Electromagnetic Radiation in Accelerators

V. M. Lopukhin and V. A. Ugarov

Modern accelerating devices (cyclotrons, betatrons, synchrotrons, and others) make it possible to reach very high energies of accelerated particles. At present, some accelerators are already in operation that can impart to particles energies of up to several hundred MeV. Thus, information has been published[^1] that the American cyclotron at Berkeley, in the laboratory of the University of California, operating in the phasotron regime (see below), accelerated $\alpha$-particles to energies of the order of 400 MeV, and deuterons—to energies of $\sim 200$ MeV. The betatron[^2] at Schenectady, near New York, in the General Electric Company laboratory, accelerated electrons to energies of $\sim 100$–$150$ MeV. Striking a target inside the apparatus, these electrons produced $\gamma$-quanta with energies of 100–150 MeV, which were emitted outward. In the same place, in Schenectady,[^3][^4] the introduction of additional magnetization in a small, 50 MeV, betatron made it possible, with its aid, to obtain $\gamma$-quanta with energies $\sim 70$ MeV. Finally, with the aid of a synchrotron,[^5] electrons were accelerated to energies $\sim 70$ MeV. The energy was extracted outward in the form of $\gamma$-quanta.

Entering the energy region $\sim 100$–$1000$ MeV, it is entirely natural to expect a large number of new physical phenomena specific to this energy region. If until now one of the main problems of nuclear—and perhaps of all modern—physics was the problem of obtaining particles with energies $\sim 100$–$1000$ MeV, now, along with the task of further increasing the energies attained, the task of comprehensively mastering the range of already obtained energies (100–400 MeV) has come fully to the fore.

The situation here is reminiscent of what took place at the beginning of this century in the study of temperatures close to absolute zero. Historically, the first task was to reach low temperatures in order to liquefy “all” gases. Indeed, in 1908 K. Onnes liquefied the last gas—helium. But then it became clear that we were dealing with an entirely new field of physics, with its characteristic phenomena and regularities (superfluidity, superconductivity, etc.). The study of this temperature region is at present in full swing.

ELECTROMAGNETIC RADIATION IN ACCELERATORS

Among the new physical phenomena discovered in connection with the attainment of high energies in accelerators, and reflected in the literature, the following may be noted.

By bombarding nuclei of light elements with $\alpha$-particles of energy 400 MeV in the laboratory of Lawrence and Seaborg at the University of California, fast neutrons with energies of $\sim 100$ MeV were obtained. Superfast $\alpha$-particles, deuterons, and neutrons proved capable of causing fission of the nuclei of such elements as platinum, titanium, lead, bismuth, and others. Nuclear fission, i.e., the splitting of a nucleus into two or several parts of approximately equal size, induced by various particles—and in a number of cases spontaneous—had hitherto been known only for uranium, thorium, protactinium, and transuranium elements, i.e., only for naturally radioactive isotopes.

Further, with the aid of $\gamma$-quanta with energies of 100–150 MeV (Schenectady, New York) and of the same $\alpha$-particles and deuterons with energies of 200–400 MeV, it proved possible to bring about multiple splitting of the nuclei of light elements with the emission of several (three to four) fragments, i.e., to obtain an explosion or evaporation reaction characteristic of collisions of cosmic particles with nuclei.

We shall devote our review to one new physical phenomenon, specific to energies of $\sim 100$ MeV. We have in mind the phenomenon of electromagnetic radiation of fast relativistic electrons in accelerators of the betatron type. It is interesting that, at the indicated energies, a noticeable part of the electron’s radiation falls in the visible region of the electromagnetic-wave spectrum.

This circumstance is quite remarkable. Quite recently the fiftieth anniversary of the discovery of the electron was celebrated. However, throughout all these fifty years it has been necessary to be content only with indirect information about the presence of electrons. All experiments with electrons have been based on secondary effects associated with the presence of electrons.

Indeed, all experiments with beams of electrons in vacuum tubes ultimately lead to the observation of a glow on a screen or on the wall of the tube. But the observed glow is by no means the glow of electrons, but the glow of molecules or atoms excited by them. The scintillation method is based on the same effect. One can speak of “observing” an electron in counters only in a very conditional sense; in counters particles are registered, “counted,” but by no means observed.

Nor do we observe particles in the Wilson chamber. In the Wilson chamber we see only the tracks of a particle. If one may resort to a comparison, then the picture observed in the Wilson chamber is completely identical with the final frames of the motion picture The Invisible Man, when we see only the footprints of an invisible man in the snow. In the method of thick-layer photographic plates of L. V. Mysovsky and

A. P. Zhdanov, we also obtain only traces; there are no direct manifestations of the particle. The particle merely “records itself” on the photographic plate.

Only quite recently was the direct glow of the electron discovered. This refers to the discovery of the Cherenkov effect. Here there is indeed a glow of the electrons themselves. But even here a reservation must be made. For the glow of a Cherenkov electron, the presence of a medium is very essential. It is, in fact, conditioned by the presence of the medium. This radiation, as is known, occurs when the velocity of the electron exceeds the phase velocity of light in the surrounding medium. The analogy of this phenomenon with certain phenomena of gas dynamics has been noted more than once. In gas dynamics, entirely special phenomena (shock waves) likewise begin when the velocities of motion exceed a certain characteristic velocity of the medium—the local velocity of sound. Thus, both for shock waves and for the Cherenkov effect, the decisive factor is the presence of a medium with definite characteristics. Therefore the name proposed by I. E. Tamm for the Cherenkov electron—“the singing electron”—seems to us very apt, since this name well emphasizes the essential significance of the surrounding medium.

The only case in which we see the “pure” radiation of the electrons themselves is the radiation of electrons in accelerators. Therefore it seems correct to us to call the direct radiation of electrons in accelerators the effect of the “glowing electron,” understanding by this, in the general case, the entire emitted spectrum, and not only the visible radiation.

The existence of radiation from fast particles in accelerators of the betatron type was predicted in 1944 by the Soviet physicists D. Ivanenko and I. Pomeranchuk^6. The further theory was developed by other Soviet, as well as American, authors^7,8,15.

Experimentally, electron radiation in a betatron was indirectly recorded by Blewett in 1946 and, finally, the “glowing electron” was observed visually by Pollock’s group in 1947. The fact that electromagnetic radiation is emitted by particles moving in accelerators also has direct practical significance.

According to classical electrodynamics, charged particles moving with acceleration lose part of their energy to radiation. As a consequence of this circumstance, particles moving in accelerators may fall out of the proper phase of the accelerating field, which will lead to a disturbance of the normal operation of the accelerator. It therefore seems of interest to determine the influence of this radiation on the character of the motion of particles in the accelerator.

We shall give a brief account of the main works in which the radiation of charged particles in accelerators is investigated, and shall present some interesting experimental data relating to this range of questions.

1. RADIATION AND THE LIMIT OF ATTAINABLE ENERGIES IN ACCELERATORS

The presence of energy losses due to radiation may restrict the range of attainable energies in an accelerator. Although the limiting energy of accelerated particles in different types of accelerators is determined by different factors, one can point to devices in which the circumstance of interest to us is decisive. Let us consider from this point of view all the principal types of accelerators.

As is known, in a simple (nonrelativistic) cyclotron intended for the acceleration of heavy particles, the limiting energy of the accelerated particles is determined by those values of the energy at which the relativistic change in the particle mass becomes appreciable. This energy depends on the mass of the particle. For a proton, for example, a change of mass by 10% already occurs at energies of the order of 100 MeV. As we shall see below, at such energies the radiation of heavy particles plays practically no role.

In a synchrocyclotron (i.e., a phasotron, which is a cyclotron with a varying frequency of the electric field, according to the idea of Veksler and later McMillan\(^{16,17}\)), by means of which relatively high particle energies are attained, radiation again proves insignificant. The limiting particle energy in this case is determined mainly by technical difficulties connected with the creation of large magnets and with frequency modulation.

For the betatron, which is a relativistic device, i.e., a device in which the relativistic change of mass does not interfere with the operation of the apparatus, the radiation of electrons moving in circular stationary orbits already plays an essential role in determining the range of attainable energies. Attention was first drawn to this circumstance in 1944 by Soviet physicists\(^{6}\).

For the linear resonance accelerator and the waveguide accelerator, the limit of attainable energies is not directly connected with the radiation of the accelerated particles. For the microtron and synchrotron (first proposed by V. I. Veksler\(^{16,18}\)) this radiation is essential. The influence of radiation on the operation of these devices will be considered below.

2. RADIATION OF A SINGLE ELECTRON

Let us derive a formula determining the radiation of an individual relativistic electron moving in a magnetic field. We proceed from the classical relativistic equation of motion of an electron containing a term that takes radiation damping into account (or, what is formally the same, from the classical nonquantum Lorentz–Dirac equation for a point electron):

\[ m \frac{d u_i}{d s} = \frac{e}{c} F_{ik} u_k + \frac{2}{3}\frac{e^2}{c^3} \left\{ \frac{d^2 u_i}{d s^2} + u_i \left( u_k \frac{d^2 u_k}{d s^2} \right) \right\}, \tag{1} \]

\[ (i, k = 1, 2, 3, 4), \]

where \(m\) is the invariant mass of the electron, \(e\) is the charge of the electron, \(c\) is the speed of light, \(u_i\) are the components of the four-dimensional velocity, \(F_{ik}\) is the electromagnetic-field tensor \([\mathbf H(F_{23}, F_{31}, F_{12}),\, i\mathbf E(F_{11}, F_{24}, F_{34})]\), and \(ds=dt\sqrt{1-\beta^2}\), where, as usual, \(\beta=\dfrac{v}{c}\). Summation is carried out over repeated indices.

Rewriting equation (1) for the values \(i=1,2,3\) and for \(i=4\), we obtain the equations for momentum and energy:

\[ \frac{d\mathbf p}{dt} = e\left(\mathbf E+\left[\frac{\mathbf v}{c}\mathbf H\right]\right) + \frac{2}{3}\frac{e^2}{c^3} \left\{ \ddot{\mathbf v} + \frac{3}{c^2}\frac{\mathbf v(\mathbf v\dot{\mathbf v})}{1-\beta^2} + \frac{\mathbf v}{c^2(1-\beta^2)} \left( \dot{\mathbf v}\dot{\mathbf v} + \frac{3}{c^2}\frac{(\mathbf v\dot{\mathbf v})^2}{1-\beta^2} \right) \right\} \frac{1}{1-\beta^2}, \tag{2} \]

\[ \frac{d\mathcal E}{dt} = e(\mathbf v\mathbf E) + \frac{2}{3}\frac{e^2}{c^3} \left( \frac{\mathbf v\ddot{\mathbf v}}{(1-\beta^2)^2} + \frac{3}{c^2}\frac{(\mathbf v\dot{\mathbf v})^2}{1-\beta^2} \right). \tag{3} \]

In these formulas \(\mathbf p=\dfrac{m\mathbf v}{\sqrt{1-\beta^2}}\) is the momentum of the particle, \(\mathbf v\) is the velocity of the particle, \(\mathcal E\) is the energy of the particle, and \(\mathbf E,\mathbf H\) are the electric and magnetic fields.

As the zeroth approximation for solving equation (3), we take the solution of the equation without taking radiation damping into account,

\[ \frac{d\mathbf p}{dt} = e\left(\mathbf E+\left[\frac{\mathbf v}{c}\mathbf H\right]\right). \tag{4} \]

We consider the problem without an electric field, i.e. when \(\mathbf E=0\). Taking into account that the magnetic field does not change the absolute value of the velocity, we shall have from (4)

\[ \dot{\mathbf v} = \frac{e}{m}\sqrt{1-\beta^2}\left[\frac{\mathbf v}{c}\mathbf H\right] \tag{5} \]

and, consequently,

\[ \ddot{\mathbf v} = \frac{e^2}{m^2}(1-\beta)^2\left[\left[\frac{\mathbf v}{c}\mathbf H\right]\mathbf H\right]. \tag{6} \]

Substituting (5) and (6) into (3), we obtain (since \(\mathbf v\dot{\mathbf v}=0\))

\[ \frac{d\mathcal E}{dt} = \frac{2e^2}{3c^3} \left\{ \frac{\mathbf v\left[\left[\dfrac{\mathbf v}{c}\mathbf H\right]\mathbf H\right]}{(1-\beta^2)^2} \frac{e^2}{m^2}(1-\beta^2) \right\} = \]

\[ = -\frac{2e^2}{3c^3}\cdot\frac{e^2}{m^2c}\cdot\frac{1}{1-\beta^2}[\mathbf v\mathbf H]^2. \]

This same expression may be rewritten in the form

\[ -r_0^2[\mathbf v\mathbf H]^2\left(\frac{\mathcal E}{mc^2}\right)^2, \tag{7} \]

bearing in mind that

\[ \mathcal{E}=\frac{mc^{2}}{\sqrt{1-\beta^{2}}} \quad\text{and}\quad r_{0}=\frac{e^{2}}{mc^{2}} \]

(the classical radius of the electron).

Formula (7), which determines the losses of an electron due to radiative braking in a magnetic field, was obtained by I. Ya. Pomeranchuk* in a work devoted to the motion of cosmic electrons in the Earth’s magnetic field (published in 1940). It can be used to the same extent for finding the radiation of a single electron when moving in an accelerator. The question of how the interaction of electrons in accelerators makes itself felt will be considered separately. Since in an accelerator (for example, to be definite—in a betatron) \(v \perp H\), and the motion takes place along a circle of radius \(R=\frac{\mathcal{E}}{eH}=\frac{c}{\omega}\) (the latter equality assumes that the velocity of the electron is close to the speed of light), from (7) we obtain the energy radiated per unit path,

\[ \mathcal{E}'=-\frac{2}{3}\,r_{0}^{2}H^{2}\left(\frac{\mathcal{E}}{mc^{2}}\right)^{2}. \tag{8} \]

For the radiation in one revolution we obtain

\[ \mathcal{E}''=-\frac{4\pi}{3}\,\frac{e^{2}}{R}\left(\frac{\mathcal{E}}{mc^{2}}\right)^{4}. \tag{9} \]

From formula (8) it is evident that the radiation of a particle (up to now we have spoken of the electron only for definiteness) per unit path is inversely proportional to the square of the particle mass. From formula (9) it is evident that the radiation in one revolution is proportional to the fourth power of the energy of the accelerated particle.

Let us estimate the maximum energy of an accelerated particle attainable in a betatron. It is evidently determined by the condition that the energy acquired from the accelerator in one revolution is wholly spent on radiation. The energy which a particle acquires in a betatron in one revolution is equal to

\[ e\oint \mathbf{E}\cdot d\mathbf{s}=e\cdot 2\pi R\cdot E. \tag{10} \]

On the other hand, according to Maxwell’s equations,

\[ \oint \mathbf{E}\cdot d\mathbf{s}=-\frac{1}{c}\,\frac{d\Phi}{dt}, \]

where \(\Phi=\pi R^{2}\overline{H}\) is the magnetic flux (\(\overline{H}\) is the mean value of the magnetic-field intensity over the area of the circle enclosed by the orbit). Since the condition for the existence of a stationary orbit is the requirement \(\overline{H}=2H_{0}\), where \(H_{0}\) is the magnetic-field intensity on the orbit \(^{11,2,3}\), (10) can be rewritten in the form

\[ \oint \mathbf{E}\cdot d\mathbf{s}=-\frac{e}{c}\,\pi R^{2}\dot{2H}_{0}, \]

whence, for the loss per unit path, we obtain the expression

\[ \mathcal{E}'=-\frac{e}{c}\,R|\dot H|. \tag{11} \]

Equating expressions (8) and (11),

\[ \frac{2}{3}\,r_0^2 H^2\left(\frac{\mathcal{E}}{mc^2}\right)^2 = \frac{e}{c}\,R|\dot H|, \]

we obtain, for the critical value of the energy \(\mathcal{E}_c\), the formula

\[ \mathcal{E}_c=mc^2\left(\frac{3eR}{2r_0^2c}\,\frac{\dot H}{H^2}\right)^{1/2}. \tag{12} \]

It follows from this that the limiting energy of electrons in a betatron increases with the rate of change of the magnetic field; for a given value of \(H\), the quantity \(\mathcal{E}_c\) is proportional to the square root of the energy acquired per unit path.

Formula (12) was obtained in 1944 by Ivanenko and Pomeranchuk\(^6\). As the authors indicate, under reasonable assumptions concerning \(R\) and \(H\), values of the order of 500 MeV are obtained for the energy \(\mathcal{E}_c\). This energy, apparently, is the maximum attainable energy for electrons accelerated in a betatron (for a refinement see \(^8\)).

It is of considerable interest to find the angular distribution and the spectrum of the radiation of an electron in a betatron. These questions are considered in the work of Artsimovich and Pomeranchuk\(^8\) and in the review by Schiff\(^9\), which was based on Schwinger’s work\(^ {10}\). The results of these works are, in the main, in agreement with one another. We shall present them, following Schiff.

The motion of an electron in a betatron is periodic. Therefore one should expect the radiation spectrum to contain harmonics that are multiples of the fundamental frequency of revolution of the electron in the orbit

\[ \omega_0=\frac{eHc}{\mathcal{E}}. \]

For the total energy going into the radiation of the \(n\)-th harmonic, we have the formula

\[ w_n=\frac{\omega_0 e^2 n}{R} \left\{ 2\beta^2 J'_{2n}(2n\beta) - (1-\beta^2)\int_0^{2n\beta} J_{2n}(x)\,dx \right\}, \tag{13} \]

where \(J_{2n}\) is the Bessel function of the first kind of index \(2n\), and \(J'_{2n}\) is its derivative. Expression (13) was obtained as early as 1911 by Schott\(^ {14}\), who, purely academically and without any application to accelerator theory, considered a number of problems relating to the radiation of moving charges, in particular the radiation of charged particles moving in a circle. Formula (13) implicitly contains the following results, obtained recently by Artsimovich and Pomeranchuk\(^8\) and by Schwinger\(^ {10}\): \(w_n\) grows as \(n^{1/3}\), with increasing \(n\) up to val—

…of order \(n\), \(\left(\dfrac{\mathcal E}{mc^2}\right)^3\), and then decreases exponentially (see Fig. 1):

\[ \left. \begin{aligned} w_n &\simeq 0.518\,\frac{\omega_0 e^2}{R}\, n^{1/3}, &&\text{for } 1 \ll n < n_0,\\ w_n &\simeq e^{-n/n_0}, &&\text{for } n > n_0,\\ n_0 &\simeq \left(\frac{\mathcal E}{mc^2}\right)^3 . \end{aligned} \right\} \tag{14} \]

From formulas (14) it is seen that the principal part of the radiation falls on high harmonics \(\left[n \sim \left(\dfrac{\mathcal E}{mc^2}\right)^3\right]\); for example, for a betatron accelerating particles to energies of 100 MeV in a field \(H \sim 10^4\) oersteds, we have \(n_0 \sim 10^7\), whence it follows directly that a considerable part of the radiation falls in the visible spectrum. Hence one may conclude that the radiation in question can also be observed visually.

Fig. 1. Dependence of the energy falling on the \(n\)-th harmonic on the number of the harmonic.

Fig. 2. Coordinate system used in formulas (14)—(16).

A few words must be said about the angular distribution of the radiation. If one introduces polar angles \(\xi\) and \(\eta\), such that \(\xi=0\) is the direction of the instantaneous velocity of the electron, and \(\eta=0\) is the plane of the orbit (see Fig. 2), then the angular distribution of the radiation will be determined by the formula

\[ \frac{\omega e^2 \beta^3}{4\pi R} \left[ \frac{1}{(1-\beta\cos\xi)^3} - \frac{(1-\beta^2)\sin^2\xi\cos^2\eta}{(1-\beta\cos\xi)^5} \right]. \tag{15} \]

One may verify that the integral of this expression over the sphere gives the value of the total radiated energy corresponding to (7). From formula (15) it is immediately clear that for \(\mathcal E \gg mc^2\) and, consequently, \(\beta \simeq 1\), the radiation is concentrated in a solid angle whose axis is the instantaneous direction of motion, with angular aperture of order \(\dfrac{mc^2}{\mathcal E}\) radians. The angular distribution of the radiation, at…

incident on one radian, averaged over the entire circumference, has the form

\[ \left(\frac{\omega e^2 \beta^3}{8\pi K}\right) \frac{\left[1+\cos^2\vartheta-\frac{\beta^2}{4}(1+3^2)\sin^4\vartheta\right]} {(1-\beta^2\sin^2\vartheta)^{7/2}}, \tag{16} \]

where \(\vartheta\) is the polar angle, chosen so that \(\vartheta=0\) perpendicular to the plane of the orbit. Under the condition \(\beta \ll 1\) we have an angular distribution determined chiefly by the usual factor \(1+\cos^2\vartheta\). Thus, the radiation of a relativistic \((\mathcal E \gg mc^2)\) single electron is concentrated mainly near the plane of the orbit, within a small solid angle embracing the direction of the electron’s motion (the electron radiating forward in the direction of motion). Since the occurrence of overtones is connected with the nonuniformity of the electron’s radiation for a given observation point at different positions of it on the orbit, the greatest number of harmonics is also concentrated in the plane of the orbit. The intensity of the radiation falling on the \(n\)-th harmonic rapidly decreases with distance from the plane of the orbit. In the direction perpendicular to the plane of the orbit, only the fundamental frequency \(\omega_0\), equal to the frequency of revolution of the electron in the orbit, is emitted.

It is also easy to determine the direction of polarization of the radiation of an electron moving in a circle. Observing the motion of the electron in the plane of its orbit, we see only its oscillations perpendicular to the direction of observation, i.e., a dipole lying in the plane of the orbit. Consequently, the radiation is polarized in the plane of motion of the electron.

3. RADIATION OF A SYSTEM OF ELECTRONS IN A BETATRON

Up to now we have considered the radiation of a single electron moving in a magnetic field. In all accelerator installations, including the betatron, we always deal with an aggregate of interacting particles. Therefore the questions quite naturally arise: how does the interaction of electrons affect their total radiation, and in what cases is the consideration applying to one electron applicable? Let us note here that it is not immediately obvious that the electron beam in a betatron will radiate, since a constant circular current does not radiate.

We shall briefly present the results of considering this question\(^{6,8,1}\). The interference effect of the radiation of \(N\) individual electrons is taken into account in the usual way: the expression for the energy falling on the \(n\)-th harmonic of the spectrum is multiplied by a factor of the form

\[ F=\left|\sum_{s=1}^{N}\exp(in\varphi_s)\right|^2, \tag{17} \]

where \(\varphi_s\) is the angular coordinate of the \(s\)-th electron. The summation is over all electrons. The factor (17) can lead to three essentially different cases.

Case 1. The particles are distributed along the circumference at equal angular distances from one another \(\left(\frac{2\pi}{N}\right)\). In this case the factor \(F\) will be equal to zero for all harmonics for which \(\frac{n}{N}\) is not an integer; this follows directly from the rule for adding vectors of complex numbers. In the case where \(\frac{n}{N}\) is an integer, the factor \(F\) is equal to \(N^2\), since each of the terms \(\exp(in\varphi_s)\) in this case is equal to unity. Thus, in the case under consideration one can reduce the radiation if one makes \(N>n_0\) (let us recall that \(n_0\) denotes the harmonic after which the exponential decrease of the radiation energy begins). We note, however, that the case \(N>n_0\) corresponds to an enormous current density in the electron beam and can hardly be realized. Moreover, it is difficult to imagine specific physical conditions under which such a uniform distribution of electrons along the circumference could exist.

Passing in formula (17) to the limit \(N\to\infty\), which practically corresponds to the case of a steady current, we arrive at the conclusion that a steady circular current does not radiate, since

\[ \int_0^{2\pi}\exp(in\varphi)\,d\varphi=0. \]

We have already noted the importance of this circumstance.

Case 2. The particles are distributed chaotically inside a beam which subtends an angle \(\varphi\) radians on the orbit, corresponding to the motion of a bunch of electrons. In this case the factor (17) will have the form

\[ F\sim N+(N^2-N)f(n\varphi), \tag{18} \]

where \(f(0)=1\) and \(f(x)\) decreases for \(x>1\), the character of the decrease depending on the mean density \(\rho\) of the electrons in the beam. The angular distance \(\varphi\) is defined as the distance over which the density of the electron bunch falls by a factor of \(e\), it being assumed that the density of electrons in the bunch varies according to a Gaussian law. Such a distribution of electrons on the orbit in the form of a bunch occurs in a synchrotron.

It follows from (18) that the large energy losses associated with the existence of a bunch are found in harmonics \(n \lesssim \frac{1}{\varphi}\) (since \(x=n\varphi \lesssim 1\)) and correspond to wavelengths \(\lambda \gtrsim\) the dimensions of the bunch.

Case 3. The electrons are chaotically distributed along the entire circumference. It is easy to see that, owing to the independence of the angular coordinates \(\varphi_s\) of the individual electrons from one another, the mixed terms in expression (17) vanish, and we obtain that the factor under consideration is

\[ F = N. \]

The physical picture corresponding to this case is as follows. In the electron beam there occur density fluctuations \(\Delta \rho\). Moreover, if the current density in the beam is small, then the fluctuations \(\Delta \rho\) will also be small, and therefore they may be regarded as independent (since they create small fields). For independent fluctuations the Poisson law, as applied to random quantities, gives

\[ \Delta \rho \sim \sqrt{N}, \]

where \(\Delta \rho\) is the fluctuation of the electron density in the beam, and \(N\) is the concentration of electrons in the beam.

The radiation of an individual fluctuation will be proportional to \((\Delta \rho)^2 \sim N\). For the complete beam, summing over all fluctuations, we likewise obtain that the radiation due to fluctuations is \(\sim N\), which coincides with the result given above. Thus it is precisely the presence of fluctuations, together with the attainment of high electron energies, that accounts for the radiation of electrons in the betatron *).

An approximate estimate of the limits of applicability of the representation of noninteracting electrons in the betatron is given in their article by Artsimovich and Pomeranchuk \(^{8}\). It is evident that the electrons may be regarded as noninteracting, and the fluctuations of their density as independent (i.e., following the Poisson law), if the maximum potentials corresponding to the fluctuations are small in comparison with the differences of the kinetic energies of the electrons moving in the orbit. The spread in kinetic energies is caused by two factors—the non-simultaneity of the release of electrons into the working space and the collisions of electrons with gas molecules (ionization losses). Comparing the magnitudes of the fluctuational potentials (which, according to Artsimovich and Pomeranchuk, amount to about 60 eV) and the magnitudes of the spread of electron velocities (the different causes producing the spread each give, on the average, about \(10^3\) eV), the authors come to the conclusion that the influence of electron interaction on their radiation in the betatron may be neglected.

4. THE INFLUENCE OF RADIATION ON THE OPERATION OF VARIOUS ACCELERATORS

1. Betatron \(^{11,12,13}\). As was indicated above, in the betatron the total loss of energy to radiation is proportional to the number of electrons in the beam. Calculations show that already at electron energies—

\[ \text{*) The role played by charge-density fluctuations in the radiation of an electron ring was noted in a discussion by D. Ivanenko and Ya. Terletsky in 1945 (at the defense of the latter’s dissertation).} \]

... of order 100 MeV one should expect a reduction of the radius of the stationary orbit. For energies \(\mathcal{E}\) of order 300 MeV (frequency of variation of the magnetic field \(\frac{\omega}{2\pi}=60\) cps, radius of the stationary orbit \(R=2\) m, field on the orbit \(H_{\max}=5000\) gauss) the losses amount to \(4.7\%\) of the total energy of the electron, if the field varies harmonically. In this case the greater part of the energy falls at wavelengths of order \(2\pi R\left(\frac{mc^{2}}{\mathcal{E}}\right)^{3}\), which corresponds to approximately 600 Å, i.e. lies in the region of the far ultraviolet.

It follows from this that the energy losses cannot be appreciably reduced by shielding. (By shielding is meant the use of a well-conducting metallic surface on which surface currents induced by the electron current arise, the radiation of which compensates the radiation of the main current.) Compensation is evidently possible only in the case of coherence of these currents, which imposes the condition on the wavelengths \(\lambda > R\).

Let us note that in a detailed review concerning the work and construction of betatrons in Germany \(^{20}\), it is indicated that, in view of the radiation of electrons in the betatron, the designers built a special model for investigating such radiation before building a 200 MeV betatron (the latter was never completed because of Germany’s defeat in the war).

1. Synchrotron \(^{16,17}\). In the synchrotron, electrons are injected into the apparatus in separate portions, each of which corresponds to the beginning of a period of increase of the magnetic field. In it the electrons move along a circular orbit in the form of a bunch of finite extent. Although at first sight it seems that in this case coherent radiation should occur and that the intensity should be proportional to the square of the number of electrons in the bunch, in fact calculations show that the maximum radiation falls at wavelengths so short (in comparison with the length of the bunch) that this radiation is again incoherent.

For bunches having an extent of the order of tens of degrees, the main part of the radiation spectrum has the same intensity as the radiation of a complete ring of electrons chaotically distributed around the circumference (let us recall that in this case the intensity is proportional to the total number of electrons). Coherent radiation falls at wavelengths of the order of the linear dimensions of the bunch. The coherent radiation per electron is proportional to the number of electrons and does not depend on the energy of the electron under the assumption \(\mathcal{E}\gg mc^{2}\). In the absence of shielding this energy is given by the formula

\[ 1.4\,\omega_{0}T\left(\frac{e^{2}}{R}\right)\left(\frac{N}{\varphi^{4/3}}\right), \tag{19} \]

where \(T\) is the electron acceleration time, \(\varphi\) is the angular extent of the bunch in radians. In equation (19) a Gaussian distribution of the density is assumed, and \(\varphi\) is measured between the points at which the density is \(1/e\) of the maximum density.

For \(\mathscr{E}\) of the order of 300 MeV, \(N \sim 10^{11}\), and \(\varphi = 0.2\) (\(\sim 12^\circ\)), the losses due to the coherent radiation of the bunch are approximately forty times greater than the losses associated with the radiation of a complete ring of chaotically distributed electrons. However, by shielding the electron beam this radiation can be considerably reduced.

In the example given above, the losses of the bunch to coherent radiation can, by means of shielding, be reduced to one tenth of the losses to radiation of a complete ring of electrons (in the absence of shielding).

In comparison with the betatron, the synchrotron has the advantage that part of the radiation losses is compensated by the automatic focusing of the electrons during motion along the orbit, which possesses phase stability. The only requirement that must be satisfied here is that the losses of the radiating electron during one period of revolution be small in comparison with the energy acquired in passing through the accelerating electric gaps. Schiff suggests that, with the aid of a synchrotron, it will be possible to attain electron energies of the order of \(10^3\) MeV.

1. Microtron\(^{18}\). The microtron is an accelerating installation consisting of an endovibrator and a magnetic field. Accelerated by the alternating potential on the endovibrator, the electrons are bent by the magnetic field along circles of increasing radii. For normal operation of the microtron it is likewise necessary that the radiation losses per revolution be smaller than the voltage on the dees. Thus here too the radiation sets a limit to the attainable energies, of approximately the same order as for the synchrotron.

2. Linear resonant accelerator\(^{5}\). A linear resonant accelerator is a system of a large number of oscillatory circuits (resonators) possessing a high quality factor, i.e. small damping. Electrons moving along such a system of resonators each time enter the accelerating phase of the corresponding resonator, which leads to an increase in their energy. While being accelerated, the electrons radiate. However, it can be shown that, when charges move in a straight line in an accelerating electric field, the ratio of the radiated energy to the acquired energy is vanishingly small. Therefore radiation losses in a linear resonant accelerator do not limit the energies attainable with its aid. Let us note, however, that to attain high energies it is necessary to create an accel-

...accelerator of very considerable dimensions: to obtain electrons with an energy of 300 MeV the size of the apparatus must be \(\sim 450\) m.

For heavy particles the length of a linear resonance accelerator has a more modest value—of the order of tens of meters for obtaining particles with an energy of \(\sim 100\) MeV. In a linear accelerator of heavy particles the role of radiation is still smaller than in a linear accelerator for electrons, since the radiation is inversely proportional to the mass of the accelerated particle.

1. Linear waveguide accelerator. The idea of operation of a linear waveguide accelerator consists in accelerating a particle by means of an electromagnetic wave propagating through the system, whose phase velocity is close to the velocity of light. The radiation losses here will also be very small for the same reasons as in the preceding case. By means of a waveguide accelerator, it will apparently be possible to obtain electrons with energies of the order of \(10^3\) MeV.

2. Synchrocyclotron. A synchrocyclotron, or phasotron, is the name given to a cyclotron operating in a regime analogous to that of a synchrotron.

In a phasotron the frequency of the alternating voltage on the dees is modulated by a somewhat lower frequency, to which correspond the periods of injection into the apparatus of the particles to be accelerated. With the proper choice of the moment of injection, the motion of the particles possesses phase stability (just as occurs in the synchrotron), despite the relativistic change in the mass of the particles. This apparatus is designed for the acceleration of heavy particles (ions, \(\alpha\)-particles, deuterons, etc.). In this case, although the relativistic effect of the change of mass does have a noticeable influence, nevertheless the velocities of the particles accelerated in it differ noticeably from the velocity of light \(c\) (in the Berkeley phasotron, which accelerates \(\alpha\)-particles to an energy of 400 MeV, their maximum velocity is \(0.45\,c\)). Therefore the radiation of charged particles in this cyclotron may be calculated by nonrelativistic formulas. This radiation has no substantial significance for the operating regime of the apparatus, if one also takes into account that the radiation is inversely proportional to the mass of the particle, which for heavy particles is relatively very large.

5. EXPERIMENTALLY OBSERVED PHENOMENA ASSOCIATED WITH ELECTRON RADIATION. “THE GLOWING ELECTRON”

Already in the work of Blewett \(^{15}\), who worked with a betatron at 100 MeV, two of the following experimental facts were pointed out. First, a contraction of the orbit of the accelerated particles was observed, so that the particles struck the target earlier than should have occurred in the absence of radiation. Second, the above-mentioned contraction of the electron orbit was one and the same for different values

of the total current in the beam. This indicates the independence of the radiation of each electron from the number of electrons in the beam. Blewett emphasized that he did not observe the radiation itself. He investigated with particular care the microwave region of the radiation, which, according to his calculations, should have accounted for the principal part of the radiation in the instrument. However, as we have already indicated, at energies of \(\sim 100\) MeV the maximum of the electron radiation falls on high harmonics corresponding to the visible or even ultraviolet region of the spectrum. It is therefore quite natural that Blewett could not detect radiation in the microwave region, although he worked with very sensitive indicators, so that radiation of \(\sim 10^{-5}\) watt would already have been noted*).

In the summer of 1947 there appeared a short note by Pollock\(^5\) and his collaborators, “On the visually observed radiation of a beam of electrons in a 70 MeV synchrotron.”

The electron orbit of the synchrotron had a radius of 29.3 cm. The radiation was visible as a small bright spot of white color on the glass surface of the vacuum tube, if one looked in the plane of the orbit toward the approaching electron.

In a normally operating synchrotron, the electrons striking the target produce bremsstrahlung X-radiation with a power of 50 roentgens/min at a distance of 1 m. At this X-ray intensity the spot was very bright; however, even at a radiation intensity equal to 1 roentgen/min, at a distance of 1 m the spot could still be observed in daylight. When coils were switched on that disturbed the electron orbits so that the electrons struck the target before the magnetic field reached its maximum, the intensity of the observed glow increased sharply with increasing electron energy, if this energy exceeded 30 MeV. If, however, the electrons struck the target after the magnetic field had reached its maximum, then the intensity of the glow did not depend on the energy with which the electrons left the beam, but was determined by the maximum energy acquired by the electrons.

The visually observed effect disappeared if the electrons struck the target with an energy less than 30 MeV. If, by means of a special resonator switched on for a short time before the magnetic field reached its maximum, the electron beam was displaced to a radius smaller than the inner radius of the target, then subsequently, as the magnetic field increased, the electron orbit would expand. In this case, instead of a small spot, the observer sees a short line stretched out in the plane of the orbit. The emitted radiation is polarized; the electric vector lies in the plane of the or-

*) A possible reason why Blewett was unable to observe visible radiation was also the circumstance that in the 100 MeV betatron with which he worked the walls of the vacuum chamber were silvered\(^ {21}\).

could be. When the Nicol prism was rotated by 90° the visually observed glow disappeared.

The authors provide no information about the spectral composition of the radiation, promising soon to give a detailed account of the experiments described.

Fig. 3. Photograph of “glowing electrons.” The synchrotron chamber is visible, as well as a bright luminous spot representing the radiation of electrons^22.

Excellent photographs of the new effect of the “glowing electron” are given in a number of recent journal issues^19,21,22. We reproduce one of them.

6. CONCLUSION

Thus, the radiation of electrons in accelerator installations of the betatron type, first predicted by Soviet physicists, was very soon confirmed by a number of experiments, culminating in direct observation of the “glowing electron” by Pollock’s group. This phenomenon is a new physical effect, characteristic of electrons with energies on the order of 100 MeV and making it possible directly to “see” the electron.

The phenomenon of the “glowing electron” is of substantial importance for electron accelerators (betatron, synchrotron). One may say that the synchrotron

and the betatron are, first, generators of bremsstrahlung ($\gamma$-quanta) and, second, generators of ordinary light (“the glowing-electron effect”), independently of their use as generators of accelerated particles.

It is now difficult to say what application this effect will find in science.

In any case, it is already beyond doubt that an experimentally interesting new physical effect has been discovered; Soviet physicists played an outstanding role in the theoretical prediction of this effect.

References

1. Grobeck, Lawrence and others, Phys. Rev., 71, 449, 1947; Perlman, Phys. Rev., 72, 342, 1947.
2. Blewett, Phys. Rev., 69, 87, 1946.
3. Westendorp, Journ. Appl. Phys., 16, 657, 1946.
4. Stockley, General Electric Review, 50, 3, 1947.
5. Elder, Gurewitsch, Langmuir, Pollock, Phys. Rev., 71, 829, 1947.
6. D. Ivanenko and I. Pomeranchuk, DAN, 44, 316, 1944; Phys. Rev., 65, 343, 1944.
7. I. Pomeranchuk, Journal of Phys., USSR, 2, 66, 1940.
8. L. Artsimovich and I. Pomeranchuk, ZhETF, 16, 270, 1946.
9. L. Schiff, Rev. Sci. Instr., 17, 6, 1946.
10. Schwinger, Phys. Rev., 70, 798, 1946.
11. D. Kerst, Phys. Rev., 60, 47, 1941.
12. Kerst and Serber, Phys. Rev., 60, 53, 1941.
13. A. P. Grinberg, UFN, 27, 30, 1945.
14. Schott, Theory of Radiation. Cambr. Univ. Press, 1912.
15. V. Ginzburg, Izv. A. N., ser. phys. 11, 165, 1946.
16. V. Veksler, Journ. of Phys., USSR, 9, 153, 1945; DAN, 44, No. 9, 1944.
17. McMillan, Phys. Rev., 68, 143, 1945.
18. V. Veksler, DAN, 43, No. 8, 1944.
19. Electronics, Febr., 1947.
20. Kaiser, Journ. Applied. Phys., 18, 1, 1947.
21. Sci. News Letter, 51, 339, 1947.
22. Gen. Electr. Rev., 50, 3, 1947.