Aim: To become familiar with the concept of the reciprocal lattice. It is very important and frequently used in the rest of the course to develop models and describe the physical properties of solids.
First we treat an alternative description of a lattice is in terms of a set of parallel identical planes. They are defined so that every lattice point belongs to one of these planes. The lattice can actually be described with an infinite number of such sets of planes. This description may seem artificial but we will see later that it is a natural way to interpret diffraction experiments in order to characterize the crystal structure. The figure below shows a 2-D lattice with three different sets of 1-D “planes” drawn.
The orientation of a plane can be specified by three integers, the so called Miller indices. The intercepts of one plane with the coordinate axes (the translation vectors of the lattice) are determined and normalized by division by the corresponding lattice constants. Then the reciprocals are taken and reduced to the set of smallest integers. In the figure above, the Miller indices for the three sets of ”planes” are given as an example. Note also that the planes in the sets with lower Miller indices are spaced further apart (shaded areas depict the interplanar spacing). The Miller indices of a single plane or a set of planes are denoted (ijk). All planes with the same symmetry are denoted {ijk}. A direction in a crystal is denoted [uvw] and all equivalent directions (same symmetry) are . In cubic crystals [ijk] is orthogonal to (ijk).
Real lattices and reciprocal lattices
Lattices are periodic structures and position-dependent physical properties that depend on the structural arrangement of the atoms are also periodic. For example the electron density in a solid is a function of position vector r, and its periodicity can be expressed as ρ(r+T)=ρ(r), where T is a translation vector of the lattice. The periodicity means that the lattice and physical properties associated with it can be Fourier transformed. Since space is three dimensional, the Fourier analysis transforms it to a three-dimensional reciprocal space. Physical properties are commonly described not as a function of r, but instead as a function of wave vector (“spatial frequency” or k-vector) k. This is analogous to the familiar Fourier transformation of a time-dependent function into a dependence on (temporal) frequency. The lattice structure of real space implies that there is a lattice structure, the reciprocal lattice, also in the reciprocal space.
Reciprocal lattice vectors
The reciprocal lattice points are described by the reciprocal lattice vectors, starting from the origin G = m´ b1 + n´ b2 + o´ b3, where the coefficients are integers and the bi are the primitive translation vectors of the reciprocal lattice. From the definition of the bi, it can be shown that exp (iGT) = 1. This condition is actually taken as the starting point by Hoffman; it is required by the condition that the Fourier series must have the periodicity of the lattice. Hence, theFourier series of a function with the periodicity of the lattice can only contain the lattice vectors (spatial frequencies) G.
Since the Miller indices of sets of parallel lattice planes in real space are integers, we can interpret the coefficients (m´n´o´) as Miller indices (ijk). This leads to a physical relation between sets of planes in real space and reciprocal lattice vectors. Hence we write G = i b1 + j b2 + k b3. The reciprocal lattice vectors are labelled with Miller indices Gijk. Each point in reciprocal space comes from a set of crystal planes in real space (with some exceptions, for example (n00), (nn0) and (nnn) with n>1 in cubic systems; nevertheless they occur in the Fourier series and are necessary for a correct physical description). Consider the (ijk) plane closest to the origin. It cuts the coordinate axes in a1/i, a2/j and a3/k. A triangular section of the plane has these points in the corners. The sides of the triangle are given by the vectors (a1/i) - (a2/j) and analogously for the other two sides. Now the scalar product of Gijk with any of these sides is easily seen to be zero. Hence the vector Gijk is directed normal to the (ijk) planes. The distance between two lattice planes is given by the projection of for example a1/i onto the unit vector normal to the planes, i.e. dijk = (a1/i)•nijk, where dijk is the interplanar distance between two (ijk)-planes. The unit normal is just Gíjk divided by its length. Hence the magnitude of the reciprocal lattice vector is given by 2π/ dijk.
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