Translational subgroup

By choosing a finite index subgroup H of the group G (i.e. such that there exist gi, , gm G with G = UigiH) of deck transformations and taking the quotient, we can obtain a bigger torus such tori have a translation subgroup, which is isomorphic to the quotient G/ff. 

On the other hand, given a torus with non-trivial translation group, there exists a unique minimal torus with the same universal cover and trivial translation subgroup. Those minimal tori correspond, in a one-to-one way, to periodic tilings of the plane. 

We obtain a ( 5,8 + , 3)-plane that is (8 + ri)R2. All above planes are periodic hence, by taking the quotient (by a translation subgroup of their automorphism group), we obtain ( 5, b, 3)-tori that are bR2. 

We now remove the inconvenience of the translation subgroup, and consequently the Bravais lattice, being infinite by supposing that the crystal is a parallelepiped of sides Aja,-where ay, j 1,2,3, are the fundamental translations. The number of lattice points, N1N2N3, is equal to the number of unit cells in the crystal, N. To eliminate surface effects we imagine the crystal to be one of an infinite number of replicas, which together constitute an infinite system. Then... 

Finally, the matrices Tk(if w) in Table 16.10 have to be multiplied by the appropriate representation rk(/i t) of the translation subgroup to give the space-group representations... 

These functions form the irreducible representations of the translational subgroup since... 

The use of symmetry—at least the translational subgroup—is essential to modem first-principles calculations on crystalline solids. Group theory is simplest for Abelian groups such as the translational subgroup of a crystal or the six-fold-rotational subgroup of the benzene molecule. For such simple cyclic groups, the irreducible representations are characterized by a phase, exp(ifc), associated with each step in a direction of periodicity. For one-dimensional (or cyclic) periodicity,... 

Let operator g = t R correspond to the coset representative t R in the coset decomposition of the crystal structure space group G over translation subgroup T (see (2.15)). In this notation the density matrix point symmetry can be written in the form... 

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