The structure of a melt: The case of liquid bismuth

This study employs Molecular Dynamics simulations and Reverse Monte Carlo modeling to characterize the atomic structure of liquid bismuth at 573 K, revealing a local arrangement dominated by deformed triangles and squares that manifest as specific peaks in Pair Distribution and Plane Angle Distributions.

The Gist

Imagine a pot of molten metal, specifically Bismuth, a silvery element that looks like a rainbow-colored crystal when it cools down. When it's hot and liquid, the atoms inside are dancing chaotically, bumping into each other like a crowded mosh pit. The big question scientists have been asking is: Even though they are moving randomly, do these atoms still hold onto some hidden shapes or patterns from when they were solid?

This paper is like a high-tech detective story where the authors used powerful computers to freeze-frame this atomic dance and look for those hidden patterns.

The Setup: A Virtual Dance Floor

The researchers built a virtual "supercell" (a tiny box) containing 216 bismuth atoms. Think of this box as a dance floor.

• They started the atoms at a cool temperature (300 K) and heated them up until they were melting and flowing (573 K).
• They ran this simulation 100 times in a row, then kept the "dance" going for another 500 steps to make sure the liquid was stable.
• To make sure they weren't just seeing a fluke, they started four different simulations with slightly different "initial pushes" (random velocities) for the atoms, just to see if the outcome was the same every time.

The Tools: Taking Snapshots of the Chaos

To understand the structure, they used two main tools:

1. PDF (Pair Distribution Function): Imagine taking a photo of the crowd and measuring the distance between every pair of people. If you see a lot of people standing exactly 3 feet apart, you get a "peak" on your graph. This tells you how far apart atoms usually sit.
2. PAD (Plane Angle Distribution): This measures the angles formed by three atoms. If Atom A, Atom B, and Atom C form a triangle, what is the angle at Atom B? This tells you the shape of the clusters.

The Big Discovery: The "Shoulder" Mystery

In the liquid state, the graph of distances (the PDF) usually has a big main peak, followed by a second peak. But in Bismuth, there is a weird "bump" or shoulder right after the first peak.

• The Controversy: Some scientists thought this shoulder was just a mistake caused by how they measured the data in real life (like a blurry photo). Others thought it was a real feature of the liquid.
• The Verdict: Since the authors created this data entirely inside a computer (no blurry photos involved), and the shoulder appeared every single time, it is real. It's a genuine feature of liquid Bismuth.

What Shapes Are Hiding in the Liquid?

By analyzing the angles and distances, the authors found that even in the chaotic liquid, the atoms aren't totally random. They are forming specific, slightly squashed shapes:

1. Deformed Triangles: The atoms like to group in threes, forming triangles. However, they aren't perfect equilateral triangles; they are squashed or stretched. This corresponds to a specific angle of about 53° to 58°.
2. Deformed Squares: The atoms also form groups of four that look like squares or diamonds, but again, they are distorted. This corresponds to angles around 85° to 90°.

The "Shoulder" Explained:
The mysterious "shoulder" bump in the distance graph is actually caused by the diagonal lines of these squashed squares. When you look at a square, the distance across the corner-to-corner (the diagonal) is longer than the side-to-side distance. In the liquid, these diagonal distances create that extra "bump" in the data.

The Conclusion

The paper concludes that liquid Bismuth isn't just a random soup of atoms. It retains a "memory" of its solid structure. Even when melted, the atoms prefer to arrange themselves into squashed triangles and squashed squares.

This explains the "shoulder" in the data: it's the fingerprint of those square-like shapes. The authors also noted that there might be even more complex shapes (like pentagons or hexagons) lurking in the data, but those are a mystery for another day.

In short: The liquid is chaotic, but it's a structured chaos, full of squashed triangles and squares that leave a distinct mark on the data, proving that the liquid structure is more organized than we thought.

Technical Summary

Technical Summary: The Structure of a Melt: The Case of Liquid Bismuth

Problem Statement
Despite bismuth (Bi) being a well-studied material with significant industrial applications and unique phase transitions (including superconductivity in amorphous and liquid states), the precise atomic structure of its liquid phase remains a subject of debate. Specifically, there are discrepancies in the literature regarding the Pair Distribution Functions (PDFs) and Plane Angle Distributions (PADs) of liquid bismuth. A central point of contention is the existence and origin of a "conspicuous pseudo-peak" (or shoulder) observed in PDFs at approximately 4.6 Å. Some researchers attribute this feature to specific atomic arrangements (such as triangular chains or square structures), while others argue it is an artifact of experimental measurement procedures or Fourier transformation of structure factors. Furthermore, the relationship between these structural features and the enhanced electronic properties (such as high-$T_c$ superconductivity) of disordered bismuth phases remains unclear. This study aims to resolve these discrepancies by investigating the atomic topology of liquid bismuth at 573 K to determine if specific geometrical structures exist that could explain the observed PDF features and potentially influence electronic properties.

Methodology
The authors employed a multi-step computational approach using Molecular Dynamics (MD) and Reverse Monte Carlo (RMC) simulations:

1. System Setup: Four supercells, each containing 216 bismuth atoms, were constructed based on a 3x3x3 multiplication of a diamond-like unit cell. This initially unstable structure was chosen to facilitate evolution into a disordered topology. The system maintained the experimental liquid density of 10.029 g/cm³ at 573 K.
2. Molecular Dynamics (MD): Simulations were performed using the Materials Studio suite (DMol3 code) with Density Functional Theory (DFT). The process involved:
   • Heating from 300 K to 573 K over 100 steps (19.4 fs each).
   • Maintaining the temperature at 573 K for 500 steps using a Nosé-Hoover NVT thermostat.
   • Four distinct initial random velocity sets ($v_0$ to $v_3$) were used to test the independence of the final structures from initial conditions.
   • Electronic calculations utilized the LDA-VWN functional, double numerical basis sets with d-function polarization (dnd), and Semi-Core Pseudo Potentials (dspp).
3. Data Analysis:
   • PDFs: Calculated for the last 1, 10, 25, and 100 steps of the MD trajectory.
   • RMC Generation: To obtain representative atomic structures, Reverse Monte Carlo was applied to the averaged PDFs of the last 100 steps for each velocity set, generating four distinct atomic configurations.
   • PADs: Plane Angle Distributions were calculated for both the MD-averaged structures and the RMC-generated structures using a custom code ("Correlation"). A bond cutoff of 4.185 Å (the minimum between the first peak and the shoulder) was defined to identify bonded atoms.
4. Validation: The stability of the simulation was verified by monitoring energy variation, which was found to be negligible. Results were compared against existing experimental data and other computational studies.

Key Results

1. Convergence and Consistency: The PDFs obtained from the four different initial velocity sets became indistinguishable when averaged over the last 100 steps, confirming that the system reached a representative liquid state independent of initial conditions. The RMC-generated structures produced PDFs identical to the MD averages, validating the RMC approach for describing the liquid phase.
2. Pair Distribution Functions (PDFs): The calculated PDFs for liquid bismuth at 573 K exhibit two prominent peaks at 3.25 Å and 6.55 Å, along with a distinct shoulder (pseudo-peak) at 4.6 Å.
   • The first peak (3.25 Å) results from the coalescence of the first two crystalline peaks (3.12 Å and 3.47 Å).
   • The shoulder at 4.6 Å corresponds to the coalescence of the third and fourth crystalline neighbor peaks (4.52 Å and 4.72 Å).
   • These results align with many experimental and computational studies that report the shoulder, though some experimental variations exist regarding its prominence.
3. Plane Angle Distributions (PADs): The PADs reveal specific angular preferences indicative of local geometrical order:
   • ~53° (MS) and ~58° (RMC): These peaks suggest the presence of deformed equilateral triangles (triads).
   • ~85° (MS) and ~90° (RMC): These peaks indicate the existence of deformed squares or rhombi.
   • 120°–140°: A broad, less pronounced region suggests the presence of more complex, higher-order geometrical structures.
4. Origin of the Shoulder: The study establishes a direct correlation between the shoulder in the PDF and the angular distribution. The interatomic distance corresponding to the diagonal of the deformed square structures (identified by the 90° angles) matches the position of the shoulder (~4.6 Å).

Significance and Claims
The paper claims to resolve the debate regarding the shoulder in the PDF of liquid bismuth. The authors assert that:

• Structural Reality: The shoulder is a genuine structural feature of liquid bismuth, not an artifact of experimental measurement or Fourier transformation errors. It arises from the statistical clustering of atoms into specific geometries.
• Geometrical Origin: The shoulder is caused by the coalescence of crystalline second-neighbor peaks and, more specifically, by the existence of deformed equilateral triangles and deformed squares within the liquid structure. The diagonal distances in these deformed squares correspond to the 4.6 Å shoulder.
• Methodological Contribution: By combining MD with RMC, the study provides a consistent set of atomic structures that reproduce both the PDF and PAD features, offering a clearer picture of the local topology than previous conflicting reports.
• Implications for Properties: While the paper does not definitively prove a causal link to superconductivity, it posits that the identification of these specific atomic arrangements (disorder, triangles, and squares) in the liquid phase provides the necessary structural context to investigate why disordered bismuth phases exhibit heightened electronic properties and superconductivity.

In conclusion, the work identifies the local atomic topology of liquid bismuth as being dominated by deformed triangular and square motifs, which directly account for the controversial shoulder in the Pair Distribution Function.