Depending on the wavelength range of their Fourier components, two types of roughness are to be distinguished:

• If the roughness period is large compared to the coherence length, we talk about macroscopic roughness.
• If the roughness is small compared to the coherence length, we talk about microscopic or atomic roughness.

Macroscopic roughness is due to facets wider than the coherence length, oriented at various angles to the x-ray beam. The symptom of macroscopic roughness is a specular reflection broader than the resolution function of the instrument (as determined by the angular width of the direct beam) or even split peaks. For x-ray mirrors this type of roughness is called slope error. Since the incident angle for such samples is not well-defined, a quantitative description of such samples is often problematic.

Microscopic or atomic roughness is on a much smaller scale than the coherence length and hence describes distortions of the wavefront of the incident wave yielding a reduction of the intensity of the scattered beam as compared to an ideal interface. Since in specular reflectivity the lateral component of the scattering vector is zero, we only have to deal with the laterally averaged roughness and not with the detailed shape of the surface defects (point defects, steps, islands, ...). An often used assumption is that the surface height function obeys a Gaussian distribution, which leads to a laterally averaged density with the shape of the error function.

One of the earliest descriptions of roughness we find in Lord Rayleigh's treatment of the reflection of acoustic waves from a hard corrugated wall. In this case the ideal reflectivity is multiplied with an exponential damping factor

R_R(q) = R_F(q)* exp(-σ²q²)

This description has some flaws: It cannot be extended easily to the case of multiple layers other than that the same roughness is assumed for each interface or some effective roughness which is some kind of mean of the various interface roughnesses. Even worse, the Rayleigh model is not good below and in the vicinity of the critical angle. This can be seen if the Rayleigh model is compared to a discrete roughness model, where the laterally-averaged density is approximated by thin discrete slabs and then treated exactly with the matrix method.
 
 

The most succesful roughness model so far has been proposed by Nevot and Croce [1], and mathematically analyzed by Vidal and Vincent [2]. We will refer to it as the NCVV model. In this model, each element of the interface matrix in the matrix method is multiplied by a factor

(A^i,i+1,rough)_j,k = (A^i,i+1)_j,k  exp{-σ² [Re(k_z,i - (-1)^j+k k_z,i+1)]² / 2}

This model compares well with the discrete model. Since the roughness factors are specific to each interface, a detailed description of a multilayer system can be given. Furthermore, the laterally-averaged density can be derived with a specific error function for each interface, as for instance in the program PARRAT32. Discrete model, Rayleigh model, and NCVV model are compared below.

References

[1] L. Nevot and P. Croce, Rev. Phys. Appl. 15, 761 (1980).
[2] B. Vidal and P. Vincent, Applied Optics 23, 1794-1801 (1984).