From WolfWikis

Representations of space groups

Reciprocal lattices

We have seen that the irreps of a 1D translation group is best looked at in terms of a series of Brillouin zones, because the points within the first BZ summarize all possible irreps of the group and the points in subsequent zones are repeated aliases of the first one. For space groups we have something very similar but of course we have to think in 3D and we have to somehow take into account the reciprocal relationship between real space and spatial frequency space (aka indirect space or reciprocal space). The best way to do that is to derive a kind of unit cell in reciprocal space by connecting the totally symmetrical irrep Γ at the origin to its closest aliases, aka the Bragg (lattice) points (hkl)= (100), (010) and (001). We have to be cautious particularly if the angles α,β and γ are not 90. In that case we need to take the unit reciprocal a* to be perpendicular to the direct vectors b and c etc. This is conveniently done by defining:

a* = b x c/ a.b x c

Likewise b* and c* are found by permutation. In the case that the angles are all 90 the a* vector and the a vector point in the same direction and their lengths are each others reciprocal, but for the triclinic, monoclinic, trigonal and hexagonal cases the relationship is more complicated and the corresponding direct and reciprocal vectors are not collinear.

Brillouin zones

We have seen that the edge of the first BZ is halfway between Γ and its first alias at k=1 in the 1D case. In the 3D (or 2D) case we must construct a halfway plane perpendicular on each vector that connects Γ to each of its nearest aliases. The volume included by these planes is the Brillouin zone. For a primitive space group this BZ will simply look like a parallellopiped 'box'. For C,F,I and R cells the box may look a bit more complicated.

First BZ of an Face centered cubic cell

Centering and systematic extinction

For example the reciprocal lattice of an F-centered cell in real space is an I-centered cell (and vice versa). If we describe the extra translations of the F cell in real space as {E|½½0}, {E|½0½}, {E|0½½} the first alias of Γ in a*-direction of reciprocal space is actually (200)*. The shortest 'whole' K-vector is (200)* and not (100)* because the vector product of (100)* and (½½0) does not produce a in integer: 1.½+0.½+1.0= ½. This implies that the alias closest to the origin of reciprocal space is (111)*, because that does give 1.½+1.½+1.0= 1 = integer.

This means that the reciprocal lattice can be considered to consist of the basis vectors (200),(020) and (002) with a body centering of (111).

In a scattering pattern this is clearly visible in that all Bragg reflections at (hkl) for which h,k and l are either all odd or all even are visible but the mixed ones remain dark. Such systematic extinctions are also tabulated in the International Tables.

(Notice that the reciprocal lattice of fcc is bcc and vice versa, but that if one is considered to have ½½0 as a translation the other has 200).

Irreps of space groups

For the pure 3D translation group, aka the triclinic space group P1, each point (pixel) inside the box-like BZ represents an irrep of the group. Its basis function is as before a Bloch function:φ= exp(ik.r) where the k vector represents the point inside the BZ in reciprocal space and r any point (xyz) in real space.

But how about P1^bar? What does an element like i do to the basis function? Well it turns (xyz) into (x^bary^barz^bar) or in vectors: r => -r so that the basis function becomes: φ'= exp(ik.- r). Interestingly we may also write that as φ'= exp(i- k.r) and imagine that the k-vector was inverted rather than the r-vector!. In other words: the rotation symmetry also applies to reciprocal space.

The star of k

So applying the inversion center in reciprocal space we end up with two functions φ and φ'. This means we end up with a two dimensional representation that is symbolized by two points in the BZ: at k and - k. Together they are know as the star of k: *k.

The story is quite general. E.g. if we have a cubic space group with 48 rotational elements we generally get a star of k composed of 48 symmetry related vectors in the BZ....

Is this 48 dimensional representation at least reducible? Unfortunately, the general answer to that is NO. The reason is that the rotational and the translational elements of the space group typically do not commute. E.g. first swapping x and y and then shifting one unit in x does not land you on the same spot at the other way around:

(0.2,0.1,0) =swap=> (0.1,0.2,0) =shift=> (1.1,0.2,0)

(0.2,0.1,0) =shift=> (1.2,0.1,0) =swap=> (0.1,1.2,0)

Special points

Fortunately there are special points in the BZ that do allow reduction.

The Γ- point

For example in point Γ, the origin of the first BZ we have k= 0. Obviously all rotations will turn this vector into itself. To arrive at basis functions for this case is essentially the same job as doing that for the point group formed by the rotational elements disregarding any translational shifts. This even holds for non-symmorphic elements, so that in point Γ the irreps of a space group like Pc will be the same as those of the point group m and for a cubic non-symmorphic like Fd3^barm (or F 4₁/d ^bar 2/m if you like the full Monti) it will look like those of O_h. (Remember your t_g and e_g's?)

The X-point at the edge

At the edge of the BZ there are also special points. Take for example the Nyqvist point in the a direction in P1^bar at k =(½00). The inversion takes this point to its inverse k =(-½00). But these two functions -albeit different as function- are each others aliases because they differ by an entire Q vector (actually a*). Thus we can take symmetric and antisymmetric combinations and reduce much like we are dealing with the point group i (or C_i). To construct a basis set for these irreps we must find a function that either has an inversion center or changes sign under it to represent the small irreps at this k-point and multiply this function with the Block-function exp(ikr) to find a suitable function for the two one-dimensional irreps in this point of the BZ.

For higher symmetries the special points at the edges do not necessarily have the full rotation symmetry as in point Γ, but anly part of it. Take the example of the point X at the edge of the BZ of a face centered cubic lattice and let's take a symmorphic space group like Fm3m. There is such a point (marked X in the diagram) at k =(100) but also at k =(010) and k =(001) and the three fold axis in the body diagonal direction of the cube that passes through the L point of the BZ takes one into the other. Clearly the star of k: *k contains the three vectors k =(100), (010) and (001). Locally in X the symmetry will be tetragonal 4/mmm (D_4h) because that is the subgroup of m3m when we leave out the threefold axes. The local small irreps will be those of this point group and one of the many books^[1] listing character tables will tell you that the irreps are either one dimensional (A_n,z or B_n,z) or two-dimensional (E_z), where n=1,2 and z=g or u.

This means that in point X we have a total or 10 irreps. As the *k contains three vectors eight of them (all A's and B's) are three-dimensional in total and two (the E's) are 6D.

Non-symmorphic trouble

At the edges of the BZ there can be a problem if the space group is non-symmorphic. In that case it can be that the reduction process cannot be applied at the edge of the BZ and the local irreps retain higher dimensionality. In that case the small irreps will not look like the irreps of a point group. In stead of 1D-A's and B's all small irreps may be 2D. (There are books that contain them e.g. by Kovalev^[2] and there is a handy website.^[3]). In band structure jargon this phenomenon is known as bands sticking together.

Band structure calculations

In band structure calculations people typically calculate the energies only in the special points and on a number of points on the edges connecting them. A band structure diagram therefore shows you how the energies evolve if you walk say from Г to X along the a*-axis and from there to point U, then to L and back to Г.

The general points in the bulk interior of the BZ are typically avoided and the reason should be clear. There is only one irrep there, which means that all wave functions of all atoms in your unit cell happily partake in an orgy of mixing in one big 48-fold degenerate state. Any attempt to simplify the problem by making use of symmetry fails because you do not have any symmetry there...

References

1. ↑ e.g. Molecular spectroscopy, Jeanne L. Hale Prentice Hall ISBN 0-13-229063-4
2. ↑ Kovalev, O; Neprivodimye predstavleniya prostranstvennykh grupp (Irreducible Transformations of Space Groups) 1961
   Representations of the crystallographic space groups O.V. Kovalev 1993 ISBN 2-88124-934-5
3. ↑ REPRES