Dynamic Properties of Dose-Rate Sensors for Gamma Radiation from Pulsed Sources

V. M. Razin

(Presented by the scientific seminar of the Department of Mathematical and Computing Instruments and Devices)

In systems for automatic regulation of the dose rate of betatron radiation, ionization chambers and photomultipliers with scintillators, or without them, are used predominantly as sensors measuring the magnitude of one of the principal output parameters of the betatron, characterizing its productivity. It would be most correct to measure the magnitude of the accelerated charge of the electrons at the end of each acceleration cycle. In this case, the influence of fluctuations of the electron energy at the moment of displacement on the properties of the sensors is completely excluded. However, the existing methods for measuring the accelerated charge have a number of serious shortcomings that hinder the use of these methods for practical purposes. The chief difficulty is created by the small value of the ratio of useful-signal magnitude to the noise level.

On the other hand, the use of ionization and scintillation methods is justified in those cases where the main factor is the ionizing capacity of the betatron rays. In the latter case, carrying out indirect relative measurements of the radiation dose rate is performed quite simply with a comparatively low noise level. Here it is only necessary to bear in mind that the radiation dose rate, as a function of energy for one and the same charge of accelerated electrons, varies in proportion to the third power of the energy. In order to reduce random fluctuations of the radiation intensity under such conditions, it is necessary to take all possible measures for precise stabilization of the energy level of the accelerated electrons at the moment of displacement.

It is known [1] that scintillation methods of recording high-energy radiation possess a whole series of advantages that have led to their wide application. In particular, these methods are the fastest-acting, practically inertia-free ones. The law of light emission by alkali-halide phosphors after excitation has, in the main, an exponential dependence

$$ I = I_0 e^{-\lambda t}. $$

The quantity \(1/\lambda\) is called the decay time.

In a scintillation sensor, the exponential character of the decay gives a corresponding exponential decrease of the current at the anode of the photomultiplier. This current is integrated by a capacitance with a time constant ...

of the time constant \(RC\), and the voltage on the capacitor is determined by the expression

\[ V=\frac{I_0R}{\lambda RC-1}\left(e^{-\frac{t}{RC}}-e^{-\lambda t}\right). \tag{1} \]

Usually \(RC\) is chosen to be large (about hundreds of \(\mu\)sec); then, for \(t\ll RC\),

\[ V\simeq \frac{I_0}{\lambda C}\left(1-e^{-\lambda t}\right). \]

The decay time of scintillators of various types lies in the range from \(10^{-9}\) to \(10^{-6}\) sec and more [1].

If the betatron radiation pulses have a long duration, of the order of units or tens of \(\mu\)sec, then by choosing a small value of the time constant \(RC\) one can obtain a practically inertia-free radiation dose-rate sensor, since the response speed of the photomultiplier itself is determined by a rise time of the order of \(10^{-9}\) sec and less. In this case the dynamic properties of the scintillation sensor will be represented by an amplifying inertia-free linear element. Here we do not touch on the question of the spectral sensitivity of the sensor under consideration, assuming that the radiation energy is stabilized.

The luminescence intensity \(I\) as a function of temperature can be determined by the relation

\[ \frac{I}{I_0}=\frac{1}{1+C_1 e^{-\frac{C_2}{T}}}, \tag{2} \]

where \(C_1\) and \(C_2\) are constants, and \(T\) is the absolute temperature.

Our experiments show that, in order to eliminate the influence of temperature on the properties of a sensor with a photomultiplier, operation of the photomultiplier without a scintillator can be practiced, placing the photomultiplier itself directly in the corresponding shielding devices, directly in a sufficiently intense beam of betatron gamma radiation. In this case the penetrating hard gamma radiation of the betatron knocks out from the photocathode and dynodes of the photomultiplier a sufficient number of electrons to form an intense voltage pulse at the photomultiplier anode, substantially exceeding in magnitude the level of interference and noise.

A significant drawback of photomultipliers is the strong dependence of the amplification properties on the voltage applied to its electrodes. The total gain of a photomultiplier is approximately proportional to a quantity \(\sim U^\kappa\), where \(U\) is the accelerating voltage between two dynodes, and \(\kappa\) is the number of dynodes.

Suppose that

\[ U=U_0\left(1+\frac{\Delta U}{U_0}\right), \]

where \(U_0\) is the mathematical expectation of the quantity \(U\), and \(\Delta U\) is a centered random variable characterized by the variance \(D[\Delta U]\) of the fluctuations of the supply-source voltage.

For

\[ \frac{\Delta U}{U_0}\ll 1 \]

we have

\[ U^\kappa\simeq U_0^\kappa\left(1+\kappa\frac{\Delta U}{U_0}\right). \]

Consequently, the variance of the centered random variable of the photomultiplier gain will exceed the variance of the random deviations of the photomultiplier supply voltage from the nominal value by a factor of \(\kappa^2\). Thus, for example, at \(\kappa=10\), \(\kappa^2=100\), and at \(\kappa=14\), \(\kappa^2=196\), i.e., the variance of the random deviations of the gain increases hundreds of times relative to the variance of the random deviations of the supply voltage. To reduce the influence of the supply voltage on the amplification properties of the photomultiplier, high-quality stabilization must be used.

Attention should also be drawn to the fact that, in many types of photomultipliers, accompanying pulses, or so-called afterpulses, are observed following the main working pulse. The amplitude of these pulses is usually considerably smaller than the amplitude of the main pulses, and they follow the main pulse after a time interval ranging from several nanoseconds to microseconds. In this case, gating must be used in order to select only the main working pulses. A somewhat more complex picture of the physical processes is observed in the ionization chamber used as a radiation dose-rate sensor [1]. Within a time of the order of \(10^{-7}\) sec after the passage of an ionizing particle through the gas of the ionization chamber, the energy of the liberated electrons decreases to several electron-volts. The liberated electrons begin to drift through the gas under the action of the applied electric field, with a velocity that depends on the magnitude of the field and on the composition and pressure of the gas. The positive ions arising during ionization move much more slowly than the free electrons. During their motion, the free electrons may be captured by positive ions, i.e., recombination is possible. In electronegative gases, electrons may be captured by neutral atoms. As a result, negative ions are formed, i.e., electron attachment takes place. Negative ions may give up an electron to positive ions, which is also recombination.

Let us examine in more detail the operation of an ionization chamber with electron collection. In such a chamber, the electron current \(j\) may be represented as the sum of two terms

\[ j=-D\,\operatorname{grad} n+nV, \tag{3} \]

where \(n\) is the electron density, \(D\) is the diffusion coefficient, and \(V\) is the drift velocity.

In what follows, to simplify the calculations, we shall assume that the electrode system of the ionization chamber is plane-parallel, that the ionization throughout the entire volume between the electrodes is isotropic, i.e., that the radiation dose rate is the same at all points of the working volume of the chamber, and that edge effects may be neglected. Furthermore, since only electron collection occurring over a short time interval is considered, in the first approximation we shall neglect electron losses due to electron attachment, recombination, and other processes. We shall assume the motion of the electrons released after ionization to be uniform, with a certain constant drift velocity \(V\). The influence of the moving charges on the potential difference between the chamber electrodes may also be neglected, because the potential of the collecting electrode changes only slightly in comparison with the total voltage on the electrodes.

Under the idealized conditions indicated above, the component of the electron current due to diffusion may be neglected, and it may be assumed that all

the electrons formed as a result of ionization are removed into the external electric circuit with a constant drift velocity.

For many gases the drift velocity can be determined by a dependence of the form

\[ V=f\left(\frac{E}{p}\right), \]

where

\[ \begin{aligned} E &\text{ is the electric-field strength, } \mathrm{V/cm},\\ P &\text{ is the gas pressure, } \mathrm{mm\ Hg}. \end{aligned} \]

From the data reported in the literature [1] it follows that, in many respects, a gas such as methane is suitable for filling the ionization chamber used as a radiation sensor for a betatron. A characteristic property of methane is the presence of a high drift velocity (on the order of \(10\ \mathrm{cm/\mu s}\)) and the practical constancy of this velocity over a wide range of variation of the parameter \(\frac{E}{P}\) (from 1.0 to \(1.4\ \mathrm{V/cm\cdot mm\ Hg}\), according to the graph in Fig. 8 in [1]). For air, in the range of values \(E/p\) from 1 to \(10\ \mathrm{V/cm\cdot mm\ Hg}\), there is [1] an empirical dependence \(V=[0.4(E/p)+0.9]\ \mathrm{cm/\mu s}\), etc.

The transfer function of an ionization chamber, as an element in an automatic-control system, can be found on the basis of the following considerations.

Let us denote by \(S\) the cross-sectional area between the electrodes, and by \(h\) the distance between them. Next, let us assume that the entire volume of the ionization chamber is subjected to ionizing gamma radiation in the form of a pulse of short duration and large amplitude, and that as a result of this, per unit time in each unit of the working volume of the chamber, a charge is formed

\[ q=\frac{Q}{Sh}, \]

where \(Q\) is the total charge formed per unit time as a result of the action of a delta pulse of radiation dose rate \(I(t)\).

We note that \(Q=F(t)S h\Delta t\delta_{\Delta}(t-\tau)=F(t)Sh\) and that

\[ I(t)=\int_{-\infty}^{+\infty} F(\tau)\delta(t-\tau)\,d\tau \quad \text{for } -\infty<t<\infty, \]

Then the current in the external circuit will be determined by the relation

\[ i(t)= \begin{cases} 0, & \text{for } t<0,\\[4pt] I_0\left(1-\dfrac{V_{(-)}}{h}t\right), & \text{for } 0\leq t\leq \dfrac{h}{V_{(-)}},\\[8pt] 0, & \text{for } t>\dfrac{h}{V_{(-)}}. \end{cases} \tag{4} \]

where \(I_0=qSV_{(-)}\) is the initial value of the current. Subsequently the current decreases linearly as the electrons depart to the anode of the ionization chamber.

Relation (4) is nothing other than the impulse transient function. It is known that the Laplace transform of the impulse transient function is the transfer function. Using the known results [2], we obtain an expression for the transfer function of an ionization chamber having plane-paral-

...plate electrodes and operating on the principle of electron collection, in the form

\[ W_n(p)=\left(\frac{V_{(-)}}{h}\right)^2 \frac{e^{-\frac{h}{V_{(-)}}p}+\frac{h}{V_{(-)}}p-1}{p^2}\,Sh. \tag{5} \]

The transfer function of the ionization chamber obtained by us determines the relationship between the ionization in the working volume and the current in the external circuit of the chamber.

Allowance for the component of the current due to ion collection may be made approximately by adding to the right-hand side of expression (5) an analogous component, but only with the ion drift velocity

\[ W_n(p)=\left[ \left(\frac{V_{(-)}}{h}\right)^2 \frac{e^{-\frac{h}{V_{(-)}}p}+\frac{h}{V_{(-)}}p-1}{p^2} + \left(\frac{V_{(+)}}{h}\right)^2 \frac{e^{-\frac{h}{V_{(+)}}p}+\frac{h}{V_{(+)}}p-1}{p^2} \right]Sh. \tag{6} \]

Let us note that the magnitude of the ion drift velocity \(V_n\) is approximately three orders of magnitude smaller than the electron drift velocity \(V\).

As a rule, in the external circuit of an ionization chamber there is an active resistance \(R\), which is the input or grid resistance of an electronic amplifier, shunted by a capacitance \(C\) between the chamber electrodes, the tube, and other elements of the amplifier input circuit.

In this case the output quantity is taken to be the voltage on the grid of the tube of the first stage of the amplifier.

The transfer function from the current in the external circuit of the ionization chamber to the voltage on the grid of the tube of the first stage of the amplifier will have the form

\[ W_c(p)=\frac{R}{1+pRC}. \tag{7} \]

Consequently, the complete transfer function from the input quantity, in the form of the radiation dose rate in the volume of the ionization chamber, to the output quantity, in the form of the voltage across the resistance in the external circuit of the chamber, may be represented by the relation

\[ \begin{aligned} W(p)&=W_n(p)W_c(p)= \left[ \left(\frac{V_{(-)}}{h}\right)^2 \frac{e^{-\frac{h}{V_{(-)}}p}+\frac{h}{V_{(-)}}p-1}{p^2} +\right.\\ &\qquad\left. +\left(\frac{V_{(+)}}{h}\right)^2 \frac{e^{-\frac{h}{V_{(+)}}p}+\frac{h}{V_{(+)}}p-1}{p^2} \right] \frac{R}{1+pCR}\cdot Sh. \end{aligned} \tag{8} \]

The expression we have obtained for the transfer function of an ionization chamber used as a sensor for measuring betatron radiation is very complex and cumbersome. In its general form it is difficult to use this expression in calculations. In connection with the foregoing, it is of interest to consider some characteristic operating modes of the ionization chamber that have practical application.

Let us consider the operation of an ionization chamber with electronic collection and a small time constant \(\tau = RC\) when recording radiation pulses from a betatron that are long in time. The design parameters of such an ionization chamber can be chosen so that the electron collection time will be considerably shorter than the duration of the radiation pulses. In this case the second term in parentheses may be neglected (ion collection is not taken into account), as may the quantity \(pCR \ll 1\). In addition, because of the smallness of the quantity \(-\frac{h}{V}p\), we use the series expansion

\[ e^{-\frac{h}{V}p} \simeq 1 - \frac{h}{V}p + \frac{1}{2}\left(\frac{h}{V}p\right)^2 . \tag{9} \]

On the basis of the above considerations we obtain

\[ W(p) \simeq \frac{1}{2} ShR . \]

From the relation obtained it is clear that, in the case under consideration, an ionization chamber with electronic collection is, in terms of its dynamic properties, a noninertial linear amplifying element. As noted above, with the appropriate design, the electron drift velocity \(V\) does not depend on, or depends only weakly on, the gas pressure and the voltage on the chamber electrodes. In this case it is possible to use the ionization chamber as a stable radiation sensor with a low noise level.

In the other case, when the duration of the radiation pulse is greater than the electron collection time but considerably less than the time constant \(\tau = RC\) of the external chamber circuit (the input circuit of the amplifier), the transfer function takes the form

\[ W(p) \simeq \frac{RSh}{pCR} = \frac{RSh}{pT}, \]

where \(T = RC\) is the time constant of the amplifier input circuit.

Consequently, for this relation of parameters, the radiation sensor is, in terms of its dynamic properties, an integrating element with respect to the electron-collection process.

In conclusion, we note that the results obtained make it possible to choose correctly the design parameters of a radiation sensor in the form of an ionization chamber and to specify operating modes corresponding to particular requirements.

Questions of amplifying the signal from the output of the ionization chamber are considered in detail in [1]. The same work indicates ways of minimizing the influence of various kinds of random noise of a fluctuation nature.

REFERENCES

1. Devices for Recording Nuclear Radiation and Their Applications. Edited by A. Snelta. Atomizdat, Moscow, 1965.

2. G. Doetsch. Guide to the Practical Application of the Laplace Transform. “Nauka” Publishing House, Moscow, 1965.