Error estimates in high count rate experiments

NUCLEAR

INSTRUMENTS

AND

METHODS

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E R R O R E S T I M A T E S IN H I G H C O U N T RATE EXPERIMENTS* B. L. CHRISMAN National Accelerator Laboratory, Batavia, Illinois 60510, U.S.A.

J. L. GROVES Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, U.S.A.

Received 20 October 1972 We have investigated the effects of pulse counting rate on chi-square calculations for the experimental arrangement used in M6ssbauer effect studies. Pulse c o u n t i n g o f nuclear a n d a t o m i c r a d i a t i o n s is used in a wide variety o f physical experiments. F o r the o b s e r v a t i o n o f physical p h e n o m e n a which have a small signal-to-noise ratio, increasingly high count rates are used; e.g., in a t t e m p t s to observe the violation o f time reversal invariance using the M 6 s s b a u e r effect1), in the study o f parity violations in nuclear g a m m a transitions2), in the o b s e r v a t i o n o f the M 6 s s b a u e r effect in proteins, etc.3). The d a t a r e d u c t i o n for such experiments usually involves a least-square fitting o f the d a t a to a theoretical function and a statistical estimate o f * Work supported in part by a grant from the National Science Foundation NSF GP 26282. 1.2

~-" <[ I.O o co0.8 i

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i ~ 0.6 c~ w ~ 0.4 t23 tad n~ 0.2 I

20

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40 60 r:t(lO 3 COUNTS/SEC.)

~r = v I ( r T ) ,

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Fig. 1. Reduced g z as a function of the input count rate. The input count rate, ri ,was varied by changing the source to detector distance. A reference count rate in the 14 keV peak was measured with a large source to detector distance where the pulse overlap problems were negligible. Then, ri was determined for each position of the source by multiplying the reference count rate by the detector solid angle ratio. A pulse height spectrum was taken for each source to detector distance and the energy window of the single-channel analyzer was set on the 14 keV line. The shaded area in the figure is the range ef reduced chi-square values expected. The probability of observing a reduced chi-square outside the shaded region is 0.02, 0.01 above and 0.01 below. 193

(1)

where r is the c o u n t rate a n d T the counting time. I t is well k n o w n t h a t eq. (1) is only valid for d a t a collected in the limit o f no pulse overlap in the detectors or electronics4). H o w e v e r , experimenters have frequently failed to consider the pulse overlap p r o b l e m s in calculating °' goodness o f fit" criteria, (e.g., Z2) a n d in estim a t i n g the statistical errors in the p a r a m e t e r s reported. To illustrate the effect we p e r f o r m e d a counting e x p e r i m e n t with detector a n d electronics set-up as in a typical M 6 s s b a u e r experiment. The e q u i p m e n t used a n d relevant settings are given in ref. 5. The multichannel analyzer was o p e r a t e d in the scalar m o d e a n d the c o u n t i n g rate was varied by changing the distance f r o m the 57Co source to the detector. W e least-square fit each " s p e c t r u m " to a straight line, a n d in fig. 1 we p l o t the reduced Z2 [calculated assuming eq. (1)] versus the c o u n t rate. The r e d u c e d Zz is defined b y Z2 = ~

_1

80

p a r a m e t e r errors based on a Poisson d i s t r i b u t i o n for a t o m i c or nuclear decay. In the limit o f a large n u m b e r o f counts per d a t a p o i n t the Poisson d i s t r i b u t i o n yields a s t a n d a r d deviation

~

[f~-F(x,)] 2 ~,

(2)

i=l

where n is the n u m b e r o f d a t a p o i n t s in the fit, m the n u m b e r o f p a r a m e t e r s in the function F ( x l ) a n d Wi is the statistical weight or the inverse variance o f the ith d a t a point. F o r the fits s u m m a r i z e d in fig. 1, Wi = 1/cr~ = 1 / Y i , and F ( x i ) = constant. F r o m fig. 1 one observes t h a t Z2 is a function o f the counting rate decreasing to values with very small statistical p r o b a bilities for high c o u n t rates. The small values o f Zz are the result o f pulse overlap o r electronic j a m m i n g which gives a d i s t r i b u t i o n o f c o u n t e d pulses different from the d i s t r i b u t i o n o f incident photons. Thus for the Zz

194

B. L. C H R I S M A N

statistic to have an absolute m e a n i n g the weights W~ in eq. (2) m u s t be modified. The modification o f W~ has been studied for various idealized detectors6). W e will give a brief sketch o f the results b u t leave the details to the references6). Basically two types o f detectors are considered; Type I a n d T y p e II. A T y p e I detector has a fixed d e a d time z such t h a t a count will be registered only if there has been no count registered in the preceding time interval A t, with A t > z. A Type I[ detector has a dead time z such that a count will be registered if there has been no i n p u t count in the preceding time interval with A t > z. The c o u n t rate in a T y p e I detector increases m o n o t o n i c a l l y as the i n p u t c o u n t rate, r I increases, a s y m p t o t i c a l l y reaching an o u t p u t c o u n t rate, ro = 1/~. A T y p e II d e t e c t o r a p p r o a c h e s zero counts out as the i n p u t c o u n t rate r~ becomes m u c h larger than 1/r. F o r T y p e l counters it can be shown that for long counting periods (T>> r) r o = q/(I +rlr),

(3)

aoz = ro/(1 q- rl't) 2,

(4)

and

where ao2 is the variance on the o u t p u t count rate. Then Wi = (aoz T ) - I -- T( [) .-~ _ . ~rl

2

(5)

v, F o r T y p e II detectors ro = rle -r~, O"2o = r o ( l _ 2 r x z e - ~ ) ,

(6) (7)

and W/ = (a2z) -1 = [ ~ } ( 1 - 2 r , - c e - r l ~ - l ) ] .

(8)

AND

J. L. G R O V E S

Typically, electronic devices are o f T y p e II whereas mechanical devices are o f Type I. However, a s t a n d a r d spark c h a m b e r violates this rule. The d a t a in fig. 1 contains the cumulative effects o f the d e t e c t o r and all the electronic equipment. Therefore the weights, Wi, for o u r experimental a r r a n g e m e n t c a n n o t be described by a single p a r a m e t e r r. Thus we believe that for a precision m e a s u r e m e n t a statistical c a l i b r a t i o n experim e n t such as we have d e m o n s t r a t e d in fig. 1 is necessary for u n d e r s t a n d i n g the weights Wi t h a t go into a fitting procedure. O n l y if the correct W i are used in the least-square analysis does ;(2 and the s t a n d a r d deviation o f the derived p a r a m e t e r s have any statistical significance.

References 1) o. c. Kistner, Phys. Rev. Letters 9 (1967) 872; Atac et al., Phys. Rev. Letters 20 (1968) 726. ~) E. M. Henley, Ann. Rev. Nu¢l. Sci. 19 (1969) 367; W. D. Hamilton, Progr. Nucl. Phys. 10 (1969) 1. 3) E. Munck, P. G. Debrunner, J. C. M. Tsibris and I. C. Gunsalus, M6ssbauer parameters of putidaredoxin and its selenium analog biochemistry 11 (1972) 855. 4) Although the warning appears in standard texts, e.g., E. Segr~, Nuclei and particles (Benjamin, New York, 1964) ch. 3 and 5; see especially the comment on p. 174. 2) All the data points shown in fig. 1 were taken with a 30 mCi 57Co source, 1,~" diameter krypton proportional detector, Ortec 109PC preamplifier, Ortec Model 410 amplifier, Ortec Model 420 single-channel analyzer, and Hewlett Packard Model 5421A multichannel analyzer. The timing constants on the amplifier were adjusted to give the largest signal-to-background in the 14.4 keV peak at low count rates. At the highest count rate the 14.4 keV peak was still observable in the pulse height spectrum. 6) A. Bharucha-Reid, Elements o f the theory o f Markov processes and their applications (McGraw-Hill, New York, 1960) p. 299.