Calculation of the Distorted Orbit of a 300 MeV Synchrotron from Magnetic Field Measurements

V. N. Eponeshnikov, G. P. Fomenko

The deviation of the accelerator magnetic field from the calculated one leads to a distortion of the instantaneous orbit of the particles and, consequently, to the loss of some portion of the working region of the magnet and to a decrease in the intensity of the accelerated-particle beam. To calculate the distorted orbit of the 300 MeV synchrotron, measurements of the magnetic field were carried out at 33 points with an interval \(\Delta \Theta = 2^\circ 30'\) in the sectors and at 18 points with an interval of \(2.07\) cm in the straight sections, at radii \(R = 95\) cm and \(R = 100\) cm. The distorted orbit was calculated by the method developed by M. S. Rabinovich [1]. The equation of motion of the electrons in the sectors, taking into account the deviation of the magnetic field from the calculated value, has the form:

\[ \frac{d^{2}\rho}{d\Theta^{2}}+\chi^{2}\rho=\chi g(\Theta), \]

where

\[ g(\Theta)=-\frac{R}{\chi}\frac{H_z(\Theta)-H_0}{H_0}, \qquad \chi=\sqrt{1-n}, \]

\[ H_0=\frac{1}{4}\left(H_{01}+H_{02}+H_{03}+H_{04}\right), \qquad H_{0N}=\frac{1}{\nu}\int_{0}^{\nu} H_{zN}(\Theta)\,d\Theta, \]

\(n\) is the magnetic-field falloff index,

\(\nu\) is the effective sector angle (Fig. 1).

A solution of this equation, more convenient for practical calculations, was obtained by K. N. Shorin [2].

\[ \rho_N(\Theta)=A_N\cos \chi(\nu-\Theta)-B_N\cos \chi\Theta+\rho_N^{*}(\Theta), \]

\[ \rho_N^{*}(\Theta)=\int_{0}^{\Theta} g_N(\xi)\sin \chi(\Theta-\xi)\,d\xi. \]

\(\Theta\) varies in each sector from \(0\) to \(\nu\).

Introduce the notation:

\[ s=\sin \chi\nu,\qquad c=\cos \chi\nu,\qquad 2p=\frac{\chi l}{R},\qquad \cos\mu=c-ps, \]

\(l_{\mathrm{eff}}\) is the effective length of the straight section.

The coefficients \(A_N\) are calculated from the formulas

\[ A_N= \frac{f_N\cos\mu+f_{N+1}+f_{N+2}\cos\mu+f_{N+3}\cos 2\mu} {2\sin\mu\sin 2\mu}, \]

\[ f_N= \frac{-a_N s-\beta_N(c-2\rho s)+\beta_{N+1}}{s} -\frac{2\rho}{\chi}\rho_N^{*}. \]

Fig. 1

\[ a_N=\eta_N+\rho_N^{*}(\nu),\qquad \beta_N=\varepsilon_\mu+\frac{1}{\chi}\rho_N^{*}(\nu),\qquad \varepsilon=\frac{R}{\chi}\delta_N, \]

\[ \delta_N=\delta_0-\frac{1}{H_0R}\int_{0}^{l_{\mathrm{ef}}} H_z(x)\,dx, \qquad \delta_0=\frac{\pi}{2}-\nu, \]

\[ \frac{1}{H_0R}\int_{0}^{l_{\mathrm{ef}}} H_z(x)\,dx \]
is the angle through which the electron is turned in the straight section.

\[ \eta_N=\frac{\delta_0 l_{\mathrm{ef}}}{2} -\frac{1}{H_0R}\int_{0}^{l_{\mathrm{ef}}} dx \int_{0}^{x} H_{zN}(\xi)\,d\xi, \]

\[ \frac{1}{H_0R}\int_{0}^{l_{\mathrm{ef}}} dx \int_{0}^{x} H_z(\xi)\,d\xi \]
is the displacement of the electron by the field of the straight section.

\(B_N\) is determined from the formula

\[ B_N=A_{N+1}-\frac{\beta_N}{s}. \]

On the basis of the shape of the magnetic field in the synchrotron it was assumed that the effective sector angle \(\nu\) is equal to \(80^\circ\), and \(l_{\mathrm{ef}}=68\ \text{cm}\).

Table

The principal characteristics of the field are \(\delta_N\), \(\eta_N\), and \(\Delta H_N\) \((\Delta H_N = H_{0N} - H_0\)—the deviation of the mean field in a sector from the mean field over all sectors). In order for the distortion of the electron equilibrium orbit to be minimal, the quantities \(\delta_N\), \(\eta_N\), and \(\Delta H_N\) must be made as small as possible.

Calculations based on measurements of the natural field at the injection level (\(H_0 = 19.8\ \text{Oe}\) in the absence of compensation) give the following values of \(\delta_N\), \(\eta_N\), and \(\Delta H_N\):

\[ \Delta H_1 = 0.17\ \text{Oe},\quad \Delta H_2 = -0.12\ \text{Oe},\quad \Delta H_3 = -0.25\ \text{Oe},\quad \Delta H_4 = 0.20\ \text{Oe}, \]

\[ \delta_1 = 11^\circ 17,\quad \delta_2 = 12^\circ 24,\quad \delta_3 = 12^\circ 30,\quad \delta_4 = 11^\circ 39, \]

\[ \eta_1 = 6.76\ \text{cm},\quad \eta_2 = 7.46\ \text{cm},\quad \eta_3 = 8.24\ \text{cm},\quad \eta_4 = 7.42\ \text{cm}. \]

The maximum deviation of the distorted orbit from the ideal one \((\Delta \rho_m)\) was found to be \(\pm 5.75\ \text{cm}\). The width of the working region of the magnet in the radial direction is \(11.5\ \text{cm}\); therefore, without correction of the magnetic field it is impossible to accelerate particles.

As a result of calculating the distorted orbit in the natural field at the moment of injection, values of the currents in the correcting windings of the synchrotron [3] were selected for which the orbit distortion is minimal. After correction of the magnetic field, field measurements were made at radii of \(95\ \text{cm}\) and \(100\ \text{cm}\) at the injection level, and at a radius of \(95\ \text{cm}\) at levels of \(100\ \text{Oe}\) and \(10000\ \text{Oe}\). The results of calculations of the distorted orbit from the magnetic measurements are given in the table.

The orbit distortion after correction at the equilibrium radius is \(\pm 1.3\ \text{cm}\) for \(H = 19.8\ \text{Oe}\), and \(\pm 0.9\ \text{cm}\) for \(H = 100\ \text{Oe}\) and \(10000\ \text{Oe}\). At a radius of \(100\ \text{cm}\), the maximum orbit distortion is \(\pm 0.6\ \text{cm}\) for \(H = 19.8\ \text{Oe}\). Thus, as a result of correction the orbit distortion decreased by a factor of 4, and throughout the entire acceleration time the loss of the working region of the magnet does not exceed 22%.

REFERENCES

1. M. S. Rabinovich. Proceedings of FIAN, 10, 1958.
2. K. N. Shorin. Dissertation, FIAN, 1961.
3. V. N. Eponeshnikov, V. N. Kuzmin. Magnetic-field correction system of the synchrotron (in the present collection).