On the Mathematical Description of the Betatron
Abstract Generated abstract
This paper proposes a practical mathematical description of betatron radiation intensity for use in control and stabilization problems. It argues that, for a given accelerator over short operating intervals, the maximum intensity is mainly determined by injection voltage and electron energy, while instantaneous intensity also depends on injection phase and injected current. Experimental results are used to express these regulatory dependences in relative coordinates, showing that their shapes remain approximately invariant under changes in injection voltage and magnetic field within experimental error. The authors derive formulas for the optimal injection phase and current, and combine them with empirical approximation functions to obtain an intensity expression suitable for analyzing regulators, stabilizers, and destabilizing effects such as supply voltage variation.
Full Text
On the Mathematical Description of a Betatron
L. M. Ananyev, M. M. Shtein
(Presented by the scientific seminar of the Department of Industrial Electronics)
Further improvement of the quality of bremsstrahlung radiation and improvement of betatron control circuits are hindered by the present absence of a mathematical description of the basic processes occurring in a betatron. In the present work an attempt is made to obtain a mathematical expression for the radiation intensity of a betatron.
As is known, the maximum accelerated charge in a betatron is determined by the aperture of the accelerating chamber, the topography of the magnetic field, and the magnitude of the injection voltage [1].
\[ q_{\mathrm{m}}=(0.1 \div 0.4)\,\pi\,10^{-11}\,\frac{(r_i-r_0)^2}{r_0}\,U_i\sqrt{\frac{1-n}{n}}. \tag{1} \]
In most practical cases of the industrial application of betatrons, it is sufficient to ensure a stable output of the accelerator over a comparatively short interval of time, which usually does not exceed several hours. During this interval of time, for a normally operating installation, certain factors, such as the geometric dimensions and vacuum of the accelerating chamber, the topography of the magnetic field, the optics and the position of the injector, may be considered unchanged.
Then the maximum possible value of the intensity for a given betatron is determined by the injection voltage and the energy of the accelerated electrons.
\[ I_{\mathrm{m}}=\kappa U_i' E^m, \tag{2} \]
where \(\kappa\) is the coefficient of proportionality. The exponent \(m\), according to data from various works [2, 3, 4], lies within the range \(2 \div 4\).
At the same time, the value of the intensity at each given moment is also determined by the instant or phase of injection \(t_i\) and by the current of the injected electrons \(J_i\). These dependences make it possible, on the one hand, to regulate the radiation intensity effectively, and on the other hand, they determine to a significant extent the instability of betatron operation.
Existing theories of the betatron do not permit obtaining, with accuracy sufficient for practice, a quantitative expression for the dependences \(I=f_1(t_i)\) and \(I=f_2(J_i)\). As a result, the search for approximating expressions for the dependence of the intensity ...
from the phase and current of injection, which would have a more general character and would make it possible to reduce to a minimum the number of experiments required in determining the approximation coefficients.
It is convenient to present the given curves in relative coordinates
\[ \frac{I}{I_0}=F_1\left(\frac{t_i}{t_0}\right) \quad \text{and} \quad \frac{I}{I_0}=F_2\left(\frac{J_i}{J_0}\right); \]
where \(I_0\) is the maximum value of the ordinate of the dependence under study, all other conditions being equal, and \(t_0\) and \(J_0\) are the abscissas of \(I_0\) and may be called, respectively, the optimal phase and the optimal injection current.
The experimental studies carried out showed that, with such a representation, these curves remain unchanged for the given betatron when the injection voltage or the magnitude of the magnetic field is varied.
The results of experiments at different values of the injection voltage are presented in Figs. 1 and 2. As can be seen from these figures, a change in the injection voltage over a wide range does not lead to any regular change in the shape of the curves. The deviations of the curves from one another that are present may be explained by experimental errors, whose calculated value is \(5 \div 7\%\). When the magnitude of the magnetic field was varied, analogous results were obtained.
Fig. 1. Dependence of intensity on the injection phase in relative coordinates
This fact makes it possible to reduce substantially the number of experiments in studying a betatron as a controlled object. Instead of the usually recorded families of curves \(I=f_1(t_i)\) and \(I=f_2(J_i)\) at different values of the induction and injection voltage, it is sufficient to have one graph for each dependence. All the remaining curves
... can be obtained by recalculation through the relative argument. In the latter case it is necessary to know the factors determining the optimal phase and the injection current.
With accuracy sufficient for practice, the optimal phase can be found from the condition that the initial electron velocity be matched with the magnitude of the magnetic field at the radius of the equilibrium orbit. For nonrelativistic injection voltages and sinusoidal variation of the magnetic flux,
Fig. 2. Dependence of intensity on injection current in relative coordinates
\[ t_0=\frac{3.37\sqrt{U_i}}{\omega B_{0m}r_0}. \tag{3} \]
Figure 3 presents the results of checking the last formula, showing satisfactory agreement between the calculated and experimental data.
It follows from the work of P. A. Cherdantsev [5] that the optimal injection current is determined only by the magnitude of the injection voltage and depends on it to the power \(3/2\). The investigations carried out confirm these propositions; however, the dependence of the optimal current on the injection voltage does not always have the power \(3/2\). If the expression for the optimal injection current is written in the form
\[ J_0=aU_i^p, \tag{4} \]
then the exponent \(p\) in our experiments lay within the range \(0.8 \div 1.5\) for different accelerating chambers. Since, when sealed-off chambers are replaced, only the position and optical properties of the injector can change, it is evident that they to a considerable extent determine the specific value of \(p\).
An experimental study of the regulatory dependences of the betatron showed that the magnitude of the injection current has a definite influence on the form of the dependence \(\dfrac{I}{I_0}=F_1\left(\dfrac{t_i}{t_0}\right)\), and vice versa. In particular, at injection currents smaller than the optimum, the dependence of the intensity on the injection phase has the form of a sharp peak in the region of the optimum phase. As the injection current is increased, the rising branch of the curve \(F_1\left(\dfrac{t_i}{t_0}\right)\) becomes flatter; however, the falling part changes only slightly. At injection currents close to the optimum or larger, the shape of the curve does not undergo substantial changes.
Fig. 3. Optimum injection phase: 1 — when changing the induction at the equilibrium orbit, \(U_i=44\) kV;
2 — when changing the injection voltage, \(B_{om}=0.23\) T.
The latter fact makes it possible, taking relation (2) into account, to write the following expression for the radiation intensity, valid for not very large changes of the phase and injection current:
\[ I=\kappa U_i E_e^m F_1\left(\frac{t_i}{t_0}\right) F_2\left(\frac{J_i}{J_0}\right). \tag{5} \]
Since the form of the regulatory dependences of the betatron is comparatively complex, it is usually not possible to choose simple approximating functions for \(F_1\left(\dfrac{t_i}{t_0}\right)\) and \(F_2\left(\dfrac{J_i}{J_0}\right)\). In all probability, the most rational approach is to find approximating functions in each specific case separately, taking into account the special features of the problems being solved. For example, in studying the operation of extremal regulators [6], when there is only a small deviation of \(t_i\) from \(t_0\) or
\(J_i\) from \(J_0\), sufficient accuracy can be provided by a quadratic trinomial. In other cases, when agreement is required over a broader range of variations in the phase and injection current, one may use an approximation by an exponential function:
\[ I=\kappa U_i E_e^m \exp -\Theta\left(\frac{t_i}{t_0}-1\right)^2 \exp -\psi\left(\frac{J_i}{J_0}-1\right)^2 . \tag{6} \]
The coefficients \(\Theta\) and \(\mu\) are determined from experimental data. Expressions (3), (4), and (6), which relate the principal quantities determining the operation of the given accelerator, can be used for the analysis of various regulators and stabilizers of radiation intensity, and also for studying the influence of destabilizing factors, in particular, changes in the supply voltage of the betatron installation.
REFERENCES
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- V. I. Gorbunov. Dissertation, TPI, Tomsk, 1958.
- Vaupel K., Stahl und Eisen, 73, 705, 1953.
- V. A. Moskalev. Dissertation, TPI, Tomsk, 1953.
- P. A. Cherdantsev. Izv. vuzov. Fizika, 6, 117, 1959.
- N. N. Balamatov, B. I. Goryachev. Proceedings of the 5th Interuniversity Conference on Electron Accelerators, 182, 1966.