Multiple exchange inHe3and in the Wigner solid
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Abstract
We use a multidimensional WKB calculation to determine the leading terms of a high-density series expansion for the multiple-exchange frequencies in solid $^{3}\mathrm{He}$. We also calculate, within the same formulation, the first terms of a low-density series expansion for exchange in the two-dimensional Wigner solid of electrons. The hierarchy between the exchange frequencies is dominated by the geometry of the lattice. Triple exchange is preponderant in the two-dimensional triangular lattice. This result holds for potentials as different as the ${(\frac{\ensuremath{\sigma}}{r})}^{12}$ (solid $^{3}\mathrm{He}$) and the Coulomb potential (Wigner solid). Three-particle exchange also dominates in the hcp lattice. We find for a hypothetical high-density bcc $^{3}\mathrm{He}$ solid the same hierarchy as that deduced from the experimental results at physical densities: The planar four-particle exchange ${K}_{P}$ and the triple exchange ${J}_{t}$ dominate, the folded four-particle exchange ${K}_{F}$ is much lower. For both ${K}_{P}$ and ${J}_{t}$ the lengths of the exchange paths and the tunneling barrier heights are practically the same. We expect these results to be qualitatively valid at physical densities and understand why ${K}_{P}$ and ${J}_{t}$ depend similarly on the molar volume.
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