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Crystallography

Field: Solid State Physics

The science that examines the arrangement of atoms in crystalline solids.

Sequence of Expressions

Let ρ(r)\rho(\mathbf{r}) be the electron density of the crystal lattice. The structure factor F(Q)F(\mathbf{Q}) for a scattering vector Q\mathbf{Q} is defined by the Fourier transform:\nF(Q)=∫unit cellρ(r)e−iQ⋅rd3r\mathbf{F}(\mathbf{Q}) = \int_{\text{unit cell}} \rho(\mathbf{r}) e^{-i \mathbf{Q} \cdot \mathbf{r}} d^3\mathbf{r} \nFor a perfect crystal, the intensity I(Q)I(\mathbf{Q}) observed at scattering vector Q\mathbf{Q} is proportional to the square magnitude of the structure factor, I(Q)∝∣F(Q)∣2I(\mathbf{Q}) \propto |\mathbf{F}(\mathbf{Q})|^2. Constructive interference occurs when Q⋅d=2πn\mathbf{Q} \cdot \mathbf{d} = 2\pi n, where d\mathbf{d} is a lattice vector and n∈Zn \in \mathbb{Z}.
Given a crystal lattice defined by basis vectors a,b,c\mathbf{a}, \mathbf{b}, \mathbf{c} and a plane intersecting the crystallographic axes at intercepts (p,q,r)(p, q, r), the Miller indices (h,k,l)(h, k, l) are defined by the reciprocal relationship:\n[hkl]=[1/p1/q1/r]\left[\begin{array}{c} h \\ k \\ l \end{array}\right] = \left[\begin{array}{c} 1/p \\ 1/q \\ 1/r \end{array}\right]\nThese indices are typically reduced to the smallest set of co-prime integers by multiplying by the least common multiple of the denominators.
A crystal structure is defined by its unit cell, characterized by the lattice parameters, which consist of the lengths of the three orthogonal basis vectors a,b,c\mathbf{a}, \mathbf{b}, \mathbf{c} and the angles between them α,β,γ\alpha, \beta, \gamma. The metric tensor gijg_{ij} relates these parameters:\ngij=vi⋅vjg_{ij} = \mathbf{v}_i \cdot \mathbf{v}_j \nwhere v1=a\mathbf{v}_1 = \mathbf{a}, v2=b\mathbf{v}_2 = \mathbf{b}, v3=c\mathbf{v}_3 = \mathbf{c}. The volume VV of the unit cell is given by:\nV=det⁡(gij)=abc1−cos⁡2α−cos⁡2β−cos⁡2γ+2cos⁡αcos⁡βcos⁡γV = \sqrt{\det(g_{ij})} = a b c \sqrt{1 - \cos^2\alpha - \cos^2\beta - \cos^2\gamma + 2 \cos\alpha \cos\beta \cos\gamma}
For monochromatic incident radiation of wavelength λ\lambda, constructive diffraction occurs when the path difference between waves scattered from adjacent crystal planes separated by interplanar spacing dhkld_{hkl} satisfies the condition:\nnλ=2dhklsin⁡θn\lambda = 2d_{hkl}\sin\theta \nwhere nn is an integer order of diffraction, θ\theta is the Bragg angle, and dhkld_{hkl} is the spacing corresponding to the Miller indices (h,k,l)(h, k, l).
The symmetry of a crystal is mathematically described by a space group GG, which is a discrete subgroup of the Euclidean group E(3)E(3). An element g∈Gg \in G is an isometry that maps the crystal onto itself. The group structure is defined by the composition of point group operations RR (rotations and reflections) and translational vectors T\mathbf{T}: \ng(r)=R(r)+Tg(\mathbf{r}) = R(\mathbf{r}) + \mathbf{T} \nwhere RR belongs to the point group and T\mathbf{T} belongs to the translational lattice Λ\Lambda. The space group GG is thus the semi-direct product of the point group and the translation group: G=P⋊Trans(Λ)G = P \rtimes \text{Trans}(\Lambda).