In a simple gas or liquid, properties are often "scalars" (just a number, like temperature). However, in a crystal, applying a force in one direction might cause a reaction in another. Rank 0 Tensors (Scalars):
The tensor and matrix representations of crystal properties have numerous applications in materials science, physics, and engineering. Some examples include:
This article provides a comprehensive overview of the physical properties of crystals and their representation by tensors and matrices. The article is suitable for researchers, students, and professionals in the field of crystal physics. In a simple gas or liquid, properties are
relates the electric field vector to the current density vector. Other examples include thermal expansion and permittivity. Higher Rank Tensors: These describe more complex interactions. Piezoelectricity is a Rank 3 tensor (relating stress to polarization), while Elasticity (Hooke’s Law) is a Rank 4 tensor. 2. Representation by Matrices While tensors provide the theoretical framework, provide the practical tool for calculation. A Rank 2 tensor is typically represented as a 3x3 matrix
The symmetry elements of any physical property of a crystal must include the symmetry elements of the crystal’s point group. Some examples include: This article provides a comprehensive
A matrix, on the other hand, is a two-dimensional array of numbers, which can be used to represent a tensor. Matrices are a convenient way to represent tensors, as they can be easily manipulated using linear algebra.
While tensors are the physical reality, are the computational tools we use to solve engineering problems. Other examples include thermal expansion and permittivity
The representation of crystal properties by tensors and matrices bridges the gap between atomic-scale symmetry and macroscopic observables. While rank-2 tensors are manageable with 3×3 matrices, higher-rank tensors demand compact notations like Voigt indexing. The power of this formalism lies in its ability to predict—using Neumann’s principle—which components vanish or become equal, drastically reducing experimental effort.