International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C. ch. 4.2, pp. 221-229

Section 4.2.4.2. Sources of information

D. C. Creaghb and J. H. Hubbelld

4.2.4.2. Sources of information

| top | pdf |

4.2.4.2.1. Theoretical photo-effect data: σpe

| top | pdf |

Of the many theoretical data sets in existence, those of Storm & Israel (1970[link]), Cromer & Liberman (1970[link]), and Scofield (1973[link]) have often been used as bench marks against which both experimental and theoretical data have been compared. In particular, theoretical data produced using the S-matrix approach have been compared with these values. See, for example, Kissel, Roy & Pratt (1980[link]). Some indication of the extent to which agreement exists between the different theoretical data sets is given in §4.2.6.2.4[link] (Tables 4.2.6.3[link](b) and 4.2.6.5[link]). These tables show that the values of [f'(\omega,0)], which is proportional to σ, calculated using modern relativistic quantum mechanics, agree to better than 1%. It has also been demonstrated by Creagh & Hubbell (1987[link], 1990[link]) in their analysis of the results of the IUCr X-ray Attenuation Project that there appears to be no rational basis for preferring one of these data sets over the other.

These tables do not list separately photo-effect cross sections. However, should these be required, the data can be found using Table 4.2.6.8[link]. The cross section in barns/atom is related to [f'(\omega,0)] expressed in electrons/atom by σ = 5636λ[\,f'(\omega, 0),] where λ is expressed in ångströms.

The values for [\sigma_{\rm pe}] used in this compilation are derived from recent tabulations based on relativistic Hartree–Fock–Dirac–Slater calculations by Creagh. The extent to which this data set differs from other theoretical and experimental data sets has been discussed by Creagh (1990[link]).

4.2.4.2.2. Theoretical Rayleigh scattering data: σR

| top | pdf |

If each of the atoms gives rise to scattering in which momentum but not energy changes occur, and if each of the atoms can be considered to scatter as if it were an isolated atom, the cross section may be written as [\sigma _R = \pi r ^2_e \textstyle\int\limits^1_{-1} (1 + \cos ^2 \varphi) \,f^2(q,Z) \,{\rm d} (\cos \varphi), \eqno (4.2.4.6)]where

  • [r_e] is the classical radius of the electron;

  • [\varphi] is the angle of scattering ([=2 \theta] if [\theta] is the Bragg angle);

  • [2 \pi \,{\rm d} (\cos \varphi)] is the solid angle between cones with angles [\varphi] and [\varphi + {\rm d} \varphi];

  • [f(q,Z)] is the atomic scattering factor as defined by Cromer & Waber (1974[link]);

  • q is [ [\sin (\varphi /2) / \lambda],] the momentum transfer parameter. Here λ is expressed in ångströms.

Reliable tables of f(q, Z) exist and have been reviewed recently by Kane, Kissel, Pratt & Roy (1986[link]). The most recent schematic tabulations of f(q, Z) are those of Hubbell & Øverbø (1979[link]) and Schaupp et al. (1983[link]). The data used in these tables have been derived from the tabulation for q = 0.02 to 109 Å−1, for all Z's from 1 to 100 by Hubbell & Øverbø (1979[link]) based on the exact formula of Pirenne (1946[link]) for H, and relativistic calculations by Doyle & Turner (1968[link]), Cromer & Waber (1974[link]), Øverbø (1977[link], 1978[link]), and high-q extensions using the Bethe–Levinger expression in Levinger (1952[link]).

As mentioned in Creagh & Hubbell (1987[link]), the atoms in highly ordered single crystals do not scatter as though they are isolated atoms. Rather, cooperative effects become important. In this case, the Rayleigh scattering cross section must be replaced by two cross sections:

  • the Laue–Bragg cross section [\sigma _{\rm LB}],

  • and the thermal diffuse scattering cross section [\sigma _{\rm TD}].

That is, [\sigma _R] is replaced by [\sigma _{\rm LB} + \sigma _{\rm TD}].

These effects are discussed elsewhere (Subsection 4.2.3.2[link]). Briefly, [\sigma _{\rm LB} = (r^2_e \lambda ^2 / 2 NV_c) \textstyle\sum\limits _H [C_p md | F | {}^2 \exp (-2M)]_H. \eqno (4.2.4.7)]In equation (4.2.4.7)[link], which is due to De Marco & Suortti (1971[link]),

  • [C_p= \textstyle {1 \over 2}(1+ \cos ^2 \varphi)];

  • [d_H] is the spacing of the (hkl) planes in the crystal;

  • [m_H] is the multiplicity of the hkl Bragg reflection;

  • [F_H] is the geometrical structure factor for the crystal structure that contains N atoms in a cell of volume [V_c];

  • [\exp (-2M)_H] is the Debye–Waller temperature factor.

It is assumed that the total thermal diffuse scattering is equal to the scattering lost from Laue–Bragg scattering because of thermal vibrations. [\sigma _{\rm TD} = (r^2_e \lambda ^2/2NV_c) \textstyle\sum\limits _H \{ C_p md | F | {}^2 [1- \exp (-2 M)] \} _H. \eqno (4.2.4.8)]This equation is not in a convenient form for computation and the alternative formalism presented by Sano, Ohtaka & Ohtsuki (1969[link]) is often used in calculations. In this formalism, [\sigma _{\rm TD}= 2 \pi r ^2 _e \textstyle\int \limits^1 _{-1} C _p\,f^2 (q,Z) \{ 1 - \exp [-2M (q)] \} \,{\rm d} (\cos \varphi). \eqno (4.2.4.9)]

The values of f(q, Z) are those of Cromer & Waber (1974[link]).

Cross sections calculated using equation (4.2.4.8)[link] tend to oscillate at low energy and this corresponds to the inclusion of Bragg peaks in the summation or integration. Eventually, these oscillations abate and [\sigma _{\rm TD}] becomes a smoothly varying function of energy.

Creagh & Hubbell (1987[link]) and Creagh (1987a[link]) have stressed that, before cross sections are calculated for a given ensemble of atoms, care should be taken to ascertain whether single-atom or single-crystal scattering is appropriate for that ensemble.

4.2.4.2.3. Theoretical Compton scattering data: σC

| top | pdf |

The bound-electron Compton scattering cross section is given by [\eqalignno{ \sigma _C = {}&\pi r \,^2_e \textstyle\int\limits^1_{-1} [1 + k (1 - \cos \varphi)] ^{-2}\cr &\times \{ + \cos ^2 \varphi + k ^2 (1 - \cos \varphi) ^2\cr &\times [1 + k (1 - \cos \varphi)] ^{-1} \} I (q,z)\, {\rm d} (\cos \varphi). &(4.2.4.10)}]Here [k = \hbar \omega /mc^2] and [I (q, z)] is the incoherent scattering intensity expressed in electron units. The other symbols have the meanings defined in §§4.2.4.2.1[link] and 4.2.4.2.2[link].

Values of [\sigma _C] incorporated into the tables of total cross section σ have been computed using the incoherent scattering intensities from the tabulation by Hubbell et al. (1975[link]) based on the calculations by Cromer & Mann (1967[link]) and Cromer (1969[link]).

References

First citation Creagh, D. C. (1987a). The resolution of discrepancies in tables of photon attenuation coefficients. Nucl. Instrum. Methods, A255, 1–16.Google Scholar
First citation Creagh, D. C. (1990). Tables of X-ray absorption corrections and dispersion corrections: the new versus the old. Nucl. Instrum. Methods, A295, 417–434.Google Scholar
First citation Creagh, D. C. & Hubbell, J. H. (1987). Problems associated with the measurement of X-ray attenuation coefficients. I. Silicon. Acta Cryst. A43, 102–112.Google Scholar
First citation Creagh, D. C. & Hubbell, J. H. (1990). Problems associated with the measurement of X-ray attenuation coefficients. II. Carbon. Acta Cryst. A46, 402–408.Google Scholar
First citation Cromer, D. T. (1969). Anomalous dispersion corrections computed from self consistent field relativistic Dirac–Slater wavefunctions. J. Chem. Phys. 50, 4857–4859.Google Scholar
First citation Cromer, D. T. & Liberman, D. (1970). Relativistic calculation of anomalous scattering factors for X-rays. J. Chem. Phys. 53, 1891–1898.Google Scholar
First citation Cromer, D. T. & Mann, J. B. (1967). Compton scattering factors for spherically symmetric free atoms. J. Chem. Phys. 47, 1892–1893.Google Scholar
First citation Cromer, D. T. & Waber, J. T. (1974). Atomic scattering factors for X-rays. International tables for X-ray crystallography, Vol. IV, edited by J. A. Ibers & W. C. Hamilton, Chap. 2.2, pp. 71–147. Birmingham: Kynoch Press.Google Scholar
First citation De Marco, J. J. & Suortti, P. (1971). Effect of scattering on the attenuation of X-rays. Phys. Rev. B, 4, 1028–1033.Google Scholar
First citation Doyle, P. A. & Turner, P. S. (1968). Relativistic Hartree–Fock X-ray and electron scattering factors. Acta Cryst. A24, 390–397.Google Scholar
First citation Hubbell, J. H. & Øverbø, I. (1979). Relativistic atomic form factors and photon coherent scattering cross sections. J. Phys. Chem. Ref. Data, 8, 69–105.Google Scholar
First citation Hubbell, J. H., Veigele, W. J., Briggs, E. A., Brown, R. T., Cromer, D. T. & Howerton, R. J. (1975). Atomic form factors, incoherent scattering functions and photon scattering cross sections. J. Phys. Chem. Ref. Data, 4, 471–538.Google Scholar
First citation Kane, P. P., Kissel, L., Pratt, R. H. & Roy, S. C. (1986). Elastic scattering of γ-rays and X-rays by atoms. Phys. Rep. 150, 75–159.Google Scholar
First citation Kissel, L., Pratt, R. H. & Roy, S. C. (1980). Rayleigh scattering by neutral atoms, 100 eV to 10 MeV. Phys. Rev. A, 22, 1970–2004.Google Scholar
First citation Levinger, J. S. (1952). Small angle coherent scattering of gammas by bound electrons. Phys. Rev. 87, 656–662.Google Scholar
First citation Øverbø, I. (1977). The Coulomb correction to electron pair production by intermediate-energy photons. Phys. Lett. B, 71, 412–414.Google Scholar
First citation Øverbø, I. (1978). Large-q form factors for light atoms. Phys. Scr. 17, 547–549.Google Scholar
First citation Pirenne, M. H. (1946). The diffraction of X-rays and electrons by free molecules. Cambridge University Press.Google Scholar
First citation Sano, H., Ohtaka, K. & Ohtsuki, Y. H. (1969). Normal and abnormal absorption coefficients of X-rays. J. Phys. Soc. Jpn, 27, 1254–1261.Google Scholar
First citation Schaupp, D., Schumacher, M., Smend, F., Rullhusen, P. & Hubbell, J. H. (1983). Small-angle Rayleigh scattering of photons at high energies: tabulations of relativistic HFS modified atomic form factors. J. Phys. Chem. Ref. Data, 12, 467–512.Google Scholar
First citation Scofield, J. H. (1973). Theoretical photoionization cross sections from 1 to 1500 keV. Report UCRL-51326. Lawrence Livermore National Laboratory, Livermore, CA, USA.Google Scholar
First citation Storm, E. & Israel, H. I. (1970). Photon cross sections from 0.001 to 100 MeV for elements 1 through 100. Nucl. Data Tables, A7, 565–681.Google Scholar








































to end of page
to top of page