On the Methodology for Designing the Poles of Simple Betatrons

K. S. Grishin

(Presented by the scientific seminar of the Physico-Technical Faculty)

In the practice of designing betatrons, difficulties usually arise that are connected with determining and choosing the relationships between the principal dimensions of the poles and the parameters of the magnetic field.

Indeed, the relationship between the electron energy, the radius \(r_o\), and the magnetic-field induction \(B_o\) on the orbit is expressed by the well-known equation

\[ A = 300 B_o r_o - 0.509 \cdot 10^6. \tag{1} \]

Equation (1) is satisfied by many possible solutions. In order that, for a given energy, the dimensions and weight of the betatron be as small as possible, it is necessary to strive to reduce \(r_o\). However, in this case the required value of \(B_o\) and the value of the average induction in the orbit circle, \(B_{\text{av}}\), will correspondingly increase; in a betatron, in the simplest case, these are related by

\[ B_{\text{av}} = 2 B_o. \tag{2} \]

Such an increase can be allowed only up to certain limits, determined by the saturation of the steel in the central part of the poles, where a homogeneous magnetic field of the greatest field-strength amplitude is usually produced. Therefore the dimensions of the central part of the poles, to a certain extent, determine the electron energy attainable in a given betatron.

There are several ways of increasing the limiting energy of a betatron for a given electromagnet weight. Usually they reduce to increasing the amplitude of the magnetic field in the central part of the poles. This is achieved either by using direct current in the excitation circuits of the betatron magnet, or by means of a phase shift between the field on the orbit and the field in the orbit circle, or by a combination of these two methods.

The use of these methods is associated with a complication of the electromagnet design, of its excitation circuit, and also with a complication of all electronic control and monitoring circuits, which, in the final analysis, leads to a considerable complication in the servicing of such betatrons. Such measures are expedient and accordingly produce an economic effect only for large betatrons intended chiefly for scientific purposes. In industry and in many scientific investigations, betatrons of up to 25–30 MeV are finding ever wider application. For the needs of industry

and medicine, it is highly desirable to have betatrons that are simple both in construction and in servicing.

Let us focus our attention precisely on the case in which the electromagnet of the betatron is excited only by alternating current, while the necessary ratio \(2:1\) is achieved by the corresponding choice of the pole profile. Let us assume that the simplest pole construction, shown in Fig. 1, has been adopted. There is a known interrelation among the quantities \(B_0\), \(B_{\text{ц}}\), \(r_0\), and \(r_{\text{ц}}\).

Fig. 1

Let us recall the reasoning used to derive the formula for this relation. If, in the working zone of the interpole space, the quantity \(n=\mathrm{const}\), then the induction \(B\) on a circumference of some radius \(r\) can be found from the expression

\[ B = B_0 \left( \frac{r_0}{r} \right)^n . \tag{3} \]

The magnetic flux through a ring of width \(r_0-r_{\text{ц}}\) in this case will be

\[ \Phi_{0-\text{ц}} = \int_{r_{\text{ц}}}^{r_0} d\Phi = 2\pi B_0 r_0^n \int_{r_{\text{ц}}}^{r_0} r^{1-n}dr = \frac{2\pi B_0 r_0^n}{2-n} \left( r_0^{\,2-n} - r_{\text{ц}}^{\,2-n} \right). \]

The flux in the central part of the poles, where \(B_{\text{ц}m}(r)=\mathrm{const}\),

\[ \Phi_{\text{ц}}=\pi r_{\text{ц}}^2 B_{\text{ц}}, \]

the magnetic flux in the orbit circle

\[ \Phi_0 = \Phi_{0-\text{ц}}+\Phi_{\text{ц}} = \frac{2\pi B_0 r_0^n}{2-n} \left( r_0^{\,2-n} - r_{\text{ц}}^{\,2-n} \right) + \pi r_{\text{ц}}^2 B_{\text{ц}}, \]

and the average induction in the orbit circle

\[ B_{\text{av}} = \frac{\Phi_0}{\pi r_0^2} = \frac{2B_0 r_0^n}{(2-n)r_0^2} \left( r_0^{\,2-n} - r_{\text{ц}}^{\,2-n} \right) + \left( \frac{r_{\text{ц}}}{r_0} \right)^2 B_{\text{ц}} . \tag{4} \]

Substituting (4) into (2), we finally obtain

\[ \frac{B_u}{B} = \frac{2}{2-n} \left[ (1-n)\left(\frac{r_0}{r_u}\right)^2 + \left(\frac{r_0}{r_u}\right)^n \right]. \tag{5} \]

This equation, sometimes called the equation of variants, has many solutions, depending on the relation among the quantities entering into it. However, one can verify that there exists a certain range of practically acceptable solutions lying within the limits of the most rational values of these quantities. Indeed, the quantities \(B_0\) and \(r_0\) are related by (1); the maximum value \(B_{um}\) depends on the grade of steel; and the quantity \(n\) may be chosen within the range \(0<n<1\), corresponding to the best conditions for electron motion during the injection and acceleration period, with allowance for the shape and dimensions of the cross section of the working zone of the interpole space. Since the quantities \(B_0\), \(r_0\), \(B_u\), and \(n\) may be specified, adopted, or determined (at least approximately), the calculation problem will include determining the quantity \(r_u\). This quantity enters the denominators of two fractions, one squared and the other raised to the power \(n\). Therefore \(r_u\) can be found either by trial-and-error or graphically. A trial-and-error solution is rather laborious and does not make it possible to investigate quickly the rationality of the results obtained.

The graphical method of solution gives much greater possibilities. In practical calculations it is convenient to use graphs constructed on the basis of equation (5) in the coordinates

\[ \frac{B_{cp}}{B_0}=f(r_u) \]

for different values of \(r_0=\mathrm{const}\) and \(n=\mathrm{const}\). In Fig. 2 such graphs are given for \(r_0=9 \div 30\ \mathrm{cm}\) and for the most frequently adopted values \(n=\tfrac{3}{4},\ \tfrac{2}{3},\ \tfrac{1}{2}\). The graph makes it possible quickly to choose a suitable \(r_u\) for any betatron with an energy from 4 to 30 MeV. In addition, this graph may be useful for a general analysis of the interdependence of the quantities entering into (5), presented here in a sufficiently visual form, which considerably simplifies the task of choosing rational variants of the basic data for the poles of betatrons in the given range of maximum energies. In particular, the graph makes it easy to trace the influence of \(n\) on the ratio of the pole dimensions.

If the width of the working zone of the interpole space is taken to be \(a=2(r_0-r_u)\), then it follows from the graph that increasing \(n\) in all cases (for \(r_0=\mathrm{const}\) or \(r_u=\mathrm{const}\)) leads to an increase in \(a\), especially at large values of \(\frac{B_u}{B_0}\). At \(\frac{B_u}{B_0}=\mathrm{const}\), for large \(r_0\), increasing \(n\) leads to a considerable increase in \(a\). For small betatrons such an increase in \(a\) is more noticeable than for large ones. We note that, for all practically acceptable betatron variants in this energy range, increasing \(n\) from \(\tfrac{1}{2}\) to \(\tfrac{3}{4}\) leads to an increase in the width of the working space by no more than \(5 \div 6\ \mathrm{mm}\). However, the quantity \(n\) noticeably affects the energy of the betatron. Indeed, if \(r_0=\mathrm{const}\) and \(r_u=\mathrm{const}\), then increasing \(n\) decreases \(\frac{B_u}{B_0}\), and this, at \(B_0=\mathrm{const}\), leads to an increase in \(B_{om}\), i.e., makes it possible, with unchanged overall dimensions of the betatron, to obtain from it a higher electron energy. From the graph it is easy to see that for small betatrons (small \(r_0\)) and at large values of \(\frac{B_u}{B_0}\) such an increase in energy is relatively greater than for large ones.

betatrons (large \(r_0\)) and small values of \(\dfrac{B_u}{B_0}\). It also follows from the graph that, for rational designs of small betatrons, the ratio \(\dfrac{B_u}{B_0}\) must always

Fig. 2.

Visible labels in the figure: \(a)\), \(b)\), \(\dfrac{B_u}{B_0}\), \(r_u\) (cm).

\[ \frac{B_u}{B_0} = \frac{1}{1-n} \left\{ (1-n)\left(\frac{r_0}{r_u}\right)^2 + \left(\frac{r_0}{r_u}\right)^n \right\} \]

be greater than for large ones. This means that, for large betatrons, higher values of \(B_0\) may be used on the orbit. It should be noted that, with \(B_{0m}=\mathrm{const}\) and \(r_0=\mathrm{const}\), increasing \(B_{qm}\) makes it possible to reduce \(r_u\), i.e., to increase the width of the working space. The effect of this phenomenon is especially perceptible for large betatrons.

In all cases, when selecting variants, the quantity \(a\) must be such that, in the inter-pole space, when the vacuum chamber with the electron gun is placed in it, there is sufficient room for the motion of the accelerated electrons.

Equation (5) was derived under the condition that there is no central axial hole in the poles and disks, which is usually made for structural reasons. However, equation (5) and the graph (Fig. 2) remain valid also for the case when such a hole is provided. In this case, the mean value of the induction in the circle of radius \(r_c\) is taken as the calculated value \(B_{cm}\). Since the induction in the steel \(B_{Fe\,m}\) is limited by the saturation of the steel, increasing the central hole leads to a decrease in the calculated \(B_{cm}\).

For small betatrons, the ratio of the radius of the hole to \(r_c\) is usually greater than for large betatrons. Accordingly, for small betatrons it is necessary to use lower values of the calculated induction \(B_{cm}\) than for large ones.

Fig. 3

The filling factor of the poles with steel has the same effect on the calculated quantity \(B_{cm}\). The average value of the filling factor of the poles with steel increases at first rapidly with increasing \(r_c\), and then more slowly. Therefore, for small betatrons it is necessary to adopt still lower values of the calculated \(B_{cm}\). The coefficient that takes into account this reduction of \(B_{cm}\) relative to the permissible values of \(B_{Fe}\) for steel is determined in each particular case in accordance with the proposed pole design, taking into account the thickness of the steel sheets and the insulation between them. In Fig. 3, for example, the ratio of the mean induction \(B_{cm}\) in the circle \(r_c\) to the maximum induction in the steel permissible for the betatron \((B_{Fe\,m})\) is shown as a function of \(r_c\) for the case in which the disks are made of steel sheets 0.4 mm thick with insulation between the sheets of 0.05 mm, with a central-hole diameter of 2 cm (without allowance for the edge effect). For large values of \(r_c\), this ratio in the present case cannot exceed 0.89.

In conclusion, we shall make some general remarks concerning the further course of designing betatron poles. To increase the economy of the betatron, it is necessary to strive to reduce the height of the inter-pole space. The limit of this reduction is the height of the space in which the accelerated electrons move,

plus the design dimensions of the vacuum chamber. The height of this space is determined by the characteristics of the magnetic field and by the conditions for admitting electrons into the chamber.

The characteristics of the magnetic field are basically specified by the shape of the pole profile, the dimensions and relative arrangement of the central disks, and are to some extent determined by the general design and manufacturing technology of the betatron.

There are theoretical methods for determining the contours of the pole profile. However, despite all their complexity and difficulty, these methods still do not yield reliable results. A profile calculated theoretically requires serious experimental correction.

In practice, an experimental method for determining the required pole profile by means of an electrolytic bath has become widespread. In this case the preliminary shape of the profiled part of the poles may be calculated from simplified formulas and, in the course of subsequent investigations, corrected in the appropriate manner.

In practice the calculation is often carried out as follows: assuming the lines of force in the working zone of the interpole space to be straight and perpendicular to the plane of the orbit, one calculates the necessary pole profile in this zone, using the relation

\[ \delta_r=\delta_0\left(\frac{r}{r_0}\right)^n, \]

where \(\delta_0\) is the distance between the poles on the equilibrium orbit of radius \(r_0\); \(\delta_r\) is the same at any other point of this zone at a distance \(r\) from the center of the pole.

The outer radius of the pole is usually taken to be \(r_n \simeq 2r_0 - r_q\). The total gap in the central part of the poles \(\delta_q=\delta_1+\delta_2+\delta_3\) is determined as

\[ \delta_q=\delta_0\frac{B_{om}}{B_{qm}}, \]

and the total height of the central disks as \(h=\delta_r-\delta_2-\delta_q\).

The number of central disks is usually taken equal to two, and the height of each disk equal to \(h/2\). The distance between the disks (\(\delta_1\), \(\delta_2\), and \(\delta_3\)), as well as the shape and dimensions of the “visor” on the peripheral part of the poles, may be approximately determined by using the method of similarity on the basis of data on the designs of operating betatrons.

More precisely, these quantities may be determined experimentally in an electrolytic bath, so that the value of \(n\) over the greater part of the working zone of the interpole space is equal to the calculated value. Let us note that, by changing the relative distance between the central disks, as well as the size and shape of the “visor,” one can always obtain such a pattern of the magnetic field for which \(n=f(r)\) will be most favorable.

Having the pattern of the lines of force of the magnetic field, one can determine the value of \(n\) for any point of the field with sufficient accuracy as

\[ n=\frac{r}{r_\ell}, \]

where \(r\) is the distance of the point under consideration from the center of the interpole space, and \(r_\ell\) is the radius of curvature of the line of force at the given point in the plane of the orbit. A final characterization of the adopted dimensions and of the shape of the pole profile can be made only after a comprehensive study of the structure of the magnetic field between the poles of the given design, or on a reduced model of these poles.