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Resolution and line shape in scintillation counters Takhar, P. S. 1963

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RESOLUTION AND LINE SHAPE IN SCINTILLATION;COUNTERS •by P.. S. TAKHAR A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF - • , .... ...... 'M^ TER'OF" SCIENCE ' " ' i n the Department of PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA In presenting this thesis in pa r t i a l fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of this thesis for financial gain shall not be alLwed without my written permission. Department of V* W^xe-C The University of British Columbia, Vancouver 3, Canada. (v) ABSTRACT An experimental study has been made of a number of factors that determine the pulse height resolution of Sc i n t i l l a t i o n Counters. A s t a t i s t i c a l model i s developed from which an analytic expression for the ideal l i n e shape i s obtained. Excellent agreement i s found with observations using an a r t i f i c i a l l i g h t pulser. An attempt to understand the noise spectra, exponential and non exponential under various conditions, has been made. A comparison i s made between non crystalline organic sc i n t i l l a t o r s and"sodium iodide crystals of similar sizes. It i s shown that for gamma-ray detection an important contribution to the line width originates with variations in the l i g h t collection efficiency from different regions of the s c i n t i l l a t o r s . (vi) ACKNOWLEDGEMENTS The author wishes to express his thanks to Dr. J . R. Prescott for suggesting this research and for his advice and supervision in performing the experiments and also for his assistance in computation and analysing the data. Thanks are also due to Dr. J. B. Warren for his interest i n this work after Dr. Prescott l e f t for the University of Alberta. ( i i ) TABLE OF CONTENTS . ....... ,_. Page I * Introduction 1 I I . . Theory 2.1 S t a t i s t i c a l Models 5 2.2 The Shape of ""the Ideal S c i n t i l l a t i o n Line 10 2.3 S t a t i s t i c a l factors i n the S c i n t i l l a t o r s 12 2.4 The Addition of Random Tube "Noise 14 I I I . Experimental Technique 3.1 S c i n t i l l a t o r s used 16 3.2 S c i n t i l l a t o r Assembling Technique 17 3.3 Arrangement f o r cooling the Photomultiplier 19 3.4 Source Holder "." 19 3.5 Radiation Sources 20 3.6 Electronics 22 IV. Procedure and Results 4.1 Noise and M u l t i p l i e r S t a t i s t i c s 28 4.2 Noise and Resolution ". 29 4.3 Non exponential single electron d i s t r i b u t i o n 34 4.4 Exponential V e r i f i c a t i o n of the shape of the Ideal S c i n t i l l a t i o n Line 38 '. 4*5 Transfer effects i n Organic S c i n t i l l a t o r s 40 4.6 Transfer effects i n sodium iodide c r y s t a l 46 V. Discussion 53 VI. Appendices ( i i i ) (Fig. No.) ILLUSTRATIONS To follow page. " 1. Liquid s c i n t i l l a t o r NE-2i9 assembly 16 2. Sodium iodide mount for gamma rays. 17 3. Sodium iodide mount for alpha-rays 18 4-. Arrangement for cooling of the photomultiplier 18 5. Source holder for alpha source. 19 6. Light pulser. 20 7. Blcok diagram of electronics. 21 8. Gathode follower for use with 704-6 photomultiplier 24 9. Cathode follower for use with 6810A photomultiplier 25 10. Relative variances as function of reciprocal pulse height at different focus settings for 704-6 and 6810A photomultipliers. 29 11. Single electron spectra i n a 704-6 photomultiplier with focus settings as a parameter. 32 12. Noise spectra i n a 704-6 photomultiplier showing the effects of cooling and continuous illumination of photo cathode. 33 13. Pulse height distributions for a pulsed l i g h t , source with a 6810A photomultiplier and a non-exponential single"electron distribution f i t t e d as described i n the text. 35 14-. Pulse height distributions for pulsed l i g h t sources with simple exponential single electron distribution i n a 704-6 photomultiplier, solid curves are computed from expression (4-) in the text. N values are given for each curve. 36 (iv) Fig. No. ILLUSTRATIONS 15. Variance vs mean for pulsed ligh t signals with a simple exponential single electron distribution i n the 704-6 photomultiplier. 37 16. Arrangement for ND 101 analyser for the study of low level pulses. 39 17. Resolution data for l i q u i d s c i n t i l l a t o r showing the contribution from photomultiplier, external alpha particle and alpha particles within the l i q u i d s c i n t i l l a t o r . 4-0 18. Resolution data for gamma-rays i n liq u i d s c i n t i l l a t o r , gamma rays resolution corrected for transfer effects, contribution from alpha particles and contribution from photomultiplier from Fig. 17 4-5 19. Resolution data for gamma-rays in Nal with experimental points, contribution from photomultiplier, corrected for in t r i n s i c resolution and corrected for delta ray function. 4-6 CHAPTER 1 1. Introduction Although the s c i n t i l l a t i o n counter i s of great value i n gamma ray*spectroscopy because of i t s high efficiency (see for example the review article by Mott and Sutton (1958) i t suffers from rather poor energy resolution. In the past there has not been any clear understanding as to what factors (for example, with sodium iodide thallium activated scintillator) set a l i m i t of about &% resolution for the f u l l width at half maximum with 0.662 mev gamma rays from a Cs - 137 source, nor of the way in which the resolution varies with gamma ray energy. Obviously a quantitative theoretical understanding of the minimum width which might be achieved with a given photomultiplier would be of considerable value i n attempts to improve the performance of the s c i n t i l l a t i o n counter. The aim of the present work was to achieve a better understanding of the factors that l i m i t resolution in s c i n t i l l a t i o n counters. In this work particular attention has been given to the role of the photomultiplier, since, other things being equal, this sets a basic l i m i t to the resolution that can be achieved. Good reviews of the previous work are to be found i n the arti c l e of Mott and Sutton, and Wright (1954) and Breitenberger (1955) who give detailed discussions of' the theory of resolution i n s c i n t i l l a t i o n counters. 2 Breitenberger (1955) described the l i g h t transfer i n terms of a four stage cascade process and gave a s t a t i s t i c a l analysis of various causes of the line width of a s c i n t i l l a t i o n counter, He described the l i g h t transfer as a four stage cascade consisting of (a) Generation of photons in the s c i n t i l l a t o r , (b) Collection of these photons in the photo cathode, (c) Photo electric conversion and (d) Collection of photo electrons into the f i r s t dynode of the photomultiplier. He pointed out that these four relations are mutually interdependent and there are i n t r i n s i c connections among these four processes. According to Breitenberger, known causes of the line width i n a s c i n t i l l a t o r counter are (i) varying transfer efficiency, ( i i ) voltage stabilization, (iii).inhomogeneous luminescence yield through-out in crystal, (iv) interaction of edge and scattering effects, (v) non proportional s c i n t i l l a t i o n response and others. Breitenberger had also shown that square of line width versus reciprocal energy could be described by straight line having certain intercept and slope, where intercept depends upon "energy dependent" transfer variance and slope on energy dependent - variance. Engelkemeir (1956) showed that l i g h t per kilovolt was not constant for different gamma ray energies. Nemilow et a l (1959) at the Academy of Science i n Moscow showed measured results on sodium iodide (Th), cesium iodide (Th) and potassium iodide (Th) which agreed with the results of Engelkemeir. mm? ^ mm -Bisi and Zappa (1958) extending Breitenberger's ideas showed that below 100 kilovolt the straight line condition does apply to reasonable spectrometers.. However, for energy above several hundred kilovolts this is not true, i.e. transfer variance decreases. This accounts for bending over of the square of line width versus reciprocal energy curve at high energies where total absorbed events are composed of multiple compton and pair production events as well as photo-electric events. Meyer and Murray (i960) gave a theory relating light-per-unit energy to the density of ionization along the particle track which provided understanding of nonproportional response. A computer program was developed by Zerby et a l (i960) for evaluating the l i g h t per unit energy for electrons starting from gamma ray data, which established li g h t per unit.energy curve for electrons. This was then incorporated in a Monte-Carlo program for calculating the pulse height spectrum for gamma ray spectrum. This was helpful in comparing the experiment and theoretical gamma-ray curves. Iredale (l96l) pointed out that i t is necessary to include the effect of delta rays as the s c i n t i l l a t i o n l i g h t produced per unit energy loss (dl'^dk) for an electron varies with the density of ionization over a wide range of energies, dl/dx increases with the increasing ionization density. Thus a further increase in the number of electrons sharing the gamma ray energy results in an increase in the light output; consequently, there w i l l be a spread in the light output, depending upon the distribution in number and energy of electrons involved in absorption. According to Iredale, two important processes which lead to variations are the number of Compton encounters of the primary gamma rays and the number of delta rays, i . e . the electrons capable of producing further ionization, produced i n the slowing down of electrons which originally share the gamma ray energy. The above effects require the use of more general expression for relative variance, for example, when l i g h t emission does not follow the poission model and the transfer probability varies from event to event, as in the case when scintillatfonss occur i n different regions of a large s c i n t i l l a t o r . The present work was aimed to investigate a l l these effects including the role of the photomultiplier i n particular. - 5 -CHAPTER 2 2. Theory of Line Shape i n Sci n t i l l a t o r Counters 2.1 S t a t i s t i c a l Models Basically," resolution and line shape i n s c i n t i l l a t i o n counters are determined, by the s t a t i s t i c a l frequency function of five separate processes:-'(a) Light production in the s c i n t i l l a t o r i t s e l f , (b) Light collection at the photo-cathode, {c) Production of electrons at the photo-cathode, (d) Collection of photo electrons at the f i r s t dynode, (e) Multiplication i n the photomultiplier dynode structure. Each of the distributions can.be described by a frequency function f(x) or by an associated generating function G (Q), moment generating function M( Q) or. characteristic function x( (9), defined as follows with the introduction of the auxiliary variable': G(<S>) = £ # f ( x ) a l l x M(&) = ^ e S x f(x) a l l x x(G) = £ e i Q x f(x) " a l l x or by corresponding integral of the frequency function i f continuous. The moment generating function i s obtained from the generating function by replacing log 9 by I'B and the characteristic function by replacing log Q by i © and vice versa. The moments of the frequency function are found a; - 6 -functions of the derivatives of G( © ), M( S ) or x( <St) evaluated at Q ±% i n the case of generating function or 8 = 6 i n the case of M( 8 ) . For example, the f i r s t moment,^u. , about the origin (the mean) of f(x) i s given by /X= M(o) and variance, xr^ (second moment about the mean) by: xr = M"(0)?—\A(«>NPrimes denote differentiation with respect to x- . The characteristic function is the Fourier transform of the frequency function which can therefore be found (at least - i n principle.) once the "characteristic function is known. Since the s t a t i s t i c a l processes-listed above are in cascade, the following property of the generating function can be used to find the generating function of the outcome of the cascade: l e t X( 0 ) , Y( 6 ), Z( 8 ) represent the generating functions of three processes i n cascade, then the generating function G( 9 ) of the output frequency function i s given by the functional: G( e) = x ^z( a) If Uy, U , U , represent the means of the frequency functions f ( x ) , Z f ( y ) , f(z) then U Q = Ux Uy Uz. Introducing the dimensionless relative o variance V = Variance/ (mean) , i . e . V = then V G = Vx + V_; + Vz (l) tt2 Ux UxUy The extension to a cascade of more than three processes i s evident. For the purposes of comparison with experiment s t a t i s t i c a l models must be sought for the five processes l i s t e d above. In the present work the following models have been adopted: (a) Light production: an average"of -v^  primary photons, having a Poisson distribution, for which the generating function i s exp (n(8 -l))» (See Appendix). This model i s plausible so long as we assume that the events i n which photons are produced are independent. In any case, Breitenberger - 7 -shows, that regardless of the precise form of the frequency function for which ti i s the mean, i t becomes effectively Poissonian provided a sufficiently small fraction of photons i s allowed to reach the photomultiplier; this was so i n the present experiments and the model appears to be just i f i e d by experimental results. (b) (c) (d) Light collection, photo-electron production and collection; In so far as each of these processes constitutes a Bernoulli t r i a l , e.g. a photo electron released from the cathode either reaches the f i r s t dynode or i t does not, each can be represented by a binomial distribution having the generating functions P(©) = 1 - p + p0 where P i s the probability of a "successful" event, e.g. a photo electron reaches the cathode. Since any number of binomial processes i n cascade generates a binomial frequency function (see Appendix), a l l three processes can be taken together into a single generating function T(&) = ( l - Y ) - V , where V i s the probability that a primary photon w i l l result in an electron arriving at the f i r s t dynode. It i s conveniently referred to as the "transfer probability". S t r i c t l y speaking, of course, the arrival of an electron at the cathode does not guarantee that a cascade w i l l reach the anode of the photomultiplier. A cascade may f a i l to start, or once started, i t may "break" at the subsequent dynode. Although this i s technically part of the multiplication process, i t may be"shown (see Appendix) to be formally equivalent to a reduction in the number of electrons reaching the f i r s t dynode, i . e . to a reduction i n V -At this stage of the argument, non uniformity of the photo cathode - 8 -i s irrelevant since the only requirement i s that V should remain constant from event to event. This i s shown formally by Breitenberger and confirmed by the observations of Hickock and Draper (1958) i n their F i g . 17 and i n the present experiments. The above property of binomial distributions i n cascade to generate another binomial distribution allows one a r t i f i c i a l l y to vary the transfer efficiency by means of f i l t e r s interposed between the l i g h t source and photo cathode, as described i n later sections. The generating function for a cascade consisting of a Poisson distribution followed by a binomial distribution i s again Poissonian (see Appendix) v i z . exp. (n (8 - l ) ) in the above notation having a mean . (e) Multiplication: The multiplier structure i s a complex system i t s e l f involving a cascade of processes i n which a single electron f a l l i n g on the f i r s t dynode produces a distribution i n the number of electrons arriving at the anode, and hence i n the voltage pulse observed there. If we make no assumptions at this stage about the model to be used for the dynode s t a t i s t i c s , then on the basis of a three-stage chain consisting of Poisson, l i g h t production, binomial transfer, and multiplication on an arbitrary model, with relative variance Vm, relation (1) takes the very simple form: (see Appendix) ' V Q = (1 +. Vm)/nv (2) Thus, regardless of the details 'of the s t a t i s t i c a l model used in - 9 -describing the multiplication, the relative variance, V of pulses at u the output of photomultiplier, is inversely proportional to n, the number of electrons arriving at the f i r s t dynode. It should be noted that the particularly simple form of relation (2) i s a consequence of the choice of a Poisson distribution for the model of l i g h t production and that i t i s only true trif: T i s constant for a l l '\ events considered. Breitenberger shows that i f T i t s e l f varies from event to event, as i s certainly the case when scintillations occur in different regions of a large s c i n t i l l a t o r , then expression (2) must be modified to read: (See Appendix) " V G = VT'+ (1 + vm)/nf (3a) where Vp i s introduced by" the non" constancy of T and the graph of VQ versus""l/nT i s linear with a positive intercept Vj> on the VQ axis. A similar result i s obtained i f instrumental resolution i s disproportionately large. I f , i n addition n i s not from a Poisson distribution but one which has a relative variance Vn, then expression (3a) becomes: V G = V T + (1 + V T) (Vn - 1) + 1 + y m n n (3b) This reduces to: V Q = (Vm - 1) + 1 + Vm (3c) n n i f there i s no variation i n *T from event to event. The effect of v a r i a b i l i t y in T or n which gives rise to the l a t t e r result should be clearly distinguished from the case where T or n i s varied deliberately i n order to vary the observed pulse height. Since expression (2) i s symmetric for n and T , either or both may be "varied for this purpose and provided that both then remain constant, expression (2) s t i l l correctly describes the v a r i a b i l i t y of the pulses observed. In expression (3), however, such variation i n v may also l i k e l y effect V T. 2.2 The shape of the ideal s c i n t i l l a t i o n line If we now choose an explicit mathematical model for the distribution of single electron pulses i n the multiplier, we can, at least, i n principle, determine the shape of the ideal s c i n t i l l a t i o n l i n e . The fact that the single-electron distribution i s nearly exponential, which appears from the experimental results i n the later part of Chapter 3 suggest choosing a simple exponential as a model. We therefore take for the frequency function of single-electron pulses G(X) = a~^ exp (-X/a), where X i s any convenient measure of pulse height." * The generating function for this frequency function i s l / ( l - a log6 ) (see Appendix) and the generating function for a cascade consisting of Poisson l i g h t production, binomial electron collection and exponential" multiplication i s G(G) =C_.(G 2(G 3(©))) Where Gj, G 2 and G^ represent the generating functions of Poisson, binomial arid exponential distributions and are given by Gj_ = exp (n(e -1)) G 2 =^9-: (1 -T")^ . G3 = 1 " 1 - a"KLpg Therefore, for the cascade of three - 11 -or = exp n a T log $ (see Appendix) 1 - a log9 Replacing logQ by i® gives characteristic function: )L[Q) = e- n exp / n ^ \ .^ 1 - ia©/ which i s the Fourier transform of the required frequency function f(x) f U) = _1_ e~±6ix <3lS - 00 The regular solution (Campbell and Foster I960) i s : f(x) = N* a"^ e" N r% exp (-X/a) I x ^2(NX/a)"^ U) Where 1^ i s a modified Bessel function'of the f i r s t kind for imaginary argument, and N replaces n T , the mean number of electrons reaching the f i r s t dynode. The solution also includes a delta function of the magnitude e~^ at the origin, which i s evident by pattern N = 0 i n the Poisson distribution which describes the f i r s t two stages, representing the cascades that break or f a i l to start. The properties of the distribution are as follows: (see Appendix) Mean Jk = Na Variance T-°'= 2Na2 hence relative variance Vx = 2aN~^ Third moment about the mean 'fe a 3N ? It has a f i n i t e value at X = 0 and i s a model for n^2. It i s skewed to the l e f t i.e., the mode occurs at lower values of x than does the mean. For large N the mode tends to (N - 2)a (Appendix) and the distribution as a whole tends to Gaussian. There i s a particularly simple , relationship between the variance and the mean viz ,3- = 2<x^ i . e . , the variance i s proportional to the mean and constant of pro-portionality i s twice the logarithmic decrement of the exponential. For suf f i c i e n t l y large values of the argument of the Bessel function (2(NK/a)^">iO the asymptotic form for the latte r i s substituted and the frequency function becomes: f(X) = Uir) ^ 1% (X/a) f exp (2(NX/a)* - x/a - N) (5) 2.3 S t a t i s t i c a l factors i n the Scintillators * In addition to the photomultiplier effects discussed above, the sc i n t i l l a t o r i t s e l f introduces additional sources of s t a t i s t i c a l fluctuations. The most obvious stems from the fact that the s c i n t i l l a t o r occupies a f i n i t e volume. For purely geometrical reasons, s c i n t i l l a t i o n events taking place in different regions of the s c i n t i l l a t o r result i n different numbers of primary photons reaching the cathode, so that the frequency function of the transfer i s no longer binomial when averaged over a number of events. An explicit calculation of this geometrical effect has been carried out by Kukushkin and Ratner (1958) for a sodium iodide crystal and Hickock and Draper (1958) have estimated i t for sodium iodide from measuring made with a variety of crystals and photomultipliers. Similar analyses have been made by Burch (1961), Barnaby and Barton (i960), and Brini et a l (1955). Other contributions to variance are flaws and inhomogeneities i n the crystal as given by Wright (1954) and Garlick and Wright (1952), edge effects, and the pos s i b i l i t y that the fluorescence mechanism in'the s c i n t i l l a t o r i s not Poissonian. Kelley et a l Breitenberger (1955) (1956) have suggested that sodium iodide has an "int r i n s i c resolution" that i s characteristic of the material regardless of i t s geometrical form. Wright and Garlick (1954) drew similar - 13 -conclusions for crystals of calcium tungstate, potassium iodide and diamond. B i s i and Zappa (1958) discuss the role played i n resolution by the multiple interactions that characteristically contribute to the full-energy peak for gamma-ray detection i n sodium iodide and Zerby, Meyer and Murray (1961) and Iredale (1961) have elegantly demonstrated that such multiple interactions combined with non-proportional energy response, Murray and Meyer (1961), result i n an "intrinsic.broadening" of gamma-ray lines which i s significant contribution to the total line width. Burch (1961), and Iredale (1961) have discussed fluctuations i n the number of delta-rays along an ionising track as a further source of line width. The above effects require the use of the more general expressions (3a), (3b) and (3c) of Chapter Two for the relative variance VQ of the sc i n t i l l a t i o n l i n e . I t ' w i l l be recalled that i f the light emission does not follow the Poisson model, then VQ = (Vn-Vn) + (1 + Vm)/n«r (6) Where V R i s the relative variance for"the production of photonsj and i f i n addition the transfer probability varies from event to event, as in the case when scintillations occur i n different regions of a large s c i n t i l l a t o r , we then have: VQ = V T - (1 - V T) (Vn" l / n) - (1 + V m ) / n T (7) where Vj, i s basically the relative variance of the frequency distribution of transfer probabilities. In a practical s c i n t i l l a t o r Vj. may also include, additively, effects int r i n s i c to the s c i n t i l l a t o r (Terby and Meyer (l96l) and Iredale (1961)) and be energy dependent. The last term in each of these expressions represents the contribution of the photomultiplier to V_. - 14 -In Section 4*6 and the following sections an attempt has been made to account for the effect of variable transfer probability on resolution, and we shall refer to this contribution to Y<j> as the "transfer variance". 2.4 The Addition of Random tube noises At room temperature, spontaneous emission of electrons from the cathode may modify the above distributions. A fraction, P of the lig h t signals w i l l have the above frequency'function but the remaining fraction ( l - P) w i l l be accompanied by one or more electrons independently (and randomly) emitted from the photo cathode. Under normal conditions the fraction ( l - P) i s small enough to make the probability of simultaneous emission of more than one electron negligible, and we consider only the t\-io alternatives of zero or one random electron accompanying the l i g h t signal. Since these two alternatives are mutually exclusive, the f i n a l frequency function i s the appropriately weighted sum of the frequency functions appropriate to zero and one extra electron respectively. The former has already been found above in expression (6). The lat t e r i s found as follows: Since the random emission and the l i g h t signal are independent, the generating function of their sum i s found as the product of the individual generating functions, v i z : e N ,_N . 1 - a log Q e x p 1^ - a log Q * - 15 -from which the frequency function F(x) i s found as before via the characteristic function, to be f + 1 (x) = a" 1 e~ N e " ^ a I c (2(NX/a))£ (8) Where I 0 i s so modified Bessel function of the f i r s t kind and the remaining symbols have their former meanings. The mean i s (N - l ) a and variance (2N - l ) a ; for large N the mode tends to N a^arid distribution as a whole to normal. Combining the two distributions (<4) and (6) gives: F(X)'= Pf(X) + (1 - p) f + 1(X) M-F = (N - 1 P)a = (2N - 1 - p 2 ) a 2 Except for small N (say less than U) the distribution f(X) and f'v±;l(X) are so similar in shape that a small admixture of one with the other does not produce much change i n the shape of the distribution. In particular, the ratio , = rF/AF i s quite insensitive to the value of P and for a l l practical purposes i t can be taken as equal to 2a. The later fact was confirmed by observation at room temperature but no frequency functions were computed. Because of the similarity of the expressions (4) and (8) no corrections were made to the distributions taken at any dry ice temperature. CHAPTER 3 Experimental Technique 3.1 Scintillators used (1) Liquid S c i n t i l l a t o r : The l i q u i d s c i n t i l l a t o r used i n the present study was NE-219 supplied by Nuclear Enterprises, Winnipeg. This highly purified monoisopropylbiphenyl employed in NE-219 i s relatively free from oxygen-quenching effects normally encountered i n liquid s c i n t i l l a t o r s . This i s a li q u i d s c i n t i l l a t o r of low v o l a t i l i t y and low toxicity and no special sample preparation i s required. This gives about 55% of the pulse height from an Anthrecene crystal phosphur under similar circumstances and has the wave length of maximum emission of 4-300 A.U. The monoisopropylbiphenyl based liqu i d s c i n t i l l a t o r gives counting efficiencies of &3%, 78% and 31*5$, for l 1 ^ , p^ 2 and C1^ beta radiation respectively. Liquid s c i n t i l l a t o r s were contained i n glass cells and were stored i n clean dry sealed containers i n the dark. (2) Plastic S c i n t i l l a t o r : Plastic scintillators were used in the present work for alpha-radiation and as well for gamma-radiation and were NE-102 of 1% inches diameter and of various thicknesses. Nail MOUNT FOR ALPHA RAYS (3) Inorganic S c i n t i l l a t o r s : The inorganic sc i n t i l l a t o r s used were sodium iodide activated cylindrical crystals of 1, if-, and 2 inches i n size, obtained from Harshaw Chemical Co. These were used for the study of resolution. For comparison with the organic s c i n t i l l a t o r mainly the 1 inch size was used. A two-inch crystal was cut i n the steps of .5, 1.0, 1.5, 2.0, 2.5, 3«0, 5.0 cms. to study the resolution with different crystal sizes. Also one thin strip was prepared for the study of resolution in Nal (Th) with alpha-particles. 3.2 Assembling technique for S c i n t i l l a t o r s : (1) Liquid S c i n t i l l a t o r Assembly: . The l i q u i d s c i n t i l l a t o r assembly used for NE-219 i s shown in Fig. i . A container of the size Ig- inches diameter and 1 inch depth was made out of pyrex. For the reflector i t was covered with a thin sheet of aluminium along the inside wall and a circular disc of aluminium with 1/8 inch hole i n the centre for the Alpha source was used at the top. These discs were always prepared specially according to the depth of the liq u i d s c i n t i l l a t o r . The container was f i l l e d with li q u i d 1% inch deep or less, depending upon the requirements, and was coupled to the photomultiplier with D.C. 200 silicone o i l . (2) Plastic S c i n t i l l a t o r Mounting: The plastic s c i n t i l l a t o r ME-102 was 'coupled to the photomultiplier with D.C. 200 silicone o i l . At the top and along the sides i t was covered with a thin sheet of aluminium with a hole of l / 8 inch i n the centre of the top for an alpha source. For gamma radiation the Nal MOUNT FOR GAMMA-RAYS 18 -plastic crystal was mounted in an aluminium container with magnesium oxide as a packing medium for the reflector. (3) Sodium Iodide mounting technique: Since the sodium iodide crystals are deliquescent, any arrangement for mounting the crystal must provide perfect protection against water vapour present i n the a i r . R. K. Swank (1952) has reported a method of mounting crystals using the highly efficient, diffuse reflecting properties of powdered magnesium oxide. This technique requires dry mounting of the crystal even free from the mineral o i l i n which they are stored, since the reflecting properties of the magnesium are destroyed i f o i l i s present. For these reasons sodium iodide crystals were mounted i n a "dry box" in which there were arrangements for removing the moisture from the atmosphere by circulation of the ai r present over a drying agent such as phosphorous-pentoxide. To start with, the crystal i s rough polished outside the dry box. This i s carried out by using emery powder and mineral o i l . The crystal i s then transferred to the dry-box, wiped free of o i l and then ground through successively finer grades of abrasive u n t i l a high polish i s achieved. I t i s then mounted i n a sealed aluminium container with a l/32 inch pyrex window and removed from the dry box and attached to the photomultiplier using bleached vaseline as optical coupling. A typical mount for Nal crystals i s shown i n Fig. 2. For the sodium iodide strip used for studying the resolution i n the case of Alpha radiation, the mount i s shown i n Fig. 3» LIQUID SCINTILLATOR NE 219 ASSEMBLY. COOLING A R R A N G E M E N T F i g . 4 F O R P H O T O M U L T I P L I E R - 19 -In addition to the above technique the following conditions must be met while mounting the sodium iodide crystals: (a) The mount must be air-tight as well as light-tight so that no moist a i r can go i n . (b) The optical coupling to the photo cathode of the photo-multiplier must be good taking care that no a i r bubble i s present in'-betweea. (c) The fluorescent l i g h t must be extracted with a uniform high efficiency from a l l parts of the crystal. (d) The X-Rays must enter the crystal with l i t t l e absorption and scattering. (e) Aluminium f o i l with a 1/8 inch hole on the top of the Nal strip was used i n the case of Alpha-particles, instead of MgO reflecting powder. 3.3 Arrangement for cooling the photomultiplier The method used for cooling the 681OA photomultiplier, for which the noise spectrum at dry ice temperature has been studied, i s shown in Fig. 4« For the cooling of the photo-cathode a simple barrel made out of Bristol board of about two inches diameter f i l l e d with powdered dry ice was used. The noise spectrum was taken as the photo-cathode was cooling down. For the cooling of the photomultiplier i t s e l f a similar belt surrounding the photomultiplier body f i l l e d with dry ice was used and the noise spectrum was again taken as described in Chapter 4. S O U R C E H O L D E R O-RING S O U R C E i ECCENTRIC N E - 2 1 9 / O - R I N G Fig. 5 20 3-U Alpha Source Holder for source position adjustment with  the liquid scintillator The checking of the uniformity of NE-219 was achieved by designing an alpha-particle source holder as shown in Fig . 5 so that the source can be moved in any position in the liquid scintillator by means of two eccentric rotating seals. The whole body of the holder rotates and the source can be placed at any radial distance by rotation of the upper seal. This arrangement was used for scanning the photo-cathode. Also the distance of the course from the photo-multiplier can be varied by moving the rod carrying the source. 3.5 Radiation Sources (1) Alpha Emitters: on n The alpha particle source consisted of Po^ deposited on a plane polished silver fo i l of 1/8 inch diameter from the solution of Polonium in 0.5N hydrochloric acid. This thin fo i l was attached to one end of a glass rod of about the same diameter and about 3 inches long. The glass rod could be mounted on the source holder rod. This source of 5.3 Mev alphas has negligibly small thicknesses as measured on the British Columbia magnetic analyser by Mr. S. Smith. For the study of transfer effects within the liquid scintillator 210 NE-219 a small alpha particle source was made by depositing Eo on the end of length of silver wire 0.030 inches in diameter for a distance of about 1 m.m. (2) Gamma Sources: Gamma Sources used were C o 5 7 , H g 2 0 3 , C s 1 3 7 , N a 2 2 , C o 6 0 , Z n 6 5 LIGHT PULSER | \ -OIJJF IOOK rvV \An 50K •5V - 21 -and RaTh 2 3 2 with energies .123; .273; . 6 6 2 ; .51 and 1.28; 1.17; 1 . 3 3 ; .511 and 2.62 MeV respectively. (3) Light Sources: Pulsed l i g h t signals were i n i t i a l l y provided by a neon lamp pulser of the type described by Hickock and Draper (1958) and by Garlick and Wright (1952) but t h i s proved rather unreliable and a pulser using P h i l l i p s DM 160 '(6977) - (Prescott and Linquist (1961)) was developed. The DM 160 i s a voltage indicator tube. I t i s a d i r e c t l y heated subminiature'tube and i t has a phosphor coated anode (Jedec, P-15) which i s dark when the tube i s biased beyond the grid bias but which glows green for negative bias l e s s than about 3V under the recommended operation conditions, i . e . plate voltage 50V, filament voltage 1.5 as shown i n F i g . 6. The i n t e n s i t y of l i g h t output i s a function of grid voltage and may r e a d i l y be varied by changing the same. Normally i n operation; the tube i s biased beyond cut-off and pulsed on by means of positive pulses of a r b i t r a r y shape of r i s e time 0.05 JO-sec applied to the g r i d . The measured r i s e time at the cathode of the photomultiplier was about 1/U. sec. The flasher continued to operate s a t i s f a c t o r i l y (though with "reduced l i g h t output) with square grid pulses even as narrow as 0.1 f\ sec. Viewed end on at a-distance of 25 cm. and with g r i d pulses 2 ft sec. wide the output pulse height was variable and stable, over a range of pulses barely distinguishable from tube noise to pulses equivalent of a one MeV electron i n Nal (Th)• Larger pulses can be obtained by increasing" the plate voltage. Over a period of hours the flasher was at l e a s t as stable as the associated Photomultiplier assembly. B L O C K - D I A G R A M O F k lPECTRCTNlCS. H.T. power Supply Preamp. Power Supply Scope C B.S. H> Scope B 2 5 6 Ch. K.S. Type pen Readout K.S.Power Supply Scope A I. B.M. Ty p e-writer Recorder F i g . r - 22 -In the end-on viewing position l i g h t from the filament produced no measurable increase i n Photomultiplier noise and i n any case may be reduced either by use of an optical f i l t e r or by under-running the filament, which operates satisfactorily at currents as low as one third of rated heater current. 3.6 Electronics and data recording A block diagram of the electric i . e . apparatus i s shown in Fig . 7 with this experiment designed to investigate, the following factors were carried out: (l) Noise and multiplier s t a t i s t i c s (Ii) Noise and resolution ( i l l ) Experiment verification of the shape of the ideal s c i n t i l l a t i o n line (IV) Non-exponential single electron distribution (v) Transfer effects i n organic s c i n t i l l a t o r s (Vi) Transfer effects in a sodium iodide s c i n t i l l a t o r (I) Photomultipliers The photomultipliers used in this study were RCA types 634-2, 6810, 681OA and 704-6, the latt e r one being a 5" tube. In preliminary studies, the behaviour of the f i r s t three types was found to be very similar and results are reported here only for 681 OA and 704-6. Both of these tubes offer independent control of the focussing conditions. In the case of the 704-6 two such controls are provided. The setting of tne control designated "Grid" was found to have insignificant effects for the purposes of the present investigation and was usually l e f t i n mid range. The distribution of voltages on the various dynodes was that recommended by the manufacturer for low light level applications. Signals on the anode of the photomultiplier were differentiated with a time constant of 4-0 /LLsec. and analysed in a Nuclear Data ND101 256 channel Kick-sorter as described l a t e r . C 1 1 ) H. T. Power Supply The regulated' high voltage model RE5001 AWI of North East Scientific Corporation, Cambridge, Mass., was used to maintain the highest s t a b i l i t y . Since the gain of the photomultiplier i s very sensitive to the voltage applied, a supply of exceptional long term s t a b i l i t y i s required. The above model gives reasonable results. (III) Preamplifier Power Supply This unit i s also a commercial regulated power supply, Model 25, made by Lambda Electric Corporation, College Point, New York, which was f a i r l y stable throughout the work. (IV) Oscilloscope A. B and C A was a Model 513 - A'"Tekronix" oscilloscope. It was used for the investigation of pulse shape and sizes at various points. No quantitative measurements were made with i t except for scanning of the photomultiplier with the lig h t pulser. Mostly i t was just for setting up the apparatus. Oscilloscopes B and C were of the type of RM 31 A supplied by Tekronix Inc. The scope B was used for displaying the spectrum from - 24 -the memory of ND 101 before the f i n a l print out and the scope C was connected to the "Busy signal" which shows the output pulse from the Kicksorter amplifier while i t i s analysing. (V) ND 101 256 Channel Kicksorter The ND 101 i s a commercially built unit supplied by Nuclear Data Inc. The analyser i s capable of handling pulses at the rate of 30,000 per sec. with maximum pulse size .3 volts and 30 - 40 jOjsec. long. A check was made with various frequencies of pulses, 2 K.C. - 30 K.C. and with various lengths of pulses before i t was used:linthe present studies. Also certain results i n the present studies were checked by a 100 channel Kicksorter i n our laboratory. The incoming pulse after amplification charges a condenser, this pulse stays f l a t for .2 Jfeec. then the condenser i s discharged with constant current. The oscillator starts as soon as the voltage on the condenser reaches maximum and continues oscillating u n t i l the capacity i s discharged. The number of oscillations depends on pulse height, these are then counted and fed to the memory in the proper channel depending upon pulse height. The whole spectrum can be displayed on an oscilloscope when in "Read Out" position, then i t can either be printed or penned, i.e. plotted according to the requirement. Gain settings were calibrated with a standard pulser and the relative gain as well as zero was found. The special arrangement used for the study of low level pulses i s shown in Fig. 10. CATHODE F O L L O W E R T O FOR U S E WITH 7#046 P H O T O M U L T I P L I E R . O U T - P O T 5MJI d=. 5 0 P F S T R A Y - C A P A C I T A N C E Fig.& - 25 -(VI) Preamplifiers f o r 7046 and 6810A Photomultipliers For the" 7046 photomultiplier a preamplifier with gain of unity using 417A tube was used to feed a 100 ohm l i n e transmitting the pulses from the photomultiplier. No amplifier stage was included, since i t was f e l t that pulses available from photomultiplier tube were s u f f i c i e n t l y large f o r analysing purposes as the amplifier of the ND 101 requires only .3 v o l t pulses. The preamplifier f o r the 7046 photomultiplier i s shown i n F i g . 8. A white cathode follower used for the 681OA with a 6BQ7A tube i s shownin the F i g . 9. This was also designed with gain of unity. A white cathode follower i s suitable for both negative and positive pulses and i t w i l l develop undistorted signals across a substantial stray capacity as with a positive going wave front the upper half of 6BQ7A can conduct strongly to change the stray capacity while on negative going wave fronts the capacity i s discharged by a heavy current i n lower part of the 6BQ7A. The action of a simple cathode folloxirer i s to maintain a constant potential difference between the grid and the cathode. S i m i l a r l y i n t h i s case i f , f o r example, the cathode or output voltage should be too negative the upper half w i l l pass more current developing a negative signal at i t s anode which causes the lower ha l f to pass less current. The changes i n both the parts tend to restore the cathode potential to i t s correct value. In the quiescent state with no input signal the cathode voltage of the upper hal f adjusts i t s e l f so that current i n the upper part i s the same as specified by the bias conditions i n the CATHODE FOLLOWER FOROJrSE WITH 6810 A PHOTOMULTIPLIER - 26 -lower half. This ci r c u i t can be regarded as a two tube feed back amplifier. (VTI) Differentiating time constant The i o XSfiF capacitance from anode to ground, Fig. 8, and the stray capacitance 6 fffiF and 47 K resistors i n series with anode and EHT gives differentrating time constant of about 40 Jl sec. in the anode unit of the photomultiplier which i s suitable for the ND 101 input c i r c u i t . Draper and Hickock (1958) and Swank and Buck (1952) have dis-cussed the effect of the choice of differentiating time constants on resolution; this i s important when comparisons are to be made between sc i n t i l l a t o r s of different decay times. A kicksorter records the maximum height reached by a pulse. It i s only electrons that arrive before the maximum is reached which contribute to the f i n a l signal and i f the differentiation time i s short compared with the decay time of the light pulse, then electrons arriving before maximum contribute to i t with unequal weights. The effect of this i s to give different variances to pulses of otherwise equal mean size from different l i g h t sources. If comparisons are to be $ade of absolute l i g h t outputs both these effects are important. For comparison of resolution, only the second matters. In the present experiments the joint choice of long photomultiplier anode time constant and delay line clipping ensures that no distortion of data occurs. Every electron arriving before the pulse height maximum i s reached i s recorded with equal weight and no electrons are recorded after pulse height maximum. Although the - 27 -f r a c t i o n of such l o s t electrons w i l l depend on the decay time of the l i g h t pulse, the remainder are s t i l l subject to Poisson s t a t i s t i c s and provided the pulses are adjusted to have the same mean height at the output, they w i l l have the same variance. The short clipp i n g time also minimizes the effects of "after pulsing". (VIII) Computations Computations were carried out on the University of B r i t i s h Columbia Awac I I I E computer and also on the I.B.M. 1620 at the University of Alberta, Calgary by Dr. J . R. Prescott. Means and variances were'computed from the complete data and not estimated, fo r example, from f u l l width at half height of a peaked frequency function. - 28 -CHAPTER IV Procedure and results 4*1 Noise and multiplier statistics The s t a t i s t i c a l contribution of the dynode structure alone can be inferred by studies of tube noise, a fraction of which, though not a l l , arises from single electrons f a l l i n g on the f i r s t dynode. Various authors - Morton and Mitchell (1948), Wright (1954) Roberts (1953) and Breitenberger (1955) have dealt with the statistics of multiplication in general terms. Assuming Poisson statistics for electron multiplication at each stage, Lombard and Martin (1961) have recently evaluated single electron pulse height distribution on a computer. The found peaked distributions, the location of the peak i n relation to the mean pulse height being determined by the stage gain. These authors remark that, in fact, the observed noise spectra i n actual photomultipliers resemble a decreasing exponential and are not consistant with the Poisson model of secondary electron emission. Observations leading'to a similar conclusion are reported by Allen (1950) and Baiker (i960) in a detailed study of tube noise, finds quasi-exponential noise spectra using R.C.A. tubes and in the present work monotonically decreasing noise spectra, have always been obtained, frequently almost exponential over several decades and usually showing a second component (see examples i n Fig. 11 and 12). Livessey (private communication) finds considerably more complex noise spectra, often having three or more distinguishable components, in an EMI type 6099B of "Venetian blind" structure. In so far as the mechanisms giving rise to noise i n photo-multipliers are s t i l l not properly understood (see e.g. Baicker (i960)) the only satisfactory way at present of constructing a model for multiplier statistics i s to make actual measurements of the noise spectrum under the prevailing conditions on the tube being used. Since the tube noise does not arise exclusively from electrons origin-ating at the cathode, i t i s necessary to distinguish this component from others by allowing a weak continuous l i g h t to f a l l on the cathode and subtraction to determind the noise distribution of interest. This i s discussed i n more detail in subsequent sections. In the following sections the foregoing analysis i s f i r s t tested with no special assumption about the stati s t i c s of multiplication process. Experimental observations of the frequency function of output pulses are then compared with predictions based on specific models for the multiplication process. In both these sections, the light pulse were produced by an a r t i f i c i a l l i g h t pulser described earlier. Finally, observations on actual s c i n t i l l a t i o n assemblies are related to the ideal cases discussed l a t e r . 4.2 Noise and resolution Measurements were f i r s t made with the l i g h t pulser described earlier to test the vali d i t y of expression (2) and hence of the i n i t i a l assumptions for the derivation of the s t a t i s t i c a l model outlined in Chapter II, (2.1). The mean voltage signal on the photomultiplier anode i s G = n T me/C where C i s the anode capacitance, m the multiplication of Relative variance as function of reciprocal pulse height at different focus settings curve (a) 7046 (b) 6 8 1 0 A (c) curve (a) corrected tor single-electron spectrum changes F i g . io RECIPROCAL PULSE HEIGHT—ARB. SCALE - 30 -the tube and e the electronic charge. Equation (2) becomes.: V* = G2/M2G = m e (1 + Vm)/GMQ (9) Thus provided Vm and m remain constant, VQ i s proportional l/MG . This relationship was repeatedly tested i n the course of experiments. In some scores of instances with different photomultipliers, widely different conditions of applied voltage and focus, and with cathodes f u l l y and partially illuminated, no case was found where V„ was not s t r i c t l y proportional to l/MG, see for example, Fig. 18. This result has, of course, been found by many observers. I t i s concluded f i r s t , that assumptions of Poisson li g h t production and binomial transfer are valid and second, that there was no significant contribution to the variance from "instrumental" resolution other than the photomultiplier i t s e l f . " When the pulse height itas varied by variation of the focus voltage above, however, the somewhat unexpected results displayed in Fig. 10, (Curves (a) and (b) were obtained). The most obvious effect of focus variation i s to change the transfer efficiency and with i t the mean pulse height. While the general trend for relative variance to decrease with increasing pulse height i s maintained, the curves particularly that for the 7046 tube, display some curious anomalies. For instance, in the 7046 the focus setting for maximum pulse height does not coincide with the setting for best resolution and the curve displays double values in this region. The plot for the 681OA has a non zero intercept on the V axis. This behaviour i s accounted for by - 3 1 -changes with focus setting of the multiplication and i t s relative variance. These can be assessed by measurements of the pulse height distribution of the tube; noise, or more accurately, that portion of the tube noise which originates at the photo-cathode. Fig. ( l l ) shows a set of noise spectra for the 704-6 obtained under conditions of weak continuous illumination from which noise under dark conditions has been subtracted. The change in the character of the distributions with focus setting i s quite evident. The relative variance Vm of each of these curves .is the Vm of expression (6) and the mean i s proportional to the multiplication m. The measured mean ranges from 1 2 . 5 channels at focus 84-0 V where Vm i s 1 . 7 . From expression ( 6 ) we define V Q = VQ/m(l + Vm) which should be proportional to MpT^- and this quantity i s plotted i n Fig. 1 0 as curve (c); the original curve (b) i s now transformed into a straight line through the origin. Such deviations from a straight line as s t i l l occur can be attributed to the fact that the behaviour of the noise spectra near the origin i s uncertain, to the extent that an extrapolation of 2 or 3 channels i s required i n this region since the Kicksorter does not record i n these channels at the moderately high counting rates involved here, about 30 ,000 C/sec. A similar transformation on the curve (b) F i g . 1 0 for 6 8 1 0 A tube also yields a straight line through the origin although i n this case changes in the focus setting principally affect m and leave Vm v i r t u a l l y unaltered. No satisfactory explanation for this behaviour of the tube noise with focus setting has been found. It appears, however, that the theoretical treatment - 32 -of Chapter II (2.1) i s well supported by experiment. Although the main concern of the investigating of the tube noise as such was to identify correctly the portion due to single electrons arriving at the f i r s t dynode from the cathode, one or two general features of the behaviour of the noise appear worthy of further comment. It has already been remarked tha;t most of the noise spectra display two clearly distinguishable components: a "steep" quasi-exponential at small pulse heights - see Fig. 12 - and a t a i l . Single electron pulses released from the photocathode by li g h t f a l l in the "steep" region of the distribution. In the tubes discussed here (R.C.A. 634-2, 6810A and 704.6) the t a i l typically contains about 2% of the total noise pulses at room temperature and has a logarithmic decrement between five and ten times larger than that for the smaller pulses. It i s important to establish whether this t a i l i s associated with illumination of the photo-cathode, for although these pulses are of relatively low abundance, they extend to quite large pulse heights and could significantly increase the relative variance Vm. Such pulses are not, i n fact associated with illumination of photo-cathode for as curve (b) in Fig. 12 shows, only the steep portion of the distribution i s increased by permitting l i g h t to leak on to the photo-cathode. Further, the temperature dependence of the t a i l makes i t doubtful whether i t even originates on the photo-cathode. When the cathode i s suddenly cooled by dumping a load of dry ice upon i t , the steep component immediately begins to decrease, reaching an equilibrium at about 4-0 minutes. Single electron spectra in a 7046 pn otomultiplier with focus settings as a parameter. - 33 -The t a i l , on the other hand, remains almost unchanged for about 15 minutes. Similarly, cooling of the body only of the photomultiplier f i r s t produces a reduction in the t a i l , while i t i s the steep part of the noise distribution that now lags behind. When the cathode of a photomultiplier (already cooled around the body) i s then cooled, no significant further reduction of the t a i l occurs, although the "steep" component i s almost suppressed. Dr. J . B. Warren suggested that i t could be residue gas effect. Curves il l u s t r a t i n g temperature effects are also shown in Fig. 12, curves (b) and (e). Changing the focus setting also changes the relative proportions of the two components in the noise. At settings remote from those which could be regarded as normal operating conditions (high gain, good resolution), the t a i l becomes much less prominent and eventually disappears. Under these conditions (see Fig. l l ) the single electron spectrum i t s e l f changes radically in character. These observations suggest that the t a i l of the noise distribution originates somewhere inside the photomultiplier, i n front of the f i r s t dynode, perhaps on the walls of tube or i n the focussing structure i t s e l f . Baieker (i960) has observed essentially similar behaviour i n a variety of R.C.A. photomultipliers. He interprets the t a i l as due to bursts of more than one electron from the photo-cathode. While the evidence does not perhaps conclusively eliminate the photo-cathode as the source, i t seems certain that i t i s not produced by l i g h t . By f i t t i n g the distribution curves discussed i n the theoretical section to the t a i l one Noise spectra in a 70 4 6 photomul t ip l i e r showing the ef fects coo l ing and c o n t i n o u s i l l u m i n a t i o n of the p h o t o - c a t h o d e . (a) room temperature phototube dark (b) " " weakly i l l u m i n a t e d (c) tube body c o o l e d to dry i c e t e m p e r a t u r e (d) en t i re tube c o o l e d F i g . & CHANNEL NO. - 3A -can estimate that at least five and possibly more electrons would be required. A few bursts of low.multiplicity (2 or 3) might be expected to occur on a basis of chance coincidence. In fact, i f an increasingly strong l i g h t leak i s used, the noise spectrum changes i t s shape in a manner quite consistant_with the increasing occurence of events of multiplicity of two. Even the strongest l i g h t source used, however, fa i l e d to affect the t a i l i t s e l f in any way. So f a r , then, as noise is concerned, a satisfactory empirical model for the stat i s t i c s of multiplication of single electrons released from the photo-cathode by l i g h t can be found and such a model gives f a i r l y good agreement. 4-.3 Non exponential Single-electron Distributions Although the spectrum of single electron pulses can be made f a i r l y accurately exponential in the case of the 704-6 photomultiplier, this i s an a r t i f i c i a l state of affairs and i n practice the frequency function of single-electrons exhibits some degree of curvature, although i t usually approximates to a simple exponential. Several alternatives for the single-electron frequency function have therefore been considered and i t seemed reasonable to choose models that contain the simple exponential as a special case^ viz f(x) = p a - 1 exp (-x/a) - ( l - p) (ab)- 1 exp (-x/ab) (9.1) f (x) = a" 1 exp (-(x/a]) / r ( l j + ) (9.2) f(x) = x^ exp (-x/a)/r( - l ) a ^ * 1 (9.3) The fi r s t two distributions reduce to simple exponentials i f p or j is unity j they are more concave than the simple exponential for p or and less concave for p or "^^1. In the present experiments, either (9.1) or (9.2) was sufficient to represent the observed single-electron distributions, and no case was found where the distribution could be satisfactorily represented by (9.3). Although the characteristic functions for the system as a whole can be written down for these models, the corresponding Fourier transforms have not so far been found. Hox^ ever, the required frequency function can be found with sufficient accuracy by Edgeworth's series expansion (Cramer (194-6)) as follows: The generating function for the output is (in the notation of Chapter 2) G(0) = exp ( -N) exp (lMGm (8 ) ) where Gm( Q) is the generating function for the multiplication process. The amount generating function is obtained as usual by replacing log G with Q . M(0) = exp (-N) exp (NJLjS)) Since G (log 8 ) = M (G), the moment generations function for m m the multiplication process i s , taking logs, Log M(Q) = -N + NMJ S)) Since the derivatives of Mm( Q) are the moments (M) about the origin of the single electron distribution itself the cumulants ^ y of the distribution are then found as the derivatives of log M( 0) with respect to $ , evaluated at 0 = o. It is seen that the cumulants are just N Pulse height distributions for a pulsed light source with a 6810A photomultiplier and a non-exponential single electron distribution; fitted as described in the text. F i g . ^ - 36 -times the moments about the origin of the single-electron distribution. If we denote the y th central moment of the complete distribution by My and the mean by JLX, then: >1 =/Ay ; %2=pi2', >3 = A 3 ; *4 =/A4. - - 3 > l a Introducing the standard variable z = (x )Ar , we can write the required frequency function (to terms of the order N _ 1 ) as: f(z) =0(°)(z) -.42 / 3 ) ( Z ) - 1 ••A-3).04(») - Jf ( 1 3 ) 2 ^ 6 ) ( Z ) + where z) i s the i t h derivative of the standard normal distribution. Using distributions (9»l) and (9*2), the vth cumulants are respectively: The parameters in these distributions are best found by t r i a l and error or direct computation from the single electron frequency function since the form of these functions does not encourage estimation of the parameter by the usual, e.g. maximum likelihood or least squares. In the course of the search for suitable models to describe non-exponential single-electron distributions, an interesting and useful approximation was discovered. Equation (6) can be rewritten G 2 / * M G = M(l+Vm) where M i s the mean pulse height for a single electron. For a simple exponential this becomes M(fifa& = 2a where a the mean single-electron distribution i s clearly non-exponential, by interpreting the ratio ^"G 2/2JUii as i f the s t a t i s t i c a l model for multiplication were a simple exponential, viz interpreting M as equal to ^G^/^ftG. This i s equivalent to.interpreting a in expression (6) merely as a scale factor given by a = ^ R G 2 / 2 ^ Q . Suppose, for example, that Vm i s less than unity. The approximation w i l l then clearly underestimate M and overestimate N = ^ Q / M > Pulsed height distributions for pulsed light source with simple exponential single electron distribution in a 7046 P.M. solid curves are computed from expression (4) in the text N values are given for each curve. ^ 1 \ 1 — I 10 100 IOOO PULSE HEIGHT ARB. LOG UNITS. the mean number of electrons at the f i r s t dynode. However, this i s exactly compensated for by the greater variance of :the exponential distribution and the resulting frequency function turns out to represent the actual frequency function to a remarkable degree of accuracy. As example i s shown i n Fig. 13 where three curves for the 681OA are shown f i t t e d in this way. The single-electron distribution for this case was of the type (9.2) with J = 2.1 and Vm = 0.53. The above procedure guarantees, of course, that at least the mean and variance are correct. In the particular examples shown i n Fig. 13, the third moment was within 10% of i t s correct value and the fourth moment within 20%. This happy accident has much relieved the tedium of calculating distributions using Edgeworth's series, ( i t s e l f an approximateion) although there i s no guarantee that i t w i l l necessarily work i n a i l cases. 4«4 Experimental verification of the shape of the ideal  S c i n t i l l a t i o n l i n e : The relations (4) and (5) i n the f i r s t chapter were tested using 7046 photomultiplier. By means of the focus control the .noise spectrum was adjusted to be as nearly a simple exponential as i t was possible to make i t . By a graphical test the distribution sas exponential over three decades. Because of loss of counts i n the f i r s t four kicksorter channels the noise spectrum was not certain in the region near the origin but i t was assumed to be exponential. Light pulses were produced by the pulse generator described i n the section "Electronics" operating at 1000 c/sec an average of less than ten electrons at f i r s t dynode, the tube pulses become confused with the noise of the photomultiplier i t s e l f . To separate the two sources of Variance vC mean for pulsed light signals with .a simple exponential single electron distribution. 7 046 photomultiplier. UJ O cr < > .A 1004-2 10 + 10 Fig. is. io _ i _ 100 1000 N 10 100 I O 0 0 channels M E A N P U L S E H E I G H T - 39 -signals, the l i g h t signals -were admitted to the kicksorter 4 by a gate opened by the same signal that drove the ligh t pulser, as shown i n F i g . 16. In this way except for those pulses occuring by chance during the time the gate i s open, noise is excluded. With a photomultiplier operated at room temperature, these randomly-gated noise pulses nevertheless constitute a problem, for although only 4.5$ of the noise pulses themselves occur while the gate i s open, the noise count-rate i s high and i n fact, some 20% of the gate pulses are accompanied by an analysed noise pulse. In terms of the complete assembly, consisting of the ligh t source, the photomultiplier and analyzer this means that a significant fraction of the signals due to l i g h t pulses are accompanied by a random noise pulse within the resolving time of the pulse height analyzer. To reduce the contribution of random noise pulses to a negligible level the photomultiplier was cooled to dry-ice temperature. Under these conditions less than 3% of gate pulses are accompanied by "an analysed noise pulse. A second function of the gate signal i s to give an estimate of the proportion of cascades that f a i l to start. For each run, the number of gate pulses was recorded on a scaler and compared with the number of pulses actually analysed. If k i s the fraction of pulses lost and i f we denote by subscripts the measured mean and variance, then the true mean and variance, M. and respectively, are given by: A = (1 - k) JUi o-2 = ( l - k ) ( ^ 2 - K ^ ) and the relative variance V by: V = (v x - k ) / ( l - k) Figure 15 shows the variance plotted against the mean for l i g h t signals. Logarithmic scales are used to cover the wide range of pulse LlGHT-PULSEi i pm Amp. A PULSE- GENERATO NO 101 ANALYSER Amp A Disc. B • 4# -V COINCIDENCE-GATE F i g . & - 40 -heights and variances encountered. At low mean pulse heights the correction has been made for cascades that f a i l to start and the extent of this correction i s indicated by the arrows attached to some of the experimental points. The points are f i t t e d by a straight line of slope unity, showing that the variance i s proportional to the mean, the proportionality constant being 11.4 1 0.1 channels, corresponding to a value -of a of 5*1 Z 1% channels. This i s to be compared with the value 6.0 * 0.2 channels obtained for a from the slope of the noise distribution. This agreement must be considered satisfactory, since measurements of a based on the noise distribution were reproducible only to about 3% during the course of the series of measurements. A scale of values of N using a = 5.7 i s shown in the figure. Actual frequency functions were calculated using 5.7 channels for the value of a and normalising the areas to the total count. These distributions are shown in F i g . 14. It i s worth observing that curves are not f i t t e d individually except insofar as the appropriate value of N for each was found from the relation = Na. Here again the agreement between the observations and the computed distributions i s very satisfactory. In F i g . 14 the logarithmic abcissa scale conceals the skew character of the distributions which i s quite evident on a linear plot even for large values ofiTN. Thus a reasonably satisfactory set of s t a t i s t i c a l models has been found to describe the performance of the photomultiplier in a s c i n t i l l a t i o n assembly. In particular, expression (4) can probably be regarded as a sufficiently good representation of the shape of the ideal s c i n t i l l a t i o n line even in cases where the simple exponential model from which i t was derived i s not applicable. It i s nevertheless "ideal" i n the sense that R E S O L U T I O N D A T A F O R LIQUID S C I N T I L L A T O R 6 ro O x 5 UJ O < 4 or < > LU ^ > 0 r— < _ l L U p Inset; Coiintours of equal response on a ver t ica l sect ion of t h e scintillator; P H O T O C A T H O D E Lower line:- contribution from photomultiplier U p p e r line: external alpha par t ic les Oban circle: a lpha-par t ic les with in the scintillator (single sample) Triangle' alpha-particle within the scintil I ators (entire volunVe) Fig. i7\ R E C I P R O C A L P U L S E H E I G H T - ARB 5 . U N I T S - 41 -i t takes no account of variations in transfer probability. However, in some cases i t does f a i r l y adequately represent the line shape i n practical s c i n t i l l a t i o n counters, for example, i n the case of alpha-particle excitation of a li q u i d s c i n t i l l a t o r and perhaps also for gamma rays of very low energy i n sodium iodide. It i s just i n these cases that the skew character of the distribution i s significant. For large signals the distribution i s effectively Gaussian. 4.5 Transfer effects i n an organic Sc i n t i l l a t o r In order to estimate the magnitude of transfer effects contribution, described i n Section 2.3> Chapter I I , the spread of pulse heights from different regions of a li q u i d s c i n t i l l a t o r , HE 219, was determined by immersing in i t a small alpha-particle source. To isolate transfer effects specifically, the response of this s c i n t i l l a t o r to an external source of alpha particles was f i r s t determined. For this purpose i t was incorporated in a glass c e l l 1.5" in diameter, f i l l e d to a depth of .0.7 cm. and optically coupled directly to the photo cathode of a 681OA photomultiplier. An aluminium reflector surrounded the ce l l inside the walls and a plane reflector was held just above the surface of the l i q u i d . Alpha-particles were incident on the free surface of the li q u i d through a"small hole i n the upper reflector, and were varied in energy by adjusting the distance of the source from the li q u i d surface. The hole in the reflector was kept sufficiently small (3 mm in diameter) to eliminate the need for solid angle corrections. I t was hoped that by confining the alpha-particle to a very small area on the top surface of a relatively large volume of s c i n t i l l a t o r that geometrical li g h t collection factors would be the same for a l l s c i n t i l l a t i o n s . The alpha-particle source - 42 -used was prepared as described earlier i n Section.3«5, Chapter I I I . In order to avoid corrections for alpha-particle range straggling, the minimum alpha-particle energy used was about 3 MeV where the contribution from straggling to the relative variance i s about" 3 x iCT 4 (Evans) (1955). The results for alpha-particles on l i q u i d s c i n t i l l a t o r are shown i n Fig. 17 (upper curve) where relative variance i s plotted'against reciprocal mean pulse height. The points are consistant with a straight line through the origin. For comparison purposes a corresponding curve for the a r t i f i c i a l l i g h t pulser, representing "ideal" resolution i s shown. It i s clear that the relative variance of the i i q u i d s c i n t i l l a t o r i s some 25$ greater than for "ideal" pulses of similar mean pulse height. Because of the source geometry i t seems unlikely that this increase could be due to variations in the l i g h t collection efficiency. Whether i t i s a consequence of the use of the heavily ionising alpha-particles or i s a property of the s c i n t i l l a t o r in the sense of an intri n s i c resolution, unfortunately cannot be determined from the present experiments. Transfer effects within the li q u i d s c i n t i l l a t o r were then investigated by f i l l i n g the s c i n t i l l a t i o n c e l l to a depth of 1 - |- inches and immersing i n i t a small alpha-particle source that could be moved from place to place as described by Cummins and Delaney (i960). This source 210 was Po deposited on the end of a length of silver wire, 0.030 inches in diameter, for a distance of about 1 mm. The source wire passed through a hole in an oversize, plane, aluminium reflector resting across the top of the c e l l . The reflector was constrained to allow late r a l movements of the source and completely covered the top of the s c i n t i l l a t o r c e l l at a l l times. - U3 — With the source within the s c i n t i l l a t o r and at the geometric centre of i t , the point represented on Fig. 17 by an open circle was obtained. Because of the small size of the source, with a single reading of this type, the transfer variance should be small and the point should l i e very close to the line obtained with external alpha-particle excitation, even though the source i s now inside the s c i n t i l l a t o r . This i s seen to be the case. Readings were now taken of mean pulse heights i n planes parallel to the surface of the s c i n t i l l a t o r from the base of the c e l l u n t i l the source began to emerge through the surface. The extreme variation between different points i n the s c i n t i l l a t o r was about 20% and the relative standard deviation 1$ (relative variance 17 x 10~^, 180 sampling points). An independent check of this figure was obtained as follows: A strong Polonium source was prepared and.left in the s c i n t i l l a t o r for 24-hours. In this period, the l i q u i d became f a i r l y heavily contaminated. The source was then removed and the pulse spectrum from the contaminated s c i n t i l l a t o r observed. This contains contributions from a l l parts of the volume and should have a correspondingly increased relative variance. This experimental point i s shown as a triangle i n Fig. 17, i t l i e s above the former curve by 18 x 10"^ which i s to be compared with 17 x 10""^  obtained i n the three dimensional survey. This agreement may be regarded as very satisfactory and i t appears that these figures provide a reasonable estimate of the transfer variance for s c i n t i l l a t i o n eventsrrandomly distributed throughout the s c i n t i l l a t o r volume. I t was observed i n the survey that the contours of equal response - 44 -on a horizontal section of the s c i n t i l l a t o r followed more or less closely the shape of the sensitivity contours of the photo-cathode obtained by scanning a small l i g h t spot over i t s surface, although the absolute variation from point to point was much less. It seems clear that greater uniformity of the photosurface would have reduced the value of the transfer variance significantly, at least i n the present s c i n t i l l a t o r . Making the cathode less uniform by adjustment of the focus control certainly worsened by resolution, even when correction had been applied for the consequent reduction in mean pulse height, sk vertical section through the s c i n t i l l a t o r showing contours of equal response i s shown as an inset to Fig. 17. It i s interesting to observe that in a l l except the l a s t measurement (contaminated scintillator) the resolution i s s t i l l close enough to ideal for the distribution curves to be noticeably skewed and for the "ideal" line shape of section 2.2 to be able to give a good f i t to them. When the s c i n t i l l a t o r i s used for the detection of gamma-rays, transfer "effects are again expected to be important since ionising events take place throughout the volume of the s c i n t i l l a t o r . The above survey was used to interpret the resolution observed for uncoliimated gamma-rays of energy 0.511, 0.622, 1.28 and 2.62 MeV i n the same s c i n t i l l a t o r , incident on the top surface. The interpretation of resolution for gamma-rays detection in an organic s c i n t i l l a t o r i s not so straight-forward as i t i s in sodium iodide, for instance, because there i s no full-energy peak and the resolution must be estimated from the shape of the Compton edge. The following procedure was adopted: Electron, energy distributions were computed for each energy: Gaussian"resolution functions Resolution data for gamma-rays in liquid scintillator Solid circles: gamma-rays resolution Opencircles: gamma-ray resolution corrected for transfer effects^ Lines A and P are respectively alpha-particle and photomultiplier taKen from fig 3 RECIPROCAL PULSE HEIGHT — ARB. UNITS were then folded in with a range of values of relative standard deviation at the Compton edge. Graphs were then constructed showing the ra t i o : abcissa at maximum/abcissa at half height versus relative standard deviation. The same ratio was found for the experimental Compton distributions and from the graph the appropriate relative standard deviation. The lat t e r figures, converted to relative variance are shown in Fig. 18 (solid circles) together with the "ideal" photomultiplier resolution l i n e , and the alpha-particle resolution curve discussed earlier. The experimental errors are unfortunately large here because of the method of estimation used and are the standard deviations of 4 measurements for the three upper points and 2 measurements for the lowest (2.62 MeV). The points for gamma-ray resolution l i e well above the curve for the same sc i n t i l l a t o r under alpha-particle excitation and the increase may be attributed to variations in the transfer probability. The three-dimensional survey of the s c i n t i l l a t o r was used roughly to estimate the expected contribution to the relative variance from such transfer effects as follows: Signals at the Compton edge arise from electrons projected straight-forward. ' The pulse heights from the three dimensional survey were therefore averaged over a distance equal to the range of such an electron parallel to the direction of incidence (perpendicular to the surface of the s c i n t i l l a t o r ) . No allowance was made for multiple scattering or edge effects. In the case of the 511 and 662 MeV gamma-rays the average was weighted to allow for the exponential absorption of the radiation. In an organic s c i n t i l l a t o r of this size, multiple interactions can.be neglected. The contributions from transfer effects computed i n this way were - 46 -subtracted from the experimental relative variances to give the points shown by open circles i n Fig. 18. Although these are more consistant with the resolution line for alpha-particles on the l i q u i d than that for the photomultiplier alone, the latter line l i e s well within the 95% confidence limits for the linear regression f i t to the data and no con-clusions can be drawn from these measurements about the existence or otherwise i f a true i n t r i n s i c resolution for this s c i n t i l l a t o r suggested by the alpha-particle measurements. However, i t i s correct to say that the major portion of the relative variance i s accounted for by a combination of photomultiplier sta t i s t i c s and transfer effects. Edge effects and scattering, neglected here, w i l l produce some further worsening i n resolution. Similar observations were made for-an Ne 102 plastic s c i n t i l l a t o r of the same dimensions, packed in magnesium oxide. Within the (rather large) experimental errors the gamma-ray resolution was the same as for the l i q u i d . Alpha-particle resolution was slightly worse (about" 8% in relative variance)"but this could be due to surface defects in the solid plastic.; 4.6 Transfer Effects i n Sodium Iodide The results for gamma-ray detection in the organic s c i n t i l l a t o r shown i n Fig. .18 are very reminiscent of similar curves for gamma-ray detection in Nal(Tl), (Kelley et a l , (1956), Bernstein (1956), B i s i and Zappa (1958), see also Fig. 19). Not'only i s the trend'of the curves with energy similar, but they l i e about the same distance above the line of ideal resolution. This originally suggested that the increased l i n e -width in sodium iodide might contain a large contribution from transfer variance and not be.due entirely to an intri n s i c resolution of the type Resolution data for gamma-rays in Nal. experimental points are solid circles. shown xio RECIPROCAL ENERGY—MEV - 47 -suggested by Kelley et al (1956). Resolution was therefore measured for gamma-rays i n a sodium iodide crystal, 1-^ -" diameter, 1-^ -" high, mounted on a 681OA photomultiplier and the results are shown i n Fig. 19 (solid c i r c l e s ) . In this instance the ordinate i s the square of the relative f u l l width at half-height of the full-energy peak and not the true relative variance used elsewhere i n this Thesis. In order to find the la t t e r i t i s necessary to have the full-energy peak sufficiently well separated from the rest of the spectrum that a plausible estimate can be made of i t s shape i n the t a i l s of the distribution which have a large influence on the variance! Although an attempt was made to find the true relative variance by estimating the shape of the curve, the results had an unacceptable scatter and the attempt was abandoned. If a l l the factors, that contribute to the line width were s t r i c t l y Gaussian, N 2 would be related to the true relative variance, by N 2 =5.56 V^. Although this w i l l not actually be the' case, as-the knowledge of these functions i s insufficient and Gaussian assumption was used, the quantitative error introduced probably does not exceed the other uncertainties and the qualitative arguments are not affected. The points in Fig. 19 are the means of six determinations and the errors are sample standard errors. The gamma-rays were of energies (in MeV) 2.62, 1.28, 1.11, 0.662, 0.511, 0.279 and 0.123 and they were uncollimated. The curve i s essentially similar to those published by Kelley et a l (1956), Bernstein (1956), and B i s i and Zappa (1958), except that i t does not extend to such low energies. The line P represents the contribution of the photomultiplier i t s e l f to N and'was obtained using the l i g h t pulser. - 48 -In addition to the contribution of the photomultiplier, there i s the contribution to N 2 from the intr i n s i c resolution discussed by Zerby, Meyer, and Murray (l96l), and Iredale (l96l). These two investigations agree well at energies above 0.5 MeV but areiin serious disagreement at lower energies. In allowing for intr i n s i c resolution the results of Zerby et a l (l96l) as being much more detailed and embodying fewer approximations were preferred. These authors publish curves showing intrinsic resolution as a function of energy and crystal size. It i s zero below about 100 KeV where the full-energy peak results almost entirely from a single photo-electric event, rises to a maximum of 5% relative f u l l width at half-height at about 4OO KeV and then f a l l s more slowly'to about 1% at 3 MeV. No . significant variations are found for differing conditions of source-crystal geometry, and for crystals greater than one inch in diameter and height, only a very slow change with crystal size i s observed and then only at gamma-ray energies over 0.5 MeV. The in t r i n s i c resolution was taken from Fig. 4 of the paper of Zerby et a l (1961,) with a reduction for crystal size from their Fig. 5. The effect of subtracting this in t r i n s i c resolution from the experimental curve i n Fig. 19 i s shown by the dashed curve, A. A further source of line-width, discussed by Iredale (I96l), i s contributed by fluctuations i n secondary electron (delta-ray) production. He has computed this contribution for a crystal slightly larger than that used i n the present experiments. It i s assumed that these figures are valid for our smaller crystal and they have been subtracted - 49 -from curve A in F i g . 19 to yield curve B. It i s unfortunate that Iredale'sir-results do not extend below 200 KeV and i t would be particularly interesting to know the behaviour at lower energies. Finally, by further subtraction of the contribution from the photomultiplier, curve C i s obtained. It i s clear that a major fraction of the line width remains to be accounted for. We interpret this as due to variable transfer probability. Curve C then represents (apart from a'constant factor) the transfer variance as a function of gamma-ray energy. It i s instructive to discuss the expected form of this curve, particularly at low energies for which experimental data i s lacking: at very low gamma-ray energies, the absorption coefficient i s sodium iodide i s so large than an over-whelming fraction of the s c i n t i l l a t i o n events take place in the top surface layers of the crystal. Even at 100 KeV, for instance, over 90% of the interactions take place i n the top 0.5 cm., mostly by the photoelectric effect. For low' energies, therefore, a l l gamma-ray energies should share a common transfer variance represented by the relative variance i n the l i g h t collection efficiency across the top surface of the s c i n t i l l a t o r , with a rise beginning i n the region of 100 KeV as the gamma-rays begin to.penetrate more deeply into the crystal. If the photoelectric effect were alone responsible for gamma-ray absorption, we should then expect the curve to continue to rise u n t i l , for a crystal of the size used here, i t levelled off at energies above about 700 KeV at a value equal to the transfer variance taken over the whole volume "of the crystal. In the region where the rise would begin, however, multiple events ;s!bart contributing to the - 50 -full-energy peak. The light production i s now the weighted average of at least two samples from the distribution of transfer probabilities, i t i s a well-known s t a t i s t i c a l result that such an average has a reduced variance. In this, case, the variance i s reduced by a factor equal to the sum of the squares of the relative energy releases. At higher energies, an increasing fraction of the full-energy peak, i s due to secondary and higher interactions. At 662 KeV, for instance, roughly 75% of the counts in the f u l l energy'peak are due to multiple events as shown by Iredale (1961) , Bergerand Doggett (1956) , and Lazeu and Davis . (1956). The transfer variance i s expected to reach a f l a t maximum and then begin to f a l l as the above averaging process takes over. In the present crystal the maximum appears to l i e at about 250 KeV. At this energy the Compton and photoelectric cross-sections are equal." At the highest energy used here, 2 .62 MeV, the contribution of multiple' interactions i s dominant. As ah example, i f the f u l l energy peak at this energy were produced exclusively by three-fold events regarded as random samples, (e.g. pair plus the capture of both annihilation quanta) the transfer variance would be reduced to about 4-0% of the value for the crystal as a whole. Although the situation i s actually a good deal more complex than this simple i l l u s t r a t i o n , this i s of about the magnitude of the observed f a l l in transfer variance between 250 KeV and 2 .62 MeV. While i t is doubtful whether any real physical meaning can be attached to an extrapolation of the transfer variance curve to even higher energies because of the increasing importance of edge and escape effects, i t i s nevertheless worthwhile to remark that even i f these latter could - 51 -be eliminated, the curve would s t i l l not extrapolate to zero at . i n f i n i t e energy. Only i f every event produced l i g h t uniformly through-out the volume of the crystal would this occur, and the discrete, character of the processes by which gamma-rays convert their energy to l i g h t precludes this p o s s i b i l i t y . B i s i and Zappa (1958) have also analysed the dependence of l i n e -width on energy. In a stati s t i c a l , sense their argument i s formally identical with that presented here. However, the data on intr i n s i c resolution were not abailable to them and their method of analysis implies that the transfer variance for a single interaction i s indepen-dent of energy. While this may very well be true for very low gamma-ray energies, i t does not seem l i k e l y that a l l energies as the survey in the l i q u i d s c i n t i l l a t o r , discussed in Section 4«6 shows. Here considerable point-to-point variations were observed in a s c i n t i l l a t o r of the same geometric size on the same photomultiplier. Insofar as the variations observed paralleled closely the non-uniformity of the photo-cathode a similar variation may be expected within the sodium iodide crystal, although because of i t s differenteoptical properties (e.g. surface f i n i s h , reflector), an exact quantitative correspondence i s not to be expected. While i t does not seem sensible to make detailed quantitative comparisons betx^een the two s c i n t i l l a t o r s , i t i s perhaps worthwhile to make a cautious semi-quantitative one. The transfer variance was measured i n the l i q u i d s c i n t i l l a t o r as "VI2 = 17 x 10-4. i f the frequency distribution of transfer probabilities were Gaussian, which i s only approximately true, this would correspond to a value for v\ 2 of about 90 x I O - 4 which i s rather larger than the data i n Fig. 19 would lead us to expect. However, since this i s an equally weighted average over the whole volume of the s c i n t i l l a t o r and makes no allowance for the unequal weights for primary events due to the exponential absorption in the s c i n t i l l a t o r , i t certainly represents an extreme upper l i m i t . A calculation based on primary events and allowing for absorption, reduces the above figure to about 60 x 10 -^ at 663 KeV and this i s consistant with the observations. The l i q u i d s c i n t i l l a t o r survey also suggests the course of curve C at lower energies. Taking the transfer variance for the top 0.5 cm. of the crystal only, we obtain a value 4.7 x 10" , i.e., a contribution of 25 x 10"^ to **\ from transfer effects at about 100 KeV and the curve C (Fig. 19) has therefore been extended to this value at 100 KeV to suggest the behaviour at lower energies. This argument also implies that the contribution from delta-rays continues to increase below 200 KeV. It should be emphasized again that the numerical values quoted here should not be taken too seriously since they are based on measurements i n a different s c i n t i l l a t o r . - 53 -DISCUSSION;-Thus this thesis presents a sufficiently coherent picture to suggesttthat a reasonably satisfactory set of s t a t i s t i c a l models has been found to describe the performance of the photo-multiplier i n a s c i n t i l l a t i o n assembly. In particular, expression f (x) = IW a""^  e~ n x~"3" exp ( -x^) l d exp ^2(N x/a^)^ can probably be regarded as a sufficiently good representation of the shape of the ideal s c i n t i l l a t i o n line even i n cases where the simple exponential model from which i t was derived i s not applicable. Expression (2), VQ = (1 - V F F I) ( n V ) _ 1 , shows that so far as the photomultiplier is' concerned, better resolution can be achieved by increasing the probability that a primary photon w i l l result in an electron arriving at the 1st dynode, (which i s well-known) or to a lesser extent and within l i m i t s , by decreasing V M , the relative variance for multiplication process. The smallest value of V M. observed in the present work i s already 0.63 for the 681OA, and i t would appear that l i t t l e improvement may be expected in this regard. For example, the smallest possible value of V M for a distribution of the type (9"^ 2) i s in fact about 0.3 with J approximately ten. It has already been remarked (Section 2.2) that the observed single-electron distributions are not consistant with a Poisson model for the stat i s t i c s of dynode multiplication. The quasi-exponential . character of these distributions invite comparison with proportional counters. Breitenberger has discussed the close formal analogy between - 54 -s c i n t i l l a t i o n and proportional counters and points out that the only essential difference i s that i n a proportional counter the transfer efficiency i s 100$. Snyder (1947), discussing proportional counters, shows that a simple duplication process leads to an exponential single-electron distribution and Frisch (quoted by Wilkinson (1950)) comes to a similar conclusion. The measurements of Curran et a l (1948), show that the single-electron distribution can be represented by a relation of the type (9.3) with ^ = 0.5. It can also be f i t t e d with a relation of type (9.2). These results suggest that the process of secondary emission in photomultipliers i s much more closely represented by a duplication model i n which the secondary electrons are more strongly correlated than the Poisson model often assumed, i n which the individual secondary electrons must be assumed to be emitted independently of each other. Comparison between non-crystalline organic s c i n t i l l a t o r s such as NE 219 and a sodium iodide crystal of similar size suggests that for gamma-rays detection, an important contribution to line width originates with variations in the l i g h t collection efficiency from different regions of the s c i n t i l l a t o r . - 55 -References 1. Allen, J.S., 1950, Proc. I.R.E. 3J, 34-6. 2. Baicker, J. A., I960. I.R.E. Trans. Nuc Sci. Ns-7, Nos. 3-4-, 76. 3. Barnaby, C. F., and Barton, J. C, I960. Proc. Phys. Soc. London, 76, 74-5.^  4.. Barlet, M. S., 1956. "Introduction to stochastic processes", Cambridge Univ. Press, p. 4- f f • 5. Bernstein, W., 1956."^  Nucleonics, 1^, No. 4-, 46. 6. Berger, M. J. , and Doggett, J., 1956. Rev. Sci. Instr. 27, 269. 7. B i s i , A., and Zappa, L., 1958. Nuclear Instr. 1, 17. 8. Brietenberger, E., 1955. "Progress i n nuclear physics", Vol. 4 ed. O.R. Frisch,Pergaman Press, London. 9. B r i n i , D., P e l i L., Rimondi, 0., and Veronesi, P., 1955. Nuovo Cim. Suppl., 2, 1048. 10. Burch, P.*R. J., 1961. Proc. Phys. Soc. London, 77, 1125. 11. Campbell, G. A., and Foster, R. M., I960. "Fourier Integrals for practical applications". Van Nostrand, New York. 12. Cramer, H., 1958. "Mathematical methods of s t a t i s t i c s ^ . Princeton Univ. Press. 13. Cummins, D";0., Delaney, C. F. G., and McAuly, I. R., I960. Sci. Proc. Roy. "Dyblin Soc, 1, 21. 14. Drapper, J. E., and Hickock, R. L., 1958. Rev. Sc i . Instr. 22, 1047. 15. Erbacher, "0., and P h i l l i p , K., 1928. Zeit. fur physik'l 51, 309. 16. Evans, R. D., 1955.' "The atomic nucleus". McGraw-Hill., New York, page 663. , 17. Garlick, C. J. F., and Wright, G. T., 1952. Proc Phys. Soc London, 65B, 415* 18. Hickock, R. L. and Drapper, J . E., 1958. Rev. Sc i . Instr. 22, 994. 19. Iredale, P.," 1961. "Nucl. Instr. and Methods 2?, 340. - 56 -20. Kelly, G. G., B e l l . , P. R., Davis, R. C , and Lazer, N. H., 1956. Nucleonics, 14, No. 4, 53. 21. Kikushkin, L. S., and Ratner, A. M., 1958. Zhur. Tekh. F i z . 28, 34-5. translation: Soc. Phys. Tech. Phys.2, 318 (1958). 22. Lazar, N. H., Davis, R. C , and B e l l . , P. R., 1956. Nucleonics 14,, No. 4-, .52. 23. Lombard, F. J., and Martin, F., 1961. Rev. Sci. Instr. 32., 200. 24-. Mahagan, W. W., 1962'. I. R.E. Trans. Nuc. Sci. ~NS-9, No. 3, 1. 25. Morton, G. A'.', and Mitchell, J . A., 194-9. Nucleonics ^ ,"No. 1, 16. 2 6 . Morton, G. A., and Mitchell, J . A., 1948. R.O.A. Review 2, 632. 27. Mott, W. E., and Sutton, R. B., 1958. " S c i n t i l l a t i o n and erenkov counter". Handbuch der physik, Vol. 45, Springerverlag, Berlin, p. 86 i 28. Murray, R. B., and Meyer, A., 1961. Phys. Rev. 122, 815. 29. Nelms, A. T., 1953.' N.B.S. Circular 5 4 2 , Washington, D.C. 30. Prescott, j . R., and Takhar, P. S., 1 9 6 2 . "I.R.E. Trans. Nuc. S c i . NS-9. No.. 3,.36. 31. Prescott, J . R., and Lindquist, D. L., 1961. Rev. Sci. Instr. 22, 990. 32. Ratner, A. M., and Kukuskin, L. S., 1958. Zhur. Tekh. F i z . 28, 1121. Translation: Soc. Phys. Tech. Phys. 3_, 1 0 4 4 . 33. Roberts, P. W., 1953. Pro6. Phys."'Soc. London", A66, 192. 34. Swank , R: K., Buck, W."L., 1952. Nucleonics 10, No. '5, 51. 35. Wright, G. T., 1954. J / S c i . Instr. 3±> 337 and 4 6 2 . 36. Wright, G. T., and Garlick, G. F. J., 1954. B r i t . J . Appl. Phys. 1, 1 3 . 37. Zerby, C. D., Meyer, A., and Murray, R. B., 196I. Nuclear Instr. and Methods 12, 115. - 57 -APPENDIX I Derivation of generating functions for Poisson, Binomial and  exponential distributions: (i) The poisson distribution i s given by where x = 0,1,2 The generating function by definition i s given which i s the generating function for Poisson distribution, ( i i ) The binomial distribution i s given by 7k= o which i s generating function for binomial distribution. - 58 -( i i i ) The exponential distribution was of the form _ O L - X QL o < °C and the generating function i s : oc to V — \__ N - 59 -APPENDIX II Finding the mean = Na and Variance = 2Na2 We have three generating functions for poisson, binomial and exponential distributions, i . e . The generating function for a l l the three processes combined is given by: For the characteristic function log & i s replaced by v. 0 and we have the characteristic function: When Q = 0, the f i r s t moment or mean i s V\1>*.<_&- \\ML \ S ^ - 60 -Hence the mean i s i ^ s HCJ.. For finding the second moment write then The second moment is Hence the variance i s <3" = 5 , K d L - 61 -Derivation of the relation VG = 1 + Vm/y^ If % >/*^>Jv>ix. represent the means of the frequency functions , ' W respectively, then the mean i s Introducing the dimensionless relative variance M"1 then s, ^  V-^ V-x. Further making no assumption about the model to be used for dynode s t a t i s t i c s , then on the basis of three-stage chain consisting of Poisson l i g h t production, binomial transfer and multiplication or arbitrary, model, \a.th relative variance Vm, the relation (l) (of Chapter 2) is taken i n the form of where V i s mean, T is the variance for the binomial distribution. Simplifying, \» = ~ V\ If the luminous mechanism i s such that then above reduces to 

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