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Off-center X-ray detection efficiencies of Si(Li) detectors
NUCLEAR
INSTRUMENTS
AND
METHODS
143 ( 1 9 7 7 )
57-60;
Q
NORTH-HOLLAND
PUBLISHING
CO.
OFF-CENTER X-RAY DETECTION EFFICIENCIES OF Si(Li) DETECTORS ZEEV B. ALFASSI and R A Y A N O T H M A N
Dept. of Nuclear Chemistry, Soreq Nuclear Research Centre, Yavne, Israel and Dept. ol Nuclear Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel Received 29 November 1976 The X-ray detection efficiencies at off-center points on the surface of Si(Li) detectors were found to be dependent on the distance from the center. T h e dependence of the efficiency e(r) on the radial distance can be approximated by the equation ~(r) e0 e x p ( ~r2) where cz is a constant, characteristic of the detector, and is a linearly decreasing function of the inverse of the energy of the X-ray. The efficiencies for h o m o g e n e o u s disc, line or rectangular radioactive sources were calculated.
1. Introduction X-ray spectrometry is widely used following Xray fluorescence 1,2) and activation analysis 3-5) for the determination of trace amounts of material. The X-rays are detected mainly by means of a lithium-drifted silicon detector. Geometric and absorption factors produce variations in the X-ray detection efficiencies for radioactive point sources placed at different places on the surface of a Si(Li) detector. Consequently, it is inaccurate to use detection efficiency data for point sources at the center of the detector for the calculation of the radioactivity of large samples counted over a large portion of the surface of the detector. This problem will not exist if the counting is done at a large distance from the detector, which makes it possible to effectively consider the sample as a point source. In many cases, however, this solution is undesirable because of the resulting low counting rates. In these cases it is preferable to maximize the counting rate by counting the sample on the surface of the detector and to correct for the variations in the off-center counting efficiencies. An analytical calculation of this correction is very difficult because of factors such as geometry and absorption which affect the detection efficiency. A simple method is to experimentally map the off-center detection efficiencies for X-ray point sources and integrate, either analytically or numerically, over the whole area of the sample. Kushelevski and Alfassi 6) measured the offcenter gamma ray detection efficiencies of cylindrical single open-ended Ge(Li) detectors for 80-1330 keV gamma rays and found that the dependence of the efficiency on the radial distance
can be approximated quite satisfactorily by the equation t ( r ) = t0exp(-/3?), where t0 is the efficiency at the center of the detector and/~ is a constant characteristic of the detector and the energy of the 7-rays. Lindstrom 7) measured the radial efficiency gradients in close-end coaxial and openend coaxial Ge(Li) detectors and found that the radial dependence of the efficiency fit a parabola somewhat better than a Gaussian. The present work was undertaken to find if a similar relation also holds for Si(Li) detectors, where the sensitive area is usually much smaller than that of the Ge(Li) detectors. 2. Experimental Two calibrated X-ray point sources obtained from Amersham were used for relative efficiency measurements: 57Co [6.4 and 14.4keVS)] and 241Am [17.7, 20.1, 26.3 and 59.5 key9)]. The two point sources were placed on the surfaces of two Si(Li) detectors (Seforad, Israel) and counted for a fixed period of time at various radial distances from the center. Measurements were made at 1.5 mm steps to an accuracy of +0.5 ram. At each step, four measurements were taken at points 90 ° apart. At each point at least 10000 counts were recorded. Detector I is 100 mm 2 in area and 4 mm in depth and detector II is 25 mm 2 in area, 3.0 mm in depletion depth; both have a 0.125 mm beryllium window. The resolution for 6.4 keV Fe K Xrays of detectors I and II, is 500 and 180 eV, respectively.
3. Results Typical counts at four points at the same distance from the center, with the radii vectors form-
58
Z. B. ALFASSI AND R. NOTHMAN
TABLE 1 Measurements of a 241Am point source at four different points at the same radial distance from the center of the detector. Energy (keV)
Radial distance (mm)
17.7 17.7 17.7 20.1
3 4.5 6 3
I
No, of counts
216554 185416 156972 52075
209168 208344 207414 176849 191899 174426 148161 156493 144255 5 0 0 3 9 5 0 1 1 8 48804
ing right angles, are given in table 1. These results show that, with slight variations within the experimental error, the efficiencies are dependent only on the distance from the center and not on the position on the detector. The radial dependence of the efficiency for different X-ray energies for the two detectors are given in table 2. The curves describing the radial dependence are generally bellshaped, as can be seen in figs. 1 and 2. Bell-shaped curves can be approximated by a modified Gaussian
g(r)
= co e x p ( - ~ r 2 ) ,
(1)
where e(r) is the efficiency at distance r from the center, eo is the efficiency at the center of the detector and ~ is a constant which determines the width of the bell-shaped curve. As can be seen in fig. 3, a linear dependence exists for l n [ e ( r ) / e o l v s r 2 but not for e(r)/eoVSr 2 or
"O
59 5 keV
01
I
I
I
I
oI
I
[
I
I
oI
I
I
I
I
o rr
0
J
I0 Distance off-center (ram)
Fig. 1. The experimental relative efficiencies as a function of the radial off-center distance together with the plot of eq. (1) using values from table 3, for detector I.
ln[e(r)/eolvsr. Thus the fit to a Gaussian 6) is much better than the fit to a parabola7). The same relationship was found for each of the X-ray energies. The constant o~ for the different X-
TABLE 2 The relative efficiencies at off-center points of the detectors for various energies (at center, efficiency = 1.0). Detector
I
Energy (keV)
1.5
3.0
6.4 14.4 17.7 20.1 26.3 59.5
0.965 0.957 0.972 0.993 0.976 0.983
0.859 0.860 0.877 0.898 0.877 0.897
Energy (keY)
1.5
6.4 14.4 17.7 20.1 26.3 59.5
0.917 0.930 0.976 0.973 0.967 0.981
Radial distance (mm) 4.5 6.0 0.703 0.761 0.759 0.793 0.768 0.817
0.529 0.624 0.631 0.676 0.663 0.724
Radial distance (mm) 3.0 4.5 0.741 0.846 0.902 0.920 0.912 0.959
0.603 0.716 0.805 0.816 0.820 0.928
7.5
9.0
0.370 0.479 0.506 0.557 0.552 0.614
0.263 0.373 0.367 0.401 0.429 0.515
6.0 0.432 0.579 0.685 0.711 0.731 0.877
X-RAY
DETECTION
L
59 5 keV I
I
L
o k9
01
b
i
L
,$
J
(,j~ D(r) e(r) d A /3
D(r) dA
where e is the efficiency for the whole sample, A is the area of the sample and D(r) is the activity per unit area at distance r from the center. a) For the case of a h o m o g e n e o u s radioactive disc of radius R, the integration will give, for the
0)
Y i4.4 keV
I
h
59
The calculated values of o~ given in table 3, indicate that o~ is a m o n o t o n o u s decreasing function of the energy of the X-rays for both detectors. Fig. 4 shows that o~ is approximately a linear function of the inverse of the energy. The efficiency of the detector for a large source counted on the surface of the detector can be calculated from eq. (3)
D e ~ e c ~ e r II
o
EFFICIEN'CIES
h
1.5
ot 0
t
2
!
'q.
L
~,
6
,w, 0.5
Distance off-center(ram)
I
Fig. 2. The experimental relative efficiencies as a function of the radial off-center distance together with the plot of eq. (1) using values from table 3, for detector II.
Oo
ray energies was determined by a least squares fit of the experimental data to the equation
~5
(2)
~ Io
where C is the number of counts. The calculated values of ~z for the different X-ray energies for the two detectors are given in table 3. The experimental data of the relative off-center efficiencies e(r)/eo together with the calculated curves of eq. (1) using the least squares values of o¢ are plotted in figs. 1 and 2. As can be seen the fit is quite satisfactory.
'~' ~ o5
ln[e(r)/eo] = ln[C(r)/Co] = --:~r 2,
5
IO
r(mm)
o,
50 rZ(mrn2)
I00
Lo
~. 0.6
TABLE 3
~z values (in m m -2) for the different X-ray energies for the two detectors. Detector
1 II
6.4
14.4
0.0170 0.0242
0.0126 0.0157
Energy (keV) 17.7 20.1
0.0124 0.0106
0.0110 0.0096
0.2 I o
26.3
59.5
0.0107 0.0090
0.0089 0.0037
50
I00
d(mm21
Fig. 3. Plot of the experimental data of relative efficiencies (at center efficiency = 1.0) for detector 1, 6.4 and 59.5 keV X-rays in three different functional forms e(r)/eoVS r 2, -In[e(r)/eo]VS r2, -ln[e(r)/eolvs r. (, 6.4keV, • 59.5 keV).
60
Z. B. ALFASSI AND R. NOTHMAN
20
source efficiencies at the center even when comparing the activities of two different nuclides in one sample, because of the different values of oc for different X-ray energies. b) For a radioactive line source of length 2a, with its center located on the s y m m e t r y axis of the detector the integration [eq. (3)] gives
Detector [
/
/
/
/
% x/~z erf(fla), e = 2--~
%
where erf stands for the error functionl°). Values for erf can be found in the tables n'n) or approximated by polynomials for digital calculationl2). /3 is the square root of o:. c) For a rectangular source of length 2a and width 2b with its center located on the s y m m e t r y axis of the cylindrical detector the total efficiency is given by
0
x
Detector IT
/
'E E v
/
/o
/ 20-
/ / o
/
/
/ io
(5)
E07~
- 4fl~ab erf (fla) erf (fib).
(6)
-
/
Y
/ /o 0
i
I
5
I0
References
I
15
lOa/E (keY") Fig. 4. The dependence of ~z on the inverse of the X-ray energy.
whole sample efficiency: eo [1 e x p ( - c~R2)], = ~R---~ -
(4)
where e0 is the efficiency at the center of the detector. Eq. (4) indicates that a large error is introduced when calculating the total activity of a large sample from the point source efficiency at the center, especially for low energy X-rays. For example, w h e n a h o m o g e n e o u s radioactive disc of 9 m m radius with 6.4 keV Fe X-rays is counted on detector I the error is about a factor of two (e/e0 = 0.543). Eq. (4) indicates also that a considerable error can be introduced by using point
1) g. O. Miller, Spectrochemical analysis by X-ray fluorescence (Plenum Press, New York, 1972). 2) T. Shiraiwa and N. Fujino, Adv. X-ray Anal. 19 (1975) 239. 3) C. Shenberg, J. Gilat and H. L. Finston, Anal. Chem. 39 (1967) 780. 4) M. Mantel and S. Amiel, Anal. Chem. 44 (1972) 548. 5) S. Amiel, M. Mantel and Z. B. Alfassi, 1976 Int. Conf. on Modern trends in activation analysis, Munich, vol. 1, p. 695. 6) A. P. Kushelevski and Z. B. Alfassi, Nucl. Instr. and Meth. 131 (1975) 93. 7) R. M. Lindstrom, 1976 Int. Conf. on Modern trends in activation analysis, Munich, vol. 2, p. 1218. 8) G. M. Lederer, J. M. Hollander and L. Perlman, Table o f isotopes, 6th ed. (J. Wiley, New York, 1967). 9) R. J. Gehrke and R. A. Lokken, Nucl. Instr. and Meth. 97 (1971) 219. 10) G. Stephenson, Mathematical methods Jor science students (Longman, London, 1961) p. 225. ll) H. E. Etherington, Nuclear engineering handbook (McGrawHill, New York, 1958). 12) M. Abramowitz and I. A. Stegun, Handbook o f mathematical ./unctions (National Bureau of Standards, Washington, D.C. 1964).
INSTRUMENTS
AND
METHODS
143 ( 1 9 7 7 )
57-60;
Q
NORTH-HOLLAND
PUBLISHING
CO.
OFF-CENTER X-RAY DETECTION EFFICIENCIES OF Si(Li) DETECTORS ZEEV B. ALFASSI and R A Y A N O T H M A N
Dept. of Nuclear Chemistry, Soreq Nuclear Research Centre, Yavne, Israel and Dept. ol Nuclear Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel Received 29 November 1976 The X-ray detection efficiencies at off-center points on the surface of Si(Li) detectors were found to be dependent on the distance from the center. T h e dependence of the efficiency e(r) on the radial distance can be approximated by the equation ~(r) e0 e x p ( ~r2) where cz is a constant, characteristic of the detector, and is a linearly decreasing function of the inverse of the energy of the X-ray. The efficiencies for h o m o g e n e o u s disc, line or rectangular radioactive sources were calculated.
1. Introduction X-ray spectrometry is widely used following Xray fluorescence 1,2) and activation analysis 3-5) for the determination of trace amounts of material. The X-rays are detected mainly by means of a lithium-drifted silicon detector. Geometric and absorption factors produce variations in the X-ray detection efficiencies for radioactive point sources placed at different places on the surface of a Si(Li) detector. Consequently, it is inaccurate to use detection efficiency data for point sources at the center of the detector for the calculation of the radioactivity of large samples counted over a large portion of the surface of the detector. This problem will not exist if the counting is done at a large distance from the detector, which makes it possible to effectively consider the sample as a point source. In many cases, however, this solution is undesirable because of the resulting low counting rates. In these cases it is preferable to maximize the counting rate by counting the sample on the surface of the detector and to correct for the variations in the off-center counting efficiencies. An analytical calculation of this correction is very difficult because of factors such as geometry and absorption which affect the detection efficiency. A simple method is to experimentally map the off-center detection efficiencies for X-ray point sources and integrate, either analytically or numerically, over the whole area of the sample. Kushelevski and Alfassi 6) measured the offcenter gamma ray detection efficiencies of cylindrical single open-ended Ge(Li) detectors for 80-1330 keV gamma rays and found that the dependence of the efficiency on the radial distance
can be approximated quite satisfactorily by the equation t ( r ) = t0exp(-/3?), where t0 is the efficiency at the center of the detector and/~ is a constant characteristic of the detector and the energy of the 7-rays. Lindstrom 7) measured the radial efficiency gradients in close-end coaxial and openend coaxial Ge(Li) detectors and found that the radial dependence of the efficiency fit a parabola somewhat better than a Gaussian. The present work was undertaken to find if a similar relation also holds for Si(Li) detectors, where the sensitive area is usually much smaller than that of the Ge(Li) detectors. 2. Experimental Two calibrated X-ray point sources obtained from Amersham were used for relative efficiency measurements: 57Co [6.4 and 14.4keVS)] and 241Am [17.7, 20.1, 26.3 and 59.5 key9)]. The two point sources were placed on the surfaces of two Si(Li) detectors (Seforad, Israel) and counted for a fixed period of time at various radial distances from the center. Measurements were made at 1.5 mm steps to an accuracy of +0.5 ram. At each step, four measurements were taken at points 90 ° apart. At each point at least 10000 counts were recorded. Detector I is 100 mm 2 in area and 4 mm in depth and detector II is 25 mm 2 in area, 3.0 mm in depletion depth; both have a 0.125 mm beryllium window. The resolution for 6.4 keV Fe K Xrays of detectors I and II, is 500 and 180 eV, respectively.
3. Results Typical counts at four points at the same distance from the center, with the radii vectors form-
58
Z. B. ALFASSI AND R. NOTHMAN
TABLE 1 Measurements of a 241Am point source at four different points at the same radial distance from the center of the detector. Energy (keV)
Radial distance (mm)
17.7 17.7 17.7 20.1
3 4.5 6 3
I
No, of counts
216554 185416 156972 52075
209168 208344 207414 176849 191899 174426 148161 156493 144255 5 0 0 3 9 5 0 1 1 8 48804
ing right angles, are given in table 1. These results show that, with slight variations within the experimental error, the efficiencies are dependent only on the distance from the center and not on the position on the detector. The radial dependence of the efficiency for different X-ray energies for the two detectors are given in table 2. The curves describing the radial dependence are generally bellshaped, as can be seen in figs. 1 and 2. Bell-shaped curves can be approximated by a modified Gaussian
g(r)
= co e x p ( - ~ r 2 ) ,
(1)
where e(r) is the efficiency at distance r from the center, eo is the efficiency at the center of the detector and ~ is a constant which determines the width of the bell-shaped curve. As can be seen in fig. 3, a linear dependence exists for l n [ e ( r ) / e o l v s r 2 but not for e(r)/eoVSr 2 or
"O
59 5 keV
01
I
I
I
I
oI
I
[
I
I
oI
I
I
I
I
o rr
0
J
I0 Distance off-center (ram)
Fig. 1. The experimental relative efficiencies as a function of the radial off-center distance together with the plot of eq. (1) using values from table 3, for detector I.
ln[e(r)/eolvsr. Thus the fit to a Gaussian 6) is much better than the fit to a parabola7). The same relationship was found for each of the X-ray energies. The constant o~ for the different X-
TABLE 2 The relative efficiencies at off-center points of the detectors for various energies (at center, efficiency = 1.0). Detector
I
Energy (keV)
1.5
3.0
6.4 14.4 17.7 20.1 26.3 59.5
0.965 0.957 0.972 0.993 0.976 0.983
0.859 0.860 0.877 0.898 0.877 0.897
Energy (keY)
1.5
6.4 14.4 17.7 20.1 26.3 59.5
0.917 0.930 0.976 0.973 0.967 0.981
Radial distance (mm) 4.5 6.0 0.703 0.761 0.759 0.793 0.768 0.817
0.529 0.624 0.631 0.676 0.663 0.724
Radial distance (mm) 3.0 4.5 0.741 0.846 0.902 0.920 0.912 0.959
0.603 0.716 0.805 0.816 0.820 0.928
7.5
9.0
0.370 0.479 0.506 0.557 0.552 0.614
0.263 0.373 0.367 0.401 0.429 0.515
6.0 0.432 0.579 0.685 0.711 0.731 0.877
X-RAY
DETECTION
L
59 5 keV I
I
L
o k9
01
b
i
L
,$
J
(,j~ D(r) e(r) d A /3
D(r) dA
where e is the efficiency for the whole sample, A is the area of the sample and D(r) is the activity per unit area at distance r from the center. a) For the case of a h o m o g e n e o u s radioactive disc of radius R, the integration will give, for the
0)
Y i4.4 keV
I
h
59
The calculated values of o~ given in table 3, indicate that o~ is a m o n o t o n o u s decreasing function of the energy of the X-rays for both detectors. Fig. 4 shows that o~ is approximately a linear function of the inverse of the energy. The efficiency of the detector for a large source counted on the surface of the detector can be calculated from eq. (3)
D e ~ e c ~ e r II
o
EFFICIEN'CIES
h
1.5
ot 0
t
2
!
'q.
L
~,
6
,w, 0.5
Distance off-center(ram)
I
Fig. 2. The experimental relative efficiencies as a function of the radial off-center distance together with the plot of eq. (1) using values from table 3, for detector II.
Oo
ray energies was determined by a least squares fit of the experimental data to the equation
~5
(2)
~ Io
where C is the number of counts. The calculated values of ~z for the different X-ray energies for the two detectors are given in table 3. The experimental data of the relative off-center efficiencies e(r)/eo together with the calculated curves of eq. (1) using the least squares values of o¢ are plotted in figs. 1 and 2. As can be seen the fit is quite satisfactory.
'~' ~ o5
ln[e(r)/eo] = ln[C(r)/Co] = --:~r 2,
5
IO
r(mm)
o,
50 rZ(mrn2)
I00
Lo
~. 0.6
TABLE 3
~z values (in m m -2) for the different X-ray energies for the two detectors. Detector
1 II
6.4
14.4
0.0170 0.0242
0.0126 0.0157
Energy (keV) 17.7 20.1
0.0124 0.0106
0.0110 0.0096
0.2 I o
26.3
59.5
0.0107 0.0090
0.0089 0.0037
50
I00
d(mm21
Fig. 3. Plot of the experimental data of relative efficiencies (at center efficiency = 1.0) for detector 1, 6.4 and 59.5 keV X-rays in three different functional forms e(r)/eoVS r 2, -In[e(r)/eo]VS r2, -ln[e(r)/eolvs r. (, 6.4keV, • 59.5 keV).
60
Z. B. ALFASSI AND R. NOTHMAN
20
source efficiencies at the center even when comparing the activities of two different nuclides in one sample, because of the different values of oc for different X-ray energies. b) For a radioactive line source of length 2a, with its center located on the s y m m e t r y axis of the detector the integration [eq. (3)] gives
Detector [
/
/
/
/
% x/~z erf(fla), e = 2--~
%
where erf stands for the error functionl°). Values for erf can be found in the tables n'n) or approximated by polynomials for digital calculationl2). /3 is the square root of o:. c) For a rectangular source of length 2a and width 2b with its center located on the s y m m e t r y axis of the cylindrical detector the total efficiency is given by
0
x
Detector IT
/
'E E v
/
/o
/ 20-
/ / o
/
/
/ io
(5)
E07~
- 4fl~ab erf (fla) erf (fib).
(6)
-
/
Y
/ /o 0
i
I
5
I0
References
I
15
lOa/E (keY") Fig. 4. The dependence of ~z on the inverse of the X-ray energy.
whole sample efficiency: eo [1 e x p ( - c~R2)], = ~R---~ -
(4)
where e0 is the efficiency at the center of the detector. Eq. (4) indicates that a large error is introduced when calculating the total activity of a large sample from the point source efficiency at the center, especially for low energy X-rays. For example, w h e n a h o m o g e n e o u s radioactive disc of 9 m m radius with 6.4 keV Fe X-rays is counted on detector I the error is about a factor of two (e/e0 = 0.543). Eq. (4) indicates also that a considerable error can be introduced by using point
1) g. O. Miller, Spectrochemical analysis by X-ray fluorescence (Plenum Press, New York, 1972). 2) T. Shiraiwa and N. Fujino, Adv. X-ray Anal. 19 (1975) 239. 3) C. Shenberg, J. Gilat and H. L. Finston, Anal. Chem. 39 (1967) 780. 4) M. Mantel and S. Amiel, Anal. Chem. 44 (1972) 548. 5) S. Amiel, M. Mantel and Z. B. Alfassi, 1976 Int. Conf. on Modern trends in activation analysis, Munich, vol. 1, p. 695. 6) A. P. Kushelevski and Z. B. Alfassi, Nucl. Instr. and Meth. 131 (1975) 93. 7) R. M. Lindstrom, 1976 Int. Conf. on Modern trends in activation analysis, Munich, vol. 2, p. 1218. 8) G. M. Lederer, J. M. Hollander and L. Perlman, Table o f isotopes, 6th ed. (J. Wiley, New York, 1967). 9) R. J. Gehrke and R. A. Lokken, Nucl. Instr. and Meth. 97 (1971) 219. 10) G. Stephenson, Mathematical methods Jor science students (Longman, London, 1961) p. 225. ll) H. E. Etherington, Nuclear engineering handbook (McGrawHill, New York, 1958). 12) M. Abramowitz and I. A. Stegun, Handbook o f mathematical ./unctions (National Bureau of Standards, Washington, D.C. 1964).