Heuristic derivation of physical laws - II

In the previous post I presented the main result of Trachenko et al. [1] concerning the speed of sound in solids and a possible fundamental upper bound for this parameter. Here, I will add a couple of observations. 

Clean derivation

The heuristic formula for the speed of sound in elemental solids, \( \frac{v_{\text{est}}}{c} = \alpha \sqrt{\frac{m_e}{2 m_p} } A^{-1/2}\) arises quite naturally; in particular, the ratio \(\frac{m_e}{m_p}\) intervenes because the Ry contains the mass of the electron, but the density is given by that of nucleons. In contrast, Press and Lightman [2] put this factor in "by hand", noting that the electron is bound, but the whole molecule vibrates (see the paragraph above Eq. (9)) and neglect the mass number dependence. This line of reasoning is also presented by Ref. [1] as a second option.

Large experimental scatter

The experimental values in Fig. 1 are rather scattered around the theoretical prediction; this is to be expected for such a simple approach, and even agreement within a factor of two for all points is remarkable; however, this should be taken into account in the following discussion. For instance, when mentioning the excellent agreement (within 3%) between the theoretically predicted maximum and the fitted value I would have expected the authors to give the uncertainty on the latter, as well as on the exponent of the variation with the mass number. How close is it to \(-1/2\)?

At what pressure?

The most serious difficulty of the universality claim has to do with the conditions under which the speed of sound is measured (or evaluated numerically). The upper value \( v_{u} = \alpha \sqrt{\frac{m_e}{2 m_p} \, c} \) should apply for solid hydrogen, and the authors further limit this to metallic hydrogen, but this putative phase only occurs at high pressure, above 400 GPa (note that the speed of sound in solid hydrogen at a few GPa is much lower than \( v_{u}\), see e.g. [3]). Simulations then yield good agreement with the \( v_{u}\), but one is now confused: why compare hydrogen at 600 GPa with all the other elements at standard pressure? Could the speed of sound in high-pressure diamond exceed \( v_{u}\)?

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¹ Trachenko, K. et al., Speed of sound from fundamental physical constants Science Advances 6, eabc8662, (2020).↩

² Press, W. H. and Lightman, A. P., Dependence of macrophysical phenomena fundamental constants Phil. Trans. R. Soc. Lond. A 310, 323-336, (1983); two lines above Eq. (10).↩
³ Guerrero, C. L. and Perlado, J. M., Speed of sound in solid molecular hydrogen-deuterium: Quantum Molecular Dynamics Approximation Journal of Physics: Conference Series 717, 012018, (2016).↩