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Polyamorph

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- TL;DR Summary
- Efficient method to compute V at given P for a non-invertible equation of state

Hello,

https://www.physicsforums.com/lib/Eqn026.pngI have the pressure ([itex]P[/itex])-volume ([itex]V[/itex]) Birch-Murnaghan equation of state coeffcients [itex](V_{0},K_{0}, K^{'}_{0}, K^{''}_{0})[/itex] for a number of different compositions. I'm interested in the volume at very specific pressures only and ideally I would like to compute [itex]V[/itex] for each composition at a given [itex]P[/itex]. My problem is the Birch-Murnaghan equation of state cannot be inverted and yields pressure for a given volume - the opposite of what I need:

[tex]P=3K_{0}f_{E}(1+2f_{E})^{5/2}\left(1+\frac{3}{2}(K^{'}_{0}-4)f_{E}+\frac{3}{2}\left(K_{0}K^{''}_{0}+(K^{'}_{0}-4)(K^{'}_{0}-3)+\frac{35}{9}\right)f^{2}_{E}\right)[/tex]

where [itex]f_{E}=\left[(V_{0}/V)^{2/3}-1\right]/2.[/itex]

So my question is, what would be the best approach to solve [itex]V[/itex] as a function of [itex]P[/itex]?

I was going to calculate [itex]P[/itex] versus [itex]V[/itex], then interpolate the result over a specific [itex]P[/itex] range. But I have a large number of compositions (~100 or so) all with different volumes and equation of state coefficients so this seems quite inefficient, particularly as the range of [itex]V[/itex] to compute [itex]P[/itex] will vary depending on composition

What would be an efficient approach to find [itex]V[/itex] for a given [itex]P[/itex]? Am I forgetting something quite trivial?

Thanks for your help!

https://www.physicsforums.com/lib/Eqn026.pngI have the pressure ([itex]P[/itex])-volume ([itex]V[/itex]) Birch-Murnaghan equation of state coeffcients [itex](V_{0},K_{0}, K^{'}_{0}, K^{''}_{0})[/itex] for a number of different compositions. I'm interested in the volume at very specific pressures only and ideally I would like to compute [itex]V[/itex] for each composition at a given [itex]P[/itex]. My problem is the Birch-Murnaghan equation of state cannot be inverted and yields pressure for a given volume - the opposite of what I need:

[tex]P=3K_{0}f_{E}(1+2f_{E})^{5/2}\left(1+\frac{3}{2}(K^{'}_{0}-4)f_{E}+\frac{3}{2}\left(K_{0}K^{''}_{0}+(K^{'}_{0}-4)(K^{'}_{0}-3)+\frac{35}{9}\right)f^{2}_{E}\right)[/tex]

where [itex]f_{E}=\left[(V_{0}/V)^{2/3}-1\right]/2.[/itex]

So my question is, what would be the best approach to solve [itex]V[/itex] as a function of [itex]P[/itex]?

I was going to calculate [itex]P[/itex] versus [itex]V[/itex], then interpolate the result over a specific [itex]P[/itex] range. But I have a large number of compositions (~100 or so) all with different volumes and equation of state coefficients so this seems quite inefficient, particularly as the range of [itex]V[/itex] to compute [itex]P[/itex] will vary depending on composition

What would be an efficient approach to find [itex]V[/itex] for a given [itex]P[/itex]? Am I forgetting something quite trivial?

Thanks for your help!