Effect of Magnetic Field Distortion at the End of the Cycle on Electron Losses in a 1500 MeV Synchrotron
Abstract Generated abstract
This paper examines how distortion of the magnetic field at high field strength near the end of the acceleration cycle affects the usable aperture and electron losses in a 1500 MeV electron synchrotron. Using measured magnetic field data from the Tomsk Polytechnic Institute synchrotron, it calculates the rms amplitude of radial betatron oscillations excited by quantum radiation fluctuations and evaluates the corresponding potential well depth, including the effect of phase oscillations. The calculations indicate a maximum rms radial betatron amplitude of about 1.2 cm and show that the potential well depth decreases by nearly a factor of two after the field exceeds about 8000 oersted. Despite this reduction, the ratio of well depth to excitation energy remains about 20, suggesting that electron losses from quantum-fluctuation-driven radial betatron oscillations should be practically negligible.
Full Text
EFFECT OF MAGNETIC FIELD DISTORTION AT THE END OF THE CYCLE ON ELECTRON LOSSES IN A 1500 MeV SYNCHROTRON
G. P. FOMENKO
In cyclic accelerators, at the end of the acceleration cycle the magnetic-field intensity in the gap reaches large values (~12000 oersted). At such high magnetic-field intensities, owing to changes in the magnetic characteristics of the steel, the distribution of the magnetic field in the gap changes, which may lead to a considerable reduction of the region of the interpole space suitable for particle acceleration. It is especially important to take this change of the working region into account in high-energy electron cyclic accelerators, where the requirements on the size of the working region are considerably higher than in proton accelerators because of the influence of quantum fluctuations of radiation on the motion of the electrons.
In the present article the results are given of a calculation of the rms amplitude of radial betatron oscillations excited by quantum fluctuations of radiation, and of the depth of the potential well of these oscillations at the end of the acceleration cycle, from measurements of the magnetic field on the 1500 MeV electron synchrotron of the Nuclear Physics Research Institute at Tomsk Polytechnic Institute.
The rms amplitude of radial betatron oscillations excited by quantum fluctuations of radiation is expressed by the following formula [1]:
\[ \overline{A_x^2} = \frac{55}{24\sqrt{3}}\, \frac{r_e \Lambda F}{\gamma} \int_{0}^{\theta} \gamma^6 \exp \left[ -2\int_{\theta'}^{\theta} \langle \zeta_x \rangle\, d\theta'' \right] d\theta', \tag{1} \]
where
\(r_e\) — classical radius of the electron;
\(\Lambda\) — Compton wavelength;
\[ \gamma=\frac{E}{E_0}; \]
\(E\) and \(E_0\) — respectively the energy and rest energy of the electron;
\(\langle \zeta_x \rangle\) — decrement of radiation damping of radial betatron oscillations;
\[ \theta=\frac{2\pi s}{\Pi}; \]
\(\Pi\) — orbit perimeter; \(s\) — arc length.
The factor \(F\) for an azimuthally symmetric magnet with gaps has the form:
\[ F=\frac{|f|_{\max}^{2}}{(1-n)^{2}}\left\langle |f'_x+i\nu_x f_x|^2\right\rangle , \]
where
\[ f_x \]
is the Floquet function for a magnetic system of this type.
Figure 1 shows the dependence curve of the rms amplitude obtained as a result of calculation by formula (1). The maximum value of the amplitude of betatron oscillations at the end of the cycle is \(\sim 1.2\) cm.
Fig. 1
In an azimuthally symmetric magnet, a potential function may be introduced for betatron oscillations analogously to how this is done for betatrons in the relativistic case in [2]:
\[ U=\frac{E_0}{e}\left[1+\frac{e^2}{E_0^2}\left(A+\frac{C}{r}\right)^2\right]^{\frac12}, \tag{2} \]
where
\[ A=A_\theta \]
is the vector potential of the magnetic field;
\[ r \]
is the radius of curvature of the orbit;
\[ C=\frac{c}{e}m_i r_i^2\dot{\theta}_i-A_i r_i \]
(the initial values of the corresponding quantities are denoted by the subscript \(i\)).
The potential function makes it possible to judge the magnitude of the focusing forces of the magnetic field and the dimensions of the working region of the accelerator. The character of the variation of \(U\) with \(r\) and \(z\) depends on the value of \(C\), which in the case of a betatron is determined by the initial values of the velocities and coordinates of the accelerated particles. In a synchrotron the accelerated particles execute phase oscillations that are slow in comparison with betatron oscillations. Therefore, the effect of phase oscillations on the potential function of betatron oscillations can be included in the constant \(C\). Since we consider the motion of electrons in the second half of the acceleration cycle, we shall neglect the dependence of the potential function on the input parameters. The potential function
the function \(U\) can be written in a form more convenient for calculation for a given field in the case \(E \gg E_0\):
\[ \frac{U}{U_0}=\frac{R}{r}\left(\frac{1}{R^2}\int_R^r xl^{-\int_R^x n\,dx'}\,dx+\frac{\Delta E}{E}+1\right), \tag{3} \]
where
\[ U_0 \]
is the value of the potential function at the equilibrium radius \(R\);
\[ \frac{\Delta E}{E} \]
is the oscillation of the electron energy due to phase oscillations;
\[ n=-\frac{d\ln H}{d\ln r}. \]
The calculation of the potential function was carried out from measurements of the magnetic field on the electromagnet of the 1500 MeV synchrotron. The depth of the potential well for equilibrium electrons decreases, beginning at 8000 oersteds, approximately according to a linear law up to the end of the acceleration cycle almost by a factor of two (see Fig. 2, curve 1). The figure plots the depth of the potential well at the radius 430.5 cm, where the inflector is located. The calculation shows that the potential function reaches a maximum at this radius only at a field strength of 12000 oersteds, so that at smaller field strengths the depth of the potential well is limited by the inflector. Curve 2 in Fig. 2 shows the change in the depth of the potential well with allowance for phase oscillations. In this calculation was made for the root-mean-square amplitude of the phase oscillations excited by quantum fluctuations of the radiation for a final amplitude of the accelerating voltage of 180 kV.
Fig. 2
In order to determine the electron losses due to betatron oscillations by Kramers’ formula [3], it is necessary to calculate the ratio of the depth of the potential well to the excitation energy, i.e., to the energy
of betatron oscillations, \(\dfrac{\Delta U}{U_{\text{exc.}}}\). The excitation energy is determined with the aid of the rms amplitude of the radial betatron oscillations calculated above. In our case, the ratio \(\dfrac{\Delta U}{U_{\text{exc.}}}\), taking account of phase oscillations (curve 2 in Fig. 2), turns out to be \(\sim 20\). At such large values of \(\dfrac{\Delta U}{U_{\text{exc.}}}\), there will be practically no electron losses due to the influence of quantum fluctuations on the radial betatron oscillations.
REFERENCES
- A. A. Kolomenskii, A. N. Lebedev. Theory of Cyclic Accelerators. Fizmatgiz, Moscow, 1962.
- P. A. Cherdantsev. Izvestiya TPI, 87, 1957.
- H. A. Kramers. Physika 7, 285 (1940).