# Crystal Structure | Geometrical structure factor

## Geometrical structure factor

The relative intensities of various reflections depend upon the following contents of unit cell i.e

i) on the number.

ii) position.

iii) electronic distribution of atoms in a unit cell.

If more atoms are present in the cell, the phase differences of the scattered wave from different atoms must be taken into account.

So, in order to consider the characteristics of the intensity of particular onder reflection and relative intensities of the various reflections, we must be take into account the contributions made by the atoms in the unit cell to the scattered amplitude in a given direction. This is done by calculating a factor F(h,k,l) for a particular reflection (h, k, l); which is known as known as geometrical crystal structure factor.

Definition :  The ratio of the amphitude scattered by entire unit cell to that scattered by a point electron at the origin for the same incident beam is known as geometrical structure factor.

Expression:

F (h,K,l) can be expressed by the following relation,

where

where

reciprocal Lattice points.

Summation over all the atoms in a unil-cell.

where

fj= atomic scattering factor of jth atom of unit cell.

rj= position vector of jth atom.

and  Φj = phase difference between the radiation scattered at the origin and that scattered by jth atom of unit cell.

Suppose the unit cell contains N atoms. If       and       be the fractional position components of jth atom then

Substituting in equation 1, we get

### Geometrical structure factor for b.c.c. lattice

Basis of the bcc structure at (0 0 0)  and (1/2 1/2 1/2)

i.e for 1st atom

for other atom

F = 2f when  h+k+l = even integer and

F = 0 when h+k+l = odd integer