Space-group operations

Space-group operations differ from point-group operations in that in addition  to the point-group operation, a non-primitive translation may be involved. We can use as our example the diamond lattice. For convenience, we will specify a space-group operation thus: {R|T}, where "R" is a point-group operation, e.g. C2(0,0,1) or S6(1,1,-1), and "T" is a non-primitive translation, e.g. ( 1/2,1/2,1/2), or (0,0,0). The term in parentheses following the point group operation indicates the axis about which the operation is to be performed. Finally, to complete the specification of a space-group operation, the point about which the operation acts would need to be defined. As this is most conveniently done in fractional unit cell coordinates, or crystal coordinates, the Cartesian coordinates are converted at this time into fractional unit cell coordinates.

 Diamond belongs to the Fd3m or Oh7 space-group, and has octahedral symmetry; its associated point-group is Oh. The 48 space-group operations of Oh7 are given in the Table.


Space-group operations for Oh7 (diamond)



{I|1/2,1/2,1/2} {Σd(0,1,-1)|000}
{C3(1,1,1)|000} {C2(1,0,1)|1/2,1/2,1/2} {S4(0,0,1)|000} {S6(1,1,1)|1/2,1/2,1/2}
{C3(1,-1,-1)|000} {C2(0,1,1)|1/2,1/2,1/2} {S4(0,1,0)|000} {S6(1,-1,-1)|1/2,1/2,1/2}
{C3(-1,1,-1)|000} {C2(0,1,-1)|1/2,1/2,1/2} {S4(1,0,0)|000} {S6(-1,1,-1)|1/2,1/2,1/2}
{C3(-1,-1,1)|000} {C2(1,0,-1)|1/2,1/2,1/2} {S42(0,0,1)|000} {S6(-1,-1,1)|1/2,1/2,1/2}
{C32(1,1,1)|000} {C2(1,-1,0)|1/2,1/2,1/2} {S42(0,1,0)|000} {S65(1,1,1)|1/2,1/2,1/2}
{C32(1,-1,-1)|000} {C4(0,0,1)|1/2,1/2,1/2} {S42(1,0,0)|000} S65(1,-1,-1)|1/2,1/2,1/2}
{C32(-1,1,-1)|000} {C4(0,1,0)|1/2,1/2,1/2} {Σd(1,1,0)|000} S65(-1,1,-1)|1/2,1/2,1/2}
{C32(-1,-1,1)|000} {C4(1,0,0)|1/2,1/2,1/2} {Σd(1,0,1)|000} S65(-1,-1,1)|1/2,1/2,1/2}
{C2(0,0,1)|000} {C43(0,0,1)|1/2,1/2,1/2} {Σd(0,1,1)|000} Σh(0,0,1)|1/2,1/2,1/2}
{C2(0,1,0)|000} {C43(0,1,0)|1/2,1/2,1/2} {Σd(1,-1,0)|000} Σh(0,1,0)|1/2,1/2,1/2}
{C2(1,0,0)|000} {C43(1,0,0)|1/2,1/2,1/2} {Σd(1,0,-1)|000} Σh(1,0,0)|1/2,1/2,1/2}

The center for each operation is not shown, however it must be present when symmetry operations are defined for use by program BZ