Debye temperature calculation formula in the QuasiharmonicDebyeApprox class

[Danil-phy-cmp-120]

Dear Friends,
Please, help me to understand the calculation of Debye temperature using debye_temperature method of QuasiharmonicDebyeApprox class.
The manual states that the formula used corresponds to Eq(6) in doi.org/10.1016/j.comphy.2003.12.001.

However, if you write down the formula from the module itself, you get:

In this expression, I don't understand where the minus sign in the degree indicators (V/n) comes from. In this case, the dimension is [K/m]. Could you please point out the error?

[janosh]

i assume you mean

pymatgen/pymatgen/analysis/quasiharmonic.py

Line 239 in fe4c7c8

debye = 2.9772e-11 * (volume / self.natoms) ** (-1.0 / 6.0) * f * np.sqrt(self.bulk_modulus / self.avg_mass)

not too familiar with this part of pymatgen myself but it could be a bug. maybe @JaGeo knows more?

[janosh]

this reads LLM generated. if so, would be good to preface with some context on what model and prompt was used. otherwise it's easy to mislead people

[hongyi-zhao]

this reads LLM generated.

Yes, as explained below:

if so, would be good to preface with some context on what model and prompt was used.

context: The complete question by @Danil-phy-cmp-120
Model: gpt-4-turbo-preview
prompt: Please give a reasonable explanation to describe the above contradiction.

[hongyi-zhao]

Now, I feed the following source code with the above question by @Danil-phy-cmp-120 to gpt-4-turbo-preview, and use the prompt "Based on the above information, provide a final reasonable explanation in English."

https://github.com/materialsproject/pymatgen/blob/fe4c7c8b7921f6d6157067563f955c50bdae71c4/pymatgen/analysis/quasiharmonic.py#L216C5-L244C1

This time, the answer given by gpt-4-turbo-preview is as follows:

The discrepancy in the Debye temperature calculation, as presented in the provided documentation and source code, primarily stems from the interpretation and transformation of the volume-per-particle term. Here's a consolidated explanation:

Theoretical Background: The Debye temperature ($\Theta$) is a fundamental concept in solid-state physics that characterizes the temperature below which quantum effects dominate the heat capacity of a crystal. The classical equation for calculating $\Theta$ involves the volume of the crystal, the number of particles (atoms), the bulk modulus (a measure of the material's resistance to compression), the average mass of the particles, and corrections for material-specific properties like the Poisson's ratio.

Equation Variants:

The original equation from the DOI provided in your query is: $$\Theta = \frac{\hbar}{k}\left(6 \pi^{2} V^{\frac{1}{2}} n\right)^{\frac{1}{3}} f(\sigma)\left(\frac{B}{M}\right)^{\frac{1}{2}}$$ This equation calculates the Debye temperature based on physical constants, the volume of the crystal ($V$), the number of atoms ($n$), the bulk modulus ($B$), the molar mass ($M$), and a function of the Poisson's ratio ($f(\sigma)$).
The formula derived from the source code provided is: $$\Theta = 2.9772 \times 10^{-11} \times \left(\frac{V}{n}\right)^{-\frac{1}{6}} \times f \times \sqrt{\frac{B}{M}}$$ This variation modifies the volume term to per-particle basis ($\frac{V}{n}$) and includes a seemingly inverse relationship, indicated by the exponent of $-\frac{1}{6}$.

Source Code Analysis:

The code snippet calculates the Debye temperature while offering the option to include anharmonic contributions through the Gruneisen parameter.
It employs a correction factor ($f$), derived from the Poisson's ratio, to adjust the calculation. This factor is more complex than a straightforward function of $\sigma$ and is calculated from the elasticity of the material (via the Poisson's ratio).
The $\left(\frac{V}{n}\right)^{-\frac{1}{6}}$ term in the source code represents a transformation intended to simplify the calculation by considering the volume available per atom, inversely related to the density. This transformation is not directly apparent in the traditional formula but is a mathematical rearrangement to facilitate computation or to reflect specific assumptions about the system's properties.

Conclusion:

The key difference lies in the transformation of the volume term and the detailed incorporation of material-specific corrections, particularly related to elastic properties (via Poisson's ratio).
The source code's approach, specifically the use of the $\left(\frac{V}{n}\right)^{-\frac{1}{6}}$ term, suggests a refinement or adjustment to the traditional calculation, possibly for computational efficiency or to accommodate additional insights into material behavior.
Both the original equation and the source code's version aim to capture the same physical principles but do so through slightly different mathematical formulations, each with its own set of assumptions and simplifications.

This analysis reveals the importance of understanding the underlying assumptions and simplifications involved in theoretical and practical applications, particularly when translating complex physical concepts into computational models.

[hongyi-zhao]

However, to fully understand the root cause of this problem, the explanation given by the LLM above still does not seem to be convincing, complete, or logically rigorous.

[hongyi-zhao]

See here for the related discussion.