Exchange in bcc Helium Mixtures

The exchange constant $J$ is calculated in helium mixtures, using the single-particle theory of Nosanow for atomic configurations likely to yield large enhancements of $J$ over the pure ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ value. ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ atoms near a pair of ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ atoms are first considered. Little relaxation (the same as around 1 ${}_{}{}^4{}_{}{}^{}\mathrm{He}$) and change in the ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ functions are found. Nonspherical Gaussian functions ${\phi }_{\mathrm{NS}}$ are then used to describe ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ surrounded by 6 ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ as first neighbors in otherwise pure ${}_{}{}^3{}_{}{}^{}\mathrm{He}$. The ${\phi }_{\mathrm{NS}}$ obtained on energy minimization is nearly spherical and little changed from its pure ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ value. From these examples, no enhanced $J$ is predicted for mixtures. This is in agreement with recent high-temperature measurements on ${}_{}{}^4{}_{}{}^{}\mathrm{He}$-rich mixtures, but disagrees with low-temperature measurements of the exchange specific heat which suggest a significantly enhanced $J$.