On the Mechanism of Electron Capture into Acceleration in a Betatron

I

B. N. Rodimov

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

Motion of Electron Beams in the Magnetic Field of a Betatron during the First Revolutions

The existing theories of the betatron \([1, 2, 3, 4, 5, 6]\) are, strictly speaking, single-electron theories; they consider the motion of an individual electron in the magnetic field of a betatron. It is clear, however, that such theories cannot give the complete picture necessary for a conscious approach to the calculation of a betatron with a high radiation intensity, since in a real betatron one is always dealing not with one electron, but with a beam of electrons.

In the present work an attempt is made to approach the problem of capturing electrons into acceleration by considering it from the very beginning as a many-electron problem.

Stationary Equilibrium Electron Beam

Let us introduce the concept of a stationary equilibrium electron beam moving in the axially symmetric magnetic field of a betatron. For the electrons of such a beam, the defocusing forces of electric repulsion are balanced by focusing forces created by a specially selected magnetic field. For an electron of such a beam, the equations of motion along the axes \(r\) and \(z\) have the form:

\[ mr\dot{\theta}^{2}-\frac{e}{c}\,r\dot{\theta}H_{z}-eE_{r}=0, \]

\[ \frac{e}{c}\,r\dot{\theta}H_{r}-eE_{z}=0. \tag{1} \]

By the vector \(\vec E\) we shall understand the strength of the electric field produced by the electrons of the beam,

\[ E_r=\frac{\partial}{\partial r}\int \frac{e\rho\,dV'}{\left|\vec r-\vec r'\right|}=-\frac{\partial \varphi}{\partial r}, \]

\[ E_z=\frac{\partial}{\partial z}\int \frac{e\rho\,dV'}{\left|\vec r-\vec r'\right|}=-\frac{\partial \varphi}{\partial r}. \tag{2} \]

Here \(\rho\) is the density of the electron beam, i.e., the number of electrons in \(1\ \mathrm{cm}^3\), and \(e\) is the absolute value of the electron charge.

As is known [3], for a stationary magnetic field one may write

\[ mr\dot\theta^{\,2}-\frac{e}{c}r\dot\theta H_z =-e\frac{\partial}{\partial r}\left[ \frac{e}{2mc^2}\left(\frac{r^2\overline H_z}{2}+C\right)^2\frac{1}{r} \right] =-e\frac{\partial V_m}{\partial r}, \]

\[ -\frac{e}{c}r\dot\theta H_r =-e\frac{\partial}{\partial z}\left[ \frac{e}{2mc^2}\left(\frac{r^2\overline H_z}{2}+C\right)^2\frac{1}{r} \right] =-e\frac{\partial V_m}{\partial r}. \tag{3} \]

Thus, instead of equations [1] we have

\[ \frac{\partial}{\partial r}\left[ \int \frac{e\rho_{\mathrm{eq}}\,dV'}{\left|\vec r-\vec r'\right|} +\frac{e}{2mc^2}\left(\frac{\frac{r^2\overline H_z}{2}+C}{r}\right)^2 \right]=0, \]

\[ \frac{\partial}{\partial z}\left[ \int \frac{e\rho_{\mathrm{eq}}\,dV'}{\left|\vec r-\vec r'\right|} +\frac{e}{2mc^2}\left(\frac{\frac{r^2\overline H_z}{2}+C}{r}\right)^2 \right]=0 \tag{4} \]

or

\[ \frac{\partial}{\partial r}\left[\varphi_{\mathrm{eq}}-V_m\right]=0, \]

\[ \frac{\partial}{\partial z}\left[\varphi_{\mathrm{eq}}-V_m\right]=0. \tag{5} \]

Here \(\varphi_{\mathrm{eq}}\) and \(\rho_{\mathrm{eq}}\) denote the potential and the density of the equilibrium beam. The expressions in square brackets may be set equal to a constant or to zero. The constant \(C\) has one value for all electrons of a given equilibrium beam. Equilibration of the forces of the magnetic field and of the electric forces is possible only for a quite definite spatial distribution of the electron density \(\rho_{\mathrm{eq}}\).

One may put

\[ V_m=-\int \frac{e\rho_{\mathrm{eq}}\,dV'}{\left|\vec r-\vec r'\right|}=-\varphi_{\mathrm{eq}}. \tag{6} \]

Then the equations of motion are written as

\[ \frac{\partial}{\partial r}\left[\varphi_{\mathrm{eq}}+\varphi_{\mathrm{eq}}\right]=0; \qquad \frac{\partial}{\partial z}\left[\varphi_{\mathrm{eq}}+\varphi_{\mathrm{eq}}\right]=0. \tag{7} \]

or

\[ \frac{\partial \varphi_{\mathrm{eqv}}}{\partial r} = - \frac{\partial \varphi_{\mathrm{rav}}}{\partial r}; \qquad \frac{\partial \varphi_{\mathrm{eqv}}}{\partial z} = - \frac{\partial \varphi_{\mathrm{rav}}}{\partial z}. \tag{8} \]

Hence

\[ \varphi_{\mathrm{eqv}}=-\varphi_{\mathrm{rav}}+K, \tag{9} \]

where \(K\) is some constant.

Thus, a magnetic field possessing focusing properties is equivalent, in its action on the electron beam, to a certain beam of fictitious positive charges distributed in space with the same density and in the same places as the electrons. For such charges

\[ \operatorname{div}\vec E_{\mathrm{eqv}}=4\pi e\rho_{\mathrm{rav}}, \]

\[ \vec E_{\mathrm{eqv}}=-\operatorname{grad}\varphi_{\mathrm{eqv}}, \tag{10} \]

\[ \frac{\partial^2\varphi_{\mathrm{eqv}}}{\partial r^2} +\frac{1}{r}\cdot\frac{\partial\varphi_{\mathrm{eqv}}}{\partial r} +\frac{\partial^2\varphi_{\mathrm{eqv}}}{\partial z^2} = -4\pi e\rho_{\mathrm{rav}}. \]

Using relation (6), one may write:

\[ 4\pi e\rho_{\mathrm{rav}} = \frac{\partial^2 V_m}{\partial r^2} +\frac{1}{r}\cdot\frac{\partial V_m}{\partial r} +\frac{\partial V_m}{\partial z^2}. \tag{11} \]

Thus, knowing the potential function of the magnetic field, we can also calculate the distribution of the equilibrium charge \(\rho_{\mathrm{rav}}\).

When \(\rho_{\mathrm{rav}}\) is equal to zero, \(V_m\) satisfies Laplace’s equation and the field will not be focusing (focusing is obtained in one direction, while in the other direction the field will be defocusing). Likewise, the field will not be focusing when \(\rho_{\mathrm{rav}}<0\) (in this case the field will be defocusing in both directions). And only when \(\rho_{\mathrm{rav}}>0\) do we have, in the region where this inequality holds, focusing forces.

It is interesting to note that although the function \(V_m\) satisfies Poisson’s equation, the usual scalar potential of the magnetic field \(V\) satisfies Laplace’s equation.

Substituting into Laplace’s equation for the scalar potential its expression in terms of the potential function \(V_m\), we obtain the identical zero [8].

For an electrostatic field, such two forms of the potential function cannot be written; therefore, with an electrostatic field one cannot obtain a focusing field of the betatron type.

Equations of motion of an electron belonging to a nonequilibrium beam of electrons

For an arbitrary electron the equations of motion will have the form:

\[ mr\dot{\theta}^{2} - \frac{e}{c}r\dot{\theta}H_z - eE_r = m\ddot r, \]

\[ -\frac{e}{c}r\dot{\theta}H_r - eE_z = m\ddot z. \tag{12} \]

We shall assume that the electron belongs to a beam with some density \(\rho\) and moves together with it in the magnetic field of the betatron. As before, the components of the electric field of the electron beam can be expressed in terms of the density \(\rho\) (\(\rho\) is in general not equal to \(\rho_{\text{eq}}\)):

\[ E_r=-\frac{\partial \varphi}{\partial r} =\frac{\partial}{\partial r}\int \frac{e\rho\,dV'}{\left|\vec r-\vec r'\right|}, \]
\[ E_z=-\frac{\partial \varphi}{\partial z} =\frac{\partial}{\partial z}\int \frac{e\rho\,dV'}{\left|\vec r-\vec r'\right|}. \tag{13} \]

We neglect the action of the force \(E_\theta\). The action of the real magnetic field is replaced by the equivalent action of the electric field of a fictitious beam of positive charges distributed in space with density \(\rho_{\text{eq}}\),

\[ E_{r\text{eq}}=-\frac{\partial\varphi_{\text{eq}}}{\partial r} =-\frac{\partial V_m}{\partial r}, \qquad E_{z\text{eq}}=-\frac{\partial\varphi_{\text{eq}}}{\partial z} =-\frac{\partial V_m}{\partial z}. \tag{14} \]

Substituting, instead of the field components, their expressions through \(\rho\) and \(\rho_{\text{eq}}\), we obtain

\[ e\,\frac{\partial}{\partial r}\int \frac{e\left[\rho_{\text{eq}}-\rho\right]\,dV'} {\left|\vec r-\vec r'\right|} =m\ddot r, \qquad e\,\frac{\partial}{\partial z}\int \frac{e\left[\rho_{\text{eq}}-\rho\right]\,dV'} {\left|\vec r-\vec r'\right|} =m\ddot z. \tag{15} \]

One may introduce the concept of an effective electric field in which the electron moves,

\[ m\ddot r=-eE_{r\text{eff}}, \]
\[ m\ddot z=-eE_{z\text{eff}}, \tag{16} \]

where, evidently,

\[ E_{r\text{eff}}=E_r+E_{r\text{eq}}, \]
\[ E_{z\text{eff}}=E_z+E_{z\text{eq}} \tag{17} \]

and

\[ \varphi_{\text{eff}}=\varphi_{\text{eq}}+\varphi, \]
\[ \rho_{\text{eff}}=\rho_{\text{eq}}-\rho. \tag{18} \]

In vector form the equations of motion will have the form:

\[ \frac{d\vec v}{dt}=-\frac{e}{m}\vec E_{\text{eff}}, \tag{19} \]

where

\[ \vec E_{\text{eff}}=-\operatorname{grad}\varphi_{\text{eff}}, \]
\[ \operatorname{div}\vec E_{\text{eff}}=4\pi e\left(\rho_{\text{eq}}-\rho\right). \tag{20} \]

We shall use the tracking method, i.e., we shall consider changes in the quantities characterizing the beam along the trajectories along which the electrons move. For an observer moving together with an electron, one may write

\[ \frac{d\vec E_{\mathrm{eff}}}{dt} = \frac{\partial \vec E_{\mathrm{eff}}}{\partial t} + (\vec v \nabla)\vec E_{\mathrm{eff}} . \]

We regard the electron beam as a hydrodynamic jet, for which the continuity equation holds

\[ \frac{\partial \rho}{\partial t} + \vec v \operatorname{grad}\rho + \rho \operatorname{div}\vec v = 0 \]

or

\[ \frac{d\rho}{dt} = -\rho \operatorname{div}\vec v . \tag{21} \]

Solving the system of equations (19), (20), (21), we would have to obtain completely the motion of the electron beam in the magnetic field, taking into account the interaction of the electrons with one another. However, it is impossible to solve it without simplifying assumptions. We shall agree, first of all, that we shall consider stationary motions, i.e., we shall always assume that

\[ \frac{\partial E_{\mathrm{eff}}}{\partial t}=0, \qquad \frac{\partial \rho}{\partial t}=0 \quad \text{etc.} \]

A beam of mutually interacting electrons in the magnetic field of a betatron

We introduce the following simplifying assumptions:

1. The potential well of the magnetic field is parabolic.
2. The radius of the toroid of the chamber is much larger than the dimensions of the chamber cross section and, consequently, than the dimensions of the effective well.
3. The electron beam is taken to have the form of a ribbon, infinite in the \(z\) direction. Only the dimension in the \(r\) (or \(x\)) direction is finite; moreover, a change in the width of the beam in this direction, or its displacement along the \(r\)-axis, does not substantially alter its character as a plane beam.
4. The density of the electron beam is a function only of the azimuthal coordinate \(y\) \((y=r\theta)\), so that \(\rho(x_2-x_1)=\rho_0(x_{20}-x_{10})\), where the subscript “0” denotes the initial values at the moment of emission from the gun.

Under these assumptions it is possible simply to express the forces acting on each electron of the beam. The forces acting from the magnetic field of the betatron will be determined by the magnitude of the equivalent magnetic-field intensity

\[ E_M = 4\pi e \rho_{\mathrm{rav}}\cdot x, \tag{22} \]

where \(x=r-R\) (\(R\) is the equilibrium radius).

This will be a focusing force. The defocusing force will be the force acting on the given electron from the other electrons of the beam. This force is determined by the intensity

\[ E_E=-2\pi e\rho\,[2x-(x_1+x_2)], \tag{23} \]

where \(x_1\) and \(x_2\) are the boundaries of the beam in the \(x\) direction.

On this basis one can immediately write the expression for the trajectory of an electron of the beam

\[ x=-\frac{e}{2m}\int\limits_{0}^{t}(t-\xi)^2\,\frac{dE_{\mathrm{eff}}}{d\xi}\,d\xi -\frac{e}{2m}E_{\mathrm{eff}}\,t^2+v_0\,t+x_0 . \tag{24} \]

In our case

\[ \frac{dE_{\mathrm{eff}}}{dt} =4\pi e\rho_{\mathrm{eq}}v -2\pi e\rho\,[2v-(v_1+v_2)] -2\pi e\,\frac{d\rho}{dt}\,[2x-(x_1+x_2)] . \tag{25} \]

If this expression is substituted into (24), we obtain a rather complicated integro-differential equation. It can, however, be converted into a differential equation if it is differentiated with respect to \(t\) three times and if certain simplifying assumptions are made.

We obtain

\[ \frac{d^2 v}{dt^2} =-\frac{4\pi e^2}{m}v(\rho_{\mathrm{eq}}-\rho) -\frac{2\pi e^2}{m}\rho(v_1+v_2) +\frac{2\pi e^2}{m}\cdot\frac{d\rho}{dt}\,[2x-(x_1+x_2)] . \tag{26} \]

If we take an electron for which \(x=\dfrac{x_1+x_2}{2}\), then for it this equation reduces to the equation of motion of a single electron in the magnetic field of the betatron, i.e.,

\[ \frac{d^2 v'}{dt^2} =-\frac{4\pi e^2}{m}\rho_{\mathrm{eq}}\,v' . \tag{27} \]

Let now

\[ v=v'+v'' . \]

Then for \(v''\), if the term containing \(\dfrac{d\rho}{dt}\) is neglected, we obtain

\[ \frac{d^2 v''}{dt^2} =-\frac{4\pi e^2}{m}v''(\rho_{\mathrm{eq}}-\rho) . \tag{28} \]

On the other hand, if we put \(v_1=-v_2\), then from equation (26), with the same neglect, we obtain this same equation.

Thus, equation (28) approximately determines the velocity of the electrons of a beam admitted directly along the equilibrium orbit, under the influence of the repulsive forces of the charges among themselves and of the focusing forces of the magnetic field.

If the beam is emitted not onto the equilibrium orbit, then this motion will be superposed on the oscillatory motion of the whole beam about the equilibrium orbit.

Under our assumptions, indicated at the beginning of the present paragraph, we can find the function \(\rho(t)\), as well as \(v''\) and \(x''\), caused by the action of the electric field of the beam and the magnetic field of the betatron.

Indeed, for a beam of electrons emitted directly onto the equilibrium orbit, we can write

\[ \begin{gathered} v_y=v_\theta=\mathrm{const},\\ \frac{\partial v''_z}{\partial r}=0,\qquad \frac{\partial v''_r}{\partial z}=0,\\ \frac{\partial \rho}{\partial r}=0,\qquad \frac{\partial \rho}{\partial z}=0 . \end{gathered} \tag{29} \]

The continuity equation gives

\[ \frac{d\rho}{dt}=v_y\,\frac{\partial \rho}{\partial y} \tag{30} \]

and

\[ \frac{d\rho}{dt}=-\rho\,\frac{\partial v''_r}{\partial r}. \tag{31} \]

Using the equality

\[ \frac{d v''_r}{dt}=v''_r\cdot \frac{\partial v''_r}{\partial r}=-\frac{e}{m}E_{r\mathrm{av}} \]

and combining it with (31), we obtain

\[ \frac{d\rho}{dt}=-\frac{\rho}{v''_r}\cdot \frac{d v''_r}{dt}. \tag{32} \]

Differentiating the last expression once more and expressing \(\dfrac{d^2 v''_r}{dt^2}\) in terms of the densities \(\rho\) and \(\rho_{\mathrm{av}}\), we obtain for the beam density the equation

\[ \frac{d^2\rho}{dt^2}-\frac{2}{\rho}\left(\frac{d\rho}{dt}\right)^2+\frac{4\pi e^2}{m}\,\rho(\rho-\rho_{\mathrm{av}})=0. \tag{33} \]

If \(\rho_{\mathrm{av}}=\mathrm{const}\), and under our assumptions this may be adopted, equation (33) can be solved, and we find \(\rho\) as a function of \(t\).

The solution has the form

\[ \rho= \frac{\rho_{\mathrm{av}}} { 1-\sqrt{1+C_1\frac{m}{4\pi e^2}\rho_{\mathrm{av}}}\, \sin\left(\sqrt{\frac{4\pi e^2}{m}\rho_{\mathrm{av}}}\,t+C_2\right) }. \tag{34} \]

If

\[ \left(\frac{d\rho}{dt}\right)_{t=0}=0, \]

i.e. the beam leaves the gun in the form of a parallel beam,

\[ C_1=-\frac{8\pi e^2}{m}\cdot \frac{1}{\rho_0} +\frac{4\pi e^2}{m}\cdot \frac{\rho_{\mathrm{av}}}{\rho_0^2}, \tag{35} \]

where \(\rho_0\) is the initial beam density. The expression under the radical is then also simplified:

\[ 1+C_1\frac{m}{4\pi e^2}\rho_{\mathrm{av}} = \left(1-\frac{\rho_{\mathrm{av}}}{\rho_0}\right)^2 = D^2. \tag{36} \]

Since at \(t=0\) \(\rho\) must be equal to \(\rho_0\), it follows that \(C_2=\dfrac{\pi}{2}\).

Finally we have

\[ \rho= \frac{\rho_{\mathrm{av}}} { 1-D\cos\sqrt{\frac{4\pi e^2}{m}\rho_{\mathrm{av}}}\,t }. \tag{37} \]

Thus, the character of the variation of \(\rho\) will be determined by the value of \(D\). In the region \(1>D>0\), after injection the electron beam first diverges, then converges. The density always remains \(<\rho_0\). In the region \(-1<D<0\), the beam first converges, then diverges, with \(\rho>\rho_0\).

In the region \(D<-1\), the beam loses its hydrodynamic character; therefore we do not consider this region.

Knowing the variation of \(\rho\) with time, we can, by virtue of our assumptions, determine the variation of the beam width in the \(r\) direction. Since

\[ l_\rho\rho=l_0\rho_0, \tag{38} \]

where \(l_t=(x_2-x_1)_t\) and \(l_0=(x_2-x_1)_0=2\Delta l_0\), then

\[ l_t=\frac{\rho_0 l_0}{\rho_{\mathrm{av}}} \left[ 1-D\cos\sqrt{\frac{4\pi e^2}{m}\rho_{\mathrm{av}}}\,t \right]. \tag{39} \]

The velocity of the edge of the beam will be

\[ v_r''= -\frac{\rho_0}{\rho_{\mathrm{av}}}\Delta l_0 \left(1-\frac{\rho_{\mathrm{av}}}{\rho_0}\right) \sqrt{\frac{4\pi e^2}{m}\rho_{\mathrm{av}}}\, \sin\sqrt{\frac{4\pi e^2}{m}\rho_{\mathrm{av}}}\,t . \tag{40} \]

If the beam moves as a whole, then the velocity of the electron is

\[ v_r=v_r'+v_r''= -\sqrt{\frac{4\pi e^2}{m}\rho_{\mathrm{av}}} \left[ x_0-\Delta l_0\left(\frac{\rho_0}{\rho_{\mathrm{av}}}-1\right) \right]\cdot \sin\sqrt{\frac{4\pi e^2}{m}\rho_{\mathrm{av}}}\,t . \tag{41} \]

Hence the trajectories of the extreme electrons of the beam are obtained:

\[ \begin{aligned} x_2&= \left[ x_0-\Delta l_0\left(\frac{\rho_0}{\rho_{\mathrm{av}}}-1\right) \right] \left[ \cos\sqrt{\frac{4\pi e^2}{m}\rho_{\mathrm{av}}}\,t-1 \right] +x_0+\Delta l_0,\\ x_1&= \left[ x_0+\Delta l_0\left(\frac{\rho_0}{\rho_{\mathrm{av}}}-1\right) \right] \left[ \cos\sqrt{\frac{4\pi e^2}{m}\rho_{\mathrm{av}}}\,t-1 \right] +x_0-\Delta l_0 . \end{aligned} \tag{42} \]

As the beam density increases, a moment may occur when the beam, in its motion in the direction \(r\), will touch the wall of the chamber, and the electrons will go to ground through the conducting layer. The percentage of electron losses is determined by the ratio

\[ \eta= \left| \frac{ X_{\mathrm{left}}-x_0-\Delta l_0\left(2\frac{\rho_0}{\rho_{\mathrm{av}}}-1\right) }{ 2\Delta l_0\left(2\frac{\rho_0}{\rho_{\mathrm{av}}}-1\right) } \right|. \tag{43} \]

\(X_{\mathrm{left}}\) is the coordinate of the left inner wall of the chamber.

From the formulas given it is evident that electrons which, without taking interaction into account, should have struck the chamber wall, owing to the interaction do not reach the wall on the first revolution.

The electron trajectories can also be calculated by a more exact method. Since

\[ x=-\frac{e}{2m}\int_0^t (t-\xi)^2\frac{dE_x}{d\xi}\,d\xi -\frac{e}{2m}E_{x0}\,t^2+v_{x0}t+x_0, \]

then, if the dependence \(\dfrac{dE_x}{dt}\) were known, the calculation would present no difficulty. However, finding \(\dfrac{dE_x}{dt}\) is rather difficult. We have

\[ v_x=-\frac{e}{m}\int_0^t (t-\xi)\frac{dE_x}{d\xi}\,d\xi -\frac{e}{m}E_{x0}\,t+v_{x0}, \]

\[ \frac{dv_x}{dt}=-\frac{e}{m}\int_0^t \frac{dE_x}{d\xi}\,d\xi=-\frac{e}{m}E_{x0}, \]

\[ \frac{d^2v_x}{dt^2}=-\frac{e}{m}\frac{dE_x}{dt} =-\frac{e}{m}v_x\frac{\partial E_x}{\partial x}, \]

\[ \frac{d^2v_x}{dt^2}=\frac{4\pi e^2}{m}\left(\rho-\rho_{\mathrm{eq}}\right)v_x. \]

Having found \(v_x\) from the last equation \(\left[(\rho-\rho_{\mathrm{eq}})\right.\) is assumed known], we find \(\dfrac{dE}{dt}\) and then \(x\). In practice, however, this method proves to be very cumbersome.

Motion of a Single Electron in the Magnetic Field of a Betatron in the Presence of an Electron Beam

The solution of this problem will serve as the basis for solving the problem of the interaction of electron beams.

The forces acting on an electron from an electron beam of width \(x_2-x_1\) are determined by the field strength produced by the beam:

\[ E_E=2\pi e\rho\,(x_2-x_1)=2\pi e\rho_0\,(x_2-x_1)_0=\mathrm{const} \tag{44} \]

(the beam is still regarded as planar).

The forces acting on the electron from the magnetic field are determined by the equivalent field strength \(E_M\)

\[ E_M=4\pi e\rho_{\mathrm{eq}}\,x \tag{45} \]

(\(x\) is the distance from the equilibrium orbit).

Obviously:

\[ \frac{dE_{\mathrm{eff}}}{dt}=\frac{dE_E}{dt}+\frac{dE_M}{dt} =4\pi e\rho_{\mathrm{eq}}\,v_r. \]

The electron velocity in the direction \(r\) will be

\[ v_r=-\frac{4\pi e^2}{m}\rho_{\mathrm{eq}}\int_0^t (t-\xi)v_r\,d\xi -\left[ \frac{4\pi e^2}{m}\rho_{\mathrm{eq}}x_0 -\frac{2\pi e^2}{m}\rho_0(x_2-x_1) \right]t+v_{r0}. \]

Differentiating twice with respect to \(t\), we obtain the previous equation for the motion of a single electron in a magnetic field

\[ \frac{d^2v_r}{dt^2}=-\frac{4\pi e^2}{m}\rho_{\mathrm{eq}}v_r. \]

However, the presence of an electron beam, which creates a field of constant strength, causes a shift of the minimum of the potential well of the magnetic field. Instead of a well of the form \(V_M=ax^2\), we obtain \(V_M=ax^2+Bx+c\). The minimum will not be at \(x=0\), but at \(x=-\dfrac{b}{2a}\). In our case

\[ a=-\frac{V_{\mathrm{inc}}}{R}(1-n_R)=-2\pi e\rho_{\mathrm{eq}}, \]
\[ b=-2\pi e\rho_0(x_2-x_1)_0, \tag{46} \]
\[ x_{\min}=-\frac{\rho_0}{\rho_{\mathrm{eq}}}\cdot\Delta l_0. \]

The motion of the electron beam will proceed along the line

\[ x_2=x_0\cos\sqrt{\lambda}\cdot t=x_0\sin\left(\sqrt{\lambda}\cdot t+\frac{\pi}{2}\right), \tag{47} \]

\[ \lambda=\frac{4\pi e^2}{m}\rho_{\mathrm{eq}} \]

(along this line, in fact, the mean electron of the beam moves).

The trajectory of an individual electron up to the point where it intersects the electron beam (more precisely, the trajectory of the mean electron of the beam) will be determined by the minimum \(A\) shifted to the left (Fig. 1)

\[ x_1=a_1\sin(\sqrt{\lambda}\cdot t+\alpha_1)-d. \tag{48} \]

The magnitude of the amplitude \(a_1\) and of the phase \(\alpha_1\) is determined from the boundary conditions at \(t=0\). The motion of the electron after the intersection is determined by the equations

\[ x_1=a_2\sin(\sqrt{\lambda}\cdot t+\alpha_2)+d, \]
\[ v_1=a_2\sqrt{\lambda}\cdot\cos(\sqrt{\lambda}\cdot t+\alpha_2), \tag{49} \]

where \(a_2\) and \(\alpha_2\) are found from the initial conditions for \(x_1\) and \(v_1\) at the point of intersection. For the beam the equation remains the same.

Fig. 1

The results of the calculation for an electron with phase \(\alpha_1=\dfrac{3}{2}\pi\) and \(x_{10}=x_0\) are presented in Fig. 1. From the figure one can clearly see the influence of the presence of the electron beam on the motion of the electron.

Motion of Two Beams of the Same Velocity on the Second Revolution

Let us explain the methods of our further investigation. We shall replace the actual picture of continuous injection of electrons and continuous rearrangement of the beam trajectories by a “stepwise” picture of stationary beams, considering the situation that in fact obtains only toward the end of the periods of revolution, which are counted according to the first electrons of the first beam. In other words, we consider the position of the beams at the moments when a new beam, which began its existence at the moment when the preceding beam completed a revolution, in turn completes a revolution. The beam that began its existence at the moment when the first beam completed its first revolution will be called the second beam, and so on. We may introduce the general notation:

\(t\) — time from the beginning of the given revolution;
\(\tau\) — time from the moment at which one beam crosses another;
\(x_0\) — coordinate of the mean point of ejection of electrons from the gun;
\({}_n^k A_m\) — amplitude of the \(n\)-th beam on the \(m\)-th revolution in the \(k\)-th section;
\({}_n^k \beta_m\) — the corresponding phase;
\({}_n^k x_m\) — the corresponding displacement;
\({}_n^k x_{m0}\) — initial displacement of the \(n\)-th beam on the \(m\)-th revolution in the \(k\)-th section.

We shall consider the motion of two electron beams of equal initial width and density. In each beam we shall consider the motion of the central electron as the motion of a single electron in the magnetic field of the betatron plus the electric field of the other electron beam. If the beams did not interact with each other, the trajectories of these electrons would, in our representation, be sinusoids with a common axis representing the position of the bottom points of the potential well created by the magnetic field. Since interaction does take place, this interaction leads to a displacement of the bottom of the potential well for the electrons of the first beam by a distance \(d = \dfrac{\rho_0}{\rho_{\mathrm{av}}}\Delta l_0\) to position \(A\) (Fig. 2). For the second electron beam the minimum is displaced by the same distance to position \(B\). After the beams cross, the positions of the new axes of oscillation of the beams correspondingly interchange.

The equations of the trajectories and velocities of beam 1 (i.e., in fact, the trajectories of those electrons of the beam which do not feel the presence of the electrons of their own beam and move as a single electron in the magnetic field and in the electric field of the other beam) will be:

\[ \begin{aligned} {}_1^1 x_2 &= {}_1^1 A_2 \sin\left(\sqrt{\lambda}\, t + {}_1^1 \beta_2\right) - d,\\ {}_2^1 x_2 &= {}_2^1 A_2 \sin\left(\sqrt{\lambda}\, t + {}_2^1 \beta_2\right) + d,\\ {}_1^1 v_2 &= {}_1^1 A_2 \sqrt{\lambda}\,\cos\left(\sqrt{\lambda}\, t + {}_1^1 \beta_2\right),\\ {}_2^1 v_2 &= {}_2^1 A_2 \sqrt{\lambda}\,\cos\left(\sqrt{\lambda}\, t + {}_2^1 \beta_2\right). \end{aligned} \tag{50} \]

The values of the amplitudes and phases are determined from the boundary conditions at the “entrance” to the second revolution, i.e., at \(t=0\).

In the second section of the second revolution we shall have the equations:

\[ \begin{aligned} {}^{2}_{1}x_{2} &= {}^{2}_{1}A_{2}\sin\left(\sqrt{\lambda}\tau_{1}+{}^{2}_{1}\beta_{2}\right)+d,\\ {}^{2}_{2}x_{2} &= {}^{2}_{2}A_{2}\sin\left(\sqrt{\lambda}\tau_{1}+{}^{2}_{2}\beta_{2}\right)-d,\\ {}^{2}_{1}v_{2} &= {}^{2}_{1}A_{2}\sqrt{\lambda}\cos\left(\sqrt{\lambda}\tau_{1}+{}^{2}_{1}\beta_{2}\right),\\ {}^{2}_{2}v_{2} &= {}^{2}_{2}A_{2}\sqrt{\lambda}\cos\left(\sqrt{\lambda}\tau_{1}+{}^{2}_{2}\beta_{2}\right). \end{aligned} \tag{51} \]

The amplitudes and phases are determined from the conditions at the point of intersection, i.e. at \(\tau_{1}=0\) \([\tau_{1}=t-t_{\text{intersection}}]\).

The results of calculations for \(n=0.7;\ 0.75;\ 0.8\) and \(d=0;\ 0.2;\ 0.5\) are presented in Fig. 2.

Fig. 2

For \(n=0.75\) the amplitude of the beam oscillations does not change; however, an increase in the effective period of oscillations occurs. This increase is the greater, the larger \(d\) is (i.e. the beam density, since \(d=\dfrac{\rho_{0}}{\rho_{\text{eq}}}\Delta l_{0}\)). For

For \(n>0.75\), on the second revolution the maximum deviation of the first bunch in the first section increases with increasing \(d\). For \(n<0.75\) this increase will not occur. In the second section (after the bunches cross), the bunches approach one another the more closely, the larger the value of \(d\).

In the figures presented, the values of \(d\) increase only up to 0.5. Larger values of \(d\) lead to a large increase in the amplitude of the oscillations and to greater losses of electrons on the chamber walls (one must also remember that a greater bunch density also leads to a greater broadening of the bunch itself).

Motion of the bunches on the third and subsequent revolutions

For the third revolution the equations of the trajectories will be constructed in complete analogy with the case of the second revolution. It is only necessary to take into account the mutual arrangement of the bunches in order to determine the “axis” of oscillation of the bunch. For example, for the first section of the third revolution the following equations are obtained:

\[ \begin{aligned} {}^{1}_{1}x_{3} &= {}^{1}_{1}A_{3}\sin\left(\sqrt{\lambda}\,t+{}^{1}_{1}\beta_{3}\right),\\ {}^{1}_{2}x_{3} &= {}^{1}_{2}A_{3}\sin\left(\sqrt{\lambda}\,t+{}^{1}_{2}\beta_{3}\right)-2d,\\ {}^{1}_{3}x_{3} &= {}^{1}_{3}A_{3}\sin\left(\sqrt{\lambda}\,t+{}^{1}_{3}\beta_{3}\right)+2d,\\ {}^{1}_{1}v_{3} &= {}^{1}_{1}A_{3}\sqrt{\lambda}\,\cos\left(\sqrt{\lambda}\,t+{}^{1}_{1}\beta_{3}\right),\\ {}^{1}_{2}v_{3} &= {}^{1}_{2}A_{3}\sqrt{\lambda}\,\cos\left(\sqrt{\lambda}\,t+{}^{1}_{2}\beta_{3}\right),\\ {}^{1}_{3}v_{3} &= {}^{1}_{3}A_{3}\sqrt{\lambda}\,\cos\left(\sqrt{\lambda}\,t+{}^{1}_{3}\beta_{3}\right). \end{aligned} \tag{52} \]

For the second section analogous equations are obtained. The results of the calculation for the values \(n=0.7;\ 0.75;\ 0.8\) and densities \(d=0;\ 0.5\) are presented in Fig. 3.

On the third revolution, symmetry of the arrangement of the bunches is not obtained even for \(n=0.75\). Therefore, on the third revolution the effect of the interaction of the bunches is more striking. For the case \(n=0.75\), bunch 2 deviates very considerably from the trajectory of this same bunch in the case where there is no interaction. The amplitude of the bunch oscillation is increased. For the first bunch an interesting peculiarity is obtained: the bunch does not tend to cross the axis \(x=0\). A forward shift of the trajectory of the third bunch takes place.

For \(n=0.7\), at high densities the second bunch departs considerably from the trajectory of the noninteracting bunch. The amplitude of the oscillation increases both for bunch 2 and for bunch 1, although the first bunch itself changes its trajectory little. The third bunch likewise does not cross the axis \(x=0\) and performs an oscillation of small amplitude about an axis parallel to the axis \(x=0\).

For \(n=0.8\) a large deviation is likewise obtained for the second bunch; the third bunch also undergoes a large deviation. The first bunch is displaced significantly forward, with a tendency toward a decrease in the amplitude of the oscillations.

Consideration of the graphs of the bunch trajectories on the first three revolutions makes it possible to draw a definite conclusion about the character of the motion of the bunches on the fourth and subsequent revolutions.

For large values of \(d\), as is seen from Figs. 2 and 3, the probability of bunches being thrown onto the chamber wall is already large on the second and third revolutions. In the throwing of bunches onto the chamber walls, a large role is played by the mutual arrangement of the bunches on the given or preceding revolutions. The greatest—

...the greatest chances of being lost on a given revolution are for those bunches which, moving in the direction of the chamber wall too steeply, remain for a long time on one side of the other bunches and therefore experience the largest repulsive forces. The smaller the value of $d$ (i.e., $\rho_0$), the smaller the probability of loss on the walls and the greater the probability of collision with the gun. The optimal values of $d$ or $\rho_0$ lie near $\rho_{\mathrm{eq}}$ within rather narrow limits. For these values of $\rho_0$ the successive

Fig. 3

bunches “fit” well. The effective period of the oscillations continually increases; the oscillations of the bunches will proceed about axes that change continuously from revolution to revolution, until the electrons begin to move along circumferences concentric with the equilibrium orbit of the betatron.

Indeed, after a sufficiently large number of revolutions the entire chamber will be filled with intersecting bunches of electrons, creating a certain mean density $\rho$. The next bunch entering the chamber thus finds itself in the field of the magnetic forces plus the elec-

tric field of the preceding bunches. The presence of this total bunch of electrons of density \(\bar\rho\) creates for the new bunch a potential well of smaller depth:

\[ \lambda'=\sqrt{\frac{4\pi e^2}{m}\left(\rho_{\mathrm{rav}}-\bar\rho\right)}. \]

The oscillations of the new bunches will begin with an ever-increasing period

\[ T=\frac{1}{\sqrt{\lambda'}}. \]

At \(\bar\rho=\rho_{\mathrm{rav}}\), focusing ceases, and upon further injection of electrons they will be thrown onto the walls of the chamber.

It should not be thought that the injection process proceeds smoothly. Under specific capture conditions, individual bunches on one revolution or another may, wholly or partially, collide with the gun or with the walls even in the most favorable case.

However, this circumstance will not be catastrophic for capture, since for the new bunch the electrostatic situation will already be different than for the “lost” bunch; therefore the capture process will continue and all losses will be compensated.\(^1\)

Conditions for the motion of stationary electron bunches with different velocities in the constant magnetic field of a betatron

In the preceding sections we dealt with electron bunches of a single velocity, for which the potential well was characterized by one value of the constant \(C\), for example \(C=0\). For electron bunches with different velocities, it is necessary to take into account the presence of a special potential well for electrons of a given velocity, i.e., a well with a definite value of the constant \(C\).

If the value of the potential function \(V_m(C,r)\) is expressed through the value of the potential function \(V_m(0,R_0)\), then, in first approximation, we obtain

\[ V_m(C,r)=V_{m0}\left[ 1+\frac{2C}{R_0^2H_{z0}} +\frac{C^2}{R_0^4H_{z0}^2} +\frac{1-n_0}{R_0^2}\Delta r^2 -\frac{2C\cdot\Delta r}{R_0^3H_{z0}} \right]. \tag{53} \]

Calculating \(\rho_{\mathrm{rav}}\) by the formula

\[ 4\pi e\rho_{\mathrm{rav}} = \frac{\partial^2 V_m}{\partial r^2} + \frac{1}{r}\frac{\partial V_m}{\partial r} \simeq \frac{\partial^2 V_m}{\partial r^2} \]

(we ignore the dependence of \(V_m\) on \(z\) and assume \(r\gg \Delta r\)), we obtain

\[ 4\pi e\rho_{\mathrm{rav}} = \frac{2V_{m0}}{R_0^2}(1-n_0). \tag{54} \]

Thus, in this approximation the equilibrium density does not depend on \(C\). The line of minima of the potential wells is determined from the condition:

\[ \frac{\partial V_m}{\partial r} = \frac{e}{2mc^2} \left[ 2H_{z0}^{\,2}(1-n_0)\Delta r - \frac{2H_{z0}}{R_0}C \right] =0. \]

The displacement of the minima relative to the minimum of the well for zero electrons is

\[ \Delta r=\frac{C}{R_0H_{z0}(1-n_0)}. \tag{55} \]

\(^1\) As we shall see below, the process of capturing electrons with the same value of the constant \(C\) for all electrons is of theoretical interest only, since in practice it cannot be carried out.

We shall further assume that the electron gun of the betatron operates in the saturation regime, i.e., at any voltage on the gun electrodes all electrons emitted from the filament are removed.

Fig. 4

For a given filament heating, the injection-current density is

\[ i=e\rho v=e\cdot 5.93\cdot 10^7 \sqrt{V}\,\rho=\text{const}. \]

Hence

\[ \rho=\frac{i}{e\cdot 5.93\cdot 10^7\sqrt{V}} =\frac{b}{\sqrt{V}} . \tag{56} \]

Let \(V_0\) be the injection voltage, in volts, corresponding to zero electrons, and let the initial density of the beam of these electrons be

\[ \rho_0=\rho_{\mathrm{eq}}\cdot k, \]

where \(k\) is some number. Then

\[ b=\rho_{\mathrm{eq}}\cdot k\sqrt{V_0}. \tag{57} \]

If the injection voltage is supplied in the form of a sinusoidal pulse, then the density of the electron beam will vary accordingly:

\[ \rho=\frac{b}{\sqrt{V_{am}\sin\omega t}}, \tag{58} \]

where \(V_{am}\) is the amplitude of the injection pulse and \(\omega\) is its frequency. The voltage of zero electrons will be

\[ V_0=V_{am}\sin\omega t_0 . \tag{59} \]

It seems obvious that the capture of electrons begins only with those electrons whose velocities correspond to those values of \(C\) for which potential wells exist. Thus, in what follows we shall be interested not in the entire curve of the injection pulse, but only in certain portions of it.

We shall consider injection under the assumption that the magnetic field remains constant; that is, the more exact pictures (Fig. 4, curves \(A\)) will be replaced by simplified ones (curves \(B\) of the same figure).

All that has been set forth makes it possible to introduce the necessary changes into the picture of the motion of electron beams with different velocities as compared with the picture of the motion of beams with a single velocity.

The change is expressed primarily in the fact that the minimum of the potential well for electrons with a smaller beam density (the subsequent beam) is shifted toward larger radii, as compared with the position of the minimum of the preceding beam, by a distance

\[ \Delta r=\Delta x_0=\frac{\Delta C}{R_0H_{z0}(1-n_0)} . \tag{60} \]

In addition, the shift of the potential well for the preceding beam 1 (Fig. 5), caused by the subsequent beam 2, will now differ by some amount \(\Delta d\):

\[ \Delta d=d_2-d_1, \]

where \(d_2\) is the displacement actually caused by beam 2, and \(d_1\) is the displacement that would have been caused by beam 2 if its density had remained equal to the density of the first beam,

\[ \Delta d=\frac{\Delta \rho}{\rho_{\mathrm{av}}}\,\Delta l_0 . \tag{61} \]

We shall assume that, over a certain interval of values of \(C\), the equilibrium density \(\rho_{\mathrm{av}}\) will remain unchanged. Both the quantity \(\Delta x_0\) and the quantity \(\Delta d\) can be expressed explicitly if the position of the voltage interval in which capture is possible is specified.

Let, for \(C=0\), the voltage be

\[ V_0=V_{am}\sin \omega t . \]

For some \(C\), the corresponding value will be

\[ V_c=V_{am}\sin \omega t_c . \]

If we assume that \(V_c\) is close to \(V_0\), then

\[ V_c=V_0\left[1+\alpha(t_c-t_0)\right], \tag{62} \]

where

\[ \alpha=\omega\,\frac{\cos \omega t_0}{\sin \omega t_0}. \]

In this case

\[ C=\Delta C \simeq \frac{cm}{e} R_i \cdot 5.93\cdot 10^7 \left(\sqrt{V_c}-\sqrt{V_0}\right) = \frac{cm}{e} R_i \cdot 5.93\cdot 10^7 \sqrt{V_0}\,\frac{\alpha}{2}(t_c-t_0). \tag{63} \]

If we take \(t_c-t_0=T\), the revolution time of one beam (in our calculations this will be the revolution time of the beam of zero electrons), then the displacement of the equilibrium orbit of the new beam relative to the preceding one may be calculated as

\[ \Delta x_0=D\frac{\sqrt{V_0}\,\alpha}{1-n_0}. \tag{64} \]

Here

\[ D=\frac{R_i}{2}\cdot 5.93\cdot 10^7 \sqrt{\frac{m}{2eV_{m0}}}\,T, \]

\(\Delta \rho\) is found from the general expression for \(C\):

\[ \rho_c=\rho_0+\Delta \rho= \frac{b}{\sqrt{V_0}} \left[1-\frac{\alpha}{2}(t_c-t_0)\right]. \tag{65} \]

Hence

\[ d_c=d_0+\Delta d = d_0\left(1+\frac{\Delta \rho}{\rho_0}\right) = d_0\left(1-\frac{\alpha}{2}T\right). \tag{66} \]

Calculation of the beam trajectories on the second revolution

For different beam velocities, equations (50) take the form

\[ {}^{1}x_2 = {}^{1}A_2 \sin\left(\sqrt{\lambda}\,t+{}^{1}\beta_2\right)-d_c, \]

\[ {}^{1}_{2}x_2 = {}^{1}_{2}A_2 \sin\left(\sqrt{\lambda}\,t+{}^{1}_{2}\beta_2\right)+d_0+\Delta x_0. \tag{67} \]

Here \(d_c\) and \(\Delta x_0\) have the values indicated in the preceding section.

The results of calculating the trajectories of beams for \(n=0.7;\ 0.75;\ 0.8\) are given in Fig. 5; \(d_0\) was taken equal to 0.5 for all three curves.

Fig. 5

Labels in the figure: \(x\); \(n=0.7\); \(n=0.75\); \(n=0.8\); first beam; second beam; of different velocities; of one velocity; \(t\cdot 10^8\) sec.

It is clear from the figures that the amplitudes on the second revolution become smaller than for beams of equal density. The steeper the rise of the voltage in the pulse, the more sharply the amplitude of the oscillations decreases. For the second beam the amplitude depends on the magnitude \(\Delta x_0\), which may turn out to be large.

If we write out the formula for the amplitude \({}_{1}^{1}A_2\), we obtain

\[ {}_{1}^{1}A_2 = \left[ x_0^2 + d_0^2 + 2d_0x_0\cos\sqrt{\lambda}\,T - d_0^2\alpha T + \frac{d_0^2\alpha^2T^2}{4} - d_0\alpha T\cdot x_0\cos\sqrt{\lambda}\cdot T \right]. \]

Thus it is seen that the amplitude of the first beam decreases in comparison with the amplitude for equal beam density, if \(\alpha>0\). For

for \(a<0\) the amplitude \({}^{1}\!A_2\) increases. For \(a<0\) the amplitude \({}^{1}_{2}\!A_2\) also increases, since \({}^{1}_{2}\!A_2=(x_0-d_0-\Delta x_0)\).

Conclusion

It is pointless to consider further revolutions, since there are fewer and fewer grounds for regarding as valid the simplifying assumptions we have made. We shall therefore try to draw a conclusion from considering the interaction of electron beams during the first revolutions, assuming that the subsequent revolutions intensify the effects in the first revolutions (the conclusions will, of course, be hypothetical in character, and their validity can be proved only by agreement with experiment of the theoretical consequences of these conclusions).

1. Owing to their mutual interaction, the electron beams acquire a tendency toward “straightening,” i.e., toward motion along a circle concentric with the equilibrium orbit.

2. The “straightened” beams can form an equilibrium beam. The equilibrium beam should determine the total number of electrons going into acceleration, i.e., ultimately, the radiation intensity of the betatron.

3. The magnetic fields of the beams themselves, which we have not considered, can only assist the process of “straightening,” but play a secondary role.

The final stage of capture, as it appears in the light of our conclusions, will be considered in the following article [7].

LITERATURE

1. D. W. Kerst and R. Serber, Phys. Rev., 60, 53–58, 1941.
2. Ya. P. Terletskii, Journ. Phys. USSR, 9, 159, 1945.
3. J. A. Rajchman and W. H. Cherry, Journ. Frankl. Inst. 243, 261–285, 1947.
4. R. Wideröe, Schweizer Archiv, 13, 225–232, 1947.
5. H. Jahn und H. Kopferman, Ann. d. Phys., 6, 305–326, 1946.
6. J. Picht. Optica, 6, 49–55, 61–97, 133–144, 1950.
7. B. N. Rodimov. On the mechanism of electron capture into acceleration in a betatron, II. Izvestiya TPI, vol. 87, 1957.
8. B. N. Rodimov. Regularities of the magnetic field of a betatron. Izvestiya TPI, vol. 87, 1957.