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)|e^{i(h)}=

i=1,N g(i, S)e ^{2i (hxi+kyi+lzi)}where N = number of atoms

Grouping symmetry-related atoms together:

F( h)=

i=asymm.

unitg(i, S)(

e ^{2i (h k l)}^{æxö çy÷ èzø}+

e ^{2i h·[Si]}^{æxö çy÷ èzø}+.... ) = |F( h)| e^{ih}

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:

[h _{2}k_{2}l_{2}] = [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(xi+Ox)+k(yi+Oy)+l(zi+Oz)}}= e^{2ih·x}e^{2ih·O}

for all atoms, and the structure factor now equals

|F| e^{i}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 P3_{1}
symmetry converts to P3_{2}).

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 P2_{1}2_{1}2_{1}, with symmetry-related positions x,y,z and
-x+½,y+½,-z, for zone (h 0 l):