Mmifo 
 
 U.S. DEPOSITORY 
 
 UNITED STATES ATOMIC ENERGY COMMISSION 
 
 AECU-715 
 (LADC-7^5) 
 
 ON THE STATISTICS OF LUMINESCENT 
 COUNTER SYSTEMS 
 
 By 
 
 Frederick Seitz 
 
 D. W. Mueller 
 
 Approved for Release: February 13, 1950 
 
 Los Alamos Scientific Laboratory 
 
 D5 
 
 Technical Information Division, ORE, Oak Ridge, Tennessee 
 
Styled, retyped, and reproduced from copy- 
 as submitted to this office. 
 
 Work performed under Contract No. W-7^05-eng-36 
 
 PRINTED IN U.S.A. 
 PRICE 10 CENTS 
 
ON THE STATISTICS OF LUMINESCENT COUNTER SYSTEMS 
 By Frederick Seitz and D. W. Mueller 
 
 1. INTRODUCTION 
 
 The type of crystal counter which depends upon the combination of luminescent crystals and a 
 photomultiplier tube shows promise of being of great service in the detection of radiations both be- 
 cause of its high sensitivity and speed of registry and recovery. This device has been developed by 
 a large number of individuals, almost too numerous to mention; however, the origin of the system 
 appears to rest with Coltman and Marshall, 1 who employed powdered luminescent materials of the 
 type used in previous commercial luminescent systems, and with Broser and Kallmann, 2 who first 
 appreciated the advantages of employing large, transparent, luminescent crystals and introduced 
 organic materials. 
 
 The purpose of this paper is to analyze some of the factors which influence the statistical be- 
 havior of luminescent counter systems, in order to evaluate the limits within which a counter may be 
 used in making a particular type of measurement. The problems of interest range over a wide spec- 
 trum of possibilities. However, the problem on which attention is focused for immediate purposes in 
 order to provide a practical objective is the following: 
 
 A crystal-counter system is employed to count the gamma rays emitted from a source in time T. 
 If N gamma rays are emitted, what is the most probable number that will be counted and what is the 
 range of variation to be expected ? An attempt is made to examine this problem in a sufficiently gen- 
 eral way that the results will have value for a much broader group of problems. 
 
 It is interesting to consider the component parts of this problem in order to be able to examine 
 the sources of statistical variations. The components are as follows: 
 
 1. The source, even if constant in the sense that it remains unchanged during the time T, will 
 contribute to the statistical variation since the gamma rays are usually emitted at random. For 
 simplicity, it is assumed that the time T is sufficiently short that variations in the source strength 
 can be neglected and that the statistical variations in emission of gamma rays can be treated on the 
 basis of a Poisson distribution. 
 
 2. Unless the source is completely surrouned by the luminescent material, some of the gamma 
 rays will not pass through this material and hence will certainly fail to be registered. The average 
 fraction which passed through the material is designated by f, so that the average number of gamma 
 rays which pass through the detecting system, if N are emitted from the source, is 
 
 v = fN (1) 
 
 If the source is isotropic, f will be determined simply by the solid angle subtended by the crystal 
 system; otherwise a somewhat more involved calculation is needed to determine f. 
 
 3. A given gamma ray may or may not produce an ionizing pulse within the luminescent crystal. 
 The possible mechanisms for producing such a pulse are the photoelectric effect, the Compton ef- 
 fect, and pair production. In the first and third cases the gamma ray transmits all its energy to the 
 crystal provided the energetic electrons produced by the gamma ray do not escape from the crystal. 
 A greater statistical variation is possible when the range of gamma-ray energy and the atomic 
 number are such that the Compton effect predominates. This would be the case, for example, if the 
 luminescent material were one of the organic types such as naphthalene or anthracene and if the 
 gamma rays had an energy in the neighborhood of 2 Mev. 
 
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 The probability of a Compton encounter may be described in terms of the mean free path X for 
 the process, namely 
 
 X = l/n e o- c (2) 
 
 where n e is the density of electrons in the luminescent material and <r c is the Compton cross section 
 per electron. If d is the thickness of the luminescent material in the direction in which the incident 
 gamma rays are traveling, the probability that a given gamma ray will pass through the system 
 without producing a Compton electron is e - * where 
 
 a = d/X (3) 
 
 The initial energy k„ of the gamma ray and the energy k after the collision are related by the 
 equation 
 
 JL I (4 ) 
 
 k,, 1 + y(l - cos 9) w 
 
 where 9 is the angle between the incident and scattered quantum and y is the energy of the incident 
 gamma ray expressed in units of the rest mass of the electron (507 kev). The energy gained by the 
 electron is t - k„ - k. From Eq. 4 the relation 
 
 k» 
 d(cos 9) = -^f-dk (5) 
 
 may be readily derived, connecting the differentials of cos 9 k. The differential cross section d<p for 
 scattering into solid angle dfl is 
 
 f dfl k 2 /ko k . , \ ... 
 
 in which r is the classical electron radius e 2 /mc 2 . If the relation dfl = 2jt sin 9 d9 is used and de is 
 replaced by dk with the use of Eq. 5 the following equation is obtained. 
 
 This relation is to be employed in the range of k varying from k„ to k„/(l + 2y) corresponding to the 
 range of 9 from to v. The quantity in parenthesis in Eq. 7 has the following values when 9 takes 
 the values 0, jt/2, and ir: 
 
 0=0 2 
 
 l + y + y 2 
 
 77/2 
 
 l + y (8) 
 
 1 + 2y + 2y 2 
 1 + 2y 
 
 For values of y not larger than about 4, this variation is sufficiently small that it is reasonably 
 good to assume that k has equal probability of falling in any part of the allowed range, or that the 
 knocked -on electron has equal probabilities of receiving any energy in the range from to 2y/(l + 2y) 
 in units of k„. For very large values of y, the sin 2 9 term in parenthesis in Eq. 7 may be neglected for 
 the most interesting collisions. It is then clear from the remaining terms in parenthesis that collisions 
 in which k is small compared with kg are preferred over those in which k is near k„. 
 
 The degree of preference is not exceedingly great for values of y in the normal radioactive range, 
 and it was assumed that the probability per unit energy range is constant within the allowed limits. 
 This gives the maximum statistical variation to be expected in a given Compton process. 
 
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 The gamma ray may conceivably make a number of Compton encounters in passing through the 
 crystal. There are two interesting extreme cases to consider which are referred to as the "thin" 
 and "thick" approximations. In the thin case, which corresponds to values of a appreciably less 
 than unity, the gamma ray has much smaller probability of making two collisions than one collision. 
 In this case it is assumed that a may be chosen to be a constant for each successive collision as if 
 its energy were not greatly affected by successive Compton encounters. In this event the probability 
 that the gamma ray will make n encounters may be described by the distribution function 
 
 P n = l*-*— 0) 
 
 " n! 
 
 for the range of n of practical interest. 
 
 In the thick approximation the gamma ray transfers all its energy to the luminescent material 
 in a succession of encounters once it has made the first encounter. Thus this case is equivalent to 
 that in which the gamma ray transfers its energy by means of the photoelectric effect or pair pro- 
 duction, provided the electrons produced do not escape. These last two cases differ from the thick 
 approximation only with respect to the geometrical distribution of points within the crystal at which 
 the electrons are released — a difference which ic not considered here. 
 
 The thin approximation is best achieved by employing a very thin crystal so that a is small 
 compared with unity and also employing soft gamma rays, for which y is 1 or less, which lose rela- 
 tively little energy in a Compton encounter. It is probably not a case which would be met in practice 
 but is interesting as one statistical extreme. It should be remarked that this limit cannot be achieved 
 by going to very soft radiation, for such radiation is scattered almost isotropically. The distance 
 which the scattered photon must traverse is usually different from that which the original photon 
 would have had to travel to pass through the crystal because it is traveling in a different direction. 
 Thus a is not a constant in this limit even though the energy of the photon is not greatly altered by 
 a Compton collision. The thick case can evidently be achieved by using a thick crystal and is the one 
 that will be met more commonly in practice. 
 
 4. The number of luminescent quanta which the crystal emits can vary even when the energy 
 transmitted to the crystal is fixed because of statistical fluctuations in the manner in which the 
 exciting radiation is distributed among the different excited states of the medium. This type of sta- 
 tistical fluctuation is partly responsible for the straggling in range of heavy ionizing particles as 
 they pass through matter. In order of magnitude the fractional variation in the number of light 
 quanta is l/Yrj , where t) is the average number. Since in this paper cases in which r] is 1000 or 
 larger are of interest, corresponding to Compton encounters in which the knocked-on electron gains 
 several hundred kev of energy, this source of statistical variation will be neglected. It could be 
 significant in cases in which the particle being detected produces very few light quanta, as for very 
 soft beta rays or x rays. 
 
 5. Only a fraction of the light quanta produced in the luminescent crystal will reach the photo- 
 electric surface. The fraction Q which does is determined primarily by geometrical factors involving 
 the angular distribution of emitted light and the angle subtended by the photosurface relative to the 
 luminescent material. Q will be in the neighborhood of 0.5 for a relatively thin layer of luminescent 
 material which is immediately adjacent to the photosurface but may be considerably smaller if the 
 luminescent crystal is somewhat farther away. It may be enhanced by placing a reflecting backing 
 
 on the luminescent material or by employing other devices which cause the light to be "funneled" 
 toward the photocathode. 
 
 6. Only a fraction p of the photons striking the photosurface will eject electrons from it. This 
 parameter appears 4 to be about 0.03 for the type of photosurface in which the photons penetrate the 
 photosurface and electrons are ejected from the back side, and it appears to be about 0.05 for the 
 type of photosurface for which electrons are ejected from the front surface. (Morton and Mitchell 4 
 have shown that the pulse -height distribution is broader than that expected on the basis of a Poisson 
 distribution of electrons at each stage.) 
 
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 7. The electrons ejected from the photocathode will give rise to pulses of various size, de- 
 pending upon the accidents which befall the primary photoelectron and the secondaries which it emits 
 from the multiplying surfaces. Actually there are two problems associated with an analysis of the 
 pulse distribution: first, the problem of determining the probability that the photoelectron will 
 actually create a measurable pulse and second, the problem of determining distribution of pulse 
 sizes when pulses are generated. If the secondary emission ratio is s, the probability that the 
 photoelectron will not eject a secondary from the first stage of the multiplier is e-s, provided it is 
 assumed that the emission of secondaries is random. This probability is of the order of a few per 
 cent for normal values of s (between 3 and 5) and is essentially equal to the probability that the 
 primary electron will not generate a pulse. Since the percentage of uncertainty in p is at least as 
 large as this, this factor may be combined with p in the following discussion, and the assumption 
 may be made that a measurable pulse is produced whenever an electron is ejected. The distribu- 
 tion of pulse sizes has been measured by Engstrom 4 using a light source. Later, his results will 
 be approximated with an appropriate mathematical function. Evidently the pulse distribution is not 
 important if the luminescent counter is employed simply as a counter of events and if a pulse of 
 arbitrary size can be employed as the signal for a significant count. Knowledge of the distribution 
 becomes important, however, if a pulse discriminator is employed so that only pulses larger than 
 a certain size are counted (as when a noise background is eliminated) or if the integrated current 
 of the photomultiplier is recorded. The first of these cases may be treated by redefining the 
 parameter p as the probability that an observable pulse is measured when a photon strikes the 
 cathode and introducing measured values of this quantity. The second case is discussed in detail. 
 
 2. THE GENERATING FUNCTION AND ITS APPLICATIONS TO THE PROBLEM 5 
 
 The generating function was introduced into probability theory very early in its development, 
 and some of its properties are described in textbooks, for example, the book by Uspensky. 5 The 
 authors have benefitted by reading a mimeographed survey of the subject by O. R. Frisch. An 
 account of some of the relations employed here is given by Jorgenson. 5 
 
 The aggregate contribution of the various unit parts of the photomultiplier system to the sta- 
 tistical variation of the system can be determined most simply with the use of generating functions 
 appropriate to each stage. If Pn is the probability that a given observation shall yield n events, for 
 example, that the source in the problem will emit n gamma rays in time T, the generating function 
 G(«) for the process of observation is defined by the series 
 
 G(«) = p €° + p.* 1 + pjt 2 + . . . + Pn« n + . . . (10) 
 
 The generating function is readily found to possess the following properties 
 
 G(0) = Po G(l) = 1 (") 
 
 The mean value m of a series of observations, namely 
 
 m = £ n Pn 
 n 
 
 is readily seen to satisfy the relation 
 
 (12) 
 
 Vd€/€ = 1 
 
 (13) 
 
 Similarly the variance of a sequence of observations, defined by the relation 
 
 v = £ (n - m) 2 Pn = L n 2 ^ - m 2 (14) 
 
 n n 
 
 is readily found to be related to the generating function by the equation 
 
 v _[*G + dG_fdGYl t= m) + m-m 2 (15) 
 
 v ld£ 2 d« Vde / Je = l Vdt 2 *-, 
 
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 In the case of the Poisson distribution 
 
 Pn=^e-« (16) 
 
 G(«) is readily found to be 
 
 G(e) ^e ^" 1 ) (17) 
 
 whence 
 
 m = a v =a (18) 
 
 In the following calculations the ratio Vv/m lS employed to provide a measure of the fractional 
 deviation from the mean or the fractional deviation. In the case of the Poisson distribution this 
 quantity is the familiar ratio 1 /Va . 
 
 Although the generating function is interesting and useful because of the properties already 
 outlined, its real service appears when the following two additional properties are considered: 
 
 I. Suppose that, instead of making one observation of the number of events of interest (such as 
 the numbei of gamma rays emitted from the source in time T), two observations are made (e.g., for 
 two time intervals T) and ask for the probability that n events are observed en toto is asked for. 
 The probability for this is the sum 
 
 PnPo + Pn-i Pi + Pn-2 P 2 + ■ • • + PoPn 
 which is the coefficient of * n in the expansion of G 2 (e). This is a special case of the more general 
 theorem: The generating function governing the probability distribution of the sum of r identical 
 observations is G r («) if G(f ) is the generating function for a single observation. 
 
 II. Suppose next that a situation is being dealt with in which each member of a set of initial 
 events that are statistically distributed (e.g., gamma rays from a source) can give rise to a series 
 of events of possibly different type (e.g., production of Compton electrons) and the statistical distri- 
 bution of the second type of event is asked for. Let Gj(e) be the generating function for the first type 
 of event (e.g., the number of gamma rays emitted by the source in a given time for the example 
 under consideration) and G 2 (t) be the generating function for the number of events of the second type 
 associated with one primary event (e.g., the number of Compton recoils produced by a single gamma 
 ray). It Is readily shown that the generating function GnU) for the number of events of the second 
 type when the statistical variation of the number of events of the first type is taken into account is 
 given by 
 
 G n U) = GjG 2 (e)] (19) 
 
 The validity of this theorem may readily be demonstrated by writing Gn in the form 
 
 %(<•) = p G?(«) + p x Gj(«) + PjG^U) + p 3 G|(«) + . . . + p n G 2 n U) + ... (20) 
 
 in which p n is the probability of n events of the first type, so that 
 
 G t (t) = p e° + p,* 1 + p 2 € 2 + . . . + p n « n + . . . 
 
 The coefficient of p n in Eq. 20 is the generating function for the total number of events of the second 
 type when it is known that n events of the first type have occurred, in accordance with theorem I. 
 This coefficient appears suitably weighted with the probability that n primary events will occur. 
 Using Eqs. 13 and 14, the mean and variance associated with the generating functions 
 
 GiU) = G*(«) Gn(e) = G 1L G 2 U)] (21) 
 
 may readily be found which occur in the cases I and II described above. The results are, respectively 
 
 mj = rm Vj = rv (22) 
 
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 in which m and v are the mean and variance associated with a single observation in case I, and 
 
 mjj = m,m 2 v n = v i m 2 + v 2 m i (23) 
 
 It is clear that, if case n were extended to that in which the second type of event can give rise 
 to a third type (e.g., if a Compton electron can give rise to ion pairs or to luminescent quanta) which 
 is statistically distributed in accordance with a generating function G 3 (e), the complete generating 
 function which takes account of the statistical variation in events of the three types is 
 
 G m = GjG 2 U)] (24) 
 
 for which the mean and variance are, by analogy with Eq. 23 
 
 m m = m n m 3 v m = v n m 3 + '^n (25 ) 
 
 The appropriate form of generating functions to be employed in each of the constituent processes 
 described in Sec. 1 will now be examined. 
 
 1. Emission from source. Since the gamma rays are emitted at random, the appropriate gen- 
 erating function is of the Poisson type, Eq. 17, namely 
 
 Gl ( £ ) = e N(«-D (26) 
 
 in which N is the average number of gamma rays emitted in the time T. 
 
 2. Passage of gamma rays into system. A given gamma ray either does or does not enter the 
 crystal. If the probability that it does is f, the generating function for this event is simply 
 
 G 2 U) = (1 - f) + U (27) 
 
 Using Eq. 19, it is readily found that the generating function G 2 («), giving the distribution of proba- 
 bilities that the gamma rays emitted at random by the source enter the crystal, is 
 
 G^e) = GjG^e)] = e«M = e^" 1 ) (28) 
 
 where v = fN. 
 
 3. Generation of photons. As stated in the introduction, the assumption is that a fixed fraction of 
 the energy which the gamma photon gives up to the crystal is transformed into light quanta. If this 
 energy is E , the number of light quanta produced is then 
 
 n = HE (29) 
 
 where |3 is a factor measuring the efficiency with which the luminescent crystal converts the ex- 
 citation energy it receives into light quanta. If hf is the average energy of the luminescent quanta 
 emitted, (3hv is the fraction of the energy of excitation which appears in the form of luminescent 
 radiation. This may be as large as 0.20 for some of the most efficient materials but can easily 
 be much smaller. According to Broser, Kallman, and Martius 6 the efficiencies of energy conver- 
 sion in zinc sulfide activated with silver and in the organic materials naphthalene, diphenyl, and 
 phenanthrene are given in Table 1. 
 
 Table 1 — Efficiency of Energy Conversion in Luminescent Materials 
 under Gamma-ray Excitation 
 
 (After Broser, Kallmann, and Martius. Values in fractions.) 
 
 Phv 
 
 ZnS:Ag 
 
 0.135 
 
 Naphthalene 
 
 0.05 
 
 Diphenyl 
 
 0.075 
 
 Phenanthrene 
 
 0.11 
 
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 The investigators find somewhat different efficiencies for beta-ray excitation. Similarly, Colt- 
 man, Ebbighausen, and Altar 7 have found the energy conversion in calcium tungstate to be 5.0 per 
 cent for x rays. The interest here is in detailed values of the efficiency in Sec. 3. 
 
 As mentioned in the introduction there are two extreme approximations that are of interest, 
 namely, those designated as "thin" and "thick." In the second case, all the energy of the gamma 
 ray is transmitted to the crystal once a first collision has occurred. The number of quanta emitted 
 is then equal to ry , the value of n when k,, is the energy of the gamma ray. If c is the probability 
 that such a collision occurs, the generating function for the number of quanta is evidently 
 
 S 3 («) = (1 - c) + ctlo (30) 
 
 If this is combined with Eq. 28, the complete generating function for the production of luminescent 
 quanta in the thick case is 
 
 S^ =exp [v cU^o - 1)] (31) 
 
 In the thin case there are two sources of statistical variation, for both the number of Compton 
 encounters and the energy transferred to the counter per collision may vary. The first of these 
 quantities is distributed in accordance with the Poisson law, Eq. 17, in the ideal thin case, for 
 which the generating function is 
 
 H 
 
 _ -a(«-l) 
 
 where a is the ratio (Eq. 3). The energy which the Compton electron receives is randomly distributed 
 between and the maximum value 2k„y/(l + 2y) in the approximation described in paragraph 3 of the 
 introduction. This means that the number of quanta generated will vary between and a maximum 
 Tj m , where 
 
 n 2kpy 
 " m = <* YTly (32) 
 
 in which |3 is the efficiency factor appearing in Eq. 29. A generating function for this random dis- 
 tribution is readily constructed by treating r; as a continuous variable and is 
 
 K 3 (e) = -J- f m e^dr, = exp (»7 m logO - 1 (33) 
 
 "mJ 7) m log€ 
 
 for which the mean and variance are V m /2 and tj^/12, respectively. The complete generating func- 
 tion for the number of quanta associated with a single gamma ray is H 3 [K 3 (£)]. 
 
 4. Emission of photoelectrons. A given light quantum either does or does not emit a photoelectron 
 from the photosurface of the multiplier. The probability that it does is Qp, so that the generating 
 function for this process is 
 
 G«(e) = [(l-flp) + flp«] (34) 
 
 As stated in paragraph 7 of the introduction, it is assumed that a measurable pulse is associated 
 with each photoelectron ejected from the cathode of the multiplier. 
 
 5. Generating function for pulse distribution. Engstrom 4 has measured the pulse-height distri- 
 bution of a typical multiplier tube. His empirical distribution is represented by the analytical 
 function 
 
 f(h) = Ah 2 exp (-h/p) (35) 
 
 in which h is the pulse height on an arbitrary scale, p is a constant measuring the width of the dis- 
 tribution, and A is a normalization factor l/2p 3 . A generating function 
 
 G 5 («) = 1/(1 -P logO 3 (36) 
 
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 may readily be constructed for this distribution. The mean and variance are 
 
 m 5 = 3p v = 3p 2 (37) 
 
 It will be seen later in this paper that it is not necessary to know p in order to determine the frac- 
 tional deviation of interest here. 
 
 3. CORRELATION BETWEEN GAMMA RAYS FROM SOURCE AND PULSES IN MULTIPLIERS 
 
 The probability that a gamma ray from the source will produce a pulse in the multiplier is dis- 
 cussed in this section. The thick and thin cases are discussed separately. 
 
 A. Thick Case 
 
 In the thick case, a gamma ray passing through the crystal has probability c of making an en- 
 counter, in which case it generates tj photons. The distribution of such encounters is random, being 
 governed by the Poisson distribution. The average number is Nfc = i/c, and the deviation is vc. Thus, 
 as far as luminescent pulses are concerned, the effective strength of the source is pc. 
 
 The probability that n of the rj light quanta will eject photoelectrons from the multiplier is given 
 by the generating function 
 
 [G 4 U)] I, °=[(l-flp) + flp«] rI ° (38) 
 
 The probability that none will eject electrons is (1 -Op) 1 ^, so that the probability of observing a 
 pulse, if one electron is sufficient to produce an observable pulse, is 
 
 p = 1 - (1 - flp^o (39) 
 
 Since ?j is usually large compared with unity, this may be approximated by 
 
 P= 1-exp (-n Qp) (40) 
 
 in which the quantity 
 
 "o^P (41) 
 
 is the average number of photoelectrons emitted from the cathode. With the use of the rule, Eq. 25, 
 for determining the mean and variance of a chain of events, ihe mean number of counts is 
 
 M = Nfc[l - exp ( - 7] flp)] = NfcP (42) 
 
 whereas the variance is 
 
 V = NfcP (43) 
 
 The fractional variance is 
 
 Yv / i y* 
 
 (44) 
 M VNfcPy 
 
 B. Thin Case 
 
 There are four statistical processes in the chain extending from the passage of gamma rays into 
 the crystal to the ejection of electrons from the photocathode, namely, those described by the gen- 
 erating functions G 2 ' , H 3 , K3, and G 4 of the preceding section. The number of electrons ejected from 
 the cathode when a single gamma ray passes through the crystal is governed by the generating 
 function 
 
 E(e)=H,{KjG 4 ( e )]] (45) 
 
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 The probability that no electron will be ejected and hence that no pulse will be recorded is E(0), 
 so that the probability of a pulse is 1 - E(0), and the generating function for pulses is 
 
 (e(0) + [1 -E(0)]e} (46) 
 
 Hence the mean and variance in the number of pulses are 
 
 M = V = Nf[l - E(0)] (47) 
 
 Since Qp is of the order of 3 per cent even when Si is unity, it is readily found that K 3 [G 4 (0)1 can be 
 approximated by the expression 
 
 K 3 [G 4 (0)]. [l - eX P ( - a yP ) ' (48) 
 
 A simple examination of E(0) shows that it approaches e-° when r) m J3p I s large compared with unity 
 and approaches exp {-ar] m Qp/2) whenT) m flp is small. It may be concluded that, in both the thick and 
 thin cases, the counts are governed by a Poisson distribution and that it is desirable to have the 
 quantities c ind 1 - e~ a as near unity as possible and the quantities %Qp and7j m £?p somewhat larger 
 than unity, although there probably is little advantage to having them as large as 10. 
 
 Suppose one is dealing with gamma rays in the vicinity of 1.5 Mev, to provide a concrete example. 
 In this case the mean free path for the Compton effect in a material such as naphthalene is of the order 
 of 15 cm. Hence if the crystal is a cube 5 cm on an edge, the factor e~ a is 0.72. The Compton elec- 
 tron will have an average energy of the order of 0.7 Mev, so that the average number of luminescent 
 quanta produced is 10,000 if the energy efficiency is taken to be 0.05. Choosing p to be 0.03, it is 
 found thatrj m £p/2 is 300 Q. Hence Q should be at least 10~ 2 if each Compton electron is expected to 
 register with reasonable faithfulness. If it is assumed that the photosurface of the multiplier has an 
 active area of about 15 cm 2 and that this surface is 5 cm from the center of the crystal, the factor Q 
 should be as large as 0.05 even if the photons are isotropically distributed, which would guarantee 
 faithful counting of Compton encounters. The same photosurface would be more nearly borderline if 
 the crystal were chosen to be a 10-cm cube and the surface were placed 10 cm from its center, for 
 then Q would be about 0.01, which is very close to the limit set above. In fact, those Compton en- 
 counters which take place at points within the crystal which are most distant from the surface may 
 fail to register if the photon distribution is isotropic. In this event it may prove profitable to employ 
 a method of light tunneling, for example, by covering all surfaces of the crystal except that opposite 
 the multiplier with a reflecting metallic covering. 
 
 4. PHOTOMULTIPLIER CURRENT 
 
 Consider next the current in the photomultiplier, or rather the charge which arrives at the 
 anode end when N gamma rays are emitted from the source. The "thick" and "thin "cases are dis- 
 cussed separately once again. 
 
 A. Thick Case 
 
 In this case the distribution of charge in the photomultiplier may be regarded as if compounded 
 of the three statistical processes which are described by the generating functions 3j , G 4 , and G 5 of 
 Sec. 2. The first of these functions gives the distribution of photons in the crystal associated with the 
 N gamma rays, th<~ second function gives the distribution of the photoelectrons from the cathode of 
 the multiplier, and the third function gives the distribution of pulses in the multiplier. The mean and 
 variance of these distributions are as follows: 
 
 Mean Variance 
 
 S3 vcr\ vcr\l 
 
 G 4 Qp Qp- (Qp) 2 = i?p 
 
 G 5 3p 3p 2 
 
M 
 
 = 3pi/CTj flp 
 
 V 
 
 = 3p 2 i/C7) flp(4 + 3n Op) 
 
 
 Vv 
 
 M 
 
 T4 + 3jj flp"|V2 
 " [ 3fC r] 0pj 
 
 10 AECU - 715 
 
 The quantity (Op) 2 may be neglected in comparison withOp since the latter is at most a few per cent. 
 
 By compounding these statistical quantities in accordance with the rule, Eq. 25, the following 
 values of the mean and variance for the charge in the multiplier are obtained: 
 
 (49) 
 
 v = op Kt r/ Mp \t + oi| .*p; 
 
 The fractional variance is 
 
 (50) 
 
 This quantity is independent of p, as pointed out previously. Moreover, it becomes independent of 
 the quantity x = T) Op when this quantity is large compared with unity. The condition placed upon x 
 for this limit to be valid is somewhat more stringent than the condition required for faithful counting 
 of luminescent pulses. That is, x must be larger than 5 for this approximation to be precise. 
 
 B. Thin Case 
 
 In this approximation the distribution of pulses is governed by a generating function that is 
 compounded of the generating functions G 2 ' , H 3 , K 3 , G 4 , and G 5 . Respectively, these correspond to 
 the distribution of gamma rays in the crystal, the distribution of Compton encounters, the distribu- 
 tion of luminescent quanta produced in the crystal, the distribution of photoelectrons from the 
 cathode, and the distribution of pulses in the multiplier. The corresponding means and variances 
 are as follow: 
 
 Mean Variance 
 
 G 2 ' 
 
 V 
 
 V 
 
 H 3 
 
 a 
 
 a 
 
 K 3 
 
 H m /2 
 
 T7 m /12 
 
 G 4 
 
 Op 
 
 Op 
 
 G 5 
 
 3p 
 
 3P 2 
 
 The mean and variance for the distribution of pulses is found to be 
 
 „ 3 
 
 P v a >? m Op 
 
 (51) 
 
 V = -p*par, m Qp [4 + - 7) m £p (4+ 3a) ] 
 
 Once again it is noticed that p drops out of the fractional variance. Whenever the quantity y = n m Op 
 is very small compared with unity, the fractional variance may be approximated by the expression 
 
 _VV = (_8/3_Y* (52) 
 
 m \ vanm apJ 
 
 In the opposite extreme, in which y is very large compared with unity, the fractional variance is 
 
 Vv _ U + 3a \ v * 
 M " V 3w* / 
 
 which approaches 2/ V3a^ if a is small compared with unity and approaches l/V^ if a is very 
 large. The latter case, in which a is large, is in contradiction with the assumptions of the thin 
 approximation; however, it is of mathematical interest. 
 
 (53) 
 
AECU - 715 11 
 
 5. FLUCTUATIONS IN CHARGE ON CONDENSOR 
 
 When dealing with a high-intensity source, it is frequently convenient to feed the current pulses 
 from the photomultiplier into a condensor which is shunted with a high resistance and measure the 
 voltage across the condensor in order to provide a measure of the average current which arrives 
 at the condensor. This voltage exhibits fluctuations because the pulses are distributed statistically 
 both in magnitude and in time. The influence of the distribution in time has been investigated by 
 Schiff and Evans 8 for the case in which the pulses are equal in magnitude. The generalization of 
 their results when the pulses vary in size is of interest here. 
 
 If the capacity of the condensor is C and the shunting resistance is R, the decay time for the 
 shunted capacity is t = RC. A charge which is fed into the condensor at time t' will have decayed by 
 a factor exp [ — (t - t') /t] by the later time t. 
 
 The assumption is that the charge associated with each pulse of the multiplier arrives in a time 
 that is short compared with the decay time of the condensor. It is also assumed that the pulses are 
 distributed in time in accordance with the distribution law governing the frequency with which gamma 
 rays enter the luminescent crystal, that is, in accordance with the generating function G 2 '(e) of Sec. 2 
 (see Eq. 28). Since interest is in specific intervals of time t, v in Eq. 28 is replaced by nt, where n 
 is the average number of gamma rays entering the crystal per unit time. Those gamma rays which 
 do not excite the crystal will give rise to pulses of size. For the purposes of this section, the 
 generating function is designated for the pulse in the photomultiplier associated with the passage of 
 a single gamma ray into the crystal by G(e). The pulse size will be assumed to be expressed in units 
 of charge. G(c ) will differ in the soft and hard approximations but may be left arbitrary for the 
 moment. 
 
 Consider the gamma rays which arrive in the time interval dt' between t' and t' + dt'. The 
 generating function associated with the current they contribute to the condensor at the time t' is 
 
 1 + ndt'[G(e) -1] (54) 
 
 which is the expansion of G 2 '[G(e)] in terms of dt' when v is replaced by ndt'. The mean value of the 
 charge associated with this generating function is 
 
 ndt'G'(l) (55) 
 
 This mean contribution will have decayed by a factor exp [(f - t)/r] by the time t. Thus the mean 
 charge at time t resulting from the accumulation for all previous times is 
 
 nG'(l) J° x exp [(t'-t)A] dt' = nrG'd) (56) 
 
 G'(l) evidently is the mean charge pulse Q in the photomultiplier associated with the entrance of a 
 single gamma ray. 
 
 Similarly, the variance in the charge on the condensor at time t is the integral of the variance of 
 Eq. 54 from t' = — oo to t' = t with a weighting coefficient exp [2(t' - t)/r] since the decay constant for 
 the square of the charge is twice as large as that for the charge. The result is 
 
 y[G"(l) + G'(l)] (57) 
 
 The quantity G"(l) + G'(l) is the mean of the square of the charge pulse associated with a single 
 gamma ray, which we shall designate as Q 2 . This is also equal to the variance of the charge pulse 
 associated with a single gamma ray plus Q 2 . 
 
 The fractional variance of the charge on the condensor is 
 
 Vv / Q 2 V/2 
 
 M \2rnQ 2 
 
 (58) 
 
12 AECU - 715 
 
 The coefficient (1/2™) y * represents the result obtained by Schiff and Evans for pulses of constant 
 amplitude. The coefficient Q 2 /Q 2 for the thick and thin cases may now be investigated. 
 
 A. Thick Case 
 
 In this case the generating function G(«) is S 3 jG 4 [G 5 (6)] . The means and variances of G 4 and G 5 
 were tabulated in the previous section. The corresponding quantities for G 3 are ti c and tiJcU - c). 
 By combining the means and variances 
 
 M = Q = 3prj cOp V = 3p 2 7? cflp[4 + 3?) flp(l - c)] (59) 
 
 are obtained. Moreover, 
 
 Q 2 = V + M 2 = 3p 2 7! cOp(4 + 3r) fip) (60) 
 
 so that 
 
 Q 2 Y* /4 + 3n flp^2 
 
 / 4 + 3T) gp \V- 
 Q*/ Isrfocflpj (61) 
 
 As should be expected, this approaches 1/ Vc whenr) £>p becomes sufficiently large, for the pulses 
 then approach the constant size and the only source of statistical variation is in the random pro- 
 duction of luminescent bursts. 
 
 B. Thin Case 
 
 In this case G(«) is H 3 (K 3 (G 4 [G 5 (e)]j) whose averages were tabulated in the previous section. 
 
 M = Q = -pa„ m Gp V = 3p 2 «77 m i2p(2+T) m i?p) (62) 
 
 Q 2 = 3p 2 «1 nf Op[2 + i| in flp(l + |a)] 
 
 /Q^ 2 = [1 2 + nmflp (1 + 3 / 4 a) f 2 
 VQ 2 / L 3 a r) m Qp J 
 
 jfip 
 
 In this case Q 2 /Q 2 approaches (4/3 + a) /a when rj m £p becomes sufficiently large. 
 
 6. CONCLUSIONS 
 
 1. The statistical variations in a counting system which consists of a source, a luminescent 
 crystal, and a photomultiplier are examined. It is assumed that the source is constant for a fixed 
 period of time, although it emits particles at random. For definiteness and to provide a maximum 
 degree of statistical variation, it is assumed that the source is a gamma emitter and that only a 
 fraction of the gamma rays fall on the luminescent crystal. The method of generating functions is 
 employed to treat the chain of events which the particles emitted from the source engender. Two 
 oppositely extreme cases are considered, namely, that in which all the energy of a gamma ray which 
 enters the crystal is transferred to the electrons and that in which the gamma ray transfers only a 
 portion of its energy in a manner that depends upon the Compton encounters it makes. The two ap- 
 proximations are referred to as the "thick" and "thin" approximations. The first can be realized 
 by using a crystal which is sufficiently thick that the gamma ray is completely absorbed. The second 
 case can be approximated by using a very thin specimen and using gamma ray energies for which the 
 Compton process predominates. 
 
AECU - 715 13 
 
 2. As might be expected the results show that the effectiveness of the system depends upon the 
 ability of the crystal to receive energy from the crystal. They also show that a measure of the ef- 
 fectiveness of the remainder of the system is provided by the quantity 
 
 x = nflp 
 
 Here r\ is equal to the number of light quanta, r\ , produced per gamma ray in the luminescent crystal 
 in the thick case and is, 77 m , the maximum number which can be produced per Compton encounter in 
 the thin case. Q is the probability that a light quantum emitted from the crystal will strike the photo- 
 surface of the multiplier, and p is the probability that a photoelectron will be emitted from the 
 cathode. The system will be a faithful counter of those gamma rays which transfer energy to the 
 crystal provided x is of the order of 5 or larger. The statistical fluctuations are then determined 
 primarily by the Poisson distribution of encounters in the crystal. If, on the other hand, the current 
 from the multiplier is measured instead of the rate of counts, the contribution of the photomultiplier 
 to the statistical error is appreciable until x is considerably larger than 5, although this error can 
 be reduced to that corresponding to the Poisson distribution of encounters in the crystal when x is 
 increased. 
 
 The statistics of the case, in which the pulses are fed into a capacitor with a time constant and 
 the voltage of the capacitor is measured, are treated from a standpoint somewhat more general 
 than that considered by Schiff and Evans. 
 
 REFERENCES 
 
 1. J. W. Coltman and F. Marshall, Phys. Rev., 72: 528 (1947); F. Marshall, J. Applied Phys., 18: 512 
 (1947). 
 
 2. I. Broser and H. Kallmann, Z. Naturforsch., 2a: 439 (1937); 642 (1947); I. Broser, L. Herforth, 
 H. Kallmann, and U. Martius, ibid., 3a: 6 (1948). 
 
 3. W. Heitler, The Quantum Theory of Radiation, Oxford University Press, 1936. 
 
 4. P. W. Engstrom, J. Optical Soc. Am., 37: 420 (1947); G. A. Morton and J. A. Mitchell, RCA Rev., 
 9: 632 (1948). 
 
 5. J. V. Uspensky, Introduction to Mathematical Probability, McGraw-Hill Book Company, Inc., 
 New York, 1937; T. Jorgenson, Am. J. Phys., 16: 285 (1948). 
 
 6. L. Herforth and H. Kallmann, Ann. Physik, 4: 231 (1949). 
 
 7. J. W. Coltman, E. G. Ebbighausen, and W. Altar, J. Applied Phys., 18: 530 (1947). 
 
 8. L. I. Schiff and R. D. Evans, Rev. Sci. Instruments, 7: 456 (1937); L. I. Schiff, Phys. Rev., 50: 88 
 (1936). 
 
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