Ising Model for the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>λ</mml:mi></mml:math>Transition and Phase Separation 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>3</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>-<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>Mixtures

Abstract

A spin-1 Ising model, which simulates the thermodynamic behavior of ${\mathrm{He}}^{3}$-${\mathrm{He}}^{4}$ mixtures along the $\ensuremath{\lambda}$ line and near the critical mixing point, is introduced and solved in the mean-field approximation. For reasonable values of the parameters of the model the phase diagram is qualitatively similar to that observed experimentally and the phase separation appears as a consequence of the superfluid ordering. Changing the parameters produces many different types of phase diagram, including as features $\ensuremath{\lambda}$ lines, critical points, tricritical points, and triple points. Certain thermodynamic features which differ from the ${\mathrm{He}}^{3}$-${\mathrm{He}}^{4}$ experiments may be artifacts of the mean-field theory.