ON THE STABILITY OF AN EQUILIBRIUM ELECTRON BEAM IN A BETATRON

P. A. CHERDANTSEV

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

To describe the motion of a single electron in the cylindrically symmetric field of a betatron, use is made of the potential function

\[ V_{mc}=\frac{e}{2mc^{2}}\left(A(r,z)+\frac{C}{r^{2}}\right)^{2}, \tag{1} \]

obtained from the equations of motion of the electron [1, 2]. Here \(m\) is the rest mass of the electron, and \(C\) is a constant determining the initial conditions of the electron. From the condition for a minimum of \(V_{mc}\) at a certain point \((R_c, Z)\), the constant is

\[ C=R_c^{2}\left(\frac{\partial A}{\partial r}\right)_{R_c}. \]

The radius \(R_c\) is the equilibrium radius for electrons with the given \(C\). The vector potential \(A(r,z)\) satisfies the equation

\[ \nabla^{2}A-\frac{A}{r^{2}}=0, \tag{2} \]

where \(A\) coincides with the component \(A_\varphi\), while the other two components, owing to the cylindrical symmetry of the field, are equal to zero. The relation between the vector potential \(\vec A\) and the magnetic-field intensity \(\vec H\),

\[ \vec H=\operatorname{rot}\vec A, \]

allows one to write

\[ H_z=\frac{2A}{r}, \]

where

\[ \overline{H}_z=\frac{2}{r^{2}}\int_{0}^{r} H_z r' \, dr' \]

is the mean value of the component of the magnetic-field intensity \(H_z\) over a circle of radius \(r\).

The study of the focusing properties of the magnetic field with the aid of the potential function (1) leads to the result that \(H_z\) must satisfy the following inequalities:

\[ 0<-\frac{r}{H_z}\frac{\partial H_z}{\partial r}<1. \tag{3} \]

The quantity \(-\dfrac{r}{H_z}\dfrac{\partial H_z}{\partial r}\) is usually denoted by \(n\). A single electron with a given \(C\) (a \(C\)-electron) in the region of the focusing forces will execute oscillations along \(r\) about the equilibrium position \(R_c\) and along the \(z\)-axis relative to the plane of symmetry \(z=0\).

If, however, in the region of the focusing forces there is a large number of moving electrons, then their collective interaction with one another substantially changes the character of the motion of each of the electrons. A magnetic field with given characteristics can hold in the region of the focusing forces a quite definite charge. In papers [2, 3] the assumption is made that an equilibrium beam is formed in the region of the focusing forces. The charge density is determined by Poisson’s equation

\[ \nabla^2 V_e=-4\pi\rho, \]

where \(V_e\) is the potential of the electric charge, and \(\rho\) is the charge density. Moreover, the potential \(V_e\) is equal to the magnetic potential \(V_{mc}\), that is,

\[ V_e=+V_{mc}. \tag{4} \]

Thus, the potential \(V_{mc}\) makes it possible to find the density \(\rho\). The charge density is computed so simply if in the region of the focusing forces there are electrons with only one \(C\).

However, this case is unrealistic. Whatever method is used to introduce the electrons, a definite set of \(C\)’s is necessarily introduced.

To compute the density of the equilibrium charge \(\rho\) with any set of \(C\)’s, one may introduce, instead of \(V_{mc}\), the so-called effective magnetic potential \(V_{\mathrm{eff}}\). In place of condition (4) there will now appear the condition

\[ V_e=+V_{\mathrm{eff}}, \]

and the charge density will be determined by a method analogous to the preceding one:

\[ \rho=-\frac{1}{4\pi}\nabla^2 V_{\mathrm{eff}}. \tag{5} \]

To find \(\rho\) and to investigate the stability of the beam it is necessary to calculate \(V_{\mathrm{eff}}\).

At each point of the equilibrium beam the focusing force due to the magnetic field \(\overline{F}_m\) is equal to the defocusing force due to the electric charge of the beam \(\overline{F}_e\). The focusing force is determined by the constant \(C\) possessed by the electron.

If there is a set of electrons with different \(C\)’s, then to each \(C\) there corresponds its own function \(V_{mc}(r,z)\). In the absence of interaction the electrons would execute oscillations about the equilibrium positions \(R_c\), determined by the minima of the functions \(V_{mc}\). If, however, we take the real case, then the electrostatic interaction of the electrons will shift the equilibrium positions \(R_c\) to certain points with coordinates \(r_c\). Moreover, the displacement \(x_c=r_c-R_c\), obviously, will be the larger the farther \(R_c\) was from the center of the beam. If the position of the center of the beam is fixed, then \(x_c\) will be determined by the value of \(R_c\), or, what is the same, by the value of \(C\).

Let us choose as the basic central orbit with radius \(R_0\) the orbit for electrons with \(C=0\). Introduce a coordinate system with center at the point \((R_0,z=0)\), with axes \(x_0,z\). Then the displacement \(x_c=r_c-R_c\) will correspond to the coordinate \(x_0=r-R_0\), and we can express the relation between \(x_c\) and \(x_0\) in the following way:

\[ x_c=f(x_0,z). \]

Here \(f(x_0,z)\) is as yet an unknown function.

The force $\bar F_e$ is a function of $x_0$ and $z$, while $F_{mc}$ is a function of $x_c$ and $z$, but $F_{mc}$ should be taken at those points where $\bar F_e=-\bar F_{mc}$, i.e., at points $x_c=f(x_0,z)$. This fact may be written as follows for the force components along $x_0$ and $z$:

\[ F_{e x_0}=(-F_{mc x})_{x_c=f(x_0,z)}, \]

\[ F_{e z}=(-F_{mc z})_{x_c=f(x_0,z)}. \]

Each of these forces is determined by the corresponding potential: $F_e$ by the electric potential $V_e$, and $(F_{mc})_{x_c=f(x_0,z)}$ by the effective potential $V_{\mathrm{eff}}$.

We may write that

\[ -\frac{\partial V_e}{\partial x_0} = \left(+\frac{\partial V_{mc}}{\partial x_c}\right)_{x_c=f(x_0,z)} = +\frac{\partial V_{\mathrm{eff}}}{\partial x_0}, \]

\[ -\frac{\partial V_e}{\partial z} = \left(+\frac{\partial V_{mc}}{\partial z}\right)_{x_c=f(x_0,z)} = +\frac{\partial V_{\mathrm{eff}}}{\partial z}. \]

The right-hand equalities give equations for $V_{\mathrm{eff}}$:

\[ \frac{\partial V_{\mathrm{eff}}}{\partial x_0} = \left(\frac{\partial V_{mc}}{\partial x_c}\right)_{x_c=f(x_0,z)}, \]

\[ \frac{\partial V_{\mathrm{eff}}}{\partial z} = \left(\frac{\partial V_{mc}}{\partial z}\right)_{x_c=f(x_0,z)}. \]

Instead of $V_{mc}$ one may substitute its explicit expression in terms of $A$ and $C$, where $A(r,z)$ depends in advance only on $x_0$ and $z$. The constant $C$ is determined by the radius $R_c$. $R_c$ may be expressed in terms of $x_0$ as follows:

\[ R_c=R_0+x_0-x_c=R_0+x_0-f(x_0,z). \]

Thus $C$ will turn out to be a function of $x_0$ and $z$, i.e., it will indicate the distribution of the given electrons $C$ in space.

The equations for $V_{\mathrm{eff}}$ take the form

\[ \frac{\partial V_{\mathrm{eff}}}{\partial x_0} = -\frac{e}{mc^2} \left[ \left(A+\frac{C}{r}\right) \left(\frac{\partial A}{\partial r}-\frac{C}{r^2}\right) \right]_{x_c=f(x_0,z)}, \tag{6a} \]

\[ \frac{\partial V_{\mathrm{eff}}}{\partial z} = -\frac{e}{mc^2} \left[ \left(A+\frac{C}{r}\right) \frac{\partial A}{\partial z} \right]_{x_c=f(x_0,z)}. \tag{6b} \]

Dividing one equation by the other, we obtain

\[ \frac{\partial V_{\mathrm{eff}}}{\partial r}\frac{\partial A}{\partial z} - \frac{\partial V_{\mathrm{eff}}}{\partial z} \left( \frac{\partial A}{\partial r} - \frac{C_{x_c=f(x_0,z)}}{r^2} \right)=0. \tag{7} \]

The solution of this equation gives $V_{\mathrm{eff}}$.

$V_{\mathrm{eff}}$ depends on the external field, specified by the vector potential $A(r,z)$. It can be shown that the solution of equation (2) is written in the form

\[ A=A_0\left\{ 1+(1-n_0)\frac{x_0^2}{2R_0^2} + \frac{n_0}{2R_0^2}z^2 + \right. \]

\[ \left. + \frac{n_0^2+2n_0-3-n'_0R_0}{3!}\frac{x_0^3}{R_0^3} - (n_0^2+n_0-n'_0R_0)\frac{z^2x_0}{R_0^3} \right\}. \tag{8} \]

where \(A_0, n_0\), and \(n'_0\) are the values of \(A\), \(n\), and \(\dfrac{\partial n}{\partial r}\) at the radius \(R_0\) for \(z=0\).

We find
\[ C_{x_c=f(x_0,z)} = \left[ R_c^2\left(\frac{\partial A}{\partial r}\right)_{R_c} \right]_{x_c=f(x_0,z)} . \]

Expressing \(R_c\) in terms of \(\dfrac{x_0-x_c}{R_0}\), and instead of \(x_c\) taking the expansion of the function \(f(x_0,z)\) in a series in powers of \(x_0,z\),

\[ f(x_0,z)=\varphi x_0+\psi\frac{x_0^2}{R_0}+\eta\frac{z^2}{R_0^2}, \]

where \(\varphi,\psi\), and \(\eta\) are dimensionless constants. After simple transformations \(C\) takes the form:

\[ \begin{aligned} C=A_0R_0\Bigg\{&(1-n_0)(1-\varphi)\delta_0 -\Bigg[(1-n_0)\psi -\frac{n_0^2+2n_0-1-n'_0R_0}{2}\,(1-\varphi)^2\Bigg]\delta_0^2 \\ &-\Bigg[(1-n_0)\eta +\frac{n_0^2+n_0-n'_0R_0}{2}\Bigg]\frac{z^2}{R_0^2}\Bigg\}. \end{aligned} \]

Here
\[ \delta_0=\frac{x_0}{R_0}. \]

Substitution of \(C\) into equation (7) leads to the following:

\[ \frac{\partial V_{\mathrm{eff}}}{\partial x_0} \left[ n_0-(n_0^2+2n_0-n'_0R_0)\delta_0 \right]\frac{z}{R_0} - \frac{\partial V_{\mathrm{eff}}}{\partial z} \left\{ (1-n_0)\varphi\delta_0 +a\delta_0^2 +(1-n_0)\eta\frac{z^2}{R_0^2} \right\}=0, \tag{10} \]

where
\[ a= \frac{n_0^2+2n_0-3-n'_0R_0}{2} + \frac{n_0^2+6n_0-7-n'_0R_0}{2}(1-\varphi)^2 + (1-n_0)(\psi+2(1-\varphi)). \]

This first-order partial differential equation is reduced to solving the equation

\[ \frac{d\delta_0}{n_0-(n_0^2+n_0-n'_0R_0)\delta_0} + \frac{\dfrac{z}{R_0^2}\,dz} {(1-n_0)\varphi\delta_0+(1-n_0)\eta\dfrac{z^2}{R_0^2}+a\delta_0^2} =0. \]

Introducing
\[ y=\frac{z^2}{2R_0^2}, \]
we obtain, after simplification,

\[ -\frac{dy}{d\delta_0} + \frac{(1-n_0)\eta} {n_0-(n_0^2+n_0-n'_0R_0)\delta_0}\,y = -\frac{(1-n_0)\varphi\delta_0+a\delta_0^2} {n_0-(n_0^2+n_0-n'_0R_0)\delta_0} \tag{11} \]

The solution of the last equation can be written in the form:

\[ y=e^{-F(\dot{\delta}_0)} \left( C_1-\int \frac{(1-n_0)\varphi \dot{\delta}_0+a\dot{\delta}_0^{\,2}} {n_0^{\,2}-(n_0^{\,2}+n_0-n'_0R_0)\dot{\delta}_0} \,e^{F(\dot{\delta}_0)}\,d\dot{\delta}_0 \right), \tag{12} \]

where

\[ F(\dot{\delta}_0)= -\int \frac{(1-n_0)\eta} {n_0-(n_0^{\,2}+n_0-n'_0R_0)\dot{\delta}_0} \,d\dot{\delta}_0 \]

\[ =\ln\left[n_0-(n_0^{\,2}+n_0-n'_0R_0)\dot{\delta}_0\right]^{ -\frac{(1-n_0)\eta}{\,n_0^{\,2}+n_0-n'_0R_0\,} }. \]

\(C_1\) is a constant of integration.

Finally, the integral of equation (10) is obtained in the form

\[ u(x_0,z)= \frac{n_0z^2}{R_0^{\,2}} +(1-n_0)\varphi\dot{\delta}_0^{\,2} +\frac{(1-n_0)\eta}{R_0^{\,2}}z^2\dot{\delta}_0 + \]

\[ +\left[a+(1-n_0)\varphi(1+n_0^{\,2}-n'_0R_0)\right] \frac{\dot{\delta}_0^{\,3}}{3} =\mathrm{const}. \tag{13} \]

The solution of equation (6) is an arbitrary function of \(u\), i.e.

\[ V_{\mathrm{eff}}=\Phi\bigl(u(x_0,z)\bigr). \]

\(V_{\mathrm{eff}}\) must have a minimum at \(z=0,\dot{\delta}_0=0\), i.e. at \(u(0,0)=0\). Proceeding from this, we expand \(\Phi(u)\) in a series in \(u\) and restrict ourselves to the linear term of the expansion,

\[ V_{\mathrm{eff}}=\Phi_0+\Phi'_0 \left\{ \frac{n_0z^2}{R_0^{\,2}} +(1-n_0)\varphi\dot{\delta}_0^{\,2} + \right. \]

\[ \left. +\frac{(1-n_0)}{R_0^{\,2}}\eta z^2\dot{\delta}_0 +a\frac{\dot{\delta}_0^{\,3}}{3} \right\}. \tag{14} \]

In addition to the constants \(\varphi,\eta,\psi\) (or \(a\)), the constants \(\Phi_0,\Phi'_0,\Phi''_0\), characterizing the function being expanded, have been added. It should be noted that in what follows it is more convenient to determine the constant \(a\), rather than \(\psi\).

The constant \(\Phi_0\) may be taken equal to \(V_0\). It remains to determine the four constants \(\varphi,\eta,a\), and \(\Phi'_0\), entering into the potential. To determine some of them we shall use condition (6b), into which \(C(x_0,z)\) must be substituted. The substitution gives the following values of the constants:

\[ \Phi'_0=V_0,\qquad \eta=n_0(1-\varphi)- \frac{n_0^{\,2}+n_0-n'_0R_0}{(1-n_0)}. \tag{15} \]

We shall find the constants \(\varphi\) and \(a\) by specifying the dimensions of the beam. Since the beam is in equilibrium, its surface is an equipotential surface, that is, on the boundary

\[ V_{\mathrm{eff}}=V_0 \left\{ 1+(1-n_0)\varphi\dot{\delta}_0^{\,2} +\frac{n_0z^2}{R_0^{\,2}} + \right. \]

\[ \left. +(1-n_0)\eta\frac{z^2\dot{\delta}_0}{R_0^{\,2}} +a\frac{\dot{\delta}_0^{\,3}}{3} \right\} =\mathrm{const}=G. \]

To the constants \(\varphi\) and \(a\) one more, \(G\), has been added. Let us specify three points on the beam boundary with coordinates \((r,0)\), \((r_2,0)\), \((R_0,z_1)\).

Upon substituting these coordinates, we obtain the following expression for the equation of the boundary surface:

\[ (1-n_0)\varphi \delta_0^2+\alpha \frac{\delta_0}{3} +\frac{n_0}{R_0^2}z^2+ \frac{(1-n_0)}{R_0^2}\eta z^2\delta_0 = \frac{n_0}{R_0^2}z_1^2 . \tag{16} \]

In addition, we obtain

\[ \varphi= -\frac{n_0 z_1^2}{(1-n_0)R_0^2}\, \frac{\delta_1^2+\delta_1\delta_2+\delta_2^2}{\delta_1^2\delta_2^2}, \tag{17} \]

\[ \alpha= -3\frac{n_0 z_1^2}{R_0^2}\, \frac{\delta_1+\delta_2}{\delta_1^2\delta_2^2}. \tag{18} \]

Let us now investigate the stability of the equilibrium electron beam. An electron will be in a state of stable equilibrium if, when it is displaced from the equilibrium position, it returns to it again and performs oscillations about the equilibrium position. At the equilibrium point two forces, \(\overline{F}_e\) and \(\overline{F}_{\mathrm{mc}}\), act on the electron and satisfy the condition \(\overline{F}_e=-\overline{F}_{\mathrm{mc}}\). These forces have different forms and are equal only at one point.

In order that stable equilibrium exist throughout the region of the focusing forces, or in part of this region, it is necessary that the curvature of the curve \(V_{\mathrm{mc}}\) at the equilibrium point be greater than the curvature of the curve \(V_{\mathrm{eff}}\) at the same point, i.e., \(K_{\mathrm{mc}}>K_e\), where \(K\) is the curvature.

The last inequality may be rewritten in the form

\[ \frac{ \left[\dfrac{\partial^2 V_{\mathrm{mc}}}{\partial x_c^2}\right]_{x_c=f(x_0,z)} }{ \left[ 1+\left(\dfrac{\partial V_{\mathrm{mc}}}{\partial x_c}\right)^2_{x_c=f(x_0,z)} \right]^{3/2} } > \frac{ \dfrac{\partial^2 V_{\mathrm{eff}}}{\partial x_0^2} }{ \left[ 1+\left(\dfrac{\partial V_{\mathrm{eff}}}{\partial x_0}\right)^2 \right]^{3/2} }. \]

But since

\[ \left(\frac{\partial V_{\mathrm{mc}}}{\partial x_c}\right)_{x_c=f(x_0,z)} = \frac{\partial V_{\mathrm{eff}}}{\partial x_0}, \]

it follows that

\[ \left[ \frac{\partial^2 V_{\mathrm{mc}}}{\partial x_c^2} \right]_{x_c=f(x_0,z)} > \frac{\partial^2 V_{\mathrm{eff}}}{\partial x_0^2}. \]

This is the condition for stable equilibrium. If all operations are carried out in the last expression, we arrive at the following condition:

\[ (1-n_0)(1-\varphi)+ \left[ n_0^2+2n_0-3-n_0'R_0+ (3-n_0)(1-n_0)(1-\varphi)-\alpha \right]\delta_0>0 . \tag{19} \]

The condition is satisfied for some \(\delta_0\) lying inside the beam, that is, \(\delta_2<\delta_0<\delta_1\). The limiting value of \(\delta_0\) may be taken as \(\delta_1\). Thus, the equilibrium condition relates the following quantities: \(n_0\), \(\delta_1\), \(\delta_2\), and \(z_1\). In particular, if \(\delta_1=\delta_2\), then for a given \(n_0\) we obtain from the equilibrium condition the permissible beam dimensions (Table 1).

Table 1

| \multicolumn{3}{c}{\(n_0 = 0.5\)} | \multicolumn{3}{c}{\(n_0 = 0.75\)} |
|---:|---:|---:|---:|---:|---:|
| \(\dfrac{z_1}{x_1}\) | \(\delta_1 = \dfrac{x_1}{R_0}\) | \(\dfrac{z_1}{R_0}\) | \(\dfrac{z_1}{x_1}\) | \(\delta_1 = \dfrac{x_1}{R_0}\) | \(\dfrac{z_{10}}{R_0}\) |
| 0.5 | 0.46 | 0.23 | 0.37 | 0.3 | 0.05 |
| 0.6 | 0.33 | 0.2 | 0.44 | 0.2 | 0.09 |
| 0.7 | 0.23 | 0.16 | 0.45 | 0.15 | 0.10 |

The study of \(V_{\mathrm{eff}}\) and of the stability condition makes it possible to draw a number of conclusions:

a) in an equilibrium beam with a single \(C\), each electron is, as it were, “free.” No forces act on it (they are balanced). Hence follows the possibility of arbitrary motions due to small external perturbations (for example, scattering by the residual gas) that do not disturb the equilibrium beam as a whole;

b) if the equilibrium beam is composed of electrons with different \(C\), then in this case the electrons with a given \(C\) execute oscillations about \(r_c\). The possibilities of arbitrary motions are limited. In this sense, a set of \(C\) in the beam is more advantageous;

c) the existence of an equilibrium beam makes possible the capture of a definite fraction of the electrons, with a given distribution in \(C\), from among those injected. The composition of the equilibrium beam in \(C\) will differ from the composition of the beam emerging from the gun;

d) knowing \(V_{\mathrm{eff}}\), one can calculate the equilibrium charge that can be held by a field with given characteristics;

e) the stability condition makes it possible to determine the dimensions of the region in which an equilibrium charge can exist (for a given \(R_0\)). As \(n_0\) increases, the dimensions of the beam decrease. For \(n_0 = 0.75\) they turn out to be those chosen in ordinary betatrons on the basis of economic and engineering considerations.

The article considers the nonrelativistic case, and the results may be regarded as valid at the beginning of the process of electron acceleration in a betatron.

References

1. Rajchman J. A. and Cherry W. H., Journ. Frankl. Inst., 243, 261—285, 1947.
2. Rodimov B. N. Regularities of the magnetic field of a betatron. Izvestiya TPI, vol. 87, 1957.
3. Rodimov B. N. On the mechanism of electron capture into acceleration in a betatron. Izvestiya TPI, vol. 87, 1957.