Neutron Scattering

v v z l 2r
V(z) = 2
-2erfC~ with erf(z) = 6J J exp(-t2)dt (6 .8)
0
is sufficient to modcl the profile . The parameter 6 is called rms-roughness and is a measure for
the root mean square width of V(z) given by the gaussian probability function
dV (z) (I
P(z) -
29L6 exp\ 262 ) .
(6 .9)
Figure 6.5 also shows that the profile does not contain any information about the lateral
structure of the rough surface . Therefore, the Rflectivity is insensitive to different in-plane
length scales ~, . It even cannot be used to distinguish between a rough interface or a density
gradient caused by e .g ., interdiffusion . It will be explained later that this can only be donc by
using diffuse scattering experiments .
Z
~i . .,l , i : P,
Figure 6 .5 : Three différent surfaces with thé sanie nns-roughness G determined by thé root
mean square width of thé probability fonction (dashed line) of thé profile V(z) (solid line) .
The in-plane structure is determined by thé lateral corrélation length ~n which is largefor thé
very left example quite smallfor thé rough surface shown in thé center and not definedfor thé
density gradient example (right) .
The reflectivity of a rough surfaces with an error fonction profile can easily be calculated.
The derivative of V(z) is determined by thé probability function P(z) [See Eq . (6 .9)] . With Eq .
(6 .5) this yields I(Qz) _ Qz4exp(-Qa2( ) . Compared with thé perfectly smooth surface thé
reflected intensity is damped by a Debye-Waller factor : The rougher thé surfaces thé less
intensity is reflected at large Qz (sec Figure 6 .61eft) .
Reflectivity scans are not only extremely sensitive to surfaces roughnesses but also to film
thicknesses of layer systems . If a thin film with thickness d and an averaged potential Vl is
deposited on a substrate with potential Vz thé density profile V(z) is given by
6 .7