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 (see reference [1]) 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 g(x)=R_g*x+t_g whose rotational part R_g sends the reflection to minus itself, i.e.:
(h,k,l) is centric if there is a symop g(x)=R_g*x+t_g in G such that R_g*(h,k,l)=(-h,-k,-l)
For example all reflection in the zone (h,k,0) are centrics in all space groups with twofold axes down c.
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 integer multiple of pi
where the vector t_g=(tx_g,ty_g,tz_g) is the translational part of the symop g that causes the reflection to be centric.
Real spaceEdit
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 an 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 positions generated by all symmetry operators that map the point onto itself). Thus, a point on a n-fold rotation axis has (maximum) occupancy of 1/n. Disorder or partial occupation will result in lower occupancy.
In space group P21, there are no special positions - the Eigenproblem has no solution.
Reciprocal spaceEdit
As mentioned above, a reflection is centric if there is a reciprocal space symmetry operator which maps it onto its Friedel mate (-h,-k,-l). 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 resulting from operator R_g = (-h, k, -l) are thus those with h,0,l. There exist space groups without centric reflections, like R3.
Other (equivalent) definitions of centric reflections:
Rupp: Centric structure factors are centrosymmetrically related reflections that are additionally related by the point group symmetry of the crystal.
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, with E being the structure factor, the statistical distribution [2] of acentric reflections is [math]\displaystyle{ P(|E|) = 2 |E| e^{-|E|^2} }[/math]
which is different from those of centric reflections; these follow [math]\displaystyle{ P(|E|) = \sqrt{\frac{2}{\pi}} e^{-|E|^2/2} }[/math]
.
Centric reflections have a special role in experimental phasing.
The moments of intensities (centric [math]\displaystyle{ \lt I^2\gt /\lt I\gt ^2=3 }[/math] ; acentric [math]\displaystyle{ \lt I^2\gt /\lt I\gt ^2=2 }[/math]) can be calculated from the above formulas [3]; the first result can e.g. be obtained with Wolframalpha using (sqrt(2/Pi) * integral ( x^4 exp(-0.5x^2) from 0 to inf )) / (sqrt(2/Pi) * integral ( x^2 exp(-0.5x^2) from 0 to inf ))^2, where the x stands for the E in the formula above.
ReferencesEdit
[1] G. Bricogne, "Fourier Transforms in Crystallography", page 68, Chapter 1.3, International Tables for Crystallography Volume B, Kluwer Publishers (2001)
[2] U. Shmueli and A. J. C. Wilson, "Statistical properties of the weighted reciprocal lattice", page 190-209, Chapter 2.1, International Tables for Crystallography Volume B, Kluwer Publishers (2006)