Effective Interaction of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">He</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>Atoms in Dilute Solutions of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">He</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">He</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>at Low Temperatures

The approximate form of the effective interaction between ${\mathrm{He}}^{3}$ atoms in superfluid ${\mathrm{He}}^{4}$ is derived from the experimental data on spin diffusion and phase seperation in dilute mixtures of ${\mathrm{He}}^{3}$ in ${\mathrm{He}}^{4}$. The interaction is weak, and attractive at long wavelengths. Calculations of the Fermi-liquid parameters for the normal state of ${\mathrm{He}}^{3}$ in solution yield results for the effective mass and spin susceptibility in agreement with experiment. The temperature for a superfluid transition associated with the ${\mathrm{He}}^{3}$ is estimated to be \ensuremath{\sim}2\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}6}$ \ifmmode^\circ\else\textdegree\fi{}K; the maximum solubility of ${\mathrm{He}}^{3}$ in ${\mathrm{He}}^{4}$ is found to be \ensuremath{\sim}6% at $T=0$. Thermodynamic and microscopic arguments are used to calculate the long-wavelength part of the effective interaction between the ${\mathrm{He}}^{3}$ atoms. The contribution arising from the exchange of a virtual ${\mathrm{He}}^{4}$ phonon is shown to be large and attractive, while the remaining part of the interaction is almost as large but repulsive; the calculated interaction at long wavelengths is thus weak and attractive and is in excellent agreement with that determined empirically; the physical origin of the weakness of the interaction is that ${\mathrm{He}}^{3}$ is an isotopic impurity. Finally, it is estimated that the application of pressure serves to weaken the effective interaction.

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