Electron capture coefficient in a 300 MeV synchrotron
Abstract Generated abstract
This paper presents an approximate calculation of the electron capture coefficient for injection and acceleration in a 300 MeV synchrotron using a one-electron approximation. The injection process is divided into two stages: capture during compression of the instantaneous orbit toward the mean orbit, evaluated with Rabinovich’s method for radial and vertical losses, and subsequent capture into the synchrotron acceleration regime while accounting for the finite buildup time of the accelerating voltage. Calculations are performed for specified beam dimensions, angular spreads, magnetic-field growth rates of 90, 175, and 300 kG/s, and resonator parameters. The results indicate that although higher magnetic-field growth improves first-stage capture, the combined capture into the acceleration regime is greatest for a field growth rate of 175 kG/s under the parameters considered.
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ELECTRON CAPTURE COEFFICIENT IN A 300 MeV SYNCHROTRON
V. K. KONONOV, V. A. SERDYUTSKII, G. P. FOMENKO
The intensity of a beam of accelerated particles is determined primarily by the efficiency of the method of injection into the accelerator. Therefore, determining the injection conditions and the requirements on the beam injected into the accelerator, under which the capture coefficient will have an acceptable value, is a necessary task in designing an accelerator.
In the present work, the electron capture coefficient for acceleration in a 300 MeV synchrotron is determined approximately. The calculation is carried out in the one-electron approximation.
The time of injection of electrons into the accelerator is divided into two stages. In the first stage the accelerating field is switched on, and the instantaneous orbit of the electrons is compressed from the injector toward the mean orbit. During one revolution time \(T\), the orbit radius decreases by the amount
\[ \Delta R = R \frac{T}{(1-n)} \frac{\dot H}{H}, \]
where \(H\) and \(\dot H\) are the magnetic field and its partial derivative with respect to time,
\(n\) is the magnetic-field fall-off coefficient,
\(R\) is the radius of the instantaneous orbit.
The time during which the first stage of injection continues can be determined from the formula
\[ \tau = (1-n)\frac{\rho}{R_0}\frac{H}{\dot H}, \]
where \(R_0\) is the mean radius of the accelerator,
\(\rho\) is the distance from the injector to the mean radius.
In the first stage of injection, \(Q_1\) particles will be captured,
\[ Q_1 = \frac{I\tau}{e}\eta_\rho \eta_z, \]
where \(\eta_\rho\) is the probability that the electron will not be lost on the injector and on the vertical walls of the chamber,
\(\eta_z\) is the probability that the electron will not be lost on the horizontal walls of the chamber,
\(\eta_1 = \eta_\rho \eta_z\) is the particle-capture coefficient in the first stage of injection.
will be captured into the synchrophasotron acceleration regime: \(Q\) particles.
\[ Q = Q_1 \eta_2 = \frac{I \tau}{e}\,\eta_1 \eta_2 = \frac{I \tau}{e}\,\eta, \]
where \(\eta_2\) is the coefficient of particle capture at the second stage of injection, and \(\eta\) is the coefficient of particle capture into the synchrophasotron acceleration regime.
For calculating the coefficient \(\eta_1\), the method developed by Rabinovich [1] is used.
Let an electron beam be admitted into the accelerator chamber, having radial size \(2\Delta\), angular spread \(2\gamma_{\max}\), and let the beam axis be deflected from the optimum direction by an angle \(\varepsilon\).
Assume that the electrons in the beam are distributed uniformly with respect to the angle of entry.
Introducing dimensionless angles
\[ \alpha = \frac{R_0 f}{\rho_i \sqrt{1-n}}\,\gamma \]
and denoting by \(\alpha_1\) and \(\alpha_2\) the minimum and maximum electron angles, we can find the coefficient \(\eta_\rho\):
\[ \eta_\rho = \frac{e}{I\tau} \sum_{k=1}^{2} \int_{0}^{\alpha_k} Q'\,\frac{d\alpha}{2\alpha_{\max}}, \]
where
\[ Q' = A \sum_{i=1}^{2} \iint_{C_i} du\,dy, \qquad A = \frac{I \rho T}{2e\,\Delta R\,\overline{O}} . \]
Fig. 1. Diagram of notation used in calculating the injection efficiency
The integration regions \(C_1\) and \(C_2\) are determined by the inequalities
\[ \frac{0<y<\overline{O}}{\sqrt{(u+y)^2+\alpha_{\varepsilon}^{2}}} -u \le a_i, \qquad \sqrt{(u+y)^2+\alpha_{\varepsilon}^{2}} \le 1, \]
where
\[ \overline{O}=\frac{2\Delta}{\rho_i}, \qquad u=\frac{\rho_0}{\rho_i}, \qquad y=\frac{x}{\rho_i}, \qquad (\text{see Fig. 1}) \]
\(a_1\) is the smaller of the two numbers \(\dfrac{3\Delta R}{\rho_i}\) and \(S\dfrac{f(\sigma_i)}{f\left(\chi-\dfrac{\pi}{4}\right)}\),
\(a_2\) is the smaller of the two numbers \(\dfrac{6\Delta R}{\rho_i}\) and \(S\dfrac{f(\sigma_i)}{f\left(\chi-\dfrac{\pi}{4}\right)}\),
\[ f(\sigma_i)=1.104,\qquad f\left(\chi-\frac{\pi}{4}\right)=1.241. \]
To select the optimum value of \(\dot H\), the calculation of \(\eta_\rho\) was carried out for the cases in which \(\dot H\) is equal to \(90\ \text{kG/sec}\), \(175\ \text{kG/sec}\), and \(300\ \text{kG/sec}\).
Fig. 2. Capture coefficient in the first stage of injection. Variation of the magnetic field \(\dot H=90\ \text{kG/sec}\) in sec
The calculation of \(\eta_\rho\) was performed for the following parameter values: \(S\rho_i=10\ \text{mm}\), \(\rho_i=50\ \text{mm}\), \(\Delta=4\ \text{mm}\); \(\gamma_{\max}\) and \(\varepsilon\) were varied from 0 to \(10^{-2}\) radians. From the results of the calculation, graphs 2, 3, and 4 were constructed. From comparison of graphs 2, 3, and 4 it can be found that, in the case when \(\dot H=300\ \text{kG/sec}\), a larger number of electrons can be captured in the first stage of injection than in the cases when \(\dot H\) is equal to \(175\ \text{kG/sec}\) and \(90\ \text{kG/sec}\), and in the case when \(\dot H=175\ \text{kG/sec}\), more than when \(\dot H=90\ \text{kG/sec}\).
The calculation of \(\eta_z\) was carried out under the following conditions: \(\Delta z=20\ \text{mm}\); \(\gamma_{z\max}\) was varied from 0 to \(10^{-2}\ \text{rad}\) and \(\varepsilon_z\) from 0 to \(10^{-3}\ \text{rad}\). As a result of this calculation it was obtained that \(\eta_z\), for these parameter values, is equal to unity.
At the moment when the instantaneous orbit reaches the position of the mean orbit, the accelerating field is switched on and the second stage of injection begins. Since in our case the period of phase oscillations is smaller-
Fig. 3. Same as in Fig. 2, but for \(\dot H = 175\ \mathrm{kg/s}\) per sec
Fig. 4. Same as in Fig. 2, but for \(\dot H = 300\ \mathrm{kg/s}\) per sec
or, compared with the possible time for establishing the amplitude of the accelerating voltage, then when calculating the electron capture coefficient at the second stage of injection \((\eta_2)\) it is necessary to take into account the finite time for establishing the amplitude of the accelerating voltage. During the time in which the amplitude of the accelerating field is established, the increase in electron energy is insufficient for the electron to move near the mean orbit; therefore all electrons of the bunch, although at different velocities, begin to move toward the inner wall of the chamber. If the time \(\tau_\nu\) for establishing the amplitude of the accelerating voltage is not very large, then some of the electrons, before reaching the inner wall of the chamber, may acquire, in one revolution, just the energy increment that is necessary, at the given rate of growth of the magnetic field, for the electron to remain at some radius. These electrons first stop at the radii at which they are at that moment, and then begin to move toward larger radii. The radius at which the electron stops depends not only on \(\tau_\nu\), but also on the initial input phase of the electron into the resonator. At the instant when the electron stops, \(\dot R\) is equal to zero.
\[ \dot R=\frac{R}{1-n}\left(\frac{\dot E}{\beta^2 E}-\frac{\dot H}{H}\right)=0, \qquad \frac{\dot E}{\beta^2 E}=\frac{\dot H}{H}. \tag{1} \]
If, in this case, \(R>R_{\mathrm{v}}\) (where \(R_{\mathrm{v}}\) is the radius of the inner wall of the chamber), then the electron will be captured into acceleration.
Let the magnetic field \(H\) and the amplitude of the accelerating voltage increase linearly with time:
\[ H=H_0+\dot H t, \qquad V=V_{\mathrm{k}}\frac{t}{\tau_\nu}, \]
where \(V_{\mathrm{k}}\) is the final value of the amplitude of the accelerating voltage.
The rate of change of the phase of the electrons relative to the accelerating field is expressed by the relation
\[ \dot\varphi=\omega_{\mathrm{p}}-q\omega. \]
Integrating, we obtain:
\[ \varphi-\varphi_0=\omega_{\mathrm{p}}t-\frac{ce}{S_1}\int_0^t \frac{H}{E}\,dt, \tag{2} \]
where \(\omega\) is the electron revolution frequency,
\(\omega_{\mathrm{p}}\) is the frequency of the accelerating field,
\(q\) is the harmonic number,
\(S_1\) is the orbit-shape coefficient,
\(E\) is the electron energy.
Having integrated under the assumption that \(E\) does not depend on time, we obtain:
\[ \varphi-\varphi_0=-\frac{1}{2}\omega_{\mathrm{p}}\frac{\dot H}{H}t^2=-at^2. \tag{3} \]
Then, using this expression for \(\varphi\), we find the dependence of \(E\) on time. In the expression for the rate of growth of the electron energy,
\[ \dot E=\frac{\omega}{2\pi}eV\sin\varphi, \]
we substitute (3) and integrate:
\[ \Delta E=E-E_0=\frac{\omega e V_{\mathrm{k}}}{4\pi a\tau_\nu} \left[\cos\left(\varphi_0-at^2\right)-\cos\varphi_0\right]. \tag{4} \]
Substitute into (2) \(E = E_0 + \Delta E\) and expand \(\frac{1}{E}\) in powers of \(\frac{\Delta E}{E_0}\), retaining only first-order terms. Integrating, we find the approximate dependence of \(\varphi\) on time:
\[ \varphi = \varphi_0 - at^2 - bt\cos\varphi_0 + b\Phi(t\varphi_0), \tag{5} \]
where
\[ b = \frac{\omega^2 e V_k}{2\pi \cdot 2a\tau E}, \]
\[ \Phi(t\varphi_0)= \sqrt{\frac{\pi}{2a}} \left[ \cos\varphi_0 \cdot C\left(t\sqrt{\frac{2a}{\pi}}\right) + \sin\varphi_0 \cdot S\left(t\sqrt{\frac{2a}{\pi}}\right) \right], \]
\(C\) and \(S\) are the Fresnel integrals.
Calculating \(\varphi\) and \(\dot E\) for specified values of \(\varphi_0\) and \(\tau\), we find the moment when condition (1) is satisfied. Then, for this moment of time, we find \(\Delta E\) and calculate the radius at which the electron is stopped.
\[ R = R_0 + \frac{R_0}{1-n}\left(\frac{\Delta E}{E\beta^2}-\frac{\Delta H}{H}\right), \tag{6} \]
where \(\beta=\frac{V}{c}\), \(V\) is the electron velocity. If \(R_{\mathrm{v}}<R\), then an electron with the selected entry phase will be captured in the acceleration process. Performing calculations for various values of \(\varphi_0\), we determine the range of useful entry-phase values. Knowing the radii at which electrons are stopped for all values of \(\varphi_0\), and assuming a uniform distribution of electron density over the chamber volume and over the amplitudes of free oscillations by the end of the first injection stage, we can calculate \(\eta_2\) for different values of \(\tau\).
The calculation was carried out for the following parameter values: \(H_0=19.8\ \mathrm{g}\), \(\dot H_1=175\ \mathrm{kg/s}\), \(\dot H_2=300\ \mathrm{kg/s}\), \(V_k=680\ \mathrm{V}\), \(R_{\mathrm{v}}=0.895\ \mathrm{m}\), \(S_1=1.402\).
Fig. 5. Capture coefficient at the second injection stage
In Fig. 5, on the basis of the results of an approximate calculation, curves are plotted expressing the dependence of the capture coefficient at the second injection stage on the time for establishing the amplitude of the accelerating voltage on the resonator at \(\dot H=175\ \mathrm{kg/s}\) (curve 1) and at \(\dot H=300\ \mathrm{kg/s}\) (curve 2).
From an analysis of Figs. 3, 4, and 5 one may conclude that the capture coefficient into the synchrophasotron acceleration regime is greatest at \(\dot H=175\ \mathrm{kg/s}\).
LITERATURE
- M. S. Rabinovich. Proceedings of FIAN. Vol. X, 1958.