International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. D. ch. 2.2, p. 295

Section 2.2.3.4. The important groups and their first classification

K. Schwarza*

a Institut für Materialchemie, Technische Universität Wien, Getreidemarkt 9/165-TC, A-1060 Vienna, Austria
Correspondence e-mail: kschwarz@theochem.tuwein.ac.at

2.2.3.4. The important groups and their first classification

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Using the Seitz operators, we can classify the most important groups as we need them at the beginning of this chapter:

  • (i) the space group, which consists of all elements [G=\{\{p|{\bf w}\}\}];

  • (ii) the point group (without any translations) [\ P=\{\{p|{\bf 0}\}\}]; and

  • (iii) the lattice translation subgroup [T=\{\{E|{\bf T}\}\}], which is an invariant subgroup of G, i.e. [T\triangleleft G]. Furthermore T is an Abelian group, i.e. the operation of two translations commute ([t_{1}t_{2}=t_{2}t_{1}]) (see also Section 1.2.3.1[link] of the present volume). A useful consequence of the commutation property is that T can be written as a direct product of the corresponding one-dimensional translations, [T=T_{x}\otimes T_{y}\otimes T_{z}.\eqno(2.2.3.15)]

  • (iv) A symmorphic space group contains no fractional translation vectors and thus P is a subgroup of G, i.e. [P\triangleleft G].

  • (v) In a non-symmorphic space group, however, some p are associated with fractional translation vectors [{\bf v}]. These [{\bf v}] do not belong to the translation lattice but when they are repeated a specific integer number of times they give a vector of the lattice. In this case, [\{p|{\bf 0}\}] can not belong to G for all p.

  • (vi) The Schrödinger group is the group S of all operations [\widetilde{g}] that leave the Hamiltonian invariant, i.e. [\widetilde {g}{\bb H}\widetilde{g}^{-1}={\bb H}] for all [\widetilde{g}\in S]. This is equivalent to the statement that [\widetilde{g}] and [{\bb H}] commute: [\widetilde{g}{\bb H}={\bb H}\widetilde{g}]. From this commutator relation we find the degenerate states in the Schrödinger equation, namely that [\widetilde{g}\varphi] and [\varphi] are degenerate with the eigenvalue E whenever [\widetilde{g}\in S], as follows from the three equations [\eqalignno{{\bb H}\varphi &=E\varphi &(2.2.3.16)\cr \widetilde{g}{\bb H}\varphi &=E\widetilde{g}\varphi &(2.2.3.17)\cr {\bb H}\widetilde{g}\varphi &=E\widetilde{g}\varphi .&(2.2.3.18)}%fd2.2.3.18]








































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