Neutron coincidence

N U C L E A R I N S T R U M E N T S AND METHODS 158 (1979)

1 8 1 - 1 8 4 ; (~) N O R T H - H O L L A N D PUBLISHING CO.

NEUTRON COINCIDENCE The detection of groups of associated pulses in a train otherwise comprised of randomly timed single pulses* C. H. VINCENT

Minhoy of DeJence, A WRE Aldermaston, Berkshire, England Received 29 June 1978 The detection and measurement of groups of associated pulses in a train otherwise comprised of randomly timed individual pulses is an important problem in nuclear instrumentation and is required for such purposes as the assay of plutonium, using the fission grouping. A solution has been found which is believed to be optimal for the condition where this is most needed, ie for low fission rates. The design of suitable hardware to implement this method appears to be quite practicable.

1. Introduction An important problem in nuclear instrumentation is the detection of groups of associated pulses in a pulse train otherwise comprised of randomly timed individual pulses. In particular, the neutron coincidence method is often used in assays for plutonium, making use of the fission grouping in the train of detector output pulses. BShnel L2) has described a method based on the use of a shift register to count all pairs of pulses (consecutive or otherwise) in a pulse train, falling within a specified time interval of each other. An alternative method, based on examination of the spectrum of pulse intervals separating consecutive pulses, has been suggested by Andrew and Tarrant and analysed by Vincent3). The purpose of the present paper is to seek an optimum method of processing the information contained in the train of input pulses and to suggest a practicable hardware system design for implementing this. 2. Theory The presence of the correlated neutrons from the fissions will modify the spectrum S(D) of pulse spacings D, which is defined as being such that the rate of occurrence of intervals between pairs of detected pulses lying in the range D to D + f D is given by S ( D ) f D , for small fD. For single-particle events only, with rate Ps and detector efficiency e, we have the simple relationship S,(D) = e2p 2.

(1)

Consider now a source which also generates fis* Copyright © Controller HMSO, London, 1978.

sions at a rate Pr, from which the neutrons are also detected by the same detector with efficiency e. Let v be the number of neutrons emitted in a fission, which may vary from one fission to another. The probability that a neutron will be detected between times t and t + f t after its emission is given for small fit by Pd(t) 6t = e~ e -~' 6t,

(2)

where cz is a fixed decay rate characteristic of the detector assembly. Under these conditions, it is clear that all the intervals concerned are between pulses occurring quite independently of each other, with the sole exception of the intervals between pairs of pulses from the same fission. This readily leads, without any approximation, to the well known result: St(D) = ~2pt2 + ½V(V--1) e2pr~ e -=°,

(3)

where Pt = Ps + ~)Pf.

(4)

Consider a measurement made by counting the number of intervals falling into each of a set of discrete channels of equal width Dr-Dr_~ = A D , where D O= 0. Let there be n channels altogether, so that the total time interval scanned is nAD = D, and let aiD be sufficiently small to give satisfactory definition of the spectrum shape, so that aiD,~ l/~z. Let the count in the rth channel of this spectrum be Cr. For specified Pr, its expected value for total counting time T would be given by Cr = A + Brpf ,

(5)

where A = e2p2tTAD,

(6)

182

c . H . VINCENT

and B, ½ v ( v - 1) e2 T [-exp(- ~O,_ 1) - e x p ( - ctD,)]. (7)

where/3
If the variable term in C, in eq. (5) is small in comparison with the fixed term, as will be the case for small ,or , the standard deviation of all the counts will have the same value flA. The best fit is therefore given by unweighted least squares, for which we seek to minimise

Wr =

=

e -ao'-'

e -aD" - e -#°'-' + e -#°',

(15)

which complies with eq. (13) for n--, ~ . The ratio of increase in the variance in this case over the minimum, assuming A D ~ I / o c and D , > 1/oq can be calculated as 0" 5

( A + B r P t - C r ) 2.

S = i

-

(8)

Ct2 e - s a t

dt

0

foo (ct e -at .

fl e-#t) 2 dt

-- [fo o 0~ e -a' (0c e -a' -- ,6' e -#') dt

r=l

Differentiating with respect to Pr and equating to zero, we obtain

2

~ + p ~-

=

~+/?

(16)

giving Br(C,-A)

,= 1

Pt" - -

n

(10)

,

z

r=l

in which the counts C, are given weightings proportional to B,. If it is assumed, for the moment, that A is known exactly, this leads to an optimum estimate of Pf having the variance

0.2 - -

A

r=l

(7 r=l

(11)

It follows that with the conditions assumed and with/3 <~ ~, a nearly optimum performance should be obtainable. A practicable method meeting all these requirements will be described. 3. C o m p a r i s o n

In one form, this method ~.2) gives a constant weighting K, say, to all intervals in a range 0 to D,, and gives the same negative weighting, - K , to all intervals in a range D~ to D~+D,,, where D~>D,,,, thus complying with eq. (13). The result corresponding to eq. (16) is then given by:

r=l

0.2

In the limit AD--,O and D,--, oo, this takes its minimum possible value, which is 2

with the register method

am

:o ct2 e-

2at

dt

{fo"

[-~)]2

e2 T

(12)

In a practical situation, it is desirable to use a weighting IV, such that (13)

r=l

which avoids the necessity for any separate determination of A. We shall seek such a weighting a which does not appreciably increase the variance of the determination of Pr. Consider the function w(t) = o: e-at-~3 e -#t,

.

K 2 dt

}

Ko~ e -al dt

~Dm = [1 - exp(-eO,)] 2 [1 - exp(-eD,,)] ~'

[v(v--(-~T)1 ] 2 82 T0~

IV, = 0,

r

,,lot

d D~

;oo~Ze-2at dt

8pt2

=

dt +

Kct e -at dt -

4pt2

0.m ~---

K 2

(14)

(17)

It is easily found that this expression has a minimum at ~,Dm= 1.25643, the value at that point being a2

2.4554 = [-1 -" exp(-~O,)] 2"

(18)

This can be reduced towards its minimum value of 2.4554, by making D~ large compared with 1/~x. A further reduction is then possible by lengthening the negatively weighted comparison interval by a factor R and reducing the magnitude of the weighting given there by the same factor, to com-

NEUTRON

ply with eq. (13). In this case O-2

~-=

1.2277 (1 + 1/R).

183

COINCIDENCE

count recorded is reduced by 1/2' of itself. This has the effect of giving an exponential decay of any number present, when there is no further input, or of maintaining a count proportional to the mean input rate, when there is input. The counter marked X in the figure has a subtraction at every clock pulse. Its decay time constant is therefore 2*T~, where T~ is the clock period. This time constant is adjusted to equal 1/a~. Each input pulse that the counter receives adds 2j+* to its count, being fed to the appropriate stage. The other counter, marked Y, has a subtraction only at every 2* clock pulses, while.each input pulse adds only 2J to its count. Its decay time constant is therefore much greater, 22* T~ = 1//3, although the mean values X' and Y' are exactly the same. (X' and Y' omit the last k digits.) The digital method used ensures an exact equality of these mean values that would not be possible in a corresponding analog system. The parameter j is adjusted in accordance with the mean input pulse rate to give a suitably large value to these equal mean values, say, 2 *-2 to 2 *-~. The X' value corresponds to the first term and the Y' value to the second term in,eq. (14).

(19)

na

Since the excess variance decreases only inversely as R and since the interval D,, = 1.25/o~ already represents a considerable number of shift-register stages for small dead time, a large number of such stages will be required to provide the interval RD,, for R >> 1, before the limit of 1.2277 is approached. 4. Practical implementation of the

optimal processing Fig. 1 shows the proposed method. The pulses are received in a clocked input circuit, which can feed one pulse per cycle (if received) into each of two binary counters. There is therefore a maximum dead time Of one clock period, exactly as in the shift register method. Each counter has 2k stages and both have the associated circuits needed to give a digital ratemeter action4.5). This action is described in detail in the references given, but may be summarised by saying that there are regular subtractions in each of which the total < >

XRATEMETER 2k-1

[_.

" E

X

INPUCT .. 'j

,NPUT I

CIRCUIT

k

~ ~ DF 'CIRCUIT FERENCE

_I

ACCUMULATO RJC I[ J

I

CLOCK I

1 ~

J

<

I

-zk

J

YRATEMETER 2k-1

F"

>

-i

_

Fig. 1. Block diagram of the system. The total binary numbers in the ratemeters are referred to as X and Y respectively. The corresponding numbers formed by the first k bits only are referred to as X' and Y". The latter is stored for 2k cycles after each Y subtraction as Y'.

184

c . H . VINCENT

The latter value is subtracted from the former by means of a fast static difference circuit. The output of the difference circuit goes to an accumulator. The actions that occur during each clock cycle are then as follows. (Two of these are conditional on an input pulse having been received during the last cycle.) 1) (Conditional on input) X'--: Y' is added to the accumulator contents. 2) X' is subtracted from X. Every 2kth cycle, Y" is substracted from Y and is also stored as Y' for the next 2 k cycles. 3) (Conditional on input) 2~+k is added to X and 2j is added to Y. As an example, consider a detector time constant (1Az) of 100/zs. For k = 6, this would require a clock period T~ of 1.56/zs. This appears to be readily feasible by the standards of modern digital electronics. The corresponding variance ratio is

l+fl/ot = 1-1-2

a2/a2m :

-k

~-

1.0156.

(20)

This is a negligible increase over the minimum variance. Shorter detector time constants could be accommodated by reducing Tc. Longer time constants could be accommodated by increasing k, say, to 7 or 8. For accumulator reading N, the fission rate is given to a close approximation by 22-JN

Pf

- _ _

V(V-- 1) e 2 T

5. L i m i t a t i o n s

.

(21)

on input pulse rates

The dead-time performance of this circuit is exactly the same as that of a shift register having the same clock rate and similar considerations therefoi'e apply at high input pulse rates. Faster digital logic circuits are likely to be available in future, permitting an even smaller dead time and a higher value of k. It is also probable that the number of stages in each counter could be increased above 2k, if 'necessary, to avoid any risk of overflow, especially in the X counter. (The separation of the stages into two distinct groups, as in the figure, would then no longer apply, as they would overlap.) The risk of an overflow is negligible, provided that the overflow point is about

ten standard deviations or more above the reading. Provided that j is increased to give a good approximation to the desired exponential response from an isolated input pulse, there is no minimum input pulse rate, other than that set by the acceptability of the rate of accumulation of data. Moderate variations of background rate during a measurement should have very little effect, since this is accurately cancelled out within a very short time (of the order of 2kAZ).

6. C o n c l u s i o n

A form of digital data processing for clocked input pulses has been found which is very nearly optimal for the measurement of the rate of included fission groups where this rate is low in comparison with the single pulse rate. The optimum processing is therefore effective for the low fission rates for which it is most necessary. The hardware requirements to implement the method do not appear to be unduly exacting. It is a particular advantage of the system that the increase in the variance of the measurement over the minimum possible can be kept to negligible proportions, using only a modest number of stages. Although the weighting will no longer be exactly optimal at higher fission rates, no rapid departure is to be anticipated and the increased rates will, in any case, give better statistical accuracy in the measurement, expressed as a percentage. The author would like to thank Mr. J. B. Parker of this Establishment for reading this paper and for his helpful comments on it.

References

l) K. BShnel, Kernforschungszentrum, Karlsruhe, Institute of Neutron Physics and Reactor Technology, Nuclear Safeguards Project, Report KFK2203 (August 1975)(in German). 2) Ditto, AWRE Translation no. 70 (54/4252) (March 1978). 3) C. H. Vincent, Nucl. Instr. and Meth. 138 (1976) 261. 4) C. H. Vincent and J. B. Rowles, Nucl. Instr. and Meth. 22 (1963) 201. 5) C. H. Vincent, R a n d o m pulse trains - their measurement and statistical properties (Peter Peregrinus, 1973).