HPGe detector photopeak efficiency calculation including self-absorption and coincidence corrections for Marinelli beaker sources using compact analytical expressions

Applied Radiation and Isotopes 54 (2001) 761–768

HPGe detector photopeak efficiency calculation including self-absorption and coincidence corrections for Marinelli beaker sources using compact analytical expressions Mahmoud I. Abbas* Physics Department, Faculty of Science, Alexandria University, Egypt Received 13 March 2000; accepted 5 June 2000

Abstract Direct mathematical methods to calculate total and full-energy peak (photopeak) efficiencies, coincidence correction factors and the source self-absorption of a closed end coaxial HPGe detector for Marinelli beaker sources have been derived. The source self-absorption is determined by calculating the photon path length in the source volume. The attenuation of photons by the Marinelli beaker and the detector cap materials is also calculated. In the experiments gamma aqueous sources containing several radionuclides covering the energy range from 60 to 1836 keV were used. By comparison, the theoretical and experimental full-energy peak efficiency values are in good agreement. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: Analytical expressions; HPGe detector; Self-absorption; Coincidence corrections; Full-energy peak efficiency

1. Introduction In the field of activity measurement by means of gamma-ray spectrometry, the need of coincidence factors which occur with radionuclides emitting more than one photon per decay, has been treated for many years by several authors (Debertin and Grosswendt, 1982; Moens and Hoste, 1983; Lippert, 1983, Wang et al., 1995; Wang et al., 1997; Wainio and Knoll, 1966; Nakamura, 1983; Rieppo, 1985; Herold and Kouzes, 1991; Komboj and Kahn, 1996; Jiang et al., 1998). For a given peak, the apparent full-energy peak efficiency (uncorrected for coincidence effects) must be multiplied by the correction factor C to get the true full-energy peak efficiency. For point sources, the photopeak efficiency is measurable without any particular difficulty (Decombaz et al., 1992). For extended sources, particularly for *Present address: INFN- Sezione di Genova, Via Dodecaneso 33, I-16146 Genova, Italy. E-mail address: [email protected] (M.I. Abbas).

large-volume sources close to the detector, the situation in this case is more difficult than the previous one. In order to evaluate the correction factors, it is necessary to know the spatial dependence of the detector efficiencies within the detector volume. For example, in the case of a simple cascade, as shown in Fig. 1, the correction factor Ca applied to photon ‘‘a’’ for point source will be Ca ¼

1 1 ÿ eTb

ð1Þ

whereas, for a circular disk source with activity homogeneously distributed, the same quantity must be written R ð2Þ A epa dA Ca ¼ R A epa ð1 ÿ eTb Þ dA Finally, for volumetric sources with activity homogeneously distributed, the correction factor Ca is given by R V epa dV R Ca ¼ ð3Þ e V pa ð1 ÿ eTb Þ dV

0969-8043/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 9 - 8 0 4 3 ( 0 0 ) 0 0 3 0 8 - 0

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M.I. Abbas / Applied Radiation and Isotopes 54 (2001) 761–768

Fig. 1. A simple decay scheme.

where, epa and eTb are the photopeak (for the gamma photon ‘‘a’’) and total (for the gamma photon ‘‘b’’) efficiencies, respectively. Note that the efficiencies in Eqs. (2) and (3) are functions of both the energy and the lateral distance. The experimental determination of the efficiency as a function of the energy and the lateral distance is tedious (Decombaz et al., 1992). To avoid these difficulties, analytical mathematical formulae can be used. The principle of this approach is described in the present work. In addition, the attenuation within the source volume (self-absorption), the core cavity and by any other materials between the source and the detector active volume will be taken into account.

2. Theoretical treatments 2.1. Determination of the photon travelling distance through the detector active volume For each photon emitted from our volumetric source the probability of striking the point where the photon actually enters the detector active volume must be known to calculate the detection efficiencies. There are three main cases to be considered, two of them having two sub-cases to find the distance a photon will travel through the detector bottom or lateral side surfaces. These five different cases and their travelling distances d (see Fig. 2) are: The striking photon may enter 1. the upper surface and emerge from the cylindrical detector active volume (a) base: L d1 ¼ ð4Þ cos y

Fig. 2. The five possible cases of photon path lengths through pffiffiffiffiffiffiffi the detector active volume: a ¼ ðr2 ÿ R2 Þ=r and b ¼ R a=r.

(b) side: Yþ h ÿ d2 ¼ sin y cos y Y ¼ r cos f 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 ÿ r2 sin2 f

ð5Þ

2. the side of the cylindrical detector active volume and emerges from (a) the opposite side: 2R0 ; cos y qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R0 ¼ R2 ÿ r2 sin2 F cos F; d3 ¼

ð6Þ

tan F ¼ tan y sin f (b) the base, ex-clouding the segment ABC: d4 ¼

L0 x0 ÿ sin y cos y

x0 ¼ r cos2 F ÿ R0 ; L0 ¼

L cos f

ð7Þ

M.I. Abbas / Applied Radiation and Isotopes 54 (2001) 761–768

763

3. the side of the cylindrical detector active volume and emerge from the segment ABC: L Yÿ d5 ¼ ÿ ð8Þ cos y sin y

we can see that these efficiencies are a sum of integrals of the type Z Z fatt ð1 ÿ eÿm:di Þsin y df dy ð12Þ

2.2. Determination of the efficiency of a cylindrical detector for a Marinelli beaker source

where di is the photon travelling distance (path length) through the detector active volume and h is the source to detector distance. Finally, the factor fatt accounts for the attenuation of gamma photons by the source itself (self-absorption) and by any other materials between the source and the detector active volume. It is indicated by P

To calculate the efficiency of a cylindrical detector for a Marinelli beaker source, one can divide the Marinelli beaker source into two parts with volumes Vi and Vii (Fig. 3). Volume Vi acts as a solid cylinder with height H and radius S, while the volume Vii acts as a thick cylindrical ring with height H0 , inner radius S0 and outer radius S. The efficiency of the first part Vi is given by 2 eM i ¼ H S2 

Z

H 0

Z

R

eðr4R; hÞ r dr þ 0

Z

S

 eðr5R; hÞ r dr dh

R

ð9Þ The efficiency of the second part Vii is given by Z H0 Z S 2 eM eðr5R; 0Þ r dr dL ð10Þ ii ¼ H 0 ðS2 ÿ S 02 Þ 0 S0 Therefore, the efficiency of a cylindrical detector arising from a Marinelli beaker source is given by eM ¼

M Vi
eM i þ Vii
eii Vi þ Vii

ð11Þ

The mathematical derivations of the efficiencies eðr4R; hÞ and eðr5R; hÞ are given in detail in previous works (Selim et al., 1998; Selim and Abbas, 2000), and

y

f

ÿ

fatt ¼ e

mj dj

j

ð13Þ

where mj is the attenuation coefficient of the jth absorber for gamma-ray energy Eg and dj is the actual path length of the gamma photon through the jth absorber (this factor will be described in the next section). The calculation of total efficiency eM T is obtained by replacing m with the total attenuation coefficient mt (excluding the coherent term) of the detector’s active medium (Hubbell and Seltzer, 1995). In the full-energy peak efficiency eM p calculation m is replaced by the peak attenuation coefficient mp of the detector’s active medium, which represents the part contributing to the photopeak only. mp is given by mp ¼ t0 þ fm so þ gn kn0

ð14Þ

where, t0 , s0 and kn0 are the photoelectric, incoherent (Compton) and pair production (in the nuclear field) coefficients corresponding to the energy of the incident photon Eg0 , respectively. The factors fm and gn give the fractions of the incoherent and pair production leading to the photopeak, respectively (Abbas and Selim, 2000).

Fig. 3. The geometric configuration of the Marinelli beaker source used in this work.

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2.3. Determination of the absorption of photons (1) By the source medium (self-absorption): For the given Marinelli beaker source and photon energy, the self-absorption is a function of the path length of the photon in the source medium. Table 1 shows that there are three different photon path lengths (through the source medium) corresponding to the main three cases of the photon travelling distances through the detector active medium.

(2) By the Marinelli beaker and the detector cap materials: The attenuation of the Marinelli beaker (with thickness t1 ) and the detector cap (end window with thickness tC and wall with thickness tCu ) materials is a function of the photon path length through these materials. Table 1 shows the three different photon path lengths (through the Marinelli beaker and the detector cap materials) corresponding to the main three cases of the photon travelling distances through the detector active medium. In the first case, attenuation acts on

Table 1 The photon path lengths through the source–detector systema d

d

dS (Source medium) dt1 (MB material)b

dtC & dtCu (DC material)c

d1, d2

H=2 cos y

t1 cos y

tC cos y

d3, d4

x0 cos y

R0 ðS0 þ t1 Þ ÿ R0 ðS0 Þ S 0 t1  0 0 sin y R ðS Þ sin y

R0 ðS þ tCu Þ ÿ R0 ðRÞ RtCu  0 sin y R ðRÞ sin y

d5

Yÿ sin y

Yÿ ðS0 þ t1 Þ ÿ Yÿ ðS0 Þ t1 ð1 þ ðr2 =2S02 Þ sin2 fÞ Yÿ ðR þ tCu Þ ÿ Yÿ ðRÞ tCu ð1 þ ðr2 =2R2 Þsin2 fÞ   sin y sin y sin y sin y

a

R0 ðRÞ ¼ R0 , Yÿ ðRÞ ¼ Yÿ . MB: Marinelli beaker. c DC: Detector cap. b

Table 2 The used radionuclides aqueous solution source data Nuclide

Eg (MeV)

Pg

A  Pga

T1=2 (days)

241

0.060 0.088 0.122 0.166 0.279 0.392 0.514 0.662 0.898 1.836 1.173 1.333

0.3600  0.00800 0.0363  0.00040 0.8560  0.00340 0.7987  0.00120 0.8148  0.00160 0.6489  0.00260 0.9840  0.00800 0.8510  0.00400 0.9400  0.00600 0.9936  0.00060 0.9986  0.00044 0.9998  0.00012

65.8717  1.0% 35.0270  3.5% 31.3675  1.8% 46.0056  3.4% 49.1424  5.8% 112.923  2.8% 231.596  6.1% 205.980  1.9% 684.856  1.9% 721.452  1.9% 256.168  3.0% 255.448  3.0%

157850  480 462.6  1.4 271.79  0.18 137.64  0.046 46.595  0.026 115.09  0.08 64.849  0.008 11020  120 106.63  0.05

Am Cd 57 Co 139 Ce 203 Hg 113 Sn 85 Sr 137 Cs 88 Y 109

60

Co a

1925.5  1.0

A is the source activity in Bq.

Table 3 The radionuclides aqueous solution volume and its corresponding height inside the standard re-entrant Marinelli beaker V ml (=Vi+Vii) 0 h0 cm (=H +H)

10 0.2

50 1.0

100 2.0

150 3.0

200 4.0

250 5.0

300 6.0

350 6.83

400 7.33

450 7.83

500 8.33

M.I. Abbas / Applied Radiation and Isotopes 54 (2001) 761–768

photons coming from the source region Vi. In the other two cases, the attenuation acts on photons coming form the source region Vii. 2.4. Coincidence effect correction for Marinelli beaker sources In addition to the correction factor of the full-energy peak for the photon ‘‘a’’ Ca (Eq. (3)); we have other two correction factors Cb and Cc for the photons ‘‘b’’ and ‘‘c’’, respectively, R V epb dV R Cb ¼ ð15Þ e ð1 ÿ ðPga =Pgb Þ eTa Þ dV pb V R V epc dV R Cc ¼ ð16Þ c ðe þ f ðP pc ga =Pgc Þ epa epb Þ dV b V where Pg is the photon emission probability and fbc is a conversion factor that gives the probability that gamma photon ‘‘a’’ is emitted, i.e. internal conversion does not take place during the transition. The conversion factor fbc can be expressed as (Wapstra et al., 1959) 1 fbc ¼ ð17Þ 1þa

765

thickness tC ¼ 0:05 cm, whereas its wall is made from the copper, with thickness tCu ¼ 0:15 cm, see Fig. 4. The Marinelli beaker source is placed in contact with the detector end cap window. The coincidence correction factors of radionuclides 60Co and 88Y were calculated using the present equations and have been taken into account to get the true full-energy peak efficiencies. In this study the core cavity, with volume of about 2 cm3, of the HPGe detector is homogenized with the active part of the crystal material, and the crystal density is reduced (from 5.323 (Hubbell and Seltzer, 1995) to 5.172 g/cm3) to account for homogenization (Haase et al., 1993).

4. Results Figs. 5–15 show the calculated and measured photopeak efficiencies of a closed end HPGe detector using

where a is the internal conversion coefficient. Determination of the correction factors Ca, Cb and Cc, in the forms such as Eqs. (3), (15) and (16), needs information about the photopeak ep and the total eT efficiencies at each differential volume element dV at lateral distance r for each counting geometry and for each specific photon energy. It is very impractical to obtain this information by experimental measurements. Now, by using Eqs. (9)–(11) these problems have been solved and Eqs. (3), (15) and (16) can be re-written as 1 Ca ¼ ð18Þ 1 ÿ eM Tb Cb ¼

1 1 ÿ ðPga =Pgb Þ eM Ta

ð19Þ

Cc ¼

1 M M 1 þ fbc ðPga =Pgc Þ ðeM pa epb =epc Þ

ð20Þ

Fig. 4. The geometric configuration of the closed end HPGe detector used in this work (dimensions in mm).

3. Experimental studies The Marinelli beaker volumetric sources are very common for low-level radioactivity measurements in activation analysis and environmental samples. The full energy peak efficiencies of mixed radionuclides (see Table 2) aqueous solutions, with volumes 10, 50, 100, 150, 200, 250, 300, 350, 400, 450 and 500 ml (see Table 3), were measured with a HPGe detector. The HPGe detector used in this work was a closed end coaxial detector with an active volume of about 75 cm3. The cap window material is a standard carbon-epoxy with

Fig. 5. True photopeak efficiency – gamma energy curve for the 10 ml source.

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Fig. 6. As Fig. 5 for the 50 ml source.

Fig. 10. As Fig. 5 for the 250 ml source.

Fig. 7. As Fig. 5 for the 100 ml source.

Fig. 11. As Fig. 5 for the 300 ml source.

Fig. 8. As Fig. 5 for the 150 ml source.

Fig. 12. As Fig. 5 for the 350 ml source.

Fig. 9. As Fig. 5 for the 200 ml source.

Fig. 13. As Fig. 5 for the 400 ml source.

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M.I. Abbas / Applied Radiation and Isotopes 54 (2001) 761–768

Table 4 M M The percentage deviation eM p:theo ÿ ep:exp =ep:theo % between the calculated and the measured photopeak coefficient for different volumes Eg MeV

V ml 10

0.060 0.088 0.122 0.166 0.279 0.392 0.514 0.662 0.898 1.173 1.333 1.836

0.02 0.03 0.02 1.72 0.01 0.01 ÿ 5.79 ÿ 0.91 0.01 ÿ 5.8 0.01 0.02

50 0.02 0.02 0.01 0.03 ÿ 3.9 ÿ 3.1 ÿ 6.9 ÿ 2.5 0.02 0.01 0.02 0.02

100

150

200

250

300

350

400

450

500

0.01 0.02 0.01 1.4 ÿ 2.3 0.04 0.02 0.03 0.01 ÿ 2.5 0.01 0.01

0.02 0.01 0.03 0.02 ÿ 6.5 4.2 0.03 0.02 0.01 ÿ 4.7 0.01 0.01

0.02 0.02 0.02 0.02 ÿ 6.3 0.02 0.03 ÿ 2.6 0.01 ÿ 6.7 0.02 0.01

0.01 0.03 0.01 0.03 ÿ 6.2 0.01 0.02 ÿ 4.6 0.02 ÿ 6.9 0.01 0.03

0.01 0.02 0.02 0.02 ÿ 6.1 0.01 0.02 ÿ 5.2 0.01 ÿ 7.1 0.03 0.01

0.03 0.01 0.01 0.03 ÿ 5.5 0.02 0.03 ÿ 5.1 0.92 ÿ 4.69 0.02 0.04

0.02 0.03 0.03 0.01 ÿ 5.3 0.03 0.05 ÿ 3.27 7.6 0.06 0.04 0.03

0.03 0.04 0.02 0.02 ÿ 3.77 0.03 ÿ 0.92 ÿ 2.67 0.04 ÿ 2.99 0.02 0.04

0.01 0.02 0.01 0.01 ÿ 3.15 0.02 ÿ 1.14 ÿ 4.11 1.85 0.07 0.03 0.02

Fig. 14. As Fig. 5 for the 450 ml source.

Fig. 15. As Fig. 5 for the 500 ml source.

radionuclides aqueous sources placed in a Marinelli beaker. Table 4 indicates the percentage deviation of the M calculated eM p:theo from the measured ep:exp photopeak efficiencies for the different used volumes. From Figs. 5–15 and Table 4, it can be clearly seen that there is a good agreement between the measured and the calculated values.

the authorities of the Center for Ionizing Radiation Metrology (CIRM), National Physical Laboratory (NPL), Teddington, UK, for making it possible to carry out the measurements.

5. Conclusions In this paper, direct mathematical expressions to calculate the photopeak efficiency, the coincidence correction and the source self-absorption have been introduced in the case of a closed end HPGe detector and Marinelli beaker sources.

Acknowledgements The author would like to thank Prof. Y.S. Selim, Alexandria University, for fruitful discussion and also

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