From time to time I find myself fascinated by the idea of deducing the relations that describe a certain phenomenon not by solving the full relevant equations but via a simplified model. I am not talking about purely dimensional analysis based on Buckingham's Pi theorem [1], but of more complicated situations, involving several parameters with the same dimensions. When presented by a gifted author (such as Weisskopf [2]), the process seems very straightforward, and one would even be tempted to teach it to undergrads. When going into the details, however, things soon become more complicated and good numerical agreement is sometimes due to the fortunate compensation of two opposite errors.
A recent paper on the speed of sound in solids [3] provides a good illustration. The authors propose a remarkably simple expression for the maximum speed of sound and support it with simulations of metallic hydrogen.
Let us try some dimensional analysis: neglecting the contribution of the shear modulus, the longitudinal speed of sound \(v = \sqrt{\frac{K}{\rho}}\), where \(K\) is the bulk modulus of the material and \(\rho\) its mass density. \(K\) is in units of \(\text{Pa} = \text{J/m}^3\), so a first estimate \(K_{\text{est}} = 1\, \text{Ry}/a_0^3\), where the Rydberg \(\text{Ry} = \frac{\alpha^2}{2} m_e c^2= 13.6\text{ eV} = 1313\text{ kJ/mol}\)
is the binding energy of the electron in the hydrogen atom, which can be elegantly expressed in terms of the fine structure constant \(\alpha\), the mass of the electron \(m_e\) and the speed of light in vacuum \(c\). The length scale \(a_0 = 0.053 \text{ nm}\) is the Bohr radius. This estimation fails miserably, because the resulting \(K_{\text{est}} = 13000\, \text{GPa}\), while the experimental values for (elemental) solids are all below 500 GPa, and most are even below 100 GPa (see the middle panel in Figure 1 of Ref. [4]). The reason is obvious if we look at the other panels of the same Figure: The Ry overestimates the cohesion energies \(E_c\) by a factor between 2 and 20, while the molar volume estimated using \(a_0\) as an interatomic distance, \(V_{\text{est}} = N_A \, a_0^3\), is almost two orders of magnitude below the real-life data. Of course, \(a_0\) is the radius so the distance between atoms is at least \(2a_0\), reducing the discrepancy by a factor of 8. This is still not enough and, furthermore, trying the estimate the numerical prefactors kind of defeats the whole purpose of dimensional analysis.
Surprisingly, estimating the speed of sound works much better! I'll follow here the reasoning in [3], although a very similar formula (diferring only by a factor of \(\sqrt{2}\)) was obtained by Press and Lightman [5]. Let us denote the (unspecified) interatomic distance by \(d\): then \(K_{\text{est}} = E_c/d^3\) and \(\rho = M_{\text{atom}}/d^3 \simeq A \, m_p/d^3\), where \(A\) is the mass number of the atom in question and \(m_p\) is the mass of the proton. Taking once again \(E_c = 1 \, Ry\), we finally obtain:
\begin{equation}
\label{eq:vA}
v_{\text{est}} = \sqrt{\frac{1 \, Ry/d^3}{A \, m_p/d^3}} = \sqrt{\frac{Ry}{A \, m_p}} = \alpha \, c \sqrt{\frac{m_e}{2 A \, m_p}} \Longrightarrow \frac{v_{\text{est}}}{c} = \alpha \sqrt{\frac{m_e}{2 m_p} } A^{-1/2}
\end{equation}
This is the first result of Ref. [3], and it works quite well, with the experimental points scattered around it within half a decade (factors of 0.6 to 2.4), see their Figure 1.
As mentioned above, Eq. \eqref{eq:vA} is not exactly new; the authors supplement it by taking the lightest element (H, with A = 1) and claiming that the corresponding value is an upper bound for the speed of sound in condensed phases:
\begin{equation}
\label{eq:vu}
v_{u} = \alpha \sqrt{\frac{m_e}{2 m_p} } c \simeq 36 \, \text{km/s}
\end{equation}
Remarkably, their DFT calculations for metallic hydrogen are in excellent agreement (within 3%) with Eq. \eqref{eq:vu}. This is a very strong conclusion: an upper limit for a physical parameter is given in terms of fundamental constants and is supported by numerical results. I do however have some reservations, which I will detail in the next post.
1 Barenblatt, G. I.
Scaling, self similarity, and
intermediate asymptotics, Cambridge University Press (1996).
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2 Weisskopf, V. F. Search for Simplicity
Am. J. Phys. 53, 19, (1985).
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3 Trachenko, K.
et al., Speed of sound from fundamental physical constants
Science Advances 6, eabc8662, (2020).
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4 Brazhkin, V. V.
et al., Harder than diamond: dreams and reality
Phil. Mag. A 82, 231-253, (2002); cited in [3] as Ref. (15).
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5 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).
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