ON THE MECHANISM OF ELECTRON CAPTURE INTO ACCELERATION IN A BETATRON

B. N. RODIMOV

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

The number of electrons captured into acceleration and its dependence on various parameters

The mechanism by which the number of electrons captured into acceleration obeys certain regularities is, in our opinion, the formation of an equilibrium electron beam circulating in the betatron chamber in the form of a closed stable ring without any radial or axial oscillations. Making such an assumption on the basis of consideration of a very rough picture of the interaction of ribbon electron beams [1], we shall try to prove its validity by means of the consequences obtained from it.

Basic conditions of injection

We shall consider the magnetic field of the betatron to vary with time, as is the case in real betatrons, without, however, taking into account phase shifts, azimuthal inhomogeneity, etc. We shall, as before, characterize the magnetic field by the potential function \(V_m\), now varying in time and determining at each instant the forces acting on the electrons.

Specifically, we shall consider the function \(V_m\) of the form given in [2]:

\[ V_m = -\frac{e}{2mc^2} \left[ \frac{\dfrac{r^2 H_z}{2}+C}{r} \right]^2 = \left[ \sqrt{V_{m0}}+\frac{C}{r}\sqrt{\frac{e}{2mc^2}} \right], \tag{1} \]

where

\[ V_{m0} = \frac{e A_1^2 r^2}{2mc^2} \left[ a J_1(kr)+bN_1(kr) \right]^2 \operatorname{ch}^2(kz) \tag{2} \]

is the potential function of “zero” electrons, for which the constant \(C=0\) (see in [2] the graphs of this function for the case of an equilibrium circle with radius \(R_0=15\text{ cm}\) and \(n_0=0.5\)).

Starting from the definition of the constant

\[ C=\frac{cmr^{2}\dot{\varphi}}{e}-\frac{r^{2}H_{z}}{2}, \]

one can write its expression in terms of the values of the quantities for the injection radius \(R_i\):

\[ C=F\sqrt{U_0\sin\omega t}\,R_i^{2}\,\frac{\overline{H}_{zam}}{2}\cdot \sin\Omega(\tau_0+t), \tag{3} \]

where \(U=U_0\sin\omega t\) is the injection-pulse voltage in volts, \(\omega\) is the frequency of the injection pulse, \(\overline{H}_{zam}\) is the amplitude value of the average magnetic-field intensity in the circle of radius \(R_i\), \(\Omega\) is the frequency of the magnetic field, \(\tau_0\) is the instant at which the injection pulse begins (the instant at which the magnetic field passes through zero is taken as the origin of \(\tau_0\); \(t\) is measured from \(\tau_0\)),

\[ F=\frac{c\cdot m}{e}\cdot 5.93\cdot 10^{7}R_i, \]

and \(c\), \(m\), \(e\) are, respectively, the velocity of light and the mass and charge of the electron.

Since \(\Omega(\tau_0+t)\) is small, one may write

\[ C=F\sqrt{U_0\sin\omega t}\;-\;R_i^{2}\,\frac{\overline{H}_{zam}}{2}\,\Omega(\tau_0+t). \tag{4} \]

We shall continue to assume that the electron gun of the betatron operates in the saturation regime, i.e., one may write the equality

\[ i=e\cdot 5.93\cdot 10^{7}\sqrt{U}\cdot \rho=\mathrm{const}, \tag{5} \]

where \(\rho\) is the density of the electron beam (the number of electrons in \(\mathrm{cm}^{3}\)).

Motion of Electrons in an Equilibrium Beam

In [1] the case of an equilibrium electron beam with a single value of the constant \(C\) was considered. An equilibrium electron beam consisting of electrons with different \(C\) can be represented if electrons with different \(C\) occupy zones at different radii. Only in this case is equality possible between the focusing forces of the magnetic field and the defocusing forces of the electric field of the electron beam. In other words, only in this case is it possible to satisfy the equality

\[ \frac{\partial}{\partial r}\int \frac{e\rho_{\text{beam}}\,dV}{|\vec r^{\,\prime}-\vec r|} = -\frac{\partial V_m}{\partial r} \]

for each kind of electron. Electrons with larger values of \(C\) are also located at larger radii. For any value of \(C\), in a real betatron it is impossible to create an equilibrium beam with electrons of only one \(C\). However, it is also impossible to create an equilibrium beam with electrons of all possible values of \(C\), since then, for the electrons of the largest \(C\), the magnetic forces would be zero; consequently the electric forces would also have to be zero, which means that the electron beam is absent. Thus, the formation of an equilibrium beam from electrons with values of \(C\) in a narrow interval \(\Delta C\) is most probable.

Boundaries of the Capture Region

It seems natural to regard as the capture region the interval of values of \(C\) for which potential wells exist.

The upper value \(C=C_{\max}\) can be determined from the following considerations. For the field at radius \(r\) one may write

\[ H_z(r)=H_z(R_0)e^{\displaystyle\int_{R_0}^{r}\frac{n(r)}{r}\,dr}. \tag{6} \]

At the points of the minima of the potential wells, for various \(C\),

\[ H_z=\frac{\bar H_z}{2}+\frac{C}{r^2}. \tag{7} \]

In particular, for \(C_{\max}\),

\[ H_z=\frac{\bar H_z}{2}+\frac{C_{\max}}{R_i^2}, \tag{8} \]

where \(R_i\) is the injection radius (at this radius \(n=1\), which corresponds to the bottom of the last upper well).

Equality (8) may be written as

\[ H_z(R_0)e^{\displaystyle\int_{R_0}^{R_i}\frac{n(r)\,dr}{r}} = \frac{H_z(R_0)}{\left(1+\dfrac{x_i}{R_0}\right)^2} \left[ 1+\frac{1}{R_0^2} \int_{R_0}^{R_i} r e^{\displaystyle\int_{R_0}^{r}\frac{n(r)\,dr}{r}} \,dr \right] +\frac{C_{\max}}{R_i^2}, \tag{9} \]

since the average field strength in a circle of radius \(r\) will be

\[ \bar H_z(r)= \frac{2H_z(R_0)}{\left(1+\dfrac{x}{R_0}\right)^2} + \frac{2H_z(R_0)}{r^2} \int_{R_0}^{r} r e^{\displaystyle\int_{R_0}^{r}\frac{n(r)\,dr}{r}} \,dr, \tag{10} \]

where \(x=r-R_0\), and \(H_z(R_0)\) is the field strength on the equilibrium circle of radius \(R_0\).

Taking for \(n(r)\) the values from [2] for the case considered there, \(R_0=15\) and \(n_0=0.5\), we obtain \(C_{\max}=29.9\cdot H_z(R_0)\).

Expression (4) for \(C\) can be transformed so that it contains the values of the quantities on the principal equilibrium radius \(R_0\). Using expression (8), we obtain

\[ C=F\sqrt{U_0\sin\omega t} - R_i^2H_{zam}(R_i)\Omega(\tau_0+t) + C_{\max}(t). \tag{11} \]

Let us replace \(H_{zam}(R_i)\) by \(H_{zam}(R_0)\) from the expression

\[ H_{zam}(R_i)=H_{zam}(R_0)e^{-\displaystyle\int_{R_0}^{R_i}\frac{n(r)}{r}\,dr}. \]

In our particular case,

\[ H_{zam}(R_i)=H_{zam}(R_0)\cdot 0.7584. \]

Thus we obtain

\[ C=73.3\sqrt{U_0\sin\omega t}-4.20\cdot 10^8(\tau_0+t). \]

The set of $C$ on the side of negative values of $C$ in our case is bounded by the inner wall of the chamber, whose radius we take to be $R_j=9\ \mathrm{cm}$. The minimum $C_{\min}$ is found in a manner entirely analogous to $C_{\max}$ from the expression

\[ H_z(R_0)e^{+\int_{R_j}^{R_0}\frac{n(r)}{r}\,dr} = \frac{H_z(R_0)} {\left(1+\frac{x_j}{R_0}\right)^2} \left[ 1-\frac{1}{R_0^2} \int_{R_j}^{R_0} r e^{+\int_{r}^{R_0}\frac{n(r)}{r}\,dr}\,dr \right] + \frac{C_{\min}}{R_j^2}, \tag{12} \]

where $x_j=R_j-R_0=-6\ \mathrm{cm}$.

Number of Electrons Captured into Acceleration and Its Relation to the Injection Current

The number of electrons $Q$ circulating in the chamber at each instant of time within the total capture period will be determined by the focusing forces both at the outer boundary of the useful zone (for which we take the gun radius $R_i$) and at the inner boundary (at the radius $R_j$ of the inner wall of the chamber). We assume that at any given instant there are electrons in the chamber only from a narrow interval of $C$ (for simplicity one may even regard it as simply a single $C$). The introduction of new electrons with a new value of $C$ is possible only at the expense of the loss of electrons of the preceding $C$’s. Therefore the interval $\Delta C$ of electrons present in the chamber will continuously move over the general interval of permissible values of the constant $C$¹.

If the dependence of the number of electrons $Q$ on time is plotted, one obtains approximately the picture shown in Fig. 3. Up to the instant $t_0$ the electron beams are formed in the region of the lower pits ($C<0$). The smallest focusing forces will be on the inner side of the chamber; therefore the quantity $Q$ will be determined precisely by these forces. As one moves toward larger $C$, these forces increase, and the number of electrons $Q$ moving in the chamber increases correspondingly (curve II). On passing into the region $C>0$, the smallest focusing forces will be the forces at the outer boundary; they will determine the amount of charge retained in the chamber (curve I). The overall course of the change in $Q$ will be represented by the curve $ABC$.

If an equilibrium beam is not formed in any of the time intervals, then $Q$ will move into the region of the largest $C$’s, and practically no charge will be captured.

If, however, at some instant $t_1$ an equilibrium beam is formed, then the number of electrons $Q_1$ present at that instant will go into acceleration. The instant of creation of an equilibrium beam $t_2$ will correspond to the charge $Q_2$, and so on (Fig. 4).

The instant of creation of an equilibrium beam is determined by the injection current $i$; namely, an equilibrium beam will be created at that instant or in that interval of time when the density $\rho_i$ of the electron beam emitted from the gun is approximately equal to the equilibrium density $\rho_{\mathrm{rav}}$ of the electrons of that $C$ which are circulating at the given instant in the chamber².

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Since

\[ \rho_i=\frac{i_{(\mathrm{amp})}}{1.6\cdot 10^{-19}\cdot 5.93\cdot 10^{7}\sqrt{U_0\sin\omega t}} . \tag{13} \]

one may write

\[ \rho_{\mathrm{av}}(t)=\frac{i}{1.6\cdot 10^{-19}\cdot 5.93\cdot 10^{7}\sqrt{U_0\sin\omega t}} . \tag{14} \]

Since the beams for different \(C\) are formed in the neighborhoods of different \(R_c\), where \(n_c\) are also different, the effect of interaction of the beams, which determines the formation of the equilibrium beam, will be the weaker the larger \(n_c\) is, for in this case the beam has a form similar to Fig. 5-b. A larger current density will be required, i.e., a larger density \(\rho_i\) for the given instant \(t\), in order to produce the necessary effect¹.

Fig. 5.
a) \(n_c<0.5\); b) \(n_c>0.5\).

Therefore we introduce the factor \(\dfrac{n_c}{1-n_c}\):

\[ \rho_{\mathrm{av}}(t)\cdot\frac{n_c}{1-n_c} = \frac{i}{1.6\cdot 10^{-19}\cdot 5.93\cdot 10^{7}\sqrt{U_0\sin\omega t}} \tag{15} \]

\(\rho_{\mathrm{av}}(t)\) is found from the formula

\[ 4\pi e\rho_{\mathrm{av}} = \frac{2V}{R_c^2} = \frac{2}{R_c^2} \left[ \sqrt{V_{\mathrm{well}}(R_c)} + \frac{C}{R_c}\sqrt{\frac{e}{2mc^2}} \right]^2 , \tag{16} \]

where \(V_{\mathrm{well}}\) is the value of the potential function at the bottom of the potential well for electrons of type \(C\) (i.e., at the radius \(R_c\)).

Introducing \(V_{\mathrm{well}}=V_0\cdot U'(R_c)\), where

\[ V_0=\frac{eA_0^2}{2mc^2} \]

and \(U'(R_c)\) is determined (see [2]) by the expression

\[ U'(R_c)=\left[aJ_1(kR_0)+bN_1(kR_0)\right]^2 , \tag{17} \]

we obtain

\[ 4\pi e\rho_{\mathrm{av}} = \frac{2V_0}{R_c^2} \left[ \sqrt{U'(R_c)} + \frac{C}{R_c\sqrt{V_0}}\cdot \sqrt{\frac{e}{2mc^2}} \right]^2 . \tag{18} \]

To calculate the value of the current \(i\) required for capture to occur at the instant \(t\), it is necessary to find the relation between \(t\) and the corresponding value of the radius \(R_c\). This relation is found in the following way. For the bottom of the well of electrons of type \(C\), one may write

\[ C=\sqrt{\frac{2mc^2}{e}}\,V_0\, \frac{R_c^2\left(\dfrac{\partial U'}{\partial r}\right)_{R_c}} {2\sqrt{U'(R_c)}} . \tag{19} \]

Since, on the other hand,

\[ C=F\sqrt{U_0\sin\omega t}-334H_{z0}, \]

then

\[ F\sqrt{U_0\sin\omega t}-334H_{z0} = \frac{R_c^2\left(\dfrac{\partial U'}{\partial r}\right)_{R_c}} {2\sqrt{U'(R_c)}}\cdot R_0H_{z0}. \tag{20} \]

Denoting by \(f(R_c)\) the expression

\[ f(R_c)= \frac{R_c^2\left(\dfrac{\partial U'}{\partial r}\right)_{R_c}} {2\sqrt{U'(R_c)}} , \tag{21} \]

we obtain

\[ f(t)= \frac{F\sqrt{U_0\sin\omega t}} {R_0H_{zam}\Omega(\tau_0+t)} -22.25 . \]

Having constructed the graphs of the functions \(f(R_c)\) and \(f(t)\), we shall find, for each instant of time \(t\), the corresponding \(R_c\), since \(f(R_c)\) must be equal to \(f(t)\). The formula

\[ i(\mathrm{amp})= \frac{5.93\cdot10^4\sqrt{U_0\sin\omega t}} {2\pi R_c^2}\,V_0 \left[ \sqrt{U'(R_c)} + \frac{C}{R_c\sqrt{V_0}} \sqrt{\frac{e}{2mc^2}} \right]^2 \frac{n_c}{1-n_c}\cdot0.33\cdot10^{-9} \tag{22} \]

gives the injection current necessary to create an equilibrium beam at the instant \(t\).

We calculate the number of captured electrons \(Q\) in the following way. We consider a beam of electrons with some value of \(C\), having the form of a toroid. At the boundary of the beam, the focusing forces of the magnetic field are equal to the defocusing forces of the electric field,

\[ \frac{\partial V_m}{\partial r}=-\frac{\partial\varphi}{\partial r}, \]

\[ \frac{\partial V_m}{\partial z}=-\frac{\partial\varphi}{\partial z}. \]

On the surface of the beam

\[ d\varphi=\frac{\partial \varphi}{\partial r}\,dr+\frac{\partial \varphi}{\partial z}\,dz=0 \]

also

\[ dV_{\mathrm{m}}=\frac{\partial V_{\mathrm{m}}}{\partial r}\,dr+\frac{\partial V_{\mathrm{m}}}{\partial z}\,dz=0 . \tag{23} \]

Hence the equation of the contour of the beam cross section is

\[ \frac{dz}{dr}=-\frac{\dfrac{\partial V_{\mathrm{m}}}{\partial r}}{\dfrac{\partial V_{\mathrm{m}}}{\partial z}} . \tag{24} \]

For simplicity we take the wells in the form of a paraboloid

\[ V_{\mathrm{mc}}=V_{\mathrm{mc}0} \left( 1+\frac{1-n_c}{R_c^2}\,x_c^2+\frac{n_c}{R_c^2}\,z^2 \right), \]

where \(x_c=r-R_c\). Then from

\[ \frac{dz}{dr}=-\frac{(1-n_c)}{n_c}\cdot\frac{x_c}{z} \]

we obtain

\[ \frac{z^2}{\left(\dfrac{1-n_c}{n_c}\right)a^2}+\frac{x_c^2}{a^2}=1 . \tag{25} \]

Putting \(a=(R_i-R_c)\), we shall have

\[ b=(R_i-R_c)\sqrt{\frac{1-n_c}{n_c}} =a\sqrt{\frac{1-n_c}{n_c}}, \tag{26} \]

where \(R_i\) is the radius of the gun, if the guns are located at the upper wells. If the guns are at the lower wells, then instead of \(R_i\) we take \(R_j\), the radius of the inner wall of the chamber.

The total number of electrons in an equilibrium beam in the wells in the vicinity of a well with constant \(C\) will be approximately equal to

\[ Q=2\pi^2 R_c(R_i-R_c)^2 \sqrt{\frac{1-n_c}{n_c}}\,\rho_{\text{[[unclear: subscript]]}} = \]

\[ =\pi(R_i-R_c)^2 \sqrt{\frac{1-n_c}{n_c}}\, \frac{V_0}{eR_c} \left[ \sqrt{U(R_c)} + \frac{C}{R_c\sqrt{V_0}} \sqrt{\frac{e}{2mc^2}} \right]^2 . \tag{27} \]

Having calculated the quantities \(Q\) for different instants of time, we can compare them with the values of the injection current for the corresponding instants of time and construct the dependence of \(Q\) on \(i\) for a given injection phase \(\tau_0\).

Analogously, \(Q\) is calculated for the inner wall of the chamber; only, instead of \(R_i\), \(R_j\) enters formula (27).

Comparison of the Theoretical Curves with the Experimental Ones

Figures 6 and 7 present the principal experimental curves. The curves in Fig. 6 give the dependence of the radiation intensity (which we take to be proportional to the number of electrons captured into acceleration)

Fig. 6

on the injection current at different injection voltages (the curves were taken while adjusting the phase of the pulse, for each change in the injection current, to the maximum radiation).

Fig. 7

Fig. 7 gives the dependence of the radiation intensity on the phase of the pulse \(\tau_0\) at different injection currents and at different injection voltages. Figure 8 presents the results of calculating \(Q\) by our formulas. The solid lines correspond to capture on the front of the pulse, the dashed lines—

capture on the tail. A characteristic feature of the curves at large phases \((\tau_0 = 21\ \mu\mathrm{sec})\): the curves \(Q\), calculated from the outer boundary, and the curves \(Q\), calculated from the inner boundary, do not intersect. This means that, in this case, capture occurs only in the lower wells.

Fig. 8

Schematically, a system of such curves for different injection phases is presented in Fig. 9 (for capture only on the front or only on the tail). The extreme right-hand points of each curve correspond to the maximum injection current capable of throwing electrons into the potential well (the existence of break-off points of the intensity is confirmed by experiment).

The dashed line drawn through the maxima of curve \(n\) and through the extreme points of the curves gives the curve corresponding to the curves of Fig. 6.

If, from the curves of Fig. 9, one constructs curves of the dependence of \(Q\) on the injection phase at different injection currents, for example, \(i_1\) and \(i_2\), one obtains curves similar to the curves of Fig. 7, with their characteristic sharp drop in intensity upon passing through a definite phase. However, at small

currents (for example, \(i_3\)) such curves are not obtained. This may be connected with the fact that precisely for small \(C\) the curves \(n(r)\) considered by us and the curves \(n(r)\) that occurred for the betatron on which the curves of Fig. 7 were taken are very different.

Fig. 9

In the schematic Fig. 10 it is clearly seen why, when the injection pulse is shifted to the right, the intensity at a given injection current, corresponding to a definite band \(\Delta C\) of the capture region, shown in the figure by thick strips, first increases and then falls sharply when the pulse leaves the \(\Delta C\) region.

Fig. 10

Thus, the theory set forth here gives a good qualitative and quantitative interpretation of the regularities of capture and opens up the possibility of purposeful experimental work to refine the regularities themselves and their theoretical interpretation.

LITERATURE

1. B. N. Rodimov. On the mechanism of electron capture into acceleration in a betatron. Izvestiya TPI, vol. 87, 1957.

2. B. N. Rodimov. Regularities of the magnetic field of a betatron. Izvestiya TPI, vol. 87, 1957.

3. R. Widevöe, Jour. Appl. Phys. 22, 362, 1951.

1. The mechanism of shaking the beams into an equilibrium beam operates when there are sufficiently clearly defined beams elongated along the \(Z\) axis. ↩↩

2. When “shaking down” into an equilibrium beam occurs, there is some loss of electrons and the beam is compressed. New capture can begin only when equality $\rho_i=\rho_{\mathrm{rav}}$ again takes place for electrons of the same $C$. ↩