LINEAR ACCELERATORS

D. W. Fry and W. Walkinshaw*)

§ 1. Introduction

The production of particles of high energy is very important for nuclear physics and for medicine (therapy and other fields). For medical purposes, particles with energies up to approximately 50 MeV are of interest. For research in nuclear physics, particles with energies of 500–1000 MeV and higher are needed. It is not surprising, therefore, that since about 1930 much attention has been devoted to the development of the most rational methods of accelerating particles.

In the early methods, particles were accelerated either by a constant electric field or by electric fields of low frequency. Lauritsen and Bennet1 used for this purpose several high-voltage transformers connected in series; Cockcroft and Walton[^3] used a series connection of a whole set of high-voltage mercury rectifiers; and, finally, Van de Graaff used the principle of the electrostatic machine for accelerating particles. All these three methods have since been widely used for accelerating particles, especially the electrostatic generator (on this subject see the review[^15]).

Such accelerators make it possible to obtain comparatively intense particle beams with a small energy spread.

However, the circumstance that the energy acquired by a particle is equal to the potential difference between the insulated electrode and the earth makes it impossible to obtain sufficiently high energies.

To overcome this difficulty, a number of new acceleration methods were proposed, for example acceleration by means of the cyclotron. In the cyclotron (Lawrence and Edlefsen[^6]) positive ions move along semicircles of increasing radius in space—

between the poles of a large electromagnet fed by direct current. The particles are accelerated in an alternating electric field of constant frequency and relatively small amplitude, which is applied between two electrodes of special shape; the potential difference between them is usually of the order of 10,000–100,000 volts. The particles are accelerated at the moments when they pass between the electrodes, i.e., twice per revolution. After a number of revolutions the energy of the particles becomes considerably greater than the potential difference applied to the accelerating electrodes. In order to accelerate positive ions to velocities close to the speed of light, Brobeck et al.^2 modulated the frequency in the cyclotron*). In this way it proved possible to obtain alpha particles with an energy of 400 MeV.

Particles can also be accelerated by another method, by successively passing them through a series of cylinders between which certain alternating potential differences are applied in a definite phase. Wideröe^94 used this method to accelerate potassium ions to 50 kV (the maximum voltage between electrodes was 25 kV). This method was later developed by Sloan and Lawrence^92 and by Sloan and Coates^91, who accelerated mercury ions to energies of 1.26 and 2.85 MeV. To accelerate particles in rectilinear motion, Beams and Snoddy^24,25 connected the electrodes not to a high-frequency generator, but to the corresponding points of a loaded long line, to which voltage pulses were applied. The speed of propagation of the pulse along the line was selected in the experiments to be the same as the speed of the particle moving through the system of electrodes. Therefore, as they passed through the gaps between the electrodes, the particles were each time accelerated by a voltage pulse from the line. In this way protons with energies of several million volts were obtained.

In the method of Beams and Trotter^25 the voltage was supplied from the generator to the electrodes by means of lines of different lengths, along which the voltage pulses propagated with a speed of the order of \(c\). The phasing of the voltage on the electrodes, ensuring continuous acceleration, was achieved by a corresponding choice of the line lengths.

Another accelerator that made it possible to avoid the difficulty of high-voltage insulation was the induction accelerator, or betatron. This device was used very successfully by Kerst^4 and many other authors for accelerating electrons. The betatron proved to be a very effective accelerator—with its aid it was possible to accelerate electrons to energies of 100 MeV^13.

In recent years several more methods of particle acceleration have been proposed, some of which have been successfully implemented—

) Frequency modulation in the cyclotron was proposed by V. I. Veksler in 1945. (Ed. note.*)

... are known\(^{9,19}\). The most important of them is the synchrotron method (Veksler\(^{16}\), McMillan\(^{8}\), Oliphant et al.\(^{10}\)). In a synchrotron the magnetic field that ensures the rotation of the particles increases with time; the acceleration is produced by a high-frequency electric field independent of this magnetic field. Among other cyclic accelerators we shall mention the microtron (Veksler\(^{16}\), Schiff\(^{15}\)) and the bevatron (Post\(^{12}\)).

The present review is devoted to the development of linear accelerators over the last three or four years. In most of the work carried out during this period on electron acceleration, centimeter and decimeter waves were used. Recently a single accelerator operating at a wavelength of \(1.5\ \mathrm{m}\), intended for the acceleration of heavy particles (protons), has been described.

In designing accelerators for heavy particles, technical problems arise that are not encountered in electron accelerators. This is connected with the fact that, at the same kinetic energy as electrons, heavy particles move with a lower velocity. These problems are briefly considered in § 10.

Recently, interest in linear accelerators has been revived in connection with the considerable successes achieved in the development of ultrahigh-frequency generators. It has become possible to build generators for frequencies of \(150\)—\(3000\ \mathrm{MHz}\) with a pulse power of \(0.1\)—\(2.0\ \mathrm{MW}\), at a repetition rate of 500 pulses per second and with a pulse duration of the order of several microseconds. Such high pulse powers and the new technique of high-frequency circuits associated with them have created new possibilities for accelerating particles by means of linear accelerators. Thus, in terms of its technical and economic characteristics this acceleration method has proved capable of competing with certain other acceleration methods\(^{13,78}\). The chief advantages of the linear-acceleration method are as follows: first, small radiation losses for relativistic particles and, second, the simplicity of introducing particles into the accelerator and extracting them from it at comparatively large currents. In addition, the cost of a linear accelerator is approximately proportional to the maximum energy of the particles, whereas the cost of a cyclic accelerator of any type is proportional, at least, to the square, and possibly to the cube, of the maximum energy. For the acceleration of electrons the first circumstance is especially important. As theory shows (see, for example,\(^{11}\)), in its motion in a circular orbit a relativistic particle loses energy to radiation:

\[ V_R=\frac{6\cdot 10^{-9}}{R}\cdot\left(\frac{V}{V_0}\right)^4\ \mathrm{eV}, \tag{1} \]

where \(R\) is the orbit radius in meters, \(V_R\) is the energy loss to radiation per 1 revolution, \(V\) is the total energy of the particle in electron-volts, and \(V_0\) is its rest energy (also in electron-volts).

At electron energies of the order of \(10^9\) eV in a cyclic accelerator of any type, for example in a synchrotron with a stable-orbit radius of \(3\) m, the radiation losses amount to about \(30\cdot 10^3\) eV. This quantity increases as the fourth power of the particle energy, and at energies greater than \(2000\)—\(3000\) MeV becomes already excessively large. Such energy losses do not arise in rectilinear motion.

There is as yet no satisfactory method for calculating the maximum current that can be injected into a linear accelerator; however, as the experiments of Fry\(^{38}\) have shown, in a two-meter electron accelerator at \(4\) MeV it is possible, for accelerating the beam, to utilize \(30\%\) of all the high-frequency power supplied to the accelerator. In accelerators of greater length, in whose calculation the losses of power for beam acceleration were not taken into account, it is possible to use a significantly smaller part of the high-frequency power, since an increase in the power consumed by the beam would lead to a lowering of the maximum particle energy. In this case a realistic and feasible figure is apparently \(10\)—\(15\%\). If, in calculating the accelerator, the load represented by the beam is taken into account, then the beam can receive an even larger share of the total power expended. In the present review it is shown (see below) that in a long accelerator the value of the effective shunt impedance is \(30\) MΩ/m. With a maximum energy of \(300\) MeV and a length of \(100\) m, the required pulse power is \(45\) MW (without taking into account the loading of the accelerator by the beam). If the beam receives \(10\%\) of this energy, then the average current in the pulse will be \(15\) milliamperes; at a duty factor of \(2500\) this corresponds to an average current of \(6\) μA.

As is known from experiment, the average output current that can be obtained in a betatron and a synchrotron is considerably smaller than this value. Blewett and others have determined the maximum current from the condition of equality between the repulsive force between charges and the focusing force of the magnetic field. The average value of this current is determined by the following formula:

\[ I_{\mathrm{av}}=\frac{mc^3}{e}\frac{\beta^3}{120}\left(\frac{\Delta}{r_0}\right)^3\frac{2\pi r_0 f}{c}\frac{1}{(1-\beta^2)}\ \text{ampere}, \tag{2} \]

where \(\frac{mc^2}{e}\) is the rest energy plus the kinetic energy possessed by the electrons at the moment of injection (in electron-volts), \(\beta\) is the ratio of the initial velocity of the electrons to the velocity of light, \(\Delta\) is the mean radius of the vacuum chamber, \(r_0\) is the radius of the stable orbit, \(f\) is the frequency of variation of the magnetic field. A \(300\) MeV synchrotron, with an initial particle energy of \(100\,000\) eV, \(\Delta=7.5\) cm, \(r_0=125\) cm and \(f=5\) cps, should give an average current of \(1.0\) μA. In practice, as has been verified for betatrons at lower energies,

for example, at 20 MeV, the average current never exceeds \(1/10\) of its theoretical value, and often it is still smaller—about \(1/100\) of the theoretical value. Thus we see that the average current of a betatron or synchrotron is from \(1/60\) to \(1/600\) of the current that can be expected from a linear accelerator.

For the reasons indicated, and also because of the expected cheapness of a linear accelerator in comparison with accelerators of other types, work has been undertaken in a number of laboratories to evaluate the possibilities opened up by the method of linear acceleration for particles with energies from several MeV up to 1000 MeV, as well as the limits of applicability of this method.

The most serious difficulty at present limiting the use of linear accelerators for obtaining particles of ultrahigh energies is the absence of sufficiently powerful pulsed tubes that would make it possible to reduce to a reasonable value the total number of all the tubes of the accelerator.

This review consists of two parts. The first part is devoted to recent achievements in the field of linear electron accelerators. The second part is devoted to heavy-particle accelerators. However, in this latter field considerably fewer papers have been published than on electron accelerators.

Part I

ELECTRON ACCELERATORS

§ 2. Dynamics of particles in high-frequency fields

2.1. Phase stability

The principal task of a linear accelerator is to create an electromagnetic wave with a varying phase velocity, less than or equal to the speed of light, whose electric vector would have a component in the direction of propagation. Particles entering such a field in the appropriate phase receive energy from the wave and are grouped into stable bunches. What is needed first of all is a method for obtaining waves of this kind. Waves of this type—with a longitudinal component of the electric vector—propagate in circular cylindrical waveguides with smooth walls; the theory of such waveguides is contained in many textbooks. However, the phase velocity of waves in a waveguide with smooth walls exceeds the speed of light. In this connection, a number of effective methods for slowing waves have been proposed. In all these methods, in addition to the desired wave, other parasitic waves also arise—higher-order

components of the Fourier expansion of the field*), which have still smaller phase velocities, some of them moving in the direction opposite to the principal wave. Since the velocity of these waves differs greatly from the velocity of the particles, on the average they do not change the energy of the particles. Considering the particle dynamics in the first approximation, all waves except the principal one may be neglected. In this approximation the phase stability of the particles, and also their radial focusing, were considered in the works of Slater and other authors ^40, 47, 64, 79.

For a start let us disregard the radial motion of the particles. The corresponding mechanism of particle acceleration is shown in Fig. 1. It shows the graph of the amplitude of the longitudinal component of the electric field for a fixed instant of time. The wave propagates from left to right with an increasing phase velocity, and of such a magnitude that a positively charged particle in position \(B\) (\(D\), etc.) is held by the wave in the constant accelerating field shown in the figure by the horizontal dashed line. The phase velocities of the particle and of the wave are the same and vary along the \(z\)-axis according to the law

Fig. 1.

\[ V = E \cdot z \cdot \sin \varphi_s + \text{constant}, \tag{3} \]

where \(V\) is the total energy (kinetic energy \(+\) rest energy) of a particle with charge \(e\), \(E\) is the field amplitude, and \(\varphi_s\) is the phase of the particle (its position at the point where the field is zero is taken as zero). Particles which are displaced from \(B\) or \(D\) to neighboring points will fall into a field which will either slow them down or accelerate them still more, and will thereby return them to the point of stable phase \(B\) or \(D\). Particles at \(A\) or \(C\) are in a position of unstable equilibrium.

Such oscillations of particles about their stable (synchronous) phase were considered by Slater ^64**) by means of the method of the quasiclassical approximation ^66. The equation of motion

*) See Slater’s paper “Design of Linear Accelerators,” Uspekhi Fizicheskikh Nauk, vol. XXXVII, issues 3–4. (Translator’s note.)

**) Uspekhi Fizicheskikh Nauk, vol. XXXVII, issue 4, p. 479. (Translator’s note.)

particle along the \(z\)-axis in the field of a traveling wave may be written in the following form:

\[ \frac{d}{dt}(m\dot z)= eE\sin\left(\omega t-\int k\,dz\right), \tag{4} \]

where \(m\) is the relativistic mass of the particle, \(e\) its charge, \(k\) the projection of the wave vector on the \(z\)-axis, depending on \(z\), \(E\) the amplitude of the wave, and \(\omega\) its angular frequency.

If the point of the stable (synchronous) phase is taken as the origin of coordinates, then the solution of the corresponding first-order equation\(^4\) gives the frequency of the phase oscillations:

\[ \frac{\omega_0}{\omega}= \left[ \frac{E\lambda \cos\varphi_s\, V_0^2}{2\pi \beta V_s^3} \right]^{1/2}, \tag{5} \]

the amplitude of which is inversely proportional to the quantity

\[ \left[ V_s^2 \beta_s^3 E \cos\varphi_s \right]^{1/4}, \tag{6} \]

where \(\varphi_s\) is the value of the synchronous phase of the particle, \(V_s\) is the total energy (rest energy plus kinetic energy) of the particle in the synchronous phase, \(\beta\) is the ratio of the particle velocity to the velocity of light, \(V_0\) is the rest energy, \(\omega\) is the angular frequency of the electromagnetic wave, and \(\lambda\) is the wavelength in vacuum. Both expressions (5) and (6) are valid only qualitatively, for the quasicalssical approximation is valid only for slow variations, and this condition is not always fulfilled in accelerators. But, as we shall see below, the amplitude of the phase oscillations decreases as the particle energy increases, while the frequency of these oscillations is considerably lower than the frequency of the wave and, in the relativistic region, tends to zero. These circumstances are extremely important in the acceleration of electrons. Let us return again to Fig. 1: suppose that particles are continuously injected into the accelerator and that their velocity is equal to the phase velocity of the wave. Then those of them whose phase lies between \(A\) and, say, \(P\), will be captured by the wave and grouped around point \(B\). Particles with too large an initial deviation from the position of the stable phase (they are shown by the points between \(P\) and \(C\)) fall out of synchronism with the wave and are not captured by it in the process of acceleration. As the average accelerating field increases, the capture region narrows. When the average accelerating field is equal to the amplitude value, the particles are not captured at all.

The synchronous phase in the initial sections of the electron accelerator must be chosen in such a way as to ensure appreciable capture of particles in the process of acceleration; however, the accelerating field must not be too small. When the electrons acquire relativistic velocities, the phase oscillations become so slow that the electrons can be accelerated on the crest of the wave, without appreciable smearing of

after the passage of particles, Johnston et al.\(^{40}\) found that the decrease in the velocity of an electron relative to the wave at high energies is approximately inversely proportional to the cube of the distance traversed by the particles. Such a rapid decrease in the particle velocity relative to the wave makes it possible to suppose that, if in the nonrelativistic region the phase velocity of the wave should be increased with distance, then in the relativistic region, starting from a certain point, it is sufficient to have in the accelerator a wave with constant phase velocity equal to the velocity of light. Slater\(^{64}\) and Walkinshaw\(^{72}\) investigated the exact solution for the case of the motion of an electron in a wave with constant phase velocity. For waves whose phase velocity is less than the velocity of light, the motion of an electron relative to the wave is similar to the oscillations of a pendulum. The electron either oscillates relative to the position of synchronous phase, in resonance with the wave, or else moves all the time in one direction—forward or backward—in the case when its initial velocity differs significantly from the velocity of the wave. Of special interest is the expressed solution for the case of a wave phase velocity equal to the velocity of light. The electron then cannot outrun the wave, and phase oscillations do not occur. If the energy of the electron is “sufficiently large” and the initial phase lies in a certain definite interval, then such an electron will be continuously accelerated by the wave. If the phase of the particle differs from the phase corresponding to the maximum of the wave by an amount not greater than \(\pm 10^\circ\), “sufficiently large” will be an energy exceeding

\[ \frac{\pi V_0^2}{2 E \lambda \cos 80^\circ} = \frac{2.3}{E\lambda}\ \text{Mev}, \]

where \(E\lambda\) and \(V_0\) are expressed in Mev.

2.2. Radial focusing. The solution of Maxwell’s equations for an axially symmetric traveling wave may be written in the following form\(^{70*}\):

\[ \begin{aligned} E_z &= -E_0 J_0(\chi \rho)\sin(hz-\omega t),\\ E_\rho &= \left(\frac{h}{\chi}\right) E_0 J_1(\chi \rho)\cos(hz-\omega t),\\ Z_0 H_\varphi &= \left(\frac{k}{\chi}\right) E_0 J_1(\chi \rho)\cos(hz-\omega t), \end{aligned} \tag{7} \]

where \(z\), \(\rho\), and \(\varphi\) are the usual cylindrical coordinates, \(\chi\) and \(h\) are the radial and axial components of the wave vector \(k\), so

*) J. A. Stratton, Theory of Electromagnetism, GTTI, 1948, p. 474. (Transl. note.)

that \(k^2=\chi^2+h^2\), and \(Z_0\) is the intrinsic impedance in vacuum. Jinston, Hansen, and Kennedy \(^{40}\), Slater \(^{64}\), and Harvey \(^{47}\), using these equations, showed that the radial force acting on a particle, equal to \(e\left(E_\rho-z\dfrac{\dot H_\varphi}{c}\right)\), can be expressed in the following form:

\[ F=\left(\frac{h}{\chi}\right)eE_0J_1(\chi\rho)\cos(hz-\omega t)\left(1-\frac{v\dot z}{c^2}\right). \tag{8} \]

In the region of phase stability, force (8) is defocusing. It should be noted, however, that as the velocity of the particle approaches the speed of light, the radial focusing force of the magnetic field approaches in magnitude the force of the electric field, and the defocusing effect is small.

Harvey \(^{48}\) points out that radial focusing can also be accomplished by means of a longitudinal magnetic field. The field strength must be greater than a certain critical value

\[ \frac{1}{300}\left[\frac{4V_0^2E_z\cos\varphi_s}{V_s}\cdot\frac{\pi}{\lambda}\cdot\frac{c}{z}\right]^{\frac12} \text{ oersted}, \tag{9} \]

where \(V_0\) is the rest energy of the particle, \(V_s\) is its total energy in electron-volts, \(E_z\) is the amplitude of the accelerating field in volts/cm, \(\lambda\) is the electromagnetic wavelength in cm, and \(\varphi_s\) is the phase of the particle relative to the point where the field is zero.

Another method of radial focusing was also proposed. It resembles the method used in Sloan and Lawrence’s ion accelerator, and is based on the action of the higher components of the expansion of the electromagnetic field, which we did not take into account in the preceding analysis.

These higher components of the field expansion localize the radial force at the entrance to and exit from the accelerating gaps. If the particles reach the middle of the accelerating gap at the moment when the field is maximal, then in the first half of the accelerating gap, where the field focuses, they will remain longer than in the second half, where the field defocuses them.

Therefore, near the maximum of the traveling wave there exists a very small region of stable phases in which phase stability and radial focusing are simultaneously provided \(^{54}\). In order, however, to keep the electrons in this small phase region, the amplitude of the longitudinal accelerating field must be so stable that this method of radial focusing can hardly be realized.

In the opinion of Schulz et al. \(^{62}\), theoretical analysis shows that in systems composed of endovibrators, electric

not coupled to one another, it is possible to achieve simultaneously phase stability and radial focusing. This assertion does not agree with the conclusions of Mallett^54. Schulz, apparently, did not take into account the radial deflection produced by the electric field, which a particle experiences upon entering and leaving the endovibrator.

Other focusing methods have also been proposed, based on distortion of the field in the accelerating gaps by means of metallic foils^79 and grids^54,39; however, these methods apparently have not been used in electron accelerators. In the region of relativistic energies the focusing problem is not so serious, and it is possible to dispense with a focusing magnetic field. Slater^64 points out that the radial component of the particle momentum, in the absence of a radial electric field, remains constant, while the mass of the particle increases with increasing energy and, as a result, the radial component of the velocity decreases. The increase of the radial deflection is equal to

\[ \delta \rho = \varphi_0 \frac{V_1}{E_z}\ln\left(\frac{V_2}{V_1}\right), \tag{10} \]

where \(V_1\) and \(V_2\) are the initial and final energies of the electron, \(E_z\) is the accelerating field, and \(\varphi_0\) is the angle between the beam axis and the initial direction of motion of the electron. Let, for example, the initial energy of the electrons be \(2\) MeV, the accelerating field \(E_z = 10\) MV/m, and the initial divergence of the beam \(\varphi_0 = 10^{-3}\) radian. Then an energy of \(1\) billion electron-volts will correspond to an additional broadening of the beam by \(1.24\) mm. It may be concluded that if, by the time the electrons attain relativistic velocities, the beam is well collimated, then its focusing will thereby be ensured subsequently as well.

§ 3. Theory of Endovibrators and Microwave Waveguides

3.1. Principal types of accelerators.

Figures \(2a\)—\(2g\) show the principal types of high-frequency systems of linear accelerators. Fig. \(2a\) depicts a circular waveguide loaded with metallic disks with apertures. Both the high-frequency energy and the electron beam pass through the apertures in the disks. With an appropriate choice of the geometrical dimensions of the waveguide, the required types of waves can propagate along it with a phase velocity less than the velocity of light.

By a corresponding choice of the load at the end, a purely traveling wave can be obtained in such a waveguide. Accelerators of this kind with a traveling wave were constructed by Fry et al.^37, Jincton et al.^40, and Starr^69.

D. U. Farn and U. Walkinshaw

Labels in the figure: electron beam; tuning plunger; \(7.5 \times 2.5\) cm copper waveguide; copper; mica; mica diaphragm; to pump; to magnetic velocity analyzer; direction of electron beam; quarter-wave transformer; section along \(A\!-\!A\); high frequency.

Fig. 2.
a) Longitudinal section of a circular waveguide loaded with iris diaphragms.
b) Cylindrical resonator.
c) Spherical resonator with input cones.
d) Resonator with several \(2/3\) accelerating gaps.
e) Rectangular waveguide with diaphragms\(^{55}\).
f) Spiral waveguide of rectangular cross section\(^{56}\).
g) Bent waveguide of rectangular cross section\(^{33}\).

Another variant is also possible: the end of the waveguide is short-circuited. Many authors have chosen this latter route: Beauvais et al. \(^{41,44,52,59,64}\), Bowen et al. \(^{26*}\), Mills \(^{53*}\), Hifford \(^{45**}\), Snoddy and Bims \(^{67***}\), successfully using systems consisting of a single endovibrator.

In addition, a number of different methods have been described for accelerating electrons by means of coupled endovibrators—works by Schultz et al. \(^{62****}\), Allen and Symonds \(^{23*****}\), Mallet \(^{55******}\), Cullen and Greig \(^{33*******}\), Hudspeth \(^{50********}\).

*) A cylindrical endovibrator with drift tubes, operating at a frequency of \(1200\ \text{Mc/s}\). Mills’ work is a development of the work of Bowen et al. One of its interesting features is that the mean current in the electron beam reached \(70\ \mu\text{A}\), with beam dimensions approximately \(0.5\ \text{mm}\) in diameter (Fig. 2b).

**) The accelerating system is an endovibrator with drift tubes, operating at a frequency of \(400\ \text{Mc/s}\). At the exit from the accelerator the electron beam is turned by a magnetic field through \(180^\circ\) and thus passes through the accelerating gap a second time. As measurements show, the energy spectrum turns out to be very broad. This circumstance is, in principle, inherent in this method and apparently limits its application.

***) A cylindrical endovibrator with drift tubes, operating at a frequency of \(400\ \text{Mc/s}\).

****) A chain of endovibrators, each of which is fed independently from its own amplifier at a frequency of \(587\ \text{Mc/s}\); there is also no electrical coupling between the endovibrators. Matching of the operating frequencies and of the phase of the oscillations in the individual resonators is carried out by the master generator. The phase of the electromagnetic field in each endovibrator is regulated by a separate phase shifter. The use of amplifiers makes it possible to overcome the difficulties of frequency stabilization inherent in self-excited generators.

*****) A system of three endovibrators with drift tubes. Coupling between them is effected through holes in the walls (see Fig. 2g). The operating frequency is \(3000\ \text{Mc/s}\), oscillations of the \(\pi\)-type. The short accelerating gap considerably reduces the flight time of the particles and increases the efficiency of the accelerator. This system compares favorably with other resonant systems.

**) A rectangular waveguide loaded with diaphragms of depth from \(\dfrac{\lambda}{2}\) to \(\dfrac{3\lambda}{4}\) (Fig. 2d). The accelerated particles pass through holes in the diaphragms at the maximum of the electric field. The efficiency of the system should be of the same order as that of a round waveguide loaded with disks with holes, but a rectangular waveguide with diaphragms is rather difficult to manufacture.

***) A rectangular waveguide folded like an accordion (Fig. 2zh). In order to obtain a \(180^\circ\) phase shift between adjacent sections, the depth of the fold is chosen equal to \(\dfrac{\lambda}{2}\). Such a construction is more difficult to manufacture than a waveguide with disks.

**) A rectangular waveguide wound into a helix (Fig. 2e); owing to this, the field phase along the path of the electron beam changes more slowly than in a straight waveguide. The difficulties arising in the design and manufacture of such a system make it less effective in comparison with other, simpler constructions.

In addition to the above-mentioned designs of end vibrators and waveguides for accelerators, on which experimental work has been carried out, a whole series of earlier designs is also known. Starr^69 describes various designs obtained from a coaxial line if its inner conductor is cut into separate sections a quarter wavelength long; the latter produce an accelerating field along the axis of the line. Harvey^46 proposed feeding from a coaxial line a certain number of coaxial half-wave resonators. Willshaw and Lamont^77 considered a circular cylindrical waveguide partially filled with dielectric. The properties of dielectric layers were also studied by Frankel^36, Brack and Wicker^29, and Oliphant^60. Most of these early projects have now been superseded by more efficient and less complicated waveguide structures.

All designers of linear accelerators devote great attention to finding the best type of waveguide or the most advantageous system of end vibrators. Hansen and Richtmyer^43 showed that, for the case of a single end vibrator with a longitudinal electric field, the most advantageous geometrical form is a biconical resonator with an aperture of \(9^\circ\) (Fig. 2b), which has the greatest shunt resistance and, consequently, for a given power produces the largest longitudinal field. On the other hand, Eckley^20 theoretically calculated the surface shape of an end vibrator in which the higher-order waves have the smallest amplitude and disturb the motion only insignificantly). Difficulties in manufacturing such optimal systems, as well as in selecting the appropriate coupling for them, have nevertheless forced most designers to use end vibrators of simpler geometrical form, chiefly cylindrical end vibrators with coupling holes*), shown in Fig. 2a. We have considerable theoretical and experimental material on end vibrators of this type. On this question there are a number of independent investigations by different authors which, to a significant extent, overlap one another. Jinkston et al.^40 and Walkinshaw^71 determined the upper limit of the energy which a relativistic particle can acquire in such a system; it corresponds to the case in which from three to four end vibrators occur per wavelength. It is assumed here, first, that the particle passes through the middle of the end vibrator at the maximum of the accelerating field, second, that each of the end vibrators receives the same power, and, third, that the dimensions

) The wave \(E_{01}\) plays the principal role. (Translator’s note.*)

**) Such a chain of end vibrators coupled to one another may be regarded as a cylindrical waveguide loaded with disks having holes on the axis.

the coupling holes between the endovibrators are so small that they may be neglected. These assumptions should be adopted in the subsequent estimates of the value of the effective shunt resistance. The effective shunt resistance per unit length, i.e. the square of the mean potential difference corresponding to the change in the particle velocity per unit length, divided by the power expended in this segment, will then be:

\[ 3.3 \cdot 10^{4}\, \frac{\left(\dfrac{\sigma}{\lambda}\right)^{1/2}}{n+2.62} \left[ \frac{\sin \dfrac{\pi}{n}}{\dfrac{\pi}{n}} \right]^{2} \ \text{ohm/meter}, \tag{11} \]

where \(n\) is the number of endovibrators per wavelength \(\lambda\), and \(\sigma\) is the conductivity of the wall metal (for copper \(\sigma = 5.8 \cdot 10^{7}\,(\text{ohm}\cdot\text{meter})^{-1}\)). For \(n = 3.5\) and \(\lambda = 0.1\ \text{m}\), according to formula (11), the shunt resistance is approximately \(100\ \text{Mohm}\). This means that, under the most favorable conditions, in a hundred-meter accelerator a pulse power of \(1\ \text{MW}\) is required to accelerate electrons up to \(100\ \text{MeV}\), and to accelerate them up to \(1000\ \text{MeV}\) a pulse power equal to \(100\ \text{MW}\) is required. The effective shunt resistance that can be achieved in practice is always less than the optimum.

It is interesting to note that if the diameter of the coupling hole between the endovibrators is much smaller than the wavelength, then, as follows from relation (11), the effective shunt resistance increases as the wavelength is shortened. For coupling holes of finite dimensions the situation is different: the shunt resistance will be greatest at some optimum wavelength. Calculation shows that, with four endovibrators per wavelength in vacuum, an optimum wavelength of \(9\ \text{cm}\) corresponds to a coupling-hole diameter of \(2\ \text{cm}\), and an optimum wavelength of \(17\ \text{cm}\) corresponds to a hole diameter of \(4\ \text{cm}\).

An excellent qualitative description of the characteristics of a chain of coupled endovibrators is given in Slater’s paper \(^{64}\). If such a system is considered as a circular waveguide with a reactive load in the form of disks with holes along the axis, then it behaves like a band-pass filter. The lower boundary of the pass band is determined by the critical frequency of the unloaded waveguide, and its upper boundary by the frequency at which neighboring endovibrators oscillate in antiphase (\(\pi\)-oscillation). At frequencies lying within the pass band, near the axis of the waveguide there may propagate, in the main, a traveling wave described by relations (7). The phase velocity of this wave can be regulated either by changing the size of the coupling holes between the endovibrators or by changing the diameter of the waveguide. Charac-

A characteristic property of this wave is that, at phase velocities less than the speed of light, as the distance from the axis of the tube increases, the longitudinal component of the electric field increases. At a phase velocity equal to the speed of light, the longitudinal component of the field does not depend on the radius.

Chu and Hansen^30 investigated, by approximate methods (analogous methods had earlier been developed by Cattler^34), the dependence of the phase and group velocities of the wave and of the losses on the frequency and the parameters of a waveguide, for the limiting case of frequencies close to the critical frequency, i.e., to the lower limit of the passband.

A considerably more accurate expression for the phase velocity in a waveguide with more than five disks per wavelength in vacuum was obtained by Walkinshaw^72 and verified experimentally by Mallet and Loach^56. In this work the higher-order components caused by the fact that in reality the spacing between the disks has a finite value were taken into account.

Slater^64 gives an exact analysis of the special case of $\pi$-oscillations at a frequency corresponding to the upper limit of the passband and for infinitely thin walls, and extends this solution, by the methods of perturbation theory, to frequencies within the passband. In all these cases numerical data needed for designing a disk-loaded waveguide were obtained.

In most designs of linear accelerators, as an acceptable compromise between the simplicity of fabrication of the accelerator and the effectiveness of its operation, a disk-loaded waveguide was chosen. In such a waveguide an accelerating field can be obtained in two ways: either by short-circuiting its end (a standing-wave system), or by terminating the waveguide in a resistance equal to its wave impedance (a traveling-wave system). The main factors that play a role in the one case and in the other are briefly considered in §§ 3.2 and 3.3; since there are many such factors, it is difficult to compare the calculated characteristics of the two systems with one another. However, in the particular case of an accelerator consisting of a single section and fed from a single source, all investigators believe that the traveling-wave method may prove more effective than the standing-wave method.

As will be shown below (§§ 3.2 and 3.3), as the particle energy for which the accelerator is designed increases, in order to maintain its effectiveness it must be divided into a number of sections (in both cases—both for acceleration by a traveling wave and for acceleration by a standing wave). In comparing the two indicated acceleration methods, it is necessary to take into account the problem of matching the phase of the field between the individual sections. If the accelerator sections are fed by separate power amplifiers, but from

if there is only one driving generator, then no serious difficulties arise with such separate feeding of the accelerator. If, however, the sources are several separate self-excited generators, for example magnetrons, the situation changes. The point is that different generators, even if tuned to one and the same frequency, will continuously change their own frequency because of tuning inaccuracies, temperature effects, and other causes; unless special measures are taken, this circumstance will lead to undesirable phase shifts between the sections. To overcome the difficulty indicated above, a whole series of methods has been proposed, but different investigators differ greatly among themselves in their comparative evaluation of these methods.

Some authors maintain that the standing-wave method has a substantial advantage over the traveling-wave method, since in this case end vibrators can also be used to stabilize the frequency of the generators. In order to assess the significance of this circumstance, it is necessary to examine carefully the physical processes of frequency stabilization.

Willshaw^74,75 and Newberry^57 considered this question and showed that at the beginning of the pulse the frequency is determined not only by the frequency of the unloaded generator, but also by the resistance and length of the line connecting the generator with the load. In the steady-state regime, when the energy is already concentrated in the end vibrator, it also determines the frequency of the generator. Thus high frequency stability is ensured. The settling time for a given system is a function of the separation of two frequencies: the natural frequency of the end vibrator and the frequency of the generator at the beginning of the pulse. When both these frequencies coincide, the settling time is the smallest. For this purpose the initial frequency of the generator must lie within the pass band of the end vibrator. The corresponding expression for the settling time was given by Slater^64. In practice the degree of frequency stability of a magnetron system can hardly provide the minimum settling time, and therefore it is considerably greater than its minimum value. Estimates of the field settling time in a standing-wave accelerator as a function of the frequency instability of the generators have still not been published, so that at present a detailed comparison, from this point of view, of the standing-wave accelerator with the traveling-wave accelerator is hardly possible.

In traveling-wave accelerators, the phases of the field in the individual sections fed by self-excited generators must be made consistent with one another. For this purpose it is necessary to provide sufficient coupling between the separate generators for their phase synchronization. It has been possible^31 to achieve phasing from two to

of four magnetrons operating into an equivalent load; apparently, this number could be increased. Nevertheless, the use of a considerable number of self-excited generators synchronized in frequency and phase is a comparatively difficult problem, and it would be desirable to design more powerful power sources, of which only a small number would be required for the accelerator.

3.2. Traveling-wave accelerator. In a traveling-wave accelerator (Fig. 3), high-frequency energy is fed into one of the ends of a waveguide with a periodic loading and, as the wave propagates along the waveguide, is gradually absorbed in the walls; the accelerating field thereby decreases exponentially. If the openings in the disks are small, then only a small part of the power passes from the preceding section*) into the next one, and therefore the absorption is considerable. As the diameter of the openings increases, an ever larger fraction of the power is transferred from section to section, and the useful length of the waveguide increases. In practice the dimensions of the coupling openings should be chosen so that the electron beam can pass freely through them. If the diameter of the openings is specified, then the length of the waveguide has a certain upper limit, above which, because of absorption of high-frequency power in the walls, the efficiency of acceleration decreases. It has been shown that at the length corresponding to this limit, approximately 90% of the power is absorbed in the walls of the waveguide, and 10% passes from the waveguide into the load. Harvey^47, Johnston et al.^40 give detailed numerical data showing the dependence of the effective shunt resistance on the length of the waveguide for various diameters of the openings in the disks, the outside diameter of the waveguide having been chosen from the condition that the phase velocity of the wave be equal to the velocity of light.

At a frequency of 3000 MHz and with openings in the disks of 2, 4, and 6 cm, the waveguide lengths determined from the conditions of absorption of high-frequency power were respectively 3.5, 40, and 130 m; there were 5 endovibrators per wavelength, and the ohmic losses in the waveguide were taken equal to their theoretical value. These lengths correspond to effective shunt resistances of 180, 1400, and 3000 MΩ.

It should be noted that although, according to these data, the total shunt resistance increases with increasing optimum length, the shunt resistance calculated per unit length decreases. Instead of making an accelerator from a single section, it is evidently more advantageous to construct a long accelerator with a whole series of sections, with the smallest practically possible openings and with the corresponding optimum section length.

*) In the case of a traveling-wave accelerator, we shall call a section the region between two adjacent disks.

Fig. 3. Diagram of a 4 MeV linear electron accelerator.

For a number of reasons, waveguide sections often cannot be made as long as would follow from the absorption conditions without reducing the efficiency of the accelerator. The efficiency of the accelerator decreases because, when the device has considerable length, it becomes impossible to keep the electrons in the accelerating phase over its entire length, since with a large section length a phase shift of the electrons relative to the wave may occur even for the smallest change in the generator frequency. The point is that a heavily loaded waveguide is characterized by a small group velocity (strong dispersion), caused by the considerable time required for establishing the field in the individual endovibrators. Therefore a small change in frequency entails a considerable change in the wavelength in the waveguide, and as a result the phase velocity of the wave changes.

The change of phase \(d\psi\), associated with a frequency deviation for a particle with coordinate \(z\), moving with velocity \(c\), according to Harvey\(^{47}\), is equal to

\[ d\psi = 2\pi \frac{z}{\lambda}\left[\frac{c}{U}-1\right]\frac{df}{f}, \tag{12} \]

where \(\frac{df}{f}\) is the relative change in frequency and \(\frac{c}{U}\) is the ratio of the group velocity of the wave to the velocity of light.

Table I gives the calculated characteristics of traveling-wave accelerators. The table shows the relative influence of the [[unclear: word continues on next page after “po-”]]

Table I

Calculated characteristics of a traveling-wave accelerator.
Wavelength 10.0 cm

Radius of the aperture in the disk in relative units \(a/\lambda\)	Characteristics group	Characteristic	0,3	0,2	0,15	0,1
Radius of the aperture in the disk in relative units \(a/\lambda\)			0,3	0,2	0,15	0,1
Characteristics calculated from absorption conditions	Characteristics calculated from absorption conditions	Optimum length	63	19	6	1,7
Characteristics calculated from absorption conditions	Characteristics calculated from absorption conditions	Shunt resistance \((\text{M}\Omega/\text{m})\)	13	19	23	2,8
Characteristics calculated from frequency-stability conditions*)	Characteristics calculated from frequency-stability conditions*)	Optimum length	12,5	4,4	2,5	0,85
Characteristics calculated from frequency-stability conditions*)	Characteristics calculated from frequency-stability conditions*)	Shunt resistance \((\text{M}\Omega/\text{m})\)	9,6	15	19	23,5

*) Frequency stability — 0.01%

absorption in the walls and frequency instability on the length of the accelerator and its effective shunt resistance.

These data were obtained under the assumption that the number of disks loading the waveguide per wavelength is five, and that the accelerating field has an amplitude value. It is assumed here that the absorption is twice the theoretical value, and that the maximum permissible phase angle between the electrons and the wave is \(2\pi/10\) radians.

The calculation of the effective shunt resistance with allowance for absorption in the walls, as well as of the group velocity, is very complicated, and it cannot be reduced to simple formulas. A comparatively simple form is possessed by the relation between the longitudinal accelerating field \(E_0\) and the energy flux along the waveguide at a phase velocity equal to the velocity of light. In the MKS system,

\[ E_0 = (480 W)^{1/2}\left(\frac{\lambda}{\pi a^2}\right). \tag{13} \]

where \(a\) is the radius of the aperture in the disk and \(\lambda\) is the wavelength of the electromagnetic wave.

3.3. Standing-wave accelerator. If the end of a waveguide loaded with disks is short-circuited, then the waveguide is transformed into an endovibrator. Neglecting the power imparted to the electron beam, one may say that all the power fed into the section of a standing-wave accelerator is absorbed by its walls, and no power enters from the waveguide into an equivalent load. The properties of such a section should be considered when oscillations are established in the stationary regime. The time for establishment of the stationary regime in a resonant system, i.e., the time during which the amplitude of the oscillations reaches a value equal to

\[ \frac{e-1}{e}\cdot A_0 \]

(\(A_0\) is the amplitude of the established oscillations), is equal to

\[ \frac{Q}{2\pi} \]

periods of the high frequency. In any system used in practice, the pulse duration must be sufficiently large, since during this time the amplitude of the oscillations must reach 90% of its stationary value[^75]. Slater[^64] showed that this time is comparable with the time for establishing oscillations for an accelerator with a traveling wave whose length is equal to the attenuation length.

In a standing-wave accelerator, only those frequencies from the pass band at which an integral number of half-waves fits along the length of the waveguide will correspond to natural oscillations (standing waves). The optimum length of an individual accelerator section is determined no longer by absorption in the walls and not by the stability of the frequencies of the power source, as was the case in an accelerator with a traveling wave, but by the spectrum of natural frequencies of the given section. Uilson[^75] gave a useful method for calculating the spectrum of frequencies for a section of a standing-wave accelerator. His method

is based on the analogy between a chain of endovibrators and a chain of equivalent shunt reactive resistances.

As the length of a standing-wave accelerator section is increased, the width of its pass band does not change; however, the number of resonance maxima in this band grows in proportion to the number of half-waves fitting along the length of the section. When such a section is fed from a self-excited generator, the need to separate neighboring resonance frequencies (neighboring natural oscillations) imposes strict limitations on its maximum length. The distance between neighboring resonance frequencies is not the same: in the middle of the pass band it is greater, and at its edges it is smaller. Therefore Louton and Hahn \(^{53}\) chose the operating point of the accelerator not at the upper boundary of the pass band (an oscillation of type \(\pi\)), but in its middle. This choice of frequency pursued the aim of using the considerable spacing between resonance frequencies in the middle of the pass band. In their accelerator there are four endovibrators per wavelength, and the phase shift between neighboring endovibrators is approximately

\[ \frac{\pi}{2} \]

(whereas for oscillations of type \(\pi\) there would be not four, but two resonance cavities per wavelength).

Such an accelerator design, in addition to providing greater separation of neighboring resonance frequencies, reduces the effect of the finite transit time of the electrons in the endovibrator; therefore the complete acceleration in each endovibrator, for a given field amplitude, is in this case greater than for oscillations of type \(\pi\).

We have already encountered an analogous result (see § 31) in the case of a straight-line chain of endovibrators. The greatest effective shunt resistance was obtained there also with three or four endovibrators per wavelength. The efficiency of a traveling-wave accelerator whose length is equal to the attenuation length, and which is fed at one end of the waveguide, amounts to \(80\%\) of the efficiency of an accelerator with uniform power distribution along its length. (In this we neglect the additional reduction in efficiency due to the finite size of the coupling apertures.) In a standing-wave accelerator the average power is distributed along the tube length according to a sinusoidal law, and its efficiency is half the efficiency of an accelerator with uniform power distribution. This is explained by the presence of the reflected wave, which does not increase the average energy of the particles, but at the same time absorbs \(50\%\) of the power introduced into the accelerator.

Although maximum efficiency is attained both in the traveling-wave accelerator and in the standing-wave accelerator with three or four endovibrators (in the case of a traveling wave—sections) per wavelength, it should be noted that a substantial circumstance,

concerning \(\pi\)-oscillations. The field in a traveling-wave accelerator may, as was indicated in § 3.1, be resolved into components—the fundamental wave and its higher harmonics with smaller amplitude (Fig. 4, \(a\)).

In a standing-wave accelerator the number of harmonics is doubled on account of the reflected waves (these waves are shown in Fig. 4, \(a\) by dashed lines). As the \(\pi\)-oscillations are approached, the spatial harmonics come closer together (Fig. 4, \(b\)), and in the accelerator four waves propagate with nearly equal amplitudes and phase velocities, but only one of them propagates in the same direction and with the same velocity as the particle itself. Finally, for \(\pi\)-oscillations both pairs of components merge, and therefore the effectiveness rapidly increases to 70% of the optimum value, corresponding to three or four sections per wavelength.

Fig. 4. \(a\)—amplitudes of the harmonics for the case in which the generator frequency lies in the middle of the pass band; \(b\)—amplitudes of the harmonics. The generator frequency is close to the upper boundary of the pass band (oscillation of type \(\pi\)).

Fig. 5.

In Fig. 5 the dependence of the shunt resistance on the number of endovibrators (sections) per wavelength is shown for both types of accelerators—with traveling and with standing wave.

The upper curve has been calculated from equation (11) for a wave of \(10\ \mathrm{cm}\); the lower curve is obtained from the upper one by halving its ordinates (because of the reflected wave). The \(\pi\)-oscillations give

greater shunt resistance; therefore Slater\(^{64}\) and many other authors preferred to make use specifically of \(\pi\)-oscillations. However, the maximum length of each of the accelerator sections (for a given spacing between neighboring resonant frequencies) is in this case shorter than in the case of \(\frac{\pi}{2}\)-oscillations. According to Slater et al.\(^{65,52,59}\), when the resonant frequencies are separated by a value on the order of 0.1%, there is no transition from one frequency to the neighboring one.

For comparison of the traveling-wave accelerator and the standing-wave accelerator, the theoretically calculated characteristics of standing-wave accelerators are given below. The corresponding characteristics of the traveling-wave accelerator are presented in Table II.

Table II

Calculated characteristics of standing-wave accelerators

Radius of the aperture in the end vibrator in relative units \(\dfrac{a}{\lambda}\)		0.4	0.3	0.2	0.1
Spacing between resonant frequencies 0.5%	Optimum length (m)	0.35	0.7	0.45	0.2
Spacing between resonant frequencies 0.5%	Shunt resistance (MΩ/m)	4.0	5.0	8.0	11.5
Spacing between resonant frequencies 0.1%	Optimum length (m)	2.1	1.6	1.0	0.45
Spacing between resonant frequencies 0.1%	Shunt resistance (MΩ/m)	4.0	6.0	9.0	13.0

Less data have been published on linear standing-wave accelerators than on traveling-wave accelerators. Therefore the data given in Table II may be less reliable. The data of Table II were obtained under the following assumptions: a) ohmic losses were taken, as is often done in practice, to be twice their theoretical value; b) the magnetron loses 25% of its power in the active stabilizing load; c) in the theoretical calculations it was assumed that the distribution of the electric field at a distance from the axis equal to the aperture radius (i.e., at \(\rho=a\)) can be obtained from a quasistatic treatment of the corresponding problem. Such an assumption may entail an error in determining the effective shunt resistance of 10 to 20%.

For a more detailed comparison of short sections of standing- and traveling-wave accelerators, one should consult the work of Newberry\(^{87}\).

§ 4. Acceleration in the Nonrelativistic Region

If the first few meters are disregarded, the electrons move in the accelerator with velocities close to the velocity of light, and much of what was said above referred precisely to this stage of the accelerator. In § 2.1 of our review we considered the motion of a particle in a wave with continuously increasing phase velocity. In this case the particles are grouped around the synchronous (stable) phase. Such a method of obtaining particles with relativistic velocities was first applied by Frahm^37 and, somewhat later, by Johnstone^40. In Frahm’s accelerator both the openings of the disks and the diameter of the waveguide are varied, so that the accelerating field remains constant along the entire length of the waveguide, while the position of the stable phase changes. A detailed calculation of the accelerator was carried out by Walkinshaw^71. His paper also gives tables of waveguide parameters. Theoretically, using this method, one can construct an accelerator on a standing wave, but in practice this is very complicated and feasible only in the case where the endovibrators have separate power feeds and are not electrically connected with one another. Otherwise a change in the geometrical dimensions in any one section of the accelerator will lead to a change in the field magnitude in all its other sections.

Louton and Hahn^52 attempted to overcome this difficulty in the following way. They divided the input part of the accelerator, where the electron velocity is less than the velocity of light, into shorter sections. Within each such section the phase velocity is constant; from section to section it changes stepwise and thus approaches the velocity of light in small steps. Such a device, however, is complicated, and moreover in this case it is unlikely that a narrow energy spectrum can be obtained from it.

Willis^75 proposed carrying out the initial acceleration by means of one or two endovibrators with strong electric fields sufficient to reach relativistic velocities.

§ 5. Construction of Accelerators

In § 3 (Fig. 2(a)—(d)) endovibrators of various geometrical forms were described. Only one of them, shown in Fig. 2(b), imparts the maximum acceleration to the particle; the remaining endovibrators shown in Fig. 2 do not possess this property. However, for them the difference in acceleration compared with the maximum is small; at the same time these “nonoptimal” types of endovibrators, for a considerable accelerator length, have the important advantage of ease of manufacture.

In constructing waveguides for accelerators, a very important role is played by manufacturing methods whose aim is to bring the electrical characteristics of the waveguide closer to their optimal values.

A round waveguide with diaphragms, length
40 cm.

Separate
section of the
waveguide

Gap

Soft
solder

A section of waveguide assembled from several
sections.

Fig. 6a. Fabrication of a waveguide by the method of
Fry et al. \(^{37}\).

Fig. 6b. Fabrication of a waveguide by the method of Johnston
et al. \(^{40}\): \(a\)—seamless tube, \(b\)—disks cut from
a single sheet, with a hole drilled in them.

Circular waveguide with diaphragms 40 cm long.

Labels in the drawing: separate waveguide sections; gap; soft rubber.

A section of waveguide assembled from several sections.

Fig. 6a. Manufacture of a waveguide by the method of Fry et al.^37

Labels in the drawing: a; b.

Fig. 6b. Manufacture of a waveguide by the method of Jenison et al.^40. a—seamless-drawn tube; b—disks cut from a solid sheet, with a hole drilled in them.

by the fact that the wave must remain in phase with the electrons over their entire path, and this phasing must be very precise. If the accelerator consists of several sections, then the phase of the wave in each of them must be adjusted so that the phase differences do not give rise, as one goes from section to section, to an increasing phase divergence between the electrons and the wave. Separate waveguide sections, whether designed for traveling-wave acceleration or for standing-wave acceleration, are highly sensitive to any changes in their operating conditions. Therefore the phase velocity of the wave changes even for the slightest deviations in the dimensions of the waveguide, in the generator frequency, and in temperature. In a traveling-wave accelerator, phase divergences in each section can be corrected by means of shortened sections with a somewhat different phase velocity of the wave, placed between the main sections for phase correction (Harvey \(^{47}\)). In a standing-wave accelerator, any phase correction can be accomplished by providing each section with a special device for precise tuning. The use, however, of a large number of tunable sections is complicated. To avoid this complication, waveguides have to be manufactured with small tolerances—of the order of \(0.015\) mm.

§ 6. Powering of Accelerators

The increase in particle energy in an accelerator \(V\) is proportional to \((PL)^{1/2}\), where \(P\) is the power fed into the accelerator, and \(L\) is its total length. Consequently, \(V\) can be increased in two ways: by increasing \(P\), or by increasing \(L\). It is difficult to find a rational method by which one could choose the proper ratio between \(P\) and \(L\), although Slater \(^{64}\) brings to the forefront considerations of an economic nature—the equal cost of the high-frequency power sources and of the accelerating system. As practice shows, for ordinary values of the accelerator length \(L\), the required power \(P\) soon begins to exceed the maximum pulsed power both of modern self-excited generators and of amplifier tubes. The problem of powering the accelerator is thus reduced to two tasks: obtaining, at a given frequency and phase, the necessary power, and distributing it uniformly over the entire length of the waveguide.

In § 3 it was shown that, in a long accelerator, a large total shunt impedance is obtained if this accelerator is divided into separate sections and, in each section, the optimal value of the coupling apertures is chosen. The questions of accelerator power supply can therefore be divided into two groups: those connected with supplying an individual accelerator section, and those connected

theoretical values. Three such methods are shown in Figs. 6a, 6b, and 6c. Fig. 6a shows a waveguide for a traveling-wave accelerator, assembled from separate sections fitted to one another and soldered externally with soft solder (Mallet and Loach^56).

Fig. 6b shows a waveguide of a similar type, but with disks manufactured by another method. They are made from disks preliminarily machined on a lathe and seated in precisely determined sections of a pipe—a honed cylinder (Johnston et al.^40). Fig. 6c shows the design of an accelerator with Louton’s standing wave^51.

In the first and third of the methods described, the measured power losses in the accelerator walls (leading to attenuation) proved to be approximately twice as large as their theoretical value. It may be expected that with the second manufacturing method the actual attenuation losses are very close to the theoretical ones.

It is also necessary to consider the question of tolerances in the manufacture of waveguides. Let us note first of all that tolerances constitute a serious problem only for long accelerators. Therefore, for most accelerators described so far in the literature, the question of tolerances does not constitute a serious problem. In long accelerators, dimensional tolerances are determined

Accelerator of two resonators, operating with oscillations of type $\dfrac{\pi}{2}$,
$\beta = 0.75,\ K = 17.75\%,\ \lambda_0 = 10.701\ \text{cm}$.

Accelerator assembled from two resonators, $\beta = 0.75,\ K = 17.75\%,\ \lambda_0 = 10.705\ \text{cm}$.
Fig. 6b. Manufacture of a waveguide by Louton’s method^51.

theoretical values. Three such methods are shown in Figs. 6a, 6b and 6v. In Fig. 6a a waveguide for a traveling-wave accelerator is shown, made up of individual sections fitted to one another and soldered on the outside with soft solder (Mallet and Loach\({}^{56}\)).

In Fig. 6b a waveguide of a similar type is shown, but with disks manufactured by a different method. They are made from disks previously turned on a lathe and set into precisely defined sections of the tube—the honed cylinder (Johnston et al.\({}^{40}\)). In Fig. 6v the construction of an accelerator with Loughton’s standing wave\({}^{51}\) is shown.

In the first and third of the methods described, the measured power losses in the accelerator walls (leading to attenuation) proved to be approximately twice as large as their theoretical value. It may be expected that, with the second manufacturing method, the actual attenuation losses will be very close to the theoretical ones.

It is also necessary to consider the question of tolerances in the manufacture of waveguides. Let us note first of all that tolerances constitute a serious problem only for long accelerators. Therefore, for most of the accelerators described up to now in the literature, the question of tolerances does not constitute a serious problem. In long accelerators, tolerances in dimensions are determined

Accelerator made of two resonators operating in oscillations of type \(\dfrac{7}{2}\), \(\beta = 0.75\), \(K = 17.75\%\), \(\lambda_0 = 10.701\ \text{cm}\).

Accelerator made of two resonators, \(\beta = 0.75\), \(K = 17.75\%\), \(\lambda_0 = 10.705\ \text{cm}\).

Fig. 6b. Manufacture of the waveguide by Loughton’s method\({}^{51}\).

with the accelerator feed composed of a number of successive sections.

Fig. 7 shows a method of introducing power from a single source into a separate section of a traveling-wave accelerator. The accelerator waveguide terminates in a properly selected equivalent resistance, which makes it possible to eliminate reflected waves carrying unused power back into the magnetron. The input impedance of the waveguide will be purely active

![Fig. 7 diagram: labels include tuning piston, magnetron, waveguides No. 1–No. 4, gaskets for lengthening the line, spectrum-analyzer probe, water load, glass window, vacuum tube, resonator, tuning masses, and manometer.]

Fig. 7. One of the feed circuits for an individual resonant section of a standing-wave accelerator (Newberry and Wilds)59.
Waveguide data

No.	Data	Mode
1	Inside diameter 0.95 cm	\(H_{11}\)
2	Conical	\(H_{11} \to H_{01}\)
3	\(0.75 \times 0.25\)	\(H_{01}\)
4	Inside diameter 10.1 cm	\(E_{0,1,24}\)

Loaded with iris diaphragms.

for the operating frequency of the system, and will not vary sharply for frequencies close to it.

Mallet and Louch56 and J. Kingston et al.40 developed effective methods of introducing high-frequency power into an ordinary rectangular waveguide. When operating at the optimum wavelength corresponding to a given size of the openings in the disks, a frequency discriminator and servomechanisms are needed for frequency regulation, since frequency stability of 0.01% and higher is required.

The length of the accelerators built up to the present time was less than the optimum; therefore frequency instability was not ...

limiting factor, and devices for precise tuning were not required. However, in connection with the construction of linear accelerators in which the limiting factor will be precisely frequency instability, there arises a need for investigations on this question as well.

In the future it will be necessary to feed into each section such a large power that even the most powerful modern generators will be unable to supply it. This difficulty can be overcome in two ways: by connecting existing tubes in parallel, or by developing new, more powerful ones; investigations have been begun in both directions. The most powerful of the generators used hitherto in linear accelerators is a magnetron with a pulse power of \(2\) MW at a wavelength of \(10\) cm, with a pulse duration of \(2\) μsec and with a pulse repetition frequency up to 500 per second. The magnetrons were connected in pairs, and in each pair matching was achieved both in frequency and in phase\({}^{32}\). Such a pair of magnetrons delivered its total power into an equivalent load; and there are no apparent reasons why not two, but a larger number of magnetrons, could not be connected in parallel. However, it is difficult to set up and adjust such a magnetron system. A more expedient way of obtaining considerable pulse powers would be the development of new super-power generator tubes, and intensive investigations are being carried out in this field. Devon\({}^{35}\) is developing a magnetron at a wavelength of \(25\) cm which is to give pulses of duration \(30\) μsec at a pulse power of \(100\) MW. Hansen\({}^{43}\) is solving the problem in another way: he is developing an amplifying klystron at a wavelength of \(10\) cm, which is to give pulses of duration \(2\) μsec with a pulse power of \(100\) MW. More modest developments in the field of magnetron technique set themselves the task of obtaining from a single magnetron at a wavelength of \(10\) cm a power of \(10\)—\(20\) MW. Even such magnetrons would render substantial assistance in the development of linear electron accelerators.

For feeding a standing-wave accelerator section, another method should be used. In order that the amplitude of the oscillations in the endovibrator should reach the value \(\dfrac{e-1}{e} A_0\), where \(A_0\) is the amplitude of the established oscillations, a time is required which is \(\dfrac{Q}{2\pi}\) times greater than the period of the high frequency. During this time the impedance of the endovibrator varies within very wide limits, starting from very small values. If the tube operates as a self-excited generator, then in order to establish in it the assigned frequency, the standing-wave accelerator section must be fed through a series-connected

active resistance. When a section of the accelerator is fed through a tuned line, then in the steady-state regime it behaves, with respect to the tube, like a parallel resonant circuit and stabilizes the frequency. The endovibrator itself determines the frequency, and, in contrast to traveling-wave accelerators, separate tuning is not required.

The problems of feeding a standing-wave accelerator have been developed by many investigators, both theoretically and experimentally. In Fig. 7 is shown the installation of Newberry and Wilson\(^{59}\) intended for this purpose. They showed that in a standing-wave accelerator consisting of 24 endovibrators

Fig. 8. Diagram of a standing-wave accelerator for 28 Mev. For synchronization, magnetron No. 11 gives a pulse 0.2 μsec earlier than the others; it is followed by magnetrons 3, 7, 15, and 19 (the advance is no longer 0.2 μsec, but 0.1 μsec).

with a quality factor \(Q = 9000\) and with the spacing between the working \(\pi\)-oscillation and the neighboring harmonic equal to \(0.2\%\), it is possible, without special difficulties, to transmit \(65\%\) of the magnetron power. The remaining \(35\%\) was dissipated in the series-connected stabilizing load. As reported by other authors\(^{23, 41, 64, 51}\), they also succeeded, with the aid of a similar installation, in transmitting \(50\text{–}60\%\) of the magnetron power to the endovibrators.

Sleter\(^{63}\) showed that still greater powers can be introduced into a separate section of a standing-wave accelerator if it is fed not from one tube, but from several tubes. The choice of the point of introduction of high-frequency power in the case considered by him of endovibrators with oscillations of type \(E_{010}\) is immaterial. Figure 8 shows five such sections. Each of them has four HK 7 magnetrons, operating at a wavelength of 10.0 cm (wavelength in vacuum), with a pulse power of 1 MW at a pulse duration of

3.4 μsec. Each such resonant section has a length of 122 cm and consists of 24 end vibrators; it contains one plunger for precise tuning and a coupling loop for the indicator. To match the phases of the magnetrons, one of them is switched on 0.1 μsec before the others and feeds a control pulse to the end vibrator. The oscillation frequencies in the sections are monitored by the envelope of the detected signals supplied from the coupling loop to an oscilloscope.

If even one of the four magnetrons operates at a frequency different from the natural frequency of the end vibrator (of an individual accelerator section), then the envelope on the oscilloscope screen appears blurred. This magnetron is then tuned slightly—until the envelope of the signal on the oscilloscope takes the form of a thin line. After this the tuning controls of all four magnetrons are coupled, and all the magnetrons together are tuned to the frequency of the required natural oscillation, so that the power introduced into the end vibrator is maximal. The magnetrons operated normally and stably already with a separation between the working π-oscillation and the nearest oscillation of 0.1% (but not 0.05%). If the magnetrons are once matched in frequency and phase, they will operate with the same frequency and phase for a very long time. An accelerator composed of a number of sections of optimum length must be supplied in all its sections with the same frequency and phase. At the power levels consumed in accelerators, it is necessary to use not one but several tubes to feed them, even if it were possible to employ the new super-powerful centimeter-wave sources now being developed. The task is therefore to feed the separate sections of the accelerator from sources that are not electrically connected with one another. The method of solving this problem depends on whether the sources of high-frequency power are self-excited generators or power amplifiers.

Slater[^65] solved this problem for the case of self-excited generators. The complete diagram of his installation is shown in Fig. 8. The accelerator consists of five sections, each of them fed by four magnetrons in the manner described above. Frequency and phase matching is achieved in the following way. One of the magnetrons is switched on 0.2 μsec before the others. A small part of the power delivered by this magnetron into the stabilizing load connected in series with it passes through protective switches into certain magnetrons of the other four sections [magnetrons 3, 7, 15, and 19 in Fig. 8 (Translator’s note)], which give a pulse with an advance of 0.1 μsec. This signal ensures the proper phasing of the magnetrons at the beginning of the pulse. When the oscillations in magnetrons 3, 7, 15, and 19 have already been established, the protective switches operate; in this way unnecessary power losses in the phasing ...

of the common line. Consequently, each section can be tuned separately by the method described above, and the oscillations in all sections begin in one phase. Signals from the various sections are fed to a crystal detector, and the envelope is observed on the screen of an oscillograph. The difference between the oscillation frequencies of the different sections can be eliminated by means of the precision-tuning plungers present in each section.

Another method of feeding sections not coupled to one another—by means of power amplifiers—was described by Schulz et al.^63 The work was carried out at a frequency of 587 Mc/s, since at this frequency it was possible to obtain as much as 600 kilowatts in a pulse from triode power amplifiers. The accelerating section consisted of two mutually uncoupled endovibrators with oscillations of type \(E_{010}\); each of them received power from a separate amplifier with a power gain of about 6–7, and both amplifiers were driven by a common master oscillator.

Power amplifiers have the advantage that, when they are used, frequency stabilization ceases to be a serious problem: all the amplifiers can be fed from a single, low-power but very stable source. From this point of view, for long accelerators the 100-Mc/s amplifier developed by Hansen^43 is of great interest. A number of other methods have also been proposed for using, in accelerators, the superpower high-frequency sources now being developed.

§ 7. Injection (admission) of particles into the waveguide

The determining characteristics of the system for injecting (admitting) electrons are: (a) the injection voltage, (b) the intensity of the electron beam, and (c) the angle of divergence of the beam.

In early work, the beam intensity was usually not of interest. The intensity was considered quite sufficient if it was possible to detect the beam and make the corresponding measurements. However, for practical purposes it is also essential to know the maximum current that can be obtained in the accelerator. To this limiting current there corresponds such a considerable decrease in the strength of the longitudinal field, associated with loading of the accelerator by the electron beam, that it reduces the maximum electron energy below a prescribed limit. Fry et al.^38 showed that in a short accelerator, where over almost the entire length there act appreciable forces that bunch the electrons, 30% of the generator power can be transferred to the electron beam. In a long accelerator, calculated without taking into account the power required by the beam, it is possible to transfer to the beam a considerably smaller fraction of the power, since an increase in the power consumed by the beam would lead to a lowering of the maximum particle energy.

It seems to us that, in the case under consideration, the power absorbed by the beam may reach 15%. If, in calculating the accelerator, the loading of the waveguide by the electron beam is taken into account, then it must be assumed that it will be possible to transfer to the electron beam a considerably larger fraction of the input power.

The questions of choosing the injection voltage and the angle of divergence of the beam are, to some extent, interrelated. Hirshfield^45, Baughn et al.^26 showed that into a standing-wave accelerator consisting of a single endovibrator one can inject an uncollimated beam of electrons with zero initial velocity. They did not calculate, however, exactly what fraction of the electrons leaving the cathode would be accelerated. On the other hand, Slater proposes injecting into a standing-wave accelerator electrons with an energy of 2 MeV from a Van de Graaff generator. The electron source of the electrostatic Van de Graaff generator supplies current in pulses (the pulse modulation is effected with the aid of a photocell controlled through a small window). The angle of divergence of the beam is of the order of \(10^{-3}\) radians. In § 2.2 it was shown that even for 1000-MeV accelerators the subsequent defocusing of the beam in the accelerator is of no substantial importance. In choosing the injection voltage it is always necessary to take into account how complicated in design and how reliable in operation the pulsed direct-current source injecting the electrons is, and to compare both these factors with the analogous characteristics of the first section in the accelerator, into which the electrons are injected at low voltage. A number of authors consider it expedient to inject electrons with an energy of about 50–75 kilovolts. As the electron source Fry et al.^37 used an electron gun with a tungsten cathode; Louton et al.^53, Johnston et al.^40 used electron sources employed in X-ray technology. The indicated range of injection voltages is also expedient for traveling-wave accelerators, since the difficulties in manufacturing waveguides with disks increase sharply as the phase velocity is reduced below \(0.4c\).

§ 8. Operating characteristics of accelerators

In the preceding sections of our review it was shown that, for practical realization of the possible advantages of linear accelerators in comparison with cyclic ones, it is necessary to solve a whole series of rather difficult technical problems. It is therefore not surprising that most of the experimental work carried out so far has been directed not toward obtaining particles of ultra-high energies, but toward solving some of the technical problems mentioned, on accelerators with comparatively small-

particle energies. Let us consider the principal physical characteristics of the installations constructed up to the present time. We shall pay special attention to the following characteristics: a) the maximum energy of the electrons in the beam of greatest intensity, b) the energy spectrum of the particles, c) the limiting value of the current in the electron beam and the duty factor in pulsed operation,

Fig. 9. Linear electron accelerator at 4 MeV; in the foreground is the electron gun.

d) the angle of divergence of the beam. Also important is the efficiency of the accelerator, which is determined by the ratio

\[ \frac{(\text{maximum particle energy})^2} {(\text{total power}\times \text{total length of the accelerator})}, \]

expressed in megohms/meter. The dimension of this parameter is the same as that of the effective shunt resistance of the waveguide per unit length (see § 3.1), but it now characterizes the efficiency not only of the waveguide, as in § 3.1, but of the entire accelerator as a whole.

Data on the accelerators that have been constructed are summarized in Table III. Figure 9 shows a photograph of a traveling-wave accelerator at

Authors	Type of accelerator	Particle energy at beam intensity in Mэв
1	2	3
Bauhen et al.26	Standing π-wave. One endovibrator with drift tubes . . . . . . .	0.6
Allen and Symonds28	Standing π-wave. One section, composed of three endovibrators with drift tubes . . . . . . . .	0.85
Hifford45	Standing π-wave. One section, which consists of one endovibrator with a drift tube. Single acceleration . . . . . . . Double acceleration*) . . . . . . .	0.65 ± 10% 1.0 (?)
Fry et al.37	Traveling wave. One section . . .	0.53
Callen and Greig33	Standing π-wave. One section, composed of three endovibrators .	0.3
Johnston et al.40	Traveling wave. One section . . .	1.7 4.5 6–6.5(2)

*) The idea of double acceleration consists in the following: the magnetic field is turned by 180° and the particles are again admitted into the accelerator (in the reverse direction), the phase of the electric

LINEAR ACCELERATORS

... let us take charged particles. Let us consider the principal physical characteristics of the installations constructed up to the present time. We shall pay particular attention to the following characteristics: a) the maximum energy of the electrons in the beam at the highest intensity, b) the energy spectrum of the particles, c) the limiting value of the current in the electron beam and the duty factor in pulsed operation,

Fig. 9. Linear electron accelerator for 4 MeV; in the foreground is the electron gun.

d) the angle of divergence of the beam. Also essential is the efficiency of the accelerator, which is determined by the ratio

\[ \frac{(\text{maximum particle energy})^2}{(\text{total power} \times \text{total length of the accelerator})}, \]

expressed in megohms/meter. The dimension of this parameter is the same as for the effective shunt resistance of a waveguide per unit length (see § 3.1), but it now characterizes the efficiency not only of the waveguide, as in § 3.1, but of the entire accelerator as a whole.

Data on the accelerators that have been constructed are collected in Table III. Figure 9 shows a photograph of a traveling-wave accelerator for

LINEAR ACCELERATORS

Table III

Width of the energy spectrum of electrons in kilovolts	Current in the pulse in mA	Duty factor	Frequency in Mc/s	Decelerating-field voltage in kV	Approximate length of the high-frequency system in cm	Power of the high-frequency source in MW; pulse in MW	Accelerator efficiency in MW: \(W/PL\)
4	5	6	7	8	9	10	11
50	50	1000	1200	0	10	—	—
—	8	2800	3000	25	12,4	0,5	12,0
300—750	70	4100	400	0	30	0,5	2,8
300—1250	—	4100	400	0	30	0,5	—
65	36	10000	3000	45	40,0	2,0	0,35
—	—	4900	2800	2	—	0,8	—
(3)	—	—	2860	78	91	0,75	4,2
(3)	—	—	2860	78	275	0,75	10,0
(3)	—	—	—	—	365	0,75	14,0

in the course of the beam of accelerated electrons from the endovibrator it loses its way. By the time the electrons pass through the field for the second time, the phase must change by \(\pi\).

Authors	Type of accelerator	Particle energy at the beam of greatest intensity, MeV
1	2	3
Louton and Hahn62	Standing $\pi/2$-wave. One section composed of: a) two endovibrators. Relative spacing between the resonant frequencies 2.3% . . . . . b) seven endovibrators. Spacing between the resonant frequencies 0.7% . . . . . . . . . . . . . .	0.39 0.57
Schulz et al.62	Standing $\pi$-wave. Two sections, each consisting of one endovibrator	1.5 (?)
Fry et al.38	Traveling wave. One section . . .	4.0
Newbern and Wilson59	Standing $\pi$-wave. One section of 24 endovibrators. Relative spacing between harmonics 0.2%.	—
Slater65	Standing $\pi$-wave. Five sections, 24 endovibrators in each. Spacing between neighboring resonant frequencies 0.1% . . . . . . . . . .	28 (calculated value)
Snoddy and Beams67	Standing $\pi$-wave. One section with one endovibrator . . . . . . . . . .	1.2
Mills68	Standing $\pi$-wave. One section with one endovibrator . . . . . . . .	1.1

Table III (continued)

Electron energy in kilovolts	Current per pulse in mA	Duty factor	Frequency in MHz	Voltage of the injection in kV	Diameter of the high-frequency system in cm	Power of the high-frequency source in pulsed MW	Accelerator efficiency in Mom / µs · \(V^{2}/\rho L\)
4	5	6	7	8	9	10	11
—	—	—	2800	58	8	0.75 (?)	2.5
—	—	—	—	57	32	0.75 (?)	1.35
—	—	—	587	—	30	—	—
220	150	2500	3000	1600	200	2.0	4.0
—	—	1000	3000	—	120	0.6	—
—	—	—	3000	2000	640	16	7.7
—	—	—	400	—	—	—	—
about 220	70	1000	1200	10	10	0.5	24

4 MeV. From Table III it is seen that the electrons are accelerated up to energies of \(0.3—6.5\) MeV; it should be expected that the accelerator built by Slater will give electrons with a maximum energy close to 28 MeV.

Most investigators measured the maximum energy of the electrons by means of rather crude magnetic analyzers, which often have low resolving power. However, even from the data published so far, a substantial difference between the energy spectra of the electrons from different accelerators is clearly visible. The narrowest spectrum was obtained by Bown et al.\(^{26}\), in whose experiments the intensity fell by a factor of two when the energy deviated by \(8\%\) from \(0.6\) MeV; Fry et al.\(^{37,38}\) obtained, at an energy of \(0.5\) MeV, a spectrum with a relative width of \(11\%\), and at \(4.0\) MeV—with a relative width of \(5.4\%\) (the boundary of the spectrum is defined by the point at which the intensity is half the maximum); Mills\(^{53}\) obtained, at an energy of \(1.1\) MeV, a spectrum with a relative width of \(10\%\)*).

In the installations of J. Johnston et al.\(^{40}\) and H. G. Hereford\(^{45}\), the energy spectrum of the electrons is considerably broader. By using a magnetic field and appropriate filters, it is, of course, possible to select and use any part of the spectrum; however, because of the loss of energy spent on accelerating electrons in the unused part of the spectrum, and also because of the considerable losses to radiation, a broad spectrum is, of course, undesirable.

The maximum intensity of the accelerated beam also varies within wide limits. According to a number of authors, in very short accelerators, as was to be expected, considerable currents can be obtained. In long accelerators this is no longer always the case, since here the phase stability of the particles and radial defocusing play an essential role. Very little data have been published on current measurements in accelerators from which one could determine the influence of these two factors not only qualitatively but also quantitatively. Fry et al.\(^{37}\) showed that in a traveling-wave accelerator the electrons can move in phased bunches without losing phasing along its entire length. These authors measured the current in the first stage of the accelerator (a 40-cm section), first without a high-frequency field and then after acceleration in the high-frequency field. In the first case the current was \(2.1\) μA, in the second—\(1.8\) μA. Fry et al.\(^{36}\) also found that, for electrons with initial energies of 45 kV and with an accelerating longitudinal electric-field strength of \(25\) kV/cm, the radial defocusing forces are completely compensated by a longitudinal magnetic field with a field strength

) According to Table III, not \(10\%\), but \(20\%\): \(\Delta E = 200\) keV at \(E = 1.1\) MeV. (Translator’s note.)*

in 540 gauss, in complete agreement with the formula given in § 2.2. In a two-meter accelerator, where the grouped electrons over the first 40 cm changed their synchronous phase with respect to the crest of the wave from 45 to 35° and retained it unchanged during further acceleration, the maximum current in the presence of a magnetic field exceeded by ten or more times the maximum current without a magnetic field. With the magnetic field switched on, it was possible to transfer to the electron beam 30% of all the high-frequency power (measured with a water calorimeter). This corresponded to a maximum pulse current of 150 ma at an electron energy of 4 Mev and an electron-beam diameter of about 7 mm. When working with intense electron beams, it is necessary, in that section of the waveguide where the electron velocities are appreciably less than the velocity of light, to create a longitudinal magnetic field that would compensate the radial component of the electric field.

Column 11 of Table III gives the efficiency of those accelerators for which sufficiently complete data have already been published. The accelerator efficiency characterized in this way shows to what extent the supplied power is used to create the longitudinal high-frequency field. As was shown in §§ 3.2 and 3.3, for each type of accelerator there is a certain greatest length of an individual section. If the section is shorter, then the accelerator efficiency is small; as the length of the section is increased up to this limit, the accelerator efficiency increases. This proposition is well illustrated by the experiments of Johnston et al. ^40, in which the length of the accelerator was increased in three successive stages and the efficiency increased, respectively, from 4.2 to 14 Mohm/m.

For standing-wave accelerators, the highest effective shunt impedance was obtained by Mills ^53—in an accelerator with one end vibrator—and by Allen and Symonds ^23 for a section of three end vibrators. The latter authors achieved a high shunt impedance by using end vibrators about a half-wavelength long with drift tubes *). With the aid of these tubes it was possible to shorten the transit time of the particles through the accelerating gap and to increase the efficiency of the accelerator in comparison with an ordinary half-wave end vibrator, but without drift tubes.

Very few authors have succeeded in constructing accelerators with such high effective shunt impedance as is given in Tables I and II. This is explained by the fact that they considered it more expedient to undertake an investigation of the basic principles of particle acceleration on less efficient sections and thereby to avoid the problems of stabilizing the generator frequency and of separating

) The scheme of the accelerating system of these authors is shown in Fig. 2. (Translator’s note.*)

eigenfrequencies of the end vibrator, which begin to play an essential role as the optimal length is approached. Therefore, in the future one should expect an increase in the efficiency of accelerators.

§ 9. Prospects for the development of electron accelerators

In addition to the increase in efficiency that may be expected from the use of waveguide systems closer to the optimal ones—as was emphasized in §§ 3.2 and 3.3—it should be anticipated that new methods will also be proposed for reducing the losses presently found in any accelerator in which microwaves are used. Two such methods for the case of traveling-wave accelerators have already been proposed by Harvey ^{48,49}.

His idea is to direct the high-frequency energy emerging from the accelerator back to the input through a phasing device and a bridge circuit. The latter must block the path of this energy back into the sources of high-frequency power. Such a bridge circuit can be assembled for almost any preassigned ratio between the power fed into the accelerator and the output power. The bridge circuit can thus be selected for any accelerator with a known attenuation.

In this way it will be possible to build accelerators with a high effective shunt resistance, free from the usual difficulties of frequency stabilization considered in § 3.2.

Another method of increasing efficiency consists in using, as the waveguide load, an anisotropic dielectric instead of metal disks. The effective dielectric constant of the dielectric must be greater in the radial direction than in the axial direction; such a waveguide load can be made, for example, in the form of a chain of ceramic disks. Such a loading method may have a number of advantages over the usual method with metal disks. The effective, “physical,” diameter of the accelerator tube would be smaller, and manufacturing a tube with ceramic disks would be cheaper than manufacturing metal disks with their complicated contacts at the place where the metal surfaces touch. By moving such ceramic disks, one can regulate the phase velocity of the wave within certain limits. In addition to the indicated technical advantages, this method makes it possible to create accelerators with higher acceleration efficiency, since the reduction of losses in copper is much more significant than the corresponding increase in dielectric losses.

Part II

HEAVY-PARTICLE ACCELERATORS

§ 10. A Proton Traveling-Wave Accelerator

In comparison with the recent successes in the field of accelerating electrons by linear methods, the problems of proton acceleration have been investigated relatively little. The only installation that has so far made it possible to obtain protons of high energies is the accelerator at Berkeley (Alvarez et al.⁷⁹), which uses the achievements of modern waveguide technology. The published information on heavy-particle accelerators is clearly insufficient.

In the main, the principles of acceleration of heavy particles are the same as for electrons, but at the same time certain new circumstances arise which make the problem of accelerating heavy particles more complicated. First, it can be shown that the type of waveguide loading which, as was shown above, is quite suitable for accelerating electrons with relativistic velocities, proves ineffective and difficult to realize for slow particles. This is readily seen by considering the rectilinear chain of endovibrators with infinitely small coupling apertures between them, described in § 3.1. In order to take into account the longer transit time of the particles through each endovibrator, the last factor in the expression for the effective shunt resistance should be replaced by the quantity

\[ \left[\frac{\sin \dfrac{\pi}{n\beta}}{\dfrac{\pi}{n\beta}}\right]^2 . \]

The optimum number of endovibrators per wavelength in vacuum is approximately \(\dfrac{3}{n\beta}\). Substituting this value from relation (11), we find that the effective shunt resistance is directly proportional to the velocity of the particles. For protons with energies below \(20\) MeV, a system of this kind will be \(5\)—\(100\) times less efficient than in the relativistic region.

In the Berkeley proton accelerator, ohmic losses in copper are reduced by using a single long endovibrator, in which its fundamental oscillation with an axial electric field and a circular magnetic field is excited *). The transit time of the particles in the endovibrator is considerably greater than the duration of a high-frequency period. In order to shield the particle from the high-frequency field at those moments when the particle enters the decelerating phase of the field, it was necessary to employ special tubes

) The \(E_{010}\) oscillation. (Translator’s note.*)

drift. The resonance system of the accelerator is shown in Fig. 10. It may be regarded as a rectilinear chain of endovibrators whose oscillations occur in one and the same phase, with the internal partitions removed (in the figure these imaginary partitions are shown by dashed lines). The power losses per unit length of the system then depend almost not at all on the number of accelerating gaps. On the other hand, the potential difference traversed by the particle is proportional to the number of accelerating gaps and, consequently, the effective shunt resistance is inversely proportional to the square of the particle velocity.

Harvie drew attention to this fundamental difference in properties between resonance systems for accelerating slow particles and systems intended for accelerating relativistic particles. He pointed out that the accelerator for mercury ions built by Sloan and Lawrence considerably surpassed contemporary electron accelerators in efficiency. This may be attributed to the circumstance that Sloan and Lawrence used only one endovibrator, while at the same time the field in it acted on the particle not once, but many times along its path.

Fig. 10. Diagram of the proton accelerator at Berkeley.

The second circumstance, essential in accelerating heavy particles, is the choice of the frequency of the accelerating field. In § 3.1 it was shown that if the openings intended for particles to pass from one endovibrator into another are small in comparison with the wavelength, then the effective shunt resistance is inversely proportional to the square root of the wavelength. This statement must be modified in order to take account of the finite size of the openings, which has the strongest effect on the efficiency of acceleration, especially in the case of small velocities. Consider an infinitely narrow accelerating gap in a small cylindrical tube. At the surface of the cylinder the field is concentrated in the gap, but as the axis of the tube is approached it decreases (the sagging of the field in the gap from the region at the surface of the cylinder toward its axis decreases as the gap is decreased), and at a distance \(1.4 r\) (where \(r\) is the radius of the tube) from the center of the accelerating gap it is equal to \(1/10\) of its maximum value (Wang \(^{17}\)). If a particle possessing-

with velocity \(v\), it must pass through this distance in a time interval smaller than half the period of the high frequency; then the radius of the cylinder must be less than

\[ \frac{\beta\lambda}{5.6}, \]

where \(\lambda\) is the wavelength in vacuum, and \(\beta\) is the ratio of the particle velocity to the velocity of light. Estimating the wavelength \(\lambda\) in this way, Kay \(^{84}\) established that for protons with an initial energy of \(1\ \mathrm{MeV}\), with drift tubes of diameter \(2\ \mathrm{cm}\), the wavelength must be not less than \(1\ \mathrm{m}\). For heavier ions the finite transit time forces one to choose an even longer operating wavelength.

In reality the particle passes through a whole series of accelerating gaps in the accelerator, and their action must be summed.

In this way we take into account the finite transit time of the particle. As was already indicated at the beginning of the review, by expanding the field in the endovibrator in a Fourier series, we obtain, among other components, also a wave with a phase velocity equal to the velocity of the particle. This wave is described by relation (7). At small phase velocities the longitudinal field near the axis is minimal and increases with increasing radius. Therefore the accelerating gaps are more effective near the surface of the tube, where the longitudinal component of the field has a large value.

The injector of protons in the accelerator of the University of California is a Van de Graaff generator of \(4\ \mathrm{MeV}\). The diameter of the drift tubes is specified by the transverse section of the proton beam and must be not less than \(5\ \mathrm{cm}\). Therefore the frequency is chosen equal to \(200\ \mathrm{MHz}\). The total length of the endovibrator is \(12192\ \mathrm{mm}^{83}\), its diameter \(96.5\ \mathrm{cm}\), the effective shunt resistance \(330\ \mathrm{M\Omega}\), and the \(Q\) factor \(Q = 70\,000\). The external diameter of the drift tubes was chosen so as to maintain the resonant frequency of the endovibrator constant over its entire length. The maximum pulse power is \(1.5\ \mathrm{MW}\) (23 pulse generators on triodes).

The loaded endovibrator has a definite spectrum of resonant frequencies and in the present case operates at the lowest of them. Separation of the resonant frequencies does not present serious difficulties \(^{89}\). A more complicated problem consists in maintaining the field distribution uniform over the entire length of the endovibrator. For this purpose it is necessary to choose definite dimensions of the drift tubes. Nor does the phasing of the individual generators \(^{81}\), which are automatically stabilized at the resonant frequency of the endovibrator, encounter difficulties. From Serber’s calculations \(^{90}\) it follows that with drift tubes provided at the entrance with tungsten grids, both phase stability and stability of transverse focusing are ensured.

The main group of generators operates with pulses of \(300\ \mu\mathrm{sec}\), with a repetition rate of 15 per second. The initial pulse

with a duration of 450 \(\mu\)sec is supplied by three generators located at the exit from the accelerator. The remaining generators are connected to the end vibrator by a 50-ohm line one wavelength long.

The proton energy at the exit from the accelerator is \(32\) MeV, with a spread of \(\pm 100\) keV. The average current at the exit is \(2 \cdot 10^{-10}\) A, the duty factor is 300, so that the average current in the pulse is \(6 \cdot 10^{-8}\) A. The current in the pulse injected by the Van de Graaff generator is 100 \(\mu\)A.

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7. E. O. Lawrence, and. M. S. Livingston, Phys. Rev. 37, 1707 (1931).
8. E. M. McMillan, Phys. Rev. 68, 143 (1945).
9. P. J. Morrison, J. Appl. Phys. 18, 133 (1947).
10. M. L. Oliphant, J. S. Gooden and G. S. Hide, Proc. Phys. Soc. 49, 666 (1947).
11. L. Page and N. J. Adams, Electrodynamics. D. Van Nostrand C°, 1940.
12. R. F. Post, Phys. Rev. 69, 126 (1946).
13. L. I. Schiff, Rev. Sci. Instrum. 17, 6 (1946).
14. R. J. Van de Graaff, Phys. Rev. 38, 1919 (1931).
15. R. J. Van de Graaff, J. G. Trump and W. W. Buechner, Reports on Progress on Physics 11, 1 (1946—1947).
16. V. K. Veksler, Journ. of Phys. 9, 153 (1945).
17. C. C. Wang, J. App. Phys. 16, 351 (1945).
18. W. F. Westendorp, and E. E. Charlton, J. App. Phys. 16, 581 (1945).
19. J. R. Woodyard, Electr. Engineering 67, 759 (1948).

Literature on electron linear accelerators

1. E. S. Akeley, J. App. Phys. 17, 1056 (1946).
2. E. S. Akeley, Phys. Rev. 69, 50 (1946).
3. E. S. Akeley, Phys. Rev. 69, 255 (1946).
4. W. D. Allen and J. L. Symonds, Proc. Phys. Soc. 59, 622 (1947).
5. J. W. Beams and L. B. Snoddy, Phys. Rev. 44, 784 (1933).
6. J. W. Beams and H. Trotter, Jr. Phys. Rev. 45, 849 (1934).
7. E. G. Bowen, O. O. Pulley and J. S. Gooden, Nature 157, 840 (1946).
8. L. Brillouin, Phys. Rev. 71, 483 (1947).
9. L. Brillouin, Phys. Rev. 74, 90 (1948).
10. G. G. Bruck and E. R. Wicker, J. App. Phys. 18, 766 (1947).

LINEAR ACCELERATORS

1. E. L. Chu and W. W. Hansen, J. App. Phys. 18, 996 (1947).
2. W. T. Cownig (1947) — unpublished work.
3. W. T. Cownig and E. J. Jones, Telecommunications Research Establishment (T. R. E.) Report A. F. 1008 (1947).
4. A. B. Cullen and J. A. Greig, J. App. Phys. 19, 47 (1948).
5. C. C. Cutler, Bell Telephone Laboratories Report MM-44-160-218 (1944).
6. S. Dewons — private communication.
7. S. Frankel, J. App. Phys. 18, 650 (1947).
8. D. W. Fry, R. B. R.-S.-Harvie, L. B. Mullet and W. Walkinshaw, Nature 160, 351 (1947).
9. D. W. Fry, R. B. R.-S.-Harvie, L. B. Mullet, and W. Walkinshaw, Nature, 162, 859 (1948).
10. D. Gabor, Nature 159, 303 (1947).
11. E. L. Ginzton, W. W. Hansen and W. R. Kennedy, Rev. Sci. Instrum. 19, 89 (1948).
12. J. Halpern, E. Everhart, R. A. Rapuano and J. C. Slater, Phys. Rev. 69, 688 (1946).
13. W. W. Hansen, report at the conference at the Massachusetts Institute of Technology. June 1948.
14. W. W. Hansen and R. D. Richtmyer, J. App. Phys. 10, 189 (1939).
15. Haxby and oth. Phys. Rev. 70, 797 (1946).
16. F. L. Hereford, J. App. Phys. 18, 956 (1947).
17. R. B. R.-S.—Harvie, T. R. E. Memorandum S3/M4601 (1945).
18. R. B. R.-S.—Harvie, Proc. Phys. Soc. 61, 255 (1948).
19. R. B. R.-S.—Harvie, Nature 162, 890 (1948).
20. R. B. R.-S.—Harvie, Nature, Proc. Phys. Soc. B. 62, 270 (1949).
21. E. L. Hudspeth, Phys. Rev. 69, 671 (1946).
22. E. J. Lawton, J. App. Phys. 19, 534 (1948).
23. E. J. Lawton and W. C. Hahn, J. App. Phys. 19, 642 (1948).
24. B. Y. Mills, Report No. 77 of Radio Physics Laboratory, University of Sydney, Australia (1948).
25. L. B. Mullet, T. R. E. Report, T. 2028 (1946).
26. L. B. Mullet, T. R. E. Report, T. 2021 (1946).
27. L. B. Mullet and Loach, Proc. Phys. Soc. 61, 271 (1948).
28. G. R. Newbery, G. E. C. Research Laboratories Memorandum of January 23, 1948.
29. G. R. Newbery, Brit. J. of Radiology (1948) — Microwave linear electron accelerator.
30. G. R. Newbery and W. E. Willshaw, Nature, 161, 519 (1948).
31. A. A. Olliner, J. App. Phys. 19, 109 (1948).
32. H. L. Schultz, report at the conference at the Massachusetts Institute of Technology. June 1948.
33. H. L. Schultz, R. Beringer, C. L. Clarke, J. A. Lockwood, R. L. McCarthy, C. G. Montgomery, P. J. Rice and W. W. Watson, Phys. Rev. 72, 346 (1947).
34. J. C. Slater, Phys. Rev. 70, 799 (1946).
35. J. C. Slater, Rev. Mod. Phys. 20, 473 (1948).
36. J. C. Slater — private information (1948).
37. J. C. Slater and N. H. Frank, Introduction to Theoretical Physics. New-York, Mc. Graw-Hill. 1933.
38. L. B. Snoddy and J. W. Beams, Phys. Rev. 74, 126 A (1948).
39. L. B. Snoddy, H. Trotter, W. Ham and J. W. Beams, J. Franklin Inst. 233, 55 (1937).
40. A. T. Starr, T. R. E. Memorandum of November 22, 1945.
41. J. A. Stratton, Electromagnetic Theory, New-York, Mc. Graw-Hill. (1941).

1. W. Walkinshaw, T. R. F. Report A E. 1005 (1947).
2. W. Walkinshaw, Proc. Phys. Soc. 61, 246 (1948).
3. W. E. Willshaw, G. E. C. Research Laboratories, England. Memorandum of December 31, 1946.
4. W. E. Willshaw, G. E. C. Research Laboratories. Report 16, 1 (1947).
5. W. E. Willshaw, G. E. C. Research Laboratories. Report 3, 3 (1947).
6. W. E. Willshaw, G. E. C. Research Laboratories. Report 1 (1948).
7. W. E. Willshaw and H. R. L. Lamont, private communication from Research Labs. of G. E. C. Ltd. England, No. 8766 B.
8. J. R. Woodyard, Phys. Rev. 69, 50 (1946).

Literature on Linear Accelerators of Heavy Particles

1. L. W. Alvarez, Phys. Rev. 70, 799 (1946).
2. L. W. Alvarez, and oth., Science, 106, 506 (1947).
3. W. R. Baker, J. V. Frank and J. D. Gow, Phys. Rev. 73, 535 A (1948).
4. J. W. Beams and L. B. Snoddy, Phys. Rev. 45, 287 (1934).
5. H. Bradner and oth., Phys. Rev. 73, 534 A (1948).
6. W. T. Cowhig, T. R. E. Memorandum S1/M178 (1946).
7. D. Gabor, Nature 160, 89 (1947).
8. E. O. Lawrence and D. H. Sloan, Proc. Nat. Acad. Sci. 17, 64 (1931).
9. F. Oppenheimer, L. H. Johnston and C. Richman, Phys. Rev. 70, 447 A (1946).
10. W. K. H. Panofsky, Phys. Rev. 70, 447 A (1946).
11. W. K. H. Panofsky, C. Richman and F. Oppenheimer, Phys. Rev. 73, 535 A (1948).
12. R. Serber, Phys. Rev. 73, 535 A (1948).
13. D. H. Sloan and W. N. Coates, Phys. Rev. 46, 539 (1934).
14. D. H. Sloan and E. O. Lawrence, Phys. Rev. 38, 2021 (1931).
15. C. Turner, B. Cork, I. Ballam and H. Gordon, Phys. Rev. 73, 534 A (1948).
16. R. Wideroe, Archiv Electrotech. 21, 387 (1928).
17. J. R. Woodyard, E. A. Martinelli, W. Toulis and W. K. H. Panofsky, Phys. Rev. 70, 447 (1946).

1. *) D. W. Fry and W. Walkinshaw, Reports on Progress in Physics, Physical Society — London, 12, 102 (1948—1949). ↩