lattice system
A lattice system is a class of lattices with the same set of lattice point groups, which are subgroups of the arithmetic crystal classes. The 14 Bravais lattices are grouped into seven lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic.
crystal system (symmetry)
In a crystal system, a set of point groups and their corresponding space groups are assigned to a lattice system. Of the 32 point groups that exist in three dimensions, most are assigned to only one lattice system, in which case both the crystal and lattice systems have the same name. However, five point groups are assigned to two lattice systems, rhombohedral and hexagonal, because both exhibit threefold rotational symmetry. These point groups are assigned to the trigonal crystal system. In total there are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic.
Unit cell
A unit cell is the most basic and least volume consuming repeating structure of any solid. It is used to visually simplify the crystalline patterns solids arrange themselves in. When the unit cell repeats itself, the network is called a lattice
Primitive cell
A primitive cell is a unit cell built on the basis vectors of a primitive basis of the direct lattice, namely a crystallographic basis of the vector lattice L such that every lattice vector t of L may be obtained as an integral linear combination of the basis vectors, a, b, c
Conventional cell
For each lattice, the conventional cell is the cell obeying the following conditions: its basis vectors define a right-handed axial setting; its edges are along symmetry directions of the lattice; it is the smallest cell compatible with the above condition.
Wigner-Seitz primitive cell
Wigner-Seitz primitive cell about a lattice point is the region of space that is closer to that point than to any other lattice point.
Result: Space groups express different possible symmetries
How to represent diferent lattices with different possible symmetries ?
1- Cubic lattice
1.1. Simple cubic SC
The simple cubic lattice is represented with only cubic symmetry because the primitive cell is cubic.
1.2. Centered cubic BC
The BC lattice may be represented with cubic and rhombohedral symmetries because the primitive cell is rhombohedral.
The body-centered cubic lattice can be considered as a rhombohedral lattice where
1.3. Face centered cubic FCC
The FCC lattice may be respresented with cubic and rhombohedral symmetries because the primitive cell is rhombohedral.
The face-centered cubic lattice can be considered as a rhombohedral lattice where .
Description of a cubic (primitive, body centred or face centred) unit cell (ac) in terms of the equivalent, primitive rhombohedral, (ar, ) and triple-primitive hexagonal, cells (ah, ch). See the following figure (from reference number 2)
References:
1- https://www.tcd.ie/Physics/study/current/undergraduate/lecture-notes/py3p03/Lecture2_2014.pdf
2- Intermettalic chemistry
2. Rhombohedral lattice
The rhombohedral lattice may be represented with rhombohedral and hexagonal symmetries.
Reference: Intermettalic chemistry
NB: The rhombohedral cell has a nearly cubic shape (αr 90°) when ch/ah ( 3/2) 1.2247…
A detailed example of the alternative descriptions of a given compound, both in terms of its hexagonal unit cell and of the corresponding rhombohedral primitive cell is presented in Chapter 4: the rhombohedral compound Mo6PbS8 (the prototype of the family of the so-called Chevrel phases) is described and unit cell constants and atomic positions are listed for its conventional hexagonal cell and for the rhombohedral primitive cell.( Reference 2)
For more information check the following video:
To convert between rhombohedral and hexagonal symmetries use the following link:
https://qpeng.org/tools/r2h.html
NB: The BC and FCC lattices may be represented with the rhombohedral symmetry and the rhombohedral lattice may be represented with the hexagonal symmetry, so we can represent the BC and FCC lattices with the hexagonal symmetry.
3. Hexagonal lattice
Description of a hexagonal unit cell (ah, ch cell edges) in terms of an ortho-hexagonal cell (equivalent orthorhombic cell a0, b0, c0) (see Fig. 3.10).
The relation between NiAs and MnP
The phase transition of the NiAs type to the MnP type has been the subject of thorough studies with a number of compounds (e.g. VS, MnAs) [129, 130]. The symmetry reduction involves two steps (Fig. 11.9). In the first step the hexagonal symmetry is lost; for this a slight distortion of the lattice would
be sufficient. The orthorhombic subgroup has a C-centred cell. Due to the centring, the cell is translationengleiche, although its size is twice as big. The centring is removed in the second step, half of the translations being lost; therefore, it is a klassengleiche reduction of index 2.
The images in Fig. 11.9 show what symmetry elements are eliminated with the two steps of symmetry reduction. Among others, half of the inversion centres are being lost. Here we have to watch out: the eliminated inversion centres of the space group Cmcm (C2/m2/c21/m) are those of the Wyckoff positions 4a (0, 0, 0) and 4b (1/2 , 0, 0), while those of the Wyckoff position 8d (1/4 ,1/4 ,0) are retained. Since the subgroup Pmcn (P21/m21/c21/n) should have its origin on a point of inversion, an origin shift is required. The shift of −1/4 ,−1/4 ,0 entails an addition of 1/4 ,1/4 ,0 to the coordinates.
After addition of 1/4 ,1/4 ,0 to the coordinates listed in Fig. 11.9 for the space group Cmcm, one obtains ideal values for an undistorted structure in the space group Pmcn. However, due to the missing distortion, the symmetry would still be Cmcm. The space group Pmcn is attained only after the atoms have been shifted away from the ideal positions. The deviations concern mainly the y coordinate of the Mn atom (0.196 instead of 1/4 )andthe z coordinate of the P atom (0.188 instead of 1/4 ). These are significant deviations, but they are small enough to consider MnP as being a distorted variant of the NiAs type.
For more information check the book of reference 1
4- Tetragonal lattice
Reference: Crystallography and the World of Symmetry Page 32
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