BETATRON1

D. W. Kerst

In nuclear physics, in order to carry out experiments on the artificial disintegration of atoms, several types of apparatus are used, differing in the way in which the bombarding particles are given the necessary large kinetic energies. Apart from the disintegration of nuclei by particles emitted by natural radioactive preparations, the installations for producing fast particles are either linear accelerators or cyclotrons. In a linear accelerator the accelerated particle is made to move from electrode to electrode along the axis of a long vacuum tube. The electrodes of such a tube are usually hollow cylinders held at potentials increasing from one end of the tube to the other. The high voltages are supplied either by large high-voltage rectifiers or by Van de Graaff electrostatic generators, whose operation is based on charging a well-insulated electrode by charges carried onto it by an endless belt of insulating material. One of the poles of the linear accelerator is connected with this high-voltage electrode, while the other pole is usually maintained at earth potential. Installations of this type have now been so perfected that they can produce potential differences of approximately \(4.5 \cdot 10^6\text{ V}\) with generator dimensions of no more than \(1.65\text{ cm}\) in diameter and \(6.6\text{ m}\) in length, provided that the generator is in an atmosphere of compressed air at a pressure of about 8 atm. (the compressed air provides good insulation of the high-voltage electrode).

A linear accelerator fed by electrostatic generators has been used to some extent for studying nuclear reactions produced by electrons and X-rays with energies up to \(3 \cdot 10^6\text{ eV}\), i.e. about \(5 \cdot 10^{-6}\) erg, but it has mainly been used for work with positive ions. Despite the fact that there is a large number of nuclear reactions arising under the action of X-rays (“photodisintegration”), their realization requires quantum energies from \(6 \cdot 10^6\) to \(8 \cdot 10^6\text{ eV}\). Thus, for these reactions it is necessary to have electron energies greater than those attainable with an electrostatic generator.

In installations of the cyclotron type, positive ions do not move along straight-line paths. They are deflected by a magnetic field and describe

spiral trajectories, being accelerated between two hollow semicircular electrodes to which a high-frequency alternating voltage is applied. The final value of the kinetic energy acquired as a result of this process can be determined as the product of the number of elementary charges of the positive ion, the number of passages from electrode to electrode, and the potential difference between the accelerating electrodes. With the aid of such installations it has been possible to accelerate deuterons to energies of \(16.5 \cdot 10^6\) eV. These values are considerably higher than the maximum potential to which the insulated electrode of an electrostatic generator can be charged.

However, the large accelerations obtained with a cyclotron are impracticable when working with electrons instead of positive ions. The reason for this is that electrons, even at comparatively very small values of kinetic energy, exhibit relativistic behavior. Every particle begins to behave relativistically when its kinetic energy approaches or exceeds the value corresponding to its mass. Thus, for example, the rest mass of the proton is equivalent to approximately \(10^9\) eV; therefore a proton with this value of kinetic energy will behave as a relativistic particle. On the other hand, an electron, with a rest mass equivalent to only \(5 \cdot 10^5\) eV, begins to display relativistic behavior already at these low energy values, since even then it appears to be moving with a velocity equal to about \(0.9\) of the upper limit of velocity—the velocity of light.

One of the conditions for the operation of a cyclotron is that the velocity of the accelerated particle increases in proportion to the square root of the number of passages between the accelerating electrodes; fulfillment of this condition for an electron, whose velocity approaches the upper limit, proves impossible. The result of this was that, with the aid of the cyclotron, it proved possible to accelerate electrons only to energy values substantially less than half a million electron-volts.

Fortunately, at present we have another instrument—the betatron—whose action does not depend on how the accelerated particle behaves, relativistically or classically. The energy which the betatron of the latest design can impart to electrons amounts to \(20 \cdot 10^6\) eV. Such high energy values make it possible to obtain correspondingly high values of the energy of X-ray quanta, so that it becomes possible to carry out nuclear reactions requiring energies of \(6 \cdot 10^6\)—\(6 \cdot 10^8\) eV.

Generally speaking, in photodisintegration the phenomenon consists in the fact that an X-ray quantum, or \(\gamma\)-photon, with an energy greater than the binding energy of a neutron in the nucleus of the atom being disrupted, reacts with this nucleus, ejecting a neutron in the same way as a light quantum tears an electron from an atom of the photosensitive material of the cathode of an ordinary photoelement with the external photoelectric effect. The new atomic nucleus produced as a result of this reaction often proves to be radioactive. In this case it is easy to determine the magnitude of the binding energy of the neutron in the initial nucleus from the value of the energy which the betatron must impart to the bombarding electrons in order for the formation of radioac-

tive substance. Fast electrons can be used to knock out a neutron directly as well, since the nucleus reacts to the passage of a fast electron with its accompanying electric field in generally the same way as to the passage of a photon of high energy.

Since the betatron is a source of very penetrating X-rays and of electrons of such high energy that they can, for example, penetrate into the human body to approximately half its thickness, applications of the betatron prove to be of interest not only from the standpoint of experiments in the field of nuclear physics, but also have direct practical significance. X-rays are widely used for industrial and therapeutic purposes, and the radiation produced by the betatron is more penetrating than that obtained by any other methods.

The high-energy electrons emitted by the betatron as a result of scattering from the target, where X-rays arise, form a very intense but weakly directed beam. These electrons, when directed into the human body, produce along their path in it ionization having the same destructive action as the X-rays now used in the therapy of deep-seated tumors. One of the shortcomings of X-rays is that they are not wholly absorbed inside. They pass through and therefore produce their effect everywhere: at the site of penetration of the tumor itself and at the exit site. To eliminate this, various techniques are used in attempts to create optimal ionization in the region of the deeply situated malignant tumor.

A beam of electrons with high penetrating power is free from these shortcomings, since the depth of its penetration is determined by the energy of the particles. Electrons with energies of \(2 \cdot 10^7\) eV penetrate into the human body to 10 cm and no deeper. Dr. F. Morrison has determined that, under these conditions, they produce maximum ionization at a distance of 7 to 8 cm from the surface. This means that, in all probability, the destruction produced by the electrons can be localized precisely in the tumor itself; the injuries at the point of entry of the beam will be small and will be completely absent beyond the tumor being irradiated. When another method is found for releasing the electron stream from the betatron (instead of scattering from the target), this stream will be more homogeneous in energy and less strongly divergent. This will make the betatron more applicable for medical purposes.

Action of the Betatron

In some respects, since the betatron is a magnetic device, it resembles a small cyclotron. Its essential difference from the cyclotron is that its operation requires the creation not of a constant, but of an alternating magnetic field. The theory of the betatron shows that the appreciable relativistic effects which appear when the velocity of the electrons approaches the speed of light do not in any way affect its operation.

Electrons from an electron gun, called an injector, are guided along a circular trajectory by a magnetic field of small intensity. During the circular motion of these electrons between the poles of the magnet, the magnetic field increases, and the change in the magnitude of the magnetic flux penetrating the orbit gives an increment of energy for one complete revolution. This gain in energy, expressed in volts, is equal,

[Figure annotations: electron orbit; tube; target; focusing box; filament; X-rays; coils expanding the orbit; magnetic flux; injector supply; curve H; 1 sec/720; 180 cycles, alternating current.]

Fig. 1. A vacuum chamber in which the acceleration of electrons takes place. The electrons entering from the injector onto the equilibrium orbit in fact, before reaching it, make many revolutions inside the chamber. The same is true of the electrons leaving the orbit and striking the target. The electrons leave the injector at instant A, and the orbit expands at instant B of each cycle.

obviously, to that instantaneous value of the potential difference which would be measured by a voltmeter connected to a single turn of wire arranged along the electron orbit.

In Fig. 1 is shown the diagram of a circular vacuum chamber in which the electrons describe a series of circular trajectories, accumulating energy at each revolution and ultimately striking a target located behind the injector, thereby giving rise to X-radiation.

The solenoids used to expand the orbit are not supplied with current until the electrons have been accelerated to the required extent. After they are switched on, they distort the distribution of the magnetic flux near the trajectory of the electrons and force them, moving along spiral paths, to strike the target.

BETATRON

target. The electrons are introduced into the magnetic field at the moment \(A\), marked on the curve \(H\) in Fig. 1, and the expansion of the orbit up to its intersection with the target takes place at the moment \(B\), when the energy of the electrons reaches its maximum. These processes are repeated during each cycle.

An increase of the induction flux through the orbit creates an additional amount of motion in the electron, so that if the field strength on the electron orbit did not increase simultaneously, the orbit would expand and very soon the electron would strike the wall of the chamber. To keep the electron on its orbit it is necessary to increase the magnetic-field strength \(H\) in proportion to the increase of the momentum \(mv\), caused by the growth of the induction flux. As we shall see, this requires a special character of the distribution of the magnetic-flux density. The radius of curvature \(r\) of the orbit is connected with the electron momentum and the magnetic-field strength by the relation

\[ mv=\frac{e}{c}Hr, \tag{1} \]

where \(e\) is the charge of the electron and \(c\) is the speed of light. According to Newton’s second law, the time derivative of the momentum is equal to the force acting on the electron, i.e.

\[ \frac{d(mv)}{dt}=f. \]

But, in turn, the force \(f\) is equal to the increase of the kinetic energy of the electron per unit length of path, which, if it is assumed that the electron orbit has an unchanged radius \(r\), is expressed as

\[ \frac{1}{2\pi r}\cdot\frac{e}{c}\Phi, \]

where \(\Phi\) is the magnetic flux piercing the orbit. Thus,

\[ mv=\int_{t_0}^{t} f\,dt =\frac{e}{2\pi rc}\int_{t_0}^{t}\dot{\Phi}\,dt =\frac{e}{2\pi rc}\int_{\Phi_0}^{\Phi} d\Phi \]

or

\[ mv=\frac{e}{c}(\Phi-\Phi_0)\frac{1}{2\pi r}; \tag{2} \]

this equality shows that the momentum of the electron is proportional to the change of flux through its circular orbit.

Combining equations (1) and (2), we have

\[ \Phi-\Phi_0=2\pi r^2H. \tag{3} \]

This means that \(\Phi_0\) is equal to zero when \(H\) is equal to zero, and that the flux \(\Phi\) is proportional to the field strength \(H\) on the orbit and must always be twice as large as that which would exist with \(H\) uniform over the entire area of the orbit.

The result just given was obtained under the assumption that \(r\) is a constant quantity. Naturally, before constructing the accelerator, it was necessary to make sure of the validity of the converse, i.e. that, for a given magnetic flux, the orbit would have a constant radius.

We do not in any way specify the character of the dependence of the magnetic flux and the field strength on time. It is necessary only that the flux \(\Phi\) increase with time and that the field strength \(H\) change proportionally. This is easy to achieve for \(H\) and \(\Phi\) in the air gap of one and the same magnetic circuit. Under real conditions, the electromagnet with its winding forms the self-inductance of an oscillatory circuit. A large number of capacitors is used to obtain resonance at the required frequency. The first betatron (built at the University of Illinois) operated at a frequency of 600 Hz, while the new installation of that university, which gives electrons with energies up to \(20 \cdot 10^6\) eV, operates at a frequency of 180 Hz.

The necessary condition for the formation of a beam of fast electrons in a betatron is that electrons deflected from their orbit as a result of collisions with molecules of the residual gases must return to their orbits under the action of focusing forces. Each scattered electron must execute oscillations about its orbit, which may be called the “equilibrium orbit,” and the amplitude of these oscillations must decrease.

Fig. 2. The force \(F_c \left[ = \dfrac{mv^2}{r} \right]\) is the centripetal force required to keep an electron on a circle of radius \(r\); \(F_m \left[ = \dfrac{e}{c}Hv \right]\) is the force with which the magnetic field actually acts on the electron. The equilibrium orbit corresponds to the abscissa \(r = r_0\).

The conditions for these oscillations can be realized by creating a suitable distribution of the magnetic field. For oscillations to arise in the axial direction, the magnetic lines of force between the poles of the electromagnet must bend outward. In such a field, an electron that has left the median plane finds itself in a magnetic field with a small radial component. This radial component is directed oppositely on the two sides of the median plane and therefore always directs the electrons toward this plane, regardless of where the electron has rolled off. To obtain the required outward bending of the magnetic lines it is sufficient to give the pole pieces such a shape that the distance between them increases as one moves away from the axis of the gap between the poles. In practice, the pole pieces are given a conical shape over almost the entire cross section. A small rim is left only at the periphery of each pole piece, in order to create a slower weakening of the field, which falls off very sharply at the edges.

The condition for the existence of oscillations in the radial direction is that the magnetic field must decrease no faster than \(1/r\). This can be understood by referring to Fig. 2, which depicts the centripetal force \(F_c\), required to keep an electron on an orbit of radius \(r\).

as a function of \(r\). This dependence is represented by a hyperbola, since \(F_c=\dfrac{mv^2}{r}\).

The force created by the magnetic field is equal to \(F_m=\dfrac{e}{c}Hv\). In existing betatrons the changes in \(v\) over several focusing oscillations are so insignificant that they may be neglected. Consequently, if \(F_m\), and hence also \(H\), has a dependence on the radius represented by the second curve in Fig. 2, it will create a centripetal force greater than the required \(F_c\) when \(r\) is greater than \(r_0\), and smaller in the opposite case. If the electron is outside the equilibrium orbit, then it finds itself in a region where the magnetic-field strength is greater than is necessary for producing a circular path. Therefore the electron will shift in the direction of the equilibrium orbit, but after crossing it will find itself in a region where the magnetic field is insufficient for producing a circular path, and it will again shift in the direction of the equilibrium orbit. These oscillations about the equilibrium orbit gradually die out because the magnetic field increases during each period of oscillation, while the amplitude of the oscillations is proportional to \(H^{-1/2}\). This effect of the increasing magnetic field on the amplitude may be likened, to a certain degree, to an increase in the stiffness of a spring on which an oscillating mass is suspended. The damping properties of the increasing magnetic field make it possible to introduce the electrons into the accelerating chamber in such a way that they arrive at a definite orbit.

The focusing effects considered lead to the formation of a narrow electron beam striking the target over a very small part of its area. Thanks to this, the X-rays emitted by the target give very sharp shadows. The target does not melt under the action of the heat released by the electron beam for the reason that the beam current is very small. A beam of intensity about \(1\,\mu\text{A}\), at an energy of \(20\cdot10^6\) eV, produces \(16\) r/min at a distance of \(1\) m. At this energy \((2\cdot10^7\ \text{eV})\) the efficiency of the beam in producing X-rays is so great that about \(65\%\) of the total energy of the beam is converted into the energy of X-ray quanta and only \(35\%\) into heat heating the target.

Fig. 3. New betatron, giving electrons with energies up to \(20\cdot10^6\) eV. The light spot on the chamber between the poles of the electromagnet is due to the injector.

The new betatron, a photograph of which is shown in Fig. 3, has pole pieces 47.5 cm in diameter and an equilibrium orbit 18.75 cm in diameter. The electromagnet is of comparatively small dimensions: 900 cm in height and 1500 cm in length; its weight, however, is 3.5 tons. The power required to generate an electron beam with electron energy of \(2 \cdot 10^7\) eV at a frequency of 180 Hz is about 25 kW. Cooling of the magnetic circuit is carried out by means of a fan, which can be seen in Fig. 3 at the lower part of the electromagnet.

The apparatus described is suitable not only for producing X-rays of high penetrating power, but also for producing artificial radioactivity, which arises in many substances as a result of photodisintegration. The energy of the electrons also proves sufficient for carrying out, on a small scale, certain experiments on cosmic rays.

Literature

1. D. W. Kerst, Phys. Rev., 60, 47, 1941; D. W. Kerst and R. Serber, Phys. Rev., 60, 53, 1941.

1. Amer. Journ. of Phys’cs, 10, 219, 1942; translated by N. S. Khlebnikova. ↩