Scientists Baffled as Most Materials Appear to Follow Mysterious ‘Rule of Four’

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A comprehensive analysis of a vast database of compounds has uncovered an intriguing pattern in the way matter organizes itself. Out of more than 80,000 electronic structures of both experimental and predicted materials examined, a striking 60 percent had a basic structural unit based on a multiple of four.

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The research team behind this discovery was puzzled by why this pattern exists—they know it’s real, but they haven’t been able to explain it. “Through an extensive investigation, we highlight and analyze the anomalous abundance of inorganic compounds whose primitive unit cell contains a number of atoms that is a multiple of four, a property that we name ‘rule of four,’” writes physicist Elena Gazzarrini, formerly at the Swiss Federal Institute of Technology Lausanne in Switzerland and now at CERN.

“The study provides a starting point for future investigations on the rule’s emergence, given that a fully satisfactory explanation of such anomalous distribution is as yet lacking.”

Researchers in the field of materials science are keen to understand the properties and behavior of different combinations of atoms. This knowledge could help develop and refine materials with specific characteristics. However, the sheer variety in how atoms can combine makes it challenging to study this field. That’s why Gazzarrini and her team became intrigued when they noticed the emerging pattern in two databases: the Materials Project (MP) database and the Materials Cloud 3-dimensional crystal structures ‘source’ database (MC3Dsource).

The majority of inorganic compounds in these databases had unit cells—the smallest possible repeating unit within a crystal structure—that were based on multiples of four. This discovery was unexpected because, theoretically, all structure types should be equally represented in these databases. A dominant pattern could indicate an unnoticed flaw in the data.

“The first intuitive reason could come from the fact that when a conventional unit cell (a larger cell than the primitive one, representing the full symmetry of the crystal) is transformed into a primitive cell, the number of atoms is typically reduced by four times,” Gazzarrini explains. “The first question we asked was whether the software used to ‘primitivize’ the unit cell had done it correctly, and the answer was yes.”

After ruling out obvious errors, the team explored other potential explanations. They considered whether the common factor was silicon, as it can bind four other atoms to its central atom. However, not all the compounds contained silicon. Similarly, there was no correlation with the formation energies of the rule of four compounds.

To better understand the rule of four, the team worked with engineer Rose Cernosky from the University of Wisconsin to create a more powerful algorithm. The algorithm grouped compounds based on similarities in their atomic properties, but even this approach did not yield a discernible pattern.

When the team used a machine learning algorithm on the data, it achieved a high level of accuracy—up to 87 percent—in predicting whether or not a compound would follow the rule of four. This suggests there may be something yet to be discovered that could explain the pattern.

Despite the difficulties in finding a definitive explanation, the findings indicate that with advanced computational techniques, researchers could make significant progress in understanding these patterns in the future.

The team’s research has been published in npj Computational Materials, offering a new avenue for exploring the underlying rules governing the structure of matter.

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