Theory¶
Scattering vector¶
The wave vector \(\vec{k}\) of a beam indicates the beam direction. Its magnitude is defined to be proportional to the spatial frequency (inverse of the wavelength \(\lambda\))
The scattering vector \(\vec{q}\) is defined as the difference between the wave vector of the scattered and incident beam
The scattering angle \(2\theta\) between incident and scattered beam is
For elastic scattering \(\lambda_s = \lambda_0\) the length of the scattering vector is related to the scattering angle as follows
We define the normal vector \(\vec{n}_H\) to a family of lattice planes \(H = \{hkl\}\) as
The scattered intensity from a crystal is maximal in the directions where the lattice planes fulfill the Laue conditions for diffraction
When only considering the vector lengths you get Bragg’s law
where \(\lambda\) and \(E\) the incident beam wavelength and energy respectively.
Strain tensor¶
The strain tensor in the direction of the scattering vector is
where \(\phi\) and \(\psi\) are the azimuth and polar angle of the scattering vector \(\vec{q}^{hkl}\) in the sample reference frame.
For angular dispersive diffraction (fix energy)
For energy dispersive diffraction (fix angle)