Theoretical comparison of the enhanced variance and twin-gate methods for monitoring plutonium in waste

NUCLEAR

INSTRUMENTS

AND M E T H O D S

165 ( 1 9 7 9 ) 5 8 9 - 6 0 5 ,

©

NORTH-HOLLAND

PUBLISHING

CO

THEORETICAL COMPARISON OF THE ENHANCED VARIANCE AND TWIN-GATE M E T H O D S FOR MONITORING PLUTONIUM IN WASTE M O DEIGHTON

AERE Harwell, Oxfordshtre, OXII ORA, England Recetved 18 June 1979 Certain plutonlum-momtonng systems, especially those used for checking and classifying drums of mixed waste, depend on detecting pa~red neutrons originating from spontaneous fissions of 24°pu Such neutrons can be detected with reasonable efficiency by suitable counters but, as they are usually accompanied by a high random background of other neutrons, the problem arises of separating the "signal", i e pairs, from the " n o i s e " or random pulses Various ways of doing this have been devised, e g variable dead-ume countersl,2), arrangements of coincidence gates 3) for measurement of the pulse interval distribution 4) A recent suggestion for doing this was to look for deviation from a true Poisson dlstnhutlon m the actual numbers of counts registered m a large number of equal time intervals Th~s paper is a theoretical analysis of the counting statistics to determine the nature and amount of the deviation and thus estimate the accuracy of fission-parr rate determinations m a few typical cases As a yardstick for comparison, we have also analysed a simple form of twin-gate method, in which total counts accumulated via a time-gate triggered by some of the pulses, are compared w~th counts m another gate triggered after a suitable delay To s~mphfy the mathematics, we have assumed all the gate-penods are separate and non-overlapping Results show that the presence of pulse-pairs mcreases the variance of the dlstnbutton above the Polsson value Using the measured excess variance y~elds an appreoably more accurate estimate of pair-rate than the twin-gate method, at high total pulse rates ( > 15 000 pulses/s), but the reverse is true at low rates ( - 1 0 0 pulses/s) With typical parameters (and a mean interval between pulse pairs of 125/is), the cross-over point would be around 1000 pulses/s

1. Introduction The usual arrangement consists of an assembly of thermal neutron detectors, smtably disposed m a moderator around the drum to be checked, and thetr combmed outputs are fed to the pulse-countmg equtpment Th~s pulse tram, effectively from a single large detector, contains a proportion of pulse pears ongmatmg from spontaneous fissions Thetr pulse separattons r have a dlstnbuUon determmed mainly by the type and amount of moderator used, and specified by a probabdtty density function p(r) In addttlon there ts a generally large random background of smgle pulses due to (~, n) events and uncorrelated wtth the fission pairs The problem lS to detect and measure the number of pulse pairs, m the presence of the background, with suffictent statlsttcal accuracy, as an lndtcation of the quantity of 24°Pu present Two ways of doing this, whtch wtll be compared, are the followmg 1 1

TWIN-GATE COMPARISON METHOD

Two gates, A and B, of equal durations T are opened m successton and route neutron pulses to two separate scalers Gate A ts opened by one of the detector pulses, whde B is opened after a fixed delay, sufficient to ensure that p(t) has decayed to a neghglble value The B-gate is effectively located at

random in ttme and the counts collected therein are a measure of the total mean pulse rate Gate A, on the other hand, has a finite probability of having been opened by the first pulse of a fission pair, m which case, with suttable choice of T, there Is a good chance it will include the second pulse This ts m addttton to counts arising from other pairs and from random background, equal to expected counts m B Thus, excluding the mtttal tnggermg pulse, one expects gate A to contam a fraction of a count more than B, on the average A measurement consists m comparing the total counts accumulated via A wtth those accumulated vm B, after many operations of both 1 2

ENHANCED VARIANCE METHOD 4t

A ttme-gate of width T is opened, at regular intervals uncorrelated with the neutron pulses, and one collects data on the frequency of occurrence of various numbers of pulses (say from 0-10) in each gate period For a purely random pulse train one expects a Potsson dlstrxbutton for the relative frequenctes of dtfferent counts, but, wtth a few trine-correlated pulse pairs present, tt can be shown that dlstortton of the Potsson dtstnbutton occurs, This method was proposed by G White of AERE A slmdar techntque appears to have been used by E J Dowdy at Los Alamos SctentlfiC Laboratory (to be published)

590

M o DEIGHTON 0 1 2 3 4

n

"~'~'arts

5 6 7

~

8 9 Count

C-ales

Shift Register

J

Fig 1 Simphfied schematic of count dlstnbuuon method

which should be detectable From the amount of distomon one can refer the mean rate of pairs The detads of the system desagn do not concern us here, but one might use a 10-stage shift register, to which the gated neutron pulses are applied Each stage of thas regaster controls a gate wath an assocmted count regaster At the start of each T-gate the shift register as reset to the 10 000 state and at the end of the T-gate a " c o u n t " pulse is routed to the appropriate count regaster vm ats gate, depending on the state of the shift register The bare essentials are sketched m fig 1 At the end of a measurement, consisting of many T-penods, the count regasters would display a rephca of the quasa-Poasson &stnbution A more sophasUcated versaon of Method (1), the "shlft-regaster" technique, whach is difficult to analyse fully (see dascussaon m sectaon 4), has been m use at AERE wath reasonable success However method (2) would appear to extract more mformataon from each gate period and maght therefore offer more efficient use of the measurement tame available, a e greater accuracy for the same overall tame or less tame consumed for the same accuracy The present purpose as to lnvest~gate the statastlcal limitations of both systems theoretacally, so as to attempt a comparison Any comparason must, of course, allow for the greater complexaty of hardware an method (2) as well as the evadent larger amount of data-processing needed to arrive at a final result Thus one as looking for an apprecmble amprovement m accuracy, to make at worthwhile We shall consader two numerical examples, one at a fairly hagh pulse rate, the other at a low rate

2. Analysis of the twin-gate method Suppose there are n pulse-pairs per second, on the average, wtth a superposed background of purely random pulses at mean rate no per second. The total pulse rate as evadently 2n+n0, gwmg on aver-

age (2n + no) T pulses m a gate of duraUon T The raUo no/n is typacally 10-20 The separataon z between the pulses of a pmr vanes somewhat at random and ts described by a (continuous) probabllaty densaty functaon p(z), but it as assumed one can choose T such that nearly all z-values are less than T, in thas case a gate opened by the first pulse of a paar as virtually certain to contain the second one If the 1/e decay tame of p(z) is around 1-25/~s, T = 400/~s, say, would be statable Wath a total rate 20 000 pulses/s, the mean counts per gate would then be about 8 We consader the small excess count an the A gate, compared wath that m B, this leads to a systematac &fference, N^-NB,between total counts accumulated over a large number, M, of operaUons of each alternately To assess the sagmficance of the result one must compare the systematac &fference wtth that which maght occur from pure chance (e g if no correlated paars of pulses were presen0 Only af the former as at least several tames the latter, would at be possable to estamate the paar rate wath any reasonable accuracy A specified accuracy sets a lower hmat to M and.thus determines the mlmmum tame reqmred for the measurement As stated above, mean counts m each B-gate as (2 n + n 0) T, so mean expected

NB = M ( 2 n + n o ) T

(1)

On average, each second of time contains no random pulses together with n first pulses of pmrs and n second pulses The probabflaty that a pulse chosen at random (a e the A-gate tngger) is actually a first one of a paar is

p = n/(2n +no),

(2)

m whach event the second pulse wall also fall wathln the gate, gwmg a mean expected count ( 2 n + n o ) T + l (excluding the trigger pulse itself). There as also a probabdaty 1 - p that thas wall not be so, m whach case the expected count as (2n+no)T only Overall, therefore, mean count per A-gate as

p [ ( 2 n + n o ) T + l] + ( 1 - p ) ( 2 n + n o ) T = (2n+no)T+p So as expected N A = M [(2n +no) T + n[(2n +no)]

(3)

From (1) and (3),

NA--NB = Mn](2n +no)

(4)

591

M E T H O D S FOR M O N I T O R I N G P L U T O N I U M IN WASTE

Turning now to random variations o f counts in each gate, the standard deviation of the count in each B-gate is + ( 4 n T - 2 n f + n o T ) ½ a result which is a little greater than the square root o f the mean counts This is due to the presence o f correlated pairs within the gate, the expression is d e n v e d in section 3 2 and indeed is the essentml basis of method (2) for measuring pair rates Here ~ signifies the mean o f all pair separations whtch are less than T The A-gate also has the same basic variance in its count, with however an additional component arising from the occasional presence o f the extra pulse The variance o f counts in a single A-gate in thus

4nT-2n~+noT + p(1-p)* and that of the total counts in M independent Agates is var(NA) = M [4nT-2ng

+n o T +

+ n(n +no)/(2n -+-/10) 2]

giving a difference N A - N B = 833___368 counts This accuracy is barely sufficient to detect the 24°pu pulse pairs reliably Indeed, with the data above, there is a proba,blhty greater than 10% that one will obtain half the correct value or less for N A - N B It is evident from eq (7) that one needs to Increase M considerably, since the count difference varies as M whereas its standard deviation varies as Increasing M to 250 000 would thus gwe a five-fold Improvement in accuracy of determination of n/(2n+no), the relative count rate o f fission pairs, to about _+9%, r m s The total time required for this measurement is 2MT or more, l e over 3 rain

Numerical example 2

Suppose n 0 = 9 5 pulses/s, n = 2 5 pulses/s so 2 n + n o = 100pulses/s total, T = 128 Us, (fixed) separation r = 100 Us If M - - 1 8 0 0 0 0 , l e total expected pulses in 30 rain, then N A = 180000 x 0 0378 _ x/180000 x 0 0373 = 6800 +

(6)

Equations (4) and (6) can be combined in the more usual form

Mn N A - - N B = 2n+n-'----~ +-

Mn(n + no)'~ 1/2 +_ 2M(4nT-2n~+noT) + ( 2 n + n o ) 2 J

N B = 60 000 ___

(5)

The final term here is generally only a few percent of the remaining ones at high pulse rates, but may be dominant at low rates Assuming the fluctuations of the two totals are independent, their variances add, when taking the difference N A - N B , and we have var(NA--NB) = 2 M ( 4 n T - 2 n ~ +noT) + + Mn (n + no)l(2 n + no) 2

W e have n 0 T = 5, nT=O 5, so (2n+no)T=6 and rrt = 1/8 The separate counts are NA = 60 833 ___6 8 ~

(7)

In general, the estimate o f the relative pair rate is simply (NA-NB)/M, and the fracUonal r m s accuracy of this result is, from eq (7),

and NB = 180000 ×0 0128 + x/180000 x 0 01294 = 2304 + x/2--~-0 So N A -- N a = 4500 + 9x/if050 = 4500_ 95 Thus

(4nT-2n~+noT) + M(-~n~no)2j

(8)

Numertcal example 1 Consider the following typical values no = 12 500 pulses/s, n = 1250 pulses/s, T = 400 Us, ~ = 100 Us, M = 10 000 The variance of a number which is occasionally 1, with probabdlty p, but otherwise 0, w~th probab]hty l - p , ~s p (1 _p)2 + (1 - p ) ( - p ) 2 = p ( 1- p )

accuracy

of

determination

of

that, owing to the low pulse rates, the gates are well separated in time and no problem o f statistical interaction occurs 2 1

x

the

n/(2n+n o) is about 2 1 °/6 here and quite good Note

OVERALL TIME FOR MEASUREMENT

In the foregoing analysis it is assumed that all the gates are independent, l e there is no interaction between successive gates, with regard to countlng statistics This requires a m i n i m u m separation o f about T between any adjacent gates in the sequence A - B - A - B Thus, for M gates o f each kind, the m i n i m u m total measurement time is 4MT, o f which only half is useful counting time It is pertinent to consider whether this time can be reduced without detriment to the results

592

M O DEIGHTON

First suppose the A - B gap is closed up (e g. the trailing edge of each A-gate provides the trigger for starting the B-gate) There is no reason to suppose there will be any change in mean counts accumulated in either gate even though they are juxtaposed In fact eq (1)-(4) remain valid However it can be shown that there is now a small positive correlation between the random errors, eA and cs say, of the counts occurring in the two parts of each AB combination This being so, one expects a slight reduction in the variance of the difference NA--NB and thus a small Improvement in system performance The main advantage, however, hes in the saving of overall ume - from 4 M T to 3MT The amount and effect of correlation are calculated as follows Accepting the results quoted above for the variance of the individual counts, we have 52 = 4 n T - 2 n f + n o T + p ( 1 - p ) 52 = 4 n T _ 2 n ~ + n o T , SO 2

~A+ea2 = 2 ( 4 n T - - 2 n ~+ no T) + p(1--p)

(9)

Also we can regard the AB combination as a single A-gate of width 2T, whose total count has an actual error eA +tB Applying the same formula~f for variance, (eA+eB)2 = 4 n 2 T - - 2 n ~ + n o 2 T + p ( 1 - p ) (10) Subtracting (9) from (10) yields +n{

~A/~B =

(11)

which signifies positive correlation, as expected It follows that the mean square difference IS

= 2(4nT-3n~+noT) + p(1-p)

from (9) and (11) For M independent gate-pairs, the variance of N A - N B would be M times greater, so the only change to our results in eqs (6), (7) and (8) is that the bracketed expression becomes ( 4 n T - 3n~ +no T), which IS not very significant Finally we ask whether one could reduce the B - A gaps appreciably, by for example triggering the A-gate from the next detector pulse following the end of the preceding B-gate The resulting gap, or t Smctly speaking, one should increase the value of f shghtly, to take account of the very small number of pairs having separatmns between T and 2 T

dead penod, would be vanable but about 0 1-0 2 T, on the average, in our example 1 Thus overall measurement time would be little more than 2MT. Unfortunately, however, the system would fail, because, although a fraction of A-gate tnggers are associated with one extra count inside the A-gate, an equal fraction of them happen to be second pulses of pairs and are therefore associated with extra counts inside the (now) immediately preceding B-gate Thus the systematic difference between N^ and Ns tends to disappear and detection of n becomes impossible The conclusion is that each contiguous pair AB of time gates must be separated from the next pair by a gap of width about T or more Thus overall measurement time cannot be less than about 3 M T , this is 5 min, for 9% accuracy, in our example 1 above 3. Analysis for count distribution method Here we measure the count distribution in timegates of length T, by collecting data on the actual numbers of pulses occurring In each of a large number M of such gates Ideally the gates must be adequately separated to avoid statistical correlation The results, plotted or displayed as a histogram, should agree with the Poisson distribution in the case of random pulses With time-correlated pulsepairs present, it is shown that the distnbution is broadened to an extent related to the number of such pairs The objective is to process the accumulated data so as to reveal any significant departure from Polsson statistics and thus estimate the pairrate as a fraction of total pulse rate The emphasis here is on the word "significant" Clearly some sophisticated problems In statistics arise, e g the variance of a variance, also the algebraic expressions for the distribution itself are difficult to handle, in particular when one has to " i n v e r t " them, i e deduce the pulse rates from the distribution

3

1

DERIVATION OF PROBABILITY GENERATING FUNCTION

First we need to obtain the probability distribution of counts in a time interval T from the mixed pairs and random pulses This is done in three stages (1) the distribution for pairs only, with a fixed separation z, (n) the distribution for pairs having assorted separations z~, z2, etc, with extension to the case of a continuous distribution of separations, and (in) the effect on this of a superposed random background of no pulses/s At each stage we are dealing with the combined count aris-

METHODS

B

~

FOR MONITORING

1

C

,B~._g___r

g2(x)= D

hme

F~g 2 T~mmg d m g r a m for finding count d~stnbutton m T for fixed-separation pairs, (a) case z < T, (b) case r > T

lng from two or more independent sources and the pmbablhty generating function (p g f ) suggests itself as a useful tool for the analysis The definition of a p g f , with a brief r6sum6 o f ~ts mare properties, is given m Appendix 1 (1) Parrs wtth fixed separatmn Let the interval BD on the horizontal t~me-axls m fig 2 represent the gate-width T Point A precedes B by an amount z (a constant) and C precedes D by the same amount W e distinguish the two cases r < T and T > 7", m (a) and (b) respectively, the former is the more important and we refer to fig. 2(a), except where otherwise stated Let the first pulse o f every pair be designated a p-pulse It is assumed the p-pulses all form a truly random train at a mean rate n per second and, m addition, every p-pulse is followed after delay T by a second or s-pulse The total of pulses occurring m T can be divided into three different groups, according to their origin, as follows (1) p-pulses o c c u m n g within AB are all outside the gate, but each has a counterpart s-pulse within the gate (2) p-pulses within BC are all within the gate and so also are all their corresponding s-pulses (3) p-pulses within CD are inside the gate, but all their s-pulses are outside Since the intervals AB, BC, CD are separate, without overlap, the fluctuations m their respective p-pulse numbers are statistically independent and so also are the fluctuations in the above three contributions to total count m T Thus, ff we find the probability generating function for each group, that for the total count Is s~mply the product The first group is a time-delayed version o f p-pulses in AB and its p g f is therefore (see Appendix 1)

g1(x) = e "'(x-~)

IN W A S T E

593

the gate, in th~s group, since there are N p-pulses and also N s-pulses Thus the p g f is

D

(a}

A

PLUTONIUM

(12)

Considering group 2, the probability of N ppulses within BC is (RN/NO e -n, where R = mean number = n ( T - T) This is the probability of counting 2 N pulses in

R -n 2N e -n x t~= o N----/e x =

x

o

(Rx2)N NV

e-Re Rx2 = e n(x:-l)

(13)

-

Note that this group can only contribute an even n u m b e r o f pulses to the total i n T, so we expect odd powers o f x to be absent from the expansion of its p g f Lastly, group 3 contributes a simple Polsson dlstrlbutton o f p-pulses in a time T, SO ItS p g f IS again, g 3 (X) - -

e "'(x- 1)

(14)

The resultant generating function defining the total count distribution in T is

a(x) = gl(x) g2(x) g3(x) _

e2nT(x - 1) eR(X 2 - 1)

le

G(x) = exp [ n ( T - ' c ) x 2 + 2n'rx - n ( T + z ) ]

(15)

W e next consider the case r > T, fig 2 (b) Here point C comes earher than B and the three nonoverlapping intervals, to be considered as independent sources o f counts in T, are AC, CB and BD Thus the breakdown into groups is different from before p-pulses in AC result in an equal number o f s-pulses in T and we have g~(x)=e "r(~-l), since AC = T Any p-pulses in CB result in no counts whatever in the gate, since these pulses are all too early and their corresponding s-pulses are too late The generating function for this special case is g2(x) = 1, i e certainty of result zero and zero probaNhty o f any other result Lastly group 3 consists of p-pulses within BD, with all the s-pulses outside the gate This group has p g f g3(x)= e "r(~-l) The resultant generating function is again the product o f these three and.

G(x) = e 2"r/x-l),

for z > T ,

(16)

which IS recognizable as that for a true random distribution (Polsson) at mean rate 2n Hence, not surprisingly, a gate of width T is incapable of recognizing pulse separations greater than this, producing merely a Polsson distribution given by the total pulse rate 2 n The conclusion can be drawn that, In order to detect any pmrs present, the gate-width T must exceed the majonty of pair separations and thus probably be at least a few times the mean

594

M 0

DEIGHTON

spacing of pairs-in short at least equal to the value used in method (1) (n) Pairs with distributed separations Bearing in mind the foregoing remarks, suppose there are nl palrs/s with separation zl, n2 palrs/s separated by r 2, and so on, where all the different values of r are less than T and n l + n 2+ n 3 -t- ----n Their separate contributions to total count in T are all independent and each has a probability distribution described by the p g f. in eq (15), with the appropnate values of n, r inserted Since each p.g f. is an exponential function, the product of them all is obtained by adding all the exponents and the composite p g f is given by

G(x) = exp {I- E n , ( T - ~ , ) ] x 2 + ['2 E n,'cl]x - ~ n, (T +z,)}

-

(17)

If ~ denotes the mean spacing of all the pairs, le

~ = (,nlxl+n2"r2+

)In,

then ~ n l ' r 1 = n ~ , also Y _ , n l = n and T is a constant Hence eq (17) becomes G(x) = exp[n(T-~)x

2 + 2 n ~ x - n(T+~)-I,

(18)

which IS the same form as (15), with T replacing z In short, the way In which the pair separations ( < T) are distributed is of secondary Importance, since only their mean value ~ enters into the 'generating function which defines the overall count distribution in T These results are true however many different values of r are present and therefore remain valid if there is a continuous distribution of pair separations as happens in the practical situation Such a distribution, often represented approximately by an exponentially decaying probability density p ( z ) = = e -=',

(19)

may well extend slightly beyond the gate-width In this case, if np denotes the total number of pulsepairs per second, the values of n and ~ to be used In eq (18) are strictly as follows n = nv

/o

p(~)d~ = np(1-e -=r)

(20)

,~ _- -np- f r zp(z) dz =[1 - (1 +ctT)e-=r]/~(1-e -=r) n J o

(21)

10 j ' r l r ,

e

08 it., i~ 06 o~

.-1~ o,~ 02

or T = T / ' r l l e

F~g 3 Variation of effectwe f with gate width T

Here cx is the reciprocal of the 1/e decay time of p(r), T would usually be from 3 to 5 times this decay time, so n IS not appreoably less than np However note that "r can still be noticeably less than rl/e, see fig 3 This IS because pairs excluded from the calculation of ~ all have large separations These pairs , n p - n m number/s, count only as added random pulses at mean rate 2 ( % - n ) , as was seen in (l) above They constitute a (usually) neghglble addition to the random background at no pulses/s, which is anyway present from other causes, and is considered next (in) E f f e c t o f a d d e d r a n d o m b a c k g r o u n d The probability generating function for the background pulses (in T) alone is of the usual form for a Polsson distribution, namely e x p [ n o T ( x - 1)] The final p g f for combined pairs and background IS therefore the product of this with the expression in eq (18), i e Gtot(X ) =

exp (ax 2 -I- b x - c)

(22)

where a = n(T-~) b = 2n¢+no T c = a+b

(23)

= n(T+~)+noT

Equation (22) is a convenient form for further manipulation Note that Gtot(1)= 1, as required 3 2

MEAN AND STANDARD DEVIATION OF DISTRIBUTION

One can readily obtmn the mean and standard deviation of the distribution represented by the p g f in eq (22) Following the method outlined in Appendix 1, we differentiate eq (22) twice giving G~ot(X) = (2 a x -I- b) exp ( a x 2 + b x - c) and G~'ot(X) = [(2 a x + b) 2 + 2 a] exp ( a x 2 + bx - c)

METHODS

FOR MONITORING

So substituting from (23)

PLUTONIUM

P2 =

/V = G~.t(1) = 2a + b = (2n + n o ) T ,

(24)

595

IN W A S T E

(a+b2/2)e -~

Pa = (ab + ba/6) e- ~etc

and = G~'ot(1)+_N = ( 2 a + b ) 2 + 2 a + 2 a + b Therefore a2 = ~-z_(N)2 = 4 a + b = 4 n T - 2 n f + n o T

(25)

Note that the mean count m the gate, ( 2 n + n 0) T, is as expected, but the vanance is somewhat larger than the mean, indicating a broader dlstnbutlon than a true Polsson one The excess variance is proportional to n and to the a m o u n t by which T exceeds In our numerical example 1 (p 591) we have a = 3 / 8 , b = 5 ¼ , c = 5 % , hence N = 6 , c r 2 = 6 7 5 Thus the variance ~s here one-eighth larger than with a Po]sson dlstnbutton or the standard deviation is about 6% greater

Perhaps the best way to obtain these coeffictents is to write out the blnomml expansions of the dtfferent powers of (ax2+ bx), in tabular form, under the approprtate powers o f x (see table I) Summing the coefficients by columns, we obtain Po = e -c Pl

= be-C

P2 = ( a + b 2 / 2 ) e -~

Pa = (ab+b3/6) e-~ p,, =

Gtot(X ) =

ab 2 b ) +-~ + e -~

P5 =

3 3 DISTRIBUTIONOF PROBABILITIES The probability Pu o f actually counting N pulses tn a gate ts obtained by expanding the p g f as a power series m x, then picking out the coefficient o f x u. Thus from (22)

(a_~

+---if-+ ( a-~

P'

=

P7

=

ps

=

P9

=

a 2b2

~--~

] e -c

a2b3 aa b 2

+--iT

(a 4 b

= [ l + b x +(a+b2/2)x 2 +

\24

] e -c.

pxo--

Hence we obtain

b5-o-~) +

a2 b"

e-C

a b6

b

)

+-if6 +

a2 b s

e

ab 7

+ - ~ 0 - +5-0-~ +

a4 bz

iT6 +

(26)

e-°

abS

aa b 3 +--~-

as

+(ab+b3/6)x 3 +

b~-~) +

+'--i-f- + ' i ~

a(~.f

+ (ax2 +bx)a/3 v+

a b4

+--X-

eaX2+bXe-C

= [1 + (ax 2 + bx) + (ax 2 + bx)2/2 ~ +

e-~

a 3 b4

+ 1-i - +

b36___~__~6 ) e-C /

ab s

a 2. b 6

+ 4o-6

+

b 1o \ + 3628800) e-c

Po = e-C

Pl = be-C

as far as N = 10

TABLE 1 Power series expansion o f Gtot0C), omitting factor e - c

x0

xI

1

b

x2

x3

x4

x5

x6

x7

x8

x9

xlO

5a4b/51 20a3b3/6 w 21a2bS/7 ~

a5/5 ! 15a4b2/6 I 35a3b4/7 I

a

b2/2 ~

2ab/2~ a2/2 ~ b3/3 w 3ab2/3 v 3a2b/3 w a3/3 e b4/4 v 4ab3/41 6a2b2/4 w b5/51

5ab4/5 w b6/61

4a3b/41 lOa2b3/51 6ab5/6 t b7/7 ~

a4/4 t lOa3b2/51 15a2b4/61 7ab6/71

b8/8 v

8ab7/8 r 28a2b6/8 t b9/9 t 9ab8/91 blO/lO*

596

M O DEIOHTON 0 3 I-

°,s I

I

(c).T:q- .oT:~

\\

0 10

1 o.z

005

(a) Pomson d,stnbul~n (nT=O,noT:2)

_o

•

N~/~

~" =0 2T

(a) Ol

o~

N~ N-(a)

(o) Po,sson d,st (nT=0 noT=6)

/ ~ } \

010

I/,~

(b) nT=O 5 noT=5

--

0o

N Fig 4 (a) Combined (random+pmrs) count dlstnbution, N = 6, = 0 2 T, (b) combmed count d~stnbutton (random+pmrs), N=6,~=05T

Fig 5 Combined count dtstnbution, N = 2, ~ = 0 2 T

" c u r v e " with h~ghest peak is the true Potsson d~stnbut~on, the other two curves bemg for two different raUos n/no In all cases the mean total count per gate (N) is 6 and fig 4(b) &ffers from 4(a) only in that the mean pmr spacmg (f) ~s 0 5 T, against 0 2 T Clearly the general nature of the d~storbon ~s s~mdar m both dmgrams, namely a broadening, but the amount is roughly proportional to T - ~ , as suggested m section 3 2 Fig 5 shows the probability distribution in one case where N = 2, compared w~th the Polsson d~stnbut~on for th~s case, th~s shows an amount of d~stort(on rather slmdar to that for curve (c) m fig 4(a)

3 4 THE F-PLOT An effectwe way to display departures of a distriNote that, ff no ttme-related pmrs are present, then a = 0 and c = b = no T, all terms of the above bution from Polsson form is to plot the funcuon polynommls &sappear except the last and the F(N) = (N + 1)pt¢+ 1/Pt~ (27a) probabdmes revert to the stmple Po~sson form The scheme of polynommls above can be ob- or, m the case of experimental observations, tained (or extended to htgher values of N if desired) (N+I)CN+I by using the following recurrence relations F(N) = CN ' (27b) (1) If 2 m Is any even integer, the polynomml for P2,. starts with a " / m ~, the remaining terms being where CN denotes the number of gates (out of some respecttvely b/2, b/4, b/6, etc t~mes the terms m large total) containing exactly N pulses It ~s easdy venfied that, for a Polsson &stnbution, F(N) is P2m-1 (2) P2.,+1 Is obtained from P2,. by muluplylng constant and independent of N, hence the plot of successwe terms in the latter by b, b/3, b/5, b/7, F(N) versus N would (theoretically) give a senes of etc., respectively points on a horizontal hne With experimental data It is ewdently a forbidding task to calculate/V or a 2 there would be a certain amount of random scatter from these probabilities, m the usual way, hence above and below the line, whose height above the the ment of the method of section 3 2 Most furth- N-axis is /V The random errors are least near the er work on th~s &stnbut~on ~s numerical, for th~s peak of the distribution, i e near N = A7 reason Any systematic departure from Po~sson usually F~gures 4(a) and 4(b) are plots of the probabdmes, Imparts a slope to the hne, posmve m the present computed from the expressions above. In each, the instance of broadening. Figures 6(a), 6(b) and (7) are

METHODS FOR MONITORING PLUTONIUM IN WASTE n : mean pmr rate (gJwng count rate 2n) no=mean random pulse rate

7jr-

/

~ : 0 2T N: 6

[.,

'F //

./~.:

: 0 5 neT:5 nZ =025 noT=5 5

s,/ .~" 5 7

//

//

_//nT

/

) -I-

1~:2 ~" : 0 2T no/n =4 I 4 h • nT: ~ ,noT: -~)

nT =1, noT=4 3

m

597

//

20

(a)

2

3o.b

/

---

it

(,) a:08 b:44 s,:o235

/

i/

I

(.} o = 0 4 b:52,SF=0125 (m) a=02. b=56,SF=00645

/

Slope of weighted least squares best fit llne : 0 2/.4 _ 22

a:Tc,b- ~

I

S F--

~2a - -0

235

N~ f=05T

N~

~=6

Fag 7 Statistics of F(N) plot for a measurement using 10 000 gates (nT = 13, no T = ~ ) nT=1,noT=4 / ~ , , . ~ nT=O5,noT =5 .... nT: 0, noT=6

(b)

Hawng determined the slope Sr of the fitted hne, one can use the approximate empirical relation (28)

Sp = 2 a/(3 a + b), ,I /

~"

/ o/

(110=05 b=50 5F:0154 (.) 0=025b:5 5 S F = 0 0 8 0

5

which appears to hold good in a number of cases we have calculated, to arrive at the number of pulse pairs present From eqs. (23) we have (2a +b)/2a = (2n + n o ) T / 2 n ( T - ~ ) , whence 1 _ 3a+b

S~

2a

(2n+no)T

-

Fag 6 F(N)=_-(N+I)P(N+I)/P(N),(a)N=6, T=02T, (b) .,V= 6, T =05 T F-plots derived from the calculated data sn Figs 4(a), 4(b) and (5) respectively The positive slopes are quite apparent, though the hnes are dstlncly curved, parttcularly for small values of N (see fig 7) This curvature, combined with the random errors present in any observaUonal data, makes precise interpretation of such data d~fficult Probably the best approach is to compute the slope of a leastsquares fitted straight hne, making due allowance for the varying stat]sUcal quahty of the points, by appropriate weighting The staUsttcal errors (r m s ) of individual points are indicated by vertical bars In fig 7 and again in fig 8 for one of the curves m fig 6(a) In both cases we assume a total of 10 000 gates

1

2n(T-'~) "12 (2n+no)T + n(T-~)

2n(T-~?) This result can be rearranged as n/(2n +no) = l f [ ( 2 / S r - 1) (1 -'~/T)],

(29)

which gives the fractional pair rate in terms of estimated Sr and T / T Note that t h s distnbutlon method, unlike method (1), does depend on a knowledge of the mean pair separation Equation (28) is seen to be at least plauslble~ by the argument following Usmg the first three results m eqs (26), we obtain F(0) = Pl/Po = b F(1) = 2p2/p I

=

(2a/b) + b

598

M O DEIGHTON

7r / /

l z

LL

]" ]" I

-~

/'J

1

fi=6 ~ ' = 0 2T noln = 10 (so nT=O 5. n°T= 5}

3 5 we,gh)ea i,ost squares best fit hne = 0 124

_ Za _ SF- ~ b - 0 125

N~

Fig 8 StaUst~cs of F(N) plot for a measurement using 10 000 gate periods m all (nT= 0 5, noT= 5) Therefore initial slope of F-plot = F ( 1 ) - F ( 0 ) = 2a/ b Also it may be observed on the curves plotted that F(N) always crosses the level N at a point close to N = N If we assume the point (N, N) lies on the curve, the mean slope o f the curve below this point is that o f the chord joining this point to the starting point (0, b) This is

(N - b ) / N

more accurate than one might guess from the individual errors The detailed statistics o f the Fplot have not been pursued much beyond this point

= ( 2 a + b - b ) / ( 2 a + b) = 2 a/(2 a + b),

which is a little less than the initial, slope, a being only a fraction of b It is then not unlikely that the slope of the tangent near N =)V is a little less still and eq (28) accords fairly closely with the slope of the least-squares fitted line, which has been computed in a few cases Note that the slope of the chord above is equal to the fractional excess variance of the distribution, calculated In section 32 Whatever the theoretical difficulties in assessing the F-plot, it does have the advantage, in practice, that it can still be used even if data for large values of N are missing (e g as in figs 4(at and 4(b)) Here the absence o f values for PN (or CN), for N > 10, would preclude any accurate estimate of tr 2 for any of the curves A n important point to note, concerning interpretation o f experimental data m F-plot form, is that random errors o f adjacent points o f the plot are highly correlated with each other This is because any error m CN enters both into the numerator o f F ( N - 1 ) and into the denommator of F(N), thus the correlation is negative This means that an estimate of the slope o f the least-squares fitted Ime ts

STATISTICS OF DATA AND ACCURACY OF RESULT

Having obtained an apparent broadening or excess vanance o f the count distribution among M gates, it still remains to determine the significance of the excess and thus also the accuracy of the deduced fission pair rate The question " H o w much might the measured variance S 2 of a Potsson distribution differ from the true value tr 29 ,, still has to be answered and there is a similar question attached to the F-plot We confine attention to the former since it is more amenable to statistical analysis It is likely that the F-plot method gives similar accuracy though this should be verified at some stage, either theoretically or experimentally Appendix 2 shows that, for a Polsson distribution, the expected variance of counts among M gates, each containing r counts on the average, is ( S 2 ) A V = r ( 1 - M -1) ~- r Also the random variation of S 2 from one experiment to another, specified by the variance of S 2, is var (S 2) = (2 r 2 + r)/M so we may w r i t e S 2 = r _-4-( 2 r 2 -]-r)l/2/M 1/2 (30) As the derivation of these results is rather complicated, It is desirable to test them, at least semi-experimentally A computer (PDP-11) and random number generator were programmed to simulate a random pulse train and pulses were counted and recorded in successive " g a t e s " Mean counts per gate was 5 5 arifl'six separate runs, of 10 000 gates each, produced six independent distributions, from which the data in table 2 were calculated TABLE 2 Measured means and variances of " r a n d o m " dlstnbutlon

Run

N

S2=N-2-(/V) 2

A B C D E F

5 4912 5 4958 5 5288 5 5138 5 4962 5 5369

5 5339 5 3932 5 3658 5 3428 5 3886 5 4866

M E T H O D S FOR M O N I T O R I N G P L U T O N I U M IN W A S T E

The lndwldual deviations of numbers m the two columns from 5 5 are generally of about the expected order, Indeed if we take the r m s values of deviations, for the six experiments, the results, namely 0 0204 and 0 106, agree quite well with the theoretical standard deviations for ~Vand S 2, respectively, which are = 00235

and

x/(2r 2+r)/M = 0081

From eqs (24) and (25), the systematic excess variance due to the presence of pairs is 2 n ( T - ~ ) , hence we deduce that the r m s fractional accuracy with which n can be determined is p = _ x/(2r z +r)/M/2n(T-~), (31) where r=(2n+no)T (This presumes that the un-' certainty In the measured S 2 is about the same, whether pairs are present or not )

Numerical examples. Going back to our first (highrate) example, section 2, and using 10 000 400ps gates, with 400ps gaps to avoid interaction, the fractional accuracy would be p = + x/ff-ff0~0 75 = + 12% which is appreciably better than the figure 45°/6 obtained by method (1) Increasing M to 250 000 would reduce the error to _+2 4% [9% by method (1)] Total times for these measurements are 2MT, against 3MT in the twin-gate system Thus the distribution method gives about four times better accuracy, with less time taken, in this example Consider, however, the low-rate case of example 2 One would need gate-widths of 50 ms here, to give an average of 5 pulses/gate Intergate gaps of approximately 400 ps would be negligible and a 30mln run would thus have M = 1800 × 20 = 36 000 Figures calculated for this case and for three smaller gate-widths are given in table 3 In no case does the accuracy approach the figure of 2 1% attained "by method (1) in section 2 TABLE 3 Accuracy of method (2) at low pulse rate (J-h run) T M r(=F/) x/(2r 2 +r)/M 2n (T-i) p

50 ms 36000 5 0 039 0 25 15 6%

10 ms 180000 1 0 0041 0 0495 8 25%

1 ms al 5×106 0 1 0 00028 0 0045 6 3%

400/~s a30>(106 004 0 00012 0 0015 8 0%

a Allowing for a 200/.is gap between adjacent gates

599

4. Discussion of results It appears from the numerical examples considered here, that the distribution method (2) offers some advantage (about 1/4 the error, for comparable measurement time) over the simple twin-gate method (1) at high counting rates, but at low rates is definitely inferior As noted at the beginning, an improved version of method (1) known as the shift-register method is currently in use, this employs a clocked shift register as a sort of extended delay line, in which signal pulses over a moderate period of time can be stored in approximately correct time-relationship to one another Whenever a pulse appears at the input, the number of pulses stored in the first k stages is added into scaler A and simultaneously the number In the last k stages is added to scaler B Effectively this arrangement resembles the two gates A, B discussed in section 2, except that every pulse closes an A-gate and the corresponding B-gate was opened some time previously, instead of later in time Thus the whole time sequence of processing is in effect reversed, which does not affect the results More important, multiple overlaps of successive gates can OCCUr

The system has the advantage, at high pulse rates, because of overlaps, that a given number of gates can be compressed into a shorter measurement time, or conversely a given time can contain many more gates of both kinds It is probably correct to assume the same systematic difference NA-NB occurs as was calculated above, see eq (4) However the statistics of both N a and NB are now by no means as straightforward as in section 2 Owing to the time-overlap of nelghbounng gates in either set, the random variations of counts in them have strong positive correlations with one another, the amount of which also varies randomly with the relative timings All that can be said at present is that the variances of the two separate totals will certainly exceed M times the individual gate variances by appreciable amounts Thus the net advantage of the system is less than appears simply from the increased M values permitted At low pulse rates (e g example 2), on the other hand, the A-gates (and B) are so well separated in time that virtually the only overlaps occurring are the two A-gates associated with each pulse-pair and their corresponding two B-gates These are only a small fraction of the total number, hence the overall statistics are probably much the same for both

600

M 0

DEIGHTON

versions of method (1) and defimtely superior to the d~strlbutlon method (2), see table 3 Only at h~gh pulse rates does method (2) offer some advantage over the simple form of method (1) and a smaller, rather doubtful advantage over the improved version of (1) Thus we conclude that the extra comphcatlon of method (2) may not be just~fled, parUcularly ff reqmred to cope with a w~de range of pulse rates

G(x) =

It ~s a pleasure to acknowledge many useful d~scuss~ons w~th G White, who originated the proposal analysed here, and with Dr J W Leake, both of A E R E , Harwell Thanks are also due to G White for programming the PDP-11 and obtainmg the data d~scussed m section 3 5

Th~s example illustrates why the product rule for p g f s gwes the correct probabdmes for the composite event, a total score of 5, for example, can arise m four different ways, namely 1 +4, 2 + 3 , 3 + 2 or 4 + 1 , and the probabdmes of these combinations are PIP4, P2Pa, PAP2, P4P~ Hence the total probablhty of score 5 from the two d~ce ~s

Appendix 1. Probability definition and uses

generating

functions-

A1 1 GENERAL Any random process r e s u l t i n g In various integral numbers, or scores, each with Its own probabdlty, may be descnbed by a probabdlty generating funcnon (p g f) g(x) If P0 is the probability of score zero, Pl the probablhty of score 1, P2 that of 2 and so on, the p g f Is defined as a polynomml in x (an auxdmry variable) m wh,ch the coefficient of x N is PN Thus, m th~s case, g(x) = p o + p ~ x + p 2 x 2 + p a x 3 +

etc

(A1 1)

A simple example 1S the throw of a single d,e, which if unbmsed may gwe any score m the range 1, 2, 3, 6, each with the same probability 1/6 Here the generating function would be g(x)

1 "-[-'~'X -1 = -~'X

"['-lx3

1 . . 4 _-r~.~ 1 ..5 + ~ x 6 +-~.~

(A1 2)

The most useful property of generating functions is that, if two or more (statistically) independent processes are combined, the total score being the sum of all the separate scores, then the p g f for the total ~s obtained by mulUplymg together the consmuent p g f's Expressing the product as a power series in x, the coefficients of x °, x ~, x 2, etc, then gwe the probablhtles of results 0, 1, 2, , respectwely For example, consider a throw of two dice (or two successive throws of one) The p g f for the total score ~s

[~-(x + x z + x 3 +x 4 + x 5 +x6)] 2

= -5~(X 2 -b2X 3 + 3 x 4 + 4 x 5 +5x 6 +6x 7 +

+ 5 x S + 4 x g + 3 x l ° + 2 x 11 +x 12)

(A1 3)

Th,s result shows that the total score may be any integer from 2 to 12, mclus~ve, the respectwe probabllmes being 1 36)

2 36)

3 36'

4 36)

5 36'

6 36)

5 36)

4 36'

3 36)

2 36'

1 36

(A1 4) which Is simply the coeffioent of x 5 m the product gwen m e q (A13) The above example, m wh,ch the two processes are identical and all the p's are equal to 1/6, can be extended to two d,sslmdar processes described by their separate p g f s Ps = PlP4+PzP3 +PAP2 +PAP1,

g l ( x ) = po + p l x +p2x2 +p3x3 +

if2(X)

=

(A1 5)

qo ']-ql x q"q2 X2 ']-q3 X3 -.I-

Here the probabdlty, P~v, of total score N is gwen by PN = PNqO+PN-lql +PN-2q2 +

+ POqN,

(A16)

which ,s ev,dently ~dent~cal to the coefficient of x N ,n the product gl(x)g2(x) So the generating funct,on for the combined score is gwen by G(x) = g l ( x ) g2(x) (A1 7) An elementary property of any p g f may be deduced from eq (A1 1) Putting x = 1, we have g(1) = Po"FPl-I"p2 -b

= 1,

(A1 8)

since the sum of the probabdmes of all poss,ble scores must be umty This provides a useful check of the accuracy of more complex forms of generating function, since ~t is true for the individual factors, it also apphes to products, e g that m eq (A1 7) A1 2 THE POISSONDISTRIBUTION A probablhty generating functlon need not be hmlted to a fimte number of terms, as in the examples above The counting of random pulses m a

M E T H O D S FOR M O N I T O R I N G P L U T O N I U M IN WASTE

given time interval T results m a total count of N pulses with probability PN, given by Polsson's formula PN = (rN[ N)) e-" (A1 9)

601

finally the standard deviation cr (or variance a 2) IS calculated from a 2 = N2-(/V) 2

(AI 13)

Example For the P o i s o n dlstnbution, g ( x ) = e r(` ~), Here r = ~T, the mean expected number of counts, from eq (AI I0) therefore, g ' ( x ) = r e "(~-~) and and ~ is the mean rate of random pulses N can g"(x) = r2e '(~ 1) have any integral value whatever, but for large K/ Therefore from (A1 11).N= r = ~T, from (A1 12), the probability tends to become vanlshmgly small ~ = r2+~V= F + r , therefore from (A1 13), For such a Polsson process, the p g f is a 2 = F + r - r 2 = r, therefore a = ~ - = O(x) = ~ pNx N = e-" ~ (rx)N = e-" e "~ = e "(~- 1) N=0

~

N)

(AI.10) In thls case the generating function is an infinite series, which sums to a simple exponential functlon Note that g(1)= I A1 3 CALCULATION OF MEAN AND STANDARD DEVIATION FROM THE P G F

Since the generating function contains the complete probability distribution, as a set of coefficients, one might expect to be able to deduce the mean value and standard deviation from it This can be done without extracting the individual probabilities, if the p g f is known Differentiation of eq (A1 l) gives g'(x) = l p l + 2 p 2 x + 3 p a x 2 +4p,~x a +

Appendix 2. The statistics of a measured Poisson distribution

A2 1 GENERAL Suppose we have a true random pulse train, at mean rate ~ pulse/s, which IS measured by counting the pulses in each of a large number, M, of equal time-gates, all of width T At the end of the measurement we have recorded Co gates with no pulses, Cl gates with one pulse in each, C2 with two pulses, and so on In accordance with Polsson's formula, one expects CN = MpN, where PN = [(nT)N/N~]e -~r,

however the actual observations would contain random errors as follows Co = Mpo + % Cx = M p l +ex

Puttmg x = 1, g'(1) = l p l +2pe+3p3 + le g'(1) = ~V,

(A1 11)

x4p4x 2 +

The set of numbers Co, Cl, etc would be the contents of the various " c o u n t registers" in fig 1 and constitute the raw data of the experiment The mean number of pulses per gate and the spread, or variance S 2, of numbers of pulses in the various gates are calculated from _

N = Yl

= (12-1)p~ + (22-2)pz + ( 3 2 - 3 ) p 3 x + + (42-4)p4x 2 + SO

- (1Pl +2p2 +3p3 +4p4 +

= N2-2~

NCN

and S2 = N2-(•) 2 ~o

= .~ u~=oU 2 C u - (.~)2

)-)

(A1 12)

where ~ denotes the mean squared value of N The mean ~V is obtained directly from eq (A1 11), this result is then used m (A1 12) to give N~ and

(A2 3)

N=0

1

g"(1) = (12p~+22p2+32p3+42p4 +

(A2 2)

C2 = M p 2 + e 2 , etc

where N denotes the mean value of N, whose distribution is defined by g(x) A second dlfferenhatlon of eq (A1 1) yields g"(x) = 1 x 2 p 2 + 2 x 3 p 3 x + 3

(A2 1)

(A2 4)

Both these quantities are expected to have the value ~T, or r for brevity, but the primary object here is to determine how the errors e0, el, e2, etc are propagated through the calculation, causing deviations of N and S 2 from their expected values

602

M O DEIGHTON

First we esttmate the magmtudes of the e's Since there ts probability PN that any gate contains N pulses exactly, the actual number of such gates occumng m a total of M is

An ldenUcal argument apphes for any two (different) values of N and we can wnte

MpN -I- [MPN (1 - PN)] 1/ 2

Th~s result lmphes negatwe correlaUon between any two of the %, el, %, errors. Such correlaUon ts merely a reflection of the fact that the algebraic sum of all the eN tS zero, since Co + C1 + C2 + to mfimty = M, exactly, m all experiments

this being a bmomml dlstrlbuUon Hence (82)AV = MpN(1 -- PN),

(A2 5)

for any value of N from 0 to oo At this point it is worthwhile to emphasize the d~stmctlon between two kinds of average occumng above, one denoted by a bar, e g N 2, slgmfies an average of a quantny (here N 2) over all the results of one e x p e n m e n t ( of M gates) An average denoted by the bracket ()A~ ~' signifies an average of some quantity m the data, e g t~, over all the possible expenments of the same sort (unhm~ted m number) that one could imagine camed out It is helpful to ~magme all the results (and subsequent processing) of one experiment as wmten out m one honzontal hne, with those of all the other (hypothetical) expenments wntten m the same way in hnes below All corresponding elements would be in .~he same vertical hne and ()AV then denotes a verttcal average, or ensemble average, whereas the bar over any quantity denotes a horizontal average, over the data of one experiment The sole exception is ~, the mean pulse rate Besides the mean square error of CN, specified by eq (A2 5), there is another quantity of ~mportance, namely the correlation between these errors Consider, for example, C2 and C5 The probability of getting e~thcr 2 or 5 pulses m a gate is P2 +Ps, again we have a binomial distribution, hence out of a total of M gates the number containing either 2 or 5 pulses ~s

(A2 8)

(/~N1 ~N2)AV = -- MPN1 PN2'

A2 2 STATISTICSOF MEAN N N Is a single number emerging from one expenment, hence reference to ~ts staUsUcs zmphes ensemble averaging For example

<'>A'-from (A2 3) I oo

= M -- ~, N (CN)AV , since the vertical average of a (honzontal) sum is the sum of the separate vertical averages

= -M1 ~o NMpN, since (eN)AV = 0, for every value of N = ~'. NpN = h T ,

(A2.9)

a relatwely trlwal result The vanance of iV, obtained from all possible experiments, is var (N) = <(i~)2>AV -- I-AV"] 2 Now 1 ~° 1 N = M~o NCN = E N(MpN+et~)

=ENpN+IENeN

C2+C s = M(p2+p5 ) +_

1

= r+~Nsn,wherer=~T.

_ {M(p2 +P5) E1 -(p2 -t-Ps)']} 1/2

Therefore,

le ((~2 -1"~5)2)AV = M [ P 2 +P5 _ p E _ 2 p 2 p 5

_p2]

( A 2 6)

(N) 2 = r 2 +

~Nsn+

N8

But, from eq (A2 5), (~22)Av = Mp2(1--P2 ) and (e2)AV = Mps(1--p5) SO addmg these,

(~ +8~>AV

2

2

M[P2 +Ps--P2 --Psi

(A2 7)

= r 2 + ~2' r~~N s N

1 ( o~ N28~+2 x +~X

× nx
Subtracting eq (A2 7) from (A2 6) y~elds

Smce

(e2 85>AV = -- Mp2 P5

(~N)AV = 0

and

(82)AV,(SN18NZ)AV

603

METHODS FOR MONITORING PLUTONIUM IN WASTE

M = 2 ) For any reasonable value of M, the difference between the expected S 2 and r is negligible Lastly we come to the main quantity required, namely

have values given earlier ((N)~).v = r 2 +0 + ~

+ 2 N,E<~2

N2MpN(1 -PN) +

var (S 2) = ( S 4 ) A V -- ( ( S 2 ) A V ) 2

NIN2(-MpNlPm))

(A2 12)

We have

= r e +-~

N2pN-~N2p~S 2=

0

$2 r2 + ~4

N2p~

-

NCN

,

from eq (A2 4)

-- 2N~<~2~ NtN2pNlPN2) =

~N 2cry-

1

N2(MpN +eN) - ~-$

N(MpI~ +~N)

=-Mo

NpN

= r z +(1/M)(r 2 + r - r 2) = r 2 +riM. Hence, using both results above var (N) = (r2+r/M) - r E = riM

+ ~-EN.

(A2 10)

(

,

This result could have been obtained more directly as follows Total counting time of one experiment = M T Therefore, total counts recorded=r-tMT_+x/-K-M-T, since it is a random pulse train Therefore, mean counts per gate = ~ T _ + ~

1 = (r e +r) + ~ - E Ne"N -- " + K E N.,,

= r +_x/ r - ~

= r+--~o ( N 2 - 2 r N ) ~ - ' ~ 2

A2 3 STATISTICSOF VARIANCE S 2 ( S 2 ) A V = (N2)AV -- ((/~)2)AV,

from eq (A2 4) Now ----

1

oD

N2 = M ~=o N2 CN 1 ~

= M~o N2(MpN+eN)

=

,(

1 2r -~2r+-~EN28N----~-ENeN

'

)2

ENeN

NeN

/

•

Since the order of magnitude of each eN is propomonal to ~/M, other things being unchanged, the second and tMrd terms above are likely to be of order M -t, M -l, respectively Hence, for M large, it is permissible, when squaring the above expression, to neglect terms with multlphers M -3, M -4 Thus we obtain $4 = r2 +--~-~ 2r (N2-2rN)eN +

Therefore, 1

(N"~)AV = ~

N2pN,

2

2r

2

since ( n N ) . v = 0

0

--_ r2+r (see Appendix 1) Hence, using result above for ((N)2},v,

Therefore, (S'*)Av = r 2 + ~ - r

(N2-2rN)eN

AV

(S2)AV = (r 2 +r) - (r 2 +r/M) = r ( 1 - M -I) (A2 11)

Th~s result, at first surprising, expresses the fact that the (internal) variance of a sample drawn at random from a population ~s on the average shghtly less than that of the parent populatmn (It is easily verified by cons~dertng extreme cases like M = 1 or

)2

Now

NeN

=

(N2e~) +2 0

E Nl
(NIN28NIBN2)

M

604

0

DEIGHTON

Therefore,

r3 = g”‘(1) = f

N(N-l)(N-2)p,

N=3

@

= $N'MBN(l-p,)

N4?~V

+ N(N-~)(N-~)PN,

= j0 + 2 N TN, NIN,(-MPNIPN,), 1

whose first 3 terms are zero anyway,

using eqs (A2.9, (A2 8), = M

(

- f

fN2P,

=f$N”-3N2+2N)pN

N2P?-2

c

NINZPNIPNZ

>

= f N3pN -3fN2p,+2fNp,

=M[;NzpN-~~NpN~]N’qN’

0

0

0

Therefore = M(r’+r-r2)

= Mr

fN3pN

Similarly

= r3+3(r2+r)-22

= r3+3r2+r

0

Likewise the fourth derivative gives $(N’-2rN)+

>

(

’

r4 = Nzo N(N-l)(N-2)(N-3)~~

= $ (N2-2rN)‘&+2

c

(Nt-2rNI)

x =

NI
=

1

X (N: -2rN,h,

&N2

Therefore

$

(N4-6N3 +I1 N2-6N)p,

zN4pN-6~N3pN+ll~N2pN-6zNpN

Hence AV

N4pN = r4 + 6(r3 f3r2

f

, I

=~(N2-2rN)2Mp,(l-p,)

i-r) - ll(r’+r)

+ 6r

0

+

= r4+6r3f7r2+r.

(N:-2rNI)(Nz-2rNz)(-MPN,P& NI
+ 2 ’ c

‘LM -

The expression above for the ensemble average of

‘&N2-2rN)2pN-~(N2-2rN)2p~( 0

(N2-2rN)EN

c

ZNTN 1

2

w:

-~~N,)(N~-~~N~)PN,PN~ >

’ >

(

thus becomes 2

= M

f(N’-2rN)‘pN-

[ 0

(

f(N’-2rN)p,

M

)]

To evaluate this expresslon we have to find the values of * 5 N3p, 0

and

CN4pN,

(N4-4rN3

c

[

-

-I-4r2N2)pN -

x(N2-2rN)PN (

=M[r4+6r3+7r2+r +

4r2(r2 +r) - (r2

)I ’

-4r(r3+3r2+r)

fr-2r2)21

0

they are fairly easily denved by an extension of the method outhned In appendix 1, section Al 3 Putting x = 1 in the 3rd derlvatlve of the probablhty geieratmg function for a Poisson dlstnbutlon gives

= M(2r2 +r) Therefore
=

r2 + c.$_

2r zMr=r’+k

-I-

METHODS FOR MONITORING PLUTONIUM IN WASTE

So, finally v a r ( S 2) = r 2 + ~ -r-

605

References r2 (

1-

M1---)2= 2 r 2M+ r

'

(A213)

using eqs (A2 12) and (A2 11) and neglecting r2/ M2

t) K P Lambert and J W Leakc, AERE-R 8300 (1976) 2) K P Lambert, AERE-R 8303 (1977) 3) K Bohnel, Dm Plutomumbestlmmung m Kembmnnstoffen mat der Neutronenkomztdenzmethode KFK 2203 Kaflsruhe (1975) 4) C H Vincent, Nucl Instr and Meth 138 (1976) 261