A personal computer based program for HPGe detector absolute-peak-efficiency calculation and calibration

Appl. Radiat. Isor. Vol. 44, No. 8. pp. 1147-I Printed in Great Britain. All rights reserved

0969-8043/93 $6.00 + 0.00 Copyright Q 1993 Pergamon Press Ltd

154,1993

A Personal Computer Based Program for HPGe Detector Absolute-peak-efficiency Calculation and Calibration TIEN-KO Department

WANG*,

of Nuclear

Engineering,

TAI-YIN

CHEN

and LIH-JEN

National Tsing Hua University, Republic of China

KANG

Hsinchu

30043, Taiwan,

(Received 8 June 1992; in revised form I February 1993) A personal computer based program is developed for HPGe detector effective solid angle calculation for various source-detector arrangements. This program uses a simplified detector model to avoid tedious computational algorithms. The source geometry can be point, disk or cylinder. The applicability of using this program for gamma-ray absolute-peak-efficiency calibration has been successfully validated various combinations of source geometry, source-detector distance and gamma-ray energy.

1. Introduction The knowledge of the absolute-peak efficiency tp for the specific source-detector arrangement is prerequisite in various fields of research and application using gamma-ray spectrometry. For the establishment of the relationship between ep and the gamma-ray energy E, for extended sources at varying sourcedetector distances, the semi-empirical method introduced by Moens et al. (1981) for Ge(Li) and high-purity germanium HPGe detectors has experienced considerable success (Moens et al., 1982; Moens and Hoste, 1983; De Corte, 1987; Smodis et al., 1988; Jovanovic et al., 1988). In short, Moens’ method of cp determination for any sample and counting geometry proceeds as follows: (1) Determine the E:’ vs E, curve by measuring point sources of various gamma-ray energies at a large reference distance (e.g. 20 cm) from the detector where the true coincidence effects are negligible (index “ref”). (2) Determine the “effective” solid angle ratio &Fo/~rr’ between a specific source and counting geometry (index “geo”) and the reference arrangement using the computer program SOLANG developed by Moens et al. (1981). (3) Determine the cr vs E, curve based on the relationship

*Author for correspondence.

with

The determination of 6;’ vs E, curve in step (1) is a standard procedure which can be done easily by measuring calibrated multigamma point sources. Since the calculation of effective solid angle ratio nw/n”’ in step (2) using the SOLANG program has to be performed on a mini- or mainframe-computer due to the computational complexities; this may cause some inconvenience to some laboratory experimentalists. Presented in this paper is a personal computer (PC) based program for calculating the effective solid angle between an (extended) source and a closed-end coaxial HPGe detector. With this program, the Moens’ method of E$ vs E, calibration can be accomplished by using the experimentalists’ own PC which may be connected to his gammaspectrometry system. Section 2 expresses the methodology of calculation and Sec. 3 the program description. The validity of this program is reported in Section 4 through a series of experimental tests. Section 5 summarizes the conclusion of this work.

2. The Mathematical

Algorithm of Solid Angle

2.1. Geometrical solid angle (a) The mathematical algorithms of the geometrical solid angle R subtended by the detector at the source position, for some typical source-detector geometries, can be expressed as follows (Moens et al., 1981; Kaplanis, 1978; Griffiths, 1971; Hubbell et al., 1961; Masket, 1957; Jaffey, 1954): 1147

I-ITS-KU WAM;

Fig. 1. Calculation of the geometrical solid angle ((1) for a coaxial

Doint source

cf

cd

FIN. 3. C‘alculatlon coaxial

(i) coaxial

point source (Fig. 1)

(iv) coaxial cylinder

s

Cl=

nf the geometrical solid angle (Q) for d&k and cylinder sources.

dR LO”lC‘ detccr<,r

fi=;$:i:

source (Fig. 3)

(Z, + /)dLiyrdr

j#:d&l”

1% x

.!

R=

2nsinOd0=2n(l

-cosr)

(2.1)

0

(ii) non-axial

2.2. The c$kr~~iw .wlid magic (Q-i)

point source (Fig. 2) ‘QIJ

n=2zr

‘d$ s0

In equation (I), the “etfective” solid angle subtended by the detector at the source position defined as (Moens CI ul., 1981):

s0 R dR

’ [R2-2R

r cos 4 + r? + zg”

(iii) coaxial disk source

(Fig.

(2.2)

3)

R dR ’ [R2 - 2Rr cos c#t+ r’+ ZG]“’

(2.3)

z t

There are two weighting factors, E‘,,, and FeH, in the integrand. The F,,,, factor accounts for the attenuation of the gamma-ray by the source itself and by any other material situated between the source and the active lone of the detector, i.e. the source container and support, the air, the Al-(or Be-) can surrounding the detector. the inactive layer on top of the detector crystal, etc. Therefore.

T(r,O,z, ) (4)

X

where ,u, is the linear, narrow-beam absorption coefficient [denoted “TOT H” by Storm and Israel (1970)] of the ith absorber and 6, the path length of the gamma-ray in the ith absorber. The other weighting factor FcR is the probability for a photon with energy E;, emitted within an angle dR impinging on the detector. to interact incoherently with the detector material before leaving it. By using the calculated 0 values, one can convert the measured 6;’ values into various c8p’” values according to the relationship: tr

Fig. 2. Calculation

0 is

of the geometrical solid angle (a) for a non-axial point source.

= (‘,‘r(ng’“/Ilr’3[(p/r)“‘“/(p/t)“’]

(5)

where the parameter lr/f is the “virtual” peak-to-total efficiency ratio which refers to the hypothetical, bare

HPGe detector absolute-peak-efficiency calculation and calibration

1149

- top l

side

,

.base

Fig. 4. Eight possible paths to be followed by a gamma-ray impinging on the detector top. and isolated

detector

crystal.

Previous

work (Moens 1983) showed that it can be assumed that p/t is a constant of the detector crystal and is independent of the change in counting geometry; thus, equation (5) can be simplified to equation (1).

et al., 1981; 1982; Moens and Hoste,

2.3. The calculation of Fef and d Most of the commercial HPGe detectors have a closed-end coaxial configuration in which part of the central core is removed. Due to the existence of this “coaxial cavity”, gamma-rays emitted from an extended source impinging on the detector top can follow eight possible paths within the detector body as shown in Fig. 4 (Moens et al., 1981). Therefore, for each straight line @ travelling from source to detector (Figs 2 and 3), its intercepts with each surface of the detector crystal and the cavity have to be calculated in order to determine to which one of the above eight paths this @ line belongs. Based on the intercepts, one can then calculate the F,, factor for a gamma-ray heading along m according to the distances between intercepts. This complexity in the Fef derivation makes the calculation of the effective solid angle a rather cumbersome task.

ffl

e Q2

T

As a remedy for the above complexity, we simplify the F,, calculation procedure by homogenizing the detector cavity with the other part of the solid crystal material. In the mean time, the density of the detector crystal is artificially reduced by the ratio of the cavity volume to the total detector volume in order to account for this homogenization. With this homogenized detector model, the calculation of the F,, factor is much simplified: (i) for a coaxial

point source (Fig. 5)

F,,=l-exp(-p~A)=I-_exp(-~.Ap.) A

=

H sece R&SC8 - z, set e

(ii) for a non-axial

if if

19< a, a, < 0 < GI* (6.1)

point source (Fig. 6)

if a CSCO- a’ set 13 if a, = tan-‘[a/(Z,

0
(6.2)

+ H)]

a2 = Ri + r2 - 2R,r cos C$

ZT

/

i ;:

/ H ,

Fig. 5. Calculation of the FcR factor for a coaxial source point.

Fig. 6. Calculation of the FeBfactor for a non-axial source point.

TIEN-KO WANG et al.

1150

3. Program Description

where the reduced attenuation coefficient & is obtained by multiplying the crystal attenuation coefficient pd by a factor of p’/p = (RiH - rih)/RiH; R,, H and r,, , h stand for the radius and height of detector and cavity, respectively; p and p’ stand for the detector (crystal) density and reduced density, respectively. With the F,,, factor as expressed in equation (4) and the F,, factor as expressed in equation (6.1) and (6.2), the effective solid angle 11, using a cylindrical source as an example, can be calculated using the equation

Based on the simplified detector model and the mathematical analogy presented in Sec. 2, a program named ESOLAN was written using FORTRAN V for the calculation of effective solid angle 0. The program reads numerical data % thesource and detector dimension, the source-detector distance, the thickness of the intervening layers and the source container walls, and the gamma-ray energies and the corresponding attenuation coefficients for each material (Storm and Israel, 1970). The numerical evaluation of the multiple integrals in the ESOLAN program is performed using the Gauss integration formula. The Gauss-Legendre base-point number n used in the integration is assigned by the program user. Although the accuracy of integration increases with increasing base-point number, our experience tells that the integration actually converges within n = 12 for point and disk sources and n = 16 for a cylindrical source. One example of convergence test is shown in Fig. 7. The motivation for using a simplified detector model is to reduce the complexity in computation so that the calculation (of a) can be performed using a personal computer and with a reasonable computing

R dR (7)

’ [R’ - 2Rr cos 4 + r2 + (Zr + 1)2]3’2

The calculation of a for point and disk sources can be performed using equation (2) to (6).

5.00

,

,

I

1

,

,

1

I

,

,

,

1

,

A-W e+~e+ c-~r+a l

-10.00

I

,

,

,

,

0

, 10

,

,

,

(

Fig. 7. Gauss integral convergence source-detector distance is 20cm. to be “correct”

,

20 Base

I

,

,

I

,

E:0.05

MeV

E:O.l E:0.5

MeV MeV

E:l.O H E:3.0

-

,

,

,

I

,

I

I

,

,

MeV MeV

,

,

,

30

point

I

number

,

,

,

( 40

,

,

,

,

, 50

(n)

test in the effective solid angle (a) calculation for cylinder source. The The n values obtained using base-point number n = 48 are assumed values and taken as the reference for comparisons.

HPGe detector absolute-peak-efficiency calculation and calibration

disk (1 .O and 2.0 cm in dia) and cylinder (3 and 6 mL in volume) were made in our laboratory. The cylindrical sources were made by encasing aqueous radionuclide solution in polyethylene vials. All sources except 24Na are coincidence-free-gamma-ray emitters. The effect of (true) y-7 coincidence summing of the two gamma-rays emitted by 24Na source was corrected (Kang and Wang, 1993) following the method suggested by De Corte (1987). Measurements were performed at eight sourcedetector distances from 3.0 to 27.5 cm. Gamma-ray spectrum analysis was made by the SAMPO 90 program (Aornio et al., 1988; Canberra Instruments, 1990). Each experimental datum point shown in this paper represents an average of three measurements; the error is controlled in the range of - 1 to 3% (in one standard deviation).

time. The ESOLAN calculation (CPU) time, for ten PC with INTEL Q values, using an IBM 80386 + 80387 (or with INTEL 80486) on MS-DOS 5.0 is: (1) - 1 s (or -0.4 s) for point sources and n = 12; (2) -3 min (or - 1 min) for disk sources and n = 12; and (3) - 1.5 h (or -0.5 h) for cylindrical sources and n = 16. 4. Experimental

Test of the ESOLAN

Program

The purpose of the ESOLAN program is to calculate the effective solid angle IT for a specific source and counting geometry. The calculated fi values can then be used, according to the relationship c go lc ;’ = @eo/~‘, f or cp calibration as described in Sec. 1. In order to test for the validity of the ESOLAN program, experiments were made to obtain cp values for various sources and counting geometries; the measured 6 r/c F’ ratios were then compared with the Qgeo/~re’ ratios obtained by ESOLAN calculations.

4.2. Results and discussion Unacceptable discrepancies were first found between the measured 6P/trf values and the ESOLAN calculated Qreo/pf values. In this work, we define a point source measured at 20.0 cm from detector as our “reference condition” where peak efficiency and effective solid angle are denoted as crf and arrf, respectively. The tp and fj obtained for all other source-detector conditions are denoted as cFm and nreo. Table l(a) lists the percentile deviation defined as (R, - RJR, for quasi-point sources, where R, = cgO/cFf and R, = ngeo/wf. The deviation between the measured results and the calculated

4.1. Experiments The experiments for cp/c;’ measurement were made with an ORTEC GMX-15185 HPGe detector. The geometric configuration of the detector and source support is shown in Fig. 8. Gamma-ray sources of “‘Trn (84.30 keV), “‘Ce (145.44 keV), “Cr (320.08 keV), *‘Sr (515.01 keV), 13’Cs (661.66 keV), 65Zn (1115.55 keV) and 24Na (1368, 63 keV and 2654.03 keV) in the forms of quasi-point (with a dimension of -0.3 cm in dia),

n

vial

1 I- I--i -4 I 0.20

cm

f

r”;

I volume or disk or point source

d ( 3-27.5

cm

)

Fig. 8. Characteristic dimensions of the ORTEC GMX 15185 HPGe detector and the counting system used in this work. AR1 44/E-F

1151

1152

TIEN-KO Table I. Percentile

WANG et al.

in terms of (R, - R,)/R, for quasi-point

deviation

sources, where R, = L~/L;'and R, = iF’/il”’

(a)

”

2

lop

p

200

2op

200

2op

2op

84.30 145.44 320.08 514.01 661.66 I 115.55 1368.63 2754.03

-10.1% -8.1% -8.6% ~ 10.4% -8.0% -10.1% -6.9% -6.4%

-1.3% -5.4% -5.7% - 5.8% -5.2% -6.1% -2.5% ~ 3.5%

-4.2% -3.8% -2.1% - 3.1% -3.3% -3.8% -1.1% - 1.7%

- I .6% -2.4% ~ 1.3% ~ I .3% - 2. I % -2.3% -2.3% -3.1%

0.6% -0.9% -1.1% 0.4% - I I% -0.6% -0.5% -0.1%

SOUPX type

Energy (kev)

3.3p 20.3p

5.3p 20.3P

7.3p 20.3p

10.3: 20.3P

“@lm ‘4’Ce S’Cf 8SSr “‘CS 6SZll 24Na *4Na

84.30 145.44 320.08 514.01 661.66 1115.55 1368.63 2754.03

- 1.3% 0.6% -0.4% -2.4% -0.1% -2.5% I .O% - 1.5%

~ 1.3% 0.5% -0.3% -0.3% 0.1% ~ I .O% 2.3% 1.7%

-0.0% 0.2% I I% 0.2% 0.3% -0.3% 2.6% 2.0%

0.9% -0.1% 0.9% I .O% -0.0% -0.3% -0.3% ~ I .O%

SOUFX

type

Energy (kev)

“@Tm 14’Ce *‘Cr *‘Sr “‘CS 6SZIl 14Na 24Na

7p

200

27.5P 2op

0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%

0.9% -0.9% -1.1% 0.3% 0.2% 0.9% 0.8% 0.0%

I .5% I I% -0.4% 0.7% 0.5% 0.2% 2.0% I I%

15.3p 20.3p

20.3p 20.3P

25.3p 20.3p

27.8p 20.3’

I .5% -0.2% -0.5% 1.3% -0.5% 0.2% 0.1% 0.0%

0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%

0.1% -1.5% ~ 1.6% 0.1% -0.3% 0.2% 0.4% -0.2%

0.7% 0.4% - 1.0% 0.2% --0.5% -0.5% 1.3% 0.7%

2op 2op

21p

0

The “reference” condition stands for the quasi-point source located 20.0 cm from the detector top. The “geometry” condition stands for swrce located 3.0, 5.0, 7.0, 10.0, 15.0, 20.0. 25.0 and 27.5cm from the detector top. The super-script p stands for quasi-point source. (a) Calculations of n were made according to “real” source-detector distances. (b) Calculations of n were performed by artificially adding 0.3 cm to each source-detector distance.

, , , ,

3.8

3.6

-

\

\

I

I

,,,,,I,

I

I

\ \

\

d =

3.2

I

Disk (D=2cm)

10

cm

---+2%

3.0

’ ’ ’ ’ 10’

I

I

1111111

Energy

1

I

-

I

(kdi;

Fig. 9. Ratios of er/c Fras a function of gamma-ray energy, where c ;‘is the peak efficiency for quasi-point source located 20.0 cm from detector top and Lr stands for the peak efficiency for disk (2 cm in dia) sources located at 10.0 cm from the detector. Also shown in the figures are the relevant curves of the P’/P’ ratio vs gamma-ray energy according to the calculations performed using source-detector distance of 10.3 cm.

HPGe detector absolute-peak-efficiency

results increases with decreasing source-detector distance. The deviation could be as large as N 10% when the source was measured at 3 cm from detector. Similar deviations were also found for disk and cylindrical sources. In order to eliminate the discrepancies between the calculation and the measurements, some modifications have to be made on ESOLAN calculations. Values of some key input parameters, including the detector crystal density p’ (or the “reduced” “reduced” crystal attenuation coefficient pi), crystal inactive-layer thickness (ref: Fig. 8) and sourcedetector distance, were changed to test their effects on the calculated n results. It was found that the most effective way is to change the source-detector distance. If this distance is changed by a finite value, the resulting change on the (calculated) solid angle will be more significant when source-detector distance is smaller; this is just the trend of the discrepancy between c;~“/c~’ and &e0/@f. Based on trialand-error adjustment, the discrepancy shown in

Table 2. Percentile

deviations

1153

calculation and calibration

Table l(a) can be practically eliminated if the sourcedetector distance is artificially increased by 0.3 cm in the calculations. The comparisons between the measured Lr/c F’ values and the newly calculated Deo/Pf values, for quasi-point sources, are listed in Table l(b). The deviations are now within the range of experimental error at all distances. Experiments and calculations were also made for disk and cylindrical sources. Similarly, if the sourcedetector distances were artificially increased by 0.3 cm, the deviations between ck/cr’ and ~~“/n”r’ could be reduced to the experimental error ranges. Results are listed in Table 2. In addition, one group of cF”/E;’ values are plotted and shown in Fig. 9, together with the calculated ngeo/arrf vs E, curves; the deviations between the two quantities are again within k 2%. In summary, peak-efficiency measurements have been made, for eight gamma-ray energies ranging from 80 keV to 3 MeV, at eight counting distances ranging from 3.0 to 27.5 cm, using sources in five

in terms of (R, - R-)/R,,

where R, = a:eo/~:’ and R, = @m/Q=’

SOWCCtv”e

Energy (keV1

3.3d’ 20.3P

5.3d’ 20.3P

7.3d’ 20.3*

10.36’ 20.30

15.36’ 20.3P

20.3d’ 20.3’

25.36’ 20.3P

27.8d’ 20.3P

6’Z” 2’Na 2’Na

84.30 145.44 320.08 514.01 661.66 1115.55 1368.63 2754.03

- 1.4% 3.5% 2.5% - 1.2% -1.1% 0.3% 0.5% -0.3%

- 1.5% 2.2% 1.1% -0.7% -0.5% -0.3% 1.3% 0.4%

-0.6% 1.9% 0.3% -0.2% 0.7% -0.2% 1.6% 1.8%

0.8% 0.7% -0.2% 0.6% -0.1% 0.4% -0.2% 0.0%

-0.6% 0.2% -0.0% 0.3% 0.4% - 0.2% 1.3% 1.0%

0.5% -0.6% -0.7% 0.3% -0.7% -0.1% 0.8% -0.2%

0.6% 0.3% -1.1% 0.1% -0.1% -0.5% 0.5% 0.3%

-0.5% -0.8% -1.0% -0.2% -0.1% 0.1% 0.4% 0.3

Source type

Energy (keV)

3.362 20.3P

5.3dz 20.3p

* 20.3P

u 20.3p

M 20.3p

m 20.3p

25.3d’ 20.3P

@ 20.3P

“?.“I We S’Cr *ssr ?n I’Na “Na

84.30 145.44 320.88 514.01 661.66 1115.55 1368.63 2754.03

- 1.2% 2.1% 1.6% -0.6% 1.8% 1.6% 3.0% 2.8%

I .4% 0.5% -0.1% 0.5% 0.3% 1.4% 0.2%

-0.5% 0.1% -0.2% 0.4% 0.5% 0.1% 0.5% 0.6%

0.2% -0.5% -0.7% 0.4% 0.1% -0.0% 0.4%

-0.5% -0.5% -3.6% 0.3% -0.8% 0.4% 0.6% 0.4%

0.7% -0.2% 1.8% -0.3% 0.0% -0.6% 0.5% -0.9%

Source type

Energy (keV)

3.3v’ 20.3p

5.3” 20.3P

7.3y’ 20.3p

10.3v’ 20.3”

15.3v’ 20.3P

20.3’3 20.3p

25.3” 20.3p

27.8” 20.3p

“@Tm ‘We Wr 8SSr “‘Cs 65Z” 14Na 2’Na

84.30 145.44 320.08 514.01 661.66 1115.55 1368.63 2754.03

-0.8% - 1.8% -2.3% -0.4% 2.1% 1.0% 3.6% 2.0%

0.5% -1.0% -2.4% -0.7% 0.9% 0.5% 2.8% 1.4%

- 1.2% -0.9% 0.1% I .O% 0.4% 1.4% 2.3%

4.4% -1.1% -1.8% -0.4% 0.5% -0.5% 1.3% 1.8%

3.0% -0.6% - 1.3% 0.1% 0.4% I .9% 0.9%

0.9% -0.7% - 0.7% 0.2% -0.2% 0.6% 2.0% 1.1%

-0.9% 0.1% -1.1% 0.1% -0.2% -0.1% -0.1% 0.1%

- 0.9% -0.5% -0.9% 0.2% -0.5% -0.8% -0.3% -0.3%

Source type

Energy (keV)

3.3y6 20.3P

5.3” 20.3P

J.J6 20.3p

10.3y6 20.3P

15.3y’ 20.3P

20.3y6 20.3p

25.3*6 20.3p

27.8y6 20.3p

“?“I We J’Cr ?Sr “‘CS 6’Z” “Na Z’Na

84.30 145.44 320.08 514.01 661.66 1115.55 1368.63 2754.03

-2.4% -1.1% 0.4% - 1.3% 1.9% 1.6% 3.8% 3.4%

-0.1% 0.5% -0.5% -0.6% 0.9% 0.2% 2.8% 0.5%

1.5% -0.6% - 1.3% -0.9% 1.6% 0.6% 2.1% 1.6%

2.1% -0.3% -1.3% 0.9% -0.0% I .2% 0.8% 1.0%

-0.3% 1.0% -1.0% -0.1% -0.0% 1.1% I .4% 1.0%

0.9% 0.5% -0.6% 0.4% -0.1% 0.4% 1.1% -0.3%

0.7% -0.2% -0.9% 0.2% 0.5% -0.1% 0.7% 0. I %

’ WS

1.3% 1.7% 1.8% 0.1% 1.9% 1.8% 2.1% 2.2%

0.5% 1.6% 0.1% 0.2% 1.5% 1.2% 0.6%

1.O%

1.4%

-

I .O%

I .O%

I .O%

-0.7% -1.3% -1.1% -0.2% -0.8% -0.4% -0.2%

n R%

The “reference” condition stands for a quasi-point source located 20.0cm from the detector top. The “geometry” condition stands for source located 3.0, 5.0, 7.0, 10.0, 15.0, 20.0, 25.0 and 27.5cm from the detector top. The super-scripts p. d,, d,, v, and vg stand for point, disk of I cm in dia. disk of 2 cm in dia. cylinder of 3 mL in volume and cylinder of 6 mL in volume, respectively. Calculations of n were performed by artificially adding 0.3 cm to each source-detector distance.

TIEN-KO WANG et al.

1154

geometries, to test the validity of using ESOLAN program for effective solid angle calculation. It was found that, with a slight adjustment (i.e. 0.3 cm) on the input value of source-detector distance, the ESOLAN program can be used successfully in our laboratory for HPGe detector efficiency conversion. It is worth mentioning here that the need for source-detector distance adjustment in the ESOLAN calculation is not caused by the simplification of our detector model. Similar adjustment is also needed when the SOLANG program, which applies an exact detector model, is used for a calculations (Moens ct al., 1981; De Corte. 1987). If an author chooses to use the ESOLAN program to establish the tp vs ET curve for his specific detector, the following tips can be used if he finds that the systematic deviation lies between the measured cpand the calculated Q-ratios. If the deviation has a trend to increase with decreasing source-detector distance, it can probably be eliminated by adjusting (i.e. adding or subtracting) the distance by a finite value in the calculations, since the influence of distance change on the value of solid angle increases with decreasing source-detector distance. On the other hand, if the deviation has a trend to increase with decreasing gamma-ray energy, then it can probably be eliminated by adjusting the crystal inactive-layer thickness, because the attenuation capability of this layer increases with decreasing gamma-ray energy. It is believed that other types of systematic deviation can also be eliminated by a trial-and-error adjustment on suitable input parameters in the ESOLAN program. One final remark needs to be made here. That is, according to our experience, the homogenization of detector cavity with other part of the solid crystal (i.e. the detector model simplification) can result in inaccurate fi values if the cavity is too large a fraction of the crystal volume (say, >7% of the crystal dimension); fortunately, most HPGe detectors have a cavity of 3% or less in crystal volume.

5. Conclusion A PC based program for calculating the effective solid angle between a source and a cylindrical HPGe detector was developed in this work using a simplified detector model. The program has been successfully tested using gamma rays in the range of 80 keV to 3 MeV, with source-detector distance varying from 3.0 to 27.5 cm, and for sources in the geometries of quasi-point, disk and cylinder. This program can be very useful in gamma-ray peak-efficiency calibration

for various source geometries. gamma-ray energies and source-detector separations. The ESOLAN program is available upon request. Otherwise, an experimentalist can also write his own program, without too much difficulty, based on the mathematical algorithms described in Sec. 2. Acknowledgements-This work was performed under the auspices of the National Science Council, Taiwan, Republic of China. The authors would also like to express their thanks to Mr C. L. Tseng who carefully prepared the gamma-ray sources used in this work.

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