AE-323 STOCKHOLM, AKTIEBOLAGET F. Photomultipliers Noise
Robben
in
the
SWEDEN
Measurement
ATOMENERGI
1968
of
Light AE-323 UDC
with
621.383.29.521.391, 535.08
S22
AE-323
NOISE IN THE MEASUREMENT OF LIGHT WITH
PHOTOMULTIPLIERS
F. Robben
ABSTRACT
In order to be able to compare measurements derived from the anode current of a photomultiplier with measurement derived from photoelectron pulse counting, a systematic investigation of the prop erties of some photomultiplier tubes has been made. This has led to a correlation of the properties of a photomultiplier based on the quan tum efficiency T|, the gain G, a photoelectron loss factor S and an effective dark rate D. In terms of these quantities the signal to noise ratio of an experimental measurement can be calculated, given the light flux and measurement technique. The fluctuations in a photomultiplier output are divided into two parts; Poisson fluctuations, and those due to excess noise. It is ex perimentally shown, from measurements on a 931A photomultiplier, that the excess noise exceeds the Poisson fluctuations only at very low frequences, or long DC measurement times (> 10 s), for both pulse counting and anode current measurements . The Poisson fluctua tions are found to be approximately the same for both pulse counting and anode current measurements, at both high light levels where the dark current, or dark pulses, are negligible, as well as at low light levels where the dark current is dominant. The excess noise is found to be somewhat greater in the case of anode current measurements. Thus both pulse counting and anode current measurement techniques have nearly identical noise properties, as far as the photomultiplier is concerned, and selection of either experimental technique depends primarly on the properties of the electronic equipment. By use of a synchronous detection technique, the variance of the pulse count was measured experimentally to an accuracy of ± 4 %, and was shown to be in agreement with that predicted by Poisson statistics.
Printed and distributed in May, 1968. LIST OF CONTENTS Page
Preface 3
1. Summary of Statistical Formulas 5 1. 1 Definition of the Average, Variance, and Standard Deviation 5 1.2 Gaussian Statistics and the Normal Error 7 1.3 Poisson Statistics 10 1. 4 Fourier Analysis of Fluctuating Quantities and the Power Spectral Density Function 11
2. A Photomultiplier Model: the Photoelectron Loss Factor S and the Effective Dark Rate D 15 2. 1 The Quantum Efficiency 71 and the Gain G 15 2.2 The Photoelectron Loss Factor S 18 2. 3 The Effective Dark Rate D 23
3. Methods for Determination of T|, G, S, and D: Fluctua tions in Measurements of Light Signals 27 3. 1 The Determination of 71, G, S, and D from the Photomultiplier Pulse Spectrum 27 3. 2 Noise in DC Measurements 28 3. 3 Noise in AC Measurements with Synchronous Detection 31 3.4 Formulas for S and D from Fluctuation Measure ments 33
4. Discriminator Design and a 931A Photomultiplier Pulse Charge Spectrum 38 4. 1 Discriminator Design 38 4. 2 931A Photomultiplier Pulse Charge Spectrum 40 4. 3 Values of S and D for the Model Photomultiplier 46
5. Fluctuation Measurements and the Inferred Values of S and D 51 5. 1 Variance of the Pulse Count for Short Gate Times 51 5. 2 Fluctuation Measurements: Inferred Values of S 53 5. 3 Fluctuation Measurements : Inferred Values of D 61
6. Fluctuation Measurements with Phase Sensitive Detection: Verification of Poisson Statistics 65
7. Conclusions and Recommendations 71
References 73 - 3 -
PREFACE
When these experiments were begun it was intended only to make a rather simple comparison between the signal to noise ratio of a pulse counting measurement and that of an anode current measurement. The equipment was built for measurement of the afterglow decay of a pulse discharge, and pulse counting was chosen for two reasons; 1) with dig ital readout, it gave a simpler electronic setup, and 2) it was believed that a somewhat better signal to noise ratio could be obtained. Very little additional equipment was required for the experiments described here, but the analysis and presentation proved to be a matter of some complexity. A report was finally attempted which would be understand able to an experimentalist who needs to use a photomultiplier, and is interested in achieving the best possible signal to noise ratio, but who does not have any special background in the theory of random signals. It is hoped that the result fulfills that goal. The shortcomings of the various data which photomultiplier tube manufacturers give for their products has long been known. For exam ple, although lighting engineers are naturally interested in lumens, to physicists who work with photon intensities this unit is most frustrating. It appears simpler to deal with quantum efficiencies, gain, and dark counting rates, since the subsequent conversion of the data to sensitiv ities in amperes/lumen, for instance, is straight forward. Thus in this report I have tried to present a set of simple and intuitively meaningful parameters to characterize a photomultiplier. Not all aspects are covered by the proposed four parameters Tt, G, S, and D; as, for in stance, there is also the pulse rise time and the photosensitive area. It is hoped that a change will be made in the future to a better presenta tion of photomultiplier tube characteristics. The equipment for these experiments was designed and constructed by Bjorn Fjallstam, who also made most of the earlier measurements. His careful work is greatly appreciated. Dr. Kurt Nygaard deserves special thanks for our many discussions of the nature of random noise, as well as for his suggestions regarding the construction of experimental equipment. The support of Dr. Josef Braun is gratefully acknowledged, and thanks are extended to Dr. Olav Aspelund for the loan of his voltage - - 4 -
to -frequency converter. Finally, I would like to thank Dr. Jacob Weitman for his detailed criticisms of an earlier, shorter version of this report. These criticisms led to a more careful development of the statistical theory, the discovery of several new relationships, and an improved presentation of the data. - 5 -
1. SUMMARY OF STATISTICAL FORMULAS
Measurements which result in a set of numbers, produced at regular invervals of time, are first treated. The relations which are more general and apply to a normal error distribution (Gaussian statis tics) are presented first. The specialization to Poisson statistics, where the measurement consists of counting a series of random events, is then made. A brief summary of the Fourier analysis of fluctuating quantities is then given. For a complete presentation ref. 1 is recommended. The power spectral density function for a signal consisting of delta functions occuring at random times is derived, thus providing a spectral descrip tion of counting a series of random events.
1.1 Definition of the Average, Variance, and
Standard Deviation
Assume that a measurement of a quantity which is constant in time, but which has fluctuations of some sort, results in a number. Let this measurement process be repeated, and let the result of each measurement be X^, i = 1, 2, ... N, where the value of i denotes the sequence of measurements. We will define the average value of by N
N £ I
If, as in this work, one is interested in obtaining an experimental value for
N-»03 . i
We might note that
2 where the mean square of X^,
In actual determinations of the variance from a finite number of measurements we will define N Var(X.) (Xi -
i= 1 where, as before, the subscript will indicate that the variance is the result of a measurement and subject to statistical fluctuations. The root mean square deviation e(X), or standard deviation, will be defined by
o(X) = [Var(X)]l/2 , (1-7) and the same convention regarding subscripts will be used. On occasion we will also refer to the normalized standard deviation, given by cr(X)/
1, 2 Gaussian Statistics and the Normal Error
If the statistical fluctuations are normally distributed, then the probability distribution of the measurements X^ is given by the Gaussian distribution function , _-(X. -
The probability that X^ lies within a certain range of values is given by the integral of Eq. (1-8) over this range. In Table I the probability that X. lies outside the range of values Table I,The Probability of Error of a Normal Distribution with Standard Deviation 1 ' Range of Error Probability that X. is no not in the range lo 32 % 2o 4. 6 3c 0. 2 4c 0.007 If the source of fluctuations in X^ is due to a large number of randomly occurring processes, then it can be shown rather generally that the probability distribution of the X^ will be given by Eq. (1-8) (from the so-called central limit theorem). Further, when the errors are random so that Eq. (1-8) is satisfied it is possible to average a number of measurements together in order to obtain a more accurate value. Thus one is often interested in checking the agreement of ex perimental measurements with an assumed Gaussian error distribu tion. The most common sources of disagreement are either drift in the source or measuring apparatus, or accidental errors of some sort which occur infrequently. Measurements with large accidental errors - 8 " can usually be detected simply by inspection, since they will deviate from the mean by several standard deviations. Drift in the measure ment can also often be detected by inspection of a plot of the data, but in the case of a large number of measurements somewhat more analysis, as described in the next paragraph, may be necessary. If a large number of measurements of have been made, then these can be divided into groups of N consecutive measurements, and the average value a( where N is the number of measurements in each group. By plotting the values of mental data, which probably agrees, in part, with this discussion. Of course, more sophisticated methods of statistical analysis may be used, but the time required for applying more elaborate methods is often too great to justify their use in experimental measurements, as the time is quite often more wisely spent in improving the quality of the individual measurements rather than in making an elaborate ana lysis of lower quality data. Var(xp, as defined by Eq. (1-6), is a fluctuating quantity, and in chapter 6 we need to know the magnitude of its fluctuations, or the variance of the variance. We will now proceed to calculate this quantity assuming a Gaussian distribution for the X^. In Eq. (1-6) for Var(X^) the mean value i=l as an approximation to the actual measured variance as given by Eq. (1-6). In order to simplify the notation let us define by Y. = X. - Z. = Var . (X.) , (1-11) J J i so that N Z. (1-12) J i=l 10 We will now calculate Var(Z) according to Eq. (1-3). Using Eq. (1-12) for Z. we obtain N = ~^r { N Y. is assumed to have a Gaussian probability distribution with zero mean, and it is shown in ref. 2 that From Eqs. (1-4), (1-13), and (1-14) it then follows that Var(Z) = 2 (2/N)1/Z (1-16) in the limit of large N. 1.3 Poisson Statistics So far no special properties of have been assumed except that the probability density function be Gaussian. If X^ consists of measure ments of the number of counts occuring in a time interval T, and these counts are randomly distributed in time, then the probability of a measurement resulting in X^ is given by the Poisson distribution p(X.) = e"*<% (1-17) where o' is the average value of X^, i. e. , a - Var(X) = It then follows that R. = Var(X.)/ 1.4 Fourier Analysis of Fluctuating Quantities and the Power Spectral Density Function In order to analyze the fluctuations present in the current output of a photomultiplier, after amplification and filtering, some results from the Fourier analysis of random signals is necessary. Further, the con cept of the power spectral density function is valuable in understanding the nature of fluctuations in a signal. First some of the formulas and functions used in the Fourier ana lysis of random signals will be stated, followed by a calculation of the power spectral density function for a signal composed of Dirac delta functions . The Fourier transform F^(tti) of a signal T(t) is defined by CO p F.(oj) = T(t)e ^ dt , (1-20) V — CO and the inversion is given by CO P F.(uj)e :'tUt d(D . (1-21) -CO From the Parseval theorem it follows that 12 CD p J fi2(t)dt = 2tt F^(w)F.*(w) d w (1-22) If F ^(uu) is the Fourier transform of the input signal of an electrical circuit, and F^(w) the transform of the output, then F->) = H(aj)F1((ti) , (1-23) where H(ou) is the complex transfer function of the circuit. Let the input of the circuit be a Dirac delta function 6(t), and let the resulting output signal be g(t) (called the time characteristic of the system). It can then be shown that g(t)e ^ dt . H(u>) = (1-24) The auto-correlation function 0^(t) of L(t) is defined by 0 (t) = lim ^(t)f.(t - r)dt , (1-25) 1 T- 2T _-T and the power spectral density function N\(u)) by N.\{u) = j' 0i(T)e ^ dT (1-26) We note that 0.(t) can be obtained from N^w) by its Fourier transform , jOJT ^i(T) = 2TT J N.(u;)e JU;i da, (1-27) and note further, by setting t = 0 and using Eq. (1-22), that T CO i p 2 i r lim 2T I fi (t)dt = N.(a))d(JU . (1-28) T-oo d Eq. (1-28) is the mean square of L(t), which can be measured, for instance , by a square law voltmeter. Thus hL(w) is just the Fourier transform of the mean square of a signal. If the signal T(t) is passed through a filter with transfer function H(w), then N2M = N^w) H(w) H*( w) , (1-29) and thus the mean square of a signal derived from f^(t) by means of filters is easily calculated. If band pass filters are used so that H(tu) differs from zero only over the frequency range Aw , Eqs. (1-28) and (1-29) provide a means of measuring N^(w), assuming it is nearly constant in the range Aw. Now let us take a signal which consists of a series of Dirac delta functions of charge q^, q^5^(t - L), where L is randomly distributed in time with an average number r per unit time. Assume that the values of q. are also random , not correlated with t., and have a mean value 1 2 1 T £i(t) = i q^t -1.) (1-30) 1= -00 The average current is given by T 1 P The mean square average of f^(t) turns out to be infinite , but after passage through an electronic circuit with limited bandpass the output signal is finite. Let us now find this value. By substituting f^(t) into Eq. (1-25) it is possible to show that 0^(T) = The first term is the correlation of each pulse with itself, propor tional only to r, while the second term is due to the net random correla- 14 - tions of the different pulses , and is proportional to r . From Eq. (1-26) it then follows that = The first term here is independent of the frequency, while the second term makes a contribution only at zero frequency, and is due to the DC component of the pulses. From Eqs. (1-28), (1-29), and (1-33) Var [f2(t) ] = 2. A PHOTOMULTIPLIER MODEL: THE PHOTOELECTRON LOSS FACTOR S AND THE EFFECTIVE DARK RATE D The model described in this chapter incorporates the usual physical ideas of how a photomultiplier functions; the only reason for its introduction is that it does not suffer from the usual defects of ex perimental devices , in that experimental devices do not behave pre cisely as they should. The experimental measurements described later are analyzed by comparison with the results predicted by this model. The quantum efficiency T, the gain G, the photoelectron loss factor S, and the effective dark rate D are defined, and formulas giving G, S, and D from the pulse charge spectrum of the photo multiplier are derived, with different values of S and D depending on whether the pulses are counted or the anode current is measured. 2. 1 The Quantum Efficiency T1 and the Gain G A photomultiplier consists of a photocathode, from which electrons are emitted due to the incident light, and a series of dynodes, which serve to amplify the number of electrons by the process of secondary emission. Electrons from the previous dynode, or the photocathode , are accelerated by a suitable electric field towards the dynode which they strike. Several low energy electrons are emitted from the dynode surface due to the collision, and an electric field accelerates these electrons to the next dynode, or towards the anode in the case of the last dynode. The group of electrons initiated by a single photoelectron do not all arrive at the anode at the same time, and as a result the pulse of charge received by the anode requires a certain time At, normally in the range _9 1-10x10 s. In this paper we are concerned with measurement times much larger than At, and it is possible to treat the pulses as Dirac delta functions. The pulse width does, however, limit the minimum time be tween countable pulses, which will result in a non-linearity in measured counting for large photo-electron rates. This effect also leads to a re duction in the variance of the number of counts at high counting rates. Current measurements do not suffer from this problem. - 16“ The photoelectron rate in electrons/s, will be given by p l - Tl(X) R(X) dX (2-1) where R(X) is the rate of incident photons of wavelength X, per unit wave length, and T|(X) is the quantum efficiency. This equation can be taken as a definition of the quantum efficiency, subject to an experimental defini tion of JL, which will be given later. The probability p^ for emission of n secondary electrons from a dynode is approximately given by the Poisson distribution, Eq. (1-17), with 1j = X^. With the average number of secondary electrons a in the range 3-5, typical for a photomultiplier , p^ is near zero for n=0, rises to a maximum near a, and then decreases rapidly for large n. The probability that there will be k electrons collected at the anode of the photomultiplier, for an initial single photoelectron, will be similar in shape to the Poisson distribution; that is, X^ = l/C should be substi tuted in Eq, (1-17), where G is the gain of the photomultiplier. To see this, consider the relative variance of the number of secondary electrons, given by Var(n)/ r = Xa(q) , (2-3) 17 - for a discriminator level setting equal to q. For very small q, a(q) will approach unity, and in this approximation Eq. (2-3) forms the basis for measuring £. That is, the limiting value of r as q approaches zero will be defined as i, and we take a(0) = 1. 0 by definition. In terms of the probability p(qr ) of a pulse of charge q’ , a(q) is given by 00 a(q) = J p(q’ )dqT , q and thus p(q) is given by P(q) = -da(q)/dq . (2-4) The average charge of the pulses is given by 00 l p G = Q = q/ = £ 2.2 The Photoelectron Loss Factor S If every photoelectron is counted, then no information is lost from the light signal once the rather inefficient conversion from photons to photoelectrons has been made. In this case the relative fluctuations in the measured signal will be at a minimum. If, however, some of the photoelectrons are not counted, or if some additional fluctuations are in troduced, such as the variations in pulse charge when measuring current, the relative fluctuations in the signal will be larger than the minimum possible (in the sense described above). It is convenient to parameterize these increased relative fluctuations by a factor S which is the effective fraction of photoelectrons counted, based on the actual signal fluctuations of the photomultiplier. Thus S= 1.0 corresponds to counting all photoelec trons (equal to T| times the number of photons), while S = 0. 5, for example, will result if the measured relative fluctuations are the same as if one counted only l/2 of the photoelectrons. A definition of S can be stated as follows. Assume a perfect photoelectron detector with equal amplification for all photoelectrons, and with a quantum efficiency and gain equal to the photomultiplier whose value of S we wish to know. Let this ideal photoelectron detector be substituted for the photomultiplier in an ex perimental measurement of the light intensity. S is then defined as the factor that the variance to signal squared ratio of the photomultiplier must be multiplied by in order to be equal to the calculated variance to signal squared ratio for the perfect photoelectron detector. As this definition is stated S can be dependent on all the para meters of the experimental measurement. However, for the photomul tiplier model described in this chapter, S has a value equal to for pulse counting measurements and for current measurements, and these values of S are functions only of the photomultiplier dynode voltage and the discriminator level. Let us find the variance to signal squared ratio for the perfect photoelectron detector, assuming a steady photoelectron rate A, which - 19 - is counted for a time T. The average number of counts Var(X) = _1_ For the model photomultiplier with a discriminator detecting all pulses with charge greater than q, the average counting rate is just a(q)j£, and the variance to signal squared ratio is given by Var(X) _ 1 From the definition of S, Eq. (2-9) should be equal to S times Eq. (2-10), P and we find that sp = a(q) . (2-11) We see that in this case corresponds to the fraction of photoelectrons which are counted, and hence the name "photoelectron loss factor". Other experimental measurements using pulse counting , such as those resulting from square or sine wave modulation of the light signal, can be shown to lead to the same value of S , and thus Eq. (2-11) is of general validity. In the case of current measurements account must be taken of the additional fluctuations introduced by the amplification process of the photomultiplier. This may be determined from knowledge of the pulse charge spectrum p(q), which in turn can be found either by direct ex perimental measurement [from a(q) as calculated according to Eq. (2-4)], or by assumption of a Poisson distribution of secondary elec trons. 2 We will first find an expression for in terms of CO q Var [q n(q) ] = q2Var [7] (q) ] (2-14) = T lq2 p(q) , so that CO Var(U) = Tl q2 p(q)dq (2-15) 0 = T i«\> Eq. (2-15) defines CO 2 P 2 and in terms of a(q) we can show that 1 0 The variance to signal squared ratio is then given by Var(U) _ 1 For the perfect photoelectron detector all pulses must contain the same charge qQ, and thus the probability of a pulse per unit charge is given by p(q) = 6(q - qQ). In this case the variance to signal squared ratio is the same as for pulse counting and is given by Eq. (2-9). From the definition of S, Eq. (2-17) when multiplied by S. should be equal to Eq. (2-9). from which we find that = As in the case of S , this result is independent of the measurement P technique. 2 Let us now find The gain of a photomultiplier of K stages is thus given by G = aK ; (2-21) from this equation the value of a can be found. For the first stage we have Var^(n) = a , (2-22) the fundamental property of a Poisson distribution. For the second stage we will have Var^(n) = (2-23) - 22 - where the first term is just Var^(n), multiplied by O' , which would be the value of Var^n) if the multiplication process did not introduce any additional fluctuation. The second term is just Vark(n) = For a ~ 4 and k ~ 10 the sum in this equation can be replaced by 1 /{a - 1) to a good approximation, and we find that 2k Vark(n) = . (2-25) From Eqs. (2-19), (2-20), (2-21), and (2-25) the final result for is obtained: 2 This is independent of the number of dynodes, since we have taken the limit of the sum in Eq. (2-24). For a=5, a typical value , will be 0.8, which will give a negligible increase in the relative standard deviation for most experimental measurements, compared to the maximum pos sible value 1.0 of S.. i Detailed measurements of secondary emission noise show that the Poisson distribution assumed above is not quite correct [3]. A more general result for the variance of the photomultiplier pulses is given by Shockley and Pierce [4]; see also Van der Ziel [ 5 ]. The power spectral density function for the photomultiplier current has already been worked out in Ch. 1, Eq. (1-33): N^uu) = The variance of any measurement of the current is proportional to N^(id ), as can be seen from Eqs. (1-28) and (1-29), and the signal itself is proportional to Nlfa> _ If all the pulses contained the same charge, this equation would 2 just be equal to l/l, and so by definition of S we find that S. = 2. 3 The Effective Dark Rate D In actual photomultipliers a dark current, or dark pulse rate, is found in the absence of light. The dark current originates from several effects, for instance, thermionic emission of electrons from the photo cathode, thermionic emission of electrons from the dynodes, ion forma tion, cosmic rays, and leakage currents. Cooling a photomultiplier reduces the dark current considerably. It is convenient to introduce an effective dark rate, D, which parameterizes the fluctuations in the signal due to the dark pulses and leakage currents. Since D and S are interlinked, it is more consistent to define these quantities together; thus in the following both quantities are defined. The meaning of S is unchanged from the previous defini tion. . A definition of both D and S can be stated as follows. Assume a perfect photoelectron detector with equal amplification for all photo electrons, with a quantum efficiency and gain equal to the photomulti plier whose values of D and S we wish to know, and with a rate D of dark pulses of the same uniform amplitude as the photoelectron pulses. Let this perfect photoelectron detector be substituted for the photo multiplier in an experimental measurement of the light intensity. In both cases, the variance of the measurement will be given by the sum of two terms, one which depends on the photoelectron pulses, and one which depends on the dark pulses. Thus the variance to signal squared - 24 - ratio (the signal is defined as that part of the measurement which is pro portional to the light intensity) will be given by the sum of the photoelec tron variance to signal squared ratio and the dark variance to signal squared ratio. According to our previous definition, S is the factor that the actual photoelectron variance to signal squared ratio must be multi plied by in order to be equal to the calculated photoelectron variance to signal squared ratio of the perfect photoelectron detector. D is defined by setting the calculated dark variance to signal squared ratio of the perfect photoelectron detector equal to the dark variance to signal squared ratio of the photomultiplier. For the model photomultiplier, D is equal to D for pulse counting measurements and for current measurements, and is only a function of the photomultiplier dynode voltage and the discriminator level. Let us now calculate for the model photomultiplier. Let the dark pulse rate, measured with a discriminator level q so that all dark pulses with charge greater than q are counted, be given by f(q). Assume a steady photoelectron rate a discriminator level q, and let be the number of counts in time T. The average value of will be just The signal Var(X) = f(q) + 1 If the perfect photoelectron detector described in the definition of S and D is used, Eq. (2-31) would be replaced by Var(X)___ D_ , _1_ 2 " 2 + (2-32) By comparison of these two equations it is seen that the second term on the right leads to Eq. (2-11) for S , and for D we find the expression - 25 Dp = f(q)/a2(q) • (2-33) Now let us evaluate D^. Let the dark pulse spectrum be s(q), where s(q) = d f(q)/dq . (2-34) Let the current be integrated for time T, resulting in charge IL. The mean charge is then given by s(q)dq + je and the signal Using the same reasoning which led to Eq. (2-15) we find that Var(U) = [ Jq2s(q)dq + l ^ q2s(q)dq Var(U) = 0______+ , For the perfect photoelectron detector described in the definition of S and D, the variance to signal squared ratio is given by Eq. (2-32). Comparison of the last term leads to Eq. (2-18) for S.. By equating the first term we find that q s(q)dq D. (2-39) i The integral in this equation can be taken over f(q) instead, and there results f max P q2df J 0 D. = (2-40) 1 2 3. METHODS FOR DETERMINATION OF 7], G, S, AND D: FLUCTUATIONS IN MEASUREMENTS OF LIGHT SIGNALS Experimental methods for determination of T], G, S, and D are discussed in this chapter. Using the formulas derived in the previous chapter, a brief summary is first given of the calculations of these parameters from measurements on the photomultiplier pulse charge spectrum. The fluctuations in a measurement of light intensity are of primary concern, as they determine the signal to noise ratio of an experiment, and the dependence of these fluctuations on S and D is worked out for some typical examples of light intensity measurement. The treatment is divided into two parts, the first dealing with measurement of steady, DC light levels, and the second dealing with the measurement of mod ulated, or AC, light signals using synchronous detection. The latter is often accomplished by use of a "lock-in amplifier"; in the case of pulse counting an exposition of this technique has appeared in ref. 7. In the last section the formulas used for calculating values of S and D from fluctuation measurements, to be described in Ch. 5, are given. 3. 1 The Determination of 71, G, S, and D from the Photo multiplier Pulse Spectrum The photoelectron counting rate l can be determined for a steady light source by extrapolation of the measured counting rate to zero pulse charge, assuming a discriminator of sufficient sensitivity. If the inten sity of the light source is known the quantum efficiency can then be found, using Eq. (2-1). Note that this will give an effective value of 7] and takes into account the possibility that some photoelectrons will not initiate a pulse charge, and are lost. The value of 7] measured under conditions assuring the collection of all photoelectrons will in general be larger than the effective value obtained in a photomultiplier. ,-28- By calibration of the discriminator for pulse charge, a(q) can be determined [Eq. (2-3)] and then the gain G can be found from Eq. (2-6). Alternatively, G can be found from measurement of the anode current by using Eq. (2-8). is just equal to a(q) [Eq. (2-11)], and can be calculated from a(q) according to Eq. (2-18). Alternatively, can be found from the gain G by use of Eq. (2-26). Upon determination of the dark pulse rate f(q) as a function of the discriminator level, can be found from Eq. (2-33) and from Eq. (2-40). 3. 2 Noise in DC Measurements The word "noise" has so far been avoided (except in the title) as it can have a variety of meanings, depending on one's point of view, and must be defined for a given experimental measurement. Noise in general is an annoyance, something which should be eliminated, and by infer ence can be eliminated. In this sense the fluctuations in a measurement of light intensity which originate in the finite number of photoelectrons are not noise, as they are an inherent part of the measurement, and in fact some measurements are based directly on these fluctuations. We will refer to this type of fluctuation as Poisson fluctuations . Included in Poisson fluctuations will be those fluctuations due to the dark pulses and to the variations in pulse charge, so that all fluctuations in a measure ment made with the photomultiplier model of Ch. 2 will be Poisson fluc tuations. These are all grouped together because their source has a power spectral density function which is independent of frequency. However, the gain, dark pulse rate, and quantum efficiency can vary in time, introducing additional fluctuations into the measurement. We will refer to this type of fluctuation as excess noise . The power spectral density function of the excess noise is not independent of frequency and generally increases at low frequencies; the effect of excess noise on measurements can generally be reduced, or elimi nated altogether. We will let the power spectral density function of the Poisson fluctuations be denoted by Np (independent of til) and that for the excess noise by N^(w), at the anode of the photomultiplier. The total value is, of course, just the sum of Np and N^(w). - 29 - Let us consider the integration of the photomultiplier current for a time T, resulting in the charge Q.. The average current will be given by X.=Q./T , (3-1) with a mean value independent of the integration time T. By Eq. (1-28), the variance will be given by CO P _1_ Var(X) = H(o))H*(cu)dc« + N (w)H(u))H*(w)dw (3-2) 2tt 2rr d e - CO where H(c«) is the complex transfer function of a filter which has the effect of integration for time T. Eq. (1-28) was written on the basis that the signal X is continuous in time, while we have taken X^ to be a sequence of measurements separated in time; however, in the limit of infinite time the variance of these two measurements is identical. We note here also that X. can just as well represent a measurement of the counting rate, as it is only necessary that CL equals unity and that Np corresponds to a signal of unit delta functions. The filter which corresponds to integration for time T with output divided by T, giving the average current X, has as a response to a 6 func tion at time t=0 the time characteristic . g(t) = l/T, 0 < t < T (3-3) = 0 otherwise. Using Eq. (1-24) we find the power transfer function to be H((B)H*(aO =I~J sin 2 kd , (3-4) and this is shown in Fig. 1 as a function of ojT. The first integral in Eq. (3-2) is easily evaluated and we find that CO p Var(X) = Np/T + ^ Ne(uu)H(a))H 4»daj (3-5) u — 00 30 Fig. 1 The power transfer function of a filter which corresponds to integration HM H w for time T. The inset shows the time characteristic g(t). 6J T Note that when X corresponds to a counting rate, then Np = r [Eq. (1-33)] and we obtain agreement with Eq. (1-18) for Poisson counting statistics. Now suppose that N^(io) is much less than Np when (jo > cjoq. Then when T « l/u) Q the integral over N^(io) will be negligible, and thus Var(X) ~ l/T . (3-6) In this case the individual values of X^ will follow a Gaussian error distri bution, as discussed in Ch. 1. However, for values of T near to or greater l/o)Q, Var(X) will be larger than predicted by Eq. (3-6), and the values of X^ will either show a drift or fluctuations, depending on the value of w at the dominant values of N^(w). This corresponds to the discussion of errors in a measurement of X^ given in Ch. 1, and we see now that the Gaussian distribution arises from a power spectral density function which is indepen dent of (jo in the range of frequencies where H(co) is not negligible, while drift and other excess noise arises from the frequency dependent N^(uo). If the photomultiplier current X is measured using an RC low pass filter with the power transfer function shown in Fig. 2, very similiar results are obtained. Neglecting the excess noise, the variance is given by Var(X) = Np/2RC (3-7) which is, interestingly enough, only half as great as that for integration for a time T=RC. ( Fig. 2 The power transfer function H((u)H*((u) of a simple RC filter. The inset shows the H(w) H M time characteristic g(t). 0 RC 2RC U RC 3, 3 Noise in AC Measurements with Synchronous Detection Suppose now that the measured output of the photomultiplier is composed of two parts; Y, which is proportional to the light intensity which we want to measure , and Z, which is a background, or zero, signal due either to dark current or to background light. If Y is much larger than Z, that is, the zero signal is negligible, the excess noise will generally only become a problem if rather precise measurements of light intensity are required. If, however, Y is smaller than Z, and particularity if Y is much smaller than Z (corresponding to a small signal in the presence of a large background) , the fluctuations in the measurement of Y due to the excess noise can often be much reduced, or eliminated altogether, by using a so-called phase sensitive detection technique. Let us first measure X=Y+Z, the sum of the two parts, and then follow this by a measurement of Z alone (by shutting off the light whose intensity we wish to know). In order to obtain Y we subtract these two measurements , and thus the variance of Y as determined by this tech nique will be given by Var(Y) = Var(X) + Var(Z) . Let us now suppose that Y is small compared to Z, so that we obtain Var(Y) = 2 Var(Z) . (3-8) - 32 - By Eq. (1-28) this can be changed to an integral over the power density spectral function and we obtain 00 CO n Var(Y) H(w)H*»dw ( g(t) = l/T , 0 < t < T = -l/T, T < t < 2T (3-10) = 0 otherwise, which has the power response function . 4 u)T H(u))H*» = (3-11) wT sin ~T shown in Fig. 3. As before, the first integral in Eq. (3-9) can be carried out and we find that Fig. 3 The power transfer func tion H(w)H*(w) of a filter corresponding to the sub traction of two consecutive integrations, each for time T. The inset shows the time characteristic g(t). 1 2 3 4 5 6 7 8 UT - 33 2N, nCO Var(Y) + 1 Ne((JU)H((ju)H*((ju)d U) (3-12) T TT -CO The contribution of Poisson fluctuations to Var(Y) is just twice as great as in Eq. (3-9). as expected. Only the values of N^(w) which lie in the band pass shown in Fig. 3 will make a contribution to Var(Y), and the very low frequency components will make no contribution at all. even if they are quite large. This is in contrast to ordinary integration which has unity response for low frequencies , as shown in Fig. 1. If now the values of Y from a large number of measurements are averaged together the power transfer function becomes peaked about U)T = tt, 2tt...... as indicated in Fig. 3 by the dashed curve , with the peaks decreasing rapidly in amplitude as uuT increases. If M is the number of cycles averaged together Eq. (3-12) becomes 2Np i * Var(Y) = + - j N>)H(u,)H*(u))du> . (3-13) Thus the Poisson fluctuations decrease with increasing M, or increasing total measurement time, and further, if N^(w) at uu = tt/T is small com pared to Np there will not be any contribution from the excess noise. In this way one is able to obtain the theoretically minimum fluctuations in a measurement of X. Rather than square wave detection of the signal, as worked out here, one can use sine wave detection as is the case with "lock-in" amplifiers. The general result is the same, except that the variance is slightly larger than for square wave detection. 3.4 Formulas for S and D from Fluctuation Measurements In the principle method used to determine S and D from the fluctua tions the photomultiplier output (current or pulses) was integrated for time T and punched into paper tape, repeating this process to obtain a series of measurements X.. Alternate measurements were then subtracted from i each other to yield the difference Yp The variance of Y^(Y^/T to be - 34 - precise) is just that given by Eq. (3-12), and thus only excess noise in the frequency range given by Fig. 3 makes a contribution to the meas ured variance. In the determination of S a large light level was used so that the dark pulses could be neglected. Using the definition of S we find a vari ance to light signal squared ratio for uniform pulses at a rate IS given by the expression Var(Y) _ 2 (3-14) This is what has been used to evaluate S as a function of T and l. Let us inquire a little more into the meaning of S as given by Eq. (3-14). Taking the case of pulse counting, let us assume that N— = la(q), as calculated for the model photomultiplier, and find Var(Y)/ 2Ne(2/T) Var(Y) (3-15) A2(q)T By comparison with Eq. (3-14) we see that the excess noise has just been absorbed into the measured value of S, -----alal — : NJ2/T) (3-16) V 1+-wj For short gate time T, = a(q) as calculated for the model photomulti plier since we have postulated that N^(uu) < Np for higher frequencies. [Note that Np = la(q). ] At long gate times Ng(2/T) may be comparable to Np, and S will be reduced. It is also informative to consider the pos sible dependence on light intensity. If the excess noise dominates in Eq. (3-15), then the variance to signal squared ratio will approach a constant if Ne(2/T) H . Under this condition S will turn out to be inversely P proportional to A. If is independent of & the variance to signal squared ratio will decrease with increasing just as with Poisson fluctuations . Similiar comments can also be made about when calculated from Eq. (3-14). In summary, S should be found to depend on both T (or au) and l, when the excess noise becomes important. was also evaluated from the variance of the current using the RC-CR filter shown in Fig. 4. This has the power transfer function (aiRC)' H(u))H*( This has been used to evaluate S. from the filter measurements. i In order to find D it is easiest to measure a variance to light signal squared ratio in the limit when the photoelectron rate is small compared to D and thus makes no contribution to the variance. We have determined the variance of the dark signal alone for the photomultiplier, and then divided by a calculated light signal. The dark signal is integrated for time T, and Var(Y.), where is the difference between two consecutive measurements , is determined (just as in the determination of S). We then assume that a small light signal is added to one of the paired measure ments, and calculate R C Fig. 4 Schematic diagram of the RC-CR CZ3- ■o filter used for fluctuation measure ments. -36- Fig. 5 The power transfer func tion H((ju)H*(uu) of the RC- CR filter. H(u)H(«) .08 For the perfect photoelectron detector used in the definition we find that Var(Y) _ 2D Let us first take the pulse counting measurements and find D^. In this case the calculated value of Var(Y) D = —5----- (3-21) P 2a2(q)T In the case of current integration measurements the calculated value of These measured values of D represent the combined effect of the Poisson fluctuations (which produce a value of D independent of the gate time T) and the excess noise within the frequency range shown in Fig. 3, as was previously shown for S. was also evaluated from the variance of the dark current using the RC-CR filter. The calculated current signal ahead of the filter is given by 6RC D. = 4. DISCRIMINATOR DESIGN AND A 931A PHOTOMULTIPLIER PULSE CHARGE SPECTRUM 4. 1 Discriminator Design Discriminators have a minimum input pulse voltage, or charge, to which they will respond, and for satisfactory performance with a photo multiplier this minimum pulse charge must obviously be smaller than BIAS VOLTAGE Fig. 6 Tunnel diode circuit. 1- Tunnel diode GE TD 252A. 2- Diode. 3- 2700. 4- 0. 1 ph. 5- Ther mistor, used for temperature stabilization. 6- 1 pf. 7- 820. -±±- 2 -te TO AMPLIFIER ttt CALIBRATE The tunnel diode discriminator circuit shown in Fig. 6 was used. A bias voltage keeps the tunnel diode close to its forward peak voltage, point 1 in Fig. 7. The discriminator level is changed by varying the bias voltage. A 270 ohm resistor connects the photomultiplier to the tunnel diode, reducing the switching time of the circuit, and diode (2) transfers the tunnel diode output pulse to the amplifier, while isolating the amplifier capacitance from the tunnel diode circuit. The 5 ns long photomultiplier pulse is somewhat attenuated by the 0. 1 gh inductance. The resulting output pulse of the tunnel diode is 5 ns long, and corre sponds to the path 1-2-3 in Fig. 7. However, reset to point 1 requires approximately 50 ns, or 10 times the output pulse duration, apparently characteristic of this type of circuit with inductance reset. A charge sensitivity calibration of this tunnel diode discriminator was made by using a 50 pps mercury relay pulse generator connected to "calibration" in Fig. 6. Due to stray wiring capacitance in the circuit the absolute value of this calibration was in error, and a suitable norma lization was later determined using a photomultiplier as the pulse source. -1 3 The minimum charge sensitivity is about 1.4x10 coulombs, as the circuit begins to free run at 18 mhz at this trigger level. In Fig. 8 the response of the tunnel diode trigger at a fixed triggering level of -13 4. 2x10 coulombs to varying pulse charge from the mercury relay pulse generator is shown. The number of counts in 10. 22 s is plotted vs. pulse charge, and varies smoothly from 0 to 511 counts. The width -13 given by the steepest straight line shown on the figure is 0. 45x10 coulombs , and can be considered to be the equivalent rms noise of the discriminator circuit. This noise is not independent of the trigger level, but shows about 50 % variation over the range 2 to 24x10 coulombs. - 40 Fig. 7 Forward voltage characteristic of a tunnel diode. v Alternatively, a Keithley Model 105 pulse amplifier followed by a discriminator was tested. The photomultiplier anode was connected directly to the high impedance unity gain stage and shunted by a 200 0 resistor, giving a time constant of 20 ns with 100 pf input capacitance. The equivalent rms noise, determined as described previously, was -13 0. 18x10 coulombs, independent of the pulse amplitude , and the minimum detectable amplitude with less than 1 noise pulse per second -13 was about 0. 5x10 coulombs. With this input circuit the Keithley amplifier has an equivalent noise 3. 5 times smaller than the direct tunnel diode trigger, as well as 5 times shorter deadtime. However, as will be seen later, a good type 931A photomultiplier tube with 100 Volts/stage has an average photoelectron pulse of about 8x10 ^ coulombs , so that either arrangement works quite satisfactorily. It is interesting to note that the equivalent noise found for the Keithley amplifier, which corresponds to 0. 18 mV, is about 2. 5 times the input rms noise as stated by the manufacturer. Since our equivalent noise corresponds somewhat to a peak-to-peak noise measurement, this result is quite reasonable. 4.2 931A Photomultiplier Pulse Charge Spectrum and Values of S and D A 6 Volt, 0.3 a tungsten filament bulb powered by a regulated supply and operated at 5 Volts was used as a light source. This light source ap peared to have a long term stability better than 3 %. As shown schemat ically in Fig. 9, this bulb illuminated the photomultiplier through a - 41 Fig. 8 Response of tunnel diode trigger at fixed level 4. 2x10 ~ * ^ coulombs to pulse charge q generated at 50 cps. N is the num ber of counts in 10. 22 s. Equivalent noise derived from maximum slope width of trigger response is 0. 45x10 coulombs . 3.6 3.8 40 4.2 4 4 4.6 4.8 50 q (1013 coulomb ) blackened tube with entrance and exit holes, a large number of baffles, and slots for insertion of various filters. A Corning CS4-102 filter isolated the wavelength band 5300 to 5700 A. Thin film neutral filters were used for attenuation, and were carefully calibrated with good agreement between the values found by measurement of the photo multiplier anode current and the values found by measurement of the pulse counting rate. Except where otherwise indicated, all pulse meas urements were taken using the tunnel diode discriminator described in section 4. 1. The 30 mhz scaler, gate, timer, and paper tape punch in dicated in Fig. 9 completed the electronic equipment used for integra tion of the photoelectron count. (A detailed description of this equipment is given in Ref. 8) Measurements of the pulse counting rate as a function of discrim inator level are shown in Fig. 10 for the highest gain 931A photomulti plier tube selected from the 3 purchased. Counting rates are shown both for light pulses and for dark pulses. The light pulse rate was nominally 100 times the dark rate, and the total voltage across the photomultiplier dynodes was varied from 800 to 1200 V. It is necessary to carefully adjust the position of the light spot (1 mm diameter) on the photocathode to obtain the maximum possible signal. In the vertical direction the response of these photomultipliers is very nearly uniform, but in the horizontal direction 1 mm motion can produce a noticeable change in - 42 - 2 3 4 1 4 5 Fig. 9 Schematic of light source and timer circuit. 1- Blackened tube with baffles. 2- Tungsten filament light source. 3- Color filter for selection of wave length. 4- Thin film neutral density filters, mounted at a SCALER TRIGGER small angle from normal to avoid internal reflections. 5- Photomultiplier. PUNCH TIMER the signal. The steep rise in counting rate at about 1x10 ^ coulombs is due to the onset of free run of the tunnel diode trigger and should be disregarded. The light counting rate was found to be very stable, but the dark counting rate varied considerably, depending on the previous voltage, temperature , and time under constant voltage. At 1000 V the dark counting rate decreased slowly by a factor 2 over a period of 1 year. In the curves shown an attempt has been made to subtract the dark counting rate from the light rate, but a relatively large error has occurred at 800 V due to drift of the dark rate. More careful measure ments show that the light pulse rate drops off approximately exponential ly with increasing discriminator level, at least to the level where the light pulse rate is only 10 % of the dark pulse rate. Casual examination of these curves shows a plateau in the light pulse rate corresponding to amplified single photoelectrons , while the dark pulse rate apparently consists primarily of electrons emitted from the dynodes with correspondingly smaller amplification, as well as a few very large pulses which probably are associated with some type of ion regeneration. All of our photomultiplier tubes had similiar charac teristics - RCA 1P21, 1P28, and 931A EMI 6095B, Hitachi R 212 (S5 response) and R 2 13 (S20 response), and Phillips XP- 1000. By extrapolation of the light pulse curves in Fig. 10 to zero dis criminator level, a photoelectron counting rate 1=1. 6x10 pps is found. There appears to be a slight dependence of 1 on the photomul tiplier dynode voltage, which may be associated with a more complete collection of the photoelectrons at higher voltages. - 43 Fig. 10 Pulse counting rate r(q) 1100 V as a function of discrim 1200 V ' inator level q for the •1000 V 93 1A with the highest pulses 900 V gain. Each dynode has 800 V an accelerating poten tial equal to 0.1 V. ,1200V 1100 V 800V .pulses 10 12 14 16 18 20 10 coulombs From measurement of the anode current the average pulse charge Fig. 11 Gain of 931A as a function of the total dynode voltage, V. The accelerating potential of each dynode equals 0. 1 V. V, volts less than that found from measurement of the anode current, which is reasonable agreement considering the various possible experimental errors. The light pulse curve shown in Fig. 12 is just la(Q), where a(Q) is the fraction of photoelectrons counted (see section 2. 1). By taking the derivative of this curve , a quantity proportional to p(q), the pulse charge spectrum [see Eq. (2-4)] is found, and this is shown in Fig. 13, plotted vs. Q. It shows a peak in the region of Q=l, but does not fall to zero at Q=0. The original data, however, is not considered accurate enough to determine p(Q) with any precision near Q=0. It is interesting to note that for a Poisson distribution p(Q) would also not approach zero at Q=0, but would have a small residual value. The dark pulse counting rate f(Q) for the 931A is also shown in Fig. 13 for the various tube voltages. The dark counting rate does not scale onto a single curve , but, at constant Q, increases for higher tube voltages. Apparently the source of the dark pulses is dependent on the value of the electric fields in the tube . The data shown in Fig. 12 is actually not the same as in Fig. 10, but was taken approximately 1 year later. During this period of time the dark pulse rate decreased almost by factor two; further, the dependence on the voltage (at constant Q) decreased as well. - 45 - 1000 V Fig. 12 Pulse counting rate r(Q) 900 V as a function of the norma 800 V lized discriminator level Q = q/ Q = q/ The linearity of the counting rate versus light level was checked by measuring the apparent attenuation of a filter at different light levels. If we assume that the discriminator is characterized by a single dead time t , the real counting rate R is given by 1 R = (4-1) 1 r t where r is the measured counting rate. Now let r^ be the counting rate without the filter r^ the counting rate with the filter. Let T(r2) = r2/rj , be the apparent transmission of the filter, and T& = R^/R^ be the actual filter transmission. One then finds that T Fig. 13 Pulse charge spectrum dr/dq as a function of the normalized discrim inator level Q, obtained by differentiation of data in Fig. 12. In Fig. 14 T(r^) is shown plotted versus r^, and good agreement with a straight line is found. The value of found from Fig. 14 is equal to the value found by measurement of the photomultiplier anode current within the experimental error, estimated to be ±2 % from the reproduc ibility of filter attenuation measurements. The deadtime of 55 ns has been referred to earlier and is in agreement with estimation from circuit measurements . If the counting rate is corrected for deadtime by use of Eq. (4-2), the dynamic range of this photomultiplier with pulse counting is nearly as great as with anode current measurement, even with the rather long deadtime of 55 ns. The linearity of the corrected counting rate is within ~ 3 % up to 1. 5x10^ pps, which corresponds to an anode current of 2.2x10 ^ a at 1000 V. Rather large fatigue effects have been found for these tubes at this level of anode current. 4. 3 Values of S and D for the Model Photomultiplier If the actual photomultiplier behaves like the model photomultiplier discussed in Ch. 2, then S and D can be determined from the pulse charge spectrum, as summarized in section 3. 1. 5^ is just equal to a(Q), according to Eq. (2-12), and a(Q) for the 931A is equal to the light pulse curve of Fig. 12 divided by i, the zero intercept. Let us now consider the selection of a suitable discriminator level for measurement of it,. With a discriminator level Q= 1. 0 S will P be about 0. 5, which results in a factor 2 increase in the variance com pared to the theoretical minimum at S^= 1. 0. A factor 2 increase in - 47 - Fig. 14 Apparent transmission T(r2) of a filter as a function of the counting rate r A deadtime T = 55 ns was derived from the slope. = 0 467 variance is a small, -but significant decrease in signal to noise ratio. On the other hand, for Q=0. 4, will be about 0. 8 and in this case the in crease in variance, and resulting loss in signal to noise ratio, will be essentially negligible in normal experimental measurements. Thus we can take 0=0. 4 as a practical working value. This means that the discriminator must have a minimum charge sensitivity of at least 0.4 11. For the Keithley 105 pulse amplifier, with a minimum sensitivity *13 6 of about 0. 5x10 Coulombs, a photomultiplier gain of only 0. 7x10 would be sufficient for a value of S =0.8. P The measured curves of a(Q) for our other photomultiplier tubes (RCA 1P121, RCA 1P28, EMI 6095B, Hitachi side window types and Phillips XP-1000) have the same general shape as shown in Fig. 12, and thus the above criterion q . =0.4 eG (4-3) unm - 48 - Fig. 15 Effective dark rate D„ for 1000 V the 931A as a function of Q 900 V and V, calculated according 800 V to the model photomultiplier. can be taken as a general requirement of the minimum discriminator sensitivity necessary in order that there is no appreciable increase in the signal fluctuations , due to undetected photoelectrons. Calculation of from a(Q) as shown in Fig. 5, using Eq. (2-18), gives the value 0. 78, independent of the photomultiplier voltage since the experimental a(Q) did not have a measurable dependence on photomulti plier voltage. Alternatively, calculation of from Eq. (2-26), which assumes a Poisson secondary electron distribution and requires only the gain, gives values ranging from 0.76 to 0.82 as the photomultiplier voltage is varied from 800 V to 1200 V. The range of experimental error in the value 0. 78 derived from the a(Q) curve is such that these values are considered to be in agreement. In actual fact more careful measurements show up small differences in a(Q) for different photomultiplier voltages. In this case a variation in S., calculated from a(Q), would be found, as is found in calculated by assumption of a Poisson distribution for the secondary electrons. The effective dark rate for pulse counting has been calculated with f(Q) and a(Q) from Fig. 12 according to Eq. (2-33), and is shown in Fig. 15 as a function of Q for various voltages. Here we note that there is a rather broad minimum in the range 0.3 < Q > 0.9, showing that, for this photomultiplier, best operation is achieved in this range of Q when the light signal H is small compared to D . Best operation - 49 - means that the relative fluctuations in the measurement of & will be at a minimum. We also note from Fig. 15 that is smaller for lower photomultiplier voltages. The minimum sensitivity of the discriminator fixes a lower bound to the photomultiplier voltage, however, so that im provement in this direction is limited. For this particular photomulti plier the value of Q (= 0. 4) previously shown to be generally satisfactory for large light level is also within the range of Q best for low light levels, and could be considered satisfactory for all light levels. Other photomulti pliers with larger values of the dark pulse rate showed a minimum in at larger values of Q, though, so that in general a value of Q greater than 0.4 is better for low light levels. The equivalent dark pulse rate for current measurement, D^, was calculated for this tube using Eq. (2-40), and is shown as a function of the photomultiplier voltage in Fig. 16. is in general about 40 % larger than D , and thus pulse counting measurements should give somewhat smaller relative fluctuations, for low light levels, than current measure ments. D. is also smaller at lower voltages, as was the case with D . i P In Table II some measurements on several different tubes are collected. The photoelectron emission l for the standard light source with a nominal filter attenuation of 100 gives the relative quantum efficiencies of these various tubes at 5500 A. The quantum efficiency of the SI photocathode of R196 is so low, and the dark pulse rate so large, that this light level could not be directly measured. Consequently, Fig. 16 Effective dark rate Dj for the 931A as a function of V, calculated for the model photomultiplier. v, volts - 50 a much larger light level, at longer wavelengths, was used for this tube. The gain shown was calculated from the pulse charge spectrum, and was in fair agreement with that found from the anode current [Eq. (2-9)], except in the case of R916 where the difference found was probably due to experimental errors resulting from the low gain and large dark pulse rate. The minimum effective dark pulse rate D , at the listed photomulti plier voltage, is given along with the value of Q for this minimum. In general it seems that the value of Q at the minimum in is larger, the larger D^. Two sets of values are given for the 931A, the second value being the higher rates found from the early measurements , and the first value the rates after one year of aging, mostly at high voltage. Values of are also given for some of the tubes , and for small 0^ (~ 1000) they are not much larger than D^. However, with large dark rates can be almost an order of magnitude larger than D , as in the case of the red sensitive R196. One Phillips XP- 1000 was also tested, but the dark rate was so high, and variable, that it was assumed that there was some defect in the tube. In general, the dark rate tends to be quite variable, and probably decreases with extended operation. Thus the values of D are not nearly so much constants of the tube, as T), G, and S are. Table II. Photomultiplier Tube Data Type of V oltage Gain Q at Min. D. i Photomultiplier (105 pps) (10&) Min. D D P P X 5400A (pps) (PPS) RCA-931A (S5) 1. 6 1000 7. 3 0. 5 620 880 (1 year earlier) I! 1 f t! 0. 66 1200 1600 RCA-1P28 (S4) 1. 1 1000 5.2 0. 7 1100 - Hitachi-R212 (S4) 1.75 1000 3.8 1.2 2100 - 4 Hitachi-R213 (S20) 2.4 1000 6. 0 1.0 4x10 - Hitachi-R196 (SI) - 1000 1.0 1. 6 5xl05 3x10 6 EMI-6097B (SI 1) 1.5 2000 40. 0 0. 5 3000 3300 - 51 5. FLUCTUATION MEASUREMENTS AND THE INFERRED VALUES OF S AND D Experimental measurements of the variance of the pulse count were used to verify the assumption of Poisson statistics. For measurements with constant light level and counting time the photomultiplier voltage and discriminator level were systematically varied in order to demon strate agreement of the results with those calculated on the basis of the model photomultiplier of Ch. 2. The counting time and light level were then systematically varied, holding the other parameters fixed, and the presence of excess noise at longer counting times, in excess of 10 s and apparently proportional to the light level, was found. These latter meas urements were repeated using current integration, and a direct compari son with the pulse counting results was made. Agreement is found with the results predicted from the model photomultiplier. The excess noise appears to be somewhat larger for current integration than for pulse counting at longer gate times. The values of S and D found have a rather large scatter, apparently a curse of fluctuation measurements , but in general show good agreement with the values calculated from the pulse charge spectrum in the previous chapter, for shorter integration times. The presence of excess noise causes S to decrease , and D to increase, for longer integration times. Systematic measurements were made only on the highest gain 931A photomultiplier , and all the data presented pertains to this tube. However, the results are supported by fragmentary measurements on several other tubes , and in some cases the conclusions are influenced by these additional measurements. 5. 1 Variance of the Pulse Count for Short Gate Times Let us first consider a light level so large that the dark pulses are negligible, i. e. , 1 » D^. With 1=1.6x10 pps, series of about 40 meas urements with a gate time of 1.26 s were made at different discriminator levels and tube voltages. The values of a(X.)/ (1-19)] are plotted in Fig. 17 as a function of the normalized discriminator - 52 - Fig. 17 Standard deviation divided 1200 V by the square root of the □ 1000V number of counts , cr(X. A 900 V cr(Xi)/ level Q. This ratio is seen to be equal to unity within the random devia tion expected for 40 measurements , except at low discriminator levels (<2x10 ^ coulombs) , where the rise is indicated by dotted lines for each voltage. This rise in R_ is a property of the tunnel diode trigger since for the same Q it is absent for higher photomultiplier gain (voltage) and consequently larger absolute discriminator level. This rise in noise occurs at the same discriminator level as the rise in counting rate shown in Fig. 10, which was attributed to the onset of free run of the tunnel diode discriminator. We can conclude from Fig. 17 that under these con ditions, the light pulses follow Poisson statistics, independently of dis criminator level and photomultiplier voltage, as long as the discriminator operates correctly. In Fig. 18 the normalized standard deviation, ct(X^)/ Fig. 18 Normalized standard o 1200 v deviation, cr(Xi)/ curve is just l/[Wa(Q)L where £ = 1. 6xl05 pps. Thus, under these conditions the photomultiplier behaves in the same manner as the model outlined in Ch. 2, within experimental error. Measurements of the variance of the dark pulse count, for a gate time of 1.26 s, give similiar results to that obtained for the light pulses. It can be concluded that the dark pulses also follow Poisson statistics, independently of discriminator level and photomultiplier voltage, as long as the discriminator detects the pulses correctly. 5.2 Fluctuation Measurements: Inferred Values of S In this section a direct comparison between the fluctuations in pulse counting and current integration is made , and the effect of excess noise at longer gate times is shown. Values of S and D are then derived from the data. Let us consider pulse counting measurements at a light level large enough that the dark pulse rate is negligible. Series of measurements of the number of counts in time T were made for different values of T, for several different light levels, while the photomultiplier voltage was held fixed at 1000 V, and the discriminator level was fixed at about 0. 4 Q. Letting be the individual measurements , and the difference be tween two consecutive measurements , as described in section 3. 3, the normalized standard deviation c^Y^/V2 gen . =___ i_ f (5-1) V 2 as follows from Eq. (3-14), and straight lines corresponding to several different values of IS are shown in Fig. 19. Consider first the pulse measurements at a light level corresponding to A = 1.8x10^ pps. For values of T = 5 s and less the normalized standard deviations fall relatively close to the line given by IS = 1.8x10^ pps, while the values at T = 20 s and 80 s are significantly above this line. Thus we infer that S^ is near unity for T < 5 s, but is appreciably smaller for T > 20 s. In Fig. 20 a plot of the data for T = 80 s is shown, where it - 1 /2 is seen that there are fluctuations well outside the range ± 2 o CURRENT = 1.6 « 10 pps 6 CURRENT = 1.6 * 10 9 CURRENT a PULSE = 9 .8 k 10 pps O PULSE = 1.8 * 10 Is = 1 6 * 10 pps s = 9.8*10 pps Is = 1.8*10 pps Fig. 19 Normalized standard deviation a(Yj.)/V2 Fig. 20 Integrated counts Xi for T = 80 s and & = 1.8x10° pps as a function of time. X| 10'cts) 1.078 ■ 80 90 time, minutes . 5 At the lower light level corresponding to l = 9. 8x10 pps similiar results are seen. In this case, the normalized standard deviations fall . 5 relatively close to the line given by IS = 9. 8x10 pps for values of T = 40 s and less, except for the point at T = 2. 5 s where some abnormal fluctuation must have occurred. This data is of somewhat lower quality than that of the higher light level, since fewer measurements at a given value of T were made. In general, it appeared that at lower light levels the effect of excess noise became apparent only at longer gate times, which is consistent with the hypothesis that the magnitude of the excess 2 noise is equal to some fraction of the light signal, i. e. , N^(cu) ~ i , at a given u), and increases with decreasing uu. For measurements of integrated current the photomultiplier current was amplified by a Keithley model 610B electrometer, which then drove a voltage-to-frequency converter. This converter is part of the standard electronic equipment developed at this company. A condenser is charged by a current proportional to the input signal, and when the voltage reaches a preset value an output pulse is generated and the condenser is discharged, from which point the cycle begins again. It operated linearly in the range 4 of output pulse rates from 0 to 10 pps, and was found to have a short 4 term stability better than 1 part in 10 . The pulses from this voltage-to- frequency converter were counted in the same scalers used for recording the pulses from the photomultiplier. In this manner largely the same equipment, and the same computer programs for data analysis, could be used for both the current integration measurements and the pulse counting measurements. -56- In all the current integration measurements the overall time constant of the voltage to frequency converter system, including the photomultiplier output circuit and the electrometer, was much smaller than the integration time, so that the system corresponded to direct integration of the photo multiplier current. Another way to say this is that the frequency response of the electronic equipment was flat over the bandpass of the differential current integration, shown in Fig. 3. The zero level fluctuations, repre senting dark current and electrometer noise, were small compared to the fluctuations with the light signal; further, the fluctuations were much larger than ± 1 count, the minimum system resolution. Eq. (5-1) was used to find the normalized standard deviation, and the results are shown in Fig. 19. Consider first the measurements at a light level corresponding to -6 = 1.6x10^ pps. For values of T = 40 s and less the normalized standard deviation follows approximately the line cor- 5 responding to IS = 1. 6x10 pps, with a rather large scatter, partly due to a small number of measurements , but also partly due to unknown sources. For T = 40 s and 160 s there appears to be significant excess noise, and plots of the integrated current measurements (made by com puter for all measurements) show significant fluctuations superimposed on a steady drift, similiar to Fig. 20. At the larger light level correspond - 5 ing to -6 = 9. 8x10 pps similiar results are obtained; however, the relative magnitude of the excess noise is larger for the longer integra tion times, and in fact even the absolute magnitude of the excess noise is greater at T = 160 s. From a comparison of the current integration measurements with the pulse counting measurements, it appears that, for integration times less than approximately 10 s, the normalized standard deviations are essentially equivalent. This is in agreement with the model photomulti plier, as it was shown in section 4. 3 that and were nearly equal. At longer integration times, though, both types of measurements indicate excess noise, and it appears that pulse counting significantly reduces the effect of excess noise. One might say that the stability of the pulse count ing technique is superior. It must be pointed out that the effect of excess noise for longer gate times is quite variable, and it is difficult to compare its value under conditions of pulse counting and current integration. - 57 — o l = 1.6 * 10 cps 10 cps 10 cps 0.05 0.1 1.0 2.0 100 200 T s Fig. 21 Photoelectron loss factor deduced from the current integration measurements, plotted as a function of the integration time T. In Fig. 21 values of deduced from Eq. (3-14) and the data used for Fig. 19 are shown. For values of T equal to 20 s and less, seems to average near unity, with a rather large standard deviation. The rather low point at T = 0. 62 s should perhaps be ignored, as a plot of the data for this point shows evidence of external disturbances, perhaps caused by some stray electrical interference. In general, if the rather large scatter of the points were due to stray electrical pickup and other noise disturbances, the values of would be expected to be significantly less than unity, which does not appear to be the case, and so the scatter is considered to be primarily due to the statistics of the measurement pro cess. If all measurements for T < 20 s are averaged together, we find that S. = 0. 95 ± 0. 3 , (5-2) and we conclude that, for T <20 s, the values of inferred from meas urements of the fluctuations of the integrated current are in agreement -58 - with the value S^ = 0. 8 calculated in section 4. 3 from the pulse charge spectrum. For values of T > 20 s there is a marked decrease in S^, which is also dependent on the value of l, as previously concluded in the discussion of Fig. 19. Some estimates of the stability of the various electronic components were made in order to try to determine the source of the excess noise measured at longer gate times, and the resulting fractional changes in the photomultiplier signal are summarized in Table III. For the lamp intensity only fluctuations in the power supply were considered, and the manufacturers specifications were multiplied by a factor 4, result ing from the measured variation in light intensity with current. For the high voltage supply the manufacturers specifications were also used. Since the photomultiplier gain varies approximately as the eighth power of the voltage, the estimated voltage fluctuations were multiplied by a factor 8 for the value listed for current integration. For pulse counting at Q = 0. 25 it was found that the counting rate varied only as the tenth root of the discriminator level, and thus the estimated high voltage fluctuations calculated for current integration were divided by 10 for pulse counting measurements . If the discriminator is set near Q = 1.0 then this latter factor is near unity. The observed daily variations in the onset of free run for the discriminator are shown; this variation seemed to be a steady drift. Short term fluctuations were not estimated. Finally, observations of the drift of the zero level of the Keithley electro meter and voltage-to-frequency converter gave the fluctuations shown. Table III. Estimated Fluctuations Due to the Electronic Components. Values Stated are Fractional Changes in Photomultiplier Signal Due to the Estimated Fluctuations of the Electronic Component. Type of Lamp intensity High voltage • Discriminator Electrometer measurement (power supply only) Pulse 1x10 ^/line volt 4x10 5/line volt 3x10 ^/day integration 4x10"4/°C 6x10 4/day Current 4x10 4/Hne volt as above 2x10 4/5 rnm. integration 6x10 3/day - 59 - With the observed line voltage fluctuations of less than 1 volt, and temperature variations less than 1 °C per hr, the fractional changes in the photomultiplier signal calculated for all the sources listed in Table III are somewhat less than the observed minimum relative standard deviations shown in Fig. 19. It then seems probable that part of the excess noise observed at longer gate times is due to the electronic components, but also that part is due to variations in the photomultiplier tube characteristics, primarily changes in the gain. The steady decreasing drift in the counting rate at larger light levels, shown in Fig. 20 for ji = 1.8x10^ pps, amounts to a fractional . -3 change equal to 2x10 per hour , and is too large to be accounted for by the electronic components. It is well known that these side window photomultiplier tubes show a fatigue effect (steadily decreasing gain) at anode currents greater than about 10 ^ a; at jt = 1.8x10^ pps the anode current was about 2. 5x10 ^ a and probably is large enough to cause a small fatigue effect. The more rapid random fluctuations shown 3 in Fig. 20, of the order of 1 part in 10 , are somewhat harder to explain on the basis of photomultiplier gain fluctuations , but at the same time are somewhat larger than that expected due to fluctuations in the electronic equipment. They are probably due , at least in part, to changes in the photomultiplier gain. A series of pulse counting measurements were also made using the Keithley 105 pulse amplifier followed by a discriminator, as discussed in section 4. 1. For short integration times results similiar to that obtained with the direct tunnel diode discriminator were obtained, but for longer integration times the normalized standard deviation was much larger than that predicted by equal to unity; in fact, larger than that found for the current integration measurements. The reason for this was not investi gated, but the photomultiplier-pulse amplifier combination had previously been found very sensitive to stray electrical pickup, and careful shielding, with special attention to the ground connections, was necessary. It then seems likely that pulses induced by stray pickup were responsible. Spe cial attention should therefore be paid to this problem if a pulse amplifier is used. On the other hand, the direct tunnel diode discriminator is very compact and built into the photomultiplier housing, thus greatly reducing the problem of electrical pickup. - 60 - Measurements were also made of the mean square fluctuations of the photomultiplier current after filtering with an RC-CR filter, previously discussed in section 3.4. The Keithley 610B electrometer was used as an input amplifier, and at higher frequencies the variance of the output was measured with a Ballantine model 320A true RMS voltmeter. Care was taken that the frequency response of the combined electrometer and photo multiplier did not appreciably change the overall power transfer function from that due to the filter alone. The part of the signal due to 50 hz line pickup, as well as the dark pulses (negligible at the light levels used for determination of S^), was determined by measurements without light illumination. Since the bandpass of the voltmeter extended only to about 5 hz in the low frequency range, a Tektronix oscilloscope with DC coupling was used for lower frequencies. The sweep of the oscilloscope was triggered at random time intervals and the signal voltage at the center of the oscil loscope screen was measured visually and recorded. After about 40 meas urements were made the variance was calculated using Eq. (1-6). It was found that the effect of 50 hz line pickup could be eliminated by triggering the oscilloscope in phase with the line voltage, but still at random phases, and this was done by using the single sweep feature with the trigger set from the line. (It might be noted that in the pulse counting and current Fig.. 22 Photoelectron loss factor Si deduced from the current mean square measurements with the RC-CR filter, plotted as a function of t = RC. T. sec -61- measurements 50 hz pickup was eliminated in the same manner since the clock for the gate time operated from the line frequency.) Good agree ment was found between oscilloscope and voltmeter results at the higher frequencies. The center frequency of the rather broad bandpass of the RC-CR filter (the power transfer function is shown in Fig. 5) is approximately the same as that for differential integration (power transfer function shown in Fig. 3) if RC is identified with T. In Fig. 22 the values of calculated using Eq. (3-18) are shown plotted vs. r = RC, and can be compared directly with the current integration results. A rather large scatter of the values of is apparent, and it was for this reason that the current integration technique was developed. However, the average value of is about equal to unity, in agreement with the current integration measurements. Also, there is an indication of some excess noise at t = 10 s, where the two low points correspond to a large value of i. 5. 3 Fluctuation Measurements: Inferred Values of D Let us first consider the dark pulse counting measurements. was calculated from the difference of two consecutive measurements using Eq. (3-21), and the results are shown plotted in Fig. 23 on doubly loga rithmic paper vs. T. These pulse measurements show a very large scatter, even for short gate times where other measurements had shown loooo o O CURRENT N =16 Fig. 23 Effective dark rate D pps o CURRENT N=64 A PULSE N=16 deduced from pulse . iS 9 5000 " V PULSE counting and current V integration measure 3000 - v ° * ments, plotted as a 2000 0. D D function of the integra tion time T. a ° 1000 A A a 500 ° , 300 0 05 0.1 0.2 0.5 10 2 0 50 10 20 50 100 200 T, sec - 62 - Fig. 24 Integrated dark counts for T = 80 s, plotted as a function of time 10 cts time, minutes a Poisson distribution and no evidence of excess noise. The data with N = 64 (N is the number of measurements at a given value of T) were early measurements when the dark pulse rate was approximately twice the value measured 1 year later, and the normalized discriminator level Q was set equal to 0. 25. At this time the value of calculated from the pulse charge measurements, as described in section 4. 3, was about 1600, and the values for T = 5 s and less are relatively near to this value. At T = 20 and 80 s the presence of excess noise is indicated because of the significantly larger values of D^, and a plot of the count number versus time is shown for T = 80 s in Fig. 24. Fluctuations con siderably larger than ± 2\j increase in the dark pulse rate. The drift shown in this figure amounts to almost 10 % per hour. The pulse measurements with N = 16 were later measurements, made with Q = 0.5, and thus should be compared with the value = = 600 shown in Fig. 15 and calculated from the pulse charge measure ments. The measured values are in general significantly larger than 600, even for short gate times where agreement was expected on the basis of other measurements. For instance, in measurements at T * 2.5 s where the voltage was varied systematically, values of in agreement with Fig. 15 were found. The larger values are apparent ly accounted for statistically by the small value of N, as well as the possibility of some external disturbances (which were not evident in plots of the measurements). The measurement at T = 160 s, though, definitely shows the effect of excess noise. - 63 - For the current integration measurements Eq. (3-23) was used to calculate D^, and the results are also shown in Fig. 2 3. Approximately the same comments made about the values of D also apply to D.. The p ' l values of calculated from the pulse charge measurements , assuming Poisson statistics, is equal to 900 (see Fig. 16), and the measured values of are on the average somewhat greater than this for T < 10 s. It is thought that this is due primarily to statistical fluctuations, with the addi tion of some external disturbances due to stray electrical pickup. The latter would not be called excess noise since it originates in the electronic equipment and we are interested primarily in the properties of the photomultiplier tube. For gate times of 80 and 160 s, however, drift and excess noise are evident in the measurements. A comparison between the pulse counting and current integration measurements shows that there is little difference between them. As cal culated from the pulse charge spectrum, is somewhat smaller than D^, but in the actual measurements of the fluctuations, the fluctuations in the results are too large to show up this difference. It is probable that more careful measurements would indicate the superiority of pulse counting, but from a practical point of view the difference is not large enough to be of concern. Measurements of the variance of the dark current using the RC-CR filter were also made. was calculated from the data using Eq. (3-26), and is shown plotted vs. t = RC on doubly logarithimic paper in Fig. 25. In the range of t covered by this graph, should have a constant value 9UUU D, Fig. 25 Effective dark rate pps deduced from the current mean square measure ments with the RC-CR. filter, plotted as a func tion of t = RC. Qnn 10'3 K)"2 10"1 1 10 T . sec - 64 - of about 1000, according to the interpretation of our current integration - 3 -2 measurements. The rather low values of D. at t = 10 and 10 s are probably due to the limited dynamic range of the rms voltmeter, as for these time constants the output is just a series of pulses. Otherwise, the values shown are in general agreement with these found from the current integration measurements. The rather large scatter in the results (many more were made under different conditions with different photomultiplier tubes) discouraged careful measurements, and, as mentioned before, led to the development of the current integration technique. - 65 - 6. FLUCTUATION MEASUREMENTS WITH PHASE SENSITIVE DETECTION: VERIFICATION OF POISSON STATISTICS The fluctuation measurements with phase sensitive detection were made primarily to demonstrate the very small normalized standard devia tion which can be achieved with this technique. Because of the lack of excess noise, and the very large number of measurements in a single run, it was possible to calculate the variance to within an error of ± 4 % (2 standard deviations), and thus to verify the assumption of a Poisson distribution for the number of counts to this degree of accuracy. As discussed in section 3. 3, phase sensitive detection filters out a band of frequencies from the photomultiplier output, centered approx imately at U)T = 2rr/T% where T* is the time for a complete cycle. The width of the band depends on the number of cycles which are averaged to gether in the measurement. If N^(u/ )«Np, Var(Y) is determined by Poisson statistics and should be equal to 2 Fig. 26 Schematic of light source and system used for fluctuation measurements with phase sensitive detection. The light source is the same as in Fig. 9. TRIGGER GATE SWITCH SCALER SCALER 1.6 ms DELAY PUNCH TIMER CLOCK so that the fluctuations and drift in this delay are also filtered out by the phase sensitive detection. Only in the switch and gate will low fre quency variations in time delay affect the results . For practical appli cations the light signal to be measured must be chopped or modulated in phase with the switching frequency, but here we are only interested in measurements of fluctuations which are not dependent on the meas ured light signal. Five runs , each extending over a period of 5-10 hours , were made with this system, and the results are summarized in Table IV. In order to be able to detect drift and slow fluctuations, the computer program used for analysis was able to divide the total number of meas urements into groups of a predetermined size, evaluate Table IV. Results from Pulse Integration Measurements with System for Phase Sensitive Petection Type of Dis c rim - Average Number of Average Fractional diff Normalized pulses inator counting measure number erence in channels variance level rate ments in of counts light (1.8 ± 1. 3) xlO'4 0. 4 1.0x10s 6200 8 .lxlO4 1,038 t . 0 36 5400 A (0.08 = 2.0) xlO"5 dark 1. 5 440 5800 368 1.024 ± .037 light 7. 04x10 5 0. 3 . 2x10 = 5000 (,. 7x10 = 0.97 - .040 5400A 8 light (1.9 - 1.0) x 10"4 0. 25 7. 7xl04 6600 1. 35x10 = 1.046 ± .035 4300A light (2.2 - 1.6) xlO"4 0. 25 7.7xl04 6800 6.7xl04 1.021 ± .0 34 4300 A Consider the first measurement of Table IV on pulses generated by 5400A light (the dark pulse rate is negligible). The fractional difference reveal any drift, or systematic fluctuations, for group sizes N varying from 50 to 500. It was also verified from these calculations that Var(Z.) 2 . J = 2 number of measurements, and the error quoted is just twice this value. - 68 - Fig. 27 Fractional difference time, hrs This normalized variance should be equal to unity in the case of Poisson statistics (it is just the analogue of Eq. (1-19) for synchronous detection), and we see that it is approximately 4 % larger than unity, just outside two standard deviations. This slight excess over unity is thought to be due to the presence of some noise pulses picked up during the measurements, which come in groups of 10-100 pulses at a time. Such groups of noise pulses are observed to be very infrequent, but not many are required. Also, it is possible that some errors may occur in the read-out cycle, resulting in a wrong value of one digit. This does not appear so likely, though, since 1) a parity check is built into the system, so that a mistake in one hole is detected, and 2) no obvious errors are found in the larger digits. The second run with dark pulses shows that the dark pulse count also obeys Poisson statistics to good accuracy. In the determinations of from pulse count statistics some rather large values were found because the variance was much larger than predicted by Poisson statis tics, but these were thought to be due either to statistical fluctuations or erroneous measurements, partly on the basis of the good agreement with Poisson statistics found here. In the third measurement with pulses generated by 5400A light a larger counting rate was used. In this case a significant assymetry showed up in the two channels. It was shown that, at higher counting - 69 - rates, one of the channels. It was shown that, at higher counting rates, one of the channels missed a small fraction of the counts , probably due to a lower maximum counting rate of one of the logic units . The fractional difference averaged over 300 measurements showed quite no ticeable drift, and thus no error is shown for the overall fractional difference. The plots of variance versus group number, however, did not show any drift, and thus the overall normalized variance should be a good number. In this case the normalized variance is somewhat less than unity, although within two standard deviations. The dead time correction for the counting rate in this case amounts to about 2. 2 %, and this leads to a correlation in the number of counts which decreases the variance. Estimation of the magnitude of this effect predicts a lowering of 1-4 % in this case. The dead time correlation effect is negligible for the other measurements. The last two measurements with pulses generated by 4300A light were carried out in order to determine if there was any dependence of the normalized variance on the wavelength, using a discriminator level which detects about 85 % of the photoelectron pulses. The values of the normalized variance are somewhat larger than unity, probably for the same reason as the first measurement, but it is evident that there is no difference between the normalized variances with 4300A light and with 5400A light, within an error of about ± 4 %. The interest in the normalized variance stems from ref. 6, where an increase in the variance over that given by Poisson statistics is pre dicted. Letting X be the number of photoelectron counts , Eq. (5b) of ref. 6 states that the normalized variance is given by Var(X)/ Rather general considerations also indicate that Eq. (6-1) is in error. Suppose that the quantum efficiency approached 100 %. In that case the photoelectrons are identical with the photons, which a priori gives Var(X)/ 7. CONCLUSIONS AND RECOMMENDATIONS The model photomultiplier described in Ch. 2 is shown to be a good approximation to the photomultiplier tested. The output of a photo multiplier consists of charge pulses of short duration, both dark pulses plus those due to light; calculations of the fluctuations in the photomulti plier signal based on the pulse charge spectrum give good agreement with the measured fluctuations. At very low frequencies, corresponding to integration times > 10 s, effects of excess noise are found, apparently resulting from drift and fluctuations in the gain and other photomultiplier parameters. For experiments with long measurement times the effect of excess noise may have to be taken into account. Measurements of the variance of the pulse count with a phase sensitive detection system show that the variance is in agreement with that calculated from Poisson statistics to within 4 %. The random nature of the pulse counts is thus indirectly verified within this error limit. Comparison of pulse counting and charge integration experiments shows little difference in the normalized standard deviation, so that either system of measurement can be expected to lead to similiar signal to noise ratios. In the case of long measurement times the pulse counting technique gives somewhat less excess noise (or better stability). It is demonstrated in section 4.2 that, with modern electronic equipment, pulse counting techniques can be used effectively up to quite high light levels, essentially as large as one normally uses with current measure ments. This requires only a small correction for dead time effects, and thus removes a major limitation of the versatility of pulse counting sys tems. Further, it appears possible to make this correction electronical ly by use of one of the "deadtimeless" discriminators now on the market. When a digital output is desired, for instance for computer processing, it is actually cheaper to construct a pulse counting system. This, com bined with the inherently better stability, makes pulse counting techniques desirable for many experimental measurements. It is recommended that photomultiplier manufacturers adopt a standard system of evaluation of photomultipliers, based on the para - 72 - meters Tl, G, S, and D as defined in this paper. Evaluation of these parameters from the photomultiplier pulse spectrum is summarized in section 3.1, and the evaluation of S and D from measurements of the fluctuations in the photomultiplier signal is summarized in section 3.4. For a practical tabulation the following system is proposed. First, the value of the quantum efficiency T| as a function of the wavelength \ is given. The assumption that 71 is independent of the photomultiplier voltage should be checked. Then, from measurements of the photomulti plier pulse spectrum, the gain G, the photoelectron loss factor for current measurements, and the effective dark rates (for pulse meas urements) and (for current measurements) can be given as a function of the photomultiplier voltage. It is shown in section 4. 3 that (for pulse measurements) should be approximately a universal function of the normalized pulse charge Q, and this assumption must be checked. Finally, from measurement of the fluctuations in the photomultiplier signal it can be checked that the values of S and D calculated according to section 3. 4 are in agreement with the values calculated from the pulse charge spectrum, at shorter integration times, as shown in chapter 5 for the 931A photomultiplier. At long integration times the value of S will decrease and the value of D will rise, due to excess noise, and the approximate integration time (or frequency) at which excess noise be comes important can be stated. The integration time at which excess noise becomes important probably depends on the light level, and this variation should be determined. These parameters must of course be supplemented by the pulse rise time, the photo cathode area, and other important characteristics of the photomultiplier. If the values listed are representative of the production tubes, then the problem of the selection of a photomultiplier would be greatly simplified, compared to the present state of affairs. Further, since quantum efficiencies and dark counting rates are al ready being quoted in advertisements for the better (and more expen sive) photomultiplier tubes, the changes recommended would be a logical step. REFERENCES KHARKEVICH A A, Spectra and Analysis. Consultants Bureau, New York, I960. BENDAT J S, Principles and Applications of Random Noise Theory. John Wiley and Sons, New York, 1958, p. 96. KURRELMEYER B and HAYNER L J, Shot Effect of Secondary Electrons from Nickel and Beryllium. Phys. 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(See the back cover earlier reports.) 287. Absolute E1, AK = O transition rates in odd-mass Pm and Eu-isotopes. By S. G. Malmskog. 1967. 33 p. Sw. cr. 10:—. 241. Burn-up determination by high resolution gamma spectrometry: spectra 288. Irradiation effects in Fortiweld steel containing different boron isotopes. from slightly-irradiated uranium and plutonium between 400—830 keV. By By M. Grounes. 1967. 21 p. Sw. cr. 10: R. S. Forsyth and N. Ronqvist. 1966. 22 p. Sw. cr. 8: —. 289. Measurements of the reactivity properties of the Agesta nuclear power 242. Half life measurements in 1ssGd. By S. G. Malmskog. 1966. 10 p. Sw. reactor at zero power. By G. Bernander. 1967. 43 p. Sw. cr. 10:—. cr. 8: —. 290. Determination of mercury in aqueous samples by means of neutron activa 243. On shear stress distributions for flow In smooth or partially rough annuli. tion analysis with an account of flux disturbances. By D. Brune and K. Jir- By B. Kjellstrom and S. Hedberg. 1966. 66 p. Sw. cr. 8: —. low. 1967. 15 p. Sw. cr. 10:—. 244. Physics experiments at the Agesta power station. By G. Apelqvist, P.-A. 291. Separtaion of "Cr by means of the Szilard-Chalmers effect from potassium Bliselius, P. E. Blomberg, E. Jonsson and F. Akerhielm. 1966. 30 p. Sw. chromate irradiated at low temperature. By D. Brune. 1967. 15 p. Sw. cr. 8: —. cr. 10:—. 245. Intercrystalline stress corrosion cracking of inconel 600 inspection tubes in 292. Total and differential efficiencies for a circular detector viewing a circu the Agesta reactor. By B. Gronwall, L. Ljungberg, W. Hiibner and W. lar radiator of finite thickness. By A. Lauber and B. Tollander. 1967. 45 p. Stuart. 1966. 26 p. Sw. cr. 8:-. Sw. cr. 10:—. 246. Operating experience at the Agesta nuclear power station. By S. Sand- 293. Absolute M1 and E2 transition probabilities in 233U. By S. G. Malmskog and strom. 1966. 113 p. Sw. cr. 8: —. M. Hojeberg. 1967. 37 p. Sw. cr. 10:—. 247. Neutron-activation analysis of biological material with high radiation levels. By K. Samsahl. 1966. 15 p. Sw. cr. 8: —. 294. Cerenkov detectors for fission product monitoring in reactor coolant water. 248. One-group perturbation theory applied to measurements with void. By R. By O. Strindehag. 1967. 56 p. Sw. cr. 10:—. Persson. 1966. 19 p. Sw. cr. 8: —. 295. RPC calculations for K-forbidden transitions in 183 W. Evidence for large 249. Optimal linear filters. 2. Pulse time measurements in the presence of inertial parameter connected with high-lying rotational bands. By S. G. noise. By K. Nygaard. 1966. 9 p. Sw. cr. 8: —. Malmskog and S. Wahlborn. 1967. 25 p. Sw. cr. 10:—. 250. The interaction between control rods as estimated by second-order one- 296. An investigation of trace elements in marine and lacustrine deposits by group perturbation theory. By R. Persson. 1966. 42 p. Sw. cr. 8: —. means of a neutron activation method. By O. Landstrom, K. Samsahl and C-G. Wenner. 1967. 40 p. Sw. cr. 10:—. 251. Absolute transition probabilities from the 453.1 keV level in 183W. By S. G. Malmskog. 1966. 12 p. Sw. cr. 8: —. 297. Natural circulation with boiling. By R. P. Mathisen. 1967. 58 p. Sw. cr. 10:—. 252. Nomogram for determining shield thickness for point and line sources of 298. Irradiation effects at 160-240°C in some Swedish pressure vessel steels. gamma rays. By C. Jonemalm and K. Malen. 1966. 33 p. Sw. cr. 8: —. By M. Grounes, H. P. Myers and N-E. Hannerz. 1967. 36 p. Sw. cr. 10:-. 253. Report on the personnel dosimetry at AB Atomenergi during 1965. By K. A. 299. The measurement of epithermal-to-thermal U-238 neutron capture rate (p28 ) Edwardsson. 1966. 13 p. Sw. cr. 8: —. in Agesta power reactor fuel. By G. Bernander. 1967. 42 p. Sw. cr. 10:—. 254. Buckling measurements up to 250°C on lattices of Agesta clusters and on 300. Levels and transition rates in 19*Au. By S. G. Malmskog, A. Backlin and B. DiO alone in the pressurized exponential assembly TZ. By R. Persson, Fogelberg. 1967. 48 p. Sw. cr. 10:—. A. J. W. Andersson and C-E. Wikdahl. 1966. 56 p. Sw. cr. 8:-. 301. The present status of the half-life measuring equipment and technique at 255. Decontamination experiments on intact pig skin contaminated with beta- Studsvik. By S. G. Malmskog. 1967. 26 p. Sw. cr. 10:-. gamma-emitting nuclides. By K. A. Edwardsson, S. HagsgSrd and A. Swens- 302. Determination of oxygen in aluminum by means of 14 MeV neutrons with son. 1966. 35 p. Sw. cr. 8: —. an account of flux attenuation in the sample. By D. Brune and K. Jirlow. 256. Perturbation method of analysis applied to substitution measurements of 1967. 16 p. Sw. cr. 10:—. buckling. By R. Persson. 1966. 57 p. Sw. cr. 8: —. 303. Neutron elastic scattering cross sections of the elements Ni, Co, and Cu 257. The Dancoff correction in square and hexagonal lattices. By I. Carlvik. 1966 between 1.5 and 8.0 mev. By B. Holmqvist and T. Wiedling. 1967. 17 p. 35 p. Sw. cr. 8: —. Sw. cr. 10:—. 258. Hall effect influence on a highly conducting fluid. By E. A. Witalis. 1966. 304. A study of the energy dependence of the Th232 capture cross section in 13 p, Sw. cr. 8: —. the energy region 0.1 to 3.4 eV. By G. Lundgren. 1967. 25 p. Sw. cr. 10:-. 259. Analysis of the quasi-elastic scattering of neutrons in hydrogenous liquids. 305. Studies of the reactivity effect of polythene in the fast reactor FRO. By L. By S. N. Purohit. 1966. 26 p. Sw. cr. 8: —. I. Tir&n and R. HSkansson. 1967. 25 p. Sw. cr. 10:-. 260. High temperature tensile properties of unirradiated and neutron irradiated 306. Final report on IFA-10, the first Swedish instrumented fuel assembly irra 20Cr—35Ni austenitic steel. By R. B. Roy and B. Solly. 1966. 25 p. Sw. diated in HBWR, Norway. By J-A. Gyllander. 1967. 35 p. Sw. cr. 10:—. cr. 8: —. 307. Solution of large systems of linear equations with quadratic or non-qua 261. On the attenuation of neutrons and photos in a duct filled with a helical dratic matrices and deconvolution of spectra. By K. Nygaard. 1967. 15 p. plug. By E. Aalto and A. Krell. 1966. 24 p. Sw. cr. 8: —. Sw. cr. 10:-. 262. Design and analysis of the power control system of the fast zero energy 308. Irradiation of superheater test fuel elements in the steam loop of the R2 reactor FR-0. By N. J. H. Scnuch. 1966. 70 p. Sw. cr. 8:-. reactor. By F. Ravndal. 1967. 94 p. Sw. cr. 10:—. 263. Possible deformed states in 11sln and mln. By A. Backlin, B. Fogelberg and 309. Measurement of the decay of thermal neutrons in water poisoned with the S. G. Malmskog. 1967. 39 p. Sw. cr. 10:—. non-1/v neutron absorber cadium. By L. G. Larsson and E. Moller. 1967. 20 p. Sw. cr. 10:—. 264. Decay of the 16.3 min. 18l Ta isomer. By M. Hojeberg and S. G. Malmskog. 1967. 13 p. Sw. cr. 10:-. 310. Calculated absolute detection efficiencies of cylindrical Nal (Tl) scintill ation crystals for aqueous spherical sources. By. O. Strindehag and B. 265. Decay properties of 1,,Nd. By A. Backlin and S. G. Malmskog. 1967. 15 p. Tollander. 1968. 18 p. Sw. cr. 10:-. Sw. cr. 10:—. 311. Spectroscopic study of recombination in the early afterglow of a helium 266. The half life of the 53 keV level in 1,,Pt. By S. G. Malmskog. 1967. 10 p. plasma. By J. Stevefelt. 1968. 49 p. Sw. cr. 10:—. Sw. cr. 10:—. 312. Report on the personnel dosimetry at AB Atomenergi during 1966. By J. 267. Burn-up determination by high resolution gamma spectrometry: Axial and Carlsson and T. Wahlberg. 1968. 10 p. Sw. cr. 10:—. diametral scanning experiments. By R. S. Forsyth, W. H. Blackladder and N. Ronqvist. 1967. 18 p. Sw. cr. 10:-. 313. The electron temperature of a partially ionized gas in an electric field. By F. Robben. 1968. 16 p. Sw. cr. 10:—. 268. On the properties of the ------>- d 3/2 transition in mAu. By A. Backlin 314. Activation Doppler measurements on U238 and U235 in some fast reactor and S. G. Malmskog. 1967. 23 p. Sw. cr. 10:—. spectra. By L. I. Tirdn and I. Gustafsson. 1968. 40 p. Sw. cr. 10:—. 269. Experimental equipment for physics studies in the Agesta reactor. By G. 315. Transient temperature distribution in a reactor core with cylindrical fuel Bernander, P. E. Blomberg and P.-O. Dubois. 1967. 35 p. Sw. cr. 10:—. rods and compressible coolant. By H. Vollmer. 1988. 38 p. Sw. cr. 10:-. 270. An optical model study of neutrons elastically scattered by iron, nickel, 316. Linear dynamics model for steam cooled fast power reactors. By H. Voll cobalt, copper, and indium in the energy region 1.5 to 7.0 MeV. By B. mer. 1968. 40 p. Sw. cr. 10:—. Holmqvist and T. Wiedling. 1967. 20 p. Sw. cr. 10:—. 317. A low level radioactivity monitor for aqueous waste. By E. J. M. Quirk. 271. Improvement of reactor fuel element heat transfer by surface roughness. 1966. 35 p. Sw. cr. 10:—. By B. Kjellstrom and A. E. Larsson. 1967. 94 p. Sw. cr. 10:—. 318. A study of the temperature distribution in UOa reactor fuel elements. By 272. Burn-up determination by high resolution gamma spectrometry: Fission pro I. Devoid. 1968. 82 p. Sw. cr. 10:-. duct migration studies. By R. S. Forsyth, W. H. Blackadder and N. Ron 319. An one-line water monitor for low level ^-radioactivity measurements. By qvist. 1967. 19 p. Sw. cr. 10:—. E. J. M. Quirk. 1958. 26 p. Sw. cr. 10:-. 273. Monoenergetic critical parameters and decay constants for small spheres 320. Special cryostats for lithium compensated germanium detectors. By A. and thin slabs. By I. Carlvik. 1967. 24 p. Sw. cr. 10:-. Lauber, B. Malmsten and B. Rosencrantz. 1968. 14 p. Sw. cr. 10:-. 274. Scattering of neutrons by an enharmonic crystal. By T. Hogberg, L. Bohlin 321. Stability of a steam cooled fast power reactor, its transients due to mode and I. Ebbsjo. 1967. 38 p. Sw. cr. 10:—. rate perturbations and accidents. By H. Vollmer. 1968. 38 p. Sw. cr. 10:-. 275. ThelAKI =1, El transitions in odd-A isotopes of Tb and Eu. By S. G. Malm 322. Progress report 1967. Nuclear chemistry. 1968. 30 p. Sw. cr. 10:—. skog, A. Marelius and S. Wahlborn. 1967. 24 p. Sw. cr. 10:-. 323. Noise in the measurement of light with photomultipliers. By F. Robben. 276. A burnout correlation for flow of boiling water in vertical rod bundles. By 1968. 74 p. Sw. cr. 10:-. Kurt M. Becker. 1967. 102 p. Sw. cr. 10:-. 277. Epithermal and thermal spectrum indices in heavy water lattices. By E. K. Sokolowski and A. Jonsson. 1967. 44 p. Sw. cr. 10:-. 278. On the transitions in odd mass Pm nuclei. By A. Backlin and List of published AES-rapporter (In Swedish) S. G. Malmskog. 1967. 14 p. Sw. cr. 10:—. 1. Analys medelst gamma-spektrometri. Av D. Brune. 1961. 10 s. Kr 6:-. 279. Calculations of neutron flux distributions by means of integral transport 2. BestrSIningsforandringar och neutronatmosfar i reaktortrycktankar — nSgra methods. By 1. Carlvik. 1967. 94 p. Sw. cr. 10:-. synpunkter. Av M. Grounes. 1962. 33 s. Kr 6:-. 280. On the magnetic properties of the K=1 rotational band in 188 Re. By S. G. 3. Studium av strackgransen i mjukt stSI. Av G. Ostberg och R. Attermo Malmskog and M. Hojeberg. 1967. 18 p. Sw. cr. 10:-. 1963. 17 s. Kr 6:-. 281. Collision probabilities for finite cylinders and cuboids. By I. Carlvik. 1967. 4. Teknisk upphandling inom reakloromrSdet. Av Erik Jonson. 1963. 64 s. 28 p. Sw. cr. 10:-. Kr 8: 282. Polarized elastic fast-neutron scattering of 1,C in the lower MeV-range. 5. Agesta Kraftvarmeverk. Sammanstallning av tekniska data, beskrivningar I. Experimental part. By O. Aspelund. 1967. 50 p. Sw. cr. 10:—. m. m. for reaktordelen. Av B. Lilliehook. 1964. 336 s. Kr 15:—. 283. Progress report 1966. Nuclear chemistry. 1967. 26 p. Sw. cr. 10:-. 6. Atomdagen 1985. Sammanstallning av foredrag och diskussioner. Av S. Sandstrom. 1966. 321 s. Kr 15:-. 284. Finite-geometry and polarized multiple-scattering corrections of experi mental fast-neutron polarization data by means of Monte Carlo methods. 7. Radiumhaltiga byggnadsmaterial ur strSIskyddssynpunkt. Av Stig O. W. By O. Aspelund and B. Gustafsson. 1967. 60 p. Sw. cr. 10:—. Bergstrom och Thor Wahlberg. 1967. 26 s. Kr 10:—. 285. Power disturbances close to hydrodynamic instability in natural circulation Additional copies available from the library of AB Atomenergi, Studsvik, Ny- two-phase flow. By R. P. Mathisen and O. Eklind. 1967. 34 p. Sw. cr. 10:—. koping, Sweden. EOS-tryckerierna, Stockholm 1968 and mean square value
(but not necessarily a Gaussian or Poisson distribution). We then have a signal
r (1-31) 2T T-co -T
2r2 . (1-32)
= qp(q)dq = qda (2-5) L 0 0 where we let a(q) be the independent variable, denoted simply by a. The gain of the photomultiplier is just
/ e , (2-6) where e is the electronic charge. It is convenient for presentation of experimental data to define the normalized pulse charge Q by
, ' (2-7) and to specify, for instance, the discriminator level in terms of Q rather than q. The gain of the photomultiplier can also be found from measurement of the average anode current. From the definition of
in Eq. (2-5) it follows that
, (2-8) and thus
can be found, and consequently G from Eq. (2-6), if £ and are known.
and
by considering a measurement consisting of the integration of the output current for a time T, resulting in the charge IF. The mean value of the number of pulses per unit charge, n(q), will be given by
, (2-13) U 0 using Eq. (2-5) for definition of
. Since n(q) is a pure number with Poisson statistics we have Var [n(q) ] =
= j q p(q)dq , 0
2 21
^/
and
, assuming a Poisson distribution for the secondary electrons with the same multiplication factor T] for each dynode. Instead of charge q let us work with
^i
/ 2 i /
, in agreement with Eq. (2-18).
]t , (2-35) ->=[J0 •
T (2-36)
2 iT
2
• - 26 -
for current measurements with the model photomultiplier. In this equa tion it is assumed that the dark pulse rate f(q) approaches the limiting value f as q approaches zero, max - 27
£T , (3-22) where the average pulse charge q is determined by experiment. Setting the measured variance divided by the calculated signal equal to Eq. (3-19), we find that Var(Y.) D. = i 2
2T (3-23) 37
jg , (3-24) where
was determined by experiment. A variance to signal squared ratio calculated according to the definition of D is given by
- 38 -
, the average photoelectron pulse charge of the photomultiplier. Further, the discriminator should have a short dead time (the minimum time between two pulses which can be detected) if large values of i are to be measured. Ideally, the dead time of the discriminator should be about equal to the length of the pulse produced by the photomultiplier. Also, the discriminator should be sufficiently stable so that the drift in the signal due to this source is negligible, if long term stability is required. ' For detection of the pulses from the photomultiplier most experi menters have used an amplifier followed by a discriminator , and general ly a rather slow amplifier and discriminator with a deadtime of the order of 1 ps since they were only interested in very weak light levels. There now exist commercial low noise pulse amplifiers, such as the Keithley model 105, which when used with a fast discriminator will have a dead time less than 10 ns and an input noise corresponding to a charge of 5 . about 10 electrons, and we have tested such an amplifier and found it to be satisfactory. However, it was considered cheaper (and more inter esting) to design a fast, high sensitivity tunnel diode discriminator which could operate directly from the photomultiplier anode , thus eliminating the expensive pulse amplifier. This goal has been realized, but at the expense of a deadtime of 50 ns, somewhat longer than obtainable with a pulse amplifier, but adequate for the present experiment. The side window photomultiplier tubes of the type 931A family have a pulse length of about 4 ns, and an ideal discriminator for use with these tubes should have a deadtime of this order. - 39 -
was found for the different voltages, using Eq. (2-8), and the result ing gain is shown in Fig. 11. The data of Fig. 10 was then replotted as a function of the normalized discriminator level Q( = q/
), and is shown in Fig. 12. In Fig. 12 all of the photoelectron pulse data, to within the exper imental error, fall on a single curve. This means that the shape of the pulse charge spectrum is approximately independent of the gain. If the secondary electron had a Poisson distribution, as discussed in section 2.2, small changes in the pulse charge spectrum would be expected, but these changes probably would lie within the scatter of data in Fig. 12. The absolute value of the discriminator calibration was found from
as calculated above, and this calibration is what was used in the preceding section on discriminator design. Subsequently, a Tektronix 545A oscilloscope with a type W preamplifier was used as a calibrated voltage discriminator with 90 pf input capacitance (including the cable and tube socket), and measurements of r vs. q for light pulses were made. The value of
at 1000 V found from this data was about 25 % - 44 -
for the 931 A. This data, taken 1 year later than that for Fig. 10, shows approximately a factor 2 reduction in dark counting rate.
= 0.4 eG, where e is the electronic charge and G is the gain of the photomultiplier. By this criterion the tunnel diode discriminator, -13 with a minimum charge sensitivity of 1.4x10 Coulombs, demands a photomultiplier gain of at least 2x10^. For the highest gain 931A photo multiplier about 830 volts is necessary for this gain, according to Fig.