FIGURE 2.1: The geometrical concept of "parallel" and "antiparallel" expressed in terms of rotations.

Furthermore, consider a given vector n0 = (n0,x, n0,y, n0,z) and the following set S of vectors nk = (nk,x, nk,y, nk,z)

S = {nk| D(2)(R) µ nk,x nk,y

¶ =

µ n0,x n0,y

¶ ,

nk,z = n0,z ± ka, k = 0, 1, 2, ...,K} . (2.4) This set consists of vectors nk that are collinear to n0 (with respect to the z axis, z = (0, 0, 1)), if in Eq. (2.4) D(2)(R)= ±I2, i.e., if for all k, R is either the identity operation E or the "inversion" i,

D(n)(E) = In , D(n)(i) = −In , n = 2 . (2.5)

If this is not the case then S is said to be non-collinear to n0. Obviously the above description is not restricted to rotations around the

z axis. The only requirement is that the three-dimensional rotation matrix can be partitioned into two irreducible parts, namely a one-dimensional and a

two-dimensional one. The one-dimensional part reflects the rotation axis. It should be noted that although these definitions already sound like a description of magnetic structures they are not: what is meant is a simple geometrical construction with no implications for physics.