Evaluation of the roughness of a crystal surface by X-ray scattering. I. Theoretical considerations

The relationship between the intensity distributions of the crystal truncation rod (CTR) scattering and the surface roughness of a crystal is discussed by developing a kinematic theory for the CTR scattering so as to reflect the two-dimensional aspect of the surface. The intensity of the CTR scattering elongated from a Bragg point is shown to be reduced by a factor |Γ(q)|2 for a surface possessing some roughness, where Γ(q) is defined by a simple Fourier summation of γp, the relative area with the same step height p on a surface, i.e. Γ(q) = p = 0∞ γp exp (2πipq), with Σp γp = 1, q being the distance in reciprocal space from the Bragg point along the CTR scattering. A pair-correlation function between the steps can, therefore, be obtained by a simple Fourier integral of the roughness damping factor |Γ(q)|2. For the case where γp has a Gaussian distribution around the average step height, |Γ(q)|2 is approximated by the well known Debye–Waller-like factor, exp (−4π2〈Δp2〉q2), where 〈Δp2〉 is the mean square deviation of step height in units of the lattice spacing. The intensity formulae proposed so far by several authors are also discussed on the basis of the above factor.