Basic Maths Definitions for Protein Crystallographers

	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^if(h) =

S

i=1,all atoms

g(i,S) e^{2pi
(hx_i+ky_i+lz_i)}

Acentric reflections

Grouping symmetry-related atoms together:

F(h)

=

S

i=asymm.
unit

g(i,S)

(
e^{2pi (h k l)}
^{æxö
çy÷
èzø}
+
e^2pi
h·[S_i]
^{æxö
çy÷
èzø}
+....

)
= |F(h)| e^if_h

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)

^{é 0  1 0ù
ê-1 -1 0ú
ë 0  0 1û}

= (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 (x_i,y_i,z_i) by (x_i+Ox, y_i+Oy, z_i+Oz), the structure factor contribution becomes

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

for all atoms, and the structure factor now equals

|F| e^ife^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 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^if(h) becomes |F(h)| e^-if(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 f_c or f_c + p:

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

		e^{2pi(h·[S_i](x))} + e^{2pi(h·[S_j](x))}
	=	e^2pif_c (e^2pif + e^-2pif)
	=	e^2pif_c (2cos if)

The phase can then only be

f_c if cosf>0, or f_c+p if cosf<0

In fact the only values f_c can take are 0, p/6, p/4, p/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):

e^2pi(hx+lz) + e^{2pi(h(½-x)-lz)}

= e^2pi(hx+lz) + e^(2pih)/2 · e^-2pi(hx+lz)

= cos(2p(hx+lz)) + i sin(2p(hx+lz)) + e^pih {cos(2p(hx+lz)) - i sin(2p(hx+lz))}

Since e^pih is 1 if h is even, and -1 if h is odd:

= 2 cos 2p(hx+lz) for h even, and all phases are 0 or p

= 2i sin 2p(hx+lz) for h odd, and all phases are p/2 or -p/2

	e^2pi(hx+lz) + e^{2pi(h(½-x)-lz)}
=	e^2pi(hx+lz) + e^(2pih)/2 · e^-2pi(hx+lz)
=	cos(2p(hx+lz)) + i sin(2p(hx+lz)) + e^pih {cos(2p(hx+lz)) - i sin(2p(hx+lz))}
Since e^pih is 1 if h is even, and -1 if h is odd:
=	2 cos 2p(hx+lz)	for h even, and all phases are 0 or p
=	2i sin 2p(hx+lz)	for h odd, and all phases are p/2 or -p/2