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)

e2i (hxi+kyi+lzi)

where N = number of atoms

Acentric reflections

Grouping symmetry-related atoms together:

F(h)	=	i=asymm. unit	g(i,S)	(	e2i (h k l) æxö çy÷ èzø	+	e2i h·[Si] æxö çy÷ èzø	+....	)
	=	|F(h)| eih

An aside: from this expression it is easy to show that the symmetry equivalent reflection h',k',l' is [h k l][S_i]. 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 (x_i,y_i,z_i) by (x_i+Ox, y_i+Oy, z_i+Oz), the structure factor contribution becomes

e^{2i{h(x_i+Ox)+k(y_i+Oy)+l(z_i+Oz)}} = e^2ih·x e^2ih·O

for all atoms, and the structure factor now equals

|F| eⁱ e^2ih·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 x_i,y_i,z_i are replaced by (-x_i,-y_i,-z_i) and none of the atoms scatter anomalously. In this case
|F(h)| e^i(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):