- Email: [email protected]
Measurement of the K X-ray absorption jump factors and jump ratios of Gd, Dy, Ho and Er by attenuation of a Compton peak
Journal of Quantitative Spectroscopy & Radiative Transfer 88 (2004) 525 – 532
www.elsevier.com/locate/jqsrt
Short communication
Measurement of the K X-ray absorption jump factors and jump ratios of Gd, Dy, Ho and Er by attenuation of a Compton peak G. Budaka , R. Polatb;∗ a
b
Department of Physics, Science and Art Faculty, Ataturk University, 25240 Erzurum, Turkey Department of Physics Education, Education Faculty of Erzincan, Ataturk University, 24030 Erzincan, Turkey Received 11 March 2003; accepted 11 April 2004
Abstract The X-ray absorption jump factor and jump ratio of Gd, Dy, Ho and Er were measured with a Si(Li) detector by attenuation, with Gd, Dy, Ho and Er foil, a Compton peak produced by the scattering of the 59:5 keV Am-241 Gamma rays. Al was chosen as secondary exciter. The experimental absorption jump factors and jump ratios are compared with the theoretical estimates of WinXcom (Radiat. Phys. Chem. 60 (2001) 23), McMaster (Compilation of X-ray cross sections UCRL-50174, 1969; Sec. II. Rev. I), Broll (X-ray Spectrom 15 (1986) 271), Hubbel and Seltzer (NISTIR (1995) 5632) and Budak (Radiat. Meas. accepted for publication). The present results constitute the Drst measurement for this combination of energy and elements, and good agreement is obtained between experiment and theory. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Mixture rule; X-ray Euorescence spectrometry; Absorption jump factor; Mass attenuation coeFcient; Absorption edge
1. Introduction X-ray Euorescence cross section, absorption jump factor, jump ratio, mass attenuation coeFcients, e.g. are important parameters in X-ray Euorescence analysis. These parameters are required in a variety of applications including, for example, in nuclear physics, radiation shielding, atomic physics, cancer therapy and industrial irradiation processing and astrophysical problems [6–9]. Absorption jump factors and jump ratio of an elements are very useful parameters for technological and many other Delds of scientiDc application. So Gd, Dy, Ho and Er elements are chosen for ∗
Corresponding author. Tel.: +90-446-214-6000; fax: +90-446-223-1901. E-mail address: [email protected] (R. Polat).
0022-4073/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2004.04.016
526
G. Budak, R. Polat / Journal of Quantitative Spectroscopy & Radiative Transfer 88 (2004) 525 – 532
the beneDt of technology usage, as, these elements are important. Gd has unusual superconductive properties and when alloyed with iron, chromium or related metals is found to greatly improve the workability and resistance to temperature and oxidation. Gd also has unique magnetic properties. It can be used as a raw material for various Euorescent compounds, absorption material in atomic reactions, magnetic bubble material, screen-sensitivity increasing material, as well as many other applications in the chemical, glass and electronic industries. Dy is relatively stable in air at room temperature, yet tarnishes in moist air. The metal can be cut with a knife or machined without sparking (if care is taken to avoid overheating). Even small amounts of impurities can greatly aMect dysprosium’s properties. Ho is relatively soft, malleable, and has unusual magnetic properties. While it oxidizes rapidly in moist air and at elevated temperatures, it is relatively stable in dry air at room temperature. It is used in producing metal halide lamps, also used as an additive of various garnets. Er is employed in foil form in radiology and is also used in vacuum deposition, as a colorant in the manufacture of pink glass. Other rare earth elements used as dopants in superconductors are also characterized by X-ray techniques. Accurate data on absorption jump factors and jump ratios Gd, Dy, Ho and Er elements are required to be used in the applications mentioned above. Also the data of these elements are needed for the correct use of spectroscopic techniques [10]. A plot of the total attenuation cross section versus incident photon energy is found to exhibit a characteristic saw tooth structure in which the sharp discontinuities, known as absorption edges, arise whenever the incident energy coincides with the ionization energy of electrons in the K, L, M shells. These sharp discontinuities are due to the fact that photoeMect interaction becomes energetically possible in the shell considered. Shell-wise photoeMect cross sections at the edges can be obtained from the absorption jumps without any theoretical assumptions. The ratio between the upper and lower edge is called the jump ratio. The diMerence between the upper and lower edge values directly gives the photo eMect cross section of that particular shell without the necessity of assuming any other partial cross section. It is important since it is a measure of the photoeMect due to the particular shell relative to other competing interaction processes. In this study, we have measured the absorption jump factors and jump ratios in selected elements for the energy range 48–58 keV. The results are compared with given theoretical values. 2. The method of computation and theoretical basis The total mass attenuation coeFcient can be obtained from the diMerences in the count value with/without the existence of a sample. The mass attenuation coeFcients are given as I = I0 exp − x: ; (1) I 1 ln ; (2) =− x: I0 where is the linear attenuation coeFcient (cm−1 ); is the density of the sample (g=cm3 ), x is the thickness of the sample (cm), I0 is the count value without the sample, and I is the count value of the radiation penetrating through the sample. Theoretical values of the total mass attenuation coeFcients have been calculated using WinXCom. This programme is a Windows version of XCOM, well
G. Budak, R. Polat / Journal of Quantitative Spectroscopy & Radiative Transfer 88 (2004) 525 – 532
527
known for calculating X-ray and gamma ray attenuation coeFcients and interaction cross section. The new program, that has been developed called WinXCom [1]. That the total mass attenuation coeFcients are calculated, is a fundamental parameter to obtain absorption jump factors and jump ratios with the connected elements. Absorption edge jump ratios r and jump diMerences are measures of that portion total absorbed X-radiation that is absorbed by a speciDed atomic energy level. For example, K and LIII jump-ratios are deDned by + + + + ··· K LI LII LIII ; (3) rK = + + + · · · rLIII =
LI
+
LIII
MI
+
LII
+
MI
MII
MII
LIII
+ ···
+ ···
:
(4)
K and LIII jump diMerences are deDned by the diMerence between the numerator and denominator in the respective equations. More simply, ; (5) r= S L − ; (6) = S L where S and L refer, respectively, to the short- and long-wavelength sides of the edge, that is, the “top” and “bottom” or maximum and minimum values of (=). The actual fraction of the total number of photoionizations that occurs in, say, the K shell is given by − (rK − 1) 1 S L = =1− : (7) JK = rK rK
S
To calculate, scattering angles have been used in the following expression by ZarasUz [11]: l ; Cos = [l + 0:25(R0 + R1 )2 ]1=2
(8)
where l is the distance from source to the sample and R0 and R1 the internal and external diameters of radioisotope source, respectively. has been calculated as 111◦ . 3. Experimental procedure The total attenuation coeFcients, K X-rays absorption jump-factors and jump ratios were determined by using transmission geometry. In the present experiment a Si(Li) detector (FWHM=160 eV at 5:96 keV) was used with an ND 66 B multichannel analyzer for detection of X-rays. This detector was coupled to a computerized 1024-multichannel analyzer through a spectroscopy automatic Dne-tuning research ampliDer. To obtain statistical sensitivity, each sample has been measured by
528
G. Budak, R. Polat / Journal of Quantitative Spectroscopy & Radiative Transfer 88 (2004) 525 – 532
Fig. 1. Experimental set-up.
collecting the spectra from selected elements for a period of 72 × 103 s. High purity (99.9%) thin foil samples of Gd, Dy, Ho and Er were using a radioactive annular source of Am-241 of strength 100 mCi and -photon energy 59:5 keV. The schematic arrangement of the experimental setup used in the present work is shown in Fig. 1. In an ideal transmission geometry Al elements have been used as secondary target. Incident Gamma rays scattering Compton from Al element have been send on absorber. In this experiment, the net counts without absorber (I0 ) and with absorber (I ) were obtained at the same time and experimental conditions. A typical Compton spectrom of Gd scattered from Al is shown Fig. 2.
4. Results and discussion The experimental and theoretical absorption jump factors and jump ratios are listed in Tables 1 and 2. It is evident from tables that the experimental results are, in general, consistent with theoretical data that were calculated using the WinXCom.
G. Budak, R. Polat / Journal of Quantitative Spectroscopy & Radiative Transfer 88 (2004) 525 – 532
529
Fig. 2. Measured Compton spectra. The coherent scattered peak is at the Am-241 energy of 59:5 keV. Curve (a) unattenuated spectrom; (b) attenuated spectrom by a Gd foil. Table 1 Experimental and theoretical absorption jump factors (Jk ) Elements
Exp.
Relative diMerence (%)
Ta
Tb
Tc
Td
Te
Gd Dy Ho Er
0:794 ± 0:019 0:765 ± 0:015 0:823 ± 0:039 0:752 ± 0:016
0.125 3.520 4.320 4.440
0.795 0.791 0.788 0.786
0.802 0.791 0.786 0.791
0.827 0.818 0.812 0.818
0.795 0.791 0.788 0.786
0.824 0.813 0.805 0.817
a
WinXcom [1],
b
McMaster [2],
c
Broll [3],
d
Hubbel and Seltzer [4],
e
Budak et al. [5].
Table 2 Experimental and theoretical absorption jump ratios (rk ) Elements
Exp.
Ta
Tb
Tc
Td
Te
Gd Dy Ho Er
4.846 4.255 5.653 4.032
4.878 4.784 4.716 4.672
5.050 4.784 4.672 4.784
5.780 5.494 5.319 5.494
4.878 4.784 4.716 4.672
5.681 5.470 5.280 5.464
a
WinXcom [1],
b
McMaster [2],
c
Broll [3],
d
Hubbel and Seltzer [4],
e
Budak et al. [5].
The experimental total mass attenuation coeFcients of Gd, Dy, Ho and Er around. K absorption edges versus incident on absorber photon energy are also graphically in Figs. 3–6. In this present method, the Compton scattering of gamma rays from low-atomic number elements in deDnition angels are possible with ideal transmission geometry to obtain energies near absorption edges of absorbers. Angle choosing is very important in ideal transmission geometry for incident
530
G. Budak, R. Polat / Journal of Quantitative Spectroscopy & Radiative Transfer 88 (2004) 525 – 532
Fig. 3. Mass attenuation coeFcients of Gd around the K absorption edge.
Fig. 4. Mass attenuation coeFcients of Dy around the K absorption edge.
G. Budak, R. Polat / Journal of Quantitative Spectroscopy & Radiative Transfer 88 (2004) 525 – 532
Fig. 5. Mass attenuation coeFcients of Ho around the K absorption edge.
Fig. 6. Mass attenuation coeFcients of Er around the K absorption edge.
531
532
G. Budak, R. Polat / Journal of Quantitative Spectroscopy & Radiative Transfer 88 (2004) 525 – 532
photon on absorber. Meanwhile, incident photon energy in selected angles for each elements can be forwarded on chosen element being absorber. After scattering Compton from Al incident gamma photons (59:543 keV) with scattering angle ( = 111◦ ) come on absorber with 51:3 keV energy. This energy is near absorption edge of Gd (Kabs = 50:23 keV) being absorber. Relative diMerence between experimental and theoretical values for Gd is 0.125%. On the occasion of this agreement near absorption edge has been conDrmed by Ayala et al. [12] for Er. Further experimental studies will be useful for various elements. References [1] Gerward L, Guilbert N, Jensen B, Levring H. X-ray absorption in matter. Reengineering XCOM. Radiat Phys Chem 2001;60:23–4. [2] McMaster WH, Kerr Del Grande N, Mallett JH, Hubbell JH. Compilation of X-ray cross sections UCRL-50174, 1969; Sec. II. Rev. 1. [3] Broll N. Quantitative X-ray Euorescence analysis theory and practice of the fundamental coeFcient method. X-Ray Spectrom 1986;15:271–85. [4] Hubbell JH, Seltzer SM. NISTIR 1995;5632. [5] Budak G, Karabulut A, ErtuWgrul M. Determination of K shell absorption jump-factor for some elements using EDXRF technique. Radiat Meas 2003;37:103–107. [6] Bambynek W, Crasemann B, Fink RW, Freumd HU, Mark H, Swift CD, Price RE, Rao PV. Mod Phys 1972;44:716. [7] Hubbell JH. NUT Rep 1989;89:4144. [8] Higgins PD, Attrix FH, Hubbell JH, Seltzer SM, Berger MJ, Sibita CH. NIST Rep 1992;92:4812. [9] Rao DV, Cesareo R, Gigante GE. Appl Phys A 1996;62:381. [10] Ottmar H, Erbele H, Matussek P, Michel-Piper I. Energy dispersive X-ray techniques for accurate heavy elements assay. Adv X-Ray Anal 1987;30:285. [11] ZarasUz A, AygXun E. A theoretical and experimental investigation of the source-sample detector geometry for an angular type radioisotope excited XRF spectrometer. J Radioanal Nucl Chem 1989;129:367–75. [12] Ayala Alejandro P, Raul Mainardi T. Measurement of the X-ray absorption jump ratio of erbium by attenuation of a Compton peak. Radiat Phys Chem 1996;47:177–81.
www.elsevier.com/locate/jqsrt
Short communication
Measurement of the K X-ray absorption jump factors and jump ratios of Gd, Dy, Ho and Er by attenuation of a Compton peak G. Budaka , R. Polatb;∗ a
b
Department of Physics, Science and Art Faculty, Ataturk University, 25240 Erzurum, Turkey Department of Physics Education, Education Faculty of Erzincan, Ataturk University, 24030 Erzincan, Turkey Received 11 March 2003; accepted 11 April 2004
Abstract The X-ray absorption jump factor and jump ratio of Gd, Dy, Ho and Er were measured with a Si(Li) detector by attenuation, with Gd, Dy, Ho and Er foil, a Compton peak produced by the scattering of the 59:5 keV Am-241 Gamma rays. Al was chosen as secondary exciter. The experimental absorption jump factors and jump ratios are compared with the theoretical estimates of WinXcom (Radiat. Phys. Chem. 60 (2001) 23), McMaster (Compilation of X-ray cross sections UCRL-50174, 1969; Sec. II. Rev. I), Broll (X-ray Spectrom 15 (1986) 271), Hubbel and Seltzer (NISTIR (1995) 5632) and Budak (Radiat. Meas. accepted for publication). The present results constitute the Drst measurement for this combination of energy and elements, and good agreement is obtained between experiment and theory. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Mixture rule; X-ray Euorescence spectrometry; Absorption jump factor; Mass attenuation coeFcient; Absorption edge
1. Introduction X-ray Euorescence cross section, absorption jump factor, jump ratio, mass attenuation coeFcients, e.g. are important parameters in X-ray Euorescence analysis. These parameters are required in a variety of applications including, for example, in nuclear physics, radiation shielding, atomic physics, cancer therapy and industrial irradiation processing and astrophysical problems [6–9]. Absorption jump factors and jump ratio of an elements are very useful parameters for technological and many other Delds of scientiDc application. So Gd, Dy, Ho and Er elements are chosen for ∗
Corresponding author. Tel.: +90-446-214-6000; fax: +90-446-223-1901. E-mail address: [email protected] (R. Polat).
0022-4073/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2004.04.016
526
G. Budak, R. Polat / Journal of Quantitative Spectroscopy & Radiative Transfer 88 (2004) 525 – 532
the beneDt of technology usage, as, these elements are important. Gd has unusual superconductive properties and when alloyed with iron, chromium or related metals is found to greatly improve the workability and resistance to temperature and oxidation. Gd also has unique magnetic properties. It can be used as a raw material for various Euorescent compounds, absorption material in atomic reactions, magnetic bubble material, screen-sensitivity increasing material, as well as many other applications in the chemical, glass and electronic industries. Dy is relatively stable in air at room temperature, yet tarnishes in moist air. The metal can be cut with a knife or machined without sparking (if care is taken to avoid overheating). Even small amounts of impurities can greatly aMect dysprosium’s properties. Ho is relatively soft, malleable, and has unusual magnetic properties. While it oxidizes rapidly in moist air and at elevated temperatures, it is relatively stable in dry air at room temperature. It is used in producing metal halide lamps, also used as an additive of various garnets. Er is employed in foil form in radiology and is also used in vacuum deposition, as a colorant in the manufacture of pink glass. Other rare earth elements used as dopants in superconductors are also characterized by X-ray techniques. Accurate data on absorption jump factors and jump ratios Gd, Dy, Ho and Er elements are required to be used in the applications mentioned above. Also the data of these elements are needed for the correct use of spectroscopic techniques [10]. A plot of the total attenuation cross section versus incident photon energy is found to exhibit a characteristic saw tooth structure in which the sharp discontinuities, known as absorption edges, arise whenever the incident energy coincides with the ionization energy of electrons in the K, L, M shells. These sharp discontinuities are due to the fact that photoeMect interaction becomes energetically possible in the shell considered. Shell-wise photoeMect cross sections at the edges can be obtained from the absorption jumps without any theoretical assumptions. The ratio between the upper and lower edge is called the jump ratio. The diMerence between the upper and lower edge values directly gives the photo eMect cross section of that particular shell without the necessity of assuming any other partial cross section. It is important since it is a measure of the photoeMect due to the particular shell relative to other competing interaction processes. In this study, we have measured the absorption jump factors and jump ratios in selected elements for the energy range 48–58 keV. The results are compared with given theoretical values. 2. The method of computation and theoretical basis The total mass attenuation coeFcient can be obtained from the diMerences in the count value with/without the existence of a sample. The mass attenuation coeFcients are given as I = I0 exp − x: ; (1) I 1 ln ; (2) =− x: I0 where is the linear attenuation coeFcient (cm−1 ); is the density of the sample (g=cm3 ), x is the thickness of the sample (cm), I0 is the count value without the sample, and I is the count value of the radiation penetrating through the sample. Theoretical values of the total mass attenuation coeFcients have been calculated using WinXCom. This programme is a Windows version of XCOM, well
G. Budak, R. Polat / Journal of Quantitative Spectroscopy & Radiative Transfer 88 (2004) 525 – 532
527
known for calculating X-ray and gamma ray attenuation coeFcients and interaction cross section. The new program, that has been developed called WinXCom [1]. That the total mass attenuation coeFcients are calculated, is a fundamental parameter to obtain absorption jump factors and jump ratios with the connected elements. Absorption edge jump ratios r and jump diMerences are measures of that portion total absorbed X-radiation that is absorbed by a speciDed atomic energy level. For example, K and LIII jump-ratios are deDned by + + + + ··· K LI LII LIII ; (3) rK = + + + · · · rLIII =
LI
+
LIII
MI
+
LII
+
MI
MII
MII
LIII
+ ···
+ ···
:
(4)
K and LIII jump diMerences are deDned by the diMerence between the numerator and denominator in the respective equations. More simply, ; (5) r= S L − ; (6) = S L where S and L refer, respectively, to the short- and long-wavelength sides of the edge, that is, the “top” and “bottom” or maximum and minimum values of (=). The actual fraction of the total number of photoionizations that occurs in, say, the K shell is given by − (rK − 1) 1 S L = =1− : (7) JK = rK rK
S
To calculate, scattering angles have been used in the following expression by ZarasUz [11]: l ; Cos = [l + 0:25(R0 + R1 )2 ]1=2
(8)
where l is the distance from source to the sample and R0 and R1 the internal and external diameters of radioisotope source, respectively. has been calculated as 111◦ . 3. Experimental procedure The total attenuation coeFcients, K X-rays absorption jump-factors and jump ratios were determined by using transmission geometry. In the present experiment a Si(Li) detector (FWHM=160 eV at 5:96 keV) was used with an ND 66 B multichannel analyzer for detection of X-rays. This detector was coupled to a computerized 1024-multichannel analyzer through a spectroscopy automatic Dne-tuning research ampliDer. To obtain statistical sensitivity, each sample has been measured by
528
G. Budak, R. Polat / Journal of Quantitative Spectroscopy & Radiative Transfer 88 (2004) 525 – 532
Fig. 1. Experimental set-up.
collecting the spectra from selected elements for a period of 72 × 103 s. High purity (99.9%) thin foil samples of Gd, Dy, Ho and Er were using a radioactive annular source of Am-241 of strength 100 mCi and -photon energy 59:5 keV. The schematic arrangement of the experimental setup used in the present work is shown in Fig. 1. In an ideal transmission geometry Al elements have been used as secondary target. Incident Gamma rays scattering Compton from Al element have been send on absorber. In this experiment, the net counts without absorber (I0 ) and with absorber (I ) were obtained at the same time and experimental conditions. A typical Compton spectrom of Gd scattered from Al is shown Fig. 2.
4. Results and discussion The experimental and theoretical absorption jump factors and jump ratios are listed in Tables 1 and 2. It is evident from tables that the experimental results are, in general, consistent with theoretical data that were calculated using the WinXCom.
G. Budak, R. Polat / Journal of Quantitative Spectroscopy & Radiative Transfer 88 (2004) 525 – 532
529
Fig. 2. Measured Compton spectra. The coherent scattered peak is at the Am-241 energy of 59:5 keV. Curve (a) unattenuated spectrom; (b) attenuated spectrom by a Gd foil. Table 1 Experimental and theoretical absorption jump factors (Jk ) Elements
Exp.
Relative diMerence (%)
Ta
Tb
Tc
Td
Te
Gd Dy Ho Er
0:794 ± 0:019 0:765 ± 0:015 0:823 ± 0:039 0:752 ± 0:016
0.125 3.520 4.320 4.440
0.795 0.791 0.788 0.786
0.802 0.791 0.786 0.791
0.827 0.818 0.812 0.818
0.795 0.791 0.788 0.786
0.824 0.813 0.805 0.817
a
WinXcom [1],
b
McMaster [2],
c
Broll [3],
d
Hubbel and Seltzer [4],
e
Budak et al. [5].
Table 2 Experimental and theoretical absorption jump ratios (rk ) Elements
Exp.
Ta
Tb
Tc
Td
Te
Gd Dy Ho Er
4.846 4.255 5.653 4.032
4.878 4.784 4.716 4.672
5.050 4.784 4.672 4.784
5.780 5.494 5.319 5.494
4.878 4.784 4.716 4.672
5.681 5.470 5.280 5.464
a
WinXcom [1],
b
McMaster [2],
c
Broll [3],
d
Hubbel and Seltzer [4],
e
Budak et al. [5].
The experimental total mass attenuation coeFcients of Gd, Dy, Ho and Er around. K absorption edges versus incident on absorber photon energy are also graphically in Figs. 3–6. In this present method, the Compton scattering of gamma rays from low-atomic number elements in deDnition angels are possible with ideal transmission geometry to obtain energies near absorption edges of absorbers. Angle choosing is very important in ideal transmission geometry for incident
530
G. Budak, R. Polat / Journal of Quantitative Spectroscopy & Radiative Transfer 88 (2004) 525 – 532
Fig. 3. Mass attenuation coeFcients of Gd around the K absorption edge.
Fig. 4. Mass attenuation coeFcients of Dy around the K absorption edge.
G. Budak, R. Polat / Journal of Quantitative Spectroscopy & Radiative Transfer 88 (2004) 525 – 532
Fig. 5. Mass attenuation coeFcients of Ho around the K absorption edge.
Fig. 6. Mass attenuation coeFcients of Er around the K absorption edge.
531
532
G. Budak, R. Polat / Journal of Quantitative Spectroscopy & Radiative Transfer 88 (2004) 525 – 532
photon on absorber. Meanwhile, incident photon energy in selected angles for each elements can be forwarded on chosen element being absorber. After scattering Compton from Al incident gamma photons (59:543 keV) with scattering angle ( = 111◦ ) come on absorber with 51:3 keV energy. This energy is near absorption edge of Gd (Kabs = 50:23 keV) being absorber. Relative diMerence between experimental and theoretical values for Gd is 0.125%. On the occasion of this agreement near absorption edge has been conDrmed by Ayala et al. [12] for Er. Further experimental studies will be useful for various elements. References [1] Gerward L, Guilbert N, Jensen B, Levring H. X-ray absorption in matter. Reengineering XCOM. Radiat Phys Chem 2001;60:23–4. [2] McMaster WH, Kerr Del Grande N, Mallett JH, Hubbell JH. Compilation of X-ray cross sections UCRL-50174, 1969; Sec. II. Rev. 1. [3] Broll N. Quantitative X-ray Euorescence analysis theory and practice of the fundamental coeFcient method. X-Ray Spectrom 1986;15:271–85. [4] Hubbell JH, Seltzer SM. NISTIR 1995;5632. [5] Budak G, Karabulut A, ErtuWgrul M. Determination of K shell absorption jump-factor for some elements using EDXRF technique. Radiat Meas 2003;37:103–107. [6] Bambynek W, Crasemann B, Fink RW, Freumd HU, Mark H, Swift CD, Price RE, Rao PV. Mod Phys 1972;44:716. [7] Hubbell JH. NUT Rep 1989;89:4144. [8] Higgins PD, Attrix FH, Hubbell JH, Seltzer SM, Berger MJ, Sibita CH. NIST Rep 1992;92:4812. [9] Rao DV, Cesareo R, Gigante GE. Appl Phys A 1996;62:381. [10] Ottmar H, Erbele H, Matussek P, Michel-Piper I. Energy dispersive X-ray techniques for accurate heavy elements assay. Adv X-Ray Anal 1987;30:285. [11] ZarasUz A, AygXun E. A theoretical and experimental investigation of the source-sample detector geometry for an angular type radioisotope excited XRF spectrometer. J Radioanal Nucl Chem 1989;129:367–75. [12] Ayala Alejandro P, Raul Mainardi T. Measurement of the X-ray absorption jump ratio of erbium by attenuation of a Compton peak. Radiat Phys Chem 1996;47:177–81.