Neutron Scattering - JUWEL - Forschungszentrum Jülich

TREFF 5
with the neutron beam we will measure the reflectivity curve step by step of the sample (see
chapter Experiment Procedure).
3.2 Data reduction
The instrument saves the number of counts as a function of scattering angle.
3.3 Data evaluation
For systems such as multilayers the scattered intensity is determined by the difference in the
potential of each layer (contrast). The potential is given by the scattering length density
ρsld = �
j bjρj with the scattering lengths bj and the particle number densities ρj. The index
j runs over all kind of atoms of the layer. The scattering length density is comparable to the
optical density in light optics.
The refraction index of each layer is given by
n ≫ 1 − λ2
2π ρsld =1− δ
.
With the angle of total external reflection Θc ∼ √ 2δ, which is usually small, it follows
kc,z = k sin(qc) ≫ kqc = 2π
�
2
λ
λ2
2π ρsld = � 4πρsld
for the critical wave vector. For a monolayer system the reflected amplitude of each interface
rf,1 and rf,2 can be calculated by the Fresnel formulae (Equation 16 in chapter 12 of the
lectures book).
Neglecting roughness at the sample surface and at the interface between layer and substrate,
for the amplitude at the surface one gets
with
rf,1 = kz,vac − kz,lay
kz,vac + kz,lay
kz,vac = k sin(q) , kz,lay =
and at the interface rf,2 = kz,lay − kz,sub
kz,lay + kz,sub
�
k2 �
z,vac − 4πρsld,lay and kz,sub = k2 z,vac − 4πρsld,sub
.
The superposition of both amplitudes yields the reflected amplitude of a monolayer sample
R =[rf,1 + rf,2 exp(2ikz,layd)]
exp(−2ikz,vacd)
[1 + rf,1rf,2 exp(2ikz,layd)]
with the film thickness d. The reflected intensity is given by re mean square of R. For
kz,vac > 3kc,z the intensity can be calculated in Born approximation by
|R| 2 ≫ π2
k4 � 2
ρsld,lay +(ρsld,lay − ρsld,sub)
z,vac
2 +2ρsld,lay(ρsld,lay − ρsld,sub)cos(2kz,vacd) �