Introduction to basic crystallography

Drenth, J.

doi:10.1107/97809553602060000808

Crystals are the indispensable objects for the structure determination of globular proteins by X-ray diffraction. They consist of building blocks (unit cells) arranged in a three-dimensional array. According to their internal symmetry, they belong to one of the 230 possible space groups. Owing to the asymmetric structure of biological macromolecules, their crystals are restricted to the 65 enantiomorphic (not superimposable on its mirror image) space groups. The diffraction of X-rays by a crystal is explained in steps, from diffraction by one electron and two electrons via an atom and a unit cell to the diffraction by a crystal. This results in the Laue conditions for diffraction and the famous law introduced by Bragg: $[2d\sin\theta=\lambda]$ . Reciprocal space is introduced as a most useful concept in constructing the directions of diffraction. The X-ray beams diffracted by a crystal are characterized by their structure factor $[{\bf F}(hkl)=|F(hkl)|\exp[i\alpha(hkl)]]$ , where [|F(hkl)|] is the amplitude of the beam and $[\alpha(hkl)]$ is its phase angle with respect to a chosen origin. $[|F(hkl)|=[I(hkl)]^{1/2}]$ , where is the intensity of the diffracted beam. This is true if some correction factors (Lorentz, polarization and absorption) are neglected. For the determination of $[\alpha(hkl)]$ various indirect and direct methods are available. Because the structure factor $[{\bf F}(hkl)]$ is the result of the scattering by all electrons in the unit cell, it can be written as $[{\bf F}(hkl)=V\textstyle\int_{x=0}^1\textstyle\int_{y=0}^1\textstyle\int_{z=0}^1\rho(x,y,z)\exp[2\pi i(hx+ky+lz)]\,{\rm d}x\,{\rm d}y\,{\rm d}z]$ , where V is the volume of the unit cell. The electron density $[\rho(x,y,z)]$ is obtained by Fourier inversion: $[\rho(x,y,z)=(1/V)\textstyle\sum_h\textstyle\sum_k\textstyle\sum_l{\bf F}(hkl)\exp[-2\pi i(hx+ky+lz)]]$ . The final result of a structure determination by X-ray diffraction is a molecular model based on the calculated electron density $[\rho(x,y,z)]$ .

International Tables for Crystallography

Access, prices and ordering

About International Tables for Crystallography