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An interesting application of scaling procedure to x-ray spectrometry
Appl. Radiat.ht. Vol. 44, No. 6, pp. 927-928, 1993 Printed in Great Britain
0883-2889/93 $6.00 + 0.00 Pergamon Press Ltd
An Interesting Application of Scaling Procedure to x-Ray Spectrometry S. A. GERASIMOV Department
of Nuclear
Physics,
Institute
of Physics, Rostov Russia
State University,
Rostov-on-Don,
344104,
(Received 3 June 1992; in revised Jorm 19 November 1992) An analytical approximation for the simple description of the photoelectric cross-section of the photon energy and atomic number is obtained using a scaling procedure.
2. Thomas-Fermi
1. Introduction A knowledge of accurate attenuation coefficients is important when an x-ray method is used for quantitative chemical or structural analysis. For these purposes simple empirical formulae are desirable. The attenuation coefficient is proportional to the total photon interaction cross-section per atom, i.e. to the sum of the cross-sections for all elementary scattering and absorption processes such as Rayleigh scattering, Compton collision and photoelectric absorption. The energy and atomic number dependencies of coherent and incoherent scattering cross-sections can be calculated using the Thomas-Fermi results for the atomic form factor and incoherent scattering function. The analogous simple calculations are absent for photoelectric absorption. The product u I E3 Zm912 varies by less than a factor of three over the range Z = l-92 and E = 10-500 keV (above or between absorption edges) (Hubbell, 1969). The simple approximation mentioned above is not sufficient for practical computations. Victoreen’s relation (Gerward, 1981) does not describe the atomic number dependence of the photoelectric cross-section Q. Of course, In ~7can be expressed as a polynomial m In E and In Z. However, only the cubic polynomial in In Z and In E gives a satisfactory fit to the photoelectric cross-section (Gerward, 1981). It is interesting to use the scaling procedure to describe the atomic number dependence of the photoelectric cross-section. The universal variables of cross-section and energy can be obtained using such an approach. Knowing the universal variables, it is easy to describe the corresponding approximate dependence. It may be the first approximation. A more exact approximation may be achieved by linear regression.
as a function
Statistical
Model
One starting point of this approach and a useful one here, is the scaling property of the ThomasFermi theory for atoms. In this theory the atomic electrons are treated as a degenerate gas obeying Fermi-Dirac statistics and the Pauli principle, with the ground-state energy of the atom equal to the zero-point energy of this gas. The average chargedensity p(r) then becomes the radial function. p(r) = &
[2me V(r)13’*
of the potential V(r) which in turn can be substituted in Piosson’s equation AI/(r) = 4ap(r), to give (Gombas, 1949). d24 (x)/dx * = ,#J3’2(x)/x “’ where V(r) = Ze 4(x)/r, Z is the atomic number, m is the mass of an electron, x = (3x/4)-2’32me2Z”3r/k2 and r is the radial coordinate. This equation is to be solved for the screening function 4(x) with the boundary conditions 4(O) = I and I = 0. It is seen that the value V(r)/Z4” is expressed in terms of a universal variable rZ’j3. In the order to derive the universal variables for a momentum p and an energy E, it is enough to consider the classical conservation law of energy E = p2/2m - eV(r). It follows from this that p/Z213 and E/Z413 are universal variables for momentum and energy. Theoretical calculations of the incoherent scattering function (Parks and Rotenberg, 1972) and the inelastic scattering cross section of charged particles (Gerasimov, 1986) show that the corresponding variables for the momentum of the scattered photon and the energy of the charged particle are the same if the Thomas-Fermi statistical model is used for the determination of the electron density and the targetelectron momentum distribution. In these cases the incoherent and inelastic scattering cross-sections have 921
S. A. GERMIMOV
928
a simple and descriptive meaning. For example, the inelastic scattering cross-section da may be treated as an area (*r2Z2/3) and in the high-energy formulation must be proportional to the number of atomic electrons. This result may be described by the transformations da + Z513du and E -+ E/Z4” (Gerasimov, 1987).
scaling procedure in describing the photoelectric cross-section as a function of the photon energy.
3. Scaling
u/Z”~ = C(E/Z4’3)
According to the previous section the transformation “energy -+ energy Zm4”” can be applied for the energy of a particle. In the atomic photoeffect the energies of the atomic and ejected electrons are related to the photon energy by the conservation law. This means that E/Z413 is a universal variable for the photon energy E. For the photoelectric cross-section the scaling procedure is more difficult. Any known result concerning atomic number dependence of the form factor or incoherent scattering function cannot be applied in this case. However, it should be noted here that for a given E . Zw4j3the total photoelectric cross-section e must be proportional to the number of atomic electrons taking part in scattering. Taking into account that u may be treated as an area one can show that cr must be transformed as a number of electrons multiplied by the square of the radius. The number of atomic electrons, n, must be transformed as n ---t n/Z; the transformation for a radial co-ordinate r is r + rZ’13. Therefore, a/Z iI3 is the universal variable for the photoelectric cross section and E . Zm4” is the variable for the photon energy. Of course, such arguments are questionable. However, on the other hand this leads to a definition of the universal variables. As an illustration Fig. 1 shows the universal energy dependence of the photoelectric cross-section. Experimental (Hubbell et al., 1974; Hubbell, 1977) and theoretical data (Hubbell, 1969) were used to demonstrate such scaled representation but the accuracy of this universal dependence is not sufficient for practical purposes. However, this figure demonstrates the validity of the
10-l
1
E/Z”‘,
10
IO2
keV
Fig. 1. Scaled energy dependence of the photoelectric cross section for carbon, argon, iron, copper and lead. Experimental results for iron and copper from (Hubbell et al., 1974; Hubbell, 1977). Theoretical data for carbon, argon
and lead from (Hubbell, 1969).Solid line is the power law.
4. Results Significant deviations from the scaling, mentioned above, are connected with the fact that the parameters o[ and C in the approximate relation
depend on the atomic number Z. It is clear that the atomic number dependence of CLcannot be sharp. Therefore, one can write In u/Z’13 = (A, + A2Z) + (A, + A,Z) In E/Z4j3 where A,, A,, A, and A, are constants which must be
defined by the least squares method. From this Q = 14.3 E-3.226+00088,5~2~4.635-0.01115-2 with a in barns and E in keV. To obtain this approximate dependence all tabulated data (Hubbell, 1969) in the range Ek < E < 500 keV and with Z 2 6 are used in this work. Here, Ek is the energy of the K-absorption edge. For this relation the worst percentage difference between fitted and tabulated values is 20% and the mean percentage difference is 3%. Note here, that for the representation In u = (B, + B,Z) + (B, + B4Z) In E the worst percentage difference can achieve 103% and more. Formally, this scaling procedure can be used for photon energies E < I$. But, from the physical point of view there are difficulties in an interpretation of such results. This is due to the fact that the transformation of the photon energy is not clear in the photon energy range E < Ek. Thus, this paper first presents an approximate approach to describe photo-absorption by various elements. The results obtained can be improved if a quadratic regression is used. References Gerasimov S. A. (1986) Semiclassical impact-parameter expansion of delta-electron spectrum. Acra Phys. Hung. 60, 279. Gerasimov S. A. (1987) Scaling in the stopping problem of fast electrons by matter. Exp. Tech. Phys. 35, 105. Gerward L. (1981) Analytical approximation for X-ray attenuation coefficients. Nucl. Instrum. Methocis 181, 11. Gombas P. (1949) Die Statistische Theorie des Atoms und ihre Anwendungen. Springer-Verlag. Wien. Hubbell J. H. (1969) Photon cross sections, attenuation coefficients, and energy absorption coefficients from 10 keV to 10 GeV. NBS Publ. NRSDS-NBS. 29, 1. Hubbell J. H. (1977) Present status of photon cross section data 1OOeV to 1OOGeV. NBS Spec. Publ. 461, 3. Hubbell J. H., McMaster W. H., Kerr Del Grande N. and Mallett J. H. (1974) X-ray cross sections and attenuation coefficients. I~tern&onalv Tables for X-ray Crysioilogra phy, Vol. 4, pp. 47-70. Kynoch Press, Birmingham. Parks D. E. and Rotenberg M. (1972) Incoherent X-ray scattering by a statistical atom. Phys. Rev. A. 5, 521.
0883-2889/93 $6.00 + 0.00 Pergamon Press Ltd
An Interesting Application of Scaling Procedure to x-Ray Spectrometry S. A. GERASIMOV Department
of Nuclear
Physics,
Institute
of Physics, Rostov Russia
State University,
Rostov-on-Don,
344104,
(Received 3 June 1992; in revised Jorm 19 November 1992) An analytical approximation for the simple description of the photoelectric cross-section of the photon energy and atomic number is obtained using a scaling procedure.
2. Thomas-Fermi
1. Introduction A knowledge of accurate attenuation coefficients is important when an x-ray method is used for quantitative chemical or structural analysis. For these purposes simple empirical formulae are desirable. The attenuation coefficient is proportional to the total photon interaction cross-section per atom, i.e. to the sum of the cross-sections for all elementary scattering and absorption processes such as Rayleigh scattering, Compton collision and photoelectric absorption. The energy and atomic number dependencies of coherent and incoherent scattering cross-sections can be calculated using the Thomas-Fermi results for the atomic form factor and incoherent scattering function. The analogous simple calculations are absent for photoelectric absorption. The product u I E3 Zm912 varies by less than a factor of three over the range Z = l-92 and E = 10-500 keV (above or between absorption edges) (Hubbell, 1969). The simple approximation mentioned above is not sufficient for practical computations. Victoreen’s relation (Gerward, 1981) does not describe the atomic number dependence of the photoelectric cross-section Q. Of course, In ~7can be expressed as a polynomial m In E and In Z. However, only the cubic polynomial in In Z and In E gives a satisfactory fit to the photoelectric cross-section (Gerward, 1981). It is interesting to use the scaling procedure to describe the atomic number dependence of the photoelectric cross-section. The universal variables of cross-section and energy can be obtained using such an approach. Knowing the universal variables, it is easy to describe the corresponding approximate dependence. It may be the first approximation. A more exact approximation may be achieved by linear regression.
as a function
Statistical
Model
One starting point of this approach and a useful one here, is the scaling property of the ThomasFermi theory for atoms. In this theory the atomic electrons are treated as a degenerate gas obeying Fermi-Dirac statistics and the Pauli principle, with the ground-state energy of the atom equal to the zero-point energy of this gas. The average chargedensity p(r) then becomes the radial function. p(r) = &
[2me V(r)13’*
of the potential V(r) which in turn can be substituted in Piosson’s equation AI/(r) = 4ap(r), to give (Gombas, 1949). d24 (x)/dx * = ,#J3’2(x)/x “’ where V(r) = Ze 4(x)/r, Z is the atomic number, m is the mass of an electron, x = (3x/4)-2’32me2Z”3r/k2 and r is the radial coordinate. This equation is to be solved for the screening function 4(x) with the boundary conditions 4(O) = I and I = 0. It is seen that the value V(r)/Z4” is expressed in terms of a universal variable rZ’j3. In the order to derive the universal variables for a momentum p and an energy E, it is enough to consider the classical conservation law of energy E = p2/2m - eV(r). It follows from this that p/Z213 and E/Z413 are universal variables for momentum and energy. Theoretical calculations of the incoherent scattering function (Parks and Rotenberg, 1972) and the inelastic scattering cross section of charged particles (Gerasimov, 1986) show that the corresponding variables for the momentum of the scattered photon and the energy of the charged particle are the same if the Thomas-Fermi statistical model is used for the determination of the electron density and the targetelectron momentum distribution. In these cases the incoherent and inelastic scattering cross-sections have 921
S. A. GERMIMOV
928
a simple and descriptive meaning. For example, the inelastic scattering cross-section da may be treated as an area (*r2Z2/3) and in the high-energy formulation must be proportional to the number of atomic electrons. This result may be described by the transformations da + Z513du and E -+ E/Z4” (Gerasimov, 1987).
scaling procedure in describing the photoelectric cross-section as a function of the photon energy.
3. Scaling
u/Z”~ = C(E/Z4’3)
According to the previous section the transformation “energy -+ energy Zm4”” can be applied for the energy of a particle. In the atomic photoeffect the energies of the atomic and ejected electrons are related to the photon energy by the conservation law. This means that E/Z413 is a universal variable for the photon energy E. For the photoelectric cross-section the scaling procedure is more difficult. Any known result concerning atomic number dependence of the form factor or incoherent scattering function cannot be applied in this case. However, it should be noted here that for a given E . Zw4j3the total photoelectric cross-section e must be proportional to the number of atomic electrons taking part in scattering. Taking into account that u may be treated as an area one can show that cr must be transformed as a number of electrons multiplied by the square of the radius. The number of atomic electrons, n, must be transformed as n ---t n/Z; the transformation for a radial co-ordinate r is r + rZ’13. Therefore, a/Z iI3 is the universal variable for the photoelectric cross section and E . Zm4” is the variable for the photon energy. Of course, such arguments are questionable. However, on the other hand this leads to a definition of the universal variables. As an illustration Fig. 1 shows the universal energy dependence of the photoelectric cross-section. Experimental (Hubbell et al., 1974; Hubbell, 1977) and theoretical data (Hubbell, 1969) were used to demonstrate such scaled representation but the accuracy of this universal dependence is not sufficient for practical purposes. However, this figure demonstrates the validity of the
10-l
1
E/Z”‘,
10
IO2
keV
Fig. 1. Scaled energy dependence of the photoelectric cross section for carbon, argon, iron, copper and lead. Experimental results for iron and copper from (Hubbell et al., 1974; Hubbell, 1977). Theoretical data for carbon, argon
and lead from (Hubbell, 1969).Solid line is the power law.
4. Results Significant deviations from the scaling, mentioned above, are connected with the fact that the parameters o[ and C in the approximate relation
depend on the atomic number Z. It is clear that the atomic number dependence of CLcannot be sharp. Therefore, one can write In u/Z’13 = (A, + A2Z) + (A, + A,Z) In E/Z4j3 where A,, A,, A, and A, are constants which must be
defined by the least squares method. From this Q = 14.3 E-3.226+00088,5~2~4.635-0.01115-2 with a in barns and E in keV. To obtain this approximate dependence all tabulated data (Hubbell, 1969) in the range Ek < E < 500 keV and with Z 2 6 are used in this work. Here, Ek is the energy of the K-absorption edge. For this relation the worst percentage difference between fitted and tabulated values is 20% and the mean percentage difference is 3%. Note here, that for the representation In u = (B, + B,Z) + (B, + B4Z) In E the worst percentage difference can achieve 103% and more. Formally, this scaling procedure can be used for photon energies E < I$. But, from the physical point of view there are difficulties in an interpretation of such results. This is due to the fact that the transformation of the photon energy is not clear in the photon energy range E < Ek. Thus, this paper first presents an approximate approach to describe photo-absorption by various elements. The results obtained can be improved if a quadratic regression is used. References Gerasimov S. A. (1986) Semiclassical impact-parameter expansion of delta-electron spectrum. Acra Phys. Hung. 60, 279. Gerasimov S. A. (1987) Scaling in the stopping problem of fast electrons by matter. Exp. Tech. Phys. 35, 105. Gerward L. (1981) Analytical approximation for X-ray attenuation coefficients. Nucl. Instrum. Methocis 181, 11. Gombas P. (1949) Die Statistische Theorie des Atoms und ihre Anwendungen. Springer-Verlag. Wien. Hubbell J. H. (1969) Photon cross sections, attenuation coefficients, and energy absorption coefficients from 10 keV to 10 GeV. NBS Publ. NRSDS-NBS. 29, 1. Hubbell J. H. (1977) Present status of photon cross section data 1OOeV to 1OOGeV. NBS Spec. Publ. 461, 3. Hubbell J. H., McMaster W. H., Kerr Del Grande N. and Mallett J. H. (1974) X-ray cross sections and attenuation coefficients. I~tern&onalv Tables for X-ray Crysioilogra phy, Vol. 4, pp. 47-70. Kynoch Press, Birmingham. Parks D. E. and Rotenberg M. (1972) Incoherent X-ray scattering by a statistical atom. Phys. Rev. A. 5, 521.