arxiv: 2603.06484 · v2 · submitted 2026-03-06 · ❄️ cond-mat.mtrl-sci

Recognition: 2 theorem links

· Lean Theorem

Structural Commonalities in Amorphous Elemental Materials

I. Rodr\'iguez , R. M. Valladares , A. Valladares , D. Hinojosa-Romero , F. B. Quiroga , S. Calder\'on-Alba , R. S. Vilchis-Peyret , A. A. Valladares

Authors on Pith no claims yet

Pith reviewed 2026-05-15 14:50 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci

keywords amorphous semiconductorsamorphous metalspair distribution functionsplane angle distributionsstructural orderelemental materialsrenormalizationmedium-range order

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The pith

Amorphous semiconductors share similar pair distribution functions with isolated first peaks, unlike metals that show bimodal second peaks and non-zero intensity between peaks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper investigates structural trends across classes of amorphous elemental materials to identify shared features and key differences. It establishes that semiconductors display consistent PDFs reflecting network-forming bonding, with a clear zero-intensity gap after the first peak. Metallic systems instead maintain consistent but distinct PDFs featuring significant intensity between peaks and a bimodal elephant-like second peak. Semi-metals present a third distinct profile. Adding plane angle distributions and a renormalization step scaled to first-peak positions allows quantitative comparison of these local geometries.

Core claim

Amorphous semiconductors exhibit similar PDFs characteristic of their network-forming nature, featuring a well-isolated first peak and zero intensity between the first and second peaks. Amorphous metallic systems display internally consistent PDF profiles different from semiconductors, with significant non-zero intensity between peaks and a bimodal second peak showing an elephant-like profile. Amorphous semi-metals display still different profiles, with PDFs for Bi similar to those for As and Sb. Plane angle distributions provide additional local geometry detail, and a renormalization approach based on first PDF peak positions quantifies the structural coincidences across classes.

What carries the argument

Pair distribution functions renormalized by the positions of their first peaks, together with plane angle distributions, to expose class-specific signatures of short-range and medium-range order in amorphous elemental solids.

If this is right

Amorphous semiconductors can be classified together by their isolated first PDF peak and zero-intensity gap, independent of the specific element.
Amorphous metals consistently share the non-zero intensity between peaks and the bimodal elephant-like second peak.
Semi-metals such as Bi exhibit PDF profiles aligned with As and Sb rather than with either semiconductors or metals.
The renormalization method enables direct numerical comparison of structural coincidences by scaling to first-peak positions.
Plane angle distributions supply complementary information on local geometry that radial PDFs alone do not capture.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The PDF class distinctions may correlate with differences in electronic transport or band gaps between amorphous semiconductors and metals.
The same classification approach could be applied to multicomponent amorphous alloys to anticipate their structural behavior.
Standardized benchmarks based on these profiles might help validate new methods for generating amorphous structures across research groups.
Medium-range order questions in related glassy systems could be addressed by testing whether similar class-specific PDF patterns appear when network versus metallic bonding dominates.

Load-bearing premise

The differences observed in PDF and PAD profiles between semiconductors, metals, and semi-metals arise from intrinsic material class properties rather than from choices in simulation method, cooling rate, or sample size used to create the structures.

What would settle it

Generating an amorphous semiconductor structure via an alternative simulation protocol whose PDF instead matches the bimodal metallic profile with non-zero intensity between peaks would falsify the class-based distinction.

Figures

Figures reproduced from arXiv: 2603.06484 by A. A. Valladares, A. Valladares, D. Hinojosa-Romero, F. B. Quiroga, I. Rodr\'iguez, R. M. Valladares, R. S. Vilchis-Peyret, S. Calder\'on-Alba.

**Figure 1.** Figure 1: Structural evolution of amorphous carbon. (a) PDFs and (b) PAD for lda-C. (c) PDF and (d) PAD for hda-C. In (a), the 256-atom model (grey) is compared to the 64-atom model (blue dots) from Alvarez et al.1 The insets in Figures (a) and (c) showcase the structure of the second peak [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 3.** Figure 3: Structural characterization of amorphous germanium (a-Ge) showing the PDFs (a) and PADs (b) obtained. In (a), the current 256-atom supercell (grey shade) is compared to the previously reported 64-atom model (blue dots) from Alvarez et al.1. The inset displays the difference between the two works for the second peak. In (b), the angle distribution has a prominent peak at 110 º and a smaller one at about 60 … view at source ↗

**Figure 4.** Figure 4: Comparative structural analysis of amorphous semiconductors (lda-C, hda-C, a-Si, and a-Ge). (a) Renormalized PDFs referred to the position of the first peaks and (b) PADs with two prominent peaks. This comparison illustrates the universal structural commonalities of the semiconductor class, characterized by distinct shell separation and tetrahedral angular distributions. The shaded gray area is the average… view at source ↗

**Figure 11.** Figure 11: Comparative structural analysis of amorphous metals (a-Al, a-Cu, a-Ag, a-Au, a-Pd, and a-Pt). (a) Renormalized PDFs and (b) PADs. This comparison illustrates the universal structural commonalities of the metal family, characterized by an evident elephant peak. The shaded gray area is the average of all others [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗

**Figure 12.** Figure 12: Structural benchmarks and local atomic environments in amorphous metals. (a) Normalized PDF: The profile illustrates in two dimensions the four primary Short- and Medium Range-Order regions identified across the metallic suite. (b) Comparison of average PADs ( [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗

read the original abstract

In the In recent times, the research community has explored diverse structures and novel fabrication methods for amorphous solids. This work investigates structural trends among different classes of amorphous materials to identify universal commonalities and fundamental differences. It is found that amorphous semiconductors exhibit similar Pair Distribution Functions (PDFs), characteristic of their underlying network-forming nature. On the other hand, amorphous metallic systems also display internally consistent PDF profiles, but different from those of the semiconducting materials. A comparative analysis of short-range and medium-range order reveals that while semiconductor structures feature a well-isolated first peak, with a zero-intensity region between the first and second peaks, metallic systems maintain a significant non-zero value between the first and second peaks. Furthermore, the second peak in metallic systems is bimodal, featuring a distinct elephant-like profile. Amorphous semi-metals display a still different profile, and the PDFs for Bi, for example, are similar to those for As and Sb. To deepen this structural comparison, we have incorporated amorphous Plane Angle Distributions (PADs), providing a more complete perspective on the local geometry. We introduce a renormalization approach that uses the positions of the first peaks in the PDFs to quantify these structural coincidences and discuss the implications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper groups amorphous elements by PDF shape—semiconductors with isolated peaks and zero gap, metals with non-zero inter-peak intensity and a bimodal elephant-like second peak—plus a first-peak renormalization and PADs, but the distinctions may reflect simulation choices more than intrinsic class differences.

read the letter

The core observation is that amorphous semiconductors show PDFs with a clean first peak separated by a zero-intensity region from the second, while metals show continuous intensity between peaks and a split, elephant-like second peak; semi-metals sit in between. The authors renormalize to the first-peak position and add plane-angle distributions to sharpen the comparison across classes. That framing and the specific second-peak descriptor are the freshest parts, even if pair-distribution work itself is routine. The comparative figures and the renormalization step give a practical way to line up short- and medium-range order without extra fitting parameters, which is useful for quick checks on new elemental glasses. The inclusion of PADs adds a local-geometry angle that complements the radial data. On the downside, the structures almost certainly come from melt-quench runs, and nothing in the abstract or summary shows that the authors held quench rate, potential, or system size fixed across semiconductor and metal examples. Those choices directly control the medium-range order that produces the inter-peak intensity and second-peak splitting, so the claimed class signatures could be protocol artifacts rather than fundamental. No error bars, sample sizes, or explicit renormalization formula appear in the provided text, which leaves the strength of the profiles hard to judge. This is for people who simulate or measure disordered elemental solids and want a compact classification scheme to compare new results against. A reader already working in the area would pick up the renormalization trick and the PAD addition, but the work stays incremental. It deserves peer review so referees can examine the actual simulation protocols and data tables; the observational core is clear enough to be worth checking even if the universality claims need more controls.

Referee Report

3 major / 2 minor

Summary. The paper claims that amorphous elemental semiconductors exhibit similar PDFs featuring an isolated first peak with a zero-intensity gap to the second peak (reflecting network-forming character), while amorphous metals display internally consistent PDFs with non-zero intensity between peaks and a bimodal 'elephant-like' second peak; semi-metals (e.g., Bi, As, Sb) show yet another profile. These distinctions are quantified via a renormalization of PDFs using first-peak positions and are supplemented by plane-angle distributions (PADs) to compare short- and medium-range order.

Significance. If the reported PDF and PAD distinctions prove robust to simulation protocol, the work would supply a practical structural taxonomy for amorphous materials that links network-forming vs. close-packed character to observable medium-range order features. The renormalization step and inclusion of PADs are positive elements that could enable quantitative cross-material comparisons, but the absence of error bars, statistical tests, and protocol controls currently limits the strength of the universality claims.

major comments (3)

[renormalization paragraph] The renormalization procedure (described after the PDF comparisons) normalizes distances by the first-peak position taken from the identical dataset under comparison. This introduces a data-dependent scaling that risks circularity when asserting 'structural coincidences'; an explicit formula and a test against an external length scale (e.g., nearest-neighbor distance from the crystalline phase) are required to establish independence.
[methods / computational details] No details are supplied on the generation of the amorphous structures (melt-quench MD parameters, quench rates, system sizes, or interatomic potentials). Because inter-peak intensity and second-peak splitting are known to be sensitive to cooling rate and potential choice, the reported semiconductor/metal distinction could be protocol-dependent rather than intrinsic; cross-class comparisons under matched protocols or within-class quench-rate sweeps are needed to support the central classification.
[PDF results section] Claims of 'similar' PDFs within each class and 'different' profiles across classes are presented without quantitative metrics (overlap integrals, Kolmogorov-Smirnov distances, or averaged profiles with standard deviations from multiple independent samples). The zero-intensity gap for semiconductors and the bimodal splitting for metals therefore remain qualitative observations whose statistical significance cannot be assessed.

minor comments (2)

[abstract] Abstract contains a repeated 'In the In recent times' phrasing that should be corrected.
[figures] Figure captions and axis labels for the renormalized PDFs and PADs should explicitly state the number of independent configurations averaged and any smoothing applied.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which have helped us identify areas for improvement in clarity and rigor. We address each major comment point by point below and will incorporate revisions to strengthen the manuscript.

read point-by-point responses

Referee: The renormalization procedure (described after the PDF comparisons) normalizes distances by the first-peak position taken from the identical dataset under comparison. This introduces a data-dependent scaling that risks circularity when asserting 'structural coincidences'; an explicit formula and a test against an external length scale (e.g., nearest-neighbor distance from the crystalline phase) are required to establish independence.

Authors: We agree that an explicit formula is necessary for transparency. In the revised manuscript, we will provide the precise mathematical definition of the renormalization (scaling all distances by the position of the first PDF peak for that material). While the first-peak position is a natural, data-derived length scale standard in PDF analysis, we acknowledge the potential concern of circularity. To address this, we will add a comparison of the renormalized peak positions against independent nearest-neighbor distances extracted from the corresponding crystalline phases, demonstrating consistency with external structural data. revision: yes
Referee: No details are supplied on the generation of the amorphous structures (melt-quench MD parameters, quench rates, system sizes, or interatomic potentials). Because inter-peak intensity and second-peak splitting are known to be sensitive to cooling rate and potential choice, the reported semiconductor/metal distinction could be protocol-dependent rather than intrinsic; cross-class comparisons under matched protocols or within-class quench-rate sweeps are needed to support the central classification.

Authors: We accept that the manuscript lacks sufficient methodological detail. The revised version will include a dedicated section or appendix with full computational parameters: system sizes, quench rates, equilibration times, and the specific interatomic potentials employed. Our structures were generated following standard protocols from the literature, but to directly address protocol dependence, we will add a brief analysis or reference to checks showing that the key features (isolated first peak with zero-intensity gap in semiconductors versus non-zero inter-peak intensity and bimodal second peak in metals) remain robust across moderate variations in quench rate within each class. revision: yes
Referee: Claims of 'similar' PDFs within each class and 'different' profiles across classes are presented without quantitative metrics (overlap integrals, Kolmogorov-Smirnov distances, or averaged profiles with standard deviations from multiple independent samples). The zero-intensity gap for semiconductors and the bimodal splitting for metals therefore remain qualitative observations whose statistical significance cannot be assessed.

Authors: We agree that quantitative support would make the classification more robust. In the revision, we will compute and report overlap integrals (or equivalent similarity measures) between PDFs within and across classes, include averaged PDFs with standard deviations derived from multiple independent samples, and apply a simple statistical test (e.g., Kolmogorov-Smirnov) to quantify the significance of distinctions such as the zero-intensity gap and second-peak bimodality. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper reports direct computational observations of PDF shapes across material classes, noting features such as isolated first peaks with zero-intensity gaps in semiconductors versus non-zero inter-peak intensity and bimodal second peaks in metals. The introduced renormalization scales the radial coordinate of each PDF by the position of its own first peak to enable shape comparisons; this is a standard, non-predictive normalization that does not derive any claimed result from itself or force distinctions by construction. No equations reduce a prediction to fitted inputs, no self-citations serve as load-bearing uniqueness theorems, and no ansatz is smuggled via prior work. The central distinctions remain empirical features of the generated structures rather than tautological outputs of the analysis method.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The analysis rests on the standard assumption that computed pair distribution functions faithfully capture the atomic arrangement in simulated amorphous samples; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

domain assumption Pair distribution functions computed from atomic coordinates accurately represent the short- and medium-range order in amorphous elemental solids.
This is the foundational premise of all PDF studies of amorphous materials and is invoked when the authors interpret the first and second peaks.

pith-pipeline@v0.9.0 · 5562 in / 1382 out tokens · 48599 ms · 2026-05-15T14:50:36.665680+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

semiconductor structures feature a well-isolated first peak, with a zero-intensity region between the first and second peaks, metallic systems maintain a significant non-zero value between the first and second peaks. Furthermore, the second peak in metallic systems is bimodal, featuring a distinct elephant-like profile.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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