COUNTERS OF PARTICLES AND QUANTA*

D. R. Corson and R. R. Wilson

INTRODUCTION

This article is a review devoted to the mechanism of action and the special properties of commonly used counters. It is intended for those interested in counting technique but without independent experience in working with all the various types of counters. The description given here should acquaint the reader with the factors that determine the suitability of different counters for particular tasks. Lack of space has not allowed us to go into the details of many questions; we have completely omitted the description of methods of calibration and measurement. The bibliography is by no means exhaustive, and the reader who turns to any particular article will find a considerably larger amount of literature.

I. IONIZATION COUNTERS

General considerations

An ionization counter is an electrical device that measures the ionization produced by a particle as it passes between two electrodes. A simple counter of this type is shown in Fig. 1. The space between two flat electrodes, located at a distance of about 1 cm, may be filled with argon at atmospheric pressure. The collecting electrode is connected to the input of a high-gain amplifier; the other electrode has a negative potential of several hundred or thousand volts relative to the first.

Suppose that an $\alpha$-particle passes through the chamber, leaving behind a typical track of intense ionization consisting of electrons and positive argon ions, as shown in Fig. 1. The electrons will move in the direction of the collecting plate with an average velocity of about $10^6$ cm/sec, while the positive ions will move in the direction of the negative, or high-voltage,

* Dale R. Corson and Robert R. Wilson, Particle and quantum counters, Rev. Sci. Inst. 19, 207 (1948). Translated by B. Bagaryatsky.

electrode with a velocity of approximately \(10^4\ \text{cm/sec}\). Since the ions move in the field of the electrodes, charges are induced on the latter, and since the collecting electrode is insulated from ground by a large resistance, a voltage pulse arises on it. This voltage pulse has a very rapid initial rise, owing to the high velocity of motion of the electrons toward the collecting plate (anode), and after this—a considerably slower rise, caused by the motion of the slow positive ions away from the anode. The typical shape of the pulse is shown in Fig. 2. After all the ions have been collected, a charge \(ne\) of electrons will be present on the collecting plate, and the final magnitude of the pulse will be

\[ V_p=\frac{ne}{C}, \]

where \(C\) is the total capacitance of the collecting electrode and the input of the amplifier.

An \(\alpha\)-particle of polonium (5.3 MeV), when stopping in the chamber, can create approximately \(2\cdot 10^5\) ion pairs. If \(C\) is about \(30\ \mu\mu F\), then the magnitude of the pulse is approximately \(10^{-3}\) volt. In order to increase the pulse to a level of 10–100 volts, at which it can be observed on an oscillograph or fed to a voltage discriminator and then recorded by means of a scaling circuit and a mechanical counter, voltage amplification of \(10^4\)–\(10^5\) times is usually employed. The shape of the pulse after amplification will, of course, depend on the electrical characteristics of the amplifier. Depending on the purpose of the counter, the shape of the electrodes may differ considerably—

Fig. 1. Typical arrangement of a plane ionization counter. The plus and minus signs denote ions along the track of an \(\alpha\)-particle.

Fig. 2. Shape of the pulse caused by the motion of the ions formed, whose initial arrangement is given in Fig. 1. The rapid rise is due to the motion of electrons; the subsequent slower rise is caused by the motion of positive ions. Note that the time scale is distorted. The dashed curve shows the effect of electron diffusion.

differ from that shown in Fig. 1. Various gases may be used; the sign of the potential may be chosen arbitrarily, and the collecting electrode may be either at high voltage or at ground potential.

Pulse Shape

Let us consider how the potential of the collecting electrode changes with time after an ionizing particle has passed through the chamber. If the collecting electrode is connected to ground through a large resistance, we may regard it as electrically isolated. The energy of the electrode system is equal to \(\frac{1}{2} C V_0^2\), where \(C\) is the capacitance of the collecting electrode with respect to the high-voltage electrode, which has the initial potential \(V_0\) with respect to the former. Suppose that at \(x=x_0\), where \(x\) is measured along a line of force, an ion with charge \(e\) has been formed. Under the action of the field \(E\) it moves to the point \(x\), and a certain amount of work is expended on this. The energy of the system will now be

\[ \frac{1}{2} C V^2 = \frac{1}{2} C V_0^2 - \int_{x_0}^{x} eE\,dx, \tag{1} \]

where \(V\) is the new potential difference between the electrodes. This may be represented in the form

\[ V_c = \frac{1}{V_0 C}\int_{x_0}^{x} eE\,dx, \tag{2} \]

where \(V_c = V_0 - V\) is the pulse voltage, small in comparison with \(V_0\). The equation written above is applicable for any electrode geometry*). For the plane configuration shown in Fig. 1, \(E = V/l\), and (2) may be rewritten as

\[ V_c = \frac{e}{Cl}(x-x_0), \tag{3} \]

and since

\[ x = x_0 + \int_0^t v\,dt, \]

then

\[ V_c = \frac{evt}{Cl}, \tag{4} \]

where \(v\) is the velocity of the ion in the constant field \(E\).

*) If the collecting electrode is connected to an amplifier, it is not difficult to show that \(C\) in equation (2) is the total input capacitance of the circuit, i.e., of the counter, the leads, and the grid.

We can now describe more accurately the pulse caused by an α-particle moving parallel to the electrodes at \(x=x_0\), as shown in Fig. 1. Equation (4) shows that the pulse will increase linearly with time, at the rate

\[ \frac{ne v_-}{Cl}, \]

up to the voltage

\[ \frac{ne x_0}{Cl}, \]

which is given by equation (3) and is marked in Fig. 2. \(v_-\) is the mean velocity of the electrons, approximately equal to \(10^6\ \text{cm/sec}\). Thus, for \(x_0=1\ \text{cm}\), the duration of the sharp initial rise is about \(10^{-6}\ \text{sec}\). The motion of the slow positive ions is the cause of the further rise of the pulse at the rate

\[ \frac{ne v_+}{Cl}, \]

where \(v_+\) is the velocity of the positive ions. The pulse increases by an additional amount

\[ \frac{ne(l-x_0)}{Cl} \]

up to the final voltage \(ne/C\), which does not depend on the point at which the ions were formed. The velocity of positive ions*) in argon is about \(10^4\ \text{cm/sec}\); consequently, the pulse reaches its final magnitude in approximately \(10^{-4}\ \text{sec}\).

Electrons and ions do not all move with the mean velocity, and, moreover, they will diffuse from the mean position. This causes a distortion of the curve in Fig. 2, as shown by the dashed line. The phenomena of diffusion, as well as recombination and attachment of electrons, which we have also neglected, will be discussed later in one of the following sections of this article.

From the foregoing we can see that proportionality of the pulse magnitude to the number of ions formed can be obtained in the case when the time constant) of the amplifier \(RC\) is large in comparison with the collection time for positive ions, i.e. \(\sim 10^{-2}\ \text{sec}\). The initial rate of rise of the pulse is also proportional to the number of ions formed, and a considerably shorter pulse, whose magnitude will be proportional to the number of ions, can be obtained by differentiating the pulse of the ionization chamber in an amplifier with a very small time constant, \(\sim 2\cdot 10^{-8}\ \text{sec}\)*).

If the time constant of the amplifier is only slightly greater than the electron collection time, then only part of the pulse caused by the motion of the electrons will be observed. In this case, as can be seen from (3), the magnitude of the pulse will depend on the place in the chamber where the ions were formed. In this case, if it is desirable to obtain

*) The velocity of positive ions is given by the expression

\[ v_+ = K\frac{E}{p}, \]

where \(K/p\) is the mobility of the ions.

**) The time constant is often called the “cutoff time” of the amplifier.

***) In the section on amplifiers it will be explained how, with the aid of a shortened delay line, a more satisfactory differentiating circuit can be obtained.

large pulses, ionization must be produced so that the electrons traverse the maximum path in the field.

The shape of the pulse caused by the electrons can be calculated from the general distribution of the initial ionization in a chamber with plane parallel electrodes. The ionization may be characterized by the charge density \(\sigma(x)\,dx\), equal to the number of electrons formed between \(x\) and \(x+dx\). The variation of \(\sigma\) with \(x\) must, generally speaking, be determined by the energy, direction of motion, and specific ionization of the particle. The induced current \(di\) to the collecting electrode, caused by the charge element \(\sigma(x)\,dx\), may be obtained from (3) by differentiation. Thus, \(di=v_{-}\sigma dx/l\), and, consequently, the total current to the electrode at time \(t\) will be

\[ i=\frac{v_{-}}{l}\int_{0}^{l-vt}\sigma(x)\,dx. \tag{5} \]

The voltage pulse on the electrode will then be equal to

\[ V_c=\frac{v_{-}}{lC}\int_{0}^{t}\int_{0}^{l-vt}\sigma(x)\,dx\,dt, \tag{6} \]

where the time integral extends only up to \(l/v_{-}\), i.e. to the moment when all the electrons have been collected.

As an example, let us consider a high-energy proton traversing the chamber almost without slowing down, so that its specific ionization does not change. Then \(\sigma(x)\) is constant, and (6), after integration, gives

\[ V_c=\frac{v_{-}\sigma}{C}\left(t-\frac{1}{2}\frac{v_{-}t^2}{l}\right). \tag{7} \]

At \(t=l/v_{-}\) the pulse increases to the value \(\frac{1}{2}\frac{\sigma l}{C}\). The remainder of the pulse, caused by the motion of the positive ions, may also be obtained from (6), using the appropriate \(v\), and must be added if the time constant of the amplifier is large. In the example given, the pulse continues to grow slowly from \(\frac{1}{2}\frac{\sigma l}{C}\) to \(\frac{\sigma l}{C}\) at \(t=l/v_{+}\), the form of the rise being given by equation (7).

Sherr and Peterson\(^1\) have shown how, by carefully measuring the shapes of pulses caused by ionizing particles which end their paths in the chamber, one can obtain information on the energy of the particles, their ranges, the angles between the directions of the tracks and the direction of the field in the chamber, and on the range–energy relation for the particles.

To determine the rate of rise of the pulse for a configuration other than plane, we must return to equation (2), which in a more convenient form may be rewritten as:

\[ V_c=\frac{V_i e}{V_0 C}. \tag{8} \]

Here \(V_i\) denotes the potential difference between the initial point of formation of the ion and the point to which the ion has moved. One of the often used configurations is concentric cylinders, in the limiting case consisting of a cylinder with a thin wire along the axis. Since the greater part of the potential drop occurs in the region near the wire, we can see from (8) that the ions collected on the wire will produce a pulse whose final magnitude will be equal to \(ne/C\), where \(n\) is the number of ions collected by the wire, which, together with the amplifier input, has capacitance \(C\). The shape of the pulse will depend on the initial distribution of the ions in the cylinder. In general, the electrons are collected at the wire, and several tenths of a microsecond pass before they all move into the region of the strong field near the wire.

Fig. 3. Modification of the simplest ionization counter. A grid of thin parallel wires shields the collecting electrode from the moving ions formed in the main volume of the chamber. As a result, the magnitude of the pulse is proportional to the charge of the electrons arriving at the grid.

A useful modification of the simple ionization chamber with plane-parallel electrodes, making special use of electron collection, is illustrated in Fig. 3. A grid of thin parallel wires shields the collecting electrode from the moving ions formed in the main volume of the chamber, and therefore the magnitude of the pulse is proportional to the charge arriving at the grid, i.e. \(ne/C\), where \(C\) is the capacitance of the system of collecting electrodes. The field between the grid and the collecting plate is usually made somewhat stronger than in the rest of the chamber, so that only rare ions are captured by the grid. Here again, careful measurement of the pulse shape can provide information about the distribution of ionization along the track of the ionizing particle, as was shown by Alfvén\(^2\) and Lewis\(^3\).

Counting gases

Before 1940 the amplifiers available to physicists for use with ionization counters were too slow to observe the rapid initial rise of the pulse caused by the motion of the electrons. Attention was directed to obtaining

pulses exactly proportional to the initial ionization, and therefore large time constants were used in the amplifiers. Under these conditions almost any gas can be used; it is only necessary to apply a sufficiently high voltage between the electrodes so that the effects of recombination and diffusion can be neglected. Usually the voltage on the chamber is raised until the pulses cease to change with a further increase of voltage, i.e. saturation is reached. For air at atmospheric pressure, in order to reach saturation, a field of several hundred volts per centimeter is required. If the path of the ionizing particle is parallel to the electric field, then a gradient approximately twice as large is necessary. This occurs, roughly speaking, as a result of recombination processes in the column, the theory of which was developed in detail by Jaffé.^4 His theory shows that the field strength required for saturation increases rapidly with increasing gas pressure. Considerably smaller field values^5–7 are needed to attain saturation in very pure gases, such as hydrogen, nitrogen, and in inert gases, in which negative ions are not formed*) by electron attachment. Clay and Marshall report that in purified argon at a pressure of the order of seven atmospheres a field of only 70 volts/cm is sufficient to achieve saturation.

Electron attachment will seriously distort the pulse shape if the mean free path for attachment is comparable with, or less than, the distance between the electrodes. Therefore electronegative gases, such as O₂ or Cl, as well as most complex molecular impurities, should be avoided.

If one is interested in fast counting or in good resolving time, as in coincidence counting, preference must be given to a rapid initial rise of the pulse caused by electron motion. This acquired practical importance when faster electron tubes were developed for television and radar amplifiers. At present it is possible to construct an amplifier that will amplify a pulse rising faster than 0.05 μsec. With such an amplifier the electron pulse from an ionization chamber can be amplified without distortion.

The rate of rise of the pulse is proportional to the electron velocity; therefore, if one wishes to obtain good resolving time, a gas having high electron mobility must be chosen. The electron velocity is a function of \(\frac{E}{p}\), and according to the data of Bradbury and Nielsen,^8,9 it is approximately \(10^6\) cm/sec for most gases in the region \(\frac{E}{p}\) from 1 to 3 volts/cm per mm Hg, and

*) Jaffé’s theory is probably applicable only to electronegative gases.

increases to a value of approximately \(10^7\) cm/sec at \(\dfrac{E}{p} \sim 20\), when spark breakdown sets in.

Simple theoretical considerations show that the electron drift velocity is given by the expression

\[ v_-=\frac{eE\lambda}{mc}, \tag{9} \]

where \(c\) is the mean velocity of motion of the electrons, and \(\lambda\) is the mean free path. From this equation we see that the drift velocity is the greater, the greater the mean free path and the smaller the energy of chaotic motion. When an electric field is applied, the energy of the chaotic motion of the electrons increases until the energy gained from the field is balanced by losses in inelastic collisions. For most gases the mean free path decreases with increasing electron energy (the Ramsauer effect). Both these effects tend to keep the electron velocity at a low level as \(E\) increases.

Allen and Rossi\(^{10}\) showed that the drift velocity of electrons in argon can be increased by adding a small amount of \(\mathrm{CO}_2\). They found in pure argon a constant drift velocity equal to \(1.0\cdot10^6\) cm/sec for \(\dfrac{E}{p} > \dfrac{1}{2}\). When \(10\%\ \mathrm{CO}_2\) was added to the argon, the drift velocity increased to \(7\cdot10^6\) cm/sec. The first excitation level of argon is 11.6 eV; consequently, the energy of chaotic motion of the electrons in pure argon is approximately 10 eV. In \(\mathrm{CO}_2\) there are many low excitation levels (about 1 eV), so that in a mixture of argon and \(\mathrm{CO}_2\) the energy of chaotic motion of the electrons is lowered, owing to inelastic collisions, to approximately one electron-volt, and the drift velocity increases, as follows from (9). The drift velocity turns out to be approximately what could have been obtained at the partial pressure of argon alone. This effect can also be observed in other inert gases, and also in non-electronegative gases, when small amounts of polyatomic gases such as alcohol vapor or methane are added to them.

Precautions must be taken so that the attachment coefficient of electrons of the polyatomic gas is not too high. Thus, if \(\mathrm{CO}_2\) is added to argon at high pressure, the electrons attach to \(\mathrm{CO}_2\) before they have time to cross the chamber. At such pressures it is necessary to use pure, purified argon alone and to be satisfied with a low collection velocity. At normal pressure the contamination of ordinary cylinder argon is sufficiently great for electron attachment to be a substantial factor, except in only the strongest fields. The addition of from three to five percent \(\mathrm{CO}_2\) to technical argon reduces the attachment effects due to impurities to a negligible value. \(\mathrm{CO}_2\) probably has a twofold effect. First, the mobility of the electrons increases,

therefore, the electron has less time in which to be captured. Secondly, the probability of attachment for many complex gases decreases with decreasing electron energy, owing to CO₂.

In general, strong ionization in the gas is desirable. It can be obtained by using argon or krypton. These gases can be purified*; alternatively, as a simpler method, a few percent of CO₂ may be added to them. If good time resolution is necessary, about 10% CO₂ should be added. For neutron counting, good gases are hydrogen or hydrogen plus methane. For counting slow neutrons, very pure BF₃ is used; Allen and Rossi¹⁰, who measured the drift velocity in it, found it equal to \(7 \cdot 10^6\) cm/sec per mm Hg. If necessary, air also is quite satisfactory, in particular if strong fields are used.

Fig. 4. Action of the amplifier. The heavy dashed curve represents the step-like input pulse. The solid curve represents the pulse at the output of the amplifier. The delay time and rise time of the amplifier are indicated.

Amplifier

As could already be seen, the amplifier is an integral part of an ionization counter. Ideally, the amplifier should reproduce exactly the course of the voltage on the collecting electrode, but at the level of 10 to 100 volts needed for the operation of an oscilloscope or discriminator. This requires voltage amplification by \(10^4\)—\(10^6\) times, depending on the initial magnitude of the ionization. The amplifier, moreover, may possess special characteristics for giving the pulse the required shape, and also for limiting it in duration or in magnitude.

For our purposes the electrical characteristics of the amplifier can best be expressed in terms of the output voltage that is obtained when a wave with a rectangular front is fed to the amplifier input. Fig. 4 shows the input (dashed curve) and output voltages (solid curve) of a typical amplifier as functions of time. The step wave at the input is applied at the instant \(t = 0\), and after some time the voltage at the output rises to a maximum and then slowly falls. As shown in Fig. 4,

* A purifier with hot calcium, described by Entchke and Pranklem in Phys. Zeits. 40, 706 (1939), is satisfactory for most gases.

“rise time” of an amplifier is defined as the interval between the points of intersection that the tangent to the output-voltage curve (taken at the point of 0.5 of the voltage maximum) forms with the abscissa axis and with the line of maximum voltage. The delay time of an amplifier is defined as the interval between the arrival of the pulse and the point of intersection of the indicated tangent with the time axis. At the end, the pulse falls with a time constant or “cutoff time,” which are usually determined by the smallest value of the quantity \(RC\) in the coupling circuits of the amplifier system*).

If it is desired to obtain a short pulse, a small \(RC\) is used in one of the transition circuits. It is still better to use a delay line with one grounded end, as will be described below. In this case the output pulse will fall almost rectangularly to zero after twice the delay time of the line. If the magnitude or shape of the pulse is to be determined, then it is also important to preserve linearity, or proportionality, between the input and output voltages. The aim of this article is not to enter into the theory of video amplifiers\(^{11, 3}\)**). If the amplifier is to be used as part of an ionization counter, it is assumed that all measurements—of rise time, cutoff time, gain factor, linearity, and delay times—are made using rectangular pulses\(^{13}\) at the input and using an oscilloscope or synchroscope at the output.

The electrical characteristics required of an amplifier depend essentially on its application. In collecting positive ions, or in “slow counting,” there is no need for the rise time to be less than \(10^{-4}\) sec, but the linearity of the amplifier must be good, and the cutoff time large (more than \(10^{-2}\) sec), so that the pulse magnitude is accurately proportional to the total ionization. An amplifier well stabilized by feedback is desirable. Several such circuits have been published in the literature\(^{3, 12, 14—16}\).

Collection of electrons is used in “fast counting.” Here one may distinguish two general methods of use. In the first, the pulse serves only as a time indicator, as, for example, in counting coincidences or in measuring time of flight. In this case one is interested in the smallest rise time and the smallest delay time, and one may safely neglect stability and linearity for the sake of speed or simplicity of the circuit. In the second method the principal consideration is the accurate determination of the magnitude and shape of the pulse.

*) The rise time is related in the following way to the upper frequency \(f_2\), whose power is cut by the amplifier by half: \(t_r \sim 1/3 f_2\). Similarly, the cutoff time is expressed as \(t_c = 1/2\pi f_1\), where \(f_1\) is the lower frequency whose power is reduced by half. The delay time depends on the number of cascades and distributed capacitances, as well as on the constants of the circuit and the tubes.

**) For an excellent review by W. K. Elmore on this question, see\(^{12}\).

Here the requirement of maximum speed must give way to the requirements of stability and linearity. The Los Alamos amplifier, model 50117, has a rise time of about 0.1 μsec and is well stabilized by feedback. The amplifier circuit recently published by Jordan and Bell^15 is almost the same, but has the additional advantage of an easy possibility of changing from a fast to a slow amplifier. If a shorter rise time is necessary, stabilizing feedback may be neglected, and the amplifier should be constructed in the simplest way, using a minimum of anode resistances^*), peaking inductances, etc. By using modern tubes (6AK5, 6AC7, 6AG7, etc.), it is possible to make an amplifier with a rise time of several units of \(10^{-8}\) sec and with a comparable delay time. Amplifiers using new traveling-wave tubes^18 should find application here. If they can be used, the resolving time will fall to \(10^{-9}\) sec. Philips EE50 and EE51 secondary-emission tubes, having transconductances of 14,000 and 28,000 micromhos respectively, should also be suitable for the construction of very fast amplifiers.

Fig. 5. Shorted delay line for giving a pulse the required shape. \(R_c\) is the characteristic impedance of the delay line.

At a high counting rate or when high resolving power is required, it is necessary, as far as possible, to reduce the pulse length. The simplest method is to choose very small values of \(RC\) in the coupling circuits, in order to clip the pulse. To avoid a back wave, the small time constant must be used in only one stage, preferably near the input. A more elegant clipping circuit, introduced into counting technique by O. R. Frisch, uses a short delay line with one grounded end, as shown in Fig. 5. A step pulse applied to the upper end propagates downward along the line, is reflected from the short-circuited end with the opposite sign, and cancels the original signal at the upper end after a time equal to twice the delay time of the line. The resulting pulse has a rectangular form, which is more desirable than the exponential pulse obtained when clipping by means of \(RC\), since it avoids long tails. The latter, when superposed, may at a high counting rate or with a large ionization background give rise to false pulses. For the operation of discriminators, a pulse with a rect—

^*) To obtain maximum speed, the optimum gain per stage is the value \(e^{1/2}\).

with an angular vertex from the delay line is more advantageous than an overhang on the front of an exponential curve. If the input signal is not rectangular in shape, but changes in magnitude during the delay time \(t_d\) of the line, the result will be a differentiated signal. Obviously, the output signal \(\Delta v\) at any instant \(t\) is expressed as

\[ \Delta v(t)=v(t)-v(t-2t_d)=\frac{dv}{dt}\Delta t, \tag{10} \]

where \(\Delta t=2t_d\). It is not difficult to construct a compact delay line for delays of several microseconds, with a rise time of less than 0.1 microsecond. Pieces of rigid coaxial cable with a shielded conductor (9UPG) give excellent results for short delays (15 meters for \(0.05\ \mu\mathrm{sec}\)).

A very substantial advantage of applying electronic collection and clipping of the pulse is the fact that, in this case, the amplifier output is almost completely freed from the microphonic effect, which causes much trouble in the collection of positive ions.

Noise

The gain limit inherent in every electronic amplifier is determined by fluctuations\(^{19–20}\) of currents in the input stage, caused by the disordered emission of electrons by the cathode (shot effect) and by thermal effects in resistances (Johnson effect\(^{21}\)). Spurious signals may also arise owing to defective parts and connections, the microphonic effect, random pickups, the flicker effect as a result of random changes on the cathode surface, and positive ion grid current caused by ionization of the gas in tubes. However, these effects can be reduced to any desired extent and will be discussed later.

The mean square voltage due to the shot effect is

\[ V_s^2=4kTR_{eq}(f_2-f_1), \tag{11} \]

where \(f_1\) and \(f_2\) are the lower and upper limits of the frequency band passed by the amplifier, \(4kT=1.6\cdot10^{-20}\) joule (for \(T=300^\circ\mathrm{K}\)), and \(R_{eq}\) is the “equivalent noise resistance”* which, when substituted into

\[ \text{*) Harris gives the following formula for determining } R_{eq} \text{ of triodes:} \]

\[ R_{eq}=2.5/g_m. \]

For pentodes:

\[ R_{eq}=\frac{i_p}{i_p+i_s}\cdot\left(\frac{2.5}{g_m}+\frac{20i_s}{g_m^2}\right), \]

where \(i_p\) and \(i_s\) are respectively the mean anode current and screen-grid current, and \(g_m\) is the slope of the grid-anode characteristic.

in place of the input, would give the very same noise, but due to the Johnson effect. For most tubes the rms value of the voltage of the shot effect is about ten microvolts.

The mean square of the voltage produced by the Johnson effect in a resistance \(R\), shunted by a capacitance \(C\), is equal to \(^{21}\)

\[ V_j^2=\frac{2kT}{\pi C}\operatorname{arctg}\frac{2\pi RC(f_2-f_1)}{1+4\pi^2R^2C^2f_1f_2}. \tag{12} \]

The noise from this effect will be negligible in comparison with the noise of the shot effect if \(RC \gg 1/2\pi f_1\). Thus, \(RC\) must be made considerably larger than the clipping time of the amplifier. With \(RC\) chosen in such a way that it lies midway between the clipping time and the rise time, a considerable noise signal can be obtained. This is sometimes useful for a rough estimate of the gain of the amplifier, since the mean-square voltage of the signal can be calculated from (12).

The first tube of the amplifier also produces false signals owing to the flicker effect, which is due to random changes in the surface of the cathode. This effect is most noticeable at low frequencies (below 5000 cycles), but it can introduce only a very small false count, even if a short clipping time is used. The effect varies from tube to tube and decreases toward the end of the service life of the tubes. Tubes can be artificially aged by applying to them, for one day, a filament power twice the normal value. Another cause of noise is grid current, caused by positive ionization in the first tube. This effect can be eliminated by selecting tubes that show a small grid current (and, consequently, have a good vacuum).

The 6AK5 tube, used as a triode, apparently is suitable as a tube for the input stage \(^{22}\).

The rise time and the clipping time should be chosen in accordance with the requirements imposed on the amplifier, in such a way as to obtain the maximum signal-to-noise ratio. If importance is attached to the pulse shape, then the rise time of the amplifier should be only slightly less than the rise time of the signal, while the clipping time should be made as large as possible, without permitting superposition of pulses. Elmore \(^{12}\) calculated that for \(t_c/t_r=50\) the signal-to-noise ratio is only 5% lower than for \(t_c/t_r=\infty\), while for \(t_c/t_r=5\) it becomes lower by approximately 20%.

Often the pulse signal is taken to correspond approximately to a step wave, and it is considered sufficient to measure the magnitude of the signal. Then, for a given width of the output pulse, the rise and clipping times must be made equal in order to obtain the maximum signal-to-noise ratio. (Under optimum conditions

noise corresponds to a few hundred elementary charges on the input capacitance.) Van Heerden^23 showed that the optimum pulse length^24,25 is approximately \(10^{-5}\) sec. The amplifier is then used as a ballistic device, and the magnitude of the output pulse is proportional to the magnitude of the input pulse, provided only that the duration of the input pulse is small in comparison with the duration of the output pulse. Unfortunately, this optimum falls in the region of the strongest microphonic noises, and it is more expedient to sacrifice somewhat the signal-to-noise ratio in order to work with shorter pulses.

Details of Construction and Operation

The typical geometry of an ionization chamber is shown in Fig. 1. The high-voltage electrode is mounted on insulators, and many materials are equally suitable: quartz, glass, porcelain, mica, sulfur, lucite, polystyrene, etc. Kovar-glass insulators are manufactured in all possible shapes and sizes and are ideal for securing electrodes and leading wires through the chamber walls. When soldering insulators into place it is better to use acid-free solder. The guard ring shown in Fig. 1 is not obligatory and may be omitted. It limits the working volume and reduces to a minimum pulses on the insulators.

The input capacitance of the amplifier is made as small as possible. To reduce the capacitance of the wires between the collector and the first tube, the first stages of the amplifier, i.e., the “preamplifier,” are usually mounted as close as possible to the counter. The simplest solution is a circuit with a cathode output at the counter chamber, feeding the cable leading to the amplifier. The cathode resistor is placed at the amplifier end of the cable and is chosen of such a value as to terminate the line appropriately. In the preamplifier it is usually desirable to have a gain factor of about 50, so that stray pulses or radio-frequency pickups in the cable are relatively smaller than the signal. The preamplifier and the cable must be well shielded. This means that the counter, the preamplifier, the cable, and the amplifier input must be surrounded by a continuous metal shield. It is better to make several ground connections and to be certain that the external shield nowhere forms a closed loop in which electromagnetic pulses could be induced. This is important, in particular, when working near such an accelerator as a cyclotron. Stray pulses may also enter the preamplifier from the 110-V alternating-current mains through the filament leads. This source of extraneous pulses can be eliminated by careful grounding and by using filters at the input of the alternating-current line. The best way of isolating the amplifier from the alternating-current mains is...

current is the use of transformers having an electrostatic shield between the primary and secondary windings. Care must be taken when connecting the amplifier to an ordinary “ground,” such as water pipes, since such grounding may accidentally prove more harmful than useful.

One of the most important advantages of using electronic collection is that the cutoff time can be made so short (~ one microsecond) that most microphone noise and filament-background noise is not passed by the amplifier. If the cutoff time is large, as in the collection of positive ions, then the input part of the circuit should be made, as far as possible, non-microphonic. This can sometimes be achieved by using grids in the counting chamber instead of plates, by increasing the volume as much as possible, and by increasing the rigidity of the construction. The use of lead plates and wires is useful because of their inelastic properties. With a large cutoff time, direct current should be used in the filament preamplifier. In any case, it is reasonable to ground the filament leg of the tube nearest to the grid leg of the input tube (thus, to ground the third leg in the 6AK5 tube). The cutoff-time constant or delay line is usually arranged in the first stage after the preamplifier, so that noise or microphonic pulses at low frequencies do not overload the subsequent tubes.

Fig. 6. A typical ionization counter in which the pulse is taken from the high-voltage electrode. \(R_2 C_2\) must be greater than \(R_1 C_1\).

The pulse may be taken from any electrode of the ionization chamber, and it is often desirable or even necessary to take it from the high-voltage electrode. This is useful, in particular, when the counter must be compact or located at some distance from the preamplifier, for example in a chain reactor.

In this case only one wire is required. Such a chamber is shown in Fig. 6. The capacitors should be chosen carefully, taking into account the possibility of false pulses due to their breakdown by high voltage. Mica capacitors of good quality, operating at a voltage equal to one half of their rated value, are sufficiently satisfactory. The input time constant \(R_1 C_0\), where \(C_0\) is the capacitance of the chamber, must be taken greater than the cutoff time of the amplifier, and \(R_2 C_2\) much greater than \(R_1 C_0\). The capacitance \(C_1\)

must be large in comparison with \(C_0\). The polarity of the voltage on the high-voltage electrode is chosen either so as to obtain a definite sign of the pulse, or, if this is not essential, so as to have the maximum electron-collection distance in the counter.

II. PROPORTIONAL COUNTERS

General considerations

A proportional counter is an ionization counter in which the initial ions are multiplied owing to collisions in a region of strong electric field. Such counters are used in cases where the initial ionization is so small that it cannot be measured with ionization counters, or when it is desirable, for purposes of simplification, to dispense wholly or partly with an electronic amplifier. Historically, proportional counters were the first electrical detectors to be used. Their geometry, in its most usual form, reduces to an axial wire (several tenths of a millimeter in diameter) placed inside a metal cylinder (several centimeters in diameter). The wire is charged positively with respect to the cylinder. An arrangement consisting of a miniature sphere placed opposite a plane electrode is called a point counter. Various other electrode configurations are also possible, including parallel planes. Almost all of the usual gases are used, at pressures ranging from several mm Hg to many atmospheres.

Gas amplification

Let us suppose that an ionizing particle produces \(n_0\) electrons in the gas of a cylindrical proportional counter. Owing to the weak field \((E \sim 1/r)\), which exists almost throughout the entire volume of the counter, the electrons will begin to move in the direction of the wire until they reach its immediate vicinity. In the very strong field near the wire, the primary electrons will, by collisions with gas molecules, produce new electrons, causing the appearance of a typical avalanche. For each primary electron there will be \(m\) electrons reaching the wire; we shall call the quantity \(m\) the gas amplification factor (multiplication factor). It is necessary to note here that the voltage pulse arises on the central wire as a result of the motion of the positive ions formed in the avalanche and moving in the direction away from the wire. This is explained by the fact that the electrons are formed too close to the wire (at a distance of several mean free paths) to cause any appreciable induced poten-

cial (cf. equation (1)). Rose and Korff\(^{26,27}\) investigated the multiplication process in detail and gave the following equation for the coefficient of gas amplification:

\[ m=\exp 2\left(\frac{3NaV_0}{\ln \dfrac{b}{a}}\right)^{\frac12} \left[\left(\frac{V_0}{V_t}\right)^{\frac12}-1\right], \tag{13} \]

where \(N\) is the number of molecules per unit volume, \(V_0\) is the voltage between the electrodes, \(V_t\) is the threshold voltage, or the voltage at which the multiplication coefficient begins to exceed unity, \(\beta\) characterizes the rate at which the cross section for ionization of molecules by collisions with electrons increases as a function of energy, and \(a\) and \(b\) are the radii of the wire and of the counter cylinder.

The approximate expression (13) is based on the assumptions that a) the photoelectric effect at the cathode may be neglected, b) secondary electron emission by positive ions at the cathode may be disregarded, c) recombination and attachment of electrons may be neglected, and d) fluctuations in the avalanche process may be neglected. The value of \(\beta\) was measured by Tate and Smith\(^{28,29*}\).

Rose and Korff experimentally verified equation (13) for large values of \(m\) (from \(10^2\) to \(10^4\)) by observing the magnitude of the pulses from particles in a counter in which the voltage, wire diameter, pressure, and kind of gas were varied. Rose and Ramsey\(^{30}\) extended the measurements to low values of \(m\) and found good agreement with (13) for values of \(m\) greater than 10, and deviations from it for lower values. They found rather a gradual increase of \(m\) with voltage than the abrupt break predicted by (13). This occurs mainly because the theory neglects fluctuations in the avalanche process. In light of this, the threshold voltage \(V_t\) in (13) may be interpreted as that which is obtained by extrapolating the rectilinear part of the curve of \(\log m\) as a function of \(V\).

Deviations at large \(m\) were found by Rose and Korff in the case of gases, chiefly simple in character, such as Ar, Ne, O\(_2\), H\(_2\), etc. For such gases the gas amplification at high voltages increases considerably faster than is indicated by expression (13). Therefore, in order to obtain large and stable gas amplification, they recommend using gas mixtures rich in polyatomic components, such as ethyl alcohol, methane, or boron trifluoride. The action of a polyatomic molecule consists in suppressing the liberation of electrons at the cathode under the influence of ultraviolet photons or positive-ion bombardment. This effect will be discussed in more detail in the chapter on Geiger counters.

* Typical values of \(\beta\) for Ar, He, H\(_2\), O\(_2\), and N\(_2\) are, respectively, \(1.81,\ 0.11,\ 0.46,\ 0.66,\) and \(0.70\cdot10^{-17}\ \text{cm}^2/\text{volt}\).

If the amplification is very large, a sufficient number of ions may be formed for the field near the filament to be distorted by the space charge. The electrostatic charge on the filament, determined by its capacitance, corresponds to approximately \(10^9\) electrons per centimeter of filament length. The electrons in the gas have time to diffuse in the longitudinal direction by about \(1\) mm before they reach the filament; consequently, the field near the filament will be distorted when approximately \(10^7\) ions are produced in the avalanche (\(10^8\) ions per centimeter). Thus, if the amplification factor is \(10^4\), then pulses caused by fewer than \(10^3\) initial ions will be amplified proportionally, while those caused by more than \(10^3\) initial ions will be weaker than the calculated value because of the weakening of the field due to the space charge. The “region of limited proportionality” extends from the counter voltages at which the strongest pulses are distorted up to the threshold of the Geiger-counting region, where the magnitude of the pulse does not depend on the magnitude of the initial ionization. Counters may be used in the region of limited proportionality if only rough discrimination is required, as, for example, of \(\alpha\)-particles from \(\beta\)-particles, and a high gas-amplification factor is required. Montgomeri and Montgomeri\(^{31}\) found that a large background ionization, such as that produced by X-rays or \(\gamma\)-rays, can reduce the magnitude of the pulse, for example, from an \(\alpha\)-particle. One can estimate the current required in order substantially to lower the field at the filament (bearing in mind that approximately \(10^8\) ions per centimeter are required). It is approximately, per centimeter of filament length:

\[ i = 10^{-13}\,\frac{K}{P}\,\frac{Ve}{a\log\frac{b}{a}}, \]

where \(\frac{K}{P}\) is the mobility of the ions. From this we may calculate that distortion of the pulses will occur when the background-ionization current exceeds approximately one milliampere. In the general case such a current will give fluctuation pulses \(V'\), determined by the expression

\[ \frac{V'}{V_0}\sim 10^{-9}\left(\frac{iRC}{e}\,m\right)^{\frac{1}{2}}, \]

where \(e\) is the charge of the electron in coulombs. Easily distinguishable fluctuation pulses of several volts in magnitude appear at the moment when the ionic current becomes sufficiently large to reduce the amplification.

Diven and Rossi\(^{32}\) measured gas amplification in proportional counters and confirmed the measurements of Rose, Korff, and Ramsey. Thus, for example, they found that the addition of a small amount of methane to hydrogen, or of \(\mathrm{CO}_2\) to argon, considerably improves the stability

counters, especially at low pressures. They also investigated the scatter in the magnitude of the pulses when the distance of the ion tracks from the wire was varied. No measurable scatter was found, which shows that electron capture did not occur. End effects were also investigated, entailing an uncertainty in the working volume of the counter. The gas-amplification coefficient begins to fall rapidly at distances from the ends of the wire or cylinder smaller than the diameter of the cylinder.

The statistical nature of the process of gas amplification also introduces its own scatter into the magnitude of the pulse. This factor was calculated by Snyder \(^{33}\), who found:

\[ (\overline{\Delta h})^{2}=\frac{2(\bar{h})^{2}}{\bar{n}}, \]

where \((\overline{\Delta h})^{2}\) is the mean square deviation of the pulse magnitude, \(\bar{h}\) is the mean pulse magnitude, and \(\bar{n}\) is the mean number of ion pairs produced in the counter by the primary ionizing particle. Snyder’s calculation leaves out of account the large fluctuations caused by delta rays, whose energy is much greater than the \(30\ \mathrm{eV}\) required to create an ion pair \(^{34}\).

Pulse shape

Since the voltage pulse is caused almost entirely by the motion of the positive ions* formed at the wire by the electron avalanche, the shape of the pulse does not depend on the position of the initial ions. Moreover, the magnitude of the pulse will be proportional to the number of initial ions even after the pulse is differentiated in the electronic amplifier.

The shape of the pulse can be calculated for the case of a cylindrical proportional counter if we assume that (a) all positive ions are formed at the surface of the wire and (b) the mean drift velocity of the positive ions toward the cylinder is given by the expression

\[ \frac{dr}{dt}=\frac{KE}{P}, \]

where \(\frac{K}{P}\) is the mobility of the positive ions, depending on the nature of the gas.

* The ratio of the voltage pulse arising from the collection of electrons to the pulse arising from the collection of positive ions is equal to \(1/\alpha_{0}\log \frac{b}{a}\), where \(\alpha_{0}\) is the value of the first Townsend ionization coefficient at the wire. Since \(\alpha_{0}\) is approximately 100 at the edge of the wire, depending, of course, on the voltage, and \(\log \frac{b}{a}\) is close to 5, the ratio is approximately \(0.2\%\).

Integrating, we now immediately find that the mean position of the ions as a function of time is given by the expression

\[ r^{2}-a^{2}=\frac{2KV_{0}t}{P\log \frac{b}{a}}. \tag{14} \]

Equation (2) may be rewritten:

\[ V_{c}=\frac{mn_{0}e}{V_{0}C}\int_{0}^{t} E_{r}\frac{dr}{dt}\,dt = \frac{mn_{0}e}{C\log \frac{b}{a}}\int_{0}^{t}\frac{1}{r}\frac{dr}{dt}\,dt, \tag{15} \]

where \(V_c\) is the voltage pulse, \(n_0\) is the initial number of ions, and \(C\) is the total input capacitance of the amplifier, including the counter. Integrating (15), we obtain, with the aid of (14):

\[ V_{c}=\frac{n_{0}me}{2C\log \frac{b}{a}}\cdot \log\left(1+\frac{t}{t_{0}}\right), \tag{16} \]

where

\[ t_{0}=\frac{a^{2}\log \frac{b}{a}}{2V_{0}\frac{K}{P}}{}^{*}). \]

When all the ions have been collected, \(V_c\) will, of course, become equal to \(n_0me/C\). Fig. 7 shows the voltage as a function of \(t/t_0\), according to equation (16). A typical proportional counter has a filament three tenths of a millimeter in diameter, coaxial with a cylinder of radius \(1\ \mathrm{cm}\). In argon the mobility \(K/P\) is about \(3\ \mathrm{cm/sec}\) per volt/cm, and, using as \(V_0\) a voltage of \(10^3\ \mathrm{V}\), we obtain \(t_0 \sim 0.015\ \mu\mathrm{sec}\). From (16) we can calculate that the pulse will rise to \(1/n\) of its final value in the time

\[ t_{1/n}=\left[\left(\frac{b}{a}\right)^{\frac{2}{n}}-1\right]t_{0}. \]

Consequently, in order for the pulse to rise to \(1/10\) of its final value, a time \(1.5t_0\), or \(0.02\ \mu\mathrm{sec}\), is required; for it to rise to \(1/4\) of its final value, \(16t_0\), or about \(0.2\ \mu\mathrm{sec}\), is required; the pulse reaches half its final value after a time \(250t_0\), or approximately \(2.5\ \mu\mathrm{sec}\). For it to rise to its full value, a time of about one millisecond is required.

In view of the fact that the mobility of the ions is not a constant quantity in the strong field near the filament, the pulse will not be described exactly by equation (16). The mobility apparently changes by about 20 percent in going from small values of \(E/p\) to very large ones,

*) This relation was first found empirically by W. E. Ramsey (see \(^{41}\)) and first derived by K. G. Montgomery and D. D. Montgomery (see \(^{42}\)).

according to Hershey’s measurements \(^{35}\) of the motion of \(K^{+}\) ions in \(\mathrm{H_2}\), He, \(\mathrm{N_2}\), and Ar. The mobility of \(K^{+}\) ions may be taken as a characteristic of the mobility of other ions. Hershey found that the mobility of \(K^{+}\) ions in \(\mathrm{H_2}\) is approximately \(16\ \mathrm{cm/sec}\) per \(\mathrm{volt/cm}\) at one atmosphere, in He about 21, and in \(\mathrm{N_2}\) and Ar about 3. Thus, with \(\mathrm{H_2}\) and He in a proportional counter, pulses should be obtained that are appreciably faster than with Ar or \(\mathrm{N_2}\).

Fig. 7. The solid curve represents the pulse shape from a proportional counter. The dotted curves represent the pulse shape after differentiation by means of \(RC\) or “clipping.”

Owing to the characteristics of the amplifier, the observed pulses will not look as they should according to (16); more often one has to observe that they are modified—usually by means of an \(RC'\) circuit. The dotted curves in Fig. 7 show the effect of differentiating the pulse by a circuit with \(RC' = 10t_0 = 0.15\ \mu\mathrm{sec}\) and with \(RC' = 2t_0 = 0.03\ \mu\mathrm{sec}\). The resulting pulses are reduced in magnitude by approximately a factor of 10 relative to the value they would have in the case of a very large value of \(RC'\), but on the other hand they have a duration of only a few tenths of a microsecond.

Resolution time

The very rapid rise of the pulse at its beginning is to some extent deceptive, since this rise does not correspond to the moment of initial ionization, but lags behind that moment by the interval required for the primary electrons to reach the wire. In pure argon this delay may

vary within a microsecond, depending on whether the primary ions were formed near the wire or near the cylinder. On the other hand, if a mixture of argon + CO₂ is used, the delay can be reduced to about 0.1 μsec, and if the ionizing particles each time strike one and the same place in the counter, an accuracy of several hundredths of a microsecond can be achieved. Hydrogen or BF₃ should also give high resolving power. In general, the smaller the diameter of the counter, the better the resolving power. In the above considerations it was assumed, of course, that the electronic apparatus connected with the counter has a comparatively short response time.

Details of Construction and Operation

Proportional counters can be used practically for any tasks. They can operate, as is now clear, with almost any geometrical configuration and with any gas. Since the voltage on the counter is high, it is usually necessary to take certain precautions with regard to insulation of the wire. Commercial kovar-glass insulators are ideal as wire holders. When installing them in place, acid soldering should be avoided. It is possible, however, to use any ceramic or plastic for insulation, and they can be fastened in place with the use of rubber or metal gaskets, or even wax.

If one wishes to obtain a small sensitive volume, it is expedient to use a point counter. Cylindrical counters are easier to manufacture and more universal in use. A typical design is shown in Fig. 12. Often, especially when high voltages are used, two kovar-glass insulators are placed in series at each end of the counter: a large insulator between the cylinder of the counter and a narrower kovar-glass tube, and then a small insulator between this tube and the wire. The kovar-glass tube is grounded, forming a kind of guard ring. The diameter of the wire is usually taken from 3 to 5 tenths of a millimeter. R. Thompson and B. Diven³⁶ made a flat proportional counter by stretching thin parallel metal wires midway between two planes formed by screens of thin wires and separated by a distance of ³/₈ inch (about 10 mm). Toroidal counters have also been used.

If possible, it is best to use an amplifier operating with such amplification that tube noise is barely observable. The remarks made concerning amplifiers in the chapter on ionization counters are equally applicable here. The voltage on the counter is raised only to such a value that the gas amplification makes it possible to observe the pulses conveniently. This

ensures operation of the counter at the lowest voltage possible, and consequently the greatest possible stability. There is no need to stipulate that the applied voltage must be stabilized either by means of electronic tubes or by the use of miniature batteries. If possible, an argon–CO₂ or hydrogen–methane mixture should be used. Both of these mixtures require a lower voltage than most other gases, and both give good resolving time. To obtain stability at high \((10^3—10^4)\) gas amplification, the polyatomic component must be increased.

III. GEIGER COUNTERS

General considerations

A Geiger counter is an ionization counter on which the voltage is so high that multiplication of the primary ions in the gas causes a discharge propagating along the entire length of the filament. The discharge ceases when the positive ions around the anode reduce the field below the multiplication threshold. Such a counter can give a count if a single electron has appeared in the working volume. Moreover, the voltage pulse obtained is so large that it can be recorded with little amplification or without any amplification at all.

For many years after their invention, Geiger counters[^37] were characterized by unreliable operation and by the impossibility of accurately predicting their behavior. Many, often contradictory, theories of counter operation and many “cookbook” recipes for their manufacture were proposed. However, over the last ten years, successful construction techniques and a fairly rigorous, to a considerable extent adequate, theory of their action have gradually led to the point where reliable counters with predetermined characteristics can now be constructed[^38–^44]. The present article gives the essential points of the theory together with some practical considerations concerning the construction of counters.

The design of counters has basically not changed since their invention by Geiger and Müller[^37]. Normally a counter consists of an axial filament in a metal cylinder enclosed in a hermetic envelope, which can be evacuated and filled with a suitable gas, generally at reduced pressure. A characteristic feature of modern counters is the use of a polyatomic gas as a constituent part of the filling. In this article the description is limited primarily to counters with such a filling.

The use of polyatomic gases (or vapors) in Geiger counters was first considered in detail by Trost[^38–^40]. He investigated...

...tested many filling mixtures and came to the conclusion that a mixture of argon (at a partial pressure of 9 cm Hg) and ethyl alcohol vapor (at a pressure of 1 cm) is optimal. Such a mixture, i.e., 9 parts argon to 1 part alcohol vapor, is usually used, although other vapors are often employed (such as amyl acetate, methane, carbon tetrachloride, acetone), and in special cases other monatomic or diatomic gases. Such a mixture gives considerably better characteristics than filling with a single monatomic or diatomic gas.

A counter properly constructed and filled with such a gas mixture behaves as an ionization counter at low voltages and as a proportional counter at higher voltages. Characteristic of a counter with a polyatomic filling gas is proportionality over a broader voltage range than in the case of filling with a single monatomic or diatomic gas.

At a sufficiently high voltage the Geiger threshold is reached—the pulses become relatively very large and all of the same magnitude. These large and uniform pulses arise because the discharge spreads along the entire length of the counter. Their magnitude does not depend on the nature of the event that caused them. If the voltage is increased further, the pulse magnitude continues to grow, and the counting intensity for any given source slowly increases. Finally, a voltage is reached at which the counter breaks down and a continuous discharge begins. In the Geiger region, the discharge associated with each individual count has a very short duration, since the space charge near the filament weakens the field below the value necessary for gas amplification.

In the Geiger region the pulses do not always begin to grow immediately after the ionizing event. In counters of ordinary dimensions there is an average delay of about one-tenth of a microsecond. After the pulse has begun, it reaches its full magnitude in a small fraction of a microsecond. The pulse length is determined by the time constant of the amplifier input and is usually several microseconds. Following the start of the pulse, the counter becomes “dead” for a period normally of several hundred microseconds, and an ionizing event occurring in the counter during this period will not be recorded. The dead time is followed by a recovery period of comparable duration, during which an ionizing event will be recorded, but in the form of a pulse of reduced magnitude.

Counters without a polyatomic gas differ in their behavior from that described above. Their Geiger threshold voltage is lower at the same pressures, and the discharge arising after a count has a tendency to persist, producing multiple counts or continuous...

...discharge, until by some means the voltage on the counter is made lower than the threshold value. In this article we shall not consider counters of this type. A detailed description of apparatus suitable for their use is available in the literature 27, 45.

Counters with polyatomic gases were often called “fast,” and those without them “slow.” The terms “self-quenching” and “non-self-quenching” seem more appropriate, though also longer. However, a polyatomic gas can increase the mobility of electrons, as was indicated in the chapter on ionization counters, and thus make the counter faster in the sense of reducing the delay time.

The role of polyatomic molecules is apparently twofold: to absorb ultraviolet quanta and thereby eliminate the cathode photoeffect, and to prevent secondary electron emission from the cathode under the action of positive ions.

Avalanche

The initial number of secondary electrons depends on the velocity and nature of the ionizing particle, the length of its path in the counter, and on the pressure and nature of the gas filling the counter. A fast electron on a path two centimeters long in a counter filled with argon at a pressure of 10 cm produces on average about 8 secondary electrons. These electrons are accelerated in the direction of the central wire. Over the course of a free path each of them acquires energy, which may be partially or completely expended in inelastic collisions with gas molecules. The mean free path is of the order of \(10^{-3}\) cm, and in the weak-field region of the counter the electrons acquire only a small amount of energy in each free path. Here only the very lowest energy states of the gas (polyatomic) molecules can be excited.

Finally, the electron reaches a point not farther than the diameter of the wire from the central wire, where the field is high enough for the electrons to acquire energy greater than the ionization potential of the gas molecules (11.3 eV for ethyl alcohol and 15.7 eV for argon). The resulting gas amplification is the same as that considered in the chapter on proportional counters. If the voltage on the counter is increased, approaching the threshold value, multiplication in the gas takes place over ever greater mean free paths, producing more ions and more quanta. The Geiger threshold is reached when a sufficient number of quanta has been produced to make the probability close to unity that at least one quantum will survive to the point where further amplification can be caused by photoionization.

Propagation of the Discharge

The discharge propagates owing to the emission and absorption of light quanta, until the ionic sheath has encompassed the entire length of the filament. This process has been studied in detail by Elder et al. ^46. The energy of the quanta is for the most part limited to values less than 15.7 eV, i.e., the ionization potential of argon. Alcohol vapor keeps the mean energy of the quanta at a level lower than that which would exist in the case of pure argon. Many quanta emitted by excited argon atoms will have energies exceeding the ionization potential of alcohol (11.3 eV), since the first excitation potential of argon is 11.6 eV. These quanta are absorbed by alcohol molecules and can cause the appearance of new electrons, which in turn can initiate the formation of new avalanches. Elder and his collaborators ^46 measured the absorption coefficient in alcohol vapor for the quanta responsible for the propagation of the discharge in an argon–alcohol counter, and found it to be 640 cm\(^{-1}\) (at a pressure of 760 mm Hg and room temperature). This means that in a counter containing alcohol vapor at a pressure of 1 cm, the number of quanta decreases to \(1/e\) of its initial value over a distance of 1.2 mm*).

In the initial avalanche the quanta are emitted in all directions (mainly from points situated near the surface of the filament). The density of the newly formed electrons decreases exponentially in all directions. These new electrons produce new avalanches (provided that they do not pass through the field weakened by the space charge of the preceding avalanches), in which new quanta arise. With each step the discharge propagates along the axis of the counter, over the maximum axial distance at which photoelectrons are still produced. It can be seen how such a stepwise process leads to the sharp Geiger threshold. Let \(\gamma\) be the number of electrons liberated by photons (capable of initiating new avalanches) per electron of the preceding avalanche. Then for the total number of electrons collected on the filament we have:

\[ N = n_{0}m + n_{0}m^{2}\gamma + n_{0}m^{3}\gamma^{2} + \cdots , \]

*) After this had been written, a paper by Liebson ^47 appeared reporting measurement results interpreted by the author as evidence that alcohol does not absorb the quanta responsible for the propagation of the discharge. However, in Liebson’s experimental arrangement the quanta passed through at least one centimeter in a mixture at normal argon pressure and a polyatomic gas before the author made his absorption measurements. After traversing such a distance the number of quanta of the type to which the propagation of the discharge is due would have been reduced by approximately \(e^{10}\) times, according to Elder’s absorption coefficient. Thus Liebson’s measurements evidently refer to another type of quanta, present in smaller numbers.

or

\[ N=\frac{n_0m}{1-m\gamma}. \]

Here \(m\) is the coefficient of multiplication of electrons in the avalanche. In the region of proportionality \(m\gamma \ll 1\). If \(m\gamma\) approaches unity, then small changes in this quantity change \(N\) very strongly. Physically it is not difficult to see that, in order for the process to continue through many stages, there must be a probability, sufficiently close to unity, corresponding to the fact that at least one quantum liberating electrons will be preserved on its path to the point where it can give rise to a new avalanche. Near the threshold the propagation may be incomplete, and as a result strong fluctuations in the magnitude of the pulse will appear. Stepwise propagation leads to a measurable propagation velocity, although the duration of each step may be no more than \(10^{-9}\) sec. This velocity varies from 2 to 20 cm/µsec, depending on the filling of the counter and the operating voltage\(^{46,48—50}\). It increases with increasing voltage and decreases with increasing pressure of the polyatomic gas. Elder’s results\(^{46}\) are shown in Fig. 8.

Fig. 8. Propagation velocity of the discharge in an argon–alcohol counter as a function of alcohol-vapor pressure \(P\) (in mm Hg) and of voltage, according to the measurements of Elder et al.\(^{46}\). The gas pressure in the counter was kept constant at 80 mm Hg.

The large absorption of ultraviolet quanta by a polyatomic gas makes the probability of a photon reaching the cathode extremely small. This was shown\(^{26}\) for the case of a proportional counter at

...the basis of the complete absence of cathode effects in the presence of polyatomic gases. Strong ultraviolet absorption is also responsible for the results of Stiver^43 and Wilkinson and Kennan^51, who showed that the discharge does not propagate through a region of weak field, as, for example, in the case of a glass bead on the counter wire. All quanta are absorbed in the vicinity of the bead, where the field is too weak for multiplication to occur. Stiver^52 constructed a “telescope,” dividing one counter into a certain number of sections by means of glass beads on the wire. The beads confine the discharge to the section through which the ionizing particle has passed, and make it possible to distinguish particles that have passed through all the sections by the magnitude of the pulse.

Pulse shape

The basic mechanism by which a Geiger pulse is formed is the same as in the case of a proportional counter. The potential of the wire falls while the positive ions left by the space charge move toward the cylinder. At the same time the motion of the electrons has little effect on the pulse, since they are formed too close to the wire*. The shape of the pulse is quite different from that for the region of proportionality, owing to the fact that a certain time is required for the discharge to spread along the length of the wire. A Geiger pulse may require (if the counter is long) several tenths of a microsecond to reach 10% of its final value, whereas a proportional counter is characterized by several hundredths of a microsecond. On the other hand, a Geiger pulse reaches any given small potential more rapidly than a proportional pulse. The time required to reach some partial value is usually, however, more significant than the time required to reach the full value of the pulse, since Geiger counters are used with small pulse amplification or with none at all, and thus relatively large pulses are required from them.

The shape of the pulse can be correctly calculated provided that the pulse is assumed to be small compared with the applied voltage \(V_0\). This assumption is strictly fulfilled only in the case of short counters operating near threshold. But we are primarily interested in the initial rise of the pulse, and here the above assumption is satisfactory. Moreover, the actual rise time can only be longer than that calculated here, since the field in which the ions move may only be weaker than is assumed.

* Positive ions do not produce new ionization during their motion. Cf.^53.

** See the footnote on p. 496.

Fig. 9 illustrates the situation after the discharge has partially propagated along the counter; its beginning is at one of the ends (the radial distribution of the space charge is greatly exaggerated). Consider an element of the space charge \(qdx\) at a distance \(r\) from the axis and \(x < vt\) from the end of the counter, where \(q\) is the linear charge density of the sheath of positive ions, and \(v\) is the propagation velocity. As it moves from \(r=a\) to \(r=r\), \(qdx\) produces a voltage pulse

\[ dV_c = 2qdx\,\frac{C_c}{LC}\log \frac{r}{a}, \tag{17} \]

where \(C_c\) is the capacitance of the counter, \(C\) is the total capacitance of the counting system, and \(L\) is the length of the counter. Assuming, as in a proportional counter, that

\[ \frac{dr}{dt}=\frac{KE}{P}, \]

where \(K/P\) is the mobility of the positive ions, we obtain

\[ \frac{r^2}{a^2}=\frac{1}{t_0}\left(1-\frac{x}{v}\right)+1 \tag{18} \]

for \(0 \leq x \leq vt\),

Fig. 9. Schematic representation of the positive ion sheath after the discharge has propagated a distance \(x=vt\) from the end of the counter. The discharge element \((qdx)\) has traversed the distance \(r-a\) from the wire.

where

\[ t_0=a^2\,\frac{\log(b/a)}{2V_0(K/P)} \]

and is approximately \(0.01\) microsecond for a Geiger counter of ordinary dimensions. Integrating (17) by means of (18) from \(x=0\) to \(x=vt\), we obtain \(V_c\) as a function of time:

\[ \frac{V_c}{V_{cf}}= \frac{t_0}{2t_s\log(b/a)} \left\{ \left(\frac{t}{t_0}+1\right) \log\left(\frac{t}{t_0}+1\right) -\frac{t}{t_0} \right\}. \tag{19} \]

The expression is valid for \(t \leq t_s\), where \(t_s=L/v\) is the time required for the discharge to propagate over the length of the counter, and

\[ V_{sf}=\frac{Lq}{C} \]

is the final magnitude of the pulse (provided that \(RC\) is so large that only a negligible part of the charge has time to leak to ground during the time under consideration). Equation (19) is shown in Fig. 10. The rate of rise increases up to \(t=t_s\), where it reaches a maximum. For \(t>t_s\), \(V_r(t)\) can be calculated in a simple manner. The rate of rise decreases rapidly for \(t>t_s\), and, as in the case of a proportional counter, about \(10^{-3}\) sec is required for the pulse to reach its full magnitude (again, if \(RC\) has no effect).

In the case where the discharge begins in the middle of the counter and propagates toward both ends, the rate of rise at any instant is twice as great, and \(t_s\) is now one half of the former-

it. The pulse is represented in Fig. 10 by the dashed line. For most pulses, the rapid rise at the beginning will then be replaced by a slower one, when the discharge, which has begun somewhere at an intermediate point, reaches one of the ends of the counter.

The pulse shape described here, i.e. for long counters containing a polyatomic gas as a component of the filling, differs from the pulse shape in so-called “slow” counters. Ramsey\(^ {41}\), who measured pulses in argon–oxygen counters, found a shape quite similar to the pulse of a proportional counter. It is reasonable to suppose that the propagation velocity in “slow” counters is high, since ultraviolet quanta can eject photoelectrons along the entire length of the counter, especially from the cathode. In this case the propagation velocity is almost immaterial for determining the pulse shape. In short self-quenching counters \(t_s\) is small, and the pulse shape is essentially the same as in the case of a proportional counter.

Fig. 10. Pulse shapes in a counter requiring one microsecond for the propagation of the discharge from one end to the other. Curve \(a\) represents the pulse when the discharge propagates from one of the ends. Curve \(b\) represents the pulse for a discharge which propagates in both directions from the center. Curve \(c\) is shown for a discharge propagating in both directions from some intermediate point. In such a counter the pulse, beginning at the center, reaches 10% of its final value (in the absence of “cutoff”) \(0.15\ \mu\text{sec}\) sooner than the pulse beginning at the end. Curve \(d\) is the pulse of a proportional counter.

Dead time

The field in the region between the positive ion sheath and the wire is reduced while the positive ions are moving outward. If this field is below the threshold required for propagation of the discharge, no counting will occur. We can define an effective potential\(^ {43}\) \(V_e\), which is equal to the integral (from \(r=a\) to \(r=b\)) of the field inside the positive ion sheath. If \(V_e = V_t\), i.e. is equal to the threshold potential, counting will occur. We can approximately calculate the dead time by considering the entire positive ion sheath to be at any instant at the same distance from the wire. In this case the field will be as shown in Fig. 11. The field is shown reduced inside the ion sheath—

wire and normal outside it. It may be assumed that at any moment the effective potential is

\[ V_e = V_{ab} - 2q \log \frac{b}{r}, \tag{20} \]

where \(V_{ab}\) is the potential difference at the counter at that moment, and \(q\) is the linear charge density in the ionic sheath. In practice the pulse is differentiated by the \(RC\) circuit, so that \(V_{ab}\) never deviates from \(V_0\) by more than a few volts. Taking \(V_{ab}=V_0\), one can solve equation (20) for \(r=r_c\), corresponding to \(V_e=V_t\), i.e. to the threshold voltage:

\[ r_c = b \exp \frac{-(V_0 - V_t)}{2q}. \]

Fig. 11. Electric field in a Geiger counter. The solid curve represents the normal field. The dotted curve represents the weakened field caused by a positive ionic sheath at a distance \(r\) from the axis.

Thus, even if the central wire is grounded, so that its potential cannot change during the discharge, the field near the wire will be too weak for counting to occur until the positive ions have moved outward to the above-indicated critical distance. Stever\(^{43}\) estimates \(r_c\) to be approximately one half the radius of the counter.

Recovery time

As soon as the field has recovered to the threshold value, counting will occur, but the pulses will be of reduced size until the field returns to its normal state. The interval from the end of the dead time to the moment when pulses of full size are obtained is called, according to Stever, the recovery time. The dead time and the recovery time are each of the order of \(2\cdot 10^{-4}\) sec. Both the dead time and the recovery time can be eliminated by reversing the potential of the counter (as soon as the pulse has begun to rise) and collecting the positive ions on the wire. This was done by Simpson\(^{54}\), as a result of which the counter could return to its sensitive state in 20 microseconds.

Suppression of secondary electrons at the cathode

If, when positive ions strike the cathode, secondary electrons are emitted, further counting will arise and the discharge will continue. Argon ions are capable of liberating secondary

electrons, but the presence of polyatomic molecules is sufficient to prevent this process. The mechanism by which this is achieved was described by Korff and Present^44. In a counter containing argon and alcohol vapor (or analogous mixtures), the argon ions do not reach the cathode, since they are neutralized as a result of collisions with alcohol molecules, and the resulting alcohol ions form a positive ionic sheath when they approach the cathode. This process is possible because the ionization potential of ethyl alcohol is 11.3 eV, while that of argon is 15.7 eV. An argon ion undergoes up to \(10^5\) collisions as it travels through the entire counter. In each collision in which an alcohol molecule participates, an electron from the neutral alcohol molecule can be transferred to the argon ion with the emission of a photon of energy 4.4 eV, which is absorbed by the alcohol molecule. Transfer of electrons in the reverse direction is energetically impossible, since the increase in the energy of the argon ions between collisions is no more than approximately 0.1 eV. Since the probability of transfer is relatively large and such a great number of collisions takes place, only alcohol ions reach the cathode. When an ion approaches the cathode to less than \(10^{-7}\) cm, its field becomes sufficient to tear an electron out of the wall; and at a distance of about \(5 \cdot 10^{-8}\) cm from the cathode the probability of neutralization becomes close to unity. The neutral molecule remains in an excited state, whose lifetime with respect to dissociation is of the order of \(10^{-13}\) sec. This is less than the time required for the molecule to come close up to the wall and transfer its excitation energy to some electron in the metal, and much less than the lifetime of the excited molecule with respect to radiation. Korff and Present established that the probabilities of the various possible processes indicate that, at the threshold of the Geiger region, approximately one count in ten should give a double pulse*). This number should increase as the potential on the counter is raised, since in that case a larger charge is collected and a larger number of ions reach the cathode. Korff and Present report that this assumption is confirmed by experiment. These double pulses, as well as the variability of the sensitive regions of the volume at the ends of the counter, lead to a slope of the counter plateau.

Random delay time

It is usually observed^56,57 that two counters do not discharge simultaneously, even if the discharges were caused simultaneously by one and the same ionizing event. Rossi and Nereson^56 and Shervin^57

*) Curran and Rae^55 measured the number of spurious pulses for various gas mixtures. They found that a mixture of argon plus alcohol vapor gives a negligible number of double pulses.

indicate an average difference in operating times of about 0.2 microseconds for counters of medium size (2.5 cm in diameter and 20 cm long) filled with standard mixtures of argon and alcohol vapor (Rossi) or argon and amyl acetate (Sherwin). This delay has two sources: 1) the time of transport of the primary electrons from the point of their formation to the wire, and 2) the variable and relatively large rise time of the pulse owing to the finite velocity of propagation of the discharge. The transport time was measured by Sherwin^57 and by den-Hartog, Müller, and Færster^58. These measurements showed that electrons require about 0.1–0.2 microseconds to cross a counter (with a polyatomic filling) of radius 1 cm. Den-Hartog et al. showed that the transport time varies in accordance with the expression

\[ t = r^2 \frac{\ln (b/a)}{2kV}, \]

where \(r\) is the distance from the wire from which the electron begins to move, \(b\) and \(a\) are the radii of the cathode and the wire, \(V\) is the voltage on the counter, and \(k\) is the electron mobility (assumed here not to depend on the field strength). For a mixture of 90 mm Hg of argon and 10 mm Hg of alcohol vapor these authors found \(k\) equal to \(1.56 \cdot 10^4\) cm/sec per volt/cm. Thus, in large counters \(t\) may amount to a microsecond or more. The transport time will produce a relative delay between two counters if the ionizing particle passes near the cylinder in one counter and near the wire in the other.

Delays also arise as a result of the requirement imposed on large pulses—to control, in the recording circuit, a discriminator or some other device operating with grid bias. This delay is variable and depends on the point in the counter at which the pulse begins, as Fig. 10 shows. In the general case the delay*) decreases when the voltage on the counter is increased and when the discriminator bias is set for operation at a small pulse amplitude (which requires increasing the voltage between the counter output and the discriminator). Lowering the grid-bias level reduces both the absolute delay time and the spread in delay times. Operation with low bias requires more careful shielding of the counter and leads than is normally practiced.

In the study of cosmic rays, counters 50 or 100 cm long are often used, and it is desirable to know how the delay times depend on the length of the counter. So long as the capacitance of the counter is small compared with the total capacitance of the counting system, as is usually the case, the delay time does not depend on the length of the counter, since

*) Sherwin^57 observed a delay depending on the voltage on the counter, which is presumably the same as that discussed here.

both the final value of the pulse and the propagation time are proportional to the length. If, however, the capacitance of the counter constitutes a significant fraction of the total capacitance, the delay time will increase with increasing counter length.

The time constant determining the differentiation of the pulse must be large in comparison with the expected delay time.

If the filling of the counter contains gases having a high electron-attachment coefficient, negative ions are formed, which move into the region of the strong field much earlier than the electron is detached and excites an avalanche. This leads to slow counting59–61, and as a result such gases as oxygen or chlorine should be carefully avoided.

Efficiency

The efficiency of a counter is defined as that fraction of ionizing particles passing through the sensitive volume of the counter which is counted. A counter may be inefficient either because the particle does not form ions, or because the particle passes through the counter during the dead time following the preceding count. The efficiency may be determined by means of a system of coincidence and anticoincidence measurements62, when the known number of cosmic particles crossing the counter in question and the number of counts are recorded. In counting cosmic particles the efficiency of an argon–alcohol counter normally exceeds 99.5%. At increased counting rates the efficiency falls because of the greater total duration of dead time per second.

In the case of special applications the efficiency of the counter may be low if the probability of ion formation in the counter is low. If the mean number of pairs of primary ions per centimeter of path at atmospheric pressure is \(n\), then the mean number of ion pairs produced along a path of length \(d\) is \(N = npd\). The probability of forming no ion at all is \(e^{-N}\), and the efficiency is \(E = 1 - e^{-N}\). In an ordinary argon counter, in which \(N\) may be equal to 8, \(E\) is very close to unity. However, in the case of small counters or filling gases with low specific ionization (for example, hydrogen), \(N\) may be, say, not greater than 2, and consequently \(E = 1 - e^{-2} = 0.865\).

Lifetime of the counter

A counter filled with a polyatomic gas has a finite lifetime, usually of the order of \(10^9\) or \(10^{10}\) counts. This is presumably explained by dissociation of the polyatomic molecules, analogous to that,

as was described above. Since each count is associated with approximately \(10^0\) ionizations, each of which leads to the dissociation of molecules of alcohol or others like them, about \(10^{10}\) molecules prove to be “used up” with each count. The filling contains on the order of \(10^{20}\) such molecules, and thus it is possible to produce about \(10^{10}\) counts before the alcohol is decomposed. It may be assumed that more complex gases give a longer lifetime because the decomposition products are themselves quenching gases. Fridland\(^{63}\) investigated with a mass spectrograph the relative content of various mass numbers in unused and used counters filled with argon and ethyl acetate. He found the content of \(\mathrm{CO_2}\), \(\mathrm{CO}\), and \(\mathrm{CH_4}\) in the decomposition products after \(10^{10}\) counts, which agrees with the dissociation theory.

Counter construction

It is not the purpose of the present article to give recipes for the construction of counters. However, there are several questions whose consideration may be useful for designing counters. For ordinary purposes a mixture of 9 or 10 cm Hg of argon and 1 cm Hg of ethyl alcohol is a sufficiently good filling. Other polyatomic

Fig. 12. Typical counter for cosmic rays.

gases, such as amyl acetate, methane, carbon tetrachloride, acetone, xylene, or ethyl acetate, may replace ethyl alcohol. In the case of methane a somewhat increased vapor concentration is required; a concentration of 25% is satisfactory. Since a polyatomic vapor is present in the filling, there is no need to treat the cathode in any special way. It is advisable, however, to make the cathode surface with the highest possible work function. This reduces sensitivity to light if the envelope is transparent and probably decreases the number of double pulses. Shiny copper cathodes can be prepared by cleaning with nitric acid and by-

by heating in the presence of hydrogen. Oxidized surfaces^45 can be obtained by cleaning with nitric acid and heating in the presence of \(NO_2\). Brass cathodes may be treated in the same way. In all cases it is well, before assembly, to polish both the filament and the cylinder. In general, the counter must be so constructed that the composition of the filling does not change with time. On this basis the parts must be degassed, and the use of materials other than glass and metal is excluded.

There exists a large number of papers^40, 45, 64–66 on the subject of constructing counters for various purposes*). For details we refer the reader to them. Fig. 12 shows a counter for cosmic rays used in the laboratory of Cornell University.

IV. ELECTRON MULTIPLIERS

General considerations

An electron multiplier is an instrument in which one or more electrons, knocked out of a metal surface by a photon or a particle, are multiplied to an easily measurable number by the emission of secondary electrons from a series of successive surfaces.

Electron multipliers have two advantages over other counters: they operate in vacuum and they are fast. Operation in vacuum is essential mainly in the case of low-energy particles, but high speed is useful in almost any counting work. Pulses from an electron multiplier rise in \(10^{-9}\) sec. or less, and the limit is practically determined by the amplifier, if one is used with the multiplier.

Slepian proposed the multiplier as an amplifier in 1919, but only in 1936 did Zworykin, Morton, and Malter^70 first give a detailed description of the instrument. The use of multipliers as particle counters is due chiefly to Bay^71–73, Allen^74–76, and Coltman and Marshall^77**).

*) On the question of the properties of counters in an unusual geometrical arrangement, see^67. A plane coparallel arrangement of the counter electrodes, giving a significantly shorter delay time and rise time of the pulse, is considered in^68.

**) The authors apparently deliberately distort the history of the invention and development of electron multipliers, completely ignoring the leading role of Soviet scientists.

In fact, the electron multiplier was invented in 1930 by the Soviet researcher L. A. Kubetskii (author’s certificate No. 24040). In June 1934, at the Television Institute, in the laboratory of L. A. Kubetskii, the first electron multiplier with an amplification factor of more than 1000 was created.

In September 1934, at the same institute, the first electron multipliers created in the USSR—the “Kubetskii tubes”—were demonstrated in operation.

Bai and Allen used a multiplier for counting particles directly, i.e., the primary particle knocked secondary electrons out of a surface, and the latter were multiplied and recorded, whereas Coltman and Marshall used a photomultiplier to detect photons emitted when the primary particles collided with a suitable phosphor.

In any multiplier the total multiplication is equal to \(m^n\), where \(m\) is the multiplication per stage and \(n\) is the number of stages. Full multiplication can be realized only if every electron leaving a given electrode reaches the next electrode. This requires the use of magnetic\(^{70}\) or electrostatic\(^{78}\) focusing. The latter was used by Bai and Allen. Multipliers with from 9 to 12 stages were employed, with multiplication from 2.5 to 5 or 6 per stage and a total multiplication from \(10^4\) to \(10^8\). Each electron, when it strikes the anode, produces a current pulse in the anode output circuit. This pulse has a finite width, and since not all electrons reach the anode at one and the same time, the total pulse is somewhat broadened. For the commercial RCA 931 photomultiplier the full width between the points corresponding to half the amplitude of the final pulse is, according to Sard’s estimate,\(^{79}\) approximately \(6\cdot 10^{-10}\) sec. This determines the ultimate limit of the resolving time of such a detector.

Dark Current

When electron multipliers are used for direct detection of particles, the principal limitation is the “dark current.” This is a current arising primarily as a result of thermionic emission at room temperature. The magnitude of the current can be calculated from Richardson’s equation

\[ I = AT e^{-\frac{e\varphi}{kT}}. \]

Among those present at the demonstration was the American scientist Dr. V. K. Zworykin (see the journal Radiofront, No. 7, April 1935, pp. 14, 18, and 21), to whom in March 1936 was issued the first British patent for the development of multistage electron multipliers by the firm RCA (see Proc. I. R. E. 24, 3, 351 (1936).

Thus, the priority of the USSR in the field of the invention and implementation of multistage electron multipliers is evident.

Likewise, the work of Soviet scientists played a prominent role in improving multipliers and developing methods for their application.

Modern electron multipliers with an antimony–cesium photocathode, created in the USSR, surpass the best American multipliers of the 1P-21 type in a number of important qualitative characteristics (photocathode sensitivity, low dark current) (see the articles by S. M. Feinstein, ZhTF 18, 1, 39 (1948), and O. Chechik, Zavodskaya Laboratoriya, 13, 12, 1440 (1947). There also see references to works by Soviet authors). The work of L. A. Kubetskii in the field of electron multipliers was awarded the Stalin Prize for 1947. (Editor’s note.)

In tube 931 the work function \(\varphi\) is about \(0.75\,\mathrm{eV}\) for an Ag—CsO\(_2\)—Cs photocathode, and \(A\) is approximately \(9.8\cdot 10^{-2}\,\mathrm{a}/\mathrm{cm}^2\). At \(300^\circ\mathrm{K}\) the dark current is thus approximately \(10^{10}\) electrons per second from the photosensitive surface. (This figure is subject to very large variations from tube to tube.) The fluctuations of this current are considerably larger than the effect produced by a single particle incident on the surface, so that detection of an individual particle proves impossible. If, however, the tube is cooled to the temperature of liquid air, the dark current falls to a few electrons per minute. This is a quite satisfactory background level, but the use of liquid air is inconvenient.

For reducing the background, more preferable than cooling the photomultiplier is the construction of tubes with a first surface having an increased work function. This method was used by Bay and Allen. The surface must be made such that its characteristics do not change on contact with air. Generally speaking, this means that the surface must be oxidized. Allen successfully used beryllium and beryllium-copper surfaces; Bay reports good characteristics for oxidized magnesium, oxidized brass, and an oxidized silver-magnesium alloy. Such surfaces give a multiplication of 4 to 6 per stage and can be exposed to air without serious deterioration of their quality.

Pulse-amplitude distribution

The number of secondary electrons per incident primary particle is small and therefore subject to large fluctuations. Analogously, the electron multiplication in the subsequent stages is also subject to fluctuations\(^{80}\). Moreover, even if there were no fluctuations in the ratio of the number of secondary electrons to the number of primary ones, the pulses would nevertheless vary in magnitude owing to the dependence of the focusing efficiency on the position on the electrode of the point at which multiplication occurred. Accordingly, the “counting intensity—bias” curve always shows a considerable change in pulse magnitude, as, for example, in Fig. 13, taken from Allen\(^{75}\). In the case of particles giving a large number of secondary electrons on the first surface, such as \(\alpha\)-particles, the “counting intensity—bias” curve is shown in Fig. 13 by the dotted curve \(B\). It guarantees that all pulses can be counted if the bias is set to some small value, but one such that noise is excluded. On the contrary, in the case of particles giving a small number of secondary electrons, there is no appreciable flat part of the integral pulse-distribution curve, and therefore it is difficult to guarantee,

that all pulses will be counted. X-rays, in particular, are difficult to count if some fraction of the pulses arises from the incidence of X-rays on the second electrode.

Voltage per Stage

Allen^75 showed that the counting intensity from a given source (X-rays) increases while the voltage per stage is raised to 320 V, after which the counting intensity remains constant. Apparently, the increase occurs as a consequence of an increase, at all stages, in the average ratio of the number of secondary electrons to the number of primary electrons when the voltage is raised. As is known, in the case of beryllium this ratio has a maximum for electrons with an energy of 400 eV. One of the integral distribution curves in Fig. 13 has a flat region in its upper part, where no further increase in the counting intensity occurs as the voltage is increased.

Fig. 13. Distribution of pulse magnitudes from an electron multiplier. Curve A, according to Allen (see^75), represents the number of pulses greater than a specified magnitude for the case of X-ray counting. Curve B shows the type of distribution curves obtained for α-particles. In this case all pulses are greater than a certain minimum magnitude.

Both Allen and Bay indicate an increase in the background level when the voltage per stage becomes too high (in Allen’s case above 350 V). Apparently this occurs as a consequence of cold emission at the edges of the electrodes.

Efficiency

The efficiency of an electron multiplier for counting particles or photons depends on the average number of secondary electrons emitted from the first surface by one particle (or photon) incident on it. Bay^73 measured the efficiency for α-particles and found it to be 100%. The average number of secondary electrons per incident α-particle in his measurements proved to be 10.

Allen^81 measured the ratio of the number of secondary electrons to the number of primary protons for a large number of targets and found it to be approximately 7.5 for protons with energies from 50 KeV to 200 KeV. Such protons should be counted with an efficiency of 100%. By

of all probability, all heavy particles or ions with energies in this interval will be counted with one-hundred-percent efficiency.

In the case of β-particles, the average number of secondary electrons per incident particle is small^82 and decreases with increasing primary energy. The result is a low counting efficiency for β-particles, although quantitative data are not available here. Allen^83 carried out extensive measurements of the counting efficiency for low-energy electrons. For beryllium-copper electrodes with multiplication by a factor of 4 per stage he found 100% efficiency for electrons with energies of about 400 eV and only 50% efficiency for particles with energies of 6000 eV.

The efficiency for X-rays and γ-rays is small and, in general, lower than the efficiency of Geiger counters for the same radiation. This occurs, in particular, because many of the secondary electrons torn from the surface have energies too high to be focused onto the secondary electrode by the weak focusing field. This effect is also partly responsible for the low efficiency in the case of β-particles. Allen studied the efficiency for X-rays in the range 0.5–1.0 Å, using a tantalum surface. He obtained 0.7 count per \(10^3\) incident photons at 0.5 Å and 1.8 counts at 1.0 Å. Einsenstein and Gingrich^84 made a direct comparison of a Geiger counter with a multiplier for counting X-rays in the range from 0.63 to 1.54 Å. According to their report, the Geiger counter is 5 to 10 times more efficient than the multiplier.

Combination of a Photomultiplier with a Phosphor

For high-energy particles or quanta, the electron multiplier has many shortcomings, including the low efficiency noted above. However, for energetic particles the method of Coltman and Marshall^77 appears promising, in which a photomultiplier is used in combination with a fluorescing material (phosphor). The particles to be counted are absorbed in the phosphor and give rise to a large number of photons, registered by the photomultiplier. Individual ionizing particles can be detected even in the presence of a large dark current, if the number of photoelectrons liberated during the phosphor reaction time is greater than the number of dark-current electrons in the same period. Coltman and Marshall obtained pulses 50 times greater than the background noise for α-particles with an energy of 5 MeV, and pulses 10 times greater than the noise for β-particles with an energy of 1.7 MeV. In these experiments a spherical mirror was used to focus photons onto the photomultiplier. The upper limit of the counting rate is determined by the decay time^85 of the phosphor, which can be made of the order of 0.1 microsecond.

The work function of the photosurface can be increased approximately twofold (from 0.75 eV to \(\sim 1.5\) eV) without changing the sensitivity of the tube

for counting purposes, since blue fluorescent materials can be used. This would reduce the dark current to a negligibly small value and would make possible the detection of particles of reduced energy.

M. Deutsch has recently reported^86 the successful application of this method to the counting of γ-rays. He used naphthalene in combination with a photomultiplier cooled to the temperature of liquid air, and obtained an efficiency of about 60% relative to the counting of γ-rays with an energy of the order of 1 MeV.

V. CRYSTAL COUNTERS

General considerations

A crystal counter is a crystal in which the pulse is produced by the motion of electrons in an electric field after the passage of an ionizing particle has caused their energy to rise to the conduction band. Successful operation of a silver-chloride crystal counter was first reported in Van Heerden’s dissertation^23 (1945). Since then only a small number of papers have appeared,^86–90 although work has been carried out in many laboratories. Crystals have the advantages of a rapid rise of pulses, absence of delay in pulse formation, absence of dead time, and high efficiency in the counting of quanta with high energy. They also make it possible to measure the energy loss of particles in passing through the crystal.

When an ionizing particle passes through a crystal, the electrons freed by ionization acquire energy and pass into the conduction band, where they can move freely in an electric field (the mobility of electrons in AgCl is several thousand cm/sec per volt/cm at the temperature of liquid air), until they are captured by an electron trap. Usually the traps are inhomogeneities or impurities in the crystal. The electrons moving in the crystal induce a current pulse on the electrodes of the circuit. The phenomenon is thus quite analogous to photoconductivity. This analogy suggests that there are probably other crystals suitable for counters. Up to the present, only silver chloride,^23 diamond,^86–88,90 and thallium bromide—thallium iodide^89 have been described as crystals that are satisfactory counters. Conductivity induced by particles has been observed in various cadmium crystals.^91 In the case of other similar crystals, such as zinc sulfide, which have so far proved unsatisfactory, questions of purity and perfection of the crystal probably play a role*). Wooldridge, Eger, and Burton^86 report,

*) Satisfactory operation of zinc sulfide as a crystal counter is reported in ^92.

that the majority of diamonds prove suitable for counting α-particles. Other investigators^88,89 note that only one diamond out of about fifty can count γ-rays. We recently tested several dozen large (from one to two carats) diamonds supplied by the Diamond Development Company of America. Among the diamonds tested there were all classes of precious stones, as well as yellow stones of various grades. We found 30 suitable diamonds out of a total of about 200 examined. In some batches up to two thirds of the stones proved suitable, whereas in other batches not a single one did. There was no correlation with the quality of the stones. Some of the lowest-grade yellow stones (15 dollars per carat) counted just as well as the very best precious stones. Considerable variations in pulse magnitude were observed between individual good diamonds.

The requirement imposed on the counter is that it be a good insulator in the absence of ionizing radiation. Diamond satisfies this requirement at room temperature, but silver chloride exhibits full conductivity unless it is cooled to the temperature of liquid air. In addition, the mobility of the electrons increases as the temperature is lowered, so that the pulses become sharper.

Photoconducting crystals*) normally have a saturation current for a given light intensity at fields of several thousand volts per centimeter. Such a characteristic is obtained when the electrons traverse the entire path through the crystal without being trapped. The same phenomenon also occurs in crystalline counters: the pulses reach their maximum value for a given mean ionization per pulse when the electrons traverse the entire path through the crystal. For AgCl the saturation field proves to be of the order of 2500 V/cm.

Silver chloride is an available crystal, since it is commercially sold. It occurs either as individual crystals or as rolled sheets. The latter are not individual crystals, and although they do work as detectors, their use for quantitative measurements of energy loss is apparently undesirable. The counting properties of AgCl depend markedly on the methods of preparing the crystals and of handling them. To obtain good pulses the crystal must first of all be thoroughly annealed. By slowly bringing the crystal to a temperature of about 400° C (over a period of several hours) and then cooling it very slowly (over a period of several days), one apparently obtains satisfactory results. Depend—

) For a detailed acquaintance with the properties of these crystals see N. F. Mott and P. W. Gurney, Electronic processes in ionic crystals*. (Oxford University Press, London, 1940.)

the dependence of the electron range on the annealing technique was clearly demonstrated by J. D. Haynes) of Bell Telephone Laboratory, who showed that, with careful annealing, electrons can be made to pass distances exceeding a centimeter. After annealing the crystal must be provided with electrodes. This can be done by coating the surfaces with a thin layer of aquadag or by silvering. Silvering may be carried out by immersing the surface for several minutes in a photographic developer*). A solution of hydroquinone and sodium carbonate in normal proportions is satisfactory.

It is essential that the silver chloride be thoroughly cleaned (for example, with the aid of a dilute solution of hyposulfite, distilled water, and ethyl alcohol), and further handling of it must take place only with rubber gloves or forceps. Touching it without gloves leads to the appearance of surface leakages, with a subsequent irregular background and pulses from the crystal***). In order to prevent undesirable chemical reactions, the metallic base must not be in contact with AgCl.

AgCl must not be exposed to light or ionizing radiation at room temperature, since this impairs its properties as a crystalline counter. The electrons liberated by these agents diffuse through the crystal until they are captured, as may be supposed, by neutral silver atoms situated in cracks of the crystal lattice. The negatively charged silver ions then attract positive silver ions and thus build up silver grains, in a process analogous to that by which the latent photographic image is built up. The silver grains are effective trapping centers, which prevent the formation of the large pulses required of the counter. If the action of the light has not been very intense, annealing restores the quality of the crystal.

If the counter operates at the temperature of liquid air, ionic conditions are absent, so that silver trapping centers are not formed. However, the positive “holes” left by the electrons do not move, and a “space charge” is formed in the crystal, which reduces the magnitude of the pulse. This space charge makes periodic annealing of the crystal necessary.

) Private communication. See also 93.
) Haynes first drew our attention to this method.
**) We are indebted to Prof. Street of Harvard University for most of the practical details given here. Concerning certain properties of AgCl used as a counter, see also 94.

Magnitude of the Pulse

To understand the way in which a crystal counter can be used, it is instructive to consider the factors determining the magnitude of the pulse.^23 Let us assume that in Fig. 14 \(N_0\) electrons have entered the conduction band, and that they are all at a distance \(x\) from the anode.

Each of these electrons, in its motion toward the anode, has a definite probability (per unit path length) of being captured, and this probability does not depend on the path already traversed. This leads to an exponential distribution of the electron paths, so that after the electrons have traveled a distance \(y\), there remain

\[ N=N_0 e^{-y/\lambda} \]

electrons, where \(\lambda\) is the mean free path. These \(N\) electrons, moving from \(y\) to \(y+dy\), will induce on the anode a charge

\[ dQ=\frac{Ne}{d}\,dy, \]

where \(d\) is the thickness of the crystal.

Fig. 14. A crystal counter of thickness \(d\), in which \(N_0\) electrons have entered the conduction band at a distance \(x\) from the anode. \(N=N_0 e^{-y/\lambda}\) electrons pass a distance \(y\) or more, and, in traversing the path \(dy\), they induce a charge

\[ dQ=Ne\frac{dy}{d} \]

on the anode.

Fig. 15. Magnitude of the pulses in a crystal counter as a function of the distance from the anode \(x\), at which \(N_0\) electrons enter the conduction band, for various path lengths \(\lambda\) traversed by electrons in the crystal. \(d\) is the thickness of the crystal. The magnitude of the pulses is represented as the ratio of the total induced charge to the total charge of the moving electrons.

The total charge pulse produced by the motion of all the electrons will be

\[ Q=\int_{0}^{x}\frac{Ne}{d}\,dy, \]

which is equal to

\[ Q=N_0 e\,\frac{\lambda}{d}\left(1-e^{-d/\lambda\cdot x/d}\right). \]

In Fig. 15, \(Q/Ne\) is plotted as a function of \(x/d\). It is evident from the figure that, if the mean free path of the electrons is large in comparison with the thickness of the crystal, the magnitude of the pulse is directly proportional to the distance from the anode at which the electrons were formed. It is assumed here that the positive holes do not move. Apparently, this corresponds to the case both of silver chloride and of diamond. Both of these crystals form a space charge for some time after the counting has taken place. If this occurs, the high voltage may be removed, and the pulses will continue as long as ionizing radiation is present, but the sign of the pulses will be the opposite. In the other case, shown in Fig. 15, when \(\lambda/d \ll 1\), most of the pulses are of approximately the same magnitude, independent of \(x/d\), but they are all small. This circumstance can be demonstrated more clearly by calculating the distribution curve of the pulse magnitudes for the case in which monochromatic \(\gamma\)-rays are absorbed uniformly throughout the entire thickness of the crystal. In the simple case, when all secondary electrons have one and the same energy and when their range is small in comparison with the dimensions of the crystal, we can calculate the number of pulses whose magnitude lies between \(q\) and \(q+dq\), where \(q=Q/N_0e\). This calculation gives:

Fig. 16. Differential distribution curve of pulse magnitudes for the case in which \(n_0\) pulses are registered. Each pulse is caused by \(N_0\) electrons that have entered the conduction band at a certain point of the crystal; the points for different pulses are distributed randomly throughout the volume of the crystal. \(\lambda\) is the mean distance traversed by \(N_0\) electrons. \(q\) is \(Q/N_0e\), and \(n(q)\) is the number of pulses whose magnitude lies between \(q\) and \(q+dq\).

\[ n(q)=\frac{n_0}{1-\frac{qd}{\lambda}}, \]

where \(n_0\) is the total number of counted pulses. In this distribution \(q\) can vary from zero, when the \(N_0\) electrons are formed at the anode, to

\[ q_{\max}=\frac{\lambda}{d}\left(1-e^{-\frac{d}{\lambda}}\right), \]

when the electrons are formed at the cathode. The function \(n(q)\) is given in Fig. 16. Thus, by me-

as \(\lambda/d\) becomes smaller and smaller, the pulses become more and more alike in magnitude, but ever smaller. What has been said can be demonstrated still more clearly if one plots the integral distribution, i.e., the fraction of pulses that are greater than some specified value. This is given in Fig. 17. A more

Fig. 17. Integral distribution curve of pulse magnitudes for the same conditions as in Fig. 16. The fraction of pulses greater than a given \(q\) is presented as a function of \(q\).

instructive curve is plotted in Fig. 18, where the fraction of pulses is given as a function of \(q_{\max}\). For \(\lambda/d = 0.2\), \(76\%\) of the pulses turn out to be greater than \(0.9\) of the maximum value.

Crystals can also be used when the ionizing particle passes its entire path through the crystal parallel to the field. In this case the magnitude of the pulse, as can be calculated, will be:

\[ Q = N_{0} e \frac{\lambda}{d}\left[1 - \frac{\lambda}{d}\left(1 - e^{-\frac{d}{\lambda}}\right)\right]. \]

\(Q\) as a function of \(\lambda/d\) is shown in Fig. 19. The magnitude of the pulses will be proportional to the number of electrons formed in the crystal, and independent of \(\lambda/d\) for \(\lambda/d \gg 1\). For such quantitative measurements, it is probably necessary to have single crystals. To achieve this, a large grown crystal must be given the required shape by methods that would not produce undesirable stresses in the crystal. Mechanical machining of crystals on a lathe is unsatisfactory. The polycrystalline structure can be detected by etching with dilute hyposulfite, which makes it possible to notice the boundaries between crystals.

We saw, in considering the pulse shape from ionization counters, that, by differentiating the pulse arising as a consequence of the motion of the electrons, one can obtain a pulse whose magnitude is proportional to the number of ions \(N_0\). If a shortened delay line is used for the differentiation, we can see (from equations (4) and (10)) that the pulse magnitude is given by the expression

\[ Q' = 2N_0 \frac{v t_d}{d}, \]

where \(t_d\) is the decay time of the line, and \(v\) is the drift velocity of the electrons. When the same method is applied to a crystal counter, \(\lambda/d\) must in all cases be made as large as possible, and the differentiation time \(2t_d\) chosen so as to be small compared with \(\lambda/v\) or \(d/v\). For \(\lambda/d \sim 1\) and \(v \sim 10^7\ \text{cm/sec}\), a delay line with \(t_d \sim 10^{-8}\ \text{sec}\) should be used. With this method the pulse magnitude is less dependent on the specific properties of the crystal. The method should find application especially when the energy losses in the crystal are large.

Fig. 18. Integral distribution curve of the pulse magnitudes, on which the fraction of pulses greater than a specified fraction of \(q_{\max}\) is represented as a function of \(q_{\max}\).

Fig. 19. Magnitude of the pulses from a crystal counter as a function of \(\lambda/d\) for the case in which the ionizing particle traverses the entire path through the crystal from one electrode to the other, transferring electrons into the conduction band with uniform density along the whole track.

Efficiency of \(\gamma\)-ray counting

The absorption coefficient of AgCl for \(\gamma\)-rays with an energy corresponding to the absorption minimum (about \(5\ \text{MeV}\)) is approximately \(0.2\ \text{cm}^{-1}\). Thus, in a crystal \(1\ \text{cm}\) thick, up to \(18\%\) of the \(\gamma\)-rays are absorbed. At energies above or below this threshold

of this magnitude the efficiency will be greater as a result of the increase in the pair-production cross section at high energies and the increase in the cross section of the Compton effect and the photoelectric effect at low energies. By using a group of crystals, it should be possible to attain an efficiency approaching unity.

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