## X‐ray Diffraction: Principles

### Abstract

The properties of biological macromolecules cannot be fully understood without knowledge of their three‐dimensional structure. For mutation of proteins and for rational drug design it is essential to know the three‐dimensional structure. X‐ray diffraction is one technique for structure determination. It requires the macromolecules to be present in a well‐organized arrangement as crystals or as fibres; amorphous material does not provide sufficient information.

Keywords: X‐rays; crystallography; crystals; diffraction  Figure 1. A unit cell in a crystal. The vectors a, b and c indicate the repeating distances in the crystal structure. Depending on relation between, and the values of, the six parameters a, b, c, α, β and γ, the crystal belongs to one of the seven crystal systems (see Table ).  Figure 2. The beams reflected by a set of parallel planes amplify each other when the path difference between reflections from successive planes is equal to 1λ, 2λ, etc. Because the path difference is also 2d sin θ, the Bragg law is obtained: 2d sin θ = n × λ. Reproduced, in modified form, from Drenth , with permission of Springer‐Verlag.  Figure 3. A wave can be represented in the ligand diagram as a vector E in a plane with horizontal and vertical axes. The wave E can be regarded as the sum of two waves, one along the horizontal axis with amplitude E cos α, and the other along the vertical axis with amplitude E sin α. The mathematical expression for E is (E cos α) + i (E sin α) = E exp(iα). Reproduced, in modified form, from Drenth with permission of Springer‐Verlag.  Figure 4. A simple system consisting of two electrons, 1 and 2. The position of electron 2 with respect to electron 1 is given by vector r. Vectors s0 and s1 indicate the direction of the incident and scattered beams. The path difference between the wave involving electron 1 and the wave involving electron 2 is p1 + p2. Reproduced, in modified form, from Drenth with permission of Springer‐Verlag.  Figure 5. Vectors s0 and s1 indicate the directions of the primary and diffracted beams respectively. Vector S = s1 − s0 is perpendicular to a plane that can be regarded as a plane reflecting the primary beam.  Figure 6. Each arrow in the ligand diagram represents the scattering by one unit cell. In the figure the scattering by nine unit cells (n = 0 to 8) is shown. In fact n is a very large number and the vectors point in all directions, the scattering by a crystal would be zero. However, in the special case that a • S is an integer, all vectors point to the right and the crystal does scatter. Reproduced, in modified form, from Drenth with permission of Springer‐Verlag.  Figure 7. The construction of the diffracted beam by means of the Ewald sphere and the reciprocal lattice. M is the center of the sphere and O is the origin of the reciprocal lattice. The primary beam passes through M and is directed towards O. If the crystal is regarded as being situated at M, a diffracted beam leaves the crystal in the direction of MP where P is any reciprocal lattice point on the surface of the Ewald sphere.

References

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