Unit cell, defining vectors INDEX

A 3D crystal has its atoms arranged such that many different planes can be drawn through them. It is convenient to be able to describe these planes in a systematic way and Fig. 4 shows how this is done. It illustrates a 2D example, but the same principle applies to the third dimension. The crystal lattice can be defined in terms of vectors a and b, which have a defined length and angle between them (it is c in the third dimension). The box defined by a and b (and c for 3D) is known as the unit cell. The dashed lines in Fig. 4A show one set of lines that can be drawn through the 2D lattice (they would be planes in 3D). It can be seen that these lines chop a into 1 piece and b into 1 piece, so these are called the 11 lines. The lines in B, however, chop a into 2 pieces, but still chop b into 1 piece, so these are the 21 lines. If the lines are parallel to an axis as in C, then they do not chop that axis into any pieces so, in C, the lines chopping a into 1 piece and which are parallel to b are the 10 lines. This is a simple rule. The numbers that are generated are known as the Miller indices of the plane. Note that if the structure in Fig. 6.4 was a 3D crystal viewed down the c axis, the lines would be planes. In these cases, the third Miller index would be zero (i.e., the planes would be the 110 planes in A, the 210 planes in B, and the 100... 

The optical indicatrix is a useful construction for visualizing the variation in the refractive index of a crystal as a function of spatial direction.It shows directions (with respect to unit-cell edges) where the refractive index is greatest and where it is the smallest. It is a three-dimensional ellipsoid with a shape defined by the ends of vectors, each with the same fixed origin. The length of each vector is proportional to... 

The unit cell reduction using Delaunay-Ito method can be easily automated as is done in the ITO indexing computer code, which is discussed in section 5.11. The Delaunay-Ito reduced unit cell, however, may not be the one with the shortest possible vectors, although the latter is conventionally defined as a standard reduced unit cell. 

Only six real parameters define the Sk,- vectors, so the phase factor is not generally needed, but it is convenient to use it when particular relations or constraints between real and imaginary vectors (i ky, 4y) are given. The magnetic moment of the atom j in the unit cell of index /, should be calculated by using Equation (44), which may be also written in this case as ... 

The wave-like modulations are generally described with reference to the reciprocal lattice of the phase rather than the direct unit cell. The modulation wave is specified in terms of a wave vector q, which is defined in terms of the reciprocal lattice vectors a, b or c. The diffraction pattern of the phase will now show a set of reflections corresponding to the subcell plus superlattice reflections due to the modulation. In cases with a single modulation wave, the indices are described, not by conventional Miller indices but by a four-index extension, so that each reflection is indexed as ha + kb + lc +mq. The value of q is obtained directly from the diffraction pattern. When there are two distinct modulations, the diffraction pattern reflections must be indexed on a five-index system ha +kb +lc +mq +mq, and for the system with modulation in three independent directs a six index notation is needed (also see Section 3.2.2). 

This mathematical theory provides a partition of the space which is analogous to the more familiar partition made in hydrology in river basins delimited by watersheds. It relies on the study of a local function F(r) called the potential function. The potential function carries the physical or chemical information e.g. the electron density, the ELF (see below), or even the electrostatic potential [56-58]. In the cases treated in the present book, the potential function is required to be defined at any point of a manifold which is either for molecules or the unit cell for periodic systems. Moreover the first and second derivatives with respect to the point coordinates must be defined for any point. Its gradient W(r) forms a vector field bounded on the manifold and determines two kinds of points on the one hand are the wandering points corresponding to W(r ) f 0. and on the other hand are the critical points for which VF(rc) = 0. A critical point is characterized by the index Ip, the number of positive eigenvalues of the second derivatives matrix (the Hessian matrix). There are four kinds of critical points in... 

Infinite one-dimensional chain of carbon 2p-orbitals with several unit cells outlined. The position of an arbitrary unit cell p is given by the vector Rp = (p — l)o, where a is the length of the repeat unit cell. The latter contains the regularly repeating motif. We can define a new index k = Ijrj/Na which runs from 0 to +7r/o such that equation 13.2 now can be recast as... 

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