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Measurement of the average path length of photons in samples by self-attenuation of X-rays
Applied Radiation and Isotopes PERGAMON
Applied Radiation and Isotopes 50 (1999) 381±388
Measurement of the average path length of photons in samples by self-attenuation of X-rays M. Korun * J. Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia Received 16 January 1998; received in revised form 16 March 1998
Abstract For homogeneous samples a relation between the average path length of photons detected in full-energy peaks in the spectra of X- and gamma-rays and the counting eciency is derived for an arbitrary sample-detector geometry. An experimental method of measuring the average path length in the samples is presented. The results of such measurements are compared to the calculated average path lengths and discussed in terms of self-attenuation factors. # 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction In gamma-ray spectrometry the activities of radionuclides contained in the sample are calculated from the count rates in the peaks occurring at the energies of the emitted gamma-rays. To perform the calculation the counting eciencies, i.e. the probabilities for registration of gamma-rays in the peaks of full-energy absorption, must be known. The counting eciency depends on the characteristics of the detector, the sample and their relative position. The material of the sample aects the counting eciency through the attenuation of gamma-rays. The in¯uence of attenuation of gamma-rays within the sample on the counting eciency can be expressed by the self-attenuation factor (Debertin and Helmer, 1988). Besides for planar geometries, this factor has also been given in analytical form, i.e. as a function of sample dimensions and the attenuation coecient, for well-type geometries by Appleby et al. (1992) and for Marinelli-type geometries by Dryak et al. (1989) and Sima (1992). All the factors were derived from one-dimensional gamma-ray transport models.
* Tel.: +386-61-177-3900; Fax: +386-61-123-2120; E-mail: [email protected].
An alternative approach is to approximate the selfattenuation factor by an assumed analytical expression of the attenuation coecient and a characteristic length and to use for that length a distance deduced from the measured spectra. This approach is analogous to the approach used by Miller (1987) for the attenuation factor for planar geometries, where an exponential function is used as the analytical expression and the half-thickness of the sample is used as the characteristic length. The use of a measured characteristic length has the advantage, that this method is not restricted to a speci®c geometry.
2. Theoretical section 2.1. Self-attenuation and the average path length In this section the relation between the counting eciency and the average path length of the photons detected in the full energy peak is derived for the case of homogeneous samples. The counting eciency for a homogeneous sample with a volume V can be expressed in the form of an integral over the sample volume (Noguchi et al., 1981; Korun and MartincÏicÏ, 1992; Overwater et al., 1993; Sima and Dovlete, 1995):
0969-8043/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 9 - 8 0 4 3 ( 9 8 ) 0 0 0 9 4 - 3
382
eV m, E
M. Korun / Applied Radiation and Isotopes 50 (1999) 381±388
1 V
V
ePS m, E, r~ dV ~r,
1
where ePS(m, E, r~) denotes the counting eciency of a point source emitting photons with energy E and located at r~ inside the sample material with the attenuation coecient m, and dV(~r) denotes the volume element in the vicinity of the point r~. The eciency of the point source in the presence of the attenuating material can be expressed as ePS m, E, r~ F m, E, r~ePS 0, E, r~,
2
where ePS(0, E, r~) denotes the point source eciency in the absence of the sample material and F(m, E, r~) the self-attenuation function which can be expressed as (Korun and MartincÏicÏ, 1992): ~ ~ eÿmsPS ~r,R dN E, r~, R VD F m, E, r~ : 3 ~ dN E, r~, R VD
V
4 The derivative of the self-attenuation function with respect to the attenuation coecient is ~ ~ ÿmsPS ~r,R ~ sPS ~r,Re dN E, r~, R @ F m, E, r~ VD ÿ ~ @m ~ eÿmsPS ~r,R dN E, r~, R
VD
~ ~ ÿmsPS ~r,R ~ sPS ~r,Re dN E, r~, R VD : sPS m, E, r~ ~ ~ eÿmsPS ~r,R dN E, r~, R
6
VD
The eciency of a volume sample depends on the attenuation coecient of the sample only through the self-attenuation function. Therefore its derivative with respect to the attenuation coecient is: @ eV m, E 1 ÿ sPS m, E, r~F m, E, r~ePS 0, E, r~ @m V V 7 dV ~r ÿsV m, E eV m, E , where sV(m, E) designates the path length in the sample averaged over all rays emitted in the sample and detected in the full energy peak: sPS m, E, r~F m, E, r~ePS 0, E, r~ dV ~r sV m, E V : F m, E, r~ePS 0, E, r~ dV ~r V
~ designates the number of rays Here dN(E, r~, R) emitted in the vicinity of r~ in the sample and absorbed or scattered in the vicinity of R~ in the detector in such a way that they are registered in the full energy peak, in the absence of attenuation within the sample. sPS(~r, ~ designates the path traversed by these photons in R) the sample and VD the sensitive volume of the detector. The self-attenuation function describes the unattenuated fraction of gamma rays emitted at r~ and registered in the full energy peak. Since F(0, E, r~) = 1, its value averaged over the sample volume is the selfattenuation factor: ePS 0, E, r~F m, E, r~ dV ~r eV m, E V F m, E, r~: eV 0, E ePS 0, E, r~ dV ~r
VD
~ ÿmsPS ~r,R
e
VD
~ dN E, r~, R
~ dN E, r~, R
ÿ sPS m, E, r~F m, E, r~,
5
where sPS(m, E, r~) designates the average length in the sample of the rays emitted at r~ and detected in the full energy peak:
8 The explicit dependence of sV(m, E) on energy describes the in¯uence of energy-dependent detector parameters on the average path length. Eq. (7) oers the possibility to relate the average path length of gamma-rays emitted in the sample and registered in the full energy peak to the counting eciency and its derivative with the attenuation coecient. In the derivation no speci®c sample-detector arrangement was assumed and it is therefore valid for any counting geometry. 2.2. Measurement of the derivative Absorption edges oer the possibility to introduce discontinuities in the attenuation coecient of the sample material at X-ray energies by introducing appropriate absorbers in the sample matrix. In this way the attenuation coecient can be changed abruptly and the derivative of the eciency can be measured by measuring the eciency on both sides of the absorption edge, using the de®nition of the derivative: @ eV m, E eV m Dm, E ÿ eV m, E @m Dm ÿ sV m, E eV m, E :
9
To perform the measurement suitable lines must appear close to the edge on each side. Since X-ray emitters radiate at closely spaced energies the derivative can be measured using suitable combinations of emitters and absorbers. The count rates in the peaks occurring at energies of E and E + DE can be expressed in terms of the eciencies as
M. Korun / Applied Radiation and Isotopes 50 (1999) 381±388
n E ab E eV m, E
10
and n E DE ab E DE eV m Dm, E DE ,
11
where a and b(E) denote the activity of the X-ray emitter and the emission probability of photons with energy E. The eciency eV(m + Dm, E + DE) can be expressed in the linear approximation as @ eV m, E DE @E @ eV m, E Dm, @m
eV m Dm, E DE eV m, E
12
and consequently, the average path length as: sV m, E
1 @ eV m, E DE eV m, E @E Dm eV m Dm, E DE =eV m, E ÿ 1 ÿ : 13 Dm
In the last equation the average path length is expressed by the partial derivative of the counting eciency with energy. This quantity is not directly measurable, since by changing the energy also the attenuation coecient changes. Therefore approximations must be introduced allowing a suitable approximation to that derivative. By dierentiation of Eq. (1) with energy the derivative of the volume eciency can be expressed as @ eV m, E 1 @ ePS 0, E, r~ F m, E, r~ @E V V @E @ F m, E, r~ ePS 0, E, r~ dV ~r: 14 @E The point-source eciency ePS(0, E, r~) can be separated into the product of the point-source eciency on the detector cap at its geometrical axis e00(E) and the spatial dependence Z(E, r~) (Korun and MartincÏicÏ, 1992): ePS 0, E, r~ e00 E Z E, r~:
15
The contribution of the derivative of Z(E, r~) to the derivative of the volume-source eciency with respect to energy is [e00(E)/V]fV[@Z(E, r~)/@E]F(m, E, r~) dV(~r). Also Z(E, r~) = 1 on the detector surface in its axis and therefore its derivative is zero there. Since the self-attenuation function has its maximum on the detector surface and falls with the distance from the detector the multiplication in the integral diminishes the in¯uence of the energy dependence of Z(E, r~). Therefore the derivative of Z(E, r~) with energy will be neglected and the derivative of the point-source eciency approximated by @ ePS 0, E, r~ de00 E Z E, r~ : @E dE
16
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The self-attenuation function depends on energy mainly through the attenuation coecient of the sample ma~ terial. The explicit dependence originates in dN(E, r~, R) and re¯ects the energy dependence of the attenuation coecient within the detector. It is this coecient that determines the average depth within the detector where the photons interact and with it the eective distance between the sample and the detector as well. At small attenuation coecients the eective distance is larger, consequently decreasing the average path length of photons in the sample. However, the resulting explicit energy dependence is weak and will be neglected. It should be noted that by introduction of these approximations the only energy-dependent detector property remaining is e00(E). Since eV(m, E) is proportional to e00(E) it follows from Eq. (7) that the average path length does not depend on e00(E). Consequently, in the framework of the approximations introduced, the average path length does not depend on the detector parameters at all. Therefore in the following, since the average path length depends on energy just via the attenuation coecient, the energy will be omitted from the list of variables determining the average path length. Taking into account the approximations introduced and using Eqs. (10), (11) and (13) the average path length can be expressed in terms of count rates and emission probabilities as sV m ÿ
n mDm, EDE b E n m, E b EDE
00 E ÿ 1 ÿ e001 E dedE DE
Dm
: 17
It can be noted that the parameter e00(E) still appears in the expression for the average path length, but just to compensate the dierent detector responses at energies of E and E + DE. Eq. (17) is rather unsuitable for use with measured data since the ®rst term in the numerator, which is close to unity and cancels to a large degree, must be known with a great accuracy in order to determine the path length. It contains the ratio of emission probabilities b(E)/b(E + DE), which may not be known with an accuracy better than some percent, so it may be not possible to determine the average path length. To avoid the use of emission probabilities, their ratio is expressed by the count rates in the measurement of another sample measured in the same counting geometry but without an absorber introducing the discontinuity in the attenuation coecient between E and E + DE. Denoting the dierence of the attenuation coecient for E + DE and E in that sample by Dm0, the count rates in the peaks at energies E +DE and E by n(m0+Dm0, E + DE) and n(m0, E) and average path length by sV(m0), the ratio of emission probabilities can be substituted yielding for the average path length
384
M. Korun / Applied Radiation and Isotopes 50 (1999) 381±388
1 DE=e00 E de00 E =dE 1 ÿ R m, Dm, m0 , Dm0 , E, DE R m, Dm, m0 , Dm0 , E, DE sV m0 Dm0 , sV m Dm where R(m, Dm, m0, Dm0, E, DE) denotes the ratio R m, Dm, m0 , Dm0 , E, DE n m Dm, E DE n m0 , E : n m0 Dm0 , E DE n m, E
18
erties, the measured dependence also describes, via the energy dependence of the attenuation coecient, the energy dependence of the average path length. 19
To calculate the average path length from Eq. (18) its value in a sample of the matrix without the absorber must be assumed. A crude value suces for the calculation, since Dm can be made much greater than Dm0 by introducing larger amounts of absorber. Alternatively, a sample containing the absorber in small concentration, just to achieve Dm = 0, can be produced. Instead of the sample without absorber that sample can be used in measurements of the average path length in samples with larger concentrations. By using that sample the term containing sV(m0) in Eq. (18) vanishes. The described approach has the advantage that the only parameter to be measured, the ratio R(m, Dm, m0, Dm0, E, DE), depends just on the ratios of count rates in peaks appearing at the same energies. Since the ratio can be measured with an arbitrary statistical accuracy, the largest uncertainties introduced are systematic, arising from the calculation of peak areas. The peaks at the energies of E and E + DE may overlap and therefore the ratio can bear systematic uncertainties from the procedure for evaluation of peak areas. Since in the ratio R only ratios between peak areas at the same energy enter, these uncertainties partially cancel because of the similarity of spectral shapes measured with the two sources. By measuring the average path length at a ®xed energy and at dierent concentrations of the absorber, the dependence of the average path length on the attenuation in the sample can be extracted. Since the average path length, in the framework of the approximations introduced, does not depend on detector propTable 1 Energies where the Ka and Kb lines occur and the energy of the absorption edge of the absorber for the three combinations of the X-ray emitter and absorber Combination
E (Kb)
E (Ka)
E (Kab)
Eu±Ce Ba±I Cd±Pd
45.48±46.82 34.97±36.01 24.94±25.60
39.52±40.12 30.63±30.97 21.99±22.16
40.45 33.16 24.35
The X-ray energies are taken from IAEA Tecdoc 619 (1991) and the energies of the absorption edges from Adams and Dams (1975).
3. Measurements The measurements were performed with an n-type detector with a beryllium window of thickness 0.5 mm and an eciency of 23% relative to that of a 7.56 cm 7.56 cm NaI(Tl) crystal at the energy of 1333 keV. The resolution was 2.0 keV at the energy of 1333 keV and 0.7 keV for the pulser peak. The sources were prepared as aqueous solutions of an X-ray emitter and absorber. Polystyrene canisters of diameter 6 cm were ®lled to the height of 3.6 cm with 100 ml of solution and placed coaxially on the detector. To test the method three combinations of X-ray emitter and absorber were measured: 152 Eu and Ce, 133 Ba and I and 109 Cd and Pd. The energies of the Ka and Kb lines, obtained from the IAEA Tecdoc 619 (1991), and the energies where the K absorption edges occur, obtained from Adams and Dams (1975), for the three combinations are listed in Table 1. Cerium was dissolved in the form of Ce(NO3)36H2O and spiked with 152 Eu; no europium carrier was needed in this case. Iodine as KI salt was dissolved in 1 M hydrochloric acid solution containing 133 Ba with about 10 mg mlÿ1 barium as carrier. Palladium as PdCl2 was dissolved in a few ml of hot aqua regia, boiled to small volume, diluted with 0.1 M hydrochloric acid, cooled and ®ltered to remove slight turbidity. This solution was then spiked with 109 Cd and cadmium carrier to about 10 mg mlÿ1. With the 152 Eu±Ce combination three samples were prepared with concentrations by mass of Ce of 0, 0.48% and 1.0%, with the combination 133 Ba±I ®ve samples with concentrations of iodine of 0, 0.21%, 0.50%, 0.99% and 2.09% and with the 109 Cd±Pd combination three samples with concentrations of palladium of 0, 0.50% and 1.48%. The attenuation coecients at the energies of the X-rays emitted are given in Table 2 for dierent concentrations of the absorber. The attenuation coecients used in the calculations are from Nemec and Gofman (1975), except for the attenuation coecient for water, where values from Hubbell (1982) were taken. The activities of the X-ray emitters used were in the order of some hundred Bq in order that the total count rate in the spectrum did not exceed some hundred counts per second. The spectra of X-rays from 133 Ba measured with various concentrations of iodine
M. Korun / Applied Radiation and Isotopes 50 (1999) 381±388 Table 2 Attenuation coecients of the solutions calculated at energies where X-ray lines occur and at dierent concentrations of the absorber Concentration m [E(Ka)] (%) (cm2/g)
m[E(Kb)] (cm2/g)
Cerium
0 0.48 1.0
0.269 0.293 0.319
0.242 0.334 0.430
Iodine
0 0.21 0.50 0.99 2.09
0.357 0.380 0.404 0.450 0.538
0.316 0.371 0.463 0.617 0.950
Palladium
0 0.5 1.02
0.678 0.758 0.919
0.536 0.820 1.389
Adsorber
The attenuation coecients were taken from Nemec and Gofman (1975) except the coecient for water, which is from Hubbell (1982).
are presented in Fig. 1 normalized to the same height of the Ka peak. Due to the ¯uorescence of iodine and scattering, the background in the vicinity of the X-ray peaks varied with the concentration of the absorber. In order to minimize variations of systematic uncertainties in the calculation of peak areas, the number of counts in the peaks occurring at the higher energy
Fig. 1. X-ray spectra from 133 Ba from samples prepared without iodine (.), with an iodine concentration of 0.50% (Q) and 2.09% (R).
385
were calculated with the same start and end channels. The peak area of the Kb rays was calculated by subtracting a constant background, extrapolated under the peak from higher energies, from the number of counts in the peak region. This method is not suitable for calculation of the number of counts in the peak belonging to the Ka rays since the background under that peak cannot be considered as constant, due to scattering and ¯uorescence of the absorber. Therefore direct peak area calculation was avoided by stripping the Ka line, measured with the sample without absorber, from the spectra of samples containing the absorber. The conditions of stripping were chosen so that the residuum showed a smooth energy dependence within the region of the Ka line. It is expected that in the systematic uncertainty of the stripping factor, resembling the systematic uncertainty of the ratio of the two peak areas, the systematic uncertainties of the areas cancel out to a large extent. The in¯uence of coincidence summing between gamma- and X-rays was neglected, since it was assumed that the presence of the absorber does not in¯uence the detection probability for gamma-rays. Consequently, the coincidence summing eects in¯uence the count rates in both X-ray lines for the same fraction and their in¯uence on the peak areas cancel in the ratio R. The in¯uence of coincidence summing among X-rays was taken into account by correcting the areas of the X-ray peaks. Twice the number of counts, multiplied by the total-to-peak ratio, in the sum peaks at the energies of 2E, 2E + DE and 2E + 2DE was added to the areas of the original peaks. This correction resulted in the change of the ratio R for 3% in the measurement of 133 Ba and for 2% in the measurement of 152 Eu in samples with the highest concentration of the X-ray absorbers. It should be mentioned that the increase in the absorber concentration increases the coincidence correction. Namely, the average depth in the sample where the detected Xrays are emitted from decreases with the concentration so the average solid angle subtended by the detector increases, increasing the probability for coincident detections. The point source eciency e00(E), necessary to correct the average path length for the dierence in the detector response at the energies of the Ka and Kb rays, was measured by point sources positioned at the geometrical axis of the detector on its surface. The eciency curve is displayed in Fig. 2. The concentrations of the absorber necessary to measure the average path length can be inferred from Table 3. Here the path lengths of the X-rays in the samples containing the emitter and absorber in various concentrations as derived from Eq. (18) are presented. The ®rst approximation, sP(m0), to the average path length in the sample without absorber, sV(m0), was cal-
386
M. Korun / Applied Radiation and Isotopes 50 (1999) 381±388
Fig. 2. Energy dependence of the eciency for a point source placed on the detector surface on its geometrical axis, presented in the energy range of the X-ray energies. The solid line denotes the energy dependence used in the calculations.
culated from the self-attenuation factor for emission normal to the sample surface (Debertin and Helmer, 1988) FP m0 , E, r~
1 ÿ eÿm0 D m0 D
20
using Eqs. (4) and (7) as sP m0
1 m Deÿm0 D 1 ÿ 0 ÿm D , m0 1ÿe 0
21
where D denotes the sample thickness. From the calculated average path lengths sV(m) new values for sV(m0) were interpolated and these were used in the second
step of the iteration procedure. The results from that step are presented in Table 3. The uncertainties of the average path lengths come from two main sources: from the uncertainty of the ratio R and the uncertainty of the average path sV(m0). In Table 3 the uncertainties due to each of the sources are presented separately, since their relative contributions vary with the absorber concentration and attenuation. They are calculated from the assumed uncertainty of 2% of the ratio R, obtained on the basis of reproducibility. For the uncertainty of the average path sV(m0) one half of the dierence of the values obtained from the ®rst and second step in the iteration procedure was taken. It can be observed that they result in the uncertainty of the average path which is at least three times smaller than the uncertainties originating in the ratio R.
4. Self-attenuation factors The feasibility of measuring average path lengths can be assessed from comparison with the path lengths obtained using calculated eciencies in Eq. (9) on the one hand, and from the use in self-attenuation factors on the other. The comparison between the measured average path lengths and the average path lengths obtained from eciencies calculated from the detector response to point sources and a simple detector model (Korun and MartincÏicÏ, 1992) is presented in Fig. 3, where both average path lengths are presented as functions of the attenuation coecient m + Dm/2. Reasonable agreement between the measured and calculated average path lengths is observed.
Table 3 The average path lengths of Ka X-rays in samples with dierent concentrations of absorber Absorber
Concentration (%)
sV(m) (cm)
DR (cm)
DsV (m0) (cm)
Cerium
0.48 1.00
0.7 1.2
0.5 0.2
0.03 0.01
Iodine
0.21 0.50 0.99 2.09
7 1.1 1.1 0.85
4 0.3 0.1 0.05
0.4 0.04 0.01 0.004
Palladium
0.50 1.48
1.1 0.76
0.3 0.04
0.1 0.01
The standard uncertainties DR present the contribution of the uncertainties of 2% of the ratio R(m, Dm, m0, Dm0, E, DE) and the uncertainties DsV(m0) the contribution of the uncertainty of sV (m0) to the uncertainty of the average path length.
M. Korun / Applied Radiation and Isotopes 50 (1999) 381±388
Fig. 3. Comparison between the average path lengths obtained experimentally (.) and from Eq. (7), where calculated eciencies were used (Q). The scattering of the calculated points originates in the uncertainty in the calculations of the derivative of the eciency.
The average path length given in Eq. (21) is not appropriate for comparison with measured average path lengths. Namely, in the experimental arrangement photons can impinge on the crystal at a variety of angles which is taken into account in the measured average path length but the analytical expression is derived supposing a parallel beam of gamma-rays. A new analytical expression for the self-attenuation factor can be introduced ad-hoc in analogy with the attenuation factor given in Eq. (20), i.e. by introducing the second term from the expansion of the exponential function into the expression and by requesting F(0, E, r~) = 1: ÿ1 mx 0 eÿmx 0 F1 m, E, r~ 2 , mx 0 2
22
where x0 denotes a characteristic length of the sample. From this factor by use of Eqs. (4) and (7) the average path length is derived as: s1 m
1 2 ÿ mx 0 ÿ 2 mx 0 eÿmx 0 : m 1 ÿ mx 0 ÿ eÿmx 0
23
At values of mx0>>1, i.e. for strong self-attenuation, the average path length assumes the value 1/m, as in the case of the one-dimensional transport model since then the rays are emitted almost perpendicularly to the surface of the sample. At values of mx0<<1, i.e. for a weak self-attenuation, the value x0 24 s1 0 3 is attained. Supposing sV(m)=s1(m) the parameter mx0 can be obtained from Eq. (23) and consequently the self-attenuation factor from Eq. (22). In Fig. 4 a comparison between the approximative attenuation factor
387
Fig. 4. The comparison between the self-attenuation factors obtained from average path lengths (.) and from Eq. (2) and calculated eciencies (Q).
F1 m, E, r~ and the attenuation factor calculated by Eq. (4) from computed eciencies (Korun and MartincÏicÏ, 1992) are presented as functions of the attenuation coecient m + Dm/2. The values s1(0) calculated from Eqs. (23) and (24) are presented in Table 4. The average over the values s1(0) agrees with the average path length extrapolated to zero self-attenuation in Fig. 3, and with the half-thickness of the sample as proposed by Miller (1987). 5. Conclusion For homogeneous samples a relation between the average path length of photons in the sample and their probability for registration in the full-energy peak is presented. The relation is used to derive a method for calculating the average path lengths from measured spectra. To implement the method spectra of samples containing an X-ray emitter and an absorber, having an absorption edge between the energies of the Kaand Kb-rays, must be performed. In the framework of Table 4 Values of the average path length of the gamma-rays in sample at negligible self-attenuation calculated at dierent self-attenuations in the sample m (cmÿ1)
s1(0) (cm)
0.31 0.37 0.43 0.53 0.84 0.79 1.15
0.621.4 1.620.4 1.620.7 1.620.3 1.820.2 3.221.7 2.520.5
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M. Korun / Applied Radiation and Isotopes 50 (1999) 381±388
the method average path lengths, independent on detector parameters, are obtained. Reasonable agreement between the average path lengths obtained from such measurements, and average path lengths derived from eciencies, calculated from a detector model and measured detector data (Korun and MartincÏicÏ, 1992), is obtained. An analytical expression for the self-attenuation factor for use with measured average path lengths is proposed. For the sample geometry used in the experiment, agreement between the self-attenuation factors obtained from calculated eciencies and measured average path lengths is demonstrated. In cases of geometries where photons impinge on the detector at a variety of angles, constructing the self-attenuation factors from measured average path lengths is shown to be a useful alternative to one-dimensional gamma-ray transport models. Acknowledgement The author thanks his colleague Dr. A.R. Byrne for preparing the sources and for the review of the manuscript. References Adams, F., Dams, R., 1975. Applied Gamma-Ray Spectrometry. Pergamon Press, Oxford. Appleby, P.G., Richardson, N., Nolan, P.J., 1992. Selfabsorption corrections for well-type detectors. Nucl. Instr. Meth. Phys. Res. B 71, 228.
Debertin, K., Helmer, R.G., 1988. Gamma- and X-ray Spectrometry with Semiconductor Detectors. North Holland, Amsterdam. Dryak, P., Kovar, K., Plchova, L., Suran, J., 1989. Corrections for the Marinelli geometry. J. Radioanal. Nucl. Chem. Lett. 135, 281. Hubbell, H.J., 1982. Photon mass attenuation and energyabsorption coecients from 1 keV to 20 MeV. Int. J. Appl. Radiat. Isot. 33, 1269. IAEA Tecdoc 619, 1991. X-ray and Gamma-Ray Standards for Detector Calibration. IAEA, Vienna, 1991. Korun, M., MartincÏicÏ, R., 1992. Eciency calibration of gamma-ray spectrometers for volume-source geometry. Int. J. Appl. Radiat. Isot. 43, 29. Miller, K.M., 1987. Self-absorption corrections for gamma ray spectral measurements of 210 Pb in environmental samples. Nucl. Instr. Meth. Phys. Res. A 258, 281. Nemec, O.F., Gofman, Yu.V., 1975. Spravochnik po Yadernoi ®zike, Izdatel'stvo Naukova Dumka. Kiev, 1975. Noguchi, M., Takeda, K., Higuchi, H., 1981. Semi-empirical g-ray peak eciency determination including self-absorption correction based on numerical integration. Int. J. Appl. Radiat. Isot. 32, 22. Overwater, D.M.W., Bode, P., de Goeij, J.J.M., 1993. Gamma-ray spectroscopy of voluminous sources. Correction for source geometry and self-attenuation. Nucl. Instr. Meth. Phys. Res. A 209. Sima, O., 1992. Photon attenuation for samples in Marinelli beaker geometry: an analytical computation. Health Phys. 62, 445. Sima, O., Dovlete, C., 1995. Linear relations for eciency calibration of g-spectrometric measurements of bulk samples. J. Radioanal. Nucl. Chem. Lett. 200, 191.
Applied Radiation and Isotopes 50 (1999) 381±388
Measurement of the average path length of photons in samples by self-attenuation of X-rays M. Korun * J. Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia Received 16 January 1998; received in revised form 16 March 1998
Abstract For homogeneous samples a relation between the average path length of photons detected in full-energy peaks in the spectra of X- and gamma-rays and the counting eciency is derived for an arbitrary sample-detector geometry. An experimental method of measuring the average path length in the samples is presented. The results of such measurements are compared to the calculated average path lengths and discussed in terms of self-attenuation factors. # 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction In gamma-ray spectrometry the activities of radionuclides contained in the sample are calculated from the count rates in the peaks occurring at the energies of the emitted gamma-rays. To perform the calculation the counting eciencies, i.e. the probabilities for registration of gamma-rays in the peaks of full-energy absorption, must be known. The counting eciency depends on the characteristics of the detector, the sample and their relative position. The material of the sample aects the counting eciency through the attenuation of gamma-rays. The in¯uence of attenuation of gamma-rays within the sample on the counting eciency can be expressed by the self-attenuation factor (Debertin and Helmer, 1988). Besides for planar geometries, this factor has also been given in analytical form, i.e. as a function of sample dimensions and the attenuation coecient, for well-type geometries by Appleby et al. (1992) and for Marinelli-type geometries by Dryak et al. (1989) and Sima (1992). All the factors were derived from one-dimensional gamma-ray transport models.
* Tel.: +386-61-177-3900; Fax: +386-61-123-2120; E-mail: [email protected].
An alternative approach is to approximate the selfattenuation factor by an assumed analytical expression of the attenuation coecient and a characteristic length and to use for that length a distance deduced from the measured spectra. This approach is analogous to the approach used by Miller (1987) for the attenuation factor for planar geometries, where an exponential function is used as the analytical expression and the half-thickness of the sample is used as the characteristic length. The use of a measured characteristic length has the advantage, that this method is not restricted to a speci®c geometry.
2. Theoretical section 2.1. Self-attenuation and the average path length In this section the relation between the counting eciency and the average path length of the photons detected in the full energy peak is derived for the case of homogeneous samples. The counting eciency for a homogeneous sample with a volume V can be expressed in the form of an integral over the sample volume (Noguchi et al., 1981; Korun and MartincÏicÏ, 1992; Overwater et al., 1993; Sima and Dovlete, 1995):
0969-8043/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 9 - 8 0 4 3 ( 9 8 ) 0 0 0 9 4 - 3
382
eV m, E
M. Korun / Applied Radiation and Isotopes 50 (1999) 381±388
1 V
V
ePS m, E, r~ dV ~r,
1
where ePS(m, E, r~) denotes the counting eciency of a point source emitting photons with energy E and located at r~ inside the sample material with the attenuation coecient m, and dV(~r) denotes the volume element in the vicinity of the point r~. The eciency of the point source in the presence of the attenuating material can be expressed as ePS m, E, r~ F m, E, r~ePS 0, E, r~,
2
where ePS(0, E, r~) denotes the point source eciency in the absence of the sample material and F(m, E, r~) the self-attenuation function which can be expressed as (Korun and MartincÏicÏ, 1992): ~ ~ eÿmsPS ~r,R dN E, r~, R VD F m, E, r~ : 3 ~ dN E, r~, R VD
V
4 The derivative of the self-attenuation function with respect to the attenuation coecient is ~ ~ ÿmsPS ~r,R ~ sPS ~r,Re dN E, r~, R @ F m, E, r~ VD ÿ ~ @m ~ eÿmsPS ~r,R dN E, r~, R
VD
~ ~ ÿmsPS ~r,R ~ sPS ~r,Re dN E, r~, R VD : sPS m, E, r~ ~ ~ eÿmsPS ~r,R dN E, r~, R
6
VD
The eciency of a volume sample depends on the attenuation coecient of the sample only through the self-attenuation function. Therefore its derivative with respect to the attenuation coecient is: @ eV m, E 1 ÿ sPS m, E, r~F m, E, r~ePS 0, E, r~ @m V V 7 dV ~r ÿsV m, E eV m, E , where sV(m, E) designates the path length in the sample averaged over all rays emitted in the sample and detected in the full energy peak: sPS m, E, r~F m, E, r~ePS 0, E, r~ dV ~r sV m, E V : F m, E, r~ePS 0, E, r~ dV ~r V
~ designates the number of rays Here dN(E, r~, R) emitted in the vicinity of r~ in the sample and absorbed or scattered in the vicinity of R~ in the detector in such a way that they are registered in the full energy peak, in the absence of attenuation within the sample. sPS(~r, ~ designates the path traversed by these photons in R) the sample and VD the sensitive volume of the detector. The self-attenuation function describes the unattenuated fraction of gamma rays emitted at r~ and registered in the full energy peak. Since F(0, E, r~) = 1, its value averaged over the sample volume is the selfattenuation factor: ePS 0, E, r~F m, E, r~ dV ~r eV m, E V F m, E, r~: eV 0, E ePS 0, E, r~ dV ~r
VD
~ ÿmsPS ~r,R
e
VD
~ dN E, r~, R
~ dN E, r~, R
ÿ sPS m, E, r~F m, E, r~,
5
where sPS(m, E, r~) designates the average length in the sample of the rays emitted at r~ and detected in the full energy peak:
8 The explicit dependence of sV(m, E) on energy describes the in¯uence of energy-dependent detector parameters on the average path length. Eq. (7) oers the possibility to relate the average path length of gamma-rays emitted in the sample and registered in the full energy peak to the counting eciency and its derivative with the attenuation coecient. In the derivation no speci®c sample-detector arrangement was assumed and it is therefore valid for any counting geometry. 2.2. Measurement of the derivative Absorption edges oer the possibility to introduce discontinuities in the attenuation coecient of the sample material at X-ray energies by introducing appropriate absorbers in the sample matrix. In this way the attenuation coecient can be changed abruptly and the derivative of the eciency can be measured by measuring the eciency on both sides of the absorption edge, using the de®nition of the derivative: @ eV m, E eV m Dm, E ÿ eV m, E @m Dm ÿ sV m, E eV m, E :
9
To perform the measurement suitable lines must appear close to the edge on each side. Since X-ray emitters radiate at closely spaced energies the derivative can be measured using suitable combinations of emitters and absorbers. The count rates in the peaks occurring at energies of E and E + DE can be expressed in terms of the eciencies as
M. Korun / Applied Radiation and Isotopes 50 (1999) 381±388
n E ab E eV m, E
10
and n E DE ab E DE eV m Dm, E DE ,
11
where a and b(E) denote the activity of the X-ray emitter and the emission probability of photons with energy E. The eciency eV(m + Dm, E + DE) can be expressed in the linear approximation as @ eV m, E DE @E @ eV m, E Dm, @m
eV m Dm, E DE eV m, E
12
and consequently, the average path length as: sV m, E
1 @ eV m, E DE eV m, E @E Dm eV m Dm, E DE =eV m, E ÿ 1 ÿ : 13 Dm
In the last equation the average path length is expressed by the partial derivative of the counting eciency with energy. This quantity is not directly measurable, since by changing the energy also the attenuation coecient changes. Therefore approximations must be introduced allowing a suitable approximation to that derivative. By dierentiation of Eq. (1) with energy the derivative of the volume eciency can be expressed as @ eV m, E 1 @ ePS 0, E, r~ F m, E, r~ @E V V @E @ F m, E, r~ ePS 0, E, r~ dV ~r: 14 @E The point-source eciency ePS(0, E, r~) can be separated into the product of the point-source eciency on the detector cap at its geometrical axis e00(E) and the spatial dependence Z(E, r~) (Korun and MartincÏicÏ, 1992): ePS 0, E, r~ e00 E Z E, r~:
15
The contribution of the derivative of Z(E, r~) to the derivative of the volume-source eciency with respect to energy is [e00(E)/V]fV[@Z(E, r~)/@E]F(m, E, r~) dV(~r). Also Z(E, r~) = 1 on the detector surface in its axis and therefore its derivative is zero there. Since the self-attenuation function has its maximum on the detector surface and falls with the distance from the detector the multiplication in the integral diminishes the in¯uence of the energy dependence of Z(E, r~). Therefore the derivative of Z(E, r~) with energy will be neglected and the derivative of the point-source eciency approximated by @ ePS 0, E, r~ de00 E Z E, r~ : @E dE
16
383
The self-attenuation function depends on energy mainly through the attenuation coecient of the sample ma~ terial. The explicit dependence originates in dN(E, r~, R) and re¯ects the energy dependence of the attenuation coecient within the detector. It is this coecient that determines the average depth within the detector where the photons interact and with it the eective distance between the sample and the detector as well. At small attenuation coecients the eective distance is larger, consequently decreasing the average path length of photons in the sample. However, the resulting explicit energy dependence is weak and will be neglected. It should be noted that by introduction of these approximations the only energy-dependent detector property remaining is e00(E). Since eV(m, E) is proportional to e00(E) it follows from Eq. (7) that the average path length does not depend on e00(E). Consequently, in the framework of the approximations introduced, the average path length does not depend on the detector parameters at all. Therefore in the following, since the average path length depends on energy just via the attenuation coecient, the energy will be omitted from the list of variables determining the average path length. Taking into account the approximations introduced and using Eqs. (10), (11) and (13) the average path length can be expressed in terms of count rates and emission probabilities as sV m ÿ
n mDm, EDE b E n m, E b EDE
00 E ÿ 1 ÿ e001 E dedE DE
Dm
: 17
It can be noted that the parameter e00(E) still appears in the expression for the average path length, but just to compensate the dierent detector responses at energies of E and E + DE. Eq. (17) is rather unsuitable for use with measured data since the ®rst term in the numerator, which is close to unity and cancels to a large degree, must be known with a great accuracy in order to determine the path length. It contains the ratio of emission probabilities b(E)/b(E + DE), which may not be known with an accuracy better than some percent, so it may be not possible to determine the average path length. To avoid the use of emission probabilities, their ratio is expressed by the count rates in the measurement of another sample measured in the same counting geometry but without an absorber introducing the discontinuity in the attenuation coecient between E and E + DE. Denoting the dierence of the attenuation coecient for E + DE and E in that sample by Dm0, the count rates in the peaks at energies E +DE and E by n(m0+Dm0, E + DE) and n(m0, E) and average path length by sV(m0), the ratio of emission probabilities can be substituted yielding for the average path length
384
M. Korun / Applied Radiation and Isotopes 50 (1999) 381±388
1 DE=e00 E de00 E =dE 1 ÿ R m, Dm, m0 , Dm0 , E, DE R m, Dm, m0 , Dm0 , E, DE sV m0 Dm0 , sV m Dm where R(m, Dm, m0, Dm0, E, DE) denotes the ratio R m, Dm, m0 , Dm0 , E, DE n m Dm, E DE n m0 , E : n m0 Dm0 , E DE n m, E
18
erties, the measured dependence also describes, via the energy dependence of the attenuation coecient, the energy dependence of the average path length. 19
To calculate the average path length from Eq. (18) its value in a sample of the matrix without the absorber must be assumed. A crude value suces for the calculation, since Dm can be made much greater than Dm0 by introducing larger amounts of absorber. Alternatively, a sample containing the absorber in small concentration, just to achieve Dm = 0, can be produced. Instead of the sample without absorber that sample can be used in measurements of the average path length in samples with larger concentrations. By using that sample the term containing sV(m0) in Eq. (18) vanishes. The described approach has the advantage that the only parameter to be measured, the ratio R(m, Dm, m0, Dm0, E, DE), depends just on the ratios of count rates in peaks appearing at the same energies. Since the ratio can be measured with an arbitrary statistical accuracy, the largest uncertainties introduced are systematic, arising from the calculation of peak areas. The peaks at the energies of E and E + DE may overlap and therefore the ratio can bear systematic uncertainties from the procedure for evaluation of peak areas. Since in the ratio R only ratios between peak areas at the same energy enter, these uncertainties partially cancel because of the similarity of spectral shapes measured with the two sources. By measuring the average path length at a ®xed energy and at dierent concentrations of the absorber, the dependence of the average path length on the attenuation in the sample can be extracted. Since the average path length, in the framework of the approximations introduced, does not depend on detector propTable 1 Energies where the Ka and Kb lines occur and the energy of the absorption edge of the absorber for the three combinations of the X-ray emitter and absorber Combination
E (Kb)
E (Ka)
E (Kab)
Eu±Ce Ba±I Cd±Pd
45.48±46.82 34.97±36.01 24.94±25.60
39.52±40.12 30.63±30.97 21.99±22.16
40.45 33.16 24.35
The X-ray energies are taken from IAEA Tecdoc 619 (1991) and the energies of the absorption edges from Adams and Dams (1975).
3. Measurements The measurements were performed with an n-type detector with a beryllium window of thickness 0.5 mm and an eciency of 23% relative to that of a 7.56 cm 7.56 cm NaI(Tl) crystal at the energy of 1333 keV. The resolution was 2.0 keV at the energy of 1333 keV and 0.7 keV for the pulser peak. The sources were prepared as aqueous solutions of an X-ray emitter and absorber. Polystyrene canisters of diameter 6 cm were ®lled to the height of 3.6 cm with 100 ml of solution and placed coaxially on the detector. To test the method three combinations of X-ray emitter and absorber were measured: 152 Eu and Ce, 133 Ba and I and 109 Cd and Pd. The energies of the Ka and Kb lines, obtained from the IAEA Tecdoc 619 (1991), and the energies where the K absorption edges occur, obtained from Adams and Dams (1975), for the three combinations are listed in Table 1. Cerium was dissolved in the form of Ce(NO3)36H2O and spiked with 152 Eu; no europium carrier was needed in this case. Iodine as KI salt was dissolved in 1 M hydrochloric acid solution containing 133 Ba with about 10 mg mlÿ1 barium as carrier. Palladium as PdCl2 was dissolved in a few ml of hot aqua regia, boiled to small volume, diluted with 0.1 M hydrochloric acid, cooled and ®ltered to remove slight turbidity. This solution was then spiked with 109 Cd and cadmium carrier to about 10 mg mlÿ1. With the 152 Eu±Ce combination three samples were prepared with concentrations by mass of Ce of 0, 0.48% and 1.0%, with the combination 133 Ba±I ®ve samples with concentrations of iodine of 0, 0.21%, 0.50%, 0.99% and 2.09% and with the 109 Cd±Pd combination three samples with concentrations of palladium of 0, 0.50% and 1.48%. The attenuation coecients at the energies of the X-rays emitted are given in Table 2 for dierent concentrations of the absorber. The attenuation coecients used in the calculations are from Nemec and Gofman (1975), except for the attenuation coecient for water, where values from Hubbell (1982) were taken. The activities of the X-ray emitters used were in the order of some hundred Bq in order that the total count rate in the spectrum did not exceed some hundred counts per second. The spectra of X-rays from 133 Ba measured with various concentrations of iodine
M. Korun / Applied Radiation and Isotopes 50 (1999) 381±388 Table 2 Attenuation coecients of the solutions calculated at energies where X-ray lines occur and at dierent concentrations of the absorber Concentration m [E(Ka)] (%) (cm2/g)
m[E(Kb)] (cm2/g)
Cerium
0 0.48 1.0
0.269 0.293 0.319
0.242 0.334 0.430
Iodine
0 0.21 0.50 0.99 2.09
0.357 0.380 0.404 0.450 0.538
0.316 0.371 0.463 0.617 0.950
Palladium
0 0.5 1.02
0.678 0.758 0.919
0.536 0.820 1.389
Adsorber
The attenuation coecients were taken from Nemec and Gofman (1975) except the coecient for water, which is from Hubbell (1982).
are presented in Fig. 1 normalized to the same height of the Ka peak. Due to the ¯uorescence of iodine and scattering, the background in the vicinity of the X-ray peaks varied with the concentration of the absorber. In order to minimize variations of systematic uncertainties in the calculation of peak areas, the number of counts in the peaks occurring at the higher energy
Fig. 1. X-ray spectra from 133 Ba from samples prepared without iodine (.), with an iodine concentration of 0.50% (Q) and 2.09% (R).
385
were calculated with the same start and end channels. The peak area of the Kb rays was calculated by subtracting a constant background, extrapolated under the peak from higher energies, from the number of counts in the peak region. This method is not suitable for calculation of the number of counts in the peak belonging to the Ka rays since the background under that peak cannot be considered as constant, due to scattering and ¯uorescence of the absorber. Therefore direct peak area calculation was avoided by stripping the Ka line, measured with the sample without absorber, from the spectra of samples containing the absorber. The conditions of stripping were chosen so that the residuum showed a smooth energy dependence within the region of the Ka line. It is expected that in the systematic uncertainty of the stripping factor, resembling the systematic uncertainty of the ratio of the two peak areas, the systematic uncertainties of the areas cancel out to a large extent. The in¯uence of coincidence summing between gamma- and X-rays was neglected, since it was assumed that the presence of the absorber does not in¯uence the detection probability for gamma-rays. Consequently, the coincidence summing eects in¯uence the count rates in both X-ray lines for the same fraction and their in¯uence on the peak areas cancel in the ratio R. The in¯uence of coincidence summing among X-rays was taken into account by correcting the areas of the X-ray peaks. Twice the number of counts, multiplied by the total-to-peak ratio, in the sum peaks at the energies of 2E, 2E + DE and 2E + 2DE was added to the areas of the original peaks. This correction resulted in the change of the ratio R for 3% in the measurement of 133 Ba and for 2% in the measurement of 152 Eu in samples with the highest concentration of the X-ray absorbers. It should be mentioned that the increase in the absorber concentration increases the coincidence correction. Namely, the average depth in the sample where the detected Xrays are emitted from decreases with the concentration so the average solid angle subtended by the detector increases, increasing the probability for coincident detections. The point source eciency e00(E), necessary to correct the average path length for the dierence in the detector response at the energies of the Ka and Kb rays, was measured by point sources positioned at the geometrical axis of the detector on its surface. The eciency curve is displayed in Fig. 2. The concentrations of the absorber necessary to measure the average path length can be inferred from Table 3. Here the path lengths of the X-rays in the samples containing the emitter and absorber in various concentrations as derived from Eq. (18) are presented. The ®rst approximation, sP(m0), to the average path length in the sample without absorber, sV(m0), was cal-
386
M. Korun / Applied Radiation and Isotopes 50 (1999) 381±388
Fig. 2. Energy dependence of the eciency for a point source placed on the detector surface on its geometrical axis, presented in the energy range of the X-ray energies. The solid line denotes the energy dependence used in the calculations.
culated from the self-attenuation factor for emission normal to the sample surface (Debertin and Helmer, 1988) FP m0 , E, r~
1 ÿ eÿm0 D m0 D
20
using Eqs. (4) and (7) as sP m0
1 m Deÿm0 D 1 ÿ 0 ÿm D , m0 1ÿe 0
21
where D denotes the sample thickness. From the calculated average path lengths sV(m) new values for sV(m0) were interpolated and these were used in the second
step of the iteration procedure. The results from that step are presented in Table 3. The uncertainties of the average path lengths come from two main sources: from the uncertainty of the ratio R and the uncertainty of the average path sV(m0). In Table 3 the uncertainties due to each of the sources are presented separately, since their relative contributions vary with the absorber concentration and attenuation. They are calculated from the assumed uncertainty of 2% of the ratio R, obtained on the basis of reproducibility. For the uncertainty of the average path sV(m0) one half of the dierence of the values obtained from the ®rst and second step in the iteration procedure was taken. It can be observed that they result in the uncertainty of the average path which is at least three times smaller than the uncertainties originating in the ratio R.
4. Self-attenuation factors The feasibility of measuring average path lengths can be assessed from comparison with the path lengths obtained using calculated eciencies in Eq. (9) on the one hand, and from the use in self-attenuation factors on the other. The comparison between the measured average path lengths and the average path lengths obtained from eciencies calculated from the detector response to point sources and a simple detector model (Korun and MartincÏicÏ, 1992) is presented in Fig. 3, where both average path lengths are presented as functions of the attenuation coecient m + Dm/2. Reasonable agreement between the measured and calculated average path lengths is observed.
Table 3 The average path lengths of Ka X-rays in samples with dierent concentrations of absorber Absorber
Concentration (%)
sV(m) (cm)
DR (cm)
DsV (m0) (cm)
Cerium
0.48 1.00
0.7 1.2
0.5 0.2
0.03 0.01
Iodine
0.21 0.50 0.99 2.09
7 1.1 1.1 0.85
4 0.3 0.1 0.05
0.4 0.04 0.01 0.004
Palladium
0.50 1.48
1.1 0.76
0.3 0.04
0.1 0.01
The standard uncertainties DR present the contribution of the uncertainties of 2% of the ratio R(m, Dm, m0, Dm0, E, DE) and the uncertainties DsV(m0) the contribution of the uncertainty of sV (m0) to the uncertainty of the average path length.
M. Korun / Applied Radiation and Isotopes 50 (1999) 381±388
Fig. 3. Comparison between the average path lengths obtained experimentally (.) and from Eq. (7), where calculated eciencies were used (Q). The scattering of the calculated points originates in the uncertainty in the calculations of the derivative of the eciency.
The average path length given in Eq. (21) is not appropriate for comparison with measured average path lengths. Namely, in the experimental arrangement photons can impinge on the crystal at a variety of angles which is taken into account in the measured average path length but the analytical expression is derived supposing a parallel beam of gamma-rays. A new analytical expression for the self-attenuation factor can be introduced ad-hoc in analogy with the attenuation factor given in Eq. (20), i.e. by introducing the second term from the expansion of the exponential function into the expression and by requesting F(0, E, r~) = 1: ÿ1 mx 0 eÿmx 0 F1 m, E, r~ 2 , mx 0 2
22
where x0 denotes a characteristic length of the sample. From this factor by use of Eqs. (4) and (7) the average path length is derived as: s1 m
1 2 ÿ mx 0 ÿ 2 mx 0 eÿmx 0 : m 1 ÿ mx 0 ÿ eÿmx 0
23
At values of mx0>>1, i.e. for strong self-attenuation, the average path length assumes the value 1/m, as in the case of the one-dimensional transport model since then the rays are emitted almost perpendicularly to the surface of the sample. At values of mx0<<1, i.e. for a weak self-attenuation, the value x0 24 s1 0 3 is attained. Supposing sV(m)=s1(m) the parameter mx0 can be obtained from Eq. (23) and consequently the self-attenuation factor from Eq. (22). In Fig. 4 a comparison between the approximative attenuation factor
387
Fig. 4. The comparison between the self-attenuation factors obtained from average path lengths (.) and from Eq. (2) and calculated eciencies (Q).
F1 m, E, r~ and the attenuation factor calculated by Eq. (4) from computed eciencies (Korun and MartincÏicÏ, 1992) are presented as functions of the attenuation coecient m + Dm/2. The values s1(0) calculated from Eqs. (23) and (24) are presented in Table 4. The average over the values s1(0) agrees with the average path length extrapolated to zero self-attenuation in Fig. 3, and with the half-thickness of the sample as proposed by Miller (1987). 5. Conclusion For homogeneous samples a relation between the average path length of photons in the sample and their probability for registration in the full-energy peak is presented. The relation is used to derive a method for calculating the average path lengths from measured spectra. To implement the method spectra of samples containing an X-ray emitter and an absorber, having an absorption edge between the energies of the Kaand Kb-rays, must be performed. In the framework of Table 4 Values of the average path length of the gamma-rays in sample at negligible self-attenuation calculated at dierent self-attenuations in the sample m (cmÿ1)
s1(0) (cm)
0.31 0.37 0.43 0.53 0.84 0.79 1.15
0.621.4 1.620.4 1.620.7 1.620.3 1.820.2 3.221.7 2.520.5
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M. Korun / Applied Radiation and Isotopes 50 (1999) 381±388
the method average path lengths, independent on detector parameters, are obtained. Reasonable agreement between the average path lengths obtained from such measurements, and average path lengths derived from eciencies, calculated from a detector model and measured detector data (Korun and MartincÏicÏ, 1992), is obtained. An analytical expression for the self-attenuation factor for use with measured average path lengths is proposed. For the sample geometry used in the experiment, agreement between the self-attenuation factors obtained from calculated eciencies and measured average path lengths is demonstrated. In cases of geometries where photons impinge on the detector at a variety of angles, constructing the self-attenuation factors from measured average path lengths is shown to be a useful alternative to one-dimensional gamma-ray transport models. Acknowledgement The author thanks his colleague Dr. A.R. Byrne for preparing the sources and for the review of the manuscript. References Adams, F., Dams, R., 1975. Applied Gamma-Ray Spectrometry. Pergamon Press, Oxford. Appleby, P.G., Richardson, N., Nolan, P.J., 1992. Selfabsorption corrections for well-type detectors. Nucl. Instr. Meth. Phys. Res. B 71, 228.
Debertin, K., Helmer, R.G., 1988. Gamma- and X-ray Spectrometry with Semiconductor Detectors. North Holland, Amsterdam. Dryak, P., Kovar, K., Plchova, L., Suran, J., 1989. Corrections for the Marinelli geometry. J. Radioanal. Nucl. Chem. Lett. 135, 281. Hubbell, H.J., 1982. Photon mass attenuation and energyabsorption coecients from 1 keV to 20 MeV. Int. J. Appl. Radiat. Isot. 33, 1269. IAEA Tecdoc 619, 1991. X-ray and Gamma-Ray Standards for Detector Calibration. IAEA, Vienna, 1991. Korun, M., MartincÏicÏ, R., 1992. Eciency calibration of gamma-ray spectrometers for volume-source geometry. Int. J. Appl. Radiat. Isot. 43, 29. Miller, K.M., 1987. Self-absorption corrections for gamma ray spectral measurements of 210 Pb in environmental samples. Nucl. Instr. Meth. Phys. Res. A 258, 281. Nemec, O.F., Gofman, Yu.V., 1975. Spravochnik po Yadernoi ®zike, Izdatel'stvo Naukova Dumka. Kiev, 1975. Noguchi, M., Takeda, K., Higuchi, H., 1981. Semi-empirical g-ray peak eciency determination including self-absorption correction based on numerical integration. Int. J. Appl. Radiat. Isot. 32, 22. Overwater, D.M.W., Bode, P., de Goeij, J.J.M., 1993. Gamma-ray spectroscopy of voluminous sources. Correction for source geometry and self-attenuation. Nucl. Instr. Meth. Phys. Res. A 209. Sima, O., 1992. Photon attenuation for samples in Marinelli beaker geometry: an analytical computation. Health Phys. 62, 445. Sima, O., Dovlete, C., 1995. Linear relations for eciency calibration of g-spectrometric measurements of bulk samples. J. Radioanal. Nucl. Chem. Lett. 200, 191.