Fix BMEOS to use Birch-Murnaghan equation instead of Murnaghan equation

[Copilot]

Plan for Fixing Birch-Murnaghan EOS

• Understand the difference between Murnaghan and Birch-Murnaghan equations
• Review current implementation of BMEOS function
• Implement the correct 3rd-order Birch-Murnaghan equation
• Add Murnaghan equation as a separate function (for compatibility)
• Update documentation to reflect the correct equation
• Test the implementation and validate correctness
• Update scan_pressures function documentation
• Run code review and fix issues
• Run security checks (no issues found)
• Final verification complete
• Make internal function naming consistent (invbmeos → invmurneos)

Summary

Successfully implemented the correct 3rd-order Birch-Murnaghan equation of state, replacing the incorrect Murnaghan equation that was previously used. All tests pass, documentation is updated, and internal naming is now consistent.

Key Changes:

1. BMEOS function: Now implements true 3rd-order Birch-Murnaghan EOS
2. MurnaghanEOS function: Added as separate function for reference/compatibility
3. Documentation: Updated in elastic.py, init.py, and lib-usage.rst
4. Internal naming: Renamed invbmeos to invmurneos for consistency
5. Validation: All unit tests pass, no security issues found

Original prompt

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This section details on the original issue you should resolve

<issue_title>Birch-Murnaghan equation of state</issue_title>
<issue_description>Thank you so much for sharing your code. I met some issues in the code. I think the equation used in file 'elastic.py' function 'BMEOS' is not Birch-Murnaghan equation.
It is Murnaghan equation: https://en.wikipedia.org/wiki/Murnaghan_equation_of_state
The correct equation is: https://en.wikipedia.org/wiki/Birch%E2%80%93Murnaghan_equation_of_state </issue_description>

Comments on the Issue (you are @copilot in this section)

@jochym You are right. I will make the correction in the text. Later, I will implement B-M version as well. (I will also accept PR if you have time ;). Tanks, @Ruoyan0103 !

• Fixes Birch-Murnaghan equation of state #71

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[jochym]

Make internal function naming consistent

[jochym]

This should also beem modified (mame). This is inverse Murnaghan not Birch- Murnaghan @copilot

[Copilot]

Renamed invbmeos to invmurneos and updated the comment on line 624 to say "Murnaghan EOS" instead of "B-M EOS" for consistency. Fixed in fd4faaa.