REPRESENTATION OF THE PLATES OF A PLANAR INFLECTOR OF A SYNCHROTRON ON THE PHASE PLANE

N. V. Trikhanova, V. A. Kochegurov

In the 1500 MeV electron synchrotron of the Nuclear Physics Research Institute of the Academy of Sciences, injection of the beam into the chamber for acceleration is carried out on a circular section; moreover, the action of the stray magnetic field and of the magnetic field in the interpolar space is compensated by a flat-shaped capacitor (inflector) consisting of several plates.

In [1] it is shown that at high energies of the injected particles the injection path of the synchrotron must be calculated with allowance for the finite value of the beam emittance. In this connection the following problem arises: to construct the image of the inflector plates on the phase plane. In the present article the image and transformation on the phase plane of slit diaphragms are considered, and it is then shown that, from the point of view of transmission capacity, the inflector is equivalent to two slit diaphragms, one of which is located at the entrance and the other at the exit of the inflector.

Fig. 1. Image on the phase plane of a slit diaphragm

A slit diaphragm is represented on the phase plane by a pair of parallel straight lines. The transformation of points in phase space is linear and is described by the expression [2]:

\[ \begin{pmatrix} x(y) \\ x'(y) \end{pmatrix} = \begin{pmatrix} a(y) & b(y) \\ c(y) & d(y) \end{pmatrix} \begin{pmatrix} x_0 \\ x'_0 \end{pmatrix} = A \begin{pmatrix} x_0 \\ x'_0 \end{pmatrix}, \tag{1} \]

where

\(y\) is the longitudinal coordinate;

\(x(y)\) and \(x'(y)\) are the displacement and the angle of inclination of the particle relative to the optical axis;

\(a(y), b(y), c(y), d(y)\) are coefficients characterizing the action of the transforming devices.

Let us consider how relation (1) transforms a pair of parallel straight lines. A slit diaphragm, symmetrically located rel-

with respect to the optical axis, is represented in the phase plane by a pair of vertical lines \(A_1A_2\) (Fig. 2, a) for a given value of the longitudinal coordinate. For another value of the longitudinal coordinate the lines \(A_1A_2\) are transformed into a pair of parallel lines characterized by the relative numbers \((m,n)\), as shown in Fig. 2, b. The transformation is carried out according to the expression

\[ \frac{m}{n}=A\left(\frac{m_0}{n_0}\right), \ldots \tag{2} \]

where

\[ m,n \]

are relative numbers characterizing the points of intersection of the straight lines with the coordinate axes;

\[ A= \begin{pmatrix} a & b\\ c & d \end{pmatrix} \]

is the transformation matrix.

Fig. 2. Representation in the phase plane of a slit diaphragm for a value of the longitudinal coordinate: a) at the place of its installation \(y=y_0\); b) for another value of the longitudinal coordinate \((y=y_1)\)

Two parallel slit diaphragms, placed at some distance from one another, are represented in the phase plane in the coordinate system of the second diaphragm by two pairs of parallel straight lines.

Let us represent the inflector plates as a series of slit diaphragms placed at some distance from one another, with the first diaphragm located at the entrance of the inflector and the last at the exit. Let the number of diaphragms be three; the second diaphragm is located in the middle of the inflector, whose length is equal to \(l\). In the coordinate system of the second diaphragm, the first and second diaphragms are represented in the phase plane in the form of the parallelogram shown in Fig. 3, a. From relation (2) we determine that the straight lines \(BC\) and \(AD\) intersect the coordinate axes at the points \(\left(\mp \frac{1}{n_0}, 0\right)\), \(\left(0,\frac{2}{ln_0}\right)\). Transform the parallelogram shown in Fig. 3, a, from the second diaphragm to the third, i.e. to the exit of the inflector, using relation (2). The transformed parallelogram is shown in Fig. 3, b. The straight lines \(B_1B_2\) and \(D_1D_2\) are the image of the diaphragm located at the exit of the inflector. The useful area in the phase plane is evidently the parallelogram \(B_2B_1D_2D_1\). It is not difficult to note that the configuration of this parallelogram is determined only by the positions of the entrance and exit diaphragms.

We arrive at the same conclusion if we increase the number of diaphragms and successively carry out the transformation from one diaphragm to another. Thus, from the point of view of the transmission capacity of the inflector plate, it should be regarded as two slit diaphragms, one of which is located at the entrance of the inflector, and the other at the exit.

Fig. 3. Representation on the phase plane: a) two diaphragms separated from one another by a distance \(l/2\), at the location of the second diaphragm; b) three diaphragms at the location of the third diaphragm

REFERENCES

1. V. A. Kochegurov, N. V. Trikhanova. Calculation of the injection path of an electron synchrotron taking into account the finite value of the beam emittance. Proceedings of the VI Interuniversity Conference on Accelerators, Tomsk, 1966.

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