INFLUENCE OF AN EXTERNAL MAGNETIC FIELD ON THE CHARACTERISTICS OF THE PHASE ELLIPSE IN AN ELECTRON-OPTICAL SYSTEM

N. V. TRIKHANOVA, V. A. KOCHETUROV

In [1] it is shown that the calculation of the motion of electron beams in transport systems must be carried out in the phase plane. In this case the characteristics of the beam along the motion are represented in the form of a phase ellipse described by the equation

\[ \gamma x^2 + 2\alpha x x' + \beta x'^2 = \varepsilon, \tag{1} \]

where \(\beta, \gamma, \varepsilon\) characterize, respectively, the half-width, the half-divergence of the beam, and the area of the phase ellipse.

The calculation reduces to determining the parameters of the ellipse \((\gamma,\alpha,\beta)\), which vary along the motion depending on the nature of the external forces acting on the electron beam. It should be noted that, in accordance with Liouville’s theorem, the area of the ellipse remains constant. For a space without external magnetic fields, the parameters of the ellipse are transformed in accordance with the equation [2]:

\[ \begin{pmatrix} \beta\\ \gamma\\ \alpha \end{pmatrix} = \begin{pmatrix} 1 & y & y^2\\ 1 & 1 & -y\\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} \beta_0\\ \gamma_0\\ \alpha_0 \end{pmatrix}. \tag{2} \]

Under the action of an external transverse magnetic field, all particles will undergo motion described by the equation

\[ \frac{d^2 x}{dy^2} = \frac{ecH_z}{\sqrt{W^2 - W_0^2}}, \tag{3} \]

where

\[ \begin{aligned} e &\text{ — charge of the electron,}\\ c &\text{ — speed of light,}\\ W &\text{ — total energy of the electron,}\\ W_0 &\text{ — rest energy of the electron,}\\ H_z &\text{ — intensity of the external magnetic field.} \end{aligned} \]

The solution of equation (3) has the form:

\[ x' = -AH_z y + x'_0; \]

\[ x = -\frac{1}{2} AH_z y^2 + x'_0 y + x_0, \tag{4} \]

where

where

\[ A=\frac{ec}{\sqrt{W^2-W_0^2}}, \]

\(y\) is the longitudinal coordinate,

\(x_0', x_0\) are the initial angle and deviation of the electron.

Solving equations (1) and (4) simultaneously for the given initial conditions \(\gamma_0, \beta_0, \alpha_0, x_0, x_0'\) and the external magnetic field, we obtain the equation of the phase ellipse at a certain value of \(y\):

\[ \gamma x^2+2\alpha xx' + \beta x'^2+2\delta x+2\eta x'=\varepsilon_1, \tag{5} \]

where

\[ \gamma=\gamma_0; \]

\[ \alpha=\alpha_0+\gamma_0 y; \]

\[ \beta=\gamma_0-2\alpha_0 y+\beta_0; \]

\[ \delta=\frac{1}{2}AH_z y^2\gamma_0+AH_z y(\alpha_0-\gamma_0 y); \]

\[ \eta=-\frac{1}{2}AH_z y^3\gamma_0+AH_z y^2(\gamma_0 y-2\alpha_0)+AH_z y\left(\frac{y\alpha_0}{2}+\beta_0\right). \]

Equation (5) is the equation of a phase ellipse with a shifted center. The coordinates of the center of such an ellipse are determined by:

\[ x_y=-\frac{ \begin{vmatrix} \delta & \alpha\\ \eta & \beta \end{vmatrix} }{ \begin{vmatrix} \gamma & \alpha\\ \alpha & \beta \end{vmatrix} } =-\frac{1}{2}AH_z y^2; \tag{6} \]

\[ x_y'=\frac{ \begin{vmatrix} \gamma & \delta\\ \alpha & \eta \end{vmatrix} }{ \begin{vmatrix} \gamma & \alpha\\ \alpha & \beta \end{vmatrix} } =AH_z y. \]

It is not difficult to see that the coefficients \(\gamma,\alpha,\beta\) in equation (5) and these same coefficients calculated from expression (2) for the drift space are, respectively, equal. From this one may conclude that the action of an external transverse magnetic field during the motion of an electron beam in the drift space does not distort the configuration of the phase ellipse, but only shifts its center. This makes it possible, when calculating electron-optical systems with allowance for external magnetic fields, to use the methodology developed in [1] with the application of analog computing techniques.

REFERENCES

1. V. A. Kochegurov, N. V. Trikhanova. Proceedings of the VI Interuniversity Conference on Electron Accelerators, 1966.

2. P. T. Kirstein, Journal of Electronics and Control. First Series, vol. XIV, No. 3, 1963.