SELF-ATTENUATION FACTORS IN GAMMA-RAY SPECTROMETRY M. KORUN * )

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1 Attenuation M. Korun in gamma-ray spectrometry SELF-ATTENUATION FACTORS IN GAMMA-RAY SPECTROMETRY M. KORUN * ) J. Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia * matjaz.korun@ijs.si The relation between the self-attenuation factors and the distribution function describing the number of photons detected in the full-energy peaks, as a function of their path length in the sample is presented. The relations between the self-attenuation factor and the moments of the distribution function, the average path length and the variance are also presented. The use of these relations is illustrated by applying them to self-attenuation factors describing attenuation in cylindrical samples. The results of the calculations are compared with the measured average path lengths and discussed in terms of the properties of the distribution function. 1 Introduction The sample material influences the counting efficiency in X- and gamma-ray spectrometry via the self-attenuation factor [1]. This factor describes the amount of radiation emitted within the sample and prevented from registration in the fullenergy peaks in the spectrum by interaction within the sample. At an energy E it is given by the ratio of two counting efficiencies: the counting efficiency for the actual sample ε V (µ,e) where µ denotes the attenuation coefficient and the counting efficiency for the same sample-detector geometry but without self-attenuation ε V(0,E): F V(µ,E) = εv(µ,e) (1) ε V (0,E) The self-attenuation factor depends in the first instance on the shape of the sample and the attenuation coefficient within the sample. It also depends on the sample-detector distance since when the sample is displaced from the detector the beam of rays from the sample impinging on the detector becomes narrower, consequently changing the self-attenuation. It depends on the detector properties as well, since at larger attenuation coefficients in the sensitive material of the detector the effective distance between the sample and the sensitive material is smaller. The influence of energydependent detector characteristics on the self-attenuation factor, reflected in its explicit energy dependence, is smaller than the influence of the implicit dependence via the attenuation coefficient in the sample. Namely, the detector parameters influence the counting efficiencies of the attenuating and of the non-attenuating sample and their influence on the self-attenuation factor cancels partially out in the ratio. As a consequence, the approximations to self- attenuation factors for samples of various shapes are given just as functions of sample parameters, disregarding the detector properties. Self-attenuation factors have been given as functions of the attenuation coefficient for planar samples in [1,2], for Marinelli beakers in [3,4] and for well-type detectors in [5] supposing one dimensional photon transport models. For planar geometry the factors are given in terms of the sample thickness, for Marinelli beakers in terms of Czech J. Phys. 49/S1 (1999) 429

2 M. Korun the average thickness of the Marinelli beaker as seen from the detector, and for well-type geometries in terms of the vial diameter and height. The complexity of the expressions, describing the relation between the sample shape and the self-attenuation factor, increases with the number of parameters determining the sample shape. Therefore, for samples of complex shapes, it is more appropriate to determine the self-attenuation factors in terms of the moments of the distribution function describing the number of photons, detected in the full-energy peak, as a function of their path lengths in the sample. 2 Theoretical section It can be shown (Appendix A) that, in the approximation where the it is assumed that the self-attenuation factors do not depend explicitly on detector parameters, the self- attenuation factor can be expressed by the distribution of photons as a function of their path lengths covered in the sample under the condition of no self-attenuation as: F V(µ) = V e µsps (s PS) V (s PS) where V denotes the sample volume, /(s PS) the distribution function, i. e. the number of photons that travel a distance between s PS and s PS + through the sample and, in the absence of self-attenuation, would be registered in the full-energy peak. It is shown in refs. [6] and [7] that in homogeneous samples the moments of the distribution of photons detected in the full-energy peaks, can be expressed in terms of the self- attenuation factor. The average path, which is a measurable quantity, can be expressed as (2) _ s v(µ) V s PS e µsps (s PS ) µsps e (s V PS) = 1 F V(µ) F V(µ) µ (3) and the variance, i.e. the second central moment, as: σ V 2 (µ) = s _ V(µ) µ (4) In the following the relations between the self- attenuation factor, the distribution function of photons over their paths in the sample and the moments of the distribution function are illustrated by self-attenuation factors for slab- shaped samples. A crude approximation to the self-attenuation factor, F 0 (µ) = e µd 2 where D denotes the sample thickness, is obtained supposing that all photons traverse equal paths in the sample, namely its half-thickness. Consequently, the corresponding distribution function of photons over their path lengths is described with a delta function: 430 Czech J. Phys. 49/S1 (1999)

3 Attenuation in gamma-ray spectrometry 0 (s PS) = δ( ds D 2) PS with the average path of _ s 0(µ)= D/2 and the variance of σ 2 0(µ)=0. By supposing emission normal to the sample surface the attenuation factor [1] F l(µ) = 1 e µd µd is obtained supposing at no self-attenuation a uniform distribution function in the interval 0 < s PS < D, where D denotes the sample thickness. By Eqs. (3) and (4) this model leads to an average path of _l(µ) s = 1 µ (1 µde µd 1 e µd) and a variance of (2 + µ D 2 ) e µd + e 2µD σ l (µ) = µ 2 (1 e µd ) 2 That model is not suitable for use with measured average paths since the various directions of the photons impinging on the detector introduce a departure from the uniform distribution function of photons. Therefore for comparing the measured values of the parameters describing the distribution function of photons over their path lengths in the sample, more realistic photon transport models and distributions functions must be introduced. The new distributions must reflect the actual experimental counting conditions: At small path lengths the distribution function exhibits a maximum, reflecting the fact that these photons are more likely to impinge on the detector. Also the distribution becomes wider, since gamma rays can cover longer distances in the sample than just the sample thickness, consequently increasing the variance. These properties are taken into account by using a linearly decreasing function for the distribution of photons over their path length: (µ) = 1 s PS D l 0 s PS D l (5) and 0 otherwise, where D 1 denotes the maximal possible path length in the sample. From Eq. (2) it follows that the self- attenuation factor is At small self-attenuation the average path and the variance assume the values of _2 s (µ) = D l 3 (1 µd l ) 3 and σ 2 2 (µ) = D l 2 18 (1 2µDl 15 ) In the limit of high attenuation the average path length and the variance assume values of 1/µ and 1/µ 2 respectively. Czech J. Phys. 49/S1 (1999) 431

4 M. Korun 3 Discussion From Eq. 2 the self-attenuation factor, when its independence of detector properties is assumed, is defined by the distribution of photons, detected in the full-energy peak, as a function of their path lengths covered in the sample. Further, the distribution function of gamma-rays is given by the shape of the sample. This is not uniform, as assumed in the case of the gamma-ray transport model for the case of cylindrical samples [1], but, for samples placed close to the detector, is asymmetrical, resembling a decreasing slope up to the maximal path length. The asymmetry is due to the fact that most photons detected are emitted at small distances from the detector and consequently they cover shorter path lengths in the sample. To obtain information about the distribution function, the average path length of gamma-rays in the sample as a function of the attenuation coefficient is measured. Then the variance can also be determined from Eq. 4. By comparison of the measured values with the values obtained from calculations with different distributions their validity can be assessed. The average path length as a function of the attenuation coefficient in the sample was measured using energy-dependent self-attenuation of X-rays. The experiment was performed with cylindrical samples with the diameter of 6 cm and thickness of 3.6 cm, placed on the detector and by measuring the count rates in the X kα and X kβ lines. The samples contained an X- ray emitter and an absorber having the absorption edge at an energy between the energies of the X-ray lines [6]. The results are presented in Fig. 1 together with the average paths calculated with a uniform distribution and the linear distribution given in Eq. 5. For the maximum path length of photons in the sample was taken the maximum path of photons emitted at the upper edge of the sample and absorbed at the upper edge of the detector crystal D 1 = D D sd + D (R s + R d ) 2 + (D sd + D) 2 where R s, R d, D sd and D denote the radius of the sample, the radius of the crystal, the distance between the sample and crystal and the sample thickness respectively. It can be seen that the average path lengths obtained from the linearly decreasing distribution agree better with the measured path lengths than the path lengths obtained from the uniform distribution. It should be mentioned that the relative difference between the self-attenuation factors calculated from the two distributions amounts to about 2 percent. The reason that the one-dimensional gamma-ray transport model combined with the uniform distribution [1] yields good values for the self-attenuation factors lies in the fact that the effects of both approximations nearly compensate. 4 Conclusions The introduction of the function describing the distribution of photons detected in full energy peaks over their path lengths covered in the sample opens the possibility to express the self-attenuation factor and the average path in a geometry-independent way. By constructing the distribution from the sample-detector geometry it is possible to calculate the self-attenuation factor as well as the average path. This was performed for the simple case of slab-shaped samples measured on top of the detector. It was shown that a linearly decreasing function adequately describes the distribution of photons over their path lengths in the sample in this geometry. The agreement 432 Czech J. Phys. 49/S1 (1999)

5 Attenuation in gamma-ray spectrometry Fig. 1. The average paths of photons detected in the full- energy peak as functions of the attenuation coefficient obtained by calculation supposing a one-dimensional photon transport model and a uniform distribution [1] ( ), linear distribution ( ) and by measurement [6] ( ). between the measured and calculated average path lengths enables the extension of the calculation of the self-attenuation factors on the basis of measured average paths on samples of more complex shapes using other distribution functions. References [1] K. Debertin and R. G. Helmer: X and Gamma-Ray Spectrometry with semiconductor detectors, North Holland, Amsterdam, [2] K. M. Miller: Nucl. Instr. And Meth. A 258 (1987)281. [3] P. Dryak, K. Kovar, L. Plchova and J. Suran: J. Radioanal. Nucl. Chem. Lett. 135 (1989) 281. [4] O. Sima: Health Phys. 62 (1992) 445. [5] P. G. Appleby, N. Richardson and P. J. Nolan: Nucl. Instr. and Meth. B 71 (1992) 228. [6] M. Korun: Appl. Radiat. Isot., in press. [7] M. Korun: to be published. [8] M. Korun and R. Martinčič: Appl. Radiat. Isot. 43 (1992) 22. Czech J. Phys. 49/S1 (1999) 433

6 M. Korun Appendix A Derivation of the relation between the self-attenuation factor and the distribution of photons, detected in the full-energy peaks, as a function of their path lengths covered in the sample Using Eq. (1) the self-attenuation factor can be expressed as F V (µ,e) = V F(µ,E,r ) ε PS (E,r ) dv(r ) V ε PS(E,r ) dv(r ) where V denotes the sample volume, µ the attenuation coefficient in the sample, ε PS(E, r ) the efficiency for a point source emitting photons with energy E and located at the point r and F(µ,E,r ) the self-attenuation function [8] F(µ,E,r e µsps(r,r ) (E,r,R ) ) = V D V D (E,r,R ) Here (E,r,R ) denotes the number of gamma-rays with energy E emitted at r in the sample and absorbed at R in the sensitive volume of the detector in such a way that they would be registered in the full-energy peak in the absence of self-attenuation, s PS(r,R ) the path in the sample of these rays and V D the sensitive volume of the detector. Introducing the approximation that the path in the sample does not depend on detector parameters, i.e. s PS(r,R )=s PS(r ), the self-attenuation function reduces to F (µ,e,r ) = e µsps(r ) With the approximation introduced the self-attenuation factor e µsps(r ) εps(r ) dv(r ) F V(µ) = V V ε PS (r ) dv(r ) e µsps(r ) (r ) = V (r V ) is obtained, where (r ) denotes the number of photons emitted in the vicinity of r in the sample and registered in the full-energy peak in the absence of self-attenuation. Since that number can be expressed as a function of the path length in the sample s PS as the independent variable (r ) = the self-attenuation factor becomes (s PS) e µsps (s PS) F V( µ) = V. (s PS) V 434 Czech J. Phys. 49/S1 (1999)