Study of Cherenkov Radiation as a Method for Determining the Velocity of Fast Charged Particles

In 1934 P. A. Cherenkov1 discovered that many liquids glow under the action of $\gamma$ rays from radioactive preparations, and showed that the resulting glow is not connected with luminescence of the liquid, but originates from fast electrons formed in the liquid upon absorption of $\gamma$ rays. A theoretical study of Cherenkov radiation, carried out by I. E. Tamm and I. M. Frank2, showed that its source

is in fact the moving electron itself, and made it possible to clarify the conditions for the occurrence of such radiation. It turned out that any charged particle moving in a medium with refractive index \(n\) at a speed greater than the phase velocity of light \(\frac{c}{n}\) in that medium becomes a source of Cherenkov radiation. This radiation is polarized and forms with the direction of motion of the charged particle an angle \(\vartheta\), determined from the following condition:

\[ \cos \vartheta = \frac{c}{vn}, \tag{1} \]

which is also the condition for the occurrence of Cherenkov radiation. Tamm and Frank also gave a formula for calculating the energy losses due to Cherenkov radiation. In the visible region of the spectrum, where dispersion may be neglected, this formula takes the form

\[ \frac{dN}{dL} = 2\pi \left(\frac{z^2 e^2}{hc^2}\right) \left(1 - \frac{1}{n^2\beta^2}\right)\Delta\nu \ \text{quanta/cm} \simeq \tag{2} \]

\[ \simeq 500\cdot \sin^2 \vartheta \ \text{quanta of visible light/cm}, \]

where \(\frac{dN}{dL}\) is the number of visible-light quanta emitted by a particle with charge \(Ze\) per unit path length, and \(\Delta\nu\) is the width of the spectral region in hertz. The properties of Cherenkov radiation were initially tested on fast electrons, whose velocity practically does not differ from the velocity of light. In 1950[^3] Cherenkov radiation was successfully used to detect fast mesons in cosmic rays: a beam of mesons passed through a vessel with distilled water, and the resulting radiation was recorded with the aid of a photomultiplier. Obviously, this method of detecting fast particles is the most inertia-free of all those currently available. The properties of Cherenkov radiation on heavy charged particles had not been verified, since until recently it had not been possible to obtain beams of heavy particles possessing velocities sufficient for Cherenkov radiation to arise in ordinary transparent media for visible light with refractive indices \(n = 1.5\text{--}2\). It is clear that formula (1) gives a very convenient method for the precise measurement of a particle’s velocity: for this purpose it is necessary to measure the angle \(\vartheta\) formed by the direction of the radiation with the direction of motion of the particle, and the refractive index of the medium \(n\). Both these quantities can be measured with high accuracy. The aim of the work being reviewed[^4] was to verify formulas (1) and (2) for a beam of fast protons with an energy of 340 MeV (\(\beta = 0.68\)), accelerated in a cyclotron, and to determine the experimental conditions necessary for the most accurate possible determination of proton velocities by formula (1). The deflecting system of the cyclotron produces a well-collimated beam of protons with a current density of about \(2\cdot 10^{-11}\) A/cm\(^2\). Cherenkov radiation was recorded with the aid of photographic plates, and at this current density the duration of the exposures used lay between 3 minutes and 1 hour, depending on the experimental conditions. A preliminary investigation of Cherenkov radiation from fast protons was carried out in an experiment whose scheme is shown in Fig. 1a. This experiment in principle repeats the first measurements made by Cherenkov and other investigators. As a medium with a large refractive index there served a transparent crystal of silver chloride with polished faces, of size \(1 \times 1 \times 1\) cm\(^3\). The crystal was placed at the center of a spherical mirror, and the cone of Cherenkov radiation intersected the mirror in two regions,

Fig. 1.

a

Labels in the diagram:

• View in the direction of the beam
• Photographic plate
• Spherical mirror
• AgCl crystal
• Proton beam
• Side view

b

Labels on the photographic image:

• Deuterons, 340 MeV
• Protons, 270 MeV
• Protons, 225 MeV

Fig. 2.

Labels in the diagram:

• “Leshka”
• Glass plate
• Protons

Fig. 3.

Labels on the image:

• without prism
• at 1 degree
• with prism

Visible scale markings: 33, 34, 38, 39, 40, 41, 42.

arranged symmetrically with respect to the proton beam. The radiation reflected by the mirror fell on a photographic plate. Fig. 1b shows an image, obtained on the photographic plate, of the luminous parts of the mirror. By photometering the photographic plate, one can, knowing its sensitivity and the exposure, determine the intensity of the Cherenkov radiation. The value thus obtained,

\[ \frac{dN}{dL}, \]

is equal to 105 quanta per 1 cm of the path of the proton beam in the crystal. Formula (2) gives for

\[ \frac{dN}{dL} \]

a value equal to 250 quanta/cm of path. Taking into account the absorption of the radiation in the crystal itself and upon reflection from the mirror, such agreement should be regarded as good and as confirming the Cherenkov character of the observed radiation. It is obvious that the considerable size of the spot on the photographic plate makes a sufficiently accurate determination of the angle \(\vartheta\) impossible. A further improvement of the measurement method is shown in Fig. 2. As a medium with a large refractive index (\(n = 1.88\)), a plane-parallel glass plate \(2/3\) mm thick was used. The radiation was recorded with a “Leica”-type camera

Fig. 4.

with improved optics. The upper part of Fig. 3 shows a photograph obtained in this experiment. On the horizontal scale on the photograph shown are plotted the values of the angle \(\vartheta\) in degrees. The considerable blurring of the light beam observed also in this experiment (the beam occupies the region of angles from 37 to 41°) is caused chiefly by the fact that Cherenkov radiation has a continuous spectrum, and therefore for different parts of the visible spectrum the refractive index of the glass plate, and consequently the angle \(\vartheta\), are different. Therefore the next step in increasing the accuracy of the measurement of the angle \(\vartheta\) was the use of an achromatic prism, by means of which radiation with wavelength about \(\lambda = 5460\) Å was focused on the photographic plate. At the same time an image of a scale divided directly into degrees of the angle \(\vartheta\) was projected onto the plate (Fig. 4). The lower part of Fig. 3 gives a photograph

of the light spot for this case. From the photograph it is seen that under these conditions the angle \(\vartheta\) can be measured with an accuracy close to \(0.2^\circ\). The observed finite width of the spot on the photographic plate is explained by the following causes: 1) the scattering suffered by the protons in the glass; this scattering changes the direction of motion of the protons and blurs the position of the spot on the photographic plate; 2) the slowing down of the protons in the glass, causing a decrease in \(\vartheta\); 3) diffraction caused by the fact that the radiation is emitted by a finite section of the plate; 4) the divergence of the primary proton beam, which is not strictly parallel; 5) chromatic effects of the second order; 6) insufficient monochromaticity of the proton beam. The distribution of errors caused by each of the listed conditions of the experiment is shown in Fig. 5, where the deviation of the angle \(\vartheta\) from the mean value, in minutes, is plotted along the abscissa axis. The combined action of all these errors leads to the proton energy being measured with a root-mean-square error \(\sigma = 3.5\) MeV. The accuracy of determining the velocity of the protons or their energy can also be estimated from the accuracy with which the angle \(\vartheta\) and the refractive index of the medium \(n\) are measured. The authors give for \(\vartheta\) the value \(19^\circ 13' \pm 25'\), and for the refractive index a value determined with an accuracy of 0.0003. Such accuracy in the determination of \(\vartheta\) and \(n\) leads to the proton velocity being determined with an accuracy of \(\pm 0.0005\), and the proton energy, correspondingly, with an accuracy of \(\sigma = 0.8\) MeV, which is four and a half times smaller than the error indicated above, \(\sigma = 3.5\) MeV. This discrepancy indicates that the accuracy of the energy measurement in this method does not correspond to the degree of monochromaticity of the beam: if the proton beam were more homogeneous in energy and underwent less scattering in the glass, its energy could be measured with an error of about \(\sigma = 0.8\) MeV. At the end of the paper the authors analyze each of the errors listed above and determine the measurement conditions necessary for the combined action of all errors to be minimal. They point out, in particular, that the error in determining the proton velocity can be reduced by a factor of two or three if, instead of glass, a polystyrene plate \(0.7\) mm thick is used, which weakens proton scattering. In general, errors 1 and 2 decrease as the thickness of the plate is reduced. However

Fig. 5.

a reduction of this thickness causes an increase in diffraction effects. Thus, for a plate of a given material there exists an optimal thickness that ensures a minimal error. In its accuracy, the method described for measuring the velocity and energy of fast protons can be compared only with the method of deflecting particles in a magnetic field, but it has, however, significant advantages over the latter: it is very simple and makes it possible to measure the energy of protons accurately without thereby weakening the intensity of the beam. This is essential in experiments in which the measurement of the energy precedes the further use of the beam for some precise measurements. The use of magnetic analysis in such a case would weaken the intensity of the beam, and the corresponding apparatus would be very cumbersome.

A. V.

References

1. P. A. Cherenkov, DAN 2, 451 (1934); S. I. Vavilov, DAN 2, 457 (1934).
2. I. E. Tamm and I. M. Frank, DAN 14, 107 (1937).
3. UFN 44, issue 3, 443 (1951).
4. R. L. Mather, Phys. Rev. 84, 181 (1951).