130 R. W. G. Wyckoff — Theory of Space Groups 



Name Symbol Translations 



1. Simple cubic T^ ± m2T^, ± ilIt,., ± p^r-^. 

 where r^ = Ty = t^ in length and are translations along the axis 

 of the subscript ; m, ?i, and 79 are any integers or zero. When 

 ')7i=:n=p = l, the translations are J|>r^>?^^7^ve translations and locate 

 the adjacent points of the lattice. 



2. Body-centered cubic T/' 2t.^; 2Ty; 2t,.; r.,, r,., t^.'^ (primitive) 



3. Face-centered cubic Vj r^, r,.; r,., t,, ; t,., t^. (primitive). 



As an example of the space and analytical represen- 

 tation of a space gronp, a simple group having the 

 symmetry of the holohedry of the monoclinic system 

 (designated Csi.^) will be considered. The point gronp 

 has already been described as having the following 

 equivalent points : 



xyz, xijz, xijz, xyz. 



The lattice underlying this space gronp is the simple monoclinic 

 lattice (Fm) having the primitive translations 2tx, 2Ty, 27^. This 

 space group is obtained by placing the point group CJ, nn- 

 chanoed, at each point of the lattice. If we choose some point 

 of the lattice, O of figure 3, as the origin, then the coordinates of 

 the points of the group about A, which is the lattice point ob- 

 tained by the primitive translation 2tx, are (from the same origin): 



X + 2tx, y, z\ 2tx— a^, y, 2; 2t^— x,y,z\x^ 2t^, y, z. 



In a similar way the coordinates of equi-oalent points about neigh- 

 boring points of the lattice, and in general about any point of the 

 space group C.i/, are given by one of the following sets: 



£c±2;?i Tx, y±:2n Xy, z±2p t-,_ ; 



±2mTx — .1', +2n Ty — y, 2±2^:>T2 ; 



±2mTx — aj, ±2nTy — y, ±2pT.^ — z; 



X ± 2r/i Tx, y ± 2n Ty, ± 2p t^ — z, where 



jy?, 71, and p are any whole numbers, including zero. 



The 230 space groups, which are all of the possible ways 

 of arranging points in space so that the resulting whole 

 will have crystallographic symmetry, can be obtained in 

 part by distributing the point groups themselves 

 according to the lattices of corresponding symmetry; the 

 rest by distributing in a similar manner groups of points 

 which are developed with the aid of such elements of 

 symmietry as '^ glide planes'' or '^ screw axes.'' 



Every crystal considered as an orderly arrangement 



^^ According to the notation of Schonflies the composite translation having 

 the components r^, r,-, r^ is rej^resented by r^ + Xy + r^. 



