Space-group operations differ from point-group operations in that in addition to the point-group operation, a non-primitive translation may be involved. Thus far, we have been using as our example the diamond lattice which is suitable for illustrating space-group operations. 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 must 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 space-group operations are given in the Table.
Table:
Space-group operations for Oh7 (diamond)|
{E|000} |
{C2(1,1,0)|1/2,1/2,1/2} |
{I|1/2,1/2,1/2} | {sd(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} | {sd(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} | {sd(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} | {sd(0,1,1)|000} | sh(0,0,1)|1/2,1/2,1/2} |
| {C2(0,1,0)|000} | {C43(0,1,0)|1/2,1/2,1/2} | {sd(1,-1,0)|000} | sh(0,1,0)|1/2,1/2,1/2} |
| {C2(1,0,0)|000} | {C43(1,0,0)|1/2,1/2,1/2} | {sd(1,0,-1)|000} | sh(1,0,0)|1/2,1/2,1/2} |