- Email: [email protected]
The effective atomic number revisited in the light of modern photon-interaction cross-section databases
ARTICLE IN PRESS Applied Radiation and Isotopes 68 (2010) 784–787
Contents lists available at ScienceDirect
Applied Radiation and Isotopes journal homepage: www.elsevier.com/locate/apradiso
The effective atomic number revisited in the light of modern photon-interaction cross-section databases S.R. Manohara a,1, S.M. Hanagodimath a, K.S. Thind b, L. Gerward c, a
Department of Physics, Gulbarga University, Gulbarga 585 106, Karnataka, India Department of Physics, Guru Nanak Dev University, Amritsar 143 005, Punjab, India c Department of Physics, Technical University of Denmark, DK-2800 Lyngby, Denmark b
a r t i c l e in f o
a b s t r a c t
Keywords: Effective atomic number Photon-interaction cross-sections Biomolecules
The effective atomic number, Zeff, has been calculated for fatty acids and cysteine. It is shown that Zeff is a useful parameter for low-Z materials at any energy above 1 keV. Absorption edges of medium-Z elements may complicate the energy dependence of Zeff below 10 keV. The notion of Zeff is perhaps most useful at energies where Compton scattering is dominating, and where Zeff is equal to the mean atomic number, /ZS, over a wide energy range around 1 MeV. & 2009 Elsevier Ltd. All rights reserved.
1. Introduction For a complex medium, the ‘‘effective atomic number’’ is in some cases a convenient parameter for representing X-ray and gamma ray interactions, e.g. in designs of radiation shielding or in calculations of absorbed dose. However, as originally stated by Hine (1952), the effective atomic number of a multi-element material is not a constant. It is varying with photon energy, depending on the relative importance of the interaction processes involved. Among early applications one may mention bone densitometry and measurements of fat content in liver (Puumalainen et al., 1977). Yang et al. (1987) measured Zeff of soft human tissues. More recently, Kumar and Reddy (1997) determined the effective atomic number for materials of dosimetric interest. Very recently, Manjunathaguru and Umesh (2006, 2007), Manohara and Hanagodimath (2007a, b), and Manohara et al. (2008a) determined the effective atomic number of some biomolecules. Grinyov et al. (2007) developed dual-energy radiography for separate detection of materials differing in their effective atomic number and local density. Similarly, Torikoshi et al. (2007) developed a dual-energy X-ray CT method using synchrotron radiation to obtain highresolution Zeff images for medical diagnosis. Thus, there is a renewed interest in the effective atomic number for applications in biology and medicine. Early calculations of the effective atomic number were based on parameterization of the photon-interaction cross-section by
Corresponding author. Tel.: + 45 4525 3146; fax: + 45 4593 2766.
E-mail address: [email protected] (L. Gerward). 1 Present address: Department of Physics, Bahubali College of Engineering, Shravanabelagola 573 135, Karnataka, India. 0969-8043/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.apradiso.2009.09.047
fitting data over limited ranges of photon energy and atomic number (Jackson and Hawkes, 1981; Bradley et al., 1986). As the present results will show, modern databases of photon-interaction cross-sections and interpolation programs (Berger and Hubbell, 1987/1999; Gerward et al., 2001, 2004) have made it possible to calculate effective atomic numbers with much improved accuracy and information content over wide ranges of energy and composition.
2. Theory Formulas for calculating the effective atomic number, valid for energies above 1 keV and for all types of materials, have been treated elsewhere (Manohara et al., 2008b), an only a brief summary is given here. The total photon-interaction cross-section per molecule, sm, can be written X sm ¼ ni si ð1Þ i
where ni is the number of atoms of the ith constituent element in the molecule (i.e. the number of formula units), and si is the total photon-interaction cross-section of element i. Essentially, the definition of the effective atomic number, Zeff, is based on the assumption that the actual atoms of a given molecule can be replaced by an equal number of identical (average) atoms, each of which having Zeff identical (average) electrons. Thus, the total photon-interaction cross-section per molecule can be written
sm ¼ nsa ¼ nZeff se
ð2Þ
where sa is the effective (average) cross-section per atom, se is the P effective (average) cross-section per electron, and n = ini is the
ARTICLE IN PRESS S.R. Manohara et al. / Applied Radiation and Isotopes 68 (2010) 784–787
total number of atoms in the molecule. It follows that the effective atomic number of a chemical compound is given by P ns Zeff ¼ Pi i si ð3Þ j j nj Zj The effective atomic number, Zeff, is closely related to the effective electron density, Ne,eff, expressed in number of electrons per unit mass. The two parameters are related through nZ Z Ne;eff ¼ NA P eff ¼ NA eff /AS i ni Ai
ð4Þ
where Ai is the atomic weight of the ith element, NA is the P Avogadro constant, and /AS= iniAi/n is the average atomic weight. Similarly, the average electron density, /NeS, can be defined as /Ne S ¼
NA X /ZS n Z ¼ NA M i i i /AS
785
Table 1 Fatty acids studied in the present work.
1 2 3 4 5 6 7 8 9 10 11
Compound
Formula
/ZS
Zeff at 1 MeV
Lauric acid Myristic acid Palmitic acid Stearic acid Arachidic acid Behenic acid Lignoceric acid Cerotic acid Montanic acid Palmitoleic acid Oleic acid
C12H24O2 C14H28O2 C16H32O2 C18H36O2 C20H40O2 C22H44O2 C24H48O2 C26H52O2 C28H56O2 C16H30O2 C18H34O2
2.95 2.91 2.88 2.86 2.84 2.82 2.81 2.80 2.79 2.96 2.93
2.95 2.91 2.88 2.86 2.84 2.82 2.81 2.80 2.79 2.96 2.93
/ZS is the mean atomic number, and Zeff is the effective atomic number calculated at 1 MeV.
ð5Þ
P P where M= iniAi is the molar mass, and /ZS= iniZi/n is the mean atomic number. Total photon-interaction cross-sections can be obtained from existing tabulations (Hubbell and Seltzer, 1995), or calculated as needed using computer programs, such as XCOM (Berger and Hubbell, 1987/1999) or its Windows successor WinXCom (Gerward et al., 2001, 2004). The latter program has the advantage of being able to export the cross-sectional data to a predefined MS Excel template, thereby facilitating subsequent graphical or numerical data treatment. For a given material, the program XMuDat (Nowotny, 1998) calculates a single-valued effective atomic number, which we will call ZX,eff: X ZX;eff ¼ ð ai Zim1 Þ1=ðm1Þ ð6Þ i
where si is the fractional number of electrons of element i, and m is a constant between 3 and 5. It is suggested that m is set to 3.6 for materials with Zeff less than 6, and 4.1 for a material with Zeff higher than 6 (Jackson and Hawkes, 1981).
3. Biomolecules Proteins are major structural components in animal and human tissues, being a key part of skin, nails, cartilage and muscles. Other proteins transport oxygen and perform further important tasks. All proteins are chemically similar, the basic building blocks being amino acids. Hydrolysis of fats or oils, which are media for energy storage, yields glycerol and carboxylic acids, called fatty acids. 3.1. Fatty acids Table 1 lists the fatty acids studied in the present work. Fig. 1 shows the calculated effective atomic number, Zeff, as a function of photon energy, E. All fatty acids behave similarly, since they consist of hydrogen, carbon, and oxygen in about the same proportions. One can distinguish three energy regions, in each of which Zeff for a given compound is about constant. The regions are approximately Eo0.01 MeV, 0.05oE o5 MeV, and E4200 MeV. Between these three energy regions, there are transition regions with a rapid variation of Zeff. The energy behavior of Zeff for total interaction mirrors the relative importance of the partial processes. The dominating process is photoelectric absorption at low energies, incoherent (Compton) scattering at intermediate energies, and pair production
Fig. 1. Calculated effective atomic numbers of fatty acids for total photon interaction. Sample numbers 1–11 are as for Table 1.
at high energies. Coherent (Rayleigh) scattering never plays any significant role in this connection, since it occurs mainly at low energies, where photoelectric absorption is by far the most important interaction process. At intermediate energies, Zeff for a given low- or medium-Z compound is about constant over a wide energy range, typically two decades between 0.05 and 5 MeV (Fig. 1). In this energy region, Compton scattering accounts for practically all photon interactions. The Compton scattering cross-section per atom is proportional to the atomic number. It can then be shown that Zeff is equal to the mean atomic number, /ZS, and Ne,eff is equal to the average electron density, /NeS. As seen in Table 1, Zeff calculated at 1 MeV agrees perfectly with /ZS. In the low- and high-energy regions, Zeff is a weighted mean, where the element with the highest atomic number has the greatest weight. Therefore, the weighted mean is larger than the simple mean. The largest value of Zeff occurs in the low-energy range, typically below 10 keV, where the Z4 dependence of the photoelectric absorption cross-section gives a heavy weight for the highest atomic number. In the high-energy range, typically above 200 MeV, Zeff is again constant but smaller than in the lowenergy range. This is due to the dominance of pair production, the cross-section of which has a weaker Z2 dependence. Hence, pair
ARTICLE IN PRESS 786
S.R. Manohara et al. / Applied Radiation and Isotopes 68 (2010) 784–787
Hubbell (1999) concludes that the envelope of uncertainty of s is of the order of 1–2% in the energy range from 5 keV to a few MeV. Discrepancies up to 25% are known to occur between 1 and 4 keV, but these low energies are generally of little interest in medical and biological applications. Thus, we conclude that Zeff, calculated at energies above 5 keV, is accurate to within a few %.
4. Conclusions
Fig. 2. Calculated effective atomic number of cysteine for total photon interaction as a function of energy. The solid squares are the experimental data points of Manjunathaguru and Umesh (2007). The dotted line indicates the single-valued effective atomic number ZX,eff provided by XMuDat. The horizontal dashed line indicates the mean atomic number /ZS, and the vertical dashed line the energy of the K absorption edge of sulfur (S).
production gives less weight to the higher-Z elements than photoelectric absorption. Our results for fatty acids are in good agreement with experimental data of Manjunathaguru and Umesh (2006, 2007), who have determined Zeff of some biologically important compounds in the energy range 6.4–1330 keV. 3.2. Cysteine For cysteine, an amino acid, the graph of Zeff versus energy (Fig. 2) exhibits about the same energy regions as for the fatty acids. At intermediate energies, here narrowed down to one decade between 0.2 and 2 MeV, where Compton scattering is dominating, Zeff is about 4.6 in good agreement with /ZS=4.57 calculated from the chemical formula, C3H7O2NS. A new feature, where cysteine differs from the fatty acids, is the discontinuous jump of Zeff at 2.472 keV. This is due to the K absorption edge of the medium-Z element sulfur (Z=16). It is obvious that the presence of the absorption edge makes the applicability of the effective atomic number somewhat problematic at low energies. The single-valued effective atomic number provided by XMuDat is only a rough approximation at energies below about 10 keV (Fig. 2). This follows from the fact that Eq. (6) is derived from a parameterization of the photoelectric absorption crosssection. Users of XMuDat should therefore treat the effective atomic number with some caution. Fig. 2 also shows that our calculations are in good agreement with the experimental data of Manjunathaguru and Umesh (2007). It is interesting to notice that these authors have determined Zeff in the transition range between photoelectric absorption and Compton scattering, where Zeff is rapidly varying with energy. 3.3. Accuracy of calculations Eq. (3) shows that the accuracy of the calculated effective number is solely determined by the elemental cross-sections, si.
The effective atomic number is a useful parameter for low- and medium-Z materials, encountered in biological and medical applications. The effective atomic number is constant and equal to the mean atomic number, /ZS, over a wide energy range around 1 MeV, where Compton scattering is the main photoninteraction process. Below 10 keV, absorption edges of medium-Z elements may complicate the energy dependence of Zeff. The single-valued effective atomic number provided by the XMuDat software is a rough approximation below about 10 keV. It should be remembered that the concept of the effective atomic number is based on an underlying theory of X-ray and gamma ray interactions with matter. Thus, Zeff reflects our experimental and theoretical knowledge of mass attenuation coefficients and atomic interaction cross-sections, as given in recent compilations and databases (Gerward, 1993; Hubbell, 1999, 2006). Finally, we would like to mention that the program WinXCom is available free of charge. Interested readers are welcome to send an e-mail request to [email protected].
References Berger, M.J., Hubbell, J.H., 1987/1999. XCOM: Photon cross sections database, web version 1.2, available at http://physics.nist.gov/xcom. National Institute of Standards and Technology, Gaithersburg, MD 20899, USA, 1999. Originally published as NBSIR 87-3597 ‘‘XCOM: Photon cross sections on a personal computer’’. Bradley, D.A., Chong, C.S., Ghose, A.M., 1986. Photon absorptiometric studies of elements, mixtures and substances of biomedical interest. Phys. Med. Biol. 31, 267–273. Gerward, L., 1993. X-ray attenuation coefficients: current state of knowledge and availability. Radiat. Phys. Chem. 41, 783–789. Gerward, L., Guilbert, N., Jensen, K.B., Levring, H., 2001. X-ray absorption in matter. Reengineering XCOM. Radiat. Phys. Chem. 60, 23–24. Gerward, L., Guilbert, N., Jensen, K.B., Levring, H., 2004. WinXCom—a program for calculating X-ray attenuation coefficients. Radiat. Phys. Chem. 71, 653–654. Grinyov, B., Ryzhikov, V., Lecoq, P., Naydenov, S., Opolonin, A., Lisetskaya, E., Galkin, S., Shumeiko, N., 2007. Dual-energy radiography of bone tissues using ZnSebased scintielectronic detectors. Nucl. Instrum. Methods A 580, 50–53. Hine, G.J., 1952. The ‘‘effective’’ atomic numbers of materials for various gammaray interactions. Phys. Rev. 85, 725. Hubbell, J.H., Seltzer, S.M., 1995. Tables of X-ray mass attenuation coefficients and mass energy-absorption coefficients 1 keV–20 MeV for elements Z= 1 to 92 and 48 additional substances of dosimetric interest. NISTIR 5632. National Institute of Standards and Technology, Gaithersburg, MD 20899, USA. Hubbell, J.H., 1999. Review of photon interaction cross section data in medical and biological context. Phys. Med. Biol. 44, R1–R22. Hubbell, J.H., 2006. Review and history of photon cross section calculations. Phys. Med. Biol. 51, R245–R262. Jackson, D.F., Hawkes, D.J., 1981. X-ray attenuation coefficients of elements and mixtures. Phys. Rep. 70, 169–233. Kumar, T.K., Reddy, K.V., 1997. Effective atomic numbers for materials of dosimetric interest. Radiat. Phys. Chem. 50, 545–553. Manjunathaguru, V., Umesh, T.K., 2006. Effective atomic numbers and electron densities of some biologically important compounds containing H, C, N and O in the energy range 145–1330 keV. J. Phys. B At. Mol. Opt. Phys. 39, 3969–3981. Manjunathaguru, V., Umesh, T.K., 2007. Total interaction cross sections and effective atomic numbers of some biologically important compounds containing H, C, N and O in the energy range 64–136 keV. J. Phys. B At. Mol. Opt. Phys. 40, 3707–3718. Manohara, S.R., Hanagodimath, S.M., 2007a. Studies on effective atomic numbers and electron densities of essential amino acids in the energy range 1 keV– 100 GeV. Nucl. Instrum. Methods B 258, 321–328. Manohara, S.R., Hanagodimath, S.M., 2007b. Effective atomic numbers for photon energy absorption of essential amino acids in the energy range 1 keV to 20 MeV. Nucl. Instrum. Methods B 264, 9–14.
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Manohara, S.R., Hanagodimath, S.M., Gerward, L., 2008a. Energy dependence of effective atomic numbers for photon energy absorption and photon interaction: studies of some biological molecules in the energy range 1 keV–20 MeV. Med. Phys. 35, 388–402. Manohara, S.R., Hanagodimath, S.M., Thind, K.S., Gerward, L., 2008b. On the effective atomic number: a comprehensive set of formulas for all types of materials and energies above 1 keV. Nucl. Instrum. Methods B 266, 3906–3912. Nowotny, R., 1998. XMuDat: Photon attenuation data on PC. International Atomic Energy Agency, Vienna.
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Puumalainen, P., Olkkonen, H., Sikanen, P., 1977. Assessment of fat content of liver by a photon scattering technique. Int. J. Appl. Radiat. Isot. 28, 785–787. Torikoshi, M., Ohno, Y., Natsuhori, M., Ito, N., Uesugi, K., Yagi, N., Tsunoo, T., De, M., Endo, M., 2007. Dual-energy X-ray CT with vertically expanded irradiation field. Nucl. Instrum. Methods A 580, 996–999. Yang, N.C., Leichner, P.K., Hawkins, W.G., 1987. Effective atomic numbers for lowenergy total photon interactions in human tissues. Med. Phys. 14, 759–766.
Contents lists available at ScienceDirect
Applied Radiation and Isotopes journal homepage: www.elsevier.com/locate/apradiso
The effective atomic number revisited in the light of modern photon-interaction cross-section databases S.R. Manohara a,1, S.M. Hanagodimath a, K.S. Thind b, L. Gerward c, a
Department of Physics, Gulbarga University, Gulbarga 585 106, Karnataka, India Department of Physics, Guru Nanak Dev University, Amritsar 143 005, Punjab, India c Department of Physics, Technical University of Denmark, DK-2800 Lyngby, Denmark b
a r t i c l e in f o
a b s t r a c t
Keywords: Effective atomic number Photon-interaction cross-sections Biomolecules
The effective atomic number, Zeff, has been calculated for fatty acids and cysteine. It is shown that Zeff is a useful parameter for low-Z materials at any energy above 1 keV. Absorption edges of medium-Z elements may complicate the energy dependence of Zeff below 10 keV. The notion of Zeff is perhaps most useful at energies where Compton scattering is dominating, and where Zeff is equal to the mean atomic number, /ZS, over a wide energy range around 1 MeV. & 2009 Elsevier Ltd. All rights reserved.
1. Introduction For a complex medium, the ‘‘effective atomic number’’ is in some cases a convenient parameter for representing X-ray and gamma ray interactions, e.g. in designs of radiation shielding or in calculations of absorbed dose. However, as originally stated by Hine (1952), the effective atomic number of a multi-element material is not a constant. It is varying with photon energy, depending on the relative importance of the interaction processes involved. Among early applications one may mention bone densitometry and measurements of fat content in liver (Puumalainen et al., 1977). Yang et al. (1987) measured Zeff of soft human tissues. More recently, Kumar and Reddy (1997) determined the effective atomic number for materials of dosimetric interest. Very recently, Manjunathaguru and Umesh (2006, 2007), Manohara and Hanagodimath (2007a, b), and Manohara et al. (2008a) determined the effective atomic number of some biomolecules. Grinyov et al. (2007) developed dual-energy radiography for separate detection of materials differing in their effective atomic number and local density. Similarly, Torikoshi et al. (2007) developed a dual-energy X-ray CT method using synchrotron radiation to obtain highresolution Zeff images for medical diagnosis. Thus, there is a renewed interest in the effective atomic number for applications in biology and medicine. Early calculations of the effective atomic number were based on parameterization of the photon-interaction cross-section by
Corresponding author. Tel.: + 45 4525 3146; fax: + 45 4593 2766.
E-mail address: [email protected] (L. Gerward). 1 Present address: Department of Physics, Bahubali College of Engineering, Shravanabelagola 573 135, Karnataka, India. 0969-8043/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.apradiso.2009.09.047
fitting data over limited ranges of photon energy and atomic number (Jackson and Hawkes, 1981; Bradley et al., 1986). As the present results will show, modern databases of photon-interaction cross-sections and interpolation programs (Berger and Hubbell, 1987/1999; Gerward et al., 2001, 2004) have made it possible to calculate effective atomic numbers with much improved accuracy and information content over wide ranges of energy and composition.
2. Theory Formulas for calculating the effective atomic number, valid for energies above 1 keV and for all types of materials, have been treated elsewhere (Manohara et al., 2008b), an only a brief summary is given here. The total photon-interaction cross-section per molecule, sm, can be written X sm ¼ ni si ð1Þ i
where ni is the number of atoms of the ith constituent element in the molecule (i.e. the number of formula units), and si is the total photon-interaction cross-section of element i. Essentially, the definition of the effective atomic number, Zeff, is based on the assumption that the actual atoms of a given molecule can be replaced by an equal number of identical (average) atoms, each of which having Zeff identical (average) electrons. Thus, the total photon-interaction cross-section per molecule can be written
sm ¼ nsa ¼ nZeff se
ð2Þ
where sa is the effective (average) cross-section per atom, se is the P effective (average) cross-section per electron, and n = ini is the
ARTICLE IN PRESS S.R. Manohara et al. / Applied Radiation and Isotopes 68 (2010) 784–787
total number of atoms in the molecule. It follows that the effective atomic number of a chemical compound is given by P ns Zeff ¼ Pi i si ð3Þ j j nj Zj The effective atomic number, Zeff, is closely related to the effective electron density, Ne,eff, expressed in number of electrons per unit mass. The two parameters are related through nZ Z Ne;eff ¼ NA P eff ¼ NA eff /AS i ni Ai
ð4Þ
where Ai is the atomic weight of the ith element, NA is the P Avogadro constant, and /AS= iniAi/n is the average atomic weight. Similarly, the average electron density, /NeS, can be defined as /Ne S ¼
NA X /ZS n Z ¼ NA M i i i /AS
785
Table 1 Fatty acids studied in the present work.
1 2 3 4 5 6 7 8 9 10 11
Compound
Formula
/ZS
Zeff at 1 MeV
Lauric acid Myristic acid Palmitic acid Stearic acid Arachidic acid Behenic acid Lignoceric acid Cerotic acid Montanic acid Palmitoleic acid Oleic acid
C12H24O2 C14H28O2 C16H32O2 C18H36O2 C20H40O2 C22H44O2 C24H48O2 C26H52O2 C28H56O2 C16H30O2 C18H34O2
2.95 2.91 2.88 2.86 2.84 2.82 2.81 2.80 2.79 2.96 2.93
2.95 2.91 2.88 2.86 2.84 2.82 2.81 2.80 2.79 2.96 2.93
/ZS is the mean atomic number, and Zeff is the effective atomic number calculated at 1 MeV.
ð5Þ
P P where M= iniAi is the molar mass, and /ZS= iniZi/n is the mean atomic number. Total photon-interaction cross-sections can be obtained from existing tabulations (Hubbell and Seltzer, 1995), or calculated as needed using computer programs, such as XCOM (Berger and Hubbell, 1987/1999) or its Windows successor WinXCom (Gerward et al., 2001, 2004). The latter program has the advantage of being able to export the cross-sectional data to a predefined MS Excel template, thereby facilitating subsequent graphical or numerical data treatment. For a given material, the program XMuDat (Nowotny, 1998) calculates a single-valued effective atomic number, which we will call ZX,eff: X ZX;eff ¼ ð ai Zim1 Þ1=ðm1Þ ð6Þ i
where si is the fractional number of electrons of element i, and m is a constant between 3 and 5. It is suggested that m is set to 3.6 for materials with Zeff less than 6, and 4.1 for a material with Zeff higher than 6 (Jackson and Hawkes, 1981).
3. Biomolecules Proteins are major structural components in animal and human tissues, being a key part of skin, nails, cartilage and muscles. Other proteins transport oxygen and perform further important tasks. All proteins are chemically similar, the basic building blocks being amino acids. Hydrolysis of fats or oils, which are media for energy storage, yields glycerol and carboxylic acids, called fatty acids. 3.1. Fatty acids Table 1 lists the fatty acids studied in the present work. Fig. 1 shows the calculated effective atomic number, Zeff, as a function of photon energy, E. All fatty acids behave similarly, since they consist of hydrogen, carbon, and oxygen in about the same proportions. One can distinguish three energy regions, in each of which Zeff for a given compound is about constant. The regions are approximately Eo0.01 MeV, 0.05oE o5 MeV, and E4200 MeV. Between these three energy regions, there are transition regions with a rapid variation of Zeff. The energy behavior of Zeff for total interaction mirrors the relative importance of the partial processes. The dominating process is photoelectric absorption at low energies, incoherent (Compton) scattering at intermediate energies, and pair production
Fig. 1. Calculated effective atomic numbers of fatty acids for total photon interaction. Sample numbers 1–11 are as for Table 1.
at high energies. Coherent (Rayleigh) scattering never plays any significant role in this connection, since it occurs mainly at low energies, where photoelectric absorption is by far the most important interaction process. At intermediate energies, Zeff for a given low- or medium-Z compound is about constant over a wide energy range, typically two decades between 0.05 and 5 MeV (Fig. 1). In this energy region, Compton scattering accounts for practically all photon interactions. The Compton scattering cross-section per atom is proportional to the atomic number. It can then be shown that Zeff is equal to the mean atomic number, /ZS, and Ne,eff is equal to the average electron density, /NeS. As seen in Table 1, Zeff calculated at 1 MeV agrees perfectly with /ZS. In the low- and high-energy regions, Zeff is a weighted mean, where the element with the highest atomic number has the greatest weight. Therefore, the weighted mean is larger than the simple mean. The largest value of Zeff occurs in the low-energy range, typically below 10 keV, where the Z4 dependence of the photoelectric absorption cross-section gives a heavy weight for the highest atomic number. In the high-energy range, typically above 200 MeV, Zeff is again constant but smaller than in the lowenergy range. This is due to the dominance of pair production, the cross-section of which has a weaker Z2 dependence. Hence, pair
ARTICLE IN PRESS 786
S.R. Manohara et al. / Applied Radiation and Isotopes 68 (2010) 784–787
Hubbell (1999) concludes that the envelope of uncertainty of s is of the order of 1–2% in the energy range from 5 keV to a few MeV. Discrepancies up to 25% are known to occur between 1 and 4 keV, but these low energies are generally of little interest in medical and biological applications. Thus, we conclude that Zeff, calculated at energies above 5 keV, is accurate to within a few %.
4. Conclusions
Fig. 2. Calculated effective atomic number of cysteine for total photon interaction as a function of energy. The solid squares are the experimental data points of Manjunathaguru and Umesh (2007). The dotted line indicates the single-valued effective atomic number ZX,eff provided by XMuDat. The horizontal dashed line indicates the mean atomic number /ZS, and the vertical dashed line the energy of the K absorption edge of sulfur (S).
production gives less weight to the higher-Z elements than photoelectric absorption. Our results for fatty acids are in good agreement with experimental data of Manjunathaguru and Umesh (2006, 2007), who have determined Zeff of some biologically important compounds in the energy range 6.4–1330 keV. 3.2. Cysteine For cysteine, an amino acid, the graph of Zeff versus energy (Fig. 2) exhibits about the same energy regions as for the fatty acids. At intermediate energies, here narrowed down to one decade between 0.2 and 2 MeV, where Compton scattering is dominating, Zeff is about 4.6 in good agreement with /ZS=4.57 calculated from the chemical formula, C3H7O2NS. A new feature, where cysteine differs from the fatty acids, is the discontinuous jump of Zeff at 2.472 keV. This is due to the K absorption edge of the medium-Z element sulfur (Z=16). It is obvious that the presence of the absorption edge makes the applicability of the effective atomic number somewhat problematic at low energies. The single-valued effective atomic number provided by XMuDat is only a rough approximation at energies below about 10 keV (Fig. 2). This follows from the fact that Eq. (6) is derived from a parameterization of the photoelectric absorption crosssection. Users of XMuDat should therefore treat the effective atomic number with some caution. Fig. 2 also shows that our calculations are in good agreement with the experimental data of Manjunathaguru and Umesh (2007). It is interesting to notice that these authors have determined Zeff in the transition range between photoelectric absorption and Compton scattering, where Zeff is rapidly varying with energy. 3.3. Accuracy of calculations Eq. (3) shows that the accuracy of the calculated effective number is solely determined by the elemental cross-sections, si.
The effective atomic number is a useful parameter for low- and medium-Z materials, encountered in biological and medical applications. The effective atomic number is constant and equal to the mean atomic number, /ZS, over a wide energy range around 1 MeV, where Compton scattering is the main photoninteraction process. Below 10 keV, absorption edges of medium-Z elements may complicate the energy dependence of Zeff. The single-valued effective atomic number provided by the XMuDat software is a rough approximation below about 10 keV. It should be remembered that the concept of the effective atomic number is based on an underlying theory of X-ray and gamma ray interactions with matter. Thus, Zeff reflects our experimental and theoretical knowledge of mass attenuation coefficients and atomic interaction cross-sections, as given in recent compilations and databases (Gerward, 1993; Hubbell, 1999, 2006). Finally, we would like to mention that the program WinXCom is available free of charge. Interested readers are welcome to send an e-mail request to [email protected].
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