Structure factor

The structure factor is a measure of the scattering power of a crystal base. The structure factor indicates the relative intensity of the Miller indices, given the reflection.

• X-ray diffraction: The dispersion of the electromagnetic radiation is of the electrons of the atoms. The structure factor is the Fourier transform of the electron distribution within a unit cell.
• Electron diffraction: the electrons are scattered by the Coulomb interaction of the orbital electrons and the nuclei. The structure factor, the Fourier transform of the charge distribution within a unit cell.
• Neutron diffraction: neutrons interact through strong interaction with the atomic nuclei and because of its magnetic moment with the magnetic moment of the atoms. The structure factor, the Fourier transform of the distribution of the core ( Nukleonenverteilung ) and of the magnetic structure within a unit cell.

The Bragg or, equivalently, the Laue condition go from pure crystal lattice ( the lattice point a point scattering center ) and indicate the direction in which diffraction peaks can be observed. The relative intensity of the reflections depends on the structure of the base, the stray capacity of the base and of atoms of the thermal motion of atoms, it is described by the structural factor.

Description

Choosing a reference point within the unit cell as an origin. Consider two infinitesimal volume elements as scattering centers, one at the reference point, one at. The wave vector of the incident radiation is he who was of the scattered. This results in the following path difference ( path difference ):

The phase difference is ( the scattering is elastic, ie ):

After the Laue condition diffraction peaks can be observed only when the change of the wave vector in the scattering process corresponds to a reciprocal lattice vector. This is used:

Now be integrated over the volume of a unit cell and the weighted phase differences with the scattering power of each volume element. The scattering power varies depending on the bending experiment (see above), the electron density, the charge density or the density of nucleation.

The diffracted wave on the crystal has an amplitude that is proportional to just the calculated size.

Is called the structure factor. This is from the Miller indices, dependent, since the reciprocal lattice vector is the same. The structure factor is thus the Fourier transform of the scattering power (for example, the electron density).

The vector can be written as a linear combination of the primitive lattice vectors. The relation can evaluate the scalar product in the exponent (equivalent ):

The structure factor is a complex quantity. As a result, a diffraction experiment observing the intensity of the diffracted wave, which is proportional to the squared magnitude of the structure factor:

Thus, all phase information is lost. Dignity as a result of a measurement is available, could be found by Fourier transformation of the desired size:

But there is only known to approximation methods, such as the Patterson method, are used to solve the phase problem.

Atomic scattering factor

The position vector is then decomposed into a part which shows the reference point to the core of the -th atom, and a vector pointing from the core of the - th atom to the observed volume element.

In the equation for the structure factor, the integral over the whole unit cell is divided into a smaller sum of integration areas, namely, the volume of the individual atoms. Here, the scattering power (eg electron density ) of the - th atom. The sum runs over all atoms of the unit cell:

The integral is the atomic scattering factor (or atomic form factor) of the - th atom named:

Thus, the structure factor writes as follows:

With the above notation introduced components:

Considering in addition, the thermal motion of atoms, it is time dependent. Now it decomposed into a mean residence ( equilibrium position, resting ) and the displacement ( time-dependent). The latter leads to the Debye-Waller factor.

Example

As an example of the structure factor is calculated for a cesium chloride structure. The grid is so primitive cubic with two polyatomic basis, the primitive lattice vectors are, ,. The one basic atom sits in at the other.

Is the sum of the Miller indices just to the diffracted X-ray having high intensity, with an odd sum, the intensity is at a minimum. If both have the same basic atoms atomic scattering factor, with an odd sum of the intensity is equal to zero; one speaks of complete extinction. This applies to the body-centered cubic lattice ( bcc lattice ), when describing it in the system of the primitive cubic lattice with two identical atoms based: