Centric and acentric reflections

Centric reflections play the same role in reciprocal space as special positions in real space: they may occur at the borders of the asymmetric unit. It makes sense to discuss the reciprocal case and the real space together.

Real space

Let's first look at real space. A special position results if there exists one or more symmetry operators, other than the trivial operator {x,y,z}, which map this position upon itself. As a example: take spacegroup P2 with its symmetry operators {x,y,z} and {-x,y,-z}. Now consider any point with x=0 and z=0, and some value of y. Obviously this point, when transformed with -x,y,-z , yields 0,y,0 - just the same point! Thus this is a special position.

Mathematically, to find special positions we have to solve the Eigenproblem A v = v where A is the symmetry operator (expressed as rotation matrix and translation vector), and v, the Eigenvector, represents the special position(s). For a given space group, we need to check all symmetry operators.

An atom at a special position usually has an occupancy of 0.5. However, it may happen that more than one symmetry operator maps the special position upon itself; in that case the occupancy is 1/(number of symmetry operators mapping point onto itself).

In space group P2_1, there are no special positions - the Eigenproblem has no solution.

Reciprocal space

In reciprocal space the situation is quite similar. A reflection is centric if there is a reciprocal space symmetry operator which maps it onto itself. Reciprocal space symmetry operators can be obtained from the real space symmetry operators by following two rules:

a) take the rotation matrix and transpose it

b) omit the translation vector

Whereas rule b) makes things slightly easier in reciprocal space, we must be aware that in reciprocal space we have additional symmetry, namely Friedel symmetry. This means for each reciprocal symmetry operator we also have to consider the Friedel-related operator (all elements of the matrix multiplied by -1).

To find centric reflections, we just solve the Eigenvalue problem A v = v, now considering each reciprocal space symmetry operator in turn. Centric reflections in space group P2 and P2_1 are thus those with 0,k,0.

Attention: there are space groups without centric reflections, like R3.