Kinematical TEM?

[argerlt]

This might be a silly question that exposes my ignorance wrt. kikuchi patterns, but:

For a recent poster at M&M 2025, I used a modified version of this kikuchipy function to simulate GaAs in a 200Kev TEM, which I compared to Bloch wave simulations using py4DSTEM, as seen below.

The line positions line up nicely, which was all I was trying to verify at the time, but the shapes i gave to the kinematic lines are just gaussians with FWHM scaled by intensity. This is probably a weak assumption, as seen by the differing line widths between the two methods.

Does anyone know of a more reasonable approximation for a kinematical TEM kikuchi line width and shape? bonus points if it can capture the enrichment AND depletion zones. My understanding from William and Carter is it's a function of excitation error, but I'm lost on the actual math.

I did find one solution was to just do all the normal setup for a Bloch wave simulation at every observation vector, but stop here, skip the expensive eigenvalue decomposition, and just return the direct beam intensity. However, this is nowhere near as fast as kikuchipy's "draw a band for every HKL" method (hours versus seconds on my desktop).

Also, for anyone staring closely at the image below: I made the Bloch wave simulations consider different subsets of HKL's for different areas, as a way to speed up computation time from weeks to hours. The result is some discontinuous lines, which is an artifact of my poor subdivision methodology, not py4DSTEM's implementation.

[hakonanes]

Interesting comparison!

How did you modify the function? If you'd think it useful, perhaps we can add your modification as a new feature? It may be a poor-mans assumption of Kikuchi bands, but infinitely better than just straight lines, I should think.

I'm afraid I haven't worked with the physics behind more advanced simulations. I know Aimo Winkelmann (@wiai) has equations for two-beam band profiles in this xcdskd code base (https://github.com/wiai/xcdskd/blob/master/src/aloe/physics/reimer.py), but I haven't looked into using them. And they do not capture excess and deficiency lines, unfortunately. No bonus points there.

Neat that you've used Py4DSTEM for these simulations! I know they originally based their Bloch wave simulations on EMsoft. I had hoped to look into using it to create dynamical EBSD simulations, but never got the time.

[argerlt]

Oh neat, I think @wiai is exactly what I wanted!

Ill give that a try, if I cannot figure it out, I'll post my objectively worse poor-man approximation, and anyone who is interested can give a shot at making it something worth adding to kikuchipy.

@hakonanes, if you want to play around with py4DSTEM's bloch wave capabilities, I learned what I did from this tutorial. For my case though of considering 300 beams over 20 million observation vectors (equivalent to xyz_hemi and xyz_reflector in kikuchipy, respectively), it was the difference between 73 hours versus 0.4 seconds for the top versus the bottom image.

[hakonanes]

if I cannot figure it out, I'll post my objectively worse poor-man approximation

I'm sure it's anyway better than no approximation!

[argerlt]

Woof, this got complicated FAST. I'm sharing here in case anyone has input.

First off, thank you @gsparks3 for introducing me to this Winkelmann et. al. paper. I found it to be a great summary, and it presents a model I think would make sense to include in kikuchipy if there is interest.

To summarize: For Kikuchi diffraction, the notion of a "global map" only makes sense when the electron source can be modeled as a coherent point emitter (CPE). This works for EBSD, but not TKD or TEM. What I modeled above in both PY4DSTEM and modded kikuchipy is just the elastic scattering, and is more akin to a composite direct-beam Large Angle Convergent Beam Electron Diffraction (LACBED) map on a thin (100nm-ish) specimen. Still interesting, but not what I need for my work.

For TEM and TKD, you need to consider a few additional effects:

1. diffraction from incoherent diffuse intensity (IDI) distributions; ie, the distribution of intensities and phases resulting from inelastic scattering events (phonons, defects, etc) when the source is directional (a STEM probe, for example).
2. excess-deficiency effects near Bragg conditions due to background intensity gradients (for CPE's, these cancel out)
3. anomalous absorption

The standard Bloch wave simulation approaches as described in De Graef's textbook and implemented in py4DSTEM and EMSOFT do not model these effects, which can lead to disagreement between simulated and measured patterns such as what is seen in Figure 1 of the paper above. Another great figure showing off specifically the 2nd effect and how it changes with specimen orientation can be seen in Figure 19.3 of Williams and Carter (apologies for not posting directly, I do not believe either are open access), There are still ways to simulate the above effects with Bloch waves, but it is computationally very expensive and I haven't seen an open source tool yet that does so, though I know De Graef's group at CMU has presented on this topic.

Incidentally, most multislice simulators have troubles modeling these effects as well, which I found surprising. Most use super-positions of different waves at differing intensities representing finite integrations over the IDI. This is computationally very demanding. AbTEM can do it with some considerable modifications, but compared to a kinematical model, this is the difference between weeks versus milliseconds in terms of calculation time.

However, this Winkelmann paper gives two equations (13 and 14) which summed together I believe give the "line profile" I was after. The only additional variables are

1. relative beam intensity
2. excitation error
3. extinction distance.

I'm not sure how best to parameterize the relative beam intensity, though this paper got some pretty believable results with linear assumptions (Figures 5 and 6).

The paper goes on to give a much more sophisticated phenomenological model that includes dynamical interactions between all the effects above, but I think equations 13 and 14 are the extent of what I need for my immediate work, and a good first step for a "poor man's approximation".

I am going to try implementing equations 13 and 14 with kikuchipy and see if I can get anything realistic, but if anyone has pointers or contributions, feel free to add a comment below.

[hakonanes]

I found it to be a great summary, and it presents a model I think would make sense to include in kikuchipy if there is interest.

I've been approached by many people who'd like to simulate patterns in Python without having to learn to use other software, like EMsoft, so there's interest in including dynamical simulations in kikuchipy. Two-beam simulations would also be of interest.

a good first step for a "poor man's approximation".

Indeed! As mentioned, anything in this direction is better than the current situtation, which is nothing.

if anyone has pointers or contributions, feel free to add a comment below

None at the moment. As always, the best way to get feedback is to open a PR early. I'd be very happy to review something like this.