METHOD FOR INJECTING ELECTRONS INTO A CYCLIC ACCELERATOR WITH A CENTRAL MAGNETIC FLUX

B. V. Okulov

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

In known methods of injection into cyclic electron accelerators with a central magnetic flux, for example into betatrons, electron injection is most often carried out from an electron gun whose electrodes are located directly in the stability region or on its boundary. At high injection voltages and in a number of other cases, the gun electrodes are placed outside the stability region, and various inflector devices are used to introduce electrons into the latter.

Fig. 1. Accelerator schematic: 1 — accelerator magnetic core; 2 — magnetizing coils; 3 — central opening; 4 — central gap; 5 — accelerating gap (interpolar space); 6 — injector cathode; 7 — injector anode; 8 — high-voltage injector insulator; 9 — cathode-heater insulator; 10 — vacuum-tight pole surface; 11 — vacuum wall of the accelerating gap; 12 — electron beam; 13 — copper plane of the accelerator

The inflector device limits the magnitude of the injection current and complicates the adjustment and operation of the accelerator [1]. In order to increase the injected and accelerated current and to provide high-voltage injection without an inflector, it is proposed that the injector electrodes be placed in the central gap of the accelerator (Fig. 1), i.e., in the region of intersection

of the central magnetic flux by the median plane [2]. In this case it is expedient to make the electron injector in the form of a cylindrical diode, and to align the symmetry axis of the injector with the symmetry axis of the central gap and of the interpole space. In this arrangement the magnetic field of the central gap performs the function of an inflector.

As can be seen from the figure, such an injector lies outside the stability region; therefore it has no clearance. Capture of electrons into acceleration in this case is accomplished by realizing, during injection, a sufficiently pronounced nonstationary process, which occurs, in particular, when the injection current exceeds a certain value [1, 3].

Fig. 2. Distribution of induction in the median plane: 1—cathode; 2—anode; 3—electron beam

Let us consider the injection conditions as applied to a betatron with an iron core (Fig. 1).

We assume that the median surface is a plane perpendicular to the accelerator axis, and that the magnetic induction in this plane depends on the radius in the following way (Fig. 2):

\[ \text{for } 0 \le r < r_{\mathrm{c}}, \quad B=0, \tag{1} \]

\[ \text{for } r_{\mathrm{c}} \le r \le r_c, \quad B=B_c, \tag{2} \]

\[ \text{for } r_c < r \le r_{\mathrm{n}}, \quad B=B_0\left(\frac{r_0}{r}\right)^n , \tag{3} \]

where

\[ r_{\mathrm{c}} \text{— radius of the central opening in the electromagnet;} \]

\[ r_c \text{— radius of the central core;} \]

\[ r_0 \text{— radius of the equilibrium orbit;} \]

$r_{\mathrm{n}}$ — outer radius of the pole tips of the magnetic circuit;
$n$ — magnetic-field index in the inter-pole space;
$B_c$ — magnetic induction in the central gap;
$B_0$ — magnetic induction on the equilibrium orbit.

Motion of Electrons in the Central Gap. Necessary Injection Condition

For simplicity, let us consider the motion of electrons only in the median plane. Since the injector operates at high voltage, the thermal velocities of the electrons upon leaving the cathode may be neglected, and it may be assumed that the initial angular velocity of the electrons is zero. Then, owing to the axial symmetry of the device, the angular velocity of an electron at radius $r > r_{\mathrm{k}}$ for any distribution of the potential along the radius is uniquely determined by Busch’s theorem [4]:

\[ \dot{\theta}=\frac{\eta}{2\pi r^2}\left(\Phi_r-\Phi_{\mathrm{k}}\right), \]

where

\[ \eta=\frac{e}{m} \]

is the ratio of the electron charge to its mass;

$\Phi_r$ — magnetic flux in a circle of radius $r$;
$\Phi_{\mathrm{k}}$ — magnetic flux penetrating the cathode.

The tangential component of the electron velocity is

\[ v_\tau=r\cdot\dot{\theta}=\frac{\eta}{2\pi r}\left(\Phi_r-\Phi_{\mathrm{k}}\right). \]

In accordance with conditions (1) and (2),

\[ \left. \begin{aligned} \Phi_r&=\pi B_c\left(r^2-r_{\mathrm{c}}^2\right),\\ \Phi_{\mathrm{k}}&=\pi B_c\left(r_{\mathrm{k}}^2-r_{\mathrm{c}}^2\right). \end{aligned} \right\} \tag{4} \]

Consequently,

\[ v_\tau=\frac{\eta B_c}{2r}\left(r^2-r_{\mathrm{k}}^2\right). \tag{5} \]

Let us introduce dimensionless radii

\[ \left. \begin{aligned} \frac{r}{r_0}&=R,\\ \frac{r_{\mathrm{k}}}{r_0}&=R_{\mathrm{k}}, \end{aligned} \right\} \tag{6} \]

and, taking into account the equality

\[ \eta r_0 B_0=v_0, \tag{7} \]

where $v_0$ is the value of the velocity required for motion of the electron at the moment of injection along the equilibrium orbit, we obtain

\[ v_{\tau}=\frac{v_0}{2}\cdot\frac{B_c}{B_0}\cdot\frac{1}{R}\left(R_i^2-R_k^2\right). \tag{8} \]

Let us find the ratio \(B_c/B_0\). We write the betatron condition

\[ 2B_0=\overline{B}_0=\frac{\Phi_0}{\pi r_0^2}, \tag{9} \]

where

\(\overline{B}_0\) is the average magnetic-field induction within the circle of the equilibrium orbit;

\(\Phi_0\) is the corresponding magnetic flux.

Taking into account conditions (1), (2), and (3), we have

\[ \Phi_0=\Phi_c+\frac{2\pi}{2-n}\,r_0^n B_0\left(r_0^{2-n}-r_c^{2-n}\right), \tag{10} \]

\[ \Phi_c=\pi B_c\left(r_c^2-r_{\mathrm{ц}}^2\right). \tag{11} \]

Introduce the dimensionless radii

\[ \left. \begin{aligned} \frac{r_c}{r_0}&=R_c,\\ \frac{r_{\mathrm{ц}}}{r_0}&=R_{\mathrm{ц}}, \end{aligned} \right\} \tag{12} \]

and, taking into account equalities (9)—(12), we obtain

\[ \frac{B_c}{B_0}=\frac{2}{2-n}\, \frac{\left(1-n+R_c^{2-n}\right)}{\left(R_c^2-R_{\mathrm{ц}}^2\right)}. \tag{13} \]

Substitute the value of the ratio \(B_c/B_0\) into equation (8) and divide both sides of the resulting equality by the velocity of the electrons \(v_i\) emerging from the injector. We obtain a dimensionless expression for the tangential velocity of the electron in the central gap

\[ \frac{v_{\tau}}{v_i}= \frac{v_0}{(2-n)v_i R} \left(1-n+R_c^{2-n}\right) \frac{\left(R^2-R_k^2\right)}{\left(R_c^2-R_{\mathrm{ц}}^2\right)}. \tag{14} \]

It is obvious that, when \(v_{\tau}=v_i\), the electron is at the turning point of the trajectory. If the turning point lies at a radius smaller than \(R_c\), then the electron will not leave the central gap and will return to the cathode (Fig. 3, trajectory 5). If, however, \(v_{\tau}=v_i\) at \(R=R_c\), then equation (14) takes the form

\[ 1=\frac{v_0}{v_i}\cdot\frac{\left(R_c^2-R_k^2\right)}{R_c^2\cdot C_1}, \tag{15} \]

where

\[ C_1=\frac{(2-n)}{R_c}\cdot \frac{\left(R_c^2-R_{\mathrm{ц}}^2\right)} {\left(1-n+R_c^{2-n}\right)}. \tag{16} \]

In this case the electron will emerge onto the circumference of radius \(R_c\) with zero radial velocity and, as a result of a fluctuation of some magni-

causes it either to return to the cathode (trajectory 6), or to enter the inter-pole space (trajectory 7).

Taking into account the equalities

\[ v_i=\sqrt{2\eta U_i},\qquad v_0=\sqrt{2\eta U_0}, \]

where \(U_i\) is the voltage on the injector, and \(U_0\) is the “required” voltage, equation (15) takes the form

\[ 1=\sqrt{\frac{U_i}{U_0}}\cdot\frac{(R_c^2-R_k^2)}{R_c^2\cdot C_1}. \tag{17} \]

From equation (17) we find the formula for determining the cathode radius at which the turning point of the electron trajectory lies at radius \(R_c\).

Fig. 3. Diagram of electron motion in the median plane of the betatron: 1 — cathode; 2 — central gap; 3 — inter-pole space; 4 — equilibrium orbit; 5, 6, and 7 — electron trajectories; \(P\) — turning point of the electron trajectory in the inter-pole space

\[ R_k=R_c\sqrt{\,1-C_1\sqrt{\frac{U_i}{U_0}}\,}. \tag{18} \]

In the special case, when \(U_i=U_0\),

\[ R_k=R_c\sqrt{1-C_1}. \tag{19} \]

The voltage on the injector corresponding to equality (17) will be called critical and denoted by \(U_{i\mathrm{cr}}\).

From equality (17)

\[ U_{i\mathrm{cr}}=\frac{U_0}{C_1^2}\left[1-\left(\frac{R_k}{R_c}\right)^2\right]^2 . \tag{20} \]

Obviously, if the voltage on the injector exceeds the critical value, then all electrons will enter the inter-pole space and form the injection current.

Thus, the necessary condition for injection of electrons into the inter-pole space has the form

\[ U_i>U_{i\mathrm{cr}}. \]

Motion of an electron in the inter-pole space

According to Busch’s theorem, the tangential component of the electron velocity in the inter-pole space can be written in the form

\[ v_\tau=r\cdot \dot{\theta}=\frac{\eta}{2\pi r}\left(\Phi_r-\Phi_k\right), \tag{21} \]

where

\[ \Phi_r=\Phi_c+\frac{2\pi}{2-n}\cdot r_0^n B_0\left(r^{2-n}-r_c^{2-n}\right). \tag{22} \]

Substituting into equalities (21) and (22) the values of the corresponding quantities from equalities (4), (6), (7), and (9–13), after transformations we obtain

\[ v_\tau=\frac{v_0}{(2-n)R}\cdot\left(R^{2-n}+C_2\right), \tag{23} \]

where

\[ C_2=\left(1-n+R_c^{2-n}\right)\frac{\left(R_c^2-R_k^2\right)}{\left(R_c^2-R_{\mathrm{ц}}^2\right)}-R_c^{2-n}. \tag{24} \]

If the space charge of the electrons in the inter-pole space is neglected, then at any instant of time the equality

\[ v_i^2=v_r^2+v_\tau^2 \]

is valid.

Whence, taking into account condition (6), we have

\[ v_r=r_0\frac{dR}{dt}=\sqrt{v_i^2-v_\tau^2}, \]

and, with equation (23) taken into account, we obtain

\[ \frac{dR}{dt}=\frac{1}{r_0}\sqrt{v_i^2-\left[\frac{v_0}{(2-n)R}\left(R^{2-n}+C_2\right)\right]^2}. \tag{25} \]

From equations (21) and (23) it follows that

\[ \frac{d\theta}{dt}=\frac{v_0}{(2-n)rR}\left(R^{2-n}+C_2\right). \tag{26} \]

Eliminating \(dt\) from equations (25) and (26), we find the equation of the electron trajectory in the inter-pole space:

\[ \theta=\int \frac{\left(R^{2-n}+C_2\right)dR} {R\sqrt{(2-n)^2\left(\frac{v_i}{v_0}\right)^2R^2-\left(R^{2-n}+C_2\right)^2}} +C_3. \tag{27} \]

Taking into account the complete axial symmetry of the device, one can obtain useful information about the motion of the electron without evaluating the integral (27). In particular, one can find the turning point of the trajectory. For simplicity, let \(n=\mathrm{const}\); then \(C_2=\mathrm{const}\), and from equation (23) we have

\[ R^{2-n}-(2-n)\frac{v_\tau}{v_0}R+C_2=0. \]

It is obvious that for \(v_\tau=v_i\) the electron trajectory has a turning point whose dimensionless radius \(R_{\mathrm{p}}\) is determined by the equation

\[ R_{\mathrm{p}}^{2-n}-(2-n)\frac{v_i}{v_0}R_{\mathrm{p}}+C_2=0. \]

In the particular case \(n=0.5\), the equation is transformed into the cubic

\[ x^3-1.5\frac{v_i}{v_0}x^2+C_2=0, \]

where \(x=\sqrt{R_{\mathrm{p}}}\).

REFERENCES

1. B. V. Okulov. Dissertation. Tomsk, TPI, 1968.

2. B. V. Okulov. Author’s Certificate No. 215355 of January 30, 1967, BI, No. 13, p. 51.

3. B. V. Okulov. Method of injecting electrons into cyclic accelerators. Parts I and II. Report of NIIYaF EA, 1967.

4. J. R. Pierce. Theory and Design of Electron Beams. Moscow, “Soviet Radio,” 1956.