DESIGN OF LINEAR ACCELERATORS

J. C. Slater*)

VII. INPUT IMPEDANCE OF TUBES WITH STANDING AND TRAVELING WAVES

We have seen that the distinction between long and short accelerators indicated in the preceding sections plays a most important role both in choosing the method of feeding the apparatus and in estimating the voltage attainable in it. In order to consider the feeding problem, it is necessary to investigate the input impedance of the accelerator (calling the input the point from which the energy is supplied). In the case of standing waves we have a discrete series of natural frequencies, and the input impedance has a peak when the external frequency coincides with one of them. As is well known, the width of a resonance peak (on the frequency scale) is related to the frequency as \(\frac{1}{Q_0}\). If this width is small in comparison with the spacing between neighboring natural frequencies, we shall say that the latter are well separated. Then, as will be shown below, one can use the resonant properties of the cavity to stabilize the frequency of the oscillator, obtaining what may be called resonant feeding. We shall show that this occurs when the linear dimensions of the accelerator are small in comparison with the attenuation length \(l_0\). On the other hand, we shall see that if the length of the accelerator is large in comparison with \(l_0\), then the natural frequencies lie very close to one another, and resonant feeding methods are inapplicable.

In a system with traveling waves there are no resonance peaks of the input impedance; the cavity cannot be used to stabilize the frequency of the oscillator, and for this purpose one must use other methods. The point is that, since we are speaking of impedance, resonance appears as a result of the interference of incident and reflected waves; the latter are absent in the system under consideration. There are only pass bands with a smoothly varying input impedance, which becomes purely imaginary in the intervals between the bands.

*) Conclusion. See UFN 37, 316 (1949); Rev. Mod. Phys. 20, 473 (1948).
Translated by V. Averbakh.

Let us trace how, in a system with traveling waves, the resonant band structure that we have for standing waves disappears. The mechanism of this is very simple, and it may be considered in two ways. First, in going over to a long accelerator the amplitude of the wave reflected from the opposite end decreases; therefore, even for standing waves the resonance peaks of the input impedance will be small and will disappear in the limiting case of a very long waveguide. Secondly, when the end of the tube changes from reflecting (standing waves) to absorbing (traveling waves), the quality factor for each natural frequency decreases. This causes a broadening of the resonance curve, and we shall show that it is just large enough to produce overlap of peaks which can no longer be resolved (in the spectroscopic sense). This assertion is essentially equivalent to what was said above. Indeed, let us denote the spacing between neighboring natural frequencies and the frequency respectively by \(\Delta\omega\) and \(\omega\); then the condition for overlap of neighboring resonance lines is: \(\frac{\Delta\omega}{\omega} \sim \frac{1}{Q_0}\); on the other hand, the condition that the broadening due to absorption at the end cause overlap of the lines is written in the form:

\[ \frac{\Delta\omega}{\omega} \sim \frac{v_g}{\omega l} = \frac{1}{Q_0}\frac{l_0}{l}. \tag{14} \]

It will be shown below that formula (14) approximately gives the difference between neighboring natural frequencies of a tube with standing waves. We see that at distances smaller than the attenuation length, the line broadening due to attenuation in the standing wave cannot cause overlap of the peaks; at the same time, for \(l > l_0\) this overlap does occur.

Let us now see how far apart the natural frequencies are from one another and show, in particular, that the difference between neighboring frequencies is approximately determined by formula (14). The natural frequencies (normal oscillations) of a resonant band with reflecting walls at the ends are determined from the condition that the length of the band must contain an integer number of half-waves \(\lambda_g/2\). This ensures satisfaction of the boundary conditions (for example, vanishing of the tangential component of the electric field) at both ends of the band. Thus, we have:

\[ l = \frac{m\lambda_g}{2}, \]

where \(m\) is an integer, i.e.

\[ \frac{1}{\lambda_g} = \frac{m}{2l}. \]

Consequently, the eigenvalues are separated from one another by the constant interval \(\frac{1}{2l}\), i.e., in Fig. 2 they correspond to a set of equal-

separated from one another by abscissas. We see that oscillations of type \(\pi\) (Fig. 2) occur when

\[ \frac{1}{\lambda_g}=\frac{1}{2L}; \]

thus, if the waveguide consists of \(N\) sections (so that \(l=NL\)), we obtain \(N\) different natural oscillations corresponding to \(m=1,2,\ldots,N\): the last of these belongs to type \(\pi\). From the periodicity of Fig. 2 it follows that higher values of \(m\) give nothing new. The frequencies of these oscillations (or the quantities \(\frac{1}{\lambda_0}\)) can be found, as shown in Fig. 6, by laying off the natural values \(\frac{1}{\lambda_g}\) and determining the corresponding values \(\frac{1}{\lambda_0}\).

Fig. 6. Calculation of the natural frequencies of a loaded resonant line.

It is clear that the natural oscillations form a group of \(N\) (or \(N+1\)) frequencies constituting one definite band; moreover, the frequencies are especially densely located at the bottom and at the top of the pass band (i.e., near oscillations of types \(0\) and \(\pi\), corresponding to \(m=0\) and \(m=N\)); in the middle of the band (near \(m=\frac{N}{2}\), which corresponds to type \(\frac{\pi}{2}\)) the intervals between frequencies are larger. Since the number of natural oscillations in the pass band is equal to \(N\), and the width of the band does not depend on \(N\), it is clear that the average distance between frequencies is equal to the width of the band divided by \(N\).

Let now the difference between the ordinates of neighboring points (Fig. 6) be

\[ \Delta\left(\frac{1}{\lambda_0}\right)=\frac{\Delta\omega}{2\pi c}. \]

Then the difference of the corresponding abscissas will be

\[ \Delta\left(\frac{1}{\lambda_g}\right)=\frac{1}{2l}. \]

Using (5), we obtain:

\[ \frac{v_g}{c}=\frac{\Delta\omega}{2\pi c}\cdot 2l;\qquad v_g=\frac{l\Delta\omega}{\pi}, \]

and consequently,

\[ \frac{\Delta\omega}{\omega}=\frac{\pi v_g}{\omega l}=\frac{\pi v_g}{\omega NL}. \]

To within a factor \(\pi\), characterizing the accuracy of the method, this coincides with equation (14), which is what we wished to prove. Hence it is also seen that, as was noted in the preceding paragraph, the difference between neighboring frequencies is inversely proportional to \(N\). Thus, the assertions stated at the beginning of this section have been proved: in a short (in our sense of the word) accelerator, the nat—

the natural frequencies of the standing waves are well separated; in a long one they overlap. The broadening of the lines caused by the absorbing end, which turns standing waves into traveling ones, is in all cases sufficiently large to cause an overlap of the peaks.

VIII. SPECIAL CASE OF OSCILLATIONS OF TYPE \(\pi\)

For oscillations of type \(\pi\),

\[ \frac{d(1/\lambda_0)}{d(1/\lambda_g)}=0 \]

and from our approximate formulas it follows that in this case the group velocity, the difference between neighboring frequencies, and the attenuation length vanish. Consequently, any apparatus of finite dimensions ought to be regarded as long. In reality, however, matters do not reach such an extreme. Taking into account the special importance of this case for the MIT and University of California accelerators, in this section we shall consider it in greater detail. (In the University of California accelerator, to be sure, oscillations of type \(2\pi\) are used, but they possess the same properties.) It is essential that, since the slope of the curve \(1/\lambda_0\), as a function of \(1/\lambda_g\), is zero, the relation between these quantities for oscillations close to type \(\pi\) may be written in the form:

\[ \frac{1}{\lambda_0}=\frac{1}{2L}-a\left(\frac{1}{\lambda_g}-\frac{1}{2L}\right)^2, \]

where \(\frac{1}{2L}\) is the value of \(\frac{1}{\lambda_g}\) for oscillations of type \(\pi\); \(a\) is some constant. As is known, oscillations of type \(\pi\) are obtained if in the general formula

\[ \frac{1}{\lambda_g}=\frac{m}{2NL} \]

one sets \(m=N\); the nearest natural oscillation to it belongs to \(m=N-1\).

Therefore we have:

\[ \frac{1}{\lambda_g}=\frac{N-1}{2NL}=\frac{1}{2L}-\frac{1}{2NL}. \]

Substituting this into the preceding equation, we find the difference of the reciprocal wavelengths of the type-\(\pi\) wave and of the nearest natural oscillation:

\[ \Delta\left(\frac{1}{\lambda_0}\right)=\frac{a}{(2NL)^2}. \]

It is not equal to zero, as might have been inferred from our elementary theory and from the vanishing of the group velocity, but decreases with increasing tube length as \(\frac{1}{N^2}\) (in contrast to \(\frac{1}{N}\), which occurs for oscillations of type \(\frac{\pi}{2}\)). For practical purposes, the attenuation length in this case may be taken to be the distance over which the width of the resonance curve for oscillations of type \(\pi\) becomes comparable with the frequency difference, computed above, between the type-\(\pi\) oscillation and the nearest natural oscillation. Oscillations of a cavity whose dimensions are small in comparison with this

length, will be well separated from the others, and the device can be fed by resonance methods. The beat velocity between oscillations of the \(\pi\)-type and those nearest to it, as a function of the difference of their frequencies, is determined by formula (5); one cannot really speak here of a group velocity at all, since it is easy to see that the beat velocity between the \(N\)-th and the \((N-k)\)-th normal oscillations is proportional to \(k\) (and is not constant!), so that the disturbance spreads out as it propagates along the tube. Nevertheless, during the time \(T_0\) required for the field to be established in the cavity, the beats between the oscillation of the \(\pi\)-type and its nearest neighbor will traverse a distance equal to the attenuation length defined above, while the beats between other pairs of normal oscillations will propagate over an integral multiple of this length; thus the disturbance can distribute itself quite uniformly throughout the entire cavity, although, unlike the usual case, this is due to multiple regular reflections. Thus, for most of our purposes, the concept of attenuation length just introduced is quite sufficient, and it is the best that can be done here. In the MTI accelerator, the attenuation length so defined turns out to be of the order of 4–5 meters, so that the proposed dimensions (6 meters) correspond to the length of the tube.

IX. POWER SUPPLY OF LINEAR ACCELERATORS

In the preceding sections we have seen that linear accelerators are divided into two essentially different classes—long and short. The distinction is determined by whether the linear dimensions of the device are large or small in comparison with the characteristic length \(l_0 = \dfrac{v_g}{\omega} Q_0\). The latter is essentially equal to the distance traversed by a traveling wave during the time \(\dfrac{Q_0}{\omega}\) required for the field to reach its maximum value. In a short accelerator it is most convenient to use standing waves, thereby turning the tube into a resonant cavity with well-separated natural frequencies. Thus, the problem reduces to introducing power into a resonant cavity. Where the power is introduced is immaterial; what is essential is only that the vibrator operate exactly at one of the natural frequencies. If it is necessary to connect a number of vibrators in parallel, then they must all operate not only at the same frequency but also in one and the same phase. We shall discuss this problem later. Its character differs depending on whether we use self-excited vibrators of the magnetron type, whose operating frequency is determined by the natural frequency of the load, or power amplifiers, whose frequency is set by the installation feeding the grid circuit. We shall see that in both cases the phasing of the vibrators is quite simple. It has been successfully carried out in practice in a number of laboratories. More difficult, though also more important, is the case of a long

accelerator. Here the difference between traveling and standing waves is immaterial; the problem of feeding is posed in essentially the same way in both cases. We have seen that a long accelerator can be divided into sections that are short in comparison with \(l_0\), and each of them can be fed separately. In the case of standing waves the sections can be physically separated from one another by reflecting walls (in which, of course, sufficiently large openings must be made for the passage of the particles), but the walls may also be purely imaginary; in the excitation of traveling waves this is even necessary, since real walls would produce a reflected wave. Each section may be fed as a short accelerator, but an additional external phasing circuit is needed so that they all operate in phase. No internal methods (which will be described for the case of a short accelerator fed by a self-excited oscillator) can establish the proper phase relations here, since the apparatus is, by definition, so long that a signal sent from one end will not have time to reach the other during the time in which the field is established. It is therefore necessary to phase the system from outside, and this is again a problem that depends essentially on what we use for feeding—self-excited oscillators or power amplifiers. To phase a system of short accelerators, a signal is required first of all. One must have some frequency standard, amplifying, if necessary, the signal sent by it so that it is sufficiently intense in any part of the system. If the accelerator is fed by self-excited oscillators, then, as will be shown in the next section, the signal intensity may be small in comparison with the power delivered by the oscillators. In any of the accelerators now being designed, one of the self-excited oscillators may simply be used to send the phasing signal. By means of suitable branches at its output, the power radiated by it can be directed to all the short sections of the accelerator. The signal must be sent sufficiently in advance of the switching-on of the remaining oscillators, so that they are all phased before the field is established. If it is necessary to isolate the phasing circuit from the large power in the accelerator, a protective switch may be placed between them. Judging from the experiments at MIT, self-excited oscillators feeding separate short sections of the accelerator can quite well be phased in this manner. In the case where the accelerator is fed by power amplifiers, any of them can supply a phasing signal of the type described. By placing several amplifier stages, the signal intensity can be brought up to the required value. At present there are no high-frequency power amplifiers, although in principle powerful klystrons or resonators can be built and used as power amplifiers; it is entirely possible that their power output will be no less than that of modern high-power magnetrons.

It was already mentioned above that at Stanford it is further intended to use klystron power amplifiers. In the Yale project, triode power amplifiers are used at a frequency of 568 megacycles (which is very close to the maximum frequency at which such amplifiers can operate).

From our discussion it is clear that the main part of the problem of feeding an accelerator from self-excited oscillators is the phasing of the latter. This is accomplished either by sending a signal from one of the oscillators, which is the case when many oscillators operate for one short accelerator, or with the aid of a phasing signal sent from outside, as in a long accelerator. We shall now consider this problem and show that, under appropriate conditions, a self-excited oscillator can be phased by a weak external signal.

X. FEEDING FROM SELF-EXCITED OSCILLATORS

The problem of feeding a resonant circuit from a self-excited oscillator, as well as the problem of phasing oscillators with the aid of an external signal, was considered in detail by the author elsewhere [3]; here we shall reproduce the main points of the reasoning. First of all, the operating frequency of a self-excited oscillator is determined by the natural frequency of the intermediate circuit or of the resonant band according to the following basic equation:

\[ \frac{g+ib}{C\omega_0} = i\left(\frac{\omega}{\omega_0}-\frac{\omega_0}{\omega}\right) + \frac{1}{Q_0} + \frac{1}{Q_{\mathrm{ext}}}(G+iB). \tag{15} \]

Let us examine the quantities entering here. In every oscillator some electron discharge occurs. It is caused by the radio-frequency voltage generated by the oscillator itself, and represents a current of the corresponding frequency. The quantity \(g+ib\) denotes the corresponding total conductance, i.e. the ratio of the discharge current to the voltage; \(g\) and \(b\) are, respectively, the ohmic and reactive conductances. Since in an electron discharge the dependence of the current on the field is nonlinear, \(g+ib\) is a function of the voltage. In reality, one of the current components in the oscillator is usually in phase with the voltage, decreasing as the latter increases; therefore \(g\) is a quite definite monotonically decreasing function of the voltage, and \(b\) is also a function of the voltage. The form of this function is of no interest to us; we note only that, since both \(g\) and \(b\) are uniquely determined by the voltage, the latter can be eliminated and a functional dependence between \(g\) and \(b\) obtained. The other quantities entering equation (15) are parameters of the circuit. It can be shown that the total electronic conductance \(g+ib\) is shunted by a parallel-connected circuit. Multiplying equation (15) by \(C\omega_0\), where \(C\) is the capacitance of this circuit and \(\omega_0\) is its natural—

frequency. We see that the first, second, and third terms on the right take the form:

\[ iC\omega,\ -iC\frac{\omega_0^2}{\omega}=\frac{1}{iL\omega}\left(\text{since } \omega_0^2=\frac{1}{LC}\right)\ \text{and}\ \frac{C\omega_0}{Q_0}, \]

i.e., they represent, respectively, the capacitive, inductive, and ohmic conductance of the circuit (\(Q_0\) determines the losses in the unloaded circuit). The remaining terms describe the total conductance of the load connected to the vibrator: \(G\) and \(B\) represent its ohmic and reactive conductances (written in dimensionless form), while the “external \(Q\)” \(Q_{\mathrm{ext}}\) describes the coupling of the vibrator to the load (by definition \(Q_{\mathrm{ext}}\) is small if the coupling is large, and conversely). It is now possible to determine the dependence of the operating properties of the magnetron on the external load. Let us divide equation (15) into its real and imaginary parts. The first of these determines the ohmic conductance of the vibrator \(g\) as a function of the load conductance \(G\) and of the circuit parameters. Since, as we have seen, there is a functional dependence between \(g\) and the radio-frequency voltage, we can find the voltage as a function of \(G\). From this, in turn, \(b\) is determined. After this, taking the imaginary part of equation (15) gives the frequency as a function of \(b\) and of the reactive conductance of the load \(B\). Assuming, for simplicity, that \(b=0\), we obtain (for \(\omega\) not too far from \(\omega_0\)):

\[ \frac{\omega-\omega_0}{\omega_0}=-\frac{B}{2Q_{\mathrm{ext}}}. \tag{16} \]

Thus, the operating frequency differs from the natural frequency of the intermediate circuit by an amount proportional to the inductive conductance of the load; moreover, this change of frequency (the so-called frequency pulling) is the smaller, the larger \(Q_{\mathrm{ext}}\) is, i.e., the weaker the coupling between the vibrator and the load. In practice \(Q_{\mathrm{ext}}\) is usually not large. Thus, for example, in ordinary magnetrons operating at a frequency of 3000 megacycles, \(Q_{\mathrm{ext}}\) is of the order of 100. Consequently, an external load with a reasonable value of inductive conductance can produce appreciable frequency pulling.

Let us now turn to the operation of a self-excited vibrator on a resonant load, the role of which may be played, for example, by the cavity of a short linear accelerator operating with standing waves. In this case \(G+iB\) represents the total input conductance of the resonant load, expressed in dimensionless units and referred to the corresponding fundamental level in the magnetron. It depends very strongly on frequency, so that the latter enters into both the real and the imaginary parts of equation (15), whose solution must therefore be sought by more refined methods. Most often one uses the complex plane, in which are plotted the graphs

of the right-hand side of (15) for all frequencies and of the left-hand side for all values of the voltage. The intersection of these two curves determines both the operating frequency and the voltage. As an example, Fig. 7 shows a case which is actually realized in the MTA accelerator design. The load here is a parallel resonant circuit connected in series with a resistance. In practice it can be realized in the form of a resonant line, by placing at its output a suitable three-terminal network (one end of which must be connected to the resistance, and the other—sufficiently long—to the vibrator). In Fig. 7

Fig. 7. Determination of the operating characteristics of a magnetron operating into a resonant load. The marks on the curve

\[ \left(\frac{\omega}{\omega_0}-\frac{\omega_0}{\omega}\right)+\frac{B}{Q_{\mathrm{ext}}} \]

as a function of

\[ \frac{1}{Q_0}+\frac{G}{Q_{\mathrm{ext}}} \]

correspond to equal increments of frequency.

two cases are shown: 1) the resonant line is tuned exactly to the magnetron frequency; 2) the frequency of the resonant line differs slightly from the magnetron frequency. We see that in the first case the curves representing the right- and left-hand sides of equation (15) intersect only once, whereas in case (2) there are three points of intersection; one of them corresponds to a frequency close to the resonant one, and the other two to shifted frequencies. In order to clarify the meaning of these three points of intersection, let us consider the process of excitation of oscillations. For the study of the nonstationary state it is convenient to introduce a complex frequency, whose real part is the ordinary oscillation frequency, while the imaginary part characterizes the rate at which the stationary regime is established. The curve representing the right-hand side of equation (15) is the locus of points corresponding to real frequencies; in the rest of the plane the frequencies are complex. If one moves along the curve of real frequencies in the direction of increasing frequency, then on the right there will be the region of increas-

of amplitude growth, and on the left—the region of attenuation. It is now easy to investigate the process of excitation of oscillations. As has already been noted, as the voltage increases the quantity \(g\) decreases, i.e. the point on the curve \(g(b)\) shifts to the left. At first, when the voltage is still very small, we are far to the right of the point of intersection of this curve with the curve of real frequencies. As the voltage grows, we move to the left along the curve \(g(b)\) and, finally, approach the curve of real frequencies. The rate of increase of the field thereby continuously decreases, and, finally, at the point of intersection a stationary regime is established. In case 1 this state is stable, since, as is seen in Fig. 7, a small displacement to the left from the point of intersection, corresponding to an increase of the voltage, brings us into the region of attenuation of amplitudes, as a result of which the equilibrium value of the voltage is restored again. The same is produced by a small displacement to the right along the curve \(g(b)\), since we then enter the region of amplitude growth. In case 2 the points \(a\) and \(c\) have the same properties, but the point \(b\) describes an unstable regime: a small displacement to the right from it, corresponding to a decrease of the voltage, brings us into the region of amplitude attenuation; therefore the system does not return to its initial state, but moves continuously to the right along the curve \(g(b)\), until it reaches the point \(c\). Similarly, a small displacement to the left from \(b\) brings the system to the point \(a\). Thus, two points of intersection describe a stable stationary regime, one—an unstable one.

It now remains to determine which of the two possible oscillation regimes—\(a\) or \(c\)—is established under the given conditions. The answer is clear: if the initial amplitude is small, i.e. if the representative point has moved along the curve \(g(b)\) from right to left, then the state described by the point \(c\) will be established. In this case, however, the resonance band is only weakly excited, while the greater part of the energy is contained in the vibrator itself, which is useless for our purposes. As the band is tuned to the frequency of the vibrator, or, conversely, the vibrator to the frequency of the band, the curves of Fig. 7 shift upward or downward (from case (1) to case (2)); the magnitude of the vertical displacement serves as a measure of the tuning. Thus, the resonance oscillations of the band that interest us are excited only in a small frequency range, in which the curve \(g(b)\) passes through a sufficiently narrow opening (the throat) of the resonance curve. This frequency interval can be increased by introducing a resistance successively into the circuit, as a result of which the throat of the resonance curve widens. For a finite distance between the magnetron and the band, the throat of the resonance curve narrows as the resistance of the circuit decreases; if the latter is too small, the system cannot operate at all in the resonant regime. At the same time, with a considerable circuit resistance the admissible frequency interval is quite large, but the ohmic losses also increase.

We see that, in the process of establishing oscillations, the frequency of the oscillator is automatically stabilized. Indeed, the marks on the curve of real frequencies, marking equal increments of frequency, are very far apart from one another in the resonance region. Therefore the frequency changes only very slightly when, in the course of tuning the oscillator, the curve of real frequencies shifts upward or downward relative to the curve \(g(b)\). The operating frequency must lie within the pass band of the resonance band, and, if the \(Q\) of the band is of the order of several thousand (as will be the case for a linear accelerator), then the frequency of the oscillator will be stabilized correspondingly. This circumstance is very important, since, generally speaking, it is not easy to tune an oscillator exactly to a prescribed frequency and to stabilize it; the resonance band of our type provides the required value of the frequency automatically.

The point is that, when the frequency of the oscillator deviates slightly from the resonance frequency, the reactive conductance of the resonator increases greatly; as a result of the pulling that occurs from this, the frequency of the oscillator returns to the resonance value.

We have considered the operation of a self-excited oscillator both with a resonant and with a nonresonant load, and have established that in the second case the frequency of the oscillator is determined by the intermediate circuit; whereas in the case of a resonant load of high \(Q\), tuned approximately to the same frequency as the intermediate circuit, the operating frequency is stabilized precisely by the load. This, however, is true only if a proper resistance is introduced into the circuit, for, as is usual in the case of two resonant circuits with close frequencies, we have here two normal oscillations, one of which can be excluded by a suitable choice of resistance. Let us now see how a self-excited oscillator operates in the presence of an external signal with a frequency close to the resonance frequency.

An external signal introduced through the output line of the oscillator and possessing exactly the latter’s operating frequency is indistinguishable from the reflected wave. Consequently, it can be taken into account by introducing into the reflection coefficient (i.e. into the load impedance) a correction term proportional to the ratio of the amplitudes of the signal and of the oscillator’s own oscillations; in other words, one more term with an arbitrary phase appears on the right-hand side of equation (15), of order of magnitude equal to \(Q_{\mathrm{ext}}^{-1}\), multiplied by the ratio of the amplitudes of the signal and of the oscillator oscillations. It acts on the oscillator in the same way as any other ohmic or reactive load and, in particular, may pull the frequency according to formula (16). If, for some phase difference, the correction term is sufficiently large, then an oscillator initially operating at a frequency somewhat different from the frequency of the signal may, for a certain phase relation, be synchronized with the latter. Investigation of the process of switching on an external signal\({}^{32}\) shows that this phenomenon indeed

takes place: as soon as the signal reaches the vibrator, the phase difference between them begins to approach the value at which the aforementioned frequency pulling occurs, and, after a very short interval of time, a stationary regime sets in—the vibrator enters into synchronism with the external signal (the phase angle between their oscillations is the greater, the larger the initial frequency difference).

Thus, one can lock the vibrator to an external signal by synchronizing their frequencies. This can be accomplished even with the aid of a comparatively weak signal.

Let, for example, \(Q_{\text{ext}} = 100\), and let the ratio of the amplitudes of the oscillations of the signal and the vibrator be \(1/10\) (consequently, the ratio of powers is \(1/100\)). Then from equation (16) it is seen that the maximum possible change in the vibrator frequency is about \(1/2000\). If the initial frequency difference of the signal and the vibrator is less than this value, then in the stationary state the phase difference between them will correspondingly decrease.

The situation changes if the vibrator frequency is, in addition, stabilized by a resonant strip; as we have seen, in this case the magnetron behaves, with respect to frequency pulling, as if it possessed the load quality factor, and not the normal value \(Q_{\text{ext}}\). Here the “locking” of the vibrator to the signal occurs over a very small frequency interval; however, more is not required, since the vibrator is already tuned rather accurately to the required frequency with the aid of the resonant strip. If the frequency of the external signal is close to the center of the resonant band of the stabilizing strip, then it will nevertheless be able to bring the vibrator into synchronism. Even if the signal is weaker than is needed for establishing synchronism, it will still be possible to force the vibrator to start operating in phase with it, and this situation will not subsequently change simply because the frequencies of the signal and the vibrator are sufficiently close. While the oscillations of the vibrator are only becoming established, their amplitude is, naturally, smaller than in the stationary state; correspondingly, the ratio of the amplitudes of the oscillations of the signal and the vibrator increases, i.e. the correction to the full conductance increases, and a greater frequency pulling becomes possible. Thus, during the war it was found\(^{12}\) that a magnetron can be made to operate in phase with a signal whose power is \(10^4\) times less than the power radiated by the magnetron in the stationary regime. Furthermore, two vibrators stabilized by resonant strips can be tuned so close to one another that during the passage of the pulse with which we are concerned in a linear accelerator, they will not go out of phase by more than a small fraction of a period. Consequently, it can be arranged so that, if two vibrators, under the action of a comparatively weak pulse, once begin to operate in phase with one another, then thereafter they will continue to operate in phase.

Up to now we have spoken about phasing a vibrator by means of an external signal. Let us now consider the phasing of one vibrator by another.

Suppose we have a resonant cavity, like the tube of a short linear accelerator, fed by a number of vibrators which, in this way, are all coupled to one another. If the cavity is short, then the time during which the signal passes from one vibrator to another is small in comparison with the time required for the field to be established in the cavity. Then each vibrator sends the others a signal that can be used for phasing. Since the coupling between the vibrators in this case is very strong, their phases will be very strictly matched. Indeed, a more detailed study shows that if the operating frequencies of the vibrators are sufficiently close to the frequency of the cavity, so that their phases are coupled to the phase of the oscillations in the cavity, then the power of the signal received by each of them is more than sufficient for phasing. There is only one difficulty here: at the very beginning of operation—immediately after the signals are sent—there occurs a short period of confusion and disorder, when each vibrator tends to establish precisely its own phase. This difficulty can be avoided by sending the phasing signal before operation begins—either from outside, or by switching on one of the vibrators earlier than the others. Then all the vibrators will at once begin to operate in phase. Subsequently this regime is maintained quite satisfactorily.

From everything that we have seen in this section, it seems to follow that a long linear accelerator can be fed from self-excited vibrators. This is accomplished in the following way: by means of real or fictitious surfaces the accelerator is divided into a number of short sections. Into each of them a phasing signal is introduced from a pulsed vibrator feeding all the sections (naturally, the length of the connecting lines must be chosen with allowance for the delay—so that the necessary phase conditions are fulfilled). Thus it turns out that each section is fed by an entire set of vibrators, triggered in succession after short intervals of time following the sending of the phasing signal.

Accordingly, oscillations synchronized with the phasing signal will be excited in the sections, and their phases will be matched with one another. The oscillation frequency is stabilized by the resonant cavity (if one is dealing with a standing-wave accelerator), and if all the short sections are sufficiently well tuned to one and the same frequency, all of them will operate in phase during the entire passage time of the pulse.

It is clear that the precise tuning of the sections plays an essential role. In the next section we shall consider the question of the permissible frequency deviations, in order to see whether it is possible to tune the sections of a long accelerator with the required degree of accuracy.

XI. Tolerances in a Linear Accelerator

In order that an electron or ion may resonate with some Fourier component of the field, the latter must propagate with a definite velocity. In the following sections we shall see that, in the acceleration of positive ions, whose velocity at almost all energies is considerably less than the velocity of light, stable groups of particles are formed which move together with the field. Therefore here there is no need to fix the wave velocity very precisely. But electrons, in all cases of interest, practically attain the velocity of light at the very beginning of the accelerator and subsequently move with this velocity. Consequently, the accelerating electromagnetic wave must propagate just as rapidly in the tube of the apparatus. Further, if the phase of the wave at the end of the accelerator differs appreciably from the required value, then the necessary phase relation between the wave and the electrons cannot be established, and the accelerator will not operate as it should. Thus only small deviations from the correct value of the phase are possible; correspondingly, stringent requirements are imposed on frequency stabilization and on the accuracy with which the dimensions of the apparatus are adjusted. In the present section we shall examine these requirements in more detail and see whether they can be satisfied.

The situation depends essentially on what type of accelerator is being considered. The difficulties are great only for long devices. We shall therefore begin our consideration with a linear accelerator divided into a number of short resonant sections, phased by a special circuit. First we shall consider the requirements imposed on the individual short sections, and then pass on to the conditions imposed on their relative phasing. Each section has a well-defined natural frequency; the width of the resonance peak is related to the frequency itself as \(1/Q_0\). Let us note first of all that there is no need to control the signal frequency with an accuracy exceeding this quantity. The reason for this can be understood by considering the duration of the pulse. As was already noted, a large part of the pulse propagation time is spent in establishing the oscillations. The time required for an electron or ion to pass through the entire accelerator after the field has been established is small in comparison with the time needed for the oscillations to build up. But, as is known, the spectrum of a short pulse of monochromatic oscillations is spread out, the line width being inversely proportional to the number of waves in the train. The latter, in our case, is of the order of \(Q_0\) (since this is the number of periods required for the establishment of the field in the cavity). Thus the spectral width of the pulse is comparable with the width of the resonance line of the cavity. But a cavity operating as a frequency stabilizer can hold the frequency of the vibrator within limits considerably smaller than the width of the resonance band. Therefore

must be easy to stabilize the frequency with the necessary degree of accuracy.

It is possible, however, that the natural frequency of the cavity will differ somewhat from the desired value; an incorrect relation of frequency to wavelength will, naturally, lead to an incorrect value of the velocity. In practice one can construct a short section whose natural frequency (for a given section length) will differ from the correct value by approximately two or three hundredths of a percent. For our purposes this is insufficient, since for \(Q_0\) of the order of 18000 (as is the case in the MIT accelerator) the relative bandwidth turns out to be smaller than this value. Therefore it is necessary to introduce into each short section a special tuning circuit in order to correct its frequency. Verification of the tuning is quite simple. It is only necessary, having created a signal of exactly the required frequency, to send it as individual pulses into one of the sections and observe whether beats occur between the two signals during the propagation time of the pulse. If there are no beats, this means that the given section and the vibrators feeding it are tuned to the given frequency with the required degree of accuracy. It is clear, further, that if all the sections are tuned in this manner, then, once they begin operating in phase with one another, they will not get out of phase, to the required degree of accuracy, during the propagation time of the pulse. And nothing more is required. In practice it is not difficult to tune stabilized vibrators with such a degree of accuracy, and from time to time to maintain the tuning.

Let us now consider how one can coordinate the operation of short sections that are far apart from one another—say, located at opposite ends of a long accelerator. It is clear that if the length of the accelerator is many thousands of times greater than the wavelength, then an error in a small fraction of the latter will indeed be negligible. Since random errors compensate one another, it should be possible to construct such a tube with an error of several hundred-thousandths of the total length of the instrument. The error in the total length of the accelerator can also be compensated by a corresponding change in the natural frequency, which can be carried out by providing each section with a tuning circuit. Let us assume, for definiteness, that oscillations of type \(\pi\) are excited in the instrument (as is the case in the MIT accelerator). Then the distance between neighboring diaphragms is equal to half the wavelength, and the total number of half-waves fitting into the tube can be found by simple counting. One may be certain that, with the mode of operation that we propose, the oscillations of each short section will belong to type \(\pi\). Therefore the operating wavelength is equal to the length of the instrument divided by the number of half-waves. Now, knowing the speed of light, it is easy to determine the necessary frequency and, by means of tuning circuits separate for each section, to tune all the short sections to it.

In the manner described, it is not difficult, with the necessary degree of accuracy, to construct an accelerator consisting of short resonant sections, the frequency and phase of each of which are regulated independently. As we have seen, in such a circuit a sufficiently sensitive stabilization of the frequency of the feeding vibrators is automatically provided. The situation, however, changes completely if we pass to a traveling-wave system. Here there is no longer a resonant band that would stabilize the frequency. Using self-oscillating vibrators, we must either rely on the constancy of the frequency of the vibrators themselves, or make use of some separate stabilizing bands. In the latter case the entire construction becomes very complicated, and many advantages of the traveling-wave system are lost. If this is not done, then a very serious question arises: can the vibrators be tuned so accurately that they remain synchronized throughout the entire time of propagation of the pulse and phased by the initial signal? The frequency of oscillations in a magnetron depends on its temperature and operating conditions, and these are difficult to control. Therefore it seems to the author practically necessary to have a stabilizing band that would smooth out frequency fluctuations arising from changes in temperature and other causes. It is noteworthy that in the reports of the English group (T. R. E.) designing a traveling-wave accelerator fed by a magnetron, the frequency variation is indicated as one of the reasons limiting the possible length of the instrument. The frequency variations assumed there are considerably larger than those encountered in the case of frequency stabilization by a resonant accelerator. This circumstance seriously limits the English estimates of the maximum attainable acceleration. Another possibility is to feed the accelerator from power amplifiers, controlling their frequency by means of a well-regulated phasing signal. This method of feeding a long traveling-wave accelerator appears to the author to be the only one having a chance of success. This opinion is also shared by the Stanford group, which connects its plans with the use of a powerful klystron power amplifier, still to be built.

In the case of an accelerator operating on traveling waves, one more problem arises: here it is impossible to divide the tube into short sections physically separated from one another. Therefore the possibility is lost of easily checking the resonant frequency of an individual section. Nor is it possible to synchronize the sections with one another by means of separate tuning circuits. Of course, in the course of constructing the instrument one can bound each section by reflecting walls, check its own frequency, and synchronize the sections by means of tuning circuits periodically introduced into the waveguide. However, after assembly of the instrument this is no longer possible. Probably, for testing an already completed instrument it will be most convenient to send in

signal parallel to the main line (such a signal may be the wave traveling in the coaxial line), and to compare the phases of the wave in the signal and in the accelerator at various points along the line. This method, although more complicated than that used in the case of standing waves, will probably nevertheless prove feasible. If powerful klystron amplifiers existed at the present time, it is quite possible that traveling-wave accelerators would turn out to be the most advantageous, since they provide a certain gain in acceleration. On the other hand, since in fact there are no powerful klystrons at present, while there are powerful magnetrons, the problem of phasing and frequency stabilization is solved much more simply in the case of accelerators operating with standing waves. It is precisely for this reason that the latter type is considered in the MIT project.

XII. DYNAMICS OF PARTICLES IN THE ACCELERATOR

Up to now we have spoken only about the mechanism by which the accelerating field is established and about the properties of the device. Let us now consider the problem of the motion of particles in this accelerating field. We shall divide it into two parts: first, we shall investigate the longitudinal motion of particles parallel to the axis of the accelerator; second, the transverse motion and the problems of focusing. Further, considering only the longitudinal motion, we shall again divide the problem into two parts, namely, motion in an external field propagating with constant and with variable velocity. Up to this point it has everywhere been assumed that the properties of the waveguide tube do not change from point to point, so that the resonating wave propagates with unchanged velocity. This assumption is, of course, valid for particles moving with the velocity of light; however, in the case of positive ions or electrons whose velocity has not yet reached the velocity of light, the wave must be accelerated in accordance with the acceleration of the particle.

We have already seen that, for a constant particle velocity, only one of the plane waves into which the field is decomposed is significant. The effect produced by all the other Fourier components vanishes upon averaging over a period. The longitudinal (i.e., parallel to the \(z\) axis) component of the electric field of this wave has the form:

\[ E_z = E \sin \omega \left( t - \frac{z}{v_0} \right), \]

where \(\omega\) is the resonant frequency, and \(v_0\) is the phase velocity. This is the value that the field assumes on the \(z\) axis (at present we are interested only in it). The equation of motion of a particle with charge \(e\) and rest mass \(m_0\) in this field will be:

\[ \frac{dp}{dt} = eE \sin \omega \left( t - \frac{z}{v_0} \right), \]

where the momentum \(p\) is given by the expression:

\[ p=\frac{m_0 v}{\sqrt{1-v^2/c^2}},\qquad v=\frac{dz}{dt}. \]

It is convenient to introduce a coordinate system moving with the velocity of the traveling wave \(v_0\). For the \(z'\)-displacement of the particles in this coordinate system, we have: \(z'=z-v_0t\).

Now the equation of motion in this coordinate system can be written in Hamiltonian form. This possibility was pointed out to the author by Dr. Mason of MIT; a similar method was also used in the works of T. R. E. Let us introduce the Hamiltonian function

\[ H=\sqrt{m_0^2c^4+p^2c^2}-pv_0-eE\,\frac{v_0}{\omega}\cos\omega\frac{z'}{v_0}. \tag{17} \]

Fig. 8. Phase space for \(v_0=\dfrac{c}{2}\).

By direct verification it is easy to see that the equation of motion can be rewritten in the form:

\[ \frac{dp}{dt}=-\frac{\partial H}{\partial z'},\qquad \frac{\partial z'}{\partial t}=\frac{\partial H}{\partial p}. \]

These are Hamilton’s equations with a Hamiltonian function that does not depend explicitly on time.

They show that \(H\) remains constant during the motion of the particles. Consequently, the straight lines \(H=\text{const.}\), drawn in the phase plane \((z',p)\), give the relation between the coordinate and the momentum of the moving particle. From this the velocity of the particle at any point of its trajectory is determined. Figure 8 shows the phase space of a particle moving with velocity

\[ v_0=\frac{c}{2}; \]

later, in Fig. 9, we shall also discuss the case \(v_0=c\). Along the abscissa axis is plotted the dimensionless quantity

\[ \frac{\omega z'}{v_0}, \]

which increases by \(2\pi\) when we advance along the \(z\)-axis by one wavelength. As ordinate the dimensionless quantity

\[ \frac{p}{m_0c} \]

is used. We also give a scale for the quantity

\[ \sqrt{m_0^2c^4+p^2c^2}-m_0c^2, \]

i.e. the ordinary kinetic energy expressed as a function of the rest energy \(m_0c^2\). The energy curves are drawn for constant values of the dimensionless quantity

\[ \frac{H}{m_0c^2}. \]

Fig. 9. Phase space for \(v_0=c\).

As is seen from equation (17), the Hamiltonian function contains a certain parameter which, in dimensionless form, has the form

\[ \frac{eE}{m_0c^2}\cdot\frac{v_0}{\omega}. \]

Its physical meaning is easy to see. Let \(\lambda_g\) be the length of the resonant wave propagating with velocity \(v_0\). Then

\[ \frac{v_0}{\omega}=\frac{\lambda_g}{2\pi}, \]

and our parameter evidently represents, in fractions of \(m_0c^2\), the energy which a particle undergoing maximum acceleration acquires over a path of length

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

In Fig. 8 this quantity is taken equal to \(0.10\); this agrees in order of magnitude with what can actually be obtained in the M.T.I. accelerator.

Considering Fig. 8, we first of all note a series of closed oval curves, representing periodic orbits. They surround points at which the potential energy entering the Hamiltonian function (17) has a minimum, while the momentum is equal to the momentum of a particle moving with velocity \(v_0\).

The force acting on the particle at these moving points of stable equilibrium is equal to zero. On leaving such a point, the particle experiences a force returning it to its former position. In other words, at the given point of space and at the given instant of time the force is zero, but it increases as the wave advances. A particle lagging behind the stable position will be accelerated until its velocity becomes greater than \(v_0\). Then it will overtake the wave, and a retarding force will begin to act on it, as a result of which the velocity of the particle will decrease to \(v_0\) and below; the particle will again fall behind, and so on. We obtain the cycle shown in Fig. 8 (points \(a, b, c, d\)). In other words, the point of phase space representing the particle describes a closed curve, moving clockwise.

On the other hand, orbits of the type of curve \(e\) in Fig. 8 are possible. They correspond to so large a velocity of the particle that, despite the retardation, it nevertheless overtakes the wave, passing from one minimum of potential energy to another, and so on. There also exist orbits of the opposite type (curve \(f\)), corresponding to a very small velocity of the particle—such that the latter lags behind the wave all the time. Curve \(g\) represents the limiting case of transition from a periodic orbit to one of the last two cases. It passes through a point of unstable equilibrium, corresponding to motion with velocity \(v_0\), but with a phase shift by \(\pi\) in comparison with the position of stable equilibrium. At this point the force acting on the particle is also equal to zero, but it does not increase; rather, it decreases with time, so that a particle, once it has overtaken the wave, enters an accelerating field and goes farther and farther away; likewise, a particle that has once fallen behind lags more and more. We see that the motion under study is physically similar to the motion of a pendulum in which a particle, under the action of a constant vertical force, oscillates in a vertical plane at a given distance from the axis. In a pendulum, motions of small amplitude are periodic, whereas at sufficiently large energy the particle describes circles, slowing down at the upper point of the orbit but not changing the direction of motion. Here too there is a limiting orbit passing through the position of unstable equilibrium; the latter corresponds to the case when the velocity of the particle at the upper point of the orbit is zero; after experiencing an infinitely small displacement, the particle will go downward, make a circle, and again stop at the top; for this, however, an infinitely long time is required. We shall show that this resemblance to a pendulum is by no means superficial, but is inherent in the very essence of the matter. The analogy with the pendulum can be used to obtain certain results concerning the periods and amplitudes of the oscillations, as well as other properties of our motion. Thus, for small amplitudes (i.e. for the small ovals in Fig. 8) the period of oscillation does not depend on the amplitude, but for large amplitudes the period increases, tending to infinity for the limiting orbit just considered.

Passing next to aperiodic orbits (corresponding to circular motion of the pendulum), we see that the time of revolution, infinite near the limiting orbit, gradually decreases as one moves away from the curve \(g\); when the velocity of the particle becomes much greater than \(v_0\) (which corresponds to rapid rotation of the pendulum), the periodic field may be regarded as a small perturbation, and the velocity of the particle is approximately constant. Hence it is easy to find the period of revolution.

This analogy can be continued and used for calculating periods and amplitudes, if our formulae are rewritten in a suitable way. The only thing that distinguishes our problem from the pendulum problem is the relativistic change of mass. For example, over a period of oscillation the energy of the particle changes from a maximum to a minimum. If this change in energy is large enough to cause a noticeable change in the mass, then our problem differs essentially from the pendulum problem. However, it can be reduced to the latter if the change in mass is small. Mathematically this is expressed as follows: let

\[ v'=\frac{dz'}{dt}=v-v_0 . \]

Expand the momentum \(p\) in powers of \(v'\). We obtain:

\[ \begin{aligned} p &= \frac{m_0 v_0}{\sqrt{1-\frac{v_0^2}{c^2}}} + \frac{m_0 v'}{\left(1-\frac{v_0^2}{c^2}\right)^{3/2}} +\ldots \\ &=p_0+m_l v'+\ldots=p_0+p'+\ldots, \end{aligned} \]

where

\[ m_l=m_0\left(1-\frac{v_0^2}{c^2}\right)^{-3/2} \]

is the longitudinal mass. In the same way we expand the Hamiltonian function (17). In doing this, as is not difficult to see, the series must be cut off not at the second term, but at the third; however, the answer obtained is very simple:

\[ H=m_0 c^2\left(1-\frac{v_0^2}{c^2}\right)^{1/2} +\frac{p'^2}{2m_l} -eE\,\frac{v_0}{\omega}\cos\frac{\omega z'}{v_0} \tag{18} \]

or

\[ H=H_0+H', \]

where \(H_0\) is the first term of (18), and \(H'\) includes everything else. In these notations the equations of motion take the form:

\[ \frac{dp'}{dt}=-\frac{\partial H'}{\partial z}; \qquad \frac{dz'}{dt}=\frac{\partial H'}{\partial p'} . \]

These are precisely the equations of oscillation of a particle of mass \(m_1\) in the field

\[ -eE\sin\frac{\omega z'}{v_0}, \]

i.e., the equations of a pendulum. Let us now draw the phase space of the particle in the variables \(z'\), \(p'\). The equilibrium position in Fig. 8 will then be shifted in the direction of the axis of abscissas (the \(z'\)-axis), but will not fall on it; the curves \(H'=\mathrm{const.}\) will be very similar to the curves \(H=\mathrm{const.}\) of Fig. 8, with small differences due to higher powers of \(v'\).

It is now possible, in the indicated approximation, to derive a number of properties of our system, using the known solution of the pendulum problem. The angular frequency of small oscillations is given by the square root of the force constant divided by the mass. The force acting on the particle in the case of small oscillations takes the form \(-eE\,\frac{\omega z'}{v_0}\) (the sine is replaced by its argument). Thus, the angular frequency of the oscillations (we shall denote it by \(\omega_0\)) is

\[ \omega_0=\left[\frac{e}{m_0}E\frac{\omega}{v_0}\left(1-\frac{v_0^2}{c^2}\right)^{3/2}\right]^{1/2}. \tag{19} \]

This quantity, of course, has nothing in common with the angular frequency of the wave \(\omega\). One can also calculate the amplitude of oscillations of the particle (in the phase of the electromagnetic wave, equal to \(\frac{\omega z'}{v_0}\)), whose energy (in the moving coordinate system) is equal to \(W\). It is easy to see that this amplitude is determined by the equation:

\[ \frac{W}{m_0c^2}=\frac{1}{2}\cdot(\text{amplitude})^2\cdot\frac{eE}{m_0c^2}\cdot\frac{v_0}{\omega}. \]

This formula is valid for small oscillations, when the energy is proportional to the amplitude. Its accuracy at large amplitudes can be estimated by considering the limiting case of an amplitude equal to \(\pi\). We have in this case:

\[ \frac{W}{m_0c^2}=\frac{\pi^2}{2}\cdot\frac{eE}{m_0}\cdot\frac{v_0}{c^2\omega}, \]

whereas the exact formula (18) gives the numerical coefficient 2.

It is important to note that \(W\)—the energy of oscillation in the moving coordinate system—is not equal to the change in the kinetic energy of the particle in the stationary coordinate system. It is easy to show that the total kinetic energy is given by the formula

\[ KE=KE_0+m_1v_0v'+\ldots, \]

where \(KE_0\) is the kinetic energy of the particle moving with velocity \(v_0\).

The maximum and minimum values of the kinetic energy of a particle oscillating with a given amplitude are:

\[ KE=KE_0 \pm m_e v_0 \left(\frac{2W}{m_e}\right)^{1/2} =KE_0 \pm (\text{amplitude})\cdot m_0 c^2 \times \]

\[ \times \frac{v_0}{c}\left[ \frac{eE}{m_0c^2} -\frac{v_0}{\omega}\left(1-\frac{v_0^2}{c^2}\right)^{-3/2} \right]^{1/2}. \tag{20} \]

In the limiting case of an orbit passing through the position of unstable equilibrium, this gives:

\[ KE=KE_0 \pm 2m_0c^2\frac{v_0}{c} \left[ \frac{eEv_0}{m_0c^2\omega} \left(1-\frac{v_0^2}{c^2}\right)^{-1/2} \right]^{1/2}. \tag{21} \]

As follows from the derivation, these formulas are valid only so long as the change in kinetic energy is so small that the change in the particle mass during the oscillations may be neglected. From formula (21) it is seen that for

\[ \frac{eEv_0}{m_0c^2\omega}=\frac{1}{10}, \]

as adopted in Fig. 8, this condition is not fulfilled very well, but even in this case the error is small. Of course, formulas (20) and (21) give the change in energy caused by the average force acting on the particle from the accelerating field over the half-period during which the particle energy increases from the minimum to the maximum value (it is determined by the product of the average force and the distance which the particle, with velocity about \(v_0\), traverses in the field during this time).

Let us now consider how the curves of Fig. 8 change when various parameters are varied. When the accelerating field \(E\) is changed, the ovals contract or expand in the vertical direction. As is seen from equations (20) and (21), the height of the ovals for a given value of \(v_0\) is proportional to \(E^{1/2}\).

The picture shown in Fig. 8 corresponds to a large acceleration, which can be obtained in the case of electrons. On the other hand, equations (20) and (21) show that for positive ions (in the same accelerating field) the ovals expand as the square root of the ratio of the masses of the electron and the ion. This corresponds to the fact that in a given field the relative change in the energy of ions is much smaller than for electrons, since the rest mass of the latter is much smaller. Another parameter at our disposal is the wave velocity \(v_0\). As it increases, a number of changes occur. First of all, the position of stable equilibrium shifts upward without bound. Indeed, the coordinate of this point is determined by the momentum of the particle with velocity \(v_0\), and it goes to infinity when \(v_0\) approaches the speed of light. At the same time, the oscillation frequency, determined by equation (19), tends to zero, i.e. the period becomes infi—

finiteness. This occurs because, when \(v_0\) approaches \(c\), even a very large change in the energy (or momentum) of the particle produces only a very small change in velocity. Therefore, in order that, owing to the difference between the velocities of the wave and the particle, the latter should lag behind the equilibrium position and again get ahead of it, a very long time is required. Connected with this is the vertical elongation of the ovals: indeed, when the period of oscillation becomes very large, the particle traverses during the half-period a very large distance (in the rest frame of the coordinates) and, consequently, during one cycle can acquire or lose a very considerable energy. Obviously, in this limiting case our approximate method of reducing the problem to the pendulum problem is no longer applicable.

When \(v_0\) becomes equal to the speed of light, the equilibrium points go to infinity, and the ovals are no longer closed. To obtain the correct solution in this case, one must return to the original, exact formulation of the problem. In Fig. 9 the same curves as in Fig. 8 are shown, but for the case \(v_0=c\) (and not \(v_0=\dfrac{c}{2}\), as before).

The orbits of Fig. 9 are divided into two, not three, classes. First, we have orbits moving together with the field (similar to the periodic curves of Fig. 8). One of them is denoted in the figure by the letters \(a, b, c\). It begins at a point with finite momentum (i.e., with a velocity less than the speed of light) and therefore at first lags behind the wave and enters an accelerating field. However, the particle can in no way attain the velocity of the wave, and therefore the phase trajectory of the representative point approaches asymptotically a certain vertical line, shifted in a definite way in phase relative to the equilibrium position. There the particle remains indefinitely long, continuing to acquire energy and momentum. The magnitude of the energy imparted to it depends on the phase of the wave associated with the asymptotic position of the particle. If this asymptotic phase is close to \(-\dfrac{\pi}{2}\) (i.e., to that phase where the acceleration is maximal), then the particle will remain indefinitely long in the field of maximum intensity \(E\) and, consequently, will acquire the maximum possible energy.

Besides these orbits moving together with the travelling wave, as a result of which the particles ultimately receive an arbitrarily large energy, there also exist orbits of another type, similar to curve \(d\) (Fig. 9). They lag behind the wave, not having sufficient velocity to keep up with the field.

Naturally, orbits of the third class, which outrun the field, are absent in the present case. Curve \(e\) of Fig. 9, whose phase tends asymptotically to \(-\pi\), represents the limiting case between the two types indicated above. At phase equal to zero it passes through a point with a certain minimum momentum (and, consequently, minimum kinetic energy). No particle with energy,

less than this minimum value cannot fall onto an orbit of the first type; for any energy greater than the minimum one can find an oscillation phase for which this is possible. Although formulas (20) and (21) are in this case already quantitatively incorrect, we can nevertheless conclude by analogy that the larger \(E\) is, the smaller this minimum kinetic energy will be. In Fig. 9 it is equal to approximately \(0.75\,\mathrm{MeV}\).

Until now we have considered the velocity of the traveling wave to be constant. Now we must consider the case when it gradually changes from point to point. In a periodic waveguide this can be accomplished by gradually changing the distance between the diaphragms (with a corresponding change of all the other dimensions, so that a field of the given frequency can be excited throughout the tube). Thus \(v_0\) is a function of position. Instead of our previous expression for the field, we must now write

\[ E_z = E \sin \omega \left( t - \int \frac{dz}{v_0(z)} \right), \]

where, for simplicity, we consider only the case when \(E\) does not depend on \(z\). How is the problem of the motion of particles in such a field to be solved? Here one cannot find the exact solution which previously led us to the Hamiltonian (17) and to the curves of Figs. 8 and 9; however, one can use an approximate method equivalent to our earlier reasoning with the pendulum. This is where we shall begin.

Previously we measured the displacement of a particle \((z')\) relative to the point with coordinate \(v_0 t\) (moving together with the wave); the initial phase was chosen so that the field at this point was always equal to zero, and the moving point \(z' = 0\) represented the equilibrium position in which the particle could remain at rest for an arbitrarily long time. Now as well, let us try to find a moving point that represents the equilibrium position, and let \(z'\) be the displacement of the particle relative to this point. Denote by \(z_0(t)\) the displacement of this equilibrium point, and by \(p_0(t)\) the momentum of the particle moving together with it; \(p_0\) depends on time, since now such a particle moves with acceleration. Obviously, we have:

\[ p_0 = \frac{m_0 v_0}{\left(1 - \dfrac{v_0^2}{c^2}\right)^{1/2}}, \]

where one must take the value of \(v_0\) at that point \(z_0\) at which the particle is located at the time \(t\). Then the equation of motion of the particle located in the equilibrium position is

\[ \frac{dp_0}{dt} = eE \sin \left( t - \int \frac{dz_0}{v_0(z_0)} \right)\omega . \]

We shall assume that the particle is at all times in one and the same phase of the accelerating field (i.e., that the velocity \(v_0\) corresponds to the motion of the particle under the action of a constant force). We then have

\[ t-\int \frac{d z_0}{v_0(z_0)}=t_0, \]

where \(t_0\) is a certain constant. This equation determines \(z_0\) as a function of \(t\). It also makes it possible to find \(t\), if the acceleration is known. If the distances between the diaphragms are chosen so as to obtain a specified acceleration, then we can find the force required for this and, equating it to \(eE \sin \omega t_0\), determine \(t_0\). We note that the latter condition can be satisfied only if the acceleration is less than \(eE\)—the maximum acceleration which the field can impart to the particle for the most favorable phase relation.

Let us now consider a particle with coordinate \(z=z'+z_0\), where \(z'\) measures the deviation of the particle from the equilibrium position, and \(v'=\dfrac{dz'}{dt}\) the rate of change of this deviation with time. We shall restrict ourselves to sufficiently small values of \(z'\), so that the velocities of the wave at the points \(z\) and \(z_0\) may be regarded as identical. We shall further restrict ourselves to sufficiently short time intervals during which \(v_0\) and the longitudinal mass do not have time to change appreciably. Then the force acting on the particle may be represented in the form

\[ eE \sin \omega \left(t-\int \frac{dz}{v_0(z)}\right) = eE \sin \omega \left(t-\int \frac{dz_0}{v(z_0)}-\frac{z'}{v_0}\right) = \]

\[ = eE \sin \omega \left(t_0-\frac{z'}{v_0}\right). \]

The change of the momentum of the particle with time is determined by the expression

\[ \frac{dp}{dt} = \frac{dp_0}{dt}+m_l\frac{dv'}{dt} = eE \sin \omega t_0+m_l\frac{dv'}{dt}. \]

Thus the equation of motion takes the form:

\[ m_l\frac{dv'}{dt} = eE\left\{\sin \omega \left(t_0-\frac{z'}{v_0}\right)-\sin \omega t_0\right\}. \]

It can be obtained from the Hamiltonian function

\[ H'=\frac{p'^2}{2m_l} - eE\frac{v_0}{\omega}\cos \omega \left(t_0-\frac{z'}{v_0}\right) + eEz'\sin \omega t_0, \tag{22} \]

where \(p'=m_l v'\), and \(m_l\) is regarded as constant.

In Fig. 10 the potential energy from equation (22) is shown as a function of \(z'\). It resembles the sinusoidal curve known from the pendulum problem, with the difference that the curve of Fig. 10 is inclined

and therefore asymmetric. It is convenient to consider the motion by means of the well-known energy integral. In Fig. 10 we draw a horizontal straight line whose ordinate is \(H'\). The vertical distance between this straight line and the potential curve gives the kinetic energy. Where \(H'\) is greater than the potential energy, the kinetic energy is positive, and motion is possible. We see that, for example, in the case \(H' = H_1\) the particle can either perform oscillations between the points \(a\) and \(b\), or, being to the left of the point \(c\), move to the right, reach the point \(c\), and then, turning back, again proceed to the left.

Fig. 10. Potential energy of the accelerated particle as a function of \(z'\).

At larger values of the energy (for example, \(H_2\)) oscillatory motion is impossible. In other words, oscillations about the point \(z' = 0\) can occur only in a narrow energy interval, the narrower the steeper the potential curve.

Fig. 11. Phase space of the accelerated particle.

From equation (22) it is seen that its steepness increases as \(\omega t_0\) approaches \(\pi/2\). This is precisely the value of the phase angle at which the particle on the equilibrium orbit experiences maximum acceleration. At \(\omega t_0^* = \pi/2\) the curve proves to be so steep that the potential wells disappear, and stable orbits no longer exist. All these relations are conveniently represented on the phase plane. It is shown in Fig. 11, similarly to Figs. 8 and 9. There are marked the trajectories of the representative points corresponding to the energies \(H_1\) and \(H_2\) of Fig. 10. The periodic character of the motion is clearly visible.

in the first case and aperiodic in the second. The connection with Fig. 8 is also clear. Aperiodic orbits in this case do not always move only ahead of or only behind the wave. The point is that for these orbits there is no resonant relation between the phases of the particle and the wave, so that the particle is not accelerated, but moves (in the rest frame) with constant velocity, though under the action of a series of superposed periodic perturbations. If at first the particle was moving faster than the wave, then the latter, accelerating, will gradually catch up with and overtake it. In a coordinate system connected with the traveling wave, it will appear that the particle begins to move with positive velocity \(v'\) and positive momentum \(p'\), gradually slows down, stops, and begins to move in the opposite direction with negative velocity and negative momentum. It is clear that such a particle can never be captured by the wave and move together with it. On the other hand, particles describing periodic orbits can be captured by the wave. They travel together with it, being continuously accelerated, like a particle on an equilibrium orbit, with the only difference that they also oscillate about the equilibrium position. This clear distinction between some particles, which are captured by the wave and resonate with it, continuously receiving energy, and others, which are not captured by the wave, justifies the assumptions made in the first section. Indeed, it was assumed there that the influence of the whole field on the particle can be replaced by the action of only one resonating plane wave, which propagates with the same velocity as the particle.

From our consideration it is evident that particles resonating with the wave differ sharply from those which do not resonate with it. And it is also quite clear that a particle can resonate only with one of the Fourier components, while all the others produce only a periodic perturbation, similar to that experienced by the aperiodic orbits of Fig. 11.

In the case of a plane wave of variable velocity, the various parameters entering into equation (22)—the longitudinal mass, the velocity \(v_0\), and in many cases also the amplitude of the electric field \(E\)—are likewise functions of the point and, consequently, change with time, although one may expect this change to be small. It is important to investigate how a slow change of these parameters affects the motion of a particle. A particle moving along an equilibrium orbit will, of course, remain there; but if there are also particles oscillating about the equilibrium position, then it is of interest to know how the amplitude of their oscillations changes when the parameters change. This question can be answered with the aid of a well-known theorem of mechanics, which in quantum theory is called the adiabatic theorem. In terms of phase space it is formulated as follows: suppose there is a periodic orbit; we compute the corresponding phase integral

\(\oint p\,dq\), i.e., the area enclosed within the trajectory of the representative point (\(p\) is momentum, \(q\) is coordinate; the integral is taken along the phase trajectory). Let the parameters characterizing the motion change slowly—so that the relative change of each of them over a period is small. The form of the trajectories of the representative point will, of course, change in this case, and it is natural to ask which of these curves in the new phase space represents the true motion. The adiabatic theorem asserts that the true motion is represented by the curve for which the phase integral has its former value. In other words, the phase integral is invariant with respect to a slow change of the parameters.

This generally means that the energy will change with the change of the parameters. For example, it can be shown that in the case of a linear harmonic oscillator the phase integral is equal to the energy divided by the frequency. Thus, if the frequency changes (as a result of a change in the mass or in the acting force), then the energy will change proportionally to it. But the energy of an oscillator whose displacement is \(A=\cos\omega_0 t\) is equal to the maximum value of its kinetic energy, \(\frac{1}{2}mA^2\omega_0^2\). Since it is proportional to the frequency \(\omega_0\), \(A\) changes proportionally to \(\frac{1}{\sqrt{m\omega_0}}\). This result can be used to investigate the change in the amplitude of small oscillations in a linear accelerator. The frequency of the oscillations, which can be obtained from the Hamiltonian function (22) (just as (19) is obtained from (17)), is given by the formula:

\[ \omega_0=\left(\frac{e}{m_e}E\,\frac{\omega}{v_0}\cos\omega t_0\right)^{\frac12}. \tag{23} \]

Since the longitudinal mass also enters into the dynamical problem, we conclude from this that the amplitude of the oscillations \(A\) is proportional to the expression \(\left(\frac{v_0}{m_e}E\omega\cos\omega t_0\right)^{-\frac14}\).

We see that when \(v_0\) approaches the speed of light and the longitudinal mass tends to infinity, the amplitude of the oscillations tends to zero. This can also be seen directly from the course of the phase curves: we have seen that, as \(v_0\) approaches the speed of light, the oval phase trajectories of the representative points are stretched in the vertical direction; consequently, their width must decrease, since the area remains constant. This also means that the particles pass onto orbits with smaller amplitude. Similarly, the amplitude of the oscillations decreases with increasing accelerating field \(E\). As we shall see in the next section, these relations can be used for very tight grouping of particles near the equilibrium position.

XIII. APPLICATION OF ELECTRON DYNAMICS TO VARIOUS TYPES OF ACCELERATORS

In the preceding section we investigated the dynamics of the longitudinal motion of particles in a traveling plane wave, both for a constant wave velocity and for the case when it varies slowly from point to point. Let us now apply these general theoretical results to the various types of accelerators of which we have spoken. The simplest of them is the M.T.I. accelerator. In this case the acceleration of electrons is involved, which are injected into the apparatus after having first been accelerated to 2 MeV in a Van de Graaff generator. The accelerator tube is constructed so that the phase velocity of the wave is equal to \(c\). We therefore have here precisely the case of Fig. 9. The electrons are injected into the apparatus in all possible phases. Thus, drawing in Fig. 9 a horizontal line with an ordinate corresponding to an energy of 2 MeV, we obtain at the entrance end a uniform distribution of particles over phase. It is seen from Fig. 9 that somewhat more than half of all phases correspond to orbits on which the particles are bound to the wave, continuously gaining energy. A particularly strong concentration of electrons will occur near the orbit (marked by the letters \(a, b, c\)) which touches the horizontal line we have drawn. The asymptotic phase for this orbit is close to \(\dfrac{\pi}{2}\), and, consequently, the particles will acquire the maximum possible energy. Thus, for the given initial energy corresponding to the value of the accelerating field chosen by us, the majority of the electrons will undergo the maximum acceleration. Dr. Mazon investigated the probable energy spectrum of the emerging electrons and found that the energies of most of them would lie in a rather narrow interval. Let us note that if \(E\) were smaller, the optimum initial energy would correspondingly increase.

In a number of other projects, including those of the General Electric Company and Stanford, it is proposed to inject into the apparatus electrons with much smaller initial energies. Therefore, in part of the tube the velocity of the wave gradually increases until the electrons have been accelerated almost to the velocity of light. Such an injection method was also developed at M.T.I.

It follows from equation (23) that, at velocities much smaller than \(c\), and at voltages of ordinary magnitude, the angular frequency of the electron oscillations lies almost in the radio-frequency range. Furthermore, the acceleration takes place so rapidly that the assumptions of a small change in the relativistic mass during one period of oscillation, or of a slow variation of the parameters with time, are not very well justified. Nevertheless, it may be supposed that our conclusions, based on the adiabatic theorem, are not altogether incorrect. Suppose that we inject electrons

with different phases into the tube, the phase velocity of the wave in which varies from point to point. The phase space in this case is shown approximately in Fig. 11. Electrons with some initial phases will be captured by the wave, while those with others will not. It is not difficult to ensure the capture of at least half of all electrons. In fact, if the velocity of the electrons introduced into the device were exactly equal to the wave velocity (i.e., if \(p'\) were equal to zero), then all electrons would be captured. As the electrons are accelerated, the amplitudes of their oscillations, according to the adiabatic theorem, must decrease; therefore the electrons cluster more closely near the points at which the acceleration is just sufficient for the particles to move together with the wave. As \(v\) approaches the velocity of light, the groups are compressed more and more, and in the limit we obtain something very similar to the case already investigated in Fig. 9. Violation of the adiabatic theorem and of our other assumptions (as a result of rapid acceleration) will probably lead to the loss of some number of particles and to a smearing of the phase of the electrons captured by the wave moving with the velocity of light; nevertheless, one may expect that a considerable fraction of the electrons admitted into the device will acquire a large energy. This prediction agrees with the experience of work on accelerating electrons in such tubes.

The problem of accelerating positive ions is much more readily amenable to analysis, since here the acceleration will certainly be very slow, owing to the large rest mass of the particles; therefore the adiabatic theorem, as well as our other assumptions, is satisfied with a high degree of accuracy. Thus, let us suppose that we introduce ions into the device with a very small initial velocity, and let the phase space of the system be similar to that shown in Fig. 11. The frequency of the traveling wave considerably exceeds the frequency of oscillations of the ions, but nevertheless the change in the particle’s energy during a period of oscillation will be much smaller than in the case of electrons. Indeed, by a simple calculation it is easy to show that the energy acquired in one period by an ion located in the equilibrium position coincides, up to a numerical factor, with the change in kinetic energy according to formula (21), although the latter was calculated for another problem.

The periods of oscillation of an ion and of an electron are related to one another as the square root of the ratio of their masses; correspondingly, the distance traversed by the ion in the field, and the energy it acquires, are greater by the same factor. However, when expressed as a fraction of the total energy, the increase in the ion’s energy turns out to be smaller than for the electron by a factor equal to the square root of the inverse mass ratio.

In accelerating positive ions it may prove advantageous to introduce them into the device under a small voltage. The field can then be increased gradually, which, as we have seen, will lead to a closer

grouping of ions around equilibrium positions in phase space (i.e., near points where the acceleration is just sufficient for the particles to move together with the field). In this way practically all ions can be gathered into stable groups. They will be accelerated together with the wave, having at every point the velocity characteristic of the wave. As the latter approaches the velocity of light, the longitudinal mass will increase, as a result of which the groups will become narrower and narrower.

Finally, at very high energies the ions will enter that section of the waveguide where the phase velocity is equal to \(c\); the phase space will acquire the form shown in Fig. 9, and the groups will prove so tight that practically all ions will experience the maximum acceleration. This circumstance definitely argues in favor of injecting into the accelerator particles with comparatively small initial velocities, since then a very close grouping of particles occurs, in which practically all of them receive the maximum acceleration. In the MIT accelerator, where the electrons are immediately captured by the wave with velocity \(c\), this does not occur. It is possible that in the future, in improved accelerator designs, the grouping effect will be used. However, with such a method of introducing electrons, very considerable difficulties arise with focusing the beam. (They will be considered in the next section.) These difficulties are so great that in the MIT project it was deemed reasonable to avoid them by using a Van de Graaff generator as the injector.

XIV. TRANSVERSE MOTION AND FOCUSING OF PARTICLES

In the preceding section we studied the properties of the longitudinal motion of particles along the axis of the accelerator and found that particles with velocities less than the velocity of light have a tendency to form stable groups surrounding equilibrium points. The latter are characterized by the fact that the accelerating field in them grows with time, while the particles are accelerated precisely so as to move together with the field.

As the velocities of the wave and the particle tend toward the velocity of light, the frequency of oscillations of the particle about the equilibrium position decreases to zero, and in the limit \(v_0 = c\) the grouping effect disappears; however, the groups formed at lower velocities do not thereby break up. We shall now study the transverse motion of the particles and the focusing and defocusing effects accompanying it. In doing so we shall make the unpleasant discovery that a stable group is, by its very nature, unstable with respect to transverse motions, and that, consequently, the grouping effect leads to defocusing of the beam.

This could have been foreseen in advance on the basis of general considerations. The point is that any wave whose velocity is less than the velocity of light can be transformed to rest. In this,

in the new coordinate system the wave is a static solution of Maxwell’s equations; the particle, situated at the point of equilibrium, is at rest, while all the others oscillate about the equilibrium position. This Lorentz transformation could also have been used for the investigation of longitudinal motion; however, the Galilean transformation used in the preceding sections leads there to the same results, and therefore we made use of it as the simpler one. But once we have obtained a static problem, Earnshaw’s theorem comes into force, according to which the electrostatic potential in empty space attains neither a maximum nor a minimum, and can have only a saddle point. Consequently, if the equilibrium is stable with respect to motion in some one direction, it must be unstable with respect to motion in a direction perpendicular to the first. Since in our case groups of particles are stable with respect to longitudinal motions, stable focusing cannot take place. Conversely, if the phases are chosen so that the beam is focused, then the groups of particles will prove unstable; this corresponds precisely to the position of unstable equilibrium of which we spoke in connection with longitudinal motion. On the other hand, at a velocity close to \(c\), the grouping effect disappears and is replaced by indifferent equilibrium. In our previous reasoning this followed from the fact that in the limit, when \(v_0=c\), the frequency of the oscillations tended to zero. Now, after we have performed the Lorentz transformation, this is matched by the circumstance that the wavelength in the waveguide tends to infinity as the velocity increases without bound, i.e., the accelerating force ceases to depend on the coordinates, and the field becomes constant in any finite region. Correspondingly, the defocusing effect also disappears when \(v_0=c\), and we have indifferent equilibrium also with respect to transverse motions.

All these assertions could be proved by relativistic methods, but it is much simpler to verify them by a direct and elementary calculation. Let us write down the formulas for the electric and magnetic fields of a traveling wave resonant with the particle, as they appear in the ordinary (resting) coordinate system. We have, as is easy to show:

\[ E_z=E\sin\omega\left(t-\frac{z}{v_0}\right)J_0(x), \]

\[ E_r=E\cos\omega\left(t-\frac{z}{v_0}\right) \frac{-jJ_1(x)}{\sqrt{1-\dfrac{v_0^2}{c^2}}}, \]

\[ B_\theta=E\cos\omega\left(t-\frac{z}{v_0}\right) \frac{v_0}{c^2}\, \frac{-jJ_1(x)}{\sqrt{1-\dfrac{v_0^2}{c^2}}}, \]

where

\[ x=-\frac{j\omega r}{v_0\left(1-\frac{v_0^2}{c^2}\right)^{1/2}} . \]

Let the particle move parallel to the \(z\)-axis with velocity \(v_0\) (here we neglect the small difference between the velocities of the particle and of the wave, which arises as a result of the oscillatory motion). The radial component of the force acting on the particle is equal to \(e(E_r-v_0B_\theta)\) (this is the \(r\)-component of the expression \(e(\mathbf E+[\mathbf v\mathbf B])\)), i.e.

\[ F_r=eE\cos\omega\left(t-\frac{z}{v_0}\right) \left(1-\frac{v_0^2}{c^2}\right)^{1/2}\{-jJ_1(x)\}. \]

In other words, the \(r\)-components of the forces with which the electric and magnetic fields act on the particle differ from one another by the factor \(\frac{v_0^2}{c^2}\); therefore, in the limit, when \(v_0\) approaches \(c\), the force acting perpendicular to the accelerator axis tends to zero, and the focusing and defocusing effects disappear.

Let us now apply these results to the investigation of the force acting on a particle for small deviations of the latter from the axis of the instrument. For small values of the argument \(x\), the Bessel functions \(J_0(x)\) and \(J_1(x)\) tend, respectively, to \(1\) and to \(\frac{x}{2}\). Thus, for small \(r\) we have:

\[ F_z=eE\sin\omega\left(t-\frac{z}{v_0}\right), \]

\[ F_r=\frac{\omega r}{2v_0}\left(1-\frac{v_0^2}{c^2}\right)eE\cos\omega\left(t-\frac{z}{v_0}\right). \tag{24} \]

We are interested mainly in the value of \(F_r\) in that phase in which the formation of stable groups of particles is possible. The latter takes place when \(F_z\) is positive and increases with time, i.e. when both \(\sin\omega\left(t-\frac{z}{v_0}\right)\) and the time derivative of this function, \(\omega\cos\omega\left(t-\frac{z}{v_0}\right)\), are positive.

It is easy to see that in this case \(F_r\) is positive and proportional to \(r\), i.e. it deflects the particles from the axis of the instrument, thereby exerting a defocusing action.

The coordinate of a particle subjected to the action of a repulsive force proportional to the distance from a given point depends exponentially on time. Writing the equation of motion of the particle under the action of the force \(F_r\) and taking into account that in this case it is necessary to use the transverse mass

\[ \frac{m_0}{\left(1-\frac{v_0^2}{c^2}\right)^{1/2}}, \]

we shall find the law of motion-

in the following form:

\[ r = r_0 e^{t/T}; \qquad \frac{1}{T} = \left[ \frac{e E_\omega}{m_0 v_0} \left(1-\frac{v_0^2}{c^2}\right)^{3/4} \cos \omega t_0 \right]^{1/2}, \tag{45} \]

where \(t_0=t-\dfrac{z}{v_0}\) (the factors \(-\dfrac{m}{\left(1-\dfrac{v_0^2}{c^2}\right)^{1/4}}\) and \(1-\dfrac{v_0^2}{c^2}\), combining, give the longitudinal mass). As is seen from (23), \(1/T\), to within the factor \(1/\sqrt{2}\), coincides with the angular frequency of oscillations about the equilibrium position. Hence we conclude that, in a time of the order of one period of oscillations about the equilibrium position, the distance of the particle from the axis of the apparatus \(r\) increases by a factor of \(e\). For the MTI accelerator this does not constitute a problem, since the period of oscillations there is practically infinite; a direct analysis of the transverse motion shows that when electrons with an initial energy of \(2\) MeV are accelerated, reaching any arbitrarily high energy, their distance from the axis increases at most by a factor of two. For comparatively small initial velocities of the electrons the problem of defocusing is also not too important. We have seen that, at those large accelerations which we have here, the electrons, already after a comparatively small number of oscillations in the bunch, reach the speed of light; consequently, although their distance from the axis increases, it does not increase much. This should not lead to an inadmissible broadening of the beam, if initially it was sufficiently narrow and well focused. Further, in the initial stage of defocusing and acceleration the electrons can easily be refocused by a longitudinal magnetic field (we shall discuss this shortly).

At the same time, for positive ions the defocusing effect proves to be exceptionally serious. It is probable that it fatally impedes the application of long ion accelerators in the energy region of about a billion eV. We have seen that a positive ion must make many oscillations about the equilibrium position before it approximately reaches the speed of light; during this time the radius of the beam will increase many times by a factor of \(e\), i.e. the beam will broaden inadmissibly. True, it should be noted that we have overlooked one focusing effect which may perhaps improve the situation somewhat. Its origin can be understood from the theory of the cyclotron. There the ions, accelerating in the gap between the dees, move in a field of the kind shown in Fig. 12. Entering this gap, the ions are focused; leaving it, they are defocused. The defocusing may be either weaker or stronger than the focusing. This is determined by two reasons. First, the field may increase with time while the ions

pass between the duants; then at the exit it will be stronger than at the entrance, and as a result defocusing of the beam will occur. It is precisely this effect that we have been considering up to now. But there is also another circumstance: in passing between the duants, the ions are accelerated. Consequently, at the exit they will be moving faster than at the entrance, and the focusing action may prove to be more effective. This phenomenon has fallen outside our consideration, since we neglected second-order terms in the change of the particle velocity.

Fig. 12. Field lines in the gap.

Thanks to this effect there exists a small region of phases in which both focusing and the formation of stable bunches are possible. This is the region near the maximum of the accelerating field, where $\cos \omega t_0'$ is almost equal to zero. The angular frequency of the oscillations, determined by formula (23), is still real here, although small, so that the bunches are still stable. At the same time the quantity $\frac{1}{T}$ from equation (25) is very small, and defocusing is almost absent. It may prove so weak that it will be masked by the other effect just described, and as a result the beam will be focused. In the English works in which these phenomena are analyzed, it is shown that there exists a small (several degrees) phase region in which the situation is precisely of this kind. Under some conditions, when the field is concentrated in very narrow gaps, the favorable phase may turn out to be rather large. It is probably to this circumstance that the first designs of linear accelerators owe their success. It is possible that, in the final analysis, it will also determine the success of ion linear accelerators.

The Berkeley group proposed another method of combating defocusing of the beam in an ion accelerator, consisting in the use of grids. It is clear that, by placing a grid at the exit into the gap between the duants, as shown in Fig. 13, one can completely eliminate the defocusing action of the field while preserving focusing. This does not in any way contradict our previous arguments concerning the inevitability of defocusing. In fact, we proceeded from an expansion of the field in a Fourier series under the assumption that the charge density throughout the region of interest to us is zero. But grids can carry charge, and in this case the potential will obey Poisson’s equation, not Laplace’s. Then it may have an absolute minimum, which provides the possibility of focusing simultaneously with the existence—

Fig. 13. Field lines in the gap with a focusing grid or foil.

by the setting of stable groups. The obvious difficulty consists in the fact that the mesh obstructs the motion of the beam.

For a short accelerator this circumstance is not essential, but for a long one it may prove fatal, since there the beam will have to pass through a whole series of meshes.

We have already mentioned that, for focusing comparatively slow electrons, one may use a longitudinal magnetic field, which will transform the electron orbits into spirals. Let us consider the influence of such a field. The problem is completely solved if one writes the Lagrange equations for a particle acted upon by the already known forces \(F_r\) and \(F_z\), and also by the forces exerted by a constant magnetic field \(B\) directed along the \(z\)-axis. It turns out that the influence of the field \(B\) on the radial motion of the particle is equivalent to the appearance of an additional potential energy proportional to \(r^2\), or, what is the same thing, of an additional attractive force proportional to \(r\). If this term is sufficiently large to balance the repulsive force \(F_r\) (which is also proportional to the distance), then stable orbits may appear—stable spirals along which the electrons move. Instead of carrying through this entire calculation, let us give a simple argument that will enable us to obtain the critical value of the magnetic field necessary for focusing.

In the absence of the force \(F_r\), a particle under the action of the magnetic field will move in a circle in a plane perpendicular to the direction of the field. As usual, we equate the force pulling the particle toward the axis of rotation to the transverse mass of the particle multiplied by the centripetal acceleration. Denoting the radius of the circle and the angular velocity by \(r\) and \(\dot{\theta}\), we have: \(eBr\dot{\theta}=m_t r\dot{\theta}^{\,2}\), whence \(\dot{\theta}=\dfrac{eB}{m_t}\), i.e., the angular velocity is equal to the Larmor frequency. On the other hand, if we also include the repulsive force \(kr\) (our force \(F_r\)), we obtain: \(eBr\dot{\theta}-kr=m_t\dot{\theta}^{\,2}r\). The solutions of this quadratic equation with respect to \(\dot{\theta}\) will be:

\[ \dot{\theta}=\frac{eB}{2m_t}\pm \sqrt{\left(\frac{eB}{2m_t}\right)^2-\frac{k}{m_t}}. \]

They represent stable circular rotation only if \(\dot{\theta}\) is real. This gives us the limiting value of \(B\):

\[ \frac{eB}{m_t}=\left(\frac{4k}{m_t}\right)^{1/2}. \]

Taking the value of \(k\) from formula (24) for \(F_r\) and making use of (23), we obtain:

\[ \frac{\gamma eB}{m_t}=\sqrt{2}\,\omega_0, \tag{26} \]

i.e., in order to compensate the defocusing action of the wave, it is necessary to turn on a longitudinal magnetic field so strong that the corresponding Larmor frequency is at least \(\sqrt{2}\) times greater than the frequency of oscillation of the electrons in the bunches.

We then obtain an indifferent equilibrium, which will pass into a stable one for any arbitrarily small increase of the magnetic field.

Now we can see what the field strength must be. We have seen that, in the case of electrons, the quantity \(\omega_0\), for large energies, can approach the radio frequency, although usually it is appreciably smaller than the latter. Consequently, the Larmor frequency must also be of the same order.

For those radio frequencies with which we are concerned here, this leads to fields of the order of several hundred, or in the extreme case several thousand, gauss. These values are readily attainable in practice, especially considering that the magnetic field need be produced only in that part of the accelerator where the velocity of the electrons is appreciably less than \(c\). On the other hand, for ions the necessary magnetic fields become inadmissibly large.

It is true that the frequency \(\omega_0\) in this case decreases by

\[ \sqrt{\frac{m_{\text{ion}}}{m_{\text{electron}}}} \]

times, but the additional factor \(m_i\) in equation (26) leads to the result that the magnetic field must increase by the same factor. Thus, in order to compensate the defocusing action of the wave, a magnetic field is required with a strength not of a thousand, but of forty to fifty thousand gauss. Moreover, it must be applied in all sections of the accelerator where the velocity of the ions is appreciably less than \(c\). In small volumes such a field can, of course, be obtained, but it is exceptionally difficult to produce it throughout an entire long accelerator.

Finally, one more problem concerning the focusing of electrons must be considered. Although particles moving with the velocity of light are not acted upon by defocusing forces, nevertheless in an accelerator of the MTI type the electrons of the primary beam will have a certain spread in directions. It is natural to ask whether this beam must be so narrow that all the electrons are aimed at the aperture of the last diaphragm at the far end of the tube.

If so, this would imply a very small, practically almost unattainable opening angle of the primary beam. Fortunately, there is no need for such strong concentration of the beam. To see this, let us consider the problem in the following very simple way. The momentum of a primary electron moving not strictly parallel to the axis of the instrument may be resolved into two components—the component parallel to the axis, \(p_z\), and the component perpendicular to it (directed along the radius), \(p_r\).

In the limiting case \(v_0 = c\), which we are now considering, the radial forces are absent and \(p_r\) remains constant. At the same time \(p_z\) increases continuously, since along the \(z\)-axis there acts

constant accelerating field. Therefore the momentum vector (and the velocity vector parallel to it) will rotate all the time toward the direction of the \(z\)-axis.

The situation is the same as in the problem of the fall of a body thrown parallel to the horizon. It is known that such a body will move along a parabola, and the distance it travels during the time of fall is equal to the product of the time of fall by the initial velocity. However, in our case, owing to the relativistic character of the motion, constancy of the velocity by no means follows from constancy of the momentum; in fact, we shall see that the radial velocity decreases as the particle advances, so that the distance traversed perpendicular to the \(z\)-axis proves to be much smaller than in the nonrelativistic case.

The solution of our problem is elementary. Let us measure the coordinate \(z\) from a certain point at which the energy, by assumption, would be equal to zero if the accelerating force were constant along the entire path of the particle. Then the energy at the point \(z\) will be equal to \(eEz\), and the \(z\)-component of the momentum (in the relativistic region) to \(\dfrac{eEz}{c}\); \(p_r=\mathrm{const}\). The true trajectory of the particle at each point is parallel to its momentum. This gives:

\[ \frac{dr}{dz}=\frac{p_r}{p_z}=\frac{p_r c}{eE}. \]

Since \(dz=c\,dt'\) (in the region where the particle velocity \(=c\)), it follows from this that the radial component of the velocity decreases inversely proportionally to \(z\).

Integrating the equation written above, we obtain:

\[ r_2-r_1=\frac{p_r c}{eE}\ln\frac{z_2}{z_1}. \]

This gives the change of the radial coordinate along the path from the point \(z_1\), where the particle entered the device, to \(z_2\), where it leaves it. It is convenient to express it through the angle \(\varphi_0\) between the axis of the tube and the initial direction of motion of the electron. Obviously, \(\varphi_0=\dfrac{p_r}{p_z}\) for \(z=z_1\). This gives:

\[ r_2-r_1=\varphi_0 z_1\ln\frac{z_2}{z_1}. \]

Substituting numerical values of the quantities entering here, we see that the broadening of the beam is very small even in a very long accelerator. Suppose, for example, that the initial energy of the electrons is \(2\,\mathrm{MeV}\), and let this correspond to \(z_1=60\ \mathrm{cm}\). If we wish to accelerate the electrons to \(2\) billion eV, we must set \(\dfrac{z_2}{z_1}=10^3\).

Let us take \(\varphi_0=10^{-3}\). This means that the beam will pass completely through a millimeter aperture at a distance of one meter

from the source; for a well-collimated beam from a Van de Graaff generator this should be possible. And we have:

\[ r_2-r_1=4\ \mathrm{mm}. \]

Even for a ten-times larger accelerator the displacement increases only by a factor of \(4/3\), i.e., it turns out to be equal to \(5.3\ \mathrm{mm}\). Thus, one may expect that it will always be possible, for any accelerator, to collimate the beam so that its width does not exceed the diameter of the holes in the diaphragms.

CITED LITERATURE

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