ELECTRON ACCELERATION

BY ELECTROMAGNETIC INDUCTION

(KERST’S BETATRON)

A. P. Grinberg

In 1940 the American physicist Donald Kerst built, at the University of Illinois (Urbana), an induction accelerator of electrons. The technical arsenal of physics was enriched by a remarkable new device, a new method. Kerst was the first successfully to put into practice the long-standing idea of using the phenomenon of electromagnetic induction to accelerate electrons, and this opened the way to the creation of installations in which the energy of electrons could be brought to extraordinarily large values.

The possibility of obtaining very fast electrons—and, moreover, by comparatively simple means—is of very great practical and scientific interest. Fast electrons find their widest application in the technique of generating X-rays. As is known, these rays arise when a beam of fast electrons is decelerated at the anticathode of an X-ray tube. X-ray technology at the present time deals with electrons of considerable velocities. Whereas in medical X-ray tubes the electrons are accelerated by a potential difference usually not exceeding 100–200 kV, in modern industrial X-ray tubes used for radiographing great thicknesses of metal (X-ray defectoscopy), the applied potential difference is much higher—it amounts to 1 MV and more. The X-rays arising in such tubes possess such hardness, i.e. such great penetrating power, that with their aid it is possible to examine steel plates up to 200 mm thick.

However, for technology—and to an even greater degree for science—the possibility of accelerating electrons to a still much higher energy would be very substantial. The study of the properties of electrons and of their interaction with matter could be extended to electrons of ultra-high energies. In addition, the possibility of having at one’s disposal a source of such electrons is of particular interest for the physics of the atomic nucleus, since with their aid, undoubtedly, it would be possible to carry out many interesting nuclear reactions and—what is still more important—it would be possible to attempt to reproduce under laboratory conditions such phenomena as, up to the present,

in time, could be observed only in the form of isolated cases in cosmic rays (the formation of “showers,” the birth of a meson). Kerst’s betatron is at present the only apparatus that makes it possible to generate electrons with record-high energy, leaving far behind everything that physicists have had at their disposal up to now.

In its fundamental importance as an instrument for scientific research, the betatron can undoubtedly be placed in the same rank as the cyclotron, and its appearance, just as the appearance of the cyclotron did in its time, opens a new era in the development of nuclear physics.

It must be noted that in many such cases as well, when a comparatively small electron energy is required, the betatron can successfully compete with the installations and methods hitherto used for obtaining fast electrons, possessing the advantages of portability and simplicity. It is highly probable that in the near future installations using the betatron method will completely displace the present ones.

PREVIOUS METHODS OF OBTAINING FAST ELECTRONS

All conceivable methods of accelerating electrons are ultimately based on one and the same principle—on the fact that an electron, being a particle possessing an electric charge, in an electric field experiences the action of a force which compels it to move in a definite direction and with a definite acceleration. The kinetic energy which a particle with a given magnitude of charge acquires due to the forces of the electric field is determined entirely by the potential difference traversed (of course, this refers to the motion of the particle in an ideal vacuum, where there is no expenditure of energy in collisions with other particles). As is well known, the kinetic energy of elementary particles is customarily expressed in electron-volts (eV)\(^1\).

In nature there exist natural sources of fast electrons—these are various radioactive substances possessing \(\beta\)-radioactivity. Thus, for example, the maximum energy of the \(\beta\)-particles of radium C (RaC) is about \(3.2\) MeV; the maximum energy of the \(\beta\)-particles of RaE is equal to \(1.2\) MeV. It was precisely with the aid of these natural sources that a number of properties of fast electrons were first studied in their time—their absorption in matter, bremsstrahlung, etc.

\(\beta\)-particles obtain their large energy at the expense of processes taking place in the radioactive nucleus. Subsequently, installations were created that made it possible to obtain “artificial \(\beta\)-particles,” i.e. electrons accelerated to energies of several million electron-

\(^1\) Let us recall that \(10^6\ \mathrm{eV} = 1\ \mathrm{MeV} = 1.6 \cdot 10^{-6}\) ergs.

volts by means of a correspondingly high potential difference. Such an apparatus is analogous to an ordinary X-ray apparatus (where the direct method of accelerating electrons is also used), differing from it chiefly in that it is designed for a much higher voltage. Such an apparatus consists of a special vacuum tube capable of withstanding high potential differences, and of a source of constant high voltage. At one end of the vacuum tube there is an “electron gun,” i.e., a source of electrons in the form of an incandescent filament, provided with the appropriate focusing electrodes, which gather the stream of electrons into a narrow beam. At the other end of the tube there is an anode, usually in the form of a ring. The electrons emerging from the gun, under the action of the electric field applied between it and the anode, move toward the anode; at the moment when they pass through the opening of the latter, they will obviously have a kinetic energy of \(E\) electron-volts, if a potential difference of \(E\) volts is applied between the anode and the cathode (i.e., the filament of the gun).

As a source of high voltage one uses either step-up transformers or special direct-voltage generators1. In recent years the electrostatic Van de Graaff generator, which has reached a high degree of technical perfection, has become the most widespread.

An installation for obtaining “artificial \(\beta\)-particles” by means of the direct method of accelerating electrons described above inevitably assumes very bulky forms, since, in connection with the use of a very high voltage, both the tube itself and the voltage generator must have correspondingly large dimensions. Let us indicate, for example, that a Van de Graaff-type generator built at the Ukrainian Physico-Technical Institute (Kharkov) and designed to produce voltages of only up to \(4\ \mathrm{MV}\), in outward appearance is a hollow metal sphere \(10\ \mathrm{m}\) in diameter, mounted on insulating columns \(8\ \mathrm{m}\) high.

With high-voltage generators it has already proved possible to obtain voltages up to \(10\ \mathrm{MV}\). Vacuum tubes for such a voltage, however, have not been built, and the greatest energy to which it has been possible to accelerate electrons has not exceeded \(4\text{—}5\ \mathrm{MeV}\). Such electrons are capable of causing the disintegration of certain atomic nuclei. It should be noted that in tubes one can obtain electron beams incomparably more intense than those delivered by radioactive substances. The intensity of the electron beam in a tube can easily reach a value equivalent to the \(\beta\)-radiation from \(20\ \mathrm{kg}\) of radium.

But ten million volts is apparently that practical limit above which modern technology for generating direct—

high voltage cannot be substantially raised, chiefly because this would require disproportionately large dimensions of the entire apparatus. Therefore the designer’s thought turned to the idea of accelerating electrons by means of some indirect method. Indeed, is it really necessary, in order to accelerate electrons to an energy of, say, \(5\ \mathrm{MeV}\), to have at one’s disposal a full voltage of 5 million volts? Could one not take a comparatively low voltage, apply it many times along the path of motion of the electron, and in this way accelerate the electron to high speeds? As is known, the method of multiple acceleration was successfully used in the so-called linear synchronous accelerator of ions. The first model of such an installation was built in 1927 by Widerøe \(^{3}\). The scheme of the linear accelerator is as follows. Ions from the corresponding source fly in vacuum successively through a straight-line series of cylindrical electrodes of increasing length. The odd-numbered electrodes in this series are connected to one pole of a high-frequency generator, and the even-numbered ones to the other. The lengths of the electrodes and the frequency of the generator are chosen in such a way that, in the gaps between the electrodes, an electric field of the required direction is obtained precisely at those moments when the ions pass through these gaps. For such synchronization it is evidently necessary that the time during which an ion flies inside a given cylinder from one of its ends to the other be equal to the period of oscillation of the voltage from the generator; thus it is necessary to satisfy the condition:

\[ l_n v_n = T = \text{const.}, \]

where \(l_n\) is the length of the \(n\)-th cylinder, \(v_n\) is the velocity of the ion on this part of its path. To obtain ions with sufficiently high energy one must arrange a large number of accelerating gaps, i.e. take a large number of cylinders, and the total length of the whole vacuum tube becomes very large. Therefore the method of the linear synchronous accelerator is practically suitable only for accelerating very heavy ions, whose velocity, at a given kinetic energy, is less than that of light ones, so that the required length of the tube is correspondingly reduced. Sloan and Coats \(^{4}\) obtained mercury ions with an energy of \(2.85\ \mathrm{MeV}\) in a tube containing 36 electrodes and having a length of \(1.85\ \mathrm{m}\). To accelerate protons to the same energy and at the same frequency \((\lambda = 29.8\ \mathrm{m},\ f \sim 10^{7}\ \mathrm{Hz})\), a tube \(25\ \mathrm{m}\) long would be required. Since a substantial increase of the frequency above \(10^{7}\ \mathrm{Hz}\) in such an installation is already practically impossible, it is clear that for electrons, because of the necessary gigantic length of the tube, this method of acceleration is entirely unsuitable.

There exists another variant of synchronous acceleration with rectilinear motion of the particles, differing in that the accelerating voltage is applied to the gaps between the cylinders not from a high-frequency generator, but from two wires forming an oscillatory system in which traveling waves are produced (the principle of the method

proposed by Ising5 in 1925). Cylindrical electrodes are connected to the corresponding series of points on the indicated wires, and the traveling wave carries its potential successively to all the accelerating gaps. Since the propagation velocity of the traveling wave is practically equal to the velocity of light, this acceleration method is also suitable for electrons, without requiring an excessively long tube. In 1934, Beams and Trotter6 obtained, in an apparatus of this type, electrons with an energy of 1.3 MeV, with a total tube length of about 2.6 m. A major drawback of this method is that only extremely small intensities of the electron beam can be obtained (we shall not dwell here on the reasons for this). Therefore experiments with accelerators of this type were abandoned.

The idea of creating an apparatus for accelerating electrons that would use the principle of multiple acceleration nevertheless continued invariably to attract the attention of physicists.7

In the best way, and with the greatest success, the method of synchronous acceleration of charged particles has been applied in the cyclotron. In this extraordinarily ingenious apparatus, first built by Lawrence and Livingston8 in 1932, instead of rectilinear motion of the accelerated particles, their motion takes place along a plane spiral, produced by means of a magnetic field. The method of synchronous acceleration described above is applied to these particles; moreover, instead of a large number of accelerating gaps, only one gap is required between two electrodes (dees), arranged along the diameter of the spiral trajectory of the particles, which thus, on their path, pass through this gap many times (twice per revolution).

With the aid of the cyclotron it has been possible to obtain powerful beams of heavy charged particles—protons, deuterons, and $\alpha$-particles (i.e., ions of hydrogen, deuterium, and helium)—with very high energies, at present reaching up to 32 MeV.

Naturally the question arises: can this very same apparatus, or at least this same method of particle acceleration, be applied also to the acceleration of electrons? Unfortunately, this question must be answered in the negative.

UNSUITABILITY OF THE CYCLOTRON FOR ACCELERATING ELECTRONS

The most important relation, constituting the basic principle of operation of the cyclotron, is that a charged particle having charge $\varepsilon$, mass $m$, and linear velocity $v$, in a transverse homogeneous magnetic field, moves in a circle (whose radius depends on the magnitude of $v$ and on the magnetic-field strength $H$), describing one revolution in the time

\[ T=\frac{2\pi mc}{\varepsilon H} \tag{1} \]

($c$ is the speed of light). What is essential in this expression is that the magni-

the quantity \(T\) does not depend on the velocity of the particle; consequently, if for one reason or another the velocity of the particle changes—for example, becomes greater and greater—this leads only to a corresponding increase in the radius of its trajectory in the given magnetic field, while the quantity \(T\) remains unchanged. It is precisely as a result of this that it becomes possible to use, for repeated acceleration of ions, a high-frequency alternating voltage with period \(\tau = T\); this is what is realized in the cyclotron. However, the right-hand side of equality (1) may be regarded as a constant quantity only so long as the mass \(m\) of the particle may permissibly be treated as a constant. In other words, \(T = \mathrm{const.}\) only in the region of nonrelativistic particle velocities \((v \ll c)\). But this means that, with the aid of a cyclotron, particles can be accelerated not to any velocity whatever, but only to a certain limiting velocity. No further increase of the velocity beyond this latter one occurs, since the mass of the particle \(m\) begins appreciably to exceed the rest mass of the particle \(m_0\) (according to the relativistic formula:

\[ m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}}, \]

and in proportion to the mass the time \(T\) also increases; by virtue of this, if at the beginning of the acceleration process the frequency \(\tau\) of the alternating voltage applied to the accelerating gap in the cyclotron was chosen in such a way that the condition \(\tau = T\) was satisfied, then after acceleration to some limiting velocity there results so appreciable an excess of \(T\) over \(\tau\) that the particle “falls out of synchronism” and, as a result, begins to slow down in the cyclotron instead of accelerating.

The condition of constancy of mass in the case of heavy particles, usually accelerated in the cyclotron—namely, protons, deuterons, and \(\alpha\)-particles—is practically satisfied up to very high energies of these ions; with the aid of a cyclotron they can be accelerated to 50–100 MeV. But the situation is entirely different in the case of electrons. The variability of the electron mass begins to have a substantial effect already at energies of the order of 25 keV. Consequently, there can be no question, for example, of accelerating electrons in a cyclotron to energies of several million electron-volts.

The circumstance that the region of relativistic velocities for electrons begins at much smaller values of kinetic energy than for heavy particles is, of course, nothing surprising. It is due to the enormous difference between the masses of the electron and the ion. Suppose, for example, that an electron and a proton have passed through a potential difference of 2 MV. Both particles will acquire the same kinetic energy of 2 MeV (since the charge of the proton is equal in magnitude to the charge of the electron). But whereas the velocity of an electron with a kinetic energy of 2 MeV is 98% of the speed of light, the velocity of a proton possessing the same kinetic energy is only 6.5% of the speed of light. The mass of such a proton exceeds by only 0.2% ...

exceeds the mass of a proton at rest, whereas the mass of an electron with an energy of 2 MeV is approximately 5 times greater than the rest mass of the electron.

It is thus quite obvious that the cyclotron—in any case in its present form—cannot serve as a generator of very fast electrons. Several modifications of the cyclotron can be proposed, whose realization would make it possible to use it for accelerating electrons. One such modification, for example, might consist in feeding to the dees of the cyclotron electric oscillations whose frequency changes periodically (frequency modulation), following the change in the quantity \(T\) during the acceleration process. Another variant, recently proposed by V. I. Veksler\(^9\), consists in greatly increasing both the frequency of the voltage applied to the dees and its amplitude. All such modifications of the cyclotron in principle solve the problem of electron acceleration, but their practical realization is apparently connected with serious technical difficulties.

ACCELERATION OF ELECTRONS BY MEANS OF ELECTROMAGNETIC INDUCTION

There is, however, yet another method of accelerating charged particles—the induction method. Its idea has long been known. During the last two decades, both theoretical proposals and attempts at the experimental realization of an induction apparatus for accelerating electrons have repeatedly been made. The most substantial calculations, accompanied also by experiment, belong to R. Wideröe\(^5\) and date from 1927. The results of these and of other experiments, which we shall discuss in more detail below, were completely negative, chiefly because of an insufficient theoretical development of the question of the necessary operating conditions of such an apparatus. In 1935 those conclusions on this question which were lacking in Wideröe’s theory were given by Steenbeck\(^ {10}\) in the text of a patent which, in addition, contained a number of valuable design proposals. Unfortunately, this information did not then become at all widely known. The experiments carried out by Steenbeck gave encouraging results, but he did not proceed beyond the preliminary stage.

Thus D. Kerst, who in 1940 built his induction accelerator\(^ {11}\), was the first physicist who succeeded brilliantly in solving in practice the problem of accelerating electrons by an indirect method, i.e. without the use of high voltage, and in particular—the problem of the induction acceleration of electrons.

The idea of the induction method of accelerating charged particles is as follows. Any variable magnetic flux induces in the surrounding space an electric field. This is a special field, a field of vortex type; its lines of force, going around the inducing-

of the magnetic flux are closed lines. In particular, if the magnetic flux possesses axial symmetry in the distribution of the magnetic-field intensity, then the lines of force of the induced electric field will be concentric circles whose plane is perpendicular to the central axis of the magnetic field, and whose center coincides with its trace on this plane.

The presence of an electric field can be used to accelerate particles possessing electric charge; and the circumstance that the lines of force of the induced electric field are closed leads to extremely important consequences. In an electric field of vortex type the potential is not a single-valued function of the coordinate; therefore, if an electron is made to describe in such a field some closed path, then the energy which the electron thereby acquires at the expense of the forces of the field need not be equal to zero, as would be the case for a constant electrostatic field. One can be convinced of this especially clearly if one imagines the motion of an electron along a closed path coinciding with one of the lines of force of the induced electric field. It is obvious that in this case, over the whole course of its motion, a force will act on the electron whose direction invariably coincides with the direction of motion of the electron, and thanks to this, on returning to the initial point, it will accordingly possess a greater velocity (i.e. greater kinetic energy) than at the beginning of its path. The gain in energy in one such revolution may be quite small, but if this process is repeated many times, then, obviously, a very large energy can be imparted to the electron.

Let us make a small calculation which will show approximately what figures may be spoken of here. The intensity \(E_r\) of the electric field induced by a magnetic flux with axial symmetry is a function of the distance \(r\) of the given point from the axis and of the time derivative of the magnetic flux \(\Phi_r\). The potential difference corresponding to one circuit around the central axis of the magnetic field along a circle of radius \(r\) will be

\[ \Delta U_r = 2\pi r \cdot E_r . \]

Suppose that over the course of \(1/1000\) sec. the magnetic flux changes uniformly \(\left(\dfrac{d\Phi_r}{dt}=A=\mathrm{const.}\right)\), and that the rate of its change \(A\) is such that the quantity \(\Delta U_r\) is equal to \(20\ \mathrm{V}\) at \(r=5\ \mathrm{cm}\). This means that if, in such a magnetic field, one coaxially places one turn of wire, taking its diameter to be \(10\ \mathrm{cm}\), then an emf of \(20\ \mathrm{V}\) will be induced in it.

Let us now assume that, instead of a turn of wire, we place in the same magnetic field a vacuum chamber, for example in the form of a flat box, in which at a distance of \(5\ \mathrm{cm}\) from the central axis there is initially a stationary free electron. Under the action of the induced electric field this electron will begin to move-

—first along the line of force of the electric field passing through the given point, and then, owing to inertia and other causes, along some trajectory that by no means coincides with this line. But let us suppose that a method has been found to make an electron, moving in an ideal vacuum under the action of the force of the vortex electric field, not leave the line of force of this field. What energy will the electron acquire under such conditions in \(1/1000\) sec.? This case represents the motion of a body under the action of a constant force, but with a continuous increase in the mass of the body in accordance with the increase of its velocity. An exact relativistic calculation shows that, in an electric field of the indicated magnitude, the length of the electron’s path in \(1/1000\) sec. will be 290 km, i.e., during this time it will make 925,000 revolutions around the central axis of the magnetic field. (So that this result may not seem surprising, let us recall that, because of the smallness of its mass, an electron already acquires an enormous velocity over even a small potential difference traversed by it. In our example, after the first revolution the electron acquires a kinetic energy of 20 eV, which corresponds to an electron velocity equal to \(2.67\cdot 10^8\) cm/sec.) Since on each revolution the electron in our case acquires, at the expense of the force of the induced electric field, an energy of 20 eV, independently of its mass, consequently, in the end its kinetic energy will be equal to \(20\cdot 925\,000 = 1.85\cdot 10^7\) eV, or 18.5 MeV.

The example given clearly proves that with the aid of electromagnetic induction it is indeed possible to impart extraordinarily large energies to electrons.

However, the idea by itself—of accelerating electrons by making them move along the lines of force of the induced electric field—would have no practical value if it were not supplemented by an indication of the method by which it is actually possible to keep the moving electron on a line of force of the vortex electric field, on an orbit closed around the magnetic flux\(^1\).

The most important result obtained by Widerøe in the theoretical analysis of the question of accelerating electrons with the aid of electromagnetic induction consists precisely in the fact that he discovered the remarkable possibility of automatic control of the motion of electrons in an induction accelerator. This possibility is as follows. It is known that in a homogeneous magnetic field the trajectory of an electron is a circle, the plane of which is perpendicular to the magnetic field (we are speaking of the case when the electron initially had a velocity \(v\), perpendicular to the magnetic field), and its radius

\(^1\) Moving in a vortex electric field, electrons can, of course, be accelerated also in the case when their trajectory is not a closed line (for example, a plane spiral). However, such a variant of an induction accelerator, as is easy to show, is in many respects worse than the variant with circular motion of the electron.

\(\rho\) depends on the magnitude \(v\) and on the intensity of the magnetic field \(H\) in the following way:

\[ \rho=\frac{mvc}{eH}, \tag{2} \]

where \(e\) is the charge of the electron, \(m\) its mass (which, in turn, depends on the velocity \(v\)). Thus, with the aid of a magnetic field one can force an electron to move along a closed path—along a circle. But precisely this, as we have seen, was the problem that confronted the designer of an induction electron accelerator. Consequently, its solution may consist in applying in the apparatus a magnetic field selected in an appropriate manner. And here an enticing idea arose: could one not use the very magnetic field that necessarily must be created in the apparatus in order to induce the electric field also as the controlling magnetic field that will keep the moving electrons all the time on one and the same closed circular orbit? The intensity \(H\) of the inducing magnetic field must vary continuously with time (otherwise \(\frac{d\Phi}{dt}=0\), and there will be no induction), and if one were dealing with an electron whose velocity is unchanging, then the trajectory of its motion in such a field, as formula (2) shows, would not be a circle, but a curve with a gradually changing value of \(\rho\), i.e. a slowly winding inward (for \(\frac{dH}{dt}>0\)) or unwinding (for \(\frac{dH}{dt}<0\)) plane spiral. As a result, after making a comparatively small number of revolutions in the vacuum chamber, the electrons would strike its walls. Since, however, during its motion in the chamber the electron continuously increases its velocity owing to the action of the induced electric field, in the magnetic field it will tend to move along an arc of a circle of ever less steep curvature. Thus, in the case of an increasing magnetic field and a simultaneous growth of the electron velocity, factors will appear that act in opposite directions: one toward a continuous decrease of the orbit radius, the other toward a continuous increase of it; and these two tendencies will always, to some extent, compensate each other. After what has been said, it will no longer seem so surprising that, as it turns out, complete balancing of these two tendencies can be achieved—conditions can be chosen under which an electron, while in an increasing magnetic field, will move along a certain definite circle of unchanged radius, continuously gaining speed. Wideroe showed as early as 1927\(^3\) what condition such an “equilibrium orbit” must satisfy. We shall give here a simple derivation of this interesting condition.

The radius of curvature \(\rho\) of the trajectory of an electron moving in a magnetic field is determined by formula (2) given above. It follows

bear in mind that in this formula \(H\) is the magnetic-field intensity at the place where the electron is located; the quantity \(\rho\) is completely independent, for example, of what the magnetic-field intensity is at the center of the electron’s circular orbit.

It follows from formula (2) that the momentum of the electron \(p(=mv)\) and the quantity \(H\rho\) corresponding to this momentum when the electron moves in a magnetic field are related by

\[ p=\frac{e}{c}H\rho \tag{3} \]

(we note in passing that this relation makes it possible to use the quantity \(H\rho\) as a convenient measure, in many cases, of the kinetic energy of the electrons).

If the problem is posed of ensuring that, despite the increase of the magnetic field \(H\), the radius of the trajectory \(\rho\) remains constant, equal, say, to \(r_0\), then, as formula (3) shows, it is necessary to make the momentum of the electron \(p\) change exactly in proportion to the change of \(H\). The momentum of the electron, however, is determined by that force of the induced electric field which accelerates the electron. In the case of an axially symmetric magnetic field, the intensity \(E\) of the induced electric field for points at a distance \(r_0\) from the central axis of the magnetic field will be equal to

\[ E=-\frac{1}{2\pi r_0 c}\frac{d\Phi}{dt}, \tag{4} \]

where \(\Phi\) is the magnetic flux through the entire area enclosed by a circle of radius \(r_0\). Thus, the intensity of the induced electric field at a given point of the orbit is determined not by the change of the magnetic-field intensity that exists at this point, but by the change of the total magnetic flux through the area of the orbit. It is precisely this circumstance that leaves the necessary freedom in choosing the conditions we need.

According to Newton’s second law (in relativistic expression),

\[ \frac{dp}{dt}=eE. \]

Since we are assuming here that conditions have been realized under which the electron moves along a circle of unchanging radius \(r_0\) in a plane perpendicular to the magnetic field, we must substitute into this equation the value of \(E\) given by expression (4). Thus,

\[ \frac{dp}{dt}=\frac{e}{2\pi r_0 c}\frac{d\Phi}{dt}. \]

Integrating this equation, we obtain:

\[ p_t-p_0=\frac{e}{2\pi r_0 c}\left(\Phi_t-\Phi_0\right), \tag{5} \]

i.e., the change in the electron’s momentum during the time \(t\) is proportional to the change in the magnetic flux through the area of the orbit during the same time. Let us now suppose that the initial value of the magnetic flux \(\Phi_0\) was chosen in such correspondence with the initial velocity of the electron that the following relation was satisfied:

\[ p_0=\frac{e}{2\pi r_0 c}\Phi_0 \tag{6} \]

(below we shall consider in more detail what this condition means). In this case, instead of formula (5) we obtain:

\[ p_t=\frac{e}{2\pi r_0 c}\Phi_t . \tag{7} \]

Let us denote the mean value of the intensity of the magnetic field inside the circumference of radius \(r_0\) by \(\overline{H}\). By definition we have

\[ \overline{H}=\frac{\Phi}{\pi r_0^2} \]

and we can rewrite equation (7) in the following form:

\[ p_t=\frac{e}{2c}\overline{H}_t r_0 . \tag{8} \]

On the other hand, we know on the basis of formula (3) that, under the condition \(\rho=r_0\), at any moment of time \(t\) the relation must be satisfied:

\[ p_t=\frac{e}{c}H_t r_0 . \tag{9} \]

Consequently, the following simple condition must hold:

\[ H_t=\frac{\overline{H}_t}{2}. \tag{10} \]

Thus, in an increasing magnetic field the electron will move along a circular orbit of constant radius in the case when the intensity of the magnetic field at the points of this orbit at every moment of time is one half of the mean intensity of the magnetic field inside the orbit; in this case the increase in the velocity (more precisely, the momentum) of the electron in the induced electric field will occur all the time proportionally to the increase in the magnetic field at the points of the orbit, so that \(\rho=\mathrm{const}\) will result.

It is obvious that, in order to fulfill condition (10), it is necessary to create in the central region inside the orbit a stronger magnetic field, which must decrease toward the periphery. The law of decrease of the field with distance from the center is not of essential significance: in any axially symmetric magnetic field, with sufficient inten-

...of the central part of it there will exist a circle (and sometimes also two concentric circles in the given plane), for the points of which the condition \(H=\dfrac{\bar H}{2}\) will be satisfied; the radius of this circle is determined entirely by the distribution of the field.

A magnetic flux with the required distribution of intensity, which will remain unchanged at any moment of time, is easy to create, for example, in the air gap between the poles of an electromagnet, by taking pole pieces in the form of obtuse truncated cones whose cut-off vertices face one another (of course, one may also use a specially chosen combination of solenoids, i.e. a magnetic system without iron).

We can now consider what is the meaning of the initial condition (6) obtained above. As we already know, in an induction accelerator the distribution of the magnetic field must be such that, for any moment of time,

\[ \bar H=\frac{\Phi}{\pi r_0^2}=2H_0. \]

Therefore condition (6) can be rewritten in the following form:

\[ p_0=\frac{e}{c}H_0 r_0. \tag{11} \]

But this is nothing other than expression (2), written for the moment of time \(t_0\). Thus the initial condition (6) has a very simple meaning: the initial magnetic-field intensity \(H_0\) (at the points of the equilibrium orbit) must have such a value that, in this field, the radius of curvature of the trajectory of an electron arriving at the equilibrium orbit tangentially to it with velocity \(v_0\left(=\dfrac{p_0}{m}\right)\) is exactly equal to the radius of the equilibrium orbit.

From the derivation given above it is easy to see that the condition \(H=\dfrac{\bar H}{2}\) is the solution of the problem posed, independently of the particular law according to which the time derivative \(\dfrac{d\Phi}{dt}\), entering into expression (4), changes with time. It is only necessary that during the process of accelerating the electron its sign not change; otherwise the sign of the intensity \(E\) of the induced electric field will also change [relations (5) and (6) will not be violated in the latter case, i.e. the electron will remain on the same equilibrium orbit, but it will be decelerated by the field \(E\), not accelerated].

Finally, it should be especially emphasized that the condition formulated above for the existence of an equilibrium orbit in an induction accelerator of electrons is completely independent of the mass of the electron. Therefore, unlike what takes place in the case of the cyclotron, the operation of the betatron is not disturbed even when the mass of the accelerated particle, having acquired a velocity close to the speed of light, increases many times.

STABILITY OF THE MOTION OF AN ELECTRON ALONG AN EQUILIBRIUM ORBIT

We showed above that, under certain initial conditions, an electron emitted onto a certain circular line existing in a nonuniform axially symmetric magnetic field will continue its further motion along this line. But will this motion along such an “equilibrium orbit” be stable under real conditions? In the numerical example we gave, it was calculated that in \(1\,1000\) sec. the electron must traverse in the vacuum chamber a colossal path of about 300 km. For comparison, let us note that in a cyclotron the path length of an ion is only of the order of 100 m. If one recalls that even under the conditions of the best technically attainable vacuum an enormous number of gas molecules will always remain in the chamber, with which the electron will often have to collide on its long path, then it becomes entirely clear what importance is acquired by the questions of stability of the motion of electrons along the equilibrium orbit, the questions of focusing, i.e. of the prolonged retention of a beam of fast-moving electrons near the geometrical line of the equilibrium orbit. The apparatus must be endowed with such properties that any accidental deviation of electrons from the equilibrium orbit gives rise to sufficiently strong forces capable of returning the deviated electrons to this orbit.

Already the first experimenters who attempted to build an induction accelerator of electrons paid great attention to the question of the stability of electron motion in the apparatus and tried to give a theoretical analysis of this question. It was also considered in theoretical works, but very briefly\(^{1}\). Its correct solution was first given by Steenbeck\(^{10}\) in 1935 (as already mentioned, the conclusions he obtained were published—in 1937—only in the text of a patent). The same results, but presented in greater detail, were obtained in their work by Kerst and Serber\(^{14}\). The theory indicated a remarkably simple and effective method for stabilizing the motion of electrons in an induction accelerator.

In solving the problem of focusing electrons in the chamber of a betatron, it is necessary to take care of two kinds of focusing. First, it is necessary to ensure the return to the equilibrium orbit of those electrons which have deviated from it in the direction toward the center or away from the center, but have not left the plane of the orbit (radial focusing). Second, it is necessary that electrons which have accidentally left the plane of the orbit upward or downward return again to the plane of the orbit (axial focusing). Both kinds of focusing are achieved by choosing

\(^{1}\) The work of V. V. Yasinskii\(^{12}\) contains plainly erroneous conclusions and assertions. In the article by Ya. P. Terletskii\(^{13a}\) the question of the stability of electron motion is considered only in passing.

Recently Terletskii has returned to this question and has examined it in detail and in a general form\(^{13b}\).

of a definite distribution of the magnetic field in the region near the equilibrium orbit. For radial focusing it is necessary and sufficient that, near the orbit, the magnetic field, as a function of the distance from the axis, decrease more slowly than \(1/r\).

Indeed, let us assume that near the orbit the law of decrease of the magnetic-field intensity is described by the expression

\[ H=\frac{A}{r^n}, \tag{12} \]

and consider two cases: \(n>1\) and \(n<1\). In Fig. 1 the curve \(F_c\) represents the centrifugal force arising when an electron moves

Fig. 1. Distribution of the magnetic-field intensity necessary for producing radial focusing of electrons.

along a circular trajectory, as a function of the radius of the latter. This curve is a hyperbola, since the centrifugal force is

\[ F_c=\frac{mv^2}{r}, \tag{13} \]

where the electron velocity \(v\) may certainly be regarded as a constant quantity over the course, say, of one revolution of the electron in its orbit.

The second force acting on the electron in the opposite direction, i.e. toward the center of the orbit, is the so-called Lorentz force, whose appearance is due to the action of the magnetic field on a moving charge. As is known, the Lorentz force is

\[ F_m=\frac{e}{c}vH. \tag{14} \]

Consequently, the course of the variation of \(F_m\) as a function of \(r\), for a given velocity \(v\), is wholly determined by the form of the dependence of \(H\) on \(r\). The curves in Fig. 1 (\(a\) and \(b\)) show \(F_m\) for the case \(H=A/r^n\), respectively for \(n>1\) and for \(n<1\). The radius of the electron orbit along which it will actually move is determined from the condition that the Lorentz and centrifugal forces balance one another [from this condition is obtained the expression (2) used by us above for the radius of curvature

trajectory of the electron when it moves in a magnetic field. Since in our case the electron moves along an equilibrium orbit, consequently, the abscissa of the point of intersection of the curves \(F_c\) and \(F_m\) in Fig. 1 is nothing other than \(r_0\)—the radius of the equilibrium orbit. Suppose that the electron has accidentally deviated from the equilibrium orbit and has fallen into a point located at a distance \(r_1 > r_0\) from the center. Then, as is easy to see from Fig. 1, in case \(a\) the centrifugal force will prove to exceed the Lorentz force, and as a result the electron will be displaced still farther toward larger \(r\), i.e., it will no longer return to the equilibrium orbit. Conversely, in case \(b\), with \(r_1 > r_0\), \(F_m > F_c\), so that the Lorentz force will return the electron toward the equilibrium orbit. Analogous circumstances also occur in the case when the electron, having left the equilibrium orbit, falls into a point located at a distance \(r_2 < r_0\) from the center: in case \(a\) it will no longer return to the equilibrium orbit, while in case \(b\) it will return to it.

Fig. 2. Origin of the magnetic forces effecting axial focusing of electrons.
\(A, B\)—section of the pole pieces of the electromagnet.

The graphical arguments presented above are, of course, not a rigorous derivation and are given only for the sake of clarity. However, an exact derivation also leads to the same result: in order to accomplish radial focusing in the betatron it is necessary that, at the points of the equilibrium orbit and in the region close to it, the exponent \(n\) in expression (12) be less than unity.

Theory shows that in this case any (not excessively strong) impulse displacing the electron from the equilibrium orbit leads only to the fact that an oscillatory motion about the line of the orbit is superposed on the main motion of the electron along the orbit, this motion damping rather quickly, so that subsequently the electron continues, as before, to move along the orbit.

As for axial focusing, it is automatically ensured by the fact that in the betatron the magnetic field is stronger at the center than at the periphery. As a consequence, the magnetic lines of force curve in the direction away from the center (“barrel-shaped field”), so that between the poles of the electromagnet there exists only one plane—the median plane \(P\) (Fig. 2)—at whose points the magnetic field has no radial component. In this plane the trajectory of the electron’s motion (the equilibrium orbit) will be located, since it is easy to see that if an electron is above this plane, then the radial component of the magnetic field (\(H_r\) in Fig. 2) will cause the appearance of a Lorentz force directed toward the median plane, while if the electron is below it, the Lorentz force will be directed upward, i.e., again toward this plane.

(note that similar axial focusing is also carried out in the cyclotron with respect to the beam of accelerated ions).

The presence of axially focusing properties in a nonuniform magnetic field also leads—in the case of displacement of the electron from the plane \(P\)—to the appearance of an oscillatory damped motion (but now in the vertical direction), which is superposed on the basic motion of the electron in its orbit. In the work of Kerst and Serber\(^{14}\) the principal parameters of the radial and axial oscillations—their amplitude, period, and damping—are calculated, and it is shown on what conditions these quantities depend.

THE INTERMITTENT CHARACTER OF BETATRON OPERATION

The process of increasing the velocity of the electrons in the betatron lasts until there exists an induced electric field \(\dot E\) directed in the required direction, or, in other words, until the increase of the magnetic flux \(\Phi\) continues (since the field strength \(E\) is proportional to \(\dfrac{d\Phi}{dt}\)). It is clear that this increase cannot last indefinitely long, since for every magnetic circuit there exists a definite, technically attainable limiting value of the flux \(\Phi\). Having reached this limit, one must stop the increase of the flux and, consequently, stop the acceleration of the electrons. Does this circumstance threaten that, within the limited time, the electron will not have time to acquire the desired velocity? Not at all. Indeed, let us consider what determines the energy of an electron, acquired by it at the expense of the force of the induced electric field in the betatron. In this device, as we know, such conditions are created that the electron is all the time on a circular orbit, making revolution after revolution on it. Using formula (9), we can determine the momentum of the electron for any instant of time \(t\). This formula shows that the momentum of the electron does not depend directly on the duration of acceleration and in the final analysis is determined by the value to which the magnetic-field strength \(H_t\) in the orbit has been brought by the given moment, and also by the design of the pole pieces of the electromagnet, since their dimensions and shape uniquely set the radius of the equilibrium orbit \(r_0\).

It is unlikely that in any of the applications of the betatron it can be important that, in the process of accelerating electrons, interruptions are unavoidable. They merely lead to the fact that fast electrons can be obtained from the betatron only in separate, intermittent portions. One can imagine two varieties of the operating regime of the betatron: either the process of growth of the magnetic flux from \(\Phi_0\) to \(\Phi_t\) is carried out once and, correspondingly, one separate portion of accelerated electrons is obtained, or this process is repeated many times one after another, and, correspondingly, portions of fast electrons arise one after another. The regime of the first type was used in his

installation of Wideröe. The regime of the second type is most simply and rationally realized by feeding the windings of the accelerator electromagnet with alternating current, which Kerst did in his betatron. In this case the magnetic flux between the poles of the electromagnet varies sinusoidally with time. Consequently, during each period there are two intervals of time during which the magnetic field increases (in absolute value), so that during this time inductive acceleration of electrons can be carried out: this occurs in the first and third quarters of each period (segments AB and CD in Fig. 3). The directions of the magnetic field during these two intervals of time are opposite to one another; therefore, if in the first quarter the electrons were accelerated while making their revolutions about the axis of the apparatus, for example clockwise, then in the third quarter of the period they will be accelerated while moving along their orbit counterclockwise. The second and fourth quarters of each period (segments BC and DE in Fig. 3) are the idle time of the betatron, since during this time the magnetic field decreases and the electrons can only be decelerated in the corresponding induced electric field.

Fig. 3. Magnetic-field strength in the betatron as a function of time.

In popular technical literature on the new generator of fast particles one often encounters the erroneous assertion that the betatron is a variety of cyclotron adapted for accelerating electrons. Everything set forth above shows that the betatron in essence has nothing in common with the cyclotron and is based on an entirely different principle. Inductive acceleration cannot even be called multiple acceleration, since the electrons in the induced field are accelerated continuously from the initial to the maximum velocity.

PREDECESSORS OF KERST

The idea of using a vortex field to accelerate charged particles was apparently first proposed by Slepian. In 1922 he patented in the USA the design of an X-ray tube[^15] in which electrons are accelerated in an induced electric field while moving in an increasing magnetic field along a spiral path. Apparently Slepian made no attempts at experimental realization of this tube.

Engineer Wideröe, in a dissertation submitted at the end of 1927, set forth a theoretical calculation of a scheme he proposed for an induction accelerator of electrons, as well as the results of his experimental work with an installation of this kind[^3]. Wideröe was the first to obtain theoretically the extremely important result concerning the possibility of an electron moving in an increasing magnetic field along a circle of unchanging radius.

Breit, Dahl, and Tuve^16 in 1927–1928 built an apparatus for the induction acceleration of electrons. It used a very rapidly increasing magnetic field, reaching its maximum value (30,000 oersteds) in \(10^{-5}\) sec. The electrons were to move along a spiral path and strike a target located at the center. Their maximum energy, according to calculation, was \(1.5\)–\(2\) MeV. In the apparatus it proved possible to obtain hard X-radiation, but its intensity was extremely small.

The physicist Walton, working in Cambridge in Rutherford’s laboratory, published in 1929 the results of his work,^17 in which he attempted to carry out the acceleration of electrons “by means of the indirect method proposed by Rutherford,” namely, by means of the induction method (Wideröe’s work is cited at the end of Walton’s paper, but apparently Rutherford proposed the idea of the method independently of Wideröe). In Walton’s apparatus the vortex field was created inside a solenoid into which a capacitor was discharged. For this case Walton gives a theoretical calculation of the conditions necessary for circular motion of the electrons; he arrives at the same result as had been found by Wideröe. In this apparatus the electrons, according to the calculation, were to be accelerated up to 540 KeV. In experiment, however, it was not possible to detect acceleration of the electrons.

Rudenberg and Steenbeck^18 in 1933 proposed a scheme for an X-ray tube device with acceleration of electrons in a vortex field. The electrons were to move along a spiral path and, at the end of the acceleration cycle, strike a target located at the center. An essential improvement over analogous earlier proposals was the use of a magnetic field decreasing from the center toward the periphery, with the aim of obtaining axial focusing of the electrons. This idea, however, was not new, since by that time it was already known that axial focusing of an ion beam in a cyclotron is achieved in this way.

In 1935 Steenbeck considered in more detail the theory of operation of the induction accelerator and, guided by the conclusions obtained, built in 1935–1936 an experimental apparatus for the induction acceleration of electrons. This apparatus has very much in common with Kerst’s betatron. In it it proved possible to obtain electrons with an energy of \(1.8\) MeV, but the beam intensity was extremely insignificant, and further work was abandoned.^19

A. Bouwers, in his monograph on superhigh electric voltages, published in 1939, proposes an original scheme for an apparatus for the induction acceleration of electrons.^20 Designers of new betatron models may yet return to this scheme of an induction accelerator in the future, since it possesses certain favorable features.

Finally, let us note that in the apparatus of Kelman, Korsunsky, and Lange^7 the induction method was also assumed as one of the possible methods of accelerating electrons.

Of the later works, one should mention the work of Ya. P. Terletsky. In his theoretical article¹³ᵃ, written in 1940, a special case of an induction accelerator was considered, namely an accelerator with a strictly parallel magnetic field. On the question of the conditions for the existence of an equilibrium orbit of electrons in a vortex field, the author obtained, of course, the same result as Widerøe. (Let us note, incidentally, that the author’s remark that Widerøe’s calculation was made inaccurately and nonrelativistically is unfounded.) The variant of an accelerator using a parallel magnetic field cannot be considered successful, since it lacks axial focusing of the electrons.

All the attempts listed above at the practical realization of an induction electron accelerator were unsuccessful. One of them, however—Widerøe’s work—is of great historical interest, and we shall give some data on his experiments. Acquaintance with Widerøe’s installation gives a clear idea of what a step forward the betatron represents in comparison with the level of 1927.

For the electromagnet of his installation Widerøe used the core of a powerful transformer. The winding of the electromagnet consisted of 16 coils of 480 turns each of wire 1 mm in diameter, all the coils being connected in parallel.

The vacuum chamber of the instrument is placed in the air gap of the electromagnet. Since it is advantageous to make the air gap as narrow as possible (in order to obtain the prescribed value of the magnetic flux with the smallest number of ampere-turns), it is desirable that the chamber, in the central part where the magnetic field must be strongest, should have a minimum thickness. Fulfilling this condition, Widerøe gave his chamber the form of a ring-shaped tube: its central part is entirely removed (this region, being inside the circular orbit of the electrons, is in any case a nonworking space of the chamber), and the pole-piece projections of the electromagnet enter the opening of the ring. A section of the latter can be seen in Fig. 4, where the entire Widerøe installation is shown schematically. The chamber is made of a glass tube 15 mm in diameter; the outside diameter of the toroid is 160 mm. A diffusion pump maintains in the chamber a high vacuum of the order of \(10^{-6}\) mm Hg.

The electron source—an electron gun—is placed in a separate branch of the chamber at a fairly large distance from the position of the equilibrium orbit of the electrons. With the aid of a system of auxiliary solenoids and a hinged screen coated with a luminescent substance (zinc sulfide), the electron beam from the gun can be focused so that it enters the chamber tangentially to the circumference of the equilibrium orbit.

By selecting the widths of the air gaps \(\delta_1\) and \(\delta_2\) (Fig. 4), Widerøe succeeded in making the condition \(\bar{H}=2H\) hold in his installation for a circle of diameter 14.5 cm. In other words, Widerøe set the radius of the equilibrium orbit \(r_0\) equal to 7.25 cm.

The further stage of preliminary adjustment of the apparatus is the establishment of the correct initial conditions, i.e. the fulfillment of relation (11), obtained by us above.

The initial velocity of the electron \(v_0\) is known, since the potential at the anode of the electron gun is known (it was about 20 kV). The required value of the initial magnetic flux was set by means of the rheostat \(R\) (Fig. 4), the electromagnet winding being supplied with direct current at a voltage of 500 V.

The operation of the installation was supposed, as Wideröe assumed, to proceed as follows. After the magnetic field (flux \(\Phi_0\)) is switched on, the previously adjusted electron gun is switched on, delivering electrons to the equilibrium orbit. Entering the magnetic field of the proper magnitude, constant in time, the electrons move in the vacuum chamber along a circle of radius \(r_0\), making on it turn after turn with constant velocity (until their initial kinetic energy is lost to a noticeable extent as a result of collisions with gas molecules in the chamber). Soon after the beam is admitted into the chamber, the switch \(E\) is closed (Fig. 4). The current in the electromagnet winding then increases rapidly, so that after only a few tenths of a second the fuses \(G\) blow and the current is switched off. During this time the magnetic flux increases from \(\Phi_0\) to some \(\Phi_f\), and the electrons, uniformly moving along the orbit, during this same time, continuing to move along the same orbit (for the condition \(\overline{H} = 2H\) is fulfilled), are accelerated by the forces of the induced electric field to the corresponding value of the momentum \(p_f\) (according to Wideröe’s calculation—to an energy of \(\sim 6\) MeV).

Fig. 4. Diagram of the arrangement of Wideröe’s installation.
\(B\) — electromagnet, \(C\) — vacuum chamber,
\(D\) — flat conducting coil.

The switch \(E\) is connected with another switch (\(F\)) in such a way that circuit \(F\) is closed 0.1 sec after the closing of circuit \(E\). The coil \(D\) included in circuit \(F\) gives an additional magnetic field, forcing the electrons to leave the equilibrium orbit and strike the outer walls of the chamber. The latter are coated on the inside with zinc sulphide, and its luminescence under the impacts of fast electrons would, according to Wideröe’s idea, make it possible to establish the fact of obtaining electrons with the expected large kinetic energy.

Wideröe’s experiments were not crowned, and could not have been crowned, with success. It is obvious, first of all, that the idea of bringing electrons to the orbit from a distant source is unsound. The trajectory of electrons in a constant magnetic field cannot have the form of a straight-line segment then passing into a closed circle. In reality, under such condi-

of the trajectories would be a symmetrical loop, so that the electrons, after making approximately one revolution around the center of the chamber, would inevitably strike its walls. Widerøe’s second crude error was his assumption that electrons with an energy of 20 keV could circulate for a long time (for example, 1 sec. or more) in the chamber along a circumference of unchanged radius, when the magnetic field was constant in time (the first stage of the experiment). It is easy to calculate that at a vacuum of the order of \(10^{-6}\) mm Hg such electrons expend all their energy on ionizing gas molecules within small fractions of a second. Finally, even if this circumstance had been taken into account, Widerøe could not have ensured that the electrons would actually move along a circumference, describing on it many revolutions one after another; in his theoretical analysis of the problem of induction acceleration of electrons, Widerøe did not find the solution to the question of the necessary conditions for stability of the motion of electrons along an equilibrium orbit, and, not knowing these conditions, did not take care of the required shape of the electromagnet pole pieces. Characteristic are the following words of Widerøe’s article, which conclude the discussion of possible ways of stabilizing the motion of electrons in the orbit: “In any case, all further investigations... must be devoted to the problem of stabilization.”

KERST’S FIRST BETATRON

Kerст assembled his first apparatus at the University of Illinois over about a year and completed its construction on July 15, 1940. This apparatus was designed to accelerate electrons approximately up to 2 MeV. It is of quite small dimensions and fits freely on a table, as can be seen from Fig. 5. Fig. 6 shows the shape and dimensions of the core of the electromagnet. The winding of the latter is supplied with alternating current. Therefore, to avoid large losses due to Foucault currents, the entire core must not be solid; like transformer cores, it is assembled from a large number of thin plates of silicon steel (“transformer iron”). The pole pieces, made from radially arranged plates, were assembled with particular care; this ensures perfect axial symmetry of the magnetic field in the air gap.

Fig. 5. General view of the first betatron.

The accelerating vacuum chamber of the apparatus is a glass doughnut, whose cross-section, corresponding to the shape of the pole

pole pieces of the electromagnet, is shown in Fig. 6. The outside diameter of the chamber is only 20 cm. By means of chemical silvering the inner surface of the chamber is covered with a thin layer of silver; this layer is grounded, and this prevents the accumulation of a static charge on the surface of the glass1.

Fig. 6. Dimensions of the electromagnet of the first betatron.
A — vacuum chamber (in section), B — leads to the electrodes of the electron gun.

The shape of the pole pieces of the electromagnet provides such a distribution of the magnetic field that the radius of the equilibrium orbit \(r_0\) is equal to 7.5 cm (Fig. 7). To characterize the exceptional effectiveness of the method of magnetic focusing of the beam in the betatron chamber, we note that the apparatus can still operate normally at such a low vacuum as \(2 \cdot 10^{-5}\) mm Hg2.

The winding of the electromagnet is fed from a small generator, of power \(\sim 4\) kW, giving an alternating current with frequency 600 Hz and effective voltage about 80 V. Owing to the large self-inductance of the electromagnet winding, directly feeding it with alternating current is associated with an extremely low value of \(\cos \varphi\) and would be highly disadvantageous. Therefore Kerst uses the resonant circuit for feeding the winding shown in Fig. 8. The capacitance of the capacitor bank is selected in such

Fig. 7. Distribution of the magnetic-field strength in the first betatron as a function of distance from the central axis.
\(r_0\)—radius of the equilibrium orbit; in the segment \(ab\) the field decreases according to the law \(r^{-3/2}\).

in such a way that the natural frequency of the resulting oscillatory circuit is equal to the frequency of the supply current (600 Hz).

For the operation of the betatron the numerical value of the frequency of the supply current plays no essential role. We have already pointed out that the final energy of the electrons accelerated in the betatron does not depend on the duration of the acceleration process. When the frequency is increased, the duration of acceleration decreases (since it is equal to \(1/4\) of a period), and the total number of revolutions made by the electron in the chamber during acceleration must decrease; but, correspondingly, the increment of energy which the electron receives in one revolution increases (since this quantity is proportional to \(\dfrac{dH}{dt}\)). As a result, the final energy of the electrons is independent of the frequency of the supply current. There are, however, a number of considerations which make it desirable to use as high a frequency as possible, i.e., a shorter duration of the working stroke of the betatron. For example, with a shorter total path length of the electron in the chamber, the number of collisions with gas molecules decreases. On the other hand, one cannot use excessively high frequencies, since as the frequency increases the losses grow strongly—chiefly in the iron—and, consequently, the problem of cooling the magnet becomes more difficult. It is characteristic that, having begun with 600 Hz, Kerst in larger models of the apparatus went over to lower frequencies of the supply current.

Fig. 8. Resonant supply circuit for the windings of the betatron electromagnet.

The source of the electrons to be accelerated is a miniature electron gun, located outside the equilibrium orbit at a distance of 9 cm from its center. The focusing electrodes of the gun are arranged in such a way that it sends into the chamber two ribbon-shaped electron beams directed in opposite directions. The electrons of one beam are accelerated during the first quarter of each period, and the electrons of the second beam during the third quarter. We have already mentioned this possibility earlier. The voltage on the anode of the electron gun, which determines the initial velocity of the electrons \(v_0\), is approximately 600 V. Let us note that this is the highest of the voltages supplied to an apparatus generating electrons with an energy of more than 2 MeV!

The fulfillment of the condition \(n < 1\) (of which we spoke above; see, for example, Fig. 1), imparting focusing properties to the magnetic field near the orbit and thereby ensuring the stability of the motion of the electrons along the equilibrium orbit, also makes it possible to solve successfully another question very important for the operation of the betatron—the question of injecting the electrons, of bringing them onto the equilibrium orbit. This problem contains two tasks. First of all, it is necessary to decide at what place in the vacuum chamber the electron gun should be placed. Obvi-

it is obvious, it is impossible to place it on the equilibrium orbit itself, for in that case the electrons, after the very first revolution, will run into the gun and their further acceleration will not occur. Consequently, the gun must be placed either outside or inside the equilibrium orbit. But in such a case, will the electrons, flying out of the gun, move in such a way as ultimately to reach the equilibrium orbit? Theoretical analysis shows that this will take place if a distribution of the magnetic field near the orbit is created for which the condition \(n < 1\) is satisfied. In this case the electrons leaving the gun in the proper interval of time and with the proper direction of the initial velocity (we shall speak of this in more detail below) will move along a complicated trajectory in the form of a pulsating flat spiral, asymptotically approaching the equilibrium orbit. This trajectory is the result of the superposition of the damped oscillatory motion of the electron about the line of the equilibrium orbit with its circular motion about the axis of the chamber. After some time the electron reaches the equilibrium orbit and thereafter moves along it (“is captured by the orbit”). In Fig. 9 there is shown schematically the approximate form of the trajectory of an electron during the first time after its departure from the gun [this drawing refers to a definite value of \(n\) \((n = 3/4)\) and to definite initial conditions].

Fig. 9. View of the initial portion of the trajectory of an electron. The equilibrium orbit is shown by a dotted line.

The second part of the problem of electron injection consists in the following: how is one to ensure in the betatron the fulfillment of the initial condition (11), which we obtained above? The electromagnet of the betatron is supplied with an alternating current; consequently, the magnetic field changes continuously, and therefore the initial condition

\[ mv_0 = p_0 = \frac{e}{c} H_0 r_0 \]

is satisfied exactly only during a certain infinitely small interval of time, in the course of which \(H\) passes through the fixed value \(H_0\). One might therefore think that the apparatus practically will not operate at all under such conditions. But here too the focusing properties of the magnetic field in the betatron make it possible to overcome the difficulty. Their presence considerably softens the initial condition (11): there exists not a single value equal to \(H_0\), but an entire range of values of the magnetic-field strength on the orbit for which electrons with the fixed entrance velocity \(v_0\) are still focused by the magnetic field of the apparatus and are delivered to the equilibrium orbit. In other words, electrons from the gun can reach the equilibrium orbit during an interval of time of finite duration. Let us turn, for an explanation, to Fig. 10. At the moment when the sinusoidally varying

the magnetic field passes through the value \(H=0\), the electrons from the gun will move rectilinearly and strike the walls of the chamber (line \(a\) in Fig. 10). At some subsequent moment the magnetic-field strength will already be different from zero, though still small, and the electrons that have emerged from the gun at that moment, under the action of this field, will fly along the curvilinear trajectory \(c\), but will likewise strike the wall of the chamber. Finally, there will come a moment when, during some small interval of time—the “working interval of injection”—the magnetic-field strength will be such that the electrons entering the chamber at this time will not reach the wall (trajectory \(d\)), but will follow a spiral path, fall onto the equilibrium orbit, and take part in the process of acceleration.

Fig. 10. Paths of electrons emerging from the gun at different moments of time.
\(AA\)—equilibrium orbit, \(B\)—electron gun, \(C\)—target.

In the following moments of time the still further increased magnetic field will already begin to bend the electron trajectories too strongly, as is shown, for example, by line \(e\). Such electrons will also strike the walls of the chamber or the electrodes of the gun and thus “drop out of the game.” Let us give some numbers. The entire interval of time from the moment when the magnetic-field strength is zero to the moment when it reaches its maximum value lasts, obviously, one quarter of a period, which at a frequency of 600 Hz amounts to \(1/2400\) sec, or 415 \(\mu\)sec (see Fig. 11). The working interval of injection lasts from the moment \(t_1=0.9\ \mu\)sec to the moment \(t_2=1.6\ \mu\)sec, i.e., in all about \(0.2\%\) of the total duration of the working cycle of the betatron!

Fig. 11. Significant moments of the time course of the working cycle of the betatron.

It is necessary, therefore, to bear in mind that in this model of the apparatus only a comparatively very small portion of electrons can be delivered to the orbit each time for further acceleration.

We have now come to the problem of extracting the electrons. An electron which at the proper time has flown out of the gun into the chamber has entered the equilibrium orbit and has moved along it for a quarter of a period, i.e. \(415\ \mu\mathrm{sec}\), continuously accelerating; having made during this time approximately 260,000 revolutions and having traversed a total path of about \(125\ \mathrm{km}\), it has acquired the maximum velocity for the given apparatus, determined by the radius of the equilibrium orbit and by the maximum value of the magnetic-field intensity on the orbit. It is now necessary to remove this electron from the equilibrium orbit and use it in one way or another, since after the moment when the magnetic field has reached its maximum and then begins to decrease, there is no point in keeping the electron on the orbit: the induced electric field at this time is already directed in the opposite direction, and the electron will begin to be decelerated in it, losing the velocity it has accumulated.

Since the processes in question are of very short duration, it is quite obvious that the extraction of the accelerated electrons from the orbit must be done automatically. In Kerst’s apparatus this problem is also solved very ingeniously. The accelerated electrons are not led outside, but are used inside the chamber: they are directed toward a small tungsten plate (\(C\), Fig. 10), which serves as a target or anticathode. Striking the target, the fast electrons are braked in it, as a result of which X-radiation arises, as in ordinary X-ray tubes. Bombarded by electrons intermittently and alternately from two sides (at the end of the first and at the end of the third quarter of each period), the target emits X-rays alternately in two opposite directions, with a frequency of 1200 very short pulses per second. Thus, the final product of the apparatus is an intermittent beam of hard X-rays freely passing outward through the walls of the vacuum chamber.

How, however, can the accelerated electrons be made at the required moment to leave the equilibrium orbit and strike the target? For this, obviously, the previous distribution of the magnetic field must be disturbed and the equilibrium orbit thereby shifted in the required direction, in this case toward a sharp decrease of its radius. At this moment the electrons will leave their former orbit and, along a rapidly contracting spiral, will tend toward the new equilibrium orbit; moving along the spiral, they will strike the target and give rise to X-radiation. Kerst carries out the automatically synchronized changes in the radius of the equilibrium orbit in the following way. To the central part of each of the pole pieces of the electromagnet there is attached a disk \(50\ \mathrm{mm}\) in diameter, made of pressed iron powder cemented with liquid glass. The magnetic properties of this material are such that in these disks magnetic saturation is reached at much smaller values of the magnetic-field intensity…

of the field than in the other parts of the core. When, at the end of a quarter of the period, the intensity of the magnetic field approaches its maximum, the field at the center becomes, in comparison with the field at the periphery, weaker than before; with this new spatial distribution of the intensity of the magnetic field the radius of the equilibrium orbit decreases, which ensures that the electrons strike the target.

According to calculation, the electrons in Kerst’s first betatron should be accelerated to an energy of 2.3 MeV (this is determined by the fact that the maximum intensity on the orbit line is 1250 oersteds and the radius of the orbit is 7.5 cm). Kerst investigated the absorption in lead of the X-rays emitted by the target, and these measurements confirmed the calculated value of the velocity attained by the electrons. This is direct proof that the apparatus really works and imparts to the electrons the energy for which it was designed.

In full agreement with the theory of X-radiation at relativistic electron velocities, the radiation from the betatron target proved to possess a sharp spatial asymmetry: the greater part of the X-ray quanta is emitted in the direction in which the electrons flew toward the target, and very few quanta are emitted in directions perpendicular to this. The presence of a sharply localized, directed beam of X-rays is a very valuable feature of the betatron. The beam of X-rays from the target of Kerst’s first betatron in the optimal direction was, in intensity, equivalent to the $\gamma$-radiation from 1 g of radium.

In summary, we may say that the model of the betatron described by us is, in essence, an extremely elegant construction of a high-voltage (but low-power) X-ray tube. Instead of constructing a cumbersome step-up transformer with a secondary winding of 260,000 turns, which could give an alternating voltage of 2.3 MV, and also constructing a rectifier and a vacuum X-ray tube for such a voltage, a compact apparatus has been created in which a step-up transformer, a rectifier, and an accelerating tube are combined into one whole; moreover, the role of the transformer’s secondary winding is played by the electron beam itself, describing along one and the same line the necessary 260,000 revolutions. Wideröe aptly called his induction electron accelerator a ray transformer (Strahlentransformator).

SUBSEQUENT MODELS OF THE BETATRON

Having obtained excellent results with his first apparatus, Kerst at once conceived the creation of a larger installation for generating electrons with an energy of 100 MeV. The firm “General Electric” agreed to provide the design and production apparatus of its research laboratories (Schenectady) for the construction of such an installation. As an intermediate stage it was decided

it was possible to construct, under Kerst’s direction, a betatron for 20 MeV^21. The construction of this installation was completed in 1941.^1)

The second betatron already has quite substantial dimensions, although, of course, they are much smaller than those of a cyclotron capable of giving protons the same energy. The general appearance of the 20 MeV betatron is shown in Fig. 12. The weight of the installation is about 3.5 tons. The electromagnet is 1.5 m long, 0.5 m wide, and 0.9 m high. The pole pieces have a diameter of 480 mm; approximately the same is the outside diameter of the glass vacuum chamber. In this betatron the radius of the equilibrium orbit is 19 cm. In each acceleration cycle the electrons make about 350,000 revolutions, traversing a path 420 km long.

Fig. 12. General appearance of a betatron generating electrons with an energy of 20 MeV.

In the first months of operation of the second betatron, electrons with an energy of 13 MeV were obtained, and by the end of 1941 it proved possible to obtain the full calculated acceleration, i.e., acceleration of electrons up to 20 MeV. The electron velocity corresponding to such an energy is less than the speed of light by only 0.03%! This is undoubtedly the greatest speed ever obtained by man. Moreover, the X-ray quanta arising when such electrons strike a target, in their energy (\(E_{\max} = 20\) MeV), exceed the hardest \(\gamma\)-quanta known and used in the experimental practice of nuclear physics, namely, the \(\gamma\)-rays of energy 17 MeV arising when lithium is irradiated with protons. In ionizing action, the sharply directed beam of X-radiation from the betatron target is equivalent to more than 1 kg of radium. It may perhaps not be superfluous to recall that the annual production of radium throughout the world is less than this amount. One may cite still another figure: the intensity of the X-radiation in the beam (at an electron energy of 20 MeV), measured with a thick-walled ionization chamber, is 50 roentgens per minute at a distance of 70 cm from the target.

The directionality of the beam of X-rays from the betatron is characterized by the fact that they produce a photographic effect only within a cone with an apex angle of 6°.

^1) The name for the new installation that appeared at that time—“reotron”—was soon replaced by its present, much more successful name.

In the 20 MeV betatron a number of technical innovations have been introduced as compared with the first model. In this betatron the gun sends a beam of electrons into the chamber in only one direction. Thus only one quarter of the period, and not two, is the working cycle of the apparatus. The target is the rear side of the gun, so that, in contrast to the first betatron, the target lies outside the orbit, and at the end of the process of accelerating the electrons the radius of the equilibrium orbit must be not decreased but increased. The method of sharply changing the orbit radius in this installation is somewhat modified. The increase of the radius (the expansion of the orbit) at the proper instants is achieved by the synchronized passage of current pulses through a special winding located in grooves on the surface of the pole pieces. A vacuum-tube radio-engineering circuit, inductively coupled with the magnetic field of the betatron, sends a current pulse into this “expanding winding” at any desired instant of the first quarter of each period, for example at the instant corresponding to point \(A\) in Fig. 11. This makes it possible to obtain from the betatron, at will, not only electrons with an energy of 20 MeV, but also electrons of any preassigned energy below this maximum, since, depending on the value of the magnetic field at the instants when the acceleration cycle ends (i.e. when the expanding winding is switched on), the energy of the electrons thrown at those instants onto the target will also change [see above, equation (9)].

The electromagnet winding is supplied, as in the first model, by means of a resonance circuit. The supply current has a frequency of \(180\ Hz\); a ferromagnetic frequency tripler, connected to the city mains, is used\(^{1}\).

The most important improvement introduced by Kerst into the operating technique of the second betatron concerns the injection of electrons into the apparatus. Experience with the first betatron showed that continuous injection of electrons into the chamber has a very unfavorable effect on the intensity of the X-radiation produced by the betatron. There is no need to clog the chamber with electrons not participating in the acceleration process; the gun must send electrons into the chamber only during the working interval of injection (i.e. from the instant \(t_1\) to the instant \(t_2\), see Fig. 11). This is what was done in the second betatron. For most of the time the gun remains without voltage, and the electron beam does not emerge from it into the chamber. By means of a radio-engineering circuit operating synchronously with the changes of the magnetic field of the electromagnet, voltage pulses are applied to the electrodes of the gun \((U_{\max}=15\text{—}20\ kV)\), so that at the proper instant the gun sends into the chamber a “short burst” of electrons; this occurs 180 times per second. The pulsed application of voltage to the electrodes of the gun undoubtedly has, as its result, one more favorable—

\(^{1}\) The American standard for the frequency of current in the mains is \(60\ Hz\).

... circumstance: the initial velocity of the electrons \(v_0\) is not fixed; it increases in accordance with the shape of the voltage pulse, and since during this same time the magnetic field also increases, this leads to a certain increase in the duration of the working interval of emission, which means an increase in the time-averaged intensity of the electron beam incident on the target. Judging from the data cited by Kerst\(^{21}\), the working emission interval in the second betatron does indeed last a much larger part of the period than in the first betatron.

There exists a more effective way of increasing the time-averaged intensity of the electron beam in the chamber: the maximum possible increase in the frequency of the current feeding the electromagnet, insofar as this is permitted by the construction of the latter. The average intensity of the electron beam grows in proportion to this frequency, provided only that a sufficient emission current from the filament of the electron gun is ensured.

With regard to the instantaneous intensity of the electron beam in the orbit, it must be noted that the principal factor limiting this intensity is the space charge, which causes Coulomb repulsion of the electrons. In Kerst’s work\(^{11}\) a rough estimate is made of the maximum current that can still occur in a given magnetic field.

In a 20 MeV betatron, according to calculation, the time-averaged current onto the target can reach up to \(1\,\mu\text{A}\).

In modern high-voltage X-ray tubes, electron currents thousands of times larger occur. Nevertheless, the intensity of the beam of X-rays from a betatron may even considerably exceed that obtained in tubes, since in going over to electrons of high energies, in addition to the corresponding increase in the quantum energy, the yield of radiation and the degree of its directionality also increase strongly.

The construction of a 100 MeV betatron was begun in the same year, 1941. Such a colossal energy as 100 MeV far exceeds everything that could be required at present for purposes of therapy or defectoscopy. But for science, the commissioning of a 100 MeV betatron promises prospects that are even difficult to assess at present.

The energy of electrons at 100 MeV already approaches the energies of cosmic electrons. Processes such as the production of a meson, known so far only from observations of cosmic particles, require an expenditure of energy of approximately 80 MeV (such is the meson’s own energy, in accordance with Einstein’s equation: \(E = mc^2\), if one assumes that the mass of the meson is 160 times greater than the mass of the electron). Thus, one may hope that the third betatron will be able to become a source of mesons artificially created in the laboratory; this may prove decisive for the complete elucidation of the nature of mesons. We shall not explain here in detail why

precisely this problem is considered one of the central problems of modern physics1.

The construction of a 100 MeV betatron has probably already been completed at present (its commissioning was planned for the spring of 1943). Detailed data on this installation are not yet available. It is known that the weight of the electromagnet of the new betatron is 125 t. The vacuum chamber is a toroid of elliptical cross section, assembled from 12 sections cast from special quartz glass[^22]; the outside diameter of the chamber is 1.8 m. The entire installation is housed in a special building with walls one meter thick around the betatron, to protect the operators from high-voltage X-rays and electrons. Control and observation of the operation of the installation are carried out from an adjoining room by means of a periscope and appropriate instruments. Figure 13 shows part of the core of the betatron under construction.

Fig. 13. Lower half of the core of the electromagnet of a betatron designed to produce electrons with an energy of 100 MeV.

Below we give a table of the basic parameters of all three betatrons.

Table

Parameter
Maximum electron energy in MeV	2,3	20	100
Radius of the equilibrium orbit in cm	~ 7,5	~ 19	~ 75
Maximum magnetic-field strength on the orbit in oersteds	1250	3600	4450
Frequency of the supply current in hertz	600	180	60
Maximum intensity of the induced electric field on the orbit in volts per centimeter	~ 0,35	0,77	1,26

Continuation of Table 1.

Maximum increase in the electron energy per one revolution, in electron-volts	$\sim 17$	92	594
Total length of the electron path during acceleration, in kilometers	125	417	1250
Weight of the electromagnet	$\sim 150\ \mathrm{kg}$	$3.5\ \mathrm{t}$	$125\ \mathrm{t}$
Power for feeding the electromagnet, in kilowatts	$<4$	26	$\sim 500$
Capacitance of the capacitor bank in the resonant supply circuit, in microfarads	400	5.5	110

MAXIMUM ELECTRON ENERGY ATTAINABLE WITH THE BETATRON

The following questions are of great interest: is it possible to construct betatrons capable of generating electrons with energies considerably exceeding 100 MeV? What difficulties arise in doing so, and what are the possible ways of overcoming them? Kerst was too optimistic in asserting that with the aid of a betatron one can obtain electrons of arbitrarily high energy. True in principle, this assertion is erroneous in practice. We have seen that, as the maximum electron energy which the betatron must deliver increases, the dimensions of the whole installation increase very rapidly. From a table-top apparatus of 2.3 MeV, weighing $\sim 150\ \mathrm{kg}$ and consuming less than 4 kW of power, in the case of generating 100 MeV electrons one has to pass to the construction of an installation weighing more than $125\ \mathrm{t}$ and consuming about 500 kW of power. Of course, even these figures do not yet represent anything unacceptable. The bulkiness and high cost of such a betatron cannot be considered a serious drawback, if one takes into account that the betatron is as yet the only generator in which record-high electron energies can be obtained. Such installations, naturally, will be unique; like the modern super-powerful cyclotron, they will be the property of large scientific-research laboratories.

However, with a further increase in electron energy, the growth in the dimensions and supply power of the betatron already leads to serious design difficulties. For example, at an electron kinetic energy $K_{\max}=300$ MeV the quantity $H\rho$ must be equal to $10^6\ \text{erst}\cdot\text{cm}$ (for relativistic electron velocities the simple formula holds: $K=300\,H\rho$, where $K$ is in eV, $H$ in oersteds, $\rho$ in centimeters). It is difficult to count on obtaining alternating magnetic fields $H$ on the orbit line exceeding 5000 oersteds (i.e. magnetic fields at the center of the orbit exceeding approximately 12–15 thousand oersteds). Consequently, the radius of the equilibrium orbit must be no less than 200 cm $\left(r_0=\dfrac{10^6}{5\cdot10^3}\right)$. In accordance with this

the weight of the electromagnet of a 300 MeV betatron must be 2–2.5 thousand tons, and the power supply about 6000 kW. For comparison, let us note that the electromagnet of the world’s only giant cyclotron in Berkeley (California), in which it is proposed to obtain deuterons with energies up to 100 MeV, weighs 4000 t, with a pole-tip radius of 234 cm, and its power supply is 2900 kW.

A very great disadvantage of the 300 MeV betatron is also the fact that, for the resonant power-supply circuit of the electromagnet winding, a colossal bank of capacitors will be required, with a total capacitance of 700 μF, capable of operating at voltages on the order of 35 kV.

The dimensions of the betatron proper and the overall bulk of the whole installation can be considerably reduced by radically changing its mode of operation. With the aid of a special generator, pulsed magnetic fields up to \(10^5\) oersteds can be obtained in a solenoid, if one confines oneself to a small area. If such a field is used for the induction acceleration of electrons, taking the radius of the equilibrium orbit to be, say, 5 cm, then we obtain \(H\rho = 5 \cdot 10^5\) oersted·cm and \(K_{\max} \simeq 150\) MeV. True, the process of accelerating the electrons could not be repeated very often, i.e. rapidly one after another. But instead of a chamber and an electromagnet of enormous dimensions, in this case we would have to deal with a miniature chamber 12–15 cm in diameter and a system of small solenoids (solenoids for such an application must possess high mechanical strength in order to avoid destruction by the forces arising when a current of enormous strength passes through them). The need for a bank of capacitors completely disappears. A machine of the type constructed by Academician P. L. Kapitsa for his investigations of the magnetic properties of bodies in superstrong magnetic fields can serve as the generator for the pulsed power supply of such a betatron.

In view of the power that will be required from the pulsed machine, it is necessary to take as small as possible a radius of the equilibrium orbit and, correspondingly, as large as possible values of the magnetic field \(H\). The use of large magnetic fields, however, encounters one difficulty of a fundamental character. Calculation shows that under these conditions the so-called radiation damping of electrons acquires substantial importance. In motion along a circle, i.e. in the presence of centripetal acceleration, the electron expends part of its energy on the emission of electromagnetic oscillations. This circumstance leads to the fact that the energy of the electron can increase in the betatron only up to a certain limit, depending on the radius of the electron’s trajectory and on the intensity of the magnetic field on this trajectory.

The question touched upon is of great interest and has not yet been discussed in the pages of journals. Therefore we shall briefly present the principal results concerning it, obtained by L. A. Artsimovich.

If we do not analyze in detail the character of the electron motion in the betatron in the presence of radiation damping, then the condition for the end of the process of electron acceleration is written very simply: the electron velocity in the induced electric field ceases to increase when the loss of energy over \(1\ \mathrm{cm}\) of path reaches a value equal to the increase of energy over \(1\ \mathrm{cm}\) of path, i.e.

\[ eE=\Delta W_{\mathrm{radiat.}} . \tag{15} \]

The formula for the radiative losses of energy of an electron was obtained by I. Ya. Pomeranchuk in his work concerning electrons in cosmic rays \(^{23a}\). This formula has the following form:

\[ \Delta W_{\mathrm{radiat.}}=\frac{2}{3}\rho_0^2\left(\frac{K}{m_0c^2}\right)^2 H^2, \]

where

\[ \rho_0=\frac{e^2}{mc^2}=2.8\cdot 10^{-13}\ \mathrm{cm} \]

(the classical radius of the electron), and \(K\) is the kinetic energy of the electron.

Substituting into condition (15) the corresponding expressions for \(E\) and \(\Delta W_{\mathrm{radiat.}}\), one can finally obtain the following formula:

\[ K_{\mathrm{lim.}}=1.3\cdot 10^{15}\frac{f}{H_{\max}^2}, \tag{16} \]

where \(K_{\mathrm{lim.}}\) is the limiting kinetic energy (in eV) which can be imparted to an electron in a betatron, \(H_{\max}\) is the amplitude value of the magnetic field on the equilibrium orbit, and \(f\) is the frequency of the current feeding the electromagnet.

However, as has already been said, condition (15) was obtained without taking into account the real conditions of motion of the electron in the betatron \(^{23b}\). In reality, the presence of radiation damping will disrupt the working process in the betatron much earlier than the equality of the increment and decrement of energy, written in the form of equation (15), is reached. Radiation damping leads to the fact that the radius of the electron orbit in the magnetic field of the betatron does not remain constant: the orbit begins to contract, and the electron moves not along a circle, but along a converging spiral. The condition determining the limiting energy of an electron attainable in the betatron can be obtained by specifying the greatest amount of contraction of the orbit radius that can still be allowed in the chamber of a given design. Calculation shows that optimal results are obtained in the following way: it is necessary to shorten the relative duration of the acceleration process, removing the electrons from the equilibrium orbit before the moment when the magnetic field reaches its maximum, i.e. before the end of the first quarter-period. This is connected with the circumstance that the increase of the electron energy (the quantity \(eE\)) becomes smaller and smaller as the indicated moment is approached, since the intensity of the induced electric field \(E\), as shown

formula (4), when the electromagnet is supplied with a sinusoidal alternating current of frequency \(f\), varies according to the cosine law:

\[ E=\frac{1}{2\pi r_{0}c}\frac{d\Phi}{dt}=E_{\max}\cdot\cos 2\pi ft=E_{\max}\cdot\cos 2\pi\frac{t}{T} \]

and, consequently, at \(t=\dfrac{T}{4}\) it becomes equal to zero.

It is convenient to characterize the relative duration of the working stroke in the betatron by the value of that phase angle \(\varphi=2\pi\dfrac{t}{T}\) at which the electrons are led away from the orbit (i.e., at which the expanding winding is switched on). Calculation gives the following formula:

\[ \frac{\Delta r}{r_{0}}= 6.4\cdot 10^{-16}\, \frac{K_{\max}H_{\max}^{2}}{f(1-n)} \frac{\dfrac{3}{8}\varphi-\dfrac{1}{4}\sin 2\varphi+\dfrac{1}{32}\sin 4\varphi}{\sin\varphi}, \tag{17} \]

where \(\Delta r\) is the reduction of the radius of the equilibrium orbit caused by the presence of radiative braking of the electrons, \(K_{\max}\) is the kinetic energy that the electron would have acquired, moving along a circumference of radius \(r_{0}\) in a magnetic field reaching the value \(H_{\max}\), if radiative braking were absent; \(n\) is the characteristic number in the law of decrease of the magnetic field in the region of the equilibrium orbit [see formula (12)].

For design reasons it is difficult to allow a relative change in the orbit radius exceeding \(20\%\). Taking \(\dfrac{\Delta r}{r_{0}}=0.2\), \(f=50\) Hz, \(n=3/4\), and \(K_{\max}=300\) MeV, one can, with the aid of formula (17), obtain Table 2 below, in which \(K=K_{\max}\times\sin\varphi\) denotes the limiting kinetic energy of the electron attainable at the corresponding magnetic-field strength on the orbit \((H_{\max})\).

Table 2

\(H_{\max}\) in oersteds	\(7.1\cdot10^{4}\)	\(4.1\cdot10^{4}\)	\(3\cdot10^{4}\)	\(2.08\cdot10^{4}\)	\(1.53\cdot10^{4}\)	\(1\cdot10^{4}\)
\(\varphi\) in degrees	23.5	30	37	44.5	53	64
\(K\) in MeV	120	150	180	210	240	270
\(r_{0}\) in centimeters	14	24.4	33.3	48	65.3	100

These figures show, in particular, that at large magnetic fields the dimensions of the chamber are, of course, considerably reduced, but it is necessary to take a small phase angle for the end of acceleration and, accordingly, to be satisfied with a final electron energy several times smaller than that which could have been obtained in the same installation in the absence of energy losses to radiation. These same figures show that if one does not go over to superstrong magnetic fields, i.e., if one reconciles oneself with the enormous dimensions acquired by the “orthodox” betatron when wishing to obtain

in it electrons of ever higher energies, the radiation braking of electrons still does not constitute a substantial obstacle at such energies as 300–500 MeV, so that in principle there are no obstacles to the construction of the corresponding installations.

It is necessary to dwell, at least briefly, on the following interesting detail in the question of the role of radiative energy losses in the operation of the betatron. A loop of wire closed in the form of a circle, through which a constant current flows, of course does not radiate any electromagnetic oscillations into space. Why, then, will an ensemble of electrons filling some circular orbit and moving in one direction along this orbit radiate these oscillations? It turns out that the reason for this lies in the fact that the density of the electron stream in the orbit is comparatively so small that fluctuations of the density play a noticeable role. If the density of the electron stream were always the same for all sections of their circular orbit, then radiation would not take place, since the interference of the oscillations associated with electrons located at diametrically opposite points of the orbit would lead to mutual cancellation of the radiation. The Coulomb mutual repulsion of the electrons is a factor leading to an equalization of the density of the electron stream in the orbit, to suppression of the fluctuations of this density. But if there are comparatively few electrons in the beam, then their Coulomb interaction may be neglected, and then the magnitude of the fluctuations is given by Poisson’s law. In this case, as the calculation made by L. A. Artsimovich shows, \(N\) electrons in the orbit lose to radiation the energy \(\Delta W = N \cdot \Delta w\), where \(\Delta w\) is the radiative loss of energy in the motion of a single electron along the same orbit.

SCIENTIFIC INVESTIGATIONS CARRIED OUT WITH THE AID OF THE BETATRON

The betatron, as an instrument for scientific investigations, is a valuable supplement to the cyclotron. The latter is adapted for the acceleration of heavy particles, but is not suitable for accelerating electrons. The betatron makes it possible to accelerate electrons, and particle energies may be obtained which the cyclotron cannot provide. In the betatron, of course, heavy charged particles, for example protons, may also be accelerated, but in accordance with the enormous difference in the masses of the proton and the electron, the energy attainable in a given installation is considerably reduced. For example, in a 20 MeV betatron, where \(H \rho = 6.82 \cdot 10^{4}\) oersted·cm (see Table 1), the final energy of protons would be only \(\sim 0.2\) MeV.

The installation, designed to generate electrons with energies up to 20 MeV, the construction of which in the laboratories of the General Electric Company was completed in the second half of 1941, was then transferred to the territory of the University of Illinois, where

and its use for scientific research began. So far only a rather small number of reports on some of the results obtained have been published. We shall briefly set them forth.

The bremsstrahlung X-radiation arising in the target of the betatron has a continuous spectrum with an upper limit equal to the energy of the electrons incident on the target. In Kerst’s laboratory several methods were developed for controlling the moment at which the expanding winding of the betatron is switched on, i.e. methods for varying the final energy of the electrons which they acquire by the moment of impact on the target. One of these methods, which makes it possible to adjust the betatron with great accuracy for the generation of electrons with a preassigned value of the maximum kinetic energy, was used in the work of Baldwin and Koch^24a, who determined, for a number of elements, the so-called threshold of the nuclear photodisintegration reaction. Such a reaction, otherwise called a reaction of the type \((\gamma, n)\), consists in the fact that a \(\gamma\)-photon possessing sufficient energy, on colliding with an atomic nucleus, tears a neutron out of it. The composition of the new nucleus formed in most cases corresponds to an unstable system of nuclear particles, i.e. the nucleus proves to be artificially radioactive. The presence of radioactive radiations, established with the aid of a Geiger counter or by another method, makes it possible to ascertain the fact that a \((\gamma, n)\) reaction has taken place. The magnitude of the energy threshold of this reaction, i.e. the smallest energy of \(\gamma\)-quanta at which the reaction still occurs, is of great interest, since it is a measure of the binding energy of the neutron in the given nucleus.

Having recorded the activity curve of a sample irradiated with X-rays from the betatron as a function of the maximum energy of the X-ray quanta (which, we repeat, can be smoothly varied in the betatron at will), it is easy to determine the threshold of the \((\gamma, n)\) reaction. Baldwin and Koch obtained the following results:

Nucleus and half-life period	\(\mathrm{C}^{11}_{6}\) (21 min.)	\(\mathrm{O}^{15}_{8}\) (2 min.)	\(\mathrm{Fe}^{53}_{26}\) (9 min.)	\(\mathrm{Cu}^{62}_{29}\) (10 min.)	\(\mathrm{Mo}^{(91,92)}_{42}\) (17 min.)
Threshold of the \((\gamma,n)\) reaction, as a result of which the given nucleus is formed (in MeV)	18.7—19.5	16.3—16.7	14.1—14.3	10.8—10.9	13.3—13.7

The authors indicate that the figures they obtained are correct to an accuracy of up to 40 keV. Such precise measurements of the thresholds of the \((\gamma, n)\) reaction—measurements in the region of such high \(\gamma\)-quantum energies as 19—20 MeV—were carried out for the first time, since only the betatron provided such a possibility. It should be noted that the results presented are very

unexpected: it is known that the binding energy of a neutron in a nucleus averages 6–8 MeV, whereas the figures indicated above all considerably exceed this value¹).

Another work carried out with the aid of the betatron is an investigation of the absorption of hard X-rays in various substances. A theoretical calculation of the absorption of such rays had been given in a number of works, but an experimental verification of the results was performed (quite recently) only for photons with energy not exceeding 2.5 MeV²⁵.

Theory shows that, as the energy of γ-quanta increases, an additional mechanism of their interaction with matter comes into play—the so-called production of electron–positron pairs; these two particles receive the energy of the γ-quantum, while the γ-quantum itself ceases to exist. Pair production begins at approximately a photon energy of 1 MeV, and with further increase in photon energy the probability of such a process increases. This leads to the result that an increase in the energy of γ-quanta does not always lead to an increase in their penetrating power: beginning with a certain optimal energy of the γ-quanta, further increase of their energy leads not to a decrease but to an increase in the absorption coefficient of these rays in matter. The predictions of the theory were checked with the aid of the betatron by Adams and Clark²⁶. The absorption of the hardest part of the bremsstrahlung spectrum of the target was studied. The detector of the hard γ-quanta that had passed through the absorber was an artificially radioactive nucleus \( \mathrm{Cu}^{62} \), formed in the reaction \( \mathrm{Cu}^{63}(\gamma,n)\mathrm{Cu}^{62} \), the threshold of which is, as we have seen, approximately 10.8 MeV. As absorbers, Al, Pb, and other substances were investigated. The following linear absorption coefficients \( \tau \) were obtained:

Absorber	\( \mathrm{Be}_{4} \)	\( \mathrm{C}_{6} \)	\( \mathrm{Al}_{13} \)	\( \mathrm{Fe}_{26} \)	\( \mathrm{Pb}_{82} \)
\( \tau \) (cm\(^{-1}\))	0.0278	0.0292	0.0601	0.234	0.635

With an accuracy of up to 1%, these figures agree with the theoretical values calculated by Heitler. Individual measurements showed, in addition, that for lead the decrease in the penetrating power of γ-rays begins at \( E_{\gamma} > 3 \) MeV.

It should be noted that the course of the curve of the γ-ray absorption coefficient as a function of their energy depends substantially on the method used to measure the intensity of the radiation that has passed through the absorber: whether a counter or an ionization chamber, etc., is used for this. This is connected with the fact that behind the absorber there are not only the primary γ-quanta that have passed through it, but also electrons and secondary γ-quanta—the products of the interaction of the radiation with matter.

¹) In a later note²⁴ᵇ the same authors give values of the nuclear photoeffect threshold for five more nuclei; these figures lie between 9.3 and 11.6 MeV.

The effect of the decrease in penetrating power that sets in at a certain value of the energy of $\gamma$-quanta would seem to destroy the hope of industrial use of high-voltage betatrons for the radiography of metal parts of such thickness as had previously already proved inaccessible to investigation by means of X-rays. Here, however, one should remember that there are two favorable circumstances: 1. the concentration of all the $\gamma$-quanta from the betatron target into a narrow beam, which occurs at high energies of the electrons incident on the target, and 2. the very high efficiency of the production of X-radiation at these energies: about 65% of the energy of the electron beam incident on the target is converted into X-ray energy when operating at 20 MeV. Owing to these circumstances, the beam of rays emerging from the betatron chamber has so great an intensity that even after very large absorption in a thick layer of material it may still remain of quite sufficient intensity to make radiography of the given object possible. The builders of the 100 MeV betatron propose to use it, in particular, for the needs of the military industry, namely, for radiography of the thickest armor plates[^22]. In addition, having at one’s disposal an installation that makes it possible to generate X-rays with energies from 1 to 100 MeV, it will be possible to establish up to what limit it still makes sense to raise the voltage in X-ray installations constructed for defectoscopic purposes.

Finally, with the aid of the betatron, an investigation has been carried out in the field of roentgenotherapy[^27]. Working with high-voltage X-rays with maximum energies of 5, 10, 15, and 20 MeV, and using as a model of the tissues of the human body a stack of wooden plates, one of which had an aperture with a small ionization chamber inserted into it, the authors obtained curves of the intensity of the ionizing action of X-rays as a function of depth, measured from the front surface of the model. In other words, the depth doses of irradiation were experimentally determined. It turned out that the distribution of ionization when such hard X-rays are used possesses a number of features very favorable for deep therapy. The maximum dose is obtained at a depth of 3–4 cm, and it may be several times greater than the dose at the surface. With increasing energy of the X-ray quanta the ionization maximum is displaced into the depth of the model. The authors point out that an even more valuable means for deep therapy would be a beam of electrons with energy 25–30 MeV. Ionization would reach its maximum at a depth of 7–8 cm, and the destructive action could be well localized inside the body, since electrons penetrate to a definite depth ($\sim 10$ cm) and no farther. However, hard electrons, in large numbers flying out of the betatron chamber owing to scattering from the electron beam and from the target, are not concentrated into a beam—they fly in all directions.

in all directions and therefore cannot be used for practical purposes.

The betatron is an excellent source of a directed beam of very hard X-ray quanta, which is extremely convenient both for purposes of nuclear research and for purposes of X-ray defectoscopy. However, obtaining a controllable beam of fast electrons is an equally urgent problem of physics. Undoubtedly, in the future the design of betatrons will make it possible to lead the electron beam out of the chamber, so that the betatron will be not only an X-ray apparatus of a special type, but also a source of a beam of electrons possessing enormous energies.

References

1. F. F. Lange and V. S. Shpinel, Izv. AN SSSR, ser. fiz., 4, No. 2, 353, 1940.
2. N. S. Khlebnikov, Uspekhi fizich. nauk, 22, 427, 1939.
3. R. Wideröe, Arch. f. Elektrotechnik, 21, 387, 1928.
4. D. H. Sloan and W. M. Coates, Phys. Rev., 46, 539, 1934.
5. G. Ising, Ark. f. Math., Astron. och Physik, 18, No. 30, H. 4, 45, 1925.
6. J. W. Beams and H. Trotter, Phys. Rev., 45, 849, 1934.
7. V. M. Kelman, M. I. Korsunskii and F. F. Lange, ZhETF, 9, 944, 1939.
8. E. O. Lawrence and M. S. Livingston, Phys. Rev., 40, 19, 1932.
9. V. I. Veksler, DAN, 43, 346, 1944. See also DAN, 44, 393, 1944.
10. M. Steenbeck, DRP No. 698867, 1940 (application 1935); USP, No. 2103303, 1937 (application 1936).
11. D. W. Kerst, Phys. Rev., 60, 47, 1941.
12. V. V. Yasinskii, ZhETF, 5, 983, 1935.
13. Ya. P. Terletskii. a) ZhETF, 11, 96, 1941; b) Journ. Phys. USSR, 9, No. 3, 1945 (in press).
14. D. W. Kerst and R. Serber, Phys. Rev., 60, 53, 1941.
15. J. Slepian, USP No. 1645304, 1927 (application 1922).
16. G. Breit, O. Dahl and M. A. Tuve. See in the article: L. A. Bauer and J. A. Fleming, Carnegie Institution of Washington, Year Book No. 27, 209, 1928.
17. E. T. S. Walton, Proc. Camb. Phil. Soc., 25, 469, 1929.
18. R. Rüdenberg and M. Steenbeck, DRP No. 656 378, 1938 (application 1933).
19. M. Steenbeck, Naturwiss., 31, 234, 1943.
20. A. Bouwers, Elektrische Höchstspannungen. J. Springer, 1939, p. 83.
21. D. W. Kerst, Rev. Sci. Instr., 13, 387, 1942.
22. H. Asbury, Popular Science, 141, No. 6, 58, 1942.
23. a) I. Ya. Pomeranchuk, ZhETF, 9, 915, 1939; b) D. Ivanenko and I. Pomeranchuk, DAN, 44, 343, 1944.
24. G. C. Baldwin and H. W. Koch, a) Phys. Rev., 63, 59 (A), 1943; b) Phys. Rev., 63, 462 (A), 1943.
25. A. A. Petrauskas, L. C. Van Atta and F. E. Myers, Phys. Rev., 63, 389, 1943.
26. G. D. Adams and R. K. Clark, Phys. Rev., 63, 60 (A), 1943.
27. H. W. Koch, D. W. Kerst and P. Morrison, Radiology, 40, 120, 1943.
28. (G. M. Almy) Sci. News Letter, 45, 372, 1944.

1. Recently a report has been published[^23] on the forthcoming construction in the USA of a “postwar betatron” for 250 MeV. ↩↩↩

2. In this apparatus, as also in the subsequent models, the chamber operates with continuous pumping, i.e. a system of vacuum pumps is provided in the installation. ↩