Introduction to Floquet theory: The calculation of spinning sideband intensities in magic‐angle spinning NMR
Bibliographic record
Abstract
Abstract In magic‐angle spinning, the Hamiltonian of the system changes periodically with time, so the spectrum shows sidebands at multiples of the rotor frequency. This changing Hamiltonian (and hence, the changing Liouvillian) complicates the simulation of the line shape considerably. A simple case is that of a spin‐½ with an anisotropic chemical shift, spinning at the magic angle. Perhaps the most famous solutions to this problem of calculating spinning sideband intensities are those of Herzfeld and Berger and Maricq and Waugh, but there are many alternatives. Floquet theory is a very powerful approach, which we describe here. We illustrate the theory by explicit calculation of the sideband patterns. Floquet theory works by expanding the periodic (due to sample spinning) Hamiltonian into a Fourier series. The time‐independent Fourier components become the blocks in a much larger, time‐independent, effective Hamiltonian. The same can be done with the Liouvillian superoperator. The effective Hamiltonian (or Liouvillian) can then be treated using familiar methods used for time‐independent problems. The considerable increase in size of the problem is normally anathema, and it offers no particular benefit in the specific case treated here. The power of the method is its generality. Regardless of the complexity of the time dependence of the Hamiltonian, the Floquet approach is the same. Therefore, learning how to calculate sideband intensities in this way gives us the tools to solve much more difficult problems. © 2001 John Wiley & Sons, Inc. Concepts Magn Reson 13: 159–170, 2001
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.003 | 0.004 |
| Insufficient payload (model declined to judge) | 0.004 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".