Elasticity and Thermal Expansivity Across Direction, Symmetry and Scale: A Thermoelastic Reassessment from Single-Crystal MgO to Cubic Ionic Solids

Elasticity and thermal expansivity are usually treated as separate materials properties, although both are derivatives of the same thermodynamic free energy and are therefore coupled by lattice anharmonicity. This work undertakes a systematic reassessment of that coupling across scalar, directional and tensorial descriptions. The analysis begins from the exact thermoelastic differential structure and distinguishes three quantities that are often conflated: thermal expansion, temperature derivatives of elastic constants, and the Anderson-Grüneisen parameter. A generalized normalized response is introduced for an arbitrary elastic modulus, while its tensorial and directional forms are written explicitly for anisotropic crystals. The cubic case is then examined using MgO, for which high-temperature single-crystal elastic constants and thermal-expansion data are available over broad temperature ranges. Published measurements show that C11, C44 and the bulk modulus do not soften at the same normalized rate. Consequently, no modulus-independent scalar "elasticity expansivity constant" exists even within one cubic crystal. The analysis further compares published Anderson-Grüneisen parameters for NaCl, KCI, CaO and MgO, finding a systematic material dependence. A central conclusion is that the physically natural object is not a single universal scalar but a family of thermoelastic response functions whose form depends on the elastic mode, crystal symmetry and thermodynamic path. The study does not claim discovery of a new universal law; rather, it identifies a precise framework in which genuinely new relations can be tested and establishes several falsifiable constraints on any proposed universal elasticity expansivity theory.