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. 243-248

Section 4.2.6.2. Theoretical approaches for the calculation of the dispersion corrections

D. C. Creaghb

4.2.6.2. Theoretical approaches for the calculation of the dispersion corrections

| top | pdf |

All the theories that will be discussed here have the following assumptions in common: the elastic scattering is from an isolated neutral atom and that atom is spherically symmetrical. All but the most recent of the theoretical approaches neglect changes in polarization of the incident photon caused by the interaction of the photon with the atom. In the event, few experimental configurations are able to detect such changes in polarization, and the only observable for most experiments is the momentum change of the photon.

4.2.6.2.1. The classical approach

| top | pdf |

In the classical approach, electrons are thought of as occupying energy levels within the atom characterized by an angular frequency [\omega_n] and a damping factor [\kappa_n]. The forced vibration of an electron gives rise to a dipolar radiation field, when the atomic scattering factor can be shown to be [f={\omega^2\over\omega^2-\omega^2_n-i\kappa_n\omega}. \eqno (4.2.6.8)]If the probability that the electron is to be found in the nth orbit is [g_n], the real part of the atomic scattering factor may be written as [{\rm Re}(\,f)=\sum_ng_n+\sum_n\displaystyle{g_n\omega^2_n\over \omega^2-\omega^2_n}.\eqno (4.2.6.9)]The probability [ g_n] is referred to as the oscillator strength corresponding to the virtual oscillator having natural frequency [\omega_n]. Equation (4.2.6.9) may be written as[{\rm Re}(f) = f_0 + f^\prime, \eqno (4.2.6.10)]where [f_0] represents the sum of all the elements of the set of oscillator strengths and is unity for a single-electron atom. The second term may be written as [f'={\int\limits^\infty_{\omega_{\kappa_i}}}{\omega'^2({\rm d}g_\kappa/{\rm d}\omega')\over\omega^2-\omega'^2}\,{\rm d}\omega' \eqno (4.2.6.11)]if the atom is assumed to have an infinite number of energy states. For an atom containing κ electrons, it is assumed that the overall value of f′ is the coherent sum of the contribution of each individual electron, whence [f'={\sum_\kappa\int\limits^\infty_{\omega_\kappa}}{\omega^{\prime 2}({\rm d}g_\kappa/{\rm d}\omega')\over\omega^2-\omega^{\prime 2}}\,{\rm d}\omega' \eqno (4.2.6.12)]and the oscillator strength of the κth electron [g_\kappa={\int\limits^\infty_{\omega_\kappa} }\bigg[{{\rm d}g\over{\rm d}\omega}\bigg]_\kappa\,{\rm d}\omega]is not unity, but the total oscillator strength for the atom must be equal to the total number of electrons in the atom.

The imaginary part of the dispersion correction f′′ is associated with the damping of the incident wave by the bound electrons. It is therefore functionally related to the linear absorption coefficient, [\mu_l], which can be determined from experimental measurement of the decrease in intensity of the photon beam as it passes through a medium containing the atoms under investigation. It can be shown that the attenuation coefficient per atom [\mu_a] is related to the density of the oscillator states by [\mu_a={2\pi^2e^2\over\varepsilon_0mc}\bigg[{{\rm d}g\over{\rm d}\omega}\bigg], \eqno (4.2.6.13)]whence [f''={\pi\over2}\omega\bigg[{{\rm d}g_\kappa\over{\rm d}\omega}\bigg]. \eqno (4.2.6.14)]

An expression linking the real and imaginary parts of the dispersion corrections can now be written: [f'={2\over\pi}\sum_\kappa P\int\limits^\infty_{\omega_\kappa}\displaystyle{\omega'f''(\omega',0)\over\omega^2-\omega'^2}\,{\rm d}\omega'.\eqno (4.2.6.15)]This is referred to as the Kramers–Kronig transform. Note that the term involving the restoring force has been omitted from this equation.

Equations (4.2.6.12)[link], (4.2.6.14)[link], and (4.2.6.15)[link] are the fundamental equations of the classical theory of photon scattering, and it is to these equations that the predictions of other theories are compared.

4.2.6.2.2. Non-relativistic theories

| top | pdf |

The matrix element for Rayleigh scattering from an atom having a radially symmetric charge distribution may be written as [M=M_1({\boldvarepsilon}_i\cdot{\boldvarepsilon}^*_f)+M_2({\boldvarepsilon}_i\cdot{\boldkappa }_f)({\boldvarepsilon}^*_f\cdot{\boldkappa}_i),\eqno (4.2.6.16)]where [{\boldvarepsilon}_i] and [{\boldvarepsilon}_f] represent the initial and final states of the photon. The matrix element [M_1] represents scattering for polarizations [{\boldvarepsilon}_i] and [{\boldvarepsilon}_f] perpendicular to the plane of scattering and [M_2] represents scattering for polarization states lying in the plane of scattering.

Averaged over polarization states, the differential scattering cross section takes the form [{{\rm d}\sigma\over {\rm d}\Omega}={ r^2_e\over2}(|M_1|{}^2+|M_2|{}^2).\eqno (4.2.6.17)]Here σ is the photoelectric scattering cross section, which is related to the mass attenuation coefficient [\mu_m] by [\sigma=(M/N_A)\mu_m\times10^{-24},\eqno (4.2.6.18)]where M is the molecular weight and [N_A] is Avogadro's number.

Using the vector potential of the wavefield A, an expression for the perturbed Hamiltonian of a hydrogen-like atom coupled to the radiation field may be written as [\hat{\scr H}=\hat{\scr H}_0 - r_ec{\bf A}\cdot{\bf P}+{r^2_e\over 2}{\bf A}^2,\eqno (4.2.6.19)]where [\hat{\scr H}_0] is the Hamiltonian for the unperturbed atom and [{\bf P}=i\hbar\nabla].

After application of the second-order perturbation theory, the matrix element may be deduced to be [M=({\boldvarepsilon}_i\cdot{\boldvarepsilon}_f^*)f_0(\Delta)+{1\over m}\langle1|T_1|1\rangle+{1\over m}\langle1|T_2|1\rangle. \eqno (4.2.6.20)]In this equation, the initial and final wavefunctions are designated as [\langle1|] and [|1\rangle], respectively, and the terms [T_1] and [T_2] are given by [T_1={\boldvarepsilon}_f\cdot{\bf P}\exp(-i{\bf k}_f\cdot {\bf r}){1\over E_1-{\scr H}_0+\hbar\omega+i\xi}{\boldvarepsilon}_i\cdot{\bf P}\exp(-i{\bf k}_i\cdot{\bf r})]and [T_2={\boldvarepsilon}_i\cdot{\bf P}\exp(-i{\bf k}_i\cdot {\bf r}){1\over E_1-{\scr H}_0+\hbar\omega+i\xi}{\boldvarepsilon}_f\cdot{\bf P}\exp(-i{\bf k}_f\cdot{\bf r})]where ξ is an infinitesimal positive quantity.

The first term of equation (4.2.6.20)[link] corresponds to the atomic scattering factor and is identical to the value given by classical theory. The terms involving [T_1] and [T_2] correspond to the dispersion corrections. Equation (4.2.6.20)[link] contains no terms to account for radiation damping. More complete theories take the effect of the finite width of the radiating level into account.

It is necessary to realize that the atomic scattering factor depends on both the photon's frequency [\omega] and the momentum vector [{\boldDelta}]. To emphasize this dependence, equation (4.2.6.7)[link] is rewritten as [f(\omega,{\boldDelta})=f_0({\boldDelta})+f'(\omega,{\boldDelta})+if''(\omega,{\boldDelta}). \eqno (4.2.6.21)]In the dipole approximation, it can be shown that [f'(\omega,0)={\textstyle2\over\pi}P\int\limits^\infty_0\displaystyle{\omega'f''(\omega',0)\over \omega^2-\omega'^2}\,{\rm d}\omega', \eqno (4.2.6.22)]which may be compared with equation (4.2.6.15)[link] and [f''(\omega,0)={\omega\over4\pi r_ec}\sigma(\omega), \eqno (4.2.6.23)]which may be compared with equation (4.2.6.14)[link].

There is a direct correspondence between the predictions of the classical theory and the theory using second-order perturbation theory and non-relativistic quantum mechanics.

The extension of Hönl's (1933a[link],b[link]) study of the scattering of X-rays by the K shell of atoms to other electron shells has been presented by Wagenfeld (1975[link]).

In these calculations, the energy of the photon was assumed to be such that relativistic effects do not occur, nor do transitions within the discrete states of the atom occur. Transitions to continuum states do occur, and, using the analytical expressions for the wavefunctions of the hydrogen-like atom, analytical expressions may be developed for the photoelectric scattering cross sections. By expansion of the retardation factor [\exp(-i{\bf k}\cdot{\bf r})] as the power series [1-i{\bf k}\cdot{\bf r}-{1\over2}({\bf k}\cdot{\bf r}){}^2+\ldots], it is possible to determine dipolar, quadrupolar, and higher-order terms in the analytical expression for the photoelectric scattering cross section.

The values of the cross section so obtained were used to calculate the values of [f'(\omega,{\boldDelta})] using the Kramers–Kronig transform [equation (4.2.6.22)[link]] and [f''(\omega,{\boldDelta})] using equation (4.2.6.23)[link]. The work of Wagenfeld (1975[link]) predicts that the values of [f'(\omega,{\boldDelta})] and [f''(\omega,{\boldDelta})] are functions of [{\boldDelta}]. Whether or not this is a correct prediction will be discussed in Subsection 4.2.6.3[link].

Wang & Pratt (1983[link]) have drawn attention to the importance of bound–bound transitions in the dispersion relation for the calculation of forward-scattering amplitudes. Their inclusion is especially important for elements with small atomic numbers. In a later paper, Wang (1986[link]) has shown that, for silicon at the wavelengths of Mo Kα and Ag Kα1, values for [f'(\omega,0)] of 0.084 and 0.055, respectively, are obtained. These values should be compared with those listed in Table 4.2.6.4[link].

4.2.6.2.3. Relativistic theories

| top | pdf |

4.2.6.2.3.1. Cromer and Liberman: relativistic dipole approach

| top | pdf |

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].

4.2.6.2.3.2. The scattering matrix formalism

| top | pdf |

Kissel, Pratt & Roy (1980[link]) have developed a computer program based on the second-order S-matrix formalism suggested by Brown, Peierls & Woodward (1955[link]). Their aim was to provide a prescription for the accurate (∼1%) prediction of the total-atom Rayleigh scattering amplitudes.

Their model treats the elastic scattering as the sum of bound electron, nuclear, and Delbrück scattering cross sections, and treats the Rayleigh scattering by considering second-order, single-electron transitions from electrons bound in a relativistic, self-consistent, central potential. This potential was a Dirac–Hartree–Fock–Slater potential, and exchange was included by use of the Kohn & Sham (1965[link]) exchange model. They omitted radiative corrections.

In principle, the observables in an elastic scattering process are momentum [(\hbar{\bf k})] and polarization [{\boldvarepsilon}]. The complex polarization vectors [{\boldvarepsilon}] satisfy the conditions [{\boldvarepsilon}^*\cdot{\boldvarepsilon}=1'\semi\quad{\boldvarepsilon}\cdot{\bf k}=0. \eqno (4.2.6.29)]

In quantum mechanics, elastic scattering is described in terms of a differential scattering amplitude, M, which is related to the elastic cross section by equation (4.2.6.16)[link].

If polarization is not an observable, then the expression for the differential scattering cross section takes the form of equation (4.2.6.17)[link]. If polarization is taken into account, as may be the case when a polarizer is used on a beam scattered from a sample irradiated by the linearly polarized beam from a synchrotron-radiation source, the full equation, and not equation (4.2.6.17)[link], must be used to compute the differential scattering cross section.

The principle of causality implies that the forward-scattering amplitude [M(\omega,0)] should be analytic in the upper half of the [\omega] plane, and that the dispersion relation [{\rm Re}\ M(\omega,0)={2\omega^2\over\pi}{\int\limits^\infty_0}{{\rm Im}\ M(\omega',0)\over \omega'(\omega'^2-\omega^2)}{\rm d}\omega'\eqno (4.2.6.30)]should hold, with the consequence that [{\rm Re}\ M(\infty,0)=-{2\over\pi}{\int\limits^\infty_0}{{\rm Im}\ M(\omega',0)\over \omega'}{\rm d}\omega'.\eqno (4.2.6.31)]This may be rewritten as [M(\omega,0)-M(\infty,0)=f'(\omega,0)+if''(\omega,0),\eqno (4.2.6.32)]with the value of [f'(\omega,0)] defined by equation (4.2.6.15)[link]. Using the conservation of probability, [{\rm Im}\ M(\omega,0)={\omega\over4\pi r_ec}\sigma_{\rm tot},\eqno (4.2.6.33)]which is to be compared with equation (4.2.6.23)[link].

Starting with Furry's extension of the formalism of quantum mechanics proposed by Feynman and Dyson, the total Rayleigh amplitude may be written as [M_n= {\sum_p} \bigg[{\langle n|T_1^*|p\rangle\langle p|T_1|n\rangle\over E_n-E_p+\hbar\omega}+{\langle n|T_2|p\rangle\langle n|T_2^*|p\rangle\over E_n-E_p+\hbar\omega}\bigg], \eqno (4.2.6.34)]where [T_1={\boldalpha}\cdot{\boldepsilon}_i\cdot\exp(i{\bf k}_i\cdot{\bf r})]and [T_2={\boldalpha}\cdot{\boldepsilon}_f^*\cdot\exp(-i{\bf k}_f\cdot{\bf r}).]The [|p\rangle] are the complete set of bound and continuum states in the external field of the atomic potential. Singularities occur at all photon energies that correspond to transitions between bound [|n\rangle] and bound state [|p\rangle]. These singularities are removed if the finite widths of these states are considered, and the energies E are replaced by [iE\Gamma/2], where [\Gamma] is the total (radiative plus non-radiative) width of the state (Gavrila, 1981[link]). By use of the formalism suggested by Brown et al. (1955[link]), it is possible to reduce the numerical problems to one-dimensional radial integrals and differential equations. The required multipole expansions of [T_1] and the specification of the radial perturbed orbitals that are characterized by angular-momentum quantum numbers have been discussed by Kissel (1977[link]). Ultimately, all the angular dependence on the photon scattering angle is written in terms of the associated Legendre functions, and all the energy dependence is in terms of multipole amplitudes.

Solutions are not found for the inhomogeneous radial wave equations, and Kissel (1977[link]) expressed the solution as the linear sum of two solutions of the homogeneous equation, one of which was regular at the origin and the other regular at infinity.

Because excessive amounts of computer time are required to use these direct techniques for calculating the amplitudes from all the subshells, simpler methods are usually used for calculating outer-shell amplitudes. Kissel & Pratt (1985[link]) used estimates for outer-shell amplitudes based on the predictions of the modified form-factor approach. A tabulation of the modified relativistic form factors has been given by Schaupp, Schumacher, Smend, Rullhusen & Hubbell (1983[link]).

Because of the generality of their approach, the computer time required for the calculation of the scattering amplitudes for a particular energy is quite long, so that relatively few calculations have been made. Their approach, however, does not confine itself solely to the problem of forward scattering of photons as does the Cromer & Liberman (1970[link]) approach. Using their model, Kissel et al. (1980[link]) have been able to show that it is incorrect to assign a dependence of the dispersion corrections on the scattering vector [{\boldDelta}]. This is at variance with some established crystallographic practices, in which the dispersion corrections are accorded the same dependence on [{\boldDelta}] as [f_0({\boldDelta})], and also at variance with the predictions of Wagenfeld's (1975[link]) non-relativistic model.

4.2.6.2.4. Intercomparison of theories

| top | pdf |

A discussion of the validity of the non-relativistic dipole approximation for the calculation of forward Rayleigh scattering amplitudes has been given by Roy & Pratt (1982[link]). They compared their relativistic multipole calculations with the relativistic dipole approximation and with the non-relativistic dipole approximation for two elements, silver and lead. They concluded that a relativistic correction to the form factor of order [(Z\alpha)^2] persists in the high-energy limit, and that this constant correction accounts for much of the deviation from the non-relativistic dipole approximation at all energies above threshold. In addition, their results illustrate that cancelling occurs amongst the relativistic, retardation, and higher multipole contributions to the scattering amplitude. This implies that care must be taken in assessing where to terminate the series that describes the multipolarity of the scattering process.

In a later paper, Roy, Kissel & Pratt (1983[link]) discussed the elastic photon scattering for small momentum transfers and the validity of the form-factor theories. In this paper, which compares the relativistic modified form factor with experimental results for lead and a relativistic form factor and the tabulation by Hubbell, Veigele, Briggs, Brown, Cromer & Howerton (1975[link]), it is shown that the modified relativistic form-factor approach gives better agreement with experiment for high momentum transfers ([\lt] 104  Å−1) than the non-relativistic, form-factor theories.

Kissel et al. (1980[link]) used the S-matrix technique to calculate the real part of the forward-scattering amplitude [f'(\omega,0)] for the inert gases at the wavelength of Mo [K\alpha_1]. These values are compared with the predictions of the relativistic dipole theory (RDP) and the relativistic multipole theory (RMP) in Table 4.2.6.2[link]. In most cases, the agreement between the S matrix and the RMP theory is excellent, considering the differences in the methodology of the two sets of calculations. Table 4.2.6.3(a)[link] shows comparisons of the real part of the forward-scattering amplitude [f''(\omega,0)] calculated for the atoms aluminium, silicon, zinc, germanium, silver, samarium, tantalum and lead using the approach of Kissel et al. (1980[link]) with that of Cromer & Liberman (1970[link], 1981[link]), with tabulations by Wagenfeld (1975[link]), and with values taken from the tables in this section. Although reasonably satisfactory agreement exists between the relativistic values, large differences exist between the non-relativistic value (Wagenfeld, 1975[link]) and the relativistic values. The major difference between the relativistic values occurs because of differences in estimation of the self-consistent-field term, which is proportional to [E_{\rm tot}/mc^2]. The Cromer & Liberman (1970[link]) relativistic dipole value is [+{5\over3}(E_{\rm tot}/mc^2)], whereas the tabulation in this section uses the relativistic multipole value of [(+E_{\rm tot}/mc^2)]. This causes a vertical shift of the curve, but does not alter its shape. Should better estimates of the self-energy term be found, the correction is simply that of adding a constant to each value of [f'(\omega,0)] for each atomic species. There is a significant discrepancy between the Kissel et al. (1980[link]) result for 62Sm and the other theoretical values. This is the only major point of difference, however, and the results are better in accord with the relativistic multipole approach than with the relativistic dipole approach. Note that the relativistic multipole approach does not include the Stibius-Jensen correction, which alters the shape of the curve.

Table 4.2.6.2| top | pdf |
Comparison between the S-matrix calculations of Kissel (K) (1977[link]) and the form-factor calculations of Cromer & Liberman (C & L) (1970[link], 1981[link], 1983[link]) and Creagh & McAuley (C & M) for the noble gases and several common metals; f′(ω, 0) values are given for two frequently used photon energies

Energy (keV)ElementRDP (C & L)S matrix (K)RMP (C & M)
17.479 (Mo [\,K\alpha_1])Ne0.0210.0240.026
Ar0.1550.1700.174
Kr−0.652−0.478 −0.557
Xe−0.684 −0.416−0.428
22.613 (Ag [\,K\alpha_1])Al 0.0320.0390.041
Zn0.260 0.3230.324
Ta−0.937 −0.375−0.383
Pb −1.910 −1.034 −1.162

Table 4.2.6.3| top | pdf |
A comparison of the forward-scattering amplitudes computed using different theoretical approaches

(a) Real part. KPR (Kissel et al., 1980[link]); C & L (Cromer & Liberman, 1970[link], 1981[link]); W (Wagenfeld, 1975[link]); and C & M (this data set).

AtomRadiation[f'(\omega,0)]
KPRC & LWC & M
19701981
13AlCr [K\alpha_1]13.32013.32813.31613.37613.326
Cu [K\alpha_1]13.20913.20413.20313.23513.213
Ag [K\alpha_1]13.03913.03213.02013.07813.041
14SiCr [K\alpha_1] 14.33314.35414.44114.365
Cu [K\alpha_1] 14.24414.24214.28214.254
Ag [K\alpha_1] 14.04214.02914.07114.052
30ZnCr [K\alpha_1]29.16129.31629.314 29.383
Cu [K\alpha_1]28.36928.38828.383 28.451
Ag [K\alpha_1]30.32330.26030.232 30.324
32GeCr [K\alpha_1] 31.53831.53830.2031.614
Cu [K\alpha_1] 30.83730.83731.9230.911
Ag [K\alpha_1] 32.22832.22832.1432.302
47AgCu [K\alpha_1]47.07546.94046.936 47.131
62SmAg [K\alpha_1]58.30756.30456.299 56.676
73TaAg [K\alpha_1]72.62572.06371.994 72.617
82PbAg [K\alpha_1]80.96680.09080.012 80.832

(b) Imaginary part [f''(\omega,0)]. KPR (Kissel et al., 1980[link]); C & L (Cromer & Liberman, 1981[link]); W (Wagenfeld, 1975[link]); and C & M (this data set).

AtomRadiation[f'(\omega,0)]
KPRC & LWC & M
13AlCr [K\alpha_1]0.5140.522 0.512
Cu [K\alpha_1]0.2430.246 0.246
Ag [K\alpha_1]0.0310.031 0.031
14SiCr [K\alpha_1] 0.6940.700.692
Cu [K\alpha_1] 0.3300.330.330
Ag [K\alpha_1] 0.0430.0470.043
30ZnCr [K\alpha_1]1.3701.373 1.371
Cu [K\alpha_1]0.6780.678 0.678
Ag [K\alpha_1]0.9320.938 0.938
32GeCr [K\alpha_1] 1.7861.841.784
Cu [K\alpha_1] 0.8860.870.886
Ag [K\alpha_1] 1.1901.231.190
47AgCu [K\alpha_1]4.2424.282 4.282
62SmCu [K\alpha_1]12.1612.218 12.218
73TaAg [K\alpha_1]4.4034.399 4.399
82PbAg [K\alpha_1]6.9376.929 6.929

In §4.2.6.3.3[link], some examples are given to illustrate the extent to which predictions of these theories agree with experimental data for [f'(\omega,0)].

That there is little to choose between the different theoretical approaches where the calculation of [f''(\omega,0)] is concerned is illustrated in Table 4.2.6.3(b)[link]. In most cases, the agreement between the scattering matrix, relativistic dipole, and relativistic multipole values is within 1%. In contrast, there are some significant differences between the relativistic and the non-relativistic values of [f''(\omega,0)]. The extent of the discrepancies is greater the higher the atomic number, as one might expect from the assumptions made in the formulation of the non-relativistic model. Some detailed comparisons of theoretical and experimental data for linear attenuation coefficients [proportional to [f''(\omega,0)]] have been given by Creagh & Hubbell (1987[link]) for silicon, and for copper and carbon by Gerward (1982[link], 1983[link]). These tend to confirm the assertion that, at the 1% level of accuracy, there is little to choose between the various relativistic models for computing scattering cross sections.

Further discussion of this is given in §4.2.6.3.3[link].

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 Brown, G. E., Peierls, R. E. & Woodward, J. B. (1955). The coherent scattering of γ-rays by K electrons in heavy atoms: method. Proc. R. Soc. London Ser. A, 227, 51–63.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. (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. & 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 Gavrila, M. (1981). Photon atom elastic scattering in inner-shell and X-ray physics of atoms and solids, edited by B. Crasemann. New York: Plenum.Google Scholar
First citation Gerward, L. (1982). X-ray attenuation coefficients of copper in the energy range 5 to 50 keV. Z. Naturforsch. Teil A, 37, 451–459.Google Scholar
First citation Gerward, L. (1983). X-ray attenuation coefficients of carbon in the energy range 5 to 20 keV. Acta Cryst. A39, 322–325.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 Hönl, H. (1933a). Zur Dispersions theorie der Röntgenstrahlen. Z. Phys. 84, 1–16.Google Scholar
First citation Hönl, H. (1933b). Atomfactor für Röntgenstrahlen als Problem der Dispersionstheorie (K-Schale). Ann. Phys. (Leipzig), 18, 625–657.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 International Tables for X-ray Crystallography (1974). Vol. IV. Birmingham: Kynoch Press.Google Scholar
First citation Kissel, L. (1977). Rayleigh scattering: elastic scattering by bound electrons. PhD thesis, University of Pittsburgh, PA, USA.Google Scholar
First citation Kissel, L. & Pratt, R. H. (1985). Rayleigh scattering: elastic photon scattering by bound electrons. In Atomic inner-shell physics, edited by B. Crasemann. New York: Plenum.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 Kohn, W. & Sham, L. S. (1965). Self consistent equations including exchange and correlation effects. Phys. Rev. A, 140, 1133–1138.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 Roy, S. C., Kissel, L. & Pratt, R. H. (1983). Elastic photon scattering at small momentum transfer and validity of form-factor theories. Phys. Rev. A, 27, 285–290.Google Scholar
First citation Roy, S. C. & Pratt, R. H. (1982). Validity of non relativistic dipole approximation for forward Rayleigh scattering. Phys. Rev. A, 26, 651–653.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 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
First citation Wagenfeld, H. (1975). Theoretical computations of X-ray dispersion corrections. Anomalous scattering, edited by S. Ramaseshan & S. C. Abrahams, pp. 12–23. Copenhagen: Munksgaard.Google Scholar
First citation Wang, M. S. (1986). Relativistic dispersion relation for X-ray anomalous scattering factor. Phys. Rev. A, 34, 636–637.Google Scholar
First citation Wang, M. S. & Pratt, R. H. (1983). Importance of bound–bound transitions in the dispersion relation for calculation of soft-X-ray forward Rayleigh scattering from light elements. Phys. Rev. A, 28, 3115–3116.Google Scholar








































to end of page
to top of page