Nuclear Spin-Lattice Relaxation in Hexagonal Transition Metals: Titanium

The nuclear magnetic resonance of $^{47}\mathrm{Ti}$ and $^{49}\mathrm{Ti}$ has been observed in hexagonal close-packed titanium metal in the temperature range $T=1\ensuremath{-}4\ifmmode^\circ\else\textdegree\fi{}$K. Measurements of the line profile and relaxation rates were carried out at 12.5 MHz by pulsed nuclear resonance techniques. Since the gyromagnetic ratios of $^{47}\mathrm{Ti}$ and $^{49}\mathrm{Ti}$ are nearly identical, the resonances of the two isotopes were superimposed. A partially resolved first-order quadrupole spectrum having a total width exceeding 8 kOe was observed, yielding a probable assignment ${h}^{\ensuremath{-}1}{e}^{2}q{Q}^{47}\ensuremath{\approx}{h}^{\ensuremath{-}1}{e}^{2}q{Q}^{49}\ensuremath{\approx}7.7$ MHz. The average Knight shift is estimated to be $K=(+0.4\ifmmode\pm\else\textpm\fi{}0.2)%$. The spin-lattice relaxation times ${T}_{1}$, which are quite long (${T}_{1}T=150\ifmmode\pm\else\textpm\fi{}20$ sec\ifmmode^\circ\else\textdegree\fi{}K), provide evidence that the conduction-electron states at the Fermi level are predominantly $d$-like. The theory of nuclear spin-lattice relaxation in hexagonal transition metals is treated in the tight-binding approximation. Contact, core-polarization, orbital, and dipolar hyperfine interactions are considered. The magnitudes of the orbital and dipolar contributions to the spin-lattice relaxation rate depend on the orientation of the magnetic field relative to the hexagonal $c$ axis. In the presence of $s\ensuremath{-}d$ mixing, the contact contribution is found to interfere destructively with one of the components of the core-polarization contribution. The predicted total relaxation rates are shown to depend more strongly on the orbital admixture coefficients than is the case in cubic transition metals. The relatively large number of parameters in the theory precludes a unique fit to the experimental results for titanium. In general, however, the calculated rates exceed the observed rate by a factor of 2-3 over a wide range of parameter values. The apparent discrepancy is attributed to the combined effects of (1) the electron-phonon enhancement of the electronic specific heat and (2) $s\ensuremath{-}d$ interference effects. The former effect causes the "bare" electron density of states to be overestimated, while the latter leads to an overestimate of the sum of contact and core-polarization contributions to the relaxation.