International
Tables for Crystallography Volume H Powder diffraction Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk © International Union of Crystallography 2018 |
International Tables for Crystallography (2018). Vol. H, ch. 3.2, pp. 252-253
Section 3.2.2.1. Peak positions^{a}Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794–3800, USA |
Starting with a single crystal, the lattice is described by non-coplanar translation vectors a, b and c, with magnitudes a, b and c, respectively, and angles α between b and c, β between c and a, and γ between a and b. Diffraction peak positions are governed by the reciprocal lattice, spanned by vectors and cyclic permutations. These have the property that , etc. (This is the `crystallographic' convention; the scattering community usually defines reciprocal-lattice vectors as larger by a factor of 2π.) The reciprocal lattice is indexed by Miller indices (hkl), so that each vector in the reciprocal lattice is given by .
Incoming and diffracted radiation are described by wave vectors k_{i} and k_{f}, both of magnitude 1/λ, and separated by the angle 2θ. The condition for a particular Bragg reflection to be observed is G_{hkl} = k_{f} − k_{i}. Note that this requires that the diffracting crystal be correctly oriented relative to the incident beam. For a given reflection G_{hkl}, the d-spacing between reflecting planes is given by d = 1/|G|. (Again, in the scattering convention, incident and diffracted beams have wave vectors of magnitude 2π/λ, and d = 2π/|G|.) In either case, this allows the Bragg condition to be written in the scalar form λ = 2d sin θ.
If the sample is a powder instead of a single crystal, some large number of crystallites will be aligned to meet the (vector) Bragg condition with the incident radiation, and then the diffracted radiation will take the form of a cone of opening half-angle 2θ.
It is often convenient to work with equations written for scalars, viz.with cyclic permutations. Here the unit-cell volume is given byNote that the volume of the reciprocal cell is , and that the equations above are valid upon exchanging starred and unstarred variables.
For powder samples, using only scalars, the Bragg condition may be written aswhere , , , , and . Crystal symmetries higher than triclinic lead to significant simplifications in the above, e.g. in the orthorhombic system, , etc., , and D = E = F = 0. See Chapter 1.1 , Reciprocal space in crystallography, in Volume B of International Tables for Crystallography for more details.