Lattice Relaxation in Solid ⁴He: Effect on Dynamics of ³He Impurities

Abstract

We present a new simplified derivation of the effect of lattice relaxation that accompanies the quantum tunneling of ³He impurities in solid ⁴He on the nuclear spin-lattice relaxation of the ³He impurities for very low impurity concentrations. As a result of the larger zero point motion of the ³He impurity compared to the ⁴He atoms, a significant lattice distortion accompanies the impurity as it moves through the lattice and the dynamics of the impurity depends on both the interaction energy between two ³He atoms and on the relaxation of the lattice for the tunneling impurity. Using a phenomenological model for the lattice relaxation we compare the nuclear spin-lattice relaxation rates observed at low temperatures with the dependence on temperature expected for a ⁴He lattice relaxation comparable to that observed by Beamish et al. (Phys. Rev. Lett. 96:195304, 2006).

Appendix A: Calculation of the Correlation Time for High 3He Concentrations

Following the method of Kubo (Cowan [22], Chap. 7), one calculates the time derivative of the autocorrelation function, $\ddot {\varGamma}_{(ij),(ij)}(t)$, and assuming an exponential decay one determines a correlation time $\tau_{c}^{-1}=G_{0}(0)^{-1}\int dt \ddot{\varGamma}_{(ij),(ij)}(t)$ The time derivative

$${\ddot{\varGamma}} =\frac{4}{x_3}\bigl\langle \bigl[T_{ij}^m,H_J \bigr]e^{-iH_Kt}\bigl[T_{ij}^{-m},H_J \bigr]e^{iH_Kt}\bigr\rangle=6J_{34}^2 \exp\biggl[ \sum _{p(ij)} i(K_{jp}-K_{ip})t\biggr] $$

with an impurity tunneling from site p to site i with spectator at site j. The energy difference K _jp −K _ip can be replaced by the average of the spatial gradient of the interaction energy $K_{ij}=K_{0}(3\cos^{2}\theta_{ij}-1)/r_{ij}^{3}$ to yield

$$\ddot{\varGamma}(t)=12J_{34}^2\cos\biggl(\sum_p\omega_pt\biggr) \quad \mbox{with}\ \omega_p=\frac{3K_0}{(R/a_0)^4}. $$

The sum of the oscillatory terms cos(ω _p t) rapidly averages to zero and only small deviations from unity of the cosine term will contribute to the overall average. Using the method of Anderson[23] we write $\ddot{\varGamma}(t)= 12J_{34}^{2}\prod_{p}{[1+(\cos\omega _{p}t -1)]}$ which can be expanded and the term cosω _p t−1 replaced by its average to yield $\ddot{\varGamma}(t)=12J_{34}^{2}\exp {[N\langle \cos\omega_{p}t-1\rangle}]$. The average

$$\begin{aligned} N\bigl\langle \cos(\omega_pt)-1\bigr\rangle =& \frac{2\pi N}{V} \int_0^{\pi/2}\!\!d\theta \sin\theta\int_0^{\infty}\!\!\!R^{-4} dR\biggl(\cos\biggl[\frac{3K_0(1-3cos^2\theta)t}{R^4}\biggr]-1\biggr)\\ =&-\frac{\pi}{2}x_3 I_1I_2(3K_0t)^{-3/4}=-0.71x_3(K_0t)^{3/4} \end{aligned}$$

where $I_{1}=\int_{0}^{\pi/2} \sin\theta d \theta|1-3\cos^{2}\theta|^{3/4}$ and $I_{2}=\int_{0}^{\infty}dx[\frac{\cos x-1}{x^{1/4}}]=2^{3/4}\varGamma(\frac {3}{4}) \times \cos(\frac{3\pi}{8})$. From Eq. (6) we find $\tau_{c}^{-1}=19\frac{J_{34}^{2}}{K_{0}}x_{3}^{-1/3}$ in reasonable agreement with Landesman [18].