Systematic extinction rules

Systematic extinction rules

The structural factor in the crystallographic diffraction is:

$F=\Sigma { f_j exp(j \vec{r} \vec{k})}$,

where $j$ runs through all the atoms in the unit cell.

When the unit cell has more than one primitive cell in it, there might be some cases that $F$ can be zero. This is called systematic extinction (absence).

For example in BCC, for every atom at position $r$, there is another identical one at $\vec{r}+(1/2,1/2,1/2)$. Then the structural factor becomes:

$F=1+exp[i\pi (h+k+l)]$,

where $(h,k,l)$ are the reciprocal indices.

Obviously, $F$ is zero when $h+k+l=2n+1$.

TABLE I, Systematic absence because of lattice type

Lattice type	Symbol	Absence
a-center (100)	A	k+l=2n+1
b-center (010)	B	h+l=2n+1
c-center (001)	C	h+k=2n+1
face center	F	h,k,l not all odd and not all even
body center	I	h+k+l=2n+1
rhomohedral	R	-h+k+l=3n+1
hexagonal	H (triple unit cell)	h-k=3n+1 or h-k=3n+2
primitive	P	no absence

Above is only the systematic absence caused by typical symmetry of lattice types. There are other systematic absences coming from existence of screw axis and glide planes. In the end, for every space group the systematic absence is more complicated than the cases shown in the above table.

To get complete information or to find out the systematic absence for every space group, one can use Bilbao server : http://www.cryst.ehu.es/. Just click on " HKLCONDReflection conditions of Space Groups" and input the space group number.