On the Question of Choosing a Rational Location for the Injector and Determining the Magnitude of the Injection Voltage for a Betatron

K. S. Grishin

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

In the practice of designing, tuning, and operating betatrons whose magnetic field has phase and static inhomogeneity, a whole series of very important questions arises, on the correct solution of which the successful operation of the betatron depends.

Many of them are usually reduced to the following:

1. What are the true conditions for the motion of electrons, and what is the actual width of the working zone in which electrons are accelerated in a magnetic field with the given characteristics?

2. How is one to determine the most advantageous location of the injector in order to obtain the maximum radiation intensity?

3. What should be the magnitude of the injection voltage, that is, what should be the phase of the beginning of injection, in order to ensure successful operation of the betatron?

4. What structure of the magnetic field is the most desirable, and what measures should be taken to improve the structure of this field in order to achieve the maximum yield of rays from the betatron?

The answer to such questions can be obtained as a result of a comprehensive, thorough investigation of the magnetic field.

The investigation of the magnetic field includes, first of all, the necessary set of measurements associated with determining the magnitude of the radius of the equilibrium orbit \(r_o\) and bringing it into agreement with the calculated value. Next come measurements of the amplitude values \(B_{\mathrm{cp}\,m}\) and \(B_m\), and the dependences \(B_{\mathrm{cp}\,m}=f(r)\) and \(B_m=f(r)\) are ascertained. Then measurements are carried out to determine the azimuthal static inhomogeneity of the magnetic field at various radii, and the overall static structure of the field in the plane of the orbit is determined; finally, measurements are made to determine the phase structure of the field.

Phase inhomogeneity is evaluated relative to the magnetic field at some point of the interpolar space of the betatron. Since the majority of the questions listed relate to the injection period, in order to solve them it is necessary to know the instantaneous picture of the field at various moments of its rise. First of all, it is necessary to investigate the question of whether always, and at all points of the interpolar space,

whether the basic conditions necessary for stable acceleration of electrons are satisfied (this refers chiefly to the value of \(n\)), i.e., whether the condition \(0<n<1\) is always and everywhere satisfied. Indeed, if, beginning from some instant during the acceleration of the electrons, the magnetic field along their path becomes such that \(n<0\) or \(n>1\), then the electrons will leave the stable acceleration regime and will be lost.

For an in-phase variation of the betatron magnetic field, for which \(n\) is not a quantity constant with radius, one may, with sufficient accuracy for individual small segments between points \(a\) and \(b\) of the curve \(B_m=f(r)\), consider the relation

\[ \frac{B_{ma}}{B_{mb}}=\left(\frac{r_b}{r_a}\right)^{n_{a-b}}, \tag{1} \]

where \(B_{ma}\) and \(B_{mb}\) are the amplitude values of the field at points \(a\) and \(b\), located respectively at radii \(r_a\) and \(r_b\) at the given azimuth \(\Theta\).

If \(r_a-r_b\to 0\), then within the limits of the given segment one may take \(n_{a-b}=\text{const}\).

If the dependence \(B_m=f(r)\) is known in the form of an experimental curve, then, for an in-phase variation of the field, for any point located at the radius \(r=\dfrac{r_a+r_b}{2}\), the value of \(n\) may be calculated from the expression

\[ n_{ma-b}= \frac{\lg\left(\dfrac{B_{ma}}{B_{mb}}\right)} {\lg\left(\dfrac{r_b}{r_a}\right)} \tag{2} \]

and, from the points, the graph \(n=f(r)\) may be constructed. It is not difficult to show that the instantaneous values of \(n\) in this case will be equal respectively to the values of \(n_m\) calculated for the amplitude values \(B\). In fact, for instantaneous values one may write

\[ n_{a-b}= \frac{\lg\left(\dfrac{B_a}{B_b}\right)} {\lg\left(\dfrac{r_b}{r_a}\right)} = \frac{\lg\left(\dfrac{B_{ma}\cdot \sin\Theta}{B_{mb}\cdot \sin\Theta}\right)} {\lg\left(\dfrac{r_b}{r_a}\right)} = n_{ma-b}, \tag{3} \]

that is, \(n_{a-b}=f(r)=n_{ma-b}\).

If at one of the points lying in the plane of the orbit, taken as the reference point of the field, the amplitude of the field strength \(B_m\) and the static field inhomogeneity \(\Delta B_a\) at some other point \(a\), relative to the reference point, are known, then the field amplitude \(B_{ma}\) at point \(a\) may be found as

\[ B_{ma}=B_m+\Delta B_a. \]

In turn, \(\Delta B_a=\Delta B_{ra}+\Delta B_{\varphi a}\), where \(\Delta B_{ra}\) and \(\Delta B_{\varphi a}\) are the radial and azimuthal components of the static field inhomogeneity at point \(a\) relative to the field at the reference point. Since, as a result of measurements, one can always obtain, for the reference azimuth on which the reference point is located, the experimental dependence \(B_{mr}=f(r)\), the field amplitude at point \(a\) may be calculated as

\[ B_{ma}=B_{mra}+\Delta B_{\varphi a}, \tag{4} \]

where \(B_{mra}=B_m+\Delta B_{ra}\) is the field amplitude on the reference azimuth at the point \(r=r_a\).

In the presence of a static inhomogeneity of the magnetic field, the dependence \(n=f(r)\) for each azimuth will in general be different, but during the time of field rise the value of \(n\) at any point of the field remains constant, and the dependence \(n=f(r)\) for any azimuth will be unchanged.

Let us consider a more general case, when the magnetic field of the betatron has both static and phase inhomogeneities. Such is the field of real betatrons.

The phase inhomogeneity of the field at any point of the interpole space can always be measured relative to the field at a reference point¹.

If the instant at which the field passes through zero at the reference point is taken as the origin of time, then the instantaneous values of the field at point \(a\) can be calculated from the expression

\[ B_a = B_{ma}\cdot \sin(\varphi+\Delta\varphi_a), \tag{5} \]

where \(\varphi\) is the phase corresponding to the time \(t\) of field rise at the reference point; \(\Delta\varphi_a\) is the phase inhomogeneity of the field at point \(a\) relative to the field at the reference point.

Similarly, for the previously considered point \(b\) we have

\[ B_b = B_{mb}\cdot \sin(\varphi+\Delta\varphi_b). \]

The instantaneous value of \(n\) in the interval \(r_a\), \(r_b\) between points \(a\) and \(b\) in this case can be found from the expression

\[ n_{a-b} = \frac{\lg\left(\dfrac{B_a}{B_b}\right)} {\lg\left(\dfrac{r_b}{r_a}\right)} = \frac{ \lg\left[ \dfrac{B_{ma}\cdot \sin(\varphi+\Delta\varphi_a)} {B_{mb}\cdot \sin(\varphi+\Delta\varphi_b)} \right] } {\lg\left(\dfrac{r_b}{r_a}\right)}. \tag{6} \]

It is not difficult to see that, in the presence of a phase inhomogeneity of the magnetic field, the quantity \(n\) changes continuously with time. This change is especially large at the beginning of the rise of the magnetic field, when \(\varphi\) is small. The law of variation of \(n\) depends strongly on the magnitude and character of the phase inhomogeneity.

In order to clarify the meaning of equation (6) and to assess the importance of the general conclusions that follow from it, it is useful to consider an example of its use for analyzing the magnetic field of a specific betatron. As such a betatron, a 5 MeV betatron excited by an industrial-frequency current was used. This betatron had \(n=2/3\), \(r_0=9.5\ \text{cm}\), pole diameter \(25\ \text{cm}\), and central insert diameter \(9\ \text{cm}\).

Figure 1 presents the results of measurements of the amplitude values \(B_{\mathrm{cp}}\) and \(B\) after setting the orbit to the calculated radius. The measurements were made in the plane of the orbit at the azimuth \(0^\circ\), taken as the reference. Also shown here are graphs of the potential function \(V_M=\left(\dfrac{B_{\mathrm{cp}}}{2}\cdot r\right)^2\) and \(n=f(r)\), constructed on the basis of these data \(B_{\mathrm{cp}}=f(r)\) and \(B=f(r)\), without taking account of the radial phase inhomogeneity.

The slope of the potential curve \(V_M=f(r)\) gives the magnitude and direction of the force acting on the electron at a given point during its oscillations relative to the equilibrium orbit. Electrons can execute oscillations

only under the condition of equality of their potential energies at the points of maximum deviations from the equilibrium orbit. If one proceeds only from this assumption, then, using the graph \(V_m=f(r)\), the zone of maximum permissible amplitudes of electron oscillations (the working zone) may be determined by the radii \(r_{\min}=6.3\) and \(r_{\max}=12.4\ \text{cm}\), where \(V_m=V_{m\min}\). Naturally, the injector can be placed only within this zone. However, if one turns to the curve \(n=f(r)\), then the zone of permissible oscillation amplitudes (correspondingly, the zone of possible injector locations) is bounded by the radii \(r_{\min}=6.2\ \text{cm}\), where \(n=0\), and \(r_{\max}=11.7\), where \(n=1\).

Fig. 1

It should be taken into account that \(B_{\mathrm{av}}=f(r)\), from which \(V_m\) was calculated, is a certain averaged quantity, measured without taking into account phase and static inhomogeneities. The true value of \(V_m\) is a function of time and depends on the degree of phase and static inhomogeneity, as well as on the difference in the law of increase of the magnetic field with time at different points of the interpole space of the betatron. It should also be borne in mind that \(n\) was calculated under the same assumptions. Therefore, the width of the working zone (the limits of permissible electron oscillations), determined from these graphs, is approximate and only to some degree corresponds to its true value at the end of electron acceleration.

To determine the true boundaries of the working zone, measurements were carried out to determine the static and phase structure of the magnetic field. Fig. 2 presents the results of measuring the azimuthal static inhomogeneity of the magnetic field in the plane of the orbit relative to the field at the main azimuth (\(0^\circ\)) and the corresponding radius. The amplitude values of the field at azimuth \(0^\circ\) are presented in the table in the same figure.

To determine the amplitude of the magnetic field at any point of the orbit plane, it is sufficient to add algebraically to the field amplitude at azimuth \(0^\circ\) and the same radius the value of the static field inhomogeneity found from the graph in Fig. 2 for this point. It should be noted that the static structure of the betatron magnetic field remains similar during the period of increase of the magnetic field, while the absolute static inhomogeneity at any point is proportional to the instantaneous value of the field.

Fig. 2

In Fig. 3 are given the results of measurements of the phase inhomogeneity of the magnetic field of the betatron under investigation in the orbit plane. The phase difference in the azimuthal and radial directions was estimated in microseconds relative to the field phase at the point where the control picker was located. If the point located in the orbit plane at azimuth \(0^\circ\) and radius \(10\ \text{cm}\) is taken as the reference point, then the scale plotted in Fig. 3 may be used to determine the phase inhomogeneity.

Figure 4 presents the phase structure of the magnetic field of the same betatron after correction of the azimuthal phase inhomogeneity by means of a system of short-circuited turns enclosing separate packages of electromagnet posts. The radial phase inhomogeneity in this case remained almost unchanged, whereas the azimuthal inhomogeneity decreased by a factor of 7.5. In addition, the transition from the phase structure (Fig. 3) to the field structure (Fig. 4) had almost no effect on the absolute magnitude and the general character of the static structure of the magnetic field. In both cases the data of Figs. 1 and 2 remained practically unchanged.

All the experimental data given above are sufficient for investigating the structure of the magnetic field in the orbit plane at any instant during its rise. For certain characteristic azimuths, at which we wish to determine the instantaneous values of \(n\), it is useful to construct the graph \(\Delta\varphi = f(r)\). Such a graph, constructed from the data of Fig. 3 for azimuths \(0^\circ\) and \(90^\circ\), is shown in Fig. 5.

Fig. 3

Fig. 4

In Fig. 6 the instantaneous values of the magnetic field at azimuth \(0^\circ\), calculated from expression (5) for various times \(t\), are presented. For small values of \(t\) one can note a considerable deviation from

Fig. 5

Fig. 6

the required law \(B=f(r)\). This deviation is especially large for \(t=0\) and \(t<0\).

In Figs. 7 and 8 the instantaneous values of \(n\) for various times \(t\), calculated from formula (6) for azimuths \(0\) and \(90^\circ\), are given.

Fig. 7

using the data of the graphs in Figs. 1, 2, and 5. Here, too, the values \(n=f(r)\), calculated for \(t=1/200\) sec, i.e., for the instant when the magnetic field at the principal point has its amplitude value, are presented. The calculation of \(n\) was carried out for points separated from one another by a distance of 5 mm \((r_1-r_2=5\ \text{mm})\).

Fig. 8

Some waviness of the curves \(n=f(r)\) for azimuth \(0\) is explained by the character of the radial static structure of the magnetic field at the given azimuth. This is clearly seen on the curve for \(t=1/200\) sec, constructed at the amplitude value of the magnetic field, when the phase nonuniformity becomes hardly perceptible; this is also shown most convincingly by the graph \(n=f(r)\) in Fig. 2, constructed without taking the phase nonuniformity into account at all. For azimuth \(90^\circ\), this waviness is intensified due to the azimuthal static nonuniformity.

For \(r<6\) cm, at any instant of time \(t\), the quantity \(n<0\), because in the median plane of the betatron under investigation there is no central gap, and the lines of force of the magnetic field in the zone directly adjoining the central insert are directed convexly toward the center of the interpolar space.

Determination of the Width of the Working Zone

Analyzing Figs. 7 and 8, one can see that, in the presence of phase nonuniformity of the magnetic field during its rise, the quantity \(n\) does not remain constant, as we observe for a field varying in phase when there is only static nonuniformity, but changes continuously. This change is especially large at the beginning of the rise of the magnetic field, when the phase nonuniformity has the greatest influence on the field structure.

For example, at azimuth \(0^\circ\) (Fig. 7), the curve of instantaneous values of \(n\) at the instant \(t=0\) rises so steeply that at the radius 6.4 cm it already passes into the region \(n>1\). At \(t=5\ \mu\)sec its steepness is considerably

decreases; at subsequent instants it begins rapidly to approach the curve constructed for the amplitude values of the field \(B\). For azimuth \(90^\circ\) (Fig. 8), the curves of the instantaneous values \(n=f(r)\) change with time at an even higher rate.

It is of interest to trace the change in the width of the working zone (where \(0<n<1\)) and in the absolute magnitude of the instantaneous values of \(n\) during the time of rise of the magnetic field. At azimuth \(0^\circ\), the width of the working zone at \(t=0\) is bounded by the radii \(r_{\min}=6.06\) and \(r_{\max}=6.4\ \text{cm}\), i.e., it is only \(0.34\ \text{cm}\), and the value of \(n\) within this interval varies with \(r\) almost linearly. At \(t=5\ \mu\text{s}\) the working zone broadens to \(1.65\ \text{cm}\), being bounded by the radii \(r_{\min}=6.1\) and \(r_{\max}=7.75\ \text{cm}\), while the value of \(n\) within this zone at first increases rapidly, and then more slowly. At \(t=15\ \mu\text{s}\), owing to the wavelike character of the curves \(n=f(r)\), there are two zones in which \(0<n<1\): one lies within the radii \(r_{\min}=6.15\ \text{cm}\) and \(r_{\max}=9.1\); in it \(n\) varies from 0 to 1. The second is bounded by the radii \(r_{\min}=9.6\) and \(r_{\max}=10.4\ \text{cm}\); in it \(n\) varies from 1 to 1. The existence of these two zones is due to the character of the static inhomogeneity of the magnetic field. At \(t=15.25\ \mu\text{s}\) these two zones merge into one; as the radius increases, the value of \(n\) at first grows rapidly and then more slowly, remaining, in the interval of radii \(8\text{–}10.5\ \text{cm}\), close to unity. During the further rise of the field, the working zone continues to expand, but already much more slowly, while the value of \(n\) inside this zone decreases more and more, gradually approaching the values of \(n\) for the amplitude field \(B\). Let us recall that all these changes in \(n\) occur because of the radial phase inhomogeneity of the magnetic field.

At azimuth \(90^\circ\), the changes in \(n\) and in the width of the working zone occur, in general, according to the same law; however, the presence of azimuthal phase inhomogeneity leads to a corresponding shift of the separate phases of the process in time.

Graphs similar to those shown in Figs. 7 and 8, constructed for other azimuths, are necessary for a detailed analysis of the conditions of electron motion at various instants of the acceleration time. For a more general analysis of these conditions it is useful, on the basis of such graphs, to construct the dependence \(r=f(t)\) for \(n=1\) and \(n=0\). Such a dependence, constructed from the data of the graphs in Figs. 7 and 8, is shown in Fig. 9. The waviness of the curves \(n=1\) is also a consequence of the above-mentioned character of the radial static inhomogeneity of the magnetic field.

With in-phase variation of the magnetic field, the value of \(n\), as we have already noted, remains unchanged throughout the entire time of variation of the field. Therefore the graphs \(n=0\) and \(n=1\) in the coordinates \((r,t)\) are expressed by straight lines \(r=\text{const}\). In our case, the presence of radial phase inhomogeneity leads to a sharp deformation of the curve \(n=1\), so that the zone \(n>1\) (shaded in the figure), at small values of \(t\), penetrates far into the region of smaller radii, greatly reducing the width of the working zone \((0<n<1)\).

The curve \(n=1\) for azimuth \(90^\circ\) is shifted still farther toward larger values of \(t\) because of the azimuthal phase inhomogeneity. The azimuthal static inhomogeneity affected only the increase in the waviness of the curve. As the width of the working zone of the betatron magnetic field one may take the width of the annular space at every point of which, at each given instant of time, the value of \(n\) lies within the limits \(0<n<1\). The graph in Fig. 9 makes it possible to determine with sufficient accuracy the width and location of the working zone of the betatron at any instant of the beginning of the rise of the magnetic field. The boundaries of this zone are determined by the dashed curves, which are the envelope lines for the cri-

curves (\(n=0\) and \(n=1\)), constructed in this case for azimuths \(0\) and \(90^\circ\). For a more accurate plotting of these boundaries, it is necessary to construct a system of curves \(r=f(t)\) for \(n=1\) for several of the most characteristic azimuths. Here it should be noted that, in order to construct such a system of curves, there is no need to carry out the complete construction of graphs analogous to Figs. 7 and 8. For this it is sufficient, using formula (6), to calculate \(n\) at those instants of time when \(n\) at the corresponding radii is close to 1; from them, construct sections of the curves \(n=f(r)\) intersecting the ordinate \(n=1\). The points of intersection give the values of \(r\) for the corresponding \(t\).

Fig. 9

In determining the width of the working zone it is also necessary to take into account the following important circumstance: the presence of phase nonuniformity of the magnetic field leads to the fact that the radius of the equilibrium orbit (\(r_0\)) does not remain constant during the rise time of the magnetic field. Indeed, from Fig. 6 it can be seen that the instantaneous values of the induction at small \(t\) are such that the betatron relation (2:1) can be satisfied for points located closer to the center of the poles than is the case for the amplitude values of the field. The instantaneous values of \(r_0\) at the beginning of the rise of the magnetic field will be smaller than those at the end of this rise. However, as the field increases, the instantaneous values of \(r_0\) rather quickly come into agreement with the value of \(r_0\) at the amplitude value of the field. The rate of this process is determined by the rate at which the influence of phase nonuniformity on the general structure of the betatron magnetic field decreases.

If the design of the betatron is such that in the median plane between the central inserts (shims) of the poles there is an air gap, then the deviations of the instantaneous values of \(r_0\) from the established radius of the equilibrium orbit will be insignificant. In addition, the dependence \(r=f(t)\) for \(n=1\) in this case will have the form of a U-shaped curve with a minimum lying in the region of the instantaneous values of \(r_0\). As we shall see below, in this case the injector may be placed in the plane of the orbit both inside and outside the circle of the orbit of radius \(r_0\), that is, \(r_i<r_0\) or \(r_i>r_0\).

Choice of the Injector Location and Determination of the Minimum Required Injection Voltage

The choice of the injector location must be made on the basis of the results of a detailed analysis of the structure of the magnetic field, especially during the injection period, taking into account the design features of the electromagnet and of the betatron chamber. In doing so it must be borne in mind that, in order to improve the conditions under which the electrons pass around the injector during their first revolutions along the orbit, the injector must be moved as far as possible away from the equilibrium orbit; i.e., the quantity \(\Delta r = r_i - r_o\) should be made as large as possible. The injector should be placed in the magnetic field of the betatron only within its working region. The most advantageous location for the injector is such a point of this region at which, at the instant injection begins, the quantity \(n\) has its greatest value \((n \simeq 1)\).

Let us return again to the graph in Fig. 9. It is clear from the figure that the inner boundary of the working region is determined by the values \(n = 0\). On the basis of the preceding considerations, it is not advantageous to install the injector at \(r_i < r_o\). It is most expedient to place the injector at some point on the outer boundary of the working region, where \(n \simeq 1\) at \(r_i > r_o\), and to carry out injection on the outside with respect to the equilibrium orbit. This is also advantageous because the instantaneous value of the radius of the equilibrium orbit \(r_{ot}\) at the moment of injection is smaller than \(r_o\), and consequently \(\Delta r\) has its greatest value. The azimuth of the most advantageous injector location for a given \(r_i\) is determined on the graph by that curve \((\theta = \mathrm{const})\) which, at the instant injection begins, is closest to touching the boundary of the working region. For example, if the radius of the injector location is taken to be \(r_i < 11\ \mathrm{cm}\), then, according to the graph in Fig. 9, the most advantageous azimuth will be \(90^\circ\); but if \(r_i > 11\ \mathrm{cm}\) is taken, then it is more advantageous to place the injector at an azimuth of \(0^\circ\). If \(r_i = 11\ \mathrm{cm}\), the injector may be placed at any azimuth, proceeding from considerations of convenience. Thus, the graph in Fig. 9 makes it possible to determine exactly the coordinates of the point of the most advantageous injector location in the magnetic field of the given structure.

In the case where, in the orbital plane, there is an air gap between the central pole disks, the working region on the graph is bounded by a U-shaped curve \(n = 1\). In this case the injector may be installed both inside the orbit circle \((r_i < r_o)\) and outside it \((r_i > r_o)\). The coordinates of the injector location point (radius and azimuth) are determined in both cases analogously to the case just considered.

Examining Fig. 9, one may note the following important circumstance, namely: in the presence of phase nonuniformity of the magnetic field, electron injection cannot be carried out for a certain time immediately after the field passes through zero. It turns out that there exists a certain minimum time \(t_{i\min}\), quite definite for a given injector-location radius \(r_i\); only after this time can the electrons admitted into the betatron be captured into a stable acceleration regime. For example, in the case where the injector is located at \(r_i = 10.5\ \mathrm{cm}\), injection may be begun \(27.5\ \mu\mathrm{sec}\) after the field passes through zero at the control point.

If injection is begun earlier than this time, for example at \(t\) between 16 and \(27\ \mu\mathrm{sec}\), the probability of capturing electrons into a stable acceleration regime is almost zero, since during this time interval at some azimuths \(n > 1\).

If injection is begun still earlier \((t < 16\ \mu\mathrm{sec})\), all electrons will undoubtedly be lost.

In the case where the injector is located at \(r_i = 11.25\ \mathrm{cm}\), injection must be begun \(55\ \mu\mathrm{sec}\) after the moment when the field at the control point passes through zero.

Since during this time \(t_{i\min}\) the induction of the magnetic field at \(r_i\) will have time to increase to a certain value, the electrons must be emitted at velocities corresponding to this value of the induction, i.e., injection must be carried out at the corresponding voltage on the gun electrodes. Consequently, in the presence of phase nonuniformity of the magnetic field, for a given injection radius there exists a certain minimum required injection voltage \(U_{i\min}\). The betatron will operate only in the case where the injection of electrons is carried out at a voltage \(U_i > U_{i\min}\). The choice of the injector location must therefore be made with allowance for the required injection voltage, so that this voltage does not exceed reasonable limits.

One more important fact should be noted. Since the phase nonuniformity, expressed in microseconds, does not depend on the amplitude of the betatron magnetic field, then, as the field amplitude is increased, the curves \(n = 1\) in Fig. 9 will shift in the direction of increasing \(t\), widening the forbidden zone \((n > 1)\). Consequently, when the amplitude of the magnetic field is increased, the minimum injection voltage required for operation of the betatron increases. Correspondingly, a reduction of the magnetic-field amplitude makes it possible to use lower injection voltages.

The effect of this phenomenon was observed in the betatron under study. When the field amplitude of the betatron under study was increased, the phase of the beginning of injection, at an unchanged value of the injection voltage, decreased to the required extent. The intensity of the radiation thereby increased, mainly due to the increase in the energy of the electrons. However, such an increase in intensity was observed only up to a certain limit, after which the intensity fell sharply to zero. Such disappearance of the radiation can be explained only by the fact that the zone \(n > 1\) penetrated into the region of the injector location and the electrons prematurely left the acceleration regime as a result of scattering and defocusing of the beam.

Choice of the Target Location

For purposes of defectoscopy it is highly desirable that the dimensions of the focal spot of the x-rays be as small as possible.

Whereas the horizontal dimensions of the focal spot in a betatron are small, chiefly because of the relatively low velocity with which the electron beam is extracted onto the target, its vertical dimensions are determined by the height of the cross section of the electron beam at the moment they strike the target, that is, by the degree of axial focusing of the beam. Since the amplitudes of the axial and radial oscillations of the electrons, and consequently the dimensions of the cross section of the electron beam in the betatron, are related by

\[ \frac{a}{b}=\sqrt{\frac{1-n}{n}}, \]

where \(a\) is the vertical and \(b\) the horizontal dimension of the beam cross section, then in order to reduce \(a\) it seems advisable, during the period of electron extraction onto the target, to make them move in a magnetic field with \(n \simeq 1\), and to install the target in the working zone at the azimuth where the field has the greatest value of \(n\).

The coordinates of the most advantageous point for placing the target can be found from the graph of Fig. 9 for the corresponding time \(t\) at which the electrons are dumped onto the target. Because the boundaries of the working zone expand toward the end of electron acceleration, the radius of the most advantageous location of the target will be greater than the radius at which the injector is located. The azimuths of the most advantageous points for locating the injector and the target may also be different. Naturally, in this case the target and the injector must be located either on opposite sides of the equilibrium orbit, or the injector must be located outside the plane—

of the orbit at the corresponding most advantageous point. In the latter case, the fact that at the instant injection begins the value of \(n\) in the region where the injector is located has the greatest value has a favorable effect on the conditions under which the electrons pass the injector during the first revolutions in the orbit. Such a placement of the injector, if necessary, will also not impede the extraction of electrons from the betatron chamber.

In practice the target is often placed on the rear wall of the injector. If the same injector is installed at the most advantageous point of the interpole space of the betatron, then, as is not difficult to see from Fig. 9, the azimuth remains the most advantageous for the target as well, although at the instant of electron impact the value of \(n\) at its location will be smaller than during the injection period. In choosing the location of the target, the design features of the electromagnet and of the betatron chamber should be taken into account. In any case, the target must be positioned so that structural parts of the electromagnet do not interfere with the free exit of the x-rays.

In addition, the final choice of the most advantageous location for the target can be made only with allowance for those changes in the structure of the magnetic field that arise as a result of the current pulses flowing in the displacement windings of the accelerated electrons onto the target.

Some measures for improving the structure of the magnetic field

Above we saw that an increase in the azimuthal phase nonuniformity leads to a shift of the curves in the graph of Fig. 9 toward larger values of \(t\), i.e., to an increase in the minimally required injection voltage \((U_{i\,min})\) and, correspondingly, to a narrowing of the working region of the magnetic field. Accordingly, a decrease in the azimuthal phase nonuniformity leads to an expansion of the working region and a lowering of the minimally required injection voltage. The curves \(\Theta = \mathrm{const}\) (Fig. 9) for different azimuths in this case drop to the level of the lower curve, with which they ultimately merge. In the latter case the injector and the target can be installed with equal effect at any azimuth.

When designing the electromagnet of a betatron and tuning its magnetic field, special attention must be paid to the questions of reducing the radial phase nonuniformity, which, as we have already seen, plays the most harmful role during the period of electron injection. Methods for correcting the azimuthal phase and static nonuniformity are sufficiently well developed. Using these methods, one can not only reduce the nonuniformities of the magnetic field, but also so influence the structure of this field as to ensure the best conditions for the motion of electrons during injection, acceleration, and their extraction onto the target. These conditions are most easily controlled by analyzing the instantaneous pattern of the magnetic field, as has been done in the present work.

In conclusion, it should be noted that the presence in the magnetic field of such components as the vacuum chamber—in the conducting layer of whose walls eddy currents of considerable magnitude may be induced—the conductors of the displacement winding, and the injector structure, introduces distortions into the structure of the magnetic field. Therefore, when investigating the instantaneous pattern of the field, it is necessary to introduce the appropriate corrections or, which is certainly more accurate, to carry out the necessary measurements in the presence of these components (or their models) in the magnetic field under study. The eddy currents arising in the conducting layer of the walls of the betatron vacuum chamber reduce, during the injection period, the value of \(n\). Such a decrease in \(n\) in the presence of

the phase nonuniformity should play a positive role, reducing \(U_{i\,\min}\) and widening the working region of the magnetic field.

Conclusion

The proposed method for investigating the instantaneous pattern of the magnetic field of a betatron makes it possible, with sufficient accuracy, to evaluate the conditions for the motion of the accelerated electrons, to find the most advantageous location of the injector and the target, and also to determine the injection start time and the minimum necessary injection voltage for a magnetic field of a specific structure under a given operating regime of the betatron.

Using this method, one can investigate the operating conditions of the betatron under other regimes as well, provided that the locations of the injector and the target remain unchanged. In addition, analysis of the instantaneous structure of the magnetic field makes it possible to identify ways of improving it; by following these, one can achieve the maximum beam intensity from the betatron.

Of course, this method of investigation cannot claim universality; however, in combination with other known methods of investigation, it can reveal more deeply the true conditions of electron motion during the injection period and thereby solve the most difficult problem confronting the development of betatrons and synchrotrons—namely, the problem of increasing radiation intensity.

1. For convenience in using the results of the investigation, it is advantageous to take the point of the proposed injector location as the reference point. If the reference point does not coincide with the point at which the injector is located, then it is necessary to recalculate the time so that at the point where the injector is located one has \(t=0\) when \(B=0\) there. ↩