MEASUREMENT OF ACCELERATED CHARGE IN A BETATRON USING AN ELECTROSTATIC SIGNAL ELECTRODE

V. A. MOSKALEV, Yu. A. GROMOV

In a betatron, the accelerated charge is estimated from the signal from a target when accelerated electrons are dumped onto it [1]. However, the results of these measurements are of a very approximate character if the energy of the accelerated electrons exceeds 1 MeV, which naturally requires the use of more accurate methods of charge measurement. One such method may be the method of measuring charge by means of electrostatic signal electrodes (ESE), which has become widely used in the set of measuring equipment on “large” accelerators [2, 3, 4]. The main positive feature of the ESE consists in the possibility of accurately calculating the absolute sensitivity of the detector and in the independence of the measurements from the particle energy.

It was shown in the literature [5] that the charge in the electron beam of a betatron can be measured by means of an ESE only during the process of beam destruction. We shall take as this moment the final stage of betatron operation—the dumping of electrons onto the target—and shall study the operation of the detector under these conditions.

For measurement by means of an ESE it is necessary to have two electrodes. Proceeding from the simplicity of the design and from the condition that electrons do not strike the main electrode during the measurement process, we shall take as the first a flat electrode introduced into the accelerating chamber, and as the second the conducting coating of the chamber. In this case the radius of installation of the measuring electrode \(R_{\mathrm{e}}\) must be greater than the radius of installation of the target \(R_{\mathrm{m}}\). Let the dimensions of the electrode in azimuth be much smaller than the beam perimeter, and let the distance \((R_{\mathrm{e}} - R_{\mathrm{m}})\) be greater than the radial size of the beam; then the section of the chamber at the place where the electrode is installed may be represented as a straightened section, and the action of the electron beam on the electrode may be considered as the action of a thin charged filament [3]. In this case the “electrode—beam” system will take the form shown in Fig. 1.

In view of the action of the strong electromagnetic fields accompanying betatron operation when electrons are dumped onto the target, we shall use the induced-current measurement mode [7], in which the measuring electrode may be loaded by a small active resistance.

In determining the induced current we shall apply the fundamental Shockley–Ramo theorem [6, 7], according to which, when a charged

particles with charge \(q\), relative to an arbitrary system of electrodes, an induced current flows in the circuit of a specified electrode:

\[ i_{\mathrm{н}}(t)=q\bigl(\vec v\cdot\nabla\varphi(x,y,z)\bigr), \tag{1} \]

where

\(\vec v\) is the velocity vector of the particle’s motion;
\(\varphi(x,y,z)\) is the potential at the point where the particle is located, determined from the condition that the given electrode has the dimensionless potential \(+1\), while the remaining electrodes are grounded.

Such an interpretation of the method for finding \(i_{\mathrm{н}}(t)\) makes it possible to apply the usual calculation or modeling of the electric field created by the electrodes.

Extending the theorem to our particular case, we obtain

\[ i_{\mathrm{н}}(t)=\tau(x_0,t)\cdot\frac{\partial x_0}{\partial t}\cdot \frac{\partial \Phi(x_0)}{\partial x_0} +\frac{\partial \tau(x_0,t)}{\partial x_0}\cdot \frac{\partial x_0}{\partial t}\Phi(x_0) +\frac{\partial \tau(x_0,t)}{\partial t}\cdot\Phi(x_0), \tag{2a} \]

\[ \Phi(x_0)=\int_{-z_{\max}}^{+z_{\max}}\varphi(x_0,z)\,dz, \tag{2b} \]

where

\(\tau(x_0,t)\) is the linear charge density of the beam,
\(x_0\) is the coordinate of its center,
\(z_{\max}\) is the coordinate above which the edge effects of the electrode have practically no influence on the magnitude of the induced current.

Fig. 1. Geometry of the “electrode–beam” system in the accelerating chamber of a betatron

Let us express (2) in terms of \(R\), the radius of the beam orbit, \(N\), the number of particles in the beam, and the radial velocity \(v\) of the beam displacement:

\[ i_{\mathrm{н}}(t)=\frac{q_e}{2\pi R(t)} \left[ v\cdot N(t)\cdot\frac{\partial \Phi(R)}{\partial R} -\frac{1}{R(t)}\,v\cdot N(t)\cdot\Phi(R) +\frac{\partial N(t)}{\partial t}\cdot\Phi(R) \right], \tag{3} \]

(\(q_e\) is the electron charge).

In expression (3), the quantity

\[ \frac{\partial N(t)}{\partial t} \]

is the current pulse on the target. Therefore, on the oscillogram of the induced current there will be seen a spike caused by the current pulse on the target.

It is known that, when the beam moves toward the target, the number of particles in it remains constant, while in the process of its interaction with the target the number of particles decreases according to a law that most closely coincides with a Gaussian curve. Experimental evidence of this is the shape of the current pulse on the target when electrons are dumped onto it at the beginning of the acceleration cycle, i.e., at an energy \(<1\) MeV. Denoting the number of particles in the beam before the moment when the electrons are dumped onto the target by \(N_0\) (this quantity, in our case, is also to be determined) and taking the above into account, po-

we obtain the following expression for the induced current:

\[ i_{\mathrm{ind}}(t)= \frac{q_e N_0}{2\pi R(t)} \cdot \left\{ v\cdot \left[ 1-\frac{1}{\sqrt{1.4\pi}\,t_{0.5}} \int_{-\infty}^{t} \exp\left(\frac{-t^2}{1.4t_{0.5}^{\,2}}\right)\,dt \right] \times \right. \]
\[ \left. \times \left[ \frac{\partial \Phi(R)}{\partial R} - \frac{\Phi(R)}{R(t)} \right] - \frac{\Phi(R)}{\sqrt{1.4\pi}\,t_{0.5}} \exp\left(\frac{-t^2}{1.4t_{0.5}^{\,2}}\right) \right\}, \tag{4} \]

where

\(t_{0.5}\) is the half-duration of the current pulse to the target at the level of 0.5 of its amplitude.

Here the time origin \(t=0\) is taken at the moment when the current pulse to the target assumes its maximum value.

As follows from (4), in order to determine the desired quantity \(N_0\) it is necessary to measure \(i_{\mathrm{ind}}(t)\) for some fixed instant of time and to determine, for that instant, all the components entering into (4).

Investigation of expression (4) shows that the only measurement instant at which all the components of (4) can be determined is the instant \(t=0\). In this case \(R(t)\) is equal to \(R_{\mathrm{t}}\), the radius of target installation, and \(i_{\mathrm{ind}}(t)\) assumes the maximum value \(i_m\). The value \(t_{0.5}\) can be estimated from \(t_{\mathrm{cut},0.5}\), the duration of the cutoff of the induced-current pulse at the level \(0.5 i_m\), while the radial displacement velocity of the electron beam can be represented as

\[ v \simeq (R_{\mathrm{t}}-R_0)\cdot \omega \cdot \operatorname{ctg}\frac{\omega}{2}\,t_m, \tag{5} \]

where

\(R_0\) is the radius of the equilibrium orbit;
\(\omega\) is the angular frequency of the current in the displacement windings;
\(t_m\) is the time interval from the moment the displacement circuit is started to the moment at which the maximum of the current pulse to the target arises, i.e., \(i_m\) in the induced-current pulse.

On this basis we obtain the final formula for determining the number of particles in the beam from the induced current on the electrode:

\[ N_0= \frac{2\pi R_{\mathrm{t}} i_m} {q_e \left\{ \frac{1}{2}(R_{\mathrm{t}}-R_0)\omega\cdot \operatorname{ctg}\frac{\omega}{2}t_m \left[ \frac{\partial \Phi(R_{\mathrm{t}})}{\partial R} - \frac{\Phi(R_{\mathrm{t}})}{R_{\mathrm{t}}} \right] - \frac{\Phi(R_{\mathrm{t}})}{\sqrt{1.4\pi}\,t_{\mathrm{cut},0.5}} \right\}} . \tag{6} \]

The values \(\Phi(R_{\mathrm{t}})\) and \(\dfrac{\partial \Phi(R_{\mathrm{t}})}{\partial R}\) are determined in calculating or modeling the electric field of electrodes of a specific geometry in the accelerating chamber of the betatron.

Thus, by measuring the induced current arising in the circuit of the measuring electrode at the moment of interaction of the electron beam with the target, one can estimate, from formula (6), the number of accelerated particles or the charge.

For an experimental check of the proposed method for measuring accelerated charge, we did the following:

1. Taking into account the above recommendations, in the chamber of one of the operating betatrons (a high-current 15 MeV betatron) a round flat electrode of diameter 46 mm was installed. The electrode installation radius was \(R_{\mathrm{e}}=235\ \mathrm{mm}>R_{\mathrm{t}}=215\ \mathrm{mm}\).

According to the corresponding dimensions of the ESE and its surroundings, a three-dimensional model was constructed, in which the electric field of the electrode was modeled by the “electrolytic baths” method.

In determining \(\Phi(R)\), integration was replaced by summation:

\[ \Phi(R) \simeq \Delta z \cdot \left[\varphi(R,z_0) + 2 \sum_{i=1}^{n}\varphi(R,z_i)\right], \tag{7} \]

where

\(n\) is the number of elements into which the beam is divided along the \(Z\) axis;

\(\Delta z\) is the length of an element along the \(Z\) axis (taken to be \(0.5\ \mathrm{mm}\)).

Fig. 2. Functions \(\Phi(R)\) and \(\dfrac{d\Phi(R)}{dR}\)

Initially, the values \(\varphi(R,z)\) were measured, and then the functions \(\Phi(R)\) and \(\dfrac{\partial \Phi(R)}{\partial R}\) were calculated. From the results of the calculations the graph in Fig. 2 was constructed. The accuracy of the modeling of \(\Phi(R)\) was \(\simeq \pm 1\%\).

II. As the electronic device to which the measuring electrode must be connected for transmitting the signal to the measuring-

Fig. 3. Oscillogram of the induced-current pulse

to the recording device—the oscilloscope—a cathode follower was used. The value of \(i_m\) was measured from the voltage signal:

\[ U_m = K \cdot R_{\mathrm{n}} \cdot i_m, \tag{8} \]

where

\(K\) is the voltage transfer coefficient of the cathode follower;
\(R_{\mathrm{n}}\) is the load resistance for the electrode.

Taking into account the magnitude of the parasitic input capacitance and the condition \(\tau_{\mathrm{in}} \ll t_{\mathrm{p}}\)—the duration of the charge variation in the beam—the electrode was loaded with an equivalent resistance \(R_{\mathrm{n}} = 38.7\ \mathrm{k}\Omega\).

Previously, without capture of electrons into acceleration, the measuring system was adjusted to a minimum of interference (the recommendations of [8, 9, 10] were used), and then, with the electron beam present in the orbit, direct measurements of \(U_m\), \(t_{\mathrm{m}}\), \(t_{\mathrm{sr}.0.5}\) were made. With the betatron tuned to an accelerated-electron energy of \(9.5\ \mathrm{MeV}\), an injection energy of \(150\ \mathrm{keV}\), and readings of the “Kaktus” roentgenometer of \(30\ \frac{\mathrm{R}}{\mathrm{min}}\) per meter, the oscillogram shown in Fig. 3 was obtained.

Here \(U_m = 0.85\ \mathrm{V}\), \(t_{\mathrm{m}} = 25.5\ \mu\mathrm{s}\), \(t_{\mathrm{sr}.0.5} = 2.5\ \mu\mathrm{s}\).

III. From the graph in Fig. 2, for the specific parameters of the electrostatic signal electrode at \(R_{\mathrm{m}} = -215\ \mathrm{mm}\), the following were determined:

\[ \Phi(R_{\mathrm{m}}) = 26.78\ \mathrm{mm}, \qquad \frac{\partial \Phi(R_{\mathrm{m}})}{\partial R} = 0.79. \]

For the betatron under study, \(R_0 = 155\ \mathrm{mm}\), \(\omega = 0.042\ \frac{\mathrm{rad}}{\mu\mathrm{s}}\).

Substitution of these values and of the measured parameters of the induced-current pulse into formulas (6) and (8) gave \(N_0 \approx 6 \cdot 10^{10}\) electrons per cycle.

IV. Analysis of formula (6) for the specific parameters of the electrostatic signal electrode in the betatron showed that the principal error in estimating the accelerated charge is determined by the error in measuring the amplitude and duration of the voltage pulse with the aid of the oscilloscope; as a result, the accuracy of the estimate of \(N_0\) was \(\approx \pm 25\%\).

Thus, the experimental test shows that the proposed method can be recommended as one of the methods for measuring accelerated charge in a betatron.

REFERENCES

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