Roughness


Another big virtue of the Fourier transform is, that it allows to analyze lattice defects in a simple, transparent way.
In the following we will focus on surfaces and thin films. So how we decribe a surface in our formalism? We introduce
the Heaviside function f(r) by

f(r) = 1  for rz<0, 0 otherwise
where we chose rz as the direction normal to the surface plane. The Fourier transform of this function exists and is
F(q) = i/qz
Now we can write a truncated lattice as
rsurf(r) = r(r) f(r)
and apply the inverse convolution theorem
Ssurf(q) = S(q) * F(q) = { Suc(q) Latt(q) } * F(q)
This means that each Bragg peak in I(q) gets an 1/qz2 tail known from dynamic theory. Note that these so-called crystal
truncation rods are per se not a dynamic effect, they naturally occur within the kinematic theory by introducing an interface in the lattice.

The above calculation assumes an arbitrarily sharp interface. However, a surface always has steps and point defects. These can be taken into account by assuming that the probability of atoms being located close to the interface plane is somewhat reduced, and that there are also atoms above the truncation plane. A simple way of modelling such a surface is by smoothing the atom density at the interface by a Gaussian:

rough(r) = (2ps2)-0.5 exp(-rz2/2s2)
The parameter s describes the width of the interface region or in other words, the root-mean-square surface roughness. Mathematically we smear out the cut-off function f(r) with the roughness Gaussian
rsurf,rough(r) = r(r) { f(r) * rough(r) }
We can interpret the function { f(r) * rough(r) } as an occupation probability for a layer at a distance z from the interface at z=0 and this rough interface is characterized by a width s.

Applying the convolution theorem we get

Ssurf,rough(q) = FT{ r(r) } * {f(r) Rough(q)} = S(q) * { F(q) Rough(q) }
where Rough(q) is simply a Gaussian in reciprocal space:
Rough(q) = exp( - s2qz2/2)
The effect of the surface roughness is, to attenuate the crystal truncation rods : rough surfaces yield weaker
diffuse scattering. If we evaluate the convolution integral, we get a sum
Ssurf,rough(q) = Suc(q) Shk exp(-s2qz2/2) / [i (qz-Ghk)]
Note that the Debye-Waller-like roughness factor falls off from each bulk Bragg peak, whereas the Debye-Waller factor (which we did not consider here, but can be included in the theory the same way as in 3D kinematic theory) falls off from q=0. This way it is possible to distinguish between the Debye-Waller factor due to thermal fluctuations and the surface roughness when scattering intensities are modeled to high precision.

The sum converges rapidly and can be evaluated numerically. Roger Cowley derived a closed analytical expression for the Gaussian roughness model which we cite without proof:

ICTR,rough(qz) = exp(- (4s/a)2 sin(a qz/2)2 ) / (4 sin(a qz/2)2 )


Roughness has a strong effect on the CTR intensity: if the s value equals the lattice constant (i.e. the FWHM is about 2 lattice constants) the intensity at the anti-Bragg condition is already too small to measure. Hence well-prepared surfaces according to surface science standards are a necessity, in order to use CTR's.