DESIGN OF LINEAR ACCELERATORS

J. C. Slater*)

CONTENTS

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
I. Properties of periodically loaded waveguides . . . . . . . . . . . . . . . . 320
II. Fourier expansion of the field, modulation coefficient, and transit-time correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
III. Group velocity in a loaded waveguide . . . . . . . . . . . . . . . . . . . 331
IV. Attenuation in a waveguide . . . . . . . . . . . . . . . . . . . . . . . . . 332
V. Input power to a waveguide . . . . . . . . . . . . . . . . . . . . . . . . . 334
VI. Influence of geometrical factors on acceleration . . . . . . . . . . . . . . 340
VII. Input impedance of tubes with standing and traveling waves. VIII. Special case of oscillations of type \(\pi\). IX. Feeding linear accelerators. X. Feeding from a self-excited oscillator. XI. Tolerances in a long accelerator. XII. Particle dynamics in the accelerator. XIII. Application of electron dynamics to various types of accelerators. XIV. Transverse motion and focusing of particles.

INTRODUCTION

A linear accelerator is an apparatus for obtaining electrons or positively charged particles of high energy, in which an alternating field is used and the particles move in a straight line (and not along a curvilinear orbit in a magnetic field, as in a cyclotron, betatron, or synchrotron). It is, in essence, a kind of loaded waveguide in which an alternating field of large amplitude is produced, such that it can be represented as a series of traveling waves, one of which propagates with the velocity of the accelerated particle. Particles having a suitable initial phase will all the time move in phase with this wave and will continuously acquire energy, as though they were in a constant field. By varying the field as the particle advances along the waveguide (in accordance with its acceleration), one can in principle obtain particles of any energy.

*) J. C. Slater, Rev. Mod. Phys. 20, 473 (1948). Translated by V. Averbakh.

From what has been said, the principal advantages and disadvantages of a linear accelerator are evident. Among the former are: the absence of the large magnet required in cyclic devices; the proportionality of the dimensions and cost of an installation of the first (and not a higher, as in a cyclotron) power of the energy attained, thanks to which a linear accelerator may prove, at least at very high energies, to be more economical; and finally, the fact that the particles are automatically emitted in a narrow beam, whereas in cyclic accelerators the extraction of the beam constitutes one of the main difficulties. A fundamental shortcoming of the linear accelerator is the circumstance that an individual particle, instead of moving in one and the same alternating field fed from a single source, must pass through a series of such fields using different energy sources. This leads to a considerable complication of the high-frequency equipment and to difficulties associated with the construction of a very long tube, requiring careful manufacture, with energy sources connected to it. It is also technically difficult to ensure agreement of the phases of the alternating fields over the entire (very great) length of the apparatus, which is necessary in order that the field vary in step with the acceleration of the particle.

There is still another constructional difficulty, absent in cyclic devices. It becomes clear after a short calculation. In any apparatus, before attaining high energy, the particles must traverse a long path—along a straight line, around a circle, or along a spiral. On this path they may depart from their ideal orbit, thus becoming lost from the beam. This “beam spreading” may be of two kinds. First, spreading in the literal sense of the word—defocusing. Second, in all accelerating devices except the betatron, particles move in groups separated from one another by one wavelength, so that each group is in phase with the accelerating wave. The particles may scatter “longitudinally,” getting ahead of their group or lagging behind it. In the cyclotron and synchrotron, particles moving in phase with the accelerating wave also turn out to be in the phase needed for focusing, so that the stability of the system is ensured automatically. In a linear accelerator, however, particles moving in the phase required for acceleration and grouping are defocused. This circumstance is very serious for a proton accelerator, but not essential for an electron accelerator, since defocusing decreases to zero as the particles reach the velocity of light.

It is still too early to analyze in detail the advantages and disadvantages of linear accelerators. Of the other electron accelerators, the practical ones are: the belt electrostatic generator up to energies of the order of 20 MeV, the betatron—in the interval from 20 to, probably, several hundred MeV, and the synchrotron—from

from several hundred MeV to about a billion eV, where, as is generally accepted,^25 radiation losses practically set a limit to any further acceleration of particles. In the range of its applicability, the electrostatic generator is an ideal laboratory instrument, giving strictly monochromatic particles in a well-collimated beam; however, a linear accelerator fed by an electron gun at a comparatively low voltage may prove cheaper and just as convenient for obtaining X-rays and for other purposes that do not require exceptional homogeneity of the particles in energy. The betatron and the synchrotron, in that energy range in which they can be used, are cheaper than the linear accelerator, though not by much. They may give more monochromatic beams, but it is much more difficult to obtain in them a well-collimated electron beam than in a linear accelerator. Their principal field of application is the production of X-rays. In the energy region above a billion eV, the linear accelerator is probably the only practically suitable one, unless the difficulties with radiation in the synchrotron are indeed as great as they seem. On the basis of our present knowledge, it does not appear impossible to build a linear electron accelerator to a billion eV, and it will be only a little more expensive than a proton synchrotron designed for the same energy range. Of course, the capabilities of linear accelerators will be easier to assess when instruments of this type now under construction, designed for energies of the order of 30 MeV, come into operation.

Of the positive-ion accelerators, the tandem electrostatic generator is good in the region up to 20 MeV, while the ordinary cyclotron encounters relativistic difficulties at about 200 MeV. The synchrocyclotron is ideally suited for energies of the order of several hundred MeV. Attempts to extend its range of applicability have shown, however, that at energies of the order of 600 MeV the design of the dees and the construction of the magnet meet with certain difficulties, which become very serious at energies around a billion eV and are probably insurmountable at still higher energies. To obtain positive ions with energies of several billion eV, the proton synchrotron is proposed—an instrument in which the magnetic field and the frequency are simultaneously modulated so as to keep the particle in phase with the electric field while maintaining the radius of its orbit constant. For this only a ring magnet is needed, the cost of which does not increase with energy as rapidly as in a synchrocyclotron with its massive solid magnet. In addition, the accelerating electrodes can be comparatively small, and the frequencies encountered here are so low that frequency modulation is carried out simply.

At present, the construction of a proton synchrotron for 2–10 billion eV is practically feasible. In comparison with such

LINEAR ACCELERATOR DESIGN

...instruments, linear accelerators have little to say in their own favor. We have already spoken of the difficulty with defocusing, which is very significant for a linear accelerator of positive ions. At present it permits proton linear accelerators to be used only for obtaining comparatively small energies (approximately in the cyclotron range). In addition, the existing proton linear accelerators are considerably larger in size (and, consequently, considerably more expensive) than electron accelerators of the same type designed for the same energy range. Thus, in cost, proton linear accelerators can hardly compete with an electrostatic generator or with cyclotrons of various types. At the time of writing this article it seems unlikely that the construction of proton linear accelerators designed for high energies will be undertaken. The construction of new instruments of this type for obtaining comparatively small energies is also unlikely.

The linear accelerator is not a new invention. Sloan \(^{20,33-35}\) and others worked in this field with moderate success during the last ten years before the war. The creation during the war of powerful sources of high-frequency energy—such as magnetrons—made it possible to feed a linear accelerator with the necessary amount of energy while operating by a pulsed method. By promising truly large accelerations, this circumstance made linear accelerators an attractive object for further work. After the war, research developed along two principal directions, closely connected with one another; both grew out of work on radars. First, Alvarez \(^{15,16,19,20,22,24,27,31}\) at the University of California (Berkeley) worked on the construction of an ion accelerator. He worked at a frequency of about 200 megacycles, using radar equipment designed for the same frequency. Second, a number of laboratories engaged in electron accelerators, using powerful magnetrons with a wavelength of about \(10\ \mathrm{cm}\), built for the purposes of radiolocation. This program was carried out in the Research Laboratory of Electronics of the Massachusetts Institute of Technology \(^{16,20,30,32}\). It arose as a direct development of ideas formed in the Radiation Laboratory of the same institute at the end of the war, but all the work and detailed design were carried out after the formation of the Research Laboratory of Electronics. Similar work in the USA was also carried out at the General Electric Company*), at Stanford \(^{10,15}\), Virginia \(^{11}\), Yale \(^{25}\), and other universities \(^{1,2,3,17}\). Abroad, extensive work was conducted in England at the Telecommunications Research Establishment (TRE)\(^{14}\), at the Polytechnic Institute in Mexico, and in a number of other places. As far as is known to the author of this article, research everywhere developed

) Articles are being printed in Journ. Appl. Phys.*

along the same lines, with minor differences of opinion at each stage of the work. The author had the opportunity to make use of preliminary reports from the laboratories at Stanford, where Hansen and his collaborators achieved significant successes, and from TRE, where the theoretical group headed by Walkinshaw and the experimental group made detailed and complete surveys of the question. The treatment of the problem in the present article and in the work of the groups mentioned above has many features in common. The present work was carried out in almost all respects independently of the others during the last two years at the Massachusetts Institute of Technology (MIT), and it seemed advisable to make this article complete, even if it overlaps with the work of other groups.

The MIT project is both theoretical and experimental. It is devoted to the study of the design features of a large accelerator and to the construction of an experimental installation for accelerating electrons to energies of the order of 20 MeV. Upon completion of the project the corresponding group will publish an experimental paper with a discussion of the results. For the time being it may be noted that a number of the results set forth in the present article have already been checked experimentally.

I. PROPERTIES OF PERIODICALLY LOADED WAVEGUIDES

The phase velocity of a wave in a waveguide is greater than the velocity of light in vacuum. A particle cannot move with such a velocity; therefore, to create a field in a linear accelerator more complicated systems are needed. All systems used in practice consist of periodically loaded waveguides of various types. In this section we shall consider the properties of such waveguides, emphasizing not so much the individual features of the various structures as their common features, which all such systems possess without exception. A considerable part of the results is already known in mathematical physics from many problems on the propagation of waves in periodic structures—such as the theory of a loaded string, electric filters, the theory of electrons in metals and crystalline solids, electron and x-ray diffraction. The interrelations of these problems are analyzed in L. Brillouin’s book Wave Propagation in Periodic Structures (1946). During the war analogous methods were applied by the author to the study of resonant oscillations in a multicavity magnetron (this is briefly touched upon in reference 31). Since these general methods are well known, we shall present many results in outline, without proof or detailed discussion. The structures most frequently used are shown schematically in Fig. 1 (a—a waveguide with diaphragms, used in most short-wave electron accelerators, including the MIT apparatus; b—a system used in an ion accel—

THE DESIGN OF LINEAR ACCELERATORS

... Alvarez’s accelerator). Turning to a concrete example, we shall usually have in mind case (a), but most of our remarks are also applicable to the Alvarez system.

The study of wave propagation in periodic structures is based on the so-called Floquet theorem. Its formulation is very simple: the wave function (in the electromagnetic case—the values of the electric and magnetic fields) corresponding to a given type of oscillation and a given frequency, when advanced along the structure by one period, is multiplied by a given complex constant. The proof of this theorem is not difficult and follows from the fact that the system, shifted along its axis (chosen as the \(z\)-axis) by one period \(L\), coincides with the original one. Consequently, the new wave function can differ from the old one only by a constant factor, which we shall write in the form \(e^{-\gamma L}\), where \(\gamma\) is a constant (generally speaking, complex). There exists a very simple function that is multiplied by \(e^{-\gamma L}\) when \(z\) increases by \(L\). This is the exponential function \(e^{-\gamma z}\). It can be shown that the most general function possessing this property is the product of \(e^{-\gamma z}\) by an arbitrary periodic function of \(z\) with period \(L\) (unchanged when \(z\) is changed by \(L\)). The latter may be represented as a Fourier series (written in complex form), i.e. as a sum of exponentials \(e^{-2\pi n i z/L}\) (\(n\) is a positive or negative integer) with suitable coefficients. Thus the wave function of the system may be written as a sum of exponential functions \(e^{-(\gamma z+2\pi n i z/L)}\) with suitable coefficients. The interpretation of this result depends on the character of \(\gamma\). Generally speaking, this constant is complex, but it can be proved that in a system without energy dissipation it must be either real or purely imaginary. In the case of real \(\gamma\), each term of the series decreases with increasing \(z\), and we have a damped wave. Such waves cannot be used for particle acceleration. In the case of purely imaginary \(\gamma\), we may write: \(\gamma=i\beta_0\). Introducing the notation:

Fig. 1.

\[ \beta_n=\beta_0+2\pi\frac{n}{L}, \tag{1} \]

we can write our exponents in the form \(e^{-i\beta_n z}\). Multiplying this by \(e^{i\omega t}\), which expresses the sinusoidal time dependence of the field, we obtain a separate Fourier component in the form \(e^{i(\omega t-\beta_n z)}\). This function is a traveling wave with angular frequency \(\omega\) and wavelength \(\dfrac{2\pi}{\beta_n}\), propagating along the \(z\)-axis with velocity

\[ v_n=\frac{\omega}{\beta_n}. \]

It is now clear from equation (1) that the disturbance produced by a periodic load may be regarded as the superposition of many traveling waves with different velocities \(v_n\), decreasing (in absolute value) for large values of \(n\). Each wave enters with its own coefficient (amplitude).

Some of these traveling waves can be used in a linear accelerator. Namely: the velocities of many Fourier components turn out to be less than the velocity of light. Consequently, one of them may be equal to the velocity of the particle being accelerated. Such a component will resonate with the particle in the sense that the phase relation between the wave and the particle will remain constant (until, of course, the particle has accelerated enough to overtake the wave; such questions will be considered later).

A particle resonating with one component of the Fourier expansion of the field will not resonate with the remaining components, since they propagate with other velocities. Indeed, in a coordinate system attached to the moving particle, the resonant component appears as a constant field: the effect it produces does not change over many periods. The other components, moving relative to the particle with high velocity, appear as fields rapidly oscillating in time, which on average have almost no effect on the motion of the particle. For almost all purposes they may be completely neglected. Thus we arrive at a very important result: only one Fourier component of the field produced by a periodic structure propagates with the same velocity as the particle; this component acts as a traveling sinusoidal wave, and only it need be taken into account when considering the motion of the particle. However, all Fourier components have finite amplitude and, consequently, carry energy, which they dissipate if the walls of the waveguide have finite conductivity. These energy losses are entirely useless from the point of view of particle acceleration. Therefore, it is necessary to find such a method of exciting the waveguide in which the amplitude of the resonant component is much greater than all the others.

To determine the velocity of each component as a function of frequency \((\omega)\), and hence to find the resonant frequency, one must know the dependence of \(\beta_0\) and, consequently, \(\beta_n\) on frequency. This is not an easy problem, which we shall discuss later. However, some general results can be obtained easily. Let us consider \(\omega\) as a function of \(\beta_0\). It is easy to show that

it is periodic with respect to \(\beta_0\), with period \(\dfrac{2\pi}{L}\). Indeed, let \(\beta_0\) increase by \(\dfrac{2\pi}{L}\). Then, according to equation (1), the quantity previously denoted by \(\beta_{-1}\) will also increase and will become equal to the former value \(\beta_0\). Similarly, all \(\beta_n\) will turn out to be equal to the former \(\beta_{n+1}\). Thus the number of each individual quantity \(\beta_n\) will change, but the whole set of them will remain unchanged. Physically this changes nothing, since the Fourier coefficients depend only on the numerical value of \(\beta_n\), and the frequency cannot change. It is also easy to show that \(\omega\) is an even function of \(\beta_0\), i.e. it does not change when the sign of \(\beta_0\) is changed. This follows from the fact that the physical picture is unchanged under the following two transformations: a) passage from the given wave function to the complex-conjugate one, when \(e^{i(\omega t-\beta_n z)}\) is replaced by \(e^{-i(\omega t-\beta_n z)}\); b) reversal of the sign of time, after which we obtain \(e^{i(\omega t+\beta_n z)}\). The final result amounts to changing the sign of all \(\beta_n\) without changing the properties of the system (in particular, the frequency).

Fig. 2. Frequency as a function of the reciprocal wavelength in a waveguide for a periodically loaded line. The slopes of the radius-vectors represent the phase velocity of the various Fourier components divided by the speed of light.

As a consequence of these two theorems, \(\omega\) as a function of \(\beta_0\) has the form shown in Fig. 2. On the graph it is convenient to plot not \(\omega\) and \(\beta_0\), but

\[ \frac{\omega}{2\pi c}=\frac{1}{\lambda_0} \quad\text{and}\quad \frac{\beta_0}{2\pi}=\frac{1}{\lambda_g}, \]

where \(c\) is the speed of light in vacuum, \(\lambda_0\) is the wavelength in vacuum, and \(\lambda_g\) is the wavelength in the waveguide corresponding to \(n=0\). We see that the latter quantity is not determined uniquely: to \(\dfrac{1}{\lambda_g}\) one may always add an integral multiple of \(\dfrac{1}{L}\) without changing \(\omega\). In other words, \(\dfrac{1}{\lambda_0}\) is a periodic function of \(\dfrac{1}{\lambda_g}\) with period \(\dfrac{1}{L}\). It is also seen that \(v_0\), the phase velocity of the Fourier component with \(n=0\), divided by the speed of light in vacuum \(c\), is given by the formula:

\[ \frac{v_0}{c}=\frac{\omega}{\beta_0 c}=\frac{\lambda_g}{\lambda_0}, \]

i.e. it is equal to the tangent of the angle of inclination of the radius-vector drawn to the given point of the curve. If this angle is greater than \(45^\circ\), then \(v_0\) is greater than \(c\); otherwise it is less than \(c\).

Plotting on the graph the values of \(\dfrac{1}{\lambda_g}\), separated from one another by the interval \(\dfrac{1}{L}\), and drawing the corresponding radius-vectors (see

Fig. 2), we shall find the phase velocities of all the Fourier components associated with the given wave function. From the figure it is clear that the phase velocity decreases as the number \(n\) increases. It is also clear that there are Fourier components with both positive and negative velocities, i.e., waves propagating both to the right and to the left.

Let us now consider qualitatively, by means of a separate example, how curves of the type shown in Fig. 2 are obtained.

As an example let us take the tube with iris diaphragms shown in Fig. 1,a.

Let us set ourselves the task of finding out how the curves change when the sizes of the openings in the diaphragms are changed, while the distance between the latter remains unchanged, from one limiting case—the complete absence of diaphragms—to the other—vanishingly small openings, when the waveguide consists of a series of cylindrical cavities separated by conducting walls*).

We shall consider only waves of the transverse magnetic type, in which the magnetic lines of force are directed along circles about the axis of the cylindrical waveguide, while the electric field lies in a plane passing through the axis and does not depend on the angle of rotation of this plane about the axis. It is precisely this type that is usually encountered in cylindrical cavities, where the electric field is directed along the \(z\) axis from one wall to the other and is proportional to \(J_0(kr)\) \(\left(k=\dfrac{2\pi}{\lambda_0}\right)\), while the magnetic field is proportional to \(J_1(kr)\). The natural frequency is determined from the condition that \(E_z\) be zero on the outer wall of the cavity, i.e., at \(r=R\), where \(R\) is the radius of the cylinder. Thus we have \(J_0\left(2\pi\dfrac{R}{\lambda_0}\right)=0\). As is known, the first zero of \(J_0(x)\) is equal to 2.405, i.e., resonance occurs at \(2\pi\dfrac{R}{\lambda_0}=2.405\). Let us now consider a sequence of cases belonging to this type of cavity oscillation, changing the radii of the openings in the diaphragms, but leaving unchanged the distances between the latter.

Let us begin with the case of the complete absence of diaphragms (the radius of the opening \(a\) is equal to the radius of the cylinder \(R\)). We have here the ordinary unloaded waveguide, in which the wavelength \(\lambda_g\) is related to the wavelength in vacuum \(\lambda_0\) by the well-known relation:

\[ \frac{1}{\lambda_0^2}=\frac{1}{\lambda_g^2}+\frac{1}{\lambda_c^2}, \tag{2} \]

where \(\lambda_c\) is the critical wavelength. The latter is defined as that wavelength in free space at which \(\lambda_g\) becomes—

\[ \_\_\_\_\_\_\_\_\_\_ \]

*) This problem has been studied in detail by the author by analytical and numerical methods, leading to quantitatively correct results. See \(^{30}\).

becomes infinite, i.e. the field in the waveguide ceases to depend on \(z\). In this case the problem becomes exactly the same as in the already considered case of a cylindrical cavity, so that the critical wavelength coincides with the value \(\lambda_0\) found there, i.e.

\[ 2\pi \frac{R}{\lambda_c}=2.405. \tag{3} \]

We see that equation (2) gives a hyperbolic relation between \(\frac{1}{\lambda_0}\) and \(\frac{1}{\lambda_g}\), as shown in Fig. 3. When \(\frac{1}{\lambda_0}\) is less than \(\frac{1}{\lambda_c}\) (i.e. when the wavelength in free space is greater than the critical value), the curve has no real points. This means that propagation of such waves is impossible; we have a purely imaginary wavelength, describing attenuation. For all higher frequencies the hyperbola lies above the bisector of the coordinate angle, indicating that the phase velocity here is greater than the velocity of light; it approaches the latter asymptotically at high frequencies. In this case, naturally, there is no periodicity in the waveguide. Consequently, there is also no periodicity in the graph of Fig. 3.

Fig. 3. Hyperbola representing \(\frac{1}{\lambda_0}\) as a function of \(\frac{1}{\lambda_g}\) in an unloaded waveguide.

Fig. 4. \(\frac{1}{\lambda_0}\) as a function of \(\frac{1}{\lambda_g}\) for a periodically loaded waveguide (with a small load). The bold part of the curve corresponds to Fig. 3.

However, even very small diaphragms (such that the aperture radius \(a\) is almost equal to \(R\)) substantially change the matter, introducing periodicity. From Fig. 4 one can see how this occurs. The change in frequency, negligible over most of the range of variation of \(\beta_0\), becomes substantial near

\[ \frac{\beta_0}{2\pi}=\frac{1}{2L}. \]

The meaning of this quantity is very simple: it corresponds to

\[ L=\frac{\lambda_g}{2}, \]

i.e. to the case when the distance between the diaphragms is equal to half the wavelength in the waveguide. Let us see how a wave of such a wave-

It may be assumed that each diaphragm causes scattering of the waves, as a result of which they are divided into transmitted and reflected parts; the latter propagates in the opposite direction. For

\[ L=\frac{\lambda_g}{2} \]

the wave reflected from the given diaphragm will be shifted in phase by \(2\pi\) relative to the wave reflected from the preceding diaphragm; the same will also occur with the transmitted waves. Thus, all the reflected waves will be able to interfere, producing a noticeable effect. In fact, they interfere so strongly that the wave cannot propagate at all in an infinitely long waveguide of this type, if there are no losses: calculation shows that the amplitudes of the reflected and incident waves are equal to one another, and a standing wave is formed. This is the same phenomenon that is known in the theory of X-ray diffraction as Bragg reflection.

The situation becomes clearer if one considers the amplitudes of the various components of the Fourier expansion of the field in our problem. Suppose that \(\beta_0/2\pi\) is equal to the value \(\beta_n/2\pi\) closest to the reciprocal wavelength that would propagate at the given frequency in an unloaded waveguide. Then for most frequencies (or \(\beta_n\)) only one Fourier component associated with \(\beta_0\) will be appreciably different from zero. In fact, it turns out that for positive \(\beta\) the reflected wave is characterized by \(n=-1\); multiply reflected waves correspond to other values of \(n\). The reflected waves, unless the Bragg condition mentioned above is satisfied, will be very weak, since waves reflected from different diaphragms cannot reinforce one another as a result of interference. Therefore the Fourier component associated with \(n=-1\) will be small, and all the others even smaller. However, as the critical value

\[ \frac{\beta_0}{2\pi}=\frac{1}{2L}, \]

is approached, the Fourier coefficient corresponding to \(n=-1\) increases and, upon reaching the critical value, becomes equal to the amplitude of the component with \(n=0\). After passing the critical value all the Fourier coefficients, except the one corresponding to \(n=0\), again decrease. At

\[ \frac{\beta_0}{2\pi}=\frac{1}{L}, \]

when the diaphragms are spaced from one another by one wavelength, the interference conditions are again fulfilled and a reflected wave appears, this time corresponding to \(n=-2\), etc.

Let us now consider the periodic structure of the curves in Fig. 4. The bold part of the curve represents the dependence of frequency on \(\beta_0\) described above. It deviates from the hyperbola characterizing the unloaded waveguide only near the critical points—

\[ \frac{\beta_0}{2\pi}=\frac{m}{2L}, \]

where \(m\) is an integer. At these points the curve has a break; two distinct waves in the waveguide correspond to one and the same wavelength.

permissible frequencies. It can be shown that they are associated with two different types of oscillations. Both of these types are standing waves (since the incident wave is completely reflected under these conditions); one of them is sinusoidal, the other cosinusoidal. Consequently, the diaphragms fall at the antinodes of one of these waves and at the nodes of the other; it is therefore not surprising that they affect the frequencies of these two types of oscillations in different ways. Waves whose frequency falls within the gaps cannot propagate; the value of \(\beta_0\) for them becomes imaginary, i.e., we obtain a damped wave. In other words, the entire frequency interval is divided into a number of pass bands, in which wave propagation is possible, separated from one another by attenuation zones. A periodically loaded waveguide is thus similar to a filter possessing an infinite number of pass bands. For a load tending to zero, we obtain the case (Fig. 3) in which the zones merge, and propagation of waves with all frequencies exceeding the critical one is possible.

In view of the already mentioned periodicity of the frequency as a function of \(\beta_0\), the curves of Fig. 4 are periodic. The parts of them drawn in bold, in the limiting case of a vanishing load, reduce, as we have seen, to the hyperbola of Fig. 3, but the other branches of the curves have an equal right to exist. They correspond to different conditions of numbering the numbers \(\beta\). Up to now the largest Fourier coefficient corresponded to \(n = 0\). The other branches of the curves correspond to numbering systems in which the Fourier coefficient for \(n = 0\) is small, but some other component is large. In the problem considered just now, the numbering system originally chosen by us is the most natural. However, with an increase in the diaphragms this is no longer so clear, since a whole series of Fourier coefficients differ noticeably from zero, and other numbering systems may prove to be just as reasonable as ours.

It is seen from Fig. 4 that now, in contrast to the case of Fig. 3, waves can propagate whose phase velocity is less than the speed of light. The curves have the form already considered in Fig. 2, and we know that to any frequency from the pass band there corresponds an infinite number of velocities, each of which is determined by the point of intersection of the curve \(\omega(\beta)\) with the straight line \(\omega = \mathrm{const}\). However, each of these velocities belongs to a definite Fourier component, and the only component noticeably different from zero is shown in Fig. 4 by a bold line, which for the greater part still lies above the bisector of the coordinate angle and, consequently, corresponds to a phase velocity greater than \(c\). In other words, although the diaphragms do lead to the appearance, in the Fourier expansion of the field, of components with small phase velocities, the amplitudes of these components for small diaphragms are small and, consequently, they cannot be used effectively for accelerating particles.

The situation, however, changes with a further increase of the diaphragms, i.e. with a decrease of the ratio \(\dfrac{a}{R}\) (\(a\) is the radius of the aperture). In this case the curves are more like those shown in Fig. 2. They may descend so that the phase velocity of the largest Fourier component becomes less than \(c\). Moreover, as a result of strong scattering of the waves at the diaphragms, the amplitudes of other components (in particular, those corresponding to small velocities) may increase noticeably. Of course, these effects are interconnected, and although now the Fourier components with small velocities, suitable for use in the accelerator, have sufficiently large amplitudes, a new difficulty arises: the other components also have appreciable amplitudes and also carry an appreciable fraction of the energy, which is useless from the point of view of accelerator operation. Suppose, for example, that the waveguide is loaded so that the phase velocity of waves of the so-called \(\pi\)-type is less than \(c\), and suppose that we use them for acceleration (waves of type \(\pi\) are those for which the phase difference corresponding to the distance from one section to the next, \(\beta_0 L\), is equal to \(\pi\); this means that \(\dfrac{\beta_0}{2\pi} = \dfrac{1}{2L}\)—the Bragg critical value). But we have already seen that in this case a reflected wave arises, of the same amplitude as the incident one. It is useless for our purpose, and all the energy contained in the reflected wave is lost. Or, let us assume that we use a wave of type \(\dfrac{\pi}{2}\), in which \(\beta_0 L = \dfrac{\pi}{2}\); \(\dfrac{\beta_0}{2\pi} = \dfrac{1}{4L}\). As is clear from the figure, in this case the waveguide must be loaded still more strongly in order for the phase velocity to be less than \(c\). Then the amplitudes of other waves propagating in the same direction will also become noticeable, and a considerable fraction of the energy will go into them. In any event, we must always “pay a penalty” for reducing the wave velocity, and this penalty is the greater, the greater the reduction achieved.

In the limit, the apertures in the diaphragms disappear completely. The waveguide then turns into a set of isolated cylindrical cavities, each of which oscillates independently of the others. To excite oscillations corresponding to various values of \(\beta_0\), we need only establish between neighboring cavities a phase difference equal to \(\beta_0 L\). In the limit the frequency does not depend on \(\beta_0\), so that the curves \(\omega=\omega(\beta)\) become straight lines parallel to the abscissa axis, and the pass bands shrink into lines.

The allowed frequencies—the ordinates of these lines—are easy to find: they are precisely the natural frequencies of the cylindrical cavity. The smallest of them is given by equation (3) and corresponds to the first pass band; the higher ones describe the case of sinusoidal propagation of the disturbance along the cavity, with the tangential component vanishing on both plane walls; thus the length \(L\) is equal to an integral number of half-waves. In this limiting case—

where it is easy to find the Fourier expansion of the field and, consequently, the magnitude of the individual Fourier components; the limiting values obtained are approximately valid also for diaphragms with very small holes \((r \ll R)\). Thus, for example, in the lower pass band the magnitude of the \(z\)-component of the field in each cavity is approximately constant and equal to \(e^{-i\beta_3 pL}\), where \(p\) is the number of the cavity. To determine the Fourier components it is only necessary to expand, by the method described above, this simple step function.

II. FOURIER EXPANSION OF THE FIELD, MODULATION COEFFICIENT AND TRANSIT-TIME CORRECTION

In the preceding section we pointed out that, in considering a loaded waveguide as a linear accelerator, it is sufficient to take into account only the Fourier component of the field that is resonant with the particle (i.e., the component whose phase velocity is equal to the velocity of the particle). Then the phase relation between the field and the particle does not change, the particle is under the action of a constant force, and the dynamical problem is very simple. On the other hand, a number of authors use another method, known from the ordinary theory of triodes and klystrons, namely, they consider the effect of the finite transit time of the particle from one side of the cavity to the other.

Let, for example, the holes in the diaphragms be small. Then two diaphragms are similar to two grids of a triode or klystron, and the field between them is approximately uniform (does not depend on \(z\)), but, of course, varies sinusoidally with time. During the motion of the particle from one diaphragm to the other, the field (varying with time) cannot constantly retain its maximum value. Calculating the mean force acting on the particle, we find that it is equal to its maximum value multiplied by a certain coefficient, called in klystron theory the modulation coefficient. The latter is equal to unity if the grids are so close to one another that the transit time may be regarded as zero, and decreases as the transit time increases. The functional dependence of the modulation coefficient on the transit time \((x)\) has the form \(\frac{\sin x}{x}\). For certain values of \(x\) this function vanishes or becomes negative; such effects have to be dealt with in the theory of high-frequency triodes, where the transit time plays an important role.

We shall now show that this modulation coefficient, or the influence of the transit time, is not for us something new, requiring separate consideration, but is obtained quite elegantly from our method of using only one Fourier component. Let the particle move with velocity \(v\) along the \(z\)-axis; its coordinate

is \(z=vt\). Let the longitudinal component of the electric field \(E_z\) be given by the formula

\[ E_z=\sum_n F_n e^{i\omega(t-z/v_n)}, \]

where \(F_n\) is the amplitude and \(v_n\) the velocity of the \(n\)-th Fourier component. As the particle moves, the field changes; this we can take into account by substituting for \(t\) the value \(t=z/v\), corresponding to the moment at which the particle is at the point \(z\). Thus the field acting on the particle at the point \(z\) is

\[ \sum_n F_n e^{i\omega\left(\frac{1}{v}-\frac{1}{v_n}\right)z}. \]

The mean (over \(z\)) value of the \(n\)-th term (we assume that the particle traverses a large distance) is equal to zero if \(v\) differs from \(v_n\), since the mean value of a harmonic function is zero. Thus the mean acting field is equal to zero unless the particle velocity coincides with the velocity of one of the Fourier components; in this latter case the mean field is equal to \(F_n\), the corresponding amplitude.

It should be noted that the result obtained is valid only if the particle traverses a large distance in the field with constant velocity. In a linear accelerator this condition is approximately (but not exactly) fulfilled; deviations from it cannot substantially alter our reasoning.

Thus we see that our method of expansion in a Fourier series automatically takes into account the finite time of flight of the particle from one pole (or diaphragm) to another. We note that this result is quite general. It does not depend on the use of a special model of a waveguide with diaphragms, but is applicable to all periodic structures. Any change in the structure leading to an increase of the amplitude of the Fourier component resonant with the particle can otherwise be described as an increase in the modulation coefficient. However, our method is much more general than the usual arguments in terms of transit time, since the latter are often carried out only for the case of a constant field between parallel grids or electrodes, whereas our method is suitable for any law of variation of the longitudinal field with the \(z\)-coordinate. By using a Fourier integral instead of a series, our method can also be applied to nonperiodic structures, to systems of the triode or klystron type. In this way one can prove the known theorems on the modulation coefficient and transit time, and also, as will be seen below, on the transverse motion of particles and on focusing. We shall not, however, make use of this.

III. GROUP VELOCITY IN A LOADED WAVEGUIDE

In the first section we showed that the phase velocity of a wave in a loaded waveguide is given by the slope of the radius vector drawn to the point representing this wave on the graph of \(\dfrac{1}{\lambda_0}\) as a function of \(\dfrac{1}{\lambda_g}\). Let us now consider the group velocity (which we shall denote by \(v_g\)) and show that it is determined by the slope not of the radius vector, but of the tangent to the curve at the given point:

\[ \frac{v_g}{c}= \frac{d\left(\dfrac{1}{\lambda_0}\right)} {d\left(\dfrac{1}{\lambda_g}\right)}. \tag{4} \]

We shall see that the concept of group velocity plays an important role in studying the motion of energy in a waveguide, as well as in the question of how large the dissipation of energy must be in order to establish a field of a specified intensity.

To derive the formula for group velocity, one usually takes the superposition of two waves with frequencies \(\omega\) and \(\omega+\Delta\omega\) and propagation constants \(\beta\) and \(\beta+\Delta\beta\), and computes the beat velocity. It is easy to show (see, for example, Slater and Frank, Mechanics, 1947, p. 168) that it is given by the expression \(\dfrac{\Delta\omega}{\Delta\beta}\). This quantity is closely connected with the slope of curves of the type shown in Fig. 2. The two waves are represented there by two points. Recalling that along the abscissa axis in Fig. 2 there is plotted

\[ \frac{1}{\lambda_g}=\frac{\beta}{2\pi}, \]

and along the ordinate axis

\[ \frac{1}{\lambda_0}=\frac{\omega}{2\pi c}, \]

we have:

\[ \frac{\text{beat velocity}}{c} = \frac{\Delta\left(\dfrac{1}{\lambda_0}\right)} {\Delta\left(\dfrac{1}{\lambda_g}\right)}. \tag{5} \]

For sufficiently close frequencies and propagation constants, the right-hand side of equation (5), representing the slope of the chord, may be replaced by the slope of the tangent to the curve, thereby obtaining the beat propagation velocity common to all waves with neighboring frequencies. This is precisely the group velocity, defined by equation (4).

Let us now, by superposing a number of plane waves of different frequencies, form a group of waves, for example, a wave packet of finite dimensions. The larger these dimensions, i.e. the greater the extent of the wave packet in space and time, the narrower the necessary interval of frequencies (or wavelengths). If the dependence of \(\dfrac{1}{\lambda_0}\) on \(\dfrac{1}{\lambda_g}\) may be regarded as linear in this frequency interval, then the beat velocity,

determined by formula (5), will be the same for all pairs of waves in the packet, and all of them will propagate with one and the same group velocity. If, on the other hand, the dimensions of the packet are small, then its Fourier decomposition covers a wide interval of wavelengths, the beat velocities of different pairs of frequencies are different, the packet will spread out, and the disturbance will propagate in a rather complicated manner.

The group velocity is the velocity of propagation of energy in the waveguide. This is most easily seen by considering a wave packet of definite length moving, for example, to the right. It is clear that in one second through a unit cross section of the waveguide there passes precisely the energy contained in a volume \(1\cdot v_g\). Thus we obtain a relation between the Poynting vector, integrated over the cross section of the waveguide (i.e. the energy flux), and the energy density per unit length: the energy flux is equal to \(v_g\) multiplied by the energy density. From Figs. 2 and 4 it is seen that the slope of the curves representing \(\dfrac{1}{\lambda_0}\) as a function of \(\dfrac{1}{\lambda_g}\) is always less than unity, so that the group velocity is always less than the velocity of light, as was to be expected from relativistic considerations. Further, we see that as the oscillations approach the \(\pi\)-type, the tangent becomes horizontal, and the group velocity tends to zero. This agrees with what has already been said about this type of oscillation. Above we saw that, as the \(\pi\)-type is approached, the reflected wave increases until, in the limit, a standing wave is formed in which the amplitudes of the incident and reflected waves are equal. In this case the resultant energy flux is absent, and \(v_g\) is equal to zero. Near the \(\pi\)-type the amplitudes of the reflected and incident waves are almost equal, the energy flux is small and, consequently, the group velocity is also very small. We see, furthermore, that when the holes in the diaphragms are small, the frequency bands contract, and even for oscillations of the type \(\dfrac{\pi}{2}\), when the group velocity reaches its maximum value, it is still very small. In the limit, when the radius of the holes decreases to zero, and the frequencies cease to depend on the wavelength in the waveguide, the group velocity also tends to zero. This is evidently in agreement with the fact that, for vanishingly small holes, energy cannot flow through the waveguide.

IV. ATTENUATION IN THE WAVEGUIDE

Up to now we have neglected attenuation in the waveguide, but it plays an essential role in the operation of a linear accelerator, and we must now consider this effect. Suppose that the energy density in a small section of the loaded waveguide is known to us. As a result of ohmic losses in the walls, energy will flow into

in them. We can relate this loss of energy to the quality factor of the unloaded waveguide \(Q_0\). The latter quantity is determined by the formula:

\[ \frac{1}{Q_0}=\frac{\text{energy dissipated in the walls in 1 sec.}}{\omega\cdot \text{stored energy}}. \tag{6} \]

This is the quality factor \(Q\) that the waveguide would have if, with the aid of ideally reflecting plates at the ends, it were turned into a resonant cavity (with no energy dissipation at the ends). Since both the energy stored in the waveguide and the energy dissipated per unit time in the walls are proportional to the length of the section under consideration, \(Q_0\) does not depend on its length.

As a result of losses in the walls, the energy of the wave propagating in the waveguide will gradually decrease. It is easy to find the magnitude of the resulting attenuation of the wave. For this purpose let us write the continuity equation for the energy flux. Denote by \(W\) the energy per unit length of the waveguide, by \(S\) the energy flux through a given cross section, and by \(D\) the power dissipation per unit length of the wall. The continuity equation states that the rate of increase of \(W\) with time is equal to minus the divergence of \(S\), minus \(D\). Since \(S\) depends only on the \(z\)-distance along the axis of the waveguide, we have:

\[ \frac{\partial W}{\partial t}+\frac{\partial S}{\partial z}+D=0. \tag{7} \]

From equation (6), however, it is seen that \(D=\omega \frac{W}{Q_0}\), and from the preceding section it is known that \(S=v_g W\). Thus equation (7) takes the form:

\[ \frac{\partial W}{\partial t}+v_g\frac{\partial W}{\partial z}+\frac{\omega}{Q_0}W=0; \qquad \frac{\partial S}{\partial t}+v_g\frac{\partial S}{\partial z}+\frac{\omega}{Q_0}S=0. \tag{8} \]

Considering the stationary state (the partial derivatives with respect to time are zero), we have: \(S=S_0 e^{-z/l_0}\); \(l_0=v_g\frac{Q_0}{\omega}\). Thus, the energy flux decreases with attenuation coefficient \(1/l_0\); \(l_0\) is the length over which the magnitude of the flux decreases by a factor of \(e\); it may be called the attenuation length. This quantity must be introduced into the expressions for the fields found in Section I; the attenuation coefficients of the electric and magnetic fields separately will be equal to one half of the attenuation coefficient of the energy flux.

The attenuation length can easily be interpreted in terms of the time required for the amplitude of oscillations in a resonant cavity to decrease by a factor of \(e\). In the resonant part of the waveguide the energy density does not depend on \(z\). Therefore the derivatives with respect to \(z\)

in equation (8) become zero, and we have \(W=W_0e^{-\frac{\omega}{Q_0}t}\). Hence it follows that the energy density decreases by a factor \(e\) in the time \(\frac{Q_0}{\omega}\). We see that during the time \(T_0\), in the course of which the energy density decreases by a factor \(e\), the energy propagates with group velocity \(v_g\) over precisely the attenuation length. The latter, consequently, increases with increasing \(Q_0\) and decreases with decreasing group velocity.

We shall see below that the concept of attenuation length plays a very large role in the theory of linear accelerators. We shall call a linear accelerator long if its length is great in comparison with the attenuation length, and short if the opposite relation holds. It is clear that in long accelerators we shall encounter difficulties. The intensity of any signal sent from one end of a long accelerator will decrease to a negligible value before it reaches the other end. Signals which we begin to transmit at a certain moment from one end of the accelerator will not even have time to reach the other end during the time \(T_0\), within which a stationary value of the field will be established near the first end (it is clear that establishing the stationary state requires the same amount of time as decay). This means that the ends of a long accelerator are essentially isolated from one another both in space and in time. However, we shall see that, for proper excitation of a long waveguide, the fields at its ends must satisfy certain phase relations. Thus we arrive at the necessity of considering the problem of excitation, for it turns out that accelerators long enough to produce particles with energies of billions of eV are long in our sense of the word as well.

V. INPUT OF POWER INTO THE WAVEGUIDE

Along the axis of a linear accelerator one must direct a very strong electric field, associated with the Fourier component whose velocity is equal to the velocity of the particle being accelerated. Knowing the entire field pattern in the accelerator, we find that the integral of the square of the field, taken per unit length of the device (and, consequently, the energy stored per unit length), is proportional to the square of this Fourier component. Indeed, let the field directed along the \(z\)-axis contain a Fourier component resonant with the particle, \(Ee^{i\omega(t-z/v)}\). The energy stored per unit length is equal to some constant, multiplied by \(\varepsilon_0 E^2\)* and by the cross section of the waveguide tube. For a given geometry of the device (i.e., given distances between diaphragms,

* \(\varepsilon_0\) is the dielectric constant of the vacuum. The author uses the MKS system. (Translator’s note.)

sizes of the holes, etc.), but for an arbitrary scale the cross section will be proportional to the square of the wavelength (in free space) at which the accelerator operates, since all linear dimensions of the cavities are proportional to it. We thus have:

\[ W=A\varepsilon_{0}E^{2}\lambda_{0}^{2}, \]

where \(A\) is a certain constant which can be determined if the character of the field is known. Above it was assumed that \(W\) is the total stored energy (which is, of course, equal to twice the electrostatic energy). Thus, in order for \(E\) to be large, a large reserve of energy is needed; consequently, in the steady state there will be considerable losses, and a large power must be supplied to the accelerator. During the process of establishing equilibrium, \(W\) will be smaller than when equilibrium has already been established; consequently, the accelerating field will also be smaller. Therefore, when working with an accelerator, one must first wait until a steady-state regime is established, and only then, when the field has reached its maximum value, introduce the particles to be accelerated. In this section we investigate what power must be supplied to the device in order to attain a field of a specified magnitude. We shall also consider the transient process by which the steady-state regime is established.

The results turn out to be different depending on whether the device is closed at its ends by ideally reflecting walls, so that a standing wave is formed in it, or whether at its ends there are walls from which the traveling waves are not reflected. The results also depend on the distribution of the power sources along the length of the accelerator. We begin with the case of reflecting walls and a uniform distribution of power sources, as occurs in the devices of MIT and the University of California. All the power supplied per unit length will then be dissipated per unit length of the walls. We already know that this dissipation is measured by the quantity \(\omega W/Q_{0}\). Consequently, denoting by \(P\) the power supplied per unit length of the device, we have:

\[ P=\frac{1}{Q_{0}}\cdot A\varepsilon_{0}^{2}E^{2}\lambda_{0}^{2}\omega . \]

Recalling that

\[ \frac{\omega}{c}=\frac{2\pi}{\lambda_{0}},\quad c=(\varepsilon_{0}\mu_{0})^{-1/2}\quad \text{and}\quad \left(\frac{\mu_{0}}{\varepsilon_{0}}\right)^{1/2}=377\ \text{ohms}, \]

we obtain from this

\[ E=a\sqrt{\frac{377PQ_{0}}{\lambda_{0}}},\quad \text{where}\quad a=\frac{1}{\sqrt{2\pi A}}. \tag{9} \]

We see that the accelerating voltage is proportional to the square root of the power supplied per unit length of the apparatus. Thus, wishing to make an accelerator with a given potential difference at the ends as short as possible (or, what is the same thing, striving to increase \(E\) as much as possible), we must supply as large a power as possible to the apparatus. On the other hand, if the reserve of power at our disposal is limited, or if it is desirable to economize on energy sources, it is necessary to turn to the other extreme, lengthening the accelerator as much as possible. To see this, let us rewrite equation (9) in a form that explicitly includes the potential difference \(El\) traversed by the particles and the total power supplied to the apparatus, \(\dot P l\) (\(l\) is the length of the accelerator).

We have:

\[ El=\alpha\left(\frac{377\,(Pl)\,Q_0 l}{\lambda_0}\right)^{1/2}. \]

Thus, the potential difference is proportional to the square root of the length of the apparatus, and, by increasing the latter without bound, one can theoretically, for a given power, accelerate particles to arbitrary energies. From the equation just written it is evident that a given potential difference at the ends of the accelerator can be obtained either by supplying a large power to a short apparatus, or by reducing the supplied power at the expense of increasing the length of the apparatus; the potential difference itself depends only on the product of the total supplied power and the length. There are no theoretical prerequisites for choosing particular values of length and power separately; therefore the decision will be determined mainly by economic considerations. The cost of a linear accelerator, apart from the specified cost of its terminal devices, consists of two parts: first, the cost of the tube of the apparatus, its evacuation, maintenance, etc. (all these quantities are proportional to the length of the accelerator); second, the cost of the energy sources and of the device for introducing it into the apparatus; these quantities are proportional to the power supplied to the apparatus, i.e., for a given potential difference they are inversely proportional to the length of the accelerator. Thus, the cost of the apparatus consists of two parts, one of which is proportional, and the other—for a given potential difference—inversely proportional to the length of the apparatus. The latter must be chosen so that the cost is minimal. But the function \(x+\frac{1}{x}\) has a minimum at \(x=1\), when both terms are equal. Consequently, the cost of the accelerator is minimal when the cost of the energy sources and of the tube (i.e., of all parts of the apparatus whose cost is proportional to its length) are equal.

Using this condition, it is easy to find the economically most advantageous distance between the energy sources, once the ...

their type and it is decided to distribute them uniformly along the apparatus: one must build a line of such length that it costs as much as the power installation.

In what follows we shall see that the distance between vibrators should not be large in comparison with the attenuation length. Consequently, the design must be such that this length exceeds the most economical distance between energy sources. It should be expected that by the time the projects now being developed at various institutes are carried out, it will be possible to estimate the cost and to find the most economical distance between vibrators; however, so far there are still no reliable figures, and it is unclear what in the end will prove most advantageous—a close spacing of the vibrators, as in the MIT project, a distant spacing, as at Stanford, or something intermediate.

In the case of traveling waves the situation changes radically. Let, as before, a definite power \(P\) be supplied per unit length of the apparatus. Now, however, it is expended in two ways: heat losses in the walls still occur \(\left(\omega \dfrac{W}{Q_0}\right.\) per unit length\()\), and in addition there appears an energy flux \(v_g W\) through the nonreflecting wall at the end of the tube. Therefore the total supplied power \(Pl\) must be equal to \(\left(\dfrac{\omega l}{Q_0}+v_g\right)W\). As a result, \(\dfrac{1}{Q_0}\) is replaced by

\[ \frac{1}{Q_L}=\frac{1}{Q_0}+\frac{v_g}{\omega l} =\frac{1}{Q_0}\left(1+\frac{l_0}{l}\right). \tag{10} \]

\(Q_L\) may be called the losses in a waveguide loaded by an absorbing wall at the end. Instead of equation (9) we obtain:

\[ E=\alpha\left(\frac{377Q_0P}{\lambda_0}\cdot\frac{l}{l+l_0}\right)^{1/2} \tag{11} \]

(it is assumed that the power is fed into the apparatus along the entire length of the walls). If the tube length \(l\) is much greater than the attenuation length, then the additional factor \(\sqrt{\dfrac{l}{l+l_0}}\) is approximately equal to 1, and the losses at the end have almost no effect on the field strength. However, when \(l\ll l_0\) this factor is much less than unity, and, for given geometry and input power per unit length, the voltage obtained in an apparatus with a traveling wave is appreciably smaller than in the case of standing waves. In other words, in a short tube it is unreasonable to use traveling waves. In a long accelerator the end losses are negligibly small in comparison with the dissipation of energy in the side walls, and the difference in voltages is insignificant. In this case the choice between standing and traveling waves is based on other considerations, which will be discussed below.

Of the existing accelerator projects, at least two (Stanford and TRE) have planned the use of traveling waves. The authors of these projects, of course, also considered feeding the waveguide from the input end. Assuming \(l \ll l_0\), we can rewrite equation (11) in a somewhat different form, expressing the energy acquired by the particle in terms of the power \(Pl\) introduced into the waveguide:

\[ El = a \left( \frac{377(Pl)}{\lambda_0} Q_0 l \frac{l}{l+l_0} \right)^{1/2} \tag{12} \]

(in this case it is immaterial whether the power is introduced continuously along the entire length of the walls or only from the end). From formula (12) it is clear that, for \(l \ll l_0\) and a constant value of \(Pl\), the energy \(El\) acquired by the particle is proportional to the length of the tube. This makes it possible to suppose that it is sufficient to feed a traveling-wave device only from one end in order to obtain arbitrary accelerations. Nevertheless, equations (11) and (12) are equivalent, and they show very clearly that, although the energy acquired by the particle is proportional to \(l\) (for small \(l/l_0\)), it is nevertheless \(\sqrt{\dfrac{l}{l+l_0}}\) times smaller than in a waveguide with a standing wave (for the same accelerator length and the same power). It is only because this factor becomes less unfavorable for large \(l\) that in this case we obtain a rapid growth of the energy with length. In addition, feeding from one end has yet another inconvenience. As we saw earlier, if \(l\) is comparable with \(l_0\), then the energy density in the waveguide decreases exponentially with \(z\), and at \(z=l_0\)—the attenuation length—it becomes very small. Consequently, after traversing this distance, the particle will cease to be accelerated. In other words, the method of feeding a traveling-wave waveguide from the end is not suitable if the accelerator length is greater than \(l_0\). At the same time, the use of standing waves is plainly more advantageous for \(l \ll l_0\). It should be noted, however, that in the case of traveling waves the value of \(a\) may turn out to be larger than for standing waves; therefore, over some interval of lengths near \(l_0\), traveling-wave accelerators may prove somewhat more advantageous. This circumstance was noted at Stanford and at TRE, where it is proposed to choose \(v_g\) so that the length of the device is approximately equal to \(l_0\).* In a long waveguide the power must be supplied along its entire length, continuously or at separate points, the distance between which is less than \(l_0\).

Let us turn to the transient processes associated with establishing oscillations in the tube. As was already noted, in the case of standing waves the time for establishing the stationary regime is of the order of

\[ T_0 = \frac{Q_0}{\omega}. \]

It is interesting to compare it with another characteristic quantity,

\[ T_1 = \frac{l}{v_g} \]

* In the original it says not \(l_0\), but \(l\). This is an obvious misprint. (Translator’s note.)

(this is the time during which the field with group velocity \(v_g\) propagates from one end of the waveguide to the other). We see that

\[ \frac{T_1}{T_0} = \frac{l}{l_0}. \]

In other words, the time in which the field propagates from one end of a long accelerator to the other is much greater than the time required to establish the steady state. While the steady state is being established, one end of the tube knows nothing about the other, and in order to ensure the proper phase relations between the ends, a certain external circuit is needed (this question is discussed in the next section). On the other hand, in a short waveguide the standing wave, during the time \(T_0\), has time to be reflected many times from both ends, and the necessary phase relations are easily established. Further, in a short waveguide it is immaterial where the energy is introduced: owing to reflections from the walls it will be distributed in the proper way throughout the whole tube, independently of where it was introduced. In a long waveguide the situation is different: there, evidently, the energy sources must be distributed uniformly along the tube, and the distance between them must not exceed the attenuation length (it is even better for it to be appreciably smaller). It is interesting to note that if, in a tube with standing waves, reflecting partitions are placed between the energy sources (of course, with small openings for the passage of particles), then the whole process of establishing the oscillations proceeds in exactly the same way as without them. In the absence of these partitions the field at a given point is established as the result of the successive superposition of a series of waves emitted by ever more distant sources; because of attenuation each succeeding wave is weaker than the preceding ones. In the presence of partitions, the place of the waves emitted by distant sources is taken by multiply reflected waves, arriving again and again at the given point. We shall see below that this circumstance in many respects simplifies our reasoning. It is possible that the actual construction of the tube will likewise be simplified if it is divided in this way into short sections.

In a tube with traveling waves the situation changes. Power can be introduced into a long tube uniformly along its whole length, as in the case of standing waves. It is worth considering the excitation of a traveling wave by an oscillator. A traveling wave may be composed of two standing waves—sine and cosine—with a phase difference of \(\pi/2\). To excite a traveling wave, one must excite separately both of these components. Consequently, power must be introduced into the waveguide at at least two points. These points must be situated at the antinodes of the corresponding components, and with the aid of some phase-shifting circuit between them a phase difference equal to \(\pi/2\) must be established. This circuit can be realized simply in the form of

of a two-phase system with a phase difference of a quarter wavelength, so that both components can be fed from one and the same oscillator.

The problem of establishing oscillations in a long tube with traveling waves, fed by uniformly distributed oscillators, is essentially no different from the analogous problem in the case of standing waves, except only that now the waves propagate in one direction only. As before, the field will not be established until the signals sent by distant oscillators reach the given point of the tube; the time for establishing the oscillations will still be of the order of \(T_0\). Only near the entrance are the excitation conditions for standing and traveling waves substantially different. In the case of standing waves, at the entrance we had reflected signals propagating in the opposite direction; in the case of traveling waves there will be none. The field at the entrance cannot be formed by the superposition of successive signals from increasingly distant sources, since they all propagate in the other direction—away from the entrance. Therefore the field at the entrance will be established sooner than in the more distant parts of the tube, but it will not be as large as there. To compensate for this, an additional source of energy will have to be placed at the entrance. Since the influence of this source is no longer felt at distances exceeding the attenuation length, we shall obtain a constant acceleration.

As for a short tube with traveling waves, where \(T_1 \ll T_0\), it is clear that the oscillations will be completely established in a time of the order of \(T_1\). Indeed, during this time a signal sent from one end of the tube will reach its other end, where it will be absorbed. Since energy flows into the waveguide only during the time \(T_1\), which is small in comparison with \(T_0\), the field will not have time to reach its maximum value; and it is clear that the stored energy and the accelerating voltage here will be smaller than in the same tube with standing waves (since in the latter the field can be reflected many times from the walls, remaining in the cavity throughout the entire time \(T_0\)). Thus we again arrive at the already considered above reduction of the accelerating voltage in a short tube when passing from standing waves to traveling waves.

VI. THE INFLUENCE OF GEOMETRICAL FACTORS ON ACCELERATION

In the preceding section we obtained equation (9) for the amplitude of the accelerating field as a function of wavelength, wall losses, and power flow per unit length. This formula is valid for any long waveguide—both for traveling and for standing waves. It is also valid in the case of a short tube with standing waves. In trying to obtain the maximum acceleration, we can vary certain quantities in this formula. These include the wavelength, the geometry of the apparatus, and the choice of operating mode, in particular the choice of standing or traveling waves. In the present section we shall examine all these factors. We shall also consider designs of various accelerators

and the results that can be expected, judging from it. Let us first examine the influence of the wavelength. It is well known that the quality factor of a resonant cavity is proportional to its volume divided by the volume of the surface layer of thickness \(\delta\), where \(\delta\) is the skin depth; the constant of proportionality is, in order of magnitude, close to unity. The skin depth as a function of the conductivity \(\sigma\) (in ohms per meter) and the wavelength in vacuum \(\lambda_0\) (in meters) is given by the expression

\[ \delta=\sqrt{\frac{\lambda_0}{377\pi\sigma}}. \]

Let us take a sequence of geometrically similar cavities corresponding to different wavelengths. The volume of each of them is, of course, proportional to \(\lambda_0^3\), and the surface area to \(\lambda_0^2\). We thus have:

\[ Q_0=\text{const}\cdot\frac{\lambda_0}{\delta}=B\sqrt{377\sigma\lambda_0}, \]

where \(B\) is a number of order unity. Substituting this into (9), we obtain:

\[ E=aB^{1/2}(377P)^{1/2}\left(377\frac{\sigma}{\lambda_0}\right)^{1/4}, \tag{13} \]

i.e., for a given power flux the accelerating field is inversely proportional to the fourth root of the wavelength. This means that, all other conditions being equal, in a linear accelerator it is more advantageous to use shorter waves. However, \(E\) varies so slowly with \(\lambda_0\) that this advantage is small, and questions of convenience in handling, availability of energy sources, and other similar considerations prove more important. It is also seen from equation (13) that \(E\) is proportional to the fourth root of the conductivity of the walls. This suggests using good conductors such as silver or copper, or working at low temperatures, at which the conductivity of metals increases. However, the advantage from this is so small that cooling the waveguide will probably cost more than increasing the input power.

We must now study the role of the geometry of the device. It affects chiefly the value of the coefficient \(a\), although \(Q_0\) and, consequently, \(B\) also depend on the geometrical conditions. By definition, \(a\) is large if the component of the field along the \(z\)-axis (the direction of acceleration of the particles) is large for a given stored energy or, conversely, if the stored energy (and, consequently, the field at all other points of the cavity) is small for a given \(z\)-component of the field. To understand how \(a\) can be increased, let us first note that the electric and magnetic fields in the wave traveling in resonance with the particle are Bessel functions (respectively \(J_0\) and \(J_1\)) of the argument \(2\pi r\sqrt{\lambda_0^{-2}-\lambda_g^{-2}}\).

Since the phase velocity of this quantity \(v=c\dfrac{\lambda_z}{\lambda_0}\) must be less than the velocity of light, we see that the argument of the Bessel function is imaginary or (in the limit, for \(v=c\)) tends to zero. But Bessel functions of an imaginary argument increase exponentially as the modulus of the latter increases, and we see that the field increases rapidly with distance from the \(z\)-axis; this growth is the more rapid the smaller the velocity. In the limiting case \(v=c\), the component of the electric field is independent of \(r\). All these conclusions are directly opposite to what occurs in an unloaded waveguide, where the phase velocity of the waves is greater than the velocity of light, the fields are given by Bessel functions of a real argument, and decrease with distance from the axis.

If only the Fourier component that is resonant with the particle were present, the greater part of the energy would be stored far from the axis, and \(\alpha\) would be small. However, as the distance from the axis increases, the other Fourier components become more and more important. They play a twofold role: first, they make it possible to satisfy the boundary conditions at the wall of the waveguide (which could not be accomplished with the aid of only a single Fourier component); second, because of them the field far from the axis is diminished, as a result of which the value of \(\alpha\) is improved. It would be extremely difficult to determine, in general form, the best structure (in the sense of increasing \(Q_0\) and \(\alpha\)). Therefore we shall consider one special example—the MTI accelerator—and ask how its construction could be improved by varying the geometry of the apparatus in various ways. In Fig. 5 the electric lines of force for this case are shown schematically. Oscillations of the \(\pi\) type are excited in the apparatus (i.e., the phase difference between neighboring diaphragms is equal to \(\pi\)) and \(\dfrac{v}{c}=1\). We see that the field in the intervals between the diaphragms is almost independent of \(r\), up to a distance approximately equal to the radius of the aperture, and begins to decrease at larger \(r\), almost as in a closed cylindrical cavity. This suggests that if the apertures in the diaphragms were smaller, then the fall of the field would begin earlier and the energy stored far from the axis would be reduced. In fact, this is so. The calculated value of \(\alpha\) for the MTI accelerator is equal to 0.48, whereas in the limiting case of vanishing apertures, when we approach closed cylindrical cavities, \(\alpha=10.3\). The apertures in the diaphragms here are made fairly large in order to guarantee unimpeded passage of the electron beam. If it should turn out that in fact this beam is more concentrated, in the future it would be possible to make the apertures of appreciably smaller radius. Correspondingly, the value of \(\alpha\) would increase. The quantity \(Q_0\) or \(B\) changes little in this case. Thus, the value of \(B\) in the MTI apparatus is approximately 0.45, while in the limiting case of vanishing apertures it amounts to ...

is 0.39. Thus, the quantity \(a\sqrt{B}\) entering equation (12) increases from 0.32 in the MIT accelerator to 0.65 in the limiting case of small apertures. Thus, by the method described, the accelerating field can be increased at most by a factor of two, and in practice by considerably less, since appreciable apertures must still be left for the passage of the electrons. In the Stanford project apertures of much smaller radius than at MIT are used, and the geometry there is probably closer to that which would be realized in an actual accelerator.

Fig. 5. Lines of force in MIT accelerators. The spacing between the lines characterizes the field strength.

The example we have considered referred to the case \(v = c\). For smaller velocities the situation changes. First, we have already seen that in this case the field by no means remains constant, but increases rapidly as one moves away from the axis. Apparently this increase continues up to a distance approximately equal to the radius of the aperture, after which the field begins to decrease. Therefore, for small velocities and large apertures the value of \(a\) may decrease appreciably. However, as the radius of the aperture is decreased the situation rapidly improves, and in the limit of vanishing radius \(a\) does not depend on the wave velocity. This is not so, however, for \(Q_0\) and \(B\). As the particle velocity decreases, the diaphragms must be placed closer and closer to one another, and the losses in the surface layer increase in comparison with the total stored energy. Therefore \(Q_0\) decreases, varying (in the limiting case of small velocities) proportionally to \(v\). Thus, our design is unsuitable for small velocities. For the acceleration of electrons, however, this circumstance is immaterial: in the MIT project it is proposed to inject into the device electrons already preaccelerated in a Van de Graaff generator to two MeV (velocity

their velocity is almost equal to \(c\); in most other accelerators the initial velocity of the electrons is also at least equal to half the speed of light. For positive ions, however, with their much smaller velocities, a waveguide with diaphragms is not suitable until the ions have been accelerated sufficiently strongly. At low velocities the system used in the proton accelerator of the University of California is much more convenient. It is shown in Fig. 1, \(b\).

Calculation shows that the values of \(\alpha\) and \(B\) in it are, respectively, 1.58 and 0.54, so that \(\alpha\sqrt{B}\) is approximately 1.16—much greater than in the MIT device and even better than for diaphragms with small apertures. However, for high velocities this system is unsuitable, since the diameters of the inner tubes in which the particles move must be the smaller, the greater the velocity. Even with the very large dimensions with which we have to deal when working at a frequency of 200 megacycles, these tubes become so narrow that the proton beam can no longer propagate freely.

Up to now it has been assumed that oscillations of type \(\pi\) are established in the waveguide. Let us now see whether matters will not be improved by operating in some other mode. One may, for example, introduce additional diaphragms into the system of Fig. 5, placing them midway between the old ones. In this case the field pattern will not change, since the lines of force will meet the new diaphragms at right angles and, consequently, the boundary conditions and Maxwell’s equations will be satisfied. There will still be resonance at the same wavelength in the waveguide, frequency, and phase velocity, but, since the diaphragms are arranged twice as frequently as before, the phase difference between them will be only \(90^\circ\), i.e. we shall obtain oscillations of type \(\frac{\pi}{2}\). We now have two possibilities to choose from: one may excite either a standing or a traveling wave (in the case of oscillations of type \(\pi\) the latter was impossible). A traveling wave can arise because, in the presence of intermediate diaphragms, we have two solutions for the field—one shown in Fig. 5 and another displaced relative to the first by one distance between diaphragms. These solutions are related to one another as sine to cosine and, being excited simultaneously with a phase difference of \(\frac{\pi}{2}\), they represent a traveling wave propagating to the right or to the left, depending on whether the indicated phase difference is positive or negative. By superposing traveling waves propagating in opposite directions, we can again obtain a standing wave. Let us consider successively both of the indicated possibilities.

In the standing wave the energy reserve and the \(z\)-component of the field are the same as for oscillations of type \(\pi\). Therefore the value of \(\alpha\) does not change. However, \(Q_0\) decreases appreciably. Indeed, now with the former energy reserve there are twice as many diaphragms in which current flows and is diss—

...is Joule heat. If the dissipation of energy occurred only in the diaphragms, this would lead to an increase of the losses (and, consequently, to a decrease of \(Q_0\)) by a factor of two in comparison with oscillations of the \(\pi\) type. The situation, however, is not so bad, since part of the energy is dissipated in the cylindrical walls of the cavity, and this quantity is the same in both cases. Calculation shows that, for oscillations of the type \(\frac{\pi}{2}\), \(Q_0\) is about 70% of the corresponding value for the case when the phase difference between diaphragms is equal to \(\pi\). The same is true for \(\overline{B}\). Thus the accelerating field (proportional to \(a\sqrt{\overline{B}}\)) is only 0.84 of the value which it has for oscillations of the \(\pi\) type. In other words, from the point of view of energy expenditure, oscillations of the type \(\frac{\pi}{2}\) are disadvantageous. It must be supposed that, since the efficiency of the device decreases so much in passing from oscillations of the \(\pi\)-type to the type \(\frac{\pi}{2}\), some deterioration will also occur in passing to any other regime. The type \(\pi\) is probably the most advantageous energetically, since it requires the minimum number of diaphragms per unit length and, consequently, leads to minimum losses in the circuit.

Nevertheless, oscillations of the type \(\frac{\pi}{2}\) also possess one advantage. It is so substantial that, at least in one project (belonging to the General Electric Company), precisely this mode of operation has been chosen. The point is that, for oscillations of the type \(\frac{\pi}{2}\), the group velocity \(v_g\) has a maximum value, whereas for the type \(\pi\) it is theoretically equal to zero. Consequently, the attenuation length \(l_0\) for oscillations of the type \(\frac{\pi}{2}\) can be quite large, while in the second case \(l_0\) is theoretically equal to zero. Therefore an accelerator of appreciable dimensions may in the first case prove to be short (in the sense that its length is less than \(l_0\)), and in the second—long. In the following section we shall consider some features of oscillations of the \(\pi\) type connected with the attenuation length, and shall show that it is possible to construct a linear accelerator of quite considerable dimensions and still make it operate as a short one; however, for the type \(\frac{\pi}{2}\) the corresponding length is much larger. It will be shown below that it is much simpler to feed a short waveguide than a long one. At the General Electric Company an accelerator of medium dimensions has been designed, and it turns out that in the case of oscillations of the type \(\frac{\pi}{2}\) it may be regarded as short, while in the case of the \(\pi\) type—as long, with the corresponding complication of the feeding problem. In the MIT project we are faced with the problem of feeding a long...

...the accelerator, and the indicated advantage of oscillations of the type \(\frac{\pi}{2}\) is absent.

Another possibility for using oscillations of the type \(\frac{\pi}{2}\) consists in exciting traveling waves. It is precisely this possibility that is used in the Stanford project. We still have here a decrease of \(B\) owing to the presence of additional diaphragms, but it is more than compensated by the increase of \(\alpha\). We shall consider the standing wave corresponding to oscillations of the type \(\pi\) as being formed by two traveling waves propagating in opposite directions. Only one of them will resonate with the electrons. If the traveling wave propagating in the other direction is excluded (which means reducing the energy reserve by half), then in a tube with a traveling wave the same acceleration will be obtained as in the case of standing waves. In other words, for the same acceleration half the power will be required, which, as is seen from equation (9), means that the quantity \(\alpha\) will increase by a factor of \(\sqrt{2}\). This is not quite exact if the traveling wave contains an appreciable reflected component, as will occur when the oscillations approach the type \(\pi\). In this case the value of \(\alpha\) decreases all the time, taking in the limit the value corresponding to oscillations of the type \(\pi\). However, for oscillations of the type \(\frac{\pi}{2}\) one may expect an increase of \(\alpha\) by a factor of \(\sqrt{2}\). Combining this with the already known decrease of \(B\), we see that in going from oscillations of the type \(\pi\) to a traveling wave of the type \(\frac{\pi}{2}\) the accelerating field increases by

\[ \sqrt{0.7\cdot 2}=1.18 \]

times. It would be worthwhile to take care of such an improvement in the properties of the apparatus if it were not accompanied by shortcomings that compensate for it. In fact, owing to the transition from standing waves to traveling waves and the reduction of the size of the holes in the diaphragms, the quantity \(\alpha\sqrt{B}\) in the Stanford project increases by approximately a factor of 1.68 in comparison with the MIT accelerator (increasing from 0.32 to 0.54). This, however, is compensated by two disadvantages. First, as we have already seen above, in a short accelerator operating on traveling waves (such as the one now being built at Stanford), the field is decreased by a factor

\[ \sqrt{\frac{l}{l+l_0}} \]

in comparison with an apparatus using standing waves. The attenuation length in the Stanford accelerator is approximately \(18\,m\), while its total length will probably be about \(6\,m\). Consequently,

\[ \sqrt{\frac{l}{l+l_0}}=\sqrt{\frac{1}{1+\frac{18}{6}}}=0.50, \]

Taking this factor into account, the numerical coefficient in the Stanford project will turn out to be only 0.27 as against 0.32 in the apparatus

MIT. The situation will, of course, improve if the device is made with a length of \(l_0\) or much greater, as is later contemplated at Stanford. We have already noted above that, in a certain range of device lengths close to \(l_0\), an accelerator operating on traveling waves may prove to be somewhat more advantageous. The second shortcoming of accelerators of this type occurs only in long devices. It is connected with the difficulty of stabilizing the frequency of the vibrators. This circumstance, which will be discussed in more detail later, is, in the author’s opinion, the most important argument in favor of using standing waves in long accelerators (provided only that the device is fed by self-excited vibrators).

It is of interest to compare the final data of all three projects—MIT, California, and Stanford. Starting from equation (13), we substitute into it the numerical value of the conductivity. For copper we take the value \(5.5 \cdot 10^7\) mho per meter. For short waves we shall take \(0.8\) of this value, since almost always the values of \(\sigma\) in an unloaded cavity at a wavelength of \(10\) cm prove to be about twenty percent less than the theoretical ones (for the California project one must even take \(0.4\) in order to obtain agreement with the observed values of \(\sigma\)). We obtain:

\[ E\ \text{(in megavolts per meter)} = 7.3\, \frac{a\sqrt{B}\sqrt{P\ \text{(megawatts per meter)}}}{\lambda_0^{1/4}\ \text{(in meters)}} . \]

For traveling waves it is necessary to add here the further factor \(\sqrt{\dfrac{l}{l+l_0}}\); instead of the coefficient \(7.3\), corresponding to the true conductivity of copper, in the case of short waves one should put \(6.9\), and for the California project, \(5.8\). Substituting the values already mentioned in the text, we have for the constant

\[ C = 7.3a\sqrt{B}\sqrt{\frac{l}{l+l_0}}:\lambda_0^{1/4}. \]

\[ C = \begin{cases} 3.9 & \text{for the MIT accelerator,}\\ 3.3 & \text{for the Stanford device,}\\ 6.1 & \text{for the University of California accelerator.} \end{cases} \]

The second figure would be \(6.6\) were it not for the correction for the small length of the device. These figures show that, despite the difference in systems, the acceleration in all three cases proves to be approximately the same (for the same power supplied per unit length). The difference in the accelerations that are expected to be obtained is due chiefly to the different powers. Thus, in the MIT accelerator there will be one magnetron for every \(32\) cm of device length. The magnetron will probably deliver about \(0.8\) MW; approximately half of this amount will go into the accelerator. This gives

\[ P=\frac{0.4}{0.32}=1.25 \]

megawatts per meter, whence \(E = 4.4\) megavolts per meter. With

With an overall accelerator length of 6.4 m, the energy imparted to the electrons is expected to be \(4.4 \cdot 6.4 = 28\) MeV. In the preliminary model of the Stanford accelerator it is proposed to use only one magnetron, with an instrument length of about 6.1 m. Since here we are dealing with traveling waves, the entire power delivered by the magnetron can be used, and, taking the same value of 0.8 MW as in the M.I.T. design, we obtain \(P = \frac{0.8}{6.1} = 0.131\) megawatt per meter, whence \(E = 1.20\) megavolts per meter. The electrons are thereby accelerated to an energy of \(6.1 \cdot 1.20 = 7.3\) MeV. On the other hand, at Stanford it is further proposed to feed the accelerator from powerful klystrons of a new design, used as power amplifiers. They will be placed periodically along the apparatus, with a period of the order of \(l_0\). It is still too early to predict what energies can be obtained here. The total power fed into the California accelerator is 2.34 MW. With an apparatus length of about 12.2 m this gives \(P = \frac{2.34}{2.12} = 0.19\) megawatt per meter, whence \(E = 2.66\) megavolts per meter, and the potential difference at the ends of the tube is equal to 32 MeV. However, since the ions are not at the peak of the wave, only 28 MeV is used for acceleration. To this is added the initial energy of the ions, equal to 4 MeV.

(To be concluded in the next issue.)