On the Calculation of the Radius of the Equilibrium Orbit of a Betatron

M. F. Filippov

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

General Considerations on the Choice of the Basic Parameters of a Betatron Electromagnet

The calculation of the electromagnet of an induction electron accelerator—a betatron—as well as that of a transformer or an electric machine, is a problem with many possible solutions. But only some of these solutions are technically and economically expedient.

In relativistic mechanics the relation between the two basic parameters of a betatron—the radius of the equilibrium orbit \((r_o)\) and the magnetic-flux density on the orbit \((B_o)\)—is, to some approximation, determined by the well-known equation:

\[ A = 3 \cdot 10^{-4} \, r_o B_o , \tag{1} \]

where \(A = W + 0.51\) is the total energy of the accelerated electrons, and \(W\) is their kinetic energy in MeV;

\(B_o\) is the magnetic-flux density on the equilibrium orbit, in gauss;

\(r_o\) is the radius of the equilibrium orbit, in cm.

Of the infinitely many pairs of values \(r_o\) and \(B_o\) satisfying the equation, only those values may be adopted which meet the following conditions:

1. The distribution of the magnetic flux in the gap between the poles must ensure, during the acceleration process, satisfaction of the so-called betatron condition \((B_k = 2B_o)\), where \(B_k\) is the average magnetic-flux density in the circle of the equilibrium orbit.

2. The betatron electromagnet must have the smallest possible weight.

3. The heating of the steel of the magnetic circuit and of the electromagnet windings during prolonged operation in the nominal regime must not exceed that permitted by the standards.

In order to obtain the smallest possible radial dimension of the pole cores, one should strive for a smaller value of \(r_o\), but in this case the magnetic-flux density on the equilibrium orbit required to obtain the specified energy of the accelerated electrons increases.

At large values of the magnetic-flux density in the interpole space of the betatron electromagnet, the influence of saturation of the steel in the central part of the magnetic circuit becomes appreciable. The influence of saturation leads to a violation of the betatron condition.

In this case, violation of the betatron relation causes a displacement of the orbit of the accelerated electrons toward the inner wall of the accelerating chamber.

In addition, an increase in the magnetic-flux density in the pole cores leads to an increase in losses due to eddy currents and hysteresis. With a nonuniform distribution of magnetic-flux density over the cross section of the magnetic circuit, the presence of large losses in the steel may create an inadmissible phase difference between the guiding and accelerating magnetic fields of the betatron, which also causes a violation of the betatron condition. Increased losses in the steel produce excessive heating of the magnetic circuit.

In order to find the most successful solution in choosing the radius of the equilibrium orbit \(r_0\), if equation (1) is used, it is necessary to consider several variants and, from the final results of the calculation, judge the suitability of one or another of them.

To introduce definiteness into the choice of the radius of the equilibrium orbit, below a calculation equation is proposed which relates the principal parameters and the radial dimensions of the interpole space of the betatron electromagnet under the condition that the betatron relation is satisfied.

Derivation of the calculation equation

Figure 1 shows the profile of the pole pieces of a betatron and the corresponding distribution curve of the magnetic-flux density \(B=f(r)\) in the median plane of the interpole space.

Fig. 1

In the working zone of the interpole space, occupied by the useful cross section of the accelerating chamber (section \(ab\)), the curve of the distribution of magnetic-flux density along the radius must satisfy the relation

\[ \frac{B_x}{B_0}=\frac{r_x^{-n}}{r_0^{-n}} \quad \text{or} \quad B_x=B_0\frac{r_0^n}{r_x^n}, \tag{2} \]

where \(B_0\) and \(B_x\) are the instantaneous values of the magnetic-flux density at the radii \(r_0\) and \(r_x\), and \(n\) is the exponent of the radial decrease of the magnetic field, which must lie within the limits \(1>n>0\) in order to provide radial and axial focusing of the beam of accelerated electrons.

The exponent of the radial decrease of the magnetic field \(n\) has different values at different radii in the working zone of the accelerating chamber (Fig. 1), but in the ring bounded by the radii \(r_0\) and \(r_c\), whose magnetic flux is taken into account by the calculation equation, \(n\) may be regarded as constant and equal to its value at the radius \(r_0\).

In the working zone, the magnetic flux through an elementary annular surface of radius \(r_x\) and width \(dr\) (Fig. 1) will be equal to:

\[ d\Phi = 2\pi r_x B_x\,dr = 2\pi B_0 r_0^n \cdot r_x^{1-n}\,dr, \]

whence the magnetic flux in the ring bounded by arbitrary radii \(r_1\) and \(r_2\) within the working zone is

\[ \Phi_{12} = 2\pi B_0 r_0^n \int_{r_1}^{r_2} r_x^{1-n}\,dr = \frac{2\pi B_0 r_0}{2-n} \left( r_2^{2-n} - r_1^{2-n} \right). \tag{3} \]

To satisfy the betatron relation, the central part of the pole tips is provided with cylindrical projections, or several cylindrical disks (shims) are placed between the poles.

The actual distribution of magnetic-flux density along the radius in Fig. 1 is represented by the curve \(abcd\). To simplify the calculations, with an accuracy sufficient for practice, the actual curve may be replaced by the curve \(abezu\), especially since the possible inaccuracy caused by such an assumption is easily corrected after assembly of the electromagnet by adjusting the gap between the poles.

Assuming that the magnetic-flux density in the ring bounded by the radii \(r_0\) and \(r_c\) (Fig. 1) varies in accordance with equation (2), while in the circle of radius \(r_c\) it is constant and equal to \(B_c\), from equation (3) we find the magnetic flux in the circle of the equilibrium orbit:

\[ \Phi_k = \frac{2\pi B_0 r_0^n}{2-n} \left( r_0^{2-n} - r_c^{2-n} \right) + \pi r_c^2 B_c . \tag{4} \]

Using, in addition to the expression obtained, the betatron condition, we find

\[ 2B_0 = \frac{2r_0^n B_0}{r_0^2(2-n)} \left( r_0^{2-n} - r_c^{2-n} \right) + \frac{r_c^2}{r_0^2} B_c, \]

whence

\[ B_0 = \frac{B_c} {2\left[ z^2 - \dfrac{1}{2-n}\left(z^2 - z^n\right) \right]}, \tag{5} \]

where

\[ z = \frac{r_0}{r_c} = \frac{r_0}{r_0-b_0}. \tag{6} \]

The quantity \(b_0\) depends on the radial size of the cross section of the accelerating chamber, i.e.,

\[ b_0 = b + (0.5 \div 1.0)\,\mathrm{cm}, \]

where \(b\) is the size of the radial semiaxis of the cross section of the accelerating chamber (Fig. 1).

From equation (6),

\[ r_0 = b_0 \frac{\gamma}{\gamma - 1}. \tag{7} \]

Substituting into equation (1) the value of \(r_0\) from equation (7) and \(B_0\) from equation (5), we obtain

\[ A = \frac{1.5 \cdot 10^{-4} B_c b_0} {\gamma \left[1 - \frac{1}{2-n}\left(1-\gamma^{\,n-2}\right)\right](\gamma-1)} \]

or

\[ A = \frac{1.5 \cdot 10^{-4} B_c b_0}{\beta}, \tag{8} \]

where

\[ \beta = \gamma \left[1 - \frac{1}{2-n}\left(1-\gamma^{\,n-2}\right)\right](\gamma-1). \tag{9} \]

The dependence of \(\beta\) on \(\gamma\) for different values of \(n\) (according to equation 9) is shown in the form of curves (Fig. 2).

Fig. 2

Equation (8), as a modification of equation (5), relates the basic dimensions and parameters of the betatron inter-pole space under the condition that the betatron relation is satisfied.

Betatron Constant

Equation (8), if equation (9) is also taken into account, contains three independent unknown quantities: \(B_c\), \(n\), and \(b_0\).

When the winding of the betatron electromagnet is supplied with an alternating sinusoidal current, the instantaneous value of the magnetic flux density on the orbit at the end of the acceleration process is

\[ B_{0p} = B_{0a} \sin \alpha_p, \]

where \(B_{oa}\) is the amplitude of the magnetic-flux density in the orbit; \(\alpha_p = 65 \div 75^\circ\) is the recommended phase for dumping the electrons onto the target.

Consequently, the maximum calculated energy of the electrons accelerated in the betatron, according to equation (8), is

\[ A=\frac{1.5\cdot 10^{-4} B_{c.m} k_c b_0 \sin \alpha_p}{\beta}, \tag{10} \]

where \(B_{c.m}\) is the maximum permissible magnetic-flux density in the steel of the central inserts, \(k_c = 0.83 \div 0.87\) is the coefficient of filling of the central inserts with steel.

When choosing the value of \(B_{c.m}\), one should take into account the magnetization curve, the specific losses in the steel of the magnetic core, and the method of cooling the electromagnet.

In betatrons with a maximum energy from 5 to 25 MeV, designed at the Tomsk Polytechnic Institute, when operating on alternating current of industrial frequency \((f = 50\ \text{cps})\) with natural cooling of the electromagnet,

\[ B_{c.m}=13500 \div 14000\ \text{gauss}. \]

For manufacturing the magnetic core of the betatrons, steel E-42 of thickness \(0.35\ \text{mm}\) was used.

The calculated value of the field fall-off index \(n\) is usually taken equal to \(n = 0.5 \div 0.75\).

The value \(b_0\) remains unknown. As indicated earlier, \(b_0\) depends on the dimensions of the cross section of the accelerating chamber of the betatron.

The dimensions of the cross section of the accelerating chamber depend on many factors. In principle, the larger the dimensions of the chamber cross section, the more probable is the capture into acceleration of a greater number of electrons emitted by the electron gun at the moment the injection voltage pulse is applied, and, consequently, the attainment of a higher intensity of gamma radiation. But with an increase in the dimensions of the chamber cross section, the dimensions of the electromagnet and the power of the capacitor bank for compensating the reactive power increase.

The dimensions of the chamber cross section are strongly affected by the design of the magnetic core, on which the phase structure of the magnetic field in the working zone of the interpolar space depends.

In betatrons with a symmetrical magnetic core, the cross section of the chamber is relatively smaller than with a U-shaped magnetic-core design.

The question of choosing the dimensions of the cross section of the accelerating chamber is not considered here, and in calculating the radius of the equilibrium orbit of the betatron, the dimension \(b_0\) must be specified.

From equation (10) we can write the following relation:

\[ \frac{b_0}{A\beta}=\frac{6.7\cdot 10^3}{B_{c.m}\cdot k_c \cdot \sin \alpha_p}, \]

where \(A\) is expressed in MeV.

The right-hand side of this equation, for the selected values of \(B_{c.m}\), \(\alpha_p\), and \(k_c\), is a constant quantity. Denoting this quantity by \(\mathrm{Б}\), we can write

\[ \frac{b_0}{A\beta}=\mathrm{Б}, \tag{11} \]

where

\[ \mathrm{Б}=\frac{6.7\cdot 10^3}{B_{c.m}\cdot k_c \cdot \sin \alpha_p} \tag{12} \]

may be called the “betatron constant.”

Determination of the radius \(r_0\)

For selected values of \(B\), \(n\), and \(\varepsilon_0\), from equation (11)

\[ \beta=\frac{\varepsilon_0}{AB}. \]

Having determined, from the curves in Fig. 2, the corresponding value of \(\gamma\), we find from equation (7)

\[ \left| r_0=\varepsilon_0\frac{\gamma}{\gamma-1}\right|. \]

As is seen from Fig. 2, the curves \(\beta=f(\gamma)\) for different exponents of field decrease \(n\), at values \(\gamma<1.5\), coincide, tending to the straight line corresponding to \(n=1\). Consequently, for values \(\gamma<1.5\), without impairing the accuracy of the calculation one may neglect the influence of \(n\) on the magnitude of the calculated radius of the equilibrium orbit.

Taking \(n=1\), from equation (9) we obtain:

\[ \beta=\gamma-1. \tag{13} \]

Substituting the obtained value of \(\beta\) into equation (11) and taking into account equality (6), we find

\[ r_0=AB+\varepsilon_0. \tag{14} \]

Equation (14) gives an accuracy quite sufficient for engineering calculations if \(\gamma<1.5\), which corresponds to betatrons with a maximum electron energy of the order of 10 MeV and higher.

The usual designs of a betatron magnetic circuit make it possible, within small limits, to adjust the radius of the equilibrium orbit if, during testing of the electromagnet, a deviation of the orbit radius from the calculated one is found.

Taking into account the possibility of adjustment, equation (14) can also be used to determine the radius of the equilibrium orbit in betatrons with a maximum energy of accelerated electrons less than 10 MeV.

If the maximum specified kinetic energy of the electrons accelerated in the betatron, \(W\), is much greater than the rest energy \((\varepsilon_0=0.51\ \text{MeV})\), then instead of equation (14) one may use the equation

\[ r_0 \simeq WB+\varepsilon_0. \tag{14a} \]

Equations (14) and (14a) make it possible, quickly and for specified conditions with a certain degree of uniqueness, to solve the question of choosing the radius of the equilibrium orbit of a betatron.

Calculation of the radius \(r_0\) in betatrons with annular grooves on the poles

A considerable part of the energy of the magnetic field of a betatron, and consequently of the reactive power consumed by the magnetizing winding of the electromagnet, falls to the share of the air gap of the central part of the interpole space in the circle of radius \(r_c\), where the magnetic-flux density has its greatest value.

The capacitance of the capacitor banks required to compensate the reactive power of the betatron electromagnet can be considerably reduced if the air gap in the central part of the interpole space is decreased or completely eliminated.

In this case an additional winding is necessary, compensating the action of the main magnetizing winding at the center (in the circle of radius \(r_c\)).

When a compensating winding is used, the pole cores must have an annular slot for accommodating the winding (Fig. 3).

Annular slots on the poles are also necessary for accommodating the compensating winding when the betatron electromagnet is magnetized by direct current.

The curve of the dependence \(B=f(r)\) in a betatron electromagnet with annular slots on the poles is shown in Fig. 3.

In this case, for determining the radius of the equilibrium orbit, equations analogous to those given above are obtained, but a new quantity \(\delta_k\) is introduced—the width of the annular slot, which can be calculated beforehand (p. 74).

Fig. 3

For betatrons with annular slots on the poles, equation (14) takes the form

\[ r_o=(\delta_o+\delta_k)+0.5\,A\mathrm{Б}+\sqrt{(0.5\,A\mathrm{Б})^2+\delta_k A\mathrm{Б}} . \tag{15} \]

In exact calculations this equation should be used. In approximate calculations, especially for revealing regularities when using the method of similitude, equation (15) is inconvenient because of its complexity.

Let us expand the last term on the right-hand side of equation (15) in a series by Newton’s binomial formula:

\[ \left[(0.5\,A\mathrm{Б})^2+\delta_k A\mathrm{Б}\right]^{0.5} = 0.5\,A\mathrm{Б}+\delta_k-\frac{\delta_k^2}{A\mathrm{Б}} \ldots , \]

Substituting the first two terms of the series into equation (15), we obtain

\[ r_o \simeq A\mathrm{Б}+(\delta_o+2\delta_k). \tag{15a} \]

Preliminary determination of the width of the annular pole slot

In calculating the radius of the equilibrium orbit in a betatron electromagnet with annular slots in the poles, the width of the annular slots must be known. At the same time, however, the final dimensions of the slots can be determined only after calculation of the winding, which is placed in the annular slots. In this case the slot dimensions must be predicted with sufficient accuracy.

For a preliminary choice of the width of the annular slot \(b_k\) (Fig. 3), one may be guided by the following considerations.

In the absence of a gap in the middle part of the interpole space of the betatron, the ampere-turns of the compensating winding may, in approximate calculations, be taken as equal to the magnetic voltage of the gap \(\delta_0\) at \(B_{0a}\), i.e.

\[ AW_k = 0.8\delta_0 B_{0a}. \]

If the gap is retained, but has a size smaller than the calculated one without the compensating winding, then

\[ AW_k = 0.8c_k B_{0a}\delta_0, \]

where

\[ c_k = \frac{\delta_c-\delta_k}{\delta_c} \]

is the degree of compensation,

\(\delta_c\) is the calculated gap at the center, without the compensating winding,

\(\delta_k\) is the gap in the presence of the compensating winding.

The cross-sectional area of the annular slot required to accommodate one coil of the compensating winding is

\[ S_k=\frac{AW_k}{2\sqrt{2}\Delta_k} =\frac{0.283c_k B_{0a}\delta_0}{\Delta_k}, \tag{16} \]

where \(\Delta_k=j_k k_z\) is the current density in the slot,

\(j_k\) is the current density in the copper of the compensating winding,

\(k_z\) is the coefficient of filling of the slot with copper, depending on the insulation of the slot and winding and on the cooling system.

Approximate values of \(\Delta_k\), depending on the voltage at the winding terminals under continuous supply of the electromagnet winding with alternating current, are given in Table 2. With pulsed supply of the winding, the permissible current density increases in accordance with the duty factor of the pulses.

Table 2

For a winding with a conductor of rectangular cross section in small betatrons, in the absence of air ducts in the winding coils, the current density in the slot may have values of the order of \(100\div120\) A/cm².

The air gap at the radius of the equilibrium orbit depends on the size \(b_0\). In an approximate calculation one may take

\[ \delta_0 \simeq 2b_0\sqrt{\frac{1-n}{n}}. \]

If the compensating winding is placed only in the annular slots, then the area occupied by the cross section of one coil, for \(h_k = m b_k\), is equal to \(S_k = m b_k^2\), and the width of the annular slot is

\[ b_k = \sqrt{\frac{S_k}{m}}, \tag{17} \]

where \(h_k\) is the height (depth) of the annular slot.

In choosing the coefficient \(m\), it should be taken into account that the larger \(h_k\) is, the smaller the calculated radius of the equilibrium orbit, i.e., the smaller the radial dimensions of the poles. At the same time, \(h_k\) cannot be greater than the pole height \((h_p)\). Depending on the design of the magnetic circuit, \(m = 1.5 \div 3\).

In the case when the compensating winding is arranged over the entire height of the central core, the cross-sectional area occupied by the entire compensating winding will be equal to

\[ 2 S_k = (2 h_k + \delta_0)b_k, \]

whence the width of the annular slot is

\[ b_k = \sqrt{\frac{2S_k}{m}} - \frac{\delta_0}{4m}. \tag{18} \]

Equation (16) contains the still unknown quantity \(B_{od}\)—the amplitude of the magnetic-flux density at the equilibrium orbit. Depending on the dimensions of the cross section of the accelerating chamber, on the degree of compensation, and on the betatron constant, \(B_{od}\) may have values in the range

\[ B_{od} = 2200 \div 4500 \text{ gauss}. \]

The lower limit applies to small betatrons operating on alternating current of increased frequency; the upper corresponds to betatrons with a maximum energy of accelerated electrons of \(100\) MeV and above.

Fig. 4

To determine the cross-sectional area of the annular slot of the pole, it is necessary first to assign a value of \(B_{od}\) and, after determining the slot width \(b_k\), then \(r_0\) and the corresponding value of \(B_{od}\), compare it with the preliminarily accepted one. If the calculated value of \(B_{od}\) differs from the pre—

[[unclear: beginning of sentence]] by more than 5%, the calculation of \(S_k\), \(\vartheta_k\), and \(r_0\) should be repeated with the calculated value of \(B_{0a}\).

For guidance in choosing the preliminary value of \(B_{0a}\), one may use the curves (Fig. 4) corresponding to the betatron constant \(\bar B = 0.6\). If the constant of the betatron being designed \((\bar B)\) is not equal to 0.6, then the value of \(B_{0a}\) found from the curve in Fig. 4 should be multiplied by \(\dfrac{0.6}{\bar B}\).

The method described gives an approximate, orienting value of \(\vartheta_k\). After the winding has been calculated, the slot dimensions sometimes have to be corrected; this is best done by changing the slot depth \(h_k\). In this case no changes in the other dimensions of the magnetic core are required.