STUDY OF AN ELECTRON BEAM INSIDE THE VACUUM CHAMBER OF A BETATRON

L. S. Sokolov

(Presented by Professor, Doctor A. A. Vorob'ev)

This paper considers the behavior of an electron beam in its orbit when the magnetic field of a betatron is disturbed by the field of pulsed currents flowing through windings placed in the inter-pole space of the betatron outside the vacuum chamber. These windings are called displacement windings. We investigated four methods of displacing the electron orbit. They differ structurally in the dimensions, shape, and arrangement of the displacement windings in the inter-pole space of the betatron (Fig. 1). The initial equilibrium orbit of the electrons is shown in Fig. 1, 1 by a dash-dotted circle.

The electrical supply circuit of the displacement windings is usually called the displacement circuit.

According to the conditions of the experimental problems, the displacement circuit and the additional equipment for it must provide:

a) pulsed supply of the displacement winding with sufficiently large currents;

b) the possibility of regulating the moment at which the displacement pulse appears in time;

c) the possibility of changing the amplitude of the displacement current;

d) the possibility of changing the duration of the current pulse;

e) the possibility of observing the displacement pulse and the radiation pulse.

The pulse circuit and the additional equipment satisfying these requirements are shown in Fig. 2. Operation of the circuit begins with the charging of capacitors \(C\) and \(C_1\) from charging transformers \(T\) and \(T_1\) through the gas-filled tubes VG-129 and VG-237; at the same time, leakage of charge through the thyratron TG15/3000 and the ignitron I-100/1000 is prevented by the fact that the thyratron is locked by a negative bias of 80–100 V on the grid, while the ignitron passes current only when the thyratron is operating.

The process of charging capacitors \(C\) and \(C_1\) usually takes place during the nonworking half of the betatron cycle, and by the moment of displacement they are already charged.

At the required moment in time, controlled by means of a phase shifter, a voltage pulse with an amplitude of 100–120 V appears on the secondary winding of the pulse transformer PT; this pulse triggers generators 1G and 2G, which deliver voltage pulses with amplitudes up to 130 V with an adjustable delay relative to the triggering pulse.

Generator 1G starts the triggered sweep circuit of the pulsed two-beam oscilloscope IO simultaneously with the pulse from the pulse transformer, while the second generator, after a certain delay, fires the thyratron. Capacitor \(C_1\) discharges through the ignitron igniter, and capacitor \(C\) through the displacement winding \(L\), the measuring resistance \(R\), and the ignitron. The voltage drop across resistance \(R\) and the current pulse produced by X-rays in the photoelectron multiplier FEU-19 are fed to the input terminals of the first and second beams of the oscilloscope.

The presence of a delay of 10–20 μsec between the moment when the waiting sweep of the oscilloscope is started and the moment when the ignitron is fired ensures clear observation of the beginning of the displacement pulse and of the beginning of the radiation pulse.

The pulse of the displacing current has the form of a sinusoidal half-wave. The amplitude of the pulse \(I\) is changed by changing the voltage on the primary winding of the transformer charging the capacitor \(C\), by means of a laboratory autotransformer \(A\). The width of the pulse \(T/2\) is adjusted by changing the capacitance of the capacitor \(C\). By analogy with the frequency of sinusoidal periodic processes, we shall call the quantity

\[ \Omega = \frac{2\pi}{T} \tag{1} \]

the frequency of the displacing current.

Passing through the winding \(L\), having \(\omega\) turns, the displacing currents produce perturbing magnetic fields, which are shown in Fig. 1, 1. The magnitudes of the perturbing magnetic fields correspond to complete removal of the electrons from the working zone of the accelerator.

Since the rate of displacement of the electron orbit depends equally on the amplitude of the pulse of the displacing magnetic flux and on its width, we introduce the quantity \(f\), characterizing the rate of increase of the displacing magnetic flux.

It is convenient to calculate the quantity \(f\) by substituting \(I\) in a and \(T\) in μsec:

\[ f = \frac{2I\omega}{T}. \tag{2} \]

Fig. 1. Layout of the displacing windings (1) and their action (2).

— unperturbed betatron field;
— betatron field perturbed by winding \(a\);
— betatron field perturbed by winding \(b\) or \(v\);
\(\cdots\) betatron field perturbed by the winding in the gap.
The fields were measured along the line \(AA\) in the mid-plane of the interpolar space.

The electrical supply circuit of the displacing windings makes it possible to vary the values of \(f\) within the limits

\[ 10 < f < 120, \]

at which complete removal of the electrons from the working zone of the betatron is possible. The moment of displacement of the equilibrium orbit can be smoothly regulated within the limits from 0 to 90 electrical degrees.

For studying the electron beam we required a special vacuum chamber with five branch tubes (Fig. 3).

An injector is located in branch tube \(A\). In order to exclude the influence of the injector during measurements, it is placed outside the plane of the equilibrium orbit.

Identical tungsten targets are placed in the diametrically located branch tubes \(B\) and \(G\). The targets are mounted on rods that can be translated without disturbing the vacuum in the betatron chamber. The general appearance of the mechanism permitting the target to be moved radially inside the vacuum chamber is shown in Fig. 4.

In branch tube \(V\) there is a tungsten filament 0.2 mm in diameter, which is attached to a rod by means of an elbow lever.

(Fig. 5). Without breaking the vacuum in the betatron chamber, the rod can be rotated together with the lever and the filament. The angle of rotation is recorded by a pointer on the transporter. When the rod is rotated, the filament moves over the surface of a circular cylinder.

To determine the radial velocity of expansion of the electron orbit, target Б is used; the radiation pulse from it is recorded by ionization chamber 1 and photomultiplier 3

Fig. 2. Bias circuit and auxiliary equipment for studying the electron beam.

(Figs. 3 and 4). Targets В and Г are removed from the working zone. On the oscillogram (Fig. 6) one can see the pulse of the bias current and the current pulse from the photomultiplier, caused by radiation from target Б. The dependence of the delay time \(t\) between the beginning of the radiation pulse and the beginning of the bias pulse on the radial position of target Б (Fig. 7) makes it possible to calculate the velocity of expansion of the orbit \(v_r\) and the distance \(\Delta r_0\) by which the orbit advances radially during one revolution of the electron in the orbit (the orbit step). For example, when target Б is moved from \(r_1 = 157.5\) mm to \(r_2 = 160\) mm, the value of \(t\) increased from \(t_1 = 3.2\) μsec to \(t_2 = 4.2\) μsec (Fig. 7). Then the average velocity of displacement of the orbit in the interval between the radii \(r_1\) and \(r_2\) is equal to:

Fig. 3. Experimental arrangement for measurements of the electron beam inside the vacuum chamber.

Fig. 4. General view of the mechanism for moving the target radially inside the vacuum chamber.

\[ v_r=\frac{r_2-r_1}{t_2-t_1}=\frac{2.5}{1}=2.5\ \text{mm}/\mu\text{sec}. \tag{3} \]

The orbit step is equal to:

\[ \Delta r_o = T_o v_r = 2.7 \cdot 10^{-3} \cdot 2.5 = 6.75 \cdot 10^{-3}\ \mathrm{mm}, \tag{4} \]

where \(T_o\) is the period of revolution of the electron in the orbit:

\[ T_o = \frac{2\pi r_o}{C} = \frac{6.28 \cdot 13}{3 \cdot 10^{10}} = 2.7 \cdot 10^{-9}\ \mathrm{sec}. \tag{5} \]

We have considered the determination of \(v_r\) and \(\Delta r_o\) in the case when winding \(v\) is used; here \(f = 19.2\ \mathrm{A}/\mu\mathrm{sec}\) and \(E = 5\ \mathrm{MeV}\). In exactly the same way one can determine \(v_r\) and \(\Delta r_o\) for other dependences obtained under

Fig. 5. Mechanism for measuring the height of the electron beam inside the vacuum chamber.

Fig. 6. Oscillogram of the pulses of the displacement current and of the radiation pulse (short pulse).

Fig. 7. Dependence of the delay time \(t\) between the beginning of the displacement pulse and the beginning of the radiation pulse on the radial position \(r\) of target \(B\), for various methods and modes of orbit displacement. Designations as in Fig. 1.

other conditions. In addition, we have chosen only one interval for measuring \(v_r\) and \(\Delta r_o\), lying between radii 157.5 and 160 mm. Let us take any dependence among those presented in Fig. 7 and determine \(v_r\) and \(\Delta r_o\) in such an interval of radii where \(t\) depends very weakly on \(r\) \((t \simeq \mathrm{const})\). In this case \(t_{i+1} - t_i \simeq 0\); consequently, \(v_r\) and \(\Delta r_o \to \infty\).

As is seen from Fig. 7, portions of the dependences where \(t \simeq \mathrm{const}\) occur for all the graphs. They begin at definite radii and continue beyond the working zone of the accelerator. The transition from finite values of \(v_r\) and \(\Delta r_o\) to infinitely large ones takes place over a small radial interval. The radii at which this transition occurs are readily determined from the dependences in Fig. 7.

They are equal to:

for windings \(a\) . . . . . . . . . . . . . . . . . . . . . \(162\ \mathrm{mm}\)
for windings \(b\) . . . . . . . . . . . . . . . . . . . . . \(165\ \mathrm{mm}\)
for windings \(v\) . . . . . . . . . . . . . . . . . . . . . \(165\ \mathrm{mm}\)
for windings \(v_{\mathrm{zaorb}}\) . . . . . . . . . . . . . \(160\ \mathrm{mm}\)

These radii correspond to those circles on which the electrons are freed from the influence of the focusing forces.

From the graphs in Fig. 7 one can draw the following additional interesting conclusions.

If the velocity of orbit displacement \(v_r\) were the same in the interval from the radius of the initial orbit \(r_o\) to the radius at which a sharp increase in \(v_r\) occurs, then the graphs of Fig. 7 in this interval would be inclined straight lines intersecting the \(r\) axis at the point \(r_o\). The condition \(v_r=\mathrm{const}\) within the indicated interval of radii is technically easy to realize by passing, for example, pulsed currents of large amplitude through the displacing windings (2–3 times greater than the current required for complete removal of the electrons from the working zone of the betatron). The dependences \(v\) \((f=29)\) and \(v\) \((E=5\ \mathrm{MeV})\) in Fig. 7 were obtained precisely under such conditions. Extrapolating them to the abscissa axis (which is not shown in Fig. 7) makes it possible to determine the radius of the initial equilibrium orbit (or the radial position of the initial orbit at the given azimuth, if it is distorted) immediately before its displacement.

By changing the instant of orbit displacement, one can trace its displacement during the acceleration of the electrons. It turned out that at an energy of \(5\ \mathrm{MeV}\) the orbit is removed from the center of the accelerator (at the azimuth of our measurement) by a distance of \(150\ \mathrm{mm}\), and at an energy of \(10\ \mathrm{MeV}\) by a distance of \(145\ \mathrm{mm}\). The calculated orbit radius, which was checked by comparing the mean field strength inside the circle of the given radius with twice the magnetic-field strength on this circle (the \(2:1\) condition), is \(130\ \mathrm{mm}\).

The discrepancy between the calculated and experimental values of \(r_o\) may occur as a result of distortion of the orbit. We shall return to this question after measurements of the radial size of the electron beam.

The contraction of the orbit with increasing electron energy is an unquestionable fact. A detailed study of orbit displacement during electron acceleration could prove very useful in clarifying the question of why some betatrons do not provide a satisfactory yield of bremsstrahlung with the maximum calculated energy.

If the radial velocity \(v_r\) of displacement of the electron orbit is known, the radial size of the beam \(\Delta r\) can be determined by Gregg’s method [1]:

\[ \Delta r = v_r \Delta t . \tag{6} \]

Here \(\Delta t\) is the duration of the radiation pulse from the target, which can be measured on the oscilloscope screen.

To determine the value of \(\Delta r\) at a given radius by formula (6), the quantities \(v_r\) and \(\Delta t\) should be measured at the same radius.

Let us consider an example of determining the radial size of the beam. In Fig. 8 are presented oscillograms of the radiation pulse and of the initial part of the displacement pulse at various radial coordinates of the target and with the use of winding \(a\).

The target coordinate was changed by \(2.5\ \mathrm{mm}\). Measuring from the oscillograms the distance \(t\) between the beginning of the displacement pulse and the beginning of the radiation pulse, we find \(v_r\) in the radius intervals \(r_1=152.5\ \mathrm{mm}\) and \(r_2=155\ \mathrm{mm}\); \(r_2=155\ \mathrm{mm}\) and \(r_3=157.5\ \mathrm{mm}\), etc.:

\[ v_{r1}=\frac{2.5}{(t_2-t_1)\alpha};\quad v_{r2}=\frac{2.5}{(t_2-t_1)\alpha};\quad v_{r3}=\frac{2.5}{(t_4-t_3)\alpha} \quad \text{and so forth,} \]

where \(\alpha\) is the coefficient reducing the denominator of \(v_r\) to microseconds.

On the oscillogram of Fig. 8 two sinusoidal quantities are shown, whose period of variation \(\tau\) is known (\(2\ \mu\mathrm{sec}\)):

\[ \alpha=\frac{\tau}{l}, \]

where \(l\) is the distance on the oscillogram corresponding to the period \(\tau\), mm.

The duration of the radiation pulse is equal to:

\[ \Delta t_i a \quad (i=1,\ 2,\ 3,\ldots). \]

In accordance with formula (6), we obtain the dependence of \(\Delta r\) on the radius (see the table).

In the table, \(J\) is the radiation from the target (in relative units, measured by ionization chamber 1) (Fig. 3). From the decrease of \(J\) at large radii one may judge that the electrons do not strike the target, being lost on the walls of the vacuum chamber. To verify the data of the table, the radial size of the beam was measured by another method.

Fig. 8. Oscillograms of the displacement pulse and the radiation pulse for various radial positions of target \(B\) (winding \(a\)). The time axis is directed from right to left.

\(1 — r = 167.5\) mm; \(2 — r = 165\) mm; \(3 — r = 162.5\) mm; \(4 — r = 160\) mm; \(5 — r = 157.5\) mm; \(6 — r = 155\) mm; \(7 — r = 152.5\) mm.

The idea of our method for determining the horizontal size of the beam during the orbit-displacement process is illustrated in Fig. 3. Target \(G\) is placed at a definite radius. Targets \(B\) and \(V\) are removed from the working zone. In the process of orbit displacement, the electron beam moves outward and, striking target \(G\), produces radiation recorded by ionization chamber 2. Target \(B\) is gradually moved inward. At some radius, target \(B\) will begin to intercept the electron beam, and with further inward motion of target \(B\) all the electrons will be stopped in it. Typical dependences of the bremsstrahlung radiation arising in targets \(B\) and \(G\) on the radial coordinate of target \(B\) are presented in Fig. 9. The radial interval in which the radiation from targets \(B\) and \(G\) changes is equal to the radial size of the beam \(\Delta r\) at the radius corresponding to the middle of the radial interval.

We note that the dependences in Fig. 9 were obtained using winding \(a\); moreover, during the first displacement of target \(B\) from a radius of 169 mm to a radius of 165 mm, target \(G\) was at a radius of 154 mm. During the second displacement of target \(B\) from \(r = 165\) mm to \(r = 163\) mm, target \(G\) was ...

was located at a radius of 151.5 mm, and with the third displacement of target B from \(r=162\) mm to \(r=161\) mm, target G was located at a radius of 149 mm. It follows from this that the equilibrium orbit of the electrons is strongly distorted.

Figure 10 presents the dependences between the coordinates of targets B and G, obtained under such conditions that one half of the electrons of the beam is stopped in target B, and the other half in target G. In other words, along the abscissa axis of Fig. 10 are plotted the radial coordinates of the orbit at the azimuth \(\Theta=\pi\), and along the ordinate axis—the corresponding radial coordinates of the orbit at the azimuth \(\Theta=0\).

Fig. 9. Change in the radial size of the beam by the two-target method.

Fig. 10. Distortion of the equilibrium orbit in the course of its expansion by symmetric displacing windings.

The dependences of Fig. 10 correspond to symmetric expansion of the orbit by windings \(a\) (solid line) and \(b\) (dashed line). The presence of distortion of the orbit cannot be the reason for a worsening of the conditions for extracting the beam from the accelerator; therefore we did not investigate the causes of the orbit distortion.

The method considered here (two targets) does not make it possible to determine the radial size of the beam when methods of asymmetric displacement of the orbit are used (windings \(v\) and \(v_{\text{gap}}\)), since one of the targets has to be placed very far inside the vacuum chamber; this leads to complete cessation of the process of electron acceleration.

Graphs of the change in \(\Delta r\) as a function of radius for symmetric displacement methods are presented in Fig. 11.

Comparing the dependences of \(\Delta r\) on \(r\) (Fig. 11) with the dependences presented in the table, we see that the values of \(\Delta r\) measured by the one-target method are 10 times greater than the values of \(\Delta r\) obtained by the two-target method.

Fig. 11. Change in the radial size of the beam with radius. The dependences were obtained by the two-target method.

We have no grounds to believe that the photomultiplier and the system for amplifying the radiation pulse increase the duration of the radiation pulse. It remains to assume that, when \(\Delta r\) is measured by the one-target method (Gregg’s method), some effects arise inside the vacuum chamber that increase the duration of the radiation pulse. When \(\Delta r\) is measured by the two-target method, these effects are eliminated. Such effects may include scattering of electrons from the target in such a way that they do not collide with the walls of the vacuum chamber and,

being subjected to the action of displacement, after some time are again dumped onto the target. With two targets present, the free space for the radial motion of the electrons is reduced, and the “lifetime” of the electrons in the vacuum chamber is shortened.

Our supposition is partly confirmed by the complex form of the radiation pulse (oscillograms in Fig. 8). In all methods of orbit displacement, a small maximum is observed at the beginning of the radiation pulse. This maximum may signify the action of the primary electron beam, whereas the remaining part of the radiation pulse is due to scattered electrons.

In measurements of $\Delta r$ we used, as a target, a tungsten plate $1\ \mathrm{mm}$ thick. The increase in the duration of the radiation pulse from a single target due to scattered electrons, when the same target is used and is placed within the region of action of the focusing forces, can be approximately taken into account by knowing the ratio of the values of $\Delta r$ measured by the two methods considered.

Fig. 12. Oscillograms of the displacement pulse and radiation pulse for different positions of target B (winding c).
The time scale is the same as in Fig. 11.
$1$ — $r = 152.5\ \mathrm{mm}$; $2$ — $r = 157.5\ \mathrm{mm}$; $3$ — $r = 162.5\ \mathrm{mm}$;
$4$ — $r = 167.5\ \mathrm{mm}$. The time axis is directed from right to left.

This ratio, obtained with symmetric methods of orbit displacement, can be used for measuring $\Delta r$ with asymmetric methods of orbit displacement, when the two-target method cannot be used.

Oscillograms of the displacement pulse and the radiation pulse for different positions of target B and with the use of winding c are shown in Fig. 12.

From the oscillograms in Fig. 12 it is seen that, when asymmetric methods of orbit displacement are used, the radiation pulse from a target at radii where the condition $n=\mathrm{const}^{*}$ is violated splits into two. The possibility of the appearance of two maxima in the radiation pulse had already been noted by Lawson [2].

On the basis of the data in Fig. 12, it may be assumed that the appearance of two maxima in the radiation pulse is due to scattering of electrons from the target. The condition for the appearance of two maxima in the radiation pulse is distortion of the equilibrium orbit. In this case the target must be located near the circumference where $n=1$, i.e., where there is a sharp change in the focusing properties of the betatron field. The effects of an increase in the duration of the radiation pulse owing to scattering of electrons from the target should be taken into account when it is necessary to obtain a short radiation pulse from the target.

The fact that the electron beam has a definite radial size, which changes in the course of orbit displacement, can be explained by the presence of radial oscillations of the electrons. These oscillations have different amplitudes and phases. Within the stable region of motion the radial size of the beam is small and amounts to $1$–$2\ \mathrm{mm}$. A sharp increase in the radial—

* $n$ is the exponent of the falloff of the betatron magnetic field with radius.

of the beam size begins together with an increase in the velocity of orbit displacement as it approaches the boundary of the stable region of motion.

To determine the beam height, a tungsten wire \(B\) is used, the inner edge of which is located at a radius of 15 cm, while targets \(Б\) and \(Г\) are removed from the working zone. The radiation that arises when the electrons are decelerated in the material of the wire is recorded by ionization chamber 1 (Fig. 4). Moving over the surface of a circular cylinder parallel to the horizontal plane, the wire intersects the electron beam in height; in this process the readings of ionization chamber 1 change. The more electrons strike the wire, the higher the readings of the ionization chamber should be. The dependence of the radiation on the vertical coordinate of the wire should give a picture of the distribution of electrons inside the vacuum chamber in height at a radius of 15 cm. A number of these dependences, obtained for different methods of orbit displacement, is shown in Fig. 13.

Fig. 13. Dependence of the readings of ionization chamber 1 (Fig. 3) on the vertical coordinate of the tungsten wire for various conditions of orbit displacement \((E = 10\ \text{MeV})\).

The radial velocity of orbit displacement, the radial size, and the height of the beam are of great interest in the design and implementation of extraction of charged particles from cyclic accelerators. The work was carried out under the supervision of Professor, Doctor of Technical Sciences A. A. Vorob’ev.

References

1. Gregg E. S., Betatron with magnetization and field forcing, Problems of Modern Physics, II, 26, 1952 (p. 140).

2. Lawson J. D., Double beam in a 30 MeV synchrotron, Problems of Modern Physics, II, 5, 1952 (p. 60).