What's That Sound?

Since I'm chugging away on a rather detailed follow-on post to the thrust & buckling post, I figured it was time for a fun one… and this time it's one that I'd been thinking about for a while.

Ever heard that metallic ping when heating a metal pan on a stove, or letting one cool? I'm willing to bet most people have.

Ever wonder where it came from?

If you recall back to my earlier posts on crystal structure and heat treatment, motion and deformation at the atomic level is intimately related to changes in the energy of the system. For example, thermal expansion as a result of heating (an elastic behavior). I also mentioned phonons, the coordinated vibration of atoms in a solid as being a part of thermal conduction. But that's not the whole story on phonons.

Phonons, to put it simply, are 'waves' in a lattice and are elastic in nature (i.e. they don't result in permanent deformation, like say dislocation motion). They can come from a variety of sources, and have two important characteristics: a frequency (measure of oscillation in time) and a wave vector (related to the spatial wave length and the wave's direction). The relationship between these two quantities is called a dispersion relation. The topic of phonon dispersion relations is pretty involved (think: several book chapters and even entire books), especially for real crystalline solids, but to give an idea of what these look like graphically:

A) Phonon spectrum in a 1d chain with 2 different particle masses. If all of the masses were the same, we'd only get the acoustic mode. B) Phonon spectrum of GaAs, demonstrating the complexity of a 3d diatomic crystal. The symbols refer to specific important wave vectors. Images courtesy of wikimedia commons.

Now let's think about what may happen when an event occurs at one point in a finite lattice (say a sudden atomic motion). Assuming the lattice is sufficiently well connected, a large atomic motion in one region can set up a traveling wave that radiates away from the disturbance. Really, this is nothing more than a group of phonons (a set of vibrations of different frequencies, sometimes called a wave packet) being created. Once the phonons reach the boundary, two major things can happen:

1. The phonon reflects (this would be the case if the lattice were in a vacuum).
2. The phonon dissipates into the surroundings (if there was some medium surrounding the lattice)

In reality, a combination of both will occur. If you imagine a fluid surrounding the lattice, then a phonon could excite a series of molecules or atoms in the fluid, leading to a wave propagating in the gas. But not all of the phonon's energy will necessarily be dissipated to the fluid, meaning that there is an impetus for the boundary atoms to rebound (rather than permanently deform). Thus the remainder results in a wave reflection at the boundary. Eventually the losses will result in the phonon decaying entirely.

Sound is how we perceive generally longitudinal vibrations in air (or other media) of a certain frequency range. In essence (though a rather simplified essence), the frequency we perceive as the pitch or tone, while the magnitude of the pressure wave we interpret (loosely) as the volume. The human hearing range is approximately 20-20,000 Hz.

In air at standard temperature and pressure (0 \(^o\)C, 1 bar) these correspond to wavelengths from 17m to 17 mm. So a disturbance of sufficient energy in air will generate a pressure pulse that we'd hear as a sound if the resulting frequency (and volume) was in the hearable range. For example, smacking the end of a tuning fork: whose vibration is nothing more than a collection of long wavelength phonons, resulting in elastic bulk motion at the fork end. This bulk motion then excites the surrounding air, and we hear the sound of the tuning fork. These particular long wavelength phonons are called acoustic phonons, because they behave in a lattice just as sound waves do in a fluid (longitudinal waves), though there are additional transverse acoustic modes in solids.

So what about our pan heating up? No one smacked it. So where's the disturbance that generates the acoustic phonons that then lead to the sound?

If you read my earlier posts on crystal structure and alloys, you may remember some discussion of lattice defects: dislocations, point defects, etc. These are examples of defects that become increasingly mobile with increasing temperature: if a large defect (or group of defects) were to suddenly move or relax, the dynamics of the event would generate phonons. Similarly, if you have a material mis-match at an interface (say a rivet and a pan), you may have some significant thermal stresses that develop and may suddenly relax. Again, the result would be phonon emission. Thus, if the phonons are acoustic type, and of sufficient energy to cause a disturbance at the surface of the body, you can generate a sound wave in the surrounding air. This is the essence of acoustic emission testing methods, though through the use of specialized equipment they are not limited to the human hearing range.

So now the fun part: how much energy is necessary to create a phonon in an elastic solid of a frequency in the human hearing range, and how does that compare to the kinetic energy available in a heated pan?

From quantum mechanics (by way of analogy to photons), we can express the energy of a single phonon of frequency \(\nu\) as follows:
\[E(\nu) = \frac{h\nu}{2} + \frac{h\nu}{e^{h\nu/k_BT} - 1}\]
Where \(h\) is Planck's constant, \(k_B\) is Boltzmann's constant and \(T\) is the temperature of the solid in Kelvin. The first term is the so-called zero point energy (sorry, Syndrome, it doesn't work that way), the energy associated with the phonon at absolute zero (zero kelvin), and the second arises from the probability of finding a phonon of frequency \(\nu\) in a crystal at temperature \(T\) (the Planck distribution, which we discussed once before). If the quantity \(E(\nu)\) is the total energy of the phonon, then the energy necessary to create it must be at least this much due to the conservation of energy.

For human hearing range (\(\nu\) = 20-20,000 Hz), assuming a pan at 100 \(^o\)C, we end up with an energy of about 0.032140 - 0.032142 eV (electronvolts are a very small unit of energy) for a single mode, which is comparable to the classical average kinetic energy of an atom in a solid at 100  \(^o\)C (\(k_BT \approx\) 0.032141 eV). Of course, what really matters is how much energy is 'released' by a particular defect relaxing, or some other relaxation event during cooling, which can be significantly larger than \(k_BT\).