On the Calculation of the Voltage on a Betatron Inflector

B. V. Okulov

(Presented by the Research Institute of Nuclear Physics, Electronics, and Automation)

There is a natural desire to reduce the magnitude of the voltage on the inflector. At a lower voltage it is easier to ensure the electrical strength of the structure, and a power-supply circuit for the inflector of smaller dimensions is required. The most obvious way to reduce the required voltage on the inflector is to decrease the distance between its plates. However, this distance cannot be reduced arbitrarily, since this would require increasing the current density in the electron beam, which is undesirable, as it may lead to an increase in electron losses in the inflector [1, 6].

In this connection, it is of interest to find general expressions for calculating the voltage on the inflector, so that subsequently, by analyzing the dependences obtained, it will be possible in each specific case to select the optimum inflector parameters.

All other conditions being equal, the magnitude of the required voltage depends on the shape of the inflector, which is conveniently characterized by the mean radius of curvature \(r_c\) (Fig. 1).

For injection into cyclic electron accelerators, deflecting plates are most often used whose center of curvature lies, relative to the injection point \(I\), on the side opposite to the axis of symmetry of the guiding magnetic field, i.e., at point \(O_1\). Such deflecting plates are called a cylindrical inflector.

Fig. 1. Schemes of injection devices with various inflectors: 1 — cylindrical inflector; 2 — flat inflector; 3 — cylindrical inflector of opposite curvature; 4 — inflector; \(r_i\) — injection radius, \(r_a\) — radius of location of the inflector anode.

A flat inflector is the simplest in construction. It may be regarded as a special case of the cylindrical inflector, when the mean radius of curvature of the latter is equal to infinity.

The electric field of a cylindrical inflector must neutralize the action on the electron of the centrifugal force and the Lorentz force, whereas the electric field of a flat inflector must neutralize the action only of the Lorentz force. Therefore, other conditions being equal, a lower voltage is required to feed a flat inflector than a cylindrical one.

If, however, the center of curvature of the cylindrical inflector is moved to the opposite side, i.e., to the side of the symmetry axis of the guide magnetic field (to point \(O_2\)), then the centrifugal force acting on an electron in the inflector will change its sign and will “work” against the magnetic field. The magnitude of the required voltage on such an “inverted” cylindrical inflector will decrease even in comparison with the voltage for a flat inflector. In what follows, the “inverted” cylindrical inflector will be called a cylindrical inflector of reverse curvature [2].

Let us find general expressions for calculating the required voltage for the three types of inflectors mentioned above.

In what follows, by the words “required voltage on the inflector” we shall mean such a voltage at which the mean electron of the beam can move along the optical axis of the inflector. We note that transverse forces of the beam’s own space charge do not act on the mean electron. We shall neglect longitudinal forces, since they are small. For simplicity, we shall also neglect edge effects in the inflector.

By the words “optical axis of the inflector” we shall mean the line of intersection of the median plane of the accelerator with the surface normal to it, every point of which is at equal shortest distances from both plates of the inflector.

Introduce the notation (Fig. 2):

\(d\) — distance between the plates of the inflector,

\(U_d\) — required voltage on the inflector.

The electrons emerge from the injector with kinetic energy \(eU_i\); therefore, for an electron moving in the inflector at a distance \(\Delta d\) from the grounded plate, the kinetic energy, taking into account the potential \(\Delta U_d\) of the space between the inflector plates, is equal to

\[ W_k=e\cdot U_i+e\cdot \Delta U_d=e(U_i+\Delta U_d)=eU_e, \]

where \(U_e\) is the “effective potential” of the electron in the inflector.

a). Flat inflector

For a flat inflector (Fig. 2a) the electric-field strength is

\[ E=\frac{U_d}{d}, \tag{1} \]

and the potential

\[ \Delta U_d=E\cdot \Delta d=\frac{U_d}{d}\cdot \Delta d. \]

Therefore

\[ U_e=U_i+\Delta U_d=U_i+\frac{U_d}{d}\cdot \Delta d. \tag{2} \]

Effective electron velocity

\[ v_{\mathrm e}=\sqrt{2\frac{e}{m_{\mathrm e}}U_{\mathrm e}}, \tag{3} \]

where \(m_{\mathrm e}\) is the effective mass of the electron.

An electron moving in a flat inflector is acted upon by the force of the electric field \(eE\) and by the oppositely directed force of the magnetic field \(ev_{\mathrm e}H\). If, for simplicity, it is assumed that during the injection time interval the magnetic-field strength in the inflector is constant in time, does not depend on the radius,* and is equal in magnitude to the magnetic-field strength at the injection radius \(H\), then

Fig. 2. Schematics for calculating the voltage on the inflector:
a — flat inflector; 1 — inflector axis; 2 — optical axis of the inflector; 3 — circumference of the injection radius; \(r_1\) and \(r_2\) — radii of curvature of the plates, \(H_i\) — magnetic-field strength at the injection radius, \(E\) — electric-field strength in the inflector.

the condition for motion of the electron in the flat inflector along a straight line parallel to the optical axis will have the form

\[ eE-ev_{\mathrm e}H_i=0, \tag{4} \]

where

\[ H_i=\frac{m_i v_i}{e r_i} =\frac{m_i}{e r_i}\sqrt{2\frac{e}{m_i}U_i} =\frac{1}{r_i}\sqrt{2\frac{m_i}{e}U_i}. \tag{5} \]

From condition (4), taking into account equalities (1), (3), and (5), we obtain

\[ \frac{U_d}{U_{\mathrm e}} = 2\frac{d}{r_i} \sqrt{\frac{m_i}{m_{\mathrm e}}\cdot\frac{U_i}{U_{\mathrm e}}}. \tag{6} \]

For comparatively small values of the ratio \(U_d/U_i\) (this case is of greatest practical interest), the value of the square root in equality (6) is close to unity.** Therefore we take

\[ \sqrt{\frac{m_i}{m_{\mathrm e}}\cdot\frac{U_i}{U_{\mathrm e}}}=1, \tag{7} \]

* The dependence of the magnetic field in the inflector on radius can be taken into account in developing the design of the inflector [1].

** The value of the square root is less than unity, and the values of \(U_d\) obtained in this approximation will be somewhat overestimated. However, a certain voltage margin on the inflector is useful, since practice shows that the beam’s own space charge, whose action on the average electron is not taken into account in this calculation, requires some increase in \(U_d\).

and thus, without introducing a significant error into the final expression, we shall simplify the subsequent calculations.

From equation (6), taking into account relations (2) and (7), it follows that

\[ \frac{U_d}{U_i}=\frac{d}{r_i}\cdot \frac{2}{\left(1-2\frac{\Delta d}{r_i}\right)} . \]

For the mean electron of the beam moving along the optical axis of the inflector, \(\Delta d=0.5d\); therefore

\[ \frac{U_d}{U_i}=\frac{d}{r_i}\cdot \frac{2}{(1-d/r_i)} . \tag{8} \]

Let us introduce the dimensionless parameters

\[ \left. \begin{aligned} \frac{U_d}{U_i}&=\overline{U}_{di},\\ \frac{d}{r_i}&=d_i, \end{aligned} \right\} \tag{9} \]

where \(\overline{U}_{di}\) is the reduced voltage (relative to the injection voltage) on the plane inflector,
\(d_i\) is the reduced width of the inflector (relative to the injection radius).

In these notations expression (8) is written in the form

\[ \overline{U}_{di}=\frac{2d_i}{1-d_i}. \tag{10} \]

If the change in the electron velocity in the inflector is neglected, i.e., if \(U_e\) is replaced by \(U_i\), then from relations (6), (7), and (9) we obtain the simpler expression

\[ (\overline{U}_{di})'=2d_i. \tag{10'} \]

The same result is obtained directly from equation (10), if \(d_i\ll 1\).

Thus, the reduced voltage on a plane inflector depends only on the reduced width of the inflector.

The result of calculations by formulas (10) and (10′) is given in the middle part of Fig. 3.

b). Cylindrical inflector

For a cylindrical inflector (Fig. 2b), at radius \(r\) the electric-field strength is

\[ E=\frac{U_d}{r\cdot \ln r_2/r_1}, \tag{11} \]

and the potential of the space between the plates is

\[ \Delta U_d = E \cdot r \cdot \ln r_2/r, \]

taking equation (11) into account,

\[ \Delta U_d = U_d \cdot \frac{\ln r_2/r}{\ln r_2/r_1}, \]

and the expression for the “effective potential” of the electron takes the form

\[ U_{\text{э}} = U_i + U_d \cdot \frac{\ln r_2/r}{\ln r_2/r_1}. \tag{12} \]

In a cylindrical inflector of a betatron, if the space charge of the beam is neglected, three forces act on an electron moving along an arc of a circle of radius \(r\): the centrifugal force \(\dfrac{m_{\text{э}}v_{\text{э}}^{2}}{r}\), the force of the electric field of the inflector \(eE\), and the force of the magnetic field of the betatron \(ev_{\text{э}}H\). Assuming, as before, that \(H = H_i = \mathrm{const}\), in a polar coordinate system with origin at point \(O\) (Fig. 2b), we have

Fig. 2b—cylindrical inflector

\[ \frac{m_{\text{э}}v_{\text{э}}^{2}}{r} = eE - ev_{\text{э}}H_i. \]

Taking into account equalities (3), (5), (7), and (11), we obtain

\[ \frac{U_d}{U_{\text{э}}} = 2\left(1+\frac{r}{r_i}\right)\ln r_2/r_1. \tag{13} \]

Substituting the value of \(U_{\text{э}}\) from equation (12), we then have

\[ \frac{U_d}{U_i} = \frac{2(1+r/r_i)\ln r_2/r_1} {1-2(1+r/r_i)\ln r_2/r}. \]

For the median electron of the beam moving along the optical axis of the inflector \((r=r_c)\), equation (13) is also valid when the space charge of the beam is taken into account, since the force of the space charge acting on the median electron is practically equal to zero. Taking into account that \(r_2=r_c+0.5d\) and \(r_1=r_c-0.5d\), from the last equality we obtain

\[ \frac{U_d}{U_i} = \frac{ 2(1+r_c/r_i)\ln \dfrac{r_c+0.5d}{r_c-0.5d} }{ 1-2(1+r_c/r_i)\ln \dfrac{r_c+0.5d}{r_c} }. \tag{14} \]

Let

\[ \frac{U_d}{U_i}=U_{di}, \]

\[ \frac{r_c}{r_i}=r_{ci}, \]

where \(U_{di}\) is the reduced voltage on the cylindrical inflector,

\(r_{ci}\) is the reduced mean radius of curvature of the inflector. In these notations, (14), with (9) taken into account, is written in the form

\[ U_{di} = \frac{ 2(1+r_{ci})\ln \dfrac{r_{ci}+0.5d_i}{r_{ci}-0.5d_i} }{ 1-2(1+r_{ci})\ln \dfrac{r_{ci}+0.5d_i}{r_{ci}} }. \tag{15} \]

If the change in the electron velocity in the inflector is neglected and \(U_{\mathrm{e}}=U_i\) is assumed, then from (13) it follows that

\[ (U_{di})' = 2(1+r_{ci})\ln \frac{r_{ci}+0.5d_i}{r_{ci}-0.5d_i}. \tag{15'} \]

The graphs of expressions (15) and (15′) are given in the right-hand part of Fig. 3.

c). Cylindrical inflector of reverse curvature

For a cylindrical inflector of reverse curvature (Fig. 2c), analogously to the preceding case, one may write

\[ E=-\frac{U_d}{r\ln r_2/r_1}, \]

\[ \Delta U_d = U_d\frac{\ln r/r_1}{\ln r_2/r_1}, \]

\[ U_{\mathrm{e}} = U_i+ U_d\frac{\ln r/r_1}{\ln r_2/r_1}. \]

The condition for motion of an electron along a circle of radius \(r\) is written in the form

\[ \frac{m_{\mathrm{e}}v_{\mathrm{e}}^{2}}{r}=eE+ev_{\mathrm{e}}H_i . \]

After the appropriate substitutions and transformations we obtain that the reduced voltage on a cylindrical inflector of reverse curvature is

\[ U_{di}^{\smile}=\frac{U_d}{U_i} = -\frac{2(1-r_i/r_i)\ln r_2/r_1}{1+2(1-r_i/r_i)\ln r/r_1}. \]

For the central electron of the beam, moving along the optical axis of the inflector,

\[ U_{di}^{\smile} = -\frac{ 2(1-r_{ci})\ln \dfrac{r_{ci}+0.5d_i}{r_{ci}-0.5d_i} }{ 1+2(1-r_{ci})\ln \dfrac{r_{ci}}{r_{ci}-0.5d_i} }. \tag{16} \]

2c — cylindrical inflector of reverse curvature

If the change in the electron velocity due to the potential in the inflector is neglected, then, by analogy with equation \((15')\), we obtain

\[ \left(U_{di}^{\smile}\right)' = -2(1-r_{ci})\ln \frac{r_{ci}+0.5d_i}{r_{ci}-0.5d}. \tag{16'} \]

The graphs of expressions (16) and \((16')\) are given in the left-hand part of Fig. 3. From Fig. 3 it follows that the correction to \(U_{di}\) due to the change in the velocity of the electrons in the inflector is substantial in most cases, and therefore it must be taken into account. It is also seen from Fig. 3 that, for a given \(d_i\), the smallest value of the required reduced voltage on the inflector is obtained for a cylindrical inflector of reverse curvature, and the largest for a cylindrical inflector. For a plane inflector, the value of the required reduced voltage assumes an intermediate value.

Thus, if one seeks to minimize the magnitude of the required voltage on the inflector, it would seem that a cylindrical inflector of reverse curvature should be used in the high-voltage injector device. However, in order to choose the optimal parameters of the inflector, it is still necessary, as shown in [1, 2], to know the dependence of the inflector length on its type, on the injection angle and on the mean radius of curvature, as well as the dependence of the inflector transmission coefficient on the specified initial conditions [1].

Fig. 3. Dependence of the reduced voltage on the inflector on the reduced mean radius of curvature. —— with allowance for \(\Delta U_d\), — — — without allowance for \(\Delta U_d\).

In calculating \(U_d\), it must also be borne in mind that in the present work formulas have been obtained for calculating the instantaneous value of the minimum required voltage on the inflector. This, in particular, corresponds to the case when injection at the maximum calculated voltage on the injector is carried out at the peak of the voltage pulse on the inflector. If it is desired (or necessary) to operate at the maximum injection voltage on one or another part of the leading or trailing edge of \(U_d\) [3], the calculated amplitude \(U_d\) must be correspondingly increased.

It follows from equations (10), (15), and (16) that the ratio of the required voltage on the inflector to the voltage on the injector depends neither on \(H_i\) nor on \(v_i\). This is very important, since it makes it possible to simplify the power-supply circuit of the high-voltage injector device [4] and thus removes the principal obstacle (the increase in the complexity of the apparatus and of the operation of the high-voltage injection system) to the broad use of high injection voltages in betatron construction. The latter, as shown in [5], in turn makes it possible to obtain a corresponding increase in the charge accelerated in the betatron.

REFERENCES

1. B. V. Okulov. Dissertation. Tomsk, TPI, 1968.
2. B. V. Okulov. Izvestiya TPI, vol. 156. Tomsk, TSU Press, 1969, 130—143.

1. B. V. Okulov. In: Electron Accelerators. Tomsk, Tomsk State University Press, 1961, 124–126.

2. B. V. Okulov. In: Electron Accelerators. Moscow, Atomizdat, issue 2, 1970, 25–28.

3. V. A. Moskalev, B. V. Okulov. ZhTF, vol. 32, 1962, 1040–1041.

4. B. V. Okulov. On the calculation of current passage through a betatron inflector. The present collection.