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Gamma-ray self-absorption corrections for Eγ>0.6 MeV and μ<3 cm−1
NUCLEAR
INSTRUMENTS
AND
METHODS
I22
0974) 399-404; ©
GAMMA-RAY SELF-ABSORPTION
NORTH-HOLLAND
PUBLISHING
CO.
CORRECTIONS
F O R E v > 0.6 M e V A N D p < 3 era- 1
Y . S. H O R O W I T Z ,
S. M O R D E C H A I
a n d A. D U B I
Department of Physics, Ben Gurion University of the Negeo, Beersheva, Israel Received 11 September 1974 The M o n t e Carlo m e t h o d was used to calculate the self-absorption correction for a large n u m b e r o f cylindrical source-cylindrical detector configurations in the energy range E~ greater t h a n 0.6 MeV a n d for values o f the total attenuation coefficient (it) up to 3 cm -z. T h e corrections are thus applicable to all possible materials a n d c o m p o u n d s a n d are presented in a functional form which allows easy calculation a n d interpolation. The error
introduced by using the self-absorption correction multiplied by a n o n - a b s o r b i n g source-detector efficiency is discussed and s h o w n to be usually negligible. T h e problem o f forwardscattered C o m p t o n ?-rays from the source contributing to the p h o t o p e a k efficiency in NaI(TI) detectors is s h o w n to be increasingly significant for E ~ < 0 . 6 MeV.
1. Introduction
coefficient up to 3 cm- ~, and are presented in a functional form which allows accurate and easy interpolation and calculation. The complete Monte Carlo program is outlined in Fortran to enable the calculation of F~ for geometrical configurations not encompassed by the present calculations. A discussion of the possible errors resulting from forward-scattered Compton 7-rays from the source contributing to the photopeak and from the use of Tp calculated for a
The problem of the detection efficiency of NaI(TI) scintillators or Ge(Li) semiconductor detectors for thick cylindrical 7-ray sources is a universal one common to many experimental configurations, For example, the absolute measurement of neutron spectra using the induced activity of threshold detectors or the many various types of absolute activation analysis all require an accurate knowledge of the overall detection efficiency. The detection efficiency for an arbitrary source-detector geometry can be defined as = Np/N
= O FsT v,
(l)
where Np is the number of counts observed in the photopeak/s, N is the strength of the source in disintegrations/s, g2 is the geometrical solid angle, Tp is the intrinsic photopeak efficiency of the detector and F~ is the self-absorption correction which accounts for the 7-rays absorbed in the source. Several authors have calculated ~ using Monte Carlo ~-4) and analytical methods s-v) for various source-detector-7-ray energy configurations while other authors have calculated F~ using the Monte Carlo a-t 1) or analytical methods 12, ~3). Although these authors have successfully illustrated the applicability of their techniques to the calculation of ~ or F~ in their specific experimental situation the almost limitless number of possible experimental configurations has limited their general usefulness. In this paper we have calculated F~ by the Monte Carlo method for as large a number of experimental configurations as possible so as to be of maximum benefit to the potential user. The calculations are carried out for 7-ray energies greater than 0.6 MeV, for source materials resulting in a total a~tenuation 399
READ, ZD, RF, RD,HF 55 C=ZD/SQRT(ZD, '2 * (RD.* RF ),'~, 2 ) G=ATAN ( I. ) 56 N H I T " 0 NESC = 0 39 DO50 1=1,5500 R=RF~SQRT(RANF(O) )
COT,,C.,,RANF(O)~,( I. 0 - C ) TE=ACOS (COT) 391
40
41
42 43 431 44 30
51
FI=G~4.*I~ANF(O)
.
.
SD=SORT(RD=~t- (R~81N (FI) ) * * 2 ) * R * COS(FI ) TEC=ATAN (SD / (Z+ZD) ) IF (TE,LT.TEC) 4 0 , 5 0 NHI T=NHIT' I $F=SQRT(RF~ ~2- (R*SIN(F I ) ),~'2) +R~CO$(F I ) 1"1[I ~ItTAN ( S F / Z ) I F ( T E . L T . T E I ) 4t ,42 EL=Z/COS (T E) GO TO 45 EL=SF/SIN (TE) W:-( I . / U ) ~MJ.OG(RAJqF ( 0 ) ) IF(WoLT. E L ) 5 0 , 4 4 N~C =NESC* I " CONTINUE A=NH IT B=NESC FSIB/A DP3 =SQRT( (I. "IS ) / ( FS*B ) ),, I00. 0 IF( DFS. GT. I. 0)39~31 PRINT 234,U,I=S, DFS~HF I .F'5. 3 ) END.
Fig. 1. The M o n t e Carlo m e t h o d in Fortran for the calculation o f the self-absorption correction.
400
Y.S.
HOROWITZ
non-absorbin9 point or disc source in eq. (1) is also presented in sections 3,1 and 3.2. 2. Method of calculation The Monte Carlo method has been used to simulate the history of many photons as they traverse the source and impinge on the detector. The 7-rays are generated randomly and isotropically throughout the source. In order to save c o m p u t a t i o n time a method due to Williams ~4) is used in which the direction of the ),-rays is sampled only in the smallest cone c o m m o n to all source points containing the detector face. For the ,,,-rays whose direction has them impinging on the detector, the distance to the first interaction is sampled from the distribution exp (-I~x), where p is the sum of the photoelectric, C o m p t o n and pair-production coefficients15). If the 7 path length in the source is larger than x the 7 is considered to have been absorbed and vice versa. The program accumulates the overall number of ~-rays aimed at the detector (N h) and the number of ?-rays which have escaped from the source (Ne). The self-absorption correction is then given by F, = N~/Nh,
(2)
with the error AFs given by
AF s = [Fs(1 - Fs)/Nh] ~.
et al.
The computer program in Fortran is shown in fig. I. Step 35 defines the c o m m o n cone, in steps 39-391 the 7-ray is generated, in steps 391-40 it is verified whether the 7-ray has hit the detector and steps 40-44 determine whether the ),-ray has been absorbed. The calculations were carried out for what we believed is the most c o m m o n type o f geometrical configuration encountered in activation analysis and activated foil counting, i.e., a disc-shaped volumetric source placed axially and symmetrically on the surface of a cylindrical detector. The actual distance between the source and the detector element was taken as 0.3 cm to account for the separation introduced by the container of the detector. The calculations include all possible combinations of the following sourcedetector geometries: Ra (radius of the d e t e c t o r ) = 3.81 cm and 10.16 cm, Rr (radius o f the source) = 0.2 cm, 0.635 cm, and 1 cm, and hf (thickness of the source) = 0.005cm, 0.01 cm, 0.05cm, 0.1 cm and 0.5 cm. 3. Results and discussion The behaviour of the self-absorption correction as a function of the total attenuation coefficient for the various source-detector configurations is displayed in
(3)
I00
0 96
0.96 ! ~
092 z
0 I-o I..d .q,¢r'
FC) U.l c~
Z
o EL 0.72 n," 0 fD
088 084-
0
080
Z o 1--
076 072 .'
o
,,./) 068 i
<~ 064 LI._ -, ..J UJ O~ 0.60 i
Rd = 1016crn
63 ,<
o 68
Rf =OZcm
U_ ..J U.I
064
(,o
o6oL
Rd=lO 16cm Rf =0 635 cm.
056
056~
052i
0 52
oz o'4o,'6o8 ioii2,4,6,'8 Zo22~,~z6 TOTAL ATTENUATION COEFFICIENT~t-(cm-O Fig. 2. The self-absorption correction as a function of the total attenuation coefficient for Ro = 10.16 cm and Rf = 0.2cm.
.....
,. . . .
0204060.8
, .....
\1,,
_
_
_
lOt2 1416 1 . 8 2 0 2 2 2 4 2 6
T O T A L A T T E N U A T I O N COEFFICIENTI, t-(cm-')
Fig. 3. T h e s e l f - a b s o r p t i o n c o r r e c t i o n as a f u n c t i o n o f t h e t o t a l a t t e n u a t i o n c o e f f i c i e n t f o r R e , = 1 0 . 1 6 c m a n d R ~ = 0 . 6 3 5 crn.
GAMMA-RAY
SELF-ABSORPTION
figs. 2-7. The results are not accurate for E~ <0.6 MeV for reasons discussed in section 3.3. For source thicknesses h~ = 0.005 cm and 0.01 cm the data were fitted with a function of the form f ( / 0 = a/~ + b, whereas for all the other thicknesses the function used was of the form f(It) = e x p . ( - 2 t 0 x (l +all~+azl~Z+aalz3). The fitting parameters are displayed in table 1 as well as the accuracy of the estimate of F,. This phenomenological fitting procedure allows accurate interpolation if the experimental configuration falls within the displayed parameters. 3.1. y-RAY FIELD CORRECTION The use of eq. (1) implies a knowledge of the detector intrinsic photopeak efficiency (T o) for the y-ray field of the absorbing source, however, most tabulations of detector photopeak efficiencies have been carried out for non-absorbing sources TM 16, 27). Only recently, Belluscio et al. 4) have evaluated To for a 3"× 3" NaI(TI) detector, for y-ray energies of 1.368, 2.754, 4.44 and 6.13 MeV from sources of aluminum, carbon and water. Since the absorption in the source is not isotropic, the y-ray field on the detector surface will differ from the v-ray field for a non-absorbing source so that the use of Tp from non-absorbing source tabulations may introduce a significant error.
fO
0
~
401
CORRECTIONS
We have attempted to evaluate this error in the following manner. For a disc source (Rf>>h¢) we have assumed that the v-ray field of the absorbing source can be approximated by the sum of a parallel beam field and an unperturbed point source field, the relative strength of which is decided by the following considerations. The number of 7-rays in the parallel beam can be approximated by
cos0,)[, exp W
where 0 a is the half opening angle of a cone defined by 0x = t a n - ' (Rjl), l is the detector length, and the exponential term is the parallel beam self-absorption correction (Fp). The remaining ?-rays are thus assigned to the point source field given by N
[ ½ ( I -- COS
02)F
~ -
1(1 - cos 01)Fo] ,
(5)
where 02 = tan-l(Rd/hO is the half opening angle of the cone subtended by the detector at the source. The average self-absorption for the point source field can then be determined from 52 = N-N- [ ½ ( c o s 01 -- c o s 0 2 ) f ' ] .
(6)
4~
,
I O0
=
096
°921 \
o
\
\
Ld Qf Q~
0 84
0
080 J
""~-o_
092 088
\
Z _00
..
{D hi ,,~
\
e:: 0 r,.D
084 080
0761 0 0,,.- 072] (3Z
n o 0") Fn
nO an
0 6 8 t Rd= IO f6crr,.
U._
I Rf" 'cm
W CO
1
\
\
o6ot
\
I
\
I.L ...J U..I 0'3
0681 064
060
056 052
~
~
T
6
~8 2[02'2 2-4.26 '
TOTAL ATTENUATION COEFFICIENTia-(cm-') Fig. 4. T h e self-absorption correction as a function o f the total a t t e n u a t i o n coefficient for Ra = 10.16 cm and Rf = 1 cm.
0~2-5~0~6os Io 12 14 L6 182.0222426 TOTAL ATTENUATION COEFFICIENTw(cm-') Fig. 5. T h e self-absorption correction as a function o f the total a t t e n u a t i o n coefficient for Rd = 3.81 cm and R~ = 0.2 cm.
402
v.s.
HOROWITZ
et al.
TABLE 1 T h e p a r a m e t e r s o f t h e s e l f - a b s o r p t i o n c o r r e c t i o n fitted to t h e f u n c t i o n f (p) = at~ + b a n d f ( # ) = e x p ( - }t#) (alpt + a21,t"2+ aa#3).
Source thickness
Parameters
Rd = 3.81
Re = 3.81
Rd = 3.81
Rr = 0 . 2
Rf = 0 . 6 3 5
Rf = 1
(cm)
(cm)
R a = 10.16 Rr = 0.2
Re = 10.16 Rr = 0.635
Rfl = 10.16 Rr = 1
Accuracy
(cm)
(cm)
(cm)
(%)
(cm)
(cm)
-6.63
-6.66
-7.13
-8.81
-9.24
-9.02
b
1.0
1.0
1.0
1.0
1.0
1.0
a x 102
- 1.14
-- 1.24
- 1.24
- 1.44
- 1.54
b
1.0
1.0
1.0
1.0
a x 103 0.005
0.5
- 1.61
0.01
0.5
0.05
0.1
0.5
}tx al x a2 x a3 x
102 102 103 10 a
+5.17 40.10 4 0.02 +0.33
+5.78 - 1.31 4 1.24 - 2.75
}t x al x a2 x a3X
10 ~ 102 10 '~ 103
+ 7.97 - 1.56 +1.54 +3.51
4 10.2 -2.98 +2.40 +4.34
)1. x at x a2X a3 x
10 102 102 103
+ 1.65 -0.313 -0,41 4 3.0
+ 2.89 -6.0 +4.30 - 8.24
+6.79 -2.88 + 2.84 - 6.60
1.0
+6.05 -0.53 + 4.0 -- 5.29
+5.40 +0.28 0 0
+ 10.8 - 1.38 40.874 -0.917
+ 8.83 40.356 -0.637 +2.14
+ 11.7 + 1.67 +1.18 -1.69
+ 12.2 - 1.75 40.80 40,248
+ 3.29 -6.68 +5.39 - 9.98
+ 1.65 - 1.73 -0,027 + 3.0
+ 2.99 - 5.10 +3.72 - 5.80
+ 3,44 -7.62 46.10 - 1.12
h~o
1.00
1.0
+6.95 -2.28 + 20.9 -4.48
0.8
1.0
2.0
1.00 '
0 96,
0924 \
\
I\ \ 0681 \ 0.84-' 0 \
eoso-
092
'"..
0887"
o I(o t,i
\
0 80 o o
0
.
7
2
o68i
z o F-
~
\
_ 0.64-
0 64"
Rd=SBlem' Rf =0655cm.
076 ~ 072 0
68.
113
N ~,~,,
u_ ...1 ILl
0.60-
Rd= 581cm Rf=l cm
064~ 060-
0.56-
X
0 56\
052,
,
,
~
,
,
,
N,
,
) . 2 0 4 0 6 08 I0 1.2 14 1.6 18202.2 2.42.6 T O T A L ATTENUATION
COEFFICIENT~-(cm-')
Fig. 6. T h e s e l f - a b s o r p t i o n c o r r e c t i o n as a f u n c t i o n o f t h e t o t a l a t t e n u a t i o n coefficient f o r R e = 3.81 c m a n d Rt = 0.635 c m .
o.52.J
0 2 Q 4 0 6 0 8 I0 t2 14 16 t 8 2 0 2 2 2 4 2 6
TOTAL ATTENUATION COEFFICIENT~t-(cm-') Fig. 7. T h e s e l f - a b s o r p t i o n c o r r e c t i o n as a f u n c t i o n o f the t o t a l a t t e n u a t i o n coefficient for R e = 3.81 c m a n d Rr = 1 c m .
GAMMA-RAY
In this manner sorbing source
and a parallel
SELF-ABSORPTION
we have resolved the field of the abinto a point source field of strength
y-ray field of strength
The overall strength
of the field is preserved
since (9)
The number of counts now be obtained from
expected
in the photopeak
can
403
CORRECTIONS
contribution to the photopeak will be subtracted away with the genera1 background. For E,<0.6MeV the Compton spectrum is increasingly peaked under the photopeak so that conventional calculations will consistently underestimate F,.At E7= 0.2 MeV the error in N, may be as large as 5%. The second complication is the effect due to the increasing importance of Rayleigh scattering below 0.6 MeV. For example, the Rayleigh cross section is 5% of the total cross section for Z = 79 at E,= 0.6 MeV. However, it has been shown”) that even at these low energies the coherent scattering effect is small since it tends to change the effective path length of the y-ray in the source in a compensating manner. In ref. 20 the effect of coherent scattering on the selfabsorption of neutrons (gcoh/crtot z 5%) is shown to be less than 1%. 4. Conclusions
and the error estimated by comparison with eq. (1). r,, and r,, are the photopeak efficiencies for point source and parallel beams respectively. Since r,, and T,,converge to a common value as the source-todetector distance increases it is obvious that the error will be greatest for the situation where the source lies directly on the detector (this is the configuration in which the anisotropy of the absorption is maximum). We have estimated the error in this configuration for an 8” x 5” NaT(Tl) crystal (currently in use in our laboratory) for two typical cases: (1)13'Cssource (E,= = 0.632 MeV, R,= 0.1 cm and h, = 0.01 cm and (2) 491n activated foil (E, = 1.1 MeV, R,= 0.635 cm and h, = 0.05 cm). In both cases the error was found to be negligible (~0.2%). In the extreme case of an activated Au foil (E, = 0.4 MeV, R,= 0.635 cm and 11~ =O.l cm) the error was still less than 7%. 3.2. SELF-ABSORPTION FOR E,
MeV
There are two effects which complicate the estimation of F,for E, 0.6 MeV the spectrum is essentially flat so that any Compton
The y-ray self-absorption correction, F,,has been calculated by the Monte Carlo method for a large number of cylindrically symmetric geometries in which the source lies directly on the face of the detector. We have shown that the calculated F,can be used with
9
14
G 12 !!I 5 t 5
IO 08
%
06
g
04
s z
02 0
0
20
30
40
50
60
70
80
90
100
(+xloo) is Fig. 8. The energy spectrum of forward-scattered Compton y-rays in the source. Note that at E, = 1.5MeV the spectrum is essentially flat at the higher energies so that the contribution to the photopeak above background will be negligible. Below 0.6 MeV the spectrum is significantly peaked in the high energy region and constitutes a significant contribution above background to the photopeak region.
404
Y.S. HOROWITZ
intrinsic photopeak efficiencies calculated for nonabsorbing sources without introducing significant error. The calculation of Fs for y-ray energies less than 0.6 MeV is hazardous and energy resolution dependent because of the increasing contribution of forwardscattered Compton 7-rays from the source contributing to the photopeak. References 1) T. Nakamura, Nucl. Instr. and Meth. 105 (1972) 77. 2) T. Nakamura, Nucl. Instr. and Meth. 86 (1970) 163. 8) L. D. Franceschi and F. Pagni, Nucl. Instr. and Meth. 70 (1969) 325. a) M. Belluscio, R. De Leo, A. Pantaleo and A. Vox. Nucl. Instr. and Meth. 118 (1974) 553. 5) R. Rieppo, Nucl. Instr. and Meth. 107 (1973) 209. 6) R. L. D. French and N. Telfer, Nucl. Instr. and Meth. 51 (1967) 159.
et al.
7) M. L. Verheijke, Intern. J. Appl. Radiation & Isotopes 15 (1964) 559. s) y . S. Horowitz, S. Mordechai and A. Dubi, Nucl. Instr. and Meth., to be published. 9) B. F. Peterman, Nucl. Instr. and Meth. 101 (1972) 611. 10) H. Gotoh, Nucl. Instr. and Meth. 107 (1973) 199. 11) j. K. Dickens, Nucl. Instr. and Meth. 98 (1972) 451. 12) W. R. Dixon, Nucleonics 8 (1951) 68. 13) M. Belluscio, R. De Leo, A. Pantaleo and A. Vox, Nucl. Instr. and Meth. 114 (1974) 145. 14) 1. R. Williams, Nucl. Instr. and Meth. 44 (1966) 160. 15) E. Storm and H. 1. Israel, Nucl. Data Tables A7 (1970) 565. 16) R. L. Heath, Scintillation Spectrometry G a m m a Ray Spectrum Catalog, 1DO-16408, Phillips Petroleum Co. Atomic Energy Division, Idaho Falls (1957). 17) C. C. Grosjean, Nucl. Instr. and Meth. 17 (1962) 289. is) R. L. Heath, Nucl. Instr. and Meth. 43 (1966) 209. 19) R. L. Heath, R. G. Helmer, L. A. Schmittroth and G. A. Cazier, Nucl. Instr. and Meth. 47 (1967) 281. 2o) K. H. Beckurts and K. Wirtz, Neutron physics (Springer Verlag, Berlin, 1967) p. 243.
INSTRUMENTS
AND
METHODS
I22
0974) 399-404; ©
GAMMA-RAY SELF-ABSORPTION
NORTH-HOLLAND
PUBLISHING
CO.
CORRECTIONS
F O R E v > 0.6 M e V A N D p < 3 era- 1
Y . S. H O R O W I T Z ,
S. M O R D E C H A I
a n d A. D U B I
Department of Physics, Ben Gurion University of the Negeo, Beersheva, Israel Received 11 September 1974 The M o n t e Carlo m e t h o d was used to calculate the self-absorption correction for a large n u m b e r o f cylindrical source-cylindrical detector configurations in the energy range E~ greater t h a n 0.6 MeV a n d for values o f the total attenuation coefficient (it) up to 3 cm -z. T h e corrections are thus applicable to all possible materials a n d c o m p o u n d s a n d are presented in a functional form which allows easy calculation a n d interpolation. The error
introduced by using the self-absorption correction multiplied by a n o n - a b s o r b i n g source-detector efficiency is discussed and s h o w n to be usually negligible. T h e problem o f forwardscattered C o m p t o n ?-rays from the source contributing to the p h o t o p e a k efficiency in NaI(TI) detectors is s h o w n to be increasingly significant for E ~ < 0 . 6 MeV.
1. Introduction
coefficient up to 3 cm- ~, and are presented in a functional form which allows accurate and easy interpolation and calculation. The complete Monte Carlo program is outlined in Fortran to enable the calculation of F~ for geometrical configurations not encompassed by the present calculations. A discussion of the possible errors resulting from forward-scattered Compton 7-rays from the source contributing to the photopeak and from the use of Tp calculated for a
The problem of the detection efficiency of NaI(TI) scintillators or Ge(Li) semiconductor detectors for thick cylindrical 7-ray sources is a universal one common to many experimental configurations, For example, the absolute measurement of neutron spectra using the induced activity of threshold detectors or the many various types of absolute activation analysis all require an accurate knowledge of the overall detection efficiency. The detection efficiency for an arbitrary source-detector geometry can be defined as = Np/N
= O FsT v,
(l)
where Np is the number of counts observed in the photopeak/s, N is the strength of the source in disintegrations/s, g2 is the geometrical solid angle, Tp is the intrinsic photopeak efficiency of the detector and F~ is the self-absorption correction which accounts for the 7-rays absorbed in the source. Several authors have calculated ~ using Monte Carlo ~-4) and analytical methods s-v) for various source-detector-7-ray energy configurations while other authors have calculated F~ using the Monte Carlo a-t 1) or analytical methods 12, ~3). Although these authors have successfully illustrated the applicability of their techniques to the calculation of ~ or F~ in their specific experimental situation the almost limitless number of possible experimental configurations has limited their general usefulness. In this paper we have calculated F~ by the Monte Carlo method for as large a number of experimental configurations as possible so as to be of maximum benefit to the potential user. The calculations are carried out for 7-ray energies greater than 0.6 MeV, for source materials resulting in a total a~tenuation 399
READ, ZD, RF, RD,HF 55 C=ZD/SQRT(ZD, '2 * (RD.* RF ),'~, 2 ) G=ATAN ( I. ) 56 N H I T " 0 NESC = 0 39 DO50 1=1,5500 R=RF~SQRT(RANF(O) )
COT,,C.,,RANF(O)~,( I. 0 - C ) TE=ACOS (COT) 391
40
41
42 43 431 44 30
51
FI=G~4.*I~ANF(O)
.
.
SD=SORT(RD=~t- (R~81N (FI) ) * * 2 ) * R * COS(FI ) TEC=ATAN (SD / (Z+ZD) ) IF (TE,LT.TEC) 4 0 , 5 0 NHI T=NHIT' I $F=SQRT(RF~ ~2- (R*SIN(F I ) ),~'2) +R~CO$(F I ) 1"1[I ~ItTAN ( S F / Z ) I F ( T E . L T . T E I ) 4t ,42 EL=Z/COS (T E) GO TO 45 EL=SF/SIN (TE) W:-( I . / U ) ~MJ.OG(RAJqF ( 0 ) ) IF(WoLT. E L ) 5 0 , 4 4 N~C =NESC* I " CONTINUE A=NH IT B=NESC FSIB/A DP3 =SQRT( (I. "IS ) / ( FS*B ) ),, I00. 0 IF( DFS. GT. I. 0)39~31 PRINT 234,U,I=S, DFS~HF I .F'5. 3 ) END.
Fig. 1. The M o n t e Carlo m e t h o d in Fortran for the calculation o f the self-absorption correction.
400
Y.S.
HOROWITZ
non-absorbin9 point or disc source in eq. (1) is also presented in sections 3,1 and 3.2. 2. Method of calculation The Monte Carlo method has been used to simulate the history of many photons as they traverse the source and impinge on the detector. The 7-rays are generated randomly and isotropically throughout the source. In order to save c o m p u t a t i o n time a method due to Williams ~4) is used in which the direction of the ),-rays is sampled only in the smallest cone c o m m o n to all source points containing the detector face. For the ,,,-rays whose direction has them impinging on the detector, the distance to the first interaction is sampled from the distribution exp (-I~x), where p is the sum of the photoelectric, C o m p t o n and pair-production coefficients15). If the 7 path length in the source is larger than x the 7 is considered to have been absorbed and vice versa. The program accumulates the overall number of ~-rays aimed at the detector (N h) and the number of ?-rays which have escaped from the source (Ne). The self-absorption correction is then given by F, = N~/Nh,
(2)
with the error AFs given by
AF s = [Fs(1 - Fs)/Nh] ~.
et al.
The computer program in Fortran is shown in fig. I. Step 35 defines the c o m m o n cone, in steps 39-391 the 7-ray is generated, in steps 391-40 it is verified whether the 7-ray has hit the detector and steps 40-44 determine whether the ),-ray has been absorbed. The calculations were carried out for what we believed is the most c o m m o n type o f geometrical configuration encountered in activation analysis and activated foil counting, i.e., a disc-shaped volumetric source placed axially and symmetrically on the surface of a cylindrical detector. The actual distance between the source and the detector element was taken as 0.3 cm to account for the separation introduced by the container of the detector. The calculations include all possible combinations of the following sourcedetector geometries: Ra (radius of the d e t e c t o r ) = 3.81 cm and 10.16 cm, Rr (radius o f the source) = 0.2 cm, 0.635 cm, and 1 cm, and hf (thickness of the source) = 0.005cm, 0.01 cm, 0.05cm, 0.1 cm and 0.5 cm. 3. Results and discussion The behaviour of the self-absorption correction as a function of the total attenuation coefficient for the various source-detector configurations is displayed in
(3)
I00
0 96
0.96 ! ~
092 z
0 I-o I..d .q,¢r'
FC) U.l c~
Z
o EL 0.72 n," 0 fD
088 084-
0
080
Z o 1--
076 072 .'
o
,,./) 068 i
<~ 064 LI._ -, ..J UJ O~ 0.60 i
Rd = 1016crn
63 ,<
o 68
Rf =OZcm
U_ ..J U.I
064
(,o
o6oL
Rd=lO 16cm Rf =0 635 cm.
056
056~
052i
0 52
oz o'4o,'6o8 ioii2,4,6,'8 Zo22~,~z6 TOTAL ATTENUATION COEFFICIENT~t-(cm-O Fig. 2. The self-absorption correction as a function of the total attenuation coefficient for Ro = 10.16 cm and Rf = 0.2cm.
.....
,. . . .
0204060.8
, .....
\1,,
_
_
_
lOt2 1416 1 . 8 2 0 2 2 2 4 2 6
T O T A L A T T E N U A T I O N COEFFICIENTI, t-(cm-')
Fig. 3. T h e s e l f - a b s o r p t i o n c o r r e c t i o n as a f u n c t i o n o f t h e t o t a l a t t e n u a t i o n c o e f f i c i e n t f o r R e , = 1 0 . 1 6 c m a n d R ~ = 0 . 6 3 5 crn.
GAMMA-RAY
SELF-ABSORPTION
figs. 2-7. The results are not accurate for E~ <0.6 MeV for reasons discussed in section 3.3. For source thicknesses h~ = 0.005 cm and 0.01 cm the data were fitted with a function of the form f ( / 0 = a/~ + b, whereas for all the other thicknesses the function used was of the form f(It) = e x p . ( - 2 t 0 x (l +all~+azl~Z+aalz3). The fitting parameters are displayed in table 1 as well as the accuracy of the estimate of F,. This phenomenological fitting procedure allows accurate interpolation if the experimental configuration falls within the displayed parameters. 3.1. y-RAY FIELD CORRECTION The use of eq. (1) implies a knowledge of the detector intrinsic photopeak efficiency (T o) for the y-ray field of the absorbing source, however, most tabulations of detector photopeak efficiencies have been carried out for non-absorbing sources TM 16, 27). Only recently, Belluscio et al. 4) have evaluated To for a 3"× 3" NaI(TI) detector, for y-ray energies of 1.368, 2.754, 4.44 and 6.13 MeV from sources of aluminum, carbon and water. Since the absorption in the source is not isotropic, the y-ray field on the detector surface will differ from the v-ray field for a non-absorbing source so that the use of Tp from non-absorbing source tabulations may introduce a significant error.
fO
0
~
401
CORRECTIONS
We have attempted to evaluate this error in the following manner. For a disc source (Rf>>h¢) we have assumed that the v-ray field of the absorbing source can be approximated by the sum of a parallel beam field and an unperturbed point source field, the relative strength of which is decided by the following considerations. The number of 7-rays in the parallel beam can be approximated by
cos0,)[, exp W
where 0 a is the half opening angle of a cone defined by 0x = t a n - ' (Rjl), l is the detector length, and the exponential term is the parallel beam self-absorption correction (Fp). The remaining ?-rays are thus assigned to the point source field given by N
[ ½ ( I -- COS
02)F
~ -
1(1 - cos 01)Fo] ,
(5)
where 02 = tan-l(Rd/hO is the half opening angle of the cone subtended by the detector at the source. The average self-absorption for the point source field can then be determined from 52 = N-N- [ ½ ( c o s 01 -- c o s 0 2 ) f ' ] .
(6)
4~
,
I O0
=
096
°921 \
o
\
\
Ld Qf Q~
0 84
0
080 J
""~-o_
092 088
\
Z _00
..
{D hi ,,~
\
e:: 0 r,.D
084 080
0761 0 0,,.- 072] (3Z
n o 0") Fn
nO an
0 6 8 t Rd= IO f6crr,.
U._
I Rf" 'cm
W CO
1
\
\
o6ot
\
I
\
I.L ...J U..I 0'3
0681 064
060
056 052
~
~
T
6
~8 2[02'2 2-4.26 '
TOTAL ATTENUATION COEFFICIENTia-(cm-') Fig. 4. T h e self-absorption correction as a function o f the total a t t e n u a t i o n coefficient for Ra = 10.16 cm and Rf = 1 cm.
0~2-5~0~6os Io 12 14 L6 182.0222426 TOTAL ATTENUATION COEFFICIENTw(cm-') Fig. 5. T h e self-absorption correction as a function o f the total a t t e n u a t i o n coefficient for Rd = 3.81 cm and R~ = 0.2 cm.
402
v.s.
HOROWITZ
et al.
TABLE 1 T h e p a r a m e t e r s o f t h e s e l f - a b s o r p t i o n c o r r e c t i o n fitted to t h e f u n c t i o n f (p) = at~ + b a n d f ( # ) = e x p ( - }t#) (alpt + a21,t"2+ aa#3).
Source thickness
Parameters
Rd = 3.81
Re = 3.81
Rd = 3.81
Rr = 0 . 2
Rf = 0 . 6 3 5
Rf = 1
(cm)
(cm)
R a = 10.16 Rr = 0.2
Re = 10.16 Rr = 0.635
Rfl = 10.16 Rr = 1
Accuracy
(cm)
(cm)
(cm)
(%)
(cm)
(cm)
-6.63
-6.66
-7.13
-8.81
-9.24
-9.02
b
1.0
1.0
1.0
1.0
1.0
1.0
a x 102
- 1.14
-- 1.24
- 1.24
- 1.44
- 1.54
b
1.0
1.0
1.0
1.0
a x 103 0.005
0.5
- 1.61
0.01
0.5
0.05
0.1
0.5
}tx al x a2 x a3 x
102 102 103 10 a
+5.17 40.10 4 0.02 +0.33
+5.78 - 1.31 4 1.24 - 2.75
}t x al x a2 x a3X
10 ~ 102 10 '~ 103
+ 7.97 - 1.56 +1.54 +3.51
4 10.2 -2.98 +2.40 +4.34
)1. x at x a2X a3 x
10 102 102 103
+ 1.65 -0.313 -0,41 4 3.0
+ 2.89 -6.0 +4.30 - 8.24
+6.79 -2.88 + 2.84 - 6.60
1.0
+6.05 -0.53 + 4.0 -- 5.29
+5.40 +0.28 0 0
+ 10.8 - 1.38 40.874 -0.917
+ 8.83 40.356 -0.637 +2.14
+ 11.7 + 1.67 +1.18 -1.69
+ 12.2 - 1.75 40.80 40,248
+ 3.29 -6.68 +5.39 - 9.98
+ 1.65 - 1.73 -0,027 + 3.0
+ 2.99 - 5.10 +3.72 - 5.80
+ 3,44 -7.62 46.10 - 1.12
h~o
1.00
1.0
+6.95 -2.28 + 20.9 -4.48
0.8
1.0
2.0
1.00 '
0 96,
0924 \
\
I\ \ 0681 \ 0.84-' 0 \
eoso-
092
'"..
0887"
o I(o t,i
\
0 80 o o
0
.
7
2
o68i
z o F-
~
\
_ 0.64-
0 64"
Rd=SBlem' Rf =0655cm.
076 ~ 072 0
68.
113
N ~,~,,
u_ ...1 ILl
0.60-
Rd= 581cm Rf=l cm
064~ 060-
0.56-
X
0 56\
052,
,
,
~
,
,
,
N,
,
) . 2 0 4 0 6 08 I0 1.2 14 1.6 18202.2 2.42.6 T O T A L ATTENUATION
COEFFICIENT~-(cm-')
Fig. 6. T h e s e l f - a b s o r p t i o n c o r r e c t i o n as a f u n c t i o n o f t h e t o t a l a t t e n u a t i o n coefficient f o r R e = 3.81 c m a n d Rt = 0.635 c m .
o.52.J
0 2 Q 4 0 6 0 8 I0 t2 14 16 t 8 2 0 2 2 2 4 2 6
TOTAL ATTENUATION COEFFICIENT~t-(cm-') Fig. 7. T h e s e l f - a b s o r p t i o n c o r r e c t i o n as a f u n c t i o n o f the t o t a l a t t e n u a t i o n coefficient for R e = 3.81 c m a n d Rr = 1 c m .
GAMMA-RAY
In this manner sorbing source
and a parallel
SELF-ABSORPTION
we have resolved the field of the abinto a point source field of strength
y-ray field of strength
The overall strength
of the field is preserved
since (9)
The number of counts now be obtained from
expected
in the photopeak
can
403
CORRECTIONS
contribution to the photopeak will be subtracted away with the genera1 background. For E,<0.6MeV the Compton spectrum is increasingly peaked under the photopeak so that conventional calculations will consistently underestimate F,.At E7= 0.2 MeV the error in N, may be as large as 5%. The second complication is the effect due to the increasing importance of Rayleigh scattering below 0.6 MeV. For example, the Rayleigh cross section is 5% of the total cross section for Z = 79 at E,= 0.6 MeV. However, it has been shown”) that even at these low energies the coherent scattering effect is small since it tends to change the effective path length of the y-ray in the source in a compensating manner. In ref. 20 the effect of coherent scattering on the selfabsorption of neutrons (gcoh/crtot z 5%) is shown to be less than 1%. 4. Conclusions
and the error estimated by comparison with eq. (1). r,, and r,, are the photopeak efficiencies for point source and parallel beams respectively. Since r,, and T,,converge to a common value as the source-todetector distance increases it is obvious that the error will be greatest for the situation where the source lies directly on the detector (this is the configuration in which the anisotropy of the absorption is maximum). We have estimated the error in this configuration for an 8” x 5” NaT(Tl) crystal (currently in use in our laboratory) for two typical cases: (1)13'Cssource (E,= = 0.632 MeV, R,= 0.1 cm and h, = 0.01 cm and (2) 491n activated foil (E, = 1.1 MeV, R,= 0.635 cm and h, = 0.05 cm). In both cases the error was found to be negligible (~0.2%). In the extreme case of an activated Au foil (E, = 0.4 MeV, R,= 0.635 cm and 11~ =O.l cm) the error was still less than 7%. 3.2. SELF-ABSORPTION FOR E,
MeV
There are two effects which complicate the estimation of F,for E,
The y-ray self-absorption correction, F,,has been calculated by the Monte Carlo method for a large number of cylindrically symmetric geometries in which the source lies directly on the face of the detector. We have shown that the calculated F,can be used with
9
14
G 12 !!I 5 t 5
IO 08
%
06
g
04
s z
02 0
0
20
30
40
50
60
70
80
90
100
(+xloo) is Fig. 8. The energy spectrum of forward-scattered Compton y-rays in the source. Note that at E, = 1.5MeV the spectrum is essentially flat at the higher energies so that the contribution to the photopeak above background will be negligible. Below 0.6 MeV the spectrum is significantly peaked in the high energy region and constitutes a significant contribution above background to the photopeak region.
404
Y.S. HOROWITZ
intrinsic photopeak efficiencies calculated for nonabsorbing sources without introducing significant error. The calculation of Fs for y-ray energies less than 0.6 MeV is hazardous and energy resolution dependent because of the increasing contribution of forwardscattered Compton 7-rays from the source contributing to the photopeak. References 1) T. Nakamura, Nucl. Instr. and Meth. 105 (1972) 77. 2) T. Nakamura, Nucl. Instr. and Meth. 86 (1970) 163. 8) L. D. Franceschi and F. Pagni, Nucl. Instr. and Meth. 70 (1969) 325. a) M. Belluscio, R. De Leo, A. Pantaleo and A. Vox. Nucl. Instr. and Meth. 118 (1974) 553. 5) R. Rieppo, Nucl. Instr. and Meth. 107 (1973) 209. 6) R. L. D. French and N. Telfer, Nucl. Instr. and Meth. 51 (1967) 159.
et al.
7) M. L. Verheijke, Intern. J. Appl. Radiation & Isotopes 15 (1964) 559. s) y . S. Horowitz, S. Mordechai and A. Dubi, Nucl. Instr. and Meth., to be published. 9) B. F. Peterman, Nucl. Instr. and Meth. 101 (1972) 611. 10) H. Gotoh, Nucl. Instr. and Meth. 107 (1973) 199. 11) j. K. Dickens, Nucl. Instr. and Meth. 98 (1972) 451. 12) W. R. Dixon, Nucleonics 8 (1951) 68. 13) M. Belluscio, R. De Leo, A. Pantaleo and A. Vox, Nucl. Instr. and Meth. 114 (1974) 145. 14) 1. R. Williams, Nucl. Instr. and Meth. 44 (1966) 160. 15) E. Storm and H. 1. Israel, Nucl. Data Tables A7 (1970) 565. 16) R. L. Heath, Scintillation Spectrometry G a m m a Ray Spectrum Catalog, 1DO-16408, Phillips Petroleum Co. Atomic Energy Division, Idaho Falls (1957). 17) C. C. Grosjean, Nucl. Instr. and Meth. 17 (1962) 289. is) R. L. Heath, Nucl. Instr. and Meth. 43 (1966) 209. 19) R. L. Heath, R. G. Helmer, L. A. Schmittroth and G. A. Cazier, Nucl. Instr. and Meth. 47 (1967) 281. 2o) K. H. Beckurts and K. Wirtz, Neutron physics (Springer Verlag, Berlin, 1967) p. 243.