Supercells of Three-dimensional Bravais Lattices

Let a (T i) (i=l,2,3) be the basic translation vectors of the initial direct lattice of type Fi and aj F2) j = 1,2,3) be the basic translation vectors of a new lattice of type F2 with the same point symmetry (symmetrical transformation) but composed 

The vectors aj(A) have well-defined orientation with respect to point-symmetry elements of the lattices that are the same for both lattices because of the symmetrical character of the transformation (4.77). Let us define the components of the vectors aj(A) by the parameters 8 assuring their correct orientation relative to the lattice symmetry elements and the correct relations between their lengths (if there are any). Then three vector relations (4.77) give nine linear nonhomogeneous equations to determine nine matrix elements / (AA) as functions of the parameters sj,. The requirements that these matrix elements must be integers define the possible values of the parameters Sk giving the solution of the problem. 

Let us demonstrate the procedure of finding the matrix of a symmetrical transformation (4.77) by the example of the rhombohedral crystal system where there is only one lattice type (R). The basic translation vectors of the initial lattice are the following  

Inserting them in (4.77) one obtains nine equations for nine elements of the matrix 1. The solution of this system is 

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