User:Ldm1954/Diff

Bulk and surface superstructures

Many materials have relatively simple structures based upon small unit cell vectors $a,b,c$. There are many others where the repeat is some larger multiple along one or more direction, for instance $Na,Mb,c$. These superstructures can arise from many reasons:#Figure X

1. Larger unit cells due to electronic ordering which leads to small displacements of the atoms. One example is Antiferroelectricity ordering.
2. Chemical ordering, that is different atom types in locations of the subcell.
3. Magnetic order of the spins. These may be in opposite directions on some atoms, leading to what is called antiferromagnetism.

Figure X: Electron diffraction from a thin silicon (111) sample with a 7x7 reconstructed surface

In addition to these occurring in the bulk, they can also occur at surfaces. When half the material is (nominally) removed to create a surface, some of the atoms will be undercoordinated. To reduce their energy they can rearrange. Sometimes these rearrangements are relatively small; sometimes they are quite large. Similar to a bulk superstructure there will be additional, weaker diffraction spots. One example is for the silicon (111) surface, where there is a supercell which is seven times larger than the simple bulk cell in two directions. This leads to diffraction patterns with additional spots as shown in Figure Y. Here the (220) are stronger bulk diffraction spots, and the weaker ones due to the surface reconstruction are marked with the red 7x7.

Aperiodic materials

Figure X: Diffraction pattern down [0100] showing an incommensurate modulation in a mixed-valent manganite oxide sulfide

In an aperiodic crystal the structure can no longer be simply described by three different vectors in real or reciprocal space. In general there is a substructure describable by three (e.g. $a,b,c$) but in addition there is some additional periodicity (one to three) which cannot be described as a multiple of the three as in a superstructure; it is a genuine additional periodicity which is an irrational number relative to the simple lattice. The diffraction pattern can then only be described by more than three indices. An example is shown in Figure X from a mixed-valent manganite oxide sulfide Sr₂MnO₂Cu_1.5−xS₂. In this material there is both variations in the occupancy of the sites with the copper atoms (vacancies) and also displacements of the atom positions. This leads to a modulation along the [100] direction of 0.2418. The diffraction spots in the Figure can be described by $g=Ah+Bk+Cl+qm$ or the four Miller indices $(hklm)$, where A, B and C are the normal reciprocal lattice vectors for a face-centered orthorhombic cell with a=0.56377 nm, b=0.56334 nm, c=1.69735 nm and the modulation has a period of about 2.3 nm; some of the extra spots due to the modulation are marked by arrows in the Figure.

An extreme example of this is for quasicrystals, which can be described similarly by a higher number of Miller indices in reciprocal space -- but not by any translational symmetry in real space.

Diffuse scattering

Figure Z: Single frame extracted from a video of a Nb_0.83CoSb sample showing diffuse intensity (snake-like) due to vacancies at the Nb sites

A further step beyond superstructures and aperiodic materials is what is called diffuse scattering. These correspond to features in the sample which can give rise to additional elastic scattering, for instance semi-random arrangements of point defects or they can be inelastic. For instance, in bulk silicon the atomic vibrations phonons are more prevalent along specific directions. One example is for a Nb_0.83CoSb sample. Because of the vacancies at the niobium sites, there is diffuse intensity with snake-like structure due to short range order originating from the repulsion between vacancies and nearest and next-nearest neighbour vacancies and the relaxation of Co and Sb atoms around these vacancies.