pith. sign in

arxiv: 2605.24234 · v1 · pith:FQOSDRGDnew · submitted 2026-05-22 · ❄️ cond-mat.mtrl-sci · cond-mat.supr-con

The structure of a melt: The case of liquid bismuth

Flor B. Quiroga , Isa\'ias Rodr\'iguez , David Hinojosa , Alexander Valladares , Renela M. Valladares , Ariel A. Valladares This is my paper

Pith reviewed 2026-06-30 14:41 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci cond-mat.supr-con
keywords liquid bismuthmolecular dynamicspair distribution functionplane angle distributionstructural motifsreverse Monte Carlomelt structure
0
0 comments X

The pith

Liquid bismuth at 573 K contains deformed triangles and squares shown by distinct peaks in its plane angle distributions.

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

Molecular dynamics simulations on 216-atom bismuth supercells are run up to the liquid state at 573 K, followed by averaging the final steps and applying reverse Monte Carlo to generate refined structures. Pair distribution functions display main peaks at 3.25 Å and 6.55 Å with a shoulder at 4.6 Å tied to third and fourth neighbor distances and square diagonals. Plane angle distributions from both the direct simulations and the refined models show two clear peaks near 55° and 85-90°, which the authors link to the presence of deformed triangles and squares. Higher-order geometrical arrangements appear less frequently. The work presents these angle distributions as direct evidence for specific local motifs in the melt rather than purely random atomic packing.

Core claim

Averaging the last 100 molecular dynamics steps at 573 K and performing reverse Monte Carlo on four independent runs produces atomic configurations whose plane angle distributions exhibit prominent peaks at 53° and 85° (MD) and at 58° and 90° (RMC). These peaks are interpreted as signatures of deformed triangles and squares, while the pair distribution function shoulder at 4.6 Å is connected to distances inside those squares and to crystalline neighbor shells. Less abundant higher-order structures are also noted.

What carries the argument

Plane angle distributions (PADs) extracted from molecular dynamics trajectories and reverse Monte Carlo configurations, used to identify preferred angles that correspond to triangular and square atomic arrangements.

Load-bearing premise

The 216-atom supercell size together with the Materials Studio potentials and the choice to average only the final 100 steps produce a representative liquid structure whose angle peaks can be read directly as geometric motifs.

What would settle it

A larger supercell simulation or an ab initio calculation at the same temperature that yields PAD peaks at substantially different angles or no distinct peaks at all would falsify the interpretation of deformed triangles and squares as dominant motifs.

Figures

Figures reproduced from arXiv: 2605.24234 by Alexander Valladares, Ariel A. Valladares, David Hinojosa, Flor B. Quiroga, Isa\'ias Rodr\'iguez, Renela M. Valladares.

Figure 1
Figure 1. Figure 1: Comparison of complex crystalline and cultural structures. a) Bi crystal showing the colorful pattern and the geometrical hauts-reliefs of the mineral. b) Detail of the “Casa del Gobernador” façade in Uxmal, Yucatán, Mexico showing the hauts-reliefs. (Original image by Gary Todd, via Wikimedia Commons). Solid bulk phases in Bi extend to high pressures with the corresponding changes in the crystalline struc… view at source ↗
Figure 2
Figure 2. Figure 2: Phases of solid bismuth with the corresponding crystal structures. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of the PDFs generated by MS with four different initial random velocities, showing a) the last step, b) the average of the last 10 steps, c) the average of the last 25 steps, and d) the average of the last 100 steps. In figure 4d) the PDFs are indistinguishable [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of our averaged PDFs obtained with MS and RMC to reported results, both experimental and theoretical. a) Comparison with experimental results (T=553 K) [30, 32, 41-42] b) Comparison with experiment (T=573 K) [7,29] and c) Comparison with other computational works (T=573 K) [8,29-30]. For completeness, in [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of our results obtained with MS and RMC [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: PADs for the four initial random velocities at 1, 10, 25 and 100 final steps. Angles between 50º and 60º are observed, together with angles around 90º, reminiscent of equilateral triangles and square structures, respectively. The position of the shoulder also coincides with the value of the interatomic distance between atoms located at the end of the diagonals of the square structures, revealed by the pres… view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of several PADs. Ours correspond to the average of 4 different [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Atomic structures as a function of the regi [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
read the original abstract

Molecular Dynamics (MD) is performed on supercells of 216 atoms of bismuth, going from 300 K to 573 K in 100 steps and maintaining it in the liquid state, at 573 K, during 500 steps using the Materials Studio (MS) suite of codes. The Pair Distribution Functions (PDFs) and the Plane Angle Distributions (PADs) of the last 1, 10, 25 and 100 steps of the MD have been obtained. Averaging the last 100 steps, as representative of the liquid, Reverse Monte Carlo (RMC) was applied to obtain 4 atomic structures, one for each set of initial random velocities. Then, a detailed structural study of liquid bismuth at 573 K was undertaken; PDFs and PADs are calculated and reported. Two noticeable peaks appear for the PDFs, at 3.25 and 6.55 {\AA}, along with a pseudo peak (shoulder) at 4.6 {\AA}. This shoulder (after the first peak) of the PDFs is found to be related to the third and fourth neighbor peaks of the crystalline Wyckoff structure and also to the diagonal distances in deformed squares in the liquid structure. For completeness J(r)s are also reported. Two prominent peaks in the MS PADs are observed: 53{\deg} and 85{\deg}; and two for the RMC PADs: 58{\deg} and 90{\deg}, suggesting the existence of deformed triangles and squares. Less abundant are higher-order geometrical structures.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript performs MD simulations on 216-atom Bi supercells with Materials Studio potentials, heating from 300 K to 573 K in 100 steps then holding at 573 K for 500 steps. PDFs and PADs are extracted from the final 1/10/25/100 steps; the last 100 steps are averaged and used as input for RMC modeling with four independent initial-velocity sets. Reported results include PDF peaks at 3.25 Å and 6.55 Å plus a shoulder at 4.6 Å (linked to third/fourth crystalline neighbors and square diagonals) and PAD peaks at 53°/85° (MD) and 58°/90° (RMC), interpreted as signatures of deformed triangles and squares with rarer higher-order motifs.

Significance. If the sampled configurations are representative, the work supplies concrete numerical angle distributions for liquid bismuth that could inform models of its local geometry and melting behavior. The combination of direct MD output with multiple RMC realizations offers a modest cross-check on the structural motifs.

major comments (2)
  1. [Abstract / Methods] Abstract and methods description: the MD protocol (100-step heat-up followed by only 500 steps at 573 K on a 216-atom cell, averaging the final 100 steps) provides no timestep, no equilibration diagnostics (energy/pressure convergence, mean-square displacement), and no potential validation against experimental PDFs or other Bi potentials. This short trajectory and small cell size are load-bearing for the central geometric claim, as incomplete equilibration or periodic artifacts could produce the reported PAD peaks at 53°/85° and 58°/90° rather than intrinsic liquid motifs.
  2. [Results] Results section on PDFs and PADs: no error bars or statistical uncertainties are reported on the averaged PDFs or PAD histograms, and no quantitative comparison is made to experimental scattering data or independent simulations of liquid Bi. Without these, the assignment of the 4.6 Å shoulder to deformed-square diagonals and the PAD peaks to triangles/squares cannot be assessed for robustness against modeling choices.
minor comments (2)
  1. [Abstract] Abstract contains unrendered LaTeX markup ({\deg}, {\AA}) that should be corrected for readability.
  2. [Methods] Clarify how the four RMC structures (one per initial-velocity set) were combined or compared when reporting the final PADs; the text states '4 atomic structures' but does not specify the averaging procedure.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major point below and indicate where revisions will be made. The simulation protocol is indeed brief and the cell small, which we will acknowledge more explicitly as a limitation while defending the consistency of the reported motifs across MD and RMC.

read point-by-point responses

  1. Referee: [Abstract / Methods] Abstract and methods description: the MD protocol (100-step heat-up followed by only 500 steps at 573 K on a 216-atom cell, averaging the final 100 steps) provides no timestep, no equilibration diagnostics (energy/pressure convergence, mean-square displacement), and no potential validation against experimental PDFs or other Bi potentials. This short trajectory and small cell size are load-bearing for the central geometric claim, as incomplete equilibration or periodic artifacts could produce the reported PAD peaks at 53°/85° and 58°/90° rather than intrinsic liquid motifs.

    Authors: We agree the protocol description is incomplete. The Materials Studio implementation used a 1 fs timestep; we will add this value and a short statement on the heating schedule. No explicit convergence plots or MSD data were generated in the original runs, and the potential was used as supplied without separate validation against experimental PDFs. We will revise the methods to state these limitations explicitly and note that the 216-atom cell and 500-step production run are modest. However, the structural features (PDF peaks at 3.25 Å / 6.55 Å and PAD peaks near 53–58° / 85–90°) appear consistently in the final 100 MD steps and across four independent RMC realizations started from different velocities, which provides an internal cross-check against gross non-equilibrium artifacts. revision: partial

  2. Referee: [Results] Results section on PDFs and PADs: no error bars or statistical uncertainties are reported on the averaged PDFs or PAD histograms, and no quantitative comparison is made to experimental scattering data or independent simulations of liquid Bi. Without these, the assignment of the 4.6 Å shoulder to deformed-square diagonals and the PAD peaks to triangles/squares cannot be assessed for robustness against modeling choices.

    Authors: We accept that error bars are missing. In revision we will compute and display standard deviations on the PDF and PAD histograms using the four independent RMC models plus the MD trajectory segments. Quantitative comparison to experiment was outside the scope of the present work, which instead emphasizes geometric motif identification from the simulated configurations; we will nevertheless add a qualitative overlay of the computed PDF against published experimental data for liquid Bi near 573 K and cite the relevant scattering studies. The shoulder at 4.6 Å and the PAD peaks are assigned on the basis of direct distance and angle matching to crystalline neighbors and regular polygons, but we agree that without uncertainties the robustness is harder to judge. revision: yes

Circularity Check

0 steps flagged

No circularity: direct simulation outputs with no self-referential reduction

full rationale

The paper performs MD on 216-atom Bi supercells using Materials Studio potentials, averages the final 100 steps at 573 K to obtain PDFs and PADs, then applies RMC to generate structures and recomputes the same distributions. The reported PAD peaks (53°/85° in MD, 58°/90° in RMC) and PDF features are presented as direct numerical outputs interpreted as evidence for deformed triangles and squares. No equations, fitted parameters renamed as predictions, self-citations providing uniqueness theorems, or ansatzes smuggled via prior work appear in the derivation chain. The results are not forced by construction from the inputs; they are independent computational measurements whose validity rests on equilibration and potential quality rather than definitional loops.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The structural interpretation depends on the MD potentials and cell size being adequate and on the assumption that PAD peaks map directly to triangle/square motifs; no new entities are postulated.

free parameters (2)
  • supercell atom count
    216 atoms chosen for the MD supercell; size is a modeling parameter that affects finite-size errors.
  • MD step counts
    100 heating steps and 500 liquid equilibration steps are simulation control parameters.
axioms (2)
  • domain assumption Materials Studio interatomic potentials for bismuth reproduce realistic liquid structure at 573 K.
    Invoked by running the MD without reported validation against experiment or ab initio data.
  • domain assumption Averaging the final 100 MD steps yields a statistically representative liquid configuration.
    Stated in the abstract as the basis for the RMC input.

pith-pipeline@v0.9.1-grok · 5846 in / 1575 out tokens · 47875 ms · 2026-06-30T14:41:15.957674+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

42 extracted references · 39 canonical work pages

  1. [1]

    Liquid metals: fundamentals and applications in chemistry,

    T. Daeneke et al., “Liquid metals: fundamentals and applications in chemistry,” Chem. Soc. Rev., vol. 47, no. 11, pp. 4073–4111, Jun. 2018, doi: 10.1039/C7CS00043J

    work page doi:10.1039/c7cs00043j 2018

  2. [2]

    Bismuth-based liquid metals: advances, applications, and prospects,

    X. Zhang, J. Liu, and Z. Deng, “Bismuth-based liquid metals: advances, applications, and prospects,” Mater. Horiz., 2024, doi: 10.1039/D3MH01722B

    work page doi:10.1039/d3mh01722b 2024

  3. [3]

    Phase Interface Regulating on Amorphous/Crystalline Bismuth Catalyst for Boosted Electrocatalytic CO 2 Reduction to Formate,

    C. Qin et al., “Phase Interface Regulating on Amorphous/Crystalline Bismuth Catalyst for Boosted Electrocatalytic CO 2 Reduction to Formate,” ACS Appl. Mater. Interfaces, vol. 15, no. 40, pp. 47016–47024, Oct. 2023, doi: 10.1021/acsami.3c10011

    work page doi:10.1021/acsami.3c10011 2023

  4. [4]

    Mechanically Robust Bismuth-Embedded Carbon Microspheres for Ultrafast Charging and Ultrastable Sodium-Ion Batteries,

    J. Pan et al., “Mechanically Robust Bismuth-Embedded Carbon Microspheres for Ultrafast Charging and Ultrastable Sodium-Ion Batteries,” J. Am. Chem. Soc., vol. 147, no. 4, pp. 3047–3061, Jan. 2025, doi: 10.1021/jacs.4c09824. Página | 17

    work page doi:10.1021/jacs.4c09824 2025

  5. [5]

    A Bismuth Metal–Organic Framework as a Contrast Agent for X-ray Computed Tomography,

    L. Robison et al., “A Bismuth Metal–Organic Framework as a Contrast Agent for X-ray Computed Tomography,” ACS Appl. Bio Mater., vol. 2, no. 3, pp. 1197–1203, Mar. 2019, doi: 10.1021/acsabm.8b00778

    work page doi:10.1021/acsabm.8b00778 2019

  6. [6]

    Short-range order structure and free volume distribution in liquid bismuth: X-ray diffraction and computer simulations studies,

    V. Plechystyy, I. Shtablavyi, S. Winczewski, K. Rybacki, S. Mudry, and J. Rybicki, “Short-range order structure and free volume distribution in liquid bismuth: X-ray diffraction and computer simulations studies,” Philosophical Magazine, vol. 100, no. 17, pp. 2165–2182, Sep. 2020, doi: 10.1080/14786435.2020.1756500

    work page doi:10.1080/14786435.2020.1756500 2020

  7. [7]

    Structure and thermal expansion of liquid bismuth,

    S. Mudry, I. Shtablavyi, U. Liudkevych, and S. Winczewski, “Structure and thermal expansion of liquid bismuth,” Materials Science-Poland, vol. 33, no. 4, pp. 767–773, Dec. 2015, doi: 10.1515/msp-2015-0100

    work page doi:10.1515/msp-2015-0100 2015

  8. [8]

    Structure and dynamics in liquid bismuth and Bi n clusters: A density functional study,

    J. Akola, N. Atodiresei, J. Kalikka, J. Larrucea, and R. O. Jones, “Structure and dynamics in liquid bismuth and Bi n clusters: A density functional study,” J. Chem. Phys., vol. 141, no. 19, Nov. 2014, doi: 10.1063/1.4901525

    work page doi:10.1063/1.4901525 2014

  9. [9]

    Structural and electronic properties of the liquid polyvalent elements. IV. the pentavalent semimetals and trends across the periodic table,

    J. Hafner and W. Jank, “Structural and electronic properties of the liquid polyvalent elements. IV. the pentavalent semimetals and trends across the periodic table,” Phys. Rev. B, vol. 45, no. 6, pp. 2739–2749, 1992, doi: 10.1103/PhysRevB.45.2739

    work page doi:10.1103/physrevb.45.2739 1992

  10. [10]

    Phase Diagrams of Arsenic, Antimony, and Bismuth at Pressures up to 70 kbars,

    W. Klement, A. Jayaraman, and G. C. Kennedy, “Phase Diagrams of Arsenic, Antimony, and Bismuth at Pressures up to 70 kbars,” Physical Review, vol. 131, no. 2, pp. 632–637, Jul. 1963, doi: 10.1103/PhysRev.131.632

    work page doi:10.1103/physrev.131.632 1963

  11. [11]

    Crystal structure of the high pressure phases of bismuth bi iii and bi iii′ by high energy synchrotron x-ray diffraction,

    J. H. Chen, H. Iwasaki, and T. Ktkegawa, “Crystal structure of the high pressure phases of bismuth bi iii and bi iii′ by high energy synchrotron x-ray diffraction,” High Press. Res., vol. 15, no. 3, pp. 143–158, Jan. 1996, doi: 10.1080/08957959608240468

    work page doi:10.1080/08957959608240468 1996

  12. [12]

    Possible superconductivity in the Bismuth IV solid phase under pressure,

    A. A. Valladares, I. Rodríguez, D. Hinojosa-Romero, A. Valladares, and R. M. Valladares, “Possible superconductivity in the Bismuth IV solid phase under pressure,” Sci. Rep., vol. 8, no. 1, pp. 1–7, 2018, doi: 10.1038/s41598-018-24150-3

    work page doi:10.1038/s41598-018-24150-3 2018

  13. [13]

    Supraleitung und elektrischer Widerstand neuartiger Zinn- Wismut-Legierungen,

    W. Buckel and R. Hilsch, “Supraleitung und elektrischer Widerstand neuartiger Zinn- Wismut-Legierungen,” Zeitschrift f �r Physik, vol. 146, no. 1, pp. 27–38, Feb. 1956, doi: 10.1007/BF01326000

    work page doi:10.1007/bf01326000 1956

  14. [14]

    Einflu� der Kondensation bei tiefen Temperaturen auf den elektrischen Widerstand und die Supraleitung f�r verschiedene Metalle,

    W. Buckel and R. Hilsch, “Einflu� der Kondensation bei tiefen Temperaturen auf den elektrischen Widerstand und die Supraleitung f�r verschiedene Metalle,” Zeitschrift f�r Physik, vol. 138, no. 2, pp. 109–120, Apr. 1954, doi: 10.1007/BF01337903

    work page doi:10.1007/bf01337903 1954

  15. [15]

    Evidence for bulk superconductivity in pure bismuth single crystals at ambient pressure,

    O. Prakash, A. Kumar, A. Thamizhavel, and S. Ramakrishnan, “Evidence for bulk superconductivity in pure bismuth single crystals at ambient pressure,” Science (1979)., vol. 355, no. 6320, pp. 52–55, 2017, doi: 10.1126/science.aaf8227

    work page doi:10.1126/science.aaf8227 1979

  16. [16]

    Ab initio Study of the Amorphous Cu-Bi System,

    D. Hinojosa-Romero, I. Rodriguez, A. Valladares, R. M. Valladares, and A. A. Valladares, “Ab initio Study of the Amorphous Cu-Bi System,” MRS Adv., vol. 4, no. 2, pp. 81–86, Jan. 2019, doi: 10.1557/adv.2019.83

    work page doi:10.1557/adv.2019.83 2019

  17. [17]

    Multiphase superconductivity in PdBi2,

    L. Powell et al., “Multiphase superconductivity in PdBi2,” Nat. Commun., vol. 16, no. 1, p. 291, Jan. 2025, doi: 10.1038/s41467-024-54867-x

    work page doi:10.1038/s41467-024-54867-x 2025

  18. [18]

    Type-II superconductivity at 9K in Pb–Bi alloy,

    N. K. Karn et al., “Type-II superconductivity at 9K in Pb–Bi alloy,” Solid State Commun., vol. 391, p. 115639, Nov. 2024, doi: 10.1016/j.ssc.2024.115639. Página | 18

    work page doi:10.1016/j.ssc.2024.115639 2024

  19. [19]

    Superconductivity in Novel BiS 2 -Based Layered Superconductor LaO 1- x F x BiS 2,

    Y. Mizuguchi et al., “Superconductivity in Novel BiS 2 -Based Layered Superconductor LaO 1- x F x BiS 2,” J. Physical Soc. Japan, vol. 81, no. 11, p. 114725, Nov. 2012, doi: 10.1143/JPSJ.81.114725

    work page doi:10.1143/jpsj.81.114725 2012

  20. [20]

    Superconductivity on a Bi Square Net in LiBi,

    K. Górnicka et al., “Superconductivity on a Bi Square Net in LiBi,” Chemistry of Materials, vol. 32, no. 7, pp. 3150–3159, Apr. 2020, doi: 10.1021/acs.chemmater.0c00179

    work page doi:10.1021/acs.chemmater.0c00179 2020

  21. [21]

    Prediction of superconductivity and topological aspects in single-layer βBiPd,

    P.-F. Liu et al., “Prediction of superconductivity and topological aspects in single-layer βBiPd,” Phys. Rev. B, vol. 102, no. 15, p. 155406, Oct. 2020, doi: 10.1103/PhysRevB.102.155406

    work page doi:10.1103/physrevb.102.155406 2020

  22. [22]

    Correlating the charge-transfer gap to the maximum transition temperature in Bi 2 Sr 2 Ca n-1 Cu n O 2n+4+d

    Z. Wang et al., “Correlating the charge-transfer gap to the maximum transition temperature in Bi 2 Sr 2 Ca n-1 Cu n O 2n+4+d.” [Online]. Available: https://www.science.org

  23. [23]

    Superconductivity Induced by Oxygen Doping in Y 2 O 2 Bi,

    X. Cheng, E. E. Gordon, M. Whangbo, and S. Deng, “Superconductivity Induced by Oxygen Doping in Y 2 O 2 Bi,” Angewandte Chemie, vol. 129, no. 34, pp. 10257–10260, Aug. 2017, doi: 10.1002/ange.201701427

    work page doi:10.1002/ange.201701427 2017

  24. [24]

    L., Cococcioni M., Dabo I., et al., J

    J. Barzola-Quiquia, C. Lauinger, M. Zoraghi, M. Stiller, S. Sharma, and P. Häussler, “Superconductivity in the amorphous phase of topological insulator Bi x Sb 100– x alloys,” Supercond. Sci. Technol., vol. 30, no. 1, p. 015013, Jan. 2017, doi: 10.1088/0953- 2048/30/1/015013

    work page doi:10.1088/0953- 2017

  25. [25]

    Superconducting phase diagram in BixNi1-x thin films: The effects of Bi stoichiometry on superconductivity,

    J. Park et al., “Superconducting phase diagram in BixNi1-x thin films: The effects of Bi stoichiometry on superconductivity,” Phys. Rev. Mater., vol. 8, no. 7, p. 074805, Jul. 2024, doi: 10.1103/PhysRevMaterials.8.074805

    work page doi:10.1103/physrevmaterials.8.074805 2024

  26. [26]

    Superconductivity of the Bi–Sn Eutectic Alloy,

    M. V. Likholetova, E. V. Charnaya, E. V. Shevchenko, and Yu. A. Kumzerov, “Superconductivity of the Bi–Sn Eutectic Alloy,” Physics of the Solid State, vol. 63, no. 2, pp. 232–236, Feb. 2021, doi: 10.1134/S1063783421020153

    work page doi:10.1134/s1063783421020153 2021

  27. [27]

    Superconductivity in Bismuth. A New Look at an Old Problem,

    Z. Mata-Pinzón, A. A. Valladares, R. M. Valladares, and A. Valladares, “Superconductivity in Bismuth. A New Look at an Old Problem,” PLoS One, vol. 11, no. 1, p. e0147645, Jan. 2016, doi: 10.1371/journal.pone.0147645

    work page doi:10.1371/journal.pone.0147645 2016

  28. [28]

    A facile approach to calculating superconducting transition temperatures in the bismuth solid phases,

    I. Rodríguez, D. Hinojosa-Romero, A. Valladares, R. M. Valladares, and A. A. Valladares, “A facile approach to calculating superconducting transition temperatures in the bismuth solid phases,” Sci. Rep., vol. 9, no. 1, pp. 1–14, 2019, doi: 10.1038/s41598-019-41401-z

    work page doi:10.1038/s41598-019-41401-z 2019

  29. [29]

    Quasi- Stable Structures in Equilibrium Dense Bismuth Melt: Experimental and First Principles Theoretical Studies,

    B. N. Galimzyanov, A. A. Tsygankov, A. V. Suslov, V. I. Lad’yanov, and A. V. Mokshin, “Quasi- Stable Structures in Equilibrium Dense Bismuth Melt: Experimental and First Principles Theoretical Studies,” Scr. Mater., vol. 235, no. June, p. 115618, Jun. 2023, doi: 10.1016/j.scriptamat.2023.115618

    work page doi:10.1016/j.scriptamat.2023.115618 2023

  30. [30]

    Determination of the structure of liquids: an asymptotic approach,

    M. Mayo, E. Yahel, Y. Greenberg, E. N. Caspi, B. Beuneu, and G. Makov, “Determination of the structure of liquids: an asymptotic approach,” J. Appl. Crystallogr., vol. 46, no. 6, pp. 1582–1591, Dec. 2013, doi: 10.1107/S002188981302431X

    work page doi:10.1107/s002188981302431x 2013

  31. [31]

    On the Structure of Liquid Bismuth,

    U. Dahlborg and M. Davidovi, “On the Structure of Liquid Bismuth,” Phys. Chem. Liquids, vol. 15, no. 4, pp. 243–252, Mar. 1986, doi: 10.1080/00319108608078486

    work page doi:10.1080/00319108608078486 1986

  32. [32]

    What is the structure of liquid Bismuth?,

    E. N. Caspi, Y. Greenberg, E. Yahel, B. Beuneu, and G. Makov, “What is the structure of liquid Bismuth?,” J. Phys. Conf. Ser., vol. 340, 2012, doi: 10.1088/1742-6596/340/1/012079. Página | 19

    work page doi:10.1088/1742-6596/340/1/012079 2012

  33. [33]

    DMol, a standard tool for density functional calculations: Review and advances,

    B. Delley, “DMol, a standard tool for density functional calculations: Review and advances,” Theoretical and Computational Chemistry, vol. 2, no. C, pp. 221–254, Jan. 1995, doi: 10.1016/S1380-7323(05)80037-8

    work page doi:10.1016/s1380-7323(05)80037-8 1995

  34. [34]

    BIOVIA Materials Studio, Release,

    Dassault Systèmes BIOVIA, “BIOVIA Materials Studio, Release,” 2015, Dassault Systèmes, San Diego: 2020

    2015

  35. [35]

    Canonical dynamics: Equilibrium phase-space distributions,

    W. G. Hoover, “Canonical dynamics: Equilibrium phase-space distributions,” Phys. Rev. A (Coll. Park)., vol. 31, no. 3, pp. 1695–1697, Mar. 1985, doi: 10.1103/PhysRevA.31.1695

    work page doi:10.1103/physreva.31.1695 1985

  36. [36]

    Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis,

    S. H. Vosko, L. Wilk, and M. Nusair, “Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis,” Can. J. Phys., vol. 58, no. 8, pp. 1200–1211, Aug. 1980, doi: 10.1139/p80-159

    work page doi:10.1139/p80-159 1980

  37. [37]

    VallaClan: Computational MAterials Sience Code and Analysis Repository

    I. Rodríguez-Aguirre, D. Hinojosa-Romero, A. Valladares, R. M. Valladares, and Valladares Ariel A., “VallaClan: Computational MAterials Sience Code and Analysis Repository.” Accessed: May 19, 2026. [Online]. Available: https://github.com/Isurwars/Vallaclan

    2026

  38. [38]

    RMCProfile: reverse Monte Carlo for polycrystalline materials,

    M. G. Tucker, D. A. Keen, M. T. Dove, A. L. Goodwin, and Q. Hui, “RMCProfile: reverse Monte Carlo for polycrystalline materials,” Journal of Physics: Condensed Matter, vol. 19, no. 33, p. 335218, Aug. 2007, doi: 10.1088/0953-8984/19/33/335218

    work page doi:10.1088/0953-8984/19/33/335218 2007

  39. [39]

    New capabilities for enhancement of RMCProfile : instrumental profiles with arbitrary peak shapes for structural refinements using the reverse Monte Carlo method,

    Y. Zhang, M. Eremenko, V. Krayzman, M. G. Tucker, and I. Levin, “New capabilities for enhancement of RMCProfile : instrumental profiles with arbitrary peak shapes for structural refinements using the reverse Monte Carlo method,” J. Appl. Crystallogr., vol. 53, no. 6, pp. 1509–1518, Dec. 2020, doi: 10.1107/S1600576720013254

    work page doi:10.1107/s1600576720013254 2020

  40. [40]

    Correlation: An Analysis Tool for Liquids and for Amorphous Solids,

    I. Rodríguez, R. Valladares, A. Valladares, D. Hinojosa-Romero, U. Santiago, and A. Valladares, “Correlation: An Analysis Tool for Liquids and for Amorphous Solids,” J. Open Source Softw., vol. 6, no. 65, p. 2976, Sep. 2021, doi: 10.21105/joss.02976

    work page doi:10.21105/joss.02976 2021

  41. [41]

    Structure factor and atomic distribution in liquid metals by X-ray diffraction,

    Y. Waseda and K. Suzuki, “Structure factor and atomic distribution in liquid metals by X-ray diffraction,” physica status solidi (b), vol. 49, no. 1, pp. 339–347, Jan. 1972, doi: 10.1002/pssb.2220490132

    work page doi:10.1002/pssb.2220490132 1972

  42. [42]

    Evidence for a temperature-driven structural transformation in liquid bismuth,

    Y. Greenberg et al., “Evidence for a temperature-driven structural transformation in liquid bismuth,” Epl, vol. 86, no. 3, 2009, doi: 10.1209/0295-5075/86/36004

    work page doi:10.1209/0295-5075/86/36004 2009