Calculation of Electron Motion in an Alternating Electric Field
Abstract Generated abstract
The paper presents a computational method for determining the motion of relativistic electrons in an alternating electric field, with application to resonator accelerators and related devices. Starting from the electron equation of motion, the authors reduce the problem to finding the exit phase by numerical integration and describe an algorithm, written in the Alfa translator input language, that iteratively locates the exit angle and computes the corresponding velocity and kinetic energy. A separate adaptive quadrature procedure is introduced to accelerate integration, including automatic step selection and error control for nonsmooth integrands. Example calculations for 250 keV electrons in an 11 cm wavelength field with a 3.4 cm gap give transit angles, output energies, and estimated fractions of output current within a specified energy spread for several voltage amplitudes.
Full Text
CALCULATION OF ELECTRON MOTION IN AN ALTERNATING ELECTRIC FIELD
R. G. Ziyakaev, A. V. Ovchinnikov
(Presented by the seminar of the High-Energy Laboratory of the Research Institute of Nuclear Physics, Electronics, and Automation)
In studying the motion of relativistic electrons in bunchers, in installations for generating millimeter and submillimeter waves, in single-resonator accelerators, etc., it is necessary to solve the equation
\[ \omega \frac{dmv}{dx} = eE_0 \sin x, \tag{1} \]
where \(\omega\) is the angular frequency of the electric field,
\(x = \omega t\) is the current phase of the voltage,
\(E_0\) is the electric-field strength.
When electrons are accelerated by a cylindrical resonator in which an \(E_{010}\)-type wave is excited, the equations of electron motion can likewise be reduced to equation (1), in the case where the radius of the electron beam is much smaller than the radius of the resonator [1], and equation (1) can be written for this case in the form
\[ \omega \frac{dmv}{dx} \simeq \frac{eU_0}{h} \sin x, \tag{2} \]
where \(h\) is the height of the resonator;
\(U_0\) is the amplitude value of the voltage along the axis of the resonator.
Integration of equation (2) leads to the equation
\[ \frac{\beta}{\sqrt{1-\beta^2}} = \frac{\beta_0}{\sqrt{1-\beta_0^2}} + \left(\frac{eU_0}{m_0c^2}\right) \left(\frac{\lambda}{2\pi h}\right) (\cos x_0-\cos x) = A(x), \tag{3} \]
where \(\beta = v/c\) is the ratio of the electron velocity to the speed of light;
\(\beta_0 = v_0/c\) is the ratio of the initial electron velocity to the speed of light;
\(E_0 = m_0c^2\) is the rest energy of the electron;
\(\lambda\) is the wavelength in the resonator;
\(x_0\) is the phase of entry of the electron into the resonator.
To determine the phase of exit of the electron from the resonator, it is necessary to solve the equation:
\[ \frac{\omega h}{c}-\int_{x_0}^{x}\frac{A}{\sqrt{1+A^2}}\,dx=0, \tag{4} \]
where the unknown exit phase \(x\) is the upper limit of integration.
For the value of \(x\) obtained by solving equation (4), it is usually necessary to know the quantity \(\beta=v/c\) and the value of the kinetic energy of the electron, which are determined from the formulas
\[ \beta=\frac{A(x)}{\sqrt{1+A^2(x)}} \tag{5} \]
and
\[ T=m_0c^2\left(\frac{1}{\sqrt{1-\beta^2}}-1\right), \tag{6} \]
where \(T\) is the kinetic energy of the electron at the moment it leaves the resonator.
Below we give an algorithm for solving equation (4), to which equation (1) is reduced. The algorithm is written in the input language of the alpha-translator [2].
Description of the Algorithm
Let us write equation (4) in the form
\[ g-\int_{x_0}^{x_0-g}\frac{A}{\sqrt{1+A^2}}\,dx -\int_{x_0+g}^{x}\frac{A}{\sqrt{1+A^2}}\,dx=0, \tag{7} \]
where \(g=\omega h/c\) is the angle of traversal of the distance \(h\) by electrons moving with the speed of light.
The first integral of equation (7) is calculated once, and to it are added the values of the integrals over subsequent intervals of length \(sh_1\) until the left-hand side of equation (5) becomes less than 0. As soon as this is achieved, the last integration interval of length \(sh_1\) is divided into two parts and the integral over the smaller interval is computed. The subdivision of the last interval is continued until the value of \(x\) is determined with the prescribed accuracy \(m\).
The value of the exit angle \(x\) is determined for ranges of entry angles from \(x_1\) to \(x_2\) with step \(sh\). If the time during which the electron remains in the field is greater than \(2\pi\), then the computation of this case is terminated and the computation for a new value of the entry angle is begun.
In the program, numerical values must be specified for the following quantities:
\[ a=\frac{\beta_0}{\sqrt{1-\beta_0^2}} \quad\text{and}\quad b=\left(\frac{eU^{\circ}}{m_0c^2}\right)\left(\frac{\lambda}{2\pi h}\right) \]
the coefficients in equation (3);
\(x_1, x_2\) — the lower and upper limits of the values of the entry angles;
\(g=\omega h/c\) — the angle of traversal of the distance \(h\) by electrons moving with the speed of light;
\(m\) — the absolute accuracy in determining the entry angles.
From punched cards, numerical values of the quantities \(a, b, sh, sh_1\) are entered, where \(a\) and \(b\) are the coefficients of equation (3);
\(sh\) — the step with which the values of the entry angles are specified;
\(sh_1\) — the initial step for finding the exit angles.
BEGIN INTEGER i; ARRAY k [1:4];
REAL a, b, x0, x, h, h1, c, p, I, r, L, beta, T;
REAL PROCED A (x);
A := a + bx (cos (x0) — cos (x));
REAL PROCED phi (x);
{real p; p := A (x); phi := p/sqrt (1 + p↑2)}
M: input (k); output (k); (a, b, h, h1) := k[ ];
FOR x0 := x1 STEP h TO x2 CYCLE
BEGIN x := x0 + d;
IKRON (x0, x, .01, .1, phi, 0, p, I);
r := 0; c := h1;
M1: L := d — I — r;
IF (L < 0) ∧ (c < m) THEN TO M3;
M2: IF x — x0 > 6.3 THEN TO M4;
IKRON (x, x + c, .01, .1, phi, 0, p, r);
IF I + r < d THEN
{I := I + r; x := x + c; TO M2}
c := c/2; TO M1;
M3 beta := phi (x);
T := .51098x (1/Sqrt (1 — beta↑2) — 1);
output (x0, x, beta, T);
M4: END; output (TRUE); TO M;
END*
Description of the IKRON procedure
In order not to make the program difficult to read, this procedure and its description are given separately from the main program. Since most of the time is spent on computing integrals, measures were taken to reduce the time required for calculating the integrals. The algorithm developed is suitable for computing integrals of the form
\[ I=\int_a^b \Phi(x)d(x) \tag{8} \]
and is recommended for nonsmooth functions \(\Phi(x)\).
No restrictions are imposed on the quantities \(a\) and \(b\); the cases \(a>b\), \(b=a\), \(b>a\) are possible. The integral is computed using “large quadrature formulas” [4]
\[ I=\sum_{k=0}^{m-1}\mathrm{H}_k \sum B_l \Phi(a\tau_k+\mathrm{H}_k V_l)+R, \tag{9} \]
where \(\mathrm{H}_i=a_{\tau i+1}-a_{\tau i}\) is the integration step;
\(R\) is the computational error.
The nodes and weights of the main and refining quadratures were taken from [3]; the order of the quadrature formulas was selected experimentally and is equal to four.
The operator for calling the procedure has the form IKRON (a, b, m, n, phi, r, p, I), where \(a\) and \(b\) are the limits of integration;
\(m\) is the minimally admissible integration step;
\(h\) — initial integration step;
\(\phi\) — the integrand expression, described in the program as a real procedure;
\(\tau\) — the required absolute error of the calculation of the integral;
\(p\) — the variable to which the value of the absolute error obtained as a result of the calculations is assigned, i.e., the actual absolute error;
\(I\) — the value of the integral.
In this procedure the computation time is reduced by means of refined quadrature formulas [3], by the nonuniform distribution of the error over the integration interval, and by automatic step selection. On each integration subinterval the current required absolute error of the calculations \(T_{\tau k}\) and the actual error of the calculations \(T_{\phi k}\) are computed; the latter is equal to the modulus of the difference of the values of the integrals on this subinterval, calculated with 5 and 9 points. If for two adjacent integrals \(\Phi_{k+1}\) and \(\Phi_k\), \(T_{\phi k}<T_{\tau k}\), then the step \(H_k\) is doubled. If \(T_{\phi k}>T_{\tau k}\), then the step is halved, and the integral \(\Phi_k\), calculated with the larger error, is recalculated. The step is reduced only in the case when it is not smaller than the minimum admissible step (i.e., always \(H>M\)); otherwise the calculations are performed with the obtained minimum step.
PROCEDURE IKRON (a, b, m, h, phi, tau, p, I),
VALUE a, b, m, tau,
BEGIN REAL y, tp, c1, c2, c3, at, vt, tt, tphi, L, r;
INTEGER k; BOOLEAN T; ARRAY y, B, Б [1:9];
T:=FALSE; L:=h; y[1]:=.01179874;
y[2]:=.069431844; y[3]:=.179856891;
y[4]:=.330009478; y[5]:=.5;
y[6]:=.669990521; y[7]:=.820143108;
y[8]:=.930568155; y[9]:=.988280125;
B[1]:=B[3]:=B[5]:=B[7]:=B[9]:=0;
B[2]:=.173927422; B[4]:=.326072577;
B[6]:=B[4]; B[8]:=B[2];
Б[1]:=Б[9]:=.031488686;
Б[2]:=Б[8]:=.085026802;
Б[3]:=Б[7]:=.133399170;
Б[4]:=Б[6]:=.163474594;
Б[5]:=.173221490;
IF b>a, THEN r:=1 ELSE
IF b<a, THEN {r:=a; a:=b; b:=r; r:=-1} ELSE
{p:=I:=0; GO TO M4};
p:=0; c1:=0; vt:=a; tp:=tau/(b-a);
M1: IF vt=b, THEN GO TO M2 ELSE at:=vt;
M3: vt:=at+L;
IF vt≥b, THEN {L:=b-at; vt:=b};
tt:=tp×L; c2:=c3:=0;
FOR k:=1, ..., 9 CYCLE
BEGIN y:=phi (at+L×y[k]);
c2:=c2+y×B[k]; c3:=c3+y×Б[k];
END;
c2:=c2×L; c3:=c3×L; tphi:=abs(c3-c2);
IF tphi>tt, THEN
BEGIN IF L/2>M, THEN {L:=L/2; GO TO M3} ELSE
BEGIN tp:=IF tau-p-tphi>0, THEN
(IF b-vt>0, THEN (tau-p-tphi)/(b-vt)
OTHERWISE tn) OTHERWISE 0
END;
END; p:=p+tf;
IF T tf<tt, THEN L:=2×J;
T:=tt−tf>0; c1:=c1+c3; TO M1;
M2: I:=c1×r;
M4:
END.
Results of the calculations
The results of calculations using this program are shown in Fig. 1. The calculation was carried out for electrons with an initial kinetic energy of 250 keV entering an electric field with \(\lambda=11\) cm and an electrode spacing of 3.4 cm. The range of angles of incidence \(x_0\) was varied from \(-\pi/2\) to \(+\pi/2\) in steps of \(10^\circ\). Figure 1 gives the dependence of the transit angle \(\varphi=x-x_0\) and of the kinetic energy \(T\) at the exit from the field on the angle of incidence \(x_0\) for different values of the voltage amplitude between the electrodes \(U\).
Fig. 1. Dependence of the kinetic energy \(T\) (solid line) at the exit of the accelerating gap and of the transit angle \(\varphi\) on the angle of incidence \(x_0\), for \(h=3.4\) cm, \(\lambda=11\) cm, \(\beta_0=0.740\), for various values of the voltage \(U\) along the axis of the accelerating gap.
At a voltage \(U=800\) kV, electrons entering in the range of input angles from \(-8\) to \(+53^\circ\) have an energy spread \(\Delta T=0.1T_{\max}\). If the beam current at an energy of 250 keV is continuous and equal to \(I\), then the current at the exit of the accelerating system for the given \(\Delta T\) will be equal to \(0.169\,I\).
For voltage values of 550, 350, and 250 kV, the corresponding output-current values will be \(0.186\), \(0.200\), and \(0.213\,I\), respectively.
In conclusion, the authors consider it their pleasant duty to express their gratitude to I. P. Chuchalin for his assistance in the work.
References
- Collection Millimeter and Submillimeter Waves. Edited by R. G. Mirimanov. Moscow, IL, 1959.
- The Alfa System. Edited by A. P. Ershov. Moscow, Publishing House of the Computing Center of the USSR Academy of Sciences, 1965.
- A. S. Kronrod. Nodes and Weights of Quadrature Formulas. Moscow, Nauka, 1964.
- Ya. S. Dymarskii, N. N. Lozinskii [et al.]. Programmer’s Handbook. Vols. 1, 2. Leningrad, Sudpromgiz, 1963, 1964.