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Helium-3's Phase diagram shows that at the right temperature and pressure combination, the solid region dips downward as temperature increases. That means that you can heat it up and it will freeze. What is going on?

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Amazingly, below 0.3K the latent heat of fusion of $^3$He is negative i.e. you need to inject heat to make the solid freeze. This is the basis of Pomeranchuk cooling.

This is miles outside my area of expertise, but I believe it's because in the liquid you get large scale alignment of the spins called para magnons, and these lower the energy of the liquid relative to the solid.

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"large scale alignment of the spins called para magnons" so they lower energy and entropy so much that the solid actually is more disordered and in a higher energy state. Strange. –  Kevin Kostlan Nov 10 '13 at 15:10
    
When it freezes you also do work. SO the He is sucking up both heat and work! –  Kevin Kostlan Nov 10 '13 at 17:40

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