Quantitative Role of Phonons and Elasticity in Tuning Uniaxial Negative Thermal Expansion of MZr₂(M = Fe, Co, and Ni)

Elucidating the mechanisms of negative thermal expansion (NTE) not only identifies the determining factors of this phenomenon but also provides guidance for the precise regulation of the coefficient of thermal expansion (CTE). However, accurately quantifying these determining factors during CTE modulation remains a critical challenge. In this Letter, we quantitatively determine the respective contributions of phonon-derived Grüneisen parameters (axial values and their difference) and elastic compliance tensors to the tuning of axial NTE in the MZr₂ system. Through temperature dependence of the neutron total scattering data (for neutron powder diffraction and neutron pair distribution function analyses), extended x-ray absorption fine structure, synchrotron x-ray diffraction, and first-principles calculations, it provides insight into the quantified roles of phonons and anisotropic elasticity in controlling the broad-range CTE. As the number of M-3d electrons grows, phonons, which are the dominant factors, weaken NTE, while elasticity, playing a cross-linking coefficient, enhances NTE. The current study not only provides deep insights into the origins of NTE in the MZr₂ system, but it also offers a novel perspective on the quantifiable control of NTE in anisotropic materials.