Analytical approach to detector efficiency in a uniform isotropic flux

Nuclear Instruments and Methods 196 (1982) 463-464 North-Holland Publishing Company

463

ANALYTICAL APPROACH TO DETECTOR EFFICIENCY IN A U N I F O R M ISOTROPIC FLUX

D. KWIAT Department o[ Nuclear Engineering, Technion

Israel Institute

of Technology, lla(fa, Israel

Received 21 September 1981

It is known that the efficiency of a detector ~(x) is connected to the escape function P( x ) by the relation c(.,) tP( \- ) [ I]. In this paper, a specific detector is considered (3"× 3" NaIL and the results obtained are compared with previously published rcsuhs [6,9,10].

1. Efficiency and escape probability

Using the same kind of argumentation as in eq. (AI) one can show that [8]

Assuming a convex geometry for the detector, we have [3,6] for the efficiency

~(E)=

fdsL.~>od~OI(s.w,E)fi'go[l--e

f

n

R,,~ l - ~ r s x P ( x ) --2~rs ~

"':):'"~)] Using the first order approximation we obtain [1].

dsL.a,>0 do) fi" go I(s, ~o, E)

Io +

(1)

c(x) = XP(x)

S

d c o f l ' ~ [ 1 _ e ~,(,~,:,,,o,].

(2)

fi-g~>0

The escape probability function is given by [2] P(x)=lfdlf(l)(l-e

"'),

(3)

where x = ~ l and [ the average chord length is given by [= 4 v/s (v-detector volume, s-detector surface area). One can then show [1] that ¢ [p,(E)] =e(x) =xP(x).

(4)

(

1

')

3XP(x) I~) + 2I I

For an isotropic and uniform flux I(s, ~o, E ) = I(E) we obtain [1]

~(E) lfdsf

I

j=l (J+l)(j+2)

" II (6)

From this expression and the fact that XP(x) ~ 1 for X>>I we see that for X>>I ¢ ( x ) = X P ( x ) independently of the anisotropy, X >> 1 is equivalent physically to either large dimensions or a large absorption coefficient for the detector. Using now eq. (8) we have after some algebra [8]: Io ÷ ~

.~(x)=xp(x)[

1

:=l ~ l+ij!

(7) 2. Deviation from isotropy In eq. (2) we made use of the isotropy approximation, where we assumed l(s~o)= I o. In practice, however, the insertion of a detector in a uniform isotropical flux introduces anistropy. In ref. 1 it was assumed that l(s~o) = I o + (h. go) Ii, where 11 is the disturbance term (i.e. deviation from isotropy, caused by the insertion of the detector). A better approximation could be obtained by using l(s, ~0) = E~:=0 (h. go)q: (!: does not depend on s here) [8]. 0029-554X/82/0000-0000/$02.75 © 1982 North-Holland

3. Comparison with earlier results Schaarschmidt and Keller [6] have calculated eq. (1) numerically for a cylindrical detector. Taking a NaI detector with dimensions H = D = 3"(3" × 3") in an isotropic 7-flux, they have plotted efficiency vs. ~R(fig. 5a in ref. 6). Recalling that X = ~ l a n d [ = 2D/3 for an H = D cylinder, we can derive efficiency vs. X and compare it with the results in table 1. The results are

D. Kwiat

464

Detector e~l~icwmlv

Table 1 Efficiency values for various geometries. ¢ ( x ) = x P ( x ) where x : ~ {

Sphere [7]

.\

0.067 0.267 2.000 5.333

0.064 0.231 0.822 0.960

Infinite [7]

Cylinder [4]

Infinite [5]

cylinder

ff= D

square cylinder

0.063 0.228 0.814 0.949

0.064 0.227 1L775 11922

~h067 0.223 0.768 0.907

Cube 141

().f)64 ).225 )748 n ~c)O

with R~ = TrsxP(x). Writing now:

Table 2 Total efficiency for a 3" × 3" NaI "/-detector. x

Table 1

Ref. 6

E(MeV)

0.267 2.000 5.333

0.227 0.775 0.922

0.775 0.922

1.20 0.18 0.13

R~_-f dsf dw(n.o~)2(1 e

....... I

i.e.. R 2 = ( n- 00)R~, where ( n - o ~ is a number between 0 and I to be determined. T o g e t h e r w i t h eqs. (2) and (3) one gets: shown in table2, and seem to approve the current approach. In refs. 9 and 10 the efficiency was measured and calculated with an isotropic point source placed at [0 cm from the detector on the crystal axis. Obviously, different results are obtained since the efficiency by its definition depends also on the flux distribution.

Appendix Let us define [1] the integral R,,:

R =f dsf,.,~>odo~(n.o~)"(1-e ""~'~').

(A1)

(2)

e

1 3xP( x ) "

References [I} D Kwiat. submitted for publication to Nucl. Sci. Eng. ( 1981L [2] D. Kwiat, Nucl. ScL Eng. 78 {19811 192. [3] R.M, Kogan, I.M. Nazarov and Sh D. Ffidman, Gamma spectrometry of natural environments and formation,,. ~Moscow. 19691 p. 153 (translated from Russian - Jerusalem c l971J Israel Program for Scientific Translations. [4] I. Carlvik. Nuct. Sci. Eng. 30 (I967) 150. [5] D. Kwiat, Submitted for publication to Nucl. Sci. Eng.

72 (19691 82. [7] D. Kwiat. Nucl. Sci. Eng. 76 11980) 255. [8] D, Kwiat. Submitted for publication to Int. J Appl. Rad

(,, .,,,)2 ~ 0

.l(,,+ ~ 0

With these assumptions one o b t a i n s : R 2 ~ , R I --½¢rs

I

(1981). [6] A, Schaarschmidt and M.J. Keller, Nucl. Instr. and Meth_

W e next a s u m e for n • ¢o ~ 1 :

(I) ,, - , ~ - -

~n. ~)=

(A2)

Isotopes 11981 ). [9] M. Giannini e~ al.. Nucl. Instr. -and Meth. 81 (1970) 104, [10] M, Betluscio et al., Nucl. Instr. mad Meth. 118 (t974) 553.