Mathematics Uncover Hidden Geometry of Imperfect Crystals

By The University of Osaka July 18, 2025

Collected at: https://scitechdaily.com/mathematics-uncover-hidden-geometry-of-imperfect-crystals/

Researchers at the University of Osaka have developed new mathematical models that offer new insights into the behavior of crystal defects.

Crystals are widely admired for their striking appearance and symmetry. However, beneath their polished surfaces lies a complex internal structure that poses significant challenges for scientists attempting to describe them with mathematical precision.

Despite these difficulties, researchers are making progress. In a recent study published in Royal Society Open Science, scientists from the University of Osaka introduced a new approach based on differential geometry. This method offers a thorough and consistent framework for understanding how crystals behave, especially when imperfections are present.

In theory, a flawless crystal would have atoms arranged in a perfectly repeating pattern. But in reality, nearly all crystals exhibit irregularities. These structural flaws might include a missing atom or an additional bond, and while they may seem minor, they can have serious mechanical effects. Such imperfections can serve as the origin of a fracture or, in some cases, actually improve the strength of the material. This makes understanding defects a critical area of study for scientists and engineers.

Dislocations, Disclinations, and the Power of Geometry

“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.”

From Empirical to Rigorous

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.

Reference: “Revisiting Volterra defects: geometrical relation between edge dislocations and wedge disclinations” by Shunsuke Kobayashi, Katsumi Takemasa and Ryuichi Tarumi, 15 July 2025, Royal Society Open Science.
DOI: 10.1098/rsos.242213

Funding: Japan Society for the Promotion of Science, Japan Science and Technology Agency