International
Tables for
Crystallography
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 3.2, pp. 252-253

Section 3.2.2.1. Peak positions

P. W. Stephensa*

aDepartment of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794–3800, USA
Correspondence e-mail: peter.stephens@stonybrook.edu

3.2.2.1. Peak positions

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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 [{\bf a}^* = {\bf b} \times {\bf c}/({\bf a} \cdot {\bf b} \times {\bf c})] and cyclic permutations. These have the property that [{\bf a}^* \cdot {\bf a} = 1], [{\bf b}^* \cdot {\bf a} = 0] 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 [{\bf G} = h{\bf a}^* + k{\bf b}^* + l{\bf c}^*].

Incoming and diffracted radiation are described by wave vectors ki and kf, both of magnitude 1/λ, and separated by the angle 2θ. The condition for a particular Bragg reflection to be observed is Ghkl = kfki. Note that this requires that the diffracting crystal be correctly oriented relative to the incident beam. For a given reflection Ghkl, 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.[\eqalign{a^* &= b c \sin \alpha / V, \cr \sin \alpha^* &= V / (abc \sin \beta \sin \gamma)\ {\rm or}\ {\rm equivalently}\cr \cos \alpha^* &= {{\cos \beta \cos \gamma - \cos \alpha}\over{\sin \beta \sin \gamma}},}]with cyclic permutations. Here the unit-cell volume is given by[V = abc(1 - \cos^2 \alpha - \cos^2 \beta - \cos^2 \gamma + 2 \cos \alpha \cos \beta \cos \gamma)^{1/2}. ]Note that the volume of the reciprocal cell is [V ^* = 1/V], 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 as[{{4 \sin^2 \theta}\over{\lambda^2}} = d^{-2} = A h^2 + B k^2 + C l^2 + D k l + E h l + F h k,\eqno(3.2.1)]where [A\! =\! a^{*2}], [B\! =\! b^{*2}], [C\! =\! c^{*2}], [D\! =\! 2b^*c^*\cos\alpha^*], [E\! =\! 2c^*a^*\cos\beta^*] and [F = 2a^*b^*\cos\gamma^*]. Crystal symmetries higher than triclinic lead to significant simplifications in the above, e.g. in the orthorhombic system, [a^*\! =\! 1/a], [A\! =\! 1/a^2] etc., [\alpha^*\! =\! \beta^*\! =\! \gamma^*\! = \!90^\circ], and D = E = F = 0. See Chapter 1.1[link] , Reciprocal space in crystallography, in Volume B of International Tables for Crystallography for more details.








































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