340 R. W. G. Wyckoff— Crystal Structures of 



tice is rhombohedral, r,, r . Calcite must then be assigned 

 to one of the two space groups D 5 3Cl or L>" 3 , which have r,. h 

 as the underlying lattice. These two space groups are 

 derived from the space group D 7 3 by multiplying it by an 

 inversion. D 7 a is obtained by placing a point group D s at 

 each point of the lattice, r,. h . If the point of inversion is 

 at the intersection of a trigonal axis and a perpendicular 

 digonal axis, B\ a results ; if the inversion is about a point 

 midway between two such intersections, D 6 3d is obtained. 



The coordinates of equivalent points of the space group 

 D 5 3d are then : 



-x y g, y z x, z x y, y x z, x z y, z y x, 



x y z,yz x, z x y, y x z, x z y, z y x. 



Those of D" 3Cl are: 



x y z; y z x; z x y; 



t x -x, Ty-y, t z -z; r x —y, r y -z, r z -x; r x -z, r y -x, r z -y; 



y x z; x z y; z y x; 



T x + y,' T y + ^ ? T z + z; t x + x, t y +$, r z + y; r x + z, T y + y,r z i-x; 



where 



is a translation of half the length of a side of the unit 

 rhombohedron (half a primitive translation) in the direc- 

 tion of the subscripted axis. 



All the special cases of these two space groups where 

 the twelve equivalent points are reduced to one, two, 

 three, four and six equivalent positions, can be obtained 

 by equating one equivalent point with each of the others. 

 (The length of the side of the unit equals one.) 



One equivalent point. 



{a) 0. (b) 1/2 1/2 1/2 



Two equivalent points. 



(a) u u u, u u it. 

 Three equivalent points. 



(a) 1/2, 1/2 0, 1/2 0. 



(b) 1/2 1/2, 1/2 1/2 0, 1/2 1/2. 



