Definitions

Creagh, D. C.

doi:10.1107/97809553602060000592

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

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International Tables for Crystallography (2006). Vol. C. ch. 4.2, pp. 242-243

Section 4.2.6.1. Definitions

D. C. Creagh^b

4.2.6.1. Definitions

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4.2.6.1.1. Rayleigh scattering

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When photons interact with atoms, a number of different scattering processes can occur. The dominant scattering mechanisms are: elastic scattering from the bound electrons (Rayleigh scattering); elastic scattering from the nucleus (nuclear Thomson scattering ); virtual pair production in the field of the screened nucleus (Delbrück scattering ); and inelastic scattering from the bound electrons (Compton scattering ).

Of the elastic scattering processes, only Rayleigh scattering has a significant amplitude in the range of photon energies used by crystallographers ( $[\lt]$ 100 keV). Compton scattering will be discussed elsewhere (Section 4.2.4).

The essential feature of Rayleigh scattering is that the internal energy of the atom remains unchanged in the interaction. The momentum $[\hbar{\bf k}_i]$ and polarization $[{\boldvarepsilon}_i]$ of the incident photon may be modified during the process to $[\hbar{\bf k}_f]$ and $[{\boldvarepsilon}_f]$ $[(\hbar{\bf k}_i,{\boldvarepsilon}_i)+A\rightarrow A+(\hbar{\bf k}_f,{\boldvarepsilon}_f).]$

4.2.6.1.2. Thomson scattering by a free electron

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From classical electromagnetic theory, it can be shown that the fraction of incident intensity scattered by a free electron is, at a position r, $[\varphi]$ from the scattering electron, $[I/I_0=(r_e/r){}^2\textstyle{1\over2}(1+\cos^2\varphi), \eqno (4.2.6.1)]$ where [r_e] is the classical radius of the electron (= 2.817938 × 10⁻¹⁵ m). The factor $[\textstyle{1\over2}(1+\cos^2\varphi)]$ arises from the assumption that the electromagnetic wave is initially unpolarized. Should the wave be polarized, the factor is necessarily different from that given in equation (4.2.6.1).

Equation (4.2.6.1) is the basis on which the scattering power of ensembles of electrons is compared.

4.2.6.1.3. Elastic scattering from electrons bound to atoms: the atomic scattering factor, the atomic form factor, and the dispersion corrections

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In considering the interaction of a photon with electrons bound in an atom, one assumes that each electron scatters independently of its fellows, and that the total scattering power of the atom is the sum of the contributions from all the electrons in the atom. Assuming that one can define an electron density ρ(r) for an atom containing a single electron, one can show that the scattering power of that atom relative to the scattering power of a Thomson free electron is $[f({\boldDelta})=\int\rho({\bf r})\exp[i({\bf k}_f-{\bf k}_i)\cdot{\bf r}]\,{\rm d}V, \eqno (4.2.6.2)]$ where $[\eqalignno{{\boldDelta}&={\bf k}_f-{\bf k}_i\equiv\hbox{ change in photon momentum}\cr &=2|k|\sin(\theta/2),}]$ θ being the total angle of scattering of the photon.

The scattering power for the atom relative to a free electron is referred to as the atomic form factor or the atomic scattering factor of the atom.

The result (4.2.6.2), which was derived using purely classical arguments, has been shown by Nelms & Oppenheimer (1955) to be identical to the result gained by quantum mechanics. If it is assumed that the atom has spherical symmetry, $[f(\Delta)=4\pi{\int\limits^\infty_0}\rho(r){\textstyle\sin\Delta r\over\textstyle\Delta r}r^2\,{\rm d}r. \eqno (4.2.6.3)]$ For an atom containing Z electrons, the atomic form factor becomes $[f(\Delta)=4\pi{\sum^{n=Z}_{n=1}\int\limits^\infty_0}\rho_n(r){\textstyle\sin\Delta r\over\textstyle\Delta r}r^2\,{\rm d}r. \eqno (4.2.6.4)]$ Exact solutions for the form factor are difficult to obtain, and therefore approximations have to be made to enable equation (4.2.6.4) to be evaluated. The two most commonly used approximations are the Thomas–Fermi (Thomas, 1927; Fermi, 1928) and the Hartree–Fock (Hartree, 1928; Fock, 1930) techniques.

In the Thomas–Fermi model, the atomic electrons are considered to be a degenerate gas obeying Fermi–Dirac statistics and the Pauli exclusion principle, the ground-state energy of the atom being equal to the zero-point energy of this gas. The average charge density can be written in terms of the radial potential function, V(r), which may then be substituted into Poisson's equation, $[\nabla^2V(r)=\rho(r)/\varepsilon_0]$ , which can then be solved for V(r) using the boundary conditions that $[\lim_{r\rightarrow\infty}V(r)=0]$ and that $[\lim_{r\rightarrow0}rV(r)=Ze]$ .

The Thomas–Fermi charge distributions for different atoms are related to each other. If the form factor is known for a `standard' atom for which the atomic number is [Z_0] then, for an atom with atomic number Z, $[f_Z({\boldDelta})=(Z/Z_0)\,f_0({\boldDelta}'). \eqno (4.2.6.5)]$ Here, $[\Delta'=\Delta(Z/Z_0)^{1/3}.]$

The most accurate calculations of wavefunctions of many-electron atoms have been made using the self-consistent-field (Hartree–Fock) method . In this independent-particle model, each electron is assumed to move in the field of the nucleus and in an average field due to the other electrons. With this approach, the charge distribution can be written as $[\rho(r)=\textstyle\sum\limits^{n=Z}_{n=1}\rho_n(r)=\sum\limits^{n=Z}_{n=1}\Psi_n^*(r)\Psi_n(r), \eqno (4.2.6.6)]$ where $[\rho_n(r)]$ is the charge-density distribution of the nth electron and $[\Psi_n(r)]$ is its wavefunction. The technique has been extended to include the effects of both exchange and correlation. Tables of relativistic Hartree–Fock values have been given by Cromer & Waber (1974). Their notation F(x, Z) is related to the notation used earlier as follows: $[f_Z(\Delta)\equiv F(x,Z),]$ where $[x={|k|\over2\pi}\sin(\varphi/2)={\Delta\over4\pi}.]$

In the foregoing discussion, the fact that the electrons occupy definite energy levels within the atoms has been ignored: it has been assumed that the energy of the photon is very different from any of these energy levels. The theory for calculating the scattering power of an atom near a resonant energy level was supposed to be effectively the same as the well understood problem of the driven damped pendulum system. In this type of problem, the natural amplitude of the system was modified by a correction term (a real number) dependent on the proximity of the impressed frequency to the natural resonant frequency of the system and a loss term (an imaginary number) that was related to the damping factor for the resonant system. Thus the scattering power came to be written in the form $[f=f_0+\Delta f'+i\Delta f'', \eqno (4.2.6.7)]$ where [f_0] is the atomic scattering factor remote from the resonant energy levels, $[\Delta f']$ is the real part of the anomalous-scattering factor, and $[\Delta f'']$ is the imaginary part of the anomalous-scattering factor. The nomenclature of (4.2.6.7) has been superseded, but one still encounters it occasionally in modern papers.

In what follows, a brief exposition of the various theories for the anomalous scattering of X-rays and descriptions of modern experimental techniques for their determination will be given. Comparisons will be made between the several theoretical and experimental results for a number of atomic species. From these comparisons, conclusions will be drawn as to the validity of the various theories and the relevance of certain experiments.

References

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

Fermi, E. (1928). Eine statistiche Methode zur Bestimmung einiger geschaften des Atoms und ihre Anwendung auf die Theorie des periodische Systems der Elemente. Z. Phys. 48, 73–79.Google Scholar

Fock, V. (1930). Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems. Z. Phys. 61, 126–148.Google Scholar

Hartree, D. R. (1928). The wavemechanics of an atom with a non-Coulomb central field. Proc. Cambridge Philos. Soc. 24, 89–132.Google Scholar

Nelms, A. T. & Oppenheimer, I. (1955). J. Res. Natl Bur. Stand. 55, 53–62.Google Scholar

Thomas, L. H. (1927). The calculation of atomic fields. Proc. Cambridge Philos. Soc. 23, 542–548.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 4.2, pp. 242-243