 Basic Maths for Protein Crystallographers Structure factor

There are many atoms in the unit cell and the reflections we see are the sum of all their diffraction waves.

F(h k l) or F(h) = |F(h)|ei (h) = i=1,N
g(i,S) e2 i (hxi+kyi+lzi)     where N = number of atoms

#### Acentric reflections

Grouping symmetry-related atoms together:

F(h) = i=asymm.unit
g(i,S) (
 e2 i (h k l) ```æxö çy÷ èzø```
+
 e2 i h·[Si] ```æxö çy÷ èzø```
+.... )
= |F(h)| ei h

An aside: from this expression it is easy to show that the symmetry equivalent reflection h',k',l' is [h k l][Si]. This means it is NOT always possible to simply replace x,y,z with h,k,l in the International Tables notations. In particular for a 3fold:

 [h2 k2 l2] = [h k l] = [k (-h-k) l]

For acentric reflections the phase for each atom is randomly distributed: If the atoms are positioned relative to a different origin, the phase of the structure factor will change but not its magnitude. Replacing (xi,yi,zi) by (xi+Ox, yi+Oy, zi+Oz), the structure factor contribution becomes

e2 i{h(xi+Ox)+k(yi+Oy)+l(zi+Oz)} = e2 ih·x e2 ih·O

for all atoms, and the structure factor now equals

|F| ei e2 ih·O

A list of alternative origins is available in \$CHTML/alternate_origins.html.

The magnitude of the structure factor is also the same if the atoms are on a different hand, i.e. all xi,yi,zi are replaced by (-xi,-yi,-zi) and none of the atoms scatter anomalously. In this case
|F(h)| ei (h) becomes |F(h)| e-i (h).

N.B.: For some space groups, changing the hand of the atoms also changes the symmetry operators, e.g. a 1/3 stepping screw axis will convert to a -1/3 stepping axis (i.e. the P31 symmetry converts to P32).

#### Centric reflections

 For centric reflections the phase for atom pairs are related such that the contributions from two atoms of a pair always equal c or c + : Each atom has a symmetry partner such that their combined contribution to the structure factor can be written as: The phase can then only be In fact the only values c can take are 0, , etc.

As an example in spacegroup P212121, with symmetry-related positions x,y,z and -x+½,y+½,-z, for zone (h 0 l): 