Normalization of CTR structure factor for CTR fitting

[tifuchs]

Normalization of CTR structure factor in CTRcalc

Currently all structure factors calculated in CTRcalc are normalized to their unit cell volume.
This causes issues (and is arguably wrong) in the calculation of bigger structures such as epitaxially grown thin films, which might have a bigger unit cell along all three directions.

Physical background

The structure factor of any material is calculated with

$$ F = \sum_i f_i (Q) e^{iQr_i}, $$

which includes all atoms $i$ in a material.

For a CTR, with a periodic lattice and a well-defined surface unit cell $F_s = \sum_i f_i (Q) e^{iQr_{s,i}}$ this can be simplified to

$$ F = A \cdot \left( \frac{F_s}{A_{s}} + \frac{F_{bulk}}{A_{bulk}} \right), $$

where $F_{bulk}$ is the CTR structure factor of the bulk crystal. Both $F_s$ and $F_{bulk}$ are calculated for a small unit cell with a surface area of $A_{s}$, or an half-infinite column of unit cells with a surface area $A_{bulk}$, respectively.

Here, the unit cell structure factors are first normalized to the unit cell surface area and then multiplied with the illuminated sample area $A$. This ensures that all atoms in the material are accounted for.

Explicit calculation of the in-plane lattice sum would be the alternative, which would provide an exact CTR width and localizes the scattering at the reciprocal lattice points in the in-plane direction. While this approach is physically more accurate, the simple normalization of F above does result in an identical structure factor at the in-plane reciprocal lattice points. Thus, calculation of the in-plane lattice sum is usually not required.

Planned implementation in orGUI

UnitCell.F_uc and WaterModel.F_uc and Film.F_uc

should return the expected structure factor without any normalization

$$ F_s = \sum_i f_i (Q) e^{iQr_{s,i}} $$

EpitaxyInterface.F_uc

Here we already have the possibility of two differently sized unit cells along the in-plane direction. Along the out-of-plane direction both unit cells must have the same size.
It probably does not matter which one to normalize to. But following the convention to normalize towards the bulk, I'd choose:

$$ F_{epit} = A_{lower} \cdot \left( \frac{F_{upper}}{A_{upper}} + \frac{F_{lower}}{A_{lower}} \right), $$

or (normalized to 1 $\mathrm{Å}^2$)

$$ F_{epit} = \frac{F_{upper}}{A_{upper}} + \frac{F_{lower}}{A_{lower}}, $$

SXRDcrystal.F

Since for CTRs we only need the structure factor at the in-plane reciprocal lattice points, the function SXRDcrystal.F could return

$$ F_a = \frac{F_s}{A_{s}} + \frac{F_{film}}{A_{film}} + \frac{F_{bulk}}{A_{bulk}} + \dots, $$

which would be a structure factor normalized to a surface area of 1 $\mathrm{Å}^2$, i.e. the $F_a$ is in units of electrons $\cdot$ $\mathrm{Å}^{-2}$.

The other option would be

$$ F_b = A_{bulk} \cdot \left( \frac{F_s}{A_{s}} + \frac{F_{film}}{A_{film}} + \frac{F_{bulk}}{A_{bulk}} + \dots \right), $$

which would be a structure factor normalized to the surface area of the bulk unit cell. Advantage of this solution is that $F_b$ has the correct units of a structure factor.

In both cases for peak shape analysis, the user could then add the in-plane lattice sums (or more likely: other in-plane distribution functions).