6.4 Numerical Calculations: Solid Nitrogen

Figure 6.23 shows the phonon density of states $g(\omega)$ of solid nitrogen at 22 K, measured by Cardini et al.[57]. With this spectrum Eq. (6.8) may be evaluated by numerical methods, assuming a particle-phonon coupling $\lambda(\omega) \propto \omega^2$, giving $\Omega_{2}(T)$ as shown in Fig. 6.24. Since we do not know the coupling constant we cannot attach an absolute scale to $\Omega_2$; only the temperature dependence is obtained. Also shown is the case where the phonon spectrum is taken to be due to a Debye-like density of states $g(\omega) \propto \omega^2$ over the entire phonon spectrum; the two outcomes are not very different.

  

Figure: Phonon density of states $g(\omega)$ of solid nitrogen at 22 K, data from Ref.[57].

  

Figure 6.24: $\Omega_{2}(T)$ for solid nitrogen, calculated from both the real (dashed line) and Debye model (solid line) phonon density of states $g(\omega)$.

At low temperatures ($T \ll \Theta_{\rm D}$ = 83.5 K), theory predicts that $\Omega_{2}(T)$ should be proportional to T⁷. This should hold up to temperatures of about 10 K, where the temperature dependence should drop off gradually to a T² dependence for $T \sim \Theta_{\rm D}$and greater. (The high temperature regime is mentioned here only to exemplify the T² behavior; s-N₂melts at 63 K).

It is not necessary that $\xi$ itself be temperature-dependent for there to be a cross-over from the low-temperature, defect-dominated behavior to the higher temperature, homogeneous regime. Since $\Omega_{2}(T)$ is always a monotonically increasing function of T, it may be that $\xi$ simply becomes negligible and the hopping behavior changes from Eq. (6.10) to Eq. (6.9) at a temperature near 20 K. At sufficiently low temperatures $\Omega_2$ will inevitably drop below $\xi$ and the behavior we expect in the limit of $T \rightarrow 0$, as long as the damping rate is sufficient that we still have incoherent tunnelling, will always be a hop rate $1/\tau_c \propto T^7$.However, there is reason to believe that $\xi$ may be temperature dependent in s-N₂. It is known that the $\alpha$-N₂lattice undergoes a transition at about 22 K, below which the molecules take on preferred orientations, rotating at a fixed angle about the diagonals of the cubic unit cell. Above this temperature they are free rotators, and the muonium atom should see the same averaged potential well at each site. The orientational disorder introduced by this transition may be responsible for creating level shifts that hinder tunnelling according to Eq. (6.10).

Impurities in the sample evidently do not affect the muonium diffusion rate at low temperatures. Experiments were performed on samples of s-N₂with CO concentrations of 0.01% and 0.1%. Carbon monoxide has the same molecular mass as N₂and freezes in the lattice as a random substitutional impurity. The results are shown in Fig. 6.25 along with the results from the most carefully annealed sample of solid ultra high purity N₂. At temperatures where muonium diffuses rapidly (between 20 and 30 K) the TF relaxation rate 1/T₂ increases with CO concentration, up to 40 ${\mu}$s^-1 with 0.1% CO, well above the rate measured for nearly static muonium in pure N₂at low temperatures. This is due to muonium diffusing to the CO where it undergoes a fast chemical reaction to form a diamagnetic species, so the relaxation rate measures the time-of-flight of the muonium to the site of the CO molecule. However, at low temperatures the relaxation rates are independent of impurity concentration. Thus the static shift $\xi$ that seems to become important at low temperatures is not due to impurites, at least at this level; its origin seems to be intrinsic to s-N₂.

  

Figure 6.25: Muonium spin relaxation rates 1/T₂ in slowly cooled, solid high-purity N₂(circles), N₂with 0.01% CO (boxes) and N₂with 0.1% CO (triangles).

The very strong temperature dependence predicted by the two-phonon model is convincingly demonstrated in the case of muonium diffusion in solid $\alpha$-N₂below 15 K. At higher temperatures, particularly 30-50 K, theory predicts a much weaker dependence than was measured, as was the case for KCl, so we cannot claim to fully understand muonium diffusion quantitatively in this temperature range. However, the rapid increase in hop rate as temperature decreases is still strong qualitative evidence that the muonium is diffusing by incoherent quantum tunnelling mediated by phonon scattering. It is possible that this quantitative failure of the model results from the breakdown of our (rather simple) assumptions regarding the frequency dependence of the coupling constants.