What Lies Between Crystal and Randomly Packed Structures? A General Characterization of Non-Periodic Order

Through an extensive study of over 7,000 ground-state structures in a 2D binary packing model, the paper reveals that while non-periodic structures dominate, approximately 35% of them exhibit "structural selectivity," a property that serves as a signature of underlying order extending well beyond the diversity limits of periodic crystals.

The Gist

Imagine you are looking at a pile of LEGO bricks. You know two extreme ways to arrange them:

1. The Crystal: You build a perfect, repeating castle. Every brick is in a specific spot, and the pattern repeats forever. It's highly ordered, simple, and predictable.
2. The Random Pile: You dump the bricks on the floor. They are jumbled, chaotic, and have no pattern. It's the definition of "messy" or "disordered."

For a long time, scientists thought that if something wasn't a perfect crystal, it had to be a random pile. They lumped everything in the middle into a big bucket called "amorphous" or "disordered."

The Big Question
Ian Douglass and Peter Harrowell asked: What actually lives in the space between the perfect castle and the random pile? Are there other ways to be organized that we just haven't noticed because we were too busy looking for perfect crystals?

To find out, they didn't use real atoms (which are messy and hard to control). Instead, they built a giant digital simulation using a 2D grid of two types of particles (let's call them Red and Blue blocks). They ran a computer experiment to find the "ground state" for thousands of different rule sets. A "ground state" is simply the most stable, lowest-energy arrangement the blocks can settle into.

They generated 7,609 different stable structures. Here is what they found:

1. The "Random" Pile is Actually the Majority

When they looked at all 7,609 structures, they found that over 96% of them were not crystals. They were non-periodic (no repeating pattern).

But here is the twist: Just because they weren't repeating crystals didn't mean they were random messes. Some of these structures were surprisingly organized.

2. Measuring "Complexity" with a "Species" Count

To tell the difference between a "messy pile" and a "complex but organized structure," the authors used a concept borrowed from ecology: Diversity.

Imagine a forest.

• If you have a forest with only one type of tree, the diversity is low.
• If you have a forest with 100 different types of trees, the diversity is high.

In their simulation, the "trees" are small local patterns of Red and Blue blocks. They counted how many different types of local patterns existed in each structure.

• Crystals usually have low diversity (only a few types of patterns repeating).
• Random piles have high diversity (every possible pattern is there).

The Discovery: They found that while crystals stop being crystals once the diversity gets too high (around 5 types of patterns), there are non-crystal structures that are highly organized even when they have up to 9 types of patterns.

3. The "Picky" Test (Structural Selectivity)

This is the most important part of the paper. How do you know if a non-crystal structure is actually "ordered" and not just a lucky accident?

The authors invented a test called Structural Selectivity. Think of it like a bouncer at a club.

• The Scenario: Imagine you have a stable structure (the club). Now, you try to sneak in a new, slightly different local pattern (a new guest) that the rules of the system could technically allow.
• The Test:
  • The "Non-Selective" (Random) Structure: The bouncer lets the new guest in. The structure just absorbs the new pattern without fighting it. It's like a pile of sand; you can add a new grain, and nothing changes. This means there is no underlying "rule" forcing the structure to be a certain way.
  • The "Selective" (Ordered) Structure: The bouncer rejects the new guest. The structure refuses to accommodate the new pattern because it would break the internal logic of the whole system. It actively excludes options.

The Result:
They found that 35% of all the non-crystal structures were "Selective."
This means that even though they don't look like repeating crystals, they are following a strict, hidden rule that forces them to reject certain arrangements. They are ordered, just not in a way we usually recognize.

4. What Do These "Hidden Order" Structures Look Like?

The paper suggests these "selective but non-crystal" structures fall into a few categories, which they illustrated with images:

• Crystals with random spots: A mostly perfect crystal with a few random "defects" sprinkled in.
• Crystals with grain boundaries: Crystals that are stitched together with messy lines in between.
• Irregular motifs: A pattern that repeats locally but doesn't line up globally (like a tiling that never quite closes the loop).
• Random networks: A maze-like structure where a specific shape repeats over and over, but it forms a complex web rather than a grid.

The Bottom Line

The paper argues that we have been too lazy with the word "disordered."

• Periodic Order: Repeating patterns (Crystals).
• Non-Periodic Order: Structures that aren't repeating but still have a "bouncer" that rejects certain patterns (The 35% found in this study).
• True Disorder: Structures that accept anything and have no underlying rules.

The authors conclude that the world of "in-between" structures is vast. About a third of the non-crystal structures they found are actually following a hidden set of rules (selectivity), proving that order exists even without a repeating pattern. They propose using "Diversity" (how many pattern types exist) and "Selectivity" (does it reject new patterns?) as better tools to describe materials than just calling them "crystals" or "glasses."

Technical Summary

Technical Summary: What Lies Between Crystal and Randomly Packed Structures?

Problem Statement
The characterization of condensed matter structures has traditionally relied on a binary distinction: periodic crystals (low complexity) and amorphous solids or glasses (maximal complexity). This dichotomy leaves a significant gap in understanding the structural landscape that exists between these two limits. Current measures of complexity, such as unit cell size ($Z$) or symmetry-based entropy, are inapplicable to non-crystalline materials. Furthermore, the term "amorphous" or "disordered" often bundles distinct classes of structures together due to a lack of metrics capable of differentiating them. The central question motivating this work is: what types of structures occupy the range of complexities between simple crystals and random packings, and how can they be characterized and distinguished?

Methodology
To explore this structural space, the authors employed the Favored Local Structure (FLS) model, a two-dimensional triangular lattice system where sites are occupied by either A or B particles. The model defines 14 distinct local structures (LS) based on the occupancy of nearest neighbors. By assigning an energy of -1 to a specific subset of these LS (the "favored" structures) and 0 to the rest, the authors generated groundstate configurations for 7,609 unique systems.

The study utilized two primary metrics:

1. Structural Diversity ($S_1$): Adapted from ecological Hill's numbers, this metric quantifies the effective number of distinct local structural "species" in a configuration. It is calculated based on the frequency distribution of the 14 local structures, providing a measure of complexity that generalizes crystallographic complexity measures to non-periodic systems.
2. Structural Selectivity: To identify order in the absence of periodicity, the authors introduced a protocol to test "selectivity." A groundstate structure is deemed selective if it excludes certain local structures that are stabilized by the Hamiltonian. This is determined by checking if a specific groundstate can be generated by a system where one or more favored local structures are explicitly excluded from the set of allowed structures. If the structure remains the groundstate despite the exclusion of a stabilizing local structure, it demonstrates an underlying ordering principle that rejects that structure.

Key Results

• Predominance of Non-Periodic Structures: Out of 7,609 unique groundstates, 96% were non-periodic. These non-periodic structures span the entire range of possible structural diversities.
• Diversity Thresholds: Periodic structures were found to be truncated at a diversity of $S_1 \approx 5$. While periodic order disappears beyond this threshold, non-periodic structures continue to exist and exhibit high diversity.
• Existence of Non-Periodic Order: A major finding is that approximately 35% of the non-periodic groundstates (2,602 structures) are selective. This indicates that these structures are not random; they adhere to an underlying ordering principle that actively excludes specific local structural options.
• Range of Non-Periodic Order: This selectivity extends up to a diversity of $S_1 \approx 9$, significantly beyond the upper limit for periodic order ($S_1 \approx 5$).
• Classification of Selective Structures: The authors preliminarily categorized these non-periodic selective structures into four types: (i) crystals with random pointwise inclusions, (ii) crystals with random grain boundaries, (iii) irregular assemblies of motifs, and (iv) random networks.
• Non-Selective "Disordered" Structures: The remaining ~65% of non-periodic structures were found to be non-selective. Even those with low diversity ($S_1 \le 5$), which exhibit repeated motifs, failed to demonstrate the capacity to exclude added favored local structures, suggesting they lack a specific ordering principle detectable by this metric.

Significance and Claims
The paper claims that the absence of crystalline periodicity does not equate to the absence of order. By introducing structural selectivity as a signature of an ordering principle, the authors demonstrate that a substantial fraction of non-periodic condensed matter possesses an underlying order that is distinct from both simple crystals and random packings.

The authors argue that the traditional term "disorder" is insufficient as it provides no information about the structure other than the absence of a specific target. Instead, they propose using structural diversity and structural selectivity as quantitative metrics to displace the term "disorder" in the analysis of non-periodic structures. These metrics allow for a more nuanced characterization of the structural landscape, distinguishing between structures that are truly random and those that are ordered through principles other than periodic repetition. The work suggests that the "middle ground" between crystals and glasses is populated by a diverse array of ordered, non-periodic states that have previously been obscured by the lack of appropriate characterization tools.