I. L. Zelmanov, A. S. Kompaneets, and Yu. S. Sayasov

On the Phase Motion of Particles in Accelerators with Fluctuating Parameters

(Presented by Academician V. N. Kondrat’ev, 16 X 1961)

The equation of phase oscillations of a particle in accelerators whose parameters undergo small random perturbations can, in most cases, be written, restricting oneself to the linear approximation, in the following form (see, for example, ((^{1}))):

[
\frac{d^2 q}{dt^2}+\omega^2(t)q=a(t)\psi(t),
\tag{1}
]

where (q) is the distance between the particle under consideration and the “ideal” synchronous particle at the moment when the latter passes through the middle of the accelerating gap; (\psi) is a random quantity characterizing the deviation of the phase of the synchronous particle from the “ideal” one as a result of perturbations of various types of parameters (for example, the lengths of the accelerating gaps, the voltage and frequency of the accelerating field, etc.); (\omega^2(t)), (a(t)) are known slowly varying functions of time, expressed in dimensionless units (t/T) ((T) is the period of the high-frequency oscillations of the accelerating field).

Let us introduce, along with (1), the equation

[
\frac{d^2 q_0}{dt}+\omega^2(t)q_0=0,
\tag{2}
]

which describes the phase oscillations of a particle in an “ideal” accelerator, and the quantities (Q=q-q_0), (\dot Q=\dot q-\dot q_0), characterizing the deviations of the position and velocity of the particle from the “ideal” ones.

The problem arises of finding the probability density (\Phi(t,Q,\dot Q)). If one assumes that (\overline{\psi(t)}=0), (\overline{\psi(t)\psi(t')}=\psi_0^2\delta(t-t')) (i.e., that the perturbations (\psi(t)) at different instants of time are statistically independent, and their mean-square value is the same for all (t) and equal to (\psi_0^2)) and that the solutions of equations (1), (2) can be found (in view of the slowness of variation of the functions (\omega^2(t)), (a(t))) in the WKB approximation, i.e.,

[
q_0\sim \omega^{-1/2}\cos\left(\int^t \omega\,dt\right),
]

then, as can be shown by the Fokker–Planck method, for (\Phi) one obtains an equation of diffusion type

[
\frac{\partial \Phi}{\partial t}
+\dot Q\,\frac{\partial \Phi}{\partial Q}
-\omega^2 Q\,\frac{\partial \Phi}{\partial \dot Q}
=
\nu(t)\left(
\frac{\partial^2\Phi}{\partial Q^2}
+\omega^2(t)\frac{\partial^2\Phi}{\partial \dot Q^2}
\right),
\tag{3}
]

where

[
\nu(t)=
\frac{\psi_0^2}{2\omega(t)}
\left(
\frac{a^2(t)}{\omega(t)}
+\omega^2(t)\int_{t_0}^{t}\frac{a^2(\tau)}{\omega(\tau)}\,d\tau
\right).
]

The function (\Phi) satisfies at the initial time (t_0) the condition

[
\Phi(t_0,Q,\dot Q)=\delta(Q)\delta(\dot Q),
\tag{4}
]

which has the meaning that at (t=t_0) the statistical spread is still absent.

With the aid of the integral transformation

$$
\Phi=\int_{-i\infty}^{i\infty}\int_{-i\infty}^{i\infty}\Psi(t,\xi,\eta)e^{\xi Q+\eta \dot Q}\,d\xi\,d\eta
\tag{5}
$$

equation (3) reduces to a first-order equation

$$
\frac{\partial \Psi}{\partial t}-\eta\frac{\partial \Psi}{\partial \xi}
+\xi\omega^2(t)\frac{\partial \Psi}{\partial \eta}
-\nu(t)\bigl(\eta^2+\omega^2(t)\xi^2\bigr)\Psi=0
\tag{6}
$$

with the additional condition (following from (4)):

$$
\psi(t_0,\xi,\eta)=-\frac{1}{4\pi^2}.
\tag{7}
$$

The solution of (6) can be found by the usual method of characteristics and leads to the formula

$$
\Psi=-\frac{1}{4\pi^2}\exp\left[-\left(\omega\xi^2+\frac{\eta^2}{\omega}\right)\int_{t_0}^{t}\nu\omega\,dt\right],
\tag{8}
$$

whence, using (5), we finally find:

$$
\Phi=\frac{1}{4\pi\displaystyle\int_{t_0}^{t}\nu\omega\,dt}
\exp\left(-\frac{Q^2}{\displaystyle\frac{4}{\omega}\int_{t_0}^{t}\nu\omega\,dt}
-\frac{\dot Q^2}{\displaystyle 4\omega\int_{t_0}^{t}\nu\omega\,dt}\right),
\tag{9}
$$

In conclusion it should be emphasized that the distribution (9) is in fact valid for any dynamical system characterized by a slowly time-varying potential (\frac{1}{2}\omega^2(t)q^2) and subject to the action of a random force (a(t)\psi(t)).

Institute of Chemical Physics
Academy of Sciences of the USSR

Received
10 X 1961

CITED LITERATURE

¹ S. Livingston, Accelerators, IL, 1956.