Neutron Scattering

F{dV(z)/dz ) to calculate the reflected intensity [sec Eq. (6.5)] . The loss of the phase
information may end up in identical reflectivities even though the potential profiles are
different . E .g ., using Eq . (6 .11) it can easily be shown, that a monolayer system with bIp2=4
and blpl=3 exactly results in the saine reflectivity as b2p2=4 and blpl=l, if the film thicknesses
are identical.
6.4 Diffuse Neutron Scattering in Born Approximation
Performing a surface scattering experiment it turns out that some intensity is also
scattered in directions with 6 6' . In this case the wave vector transfer Q has a component in
Q,-direction (see Fig . 6.4) . This off-specular signal is called diffuse scattering and is caused by
lateral structures (in-plane, in the (x,y)-plane) of the sample [4] . If the samples are perfectly
smooth or if there is no lateral structure (see Fig . 6.5 right) no diffuse scattering is expected.
In general, for rough layer systems the diffuse scattering is sensitive to the correlation
fonction C;k(R) between two interfaces j and k where R is an in-plane vector (x y) . The
correlation function is defined by
Cjk (A) - fz; (r)zk (r + R)d2r (6 .12)
with the local deviation z;(r_) from the averaged position of the interface j . The correlation
function between two different interfaces is usually called `cross-correlation' . If j=k holds
C;k(R)=C;;(R) is called `auto-correlation' . Qualitatively, C;k(R) is large if two areas of the
interfaces j and k, which are R apart from each other, `look similar' . E .g ., if the two interfaces
contain a periodic structure with the saine periodic distance D the correlation fonction exhibits
maxima at D,2D,3D . . . . For an auto-correlation function of a single rough surfaces one gets a
monotone decreasing function : For very small distances the parts of the surface look sirnilar,
the larger the distance the more different they become . The width of the curve is connected to
the lateral correlation length ~,,, of the interface (see e .g ., Fig 6.5) [5] .
The diffuse scattering can be used to investigate periodic in-plane structures, in-plane
correlation lengths of a single rough interface and correlations between two different
interfaces . Figure 6.7 depicts some examples . The complete mathematical formalism to deduce
the diffusely scattered intensity is quite complicate [6] . Therefore, the full theory is omitted in
this section . Instead, the diffuse scattering is explained using a simple monolayer system.
6.9