Neutron Scattering

6.3 Specular Neutron Reflectivity in Born Approximation
For specular reflectivity measurements the exit angle 0' is
always identical to the incident
angle 0 .
Therefore, the Q~-component is equal to zero and the reflectivity data does not contain
any particular information about the in-plane structure of the Sample . In first order Born
approximation [1] the scattered intensity is given by
I (Q` )
1 ~f d
Qa dz (Z) exP(-iQ`z)dz
2
(6 .5)
which means, that the specular reflectivity is basically determined by the Fourier transformation
of the gradient of the potential profile perpendicular to the sample surface.
The averaged
(continuons) potential of a particular material with N components is defined by
N
2nh,2
V = ~bjpj (6 .6)
mn J=1
where the b; are the scattering lengths and the pi are the particle number densities of the
components . A one-component sample with a perfectly smooth and flat surface which is
oriented in the (x y)-plane would yield a step function for the z-dependent potential :
z 2n h Zbp 1 1 0 : > 0
V (z) = (2 - 2 o(z)) _
mn C
{2nh 2bp / mn : Z :9 0
(6.7)
The derivative of V(z) is a delta-function dV(z)/dz-S(z) . With Eq . (6 .5) one gets I(Q^) _ Q^4
because the Fourier transformation of a delta-function at z=0 is identical to 1 .
However, a perfectly smooth and flat surface does not exist. Instead surface roughness or
density gradients have to be taken into account [2,3] . As shown in Fig . 6 .5 roughness means
that thé z-position of thé surface is locally différent from thé mean position at z=0 .
Averaging
thé density in thé (x y)-plane at each z-coordinate gives a smooth profile V(z) perpendicular to
thé surface.
The exact shape of thé profile depends on thé actual physical and chemical
properties close to thé surface.
For simple rough surfaces in good approximation an errorfunction
6 .6