Study of the Longitudinal Stability of an Electron Beam in a Betatron

O. V. Sokolov, B. L. Chastyulenko

(Presented by the scientific seminar of the Research Institute of Electron Introscopy)

It is known that during the injection of electrons into the accelerating chamber of a betatron, conditions are created for the short-term existence of the so-called nonstationary current. In works [1—4] a mechanism was proposed for the formation of a nonstationary current, based on the resonant interaction of circulating electrons with an inhomogeneity in the azimuthal distribution of the space-charge density created by the emission current from the injector. Experimental verification confirms the resonant mechanism of formation of the nonstationary current [3, 4]. The loss of electrons of the nonstationary current is also explained by the resonant mechanism. However, the influence of other factors on the decay of the nonstationary current is not excluded, since the nonstationary current is quite large in magnitude. In the present work the longitudinal stability of an electron beam of nonstationary current is investigated.

The finite resistance of the walls of the accelerating chamber, under certain conditions, can lead to self-bunching of a continuous circular current into bunches, i.e., to a violation of the longitudinal stability of the beam [6—13]. It is known that in betatrons (with the exception of installations with a constant guiding field and a variable gradient) this effect manifests itself in the so-called “negative mass” effect. The essence of this effect is that, in the presence of some local region with increased charge density, the electrons located ahead of this region experience the action of an additional accelerating force and acquire greater energy than the others. And since in a weak-focusing accelerator the revolution frequency of the particles decreases with increasing energy [5, 6], particles with greater energy move radially outward, lose azimuthal velocity, and fall back toward the center of the bunch. In a similar manner, particles at the other end of the bunch move radially inward, increase their azimuthal velocity, and move forward toward the center of the bunch. A beam broken up into bunches generates electromagnetic waves with frequency

\[ \omega = m\omega_0, \tag{1} \]

where \(m\) is the number of bunches in the beam.

A factor stabilizing the beam in the longitudinal direction is the presence of a sufficiently large energy spread in the beam. The energy spread in a beam of electrons forming a nonstationary current can be determined according to [5]

\[ \frac{\Delta T}{T}=\frac{\Delta r_0 \beta^2(1-n)}{r_0}, \tag{2} \]

where \(\Delta r_0\) is the radial size of the beam (cm);
\(n\) is the magnetic-field falloff index;
\(r_0\) is the radius of the equilibrium orbit (cm);

\[ \beta=\frac{v}{c}=\sqrt{1-\left(\frac{E_0}{E}\right)^2} \]

is the relativistic factor.

Let us estimate the danger posed by longitudinal oscillations in a beam of electrons with nonstationary current. We introduce the following notation:

\(N_1\) is the limiting (equilibrium) number of electrons at which the defocusing forces of the space charge are in equilibrium with the focusing forces of the magnetic field.

\(N_2\) is the smallest number of electrons at which longitudinal oscillations begin to arise; however, these oscillations do not lead to the loss of circulating electrons.

\(N_3\) is the smallest number of electrons at which intense loss of electrons on the chamber walls begins owing to the development of longitudinal instability.

It appears possible to estimate, approximately, the values of \(N_1\), \(N_2\), and \(N_3\) [4, 14–15]:

\[ N_1=2.18\cdot 10^{13}\cdot \frac{a^2}{r_0}\cdot E\cdot \frac{\beta^2}{1-\beta^2}\,n; \tag{3} \]

\[ N_2=\pi\cdot \frac{r_0}{r_e}\cdot \frac{T}{E_0}\cdot \frac{n}{1-n}\left(\frac{\Delta T}{T}\right)^2, \tag{4} \]

\[ N_3=\pi a^2\cdot \frac{T}{E_0}\cdot \frac{n(1-n)}{r_0 r_e}\left(1-\frac{x_m}{a}\right)^2; \tag{5} \]

where \(a\) is the radius of the electron beam of equivalent circular cross section (cm);

\(\alpha\) is the radius of the equivalent circular cross section of the accelerating chamber (cm);

\(E_0+T=E\) are, respectively, the electron rest energy, kinetic energy, and total energy (MeV);

\(r_e=2.82\cdot 10^{-13}\) is the classical electron radius (cm);

\(x_m\) is the amplitude of radial betatron oscillations (cm).

The growth rate (growth constant) of the longitudinal instability can be estimated by the expression [9]

\[ \tau \simeq \frac{4\pi}{\beta\omega^3 n} \left( \frac{\delta a^2 \lambda}{r_e r_0 |K| \gamma N} \right)^{1/2}, \tag{6} \]

where \(\omega=2\pi f_0\) is the angular frequency of revolution of the electrons in an unperturbed beam (1/s);

\(\delta\) is the conductivity of the inner coating of the accelerating chamber (1/s);

\(\lambda\) is the ratio of the radius of the beam of equivalent circular cross section to the radius of the equivalent circular cross section of the accelerating chamber;

\(K\) is a coefficient; for a betatron, \(K=1/\gamma\) [4];

\(N\) is the number of particles in the beam.

The values of \(N_1\), \(N_2\), and \(N_3\) for various injection voltages \((U_i)\), calculated for a 15-MeV betatron, are presented in Fig. 1. Also shown there are the values of \(N_{\mathrm{nt}}\) (the number of electrons, correspond-

corresponding to the nonstationary current under optimal capture) and \(N_{\mathrm{capt}}\) (the number of electrons captured into acceleration) [3, 4], measured on the indicated betatron, which has the following parameters:

\[ r_0=13\ \text{cm};\quad n=0.63;\quad \Delta r_0=2.7\ [4];\quad \lambda=0.755;\quad \alpha=2.83\ \text{cm}; \]
\[ \sigma \approx 1.3\cdot 10^{15}\ \text{s}^{-1} \quad \text{(conducting layer of the chamber—Aquadag)};\quad a=2.14\ \text{cm}. \]

It is seen from Fig. 1 that, at injection voltages up to 40 kV, longitudinal oscillations exist not only during injection, when a nonstationary current circulates, but also for some time after the end of the capture process, since \(N_2<N_{\mathrm{capt}}<N_{\mathrm{nt}}\). This result, following from the calculations performed, agrees well with the results of the experimental work [8]. In that work it is noted that high-frequency oscillations, indicating bunching of the electrons into clumps, are observed far beyond the trailing edge of the injection pulse (up to hundreds of microseconds), and their intensity is 10–30% of the intensity during injection. Since \(N_{\mathrm{capt}}<N_3\) (Fig. 1), the longitudinal instability should not cause losses of circulating electrons after the end of injection. During the existence of the nonstationary current, although the condition for intense electron loss due to longitudinal instability is satisfied (\(N_{\mathrm{nt}}>N_3\)), apparently it does not have time to develop because of the large time constant of its development \((\tau)\). The value of \(\tau\), according to (6), is, for a 15 MeV betatron at injected-electron energies of 20 and 100 keV, 826 and 44 μs, respectively, whereas the lifetime of the nonstationary current does not exceed fractions of a microsecond [3, 4]. It is also seen from Fig. 1 that, at an injection voltage greater than \(\sim 90\) kV, the electron beam becomes stable in the longitudinal direction (since \(N_3\) becomes smaller than \(N_2\)).

In works [8, 11–13], high-frequency electromagnetic oscillations accompanying the injection process were detected, which indi-

Fig. 1. Dependence of threshold values of the numbers of electrons on the injection voltage for longitudinal oscillations.

Fig. 2. Dependence of the nonstationary current, the intensity of the bremsstrahlung radiation, and the square of the amplitude of h.f. oscillations on the emission current from the injector.

Table 1

indicate a violation of the longitudinal stability of the electron beam. For each injection voltage, a discrete set of frequencies was found that is in a definite relation to the electron revolution frequency. The frequency of each type of oscillation is proportional to the square root of the injection voltage. The rf oscillations arise at a quite definite threshold emission current from the injector; their amplitude increases with increasing emission current.

Figure 2 presents, as functions of the emission current from the injector \((I_{\rm em})\), the values of the nonstationary current \((I_{\rm n})\) [3, 4], the intensity of bremsstrahlung X-radiation \((I_\gamma)\), and the square of the amplitude of the rf oscillations \((A^2)\) [11]. It is seen from Fig. 2 that the dependences \(I_{\rm n}\) and \(A^2\) are in many respects analogous. The rf oscillations \((A^2)\) and the nonstationary current \((I_{\rm n})\) arise at one and the same threshold emission current, at which capture \((I_\gamma)\) begins. The values of \(I_{\rm n}\) and \(A^2\) increase with increasing \(I_{\rm em}\), tending toward saturation. At the optimum emission current (when \(I_\gamma\) is maximal), \(I_{\rm n}\) and \(A^2\) amount to approximately one half of their maximum value. All this indicates that the source of the rf oscillations is the nonstationary current.

Table 1 gives the values of \(r_0\), \(\alpha\), and also

\[ \frac{\Delta T}{T}, \quad N_1,\; N_2 \text{ and } N_3 \]

for two injection voltages, 20 kV (a) and 100 kV (b), calculated from formulas (2—5), for betatrons designed by the Scientific Research Institute of Electron Introscopy: B-15, B-18.5, B-25/10, and B-35/8. Also indicated there is the value of the limiting injection voltage \((U_i^*)\), above which the beam becomes stable in the longitudinal direction. Comparison of the data given in Table 1 and Fig. 1 shows that the character of the dependences \(N_1\), \(N_2\), and \(N_3\) is analogous for all betatrons. Longitudinal oscillations arise most readily in the B-18.5 and B-25/10 betatrons, which have the smallest chamber aperture and the smallest energy spread in the beam.

Comparing the results of the estimate made for the possibility of a violation of the longitudinal stability of the beam in various betatrons with the results of studies of the nonstationary current [3, 4] and of rf oscillations during injection [8, 11—13], the following conclusions may be drawn.

1. The source of intense rf oscillations during injection is the nonstationary current existing at that time, which reaches a considerable magnitude.

2. The comparatively small energy spread in the beam and the low injection voltage in ordinary (not high-current) betatrons lead to a violation, during injection and in the initial phase of acceleration, of the longitudinal stability of the beam, which leads to rf oscillations. In the betatrons considered, with energies of 15, 18.5, 25, and 35 MeV, no violation of the longitudinal stability of the beam occurs if the energy of the injected electrons exceeds, respectively, 90, 110, 300, and 70 keV.

3. Longitudinal instability cannot cause any substantial loss of electrons either during the existence of the nonstationary current (because of the long development time \(\tau\)) or in the initial stage of acceleration (since \(N_{\rm capt} \ll N_3\)); i.e., in the longitudinal direction the electron beam in an ordinary (not high-current) betatron is stable.

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