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).↩

Heuristic derivation of physical laws - I

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.

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¹ Barenblatt, G. I. Scaling, self similarity, and intermediate asymptotics, Cambridge University Press (1996).↩

² Weisskopf, V. F. Search for Simplicity Am. J. Phys. 53, 19, (1985).↩
³ Trachenko, K. et al., Speed of sound from fundamental physical constants Science Advances 6, eabc8662, (2020).↩
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).↩
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).↩

Solving tan(x) = x

[UPDATE: 25/09/2014 with the iterative method] This kind of transcendental equation is often encountered in physics. Undergraduate students are usually shown (or asked to draw) the graphical solution:

The numerical solutions are easily found by an iterative method using a scientific calculator (see below), but how far can one go with only pen and paper?

Expansion

Aside from the trivial solution \(x_0 = 0\), one clearly has \( x_k \simeq \frac{(2k +1) \pi}{2}\) (\(k \geq 1 \)), so we can write: \[ x_k = \frac{(2k +1) \pi}{2} - \varepsilon _k, \quad \mathrm{with} \quad \varepsilon _k < 1\] One would like to do an expansion in \( \varepsilon _k\), but of course this will not work for the tangent around its divergence points. We can however use the cotangent, since \( \tan (x_k) = x_k \Rightarrow \cot (x_k) = 1/x_k \). Using standard substitution formulas for the sine and cosine yields: \[\cot \left [ \frac{(2k +1) \pi}{2} - \varepsilon _k \right ] = \tan(\varepsilon _k) \simeq \varepsilon _k \simeq \frac{2}{(2k + 1) \pi}\] where in the last equality we neglected \( \varepsilon _k\) in the denominator. One can include it for a more rigorous treatment. Finally, we have: \[ x_k \simeq \frac{(2k +1) \pi}{2} - \frac{2}{(2k + 1) \pi}, \quad \mathrm{for} \quad k \geq 1 \, ,\] giving for the first three solutions 4.5002, 7.7267, and 10.9046, to be compared with the "exact" values 4.4934…, 7.7253…, and 10.9041…. The quality of the approximation increases with the order \(k\), since \(\varepsilon _k\) decreases (the intersections are closer and closer to the vertical asymptotes).

Iteration

Let us rewrite the initial equation by applying the arctangent to both members:
\[x = \arctan (x) \tag{1}\]
For the \(k\)-th solution, the initial estimate is: \( x^0_k = \frac{(2k +1) \pi}{2} \). Let us plug it in the right-hand side of Eq. (1) to obtain the first order estimate \( x^1_k\) and then iterate. Note that the arctangent is a multi-valued function, and the standard implementation reduces it to the first branch (the one going through the origin). We are looking for the solution sitting on the \(k\)-th branch, so we need to add \(k \pi\) each time:
\[x^{i+1}_k = \arctan (x^i_k)  + k \pi \tag{2}\]
 For the first non-trivial solution (\(k = 1\)), the sequence is: 4.71239, 4.50328, 4.49387, 4.49343, 4.49341,... with the second iteration already reaching an excellent precision!