On the question of electron bunching in a synchrotron
Abstract Generated abstract
This paper examines electron bunching in a synchrotron by analogy with microwave electron devices, focusing on radial phase oscillations while neglecting betatron effects and interelectron repulsion. In a kinematic approximation, the phase motion is reformulated from an elliptic integral into canonical Weierstrass form, yielding an expression for the electron phase as a function of injection phase and time, along with the oscillation period. Using charge conservation, the author derives the instantaneous current distribution for an initially rectangular injection pulse and shows that nonlinear phase oscillations produce a nonstationary phase distribution with formal current singularities at caustic phases. The analysis argues that these singularities would be regularized by Coulomb forces, but that the kinematic treatment remains useful for estimating the fundamental current harmonic.
Full Text
On the Question of Electron Bunching in a Synchrotron
A. S. Chumakov
The question of electron bunching in a synchrotron is, in a certain sense, analogous to that encountered in the theory of microwave electronic devices. Common to these two types of electronic devices is the process of producing oscillations of the electron energy and, as a consequence, a change in the distribution function of the electrons with respect to the phases of the rf voltage. Electron bunching in a synchrotron is closely connected with radial-phase and betatron oscillations. However, the action of these kinds of oscillations on the distribution function is not the same. Betatron oscillations are not associated with a change in the energy of the particles, and the phase change caused by them in ordinary accelerators is very small. Therefore, in investigating bunching one may confine oneself to considering only radial-phase oscillations.
For the analysis of the process of electron bunching in a synchrotron it is expedient to use the methods developed in the theory of the klystron, the traveling-wave tube, etc. In particular, to solve the problem posed it is necessary to solve the equations of motion of the electrons and, using the law of conservation of charge, from the known initial distribution to find the distribution function of the electrons at any instant of time. In the present work this question is considered in the kinematic approximation, i.e., the action of the repulsive forces between electrons is neglected.
In work [1] a formula was obtained for the phase of any electron at any instant of time, expressed in terms of Jacobi elliptic functions. However, for our purposes it is more convenient to express the electron phase in terms of Weierstrass elliptic functions. For this it is necessary, in the expression for the phase velocity [2],
\[ \dot{\varphi} = \omega_0 \sqrt{ \frac{e V_0 K F q}{\pi E} \left[ \left(\sin \varphi-\varphi \cos \varphi_0\right) - \left(\sin \varphi_{\mathrm{in}}-\varphi_{\mathrm{in}}\cos \varphi_0\right) \right] } \]
to expand \(\sin \varphi\) in a Taylor series in the deviation from the equilibrium phase \(x=\varphi-\varphi_0\) and to restrict oneself to five terms of the expansion. Then the second integral of the phase equation reduces to an elliptic one:
\[ \int_0^t \sqrt{ \frac{e V_0 K F q \sin \varphi_0}{24 \pi E} } \cdot \omega_0\, dt = \int_{a_{\mathrm{in}}}^{a} \frac{da}{\sqrt{P(a)-P(a_{\mathrm{in}})}} \, , \]
where
\[ P(a)=a^4-4a^3\operatorname{ctg}\varphi_0-12a^2+24(1-\varphi_0\operatorname{ctg}\varphi_0), \]
\[ P(a_{\mathrm{in}})=a_{\mathrm{in}}^4-4a_{\mathrm{in}}^3\operatorname{ctg}\varphi_0-12a_{\mathrm{in}}^2+24(1-\varphi_0\operatorname{ctg}\varphi_0). \]
By means of the fractional-linear transformation [3]
\[ a=a_{\mathrm{in}}+\frac{P'(a_{\mathrm{in}})} {4\left\{z-\frac{1}{24}P''(a_{\mathrm{in}})\right\}} \]
(\(P'(a_{\mathrm{in}})\), \(P''(a_{\mathrm{in}})\) are the derivatives of \(P(a)\) at \(a=a_{\mathrm{in}}\)), the last integral is reduced to the canonical Weierstrass form
\[ \int_0^t \sqrt{\frac{eV_0KF\sin\varphi_0}{24\pi E}}\cdot \omega_0\,dt = \int_z^\infty \frac{dz}{\sqrt{4z^3-g_2z-g_3}}; \tag{1} \]
inversion of the elliptic integral (1)
\[ z=\gamma(u);\qquad u=\sqrt{\frac{eV_0KFq\sin\varphi_0}{24\pi E}}\cdot \omega_0 t \]
allows us to write the phase of the electron in the form
\[ a=a_{\mathrm{in}}+ \frac{P'(a_{\mathrm{in}})} {4\left\{\gamma(u)-\frac{1}{24}P''(a_{\mathrm{in}})\right\}}, \tag{2} \]
where \(\gamma(u)\) is the Weierstrass function, whose invariants are equal to
\[ g_2=12-P(a_{\mathrm{in}}); \]
\[ g_3=P(a_{\mathrm{in}})\cdot(2+\operatorname{ctg}^2\varphi_0)+8. \]
The period of the function \(\gamma(u)\) and the period of the phase oscillations \(T\), respectively, are equal to
\[ 2\omega_1=\frac{2K}{\sqrt{e_1-e_3}};\qquad T=\frac{2\omega_1} {\sqrt{\dfrac{eV_0KFq\sin\varphi_0}{24\pi E}}\cdot\omega_0}, \]
where \(K\) is the complete elliptic integral of the first kind, calculated for the modulus
\[ k^2=\frac{e_2-e_3}{e_1-e_3}. \]
In these expressions \(e_1,e_2,e_3\) are the roots of the equation \(4z^3-g_2z-g_3\), arranged in decreasing order. In Fig. 1, plots are shown of the dependence of the electron phase on the input phase at different instants of time for the case
\[ \varphi_0=60^\circ \quad\text{and}\quad \sqrt{\frac{eV_0KFq\sin\varphi_0}{24\pi E}}\cdot\omega_0=10. \]
It is evident from the plots that, in the general case, \(a_{\mathrm{in}}\) is a multivalued function of \(a\).
Taking the input current to be constant, \(i_{\mathrm{in}}\), on the basis of the charge-conservation principle one may write that, in the phase interval \(\Delta z\), the electrons carry the charge
\[ i_{\mathrm{in}}\cdot \sum_{\alpha_{\mathrm{in}}(z)} \Delta z_{\mathrm{in}} = i_{\mathrm{in}}\cdot \Delta z \cdot \sum_{\alpha_{\mathrm{in}}(\alpha)} \left|\frac{d\alpha_{\mathrm{in}}}{dz}\right|, \]
where the summation is carried out over all phase intervals corresponding to one and the same fixed phase \(\alpha\). Since the total charge is \(i\Delta\alpha\), for the instantaneous value of the current \(i\) at any time one may write the expression [4]
Fig. 1. Dependence of the electron phase on the input phase at various instants of time. Curve \(a\) is for \(t=1\cdot 10^{-7}\) sec; \(b\) for \(t=2\cdot 10^{-7}\) sec; \(c\) for \(t=3\cdot 10^{-7}\) sec; \(d\) for \(t=4\cdot 10^{-7}\) sec.
\[ i=i_{\mathrm{in}}\cdot \sum_{\alpha_{\mathrm{in}}(\alpha)} \left|\frac{d\alpha_{\mathrm{in}}}{dz}\right|. \tag{3} \]
Equation (3) says that at any instant \(t\) the current \(i\) is expressed in terms of the input current \(i_{\mathrm{in}}\) and the sum of the absolute values of the derivatives \(\dfrac{d\alpha_{\mathrm{in}}}{d\alpha}\) corresponding to the given value of \(z\). The derivatives \(\dfrac{d\alpha_{\mathrm{in}}}{d\alpha}\) can be determined graphically with the aid of Fig. 1.
Suppose that the electrons are injected into the synchrotron by a single rectangular pulse of duration corresponding to the region of capture into acceleration. Then, with the aid of equation (3), by summing the derivatives \(\dfrac{d\alpha_{\mathrm{in}}}{d\alpha}\) corresponding to a given \(z\), one can calculate the dependence of the instantaneous values of the current on the phase of the rf voltage at various instants of time. Such graphs for the above-mentioned case are shown in Fig. 2.
Analyzing the results obtained, it should be noted that, owing to the nonlinearity of the period of phase oscillations, the picture of the distribution of electrons over the phases of the RF voltage is nonstationary. Infinite peaks in the current pulses correspond to phases where
\[ \frac{d\alpha}{d\alpha_{\text{RF}}}=0. \]
The equality \(\frac{d\alpha}{d\alpha_{\text{RF}}}=0\) means that electrons emitted in an interval of phases \(d\alpha_{\text{RF}}\), at some instant of time, fall into an interval of phases \(d\alpha\) that is infinitesimally small, of higher order, and
Fig. 2. Current waveform at different moments of time.
therefore infinitely large instantaneous values of the current, carrying a charge of finite magnitude, are obtained. The presence of infinite peaks in the current pulses is explained by the fact that we have not taken into account the repulsive forces between electrons. In reality, as a result of the Coulomb interaction, the charge density everywhere retains a finite value. However, the infinite current peaks begin to play an important role when higher harmonics are considered. The contribution of these peaks to the fundamental harmonic of the current, which is of greatest interest, is insignificant. Therefore, in problems of determining the fundamental harmonic of the current, the kinematic theory of electron bunching may be used.
REFERENCES
- A. S. Chumakov, Izv. VUZov, Fizika, in press.
- M. S. Rabinovich, Proceedings of FIAN, X, 1958.
- N. I. Akhiezer, Elements of the Theory of Elliptic Functions, Gostekhizdat, 1948.
- Klystrons, edited by E. D. Naumenko, Sov. Radio Publishing House, 1952.