On the Methodology for Calculating a Betatron
Abstract Generated abstract
This paper examines betatron design methodology for achieving a specified bremsstrahlung dose without relying primarily on higher final electron energy, which increases accelerator size, shielding, cost, and may reduce irradiation quality for some applications. It develops a calculation based on the critical charge that can be confined during injection, introduces the characteristic charge as a comparative measure of the capacity of the guide magnetic field, and combines this with an empirical dose relation for bremsstrahlung. The analysis shows that raising the injection voltage can substantially increase the accelerated charge and permit the same pulse dose at lower final energy. The author argues that injection voltages of about 200 to 300 kV are technically practicable and should be used more widely to design more economical betatrons matched to irradiation requirements.
Full Text
Proceedings of the Tomsk Polytechnic Institute
Named after S. M. Kirov, Holder of the Order of the October Revolution
and the Order of the Red Banner of Labor
Volume 279 1974
On the Methodology for Calculating a Betatron
B. V. Okulov
(Presented by the Research Institute of Nuclear Physics, Electronics, and Automation)
Usually, in order to increase the bremsstrahlung dose rate of a betatron to a specified value, the maximum energy of the betatron is increased, since the bremsstrahlung dose grows rapidly with increasing energy of the accelerated electrons. However, in this case, along with the increase in radiation dose, the weight of the electromagnet and its dimensions increase [1], the capacitance of the capacitor bank increases, the weight of the biological shielding increases, etc. In addition, as the energy of the accelerator is increased, the wavelength of the bremsstrahlung decreases, which in a number of cases is undesirable. For example, in flaw detection, when the wavelength of bremsstrahlung decreases, the contrast of the photographic image may decrease; and when certain materials are irradiated with high-energy quanta, their activation and undesirable disturbances of their structure may occur. As the accelerator energy is increased, the useful irradiation field also decreases. Thus, in this case an increase in the bremsstrahlung dose of the accelerator is obtained at a comparatively high cost. At the same time, a considerable increase in the bremsstrahlung dose can be obtained by a corresponding increase in the injection energy [2, 3], i.e., without increasing the final energy of the accelerator and without the undesirable consequences associated with this.
Indeed, the “critical charge” [4], which can be confined by the magnetic field of the betatron during injection, according to [2], in the CGSE system of units is determined by the expression
\[ Q_{\mathrm{cr}}=\frac{E_i}{2eR_0}\left[\left(\frac{E_i}{E_0}\right)^2-1\right]\cdot s, \tag{1} \]
where \(E_i\) is the total energy of the injected electron,
\(E_0\) is the rest energy of the electron,
\(e\) is the electron charge, \(R_0\) is the radius of the equilibrium orbit, and \(s\) is the area of the cross section of the stability region.
Formula (1) is valid for an accelerator with magnetic-field index at the equilibrium orbit \(n_0=0.5\). In the more general case, formula (1) can be written in the form [5]
\[ Q_{\mathrm{cr}}=f_1(n_0)\frac{m_0c^2}{2eR_0}\cdot s\cdot\frac{E_i}{E_0} \left[\left(\frac{E_i}{E_0}\right)-1\right], \tag{2} \]
where
\[ f_1(n_0)=\sqrt{\frac{1-n_0}{n_0}}. \]
Let us denote
\[ f_1(n_0)\cdot \frac{m_0 c^2}{2e}\cdot \frac{S}{R_0}=\beta, \tag{3} \]
\[ \frac{E_i}{E_0}\left[\left(\frac{E_i}{E_0}\right)^2-1\right]=f_2(U_i), \tag{4} \]
where \(U_i\) is the injection voltage; then
\[ Q_{\mathrm{cr}}=\beta\cdot f_2(U_i). \tag{5} \]
As is clear from equality (3), \(\beta\) is a function of \(S\), \(R_0\), and \(n_0\)—important parameters of the accelerator. For each given accelerator the quantity \(\beta\) is constant and, according to equation (5), represents the critical charge that the focusing forces of the guide magnetic field can hold at a “unit” injection voltage, when \(f_2(U_i)=1\). This value of the critical charge, which uniquely characterizes the “capacity” of the accelerator’s guide magnetic field, is conveniently called the characteristic charge [5] and used to estimate (compare) the “capacities” of the guide magnetic field of various betatrons. In the particular case of betatrons with \(n_0=0.5\), the “unit” injection voltage is equal to \(166\ \mathrm{kV}\).
Fig. 1
Figure 1 shows the dependence of \(\beta\) on the cross section of the stability region for various radii of the equilibrium orbit. Figure 2 gives the dependence (for \(n_0=0.5\)) of the critical charge on the injection voltage and on the value of \(\beta\).
In [5] it is shown that in betatrons, at the injection voltages usually employed—several tens of kilovolts—the charge reaching the target is \(\sim 0.01 Q_{\mathrm{cr}}\). In [5] it is also shown that in a high-current betatron at a high injection voltage (\(\sim 200\ \mathrm{kV}\)), when electron scattering by residual gas and by magnetic-field inhomogeneities is reduced, when the angular characteristics of the injected beam are improved through the use of an inflector, and when, finally, relativistic effects begin to play a substantially positive role in electron capture, it is possible to accelerate a charge close to the critical one. Thus, for example, in a high-current pulsed two-chamber stereobetatron [6], a charge \(\sim 0.24 Q_{\mathrm{cr}}\) was accelerated [5]. This fact, as well as the work [3], indicates that formula (1) for calculating the critical charge can be successfully used in designing betatrons for a specified radiation intensity.
Fig. 2
Labels in the figure: vertical axis, \(Q_{\mathrm{cr}}\cdot 10^{-6}\), C; horizontal axis, \(U_i\), kV; parameter \(\beta\), C/kg. The plotted parameter values are \(0.04\cdot 10^6\), \(0.07\), \(0.1\), \(0.2\), \(0.4\), \(0.7\), \(1.0\), \(2.0\), and \(4.0\).
In [7], on the basis of an analysis of experimental data, the following dependence was obtained for the dose of bremsstrahlung \(\gamma\)-radiation on the energy of the accelerated electrons:
\[ D = K_0 \cdot 10^{-16} Q_{\mathrm{m}} \cdot E^3, \tag{6} \]
where \(Q_{\mathrm{m}}\) is the charge of the electrons delivered to the target, \(E\) is the electron energy in MeV, and \(K_0\) is the target optimality coefficient.
It is useful to consider equations (1) and (6) jointly, taking into account a certain coefficient \(K_y\), which reflects what fraction of the electrons from \(Q_{\mathrm{cr}}\) can be accelerated and delivered to the target in a given accelerator. Thus,
\[ Q_{\mathrm{m}} = K_y \cdot Q_{\mathrm{cr}}. \tag{7} \]
From equations (5), (6), and (7) we have
\[ D = 10^{-16} K_0 \cdot K_y \cdot f_1(n_0) \cdot \beta \cdot f_2(U_i) \cdot E^3, \tag{8} \]
or
\[ E\left(\frac{A}{D}\right)^{1/3}=\left[f_2(U_i)\right]^{-1/3}, \tag{9} \]
where \(A=10^{-16}\cdot K_0\cdot K_y\cdot \beta\cdot f_1(n_0)\).
From equation (9) and the corresponding curve of equal pulse doses (Fig. 3), it follows that one and the same dose can be obtained with various definite combinations of the final energy of the accelerated
Fig. 3
electrons and the injection voltage. As the injection voltage is increased, the required total energy of the accelerator decreases. For example, when the injection voltage is raised from 60 to 260 kV, the final energy of the accelerator can be reduced by a factor of 2. Correspondingly, the dimensions of the electromagnet, its weight, cost, etc., can be reduced. This must be taken into account in developing new, more economical betatrons.
A number of designs of high-voltage injector devices for voltages up to 200–300 kV have already for a long time been successfully used in betatrons [5, 8–10], so that this range of injection voltages may be regarded as mastered. The matter is only that such injection voltages should be used in as large a num-
number of betatrons, both those already built and those being designed. Simple calculations show that increasing the injection voltage from the usual value of 40–50 to 200–300 kV (all other conditions being equal) leads to a severalfold reduction in the cost of the accelerated electrons, or of the corresponding bremsstrahlung radiation.
Thus, it makes sense to choose the maximum energy of the accelerator on the basis of the optimal conditions for the interaction of the bremsstrahlung radiation with the irradiated object. The required pulse dose, however, is advantageously obtained by means of a sufficiently high injection voltage.
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