Neutron Scattering

ut is completely reflected . The critical angle can be estimated by and is on the order
of some 0 .1 degree depending on the density, the scattering lengths of the atours and the
wavelength of the neutrons . For 0 values beyond 0, the beam can penetrate the sample and is
only partly reflected. At the sample surface the reflection and transmission coefficients rf and tf
are obtained by the Fresnel formulars
k,
_ k
rf k- + k,
and
2k_
k_ + k
(6 .18)
with k, =ksin0 and k, = k,sin0, = k(n,2-cos20)'n (see Fig . 6.4) . The specular reflected intensity
I=Irla is determined by the absolute square of the reflection coefficient . It shows the typical
plateau below the critical angle, the regime of the total extenal reflection, and the rapidly
decreasing intensity beyond 0, . With appropriate approximations one gets I=8,Z/(4sin40)_Qz4
for incident angles 0 larger than 38, which is the confirmation of the Born approximation (see
Fig . 6 .9) .
Figure 6 .9 : Effect of the
total external reflection .
The large figure shows a
reflectivity of a smooth
surface in Born approximation
(Symbols) and the
exact Fresnel form (solid
line) . The dashed line
marks the critical Qz .
The left inset displays
absorption effects in the
reflectivity on a linear
scale. The same is depicted
for the transmitted
intensity in the right inset.
Not addressed yet in this section is the absorption ß, of the neutrons inside the Sample.
For most materials such as silicon the absorption is negligible but this is not the case for e .g .
cadmium or indium . Most straightforward, it can be introduced by including an imaginary part
to the reflection index
6 . 13