ANALYSIS OF ELECTRON MOTION IN THE CONTROL FIELD OF A BETATRON WITH AZIMUTHAL VARIATION

A. A. Zvontsov, V. L. Chakhov

(Presented by the scientific seminar of the Laboratory of Portable Small-Sized Betatrons)

At present, betatrons mainly use an azimuthally symmetric field decreasing with radius. The equations of motion of particles in such a field are:

\[ \frac{\partial^2 r}{\partial t^2}+\omega^2(1+n)r=0; \]

\[ \frac{\partial^2 z}{\partial t^2}-\omega^2 n z=0; \tag{1} \]

where \(r\) and \(z\) are small deviations from the equilibrium orbit of radius \(R\), and

\[ n=-\frac{R}{H}\frac{\partial H}{\partial r} \]

is the field gradient.

The further development of accelerators, in order to increase the focusing forces, proceeded along the path of abandoning azimuthal symmetry of the field.

In the Laboratory of Small-Sized Betatrons of TPI, a design of a betatron with an azimuthally nonsymmetric magnetic field was proposed.

The poles of such a betatron were made in the form of \(N\) radial ridges \((N=6)\). The poles may also be made in the form of \(N\) spiral ridges. The magnetic field of such a betatron has the following features:

1. Under a ridge, the magnetic field may decrease or increase with radius, or may not vary with radius. The sign and magnitude of \(n\) will depend on the variation of the interpole gap \(\delta\) with radius.

2. In the sectors between the ridges, owing to the small overall dimensions, the magnetic field is created by scattering of magnetic fluxes from the ridges. Since the ridges diverge along the radius, the field between the ridges will decrease with radius.

According to magnetic-measurement data, the field shape produced by such a design is complex, and a general analysis of electron motion in such a field is difficult. For a preliminary estimate of the focusing properties, the field shape in the region of the equilibrium orbit and at \(z=0\) may be represented in the form of a Fourier series:

\[ H(v,\theta)=\overline{H}(v)\left\{1+\sum A_{Nk}\cos Nk[\theta-\psi_{Nk}(v)]\right\}, \tag{2} \]

where \(\widetilde{\vec H}\) is the field averaged over azimuth,
\(A_{Nk}\) is the depth of variation of the \(N_k\)-th harmonic of the field,
\(\psi_{Nk}\) is the initial phase of the \(N_k\)-th harmonic of the field.
Taking into account only the first harmonic, for radial ridges the field can be specified in the form:

\[ H_z(v,\theta)=\overline{H}_z(v)[1+f(v)\cos N\theta], \tag{3} \]

where \(f\) is the depth of variation, equal to

\[ f=\frac{\overline{H}-H_{\min}}{\overline{H}}=\frac{H_{\max}-\overline{H}}{\overline{H}}. \tag{4} \]

Let us consider the field near a point with coordinates \(r_0,z_0\). For small deviations from this point, \(H_r\), \(H_\theta\), and \(H_z\) may be approximately represented by the first nonzero terms of the Taylor expansion [1]

\[ H_z=H_z(r)+\frac{\partial H_z}{\partial r}\rho+\frac{1}{2}\frac{\partial^2 H_z}{\partial r^2}\rho^2+\ldots; \]

\[ H_r=\frac{\partial H_z}{\partial r}z=\frac{\partial^2 H_z}{\partial r^2}\rho\cdot z+\ldots; \tag{5} \]

\[ H_\theta=\frac{\partial H_z}{\partial \theta}\cdot \frac{z}{r_0} +\frac{\partial^2 H_z}{\partial r\cdot \partial \theta}\cdot \frac{\rho\cdot z}{r_0}+\ldots, \]

where \(\rho\) is a small deviation from \(r_0\) in the median plane;
\(z\) is a small deviation from the median plane.
By virtue of the symmetry of the magnetic lines of force (they intersect the median plane at a right angle), in the median plane \(H_r=H_\theta=0\).

In the cylindrical coordinate system \((r,\Theta,z)\), the motion of a particle in a magnetic field is determined by the equations:

\[ m(\ddot r+r\dot\theta^2)=-\frac{e}{c}(\dot z H_\theta-r\dot\theta H_z); \]

\[ m(r\ddot\theta+2\dot r\dot\theta)=\frac{e}{c}(\dot r H_z-\dot z H_r); \tag{6} \]

\[ m\ddot z=-\frac{e}{c}(\dot r H_\theta-r\dot\theta H_r), \]

where \(H_r,H_\theta,H_z\) are the components of the field. The dot \((\cdot)\) denotes differentiation with respect to \(t\). We assume that during the acceleration process, over the time of several particle revolutions, the field does not change. The particle velocity \(v\) is also assumed constant. In betatrons with an azimuthally symmetric field, the equilibrium orbit is a circle of radius \(R\), determined from the equation:

\[ mv=\frac{e}{c}R\cdot H_z. \tag{7} \]

Replacing \(H_z\) by the averaged field \(\overline{H}_z\) (2), we obtain the averaged radius of the equilibrium orbit:

\[ R=\frac{mvc}{e\overline{H}_z(p)}. \tag{8} \]

We shall represent the equilibrium orbit in the form:

\[ r=R(1+\eta), \tag{9} \]

where \(R\) is the averaged radius of the equilibrium orbit;
\(\eta\) is the deviation from the averaged radius.

In the first approximation, taking into account only the first harmonic of the field variation, the quantity \(\eta\) is determined by the expression [1]:

\[ \eta=\frac{f}{N^2-n-1}\cdot \cos N\theta, \tag{10} \]

where \(n\) is the azimuthally averaged field gradient, equal to

\[ n=\frac{1}{2\pi}\int_0^{2\pi} n(\theta)\,d\theta . \tag{11} \]

With sufficient accuracy one may take

\[ r=R\left(1+\frac{f}{N^2}\cos N\theta\right). \tag{12} \]

The maximum deviation from the averaged radius of the equilibrium orbit is equal to

\[ a=\frac{fR}{N^2}. \tag{13} \]

The quantity “\(a\)” may be called the lost part of the aperture.

From (13), for a known depth of variation, one can determine the minimum number of ridges, if it is assumed that the maximum deviation from the averaged radius \(R\) must not exceed the quantity “\(a\).”

For the experimental betatron, the maximum deviation from \(R\) at \(N=6\), \(R=45\) mm, and \(f=0.25\) is \(0.312\) mm. It can be shown that the oscillations of particles relative to the closed orbit are described by the Mathieu–Hill equation [1]:

\[ \begin{aligned} r''-r(a_r+2g_r\cos 2\xi)&=0;\\ z''+z(a_z-2g_z\cos 2\xi)&=0, \end{aligned} \tag{14} \]

where

\[ a_r=\frac{4}{N^2}\left[1+n+\frac{3f^2(n+1)}{4N^2}\right]; \]

\[ 2g_r=\frac{4}{N^2}\left[(1.5+n)f+\frac{2n+n'}{N^2}f\right]; \]

\[ a_z=\frac{4}{N^2}\left\{-n+\frac{1}{2}f^2+\frac{f^2}{2N^2}\left[\frac{(n+1)n-n'\bigl[(n'+1)+2n\bigr]}{2(n+1)}\right]\right\}; \]

\[ 2g_z=\frac{4}{N^2}\left[f\left(n+\frac{2n+n'}{N^2}\right)\right]. \]

\(2\xi\) is determined from \(\theta=\dfrac{2\xi}{N}\), and where

\[ n'=\left.\frac{\partial^2 \overline{H}_x}{\partial r^2}\cdot \frac{R^2}{\overline{H}(R)}\right|_{r=R}. \]

From equation (14) one can obtain an approximate expression for the frequencies of the betatron oscillations. With sufficient accuracy they may be taken in the form:

\[ \begin{gathered} \nu_r^2=1+n+\frac{1}{2}f^2\left[\frac{1}{N^2}+\frac{3(n+1)}{2N^2}+\frac{n^2}{N^2}\right] +\frac{1}{2}f\left(R\frac{\partial f}{\partial r}\right)\cdot \frac{2(1+n)}{N^2}+\frac{1}{2N^2}\left(R\frac{\partial f}{\partial r}\right)^2;\\ \nu_z^2=-n+\frac{1}{2}f^2\left\{1-\frac{1}{N^2}\left[\frac{2+n}{n}+\frac{2n+n'}{2(n+1)}+\frac{n(n+1)}{N^2}\right]\right\}-\\ -\frac{f\left(R\frac{\partial f}{\partial r}\right)}{N^2} \left[(n+1)-\frac{2n+n'}{2(n+1)}\right] +\frac{\left(R\frac{\partial f}{\partial r}\right)^2}{2N^2}. \end{gathered} \tag{15} \]

Since \(\nu^2\) is proportional to the focusing forces, from (15) one can estimate the magnitudes of the focusing forces acting on an electron in a betatron with an azimuthally asymmetric field:

\[ F_r=1+n+\frac{f^2}{N^2}(1+n^2)+\frac{3f^2(n+1)}{4N^2}+\ldots; \]

\[ F_z=-n+\frac{1}{2}f^2-\frac{f^2}{2N^2} \left[(2+n)+\frac{2n+n'}{2(n+1)}\right]+\ldots . \tag{16} \]

It can be shown that, in the case of a magnetic field of “spiral” structure, the increase in the focusing forces can be determined from the formulas:

\[ F_r^c=\frac{f^2N^2\tg^2\gamma}{N^2-4(n+1)} -\frac{f^2N^2\tg^2\gamma}{N^2-(n+1)} \simeq \frac{3(n+1)f^2\tg^2\gamma}{N^2}; \]

\[ F_z^c=\frac{f^2N^2\tg^2\gamma}{2[N^2+4n]} -\left(-\frac{f^2N^2\tg^2\gamma}{2[N^2-(n+1)]}\right) \simeq f^2\tg^2\gamma . \tag{17} \]

Formulas (16) and (17) show that the increase in the focusing force in the radial plane is insignificant, whereas in the axial plane the focusing force increases by \(\frac{1}{2}f^2\), which amounts to \(1\text{–}5\%\) of the magnitude of the focusing force acting in the \(z\)-plane for a betatron with an azimuthally symmetric field. The addition due to the “spirality” of the field (17) may also be appreciable. In betatrons designed for low energies (3–10 MeV), it is apparently possible to confine oneself to radial ridges, since spiral poles are more complicated to manufacture than radial ones.

In betatrons with an azimuthally symmetric control field, the ratio of the magnetic-field strength at the radius of the equilibrium orbit to the mean magnetic-field strength at the radius of the equilibrium orbit is known to be \(1:2\). It is of interest to consider whether this ratio is preserved in a betatron with a control field possessing azimuthal variation.

In the general case the betatron acceleration regime is characterized by the following equation [2]:

\[ \frac{d}{dt}\left[\Pi\cdot \overline{H}_z\right] = \frac{2\pi S\,d\langle H_z\rangle}{\Pi\,dt}, \tag{18} \]

where \(\Pi\) is the perimeter of the orbit; \(S\) is the area enclosed by the orbit; \(H_z\) is the field averaged along the orbit; \(\langle H_z\rangle\) is the field averaged over the surface enclosed by the orbit.

Since the conditions must be chosen so that the orbit remains unchanged during the acceleration process, i.e., so that \(\Pi=\mathrm{const}\), then

\[ \frac{d\overline{H}_z}{dt} = \frac{2\pi S}{\Pi^2}\cdot \frac{d\langle H_z\rangle}{dt}. \tag{19} \]

Since the equilibrium orbit is specified in the form (12), the known expressions can be used to determine the perimeter of the equilibrium orbit and the area enclosed by it. Then expression (19) will have the form:

\[ \frac{d\overline{H}_z}{dt} = \frac{1}{2}\cdot \frac{\left(1+\dfrac{f}{N^2}\right)^2} {\left[1+\dfrac{f^2}{4N^2} \left(\dfrac{11+12n}{2+3n}\right)\right]^2} \cdot \frac{d\langle H_z\rangle}{dt}. \tag{20} \]

Thus, for a betatron with azimuthal variation of the magnetic field, the betatron number has the form:

\[ \frac{\overline{H}_z(t)}{\langle H_z\rangle(t)} = \frac{1}{2}\cdot \frac{\left(1+\dfrac{f}{N^2}\right)^2} {\left[1+\dfrac{f^2}{4N^2} \left(\dfrac{11+12n}{2+3n}\right)\right]^2}. \tag{21} \]

In practice,

\[ \frac{\overline{H}_z(t)}{\langle H_z\rangle(t)}=\frac{1}{2}, \]

since

\[ \frac{\left(1+\dfrac{f}{N^2}\right)^2} {\left[1+\dfrac{f^2}{4N^2} \left(\dfrac{11+12n}{2+3n}\right)\right]^2} \simeq 1. \]

For example, for the experimental betatron this factor was characterized by the value

\[ \frac{(1.007)^2}{(1.0045)^2}\simeq 1. \]

On the basis of the preliminary estimates made of the properties of the guide field of a betatron with azimuthal variation, the fundamental possibility of a betatron acceleration regime in such designs has been shown. The analysis also showed that the focusing properties of such fields are higher than in the azimuthally symmetric fields used in ordinary betatrons. The startup of the experimental betatron for radiation confirmed the correctness of the conclusions.

REFERENCES

1. Yu. G. Basargin, V. P. Belov. Some problems in the dynamics of motion of particles in a cyclotron with spatial variation of the magnetic field. Electrophysical Apparatus, No. 3, 1965.

2. A. A. Kolomenskii, A. N. Lebedev. Theory of Cyclic Accelerators, Fizmatgiz, 1962.