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Crystals are known far and wide for their beauty and elegance. But even though they may appear perfect on the outside, their microstructure can be quite complicated, making them difficult to model mathematically.

But there are people rising to the challenge. In an article this month in Royal Society Open Science, researchers from The University of Osaka used differential geometry to provide a robust, rigorous, and unified description for the mechanics of crystals and their defects.

In an ideal crystal, each atom is arranged in a perfectly periodic pattern. However, most crystals, upon closer examination, are not perfect. They contain small defects in their structure—a missing atom here, an extra bond there. These defects have important mechanical consequences—they could be the starting point of a fracture, for example, or they could even be used to strengthen materials. Understanding defects and their phenomena is thus very important to researchers.

"Defects come in many forms," explains lead author of the study Shunsuke Kobayashi. "For example, there are so-called dislocations associated with the breaking of translational symmetry and disclinations associated with the breaking of rotational symmetry. Capturing all of these kinds of defects in a single mathematical theory is not straightforward."

Indeed, previous models have failed to reconcile the differences between dislocations and disclinations, suggesting that modifications to the theory are needed. New mathematical tools using the language of differential geometry proved to be exactly what the team needed to address these issues.

"Differential geometry provides a very elegant framework for describing these rich phenomena," says Ryuichi Tarumi, senior author. "Simple mathematical operations can be used to capture these effects, allowing us to focus on the similarities between seemingly disparate defects."

Using the formalism of Riemann–Cartan manifolds, the research team was able to elegantly encapsulate the topological properties of defects and rigorously prove the relationship between dislocations and disclinations; previously, only empirical observations existed, and their rigorous mathematical forms were a mystery. In addition, they were able to derive analytical expressions for the stress fields caused by these defects.

The team hopes that their geometric approach to describing the mechanics of crystals will eventually inspire scientists and engineers to design materials with specific properties by taking advantage of defects, such as the strengthening of materials that is seen with disclinations. In the meantime, these results are yet another example of how beauty in mathematics can help us understand beauty in nature.