Amplitude and Time Measurement in Nuclear Physics

Amplitude and Time Measurement in Nuclear Physics E. BALDINGER University of Basel, Basel, Switzerland AND

W. FRANZEN University of Rochester, Rochester, N . Y . Page I. Introduction. . . . . . ......... . . . . 256 11. Amplitude Measurement of Signals of Variable Duration. . . . . . . . . . . . . . . . . 256 1. Introduction.. ...................... . . . . . . . . . . . . . . . . . . . 256 2. The Ballistic Deficit.. ............................................ 257 3. Applications of the Theory of the Ballistic Deficit.. . . . . . . . . . . . . . . . . . . 262 a. Rectangular Current Pulse Applied to RC-RC Amplifier.. . . . . . . b. Ionization Chamber Pulses. ................................. c. Delay-Line Clipped Pulses from Proportional Counters and Scintillation Counters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 111. Signal and Noise in Amplifiers and Physical Instruments.. . . . . . . . . . . . . . . . 268 ............................................... 268 1. Introduction 2. Noise Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 a. Thermal Noise. . . b. Reduced Shot Eff c. Grid-Current Noise.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 d. Induced Grid Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 e. Excess Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 f. Gamma-Ray and Recoil Nuclei Background Considered as Noise g. Some Remarks Concerning Correlated Noise Sources.. . . . . . . . . . . . . . 273 3. The Input Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 4. Linear Network Used to Achieve an Optimum Signal-to-Noise Ratio in Counting Applications.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 5. Practical Networks.. . a. RC-RC Amplifier. . b. Remarks on Noise Due to Background Pulses.. . . . . . . . . . . . . . . . . . . . 285 c. Delay-Line Clipping. ...... . . . . 286 IV. The Timing of Nuclear Events.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 1. Survey o f t h e Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 2. Characteristics of Scintillation and Cerenkov Counters. a. Statistical Fluctuations in the Decay of Scintillators.. . . . . . . . . . . . . . . 292 b. Conditions for Obtaining Optimum Time Resolution with Scintilla297 tion Counters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c. Transit-Time Dispersion in Photomultiplier Tubes. . . . . . . . . . . . . . . . . 301 255

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Page 3. Operation and Classification of Coincidence Circuits. . . . . . . . . . . . . . . . . . 302 a. Parallel Coincidence Circuits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 b . Series Coincidence Circuits.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 c. Bridge Coincidence Circuits. . . . . . . . . . . . . . . . . . 309 4. The Measurement of the Lifetime of Short-Lived Excited States. . . . . . . . 31 1 General Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

I. INTRODUCTION The problems of particle detection in nuclear physics have frequently stimulated significant developments in electronics. One reason for this is that the interaction of particles with matter, even in the case of quite energetic particles, gives rise to comparatively minute electrical effects. The observation and detailed analysis of these effects often require us to approach the ultimate limits of measurement. Thus nuclear physicists have made important contributions to the study of noise in physical instruments, t o the development of accurate pulse-height analyzers, to the design of linear amplifiers and stabilized power supplies, to the subject of timing devices and precision current integrators, and several other fields of electronics. However, the emphasis in nuclear physics has always been on electronics as a means to an end, rather than on the study of electronics as an end in itself. It is this emphasis which lends a certain characteristic, perhaps somewhat unsystematic but always stimulating flavor to the subject. The variety of problems of measurement which arises in nuclear physics is very great, and many different approaches t o these problems have been envisaged. For this reason, it is not possible in a summary of this sort to present more than a limited selection of topics. Evidently, such a selection is influenced a great deal by the experience and personal inclinations of the authors. Our presentation should therefore not be taken to be an exhaustive account of electronics as used in nuclear physics, but rather as a discussion of those topics which, in our own experience, we have found useful and challenging. For a more complete account of the subject, the reader is referred to the books and review articles in the General Bibliography. 11. AMPLITUDEMEASUREMENT OF SIGNALS OF VARIABLEDURATION 1. Introduction

Nuclear particle detectors that are employed as energy-measuring devices generally produce small signals th at must be amplified before they can be analyzed and recorded. This is true of primary detectors, such as ionization chambers, as well as of detectors in which a n electrical multi-

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plication process is utilized to increase the size of the signal, as in proportional counters and in scintillation counters with photomultipliers. I n the case of the multiplication devices, the charge avalanche arriving on the collector electrode must not exceed a certain limiting size beyond which space charge effects destroy the proportional relation between the original and the multiplied signals. On the other hand, the circuits commonly employed for the accurate measurement of voltage-pulse amplitude require sizable signals, of the order of several tens of volts. Thus the need for amplification arises. The pulse amplification necessary in energy measurements of this type is peculiar in that it is usually not necessary to reproduce the original shape of the electrical signal. Instead, what is desired is a ballistic signal, that is, a voltage signal whose amplitude is proportional to the total charge that appeared on the collector electrode of the detector. The actual shape of the output signal is then determined primarily by such considerations as the need for achieving a high signal-to-noise ratio, or for avoiding excessive overlap of successive signals. Our discussion thus will be restricted to a n analysis of the conditions under which electrical measurements with pulse amplifiers can be considered to be ballistic measurements. I n particular, we should like to investigate the effect of the interference between the inherent time constants of the amplifier and the duration of the signal. Signals of variable duration are primarily encountered in ionization chambers because of the varying location and orientation of the ionization tracks as well as the different mobilities of its electron and positiveion components. However, a certain amount of fluctuation in pulse duration is encountered in all nuclear particle detectors. Our theory thus is generally applicable to all types of ballistic charge measurements (including measurements made with ballistic galvanometers), as we shall show. 2. The Ballistic Deficit The signal arriving a t the collector electrode of a particle detector as a consequence of a primary event consists of a short current pulse I ( t ) . For the purpose of an energy measurement, we should like to determine the time integral of this transient current. An input circuit of the type T pictured in Fig. l a will produce a voltage signal proportional t o I(t)dt with arbitrarily high accuracy provided the time constant T~ = RoCo is made sufficiently large. I n practice, however, it is necessary to keep the output pulses as short as possible in order to avoid overlap of successive pulses, and the d.c. component of the signal pictured in Fig. l b must be removed t o prevent bias shifts. Furthermore, the bandwidth of the subse-

/o

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E. BALDINGER AND W. FRANZEN

quent amplifier (necessary for the reasons stated previously) must be limited in order t o avoid an excessively poor signal-to-noise ratio. Thus, practical considerations demand use of a network having transient properties considerably different from those pictured in Fig. 1. Since the time constants of the network in a practical case may not be very much longer than the duration of the signal, the part of the output signal stimulated by the early part of the current pulse will have started to decay before the current pulse has ended. As a result, the amplitude of the COLLECTOR ELECTRODE

(a)

INPUT

CURRENT

OUTPUT SIGNAL s ( t )

I(t)

v r 1 f T ) d r if

0

RoCo

>>T

t-

FIG. 1. (a) Illustration of the simplest type of charge integrating network for total charge determinations. (b) Input current and output voltage as a function of time. I n actual use, the condition RoCo>> T cannot be satisfied because of the excessively long duration of the voltage signal.

output signal is no longer exactly proportional to the charge arriving a t the collector, as illustrated in Fig. 2. The difference between the voltage amplitude resulting from a current pulse of finite duration and the amplitude resulting from an infinitely short current pulse carrying the same total charge is termed the “ballistic deficit’’ of the measurement ( I ) . It is of interest to determine the quantitative relation between the ballistic deficit and the duration of the signal. For this purpose let us assume that g(t) is the amplifier output signal resulting from the application of a current pulse of unit charge but indefinitely short duration (i.e., a delta-function current pulse) to the input a t t = 0. Then g(t - 7) is the

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output resulting from a delta-function input a t t = 7 . If the amplifier is linear, the response to an indefinitely short pulse of charge q a t 2 = T is therefore q . g(t - T ) .

FIG.2. Illustration of the concept of the ballistic deficit. We compare the output signal g ( t ) resulting from a delta-function-like input current pulse with the output s ( t ) resulting from the actual current pulse of duration T and carrying the same charge &. The maxima of g ( t ) and s ( t ) are reached at times t o and tm, respectively. The ballistic deficit is defined as A0 - A,.

Now suppose that an actual current signal I ( t ) of finite duration T is applied t o the input. We can subdivide I ( t ) into an arbitrarily large number of delta-function-like pulses. Thus a pulse of infinitesimal duration d7 at t = 7 carries a charge p = I ( T ) and ~ gives T rise to an output signal I ( T )Q T )(~~ T The . total output amplitude at t is obtained by integrating this expression from 0 to t : s(t) =

If I ( t ) = 0 for t

> T , then s(t) =

Ju

t

1(7)g(t -

(2.1)

7)dT

jOTI(7)g(t -~)dr,t

>T

(2.2)

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E. BALDINGER AND W. FRANZEN

Thus the output signal is a convolution of the response to a delta-function input with the actual current signal. Now it is possible to distinguish between two cases: the maximum of s ( t ) may occur either before or after the current pulse has ended. I n the notation of Fig. 2, either t, > T or t, < T . If the charge measurement is to be nearly ballistic, that is, performed in such a way that the ballistic deficit is small, we should usually expect* that t, > T . I n that case, t, is given by solving the expression

for t, and the maximum amplitude is A ,

= s(t,).

FIG.3. Illustration of the type of current pulse variation assumed in the discussion of the ballistic deficit. The current pulse a t right differs from that a t left by a stretching of the time axis by a factor k , while the amplitude of the current is reduced by the same factor. The total charge is Q in both cases.

To investigate the influence of the variation in pulse duration T on A,, we shall make the assumption that a change in T contracts the current and distends the time axis, or vice versa, by the same factor k , while the total charge Q remains constant, as illustrated in Fig. 3. Thus, I ( k t ) = ( l / k ) * I ( t ) and

Q

=

/o

T 1(7)dT

=

/oT k ~ ( k T ) d T

(2.4)

It is convenient t o set the charge Q equal to unity. Since k can be chosen arbitrarily, we may set k = 1 / T . The normalization condition (2.4) may then be written

* This condition is not necessarily satisfied if the current signal decreases slowly. An extreme case would be an exponentially decreasing current signal, as obtained for example in scintillation counters. In that case, T = m , but the ballistic deficit is nevertheless small provided t, > U , where u is the time constant associated with the exponential decay of the signal, as we shall show later on.

261

AMPLITUDE AND TIME MEASUREMENT IN NUCLEAR PHYSICS

where z = r / T . The expression for the output signal (2.2) becomes

s(t)

=

h1

I(z)g(t - Tz)dz

and t, is now defined by

Evidently t, is a function of T only, and we can expand it in a Taylor series about the point T = 0:

t,(T)

= to

+T

(T;)

-

T=O

+ 81 T 2 ( “dT2 -) + T-0

Here t o = t,(O) is the time a t which the maximum A0 of the response function g(t) occurs. To find the value of (dt,/dT)T,o, consider the function

Evidently, .. and, therefore,

This expression shows that t o a first approximation the output pulse maximum of a current pulse of finite duration is delayed by a n amount t, - t o Z T a l l which is equal to the centroid of the pulse I ( t ) . The amplitude A , = s(tm) is also a function of T alone, so th a t we can write

It is easy t o see that (dA,/dT)T=o are given by

=

0 always. The higher derivatives

(2.10)

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E. BALDINGER AND W. FRANZEN

It is interesting to observe* that these two coefficients no longer involve convolution integrals, but merely products of terms t h a t depend on the current distribution I ( z ) and on derivatives of the response function g(t). Of principal importance in practice is the coefficient of the T 2 term, as given by (2.9). This coefficient is necessarily negative, since g”(t0) must be negative if g(t) has a maximum at t = t o and since 1 a2 = I ( z ) ( a l - z)2dz, the second moment of the current distribution, is a positive definite quantity. Thus to order T 2 the ballistic deficit is given by

L

A O- A ,

IOTI ( r ) ( a l -

-+~‘T2g”(to)

r)2dr

(2.11)

T

with a1 = rI(r)dr. For convenience in practical computation, we have used as a variable of integration the time ( 7 ) rather than z = r / T used previously.

3. Applications of the Theory of the Ballistic Dejicit

a. Rectangular Current Pulse Applied to RC-RC Amplijier. As an application of the theory just presented, consider a rectangular current pulse of duration T and unit charge which is first integrated by means of a network similar to t h a t shown in Fig. l a and is then amplified by means of an RC-RC amplifier. An amplifier of this type is characterized by the presence of two time constants 71 = R I C l and r2 = R2C2, which limit its low-frequency response and high-frequency response, respectively (see Section III,5 and Fig. 18). I n general (Z),

is the response to a unit delta-function current pulse applied t o the input capacity COa t t = 0. G is a constant characteristic of the gain of the amplifier. I n order to achieve an optimum signal-to-noise relation, one would usually choose 7 1 = 7 2 = rp, as explained in Section III,5. I n that case, we obtain for the response function (2.12)

* It should be noted that the series does not continue to have the simple form indicated for the coefficients (2.9) and (2.10). The next term is already considerably more complicated:

AMPLITUDE

AND TIME MEASUREMENT IN NUCLEAR PHYSICS

263

This function attains a maximum of A0 = G/eCo at t o = 7p. Direct application of formula (2.11) then leads to the following expression for the relative ballistic deficit :*

Ao

(2.13)

I n Fig. 4 a n exact calculation of the output pulse amplitude and the prediction of (2.13) are compared. It is interesting to observe th a t the ballistic deficit is only about 4% of the maximum pulse amplitude when the duration T of the current pulse and the equal time constants r p of the amplifier are equal. If the ballistic deficit is to be less than >d%, then T should be about three times smaller than rp. This case has been analyzed from a somewhat different point of view by Gillespie ( 3 ) . b. Ionization Chamber Pulses. Rectangular current pulses are obtained frequently in parallel-plate ionization chambers with grid, used as electron pulse chambers (4, 5 ) . If the ionization track of an energetic charged particle which comes t o rest in the gas of the chamber is much longer than the spacing between the collector electrode and grid, the current pulse produced by electrons drifting through the grid wires will be approximately rectangular in shape, but its duration will depend strongly on the direction of the track relative t o the grid plane. Furthermore, for a properly designed grid chamber, individual current pulses will carry the same charge if they originate from monoenergetic events. Thus all the conditions of the example above are fulfilled, and we should expect a varying ballistic deficit, depending on the inclination of each track. This variation should contribute to the width of the experimentally observed pulse-height distribution. The fluctuation in signal amplitude due to varying pulse duration represents a particularly serious problem in slow ionization chambers in which the charge carriers are heavy positive and negative ions. Since these particles move much more slowly under the influence of a n electric field than electrons, the low-frequency limiting time constant of the amplifier used with the chamber must be correspondingly longer than in the case of a n electron pulse chamber. On the other hand, in slow

* We note that the third-order term given by Eq. (2.10) vanishes in the case of a rectangular pulse. It is evident from inspection of (2.10) that any current pulse which is symmetrical about its centroid 011 will give a vanishing third-order term. The next nonvanishing term of the expansion (2.8) for this case is of fourth-order in TIT,, so that to this order Ao The additional term is, however, usually negligible in a practical case.

264

E. BALDINGER AND W. FRANZEN

ionization chambers the loss of charge caused by initial recombination is usually significant since ions of both signs spend a comparatively long time near each other. To correct for this loss, it is customary to study the size of the amplified signal as a function of the electric field intensity and then to extrapolate to zero loss by use of the Jaff6 theory of recombination (6, 7). However, if the ballistic deficit is appreciable, the change 50 -

20.

z I-

E 10w (r

a

z

n

0 c

2

I-

U A

m

W

20.5.

2a W

at I

ai

0.2

05

ID

2.0

T - 5.0 7.

FIG.4. Comparison between the relative ballistic deficit as a function of T / T O calculated exactly (solid line) with the value obtained approximately (dashed line) by use of Eq. (2.13) for the case of a rectangular current impulse of duration T amplified by means of a n RC-RC amplifier having equal time constants T O .

in field intensity will affect the size of the ballistic deficit as well as the recombination loss. Haeberli et al. (8) have shown how a correction can be made for both effects so th at saturation curves in agreement with the Jaff6 theory can be obtained, as illustrated in Fig. 5. c. Delay-Line Clipped Pulses from Proportional Counters and Xcintillation Counters. Proportional counters and scintillation counters with photomultiplier tubes utilize a cascade multiplication process to increase

AMPLITUDE

AND TIME MEASUREMENT IN NUCLEAR PHYSICS

265

the size of the current signal at the collector electrode. The resulting signals are usually so large that amplifier noise is no longer a n important factor in determining the spread of experimental pulse-height distributions. Nevertheless, as explained previously, amplification is usually still necessary, and for a short low-frequency limiting time constant (used t o obtain a short pulse duration), a ballistic deficit is to be expected.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7 cmlkv

1 E

FIG. 5. Saturation curves of Po-alpha particles in a COz-filled parallel-plate ionization chamber obtained by Haeberli et al. (8). A is the amplitude of the output pulse and E is the electric field strength. The Jaff6 theory of recombination predicts a linear relation between the reciprocal of the total charge Q and the reciprocal of the field strength. Curves I and 2 show the relationship obtained experimentally for two different amplifier time constants 7 0 , while curve 3 shows the same relationship after correction for the ballistic deficit.

It should be noted that the multiplication process has the effect of uniformizing, t o a large extent, the shape of the individual current signals. Therefore, the ballistic loss will have the same relative effect on all signals. We are primarily interested in investigating then to what extent the ballistic deficit limits the minimum pulse duration a t the output of the amplifier. A characteristic feature of both proportional counters and scintillation counters is the slow decay of the current signal I ( t ) . I n the case of a proportional counter, this slow decay is principally caused by the slow motion of the positive ions (formed during the multiplication process in the high field region near the central electrode) toward the outer electrode. I n scintillation counters, the current signal decays a t a rate determined by the rate of light emission in the scintillating crystal, provided we are

266

E . BALDINGER AND W. FRANZEN

dealing with a relatively slow crystal such as sodium iodide. Thus in both cases our analysis requires modification inasmuch as the condition T < t, [see derivation of Eq. (2.3)] is no longer satisfied. We can nevertheless draw useful conclusions if we assume a specific response function for the amplifier. Since amplifier noise is no longer a n important consideration, the response network of the amplifier can be selected with a view to achieving the smallest possible ballistic deficit combined with minimum signal duration. These two requirements are best met with a delay-line clipped amplifier having a single RC time constant of integration. An amplifier of this type is shown in Fig. 21 (page 287). The high-frequency response of the amplifier is limited b y the time constant r 2 = R2C2, whereas its low-frequency response is limited by a shorted delay line. I n a delay line of this sort, the signal is applied t o one end of the line where the line is terminated in a resistance equal to its characteristic impedance. At the other end of the line, a traversal time T d away, there is a short circuit which reflects any incoming signal with reversed phase. At a time t = 2 r d , therefore, the reflected signal will begin t o cancel the applied signal. I n addition t o these circuit elements, a coupling time constant r1 = RlCl in general is also present. The response function of the amplifier is then evidently

I n this expression, COis the capacitance of the input capacitor and G is a constant characteristic of the gain of the amplifier. The output signal resulting from a n input current signal I ( t ) is then, in accordance with Eq. (2.11,

It is possible t o find the time t, at which the maximum value of this function occurs by setting ds(t)/dt = 0 and solving for t. The general solution of this problem depends in detail on the form of I ( t ) and cannot be formulated as simply as in the cases discussed previously. If we are interested, however, in achieving a minimum signal duration for a given I ( t ) independent of amplifier noise considerations, r1 and r2 are generally chosen in such a way that the rise time of the output signal is entirely

AMPLITUDE AND TIME MEASUREMENT IN NUCLEAR PHYSICS

267

determined by the rate of decay of I ( t ) and the decay of the signal is determined by the delay time 2 T d . Thus we would set r 1 >> T d and r2 << T d . I n that case, we can approximate the response function g ( t ) by gl(t) =

and gz(t)

=

0 for t

> 27d.

G

C, for 0 < t < 2 7 d

The output signal then becomes

and

If I ( t ) is a monotonically decreasing function of time, as is true for the multiplication devices, s ( t ) under these conditions will reach its maximum value always at t, = 2Td. This can be seen by considering that dsl/dt = (G/Co)I(t)is necessarily positive, while

is necessarily negative if I ( t ) decreases monotonically. * The maximum value of s(t) is then

A,

= s(tm) = -

co

I(T)dT

so that the relative ballistic deficit is (2.16) For a cylindrical proportional counter, the multiplication process can be assumed to take place in the immediate vicinity of the central wire. I n that case, the current signal caused by the motion of the positive

* It is to be noted that the choice of time constants made above is quite different from the choice recommended in Section III,5 for a delay-line clipped amplifier on the basis of signal-to-noise considerations. As we have explained, these considerations are not important here owing to the size of the original signals. The attainable pulse-height resolution is then determined by statistical fluctuations in the number of primary events in the counter and by the statistics of the multiplication process. I n order to include as many primary events in the charge integration as possible, the ballistic loss should be small.

268

E. BALDINGER AND W. FRANZEN

ions is given by (9, 10) (2.17)

I n this expression, T is the time taken by the positive ions t o reach the outer cylinder of radius b. The quantity a is the radius of the central electrode. I' can be shown to be given by

where P is the mobility of the positive ions, p is the pressure of the counter gas, and Vo is the applied potential. Generally, T is quite long (of the order of several hundred microseconds). If we apply a current signal of the type given by Eq. (2.17) to the delay-line clipped amplifier, the relative ballistic loss will be

If we assume that b >> a, as is usual in practice (a common value for b / a is lOOO), the loss of signal will be 50% when 2r,/T = a/b. For a small ballistic loss we would therefore require that 2rd >> (a/b)T. I n scintillation counters, the current signal at the collector electrode of the photomultiplier tube can be described by a n expression of the form (11) I ( t ) = Ioe-t/u,provided the decay constant (T is much larger than the spread in transit time of the electron avalanche. This is true for Na I crystals which are extensively used for gamma-ray spectroscopy. I n th a t case, Q = 0.25 psec (16). The ballistic loss is then simply

Thus, we require 2rd >> u in order to have a small ballistic loss. 111. SIGNALAND NOISE IN AMPLIFIERSAND PHYSICAL INSTRUMENTS 1. Introduction

Many physical measurements can be reduced to the problem of determining the properties of electrical signals. The precision of measurement is limited by the unavoidable presence of noise. By the term "noise" we shall understand any unwanted part of the observed electrical signal. If we consider the electrical signal appearing a t the output of a n amplifier, i t is clear th at it is always possible to pass this signal through a

AMPLITUDE

AND TIME MEASUREMENT IN NUCLEAR PHYSICS

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linear network which will contribute a negligible amount of noise provided the preceding amplification can be made as large as we wish. Thus we may describe our electrical measuring system as consisting of a n input circuit, an amplifier, and a subsequent linear network. We are interested in discussing the factors influencing the design of these circuit elements when the best determination of a specific property of the electrical signal is desired. I n this connection, several considerations are of importance. As a n introduction, a brief review of different noise sources will be given, and the requirements to be fulfilled by the input circuit for the best signalto-noise ratio will be analyzed. I n certain respects, this analysis can be made independent of any special assumptions concerning the signal, or the specific property of the signal to be measured. I n the following section, some general remarks concerning the achievement of a desired overall transmission function consistent with the best signal-to-noise. ratio will be made. Finally, we shall discuss the design of the input circuit and linear network which will yield the best signal-to-noise relation in energy measurements with nuclear particle detectors. 2. Noise Sources

The number of possible noise sources is large. Among the most important types of noise are the following: 1. Thermal noise. 2. Shot effect and reduced shot effect. 3. Grid-current noise. 4. Noise due to induced grid current. 5 . Excess noise in thermionic tubes (flicker effect) and carbon resistors. 6. Fluctuations caused by random division of current between electrodes. 7. Noise attributed to faulty tube construction (hum, poor insulation, vibration, varying wall charges). 8. Background caused by gamma-rays, recoil nuclei, etc., in particle detectors. 9. Fluctuations in the emission of thermal electrons from photocathodes. Fortunately a number of excellent reviews are available on the subject of noise and spontaneous fluctuations (13-16). For this reason, we shall restrict our discussion of noise sources to a brief survey. a. Thermal Noise. A fluctuation is associated with every dissipative process (17, 18). Thus, the complete description of the losses in an electrical circuit involves a resistance R with which a noise-current generator

270

E. BALDINGER AND W. FRANZEN

or a noise-voltage generator is associated. The symbol R represents the systematic and the noise generator the statistical aspect of the dissipation. Consider the noise-current generator connected with the resistance R in Fig. 6a. I t s noise spectrum can be described by a mean-square noise current per unit frequency range given by

where T is the absolute temperature of R. As was shown rather early (19, do), this relation still holds if different resistors in a circuit have different temperatures.

FIG.6. Two equivalent descriptions of the losses and associated fluctuations in an electric circuit.

b. Reduced Shot Eflect. I n a triode, statistical fluctuations of the plate current I , give rise t o a mean-square noise current per unit frequency range

-

is2= 2eI,r2

(3.2)

where the factor r is a measure of the “smoothing” action of the space charge and e is the charge of the electron. As an example, let us consider a 6J5 tube. Assuming a plate current of 9 ma, the factor r2turns out to be -0.04. The finite transit time of the electrons from cathode to anode causes the noise spectrum to decrease a t high frequencies. Pentodes are noisier than triodes. The additional noise in pentodes is caused by the random division of the smoothed cathode current between plate and screen grid. The fluctuations in the number of electrons arriving a t the screen per unit time can be described to a good approximation by a Poisson distribution. Owing to the fact that the cathode current is smoothed by space-charge effects, these fluctuations are partly correlated with fluctuations in plate current. c. Grid-Current Noise. Electrons and ions arrive a t the control grid almost completely a t random. The mean-square noise current per unit frequency range is therefore

iO2 = 2eI,

(3.3)

AMPLITUDE AND TIME MEASUREMENT IN NUCLEAR PHYSICS

271

where I , is the sum, without regard t o sign, of the positive and negative components of the grid current. Assuming only singly charged ions, e is the elementary charge. This type of noise is important when a floating grid is employed in the first stage of amplification. d. Induced Grid Current. At very high frequencies the input to vacuum tubes has a conductive admittance. This effect is a consequence of the finite transit time of the electrons passing through the grid wires and can be described as follows. A single electron approaching the grid structure will induce image charges. The changing image charge represents a current which flows in one direction during the approach of the electron and in the opposite direction later, the time integral of the current being zero. The combined effect of all the electrons traveling through the vacuum tube is then described as the induced grid current. The induced grid current is subject to fluctuations which cause a fluctuating voltage t o be developed across the grid-circuit impedance, and this in turn will cause fluctuations in the plate current. It is interesting to note that these fluctuations are partly correlated with fluctuations that exist in the anode current independent of the induced grid current (i.e., t ha t exist when the grid-circuit impedance is made zero). Tubes like the 6V6 (16) with a large spread in electron orbits show no correlation of this sort. It is possible to eliminate most of the correlated part of this extra tube noise, as has been shown experimentally (16, 21, 22). The mean-square induced grid current increases with the square of the frequency up t o frequencies of a few hundred megacycles per second. e. Excess Noise. Flicker effect is a term used t o describe the large amount of low-frequency noise generated in tubes with oxide-coated cathodes. The mean-square noise current per unit frequency range can be described by a formula of the type I" i,z P w = 2TV (3.4) wb ' N

where a is usually close to 2 and b is close to 1 over a rather wide frequency range. A similar formula holds for the excess noise in carbon resistors, crystal diodes, photoconductors, and transistors. The occurrence of a similar noise spectrum in all these cases suggests a similar physical mechanism. For a discussion of excess noise, the reader is referred t o Van der Ziel's book (16). Good-quality wire-wound resistors produce only thermal noise and no excess noise. I n vacuum tubes, the magnitude of the flicker effect varies from tube to tube. The best tube and the best operating point must be found experimentally. Figure 7 shows the equivalent noise resistance a t low frequen-

272

E . BALDINGER AND W. FRANZEN

cies for a Phillips E80F tube selected from about thirty tubes for minimum excess noise (23).The rise of the equivalent noise resistance at low frequencies is due to flicker effect. f . Gamma-Ray and Recoil Nuclei Background Considered as Noise. As an example, suppose we consider the measurement of nuclear reaction energies by means of an ionization chamber in the presence of a background of pulses produced by gamma-ray secondaries. Since we wish to perform a ballistic charge measurement, we may consider the background current pulses as indefinitely short; th at is, they have the character of delta-functions in time. (This assumption is valid if the ballistic deficit, as defined in Section 11, is small.) Such delta functions have a frequency

.az I 0

1

Reduced shot -effect

2

I

3

5

4

i 6

. 7 v( kc)

8

FIG.7. Equivalent noise resistance of a Phillips E80F tube (triode connected). Operating conditions: heater, 6.3 v; plate current, 0.5 ma, grid bias, -1.5 v.

spectrum independent of frequency (a “white” frequency spectrum) , similar to the frequency spectrum of shot noise in the first tube of a n amplifier. Therefore, we can apply an expression for the mean-square noise current per unit frequency range similar t o the expression used for the shot effect:

-

iT2= 2

1q d I , / =

2

q2N(q)dq

(3.5)

Here q = n e is the charge released in a single background event (e = elementary charge) and N ( q ) d q is the number of events per second with released charge between q and q dq. If we set p = J N ( q ) d p = total number of background events per second, so that J q 2 N ( q ) d q= p? = p e 2 G , then

+

i,2

= 2pe22

(3.6)

AMPLITUDE AND TIME MEASUREMENT IN NUCLEAR PHYSICS

273

This shows that the mean-square noise current per unit frequency range is proportional t o the number of background events per second p, and to the mean square number of charges n’i released per event. By a similar argument, background due to recoil nuclei in neutron experiments with ionization chambers may be considered as noise. g. Some Remarks Concerning Correlated Noise Xources. Let us assume an amplifier with two separate outputs and one input as shown in Fig. 8. We shall assume further that the outputs have different signal-to-noise ratios and t ha t their fluctuations are completely correlated. By observing the voltages at both outputs, it is evidently possible to eliminate the fluctuations completely. If the signal-to-noise ratio a t the two outputs is the same, no cancellation of the noise is possible without cancellation of the signal. OUTPUT I

INPUT

OUTPUT 2

FIG.8. Amplifier with one input and two outputs.

As another limiting case we may assume that the fluctuations are independent of each other. On this assumption, a more precise value for the signal may be achieved by observing both outputs a t the same time. It is easy to discuss mixed cases, where the noises are partly correlated. Instead of improving on the accuracy of measurement by observing both output voltages and using calculations to get the most probable value for the signal, i t is always possible to invent a device which does this calculation automatically. It can be shown th at observation of output 1 alone and using proper feedback from output 2 t o the input will just do the desired job (94). A pentode is a good example for such a system, inasmuch as one output is the anode and the second output the screen grid. Both outputs may have different signal-to-noise ratios and partly correlated fluctuations. 3. The I n p u t Circuit

Let us assume that the signal source has an infinite internal impedance and may therefore be represented as a current generator. This is true t o a good approximation for most signal sources used in nuclear physics (ionization chambers, proportional counters, scintillation counters, and detectors used in nuclear induction experiments). The input circuit will be assumed to consist of an impedance 2 = 1/Y paralleled by a parasitic capacitance C , which includes the capacitance of the signal source and the input capacitance of the amplifier, as sketched in Fig. 9.

274

E. BALDINGER AND W. FRANZEN

The thermal agitation in the input circuit is indicated by the current generator z 2 = 4IcT@(Y),where @(Y)indicates the real part of the input admittance Y = 1/Z. The fluctuations due to the amplifier are represented by the current source 3,which represents th a t part of the fluctuations acting directly on the input circuit (for example, grid-current fluctuations), and by the voltage source 2, which represents the other noise sources, reduced to the input grid. For simplifying the discussion, we shall exclude any relevant correlation of 2 with other noise sources. r-----------

I

I I

I I

I I

I I

yp

___________

L

J

SIGNAL SOURCE

-

I I I

INPUT CIRCUIT

I

i', I

I I

AMPLIFIER

FIG.9. Amplifier input circuit with signal and noise sources.

Considering thermal agitation alone, the signal-to-noise ratio (where we define the noise in terms of the r.m.s. noise current) is evidently inversely proportional to the square root of the real part of the input admittance:

(3.7)

A good signal-to-noise ratio requires a small value of 6i(Y ) .This means

that the losses of the circuit should be as small as possible. Small losses result in a good power transfer from the signal source to the input circuit, whereas the total thermal fluctuating energy of the capacitor C integrated over all frequencies (45 IcT) is constant. The purpose of the input circuit can be described as consisting of the transformation of the original current signal into a voltage signal which should be as large as possible in order to minimize the relative importance of noise sources in the amplifier, represented by 2. Therefore, we are dealing with a network of maximum gain type (interstage networks in wide-band amplifiers would also represent such a type), and C should be as small as possible. For a proof of this statement and further discussion, the reader is referred t o Bode's book (25). As a n example, consider a signal with a frequency spectrum centered around COO, as pictured in Fig. 10. (This would correspond to the type of signal encountered in nuclear induction experiments.) A simple model of a n input circuit suitable for the detection of such a signal is the resonant

AMPLITUDE AND TIME MEASUREMENT IN NUCLEAR PHYSICS

275

circuit sketched in Fig. 11. This circuit is tuned to the frequency

where the signal spectrum has its maximum. The impedance of the input circuit can be written

z =1 + jwRC[(w2 R - wo2)/w2I

(3.8)

Evidently, the signal-to-noise ratio for any frequency interval is highest and the additional noise sources 2 in the amplifier are relatively

FIG.10. Frequency spectrum to be measured with the circuit of Fig. 11.

least important when the losses are as small as possible (i.e., the resistance R is as large as possible) and when G is as small as possible. Any deviation from this condition, as represented for example by a parallel resistance (used to increase the bandwidth), will result in a poorer signalto-noise ratio for any frequency interval.* Now suppose that for some special reason (connected for example with the type of information to be derived from the signal) the frequency band to be transmitted is to be made larger than Ao = 1/RC, the band-

* In the circuit of Fig. 11, the conductive admittance of the input circuit for an infinitely high frequency is not zero, but has the value (R(l/Z) = l/R, = 1/R, where (R(l/Z) denotes the real part of the admittance. This means that the circuit pictured in Fig. 11 will not give the best over-all gain possible, for, according to Bode (66),one can represent the difference in gain-bandwidth area between a circuit with C alone and a circuit of impedance Z by means of the integral O0

[log

(&) - log IZI]

dw =

kc

In our case, R contains only the unavoidable losses and is therefore presumably a large resistance. Thus, practically nothing can be gained in gain-bandwidth area by using a more sophisticated network, whereas the losses would certainly increase with any added component.

276

E. BALDINGER AND TV. FRANZEN

width of our sharply tuned resonance circuit. From our discussion, it is clear that this increased bandwidth should not be achieved b y damping the input circuit by means of a parallel resistance. It is much better to use instead a linear network a t the output of the amplifier. It is interesting to note that even an additional parallel damping resistor cooled t o absolute zero is not as good as the arrangement just

----------- ______

*-

-0

<< I

C

8 -_________ J _______

-0

OUTPUT

0

INPUT CIRCUIT

FIG.11. Input circuit for the detection of the signal shown in Fig. 10.

mentioned. Cooling would help from the point of view of signal-to-noise ratio, however, if no parallel resistor is used, but all the unavoidable losses in the circuit are cooled down. As another method for obtaining a large bandwidth, consider the feedback circuit pictured in Fig. 12. We assume th a t the feedback may be represented as a current generator, the current being controlled by the output voltage in a prescribed manner. From the point of view of I

-INPUT - - - ---CIRCUIT -- -___-__ -

AMPLIFIER

FEEDBACK: If.Gf(w)Y.

FIG.12. Amplifier using feedback to achieve a desired frequency response.

signal-to-noise ratio, this system is just as good as our original system, provided the feedback network adds no losses to the input circuit and does not increase the input capacitance C. In practice, additional losses and some increase in C are unavoidable. Thus, we may say that a feedback system can be almost as good as our original arrangement, but cannot improve on it. However, in many cases where the slight loss in signal-tonoise ratio can be tolerated, feedback is a simple means for achieving a desired over-all frequency response.

AMPLITUDE AND TIME MEASUREMENT IN NUCLEAR PHYSICS

277

It may be noted here that a feedback system can really be a refrigerator. The total thermal fluctuating energy of the input capacitor is, according t o the principle of equipartition, >.jkT = >5C7, so that 2 = k T / C is the mean-square fluctuating noise voltage integrated over all frequencies. Negative feedback will reduce these fluctuations to a smaller value, provided the noise sources in the amplifier are small enough.* Thus such a feedback system is equivalent to the additional cooled damping resistor discussed earlier. The remarks made here are valid not only for purely electrical measuring systems, but are true quite generally. Thus suppose we consider a n electrometer sufficiently sensitive to permit observation of Brownian motion. Let us further assume that for practical reasons we wish the system t o have a n aperiodic behavior. It would be poor practice to increase the air damping until the system is critically damped. (The air damping corresponds to the introduction of a dissipation into the input circuit.) On the contrary, air damping should be made as small as possible. This could be accomplished b y placing the electrometer into a vacuum vessel. Then the desired aperiodic response can be achieved either by means of a correcting network added to a n electrical signal pickup system which records the position of the electrometer light-spot, or by feedback. Milatz (26, 27) has described a feedback system of this type in which the position of the light spot is observed by means of a photocell. The photocell is followed b y an amplifier which feeds a signal back to the input of the electrometer in such a way that the desired frequency response is achieved. Since air damping was rather small, the background due t o Brownian motion could be reduced very substantially by this method.

4. Linear Network lised to Achieve a n Optimum Signal-to-Noise Ratio in Counting Applications

The example of the electrometer shows that our conclusions regarding the design of the input circuit are quite useful. However, they are not

* The following objection to this argument could be made. From the excellent work of Bode (25) it is known that it is not possible to maintain negative feedback in a multistage feedback system over an infinite frequency region. Regions with positive feedback must exist. I n these frequency regions the fluctuations in the input circuit will be larger with feedback than without it. Thus, one may ask whether the effect of the introduction of feedback is merely to redistribute the fluctuations in the input circuit over different frequency regions, without changing the total fluctuation energy integrated over all frequencies. Calculation shows, however, that the total fluctuating energy can be “cooled” down so that the input circuit is no longer in thermal equilibrium with its surroundings. This, of course, requires expenditure of energy, which in this case is supplied by the amplifier power supply.

278

E. BALDINGER AND W. FRANZEN

sufficient in general to design the input circuit which is best from the point of view of signal-to-noise ratio. For a further discussion, it is necessary t o define the specific property of the signal to be measured, and it is also desirable to take advantage of all known properties of the input signal. We may distinguish several measuring problems such as the following: 1. The signal as a function of time is unknown and should be observed. The problem of giving the best prediction of the input signal is a subject of information theory and is treated for example by Bode and Shannon (68). 2. The time-difference between two signals should be measured as accurately as possible. 3. The signal is of the form y = A.s(t), where s ( t ) is a known function of time, and the quantity to be measured is the amplitude A . SIGNAL Aisi(t)

I

/

-

OUTPUT

GI (jw)

NOISE NOISE' N~

FIG.13. Arrangement used to calculate the optimum linear network for a measurement of A o .

Problem (3) often occurs in nuclear physics and will be discussed in this section. I n many applications, we assume th a t the signal consists of the sudden appearance of a charge Q on the capacitance associated with the input circuit of the amplifier. (This corresponds to a ballistic measurement of charge with small ballistic deficit.) The charge Q produces a small voltage step a t the input, and this step is amplified and transformed in a linear network. The amplitude of the resulting output pulse is proportional t o Q except for fluctuations due to noise in the input circuit and the amplifier. Since the object of the measurement is the determination of Q, and the shape of the individual voltage signals is similar, we have a measuring problem of type (3), as defined above. The arguments to be given below are, however, still valid when the shape of the input signal so(t) is some general function of time, and it does not need to be a step function as in the example just given. We shall assume th a t we know the shape of the signal so(t) and the frequency spectrum of the noise N o ( w ) reduced t o the input circuit, as shown in Fig. 13.

AMPLITUDE AND TIME MEASUREMENT IN NUCLEAR PHYSICS

279

The choice of the optimum network is simplified by dividing it into two parts. The first part is used t o transform the noise spectrum N o ( w ) into a “white” noise spectrum N1, i.e., into a spectrum independent of frequency. If G l ( w ) is the transfer function of the first linear network, then IGl(w)lz = N 1 / N o ( o ) .Usually for any reasonable form of N o ( w ) such a linear network can be found. At the same time, the network trans) a signal sl(t). forms the signal s ~ ( t into The second linear network should now be designed in such a way as t o give the best measurement of the amplitude A1 of a signal Alsl(t)in the presence of a white noise spectrum. It is of interest in this connection t o discuss the following rather special example. Assume th a t we have two sharp pulses, as sketched in Fig. 14. The pulses have a fixed known time separation T , and the amplitude of the second pulse aA1 has a fixed SIGNAL

I t

T

FIG.14. Signal consisting of two sharp pulses.

relation a t o the amplitude of the first pulse A1. Our object is t o measure A1 as accurately as possible. The best measurement would evidently be obtained if the amplitudes of both peaks were observed and used t o calculate the most probable value of A1. What relative weights should be given t o the two measurements? Addition of the two amplitudes with the relative weight /3 gives A 1 a/3A1. Thus, the square of the combined signal divided b y meansquare noise is A12(1 ~ & ~ / ( 1 p z ) N 1 .If we maximize this expression with respect t o p, we get the condition cr = p for the best signal-to-noise relation. This means th at the weight to be assigned to each pulse should be proportional t o its amplitude. As was pointed out earlier, it is usually possible t o find a linear network which will perform a calculation of this sort. Figure 15 shows an arrangement which will add the two impulses with their proper weighting factors a t the time T. We can generalize this conclusion and apply it t o a n arbitrary signal by regarding the signal as consisting of a series of successive delta-func-

+

+

+

280

E . BALDINGER AND W. FRANZEN

tion pulses. Thus assume th at we wish to measure the amplitude A of the signal Al.sl(t) shown in Fig. 16, and th at we wish to make use of all the information contained in the signal up t o the instant t = T,. The part of the signal for which t > T , (hatched area in Fig. 16a) is not used in the measurement.

--cz€tDELAY T

SIGNAL

ADDITION CIRCUIT

OUTPUT

ATTENUATION &a

FIG.15. Optimum linear network used to measure the amplitude of the signal of Fig. 14.

If we extend the argument given for the case of two successive deltafunction pulses t o this case, it is evident th at we must add, a t the instant t = T,, all the delta-function pulses into which we have decomposed our signal A1.sl(t),assigning a weight sl(t) to a pulse occurring a t the time t. A network which will perform this operation is shown in Fig. 17. This network is a generalization of the network shown in Fig. 15.

t

t

FIG.16. (a) Illustration of the input signal sL(t)having an arbitrary shape and (b) the response function g 2 ( t ) of the circuit (No. 2 in Fig 13) used to observe it.

We can characterize the network by its response to a single deltafunction pulse applied a t t = 0; the output signal in this case would be gz(t) = si(T, gz(t) = 0 , t

- t), 0 < t < T, < 0 and t > T ,

(3.9)

The response function g2(t) is thus a mirror image of the shape function sl(t) of our original signal A1 * sl(t). This relationship is illustrated by Fig. 16b. We may note here th at the transfer function Gz(w) of our

AMPLITUDE AND TIME MEASUREMENT I N NUCLEAR PHYSICS

281

network is the Fourier transform of g z ( t ) (3.10)

Thus, we may express the output signal resulting from the input A1.sl(t) as a convolution of the input signal with g2(t) : (3.11)

These results have been derived more formally by Halbach (99). The same problem was treated in a less general form by van Heerden (30) DELAYS INPUT SIGNAL

ATTENUATORS

B=s,co,

OUTPUT SIGNAL

-

FIG.17. Optimum linear network for observation of the signal sketched in Fig. 16.

and further by den Hartog and Muller (31). I n going to the limit T, -+ co, one can show that the best possible signal-to-noise ratio which can be obtained for a signal Aoso(t) with physically realizable circuits using linear elements is given by (3.12)

where N o ( w ) is the mean-square noise per unit frequency interval (reduced to the input) and Xo(w) is the Fourier spectrum of so(t):

An ideal amplifier followed by the two networks which we have described (networks 1 and 2 in Fig. 13) will realize the signal-to-noise

282

E. BALDINGER AND W. FRANZEN

ratio given by this equation. (Note, however, th a t this is true only if the measuring time is made infinite. I n the description of our second network we assumed a finite measuring time T,.) The usefulness of these arguments rests on the fact that we can compare practical circuits in their performance, as regards signal-to-noise ratio, with the ideal arrangement just described. As a n example, let us consider an input signal consisting of a unit voltage step at t = 0. This signal is to be observed in the presence of a mean-square noise spectrum

No(w) = a2

+ g2 -z W

(3.13)

I n this case, the Fourier spectrum of the input signal is given by SO(W) = l / j w , so t hat the best possible signal-to-noise ratio is

(3.14)

For a finite measuring time T,, the signal-to-noise ratio will be smaller, and we can obtain a value for it b y applying our two networks (networks 1 and 2 in Fig. 13) successively. Network 1, which is designed t o transform the noise spectrum No(w) into a white noise spectrum N1, turns out t o be a simple RC coupling network with coupling time-constant RC = a/g. The input voltage step is thus transformed into a n exponentially decreasing signal sl(t) = e - t / R C . The second network is therefore defined by its response t o a delta-function input: g2(t) = e-(Tm-t)/RC1

for 0

< t < T,

(3.15)

Application of the second network, taking account of the finite measuring time T,, then leads to the following expression for the signal-to-noise ratio:

(3.16) Evidently, if T , > a / g , we cannot improve the value of 7 very much by making the measuring time infinite. This corresponds to the fact that s ~ ( t contains ) little information when t is large. 5. Practical Networks

It is of interest to compare the ideal network discussed in the last section with the two principal types of practical networks and to evaluate their performance as regards signal-to-noise ratio.

AMPLITUDE

AND TIME MEASUREMENT IN NUCLEAR PHYSICS

283

a. RC-RC AmpliJier. The RC-RC amplifier sketched in Fig. 18 is characterized by the presence of two RC time-constants. One of these timeconstants is used to limit the high-frequency response of the amplifier, and the other one is used to limit its low-frequency response. I n accordance with general practice and with the conclusions drawn in our discussion of input circuits, we shall assume th at the input circuit consists of an unavoidable parasitic capacitance COand a parallel resistance ROhaving such a high value th at its influence on the over-all transmission function of the system may be neglected. As a signal source, we have chosen

FIG.18. RC-RC amplifier used with a n ionization chamber. The low-frequency limiting time constant is r1 = RIC1, and r 2 = R2Czis the corresponding high-frequency limiting time constant.

an ionization chamber or similar instrument which produces a short current pulse l o ( t ) . If we wish to measure the total charge Q = JIo(t)dt ballistically (that is, in such a way th at the ballistic deficit is small), we can assume with little error th at the voltage signal so(t) appearing at the input to the amplifier is a voltage step of amplitude Q/C appearing at t = 0, and we can regard the rise-time of this step as negligibly short. The mean-square noise voltage per unit frequency range a t the input to the amplifier can then be written

(3.17) The problem of signal-to-noise ratio in RC-RC amplifiers has been treated by a number of authors (%'-@), usually omitting the last term of E q . (3.17).* This term corresponds to flicker effect in the first tube. If we set 7 1 = R1C1 and r2 = R2C2,the input signal so(t) will give rise t o a n output signal

* Compare further the useful monograph of

Gillespie (S) and Reference 1 .

284

E. BALDINGER AND W. FRANZEN

The signal s2(t) reaches its maximum value a t the time 71

-

(3.19)

72

The square of the noise-to-signal value can then be written 1

a2712

If we set X

=

72/71

1

and integrate, this becomes

4&2 (m) [G + c 2

=

XZX/(X-l)

a2

dv

"1

2d2 log - 1)

g271 + T(X

(3.20)

(3.21)

The investigation of this function is simplified by introducing the condition 7 1 ~ = 2 constant = K 2 . This allows us to eliminate T~ so

2

8

t

1.3

SHOT EFFECT AND GRID CURRENT,

1.2

1.0 .I

I

-

k . 3

71

10

FIG.19. Square of the noise-to-signal ratio 1 / q * as a function of amplifier time constant ratio X = T ~ / T ~ .

that in Eq. (3.21) only X will occur as a n independent variable, and the terms containing a2 (shot effect) and g2 (grid-current fluctuations) differ now only by a constant factor. Curves showing the relative change in the noise-to-signal ratios for the three terms of Eq. (3.21) and for the resolving time are plotted in Figs. 19 and 20. From these two curves it can be seen that the noise-to-signal ratio and the resolving time reaches a minimum value when T~ = r 2 . This implies th at if we consider 1/q2 as a surface in three dimensions (where the three coordinate axes are l / q 2 , T ~ , and n),the extremum of 1/q2 must lie on the plane of symmetry T~ = T ~ . With this in mind, it is easy to show th at the minimum of the surface l / q 2 lies a t the point 71

=

72

a

= 70 = -

9

It is interesting t o note that ro is independent of flicker effect.

(3.22)

AMPLITUDE AND T IME MEASUREMENT I N NUCLEAR PHYSICS

285

If we substitute ro = a / g in the expression for l/v2, we obtain for the best singal-to-noise ratio possible with an RC-RC amplifier (3.23) where e is the base of natural logarithms. Of the noise sources which enter into the expression for N o ( w ) = a2 4- ( g 2 / w 2 ) -k d 2 / w , the term d 2 / w due to flicker effect is generally the least important. Its influence is restricted to low frequencies and is usually small compared with ( g 2 / w 2 ) 4- a2.This is true particularly since g2/u2is often increased considerably by background pulses, as we shall show below.

FIG.20. Resolving time T , = T I X A / X - ~ as a function of the amplifier time-constant ratio A. The resolving time is defined as the width of the output pulse obtained by dividing the area of the pulse by its maximum amplitude.

For this reason, we are justified in comparing the value of the signalto-noise ratio for a n RC-RC amplifier without flicker effect with its value for the best linear network discussed in the last section. (Note th a t we must set &/C = 1 to make this comparison.) For a n infinite measuring time, the best network is 36% better than the RC-RC network with optimum time constants 7 1 = r2 = a / g . On the other hand, for a measuring time T , equal t o the optimum time constant 7 0 = a/g, the figure of improvement is only 26%. This example shows that practical cricuits come fairly close t o realizing the theoretical maximum precision. No great improvement would be obtained by using more sophisticated networks. b. Remarks on Noise Due to Background Pulses. According to the argument made in our discussion of noise sources, background pulses due to gamma-rays and similar causes can be regarded as giving rise to a n extra shot effect in the detector, thereby increasing the total noise level. Pro-

286

E. BALDINGER AND W. FRANZEN

vided the charges are measured with a small ballistic deficit, this means that the term g 2 / w 2 in the expression for No(w) will increase and the optimum time constant T O = a / g will decrease. Thus, the time constant t o be used in a n RC-RC amplifier in an actual experiment depends on details of the experiment. Note that the shortening of T O corresponds to the fact that we should keep the pulse duration short in order to avoid excessive overlap of background pulses with pulses to be measured. The optimum time constant can be calculated if the number of background pulses per second and their size distribution is known, and if the noise characteristics of the input circuit and amplifier are known. On the other hand, it is frequently more convenient to determine the optimum time constant experimentally by analyzing the output obtained with a pulse-height analyzer when uniform test pulses are applied to the input in the presence of background pulses. The value of the theory in this case lies in the fact th a t we can restrict our observation to equal time constants. So far we have treated the fluctuations due t o background pulses in the same way as the fluctuations caused by grid current of the first tube of the amplifier. I n both cases we have assumed that the individual current pulses arriving completely a t random can be taken to be &functions. I n one respect, however, there is a difference between the two fluctuating phenomena. As is well known, the amplitude distribution of the output fluctuations caused by grid current (or shot effect) will be a Gaussian one. This is because all output pulses are stimulated by a large number of electrons at the input. For fluctuations caused b y background pulses from yr a ys, this number of primary events will be rather small. As was shown by Gillespie (S), a Gaussian distribution may practically still be assumed if more than about 20 primary events are necessary to stimulate a n output pulse. If a smaller number is involved, the distribution departs more and more from a Gaussian distribution. However, the mean-square fluctuating voltage at the output as calculated in Section 111,2 will in most cases still be a good measure of the spread of a sharp line in a n amplitude distribution. c. Delay-Line Clipping. Let us consider an input signal consisting of a unit voltage step to be observed in the presence of a mean square noise voltage No(w) = a2f g2/w2. The arrangement to be discussed is sketched in Fig. 21. The time constant T I = RIC1 = a/g is so chosen th a t the input noise spectrum is converted into a white frequency spectrum. The second part of this arrangement consists of a shorted delay-line followed by a single time constant T Z = R z C z of integration (34). Provided [ T I T Z / ( ~ I- 7 2 1 1 log ( T I / T Z ) > t, [compare formula (3.19)], the maximum of the output signal occurs a t the time t = t, corresponding to the arrival of the reflected signal a t the input of the delay-line. I n this case, the sig-

287

AMPLITUDE AND TIME MEASUREMENT IN NUCLEAR PHYSICS

nal-to-noise ratio turns out to be given by

Assuming for the moment an exact integrating network expression reduces to

(72

-

(3.24) w),

this

Solving for the optimum signal-to-noise ratio, we get t m / q = 1.25. We may compare this with the best linear network described by Eq. (3.16). For infinite measuring time, the best linear network is 11 % better. This figure of improvement is reduced to 6% if we assume a finite measuring time equal t o t,.

DELAY LINE IMPEDANCE

tln

Z . )

-

SIGNAL

FIG.21. Pulse-shaping with a shorted delay line and a single time constant of integration.

Since only a finite time constant r2 of integration is possible, we have in practice some loss in precision, which will amount to 28, 16, and 8.5%, corresponding t o values of r2/tmequal to 1, 2, and 4, respectively. Concerning signal-to-noise ratio alone, delay-line clipping does not give a significant improvement compared with the RC-RC amplifier in the practical case where r 2 is not infinite. Delay-line clipping, however, is certainly indicated when a high counting rate is desired and overlapping of pulses becomes a serious problem. A short pulse duration is required in this case, and the trailing edge of the output pulse deserves special consideration. I n consequence of the short pulse duration desired, higher frequencies have to be transmitted, so that the term g 2 / w 2 in the noise spectrum can normally be neglected. The signal-to-noise ratio for a white noise spectrum is easily obtained from formula (3.24) by going to the limit r1--+ w . It is interesting t o note th at delay-line clipping followed by exact integration ( T ~ w ) realizes in this case the best linear network possible, as may be seen from Fig. 17. Using a n integration time N

288

E. BALDINGER AND W. FRANZEN

constant T~ = 1.25tm will give a loss in signal-to-noise ratio of 17% as compared with the best network just mentioned. Calculations show that exactly the same signal-to-noise ratio is obtained with a RC-RC amplifier whose two time constants are equal to 1.25tm( 3 ) .Figure 22 shows the output pulse forms of the RC-RC amplifier and delay-line clipping amplifier when both give the same signal-to-noise ratio. It is obvious that delay-line clipping provides a shorter pulse duration and a more rapidly decaying trailing edge for the same signal-to-noise ratio than the RC-RC network.

FIG.2 2 . Response of an amplifier to a step input. The solid line represents the response of a n RC-RC amplifier with equal time-constants T~ = T Z = 7 0 , and the dashed line the response of a delay-line clipped amplifier with a single time constant of integration equal to 1.25 times t,, where 1, is twice the length of the delay line and also the time a t which the maximum of the delay-line clipped pulse occurs. The relationship between tn and T O assumed here is tn = ~ 0 / 1 . 2 5 .

I n this connection we should like to refer to the possibility, mentioned by Maeder (35), of improving the signal-to-noise ratio by use of a variable measuring time for each signal. That is, the measuring time is made dependent on the time interval between successive signals, so th a t all of this interval is used to form a good average value of the signal amplitude. One should remark in this regard that such a method will lead to a n improvement only provided that the g2/w term in the noise spectrum may be neglected for time intervals of interest (see discussion a t end of Section III,4).

AMPLITUDE

AND TIME MEASUREMENT IN NUCLEAR PHYSICS

289

OF NUCLEAR EVENTS IV. THE TIMING

I. Survey of the Problem An important experimental problem in nuclear physics is the establishment of the time relation between two events. An “event ” is identified by the passage of some characteristic form of ionizing radiation through a nuclear particle detector. The timing itself can be carried out either directly, by establishing the time relation between two electrical signals, or indirectly, by observation of some property of the radiation. An example for the indirect method is the observation of the lifetime of a nuclear excited state by observation of the Doppler shift of the emitted gamma radiation (36).Such indirect methods are usually employed when the time separation of the events to be registered is too small for direct measurement. An excellent survey of such methods has been presented by Devons (37). We shall, however, be concerned with the direct timing of two separable electrical signals exclusively. Furthermore, it is useful to contrast two different types of experimental situations. The first type is characterized by the occurrence of two physically associated events which are simultaneous for all intents and purposes. By this is meant either that their time separation is immeasurably small or else that it is not an important item of information. The task of the experiment is simply to establish the occurrence of the double event by means of a coincidence circuit. The output signal from this circuit may itself be recorded, or it may be used as a trigger for recording other information. I n the second type of problem, the events are actually separated physically by a finite time interval t h a t we wish to measure. Such a problem arises in the determination of the lifetime of a nuclear excited state by timing the successive emission of two gamma-rays, or in a measurement of the velocity of a particle by observation of its time of flight through a known distance. Experimentally, this problem can be made identical with the previous one by delaying the earlier signal by means of a delay line until it is in coincidence with the later signal. The two signals can then be mixed in a coincidence circuit, and we have the additional task of measuring the delay accurately. Thus the heart of a direct timing system is a coincidence circuit. There are occasions when a coincidence between more than two events is to be registered, so that a multiple coincidence circuit must be employed. I n other cases, it is necessary to employ anticoincidence circuits to ascertain the nonoccurrence of an event simultaneous with one or more other events. Multiple systems of this sort in general employ the same principle of operation as ordinary double coincidence circuits.

290

E. BALDINGER AND W. FRANZEN

A further important aspect of the general timing problem concerns the operation of the particle detector. I n general, the information contributed by the ionizing particle suffers a dispersion in time before it reaches the coincidence stage. We are interested in analyzing this time dispersion and the resulting loss of information. I n this respect, the newer types of particle detectors (scintillation counters and Cerenkov counters) using photomultiplier tubes represent a considerable advance over the older detectors, which depended on the direct measurement of ionization (proportional counters and ionization chambers). I n the ionization devices, a n electrical signal is derived from the drift of electrical charges under the influence of a n electrical field through a medium of some sort (usually a gas). The time jitter which arises from fluctuations in the speed of this diffusion process is approximately two orders of magnitude larger than the time uncertainty caused by the transit-time spread of the electron avalanche in photomultiplier tubes. For this reason, scintillation or Cerenkov counters are universally used a t the present time in direct timing experiments, and we shall not discuss the limitations of the ionization devices in detail. Sometimes it is important to measure both the time of occurrence and the amplitude of an electrical signal. It is usually advisable to separate the two types of measurement because the circuit properties desirable for optimum timing application are often undesirable for a n accurate amplitude measurement. A separation of circuit functions into independent timing and amplitude channels can practically always be accomplished. I n this connection, it is important to examine to what extent an associated selection in amplitude affects the operation of a coincidence circuit. Combined systems which select signals on the basis of certain time and amplitude criteria have recently found wide application. 2. Characteristics of Scintillation and Cerenkov Counters

An energetic charged particle in the process of stopping in a solid or liquid scintillator produces optically active states along its path. These states subsequently decay with the emission of light, which is then partially converted into photoelectrons by a photomultiplier tube. The electrons initiate a n avalanche in the electron multiplier structure, resulting in the arrival of a short pulse of current a t the anode. Thus we can divide the physical process leading to the production of a n observable signal into four parts: (1) the stopping of the charged particle; (2) the decay of the optically active states; (3) light collection; (4) formation of the electron avalanche. I n the case of a Cerenkov counter, step ( 2 ) is missing, since light is emitted directly during the slowing-down of the part,icle. Moreover, in such a counter the particle ceases to emit light as soon as

AMPLITUDE AND TIME MEASUREMENT IN NUCLEAR PHYSICS

291

its velocity falls below the velocity of light in the medium traversed, so that the effective stopping time is shorter than for a scintillation counter. Since a Cerenkov counter is usable only for particles of high velocity and the time uncertainties, with the exceptions just noted, are identical with those arising in scintillation counters, we shall concentrate our attention on the time dispersion of the electrical signal in a scintillation counter. 1000-

Relative Coincidence Rate (Logarithmic Scale)

100-

10-

+-I5

'

-I0

-5

5 I0 Inserted Delay( 1 0-9 sec.Units)

0

I5

FIG.23. Demonstration of irregular optical delays in long light pipes used with scintillation counters, as described by Lundby (38). The three-coincidence rate us delay curves shown above were obtained by exposing two 1P21 photomultipliers t o light flashes from a terphenyl crystal. For curve A , the two multipliers were both in direct optical contact with the same crystal. For curve B, a one-foot Lucite light pipe was introduced between one of the two multipliers and the crystal, while for curve C a 3-ft long rod was introduced. The distortions of the delay curves are caused by multiple reflections in the light pipes.

Stopping times for charged particles in solid matter can be estimated from their initial velocity and their range in the solid. Except for particles of high energy having ranges of the order of 10 cm or more, stopping times are of the order of 10-ll to l O - l 4 sec and therefore are generally negligible as factors contributing to the time spread of the output signal from a scintillation counter. [For methods of making simple esti-

292

E. BALDINGER AND W. FRANZEN

mates of stopping times, see Devons (37).] The uncertainty in time arising from the finite decay constant of the scintillator will be discussed in detail below. As regards the light collection time, it is clear from the remarks made concerning stopping times that the time spread arising from this cause is in general unimportant. It becomes important in the case of scintillators of large size, in which the light may have to traverse appreciable distances to reach the photocathode, and when long light pipes are employed. Under these circumstances, the light path usually involves multiple reflections, so th at there may be a variation of light path of the order of tens of centimeters between different components of the collected light. Such effects have been studied by Lundby (38), as illustrated in Fig. 23. It remains for us to discuss the other two major factors of time dispersion. a. Statistical Fluctuations in the Decay of Scintillators. Let us assume that the emission of photons from the optically active states of a scintillator is characterized by a decay constant X so th a t the probability of decay of an excited optical center in the infinitesimal time interval dt is given by Xdt. We assume th at the probability of decay of the center is not influenced by its environment. Let s ( t ) be the probability that the state has decayed in the time t. Then the probability th a t it has decayed in the time t dt is given by s ( t d t ) = s ( t ) dt(ds/dt)t. But this is just equal t o the probability of decay in time t , plus the probability of nondecay in this time times the probability of decay in the time d t :

+

+

+

+

~ ( t ) dt(ds/dt) = s(t)

+ [I - ~ ( t ) ] X d t

Integration of this equation then yields

for the normalized probability for decay of a single excited state between 0 and t. The differential probability of decay between t and t 4- dt is

p(t)dt

=

ds(t)

=

Xe-At&

From these relations, we can in principle derive the statistical fluctuations in light emission for the case where not one but N excited states have been produced by the passage of a charged particle. The N photons arising from the decay of these states will then give rise to R photoelectrons on the average. Several comments are of interest here. I n the first place, the initial population of N excited states is subject to statistical fluctuations. These fluctuations are however usually negligible, since the photoelectric conversion efficiency of a scintillation-counter photomulti-

AMPLITUDE AND TIME MEASUREMENT IN NUCLEAR PHYSICS

293

plier system is of the order of 5%.* Thus N is a number of the order of 20 times larger than the average number of photoelectrons R, and the relative size of the fluctuations in N is unimportant compared with the fluctuations in R. This statement, of course, does not hold in a case where the charged particle traverses the scintillator without stopping. Under these circumstances, the fluctuations in N are given by a Landau distribution (40) and therefore may be appreciably larger than the fluctuations given by a normal distribution. For our purposes we shall, however, neglect the possible variations in N . It is further of interest to note the existence of some evidence th a t in many scintillating substances the decay of the optically active states is characterized by not one decay constant but by several (41). This has been established for such slow phosphors as silver-activated zinc sulfide, but for the faster scintillators our knowledge on this subject is quite incomplete. For simplicity we shall therefore concentrate our analysis on scintillators having a single decay constant. If the time spread caused by the variation of the optical path between particle track and photocathode is neglected, the emission of the photoelectrons is described by the same decay constant X th a t describes the decay of the excited states of the scintillator. We can regard these photoelectrons t o a high degree of approximation as a random selection from the much larger population of photons. (The selection takes place in two stages, a n optical one, in which a certain proportion of the photons is selected t o impinge on the photocathode, and a photoelectric one. Both processes can be described as random.) Let p be the total (optical and photoelectric) conversion efficiency, that is, p is the probability th at a photon will release a photoelectron. Then the probability that an excited state will give rise to a photoelectron in time t is, in view of (4.1), given by v ( t ) = p(1 - e-xt), while the probability t ha t it does not give rise to a photoelectron in this time is 1 - v = 1 - p(1 - e-xt). The probability th at out of a total population of N excited states q photoelectrons are produced in time t is then given by the binomial distribution

Pdt)

=

(N

N!

v*(1 -

- q)!q!

v)N--9

(4.3)

We can also derive the probability W,(t)dt th a t the qth photoelectron is emitted between t and t dt. For this purpose me divide the population N into three classes, consisting of a class of q - 1 members that give

+

* This estimate is based on a photosensitivity of 40 p amp/lumen for the photocathode and a mechanical equivalent of light intensity of 1.61 X 10-3 w/lumen (S9). Actual conversion figures will be smaller because of optical losses.

294

E. BALDINGER AND W. FRANZEN

rise t o a photoelectron between 0 and t, a class consisting of a single member which gives rise to a photoelectron between t and t dt (with probability d v = pXe-xtdt), and a class of N - q members which do not give rise t o a photoelectron in the time specified. W,(t)dt is then given b y the trinomial distribution*

+

W,(t)dt =

N! v ~ - ' ( l - v)N-qdv ( N - p)!(q - l ) !

(4.4)

The relation between P J t ) and W,(t)dt is specified by

The mean number of photoelectrons emitted between 0 and t is then given by writing

2 N

Q

=

qPg(t)= v

a ay ( v + y ) N = VN

q=o

(where we set y = 1 - v ) , as we would expect. The mean number of photoelectrons emitted altogether is R = p N . The variance in this number is R2 - Zz = pN(1 - p) = R ( l - p) which is slightly smaller than we should expect to find if the statistical distribution in R were a Poisson distribution. t The total probability th at the qth photo-electron will be emitted a t W,(t)dt. Instead of carrying out the integration, we all is given by note that

Am

But

Therefore,

2

k=O

2 N

m

Pk(W) =

($t(l

- p)N-k

=

1

k=O

* This argument is due to E. Merzbacher

(private communication). m predicts the shape of the pulseheight distribution obtained from a scintillation counter. I n practice, an additional spread arising from statistical fluctuations in the size of the electron avalanche must be taken into account.

t It is interesting to note that Eq. (4.3) with t +

AMPLITUDE AND TIME MEASUREMENT IN NUCLEAR PHYSICS

295

We can obtain a continuous approximation to (4.4)by assuming th a t the initial number of excited atoms N is a large number and th a t the photoelectric conversion efficiency 1.1 is very small, while the product R = pN is a finite number. It is clear from our earlier discussion th a t this assumption is justified, since N is of the order of 20 times larger than R and p is of order 0.05. Thus we write Lim NP'O --rm

((N

N! - q)!(q

( N v ) q--le--Nv

- l)!

Ndv

Setting f ( t ) = Nv = R ( l - e@) for the mean number of photoelectrons emitted in time t, we then obtain the modified Poisson distribution

This expression was first derived by Post and Schiff (4%'). With the aid of these expressions, we can compute the spread in arrival time of the photoelectrons due to the finite decay constant of the scintillator. Consider, for example, a case in which the gain of the photomultiplier is so large that the arrival of the electron avalanche from a single photoelectron triggers one channel of a double coincidence circuit. Evidently no information regarding the timing of a nuclear event can be transmitted prior to the arrival of this first avalanche. Fluctuations in the arrival time cause a corresponding uncertainty in the determination of the time of occurrence of the nuclear event. One should note in this connection th at the electron avalanche due to a single photoelectron has a continuous amplitude distribution. Therefore, only a certain fraction of the single electron avalanches will succeed in triggering one channel of the coincidence circuit. Furthermore, if the ratio of the decay time constant 1/X to the transit-time spread of the electrons in the multiplier T~ is of the same order of magnitude or smaller than the mean number of photoelectrons R, there will be a n appreciable probability of overlap of individual electron avalanches. If we ignore these details for the moment, we can derive an expression for the mean time delay 6 in the arrival of the first photoelectron and the variance in this delay - G2 with the aid of Eq. (4.7). This can be done simply if we assume that 6 << 1/X, so that we can write

f(t)

=

R(1 - e-xt) E RXt

(Strictly speaking, we require that X'i;/2 << 1, since we consider th a t X2t2/2! in the series 1 - e-Xt = X t - X2t2/2! * * is negligible compared with At.) To realize the significance of this assumption for com-

+

296

E . BALDINGER AND W. FRANZEN

monly used scintillation materials, one should consider that for stilbene sec, whereas for terphenyl l / X = 4.2 X see. Then l/X = 6 X W l ( t ) d t g e-RXtRXdt is the probability that the first photoelectron ardt. This probability function has the same form, rives between t and t but falls off R times as fast as the number of photons per unit time d f / d t = RXe-xt under the assumption made here. The mean time delay in the arrival of the first photo-electron is then

+

and the variance in the mean time delay is given by

Thus if the mean number of photoelectrons emitted is 10, the uncertainty in the arrival time of the first photoelectron is approximately equal to the square root of the variance, or equal to one-tenth of the decay constant of the crystal. How long must one wait so th at the probability of arrival of the first photoelectron is 0.95, i.e., so that the coincidence efficiency is 95%? This question has been examined by Bell, Graham, and Petch (43).Evidently we would require that

Jd W1(7)d7

=

1 - ecRX1= 0.95

so that t 3/RX = 3G. A symmetrical time resolution curve (a curve of coincidence rate us delay introduced in one channel) must have a width equal t o twice this quantity, or SK, for 95% coincidence efficiency. Post and Schiff (4.2) have shown by means of a n asymptotic expansion that application of (4.7) leads to the general expression

for the mean time delay in the arrival of the qth photoelectron. The variance in the mean time delay is then (4.10)

AMPLITUDE AND TIME MEASUREMENT IN NUCLEAR PHYSICS

297

It should be noted that the expansion used converges only when the mean number R of photoelectrons is a large number. It is clear th a t when R is small, say of order unity or smaller, the mean time delay must approach 1/X and will never become larger than this quantity. I n fact, for p = 1 and R--t 0, we get from either (4.2) or (4.7) Wl(t)dt-+ XRe-"at, so that & -+ 1/X. Equation (4.10) expresses the natural limit of precision of time measurements with scintillation counters. We note that the time uncertainty is inversely proportional to R and proportional to the decay time constant of the scintillator 1/X. This demonstrates the value from the point of view of timing of the comparatively large photoelectric conversion efficiency p achieved in front-surface photomultipliers, as compared with the older types of multiplier tubes in which the photocathode cannot be brought into direct optical contact with the scintillator. b. Conditions for Obtaining Optimum Time Resolution with Xcintillation Counters. T o take full advantage of the inherent time resolution attainable with a scintillation counter, it is necessary for the coincidence circuit t o respond to the pulse from a single photoelectron released a t the cathode of the photomultiplier, as we have seen. There is a danger, however, in this method of operation, as we shall demonstrate. We must require in addition that each channel of the coincidence circuit be paralyzed for a time of the order of several times the phosphor decay time 1 / X after the receipt of a single photoelectron avalanche pulse. If the circuit does not have a dead time of this sort, it is possible for the individual photoelectron avalanches from one counter to overlap in time with those from another counter as long as the events in the two counters are separated by a time of the order of 1/X. Thus consider two scintillation counters A and B. A particle is stopped in A a time 6 1/X before another particle stops in B. Even though the single pulse resolving time of the coincidence circuit is much shorter than 6 , it is nevertheless possible for a Coincidence to be registered provided an electron avalanche, caused, say by the mth photoelectron released in A , coincides in time with the first (or a later) avalanche in B. This is illustrated in Fig. 24. On the other hand, if channel A is blocked for a time of the order of several times 1/X after the arrival of the first avalanche, such an eventuality will not occur. (This limitation does not apply t o Cerenkov counters, in which all the individual photoelectron avalanches effectively overlap in time.) Let us assume that we have a (nonideal) coincidence system without dead time which responds to pulses from single photoelectrons. I n this case, we can explain the shape of the coincidence rate us delay curve entirely by the overlap phenomenon described above, as Lundby has shown

-

298

E. BALDINGER AND W. FRANZEN

(38).If the decay time of scintillator B is T B = 1 / b , the probability th a t any photoelectron pulse will begin between T and T dT after the passage of a particle is given in the continuous approximation b y the expression*

+

FIG.24. Diagram illustrating that a zero dead-time coincidence circuit which responds to single photoelectron avalanche pulses (assumed to be resolved from each other) may register a coincidence when used with two scintillation counters, even though the events in the two counters are separated by a time T = 1/k much longer than the resolving time of the circuit. As shown above, a particle has passed through counter A a time T before another particle passes through B. Nevertheless, a coincidence is registered because of the superposition of a late pulse in A with a n early one in B.

Let the time width of the individual electron avalanche pulses be a. (It must be assumed that a is much shorter than TB. This would require a photomultiplier with very small transit-time spread, such as a 1P21.) A pulse of this width begins in channel A a t time T after the passage of a charged particle through scintillator A. Under these conditions, a pulse in channel B having the same width and beginning in the time interval

* G ( T )is the “density of events” in the terminology of Jost (44).In the continuous approximation G ( T ) is related to the probability functions previously defined by W,(t)dt = P,-l(t)G(t)dt. Evidently, G ( T ) is just R times as large as the differential decay probability for a single excited state of the scintillator, given by Eq. (4.2), as we would expect.

AMPLITUDE AND TIME MEASUREMENT IN NUCLEAR PHYSICS

299

+

T - a to T a will overlap with pulse A , so t h a t a coincidence will be recorded. Now suppose that we delay the pulse A a time Td 2 a by means of a delay line before mixing it with pulses from B. The probability th a t a B channel pulse will occur a time T Td - a to T Td a after the passage of a particle through B a t time zero is

+

+ +

The total probability for recording a coincidence between the two events for delay Td in channel A is then &l(Td) =

RBe-rd/rs(ea/rr,

-

e-a/rB

)

hme-T/rBGA(T)dT

(4.12)

where GA(T ) is the density of events in channel A . This expression demonstrates incidentally, as Lundby has pointed out (38),that the variation of coincidence rate with delay in A is independent of the distribution in time of the events in A . If we set

GA( T )d T

=

e-T’7AdT 74

where RA is the average number of photoelectrons emitted per event in scintillation counter A and T A is the decay time of the scintillator A , then

(4.13) gives the shape of the coincidence rate vs delay curve for the two counters under the conditions specified. If the same type of scintillator is used in both counters, T A = T B = T, and

Thus the delay curve falls off exponentially a t a rate determined b y the ratio of the delay Td to the decay time of the scintillators T = l/&. This is true regardless of the shortness of the resolving time of the coincidence circuit, provided the current pulses from individual photoelectron avalanches are resolved from each other. For delays shorter than the individual pulse width, th a t is for Td < a, our computation must be modified slightly, and we obtain in place of Eq. (4.14) &i(Td) = ‘ARB 7 [2 - e-ar(erd/r + e - r d / r ) ] (4.15)

300 Thus, for Td

E. BALDINGER AND W. FRANZEN

=

0, the total coincidence probability is

Ql(0) = R A R B ( ~-

ecaIr)

(4.16)

This is the coincidence efficiency of the circuit. Evidently the efficiency can be larger than unity. Th at is, the occurrence of many individual photoelectron pulses may lead to an average of more than one coincidence per physical event. Ordinarily this danger is avoided by following the COincidence circuit with a comparatively slow amplifier which is unable to resolve voltage signals separated by time intervals of the order of T = 1/X. The limitation on the width of the coincidence us delay curve (that is, the limitation on the resolving time of the system when used with scintillation counters) just presented can in principle be avoided by paralyzing each channel of the double coincidence circuit after the arrival of the first single electron avalanche pulse, as stated earlier. I n parallel or series coincidence circuits, a dead time of this sort can be introduced by integrating the current pulse a t the anode of the photomultiplier with a time constant somewhat longer than the decay time of the scintillator, so th a t the voltage signal arriving a t the input of one channel of the coincidence circuit consists of a series of voltage steps. If the gain of the system is sufficiently large, the first step (due to the first photoelectron) will cut off the vacuum tube or diode in the case of the parallel circuit, or gate on one of the grids of a 6BN6 series coincidence circuit. A voltage step then appears across the common anode load of the coincidence pair, and a shorted delay line is used to convert the step into a short pulse. When two such pulses overlap in time, a coincidence is recorded. A system of this sort has been described by Bell and Petch (43) who were able to obtain by this means a time resolution approaching the theoretical minimum resolving time given by Eq. (4.10). The circuit used by these authors will be described in more detail later on (Section IV,3). Coincidence systems of the type discussed above are characterized by pulse shaping in the coincidence circuit itself. I n general, the differentiating time constant th at can profitably be employed is limited b y the transit-time spread of single electron avalanches which arrive a t the anode of the photomultiplier tube. If a time constant smaller than this spread is employed, the pulse duration a t the mixing stage will not change, but there will be a loss in amplitude. This applies to all coincidence systems regardless of the location of the pulse-shaping network. It should be emphasized that the mechanism of pulse formation th a t we have described is highly idealized. Many commonly used photomultipliers have a transit-time spread of the same order of magnitude as the decay time of the fastest scintillators (45). I n that case, the multiplier output current is a convolution of the exponentially decaying current den-

AMPLITUDE AND TIME MEASUREMENT IN NUCLEAR PHYSICS

301

sity G(t) = XRe-Xt from the scintillator and the (approximately Gaussian) current signal from a single electron avalanche. This effect has been analyzed by Lewis and Wells (46). Evidently under these circumstances it is not possible to resolve single electron pulses. The width of the time resolution curve is then strongly influenced by the characteristics of the multiplier, as exemplified by the spread in transit time, rather than by the statistics of light emission. G. Transit-Time Dispersion in Photomultiplier Tubes. The electrons cascading through the dynode structure of a multiplier tube in general do not all experience the same electric fields, nor do they follow identical paths from one dynode to the next. As a result the output current pulse is dispersed in time. We have seen above that the resulting time spread may be a n important limitation in timing spplications. It is generally assumed that the distribution of the anode current density in time is approximately described by a normal distribution T I

.\

Se

r

/t

-

TW

where we define T, as the transit time spread (rms deviation from the mean transit time T ); S is the current gain of the photomultiplier tube. It is clear that both the transit time T and the spread in transit time T, are functions of the geometry of the dynode structure as well as of the over-all potential applied. As a rule of thumb, the multipliers with the longest transit time also seem to have the largest transit time spread. The shortest transit time spreads seem to have been achieved with the very compact RCA 1P21 tubes. Bay (47) has measured approximately 5X sec by using light from Cerenkov counters, while Post (48) has been able to reduce this figure to about 2.5 X 10-lo sec by applying very high potentials t o the dynodes for short periods of time by a pulsing technique. Allen and Engelder (49) have measured transit time spreads of -8.0 X sec in EM1 5311 tubes. This figure may be considered characteristic of the venetian-blind type of dynode structures. It is evident that a considerable improvement in transit-time spread should be possible by introduction of new types of multiplier structures. What is required for timing applications is a compact dynode structure to which high potentials can be applied and which is coupled to a n endwindow photocathode. I n addition, the tube should be capable of delivering large peak currents without saturation, and it should have a gain of the order of lo8 to 109. A number of new developments in the multiplier field in the United States have recently been described by Morton (50). It is doubtful that any of the new or contemplated designs fulfill all the requirements enumerated above.

302

E. BALDINGER AND W. FRANZEN

3. Operation and Classification of Coincidence Circuits

A coincidence circuit is a device in which two voltage pulses from separate channels are mixed so as to produce a large output signal when the pulses are simultaneous and a small output when they are not. An essential element of the circuit is therefore an amplitude discriminator which follows the mixing stage. If two pulses are of equal and uniform amplitude, i t would be possible to construct a coincidence circuit based on the principle of linear superposition. An output signal having double the amplitude of each pulse would then indicate a coincidence. I n practice, however, linear superposition is an undesirable procedure because of the distribution in amplitude of actual pulses. I n a linear superposition device, a single unusually large pulse may simulate a coincidence. For this reason, all practical coincidence circuits are nonlinear devices which perform a nonlinear superposition of the individual signals. The nonlinearity can be introduced in various ways. The simplest of these is a pulse-limiting device introduced ahead of the mixing stage. This device limits as soon as an input pulse exceeds a certain minimum amplitude. For this purpose, a diode (or a vacuum tube) may be employed which is cut off by a pulse of sufficient amplitude. Or we may interpose a trigger circuit between the output of the detector and the input to the coincidence circuit. The trigger generator fires and produces a uniform output pulse whenever a n input signal exceeding a certain minimum amplitude arrives. The output signals from two such trigger generators are then mixed in a coincidence circuit. I n the Garwin modification of the Rossi coincidence circuit (51, 52), a nonlinearity is introduced twice in order to sharpen the action of the circuit. I n series coincidence circuits based on the 6BN6 multigrid tube, the nonlinearity is a tube characteristic. Other examples can be given. I n general, the nonlinear behavior of practical coincidence circuits makes i t impossible for a single large pulse to simulate a coincidence, and it lessens the dependence of resolving time on pulse amplitude. The choice of different types of coincidence circuits for various applications has always been partly a matter of taste. However, there are a number of distinctions between the operation of commonly used circuits which may be of considerable importance in practice. The classification of coincidence circuits used here is due to Bell (53). a. Parallel Coincidence Circuits. A parallel coincidence circuit may be described as a device consisting of two switches in parallel which share a common load. When one switch is opened, the voltage drop across the load does not change appreciably, but when both are opened simultane-

AMPLITUDE

303

AND TIME MEASUREMENT IN NUCLEAR PHYSICS

ously, there is a large change of potential in the load circuit. The principle of this circuit is due to Rossi (51). As a switching device, either a vacuum tube or a germanium (or silicon) diode may be employed. The use of diodes for this purpose was suggested by Howland, Schroeder, and Shipman (54). A simple parallel coincidence circuit of the type suggested by these authors is shown in Fig. 25. I n Fig. 25a, the individual diodes are cut off by applying a positive pulse t o the cathode circuit; the common load resistor is then in the anode part of the circuit. It is also possible to cut off the diodes by applying a negative pulse t o the anodes, so that the signal is developed across a common cathode load resistor, as shown in Fig. 25b.

I

I *", 1 I

I

A.

--u

Input I

I t -

lnph 2

u- output

FIG.25. Simple diode coinciLJnce circuit: (a Arranged for positive pulse input; (b) arranged for negative pulse input. [After Howland, Schroeder, and Shipman (64).]

Let us consider the operation of the circuit in detail. Assume th a t a single positive rectangular pulse of duration T is applied to the cathode of diode DIin Fig. 25a. If the common load resistor R I is chosen so th a t Ra >> R1>> R, 4- Rz, the process of cutting off DI will have only a small effect on the current through RI. Here Rb is the backward resistance of the diode (usually of the order of a few megohms), R, is its forward resistance in its normal state of conduction (100 ohms or less), and Rz is the cathode load resistor, chosen so as to terminate the coaxial cable through which the signal is brought in. I n the state of conduction, the total resistance from B+ to ground is R I 4- (RI Rz)/2, so th a t the steadystate current flowing through R1 is given by

+

304

When

E. BALDINGER AND W. FRANZEN D1

is cut off, the current tends to change to a new value

so that the switching of the diode tends to produce a change in potential of amount

On the other hand, when both diodes are switched off simultaneously, the output potential tends to rise exponentially to a value V,( 1 - e--T/RIC1).

Time-

(b)

FIG.26. Picture of input and output pulse shapes for a diode coincidence circuit. In each of the two cases shown, the two input pulses (assumed to be simultaneous and identical in shape) are represented by the full pulse (shaded plus unshaded portions), whereas the output pulse is represented by the shaded portion. (a) Long rectangular input pulses; (b) short input pulses of Gaussian shape.

Here CI is the total capacitance from anode to ground. This signal is much larger than the single pulse output provided T/RIC1 >> (Rf Rz)/2R1. The output pulse shape under these conditions is illustrated in Fig. 26a. If a high degree of time resolution is desired, it is evidently desirable to shorten the input pulse duration T. The limitation on the possible shortening is presented by the rise time of the input pulses. Short pulses are thus in general no longer rectangular in shape. I n that case, the size and duration of the output pulses from the coincidence circuit decrease as the amplitude of the coincident input pulses is made smaller, as illustrated in Fig. 26b. It is easy to extend the argument just given to the case where the two input pulses have different amplitudes. Thus in general we obtain an amplitude distribution of output pulses which depends on the amplitudes and relative timing of the two input pulses. The resolving time that can be attained is determined largely by

+

AMPLITUDE

AND TIME MEASUREMENT IN NUCLEAR PHYSICS

305

the duration of the input pulses and the setting of the discriminator (usually a biased diode) into which signals are fed from the commonanode. I n Rossi’s original circuit (51), vacuum tubes were used as switches instead of diodes. Garwin (52) has modified the Rossi circuit by use of a nonlinear anode load as shown in Fig. 27. The operation of the circuit can be described as follows. Ordinarily, the two pentodes are in a state of strong conduction. The common plate load consists of the resistor R, in parallel with the series combination of another resistor Rb and the germanium diode D.However, since Rb is bypassed by the capacitor Cz, the a.c.

I

E

-

B+

F

FIG. 27. Parallel coincidence circuit designed by Garwin (62). Here E and F indicate individually adjustable bias voltages for the control grids of VI and V2 (both are 6AH6’s). The diode D is a General Electric germanium diode type G-7A. The values of the two resistors are R, = 3.9 kilo-ohms; Rb = 3.3 kilo-ohms. CZis a noninductive bypass capacitor.

load consists simply of R, in parallel with the diode. Slightly more than half the common plate current of the two tubes flows through Rt, and the diode. The attainment of this condition is insured by choosing a smaller value for Rb than for R,. Furthermore, both resistors are large compared to the forward resistance of the diode (but small compared with its backward resistance). If now one tube is cut off by a single incoming negative pulse, the current tries to drop from 2i, to i,. Since the bias on the diode D is just equal t o its forward voltage drop, the plate voltage rises to the vicinity of the region where the diode resistance begins to rise significantly. This increase in resistance reduces the current through the diode branch of the plate load to a value such that the total current through both branches is equal t o i,.

306

E. BALDINGER AND W. FRANZEN

We can discuss current relations simply by disregarding the change in current through Ra in response to a single pulse, and by neglecting the forward resistance of the diode in the conducting state in comparison with Rb. (These assumptions are justified t o a high degree of approximation by Garwin’s choice of R, and Rb; see Fig. 27.) I n that case, the initial current through Ra is 2ip[Rb/(Ra4- Rb)],and this is equal to the final current through this resistor after the arrival of a single pulse, according to the assumption just made. Since the total current after occurrence of a single pulse must drop to i,, a current

is left flowing through the diode when a single pulse has arrived. The singles voltage pulse a t the plate is thus slightly less than the forward voltage drop of the diode in the steady state. When two pulses arrive simultaneously, the anode voltage will rise a t the rate dv/dt = 2ip/C1 until the diode cuts off. Here C1 is the capacitance from plate to ground. At that moment, the charging current drops t o 2i,[Rb/(R,+ R b ) ] ,which is a little less than half its previous value, and continues to flow into C1 until the tubes are turned on again, a time T later. T o distinguish between singles and doubles, a biased diode discriminator follows the circuit described. It is evident that through the introduction of the diode in the anode circuit, Garwin has effectively introduced a third switch into the circuit which is not opened unless the other two parallel switches (the two vatuum tubes V1 and V,) are opened simultaneously. This sharpens the action of the circuit by enhancing the singles-to-doubles output pulse ratio. Except for the additional sharpening, the operation of the circuit is similar t o that of the diode circuit described earlier. Madey, Bandtel, and Frank (55) have applied the Garwin nonlinear load principle to a diode coincidence circuit. Diode parallel coincidence circuits have the advantage of lower capacitance over vacuum-tube circuits. This means that the rate of rise of the signal in the anode can be made faster than for vacuum-tube circuits. Furthermore, diodes are able to carry a larger steady-state forward current than commonly used sharp cut-off low-capacitance miniature pentodes. This means that the anode load resistor can be made smaller for a given desired coincidence output pulse than in the case of pentodes, thus further contributing t o the speed of the anode voltage rise. On the other hand, the output signal from a diode circuit can never be larger than the input signal. It is clear that as soon as the anode poten-

AMPLITUDE AND TIME MEASUREMENT IN NUCLEAR PHYSICS

307

tial ri'ises t o the same value as the cathode potential, the diodes are turned on again. I n pentodes, the coincidence anode signal can be made much larger than the separate amplitudes of the two input pulses because the action of the control grid is essentially independent of plate potential. One would also expect vacuum tubes t o have stabler steady-state current flow and cut-off characteristics than germanium diodes. This would contribute to stability of resolving time. (This remark may not apply to silicon diodes.)

+ 2000

1

obiput Signal

FIG.28. Coincidence circuit described by Bell, Graham, and Petch (43).In each of the two channels, a fast-rising (but slowly decaying) negative pulse from the anode of a 1P21 photomultiplier tube is used to cut off a 6AK5 tube. The resulting positive step in the plate circuit travels down a terminated 100-ohm cable which constitutes the plate load of the 6AK5. At the center of this cable, a T-joint branches off into a shorted length of 50-ohm cable. If both 6AK5's are cut off simultaneously, the pulses (formed by superposition of the reflection from the 50-ohm cable-short on the original steps) will overlap in time. This condition is detected by means of a biased germanium diode connected to the junction. In actual use, slow amplifiers connected to the last dynode of each 1P21 followed by a pulse-height analyzer are used to make an associated selection in amplitude of the photomultiplier pulses.

I n the circuits just described, the resolving time is ultimately limited to the input pulse duration. Bell, Graham, and Petch (43) have described a circuit in which pulse shaping is done essentially in the anode load circuit of the two parallel switches, as mentioned earlier (see Section IV,2,b). A similar circuit has been described by Wells (66),and a modification of it has been introduced by McClusky and Moody (67). The very considerable advantages of this circuit for timing experiments with scintillation counters have been mentioned in our earlier discussion. The operation of the circuit is explained in Fig. 28.

308

E. BALDINGER AND W. FRANZEN

Another variation of the parallel coincidence principle is due t o De Benedetti (58). b. Series Coincidence Circuits. A series coincidence circuit can be described as consisting of two switches in series connected to a power source through a common load resistor. When the two switches are closed simultaneously, a current will flow through the resistor. If either one is closed

L Output

InputI

r-

GiF5) 2.5K

*

(b)

FIG.29. (a) Internal construction of the multigrid 6BN6 vacuum tube. GI and GI are independent control grids; G2 is the accelerator grid [after Fischer and Marshall (69)j.(b) Series coincidence circuit used by Fischer and Marshall (59).

separately, no current flows. Ordinarily a multiple control grid vacuum tube serves as double switch. Series coincidence circuits have not been used very widely until recently on account of the unavailability of a suitable vacuum tube. The introduction of the gated-beam multiple-grid 6BN6 tube has removed this obstacle. A diagram of the arrangement of the grid structure in this tube [taken from a recent article by Fischer and Marshall (59)] is shown in Fig. 29a.

AMPLITUDE

AND TIME MEASUREMENT IN NUCLEAR PHYSICS

309

Each of the two control grids G1 and G3 has a sharp plate current cutoff characteristic. The effect of the two grids on the space current in the tube is not exactly identical, but fairly nearly so. One important distinction between the two grids is the fact that G3 has a capacitance relative to the plate, while G1 has effectively zero capacitance to the plate because of the shielding effect of the intermediate electrodes. Ordinarily, the tube is cut off as far as the current to the plate is concerned. There is, however, a fairly large steady current to the accelerating electrode and the beam-forming plates connected to it (see Fig. 29a), regardless of the control-grid potentials. Thus the cathode current is not cut off, and selfbias can be used for the tube. Moreover, it is evident from the construction of the tube that above a certain level of control-grid potential, the plate current becomes almost independent of the grids, except for grid current, because the cathode is nearly electrostatically shielded from the control grids. On the other hand, if either one of the control grids is biased beyond its individual cut-off potential, no plate current will flow, regardless of the potential of the other control grid. A circuit based on this tube has been investigated in detail by Fischer and Marshall (59) and is shown in Fig. 29. It is evident that the application of simultaneous positive pulses to the two grids GI and G3 will allow a short burst of current to pass through to the plate. (Actually, the two pulses should not be exactly simultaneous, but should be separated by a time interval equal to the transit time of the electrons from GI to GD, which is of the order of 10-9 sec.) The arrival of this burst of current on the plate capacitance causes a potential step t o appear there. The amplitude of this step is proportional to the amount of charge received and inversely proportional to the plateto-ground capacitance. It is advantageous t o use a large plate resistor because of the effect of the capacitance between the second control grid (G3) and the plate. The large plate resistance gives the plate circuit a long time constant in comparison with the duration of the feed-through pulse from Ga, which incidentally is of opposite polarity. This circuit is evidently a very useful device and should be capable of high-speed operation. It suffers from a disadvantage (shared by other normally cut-off vacuum-tube circuits) due to drift-tube effects which become important when the input-pulse duration approaches the electron transit time (-10-9 sec). Thus, for example, the pulse amplitude a t the plate does not exactly go to zero when two very short pulses are shifted relatively so t ha t they no longer overlap in time. These effects have been discussed a t length by Fischer and Marshall (59). c. Bridge Coincidence Circuits. I n a bridge coincidence circuit, a number of ohmic and nonlinear elements are arranged in the form of a bal-

310

E. BALDINGER AND W. FRANZEN

anced bridge. A single pulse applied to one end of the bridge results in no output pulse, while the simultaneous application of a second pulse to a different point of the bridge produces an unbalance and therefore a n output pulse. Circuits of this type have been described b y Baldinger and Meyer (60-62), Bay (6.3))Strauch (64), and others. A large variety of different arrangements of the elements of the bridge and of the points of application of the pulses have been proposed. Perhaps the most straightforward description of the operation of a bridge circuit has been presented by Strauch (64), whose coincidence arrangement is shown in Fig. 30. I

I

I

+300V

FIG. 30. Diode ,bridge coincidence circuit used by Strauch (64). The following component values were used: R1 = 1800 ohms, Rz = approx 1800 ohms, Rs = 100 ohms, R4 = 50 ohms, R6 = 20 kilo-ohms. D I and D Zare IN56 germanium diodes. The use of a 6AK5 difference amplifier at the bridge output is to be noted.

The bridge consists of the ohmic elements R1, Rz, Rt, and R4 and the nonlinear elements (germanium diodes) D1 and Dz. A single negative pulse applied t o the top input will encounter equal impedances to ground on both sides of the bridge, provided the two diodes have matched characteristics over a range of potentials. Thus the points P and Q show no voltage difference. On the other hand, if we apply a negative pulse simultaneously t o the side input, the diode D1 will be shifted to a different portion of its characteristic, so that the bridge as seen from the top input is now unbalanced and an output signal across PQ results. Another way of describing the operation of the circuit is to say th a t the side input cuts off D,so that the negative pulse applied at the top encounters a higher resistance on the left side of the bridge than on the right side. As a result, P goes down in potential relative to Q. I n practice, it is difficult to balance the diode characteristics exactly. To avoid false signals from this cause, the bridge is deliberately unbal-

311

AMPLITUDE AND TIME MEASUREMENT IN NUCLEAR PHYSICS

anced by choice of Rz so that the potential at Q is lowered relative t o P for top input singles. Secondly, an additional resistor Rg is introduced in order to lower potential Q relative to P for single negative pulses applied to the side input. Thus single pulses will produce signals having a polarity opposite t o the polarity of coincidence output signals. Bridge circuits of this type respond t o quite small signals (of the order of tenths of volts) and they have a fast recovery time. This is a very desirable feature when they are used with pulsed accelerators, although there is a danger that as a result of the fast recovery, repeated coincidences will result from the passage of a single charged particle through a scintillation counter, as discussed earlier (Section IV,2,b). Strauch’s method of eliminating single-pulse feed-through is successful provided the input pulses have no positive overshoot.

4. T h e Measurement of the Lifetime of Short-Lived Excited States An interesting application of fast-coincidence techniques is the measurement of the decay time of nuclear excited states. Let us assume t h a t two successive gamma-rays are emitted in cascade from an excited nucleus. Our object is to measure the lifetime of the intermediate state. For this purpose, the variation of coincidence rate with delay for two simultaneous events is compared with the variation of coincidence rate observed in the case of the two successive gamma-rays. Owing to the delay in the emission of the second gamma-ray, the coincidence curve is shifted asymmetrically relative to the prompt coincidence curve. A comparison of the two curves then yields the decay constant of the intermediate excited state of the nucleus. This method makes it possible to measure decay times much shorter than the resolving time of the coincidence circuit. Let us assume we have three related events in time: event A corresponds to the passage of the first gamma-ray through a scintillation counter; event B corresponds t o the passage of the second gamma-ray a time t later. The probability that B will occur between t and t dt after A is f ( t ) d t . Event C corresponds to the formation of a coincidence output pulse. We assume that C is delayed a time t’ relative to A . (This corresponds to the insertion of a delay line of length t’ in channel A . ) The probability that C will occur at all is a function of its time separation 7 = 2’ - t (which may be either positive or negative) relative t o B. We denote this probability by g(7)dr. What is the probability F(t’)dt’ under these conditions that event C will occur when the delay t’ has a value between t’ and t’ dt’? It is clear that

+

+

F(t’)

=

f(t)g(t’ - t)dt

(4.17)

312 Here

E. BALDINGER AND W. F R A N Z E N

f(t)dt

= ue-%t,

for t

> 0, and f ( t ) d t = 0, for t < 0

represents the decay probability of the excited state, where u is the decay constant associated with the disintegration of the intermediate excited state, and g(r)dr reproduces the shape of the prompt coincidence

I Coincidence

FIG.31. Relationship between prompt coincidence curve g ( t ’ ) and delayed coincidence curve F(t’) as analyzed by T. D. Newton (65). The decay constant of the nuclear excited state associated with the delayed coincidence curve is given by dividing F ( ~ A ’ ) F(ts’) by the shaded area. Note that the maximum of F(t’) occurs where F(t’) and g(t’) intersect.

curve. This analysis is due to Newton (65). Substitutingf(t) t > 0 in Eq. (4.17) gives us

= ae-gt

for

Differentiation of this expression with respect to t’ then leads to

dF dt’

=

- F(t’)l

(4.19)

This shows that the maximum of F(t’) occurs when F(t’) = g(t’). Integration of Eq. (4.19) over a limited range of time from t ~ to ’ tg‘ gives us (4.20)

AMPLITUDE AND TIME MEASUREMENT IN NUCLEAR PHYSICS

313

Thus, t o get the decay constant, we divide the difference between the values of the delayed coincidence curve a t points td’ and tB‘ by the difference between the areas of the delayed and prompt coincidence curves enclosed between tA’ and tB’. These relations are illustrated in Fig. 31. Newton’s analysis has been applied by Bell and Graham to the measurement of nuclear half-lives in the 10-9 to 10-10-sec region. A reproduction of experimental prompt and delayed coincidence curves obtained by

Relative Coincidence Rate

FIG.32. Experimental curves of coincidence rate vs delay obtained by Bell and Graham (6‘7). I n the case of Au198, coincidences were observed between the 73-kev nuclear beta-rays of Au198, which lead to a n excited state of HglQ8,and the 411-kev conversion electrons from this state. The lifetime of the excited state was previously ascertained to be less than 2 X 10-10 sec. In the case of the TmI70source, coincidences were observed between nuclear beta-rays from Tm17O leading to a n excited state of Yb”0 and 73-kev L-conversion electrons from this state. Analyzing the curve by the methods of Newton and Bay (65,67) established a half-life of 1.6 X 10-9 sec for the excited state of Yb170.

these authors (66) in the case of the 83-kev transition in YbI7O is shown in Fig. 32. It is obvious that all features of Newton’s analysis are reproduced, as a comparison of Figs. 31 and 32 clearly shows. It is clear that by means of this method one can measure lifetimes which are considerably shorter than the resolving time of the coincidence system used. (The resolving time is of the same order of magnitude as the half-width of the prompt coincidence curve.) The reason for this is th a t the delay curves represent the statistical distribution in time of the co-

314

E. BALDINGER AND W. FRANZEN

incidence probability for a large number of events. The error of measurement is thus much smaller than it would be for an individual observation. Evidently the coincidence circuit must be very stable over long periods of time for measurements of this type. GENERAL BIBLIOGRAPHY Elmore, W. C., and Sands, M., “Electronics, Experimental Techniques.” National Nuclear Energy Series, McGraw-Hill, New York, 1949. Gillespie, A. B., “Signal, Noise and Resolution in Nuclear Counter Amplifiers.” Pergamon Press, London, 1953. Lewis, I. A. D., and Wells, F. H., “ Milli-Microsecond Pulse Techniques.” Pergamon Press, London, 1954. Elmore, W. C., “Electronics for the Nuclear Physicist.” Nucleonics 2, No. 2, 4; No. 3, 16; No. 4, 43; No. 5, 50 (1948). Bode, H. W., “Network Analysis and Feedback Amplifier Design.” Van Nostrand, New York, 1945. Moody, N. F., Howell, W. D., Battel, W. J., and Taplin, R. H., Rev. Sci. Instr. 22, 439, 551 (1951).

Chance, B., Hughes, V., MacNichol, E. F., Sayre, D., and Williams, F. C., “Wave Forms.” Radiation Laboratory Series, McGraw-Hill, New York, 1948. Scarrott, G. G., Progr. Nuclear Phys. 1, 73 (1950). REFERENCES 1. Baldinger, E., and Haeberli, W., Ergeb. ezakt. Naturw. 27, 248 (1953). 2. Elmore, W. C., Nucleonics 2, No. 3, 16 (1948). 3. Gillespie, A. B., “Signal, Noise and Resolution in Nuclear Counter Amplifiers.” Pergamon Press, London, 1953. 4. Bunemann, O., Cranshaw, T. E., and Harvey, J. A., Can.J. Research27,191(1949). 5 . Herwig, L. O., Miller, G. M., and Utterback, N. G., Rev. Sci. Instr. 26, 929 (1955). 6. JaffB, G., Ann. Physik 42, 303 (1913). 7 . Jaff6, G., Physik. 2. 30, 849 (1929). 8. Haeberli, W., Huber, P., and Baldinger, E., Helv. Phys. Acta 26, 145 (1953). 9. Curran, S. C., and Craggs, J. D., “Counting Tubes, Theory and Applications.” Academic Press, New York, 1949. 10. Franzen, W., “Theory and Use of Pulse Ionization Chambers.” Preliminary

Report of the Committee on Nuclear Science of the National Research Council, Washington, D.C., in press. 11. Birks, J. B. “Scintillation Counters.” McGraw Hill, New York, 1953. 12. Hofstadter, R., Phys. Rev. 76, 796 (1949). 13. Barnes, R. B., and Silverman, S., Revs. Mod. Phys. 6, 162 (1934). 14. MacDonald, D. K. C., Repts. Progr. Phys. 12, 56 (1948-49). 16. Van der Ziel, A., Advances i n Electronics 4, 109 (1952). 16. Van der Ziel, A., “Noise.” Prentice-Hall, New York, 1954. 17. McCombie, C. W., Repts. Progr. Phys. 16, 266 (1953). 18. Callen, H. B., and Welton, T. A., Phys. Rev. 83, 34 (1951). 19. Ornstein, L. S., Burger, H. C., Taylor, J., and Clarkson, W., Proc. Roy. SOC.

A116, 391 (1927). 20. Williams, F. C., J. Inst. Elect. Engrs. (London) 83, 76 (1938). 21. Van der Ziel, A,, Wireless Eng. 28, 226 (1951).

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Rothe, H., Telefunken Roehre No. 31, 255 (1953). Baldinger, E., and Leuenberger, F., 2. angew. Math. u. Phys. 6, 420 (1955). Strutt, M. I. O., “Verstaerker und Empfaenger.” Springer, Goettingen, 1951. Bode, H. W., “Network Analysis and Feedback Amplifier Design.” Van Nostrand, New York, 1945. 26. Milatz, J. M. W., and Van Zolingen, J. J., Physica 19, 181 (1953). 27. Milata, J. M. W., and Van Zolingen, J. J., Physica 19, 195 (1953). 68. Bode, H. W., and Shannon, C. E., Proc. Inst. Radio Engrs. 38, 417 (1950). 69. Halbach, K., Helv. Phys. Acta 26, 65 (1953). SO. van Heerden, P. J., “The Crystal Counter.’’ Thesis, University of Utrecht, 194.5. 31. den Hartog, H., and Muller, F. A., Physica 13, 571 (1947). 32. AlfvBn, H., 2. Physik 99, 24, 714 (1936). 33. Keller, K. J., Physica 13, 326 (1947). 34. Elmore, W. C., Nucleonics 2, No. 3, 16 (1948). 35. Maeder, D., Helv. Phys. Acta 21, 174 (1948). 36. Bell, R. E., and Elliott, L. G., Phys. Rev. 76, 168 (1949). 37. Devons, S., Proc. Phys. SOC.(London)A68, 18 (1955). 38. Lundby, A., Rev. Sci. Znstr. 22, 324 (1951). 39. “Handbook of Chemistry and Physics,” 32nd ed., p. 2245. Chemical Rubber Publishing Company, Cleveland, 1950. 40. Landau, L., J . Phys. U.S.S.R. 8, 201, 1944. 4i. Curran, S. C., “Luminescence and the Scintillation Counter.” Academic Press, New York, 1954. 42. Post, R. F., and Schiff, L. I., Phys. Rev. 80, 1113 (1952). 43. Bell, R. E., Graham, R. L., and Petch, H. E., Can. J. Phys. 30,35 (1952). 44. Jost, R., Helv. Phys. Acta 20, 173 (1947). 46. Bittman, L., Furst, M., and Kallmann, H., Pkys. Rev. 87, 83 (1952). 46. Lewis, I. A. D., and Wells, F. H., “Milli-Microsecond Pulse Techniques.’’ Pergamon Press, London, 1954. 47. Bay, Z., Cleland, M. R., and McLernon, F., Phys. Rev. 87, 901 (1952). 48. Post, R. F., and Shiren, N. S., Phys. Rev. 78, 81 (1950). 49. Allen, J. S., and Engelder, T. C., Rev. Sci. Instr. 22, 401 (1951). 60. Morton, G. A., “Recent Developments in the Scintillation Counter Field.” Report of the International Conference on the Peaceful Uses of Atomic Energy, Geneva, 1955. 61. Rossi, B., Nature 126, 636 (1930). 62. Garwin, R. L., Rev. Sci. Instr. 24, 618 (1953). 63. Bell, R. E., Ann. Rev. Nuclear Sci. 4, 93 (1954). 64. Howland, B., Schroeder, C. A., and Shipman, J. D., Rev. Sci.Instr. 18,551 (1947). 66. Madey, R., Bandtel, K. C., and Frank, W. J., Rev. Sci. Znstr. 26, 537 (1954). 66. Wells, F. H., Brit. Inst. Radio Engrs. 11, 491 (1951). 67. McClusky, G . R. J., and Moody, N. F., Electronic Eng. 24, 330 (1952). 68. De Benedetti, S., and Richings, H. J., Rev. Sci. Instr. 23, 37 (1952). 69. Fischer, J., and Marshall, J., Rev. Sei. Instr. 23, 417 (1952). 60. Baldinger, E., Huber, P., and Meyer, K., Helv. Phys. Acta 22, 420 (1949). 61. Meyer, K., Baldinger, E., and Huber, P., Helv. Phys. Acta 21, 188 (1948). 62. Baldinger, E., Huber, P., and Meyer, K., Rev. Sci. Instr. 19, 473 (1948). 63. Bay, Z., Rev. Sci. Znstr. 22, 397 (1951). 64. Strauch, K., Rev. Sci. Instr. 24, 283 (1953). 66. Newton, T. D., Phys. Rev. 78, 490 (1950). 66. Bell, R. E., and Graham, R. L., Phys. Rev. 78, 490 (1950). 22. 23. 84. 66.