CCP4 web logo Basic Maths for Protein Crystallographers
Structure factor
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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)|eiphi(h) =

large capital Sigma
g(i,S) e2pii (hxi+kyi+lzi)     where N = number of atoms

Acentric reflections

Grouping symmetry-related atoms together:

F(h) =

large capital Sigma
g(i,S) (
e2pii (h k l)
e2pii h·[Si]
+.... )
= |F(h)| eiphih

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] matrix for 3fold symmetry = [k (-h-k) l]

For acentric reflections the phase for each atom is randomly distributed: phase sum in vector representation

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

e2pii{h(xi+Ox)+k(yi+Oy)+l(zi+Oz)} = e2piih·x e2piih·O

for all atoms, and the structure factor now equals

|F| eiphi e2piih·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)| eiphi(h) becomes |F(h)| e-iphi(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 phic or phic + pi: phase sum in vector representation for
centric reflections

Each atom has a symmetry partner such that their combined contribution to the structure factor can be written as:

structure factor expressions for centric reflections

The phase can then only be

phases for centric reflections (phi or phi+pi)

In fact the only values phic can take are 0, pi/6, pi/4, pi/3, 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):

example of structure factor and phase calculation