Nuclear-acoustic-resonance determination of the strain electric-field-gradient tensorSand its temperature dependence in Ta single crystals

The first quantitative nuclear-acoustic-resonance study of the temperature dependence of the tensor $\mathit{S}$ is presented for a cubic metal. The fourth-rank tensor $\mathit{S}$ relates the electric field gradient (EFG) at a nuclear site to the elastic strain field. For cubic metals, its temperature dependence allows a conclusive test of actual theoretical EFG models. A comprehensive discussion of the tensor $\mathit{S}$ is given and, by applying the screened potential approach of Nishiyama et al. [Phys. Rev. Lett. 37, 357 (1976)] to a lattice with both thermal and ultrasonic vibrations, simple expressions are derived for $\mathit{S}$ and its temperature dependence. In addition, the rotational contribution to the ultrasound-induced EFG is discussed in an appendix. With an internal calibration by the Alpher-Rubin effect the following experimental results were obtained for Ta at 300 K: $\left|S_{44}\right|=(6.33\mathrm{}\pm 0.29)\times {10}^{22}$ V/${\mathrm{m}}^2$ and $\frac{(S_{11}-S_{12})}{2S_{44}}=-0.65\mathrm{}\pm 0.03$. For both components of $\mathit{S}$ a decrease with temperature is observed. With regard to the empirical $T^{1.5}$ law of the static EFG the decrease can be described by $S\left(T\right)=S\left(0\right)(1-B{T}^{1.5})$ with $B=(7\mathrm{}\pm 3)\times {10}^{-6\mathrm{}}$ ${\mathrm{K}}^{-1.5\mathrm{}}$. This result strongly supports the "phonon model" of Nishiyama et al. which attributes the temperature dependence of the nuclear electric quadrupole interaction in metals to the effect of lattice vibrations.