Double space groups and their representations

Janssen, T.

doi:10.1107/97809553602060000629

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. D. ch. 1.2, pp. 50-51

Section 1.2.3.5. Double space groups and their representations

T. Janssen^a ^*

^a Institute for Theoretical Physics, University of Nijmegen, 6524 ED Nijmegen, The Netherlands
Correspondence e-mail: ted@sci.kun.nl

1.2.3.5. Double space groups and their representations

| top | pdf |

In Section 1.2.2.9, it was mentioned that the transformation properties of spin- $[{{1}\over{2}}]$ particles under rotations are not given by the orthogonal group O(3), but by the covering group SU(2). Hence, the transformation of a spinor field under a Euclidean transformation g is given by $[g\Psi ({\bf r}) = \pm U(R)\Psi (R^{-1}({\bf r}-{\bf a})) \quad \forall \,\, g = \{R|{\bf a}\} \in E(3), \eqno (1.2.3.27)]$ where the SU(2) operator [U(R)] is given by $[U(R) = E\cos (\varphi /2) + ({\boldsigma}\cdot{\bf n})\sin (\varphi /2) \eqno (1.2.3.28)]$ when the rotation R has angle $[\varphi]$ and axis $[{\bf n}]$ . When R does not belong to SO(3) one has to take [U(-R] ).

For an ordinary space group, one can construct the double space group by $[\{R|{\bf a}\} \rightarrow \{\pm U(R)|{\bf a}\} \eqno (1.2.3.29)]$ with multiplication rule $[\{U(R)|{\bf a}\}\{U(S)|{\bf b}\} = \{U(R)U(S)|{\bf a}+R{\bf b}\}. \eqno (1.2.3.30)]$ An invariant subgroup of the double space group is the translation group A. The factor group is the double point group $[K^{d}]$ of the point group K.

The representations of the double space groups can be constructed in the same way as those of ordinary space groups. They are characterized by a vector $[{\bf k}]$ in the Brillouin zone and a label for an irreducible, generally projective, representation of the (double) point group $[K_{{\bf k}}^{d}]$ of $[{\bf k}]$ , which is the double group of $[K_{{\bf k}}]$ . Again, for nonsymmorphic space groups or wavevectors $[{\bf k}]$ inside the Brillouin zone, the relevant irreducible representations of $[K_{{\bf k}}^{d}]$ are ordinary representations with a trivial factor system.

For an element g of the space group G, there are two elements of the double space group $[G^{d}]$ . If one considers an irreducible representation $[D(G^{d})]$ for the double space group and takes for each $[g\in G]$ one of the two corresponding elements in $[G^{d}]$ , the resulting set of linear operators forms a projective representation of the space group. It is also characterized by a vector $[{\bf k}]$ in the Brillouin zone and a projective representation of the point group (not its double) $[K_{{\bf k}}]$ . This projective representation does not have the same factor system as discussed in Section 1.2.3.3, because the factor system now stems partly from the nonprimitive translations and partly from the fact that a double point group gives a projective representation of the ordinary point group $[K_{{\bf k}}]$ .

The projective representations of a space group corresponding to ordinary representations of the double space group again are characterized by the star of a vector $[{\bf k}]$ . The projective representation of the group $[G_{{\bf k}}]$ then is given by $[P_{{\bf k}}(\{R|{\bf a}\}) = \exp (i{\bf k}\cdot{\bf a})\Gamma (R), \eqno (1.2.3.31)]$ where the projective representation $[\Gamma (K_{{\bf k}})]$ has the factor system $[\eqalignno{\Gamma (R)\Gamma (S) &= \omega_{s}(R,S) \exp [-i({\bf k}-R^{-1}{\bf k})\cdot{\bf a}(S) ] \Gamma (RS) &\cr&= \omega (R,S)\Gamma (RS), & (1.2.3.32)}]$ where $[\omega_{s}]$ is the spin factor system for $[K_{{\bf k}}]$ and $[{\bf a}(S)]$ is the nonprimitive translation of the space-group element with orthogonal part S. The factor system $[\omega]$ can be characterized by the defining relations of $[K_{{\bf k}}]$ . If these are the words $[W_{i}(A_{1},\ldots, A_{p}) = E,]$ then the factor system $[\omega]$ is characterized by the factors $[\lambda_{i}]$ in $[W_{i}(\Gamma (A_{1}),\ldots, \Gamma (A_{p})) = \lambda_{i}E. \eqno (1.2.3.33)]$ The factors $[\lambda_{i}]$ are the product of the values found from the spin factor system $[\omega_{s}]$ and those corresponding to the factor system for an ordinary representation [equation (1.2.3.26)].

References

International Tables for Crystallography (2006). Vol. D. ch. 1.2, pp. 50-51