Scientists Puzzled by 'Rule of Four' Pattern Found in Most Materials

A bizarre "Rule of Four" has been identified in the basic structure of the majority of inorganic materials—and scientists are stumped as to why.

The pattern is found in the so-called "unit cell" of the materials—the smallest possible repeating part of each molecular structure.

Researchers at the Swiss Federal Institute of Technology in Lausanne (EPFL) analyzed two massive databases that record the structural patterns of more than 80,000 known and predicted materials. They were surprised to discover that some 60 percent of materials had a unit cell in which the total number of atoms was a multiple of four.

The full findings of the study were published last week in the journal npj Computational Materials.

As scientists are prone to do, the team set out to look for an explanation for the unexpected pattern. At first, they thought it might be just a computing bug relating to how the material structures are classified and recorded in the databases.

"A 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," Elena Gazzarrini, a former EPFL researcher currently at CERN in Geneva, said in a statement. "The first question we asked was whether the software used to 'primitivize' the unit cell had done it correctly, and the answer was yes."

Next, the team turned to chemistry for answers. They remembered that the element silicon has a "coordination number" of four—meaning that each atom is able to bind to four others.

"We could expect to find that all the materials following this Rule of Four included silicon," Gazzarrini said. "But again, they did not."

Additionally, considering the amount of energy required to form bonds between atoms couldn't explain the "Rule of Four" either.

"The materials that are most abundant in nature should be the most energetically favored, which means the most stable ones, those with negative formation energy," Gazzarrini said. "But what we saw with classic computational methods was that there was no correlation between the Rule of Four and negative formation energies."

Finally, the researchers created a special algorithm to determine if there was any pattern that linked a material's atomic properties and whether or not it had a structure that followed the "Rule of Four," but this also found no clear result.

In the end—while the team has managed to rule out several possible explanations—they have had to report that the mystery remains unsolved.

The team did make one promising finding, however, which may help future studies into the mysterious structural phenomenon. When the researchers applied a form of artificial intelligence (AI) to the problem, they found that it could predict if a compound will follow the Rule of Four with an 87 percent success rate.

"This is interesting because the algorithm uses only local rather than global symmetry descriptors," Gazzarrini said.

This, she concluded, "suggests that there may be small chemical groups in the cells (still to be found) that may explain the Rule of Four."

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