Potential Function of the Focusing Forces of the Betatron Magnetic Field and Its Practical Application in Accelerator Tuning

D. A. Noskov

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

The potential function as the most graphic characteristic of the focusing properties of a magnetic field

The focusing properties of the guide fields of betatrons and synchrotrons are determined by the configuration of the magnetic field and by its intensity. A characteristic of the field shape is the exponent of the decrease of the magnetic-field intensity with radius, “$n$.”

In the practice of betatron construction this characteristic is widely used as a criterion of the focusing properties of the magnetic field, as a criterion of the suitability of the betatron guide field for focusing and accelerating an electron beam. However, this characteristic, while giving an approximate idea of the relation between the axial and radial focusing forces, does not give a complete picture of the focusing properties of the field, of the character and magnitude of the focusing forces acting on the beam of accelerated electrons.

A more graphic and complete characteristic of the focusing properties of a magnetic field is the potential function of the focusing forces of the given field [1]^1). The surface described by the potential function of the focusing forces of the guide magnetic field is customarily called the potential well. The shape and magnitude of the potential well determine the magnitude of the charge that can be focused and accelerated in the given field and, consequently, also determine the radiation intensity of the accelerator [4]. Therefore, in order to obtain a clearer idea of the focusing properties of the guide magnetic field, it is necessary to construct the potential function of the focusing forces of this field in the form of equipotential lines or in the form of sections of the potential well in the plane $z = 0$ and in the plane $r = \mathrm{const}$.

Method for calculating and constructing the potential function

The section of the potential function in the plane $z = \mathrm{const}$ is expressed as the integral of the radial component of the force of the magnetic field:

\[ V_{\mu}(r, z = \mathrm{const}) = - \int_{r_0}^{r} F_r \cdot dr . \tag{1} \]

^1) Here the potential function of “zero” electrons is considered [4].

The radial component of the focusing force, to an accuracy of 1%, is expressed by equation [2]:

\[ F_r = - \frac{e^2}{m}\,(Hz_0 \cdot r_0)\left(Hz - Hz_0 \frac{r_0}{r}\right). \tag{2} \]

Substituting this expression into equation (1), we obtain the expression for the cross section of the potential function in the plane \(z=\mathrm{const}\) in the following form:

\[ V_m(r, z=\mathrm{const}) = \frac{e^2}{m}(Hz_0 \cdot r_0) \int_{r_0}^{r} \left[ Hz - Hz_0 \frac{r_0}{r} \right]\,dr. \tag{3} \]

This integral is solved graphically. The values \(Hz\), \(Hz_0\), and \(r\) are taken from experimental data; from them, on the same scale, the curves \(Hz=f(r)\) and \(Hz_0 \dfrac{r_0}{r}=f(r)\) are plotted (Fig. 1a′); then the areas enclosed between these curves for various values of \(r\) are calculated and plotted on a separate graph. This is the cross section of the potential well in the plane \(z=\mathrm{const}\) in relative units (Fig. 1b′).

To construct the complete potential surface and equipotential lines of the potential function, it is necessary to construct sections of the potential wells in several planes \(z=\mathrm{const}\), by the method described above, and to construct a section of the potential well in the plane \(r=\mathrm{const}\) (or \(r=r_0\)), or to construct one section in the plane \(z=0\) and several sections in the plane \(r=\mathrm{const}\) [2].

The section of the potential well in the plane \(r=\mathrm{const}\) is calculated and constructed in the same way, by graphical integration of the expression for the potential function through the axial component of the focusing force:

\[ V_m(z, r=\mathrm{const}) = \int_{0}^{z} F_z \cdot dz = - \frac{e^2}{m}(Hz_0 \cdot r_0) \int_{0}^{z} H_r(z, r=\mathrm{c}) \cdot dz. \tag{4} \]

The magnitude of the radial component of the magnetic-field intensity is found by graphical integration of the expression:

\[ H_r = - \frac{1}{r}\int_{0}^{z} n \cdot Hz \cdot dz. \tag{5} \]

Knowing the spatial distribution of the magnetic field in the working region, \(H_z=f(r,z)\) and \(n=f(r,z)\), one can easily and with sufficient accuracy construct the equipotential lines of the potential function of the focusing forces of the control magnetic field [3].

This method can be used for calculating and constructing potential functions of focusing fields of any shape, regardless of how the field fall-off index varies as a function of \(r\) and \(z\); moreover, no additional instruments are required for the calculations and construction.

It is most convenient to carry out the calculations and construction of the equipotential lines in the following order:

1. Record the dependences of the magnetic-field intensity on the radius in different planes, \(H_z=f(r)\).

1. Calculate or measure the values of the field fall-off index “\(n\)” and construct the curve of the dependence of the index \(n\) on \(z\), \(n=f(z)\), for the radius of the equilibrium orbit (Fig. 1a).

2. By graphical integration, construct the dependence

\[ H_r=f(z) \]

(Fig. 1b).

1. From the curve \(H_r=f(z)\), by graphical integration construct the section of the potential well in the plane \(r=r_0\) (Fig. 1c).

2. By graphical integration, calculate and construct the section of the potential well in the planes \(z=0\), \(z=1\), \(z=2\), etc., within the working zone of the accelerator on one side of the central plane, since the field is symmetric and there is no need to construct sections over the entire volume.

The calculated curve

\[ H_{z_0}\frac{r_0}{r}=f(r) \]

remains unchanged for all sections, since \(H_{z_0}\) changes only slightly with \(z\).

Fig. 1

The construction of sections of the potential wells may be carried out in relative units; then the minimum of the well section in the plane \(z=0\) is taken as the origin for the value of the function, while the minima of the sections in other planes will be shifted upward by an amount corresponding to the value of the potential function in the section in the plane \(r=r_0\).

1. Construct the equipotential lines of the potential function of the focusing forces of the field from the known values of the potential function at various points of the field (Fig. 1 \(a'\), \(b'\)).

Figures 1a, b, c show the various stages of calculating and constructing the equipotential lines of the focusing forces of the betatron field.

Application of the Potential Function in Betatron Tuning

From the shape and magnitude of the potential well one can judge with sufficient accuracy the focusing properties of the magnetic field, the magnitude of

charge that can be accelerated and, consequently, also about the intensity of the accelerator radiation.

For example, if we have several betatrons with different field shapes, then, by constructing the potential functions of these fields, one can say which of these accelerators will give radiation of greater intensity.

Fig. 2

With the aid of the potential function it is possible to establish in advance the optimum radius of the equilibrium orbit. From the form of the potential function one can determine the optimum position of the electron gun. For example, in the investigation of a betatron without an iron yoke and poles, it became necessary to compare the focusing properties of control fields of various configurations.

Fig. 3

From the curves \(n=f(r)\) (Fig. 2) it is difficult to say anything about the focusing properties of a given field. Having constructed the potential functions in the form of equipotential lines (Figs. 3a, b, c, d, e), one can state with confidence that the fields with \(n\) on the orbit equal to 0.62 and 0.72 possess the best focusing properties. The focusing properties with \(n=0.25\) and 0.36 are much worse than the former. Investigations of the dependence of the radiation intensity on the form of the guiding field confirm this conclusion. From Fig. 4 it is seen that radiation in fields with \(n=0.25\) is almost absent, is maximal at \(n=0.6\) and 0.72, and is small at \(n=0.36\).

Fig. 4

Fig. 5

Preliminary determination of the optimal position of the radius of the equilibrium orbit within the limits of the working region with the aid of the potential function has also been verified experimentally. It has been established that the optimal position of the equilibrium orbit is that for which the potential function has identical values at the inner and outer boundaries of the accelerating chamber.

Fig. 6

Thus, by constructing potential wells for several orbits, one can find the optimal position of the equilibrium orbit. For example, for a betatron without an iron core and poles, from the potential function it was established that the optimal equilibrium orbit is at a radius of \(9.5\ \text{cm}\) (Fig. 5).

The experimental dependence of the radiation intensity on the position of the equilibrium orbit (Fig. 6) shows that changing the radius of the equilibrium orbit from 9 to 9.7 cm has only a slight effect on the radiation. Consequently, the orbit with radius 9.5 cm is very close to the optimum. The potential functions for radii 9.5 and 9.3 differ very little from one another.

References

1. J. A. Rajchman and W. H. Cherry. The electron mechanics of induction accelerator. J. Franklin Inst. 243, 1947.

2. V. N. Rodimov. On the mechanism of electron capture and acceleration in a betatron. Izvestiya TPI, vol. 87, 1957.

3. G. I. Dimov. Candidate dissertation. TPI, 1954.

4. D. A. Noskov. Investigation of the features of operation of a betatron without an iron yoke. Candidate dissertation. TPI, 1954.