Breakdown of Magic Numbers in Spherical Confinement
Bibliographic record
Abstract
Magic numbers in finite particle systems correspond to specific system sizes that allow configurations with low free energy, often exhibiting closed surface shells to maximize the number of nearest neighbors. Since their discovery in atomic nuclei, magic numbers have been essential for understanding the number-structure-property relationships in finite clusters across different scales. However, as the system size increases, the significance of magic numbers diminishes, and the precise system size at which magic number phenomena disappear remains uncertain. In this study, we investigate colloidal clusters formed through confined self-assembly. Small magic number clusters display icosahedral symmetry with closed surface shells, corresponding to pronounced free energy minima. Our findings reveal that beyond a critical system size, closed surface shells disappear, and free energy minima become less pronounced. Instead, we observe a distinct type of colloidal cluster, termed a football cluster, which retains icosahedral symmetry but features lower-coordinated facets disconnected by terraces. A sphere packing model demonstrates that forming closed surface shells becomes impossible beyond a critical system size, explaining the breakdown of magic numbers in large confined systems.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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 teacher head, 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".