Inverting the Birch-Murnaghan Equation of State

• I
• Polyamorph
In summary, the conversation discusses the pressure-volume Birch-Murnaghan equation of state and finding the volume at specific pressures. The equation cannot be inverted, so the best approach is to solve for V as a function of P using interpolation or the Newton-Raphson method. A suggestion is also made to use a first order solution or Taylor series for f_E. Ultimately, the Newton-Raphson method is determined to be the most efficient approach.
Polyamorph
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 ($P$)-volume ($V$) Birch-Murnaghan equation of state coeffcients $(V_{0},K_{0}, K^{'}_{0}, K^{''}_{0})$ for a number of different compositions. I'm interested in the volume at very specific pressures only and ideally I would like to compute $V$ for each composition at a given $P$. 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:
$$P=3K_{0}f_{E}(1+2f_{E})^{5/2}\left(1+\frac{3}{2}(K^{'}_{0}-4)f_{E}+\frac{3}{2}\left(K_{0}K^{''}_{0}+(K^{'}_{0}-4)(K^{'}_{0}-3)+\frac{35}{9}\right)f^{2}_{E}\right)$$
where $f_{E}=\left[(V_{0}/V)^{2/3}-1\right]/2.$

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

I was going to calculate $P$ versus $V$, then interpolate the result over a specific $P$ 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 $V$ to compute $P$ will vary depending on composition

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

I'm puzzled by the first term that contains ## f_E ##. (## 3K_o f_E ##). If ## V \approx V_o ##, then ## f_E \approx 0 ##. Does this mean that ## P \approx 0 ##? ## \\ ##

Does this mean that ## P \approx 0 ##? ## \\ ##

Thanks for your reply. Yes at V=V0 the pressure is zero (ambient pressure).

What that means is the equation really reads ## P=P_o+## "what you have as ##P ##". ## \\ ## You may try doing a first order solution of ## f_E ## in terms of ## P-P_o ##, neglecting the ## f_E^2 ## and higher terms. It wouldn't be the perfect solution, but it would be a start. ## \\ ## Alternatively, try something like ## f_E=A(P-P_o)+B(P-P_o)^2+... ##, and see if you can determine ## A ## and ## B ##. (## A ## and ## B ## would come from Taylor series type calculations).

Last edited:
Polyamorph
You are "interested in the volume at very specific pressures only". So use the Newton–Raphson method.

Polyamorph
A follow-up to post 4: If ## x=At+Bt^2 ##, if my algebra is correct, ##t=\frac{x}{A}-\frac{Bx^2}{A^3} ##, neglecting 3rd order and higher terms.

Polyamorph
David Lambert said:
You are "interested in the volume at very specific pressures only". So use the Newton–Raphson method.
I've done it this way, it works! Thanks to both of you for your help.

1. What is the Birch-Murnaghan Equation of State?

The Birch-Murnaghan Equation of State is a mathematical equation used to describe the relationship between pressure, volume, and energy in a material. It is commonly used in the study of materials under high pressure.

2. Why is it important to invert the Birch-Murnaghan Equation of State?

Inverting the Birch-Murnaghan Equation of State allows scientists to determine the pressure and energy of a material given its volume. This is useful in understanding the behavior of materials under different conditions, such as extreme pressure.

3. How is the Birch-Murnaghan Equation of State inverted?

The Birch-Murnaghan Equation of State can be inverted using a variety of numerical methods, such as the Newton-Raphson method or the Levenberg-Marquardt algorithm. These methods involve solving a system of equations to determine the pressure and energy values.

4. What materials can the Birch-Murnaghan Equation of State be applied to?

The Birch-Murnaghan Equation of State can be applied to a wide range of materials, including solids, liquids, and gases. It is commonly used in the study of materials such as minerals, metals, and polymers.

5. What are the limitations of the Birch-Murnaghan Equation of State?

The Birch-Murnaghan Equation of State is based on certain assumptions, such as the material being isotropic and undergoing elastic deformation. It may not accurately describe the behavior of materials that do not meet these assumptions, such as highly anisotropic materials or those undergoing plastic deformation.

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