Dead-time correction in multichannel analyzers

NUCLEAR

INSTRUMENTS

AND

METHODS

93

097I)

I3I--I32;

C~ N O R T H - H O L L A N D

PUBLISHING

CO.

DEAD-TIME C O R R E C T I O N IN M U L T I C H A N N E L ANALYZERS J. S I M M E R

Laboratory of Algology, Institute of Microbiology, Czechoslovak Academy of Sciences, T~ebo~, Czechoslovakia Received 24 September 1970 and in revised form 10 December 1970

Carloni et al. a) proposed an original method of dead-time correction in multichannel analyzers, by means of a calculation procedure based on the distribution moments of recorded impulses. Although their basic idea is correct and can be of much use, the authors, unfortunately, failed to present a correct solution of the whole problem, having made a wrong mathematical expression of the dead-time effect. The incorrectness rests in the assumption that the probability density of any of the poissonian distributed events occurring depends on the number of those previously observed [see eq. (1) in ref. 1]. There is, however, an exact solution of this problem. Let us assume: i) a source of quite randomly distributed and infinitely short impulses with the mean frequency Fo; ii) a detector counting each impulse which occurs after the dead-time t following every recorded impulse; iii) a multichannel analyzer accumulating in the kth channel the number nk of the time intervals G (gatetime) during which k impulses were exactly recorded. It follows from i) and ii) that the detector counts one impulse if the source sends at least one impulse during the time interval of length t. The probability p of such an event is p = 1-exp(-tFo)

(1)

as it was extensively proved in ref. 2. The probability

Pk(G) that this event will occur in the gate-time G

hence

t = (G/M) (1 -- D/M). Combining (1), (3) and (4) we get finally

G(M-D)

p = Mt/G.

lim Fo = M/G.

using the relations (4) and (5), t and Fo are calculated with an exactness limited only by the statistical error in estimating the values of M and D. The best estimation is obtained by using the well-known formulae M = 1 a# ~ knk -

and 1

G/t

( k - M ) 2 nk,

D ~-.--

n--1 k=O where n Y~nk. The standard errors E[x] of the sample values of M and D are defined by the relations =

=

(D/n)~

and

(1-6D/M.(1-D/M)) =

D(2)~"

½

1

By means of these values we get g [Fo] as E[fo]

D = (G/t) p ( 1 - p )

-

r/ k = l

(3)

Variance D of the binomial distribution (2) is

(5)

D

D~M

EED]

since each event occurs independently. If M is the mean number of impulses recorded during G, then

M In--.

For D ~ M holds in accordance with the intuitive meaning

EEM] (2)

M2 -

Fo = -

just k-times is obviously Pk (G) = (G~t) pk ( l -- p)a/t-k ,

(4)

=

,,--

2_ t _

,

L\aM

where

or, in terms of (3)

8Fo=

aM

D = M(1--MUG ) 131

M [1-(~-G(M-D) --D

2) l l nM

132

J. SIMMER

and

OFo OD

lim oF° D-.M OD

G(M-D)

M-D

D

For the reason mentioned above, Carloni et al. arrived at quite different results. Let us compare the values of F6 and t', obtained by using the asymptotic relations (15') and (16') in ref. 1, with F 0 and t according to (5) and (4) in this paper. The comparison, presented in fig. 1, shows that (15') nearly gives the exact values if D / M > 0.7. The same graph, however, shows that t' is approximately equal to t/2 for the same range of DIM. This fully corresponds with the conclusions presented in ref. 2 that the real dead-time is twice as long as that calculated (for tF ~. t) from the usual (but wrong) formula

I f D --* M, then lim OFo D--,~ c3M

2.0

.

.

.

.

3

2G '

j

.

.

.

1 2G

.

F = Fo/(l + tFo), "LO

O0 O0

which is identical with the asymptotic formula (13) in ref. 1; F is the frequency of recorded impulses, i.e. in our case F = M/G. The agreement between theory and experiment, presented by Carloni et al., is therefore only apparent. I

,

.

,

o

t0 ~M Fig. 1. Explanation see in the text.

R eferences

1) F. Carloni, A. Corberi, M. Marseguerra and C. M. Porceddu, Nucl. Instr. and Meth. 78 (1970) 70. 2) j. Simmer, Naturwissenschaften 56 (1969) 633.