Structural-Order Analysis Based on Applied Mathematics

The development of experimental and simulation technologies has afforded us access to material science data on a more massive scale than that in the past. For such large-scale data on various materials, effective and efficient analysis methods have been developed using applied mathematics and data science. For analyzing structural order, statistical analysis approaches based on chemical bonding are used. This approach can evaluate the structural order on a short-range scale. However, it cannot analyze structural order on the scale over the length of the chemical bonding referred to as the intermediate range, which is essential for understanding amorphous or glassy materials. This chapter introduces two useful characterization approaches based on applied mathematics for a geometric structure, which are useful to identify hyperordered structures in intermediate-range scale. The first approach is based on the topological structure of an atomic configuration (or point cloud) based on persistent homology. The second approach is based on the network topology of chemically bonded atoms based on rings. This chapter introduces the applied analyses of amorphous materials using these methods.