396 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1960 



• TWO FOLD AXIS 

 A THREE FOLD AXIS 

 ■ FOUR FOLD AXIS 



Figure 6. — The rotational symmetries of a cube. Except for the magnetization itself, 

 which does define a unique direction, the ferrimagnetic garnets have these symmetries. 



We will discuss each of these and show the relation to domain struc- 

 tures observed in the transparent magnetic garnets. 



The exchange energy is a minimum if all the net moments of 

 neighboring unit cells are exactly parallel to each other. It would 

 be prohibitively large if neighboring cells were to have their moments 

 antiparallel. The result is that if for some other reasons the mag- 

 netization must lie along different directions in different parts of the 

 crystal, the transition fi'om one direction to the other is spread over 

 many cells. The exchange forces are short-range forces. Very little 

 energy is involved if parts of the crystal with different directions of 

 the magnetization are far removed from each other. 



The garnet structure has the symmetry of a cube. A cube has 

 several rotational axes of symmetry. It can be rotated one-quarter 

 of a turn about any of three mutually perpendicular axes, and it 

 will come back to a position indistinguishable from the original. 

 These fourfold axes are parallel to the cube edges. A cube can be ro- 

 tated one- third of a turn about any of the four body diagonals, and it 

 will come back on itself. These are the threefold axes. This is only a 



