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. 245-246

Section 4.2.6.2.3.1. Cromer and Liberman: relativistic dipole approach

D. C. Creaghb

4.2.6.2.3.1. Cromer and Liberman: relativistic dipole approach

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It is necessary to consider relativistic effects for atoms having all but the smallest atomic numbers. Cromer & Liberman (1970[link]) produced a set of tables based on a relativistic approach to the scattering of photons by isolated atoms that was later reproduced in IT IV (1974[link]). Subsequent experimental determinations drew attention to inaccuracies in these tables in the neighbourhood of absorption edges owing to the poor convergence of the Gaussian integration technique, which was used to evaluate the real part of the dispersion correction. In a later paper, Cromer & Liberman (1981[link]) recalculated 34 instances for which the incident radiation lay close to the absorption edges of atoms using a modified integration procedure. Care should be exercised when using the Cromer & Liberman computer program, especially for calculations of [f'(\omega,0)] for high atomic weight elements at low photon energies. As Creagh (1990[link]) and Chantler (1994[link]) have shown, incorrect values of [f'(\omega,0)] can be calculated because an insufficient number of values of [f''(\omega,0)] are calculated prior to performing the Kramers–Kronig transform. In a new tabulation, Chantler (1995[link]) presents the Cromer & Liberman data using a finer integrating grid. It should be noted that the relativistic correction is the same as that used in this tabulation.

These relativistic calculations are based on the scattering formula developed by Akhiezer & Berestetsky (1957[link]) for the scattering amplitude for photons by a bound electron, viz: [S_{i\rightarrow f}=-2\pi i\delta(\varepsilon_1+\hbar\omega_1-\varepsilon_2-\hbar\omega_2)\, \displaystyle{ 4\pi(e\hbar c){}^2\over 2mc^2\hbar(\omega_1\omega_2){}{^{1/2}}} f. \eqno (4.2.6.24)]Here the angular frequencies of the incident and scattered photons are [\omega_1] and [\omega_2], respectively, and the initial and final energy states of the atom are [\varepsilon_1] and [\varepsilon_2], respectively. The scattering factor f is a complicated expression that includes the initial and final polarization states of the photon, the Dirac velocity operator, and the phase factors [\exp(i{\bf k}_1\cdot{\bf r})] and [\exp(i{\bf k}_2\cdot{\bf r})] for the incident and scattered waves, respectively. Summation is over all positive and negative intermediate states except those positive energy states occupied by other atomic electrons. The form of this expression is not easily related to the form-factor formalism that is most widely used by crystallographers, and a number of manipulations of the formula for the scattering factor are necessary to relate it more directly to the crystallographic formalism. In doing so, a number of assumptions and simplifications were made. Cromer & Liberman restricted their study to coherent, forward scatter in which changes in photon polarization did not occur. With these approximations, and using the electrical dipole approximation [[\exp(i{\bf k}\cdot{\bf r})=1]], they were able to show that [f(\omega,0)=f(0)+f^+(\omega,0)+{\textstyle{5\over3}}{ E_{\rm tot}\over mc^2}+if''(\omega,0).\eqno (4.2.6.25)]In equation (4.2.6.25)[link], f(0) is the atomic form factor for the case of forward scatter [({\boldDelta}=0)], and the term [[+{5\over3}(E_{\rm tot}/mc^2)]] arises from the application of the dipole approximation to determine the contribution of bound electrons to the scattering process. The term [f''(\omega,0)] is related to the photoelectric scattering cross section expressed as a function of photon energy [\sigma(\hbar\omega)] by [f''(\omega,0)={mc \over 4\pi\hbar e^2}\hbar\omega\,\sigma(\hbar\omega) \eqno (4.2.6.26)]and [f^+(\omega,0)=\bigg({1\over 2\pi^2\hbar r_ec}\bigg)P{\int\limits^\infty_{mc^2}}{(\varepsilon^+-\varepsilon_1)\sigma(\varepsilon+-\varepsilon_1)\over (\hbar\omega)^2-(\varepsilon^+-\varepsilon_1)^2}{\,{\rm d}}\varepsilon^+.\eqno (4.2.6.27)]These equations may be compared with equations (4.2.6.23)[link] and (4.2.6.22)[link], respectively. But equation (4.2.6.25)[link] differs from equation (4.2.6.21)[link] by the term [{5\over3}(E_{\rm tot}/mc^2)], which is constant for each atomic species, and is related to the total Coulomb energy of the atom. Evidently, to keep the formalism the same, one must write[f'(\omega,0)=f^+(\omega,0)+\textstyle{5\over3}\displaystyle{ E_{\rm tot}\over mc^2}.\eqno (4.2.6.28)]In Table 4.2.6.1[link], values of [E_{\rm tot}/mc^2] are set out as a function of atomic number for elements ranging in atomic number from 3 to 98.

Table 4.2.6.1| top | pdf |
Values of Etot/mc2 listed as a function of atomic number Z

ZSymbol[E_{\rm tot}/mc^2]
3Li−0.0004
4Be−0.0006
5B−0.0012
6C−0.0018
7N−0.0030
8O−0.0042
9F−0.0054
10Ne−0.0066
11Na−0.0084
12Mg−0.0110
13Al−0.0125
14Si−0.0158
15P−0.0180
16S−0.0210
17Cl−0.0250
18Ar−0.0285
19K−0.0320
20Ca−0.0362
21Sc−0.0410
22Ti−0.0460
23V−0.0510
24Cr−0.0560
25Mn−0.0616
26Fe−0.0680
27Co−0.0740
28Ni−0.0815
29Cu−0.0878
30Zn−0.0960
31Ga−0.104
32Ge−0.114
33As−0.120
34Se−0.132
35Br−0.141
36Kr−0.150
37Rb−0.159
38Sr−0.171
39Y−0.180
40Zr−0.192
41Nb−0.204
42Mo−0.216
43Tc−0.228
44Ru−0.246
45Rh−0.258
46Pd−0.270
47Ag−0.285
48Cd−0.300
49 In−0.318
50Sn−0.330
51Sb−0.348
52Te−0.363
53I−0.384
54Xe−0.396
55Cs−0.414
56Ba−0.438
57La−0.456
58Ce−0.474
59Pr−0.492
60Nd−0.516
61Pm−0.534
62Sm−0.558
63Eu−0.582
64Gd−0.610
65Tb−0.624
66Dy−0.648
67Ho−0.672
68Er−0.696
69Tm−0.723
70Yb−0.750
71Lu−0.780
72Hf−0.804
73Ta−0.834
74W−0.864
75Re−0.900
76Os−0.919
77Ir−0.948
78Pt−0.984
79Au−1.014
80Hg−1.046
81Tl−1.080
82Pb−1.116
83Bi−1.149
84Po−1.189
85At−1.224
86Rn−1.260
87Fr−1.296
88Ra−1.332
89Ac−1.374
90Th−1.416
91Pa−1.458
92U−1.470
93Np−1.536
94Pu−1.584
95Am−1.626
96Cm−1.669
97Bk−1.716
98Cf−1.764

To develop their tables, Cromer & Liberman (1970[link]) used the Brysk & Zerby (1968[link]) computer code for the calculation of photoelectric cross sections, which was based on Dirac–Slater relativistic wavefunctions (Liberman, Waber & Cromer, 1965[link]). They employed a value for the exchange potential of 0.667[\rho({\bf r})^{1/3}] and experimental rather than computed values of the energy eigenvalues for the atoms.

The wide use of their tables by crystallographers inevitably meant that criticism of the accuracy of the tables was forthcoming on both theoretical and experimental grounds. Stibius-Jensen (1979[link]) drew attention to the fact that the use of the dipole approximation too early in the argument caused an error of [-{1\over2}Z(\hbar\omega/mc^2)^2] in the tabulated values. More recently, Cromer & Liberman (1981[link]) include this term in their calculations. Some experimental deficiencies of the tabulated values of [f'(\omega,0)] have been discussed by Cusatis & Hart (1977[link]), Hart & Siddons (1981[link]), Creagh (1980[link], 1984[link], 1985[link], 1986[link]), Deutsch & Hart (1982[link]), Dreier, Rabe, Malzfeldt & Niemann (1984[link]), Bonse & Hartmann-Lotsch (1984[link]), and Bonse & Henning (1986[link]).

In the latter two cases, the Kramers–Kronig transformation of photoelectric scattering results has been performed without taking into account the term that arises in the relativistic case for the total Coulomb energy of the atom. Although good agreement with the Cromer & Liberman tables is claimed, their failure to include this term is an implied criticism of the Cromer & Liberman tables. That this is unjustified can be seen by references to Fig. 4.2.6.2[link][link] taken from Bonse & Henning (1986[link]), which shows that their interferometer results [which measure [f'(\omega,0)] directly] and the Kramers–Kronig results differ from one another by [\sim E_{\rm tot}/mc^2] in the neighbourhood of the K-absorption edge of niobium in the compound lithium niobate.

[Figure 4.2.6.1]

Figure 4.2.6.1| top | pdf |

The relativistic correction in electrons per atom for: (a) the modified form-factor approach; (b) the relativistic multipole approach; (c) the relativistic dipole approach.

[Figure 4.2.6.2]

Figure 4.2.6.2| top | pdf |

Measured values of f′(ω, 0) at the K-edge of Nb in LiNbO3 and the Kramers–Kronig transformation of f′′(ω, 0). The curve is obtained by transformation and the points are measured by interferometry. For (a), the polarization of the incident radiation is parallel to the hexagonal c axis, and for (b) it is at right angles to the hexagonal c axis. After Bonse & Henning (1986[link]). Note that the distortion of the dispersion curve is due to X-ray absorption near-edge structure (XANES) effects (Section 4.2.4[link]).

Further theoretical objections have been made by Creagh (1984[link]) and Smith (1987[link]), who has shown that the Stibius-Jensen correction is not valid, and that, when higher-order multipolar expansions and retardation are considered, the total self-energy correction becomes [E_{\rm tot}/mc^2] rather than [{5\over3}E_{\rm tot}/mc^2]. Fig. 4.2.6.1[link] shows the variation of the self-energy correction with atomic number for the modified form factor (Creagh, 1984[link]; Smith, 1987[link]; Cromer & Liberman, 1970[link]).

For the imaginary part of the dispersion correction [f''(\omega,0)], which depends on the calculation of the photoelectric scattering cross section, better agreement is found between theoretical results and experimental data. Details of this comparison have been given elsewhere (Section 4.2.4[link]). Suffice it to say that Creagh & Hubbell (1990[link]), in reporting the results of the IUCr X-ray Attenuation Project, could find no rational basis for preferring the Scofield (1973[link]) Hartree–Fock calculations to the Cromer & Liberman (1970[link], 1981[link]) and Storm & Israel (1970[link]) Dirac–Hartree–Fock–Slater calculations.

Computer programs based on the Cromer & Liberman program (Cromer & Liberman, 1983[link]) are in use at all the major synchrotron-radiation laboratories. Many other laboratories have also acquired copies of their program. This program must be modified to remove the incorrect Stibius–Jensen correction term, and, as will be seen later, the energy term should be modified to be [E_{\rm tot}/mc^2].

References

First citation Akhiezer, A. I. & Berestetsky, V. B. (1957). Quantum electrodynamics. TESE, Oak Ridge, Tennessee, USA.Google Scholar
First citation Bonse, U. & Hartmann-Lotsch, I. (1984). Kramers–Kronig correlation of measured f′(E) and f ′′(E) values. Nucl. Instrum. Methods, 222, 185–188.Google Scholar
First citation Bonse, U. & Henning, A. (1986). Measurement of polarization isotropy of the anomalous forward scattering amplitude at the niobium K-edge in LiNbO3. Nucl. Instrum. Methods, A246, 814–816.Google Scholar
First citation Brysk, H. & Zerby, C. D. (1968). Photoelectric cross sections in the keV range. Phys. Rev. 171, 292–298.Google Scholar
First citation Chantler, C. T. (1994). Towards improved form factor tables. Resonant anomalous X-ray scattering, edited by G. Materlik, C. J. Sparks & K. Fischer, pp. 61–79. Amsterdam: North Holland.Google Scholar
First citation Chantler, C. T. (1995). Theoretical form factor, attenuation and scattering tabulation for Z = 1–92 from 1–10 eV to E = 0.4–1.0 MeV. J. Phys. Chem. Ref. Data, 24, 71–643.Google Scholar
First citation Creagh, D. C. (1980). X-ray interferometer measurements of the anomalous dispersion correction f′(ω, 0) for some low Z elements. Phys. Lett. A, 77, 129–132.Google Scholar
First citation Creagh, D. C. (1984). The real part of the forward scattering factor for aluminium. Unpublished.Google Scholar
First citation Creagh, D. C. (1985). Theoretical and experimental techniques for the determination of X-ray anomalous dispersion corrections. Aust. J. Phys. 38, 371–404.Google Scholar
First citation Creagh, D. C. (1986). The X-ray anomalous dispersion of materials. In Recent advances in X-ray characterization of materials, edited by P. Krishna, Chap. 7. Oxford: Pergamon Press.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. (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. & 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. & Liberman, D. A. (1981). Anomalous dispersion calculations near to and on the long-wavelength side of an absorption edge. Acta Cryst. A37, 267–268.Google Scholar
First citation Cromer, D. T. & Liberman, D. A. (1983). Calculation of anomalous scattering factors at arbitrary wavelengths. J. Appl. Cryst. 16, 437.Google Scholar
First citation Cusatis, C. & Hart, M. (1977). The anomalous dispersion corrections for zirconium. Proc. R. Soc. London Ser. A, 354, 291–302.Google Scholar
First citation Deutsch, M. & Hart, M. (1982). Wavelength energy shape and structure of the Cu Kα1 X-ray emission line. Phys. Rev. B, 26, 5550–5567.Google Scholar
First citation Dreier, P., Rabe, P., Malzfeldt, W. & Niemann, W. (1984). Anomalous X-ray scattering factors calculated from experimental absorption factor. J. Phys. C, 17, 3123–3136.Google Scholar
First citation Hart, M. & Siddons, D. P. (1981). Measurements of anomalous dispersion corrections made from X-ray interferometers. Proc. R. Soc. London Ser. A, 376, 465–482.Google Scholar
First citation International Tables for X-ray Crystallography (1974). Vol. IV. Birmingham: Kynoch Press.Google Scholar
First citation Liberman, D., Waber, J. T. & Cromer, D. T. (1965). Self-consistent field Dirac–Slater wavefunctions for atoms and ions. I. Comparison with previous calculations. Phys. Rev. A, 137, 27–34.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 Smith, D. Y. (1987). Anomalous X-ray scattering: relativistic effects in X-ray dispersion analysis. Phys. Rev. A, 35, 3381–3387.Google Scholar
First citation Stibius-Jensen, M. (1979). Some remarks on the anomalous scattering factors for X-rays. Phys. Lett. A, 74, 41–44.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








































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