The diffraction effect which occurs when X-rays are scattered from a crystalline material may be formulated in a simple manner by considering the diffraction of a plane polarised X-ray from a small crystal placed at the origin made up of unit cells defined by the three vectors , and . The source is a large distance away form the crystal so that the incident X-rays can be assumed to be plane waves with wavelength . Similarly the point of observation is a large distance from . Fig. shows the geometry of the scattering. The incident and scattered wave vectors are labeled and respectively. The electric field vector of the incident X-ray is assumed to be normal to the plane of the paper. The th atom in the th unit cell has position vector and atomic scattering factor . is given by
where is the atoms position vector with respect to the origin of any unit cell and , and are fractional coordinated of the th atom in the unit cell.
Figure: A plane wave represented by the beam vector is incident on a small crystal containing N unit cells defined by the vectors , and . The th atom in the th unit cell is represented by its position vector . The scattered radiation represented by is observed at a point P, a large distance away, such that . (taken from Warren )
The instantaneous electric field at is given by
since for , . The distances and are equal to and respectively. Thus by expanding with Eq. we may write as
To obtain the total electric field at we must sum over all atoms in the unit cell and over all unit cells. Assuming the crystal to be a simple parallelopipedon with sides of length , and where , the total field is given by
The first summation term in Eq. is called the structure factor, , since it is dependent solely on the positions of the constituent atoms in the unit cell.
Each of the last three summation terms in Eq. take the form of the geometric progression with
Thus the sum in the -direction can then be written
with similar expressions for the and summations. If we now multiply in Eq. by its complex conjugate then with some manipulation of the expanded summation terms we obtain an expression for the intensity of the scattered radiation for a polarised incident beam
For very large values of , and the intensity will be have very sharp maxima when the equations
are satisfied. These are the Laue equations stipulating the diffraction condition. The integers , and are the corresponding Miller indices of the Bragg plane of reflection.
By considering a general reciprocal lattice vector it can be shown that the Laue equations are equivalent to the Bragg equation which in vector form may be written
where is the reciprocal lattice vector corresponding to the Bragg plane with Miller indices . Given Eq. and our definition of we can now rewrite the structure factor (Eq. ) in the more commonly seen form
In practice the incident X-ray beam is not parallel and the sample crystals are in general mosaic meaning that they diffract as though they were made up of a very large number of smaller crystals packed together at slightly differing orientations. The diffraction maxima are therefore much broader and lower than predicted by Eq. . The measurable quantity of a diffraction experiments is the integrated intensity, i.e. the total intensity resulting from integrating under a finite width diffraction profile.
The definition of the structure factor in Eq. is valid for a crystal in which all the atoms are stationary and present in every single unit cell in the crystal. In reality the atoms are continually vibrating about an average position due to the effects of temperature and particularly for the case of heavy atoms introduced into protein crystals, the heavy atom may not be taken up by every protein molecule. Thermal vibration causes a reduction in intensity of the Bragg peaks. If the th atoms thermal motion is described by an isotropic mean squared displacement about its mean position then we can define a temperature factor for that atom such that
The second effect is accounted for by an atomic occupancy factor which is the fraction of unit cells in the crystal which contain the atom in question. The structure factor equation including the effects of isotropic thermal vibration and the atomic occupancy is then written