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Accurate calculation of total efficiency of Ge well-type detectors suitable for efficiency calibration using common standard sources
Nuclear Instruments and Methods in Physics Research A 450 (2000) 98}108
Accurate calculation of total e$ciency of Ge well-type detectors suitable for e$ciency calibration using common standard sources O. Sima* Physics Department, Bucharest University, P.O.Box MG-11, Bucharest}Magurele, RO-76900, Romania Received 10 September 1999; received in revised form 7 December 1999; accepted 21 December 1999
Abstract A new method for the calculation of coincidence summing out corrections appropriate for c-spectrometry measurements in the Ge well-type detector geometry is proposed. The method is based on a simple formula for the total e$ciency of the detector, using an exact analytical relation for the mean detector thickness, combined with approximate formulas describing the interactions in the source and in the materials interposed between the source and the detector. The method can be easily applied for establishing the corrected, nuclide independent, e$ciency calibration curves of Ge well-type detectors using experimental calibration data obtained with the common mixed c-ray standard source containing Co and Y. 2000 Elsevier Science B.V. All rights reserved. PACS: 29.30.Kv Keywords: Well-type detectors; True coincidence summing; Total e$ciency; Ge detectors; c-Ray spectrometry
1. Introduction Well-type detectors are frequently used when low activity, small volume samples are to be analysed by c-ray spectrometry. Positive features of such measurements are: (a) very high e$ciency; (b) reduced dependence of the e$ciency on the measurement geometry; (c) small self-attenuation corrections. The main drawback of the measurements in the well-type detector geometry is the presence of high
* Tel.: #40-1-7805-385; fax: #40-1-4208-625. E-mail address: [email protected] (O. Sima).
coincidence summing e!ects in the case of multiphoton emitting nuclides. Such nuclides are frequently encountered in typical measurements. True coincidence summing e!ects occur when two or more than two photons produced subsequently in the same disintegration act, interact with the detector in a time interval which is smaller than the time required by the detector system to produce separate signals. In this case, a single global signal is produced, instead of a number of signals, associated each with a speci"c photon [1]. Consequently, certain signals may be lost from the corresponding peaks (summing out ewects); certain signals may be added to sum peaks (summing in ewects). Coincidence summing out e!ects depend on the probability that the photon of interest is
0168-9002/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 0 ) 0 0 2 5 4 - 0
O. Sima / Nuclear Instruments and Methods in Physics Research A 450 (2000) 98}108
accompanied by other photons, multiplied by the corresponding total detection e$ciencies e . Coin2 cidence summing in e!ects are proportional to the probability of simultaneous emission of all the photons involved, multiplied by the corresponding full-energy peak (FEP) e$ciencies e. The count rate R(E ) in the peak i (energy E ) of G G a nuclide X is modi"ed by coincidence summing e!ects. The coincidence summing correction factor F (E , X) is de"ned by G R(E , X) G F (E ; X)" . G y e(E )A(X) G G
(1)
In the equation above A(X) is the nuclide disintegration rate and e(E ) is the FEP e$ciency, as G obtained from a calibration measurement using nuclides showing negligible coincidence summing e!ects. With the exception of pure sum peaks, y represents the emission probability of the correG sponding photon; in the case of pure sum peaks y is set equal to 1 by convention (instead of y "0) G G and the multi-photon emission probability is embedded into the value of F . When coincidence summing e!ects are negligible F "1. In the presence of coincidence summing e!ects the value of F is nuclide- and peak-depen dent. Consequently, the problems related to such e!ects and to the evaluation of Eq. (1) may be avoided only by applying nuclide-speci"c calibrations. This procedure requires tedious experimental work and can be applied only in the case of particular matrices of the source. If nuclide-speci"c calibration is not available, Eq. (1) with appropriate values of F should be applied for establishing general purpose, nuclideindependent, calibration curves e(E). Indeed, the commonly used calibration sources contain Co and Y for e$ciency calibration in the high-energy part of the spectrum. The important coincidence e!ects (purely of the summing out type) present in the peaks of these nuclides should be accounted for when constructing the curves e(E). The speci"c values of F (E ; X) are also necessary when the G activity of nuclide X is to be assessed in an unknown sample on the basis of the measured count rate by applying nuclide-independent e$ciency calibration.
99
For the computation of F the conditional emis sion probability of various sets of photons (depending only on the decay scheme of the nuclide X) should be combined with the total and full energy peak e$ciencies [1]. Both e$ciencies depend on the experimental set up and on the matrix of the source; they also depend on the energy of the photon. Interesting methods for the computation of the coincidence summing corrections appropriate to well-type HPGe detectors have been recently proposed [2}4]. In the method developed by Blaauw [3] two experimental spectra (one for a standard calibration source, the other for a nuclide (Br) emitting many coincident photons) are "tted for obtaining peak e$ciency and peak-to-total ratio. The resulting functions are used afterwards for the analysis of other measurements. The di$culty of the method lies in the preparation of the Br source in the corresponding geometry and matrix. Wang et al. [4] use experimentally determined peak and total e$ciency curves for the computations of the coincidence summing corrections; they also introduce an approximate correction for the case of extended sources, based on the concept of e!ective solid angles [5]. Sima and Arnold [2] developed a realistic Monte Carlo code for the computation of the coincidence summing corrections; this code, enlarged with procedures for the computation of the self-attenuation corrections [6] and provided with user friendly interfaces is called germanium spectroscopy correction factors (GESPECOR). Contrary to the methods referred to above [3,4], in GESPECOR no experimental calibration data are required, if accurate values for the parameters of the detector system are available. Volume sources, with any matrix and density, can be as accurately and as easily modelled as the point sources. The main drawback of the method lies in the long computing time required by the Monte Carlo simulation procedure. Problems may also arise if detector data are inaccurate or if incomplete charge collection a!ects the detector signal; in such cases, a comparison between computed and measured values of the peak e$ciency for point sources may be used as a check of the model, or to improve the input data, when necessary. Of course, in the case of the methods [3,4], relying
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O. Sima / Nuclear Instruments and Methods in Physics Research A 450 (2000) 98}108
on experimentally determined parameters, the above problems are largely circumvented. In this paper a new theoretical method, much faster than the Monte Carlo method, is proposed. In this method the total e$ciency e (E) of Ge 2 well-type detectors is computed by an analytical relation based on an exact formula for the mean detector thickness. The e!ect of photon interactions in the source and in the materials interposed between the source and the sensitive volume of the detector on the total e$ciency is included by means of approximate analytical relations. The e (E) for2 mula can be directly applied for the evaluation of coincidence summing out e!ects. Thus, a simple method to obtain nuclide-independent e$ciency calibration curves e(E) using sources containing Co, Y, Ce is provided. The formula for e (E) supplemented by e(E) curves can be used for 2 the evaluation of the complete coincidence summing e!ects.
2. The model In this section the formula for the computation of e (E) will be derived. First, a simple case, but 2 incorporating the most important features of the model, will be analysed. Then several correction factors will be discussed and the "nal formula for e (E) will be presented. 2 2.1. The case of a point source and a bare detector We consider "rst the case of a point source P placed on the axis in the well of a simpli"ed model of the detector. In this model the end cap, the germanium dead layer and the detector holder are neglected. Photon interactions within the source and in the container of the source are neglected too. The above assumptions will be partly removed in the next sections. We consider also that the scattering of the photons from the shield and other surrounding materials back to the detector may be neglected. Due to the special geometry of the measurement in the case of the well-type detectors, this approximation should be rather good and will be maintained throughout this paper. The assumption of
cylindrical symmetry of the experimental set up will be applied as well. Suppose that the source has an emission rate of N photons per second, with energy E. In the simpli"ed model adopted, the count rate R in 2 the total spectrum of this source will be given by the following integral over the solid angle *X of the detector as seen from the source: N R " 2 4p
X
[1!exp(!k(E) l(X))] dX"N e (E). 2 (2)
In Eq. (2) k(E) is the total attenuation coe$cient (excluding coherent scattering) of the photons with energy E in the detector, l(X)"l(h) is the thickness of the sensitive volume of the detector for a photon trajectory with angular coordinates X"(h, ). It is supposed that low-energy spectrum cut-o! is not important and that charge collection problems, if present, do not a!ect the count rate in the total spectrum. For the evaluation of the integral of the exponential function we consider l(h)"t#m(h)
(3)
where t is a suitable mean value to be determined later, and develop the exponential function:
[1!exp(!kl(X))] dX X "*X!exp(!kt)
;
[1!km(X)#km(X)2] dX. X If t is chosen such that
X
m(X) dX"0
(4)
(5)
then the linear term in m under the integral in Eq. (4) vanishes. With this choice t is de"ned as the mean detector thickness as seen from the source: 1 t" *X
l(X) dX. (6) X Due to the fact that in the case of usual well-type detectors the distribution of l(X) is concentrated in the vicinity of t, the terms in higher powers in m are
O. Sima / Nuclear Instruments and Methods in Physics Research A 450 (2000) 98}108
101
expected to have a small contribution to the integral in Eq. (4) and will be neglected. Consequently, an approximate formula for the total e$ciency is e (E)"f [1!exp(!k(E)t)] 2 % where f is the geometric factor % *X f " . % 4p
(7)
(8)
Eq. (6) can be exactly evaluated in the case of the well-type detector geometry (Appendix), with the result 2p t" [F(R , H )#F(R , H )
*X !F(R , H )!F(R , H )]
where F(R, H)"R arctan
(9)
H R H # ln 1# . 2 H R
(10)
In the above equations R and R are the outer and
inner radii of the sensitive volume of the detector crystal and the H parameters are de"ned by V H "Z !Z (11) V V with x equal to t, w and b, respectively (Fig. 1). For negative values of H the symmetry property of the function F can be applied in Eq. (9): F(R,!H)"!F(R, H).
(12)
In the common case when the point source is placed inside of the well of the detector, the solid angle *X is given by
H *X"2p 1# . (13) (H#R In the case when the point source is placed above the detector crystal, that is H (0, R should be replaced by R in Eq. (13).
Eqs. (7)}(9) and (13) provide a simple analytical formula for the computation of the total e$ciency of a bare well-type detector for a point source. 2.2. A more realistic case In this section the model will be extended to include the detector end cap (EC), the germanium
Fig. 1. The geometric parameters of a well-type HPGe detector system.
dead-layer (DL) and the container of the source (CS). Also, the case of a volume source will be considered. Frequently, the interactions of the photons in the EC, CS or DL may be neglected in the well-type detector measurements, because usually these materials are very thin. The above assumption is applied in the model of the bare detector. But this is not always a good approximation, especially in the case of low-energy photons. For example, in the case when the well is lined with a high Z layer (a practice which is bene"cial for reducing the coincidence summing with X-rays and low-energy gammas and for suppressing the e!ects dependent on the matrix of the sample [7]), it is certainly inappropriate to neglect the interactions of low-energy photons in the lining material, because this material was intentionally introduced to attenuate such photons. Small volume samples, of the order of 0.5}3 cm, are typically measured in the well-type detector geometry. Despite the small volume, the spatial
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O. Sima / Nuclear Instruments and Methods in Physics Research A 450 (2000) 98}108
extension of the source (especially along the symmetry axis of the well) may be relevant for the computation of the geometric factor f and should % be taken into account. In the case of highly attenuating matrices, photon interactions within the source should also be included. In the Monte Carlo calculations reported in Ref. [2] all the above e!ects are realistically modelled. In the present paper simpler formulas will be proposed. 2.2.1. Transmission through the EC, CS and DL The probability P of a photon incident on the end cap to be able to interact further with the detector is approximated by P "exp(!k t )#[1!exp(!k t )]
0.5 k #k _! _. k (14)
The "rst term represents the probability of transmission without interaction in the EC; k is the total interaction coe$cient of the photon in the EC material (subscript e) and t is the mean thickness of EC. The second term is an approximate evaluation of the probability that a secondary particle resulting from a photon interaction in the EC can interact further with the detector. It is supposed that after pair production interactions (subscript Pp) the annihilation radiation will reach the detector. In the case of Compton interactions (subscript Co) the chance of the scattered radiation to reach the detector is considered approximately equal to 50%; this value takes into account the fact that the backscattered photons have lower energy and usually have to be transmitted again through the source and the other materials before reaching the detector and consequently their chance to interact with the detector is much reduced. Photoelectric interactions in the EC material are considered as absorption processes which do not contribute to the total e$ciency. The mean thickness t of the walls of the EC may be approximated by t t " d R !R
(15)
where t is given in Eq. (9) and d is the thickness of the walls of the end cap within the well. Similar relations should be applied for the probability P and P of `transmissiona through the wall of the container of the source (subscript c) and through the dead layer of the detector (subscript d). 2.2.2. Spatial extension of the source Due to changes both in *X and in t and the other mean thicknesses t (end cap), t (container's walls), t (Ge dead layer), the total e$ciency depends on the position of the point source within the well of the detector. While displacement in a plane perpendicular to the axis of the well practically does not change the total e$ciency, the displacement along the axis has a more important e!ect. In Fig. 2, the dependence of the geometric factor f (Eq. (8)) and % of the mean detector thickness t (Eq. (9)) on the position of the point source along the axis of the well is represented. The quantity 1!exp(!k(E)t) is also represented, for several values of the energy
Fig. 2. The dependence of the geometric factor f , of the mean % detector thickness t and of the quantity (1!exp(!l(E) t) on the position of the point source inside the well of the detector. The relative coordinate f is equal to 0 for the bottom of the crystal well and is equal to 1 for the top of the crystal. All the curves are normalized to the corresponding values for f"0.
O. Sima / Nuclear Instruments and Methods in Physics Research A 450 (2000) 98}108
E. The relative coordinate inside the well is de"ned by f"(Z !Z )/(Z !Z ), where Z and Z rep resent the ordinates of the bottom and of the top of the well (Fig. 1). All the quantities represented are normalized to the values corresponding to the point source placed at the bottom of the well, Z "Z . As the solid angle is the most sensitive factor, the e!ect of the spatial extension of the source may be included in the model by computing the geometric factor f as a suitable mean value: % 8 *X(z) 1 1 f " dz" [1#g(Z )!g(Z )]
% Z !Z 4p 2
8 (16)
where Z and Z are the lower and upper ordinates
of the active material in the source and ((Z !z)#o!o g(z)" . (17) Z !Z
In the equation above the parameter o takes the value o"R if z(Z and o"R if z'Z .
In the common case when the source is completely contained within the well, Eq. (16) reduces to f " %
1 ((Z !Z )#R!((Z !Z )#R
. 1# 2 Z !Z
(18) The dependence of t and of 1!exp(!k t) on the extension of the source may be approximated by evaluating t in Eq. (9) for the centre of the source, Z #Z . Z" 2
(19)
2.2.3. Interactions in the source In general, terms of two types contribute to the total e$ciency of a given experimental set-up. The direct (also called unscattered) contribution is represented by the photons which interact with the detector without prior interaction within the source. The scattered contribution is represented by photons which interact with the detector after
103
scattering in the source or other materials. In order to evaluate the probability P of a photon emitted by nuclear decay to escape from the source (subscript s), either without interactions, or after scattering in the source, the probability P of a photon _L to undergo exactly n interactions within the source is required. Obviously P is equal with the prob_L ability of the photon to be absorbed in the source at the nth interaction plus the probability of the photon to escape from the source after the nth interaction. The photoelectric interactions within the source are considered as absorption processes. Consequently, the probability of absorption at the nth interaction is equal to the ratio of the photoelectric to the total attenuation coe$cient at the energy of the photon before the interaction. Then
k (E ) P "P # P 1! _. L\ . _ _L k (E ) L\ L In this equation, E represents the energy of the L\ photon before the nth interaction; the subscript Ph indicates photoelectric interaction. The "rst term in the equation represents the probability of escaping without interactions in the source and this is the dominant term in the case of well-type detector measurements. In the same case it is a reasonable approximation to neglect in the sum all terms with n higher than 1. Indeed, at energies of the order of hundreds of keV the source dimensions are small in comparison with the photon mean free path and the probability of two or more than two collisions in the source is low. At lower energy, the dominant interaction is the photoelectric e!ect and the photon is absorbed at the "rst interaction. Using this approximation P can be related to P by _ _ P "1!P . _ _ Introducing the above equation for P and rear_ ranging the terms results in the following "nal formula for the probability P of a photon emitted by nuclear decay to escape from the source: k (E) P "1!(1!P ) _. . _ k (E)
(20)
The probability P of photon escaping from the _ source without interactions is approximated by the
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O. Sima / Nuclear Instruments and Methods in Physics Research A 450 (2000) 98}108
Wigner formula [8] as modi"ed by Rothenstein [9]: 1 P " (21) _ 1#k (E)l /3
where l is the mean chord in the source,
2R (Z !Z ) 4< . (22) l " "
S R #(Z !Z )
< is the volume and S is the area of the surface of the cylindrical source. (The Appleby formula [10] could also be used for the computation of P , but _ the e!ect on the total e$ciency obtained with these di!erent formulas is very similar.) 2.3. The total ezciency formula Combining the results from the previous sections the "nal formula for the total e$ciency is e (E)"f [1!exp(!k(E)t)]P P P P . (23) 2 % The mean detector thickness t given in Eq. (9) as well as the mean thicknesses of the end cap t (Eq. (15)), of the walls of the container t and of the germanium dead layer t required for the estima tion of P , P and P (Section 2.2) should be evalu ated for a point placed in the centre of the source. The geometric factor f should be evaluated by % Eqs. (8) and (13) in the case of a point source or by Eq. (18) in the case of an extended cylindrical source. In the case of an extended source the value of P should be computed by Eq. (20). The photon interaction coe$cients for germanium, for EC and CS materials and for the matrix of the source required for the evaluation of Eq. (23) can be conveniently computed by the XCOM program of Berger and Hubbell [11]. It should be mentioned that in common applications the last 4 factors in Eq. (23) represent small correction factors.
3. Validation of the model In order to check the validity of the model the values of the coincidence summing correction
factors F computed by the proposed method were compared with published data and with values computed by other methods. The computations were carried out for two HPGe well-type detectors, with the parameters listed in Table 1. The D1 detector [2] is a 350 cm detector operated at the Physikalisch}Technische Bundesanstalt (PTB), Braunschweig (Germany). The other detector, D2, lined with a gold layer [3], is operated at the Interfaculty Reactor Institute (IRI), Delft (Netherlands). The parameters of the sources for which the comparisons were done are listed in Table 2. The "rst source, S1, is a very small volume source. The sources S2(h) have increasing volumes, as the "lling height h increases from 0.8 to 5.0 cm. The sources S1 and S2 have low attenuation matrices. The S3 source is chosen as an example of a highly attenuating matrix. The rather involved decay data required by the computation of the F factors were prepared by a specialized subroutine included in the GESPECOR software. The primary decay scheme data were taken from the KORDATEN "le developed and maintained at PTB as input "le for the KORSUM program [12]. Alternative procedures, such as the matricial technique [13,14] or the Monte Carlo simulation of the decay chains [15] could be used to prepare the necessary input data.
Table 1 Parameters of the HPGe well-type detectors D1 and D2 used in this work Parameter
Det. D1 [2]
Det. D2 [3]
Crystal radius (cm) Crystal length (cm) Radius of the crystal well (cm) Length of the crystal well (cm) Ge dead layer in the well (lm) End cap in the well: material End cap in the well: thickness (mm) End cap radius in the well (cm) End cap length in the well (cm) Distance crystal well-bottom of end cap (cm) Lining in the well: material Lining in the well: thickness (mm)
3.75 8.20 0.80 5.65 0.30 Al 0.75
2.275 6.22 0.81 4.70 0.30 Al 0.75
0.50 6.15 0.30
0.46 4.70 0.30
* *
Au 0.40
O. Sima / Nuclear Instruments and Methods in Physics Research A 450 (2000) 98}108
105
Table 2 Parameters of the sources S1, S2 and S3 used in this work Parameter
Source S1 [3]
Source S2 [2]
Source S3
Radius (cm) Height h (cm) Matrix Container: material Container: thickness (mm)
0.40 0.15 Filter paper Polyethylene 1.0
0.40 0.8; 3.0; 5.0 Water Polyethylene 0.7
0.40 5.0 Copper * *
Table 3 Coincidence summing correction factors F appropriate to the D2 detector and the S1 source Nuclide
Energy (keV)
Present work
Sc-46 Sc-46 Ag-110 Ag-110 Ag-110 Ag-110 Ag-110 Ag-110 Ag-110 Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Sb-124 Sb-124 Sb-124 Sb-124 Sb-124 Sb-124
889.25 1120.51 657.76 677.62 687.01 706.68 763.94 884.68 937.49 475.36 563.23 569.31 604.71 795.87 801.95 602.73 645.86 713.78 722.78 968.19 1690.97
0.636 0.606 0.343 0.201 0.275 0.239 0.332 0.326 0.330 0.384 0.332 0.335 0.533 0.520 0.384 0.715 0.417 0.359 0.434 0.369 0.568
m m m m m m m
Ref. [3] 0.63 0.61 0.34 0.20 0.27 0.23 0.32 0.32 0.33 0.37 0.33 0.33 0.53 0.51 0.37 0.71 0.41 0.35 0.43 0.37 0.56
First, the case of the small volume source S1 was considered. In Table 3, the F factors computed for several nuclides by the present method are compared with the values published by Blaauw [3]. These values correspond to the measurement of the S1 source with the D2 detector. As in the present method only the total e$ciency is computed, the peaks which have a contribution from summing in e!ects amounting to more than 3% were not included in Table 3. The agreement between the results computed in this work and the published
values is very good, the highest discrepancy being less than 4%. Next, the computations were carried out for extended water sources with di!erent "lling heights h. The experimental values [2] of the F factors for the measurement of Cs and Co solutions with the D1 detector depend on the "lling height. These values are well reproduced by GESPECOR calculations. In Table 4, the results predicted by the present model are compared with the results computed by GESPECOR for the D1 detector and S2 sources, with h"0.8, 3.0 and 5.0 cm. In Table 4 peaks with important summing in contributions (e.g. the 1401 keV peak of Cs) have been included besides peaks for which summing in contributions are negligible; the summing in contribution was evaluated using the FEP e$ciency computed by GESPECOR. The highest discrepancy between the F values predicted by the present method and the results computed by GESPECOR is about 2% for h"0.8 cm, 3% for h"3.0 cm and 5% for h"5.0 cm. Finally, the case of an extended source with a highly attenuating matrix was considered. In Table 5, the results computed by the present method for the S3 source (matrix: copper) and the D1 detector are compared with the results computed by GESPECOR. Even in this extreme case the highest discrepancy is less than 6%. The results presented in Tables 3}5 prove that the simpli"ed model developed in this paper is reasonably accurate. In fact, in many cases even the simpler model of a point source and a bare detector, Eq. (7) is good enough for the computation of coincidence losses. Only when low-energy photons are involved, or when the source has a
106
O. Sima / Nuclear Instruments and Methods in Physics Research A 450 (2000) 98}108
Table 4 Coincidence summing correction factors F in the case of the measurement of the sources S2(h) with the detector D1. The "lling height is denoted by h (cm). Results presented in columns labelled by (a) are from this work, those in columns labelled by (b) are computed by GESPECOR Nuclide
h"0.8
Energy (keV)
Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Co-60 Co-60
475 563 569 605 796 802 1039 1168 1365 1401 1173 1332
h"3.0
h"5.0
(a)
(b)
(a)
(b)
(a)
(b)
0.155 0.099 0.103 0.289 0.261 0.156 0.499 1.033 1.335 0.380 0.436 0.414
0.156 0.099 0.103 0.289 0.261 0.156 0.507 1.053 1.363 0.388 0.434 0.413
0.162 0.105 0.109 0.297 0.269 0.162 0.513 1.054 1.366 0.377 0.444 0.423
0.165 0.108 0.112 0.300 0.273 0.164 0.520 1.061 1.378 0.374 0.445 0.424
0.189 0.131 0.135 0.332 0.305 0.189 0.548 1.066 1.380 0.329 0.478 0.457
0.195 0.138 0.142 0.336 0.311 0.195 0.559 1.093 1.419 0.346 0.477 0.458
Table 5 Coincidence summing correction factors F in the case of the measurement of the S3 source (source matrix: copper) with the detector D1 Nuclide
Energy (keV)
Present work
GESPECOR
Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Co-60 Co-60
475 563 569 605 796 802 1039 1168 1365 1401 1173 1332
0.194 0.137 0.140 0.337 0.314 0.194 0.494 0.866 1.090 0.231 0.479 0.458
0.191 0.135 0.138 0.326 0.308 0.191 0.489 0.858 1.090 0.245 0.457 0.439
strongly attenuating matrix, signi"cantly better results are obtained with the complete formula Eq. (23).
4. Conclusions A simple formula for the computation of the total e$ciency of a well-type HPGe detector was
proposed. In this formula the geometric parameters of the detection system are combined with photon interaction coe$cients in expressions which are easily evaluated. The total e$ciency is useful for the calculation of coincidence losses from the peaks (summing out e!ects) which inevitably occur in the measurements with well-type detectors. In the case of the peaks with negligible summing in contribution, the total e$ciency can be immediately used for the computation of the coincidence summing correction factors, F . Numerous peaks are of this type. Especially important examples are the main peaks of Co, Y and Ce, which are frequently used for e$ciency calibration. Using the appropriate F fac tors and Eq. (1) an ideal, nuclide independent e$ciency curve can be obtained on the basis of the experimental calibration data measured with the common mixed c-ray standard source, containing the above-mentioned nuclides. Total e$ciency combined with full-energy peak e$ciency can be used for the evaluation of F in the general case, when both coincidence summing out and coincidence summing in e!ects are present. The method obtained in this way is a very easy and very fast procedure for the computation of the coincidence summing corrections for
O. Sima / Nuclear Instruments and Methods in Physics Research A 450 (2000) 98}108
measurements with well-type HPGe detectors. The associated computational algorithm was implemented in the GESPECOR software, as an alternative to the more accurate but time consuming default Monte Carlo procedure.
107
The integrals can be explicitly computed, using the equality
arctan
z z do"o arctan o o
z z#o # ln #const z 2 Appendix
"F(o, z)#const
A.1. The mean detector thickness t
where the function F is de"ned in Eq. (10). This function has the properties:
First, Eq. (6) is transformed into an integral over the volume < of the detector in a spherical coordinate system r, h, with the origin in the position of the point source: 1 t" *X
P X 1 1 d< l(X) dX" dX dr" *X *X r X X P X 4 (A.1)
where r (X) and r (X) represent the radial coor
dinates of the intersection of a photon trajectory characterized by angular coordinates X"(h, ) with the sensitive volume of the detector. Obviously,
P X
dr"l(X).
(A.2)
P X
Next the integral in Eq. (A.1) is transformed to the cylindrical coordinates o, a, z represented in Fig. 1: 1 t" *X
o do da dz . o#(z!Z ) 4
(A.3)
Here Z is the ordinate of the point source in the cylindrical coordinate system. Consider the integral
I(o , o , z , z )" "
M X
M
M
M
o do dz o#(z!Z ) X
z !Z arctan o
z !Z !arctan o
(A.5)
do.
(A.4)
lim F(o, z)"0 M
(A.6)
lim F(o, z)"0 X F(o,!z)"!F(o, z).
(A.7) (A.8)
Accordingly, the function I(o , o , z , z ) can be written as I(o , o , z , z ) "F(o , z !Z )!F(o , z !Z ) !F(o , z !Z )#F(o , z !Z ). (A.9) The domain of integration < is divided into two cylindrical domains < (o "R , o "R ,
z "Z , z "Z ) and < (o "0, o "R ,
z "Z , z "Z ). The integral over < in Eqs. (A.1) and (A.3) is replaced by the sum of the integrals over < and < . Using Eqs. (A.3), (A.4) and (A.9) and the above properties of the function F, the "nal formula for the mean detector thickness t (Eq. (9)) is obtained.
References [1] K. Debertin, R.G. Helmer, Gamma- and X-ray Spectrometry with Semiconductor Detectors, North-Holland, Amsterdam, 1988. [2] O. Sima, D. Arnold, Appl. Radiat. Isot. 47 (1996) 889. [3] M. Blaauw, Nucl. Instr. and Meth. A 419 (1998) 146. [4] T.K. Wang et al., Nucl. Instr. and Meth. A 425 (1999) 504. [5] L. Moens et al., Nucl. Instr. and Meth. 187 (1981) 451. [6] O. Sima, C. Dovlete, Appl. Radiat. Isot. 48 (1997) 59. [7] M. de Bruin, P.J.M. Korthoven, P. Bode, Nucl. Instr. and Meth. 159 (1979) 301.
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[8] A.M. Weinberg, E.P. Wigner, The Physical Theory of Neutron Chain Reactors, The University of Chicago Press, Chicago, 1958. [9] W. Rothenstein, Nucl. Sci. Eng. 7 (1960) 162. [10] P.G. Appleby, N. Richardson, P.J. Nolan, Nucl. Instr. and Meth. B 71 (1992) 228. [11] M.J. Berger, J.H. Hubbell, XCOM: photon cross sections on a personal computer, NBSIR 87-3597, 1987.
[12] K. Debertin, U. SchoK tzig, Nucl. Instr. and Meth. 158 (1979) 471. [13] T.M. Semkow, G. Mehmood, P.P. Parekh, M. Virgil, Nucl. Instr. and Meth. A 290 (1990) 437. [14] M. Korun, R. Martincy icy , Nucl. Instr. and Meth. A 355 (1995) 600. [15] M. Decombaz, J.L. Laedermann, Nucl. Instr. and Meth. A 369 (1996) 375.
Accurate calculation of total e$ciency of Ge well-type detectors suitable for e$ciency calibration using common standard sources O. Sima* Physics Department, Bucharest University, P.O.Box MG-11, Bucharest}Magurele, RO-76900, Romania Received 10 September 1999; received in revised form 7 December 1999; accepted 21 December 1999
Abstract A new method for the calculation of coincidence summing out corrections appropriate for c-spectrometry measurements in the Ge well-type detector geometry is proposed. The method is based on a simple formula for the total e$ciency of the detector, using an exact analytical relation for the mean detector thickness, combined with approximate formulas describing the interactions in the source and in the materials interposed between the source and the detector. The method can be easily applied for establishing the corrected, nuclide independent, e$ciency calibration curves of Ge well-type detectors using experimental calibration data obtained with the common mixed c-ray standard source containing Co and Y. 2000 Elsevier Science B.V. All rights reserved. PACS: 29.30.Kv Keywords: Well-type detectors; True coincidence summing; Total e$ciency; Ge detectors; c-Ray spectrometry
1. Introduction Well-type detectors are frequently used when low activity, small volume samples are to be analysed by c-ray spectrometry. Positive features of such measurements are: (a) very high e$ciency; (b) reduced dependence of the e$ciency on the measurement geometry; (c) small self-attenuation corrections. The main drawback of the measurements in the well-type detector geometry is the presence of high
* Tel.: #40-1-7805-385; fax: #40-1-4208-625. E-mail address: [email protected] (O. Sima).
coincidence summing e!ects in the case of multiphoton emitting nuclides. Such nuclides are frequently encountered in typical measurements. True coincidence summing e!ects occur when two or more than two photons produced subsequently in the same disintegration act, interact with the detector in a time interval which is smaller than the time required by the detector system to produce separate signals. In this case, a single global signal is produced, instead of a number of signals, associated each with a speci"c photon [1]. Consequently, certain signals may be lost from the corresponding peaks (summing out ewects); certain signals may be added to sum peaks (summing in ewects). Coincidence summing out e!ects depend on the probability that the photon of interest is
0168-9002/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 0 ) 0 0 2 5 4 - 0
O. Sima / Nuclear Instruments and Methods in Physics Research A 450 (2000) 98}108
accompanied by other photons, multiplied by the corresponding total detection e$ciencies e . Coin2 cidence summing in e!ects are proportional to the probability of simultaneous emission of all the photons involved, multiplied by the corresponding full-energy peak (FEP) e$ciencies e. The count rate R(E ) in the peak i (energy E ) of G G a nuclide X is modi"ed by coincidence summing e!ects. The coincidence summing correction factor F (E , X) is de"ned by G R(E , X) G F (E ; X)" . G y e(E )A(X) G G
(1)
In the equation above A(X) is the nuclide disintegration rate and e(E ) is the FEP e$ciency, as G obtained from a calibration measurement using nuclides showing negligible coincidence summing e!ects. With the exception of pure sum peaks, y represents the emission probability of the correG sponding photon; in the case of pure sum peaks y is set equal to 1 by convention (instead of y "0) G G and the multi-photon emission probability is embedded into the value of F . When coincidence summing e!ects are negligible F "1. In the presence of coincidence summing e!ects the value of F is nuclide- and peak-depen dent. Consequently, the problems related to such e!ects and to the evaluation of Eq. (1) may be avoided only by applying nuclide-speci"c calibrations. This procedure requires tedious experimental work and can be applied only in the case of particular matrices of the source. If nuclide-speci"c calibration is not available, Eq. (1) with appropriate values of F should be applied for establishing general purpose, nuclideindependent, calibration curves e(E). Indeed, the commonly used calibration sources contain Co and Y for e$ciency calibration in the high-energy part of the spectrum. The important coincidence e!ects (purely of the summing out type) present in the peaks of these nuclides should be accounted for when constructing the curves e(E). The speci"c values of F (E ; X) are also necessary when the G activity of nuclide X is to be assessed in an unknown sample on the basis of the measured count rate by applying nuclide-independent e$ciency calibration.
99
For the computation of F the conditional emis sion probability of various sets of photons (depending only on the decay scheme of the nuclide X) should be combined with the total and full energy peak e$ciencies [1]. Both e$ciencies depend on the experimental set up and on the matrix of the source; they also depend on the energy of the photon. Interesting methods for the computation of the coincidence summing corrections appropriate to well-type HPGe detectors have been recently proposed [2}4]. In the method developed by Blaauw [3] two experimental spectra (one for a standard calibration source, the other for a nuclide (Br) emitting many coincident photons) are "tted for obtaining peak e$ciency and peak-to-total ratio. The resulting functions are used afterwards for the analysis of other measurements. The di$culty of the method lies in the preparation of the Br source in the corresponding geometry and matrix. Wang et al. [4] use experimentally determined peak and total e$ciency curves for the computations of the coincidence summing corrections; they also introduce an approximate correction for the case of extended sources, based on the concept of e!ective solid angles [5]. Sima and Arnold [2] developed a realistic Monte Carlo code for the computation of the coincidence summing corrections; this code, enlarged with procedures for the computation of the self-attenuation corrections [6] and provided with user friendly interfaces is called germanium spectroscopy correction factors (GESPECOR). Contrary to the methods referred to above [3,4], in GESPECOR no experimental calibration data are required, if accurate values for the parameters of the detector system are available. Volume sources, with any matrix and density, can be as accurately and as easily modelled as the point sources. The main drawback of the method lies in the long computing time required by the Monte Carlo simulation procedure. Problems may also arise if detector data are inaccurate or if incomplete charge collection a!ects the detector signal; in such cases, a comparison between computed and measured values of the peak e$ciency for point sources may be used as a check of the model, or to improve the input data, when necessary. Of course, in the case of the methods [3,4], relying
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O. Sima / Nuclear Instruments and Methods in Physics Research A 450 (2000) 98}108
on experimentally determined parameters, the above problems are largely circumvented. In this paper a new theoretical method, much faster than the Monte Carlo method, is proposed. In this method the total e$ciency e (E) of Ge 2 well-type detectors is computed by an analytical relation based on an exact formula for the mean detector thickness. The e!ect of photon interactions in the source and in the materials interposed between the source and the sensitive volume of the detector on the total e$ciency is included by means of approximate analytical relations. The e (E) for2 mula can be directly applied for the evaluation of coincidence summing out e!ects. Thus, a simple method to obtain nuclide-independent e$ciency calibration curves e(E) using sources containing Co, Y, Ce is provided. The formula for e (E) supplemented by e(E) curves can be used for 2 the evaluation of the complete coincidence summing e!ects.
2. The model In this section the formula for the computation of e (E) will be derived. First, a simple case, but 2 incorporating the most important features of the model, will be analysed. Then several correction factors will be discussed and the "nal formula for e (E) will be presented. 2 2.1. The case of a point source and a bare detector We consider "rst the case of a point source P placed on the axis in the well of a simpli"ed model of the detector. In this model the end cap, the germanium dead layer and the detector holder are neglected. Photon interactions within the source and in the container of the source are neglected too. The above assumptions will be partly removed in the next sections. We consider also that the scattering of the photons from the shield and other surrounding materials back to the detector may be neglected. Due to the special geometry of the measurement in the case of the well-type detectors, this approximation should be rather good and will be maintained throughout this paper. The assumption of
cylindrical symmetry of the experimental set up will be applied as well. Suppose that the source has an emission rate of N photons per second, with energy E. In the simpli"ed model adopted, the count rate R in 2 the total spectrum of this source will be given by the following integral over the solid angle *X of the detector as seen from the source: N R " 2 4p
X
[1!exp(!k(E) l(X))] dX"N e (E). 2 (2)
In Eq. (2) k(E) is the total attenuation coe$cient (excluding coherent scattering) of the photons with energy E in the detector, l(X)"l(h) is the thickness of the sensitive volume of the detector for a photon trajectory with angular coordinates X"(h, ). It is supposed that low-energy spectrum cut-o! is not important and that charge collection problems, if present, do not a!ect the count rate in the total spectrum. For the evaluation of the integral of the exponential function we consider l(h)"t#m(h)
(3)
where t is a suitable mean value to be determined later, and develop the exponential function:
[1!exp(!kl(X))] dX X "*X!exp(!kt)
;
[1!km(X)#km(X)2] dX. X If t is chosen such that
X
m(X) dX"0
(4)
(5)
then the linear term in m under the integral in Eq. (4) vanishes. With this choice t is de"ned as the mean detector thickness as seen from the source: 1 t" *X
l(X) dX. (6) X Due to the fact that in the case of usual well-type detectors the distribution of l(X) is concentrated in the vicinity of t, the terms in higher powers in m are
O. Sima / Nuclear Instruments and Methods in Physics Research A 450 (2000) 98}108
101
expected to have a small contribution to the integral in Eq. (4) and will be neglected. Consequently, an approximate formula for the total e$ciency is e (E)"f [1!exp(!k(E)t)] 2 % where f is the geometric factor % *X f " . % 4p
(7)
(8)
Eq. (6) can be exactly evaluated in the case of the well-type detector geometry (Appendix), with the result 2p t" [F(R , H )#F(R , H )
*X !F(R , H )!F(R , H )]
where F(R, H)"R arctan
(9)
H R H # ln 1# . 2 H R
(10)
In the above equations R and R are the outer and
inner radii of the sensitive volume of the detector crystal and the H parameters are de"ned by V H "Z !Z (11) V V with x equal to t, w and b, respectively (Fig. 1). For negative values of H the symmetry property of the function F can be applied in Eq. (9): F(R,!H)"!F(R, H).
(12)
In the common case when the point source is placed inside of the well of the detector, the solid angle *X is given by
H *X"2p 1# . (13) (H#R In the case when the point source is placed above the detector crystal, that is H (0, R should be replaced by R in Eq. (13).
Eqs. (7)}(9) and (13) provide a simple analytical formula for the computation of the total e$ciency of a bare well-type detector for a point source. 2.2. A more realistic case In this section the model will be extended to include the detector end cap (EC), the germanium
Fig. 1. The geometric parameters of a well-type HPGe detector system.
dead-layer (DL) and the container of the source (CS). Also, the case of a volume source will be considered. Frequently, the interactions of the photons in the EC, CS or DL may be neglected in the well-type detector measurements, because usually these materials are very thin. The above assumption is applied in the model of the bare detector. But this is not always a good approximation, especially in the case of low-energy photons. For example, in the case when the well is lined with a high Z layer (a practice which is bene"cial for reducing the coincidence summing with X-rays and low-energy gammas and for suppressing the e!ects dependent on the matrix of the sample [7]), it is certainly inappropriate to neglect the interactions of low-energy photons in the lining material, because this material was intentionally introduced to attenuate such photons. Small volume samples, of the order of 0.5}3 cm, are typically measured in the well-type detector geometry. Despite the small volume, the spatial
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O. Sima / Nuclear Instruments and Methods in Physics Research A 450 (2000) 98}108
extension of the source (especially along the symmetry axis of the well) may be relevant for the computation of the geometric factor f and should % be taken into account. In the case of highly attenuating matrices, photon interactions within the source should also be included. In the Monte Carlo calculations reported in Ref. [2] all the above e!ects are realistically modelled. In the present paper simpler formulas will be proposed. 2.2.1. Transmission through the EC, CS and DL The probability P of a photon incident on the end cap to be able to interact further with the detector is approximated by P "exp(!k t )#[1!exp(!k t )]
0.5 k #k _! _. k (14)
The "rst term represents the probability of transmission without interaction in the EC; k is the total interaction coe$cient of the photon in the EC material (subscript e) and t is the mean thickness of EC. The second term is an approximate evaluation of the probability that a secondary particle resulting from a photon interaction in the EC can interact further with the detector. It is supposed that after pair production interactions (subscript Pp) the annihilation radiation will reach the detector. In the case of Compton interactions (subscript Co) the chance of the scattered radiation to reach the detector is considered approximately equal to 50%; this value takes into account the fact that the backscattered photons have lower energy and usually have to be transmitted again through the source and the other materials before reaching the detector and consequently their chance to interact with the detector is much reduced. Photoelectric interactions in the EC material are considered as absorption processes which do not contribute to the total e$ciency. The mean thickness t of the walls of the EC may be approximated by t t " d R !R
(15)
where t is given in Eq. (9) and d is the thickness of the walls of the end cap within the well. Similar relations should be applied for the probability P and P of `transmissiona through the wall of the container of the source (subscript c) and through the dead layer of the detector (subscript d). 2.2.2. Spatial extension of the source Due to changes both in *X and in t and the other mean thicknesses t (end cap), t (container's walls), t (Ge dead layer), the total e$ciency depends on the position of the point source within the well of the detector. While displacement in a plane perpendicular to the axis of the well practically does not change the total e$ciency, the displacement along the axis has a more important e!ect. In Fig. 2, the dependence of the geometric factor f (Eq. (8)) and % of the mean detector thickness t (Eq. (9)) on the position of the point source along the axis of the well is represented. The quantity 1!exp(!k(E)t) is also represented, for several values of the energy
Fig. 2. The dependence of the geometric factor f , of the mean % detector thickness t and of the quantity (1!exp(!l(E) t) on the position of the point source inside the well of the detector. The relative coordinate f is equal to 0 for the bottom of the crystal well and is equal to 1 for the top of the crystal. All the curves are normalized to the corresponding values for f"0.
O. Sima / Nuclear Instruments and Methods in Physics Research A 450 (2000) 98}108
E. The relative coordinate inside the well is de"ned by f"(Z !Z )/(Z !Z ), where Z and Z rep resent the ordinates of the bottom and of the top of the well (Fig. 1). All the quantities represented are normalized to the values corresponding to the point source placed at the bottom of the well, Z "Z . As the solid angle is the most sensitive factor, the e!ect of the spatial extension of the source may be included in the model by computing the geometric factor f as a suitable mean value: % 8 *X(z) 1 1 f " dz" [1#g(Z )!g(Z )]
% Z !Z 4p 2
8 (16)
where Z and Z are the lower and upper ordinates
of the active material in the source and ((Z !z)#o!o g(z)" . (17) Z !Z
In the equation above the parameter o takes the value o"R if z(Z and o"R if z'Z .
In the common case when the source is completely contained within the well, Eq. (16) reduces to f " %
1 ((Z !Z )#R!((Z !Z )#R
. 1# 2 Z !Z
(18) The dependence of t and of 1!exp(!k t) on the extension of the source may be approximated by evaluating t in Eq. (9) for the centre of the source, Z #Z . Z" 2
(19)
2.2.3. Interactions in the source In general, terms of two types contribute to the total e$ciency of a given experimental set-up. The direct (also called unscattered) contribution is represented by the photons which interact with the detector without prior interaction within the source. The scattered contribution is represented by photons which interact with the detector after
103
scattering in the source or other materials. In order to evaluate the probability P of a photon emitted by nuclear decay to escape from the source (subscript s), either without interactions, or after scattering in the source, the probability P of a photon _L to undergo exactly n interactions within the source is required. Obviously P is equal with the prob_L ability of the photon to be absorbed in the source at the nth interaction plus the probability of the photon to escape from the source after the nth interaction. The photoelectric interactions within the source are considered as absorption processes. Consequently, the probability of absorption at the nth interaction is equal to the ratio of the photoelectric to the total attenuation coe$cient at the energy of the photon before the interaction. Then
k (E ) P "P # P 1! _. L\ . _ _L k (E ) L\ L In this equation, E represents the energy of the L\ photon before the nth interaction; the subscript Ph indicates photoelectric interaction. The "rst term in the equation represents the probability of escaping without interactions in the source and this is the dominant term in the case of well-type detector measurements. In the same case it is a reasonable approximation to neglect in the sum all terms with n higher than 1. Indeed, at energies of the order of hundreds of keV the source dimensions are small in comparison with the photon mean free path and the probability of two or more than two collisions in the source is low. At lower energy, the dominant interaction is the photoelectric e!ect and the photon is absorbed at the "rst interaction. Using this approximation P can be related to P by _ _ P "1!P . _ _ Introducing the above equation for P and rear_ ranging the terms results in the following "nal formula for the probability P of a photon emitted by nuclear decay to escape from the source: k (E) P "1!(1!P ) _. . _ k (E)
(20)
The probability P of photon escaping from the _ source without interactions is approximated by the
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O. Sima / Nuclear Instruments and Methods in Physics Research A 450 (2000) 98}108
Wigner formula [8] as modi"ed by Rothenstein [9]: 1 P " (21) _ 1#k (E)l /3
where l is the mean chord in the source,
2R (Z !Z ) 4< . (22) l " "
S R #(Z !Z )
< is the volume and S is the area of the surface of the cylindrical source. (The Appleby formula [10] could also be used for the computation of P , but _ the e!ect on the total e$ciency obtained with these di!erent formulas is very similar.) 2.3. The total ezciency formula Combining the results from the previous sections the "nal formula for the total e$ciency is e (E)"f [1!exp(!k(E)t)]P P P P . (23) 2 % The mean detector thickness t given in Eq. (9) as well as the mean thicknesses of the end cap t (Eq. (15)), of the walls of the container t and of the germanium dead layer t required for the estima tion of P , P and P (Section 2.2) should be evalu ated for a point placed in the centre of the source. The geometric factor f should be evaluated by % Eqs. (8) and (13) in the case of a point source or by Eq. (18) in the case of an extended cylindrical source. In the case of an extended source the value of P should be computed by Eq. (20). The photon interaction coe$cients for germanium, for EC and CS materials and for the matrix of the source required for the evaluation of Eq. (23) can be conveniently computed by the XCOM program of Berger and Hubbell [11]. It should be mentioned that in common applications the last 4 factors in Eq. (23) represent small correction factors.
3. Validation of the model In order to check the validity of the model the values of the coincidence summing correction
factors F computed by the proposed method were compared with published data and with values computed by other methods. The computations were carried out for two HPGe well-type detectors, with the parameters listed in Table 1. The D1 detector [2] is a 350 cm detector operated at the Physikalisch}Technische Bundesanstalt (PTB), Braunschweig (Germany). The other detector, D2, lined with a gold layer [3], is operated at the Interfaculty Reactor Institute (IRI), Delft (Netherlands). The parameters of the sources for which the comparisons were done are listed in Table 2. The "rst source, S1, is a very small volume source. The sources S2(h) have increasing volumes, as the "lling height h increases from 0.8 to 5.0 cm. The sources S1 and S2 have low attenuation matrices. The S3 source is chosen as an example of a highly attenuating matrix. The rather involved decay data required by the computation of the F factors were prepared by a specialized subroutine included in the GESPECOR software. The primary decay scheme data were taken from the KORDATEN "le developed and maintained at PTB as input "le for the KORSUM program [12]. Alternative procedures, such as the matricial technique [13,14] or the Monte Carlo simulation of the decay chains [15] could be used to prepare the necessary input data.
Table 1 Parameters of the HPGe well-type detectors D1 and D2 used in this work Parameter
Det. D1 [2]
Det. D2 [3]
Crystal radius (cm) Crystal length (cm) Radius of the crystal well (cm) Length of the crystal well (cm) Ge dead layer in the well (lm) End cap in the well: material End cap in the well: thickness (mm) End cap radius in the well (cm) End cap length in the well (cm) Distance crystal well-bottom of end cap (cm) Lining in the well: material Lining in the well: thickness (mm)
3.75 8.20 0.80 5.65 0.30 Al 0.75
2.275 6.22 0.81 4.70 0.30 Al 0.75
0.50 6.15 0.30
0.46 4.70 0.30
* *
Au 0.40
O. Sima / Nuclear Instruments and Methods in Physics Research A 450 (2000) 98}108
105
Table 2 Parameters of the sources S1, S2 and S3 used in this work Parameter
Source S1 [3]
Source S2 [2]
Source S3
Radius (cm) Height h (cm) Matrix Container: material Container: thickness (mm)
0.40 0.15 Filter paper Polyethylene 1.0
0.40 0.8; 3.0; 5.0 Water Polyethylene 0.7
0.40 5.0 Copper * *
Table 3 Coincidence summing correction factors F appropriate to the D2 detector and the S1 source Nuclide
Energy (keV)
Present work
Sc-46 Sc-46 Ag-110 Ag-110 Ag-110 Ag-110 Ag-110 Ag-110 Ag-110 Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Sb-124 Sb-124 Sb-124 Sb-124 Sb-124 Sb-124
889.25 1120.51 657.76 677.62 687.01 706.68 763.94 884.68 937.49 475.36 563.23 569.31 604.71 795.87 801.95 602.73 645.86 713.78 722.78 968.19 1690.97
0.636 0.606 0.343 0.201 0.275 0.239 0.332 0.326 0.330 0.384 0.332 0.335 0.533 0.520 0.384 0.715 0.417 0.359 0.434 0.369 0.568
m m m m m m m
Ref. [3] 0.63 0.61 0.34 0.20 0.27 0.23 0.32 0.32 0.33 0.37 0.33 0.33 0.53 0.51 0.37 0.71 0.41 0.35 0.43 0.37 0.56
First, the case of the small volume source S1 was considered. In Table 3, the F factors computed for several nuclides by the present method are compared with the values published by Blaauw [3]. These values correspond to the measurement of the S1 source with the D2 detector. As in the present method only the total e$ciency is computed, the peaks which have a contribution from summing in e!ects amounting to more than 3% were not included in Table 3. The agreement between the results computed in this work and the published
values is very good, the highest discrepancy being less than 4%. Next, the computations were carried out for extended water sources with di!erent "lling heights h. The experimental values [2] of the F factors for the measurement of Cs and Co solutions with the D1 detector depend on the "lling height. These values are well reproduced by GESPECOR calculations. In Table 4, the results predicted by the present model are compared with the results computed by GESPECOR for the D1 detector and S2 sources, with h"0.8, 3.0 and 5.0 cm. In Table 4 peaks with important summing in contributions (e.g. the 1401 keV peak of Cs) have been included besides peaks for which summing in contributions are negligible; the summing in contribution was evaluated using the FEP e$ciency computed by GESPECOR. The highest discrepancy between the F values predicted by the present method and the results computed by GESPECOR is about 2% for h"0.8 cm, 3% for h"3.0 cm and 5% for h"5.0 cm. Finally, the case of an extended source with a highly attenuating matrix was considered. In Table 5, the results computed by the present method for the S3 source (matrix: copper) and the D1 detector are compared with the results computed by GESPECOR. Even in this extreme case the highest discrepancy is less than 6%. The results presented in Tables 3}5 prove that the simpli"ed model developed in this paper is reasonably accurate. In fact, in many cases even the simpler model of a point source and a bare detector, Eq. (7) is good enough for the computation of coincidence losses. Only when low-energy photons are involved, or when the source has a
106
O. Sima / Nuclear Instruments and Methods in Physics Research A 450 (2000) 98}108
Table 4 Coincidence summing correction factors F in the case of the measurement of the sources S2(h) with the detector D1. The "lling height is denoted by h (cm). Results presented in columns labelled by (a) are from this work, those in columns labelled by (b) are computed by GESPECOR Nuclide
h"0.8
Energy (keV)
Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Co-60 Co-60
475 563 569 605 796 802 1039 1168 1365 1401 1173 1332
h"3.0
h"5.0
(a)
(b)
(a)
(b)
(a)
(b)
0.155 0.099 0.103 0.289 0.261 0.156 0.499 1.033 1.335 0.380 0.436 0.414
0.156 0.099 0.103 0.289 0.261 0.156 0.507 1.053 1.363 0.388 0.434 0.413
0.162 0.105 0.109 0.297 0.269 0.162 0.513 1.054 1.366 0.377 0.444 0.423
0.165 0.108 0.112 0.300 0.273 0.164 0.520 1.061 1.378 0.374 0.445 0.424
0.189 0.131 0.135 0.332 0.305 0.189 0.548 1.066 1.380 0.329 0.478 0.457
0.195 0.138 0.142 0.336 0.311 0.195 0.559 1.093 1.419 0.346 0.477 0.458
Table 5 Coincidence summing correction factors F in the case of the measurement of the S3 source (source matrix: copper) with the detector D1 Nuclide
Energy (keV)
Present work
GESPECOR
Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Cs-134 Co-60 Co-60
475 563 569 605 796 802 1039 1168 1365 1401 1173 1332
0.194 0.137 0.140 0.337 0.314 0.194 0.494 0.866 1.090 0.231 0.479 0.458
0.191 0.135 0.138 0.326 0.308 0.191 0.489 0.858 1.090 0.245 0.457 0.439
strongly attenuating matrix, signi"cantly better results are obtained with the complete formula Eq. (23).
4. Conclusions A simple formula for the computation of the total e$ciency of a well-type HPGe detector was
proposed. In this formula the geometric parameters of the detection system are combined with photon interaction coe$cients in expressions which are easily evaluated. The total e$ciency is useful for the calculation of coincidence losses from the peaks (summing out e!ects) which inevitably occur in the measurements with well-type detectors. In the case of the peaks with negligible summing in contribution, the total e$ciency can be immediately used for the computation of the coincidence summing correction factors, F . Numerous peaks are of this type. Especially important examples are the main peaks of Co, Y and Ce, which are frequently used for e$ciency calibration. Using the appropriate F fac tors and Eq. (1) an ideal, nuclide independent e$ciency curve can be obtained on the basis of the experimental calibration data measured with the common mixed c-ray standard source, containing the above-mentioned nuclides. Total e$ciency combined with full-energy peak e$ciency can be used for the evaluation of F in the general case, when both coincidence summing out and coincidence summing in e!ects are present. The method obtained in this way is a very easy and very fast procedure for the computation of the coincidence summing corrections for
O. Sima / Nuclear Instruments and Methods in Physics Research A 450 (2000) 98}108
measurements with well-type HPGe detectors. The associated computational algorithm was implemented in the GESPECOR software, as an alternative to the more accurate but time consuming default Monte Carlo procedure.
107
The integrals can be explicitly computed, using the equality
arctan
z z do"o arctan o o
z z#o # ln #const z 2 Appendix
"F(o, z)#const
A.1. The mean detector thickness t
where the function F is de"ned in Eq. (10). This function has the properties:
First, Eq. (6) is transformed into an integral over the volume < of the detector in a spherical coordinate system r, h, with the origin in the position of the point source: 1 t" *X
P X 1 1 d< l(X) dX" dX dr" *X *X r X X P X 4 (A.1)
where r (X) and r (X) represent the radial coor
dinates of the intersection of a photon trajectory characterized by angular coordinates X"(h, ) with the sensitive volume of the detector. Obviously,
P X
dr"l(X).
(A.2)
P X
Next the integral in Eq. (A.1) is transformed to the cylindrical coordinates o, a, z represented in Fig. 1: 1 t" *X
o do da dz . o#(z!Z ) 4
(A.3)
Here Z is the ordinate of the point source in the cylindrical coordinate system. Consider the integral
I(o , o , z , z )" "
M X
M
M
M
o do dz o#(z!Z ) X
z !Z arctan o
z !Z !arctan o
(A.5)
do.
(A.4)
lim F(o, z)"0 M
(A.6)
lim F(o, z)"0 X F(o,!z)"!F(o, z).
(A.7) (A.8)
Accordingly, the function I(o , o , z , z ) can be written as I(o , o , z , z ) "F(o , z !Z )!F(o , z !Z ) !F(o , z !Z )#F(o , z !Z ). (A.9) The domain of integration < is divided into two cylindrical domains < (o "R , o "R ,
z "Z , z "Z ) and < (o "0, o "R ,
z "Z , z "Z ). The integral over < in Eqs. (A.1) and (A.3) is replaced by the sum of the integrals over < and < . Using Eqs. (A.3), (A.4) and (A.9) and the above properties of the function F, the "nal formula for the mean detector thickness t (Eq. (9)) is obtained.
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