Coexistence of close packed structures in large substrate-free Ar-Kr clusters according to THEED data

This study utilizes in-situ transmission electron diffraction to demonstrate that large, substrate-free Ar-Kr clusters formed via supersonic expansion exhibit a size-dependent coexistence of fcc and hcp phases, with the hexagonal fraction increasing with cluster size and peaking at equimolar composition, supporting a thermally activated diffusion mechanism for hcp nucleation.

The Gist

Imagine you have a giant bucket of tiny, invisible marbles made of Argon and Krypton. These aren't just any marbles; they are frozen atoms of noble gases. When you spray them out of a nozzle into a vacuum, they cool down instantly and clump together to form "clusters"—tiny, floating snowballs of atoms.

This paper is about figuring out what shape these tiny snowballs take as they grow bigger.

The Big Puzzle: The "Shape" Dilemma

In the world of atoms, there are two main ways to pack these marbles tightly together, like stacking oranges in a grocery store:

1. The "Cube" Stack (fcc): Imagine stacking oranges in a perfect square grid. This is the most common shape for these gas clusters when they are small.
2. The "Hexagon" Stack (hcp): Imagine stacking them in a honeycomb pattern. Physics theories say this shape is actually slightly more efficient and "happier" for the atoms, but in the real world, big blocks of these gases usually stick to the square stack unless you squeeze them with massive pressure.

The Mystery: Scientists have long wondered: When does a tiny cluster decide to switch from the square stack to the hexagon stack? And does mixing two different gases (Argon and Krypton) change the rules?

The Experiment: A High-Speed Freeze-Frame

The researchers created these clusters by shooting a super-cooled gas mixture through a tiny nozzle into a vacuum. It's like opening a pressurized soda can in space; the gas expands, cools, and instantly turns into a mist of tiny clusters.

They used a powerful electron camera (a technique called THEED) to take "snapshots" of these clusters while they were still floating in the air. They looked at clusters ranging from very small (about 2,000 atoms) to quite large (100,000 atoms) and tested them with different mixtures of Argon and Krypton.

What They Found: The "Size" Switch

Here are the main discoveries, explained simply:

1. The "Magic Size" Threshold
It turns out the mix of gases doesn't matter for the start of the change. Whether the cluster is pure Argon, pure Krypton, or a 50/50 mix, they all behave the same way initially.

• The Rule: As long as the cluster is smaller than a certain "magic size" (about 10,000 atoms), it stays in the square (fcc) shape.
• The Switch: Once the cluster grows bigger than that magic size, it starts to develop the hexagonal (hcp) shape. It's like a child growing tall enough to finally reach the top shelf; the size is the trigger, not the ingredients.

2. The Two-Phase Snowball
Here is the most surprising part: The clusters don't just flip from square to hexagon. They become hybrids.

• Think of a cluster as a snowball that is half-square-packed and half-hexagon-packed at the same time.
• As the cluster gets even bigger, the hexagonal part grows, but the square part doesn't disappear. Both shapes live together inside the same tiny snowball.
• Even in the largest clusters they tested (100,000 atoms), they never saw a cluster that was 100% hexagonal. It's always a mix.

3. The "Perfect Mix" Effect
While the start of the change depends only on size, the amount of hexagonal shape depends on the recipe.

• If you mix Argon and Krypton in equal amounts (a 50/50 split), the cluster loves to form the hexagonal shape the most.
• It's as if the two different-sized atoms (Argon is smaller, Krypton is larger) create a little bit of "stress" or "wobble" in the square structure. This wobble makes it easier for the atoms to rearrange into the hexagonal shape. The more "wobble" (which happens at the 50/50 mix), the more hexagonal structure appears.

Why Does This Happen?

The researchers believe this happens because of how the clusters grow.

• The Old Theory: Some thought the jet might contain two separate groups of clusters: some that are square and some that are hexagonal.
• The New Evidence: The data suggests that inside a single cluster, both shapes are growing side-by-side. As the cluster grows from a liquid droplet, it starts forming a square core, but as it gets bigger, a hexagonal "seed" starts growing inside it. They grow together, like two different flavors of ice cream swirling in the same cone, rather than two separate cones.

The Bottom Line

This study shows that for these tiny, floating gas clusters:

1. Size is King: You need to be big enough (over 10,000 atoms) before the hexagonal shape even tries to appear.
2. Mixing Helps: If you mix Argon and Krypton equally, the hexagonal shape becomes much more dominant.
3. Coexistence is Normal: These clusters are rarely just one shape; they are usually a stable mix of both the square and hexagonal structures living together.

It's a bit like a crowd of people: when the group is small, everyone stands in a square formation. But once the crowd gets huge, a section of the crowd naturally shifts into a hexagonal pattern, and both patterns end up standing together in the same space.

Technical Summary

Technical Summary: Coexistence of Close-Packed Structures in Large Substrate-Free Ar-Kr Clusters

Problem Statement
The paper addresses the long-standing "fcc–hcp dilemma" in the physics of solidified rare gases: the competition between face-centered cubic (fcc) and hexagonal close-packed (hcp) structures. While theoretical calculations suggest a slight energetic advantage for the hcp structure, bulk rare gas crystals (except helium) typically stabilize in the fcc phase under normal pressure, with hcp formation usually requiring ultrahigh pressure or specific thermal annealing in thin films. However, recent observations indicate that large rare gas clusters can exhibit a stable hcp phase. A critical gap in understanding remains regarding the phase behavior of binary rare gas clusters (specifically Ar–Kr) across the full range of component concentrations. Unlike bulk Ar–Kr mixtures, which form exclusively fcc substitutional solid solutions, it is unclear how the component ratio influences the nucleation and growth of the hcp phase in free clusters, and whether large clusters exist as single-phase aggregates or as coexisting two-phase systems.

Methodology
The study utilized Transmission High-Energy Electron Diffraction (THEED) to perform a quantitative phase analysis of substrate-free single-component and binary Ar–Kr clusters.

• Cluster Generation: Clusters were produced via the adiabatic expansion of pre-cooled gas (or gas mixtures) through a supersonic nozzle into a vacuum. The average cluster size ($\bar{N}$) was controlled by adjusting the inlet pressure ($0.1–0.6$ MPa) and temperature ($100–300$ K), yielding sizes ranging from $2\times10^3$ to $1\times10^5$ atoms per cluster.
• Composition Control: The study covered single-component Ar and Kr, as well as binary mixtures with krypton mole fractions of $0.25$ and $0.8$. The initial gas mixture composition was adjusted using a semi-empirical equation to account for the "enrichment effect," where the cluster composition differs from the gas mixture due to differences in intermolecular binding energy.
• Data Analysis: Diffraction patterns were recorded in-situ at a distance of 100 mm from the nozzle. The integrated intensities of diffraction maxima for hcp (101) and fcc (200) planes were approximated using Lorentzian functions to determine the relative volumes ($V_{hcp}/V_{fcc}$) and domain sizes of the crystalline phases.

Key Results

1. Threshold Size Independence: The study revealed that the threshold cluster size for the onset of hcp phase formation is independent of the component composition. For both single-component and binary Ar–Kr clusters, the hcp phase begins to form when the average cluster size reaches approximately $90$Å ($\bar{N} \approx 1\times10^4$ atoms/cluster).
2. Two-Phase Coexistence: Clusters larger than this threshold do not transition entirely to a single hcp phase. Instead, they exhibit a two-phase fcc-hcp structure. Crucially, the concentration of chemical components (Ar and Kr) is identical in both the fcc and hcp phases within a single cluster.
3. Composition Dependence of Phase Fraction: While the onset of the hcp phase is size-dependent only, the fraction of the hcp phase in larger clusters depends on the component concentration. The fraction of the hexagonal phase increases with cluster size and is maximized in clusters with an equimolar (1:1) Ar–Kr composition. This is attributed to static distortions in the crystal lattice caused by the geometric size mismatch between Ar and Kr atoms, which facilitate diffusion processes.
4. Domain Evolution: As clusters grow beyond the threshold, the size of the hcp domains increases rapidly before slowing down. Simultaneously, the fcc domains continue to grow steadily without shrinking, even in equimolar mixtures. This indicates that the fcc phase does not disappear but coexists with the growing hcp phase.
5. Formation Mechanism: The data supports the hypothesis that these are mixed two-phase clusters formed within a single aggregate during the supersonic jet expansion, rather than separate single-phase fcc and hcp clusters. The simultaneous growth of both phases suggests a thermally activated diffusion mechanism for hcp nucleation, likely initiated by deformation-type stacking faults inherent in the fcc lattice of rare gas clusters.

Significance and Claims
The paper claims to provide the first quantitative phase diagram for substrate-free Ar–Kr clusters across the entire concentration range. The primary significance lies in establishing that the transition from fcc to a two-phase structure in free clusters is a pure size effect, distinct from the temperature-dependent mechanisms observed in thin films.

The authors argue that the observation of simultaneous growth of fcc and hcp domains, along with the constant component concentration in both phases, serves as indirect confirmation that large free clusters are inherently two-phase aggregations. This finding aligns with a thermally activated diffusion model for hcp nucleation, where the process begins at the crystallization stage of pre-condensed aggregates. The results also clarify that even in large clusters ($\bar{N} \approx 1\times10^5$), a single-phase hcp state is not achieved, mirroring the behavior of bulk rare gas cryocrystals under ultrahigh pressure where a single-phase hcp state is also unattainable.