Saturday, October 10, 2020

How fast can sound be in a solid or liquid?

There is a new paper here that argues through dimensional analysis for an upper limit to the speed of sound in solids and liquids (when the atoms bump up against each other).  The authors derive that the maximum speed of sound is, to within numerical factors of order 1, given by \(v_{\mathrm{max}}/c = \alpha \sqrt{m_{e}/(2m_{p})} \), where \(\alpha\) is the fine structure constant, and \(m_{e}\) and \(m_{p}\) are the masses of the electron and proton, respectively.  Numerically, that ends up being about 36 km/s.  

It's a neat argument, and I agree with the final result, but I actually think there's a more nuanced way to think about this than the approach of the authors.  Sound speed can be derived from some assumptions about continuum elasticity, and is given by \(v_{s} = \sqrt{K/\rho}\), where \(K\) is the bulk modulus and \(\rho\) is the mass density.  Bulk modulus is given by (negative) the inverse fractional change in volume of a substance when the pressure on the substance is increased.  So, a squishy soft substance has a low bulk modulus, because when the pressure goes up, its volume goes down comparatively a lot.

The authors make the statement "It has been ascertained that elastic constants are governed by the density of electromagnetic energy in condensed matter phases."  This is true, but for the bulk modulus I would argue that this is true indirectly, as a consequence of the Pauli principle.  I wrote about something similar previously, explaining why you can't push solids through each other even though the atoms are mostly empty space.  If you try to stuff two atoms into the volume of one atom, it's not the Coulomb repulsion of the electrons that directly stops this from happening.  Rather, the Pauli principle says that cramming those additional electrons into that tiny volume would require the electrons to occupy higher atomic energy levels.  They typical scale of those atomic energy levels is something like a Rydberg, so that establishes one cost of trying to compress solids or liquids; that Rydberg scale of energy is how the authors get to the fine structure constant and the masses of the electron and proton in their result.  

I would go further and say that this is really the ultimate limiting factor on sound speed in dense material.  Yes, interatomic chemical bonds are important - as I'd written, they establish why solids deform instead of actually merging when squeezed.  It's energetically cheaper to break or rearrange chemical bonds (on the order of a couple of eV in energy) than to push electrons into higher energy states (several eV or more - real Rydberg scales).  

Still, it's a cool idea - that one can do intelligently motivated dimensional analysis and come up with an insight into the maximum possible value of some emergent quantity like sound speed.  (Reminds me of the idea of a conjectured universal bound on diffusion constants for electrons in metals.)

1 comment:

John Vann said...

Does this result include degenerate matter or neutronium?
Also, is degenerate matter just regular matter with a disaffected attitude?