Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 1.2, pp. 10-11   | 1 | 2 |

Section 1.2.4. The isolated-atom approximation in X-ray diffraction

P. Coppensa*

aDepartment of Chemistry, Natural Sciences & Mathematics Complex, State University of New York at Buffalo, Buffalo, New York 14260-3000, USA
Correspondence e-mail:

1.2.4. The isolated-atom approximation in X-ray diffraction

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To a reasonable approximation, the unit-cell density can be described as a superposition of isolated, spherical atoms located at [{\bf r}_{j}]. [\rho_{\rm unit\; cell} ({\bf r}) = {\textstyle\sum\limits_{j}} \rho_{{\rm atom}, \, j} ({\bf r})\ast\delta ({\bf r} - {\bf r}_{j}). \eqno(] Substitution in ([link] gives [{F({\bf H}) = {\textstyle\sum\limits_{j}} \hat{F} \{\rho_{{\rm atom}, \, j}\} \hat{F} \{\delta ({\bf r} - {\bf r}_{j})\} = {\textstyle\sum\limits_{j}}\; f_{j} \exp (2\pi i{\bf H}\cdot {\bf r}_{j})} \eqno(] or [\eqalignno{F(h, k, l) &= {\textstyle\sum\limits_{j}}\; f_{j} \exp 2\pi i(hx_{j} + ky_{j} + lz_{j})\cr &= {\textstyle\sum\limits_{j}}\; f_{j} \{\cos 2\pi (hx_{j} + ky_{j} + lz_{j})\cr &\quad + i \sin 2\pi (hx_{j} + ky_{j} + lz_{j})\}. &(}] [f_{j}(S)], the spherical atomic scattering factor, or form factor, is the Fourier transform of the spherically averaged atomic density [\rho_{j}(r)], in which the polar coordinate r is relative to the nuclear position. [f_{j}(S)] can be written as (James, 1982[link]) [\eqalignno{f_{j}(S) &= {\int\limits_{\rm atom}} \rho_{j} (r) \exp (2\pi i{\bf S}\cdot {\bf r}) \;\hbox{d}{\bf r}\cr &= {\int\limits_{\upsilon = 0}^{\pi}}\; {\int\limits_{\varphi = 0}^{2\pi}} \;{\int\limits_{r = 0}^{\infty}} \rho_{j} (r) \exp (2\pi i Sr \cos \vartheta) r^{2} \sin \vartheta\; \hbox{d}r \;\hbox{d}\vartheta \;\hbox{d}\varphi\cr &= {\int\limits_{0}^{r}} 4\pi r^{2} \rho_{j} (r) {\sin 2\pi Sr \over 2\pi Sr}\; \hbox{d}r \equiv {\int\limits_{0}^{r}} 4\pi r^{2} \rho_{ j} (r) j_{0} (2\pi Sr)\; \hbox{d}r\cr & \equiv \langle \;j_{0}\rangle, &(}] where [j_{0} (2\pi Sr)] is the zero-order spherical Bessel function.

[\rho_{j}(r)] represents either the static or the dynamic density of atom j. In the former case, the effect of thermal motion, treated in Section 1.2.9[link] and following, is not included in the expression.

When scattering is treated in the second-order Born approximation, additional terms occur which are in particular of importance for X-ray wavelengths with energies close to absorption edges of atoms, where the participation of free and bound excited states in the scattering process becomes very important, leading to resonance scattering. Inclusion of such contributions leads to two extra terms, which are both wavelength- and scattering-angle-dependent: [f_{j} (S, \lambda) = {f_{j}}^{0} (S) + f'_{j} (S, \lambda) + if''_{j} (S, \lambda). \eqno(]

The treatment of resonance effects is beyond the scope of this chapter. We note however (a) that to a reasonable approximation the S-dependence of j′ and j″ can be neglected, (b) that j′ and j″ are not independent, but related through the Kramers–Kronig transformation, and (c) that in an anisotropic environment the atomic scattering factor becomes anisotropic, and accordingly is described as a tensor property. Detailed descriptions and appropriate references can be found in Materlick et al. (1994)[link] and in Section 4.2.6[link] of IT C (2004)[link].

The structure-factor expressions ([link] [link] [link] [link] can be simplified when the crystal class contains non-trivial symmetry elements. For example, when the origin of the unit cell coincides with a centre of symmetry [(x, y, z \rightarrow -x, -y, -z)] the sine term in ([link] cancels when the contributions from the symmetry-related atoms are added, leading to the expression [F({\bf H}) = 2 {\textstyle\sum\limits_{j = 1}^{N/2}}\; f_{j} \cos 2\pi (hx_{j} + ky_{j} + lz_{j}), \eqno(] where the summation is over the unique half of the unit cell only.

Further simplifications occur when other symmetry elements are present. They are treated in Chapter 1.4[link] , which also contains a complete list of symmetry-specific structure-factor expressions valid in the spherical-atom isotropic-temperature-factor approximation.


International Tables for Crystallography (2004). Vol. C. Mathematical, physical and chemical tables, edited by E. Prince. Dordrecht: Kluwer Academic Publishers.
James, R. W. (1982). The optical principles of the diffraction of X-rays. Woodbridge, Connecticut: Oxbow Press.
Materlik, G., Sparks, C. J. & Fischer, K. (1994). Resonant anomalous X-ray scattering. Theory and applications. Amsterdam: North-Holland.

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