Four layer neutron reflectivity

Calculates specular reflectivity for upto 4 slab-like layers on a substrate. Follows Parratt formulism[1]:

\[
R_n=\frac{r_{n,n+1}+R_{n+1}\exp{2id_{n+1}k_{z,n+1}}}{1+r_{n.n+1}R_{n+1}\exp{2id_{n+1}k_{z,n+1}}} \]

Each layer is characterised by: scattering lenght density [A^-2], Thickness [A], and Roughness[A].

SLD of capping medium can be set. As can SLD and roughness of substrate.

Effect of roughness follows Nevot and Croce[2]. e.g.
\[
r_{ij}=\frac{k_i-k_j}{k_i+k_j}\exp -2k_ik_j\sigma^2\]

where $r_{ij}$ is the fresnel reflection term between layers $i$ and $j$, and $sigma$ is the roughness of this interface.

Validation
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Validation of the code was done by comparing the output of the model to the output of the online neutron reflectivity calculator produced by NIST (https://www.ncnr.nist.gov/instruments/magik/calculators/reflectivity-calculator.html). The NIST software uses an alternative method to generate the relfectivity (Abeles). This model appears to match the NIST one well.



References
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[1] "Surface Studies of Solids by Total Refiection of X-Rays", L. G. Parratt, Physical Review, V95(2), 1954, pp 359-369.
[2] L. Nevot and P. Croce, Appl. Phys. 15, 761 1980.

Example Data: