Use of Electrodes as an Accelerating Device of a Synchrotron
Abstract Generated abstract
This paper examines the use of ring electrodes, rather than coaxial resonators, as the accelerating device in an electron synchrotron. It argues analytically that acceleration by such electrodes is possible despite the presence of return fields, since the net energy gain per revolution equals the gain in the accelerating gap, while losses from nonuniform fields are small in practice. The equivalent circuit including the chamber coating is analyzed, showing that resonance effects in the coating can generally be neglected and that the electrode capacitance can be compensated with a two-wire line. The proposed arrangement is presented as simpler, more easily tunable, and capable of higher gap voltage for a given generator power, although it may require shielding to reduce electromagnetic radiation.
Full Text
USE OF ELECTRODES AS AN ACCELERATING DEVICE FOR A SYNCHROTRON
B. A. Solntsev
(Presented by the scientific seminar of the Faculty of Physics and Technology)
In electron synchrotrons having a ring-type electromagnet, coaxial resonators are usually used as the accelerating device. Resonators of this type are placed in the interpole space of the electromagnet and therefore cannot have a sufficiently large distance between the inner and outer conducting surfaces. This distance is always considerably less than the optimum, and the quality factor of the resonators proves to be lower than that of ordinary coaxial resonators.
In tuning the high-frequency system of a synchrotron, it is very important to be able to vary the frequency of the accelerating electric field, since in many cases the maximum radiation intensity is obtained at an electric-field frequency different from the calculated one. Coaxial resonators intended for use in a synchrotron are difficult to retune in frequency; the possible limits of frequency variation of such resonators scarcely reach 2–3%.
Considerably simpler and more convenient than coaxial resonators is an accelerating device in the form of two ring electrodes embracing the accelerating chamber along a slit in its conducting coating, between which the accelerating electric field is created. Such a device is shown in Fig. 1.
Fig. 1
At present, by analogy with a potential electric field, it is commonly believed that the process of acceleration in a field formed by such electrodes is impossible.
To examine this question, let us imagine an electron moving in a constant magnetic field in a circular orbit of radius $r_0$. Let the electron pass through the accelerating gap between the electrodes instantaneously, at the moment when the phase of the electric field is equal to $\varphi_s$. We shall assume that the electric-field intensity outside the accelerating gap is constant along
orbit. The expression for the electric-field intensity outside the accelerating gap in this case has the form:
\[ E=\frac{V_m}{2\pi r_0}\sin(\omega t-\varphi_s), \tag{1} \]
where \(V_m\) is the amplitude of the accelerating voltage across the gap between the electrodes,
\(\omega\) is the angular frequency of the electric field.
The increase in the electron energy during one revolution is
\[ \delta A=eV_m\sin\varphi_s-\int_0^{2\pi r_0} e\,\frac{V_m}{2\pi r_0}\sin(\omega t-\varphi_s)\,dl . \tag{2} \]
Here \(e\) is the electron charge,
\(l\) is the path traversed by the electron.
The path \(l\) is proportional to the electron velocity \(v\); therefore \(dl=vdt\), and
\[ \delta A=eV_m\sin\varphi_s-\int_0^{kT}\frac{eV_m v}{2\pi r_0}\sin(\omega t-\varphi_s)\,dt, \tag{3} \]
where \(k\) is the coefficient of multiplicity of the frequency of the electric field with respect to the frequency of revolution of the electron,
\(T\) is the period of oscillation of the electric field.
As a result of integration we obtain
\[ \delta A=eV_m\sin\varphi_s. \tag{4} \]
Thus, the total increase in energy per revolution is equal to the increase in energy in the accelerating gap and does not depend on the forces acting on the electron as it moves along the orbit. After leaving the accelerating gap, during the first quarter of the path the electron is decelerated; during the next two quarters of the path it is accelerated; and during the last quarter of the path it is again decelerated (for \(k=1\)), with the amount of energy given up and the amount of energy received by the electron proving to be equal. If the magnetic field increases with time, we obtain the same expression for \(\delta A\), the only difference being that \(\varphi_s\) will change in accordance with the change in the magnetic field.
In reality, the electric-field intensity cannot be regarded as identical at every point lying on the orbit. As a result, during the time between two successive passages through the accelerating gap the electron loses energy; however, as experiment has shown, the energy loss is small. When electrodes are used, the voltage across the accelerating gap has to be increased by no more than 20–30% in comparison with the voltage that must be provided in the case of coaxial resonators.
If the presence of the conducting coating of the chamber is taken into account, then the configuration of the electric field inside the chamber will not differ from the form of the field produced by coaxial resonators—the conducting coating shields the electron orbit from the return field of the electrodes.
In a strict treatment, the potentials of the sections of the conducting coating located under the electrodes cannot be considered identical, since the coating represents a certain inductance. Figure 2 presents an equivalent circuit of the accelerating device taking account of the conducting coating of the chamber. In Fig. 2, \(L_k\) is the inductance of the chamber coating,
$R_k$—ohmic resistance of the coating, $C_k$—capacitance of the gap in the coating, $C_{\mathrm{e}}$—capacitance between the ring electrode and the coating.
From consideration of the equivalent circuit it is seen that, under certain conditions, voltage resonance is possible in this circuit. It is known that in the case of series voltage resonance the voltage on the capacitance and on the inductance exceeds the voltage of the power source by a factor of $Q$, where $Q$ is the quality factor of the system. Consequently, with a proper choice of the capacitance $C_{\mathrm{e}}$, one may expect the voltage across the gap in the conducting coating of the chamber to exceed the voltage applied to the electrodes. The resonance condition for the case under consideration is written in the form
Fig. 2
$$ \frac{1}{\omega C_{\mathrm{e}}}=\omega L_k-\frac{1}{2\omega C_k}. \tag{5} $$
For this condition to be fulfilled, the inductive reactance of the conducting coating must be greater than the capacitive reactance of the gap in it. Such a relation between $L_k$ and $C_k$ does in fact occur. The capacitance between the electrode and the coating can be regulated over wide limits by changing the width of the electrodes.
In synchrotrons for energies of 15–25 MeV the inductance of the coating is tenths of a microhenry, while the capacitance of the gap in it is several micromicrofarads. The active resistance reaches hundreds of ohms. Determining the quality factor of the entire system from the formula
$$ Q=\frac{2\omega L_k-\dfrac{1}{\omega C_k}}{2R_k}, $$
we find that it does not exceed unity; therefore raising the voltage by means of series resonance in synchrotrons for 15–25 MeV is not possible. In synchrotrons for higher energies, $L_k$ and $R_k$ increase approximately in proportion to the increase in the radius of the orbit. The angular frequency of the electric field, however, is inversely proportional to the radius. Consequently, in synchrotrons with electron energies above 25 MeV the quality factor of the accelerating system is also less than unity. Thus, the influence of the conducting coating may be neglected in all cases.
The ring electrodes present a capacitive load for the high-frequency generator. Meanwhile, the generator delivers maximum power when the load is active in character. Therefore the reactance of the accelerating device must be compensated by a reactance of the opposite sign. Most often in such cases reactive stubs connected to the supply feeder near the load are used. In the present case, however, it is more convenient to connect, in parallel with the capacitance of the electrodes, a section of
of a two-wire line of corresponding length, as shown in Fig. 3. A section of a two-wire line loaded by a capacitance \(C_\mathrm{e}/2\) will have an active resistance if the relation
\[ \frac{2}{\omega C_\mathrm{e}} = Z_0 \operatorname{tg} ml \tag{7} \]
is satisfied in the case of a short-circuited line, and
\[ \frac{2}{\omega C_\mathrm{e}} = Z_0 \operatorname{ctg} ml \tag{8} \]
for an open line.
Here \(Z_0\) is the characteristic impedance of the line,
\[ m=\frac{2\pi}{\lambda} \]
is the phase constant,
\(\lambda\) is the wavelength. The resonant impedance of an accelerating device of this type is expressed by the formula [1]
\[ Z_p=\frac{8Z_0^2}{R\lambda}, \tag{9} \]
where \(R\) is the active resistance per unit length of the line.
Fig. 3
Since, when electrodes are used, the geometrical dimensions of the accelerating device are not limited by the size of the interpole space of the electromagnet, it is possible to satisfy all optimal conditions, and the resonant impedance is substantially higher than that of the coaxial resonators used in synchrotrons.
An accelerating device in the form of electrodes is easily tunable over wide frequency ranges. Tuning is carried out by simply moving the short-circuiting bridge along the length of the line.
The device of the type considered is a symmetrical system and is therefore very convenient in the case when the high-frequency generator is assembled according to a push-pull circuit, since it does not require symmetrizing devices. Matching the input impedance of the device to the feeding feeder is performed by moving the point of connection of the feeder along the length of the line. Finally, such a device is considerably simpler to manufacture than a coaxial resonator of any design.
Among the disadvantages one must include appreciable radiation of electromagnetic energy, but it can be greatly reduced by appropriate shielding.
The basic operating characteristics of a synchrotron when electrodes are used as the accelerating device do not differ in any way from the characteristics when coaxial resonators are used; at the same time,
the voltage across the accelerating gap, for the same power delivered by the generator to the load, is considerably higher when electrodes are used.
References
- R. King, H. Mimno, A. Wing, Transmission Lines, Antennas, and Waveguides. Gosenergoizdat, 1948.