paracrystalline model simulation

[pkienzle]

For checking implementation of #284 corrections we can simulate a lattice of spheres and compare the resulting patterns.

The dnn parameter as currently implemented represents lattice spacing. For example, the following shows that the peak positions are aligned when the lattice spacing (a=500) matches dnn in the code.

$ python explore/realspace.py -w 0.06 -t fcc -n 20,20,20  -s 10000 -m 500 -z 500,500,500 sphere radius=10

The lineshape broadening appears to be less in the theory than in the simulation for a given distortion factor but I have not investigated different distortion factors and different simulation sizes. It may be that the distortion model (gaussian around the lattice point) is not implemented correctly or does not match what the paracrystal models are using for distortion.

Ignore details at low q due to finite lattice size.

The realspace.py is using an incorrect scaling factor, so ignore offset in y.

The basic simulation of a 2D pattern does not have enough resolution to verify the details of the simulation:

$ python explore/realspace.py -d 2 -w 0.06 -t fcc -n 10,10,10  -s 10000 -m 500 -z 250,250,250 -q 0.2 -v 20,30,40 sphere radius=10

Increasing the lattice size did not help. Increasing the name of samples to s=150000 has not yet completed running since the algorithm is O(n²).

[butlerpd]

Now that the paracrystalline models have been corrected, is this still relevant @pkienzle? If so is it correct given the fix is different from what was supposed in #284?

[pkienzle]

The simulation code is still relevant in as much as it can be used to cross check the analytical results. It may also be useful in the development of models of different types of distortion. It is not part of the released packages so the code does not need careful checking.

[pkienzle]

The powder peak positions now align with the analytic forms for fcc, bcc and sc, so this code can be merged.

The peaks have the correct width and relative scale gives okay results with small distortion value (-w 0.04):

python explore/realspace.py -w 0.04 -t fcc -n 50,50,50  -s 30000 -m 1000 -z 250,250,250 sphere radius=10

There are still improvements to be done.

• Larger distortions (-w 0.1) do not match, with narrow the peaks in the simulation and broad peaks in the analytic model.
• For 2D models the simulation does not show the starburst around each peak that appears in the analytic calculation with large distortions.
• The calculation is too slow and the accuracy is too poor

Attempts to improve accuracy with larger lattices will not work. Too many of the n² pairs will be far apart and not contributing to the accuracy of the q region of interest. Instead we need a virtual lattice, with the appropriate pair-wise sampling to capture dispersity in position, orientation and size that can be summed over many pairs. This can be done with a separate PR.