Binding energy of $^3$He to dislocations in solid $^4$He

Recent heat capacity experiments on solid $^4$He [1] show a peak in the specific heat which is interpreted as the signature of the supersolid transition. We pursue an alternative explanation for the heat capacity feature in which $^3$He impurities desorb from dislocations in solid $^4$He; the peak temperature scales with the binding energy of $^3$He to dislocations in $^4$He. Within a continuum elastic model for solid $^4$He, we make quantum mechanical estimates for the binding energy, using a combination of variational and numerical methods. We find for a short distance cut-off of one lattice constant of $^4$He, the binding energy is about 70 mK for edge and 60 mK for a screw dislocation.\newline\newline \noindent [1] X. Lin, A. C. Clark, and M. H. W. Chan, Nature {\bf 449}, 1025 (2007).