Crystallography refers to the analysis of atoms in crystals and topology refers to the study of distortion and invariant connectivity characteristics of mathematical objects. Thus, crystallographic topology deals with association of the two attributes together. The topology of crystallographic groups is approached using orbifolds, and that of simple crystal structures using Morse functions on orbifolds.
W. D. Dunbar is considered as the father of orbifold as he started the first study of orbifolds in 1981.Crystallographic orbifold O is defined as the quotient space of either a sphere SP, or Euclidean EU, space modulo a discrete crystallographic symmetry group, G and is denoted as
O= (SP or EU)/G
An orbifold comprises a basic topological space along with a singular set embedded in it. Moreover, an interesting fact is that a properly bound fundamental domain within a space group’s unit cell is an orbifold.
Crystallographic Orbifolds Types
There are three types of groups in which general crystallography could be divided in terms of crystallographic orbifolds. They are:
This is used for explaining the symmetry of special positions within a space group and is also termed as Wyckoff site symmetries.
The space groups projected along their primary axes of symmetry become plane groups.
In case of space groups, all of them have a parent point group. The main use of space groups are for classification of schemes. It is important to note that order of a space group is always infinite.
The crystallographic orbifolds associated respectively with the above groups are elliptic 2-orbifolds,Euclidean 2-orbifolds, and Euclidean 3-orbifolds. Thus, the above are referred to as point orbifolds, plane orbifolds, and space orbifolds respectively.
Critical Nets and Orbifolds
Morse functions are the basis of critical net in crystallography and this plays a vital role in crystal chemistry and crystallographic topology. Critical net is defined as mathematical mapping from Euclidean 3-space to Euclidean 1-space. This could be used to orbifold so that the Euclidean 1-space of density is deformed vertical in the page. The critical-net-on-orbifold model features the conventional crystallographic invariant lattice complexes and permits concise quotient-space topological figures to be drawn without any repetitions that are attributed to normal crystal structure figures.
Lattice Complex takes a vital role in crystallography and the history of lattice complex started many years before in the branch of crystallography. Lattice complexes refer to the configurations of points that recur at least once but generally repeatedly throughout the family of all space groups. It is important to note that points on symmetry elements have smaller total unit cell occupancy and this is called the Wyckoff site multiplicity.
The critical points are best described as representing 0-, 1-, 2-, and 3-dimensional cells in a topological Morse function. In this, generally non-degenerate crucial points are taken into consideration here because a degenerate crucial point can at all times be distorted into a series of non-degenerate ones via the morsification process. A degenerate critical point will have a singular second derivative matrix with one or more zero or nearly zero eigen values. In fact, the critical points are present where the first derivative with respect to global density will be zero. Also, a 3-3 symmetric matrix occurs as the second derivative at that particular point. Moreover, purely when the critical point is fully non-degenerate, will this have a non-zero determinant.
Color Crystallographic Groups
The group or normal-subgroup relationships could be well defined by using the concept of color crystallographic groups and this concept began in early 1984. The color crystallographic groups possess both symmetry and anti symmetry operators which is used for defining the above relationships in a well structured manner. The crystallographic bicolor group set belonging to each group helps in explaining the one index-2 group or subgroup pair for regular crystallographic groups. In the bicolor crystallographic group, each element has an associated even or odd binary parity flag. This is calculated and arrived based on the product of group generator parities that produce the element.
The concept of anti symmetry plays a vital role in color crystallographic groups. The term anti symmetry was coined by Heesch and Shubnikov. Bicolor groups also referred to as magnetic groups helps to study and explain in detail concurrently the arrangement of atoms also refered as regular symmetry along with the up or down magnetic spin vector orientations which is referred to as anti symmetry for magnetic atoms in any crystal. The use of critical net on orbifold drawing expands in more areas and to name one in this direction is that it is used for explaining the complete summary of the structure’s local and global topology if along with the critical net on orbifold the lattice complex information for each critical point site is added and also the Wyckoff site multiplicities being recorded on the same drawing.
There are totally thirty six cubic crystallographic space groups and 194 space groups. However, the thirty six cubic crystallographic space groups are unlike the 194 space groups. This is because each of them has body diagonal 3-fold axes that arise from their tetrahedral and octahedral point groups. Also, the cubic groups’ orbifolds are uncomplicated as compared to simpler space groups that are derivatives of cyclic and dihedral point groups. The different crystal shapes taken by different crystals is because of the prototypes associated, for instance, the different crystal shapes in muscovite are due to stacking sequence shifts and not due to different atomic structures. The technical term used to refer the different shapes occupied by crystal is polytype. Sometimes, it is also possible to get polytype occurred when substitution causes distortion in the shape. Generally, structural distortion takes place in compounds that crystallize at different temperatures and or pressures.
Thus, the benefits of orbifolds and critical nets on crystallographic orbifolds are that it gives a detailed and concise closed-space portrait of the topology for crystallographic groups and simple crystal structures. In fact, a well detailed crystallographic orbifold atlas, if prepared, would help in giving and projecting the complete tabulation of the topological properties of crystallographic orbifolds. This would help and would be useful to crystallographers in various ways for determining the attribute of crystals.