Investigation of Isotopic Effects in Debye Temperature and EXAFS Cumulants of Zirconium

Isotopic effects in the Debye temperature and in the temperature-dependent extended X-ray absorption fine structure (EXAFS) cumulants have been predicted based on the developed anharmonic Einstein model in the case of non-ideal axial ratio of hexagonal close-packed (hcp) zirconium isotopes. We derive analytical expressions for the force constants, the Debye temperature, and the first three EXAFS cumulants as functions of the axial ratio e = c/a. Numerical calculations are conducted for the isotopes 90Zr, 91Zr, 92Zr, 94Zr, and 96Zr up to a temperature of 700 K. Our findings reveal that the effects of isotopic mass differences are clearly evident at low temperatures, particularly below 100 K, where quantum effects predominate. At higher temperature, the anharmonicity caused by thermal lattice vibrations significantly influences the EXAFS cumulants, and reduces the isotopic effects in EXAFS cumulants. Furthermore, our derived Debye temperatures from the developed model exhibit reasonable agreement with experimental value measured by X-ray diffraction.