X-ray Crystal Truncation Rod - Theory

Theory

A particle incident on a crystalline surface with momentum will undergo scattering through a momentum change of . If and represent directions in the plane of the surface and is perpendicular to the surface, then the scattered intensity as a function of all possible values of is given by

Where is the penetration coefficient, defined as the ratio of x-ray amplitudes scattered from successive planes of atoms in the crystal, and, and are the lattice spacings in the x, y, and z directions, respectively.

In the case of perfect adsorption, and the intensity becomes independent of, with a maximum for any (the component of parallel to the crystal surface) that satisfies the 2D Laue condition in reciprocal space

for integers and . This condition results in rods of intensity in reciprocal space, oriented perpendicular to the surface and passing through the reciprocal lattice points of the surface, as in Fig. 1. These rods are known as diffraction rods, or crystal truncation rods.

When is allowed to vary from 0, the intensity along the rods varies according to Fig. 2. Note that in the limit as approaches unity, the x-rays are fully penetrating, and the scattered intensity approaches a periodic delta function, as in bulk diffraction.

This calculation has been done according to the kinematic (single-scattering) approximation. This has been shown to be accurate to within a factor of of the peak intensity. Adding dynamical (multiple-scattering) considerations to the model can result in even more accurate predictions of CTR intensity.