Calculated total efficiencies of coaxial Ge(Li) detectors

NUCLEAR INSTRUMENTS AND METHODS

91 (1971) 377-379; ©

NORTH-HOLLAND PUBLISHING CO.

CALCULATED TOTAL EFFICIENCIES OF COAXIAL Ge(Li) DETECTORS R. GRIFFITHS

NI/clear Physics Laboratory, University of Oxford, Eng/and

Received 12 October 1970 A program has been written to compute the total efficiences of coaxial Ge(Li) detectors. Graphs are presented which show how the efficiency of four typical detectors of active volumes 20, 30,

50 and 80 cm3 varies for gamma rays of energies 0.0 I to 10.0 MeV for source to detector distances in the range 1.0 to 25.0 cm.

1. Introduction

where S is the slant thickness of the detector, .u is the total absorption coefficient in germanium for the gamma ray energy in question and dQ is an element of solid angle over which S may be considered constant. The integration is carried out over the whole of the solid angle subtended at the source by the detector. For the coaxial geometry it is simplest to define dQ as the solid angle between two cones with apexes at the source whose intersections with the front face of the detector are circles ofradii rand r+ dr, see fig. 1. Then

The efficiencies of rectangular planar Ge(Li) detectors have been computed by Hotz, Mathiesen and Hurleyl). In many applications coaxial detectors are used in preference to planar detectors because of their larger sensitive volumes and consequent higher efficiencies while still retaining good timing characteristics. Therefore a program has been written which will compute the total efficiency of a given coaxial detector for a specified gamma ray energy and source to detector distance.

dQ =

2. Analysis

The total efficiency (e) is defined as the ratio of the total number of counts to the number of gamma rays emitted and can be expressed as

e = (1/41t)

f

[l-exp(-.uS)]dQ,

(1)

dr

.

(2)

Fig. 2 shows a cross section of the detector. = source to detector distance; R1 radius of p-type core; R z = radius of detector; t length of detector. Four distinct regions occur:

D

(1)

source

21trD

(r 2 +D 2 )t

o<

(J

<

(Jl

i.e. 0 < r < R 1 DJ(D + t).

p-type core

,. intrinsic region Fig. I. Geometry of detector showing coordinate system used.

377

Fig. 2. Cross section of detector showing regions considered.

378

R. GRIFFITHS

The gamma flux passes entirely through the non-sensitive central p-type core.

et

(2)

<

e<

(J2

i.e. R. Dj(D+ t) < I' < R t

•

The gamma flux is first attenuated by a factor (X in the p-type core and then passes into the intrinsic region and out of the bottom face of the detector.

ex = exp [-/-l(R t -I') (r 2 +D 2 )tjr], 5 0 , .... 02

=

2

2

(3)

2

t(r +D yijD - (R t -I') (r 2 +D )tjr.

(4)

The gamma flux enters the intrinsic region through the front face and leaves through the bottom face. (5)

>u

z

UJ

Q

I.L. I.L. UJ

.-J

«

I-

o

I-

1~.01

0.1 1.0 ENERGY (MeV)

10.0

0.

10 01

Fig. 3. Efficiency curves for detector 1.

0.1

1.0

10.0

ENERGY (MeV) Fig. 5. Efficiency curves for detector 3.

>u

~

Z

ILl

UJ

U

U

u: I.L.

u: I.L.

UJ

UJ

1cr~"".01---L......L...LJ-I.J.LLI0_..L...IL.....L.JUJ.LU-.-L....L..1...L.L1.UJ

.1

1.0

ENERGY (MeV) Fig. 4. Efficiency curves for detector 2.

10.0

0.01

0:1

1.0

ENERGY (MeV) Fig. 6. Efficiency curves for detector 4.

10.0

CALCULATED TOTAL EFFICIENCIES OF COAXIAL

(4)

83 < 8 < 94 i.e. DR 2 /(D+t) <

I'

379

DETECTORS

TABLE I

< R2 •

The detectors' dimensions.

The gamma flux enters the intrinsic region through the front face and leaves through the side.

Rl

(6)

Combining eqs. (I), (2), (3), (4), (5) and (6) and simplifying gives,

(7) The first of the integrals in eq. (7) may be solved using the substitution r = D tan ¢ to give

f:,2 (r2;~2)tdl' = (Ri+D2)-t-(R~+D2)-t.

Ge(Li)

(8)

The other integrals in eq. (7) have no simple solution and numerical methods must be employed. A program was written which computed e from eqs. (7) and (8). Using a PDPlO computer e was found as a function of energy for four typical detectors whose dimensions are given in table 1. Values of attenuation coefficients were obtained from the data ~f Storm et al. 2). Source to detector distances of 1, 3, 5, 10, 15 and 25 cm were considered. The results are displayed in figs. 3-6. 3. Discussion

Using the program described in this paper the total detection efficiencies of coaxial Ge(Li) detectors may be calculated. By setting R 1 = 0 the total efficiency of

Detector Detector Detector Detector

I 2 3 4

(em)

(em)

R2

t (em)

Active volume (cm 8)

0.3 0.3 0.3 0.7

1.5 1.5 1.8 2.1

3.0 4.4 5.1 6.5

20 30 50 80

a cylindrical planar detector may be computed. The analysis has assumed that the point source is situated along the axis of symmetry, the case which usually occurs in experiments. Measurements by Michaelis 3 ) on particular detectors in other irradiation 'geometries have shown a decrease in efficiency. Full-energy peak efficiencies have not been calculated as it has been shown by Gonidec et al. 4) that a considerable contribution to the fulI·energy peak is due to multiple Compton events and so to compute full-energy peak efficiencies a Monte Carlo program must be employed in a manner similar to that of Waino and Knoll~). The author wishes to thank Professors D. H. Wilkinson, F. R. S. and K. W. Allen for the use of the Oxford Nuclear Physics Laboratory PDP 10 computer on which the calculations were performed. References 1) H. P. Hotz, J. M. Mathiesen and J. P. Hurley, Nuc!. Instr.

and Meth. 37 (1965) 93.

2) E. Storm, E. Gilbert and H. Israel, Los Alamos Sci. Lab.

Report LA-2237 (1958).

8) W. Michaelis, Nuc!. Instr. and Meth. 70 (1969) 253.

4) J. P. Gonidcc, G. Walter and A. Coche, Nuc\. Instr. and Meth.

53 (1967) 185.

6) K. M. Waino and G. F. Knoll, Nuc\. Instr. and Meth. 44

(1966) 213.