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.

A definition and a theorem about centric reflections are stated here before the role of centrics is examined.

Definition: A reflection (h,k,l) is said to be centric if in the space group there is at least one symmetry operation whose rotational part sends the reflection to minus itself, i.e.:

(h,k,l) is centric if there is a symop g in G such that R_g*(h,k,l)=(-h,-k,-l)

Theorem: The phase of a centric reflection is restricted to phi(h,k,l)=pi*(h*tx_g+k*ty_g+l*tz_g) plus or minus any number of pi

where the vector (tx_g,ty_g,tz_g) is the translational part of the symop g that causes the reflecion to be centric.

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. Generally, positions on n-fold symmetry axes are special.

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 (at most) 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). Ths, a point on a n-fold rotation axis has (maximum) occupancy of 1/n.

In space group P2₁, 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 P2₁ are thus those with 0,k,0. There exist space groups without centric reflections, like R3.

Properties: centric reflections have only two phase possibilities, e.g. 0° and 180° (but in any case 180° apart), and centric reflections do not have an anomalous signal (can these properties be easily derived here?).

Furthermore, the "intensity statistics" of centric reflections ([math]\displaystyle{ P(|E|) = 2 |E| e^{-|E|^2} }[/math] ) <gnuplot> set xrange [0:5] plot 2*x*exp(-x*x) </gnuplot> are different from those of acentric reflections ([math]\displaystyle{ P(|E|) = \sqrt{\frac{2}{\pi}} e^{-|E|^2/2} }[/math] ) <gnuplot> set xrange [0:5] plot sqrt(2/3.14)*exp(-x*x/2) </gnuplot> .

Centric reflections have a special role in experimental phasing.