Structure determination

Definition

Structure determination in crystallography refers to the process of elaborating the three-dimensional positional coordinates (and also, usually, the three-dimensional anisotropic displacement parameters) of the scattering centres in an ordered crystal lattice. Where a crystal is composed of a molecular compound, the term generally includes the three-dimensional description of the chemical structures of each molecular compound present.

Experimental techniques

Owing to the highly ordered arrangement of atoms as scattering centres in a crystal lattice, most structure determination techniques involve the diffraction of electromagnetic or matter waves of wavelengths comparable to atomic dimensions. Bragg's Law specifies the condition for plane waves to be diffracted from lattice planes. The diffracted radiation passing through a crystal emerges with intensity varying as a function of scattering angle. This variation arises from constructive and destructive interference of scattered beams from the planes associated with the different atoms present in the lattice. The result is seen by an imaging detector as a pattern of diffraction spots or rings.

Among diffraction-based techniques are:

* single-crystal X-ray diffraction
* X-ray powder diffraction
* X-ray fibre diffraction
* neutron powder diffraction (occasionally neutron single-crystal diffraction)
* polymer electron diffraction

Other techniques for three-dimensional structure determination that are complementary to diffraction methods include

* electron microscopy
* nuclear magnetic resonance spectroscopy (used largely for biological macromolecules in solution)

Methodology

The following summary applies to single-crystal X-ray diffraction. A crystal, mounted on a goniometer, is illuminated by a collimated monochromatic X-ray beam, and the positions and intensities of diffracted beams are measured. The measured intensities [math]I_{hkl}[/math] (corresponding to scattering from a lattice plane with Miller indices [math]h, k, l[/math] are reduced to structure amplitudes [math]F_{hkl}[/math] by the application of a number of experimental corrections:

[math]F^2_{hkl} = I_{hkl}(k \mathrm{Lp} A)^{-1}[/math]

where [math]k[/math] is a scale factor, Lp the Lorentz--polarization correction, and [math]A[/math] the transmission factor representing the absorption of X-rays by the crystal. The structure amplitude represents the amplitude of the diffracted wave measured relative to the scattering amplitude of a single electron.

See also

Dynamical theory of X-ray diffraction A. Authier. International Tables for Crystallography (2006). Vol. B, ch. 5.1, pp. 534-551 doi:10.1107/97809553602060000569

Dynamical theory of electron diffraction A. F. Moodie, J. M. Cowley and P. Goodman. International Tables for Crystallography (2006). Vol. B, ch. 5.2, pp. 552-556 doi:10.1107/97809553602060000570

Dynamical theory of neutron diffraction M. Schlenker and J.-P. Guigay. International Tables for Crystallography (2006). Vol. B, ch. 5.3, pp. 557-569 doi:10.1107/97809553602060000571