Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). 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}) \,{\rm 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\,{\rm d}r \,{\rm d}\vartheta \,{\rm d}\varphi\cr &= {\int\limits_{0}^{\infty}} 4\pi r^{2} \rho_{j} (r) {\sin 2\pi Sr \over 2\pi Sr}\,{\rm d}r \equiv {\int\limits_{0}^{\infty}} 4\pi r^{2} \rho_{ j} (r) j_{0} (2\pi Sr)\,{\rm 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. [Resonance scattering is referred to as anomalous scattering in the older literature, but this misnomer is avoided in the current chapter.] 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 [f'_{j}] and [f''_{j}] can be neglected, (b) that [f'_{j}] and [f''_{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 nontrivial 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 expresions 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: 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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