STABILIZATION OF BETATRON RADIATION

V. M. RAZIN

(Presented to the scientific seminar of the Faculty of Physics and Technology)

The stability of betatron radiation depends on many causes. Let us note some of them.

1. Fluctuations of the line voltage affect the stability of the processes of electron capture, i.e., the number of electrons captured in each acceleration cycle changes from cycle to cycle, as a result of which the mean value of the radiation also changes.

2. If the dumping of electrons onto the target is carried out at a constant phase relative to the magnetic field of the betatron, then the presence of fluctuations in the line voltage leads to a change in the energy of the electrons at the moment of dumping. Since the dose rate of the radiation at a constant mean current of accelerated electrons depends strongly on their energy at the moment of braking (the dependence approaches cubic), even small changes in energy are the cause of considerable changes in the magnitude of the radiation.

3. The sensor of the synchronizing pulses for the injection circuit is usually a coil with a permalloy core placed in the magnetic field near the working zone of the betatron. Imperfection in the mechanical fastening of this coil can lead to disturbance of the synchronization processes and to a change in the radiation.

4. The vacuum chamber with the electron gun is subjected to the action of mechanical vibrations of the betatron electromagnet, as a result of which the position of the vacuum chamber and gun in the magnetic field of the betatron may change, and the relative arrangement of the gun electrodes may also change. All this also affects the capture process and, consequently, the intensity of the radiation.

5. Changes in the emission properties of the cathode of the electron gun affect the capture process; imperfection of the vacuum affects the processes of capture and acceleration, etc.

In this far from complete list, the first cause has the most substantial influence on the stability of the radiation. This is explained by the fact that, for normal operation of the betatron, the accuracy of timing of the injection pulse with respect to the change in the magnetic field is measured by a quantity of the order of tenths and even hundredths of a microsecond (at a supply frequency of 50 cps). The slightest disturbance of precise timing is the cause of the appearance of considerable changes in the radiation intensity.

In order to increase the stability of operation of the betatron, one may carry out complete stabilization of the entire power supply of the installation, including the supply of the electromagnet. In our opinion, this method is not the best,

since the technical implementation of high-power stabilization (the electromagnet consumes power on the order of ten or more kilowatts) is difficult and in some cases involves considerable power losses.

It should be considered more expedient, for stabilization purposes, to use various kinds of automatic synchronization and regulation circuits in the electrical circuits of the betatron, while leaving the power supply of units that consume large power unstabilized. Here one may use circuits for automatic adjustment of the phase of generation of the voltage pulse on the electron gun [1], automatic regulation circuits connected directly with the radiation, as proposed in [2], etc.

Before proceeding to describe the system we have developed for stabilizing betatron radiation, let us consider the theoretical aspect of the question.

For injection voltages on the order of 30–80 kV, the nonrelativistic relation may be used for calculations:

\[ \sqrt{U}=0.297\, r H_m \sin \omega t, \tag{1} \]

where

\[ \begin{aligned} U &\text{ — injection voltage in volts,}\\ H_m &\text{ — amplitude of the magnetic-field intensity on the equilibrium orbit}\\ &\text{\quad of the betatron in oersteds,}\\ r &\text{ — radius of the equilibrium orbit in cm,}\\ \omega &\text{ — angular frequency of the current feeding the electromagnet,}\\ t &\text{ — time in sec.} \end{aligned} \]

Taking into account that, for most betatrons, injection is carried out at the moment \(\omega t \le 1^\circ\), one may, with a high degree of accuracy (up to \(0.025\%\)), adopt the approximation

\[ \sin \omega t = \omega t. \]

Assuming that for the given betatron \(r=\mathrm{const}\), we write the expression in the form:

\[ U=kH_m^2 \omega^2 t^2, \tag{2} \]

where \(k\) is a proportionality coefficient.

Let us consider possible variants for fulfilling condition (2).

1. The injection-phase regulation circuit may be arranged in such a way that the equality \(\omega t=\mathrm{const}\) is satisfied. In this case, for relation (2) to be fulfilled, the injection voltage and the amplitude of the magnetic-field intensity must be connected by the condition

\[ U=k_1 H_m^2. \tag{3} \]

Here, as in all other cases, we shall assume that the amplitude of the magnetic-field intensity on the orbit is related by direct proportionality to the line voltage \(U_{\text{line}}=k_2H_m\), for which one may write the equality:

\[ \frac{dU_{\text{line}}}{U_{\text{line}}}=\frac{dH_m}{H_m}. \tag{4} \]

Differentiating (3) and dividing by the original equality, we obtain

\[ \frac{dU}{U}=2\,\frac{dH_m}{H_m} \]

or, in finite increments:

\[ \frac{\Delta U}{U}\simeq 2\,\frac{\Delta U_{\text{line}}}{U_{\text{line}}}=2\,\frac{\Delta H_m}{H_m}. \tag{5} \]

This method of carrying out the matching cannot be considered advisable, since the practical fulfillment of condition (5) is difficult and, moreover, a considerable increase in the injection voltage with some increase in the mains voltage is undesirable in view of the danger of breakdown of the gun, if one takes into account that, in order to obtain maximum radiation intensity, it is almost always necessary to operate at the limit of the electric strength of the gun.

The only advantage of such a method is, perhaps, the better preservation of the matching of the rate of change of the injection voltage at the moment of capture with the rate of change of the required voltage, determined by condition (2).

1. If the charging transformer of the injection circuit is supplied from the same voltage as the electromagnet, or, still better, from a special winding on the electromagnet, then the injection voltage will be proportional to the supply voltage:

\[ U = \kappa_3 U_{\text{mains}} = \kappa_4 H_m . \tag{6} \]

In this case, on the basis of (2) and assuming that \(\omega = \mathrm{const}\), one may write:

\[ \kappa_4 H_m = \kappa_5 H_m^2 t^2,\quad \text{i.e. } H_m t^2 = \kappa_6 . \tag{7} \]

Differentiation of (7) gives:

\[ \frac{dt}{t} = -\frac{1}{2}\frac{dH_m}{H_m}. \]

In finite increments, taking (4) into account, we have

\[ \frac{\Delta t}{t} \simeq -\frac{1}{2}\frac{\Delta U_{\text{mains}}}{U_{\text{mains}}} = -\frac{1}{2}\frac{\Delta H_m}{H_m}. \tag{8} \]

Consequently, with this method of synchronization, in order to satisfy condition (2) under fluctuations of the mains voltage, it is necessary, in accordance with equation (8), to change the phase of generation of the injection-voltage pulse correspondingly. The minus sign indicates the fact that an increase in the mains voltage must be accompanied by a decrease in the delay time of the injection pulse relative to the instant of zero value of the magnetic field on the orbit.

The practical implementation of this method presents no particular difficulties [1].

1. When the charging transformer of the injection circuit is supplied from a ferroresonance stabilizer, it appears possible, under fluctuations of the mains voltage, to fulfill the condition \(U = \mathrm{const}\). Condition (2) here reduces to the following relation:

\[ H_m^2 t^2 = \kappa_7,\quad \text{or } H_m t = \kappa_8 . \tag{9} \]

As a result of differentiating (9) we obtain

\[ \frac{dt}{t} = -\frac{dH_m}{H_m} \]

or, in finite increments,

\[ \frac{\Delta t}{t} \simeq -\frac{\Delta H_m}{H_m} = -\frac{\Delta U_{\text{mains}}}{U_{\text{mains}}}. \tag{10} \]

The practical execution of the method is easily realizable. The constancy of the injection voltage in this case reduces the probability of breakdowns of the electron gun in the presence of fluctuations of the mains voltage.

It should be noted that in all these three cases one considers the conditions for preserving the optimal operating regime of the injection scheme (in the sense of ensuring the maximum productivity of the installation). Since in some methods a change in the injection voltage is allowed, this thereby permits the possibility that oscillations of the radiation intensity may arise. Consequently, by the concept “stabilization of radiation” we shall here mean, properly speaking, stabilization of the optimum conditions for the capture of electrons into acceleration, i.e., automatic maintenance of the maximum productivity of the installation. Such an approach to the consideration of the question is practically justified by the fact that in most cases of betatron use the determining factor is the integral action of the radiation.

Of all the methods considered, preference should, in our opinion, be given to the third, whose practical implementation differs little from that of the second.

Let us note that, when the injection voltage is constant, the matching conditions do not change even for relativistic velocities of the electrons emitted by the gun.

Indeed, the nonrelativistic relation is written in the form

\[ r H_m \sin \omega t = \frac{10^4}{2.97}\sqrt{U}, \]

where the injection voltage \(U\) is expressed in megavolts.

The relativistic expression has the form:

\[ r H_m \sin \omega t = \frac{10^4}{3}\sqrt{U(U+1.02)}. \]

From consideration of these equalities it is seen that in both cases, for \(U=\mathrm{const}\), \(\omega=\mathrm{const}\), \(r=\mathrm{const}\), and \(\sin \omega t \simeq \omega t\), the matching conditions are identical,

\[ H_m t = \mathrm{const}. \]

Under practical conditions, when working with a synchronizing-pulse pickup in the form of a coil with a permalloy core, it is necessary to take into account the fact that the coil is not installed directly on the orbit, but outside the vacuum chamber, near the working region at the poles. Here the change in the magnetic field, owing to the demagnetizing action of the copper of the magnetizing coils, lags somewhat in phase behind the change of the field on the orbit.

To obtain the required delay of the control pulse generated in the coil with a permalloy core, a “constant” current is passed through this same coil (constant in the sense that its value at the moment of injection does not depend on oscillations of the voltage of the supply network), from a storage-battery bank or a ferroresonant stabilizer.

If a ferroresonant stabilizer is used as the source of the constant bias, then a large active resistance must be included in the circuit in order to reduce to a minimum the phase shift between current and voltage [1].

In the presence of such a constant bias, the phase of generation of the control pulse will, during oscillations of the line voltage, automatically change in such a way that, with the proper choice of parameters, the conditions (8) or (10) will be satisfied at all times, depending on the method of regulation.

In order to carry out regulation by the second method, and when using the third method to be able to take into account the phase shift in the change of the magnetic fields at the orbit and at the place where the sensor is installed, it is necessary to introduce into the regulation circuit a “variable” bias magnetization (variable in the sense that this component of the bias magnetization at the moment of injection must be proportional to the voltage on the electromagnet).

Fig. 1

The alternating component of the bias current can be obtained from a transformer whose voltage is in phase with, and proportional to, the voltage on the electromagnet. To reduce the phase shift between the current and the voltage, a large active resistance is included in the circuit.

In the simplest case the control and regulation circuit has the form shown in Fig. 1 [1].

The magnitude of the constant and alternating components of the bias magnetization may be determined on the basis of the following considerations. Let the magnetic field at the place where the sensor is installed vary according to the law:

\[ H = H_{m_1}\sin(\omega t - \varphi) \simeq H_{m_1}(\omega t - \varphi). \tag{11} \]

By means of the bias currents, a magnetic field is created at the same place

\[ H_{\text{bias}} = - (H_0 - k_9 H_{m_1}), \tag{12} \]

where

\[ \begin{aligned} H_0 &\text{ — the field due to the constant component of the bias magnetization,}\\ k_9 H_{m_1} &\text{ — the field from the alternating component,}\\ k_9 &\text{ — a constant coefficient, with } |k_9| \ll 1. \end{aligned} \]

Obviously, the instant \(t_0\) of generation of the control pulse is determined from the condition \(H + H_{\text{bias}} = 0\), that is,

\[ H_{m_1}(\omega t_0 - \varphi) = H_0 - k_9 H_{m_1}. \]

We bring this expression to the form

\[ \omega t_0 - \varphi = \frac{H_0}{H_{m_1}} - k_9 \tag{13} \]

and differentiate, considering \(t_0\) and \(H_{m_1}\) to be variable (in the presence of fluctuations of the mains voltage),

\[ \omega\,dt_0 = - \frac{H_0}{H_{m_1}^{2}}\,dH_{m_1}. \tag{14} \]

When the second method of regulating the injection phase is used, the condition

\[ \frac{dt_0}{t_0} = -\frac{1}{2}\frac{dH_m}{H_m} = -\frac{1}{2}\frac{dH_{m_1}}{H_{m_1}}, \tag{15} \]

must be satisfied, since it may be assumed that the amplitude of the magnetic field at the place where the sensor is installed, for small changes of the mains voltage, will be proportional to the field amplitude at the orbit, i.e. the equality

\[ \frac{dH_m}{H_m}=\frac{dH_{m_1}}{H_{m_1}}. \]

As a result of dividing (14) by (15), we obtain:

\[ H_0=\frac{1}{2}H_{m_1}\omega t_0 \]

and from (13) we find, for this method, the value

\[ \kappa_9=\varphi-\frac{\omega t_0}{2}. \tag{16} \]

Since in practice \(\varphi < \frac{\omega t_0}{2}\), in this case the alternating and constant components of the biasing must be directed, according to (12), in one and the same direction (opposite to the main field).

For the third method we take into account the condition

\[ \frac{dt_0}{t_0}=-\frac{dH_m}{H_m}=-\frac{dH_{m_1}}{H_{m_1}} \]

and in an analogous way find:

\[ H_0=H_{m_1}\omega t_0, \tag{17} \]

\[ \kappa_9=\varphi. \tag{18} \]

Here the alternating and constant components of the biasing have opposite directions (the constant component is directed opposite to the main field, the alternating component—in the direction of the main field).

From (12) and the last relations it is seen that the alternating component of the biasing compensates the phase shift \(\varphi\), whereas the constant component ensures automatic fulfillment of condition (10) under fluctuations of the mains voltage within certain limits.

We shall determine the magnitude of the permissible deviations of the mains voltage from the nominal value for one particular case when regulation is carried out by the third method.

For this case it is necessary to satisfy the condition

\[ U_{\text{net}}t_0=\text{const}=\kappa_{10}, \tag{19} \]

where \(t_0\) is the injection time.

In the third method it is practically easy to realize fulfillment of the approximate matching condition

\[ \frac{\Delta t_0'}{t_0}\simeq-\frac{\Delta U_{\text{net}}}{U_{\text{net}}}. \tag{20} \]

For exact fulfillment of the matching condition (19), the relation

\[ \left(U_{\text{net}}+\Delta U_{\text{net}}\right)\left(t_0+\Delta t_0''\right)=\text{const}=\kappa_{10} \tag{21} \]

must be satisfied.

As a result of dividing (21) by (19), we obtain the condition for exact fulfillment of matching for finite increments \(\Delta t_0''\) and \(\Delta U_{\text{net}}\):

\[ \frac{\Delta t_0''}{t_0} = \frac{1}{1+\frac{\Delta U_{\text{net}}}{U_{\text{net}}}} -1 = -\frac{\frac{\Delta U_{\text{net}}}{U_{\text{net}}}}{1+\frac{\Delta U_{\text{net}}}{U_{\text{net}}}}. \tag{22} \]

Subtracting (20) term by term from (22), we determine the relation between the phase error and the deviation of the line voltage from its nominal value:

\[ \frac{\Delta t_0''-\Delta t_0'}{t_0} = \frac{\left(\dfrac{\Delta U_{\text{line}}}{U_{\text{line}}}\right)^2} {1+\dfrac{\Delta U_{\text{line}}}{U_{\text{line}}}}. \tag{23} \]

For a series of betatrons of \(15\ \text{MeV}\) manufactured at the Tomsk Polytechnic Institute, \(t_0 \simeq 50\ \mu\text{sec}\) at an injection voltage of about \(40\ \text{kV}\). The permissible error, determined on the basis of experimental studies for radiation fluctuations within \(5\%\), is measured by a value of \(0.1\ \mu\text{sec}\). Consequently, for this particular case

\[ \frac{\Delta t_0''-\Delta t_0'}{t_0} = \frac{0.1}{50} = 0.002. \]

Solving equation (23) with respect to \(\dfrac{\Delta U_{\text{line}}}{U_{\text{line}}}\) for this case, we obtain:

\[ \frac{\Delta U'_{\text{line}}}{U_{\text{line}}} = +4.6\% \quad \text{and} \quad \frac{\Delta U''_{\text{line}}}{U_{\text{line}}} = -4.4\%, \]

i.e., the exact matching condition (22) is fulfilled with the required relative accuracy of \(0.2\%\) only when the line voltage varies within the limits from \(95.6\) to \(104.6\%\) of the nominal value. For such calculations one may also use the simplified formula obtained from (23), if in the denominator one neglects the quantity \(\dfrac{\Delta U_{\text{line}}}{U_{\text{line}}} \ll 1\):

\[ \frac{\Delta U_{\text{line}}}{U_{\text{line}}} \simeq \pm \sqrt{\frac{\Delta t_0''-\Delta t_0'}{t_0}}. \tag{24} \]

For our particular case, using the approximate formula we obtain

\[ \frac{\Delta U_{\text{line}}}{U_{\text{line}}} = \pm 4.5\%. \]

For the second method of matching, by analogous calculations it is not difficult to obtain the relation

\[ 1+\frac{\Delta t_0''-\Delta t_0'}{t_0} = \frac{1}{\sqrt{1+\dfrac{\Delta U_{\text{line}}}{U_{\text{line}}}}} + \frac{1}{2}\, \frac{\Delta U_{\text{line}}}{U_{\text{line}}}. \tag{25} \]

Solving this cubic equation by an approximate method, for our particular case we determine the permissible fluctuations of the line voltage for the second method of matching:

\[ \frac{\Delta U_{\text{line}}}{U_{\text{line}}} \simeq \pm 8\%. \]

Using expression (23) or (25), depending on the method of matching, one can determine the permissible fluctuations of the line voltage for any other cases.

In conclusion of the theoretical consideration of the question, we note that in all our reasoning it was tacitly assumed that the capture of electrons, with changing phase relations, always occurred in one and the same part of the injection-voltage pulse (let us say, always at the peak or, what is closer to reality, on one and the same section of the front). Consequently, the influence of the shape of the injection pulse on the matching conditions is not taken into account here. This circumstance makes the entire analysis approximate to a certain extent.

The block diagram we have developed for the entire system for stabilizing the optimum capture conditions has the form shown in Fig. 2. According to this diagram, the alternating voltage stabilized by means of a ferroresonant stabilizer is used to supply all electronic circuits, and also to supply the filament circuits of the bias and injection circuits. When the third method of matching is used, from this same stabilizer—and not from the winding on the magnet, as is shown in Fig. 2 for the second method of matching—the charging transformer of the injection circuit must also receive voltage.

Fig. 2

Block diagram of the stabilization system

1—special winding on the betatron magnet; 2—ferroresonant voltage stabilizer; 3—pulsed electronic circuit; 4—stabilizer of the gun emission current; 5—electronic circuit for controlling the reset phase; 6—electronic direct-current stabilizer; 7—circuit to the charging transformer of the injection circuit; 8—circuit to the thyratron grid of the injection circuit; 9—filament supply of the injection and bias circuits; 10—circuit from the pulse transformer of the injection circuit (from the cathode of the electron gun); 11—gun filament; 12—circuit from the conducting layer of the vacuum chamber; 13—circuit to the thyratron grid of the bias circuit; 14—sensor in the form of a coil with permalloy.

The total power consumption from the ferroresonant stabilizer in this case is measured by a value of the order of 500–600 VA, so that the dimensions of the stabilizer will not be large. A voltage of the order of 220 V from a special auxiliary winding on the electromagnet is used to supply

of transformer \(T_1\), which creates the alternating component of the bias magnetization in order to obtain a constant phase displacement of the injection pulse. The charging transformer of the injection circuit is fed from the same winding if the second matching method is used. A voltage proportional to the voltage on the electromagnet is also required for the electronic control circuit of the phase of extraction, if, for automatic adjustment of the phase that ensures constancy of the energy of the electrons extracted onto the target, a circuit with an integrating network is used [3, 4]. If, however, in the displacement circuit an electronic circuit with a constant delay is used, then as the reference pulse one should use the voltage pulse from the pulse circuit in the grid circuit of the thyratron of the injection circuit.

The electronic pulse circuit generates short control pulses with a steep leading edge (to reduce the spread in the firing of the thyratron in the injection circuit), approximately coinciding with the midpoint of the voltage pulse from the synchronizing-pulse sensor, which makes it possible to reduce the influence of the nonlinearity of the sensor parameters.

This circuit is similar to the circuit of Fig. 9 in one of the published papers [1], and therefore a detailed description of the pulse circuit is not given here.

A description of the stabilizer of the pulsed emission current of the electron gun (more precisely, of the mean value of the electron current to the conducting layer of the vacuum chamber) is given in a separate paper [5].

To create the dc component of the bias magnetization, the circuit uses an electronic dc stabilizer, whose principal circuit diagram is shown in Fig. 3. The use of an electronic stabilizer instead of a ferroresonant one made it possible to eliminate the possibility that the frequency dependence of the stabilized voltage of the latter would affect the operation of the circuit.

Fig. 3

The capacitance \(C_4\) at the output of the stabilizer serves to pass the current of the alternating component of the bias magnetization. The voltage divider \(D\) (Fig. 2) is connected to the cathode of the electron gun. After the isolating pulse transformer \(T\), the voltage is rectified (only the voltage corresponding to the amplitude of the operating pulse is passed) by diode \(L_6\) and averaged in the circuit \(C_5R_6\). The averaged voltage is fed to the grid circuit of the regulating tubes 6P3 (Fig. 3) and serves to correct the constant

component of the bias current as a function of changes in the injection amplitude.

The need for such correction is explained by the fact that a change in the injection voltage is determined not only by a change in the line voltage. Thus, when the third method of synchronization is used, the injection voltage stabilized by a ferroresonant stabilizer will depend on the frequency, unless special types of stabilizers are used. Further, as experience in operating betatrons shows, the injection voltage may change as a result of changes, under the action of mechanical vibrations, in the distribution of the gun-cathode emission current between the anode and the conducting layer of the vacuum chamber.

The emission-current stabilizer maintains a constant electron emission current to the conducting layer, since only at a constant value of this current is the maximum output of the installation ensured. Redistribution of the currents between the layer and the anode obviously changes the magnitude of the total pulse current of the cathode and, consequently, affects the value of the injection voltage.

The presence of the correction circuit makes it possible to allow automatically for these changes in the injection voltage. Otherwise, the operation of the electronic direct-current stabilizer is similar to that of the circuits described in the literature [6] and requires no detailed explanation.

Practical adjustment of the radiation-stabilization circuit is carried out as follows. With the correction circuit open (tube L6 in the circuit of Fig. 3 is absent), the constant and alternating components of the bias are selected in such a way that the radiation is at all times maximal when the voltage on the magnet is varied by \(\pm 10\%\). After this the correction circuit is switched on, and, at constant voltage on the magnet, the constant and alternating components of the control voltage in the circuit of the electronic current stabilizer are selected (with the aid of potentiometers \(r_1\) and \(r_6\) of the circuit in Fig. 3) so that the radiation is at all times maximal when the injection voltage is varied by \(\pm 10\%\).

It should be noted that the method described above for stabilizing the optimum capture conditions will also be effective in the presence of changes in the frequency of the supply voltage, since the capture moment is determined by the value of the parameter \(\omega H_m\), independently of what has caused its change (the value of \(\omega\) or \(H_m\)).

The application of the stabilization circuit described here made it possible to achieve quite effective stabilization of betatron radiation by comparatively simple means.

If there is a need to stabilize the intensity of betatron radiation in the strict sense of the word, then electronic circuits associated with direct measurement of the radiation must be used for the regulating circuit.

REFERENCES

1. T. W. Dietze, T. M. Dickinson, Proceedings of the IRE, 96, 1171—1178, 1949.
2. D. W. Fry, J. Dain, H. H. H. Watson, H. F. Payne, Proceedings of the IEE, 97, part I, 305—319, 1950.
3. I. Paul, T. J. Wang, Electronics, 19, 156, 1946.
4. L. Katz et al., Canadian J. of Research, 28-A, 113, 1950.
5. Razin B. M. Emission-current stabilizer. Izvestiya TPI, 87, 1957.
6. Bonch-Bruevich A. M. The Use of Electron Tubes in Experimental Physics, GITTL, 1954.