International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C. ch. 4.2, pp. 221-229
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Of the many theoretical data sets in existence, those of Storm & Israel (1970), Cromer & Liberman (1970
), and Scofield (1973
) 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
). Some indication of the extent to which agreement exists between the different theoretical data sets is given in §4.2.6.2.4
(Tables 4.2.6.3
(b) and 4.2.6.5
). These tables show that the values of
, 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
, 1990
) 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. The cross section in barns/atom is related to
expressed in electrons/atom by σ = 5636λ
where λ is expressed in ångströms.
The values for 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
).
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 where
Reliable tables of f(q, Z) exist and have been reviewed recently by Kane, Kissel, Pratt & Roy (1986). The most recent schematic tabulations of f(q, Z) are those of Hubbell & Øverbø (1979
) and Schaupp et al. (1983
). 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
) based on the exact formula of Pirenne (1946
) for H, and relativistic calculations by Doyle & Turner (1968
), Cromer & Waber (1974
), Øverbø (1977
, 1978
), and high-q extensions using the Bethe–Levinger expression in Levinger (1952
).
As mentioned in Creagh & Hubbell (1987), 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:
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That is, is replaced by
.
These effects are discussed elsewhere (Subsection 4.2.3.2). Briefly,
In equation (4.2.4.7)
, which is due to De Marco & Suortti (1971
),
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It is assumed that the total thermal diffuse scattering is equal to the scattering lost from Laue–Bragg scattering because of thermal vibrations. This equation is not in a convenient form for computation and the alternative formalism presented by Sano, Ohtaka & Ohtsuki (1969
) is often used in calculations. In this formalism,
The values of f(q, Z) are those of Cromer & Waber (1974).
Cross sections calculated using equation (4.2.4.8) 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
becomes a smoothly varying function of energy.
Creagh & Hubbell (1987) and Creagh (1987a
) 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.
The bound-electron Compton scattering cross section is given by Here
and
is the incoherent scattering intensity expressed in electron units. The other symbols have the meanings defined in §§4.2.4.2.1
and 4.2.4.2.2
.
Values of 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
) based on the calculations by Cromer & Mann (1967
) and Cromer (1969
).
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