two positions at an angle of \(45^\circ\) with respect to the electron beam

and corrections for refraction at the surface of the film were introduced into the observed positions of the maxima.

2) Radiation spectra were obtained in the case of water, alcohol, and benzene (Fig. 7, e). For these purposes the liquids were placed in a very thin quartz sphere introduced into the electron beam. The spectra were photographed on a quartz spectrograph with a dispersion of 30 Å/mm at 3500 Å. For all three liquids the spectra were the same: intense in the blue-violet region, containing no lines or band structure, even when examined visually with a spectroscope with a dispersion of 45 Å/mm and a narrow slit.

3) Absolute measurement of the intensity of the radiation from a quartz sphere, \(d = 0.5\) cm, filled with water, was carried out with a photoelement calibrated in microamperes per watt. These results indicate that an electron with an energy of 1.9 MeV, stopped in water, gives up to 40 quanta in the wavelength range 4000–6700 Å; these figures agree with those calculated from the Frank and Tamm equation (12) to within a factor of two.

Wyckoff and Henderson \(^{23}\) set up similar experiments to check the dependence of \(\theta\) on \(\beta\). They took thin mica films (with thicknesses 0.00013–0.0025 cm), fastened them inside a conical mirror, and used photographic plates to record the light. The experiments were carried out with an electron current of \(\sim 1\) μA, with electron energies of 240–815 keV. The results are given in Fig. 8. The curve was calculated from the known Cherenkov relation (1) with the value 1.59 for the refractive index of mica. Wyckoff and Henderson understood that, in principle, two cones of radiation should arise from mica, since it is a biaxial crystal, but the resolution was insufficient for these cones to be separated. In concluding the review of experiments of this type, one should mention the short note by Harding and Henderson \(^{24}\), which describes experiments with electrons passing through mica at energies close to the threshold of the Cherenkov effect. From these experiments they concluded that there is indeed a definite threshold, as is predicted by the theory; that the light at this threshold is not polarized and that it rapidly becomes polarized as the electron energy increases; and, finally, that the total energy of the radiation increases approximately linearly with the energy of the electrons above the threshold, at least up to energies twice the threshold.

Fig. 8. Dependence of \(\theta\) on \(\beta\) \(^{23}\).

Getting \(^{42}\) was the first to propose using a photomultiplier to detect Cherenkov radiation. In his short article

he calculates the photon yield that should be expected for a relativistic particle passing through a cylindrical block of lucite 20 cm long. On the basis of considerations of the photoelectric yield of the cathode of a photomultiplier and of the dark-current pulses (which are caused mainly by thermionic emission from the photocathode), he estimated that the signal-to-noise ratio should be sufficiently high for such a system with a multiplier at a bandwidth of 10 megacycles per second. Getting also proposed an optical system which, in the case of a parallel beam of particles, was to concentrate in one direction all the light arising at the given Cherenkov angle in an extended “converter.” One such scheme is shown in Fig. 9. The particles enter

Fig. 9. Detectors using a photomultiplier, proposed by Getting^42.

a solid lucite cone through the vertex and propagate along the axis. If the half-angle at the vertex is equal to \(\phi = \frac{1}{2}\theta\), where \(\theta\) is the Cherenkov angle, then the light undergoes total reflection and emerges parallel to the axis; consequently, it can be focused through a diaphragm onto the cathode of the photomultiplier. In the second variant of this scheme (Fig. 9, b) the converter was lengthened by attaching a lucite cylinder, in order to increase the path length, and the light was directed by successive reflections into the conical part.

b) Cherenkov radiation from cosmic-ray particles

Dicke^43 attempted to detect cosmic rays by means of an apparatus similar to that proposed by Getting (Fig. 9, b), mounted vertically above a Geiger counter in a coincidence circuit. This experiment gave a negative result, for reasons unknown. The very same detector was subsequently used with partial success for detecting secondary electrons generated in a lead plate by X-rays from a 20-Mev betatron.

Soon after these experiments, Weiss and Anderson^44, using a very simple device shown in Fig. 10, a, attempted to register cosmic rays, employing in their scheme a photosensitive-

sensitive Geiger—Müller counters of special construction. These counters had gold-coated cathodes, were very sensitive to the ultraviolet region \(2000\)—\(3000\) Å, and gave a quantum yield of the order of \(2\cdot 10^{-4}\)—\(60\cdot 10^{-4}\) counts per quantum. The experiment consisted in measuring the counting rate with the counter immersed first in pure water, and then in a solution containing \(1\%\) (by weight) of hydroquinone.

The fraction of counts associated with Cherenkov radiation in pure water can be excluded by using a hydroquinone solution, which has strong absorption in the ultraviolet region of the spectrum. The results of three separate series of experiments with a counter of exceptionally high sensitivity are given in Table III.

Table III

Series	Pure water, counts per minute \(N_1\)	Hydroquinone solution, counts per minute \(N_0\)	Possible fraction due to the Cherenkov effect, counts per minute \(N_1-N_0\)	Relative fraction of the Cherenkov effect, % \((N_1-N_0)/N_0\)
1	\(232\pm5\)	\(210\pm5\)	\(22\pm7\)	\(10\pm4\)
2	\(254\pm5\)	\(224\pm6\)	\(30\pm8\)	\(13\pm5\)
3	\(263\pm7\)	\(224\pm5\)	\(41\pm9\)	\(18\pm6\)

These results indicate that individual cosmic-ray particles were recorded with low efficiency.

The average fraction of the counting rate for light radiation in pure water, amounting to as much as \(\sim 14\%\) of the total number of Geiger counts, is comparable with the figure \(\sim 10\%\), calculated from the geometry of the apparatus and the quantum yields.

The same investigators extended their experiments, using solutions of thorium nitrate (and its decay products), in the hope of detecting Cherenkov radiation caused by the passage of \(\beta\)-particles through the substance (Fig. 10, b). In this experiment no clear results were obtained, which the authors attributed to the small fraction of particles for which the condition \(\beta n > 1\) would be satisfied. Positive

Fig. 10. Apparatus used by Weiss and Anderson\(^{44}\) to detect Cherenkov radiation in water with photosensitive Geiger counters: \(a\)—arrangement for detecting cosmic rays; \(b\)—detection of \(\beta\)-rays of thorium. \(1\)—Geiger—Müller counter with shielded cathode and ultraviolet glass; \(2\)—pure water and a substance absorbing ultraviolet; \(3\)—5000-ml vessel; \(4\)—ebonite vessel.

tive result in their first experiment cannot definitely be ascribed to the Cherenkov effect, since the possibility of ultraviolet recombination radiation accompanying ionization cannot be excluded.

Fig. 11. Jelly’s water detector, used for counting cosmic-ray \(\mu\)-mesons\({}^{45}\).
\(1\)—Geiger–Müller counters; \(2\)—to the cathode follower; \(3\)—glass plate with black paper; \(4\)—distilled water; \(5\)—silvered vessel on the outside; \(6\)—light-tight casing; \(7\)—light-collecting cone, coated with MgO; \(8\)—photomultiplier; \(9\)—to the main amplifier and discriminator; \(10\)—amplifier.

The first experiments in which not electrons but other particles were detected with the aid of the Cherenkov effect were carried out by Jelly\({}^{45}\), who counted individual \(\mu\)-mesons of cosmic rays in distilled water. The water detector (Fig. 11) at first operated in combination with a system of Geiger counters in a coincidence circuit, acting as a simple telescope for selecting particles close to the zenith. Since almost all cosmic-ray particles pass through the instrument from top to bottom, the Cherenkov radiation will therefore also be directed downward, so that the coincidence counting rate should be greater when the photomultiplier is in position \(8\) under the vessel than in the case when it is in the upper position \(K'\).

For the investigation, various series of counts were carried out at different shifts of the discriminator connected to the amplifier, with the photomultiplier in the two positions \(K\) and \(K'\); this was accomplished by rotating the entire detector about its central point \(S\). Both the background of accidental coincidences and the background arising from the fact that the photomultiplier can record, although with low efficiency, particles passing through it were taken into account. To take account of this second source of background, the water was removed from the vessel. The results of the experiment are given below, in Table IV, from which it can be seen that the light intensity \(I_B\) is greater than \(I_A\), where \(B\) and \(A\) refer respectively to the upper and lower positions of the photomultiplier relative to the vessel. In this table \(P\) denotes the counting rate for the photomultiplier and the empty vessel, and \(W\) the same in the case when the vessel is filled with water. All figures are given in counts per minute.

The fact that the ratio \(I_B/I_A\) is not infinite is interpreted as the result of a certain internal reflection of light upward when the photomultiplier is in position \(A\). Control-

... an experiment was carried out with a 0.5% solution of terphenyl in xylene—a scintillation substance discovered by Reynolds et al. ^46, for which the light should be emitted isotropically. In this case the ratio \(I_B/I_A\) was \(1.26 \pm 0.06\) at a bias of 2 V. The absolute efficiency of the detector used in this way proved to be about 50% (at a bias of 2 V), when measured by comparing the counting rate of 3.35 per minute for water in position \(B\) with the calculated counting rate of 7 per minute for the known flux of cosmic-ray particles and the geometry of the system.

Table IV

| | \multicolumn{6}{c}{Bias in the discriminator} |
|---|---:|---:|---:|---:|---:|---:|
| | \multicolumn{2}{c}{2 volts} | \multicolumn{2}{c}{5 volts} | \multicolumn{2}{c}{15 volts} |
| | \multicolumn{2}{c}{photomultiplier position} | \multicolumn{2}{c}{photomultiplier position} | \multicolumn{2}{c}{photomultiplier position} |
| | \(B\) | \(A\) | \(B\) | \(A\) | \(B\) | \(A\) |
| \((P+W)\) | \(4.14 \pm 0.03\) | \(2.29 \pm 0.10\) | \(4.01 \pm 0.11\) | \(1.73 \pm 0.04\) | \(2.54 \pm 0.08\) | \(0.71 \pm 0.03\) |
| \(P\) | \(0.79 \pm 0.03\) | \(1.26 \pm 0.02\) | \(0.47 \pm 0.01\) | \(0.98 \pm 0.03\) | \(0.15 \pm 0.01\) | \(0.40 \pm 0.02\) |
| \(W\) | \(3.35 \pm 0.04\) | \(1.03 \pm 0.10\) | \(3.54 \pm 0.11\) | \(0.75 \pm 0.05\) | \(2.39 \pm 0.08\) | \(0.31 \pm 0.04\) |
| Ratio \(I_B/I_A\) | \(3.25 \pm 0.32\) | \(3.25 \pm 0.32\) | \(4.72 \pm 0.35\) | \(4.72 \pm 0.35\) | \(7.7 \pm 1.0\) | \(7.7 \pm 1.0\) |

It was found that most of the particles responsible for the coincidences were \(\mu\)-mesons. This was shown by placing a lead absorber 10 cm thick above the apparatus, after which \(W\)—the counting rate under the vessel—fell from \(3.35 \pm 0.04\) to \(2.67 \pm 0.11\) per minute. This decrease, amounting to \(\sim 20\%\), approximately corresponds to the fraction of electrons in the vertical flux of cosmic rays. Since \(\sim 98\%\) of the penetrating component at sea level consists of \(\mu\)-mesons, there is no indication that the detector counts protons or other types of mesons.

Jelley later found that, at high bias values, at which the counting of photomultiplier dark-current pulses is sufficiently reduced, the setup becomes sensitive to its orientation and can detect particles with good resolution relative to the background without the need to use a coincidence circuit. In Fig. 12, a are shown the curves obtained with the water detector. Curves b and c refer to the detector counting rates in positions \(B\) and \(A\), respectively, whereas curve a refers to the empty vessel (in both positions). The ratio of curves d and e for water alone is plotted in the inset graph.

of the figure and shows that the ratio of the intensities reaches 70/1, which should have been obtained for the ratio of the intensity of the light passing downward to the intensity of the light going upward.

In this case the extrapolated counting rate at zero displacement (projection of curve b), amounting to \(\sim 170\) per minute,

Fig. 12. Results of Jelley’s experiments:
a) bias-voltage curves illustrating the directional properties of the radiation;
b) changes in the counting rate with zenith angle.
a — curve for the photomultiplier in both positions; b — photomultiplier in the lower position under 20 cm of water; c — photomultiplier above 20 cm of water; f — photomultiplier under 100 cm of water.

is comparable with the total flux of cosmic rays falling on the detector from the upper hemisphere. A spatial characteristic of such a detector of simple cylindrical form was obtained

and compared with the distribution \(\cos^2 \theta\), corresponding to the law of variation of the intensity of \(\mu\)-mesons with zenith angle \(\theta\) (Fig. 12, b).

Attempts to determine the energies of individual cosmic-ray particles by measuring the Cherenkov angle \(\theta\) and, consequently, \(\beta\), required very small solid angles of particle acceptance in order to obtain the necessary collimation, which, in turn, led to excessively low counting rates. Taking this into account, Bassi\(^{47}\) proposed determining the energy by measuring the light intensity, using relation (12). Bassi was the first to make a water detector, very similar to that used by Jelly, and, in a triple-coincidence arrangement with the detector placed between two Geiger counters, obtained almost 100% efficiency for counting particles passing through his detector. His attempts to measure energy by this method encountered obstacles in the statistical fluctuations of the number of electrons emitted in the photomultiplier. If one takes the flux to be 3000 photons per particle (for particles with \(\beta = 1\)) and the photocathode yield to be one electron per 100 photons, then fluctuations of 30 electrons should amount to about 20%, which, in turn, should lead to a large error in \(\beta\) and therefore in the determination of particle energies. We shall return to this problem when considering practically usable detectors.

In a later series of experiments Bassi et al.\(^{48}\) measured the relative intensities of Cherenkov radiation (in a Plexiglas converter) produced by cosmic-ray particles at sea level. The Cherenkov detector was placed between Geiger counters in an anticoincidence arrangement with lead absorbers. With this device Bassi and his collaborators observed changes in the intensity of Cherenkov radiation as a function of particle range and did not find agreement with the Frank and Tamm theory at very high energies; the light yield increased with increasing energy. The particles were selected in three range intervals: 40–100, 140–500, and 500–\(\infty\) g/cm\(^2\) of air equivalent. The corresponding figures for the light intensity \(W\), in relative units, were: \(22.5 \pm 0.7\), \(24.9 \pm 0.8\), and \(32.2 \pm 0.7\), respectively. This increase at high energies cannot be explained either by a change of \(\theta\) with energy or interpreted in connection with knocked-out electrons. This result is compatible with Budini’s predictions mentioned in the preceding section.

c) Radiation in aqueous solutions of radioactive isotopes

The works mentioned up to now concerned chiefly Cherenkov radiation observed in organic liquids and solids, on the one hand, and in distilled water, on the other. Recently, however, Belcher\(^{49}\) carried out detailed studies of the weak luminescence observed in aqueous

in solutions of radioactive isotopes, and showed that almost all the observed light can be attributed to the Cherenkov effect; moreover, he showed that the relative and absolute intensities are in good agreement with the theory of Frank and Tamm. He also found that in the case of solutions of $\alpha$-emitters and $\beta$-emitters with $\beta$-particles below the Cherenkov threshold, there is nevertheless a weaker luminescence associated with excitation of the solvent. Belcher’s apparatus is shown schematically in Fig. 13. In this work, owing to the very low values of the energy obtained from the isotopes studied and, consequently, to the correspondingly short path lengths, the light pulses were extremely weak in comparison with those observed by Jelly in his experiments with $\mu$-mesons and water. In general outline Belcher’s experiments were carried out as follows: the radioactive solution under investigation was placed in a small cuvette $E$ made of Perspex (or duralumin), provided with a light guide $D$ of Perspex, mounted opposite the photocathode of a photomultiplier (type RGA, IP21). Taking into account the very low intensities, precautions were taken to lower the frequency of dark-current pulses from the photomultiplier cathode by cooling the entire system with liquid nitrogen $C$, placed in the upper part of a heavy brass vessel $B$, surrounded on all sides by cotton wool for thermal insulation. Under these conditions the thermal background of the photomultiplier was always less than one count per second.

Fig. 13. Belcher’s apparatus for studying solutions of radioactive isotopes\({}^{49}\).

For the work, low concentrations were used (always lower than those which produced a noticeable coloration of the solutions) of eight samples of nuclei in simple chemical forms. Six of them had $\beta$-spectra with maxima lying above the Cherenkov threshold (260 keV for electrons in water), and six emitted $\gamma$-rays. Table V shows the results obtained with a duralumin cuvette and with various substances. The observed counting rate was reduced to a value of 1 microcurie per milliliter, corrected for various background effects, which included also the effects of accompanying $\gamma$-rays, if any were present. The isotopes emitting $\gamma$-rays are marked with an asterisk.

Two samples emitting $\alpha$-particles were investigated: the isotope polonium with mass 210 and natural uranium. The counting rates for them (in the same units as in Table V) were $14 \pm 1.0$ and $250 \pm 20$ counts per second.

The results of these experiments are presented in Fig. 14, where the mean radiation intensity, expressed in quanta (between 3000 and 7000 Å) per β-particle, is plotted as a function of the maximum energy of the β-particle in megaelectronvolts. The solid line is constructed according to the Frank and Tamm theory, taking into account: a) the change of β along the path of the particle, since it slows down until its velocity falls below the Cherenkov threshold; b) the distribution of electron energies in β-spectra; c) wall effects, i.e., the losses of particles emitted near the boundaries of the vessel, and \(d\)-effects associated with the self-absorption of γ-rays, if they are present.

Fig. 14. Results of Belcher’s experiment (continuous curve constructed according to the Frank and Tamm theory).

The agreement of experiment with theory is very good, with the possible exception of the region of small energies near the threshold. A value was obtained for the absolute intensity; it amounted to as much as \(\sim 90\) photons

Table V

	Chemical formula	Maximum β-energy, MeV	Counts per second
\(^{35}\mathrm{S}\)	\(\mathrm{SO}_4^{2-}\)	0.168	\(0.25 \pm 0.017\)
\(^{45}\mathrm{Ca}\)	\(\mathrm{Ca}^{2+}\)	0.254	\(0.55 \pm 0.038\) Cherenkov threshold
\(^{60}\mathrm{Co}^{*}\)	\(\mathrm{Co}^{2+}\)	0.31	\(85 \pm 5.9\)
\(^{59}\mathrm{Fe}^{*}\)	\(\mathrm{Fe}^{3+}\)	(0.26)	\(72 \pm 5.0\)
\(^{131}\mathrm{I}^{*}\)	\(\mathrm{I}^{-}\)	(0.46) (0.31)	\(64 \pm 4.5\)
\(^{198}\mathrm{Au}^{*}\)	\(\mathrm{Au}\)	(0.96)	\(280 \pm 20\)
\(^{32}\mathrm{P}\)	\(\mathrm{PO}_4^{3-}\)	1.69	\(1440 \pm 100\)
\(^{42}\mathrm{K}^{*}\)	\(\mathrm{K}^{+}\)	(2.07) (3.57)	\(3100 \pm 210\)

in the interval 3000–7000 Å for a 1 MeV electron stopping in an aqueous medium. This constitutes an energy loss in this

of the wavelength region, equal to \(\sim 0.20\%\) of the total energy scattered by the particle in the medium.

The weak glow emitted by \(\alpha\)-particles of \(^{210}\mathrm{Po}\), which is not accompanied by \(\beta\)-radiation, amounts to up to \(\sim 0.5\) photon per \(\alpha\)-particle stopping in the medium (photons of the same wavelength intervals).

In conclusion, it may perhaps be mentioned that phenomena of luminescence in aqueous solutions of radioactive \(\alpha\)-emitters are of special interest, since Dee and Richards \(^{50}\) expressed the opinion that it is precisely photon emission that may be responsible for the biological effects of radiation.

Dainton \(^{51}\) considered the possible role of the Cherenkov effect in radiation chemistry.

Recently Greenfield et al. \(^{52}\) studied the spectrum of light emitted in distilled water when it was irradiated by Ra \(\gamma\)-rays and \(P^{32}\) \(\beta\)-rays, and found good agreement with the theory of Frank and Tamm.

d) Cherenkov radiation in the atmosphere

In 1948 Blackett \(^{70}\) predicted that some part of the mean light in the night sky should be connected with the passage of ultrarelativistic cosmic-ray particles through the atmosphere. This part of the radiation is estimated at \(\sim 10^{-4}\) of the total intensity. Recently Galbraith and Jelley \(^{53}\) detected light pulses of duration \(<0.2\,\mu\mathrm{s}\), which correlated with strong air showers.

In considering the relative intensity it could be assumed that these light pulses may more likely be attributed to Cherenkov radiation from shower particles than to ionization processes.

The apparatus used in these experiments consisted of a simple telescope assembled from a parabolic mirror 10 inches in diameter (\(f=0.5\)), at the focus of which was mounted a standard photomultiplier with an end window, connected to a fast amplifier and recording system. The mean counting rate of light pulses, at a voltage on the amplifier sufficient to reduce the counting rate from background values (from the mean light of the night sky) to very small values, was about one pulse per minute with this optical system. Fig. 15 shows the differential distribution of pulses with altitude, obtained on an exceptionally clear night; the main features of the telescope arrangement are shown within the drawing.

By the time this article was written, the experiments had only just begun, and in the course of the work only a very approximate value of the absolute intensity of the light flashes had been obtained, corresponding to a flux of about 3 photons/\(\mathrm{cm}^2\)/pulse \(^{53}\).

In attempts to separate Cherenkov radiation from radiation of other sources, it is proposed to set up polarization experiments, using two parallel light receivers with polarizers on photomultipliers, oriented at 90° relative to one another; in this way only the radial characteristics of the polarization effect can be detected*).

By contrast with the phenomenon in a solid or in a liquid medium, in a gas at atmospheric pressure light is emitted at a very small angle to the trajectory of the particle. Thus, for air at sea level the refractive index is \(n = 1.00029\), and therefore for an ultrarelativistic particle (for which \(\beta = 1\)) in this case equation (1) gives \(\theta \sim 1^\circ\).

From this it follows that the Cherenkov effect makes it possible to realize an instrument for the study of cosmic rays that possesses a high degree of directionality with a moderate area. This instrument may find application in searches for localized sources of extended atmospheric showers and, consequently, in searches for primary particles.

Fig. 15. Distribution with height of light pulses in the atmosphere associated with cosmic rays, obtained by Galbraith and Jelley \({}^{53}\) with a simple reflecting telescope and photomultiplier.
\(Ph\)—photomultiplier (type 6260); \(M\)—mirror 10 inches in diameter, focal length 46 inches.

IV. PRACTICAL APPLICATIONS OF CHERENKOV DETECTORS

a) General considerations

In the practical development of radiation detectors based on the preceding experimental and theoretical work on Cherenkov radiation, two methods of observation and measurement of light may be used, namely: photographic registration of radi—

*) Since the writing of this article, Galbraith and the author, working with light receivers at the Pic-du-Midi Observatory, have definitely shown that these light pulses are indeed associated with Cherenkov radiation. This was established by observing the polarization of the light; additional data were obtained from experiments with two light receivers; from these data it is evident that the light travels in a narrow beam, of width \(\sim 4^\circ\) or less. This work will be published.

tion and the use of a photomultiplier. Visual observations must be discarded because of their inaccuracy and subjective errors. The same can be said of photosensitive Geiger counters, which have low intrinsic efficiency and do not have the ability to distinguish a flash of light from the direct incidence of charged particles passing through the counter itself. In practice, vacuum photocells can also be used, but this would lead to the need to use tube amplifiers with exceptionally high gain and a low noise level.

Basically, the applications depend on two main characteristics of Cherenkov radiation, namely: on the fact that the light is emitted forward inside a cone whose generatrix makes an angle \(\theta\) with the particle trajectory, where \(\theta\) is determined by equation (1), and on the fact that there is a definite threshold value of \(\beta\), below which the radiation is absent.

From these features of the phenomenon there also follow those applications which this relatively new detector may have. These possibilities are listed below:

1) The development of detectors possessing exceptional speed and high output for a moderate area, i.e., detectors capable of high counting rates.

2) Direct determination of the velocity of a charged particle within limited ranges above the threshold, which, in turn, makes possible the direct determination of the particle energy if the particle mass is known.

3) The possibility of distinguishing particles of different masses having one and the same energy or the same range in an absorber.

4) The ability to determine in what direction an ultrarelativistic particle is moving for the case of energies belonging to a region where the rate of change of ionization along the path is negligible, or where the curvature of the particle path in a magnetic field is insignificant or gives an ambiguous answer when the sign of the particle charge is not fully established.

5) The possibility of determining the charges of ultrarelativistic particles (i.e., of primary cosmic radiation) owing to the dependence of the light output on \(e^2\) (see (12)).

Not all these possibilities are provided by Cherenkov detectors alone; for example, the properties listed in item 1) are also possessed by scintillation counters, as well as by some spark counters.

The dependence on \(e^2\) (item 5)) is also used in ionization detectors, photographic methods, and the Wilson chamber, and only the possibilities listed in items 2), 3), and 4) are characteristic solely of the Cherenkov detector.

The technique employed depends substantially on the special conditions in each type of experiment. For example, in constructing an instrument

must be taken into account: the intensity and energy of the radiation, regardless of whether the particles will travel as a parallel beam, etc. For example, photographic techniques can be applied only in those cases where there is a sufficiently powerful flux of particles, where good collimation is possible, etc. An exact determination of particle velocities, based on measurements of the angle \(\theta\), requires an especially high degree of collimation and limits the application of this method only to the case of high-energy accelerators.

The possibility of exact measurements of \(\beta\) for individual cosmic-ray particles is strictly limited for the following reasons: first, the required collimation of the particles must lead to counting rates that are too low, and, second, the range of energies associated with cosmic rays, even in the case of phenomena of one and the same type, is so large (from \(\sim 10^7\) to \(\sim 10^{16}\) eV) that throughout this entire range the values of \(\beta\) will be close to unity, and \(\theta\) will therefore be insensitive to changes in \(\beta\) (for condensed media), as follows from the equation \(\cos\theta = 1/\beta n\). The limits can be extended by the use of gaseous media.

The counting speed characteristic of Cherenkov counters is, in itself, higher than that of all other known types of counters. The delay is associated with the collection of light; the scatter of this delay depends on the type of optical system used and on the dimensions of the “radiator” (another designation for the term “converter,” used to denote the medium in which the radiation is excited). In practical detectors this delay may be about \(10^{-11}\)—\(10^{-9}\) sec. This extremely rapid response, unfortunately, cannot be used, since modern photomultipliers and the subsequent circuits increase the delay to \(10^{-8}\) sec. Recently, however, Bayot et al.\(^{69}\), using a coincidence circuit with two photocells, succeeded in obtaining a resolving time of \(\sim 2\cdot 10^{-10}\) sec.

The Cherenkov counter also possesses two further properties which may be useful in certain applications in work with pulsed beams of high intensity: first, there are no quenching effects here, and, second, there is no afterpulse or afterglow, which, for example, sometimes occurs in scintillating media. However, once again, these advantages are partly lost as a result of a considerable loss of sensitivity of the photomultiplier under the action of intense irradiation and the presence of additional pulses\(^{54}\) following the large pulse.

Since the photomultiplier is an important connecting link in most Cherenkov detectors used at the present time, it will be useful to recall certain characteristics of photomultipliers in connection with their application in these counters.

b) Photomultiplier

Morton[^55] considered the characteristics of several types of photomultipliers from the standpoint of their applicability in scintillation counters; many features of multipliers that are essential for a scintillation counter are also necessary for a Cherenkov counter. Although the data given in Morton’s work are obsolete, it is nevertheless useful to refer to them.

A photomultiplier must have the following properties in order to ensure the best qualities of the detector:

1) The photocathode must have a high photon yield, so that a given light signal gives the maximum number of electrons.

2) The spectral sensitivity curve of the photomultiplier must be as close as possible to the curve of the Cherenkov radiation spectrum, i.e., the multiplier must be sensitive to the blue end of the spectrum and, as far as possible, to the ultraviolet.

3) The photocathode must be semitransparent, deposited on the envelope of the photomultiplier, so that optical contact between the photomultiplier and the radiator can be achieved with maximum optical efficiency.

4) The total output must be high enough that relatively small amounts of light at the cathode can be measured without resorting to large amplifications after the photoelement.

5) A photomultiplier with a low level of shot noise must be chosen.

6) For precise time measurements, an instrument with as short a transit time as possible is needed. A small spread in transit times, which, generally speaking, is a consequence of a short transit time, is also desirable.

The conditions formulated above are, generally speaking, stricter than those which a scintillation counter must satisfy, since the light output in the case of a Cherenkov counter may be only \(\sim 1\%\) of the output of a scintillator of the same size. However, some compensation is often possible when the path of the particle in the radiator can be lengthened by increasing the dimensions of the latter. This is explained by the fact that, in the case of the Cherenkov effect, the light can, by means of a suitable arrangement, be more easily directed in one direction toward the multiplier, and also by the fact that most scintillating media strongly absorb their own radiation, thereby imposing restrictions on the appropriate dimensions of the radiator.

In a photomultiplier, the statistical spread in the magnitude of the output pulse depends, for a given flux of photons incident on the cathode, on the number of electrons emitted by the latter, and also on the electron yield per stage, since there are fluctuations

Cherenkov Radiation

in secondary emission, which, in turn, create a certain broadening in the distribution of pulse magnitudes. Consequently, generally speaking, it is impossible to measure with any accuracy the amount of light from a single particle, and it is necessary to observe a large number of events of the same type and obtain the distribution of pulses by magnitude in order to indicate the magnitude of the primary pulse.

Morton\(^{55}\) showed that if \(\delta\) is the number of electrons emitted at the photocathode, \(\sigma\) is the mean value of the secondary-emission yield, and \(m\) is the number of multiplier stages, then the relative fluctuation of the output \(P\) will be approximately equal to

\[ \frac{\overline{(\Delta P)^2}}{P^2} = \frac{1}{\delta}\, \frac{\sigma^{m+1}-1}{\sigma^m(\sigma-1)} \sim \frac{1}{\delta}\left(\frac{\sigma}{\sigma-1}\right). \tag{36} \]

In this equation it is assumed that the photoelectric and secondary electron emission obey the Poisson distribution. It is further assumed that no loss of electrons occurs between adjacent stages.

Let us consider a numerical example. If we have a glass radiator \((n=1.50)\) of thickness \(10\ \mathrm{cm}\) and irradiate it with ultrarelativistic electrons, then in the visible part of the spectrum there will arise \(\sim 2500\) photons for each incident electron. Suppose that all of them reach the photocathode, and assume that its sensitivity is equal to \(30\ \mu\mathrm{A}/\mathrm{lm}\). The quantum yield in this case is \(\sim 6\%\), and therefore 2500 photons will liberate 150 electrons from the cathode. A typical 11-stage multiplier can give a total amplification of \(\sim 10^7\), and since the yield \(G\) is approximately equal to

\[ G=\sigma^m, \tag{37} \]

we obtain

\[ \sigma=4.4. \]

Thus, from formula (36) we find:

\[ \frac{\sqrt{\overline{(\Delta P)^2}}}{P}\sim 9\%. \]

As has already been indicated, Cherenkov signals are generally considerably smaller than the signals from scintillating crystals, so that it is especially important to distinguish them from the background of the dark current of the multiplier, which, in turn, itself has a very broad distribution of pulses. To facilitate this discrimination, it is often useful to use two multipliers operating in fast coincidence circuits. Whether this system is used or not, in any case it is necessary to obtain the best signal-to-noise ratio. This is achieved by using an amplifier with the greatest possible bandwidth, which, in other words, means using

integrating and differentiating cells with small time constants. In this way, individual noise pulses can be resolved and their influence reduced to a minimum. In the USA it is now accepted practice to use amplifiers with a passband of 200 Mc/s, whereas in England, where such amplifiers are generally not available, it has usually proved expedient to use a passband of 5 Mc/s.

What has been said is sufficient to characterize some of the most important properties of photomultipliers, and we now turn to a review of the practical forms of Cherenkov detectors that have been developed up to the present time and are used as a new experimental tool in nuclear physics and cosmic-ray physics.

c) Counters with focusing properties

Marshall[^56] was apparently the first to construct a practical detector in which a photomultiplier was used and which possessed focusing properties. Here we shall call a focusing counter such a counter in which light, emitted at a definite Cherenkov angle by a radiator of finite size, can be focused onto a small area, from which it can be directed onto the photocathode of a photomultiplier, while light directed at other angles will not reach it. The same author, in a later and more complete work[^57], considers various types of detectors for measuring the energies of beams of various particles produced by high-energy accelerators.

The designs of all these detectors are based primarily on the use of a cylindrically symmetric arrangement of the radiator and optical parts, and it is assumed that in all the instruments beams are used in which the particles are directed parallel to one another so as to enter the receiver parallel to its axis.

From general considerations it can be shown that it is impossible to focus all the light to a point if the radiator has a finite diameter, even in the case of a definite Cherenkov angle. As we shall see, all that can be achieved in this direction is to obtain an image whose dimensions are comparable with the dimensions of the radiator.

Let a particle pass parallel to the axis of a cylinder of dielectric, but at a distance \(d\) from the axis. Since Cherenkov radiation is emitted at an angle \(\theta\) from all points of the trajectory, consequently the greater part of the light goes at an angle to the axis. It follows from this that the photons will have an angular momentum with respect to the axis. This angular momentum is conserved in any optical system of this kind and can never be changed, whatever reflections or refractions it may undergo at cylindrically symmetric surfaces of the section. This sets a limit to the sharpness

focus. For the most deflected ray inside the dielectric, the angular momentum of the photon relative to the axis of the system is equal to \(d(\sin \theta)h\nu/c\). The same photon, upon leaving the dielectric, will have angular momentum \(D(\sin \theta')h\nu/c\), where \(\theta'\) is the angle of scattering of the light outside the medium and \(D\) is the closest approach of the photon to the axis, which ideally should be attained in the focal plane. Conservation of angular momentum leads to the condition

\[ D = nd \frac{\sin \theta}{\sin \theta'} ; \tag{38} \]

\(D\) can never be smaller than \(d\), but usually exceeds it. If \(D\) is determined by the dimensions of the photocathode, then, consequently, the upper limit is determined by the size \(d\) of the radiator of the Cherenkov detector, provided effective collection of the light is achieved.

On the basis of these considerations Marshall developed a series of detectors, which will be described below. In one of his first instruments (Fig. 16) the particles enter a cylindrical radiator made of

Fig. 16. One of Marshall’s first focusing detectors. \(P\) is the path of the Cherenkov ray possessing the angular momentum largest among those possible in the given radiator.

lucite, in which Cherenkov radiation is excited. This light is then directed by total internal reflection toward the center of a hemispherical lens (also made of lucite). Since the refractive index of this material is 1.50, the lens had a focal length twice as large as the radius of curvature. The light is thus focused into a sharp ring: for particles with one and the same value of \(\beta\), at a distance of three radii from the center of curvature of the lens. A cylindrical mirror (with radius equal to half the radius of this ring) reflects the image back toward the center and gives

point focus for rays coplanar with the axis of the system. Particles parallel to it, but displaced from the axis, give rays traveling at an angle (with respect to the axis), and in this case condition (38) will be applicable; in this case the light is focused into a ring, as, for example, at \(A\). The radiation path from the particle in the most extreme case, i.e., when the particle path lies at the boundary of the cylindrical radiator, is shown by the dotted line in Fig. 16. In the case under consideration the angle of convergence of the light to the focal disk is equal to the Cherenkov angle, and the diameter of the image is 1.5 times greater than the diameter of the radiator for \(n=1.5\) in Lucite. The light is collected onto a photomultiplier placed behind an annular diaphragm, and rays of different Cherenkov angles are selected by moving this diaphragm along the axis of the system. If the aperture radius is \(r\) and that of the radiator is \(\rho\), then it can be shown that, for the case of best focusing, the fraction \(f\) of the light collected by this aperture is expressed by the formula

\[ f=\frac{2}{\pi}\arcsin\left(\frac{r}{n\rho}\right), \tag{39} \]

in which we neglect spherical aberration in the lens, dispersion, and the small angle of multiple Coulomb scattering.

The arrangement shown in Fig. 16 has the disadvantage that the photomultiplier is located directly in the particle beam; this leads to a considerable background caused by one or more effects: scintillation pulses, local Cherenkov radiation in the glass envelope of the photomultiplier, and emission of \(\delta\)-rays from the photocathode as a result of ionization associated with the passage of the particle through the glass. Consequently, it is better to place the photomultiplier outside the main beam, and further, if two such photomultipliers are used in a fast coincidence circuit, better discrimination against false pulses can be obtained and, in this case, operation at lower discriminator levels, down to the noise level, is possible in order to increase the absolute efficiency. A measured detector scheme having these features is shown in Fig. 17. In this device the radiator and lens are combined, so that the Cherenkov process occurs

Fig. 17. Detector with two photomultipliers, assembled according to the coincidence scheme. \(L\)—Lucite-radiator lens; \(M\)—cylindrical mirror; \(Ph\)—photomultipliers of type 1P28.

in the lens itself. The resolution obtained with this device is illustrated by the curves in Fig. 18, where the light intensity is plotted as a function of the position of the radiator and of the Cherenkov angle. These curves refer to a beam of negative \(\pi\)-mesons with an energy of \(145\) MeV, obtained from the 170-inch synchrocyclotron: \(a\)—observed directly, and \(b\)—after passage through a layer of graphite absorber \(12\ \mathrm{g/cm^2}\) thick, which lowers their energy to \(121\) MeV. The measured angles, \(39.9^\circ\) and \(38.0^\circ\), are close to the calculated values \(40.4^\circ\) and \(38.1^\circ\), respectively. This comparatively good resolution was obtained under conditions of a rather large shift in the coincidence circuit. Higher internal yields were obtained at lower values of the shift, but with a loss in resolving power. Marshall also discusses other combinations of radiator and optical system, some of which were tested and others not.

Fig. 18. Curves illustrating the resolution obtained with the aid of the instrument shown in Fig. 17 for a beam of \(\pi\)-mesons\({}^{57}\): \(a\)—a beam of negative \(\pi\)-mesons with an energy of \(145\) MeV; \(b\)—the same after passage through a graphite absorber.

These detectors were used in the following cases: a nonfocusing water counter was constructed for counting electrons arising under the action of the \(\gamma\)-rays from \(\pi^\circ\)-meson decay, and a nonfocusing lucite counter was used to detect neutrons in the energy range \(360\)–\(450\) MeV, where the neutrons produced recoil protons in a paraffin block. Counting of \(450\)-MeV protons relative to the background of protons of lower energy was also carried out in scattering experiments. For \(450\)-MeV protons \(\beta=0.74\), and \(n\) must be greater than \(1.35\), whence it is evident that lucite was the most suitable material. Counting and energy measurements of \(\pi^\pm\)-mesons were also performed, as described above, down to energies of \(70\) MeV, with a lucite radiator.

d) High-precision photographic apparatus

Mather\({}^{7}\) was the first to observe Cherenkov radiation from protons and used this for measurements of the energy of a beam emerging from the 184-inch cyclotron at Berkeley (California). He developed

precise photographic instrument, extremely simple and elegant, and used it for absolute calibration of the velocity of a proton beam. By the time his work was published, Mather estimated the total error (standard deviation) in the determination of energy at \(\sim 1\%\) for an energy of \(340\) MeV, although after a careful analysis of the methods with the aim of improving the accuracy, it is apparently possible to say that the error can be reduced to \(\sim 0.1\%\) for the same energies.

The first detection of the Cherenkov effect with protons was obtained using a cube with a side of \(1\) cm made of transparent silver chloride \((n = 2.07)\) with polished surfaces. This cube (Fig. 19) was surrounded by a silvered spherical mirror, which focused the light emerging in a horizontal

Fig. 19. One of Mather’s early photographic detectors, used to measure the energy of a proton beam of \(340\) MeV: \(a\)—side view; \(b\)—top view; \(Ph\)—photographic plate; \(M\)—spherical mirror; \(C\)—cube of AgCl; \(B\)—proton beam.

Fig. 20. Mather’s precision photographic instrument\({}^{7}\), intended for accurate measurements of proton velocities: 1—protons; 2—glass plate; 3—aluminized plate; 4—glass prism; 5—scale; 6—light-tight box; 7—camera; 8—35-mm film.

plane into a ring at the place where the photographic plate was located. After this experiment Mather proceeded to improve the optical system in order to obtain a higher resolution in measurements of the Cherenkov angle \(\theta\). The first step in this direction was the use of a uniform thin glass plate as the source of radiation, inclined with respect to the proton flux, as shown in Fig. 20, so that part of the light cone would emerge normally to the surface; this would eliminate first-order refraction effects and spherical aberration. In order that the camera intended for recording the light should be placed sufficiently close to the glass plate and yet not be too close to the proton beam, it was convenient to aluminize one of the surfaces of the plate and observe the emitted light on its return path through the plate. Avoiding refraction phenomena in the converter, the emerging parallel beam is focused by the camera lens onto the film.

To avoid the dispersion following from expression (1), Møller ingeniously introduces a small prism, shown in Fig. 20, with an angle \(\alpha\), chosen so that its dispersion cancels the first-order dispersion of the Cherenkov rays.

Denoting by \(\psi\) the direction of the ray visible to the camera, Møller obtains the following conditions for the achromatization of the system and for the required prism angle:

\[ \frac{d\psi}{d\lambda} = \frac{d\psi}{dn}\frac{dn}{d\lambda} = \left[ \frac{1}{(n^2\beta^2-1)^{\frac12}} - \frac{2\sin \frac12 \alpha}{\left(1-n^2\sin^2 \frac12 \alpha\right)} \right] \frac{dn}{d\lambda} =0 \tag{40} \]

and

\[ \sin \frac12 \alpha = \frac{1}{\sqrt{\,n_0^2+4\left(n_0^2\beta_0^2-1\right)\,}}, \tag{41} \]

where \(n\) is the refractive index corresponding to the “effective” wavelength of the entire system.

The optical system contains an illuminated scale and a collimator, making it possible to measure accurately the position of the Cherenkov image. Fig. 21 shows a microphotogram of the image obtained with this instrument, with the scales of \(\theta\), \(\beta\), and the kinetic energy of the protons \(E\) plotted on it; it can be seen that the total error does not exceed \(1\%\); the components producing this error are listed in Table VI.

Fig. 21. Microphotometric record of an achromatic image of Cherenkov radiation from protons.

The values of the kinetic energy \(E\) were obtained from the measured quantities \(n\) and \(\theta\)—by formula (1) and the equation

\[ E=mc^2\left[\frac{1}{\sqrt{1-\beta^2}}-1\right], \tag{42} \]

where \(m\) is the rest mass of the particle, so that

\[ E=mc^2\left[\frac{n\cos\theta}{(n^2\cos^2\theta-1)^{1/2}}-1\right]. \tag{43} \]

Since the energy measurements carried out by Maeder are the most accurate results obtained with the aid of a Cherenkov detector, we shall consider in more detail the various sources of errors and discuss how, in future instruments, they might be reduced so as to approach the theoretical limit of resolution.

Table VI

		\(\Delta\beta\)	\(\Delta E,\ \mathrm{Mev}\)
Error in \(\theta\) . . . . . . . .	\(\pm 2.5'\)	\(\pm 0.00040\)	\(\pm 0.6\)
Error in \(n\) . . . . . . . .	\(\pm 0.0003\)	\(\pm 0.00012\)	\(\pm 0.2\)
Error in the reading of \(\vartheta\) . . . .	\(\pm 1.6'\)	\(\pm 0.00025\)	\(\pm 0.4\)
Total error . . . . . . . .	—	\(\pm 0.0005\)	\(\pm 0.8\)

There are six effects that are sources of spread in the intensity distribution of Cherenkov rays in the achromatic system described above; these effects are the following:

1) Coulomb scattering of the proton beam in matter.
2) Change of velocity, since the material slows the protons.
3) Diffraction effects associated with the finite path length of the protons in the glass.
4) Second-order chromatic effects.
5) Internal divergence of the incident proton beam.
6) Spread of velocities in this beam.

Let us consider all these effects separately.

Scattering. The mean square scattering angle for a particle of charge \(ze\) in passing through a medium with atomic number \(Z\) is given by Scott (1949):

\[ \overline{\delta^2}=\frac{8\pi e^4 Z^2 z^2 NX}{p^2 v^2}\ln\frac{150p}{\mu c Z^{1/3}}. \tag{44} \]

Here \(N\) is the number of nuclei per \(1\ \mathrm{cm}^3\), \(X\) is the path traversed in the medium, and \(\mu\), \(p\), and \(v\) are the mass, momentum, and velocity of the particle. In Maeder’s experiments, where \(X\) has the value \(0.48\ \mathrm{g/cm^2}\) along the direction of the beam, scattering led to a constant deviation in the angular intensity distribution of the order of \(\Delta\theta=\pm 11'\).

Slowing down. The broadening of the image, introduced by the change in the value of \(\beta\) as the proton passes through the glass plate, is calculated from the expression for the energy loss given by Bethe\(^{58}\) and Bloch\(^{59}\), which leads to a value of \(0.99\) MeV. Direct measurement gives the figure \(0.72\) MeV. As a compromise, the mean value \(0.86\) MeV was adopted, and \(\Delta \theta\) was calculated from the expression

\[ \Delta \theta=\frac{1}{2}\left(\frac{d\theta}{dE}\right)\left(-l\frac{dE}{dx}\right). \tag{45} \]

In the instrument that was used, \(d\theta/dE=3.94\) min/MeV; consequently, \(\Delta \theta=1.7\) angular minutes.

Diffraction. The width of the Cherenkov image is increased as a result of diffraction, associated with the finiteness of the path length \(l\), by an amount given by the expression

\[ \Delta \theta=\frac{0.38\lambda_{\mathrm{cp}}}{nl\sin\theta}, \tag{46} \]

which in the case under consideration is \(\pm 0.68\) for \(\lambda_{\mathrm{cp}}=5000\) Å.

Chromatic effects. Taking into account the spectral distribution of the Cherenkov radiation (see (17)) and the spectral sensitivity of the film, and using formulas (40) and (41), a spread of \(\pm 3.2\) in the values of \(\psi\) was calculated, which, in turn, leads to a value of \(\pm 1.7\) for \(\theta\).

Fig. 22. Individual curves of the intensity distribution over the angles for Cherenkov rays in Mazera’s achromatic instrument: 1 — scattering; 2 — slowing down; 3 — diffraction; 4 — divergence; 5 — chromatic effects; 6 — energy spread; 7 — all six effects together.

Properties of the beam. The divergence of the proton beam and the spread in particle energy make their own contribution to the effective width of the Cherenkov image recorded by the film in Mazera’s experiments, although these effects are not connected with the detector itself. The beam divergence gives a spread in \(\theta\) of the order of \(\Delta \theta=\pm 4'\), while the energy spread in the beam, \(\sim 1.8\) MeV, leads to an error \(\Delta A\sim \pm 7'\).

From these data it is evident that the spread introduced by scattering is the principal source of errors in measurements of \(\theta\).

The distribution and relative magnitudes of each of these effects, as well as the total curve, are shown in Fig. 22. (The abscissas are given in angular minutes of \(\theta\).) From the total curve (Fig. 22, 7) it is seen that the standard deviation \(\Delta A_{\text{total}}\sim \pm 14\) angular minutes. This value agrees almost exactly with the value obtained from the microphotogram of Fig. 21, although such a direct comparison may be considered

justified only in the case where the characteristics of the photographic film are taken into account.

Mather then discusses methods for choosing the parameters, for example the refractive index and the thickness of the converter, in order to obtain maximum accuracy in an instrument of this type. The results of these calculations are given in the form of curves in Fig. 23, where the values of \(\Delta E\) (Mev) are plotted as a function of the thickness of the plate in which Cherenkov radiation arises, or, more precisely, as a function of the ratio \(l/\beta n\), expressed in millimeters. It should be noted that these curves include the effects of beam divergence, although, as has already been said, they are not connected with the properties of the instrument itself. If the beam does not diverge, then in the case of a polystyrene converter \((n = 1.59)\) an energy resolution of \(\sim 0.33\) Mev can be achieved at an energy of \(\sim 340\) Mev, i.e. \(\sim 0.1\%\).

Fig. 23. Computed energy resolution as a function of the thickness of the substance in millimeters for instruments of the Mather type.

In order to give some idea of the total sensitivity of the device, it should be said that the exposures were “less than an hour,” but in exceptional cases amounted to only 3 min when using Atsco Priple S film, a proton current density of \(2 \cdot 10^{-11}\) a/cm\(^2\), and an effective converter area of \(\sim 1\) cm\(^2\).

d) Proton selector

Duerden and Hyams\({}^{60}\) developed a water detector for separating out a very weak flux of cosmic-ray protons at sea level under conditions of a much more intense background of \(\mu\)-mesons and electrons. If a particle has speed \(\beta c\) and rest mass \(M\), it can be shown that its residual range \(R\) is related to these quantities by the following expression\({}^{61}\):

\[ R = \frac{Mc^2}{A}\left[1 - (1-\beta^2)^{1/2}(1-\beta)^{-1/2}\right]\ \text{g/cm}^2, \tag{47} \]

where \(A = 1.2\), if \(Mc^2\) is expressed in megaelectronvolts.

Further, if \(\beta c\) is less than or equal to the Cherenkov critical speed for water \(\beta_c c\), where \((1/\beta_c = n = 1.33)\), then it follows from this that

\[ Mc^2 \geq 6.85R\ \text{Mev}. \tag{48} \]

Thus, if a charged particle has a range \(>15\ \mathrm{g}/\mathrm{cm}^2\) and at the same time does not give Cherenkov radiation, then it must have a residual energy \(>100\ \mathrm{MeV}\) and therefore must be heavier than a \(\mu\)-meson.

The curves calculated for the total intensity of Cherenkov radiation for protons and \(\mu\)-mesons in water are shown in Fig. 24, in which the particle energies have been converted into ranges expressed in grams per square centimeter. In both cases there is a small contribution from knocked-out electrons, which is also shown in the figure.

Fig. 24. Theoretical curves for the intensity of protons as a function of range for heavy particles in water\(^{60}\). \(I_e\) is the part of the curve corresponding to knocked-out electrons.

Fig. 25. Water detector, constructed by Duerden and Hiam, in which diffuse reflection of light is used\(^{60}\).

The main details of the detector are shown in Fig. 25. In the outer metal box \(H_2\) there is placed a photomultiplier \(P\) (EMI, type VX 5045). The inner box \(H_1\), made of Perspex, has double walls on both sides, the spaces between which are filled with magnesium carbonate, and inside it is filled with distilled water.

Duerden and Hiam used the large mean free path of visible light in water and the multiple reflection of light from diffusely scattering surfaces, as opposed to the use of the directional properties of the radiation. In this way a very high efficiency was achieved for detecting relativistic particles falling on a surface of \(400\ \mathrm{cm}^2\) inside a cone with half-angle \(45^\circ\) (selected by a telescope of Geiger counters). With this detector, placed in the telescope, the height distribution of pulses from individual mesons was measured (Fig. 26), having a relatively small width.

From statistical considerations and from the width of this distribution it can be shown that the mean signal corresponded to the emission of \(\sim 30\)

electrons from the photocathode, thereby giving an optical efficiency of \(\sim 15\%\) for the full fraction of the exciting light incident on the photocell.

If we denote by \(A\) and \(B\) the firings of the Geiger telescope above and below the water detector \(W\), then “heavy particles” will be selected by recording coincidences of the type \(A+B-W\), i.e., coincidences in the Geiger telescope not accompanied by a pulse in the Cherenkov detector. Selection of particles by range is carried out by placing a lead plate \(20\ \mathrm{cm}\) thick between \(B\) (Fig. 25) and the following position \(C\) below the whole apparatus. From the observed ratio of counting rates

\[ \frac{A+B-W}{A+B}=\frac{2.15\pm 15}{1000} \]

and after insertion of the range selector \(C\)

\[ \frac{A+B+W+C}{A+B+C}=\frac{0.32\pm 0.1}{1000} \]

we see that 80% of the counts of “heavy particles” are associated with protons. The remaining 20% are apparently due to knock-on electrons and (or) other light particles passing through the apparatus in coincidence with protons. The magnitude of the momentum of the protons selected by the anticoincidence method was 700–1100 MeV/\(c\); the lower limit was obtained with the aid of the range selector, and the upper limit by determining the Cherenkov threshold in water.

Fig. 26. Experimental distribution of Cherenkov pulses by height in the case of individual \(\mu\)-mesons.

In experiments with this apparatus it was found that the absolute flux of protons with this value of the momentum at sea level is

\[ (1.5\pm 0.1)\cdot 10^{-5}\ \mathrm{cm}^{-2}\cdot \mathrm{sec}^{-2}\cdot \mathrm{sterad}^{-1}. \]

The apparatus was placed under a Wilson chamber with a lead absorber inside, in order to facilitate identification of heavy particles with protons. It was found that \(<5\%\) of these counts of “heavy particles” should be attributed to the inefficiency of Cherenkov detection of relativistic particles.

The selector described here primarily records protons at sea level with velocities greater than is possible by any other method; the absolute counting rate was 3.5 protons per hour. In terms of specific ionization, this apparatus should identify protons entering it with an ionization smaller by a factor of 1.3.

During the writing of this article, Hyams\(^{62}\) installed such a detector at the Pic-du-Midi observatory in combination with a Wilson chamber

with the aim of searching for the negative proton. At the same time it was to be used for searches for unstable heavy mesons and for measuring the lifetime of charged \(x\)-mesons; it was proposed to use the second Cherenkov counter for detecting relativistic decay products.

e) The albedo of cosmic rays

An interesting application, in which the ability of a Cherenkov counter to determine the direction of motion of a fast particle is used, was made by Winkler\({}^{63}\). By this method he attempted to measure the “albedo” of cosmic rays in a balloon flight at high altitudes; he used the simple device shown schematically in Fig. 27. The albedo was defined as the ratio of the number of particles passing from below upward to the number of particles going from above downward.

The lucite block \(L\), taken as the Cherenkov medium, was placed between two assemblies of Geiger counters \(G_1\) and \(G_2\). Particles were recorded by the counters in a simple coincidence circuit when they passed upward or downward through the instrument within small angular limits relative to the zenith. For particles going downward, along the path indicated by the dotted line, the radiation emitted at the Cherenkov angle \((n = 1.50,\ \theta = 48^\circ 10')\) relative to the path is directed downward by internal reflection until it reaches the bottom, where it passes through the beveled surface of the block. The light emerging through one of the cuts falls on the cathode in the end of the photomultiplier \(P\), which is optically coupled to the lucite. Particles flying along the same path upward give Cherenkov radiation which is likewise directed upward. At the upper end the light is absorbed by the light trap \(T\), filled with a black absorbing substance.

Fig. 27. Winkler’s instrument, designed for measuring the “albedo” of cosmic rays in the upper layers of the atmosphere\({}^{64}\).

The total output was measured from the ratio of the counts of the Cherenkov telescope to the counts of the Geiger telescope, i.e., the ratio of triple coincidences to double coincidences \(G_1LG_2/G_1G_2\) was taken. Some results of this experiment are given in Table VII, where \(N_1\) and \(N_2\) refer respectively to the cases when the instrument was oriented as shown in the figure, and in the reverse position, when the detector was rotated by \(180^\circ\) about the horizontal axis (through the center). The figures in the table refer to extrapolated values at zero bias on the discriminator following the Cherenkov detector.

From these results Winkler obtained that the albedo for relativistic particles (i.e., for which \(\beta > 0.7\)) with a Cherenkov threshold in lucite at a zenith angle of \(60^\circ\), at an atmospheric depth of \(17\ \mathrm{g/cm^2}\), is \(0.03 \pm 0.02\). The results of this experiment, even if better statistics are accumulated, are not very satisfactory. First of all, although at sea level the detector had a high efficiency,

Table VII

The figures correspond to the ratio \(G_1LG_2/G_1G_2\), in %

At sea level, \(0^\circ\) zenith	At sea level, \(0^\circ\) zenith	\(120\)—\(150\ \mathrm{g/cm^2}\), \(0^\circ\) zenith	\(120\)—\(150\ \mathrm{g/cm^2}\), \(0^\circ\) zenith	\(17\ \mathrm{g/cm^2}\), \(0^\circ\) zenith	\(17\ \mathrm{g/cm^2}\), \(0^\circ\) zenith	\(17\ \mathrm{g/cm^2}\), \(60^\circ\) zenith	\(17\ \mathrm{g/cm^2}\), \(60^\circ\) zenith
\(N_1\)	\(N_2\)	\(N_1\)	\(N_2\)	\(N_1\)	\(N_2\)	\(N_1\)	\(N_2\)
94	21	70	16	93	18	69	16

94%, for particles going downward, this figure fell to 70% at \(130\ \mathrm{g/cm^2}\) and then again rose to 93% at \(17\ \mathrm{g/cm^2}\). This indicates that the ratio of the number of relativistic particles to the number of nonrelativistic particles changes during passage through the atmosphere. In itself this fact is not surprising, since it is now known that the structure of cosmic rays is a complicated function of the depth of penetration into the atmosphere. It follows from this that an unambiguous interpretation of albedo measurements with Cherenkov counters by this method would be possible only if other data were taken into account in analyzing the results (for example, data on the masses and energy spectra of the particles). Secondly, we see from Table VII that at sea level the detection efficiency is no more than 21% if the apparatus is turned over. Since, as is known, the flux of relativistic particles going upward from the surface is much smaller—\(\sim 0.23\)—than the flux going from above downward, we see that the ratio of the detection efficiencies of the latter in the two positions of the detector is only \(4.5/1\). Part of the “reverse” efficiency can be explained by lateral showers penetrating into the instrument, although it would be difficult to explain everything by this.

A ratio of the direct output to the reverse one, amounting to \(10/1\), was obtained by Winkler at higher bias values, but, as Jelly\({}^{45}\) found, this was obtained at the expense of absolute efficiency. For example, for \(N_1\) and \(N_2\) values of 62 and 6% were found (see the first column in Table VII) at a bias level of \(25\ \mathrm{V}\).

Winkler and Anderson\({}^{64}\) recently improved this receiver by introducing a new type of photomultiplier (RCA, type C7175), having a very large cathode surface, \(85\ \mathrm{cm^2}\), deposited on a cylindrical glass envelope. The converter made of lucite was made

concave at one end and optically coupled to a photomultiplier. With this detector, placed between Geiger counters, as was mentioned, and adjusted to the vertical flux of cosmic rays, in an underground laboratory they obtained a remarkably narrow maximum in the height distribution of pulses, with a full width of 23% at half-maximum (Fig. 28). This width agrees with the calculated value of an emission of 6500 quanta per particle and a cathode yield of from 5 to 10%. The efficiency of such a receiver reached 100%, and a very high ratio \(N_1/N_2\) was found. This receiver was soon applied to the “albedo” problem.

Fig. 28. Distribution of pulses of cosmic particles by height, obtained underground with the aid of a detector constructed by Winker⁶⁴.

V. CONCLUSION

The Cherenkov detector has been tested and has proved to be a very useful instrument in nuclear physics and cosmic-ray physics, since it provides high speed, high efficiency, and possesses appreciable directional properties. In addition, it is simple in construction and does not require rare materials for its manufacture. Its applications, however, are to some extent limited by the very nature of Cherenkov radiation, although, apparently, in the near future its use should expand, since new types of accelerators will yield particles of ever greater and greater energies.

REFERENCES CITED

1. P. A. Cherenkov, DAN SSSR 2, 451 (1934).
2. I. M. Frank and I. E. Tamm, DAN SSSR 14, 109 (1937).
3. L. Mallet, C. R. Acad. Sci. (Paris), 188, 445 (1929).
4. S. I. Vavilov, DAN SSSR 2, 457 (1934).
5. G. Collins and V. Reiling, Phys. Rev. 54, 499 (1938).
6. W. Heitler, The Quantum Theory of Radiation (Oxford University Press), 1944.
7. R. L. Mather, Phys. Rev. 84, 181 (1951).
8. P. A. Cherenkov, DAN SSSR 3, 413 (1936).
9. P. A. Cherenkov, DAN SSSR 14, 101 (1937).
10. P. A. Cherenkov, DAN SSSR 14, 105 (1937).
11. P. A. Cherenkov, Phys. Rev. 52, 378 (1937).
12. P. A. Cherenkov, DAN SSSR 21, 116 (1938).

1. P. A. Cherenkov, Dokl. Akad. Nauk SSSR 21, 319 (1938).
2. P. A. Cherenkov, Izv. Akad. Nauk SSSR, Ser. Fiz. 4–5, 455 (1937).
3. I. E. Tamm, J. of Phys. USSR 1, 439 (1939).
4. A. Sommerfeld, Göttingen Nachrichten 99, 363 (1904).
5. F. Klein and A. Sommerfeld, Theorie d. Kreisels, Leipzig IV, p. 925 (1910).
6. V. L. Ginzburg, J. of Phys. USSR 3, 101 (1940).
7. K. Tanaka, UCRL, Report No. 1286 (1951).
8. I. M. Frank, Dokl. Akad. Nauk SSSR 42, 341 (1944).
9. Li, Yin-Yuan, Phys. Rev. 80, 104 (1950).

10. L. I. Schiff, Quantum Mechanics, p. 264 (McGraw-Hill, New York), 1949.

11. H. Wyckoff and J. E. Henderson, Phys. Rev. 64, 1 (1943).

12. J. M. Harding and J. E. Henderson, Phys. Rev. 74, 1560 (1948).
13. Li, Yin-Yuan, Phys. Rev. 82, 281 (1951).
14. K. G. Dedrick, Phys. Rev. 87, 891 (1952).
15. E. Fermi, Phys. Rev. 57, 485 (1940).
16. G. Beck, Phys. Rev. 74, 795 (1948).
17. T. Taniuti, Progr. Theor. Phys. Japan 6, 207 (1951).
18. N. Bohr, Dann. Mat. Fys. Med. 18 (8) (1948).
19. M. Schönberg, Nuovo Cimento 9 (2), 210 (1952).
20. P. Budini, Phys. Rev. 89, 1147 (1953).
21. P. Budini, Nuovo Cimento 10 (3), 236 (1953).
22. R. M. Sternheimer, Phys. Rev. 89, 1148 (1953).
23. V. L. Ginzburg, J. of Phys. USSR 2, 441 (1940).
24. R. T. Cox, Phys. Rev. 66, 106 (1944).
25. J. M. Jauch and K. M. Watson, Phys. Rev. 74, 1485 (1948).
26. J. Jauch and K. M. Watson, Phys. Rev. 75, 1249 (1949).
27. V. L. Ginzburg, Izv. Akad. Nauk SSSR, Ser. Fiz. 11 (2), 165 (1947).
28. V. L. Ginzburg, Dokl. Akad. Nauk SSSR 56 (7), 699 (1947).
29. M. Abele, Nuovo Cimento, Supplement No. 9, 207 (1952).
30. I. A. Getting, Phys. Rev. 71, 123 (1947).
31. R. H. Dicke, Phys. Rev. 71, 737 (1947).
32. P. B. Weisz and B. L. Anderson, Phys. Rev. 72, 431 (1947).
33. J. V. Jelley, Proc. Phys. Soc. A64, 82 (1951).
34. G. T. Reynolds, F. B. Harrison and G. Salvini, G., Phys. Rev. 78, 488 (1950).
35. P. Bassi, Nuovo Cimento 8 (10), 807 (1951).

36. P. Bassi, A. M. Bianchi and C. Manduchi, Nuovo Cimento 9, 861 (1952).

37. E. H. Belcher, Proc. Roy. Soc. A216, 90 (1953).

38. P. I. Dee and W. T. Richards, Nature 168, 736 (1951).
39. F. S. Dainton, Ann. Rep. Chem. Soc. 45, 5 (1949).
40. M. A. Greenfield, A. Norman, A. H. Dowdy and P. M. Kratz, J. Opt. Soc. Amer. 43 (1), 42 (1953).
41. W. Galbraith and J. V. Jelley, Nature, Lond. 171, 349 (1953).
42. T. N. K. Godfrey, F. B. Harrison and J. W. Keuffel, Phys. Rev. 84, 1248 (1951).
43. G. A. Morton, RCA Review 10, 525 (1949).
44. J. Marshall, Phys. Rev. 81, 275 (1951).
45. J. Marchall, Phys. Rev. 86, 685 (1952).
46. H. A. Bethe, Handb. der Physik 24, 522 (1933).
47. F. Bloch, Ann. Phys. Lpz. 16, 285 (1933).
48. T. Duerden and B. D. Hyams, Phil. Mag. 43, 717 (1952).
49. L. Janossy, Cosmic Rays (Clarendon Press, Oxford), p. 127, 1948.
50. B. D. Hyams, private communication, 1953.
51. J. Winckler, Phys. Rev. 85, 1054 (1952).

1. J. Winckler and K. Anderson, Rev. Sci. Instr. 23, 765 (1952).
2. P. A. Cherenkov, DAN SSSR 20, 651 (1938).
3. S. Robin, J. Phys. Radium (Paris), 11 (1950).
4. W. T. Scott, Phys. Rev. 76, 212 (1949).
5. A. Battig, Instituto de Fisica. Universidad Nacional del Tucuman, Argentina, Publication No. 591, vol. 30 (1951).
6. Z. Bay, M. R. Cleland and F. McLernon, Phys. Rev. 87, 901 (1952).
7. P. M. S. Blackett, Gassiot, Comm. Rep. Phys. Soc. 34 (1948).
8. F. X. Eder, Funk u. Ton. 3, 1949, 67 (1949).
9. E. Maurer and H. Kolz, Zeits. angew. Phys. 2, 223 (1950).