Bibliographic record
Abstract
We have presented a detailed analysis of the higher-order anharmonic contributions to the average square atomie displacement (u 2) or Debye-Waller factor by the diagrammatic method and the Zubarev type Green's function method. The Hamiltonian employed in the detailed analysis contains the anharmonic terms up to O(λ4)that is the cubic, quartic, quintic and sextic terms which arise from the Taylor expansion of the potential energy. Thus the various contributions to (u 2) of O(λ2) and O(λ4), presented in the diagrammatic language, arise from the first-, second-, third- and the fourth-order perturbation theory. This analysis establishing interrelationships between the Helmholtz free energy (F), self-energy and (u 2) reveals that a total of 16 diagrams contribute to (u 2) for a Bravais lattice or a lattice with a basis where every atom is at a site of inversion symmetry. A simple prescription is given for the derivation of (u 2) diagrams from the F diagrams. Out of 16 diagrams, two are of O(λ2) and 14 of O(λ4), where λ is a perturbation expansion parameter. It is also shown that at least up to O(λ4) the contributions to (u 2) from nine out of 16 diagrams are included in the Green's function iterative solution if the latter is generated from the anharmonic Hamiltonian containing the cubic and auartic terms. Exact and approximate numerical magnitudes for all the diagrams are obtained for a nearest-neighbour 6-12 Lennard-Jones fec solid by extrapolating the Brillouin zone (BZ) sums to the q → 0 limit. A disproportionate contribution to the BZ sums comes from this region of q space in the calculation of (u 2) In the context of (u 2) the importance of the various self-energy diagrams involving the same type of anharmonic vertices is established. It is shown that there exists a heavy cancellation among these 14 diagrams; nevertheless their total contribution to (u 2) is sufficient to bring it in almost exact agreement with the classical Monte Carlo results, which are also obtained for the same model.
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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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.010 | 0.001 |
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".