On the Calculation of the Length of a Betatron Inflector
Abstract Generated abstract
The paper derives geometric formulas for calculating the length of electrostatic inflectors used in high voltage injection systems for betatrons and electron synchrotrons, where excessive inflector length increases electron losses on the plates. It treats three simple configurations, a flat inflector, a cylindrical inflector, and a cylindrical inflector of reverse curvature, expressing their lengths in terms of injection radius, inflector radius, injector dimensions, anode position, and injection angle. The resulting dimensional and reduced formulas include both general cases and the special case of zero injection angle. A comparison of reduced lengths indicates that the cylindrical inflector is shortest but requires the highest supply voltage, while the reverse curvature cylindrical inflector is longest but requires the lowest voltage, leading to usage recommendations depending on injection energy and voltage constraints.
Full Text
On the Calculation of the Length of a Betatron Inflector
B. V. Okulov
In recent years, in betatrons and electron synchrotrons, high-voltage injection with inflector input has been used increasingly often [1, 2, 3, 4, 5]. In this connection, the basic relations for calculating the parameters of the inflector device and, in particular, the length of the electrostatic inflector, are of interest. The length is an important characteristic of the inflector [6]. The greater the length of the inflector, the more electrons, other conditions being equal, are lost on its plates, and the fewer electrons are injected into the accelerating chamber [7].
Let us consider three types of inflectors having the simplest geometrical form: a flat inflector, a cylindrical inflector, and a cylindrical inflector of reverse curvature (Fig. 1).
Introduce the following notation:
\(l^{-}\) — length of the flat inflector;
\(l^{\circ}\) — length of the cylindrical inflector;
\(l^{\sim}\) — length of the cylindrical inflector of reverse curvature;
\(r_i\) — injection radius;
\(r_c\) — mean radius of the inflector;
\(r_a\) — radius at which the point of the electron gun anode nearest to the circumference of the injection radius is located;
\(\delta_p\) — “radial” dimension of the gun anode;
\(\varphi\) — central angle of the inflector;
\(\varepsilon\) — injection angle (the angle between the tangent to the circle of radius \(r_i\) and the tangent to the circle of radius \(r_c\)).
Let us find formulas for calculating the length of the three types of inflectors indicated above.
Let, in Fig. 1b, \(OB\) be parallel to the axis of the inflector, while \(IC\) and \(AB\) are perpendicular to it. Then \(\angle OIC = \varepsilon\), \(CB = l^{-}\), \(IC = \delta_p + AB\), \(OB = OC + l^{-}\). From \(\triangle OAB\), \(OB^2 = r_a^2 - BA^2\), or \((OC + l^{-})^2 = r_a^2 - (IC - \delta_p)^2\). But \(OC = r_i \cdot \sin \varepsilon\), and \(IC = r_i \cdot \cos \varepsilon\); therefore \((r_i \cdot \sin \varepsilon + l^{-})^2 = r_a^2 - (r_i \cdot \cos \varepsilon - \delta_p)^2\), and finally, for the length of the flat inflector, we have
\[ l^{-} = \sqrt{r_a^2 - (r_i \cdot \cos \varepsilon - \delta_p)^2} - r_i \cdot \sin \varepsilon . \tag{1} \]
For \(\varepsilon = 0\)
\[ l_0^{-} = \sqrt{r_a^2 - (r_i - \delta_p)^2}. \tag{2} \]
Fig. 1. a — cylindrical inflector, b — flat inflector, c — cylindrical inflector of reverse curvature; 1 — deflecting plate of the inflector, 2 — grounded plate of the inflector, 3 — injector anode, 4 — focusing electrode, 5 — cathode, 6 — tangent to the circle of injection radius, 7 — tangent to the circle of the mean radius of the inflector.
If \(\varphi\) is taken in radians, then from Fig. 1a
\[ \overset{\frown}{l}=r_c\cdot\varphi . \tag{3} \]
Let us find \(\varphi\).
By the law of cosines, from \(\triangle OAB\),
\[ r_a^2=OB^2+BA^2-2OB\cdot AB\cdot\cos(\varphi+\psi). \tag{4} \]
From \(\triangle OИB\),
\[ OB^2=r_i^2+r_c^2+2r_i\cdot r_c\cdot\cos\varepsilon . \tag{5} \]
From Fig. 1a,
\[ BA=r_c+\delta_{\mathrm{п}} . \tag{6} \]
From (4), (5), (6) it follows that
\[ \cos(\varphi+\psi)= \frac{r_i^2+r_c^2+2r_i r_c\cdot\cos\varepsilon+(r_c+\delta_{\mathrm{п}})^2-r_a^2} {2(r_c+\delta_{\mathrm{п}})\sqrt{r_i^2+r_c^2+2r_i r_c\cdot\cos\varepsilon}} =N^\circ . \tag{7} \]
or
\[ \varphi+\psi=\arccos N^\circ . \]
From \(\triangle OИB\),
\[ r_i^2=OB^2+r_c^2-2OB r_c\cdot\cos\psi . \]
Taking (5) into account,
\[ \cos\psi= \frac{r_c+r_i\cdot\cos\varepsilon} {\sqrt{r_i^2+r_c^2+2r_i r_c\cdot\cos\varepsilon}} =M^\circ . \tag{8} \]
Whence
\[ \psi=\arccos M^\circ . \]
Then
\[ \varphi=\arccos N^\circ-\arccos M^\circ , \tag{9} \]
and equation (3), taking (7), (8), (9) into account, takes the form:
\[ \overset{\frown}{l}=r_c\left[ \arccos \frac{r_i^2+r_c^2+2r_i r_c\cdot\cos\varepsilon+(r_c+\delta_{\mathrm{п}})^2-r_a^2} {2(r_c+\delta_{\mathrm{п}})\sqrt{r_i^2+r_c^2+2r_i r_c\cdot\cos\varepsilon}} - \arccos \frac{r_c+r_i\cdot\cos\varepsilon} {\sqrt{r_i^2+r_c^2+2r_i r_c\cdot\cos\varepsilon}} \right]. \tag{10} \]
In the special case, when \(\varepsilon=0\),
\[ \overset{\frown}{l}=r_c\arccos \frac{(r_c+r_i)^2+(r_c+\delta_{\mathrm{п}})^2-r_a^2} {2(r_c+r_i)(r_c+\delta_{\mathrm{п}})} . \tag{11} \]
From Fig. 1c,
\[ \overset{\frown}{l}=r_c\cdot\varphi . \tag{12} \]
Let us find \(\varphi\).
By the law of cosines,
\[ r_a^2=OB^2+BA^2-2OB\cdot BA\cdot\cos(\varphi+\psi), \tag{13} \]
but
\[ OB^2=r_c^2+r_i^2-2r_c\cdot r_i\cdot\cos\varepsilon \tag{14} \]
and
\[ BA=r_c-\delta_{\mathrm{п}} . \tag{15} \]
From (12), (13), (14), (15), after transformations analogous to the transformations in calculating the cylindrical inflector, we obtain:
\[ \overset{\frown}{l}=r_c\left[ \arccos \frac{r_c^2+r_i^2-2r_c r_i\cdot\cos\varepsilon+(r_c-\delta_{\mathrm{п}})^2-r_a^2} {2(r_c-\delta_{\mathrm{п}})\sqrt{r_c^2+r_i^2-2r_c\cdot r_i\cdot\cos\varepsilon}} - \arccos \frac{r_c-r_i\cdot\cos\varepsilon} {\sqrt{r_c^2+r_i^2-2r_c\cdot r_i\cdot\cos\varepsilon}} \right]. \tag{16} \]
In the special case, when \(\varepsilon=0\),
\[ l_o^{\smile}=r_c\cdot \arccos \frac{(r_c-r_i)^2+(r_c-\delta_{\mathrm{п}})^2-r_a^2} {2(r_c-r_i)(r_c-\delta_{\mathrm{п}})} . \tag{17} \]
Introduce the dimensionless quantities:
\[
r_{ai}=\frac{r_a}{r_i}
\]
— the reduced minimum radius of the anode position of the injector,
\[
r_{ci}=\frac{r_c}{r_i}
\]
— the reduced mean radius of the inflector,
\[
\delta_{\mathrm{п}i}=\frac{\delta_{\mathrm{п}}}{r_i}
\]
— the reduced radial dimension of the injector,
\[
l_i^{-}=\frac{l^{-}}{r_i}
\]
— the reduced length of the flat inflector,
\[
l_i^{\frown}=\frac{l^{\frown}}{r_i}
\]
— the reduced length of the cylindrical inflector,
\[
l_i^{\smile}=\frac{l^{\smile}}{r_i}
\]
— the reduced length of the cylindrical inflector of reverse curvature.
Then we obtain, respectively:
in the general case,
\[ l_i^{-}=\sqrt{r_{ai}^2-(\cos\varepsilon-\delta_{\mathrm{п}i})^2}-\sin\varepsilon, \tag{18} \]
\[ l_i^{\frown}=r_{ci}\left[ \arccos \frac{r_{ci}^2+2r_{ci}\cdot\cos\varepsilon+1+(r_{ci}+\delta_{\mathrm{п}i})^2-r_{ai}^2} {2(r_{ci}+\delta_{\mathrm{п}i})\sqrt{r_{ci}^2+2r_{ci}\cdot\cos\varepsilon+1}} - \arccos \frac{r_{ci}+\cos\varepsilon} {\sqrt{r_{ci}^2+2r_{ci}\cdot\cos\varepsilon+1}} \right], \tag{19} \]
\[ l_i^{\smile}=r_{ci}\left[ \arccos \frac{r_{ci}^2-2r_{ci}\cdot\cos\varepsilon+1+(r_{ci}-\delta_{\mathrm{п}i})^2-r_{ai}^2} {2(r_{ci}-\delta_{\mathrm{п}i})\sqrt{r_{ci}^2-2r_{ci}\cdot\cos\varepsilon+1}} - \arccos \frac{r_{ci}-\cos\varepsilon} {\sqrt{r_{ci}^2-2r_{ci}\cdot\cos\varepsilon+1}} \right], \tag{20} \]
In the special case, when \(\varepsilon=0\),
\[ l_{io}^{-}=\sqrt{r_{ai}^2-(1-\delta_{\mathrm{п}i})^2}, \tag{21} \]
\[ l_{io}^{\frown}=r_{ci}\cdot \arccos \frac{(r_{ci}+1)^2+(r_{ci}+\delta_{\mathrm{п}i})^2-r_{ai}^2} {2(r_{ci}+1)(r_{ci}+\delta_{\mathrm{п}i})}, \tag{22} \]
\[ l_{io}^{\smile}=r_{ci}\cdot \arccos \frac{(r_{ci}-1)^2+(r_{ci}-\delta_{\mathrm{п}i})^2-r_{ai}^2} {2(r_{ci}-1)(r_{ci}-\delta_{\mathrm{п}i})}. \tag{23} \]
From a comparison of the reduced lengths of the inflectors it is seen that, for given \(r_{ai}\), \(\delta_{\mathrm{п}i}\), the cylindrical inflector has the minimum length. However, its supply requires the maximum voltage. Under the same conditions, the cylindrical inflector of reverse curvature has the greatest length, but its supply requires the minimum voltage. On the basis of what has been said above, for comparatively low injection voltages, when the influence of space charge on the beam divergence is large, and the voltages on the inflector are easily attainable in absolute magnitude, it is possible to recommend for use
cylindrical inflector. At very high injection energies, when the role of space charge in the beam divergence decreases because of relativistic effects, while the required voltage on the inflector becomes difficult to attain in absolute magnitude, a cylindrical inflector of reverse curvature may be recommended for use.
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