On the Dependence of the Intensity of an Accelerated Ion Beam in a Cyclotron on the Accelerating Voltage
Abstract Generated abstract
This paper derives the dependence of accelerated ion beam intensity in a pulsed cyclotron on the amplitude of the accelerating voltage, motivated by the need for mathematical models for accelerator control systems. Using relations for particle phase, final energy, trajectory step, and deflector transmission, it obtains an expression for the relative beam intensity, \(I/I_0=\sqrt{V_m^2-V_0^2}/V_m\), where \(V_0\) is the voltage at which acceleration ceases. The formula is compared with experimental results for the Tomsk Institute of Nuclear Physics cyclotron accelerating deuterons to 13.5 MeV. The authors also analyze how voltage changes alter the integer number of particle revolutions to the final radius and identify discrete accelerating-voltage values at which beam alignment with the detector slit is preserved.
Full Text
ON THE QUESTION OF THE DEPENDENCE OF THE INTENSITY OF A BEAM OF ACCELERATED IONS IN A CYCLOTRON ON THE ACCELERATING VOLTAGE
V. A. KOCHEGUROV, Yu. P. KHARITONENKO
Determining the dependence of the beam intensity in accelerators on various adjustable parameters is necessary in the calculation and design of automatic control systems for these installations. The problem solved in the present work arises, in particular, in developing a mathematical model of a cyclotron as a controlled object, needed for the synthesis and investigation of a feedback loop.
If the cyclotron operates in pulsed mode, then the beam intensity [2]
\[ I=\frac{ZeN_{\tau}}{t}, \tag{1} \]
where \(Z\) is the charge multiplicity of the particle;
\(e\) is the electron charge;
\(t\) is the pulse duration;
\(N_{\tau}\) is the number of accelerated particles (per pulse) that have reached the target at the exit from the accelerating chamber.
Let us denote the injection current by
\[ I_0=\frac{ZeN_0}{t}. \]
Here \(N_0\) is the number of injected particles per pulse. Then the beam intensity at a remote target is
\[ I=I_0\frac{N_{\tau}}{N_0}. \tag{2} \]
Taking into account the commensurability of the thickness of the deflector cutoff plate and the trajectory step \(\Delta r\) of the accelerated particles at the final radius in the range of actually permissible accelerating voltage \(V_m\), one may write
\[ N_{\tau}=F(V_m)\Delta r N_0, \tag{3} \]
where
\[ \Delta r=\frac{2E_0V_m}{300ZH\sqrt{2E_0E}}\sin\varphi. \]
Here \(E_0\) is the rest energy of the ion;
$H$ — magnetic-field strength;
$E$ — ion energy at the final radius;
$\varphi$ — phase of the accelerating voltage at the moment when the ion crosses the accelerating gap, measured from the instant at which the voltage passes through zero.
A qualitative analysis of formula (3) shows that the number of accelerated particles $N_\tau$ cannot be greater than the number of injected ions $N_0$ and is equal to zero when $\Delta r = 0$. From these conditions,
\[ F(V_m)=\frac{300ZH\sqrt{2E_0E}}{2E_0E}, \qquad N_\tau=N_0\sin\varphi . \tag{4} \]
Taking (4) into account, formula (2) assumes the form:
\[ I=I_0\sin\varphi . \tag{5} \]
We shall next show the dependence of the phase $\varphi$ on the accelerating voltage. The energy at the final radius [2] is
\[ E=2\sqrt{\frac{2E_0 Z V_m}{\pi}\left(\cos\varphi-\cos\varphi_0\right)}, \tag{6} \]
where $\varphi_0 \simeq \dfrac{\pi}{2}$ is the phase of the high-frequency voltage at the beginning of the acceleration cycle (after the first half-turn) [1].
From equality (6) we find:
\[ \cos\varphi=\frac{\pi E^2}{4E_0 Z V_m}. \tag{7} \]
Substituting $\varphi=0$ in (7)—the phase value at which acceleration ceases—we find the voltage of zero intensity:
\[ V_0=\frac{\pi E^2}{4ZE_0}. \tag{8} \]
Substituting (8) into (7), we find
\[ \cos\varphi=\frac{V_0}{V_m};\qquad \sin\varphi=\frac{\sqrt{V_m^2-V_0^2}}{V_m}. \]
Thus, the dependence of the beam intensity on the accelerating voltage is expressed by the formula
\[ I=I_0\frac{\sqrt{V_m^2-V_0^2}}{V_m}. \tag{9} \]
For the cyclotron of the Institute of Nuclear Physics (Tomsk), designed to accelerate deuterons to an energy of $13.5$ MeV at a radius $r=52.5$ cm,
\[ V_0=75\ \text{kV}, \]
the dependence (9) is shown in Fig. 1, where the points denote the experimental results.
Dependence (9) is valid in the case where the beam trajectory passes in such a way that the deflector “knife” is located exactly midway between the last and next-to-last orbits. In the real case, a change in the amplitude $V_m$ of the accelerating voltage leads to a change in the number of revolutions $(n)$ of the particles reaching the final radius.
If the energy increment per revolution is [1]
\[ \frac{dE}{dn}=2ZV_m\sin\varphi, \tag{10} \]
then the total energy at the final radius is
\[ E=2ZV_m\int_0^n \sin\varphi\,dn =2ZV_m n\,\overline{\sin\varphi}, \tag{11} \]
Fig. 1
where
\[ \overline{\sin\varphi} =\frac{1}{n}\int_0^n \sin\varphi\,dn \simeq \frac{1+\sin\varphi_n}{2}, \tag{12} \]
\(\varphi_n\) is the phase at the final radius.
Fig. 2
Now, from (11), taking (12) into account, it is not difficult to obtain the number of revolutions of the ion as a function of the accelerating voltage:
\[ n=\frac{E}{2ZV_m\overline{\sin\varphi}} =\frac{E}{ZV_m(1+\sin\varphi_n)} =\frac{E}{V_m+\sqrt{V_m^2-V_0^2}}. \]
Taking for \(n\) a series of integers, we obtain from the graph (Fig. 2) a series of values of \(V_m\) for which the final radius is exactly equal to the specified one (Table 1).
Table 1
| \(n\) (rev.) | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 |
| \(V_m\) (kV) | 133.5 | 131.3 | 129.3 | 127.4 | 125.5 | 123.9 | 122.3 | 120.8 |
| \(n\) (rev.) | 64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 |
| \(V_m\) (kV) | 119.3 | 117.7 | 116.3 | 114.8 | 113.6 | 112.4 | 111.2 | 109.9 |
| \(n\) (rev.) | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 |
| \(V_m\) (kV) | 108.8 | 107.7 | 106.6 | 105.5 | 104.4 | 103.6 | 102.8 | 102 |
From Table 1 it is not difficult to see the necessity of stabilizing the accelerating voltage at the discrete values of the series obtained; moreover, at larger values of \(V_m\), stabilization requires less accuracy. In fact, the necessity of stabilizing the accelerating voltage is dictated by the need to preserve an integer number of revolutions of the accelerated particles, since otherwise the center of the beam cross section does not fall at the center of the cross section of the detector entrance slit. In this case, in order to change the number of ion revolutions by one, a change in the accelerating voltage will be required: at 102 kV, by 0.785%, and at 133 kV, by 1.65%.
REFERENCES
- J. D. Livingood. Principles of Cyclic Particle Accelerators, IL, 1963.
- E. G. Komar. Charged-Particle Accelerators. Atomizdat, 1964.