Second sound

In condensed matter physics, second sound is a quantum mechanical phenomenon in which heat transfer occurs by wave-like motion, rather than by the more usual mechanism of diffusion. Its presence leads to a very high thermal conductivity. It is known as "second sound" because the wave motion of entropy and temperature is similar to the propagation of pressure waves in air (sound).^[1] The phenomenon of second sound was first described by Lev Landau in 1941.^[2]

Normal sound waves are fluctuations in the displacement and density of molecules in a substance;^[3][4] second sound waves are fluctuations in the density of quasiparticle thermal excitations (rotons and phonons^[5]). Second sound can be observed in any system in which most phonon-phonon collisions conserve momentum, like superfluids^[6] and in some dielectric crystals^[1][7][8] when Umklapp scattering is small.

Contrary to molecules in a gas, quasiparticles are not necessarily conserved. Also gas molecules in a box conserve momentum (except at the boundaries of box), while quasiparticles can sometimes not conserve momentum in the presence of impurities or Umklapp scattering. Umklapp phonon-phonon scattering exchanges momentum with the crystal lattice, so phonon momentum is not conserved, but Umklapp processes can be reduced at low temperatures.^[9]

Normal sound in gases is a consequence of the collision rate τ between molecules being large compared to the frequency of the sound wave ω ≫ 1/τ. For second sound, the Umklapp rate τ_u has to be small compared to the oscillation frequency ω ≪ 1/τ_u for energy and momentum conservation. However analogous to gasses, the relaxation time τ_N describing the collisions has to be large with respect to the frequency ω ≫ 1/τ_N, leaving a window:^[9]

${\displaystyle {\frac {1}{\tau _{\rm {u}}}}\ll \omega \ll {\frac {1}{\tau _{N}}}}$

for sound-like behaviour or second sound. The second sound thus behaves as oscillations of the local number of quasiparticles (or of the local energy carried by these particles). Contrary to the normal sound where energy is related to pressure and temperature, in a crystal the local energy density is purely a function of the temperature. In this sense, the second sound can also be considered as oscillations of the local temperature. Second sound is a wave-like phenomena which makes it very different from usual heat diffusion.^[9]

Second sound is observed in liquid helium at temperatures below the lambda point, 2.1768 K, where ⁴He becomes a superfluid known as helium II. Helium II has the highest thermal conductivity of any known material (several hundred times higher than copper).^[10] Second sound can be observed either as pulses or in a resonant cavity.^[11]

The speed of second sound is close to zero near the lambda point, increasing to approximately 20 m/s around 1.8 K,^[12] about ten times slower than normal sound waves.^[13] At temperatures below 1 K, the speed of second sound in helium II increases as the temperature decreases.^[14]

Second sound is also observed in superfluid helium-3 below its lambda point 2.5 mK.^[15]

As per the two-fluid, the speed of second sound is given by

${\displaystyle c_{2}=\left({\frac {TS^{2}}{C}}\,{\frac {\rho _{s}}{\rho _{n}}}\right)^{1/2}}$

where ${\displaystyle T}$ is the temperature, ${\displaystyle S}$ is the entropy, ${\displaystyle C}$ is the specific heat, ${\displaystyle \rho _{s}}$ is the superfluid density and ${\displaystyle \rho _{n}}$ is the normal fluid density. As ${\displaystyle T\rightarrow 0}$, ${\displaystyle c_{2}=c/{\sqrt {3}}}$, where ${\displaystyle c=(\partial p/\partial \rho )_{S}\approx (\partial p/\partial \rho )_{T}}$ is the ordinary (or first) sound speed.

Second sound has been observed in solid ⁴He and ³He,^[16][17] and in some dielectric solids such as Bi in the temperature range of 1.2 to 4.0 K with a velocity of 780 ± 50 m/s,^[18] or solid sodium fluoride (NaF) around 10 to 20 K.^[19] In 2021 this effect was observed in a BKT superfluid^[20] as well as in a germanium semiconductor^[21][22]

In 2019 it was reported that ordinary graphite exhibits second sound at 120 K. This feature was both predicted theoretically and observed experimentally, and was by far the highest temperature at which second sound has been observed.^[23] However, this second sound is observed only at the microscale, because the wave dies out exponentially with characteristic length 1-10 microns. Therefore, presumably graphite in the right temperature regime has extraordinarily high thermal conductivity but only for the purpose of transferring heat pulses distances of order 10 microns, and for pulses of duration on the order of 10 nanoseconds. For more "normal" heat-transfer, graphite's observed thermal conductivity is less than that of, e.g., copper. The theoretical models, however, predict longer absorption lengths would be seen in isotopically pure graphite, and perhaps over a wider temperature range, e.g. even at room temperature. (As of March 2019, that experiment has not yet been tried.)

Measuring the speed of second sound in ³He-⁴He mixtures can be used as a thermometer in the range 0.01-0.7 K.^[24]

Oscillating superleak transducers (OST)^[25] use second sound to locate defects in superconducting accelerator cavities.^[26][27]

• Zero sound
• Third sound

• Sinyan Shen, Surface Second Sound in Superfluid Helium. PhD Dissertation (1973). http://adsabs.harvard.edu/abs/1973PhDT.......142S
• V. Peshkov, "'Second Sound' in Helium II," J. Phys. (Moscow) 8, 381 (1944)
• U. Piram, "Numerical investigation of second sound in liquid helium," Dipl.-Ing. Dissertation (1991). Retrieved on April 15, 2007.