Second sound

Short description: Quantum mechanical phenomenon in which heat transfer occurs by wave-like motion

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[1] is similar to the propagation of pressure waves in air (sound). 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 particle-like 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. (Umklapp phonon-phonon scattering exchanges momentum with the crystal lattice, so phonon momentum is not conserved.)

## Contents

* 1 In helium II
* 2 In other media
* 3 Applications
* 4 See also
* 5 References
* 6 Bibliography

## In helium II

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).[9] Second sound can be observed either as pulses or in a resonant cavity.[10]

The speed of second sound is close to zero near the lambda point, increasing to approximately 20 m/s around 1.8 K,[11] about ten times slower than normal sound waves.[12] At temperatures below 1 K, the speed of second sound in helium II increases as the temperature decreases.[13]

Second sound is also observed in superfluid helium-3 below its lambda point 2.5 mK.[14]

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

[math]\displaystyle{ c_2 = \left(\frac{TS^2}{C}\,\frac{\rho_s}{\rho_n}\right)^{1/2} }[/math]

where [math]\displaystyle{ T }[/math] is the temperature, [math]\displaystyle{ S }[/math] is the entropy, [math]\displaystyle{ C }[/math] is the specific heat, [math]\displaystyle{ \rho_s }[/math] is the superfluid density and [math]\displaystyle{ \rho_n }[/math] is the normal fluid density. As [math]\displaystyle{ T\rightarrow 0 }[/math], [math]\displaystyle{ c_2=c/\sqrt{3} }[/math], where [math]\displaystyle{ c=(\partial p/\partial \rho)_S\approx (\partial p/\partial \rho)_T }[/math] is the ordinary (or first) sound speed.

## In other media

Second sound has been observed in solid 4He and 3He,[15][16] 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,[17] or NaF around 10 to 20 K.[18]

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.[19] 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.)

In 2021 this effect was observed in a BKT superfluid[20] as well as in a germanium semiconductor[21][22]

## Applications

Measuring the speed of second sound in 3He-4He mixtures can be used as a thermometer in the range 0.01-0.7 K.[23]

Oscillating superleak transducers (OST)[24] use second sound to locate defects in superconducting accelerator cavities.[25][26]

## See also

* Third sound

## References

1. ↑ 1.0 1.1 Srinivasan, R (June 1999). "Second Sound: The Role of Elastic Waves". Resonance 4: 15–19. doi:10.1007/bf02834631. http://www.ias.ac.in/resonance/June1999/pdf/June1999p15-19.pdf.
2. ↑ Landau, L. (1941). Theory of the superfluidity of helium II. Physical Review, 60(4), 356.
3. ↑ Feynman, Richard (4 October 2011). Feynman Lectures on Physics. Basic Books. ISBN 978-0465024933.
4. ↑ Feynman. "Sound. The wave equation". Caltech. https://feynmanlectures.caltech.edu/I_47.html.
5. ↑ Smith, Henrik; Jensen, H. Hojgaard (1989). "Section 4.3: Second Sound". Transport Phenomena. Oxford University Press. ISBN 0-19-851985-0. https://books.google.com/books?id=xPt9AAAAIAAJ.
6. ↑ Srinivasan, R (March 1999). "Second Sound: Waves of Entropy and Temperature". Resonance 3: 16–24. doi:10.1007/BF02838720. http://www.ias.ac.in/resonance/Mar1999/pdf/Mar1999p16-24.pdf.
7. ↑ Prohofsky, E.; Krumhansl, J. (1964). "Second-Sound Propagation in Dielectric Solids". Physical Review 133 (5A): A1403. doi:10.1103/PhysRev.133.A1403. Bibcode: 1964PhRv..133.1403P.
8. ↑ Chester, M. (1963). "Second Sound in Solids". Physical Review 131 (5): 2013–2015. doi:10.1103/PhysRev.131.2013. Bibcode: 1963PhRv..131.2013C.
9. ↑ Lebrun, Phillipe (July 17, 1997). ﻿Superfluid helium as a technical coolant﻿ (LHC-Project-Report-125). CERN. p. 4. http://cdsweb.cern.ch/record/330851/files/lhc-project-report-125.pdf.
10. ↑ Van Der Boog, A. G. M.; Husson, L. P. J.; Disatnik, Y.; Kramers, H. C. (1981). "Experimental results on the velocity of second sound and the viscosity in dilute 3He-4He mixtures". Physica B+C 104 (3): 303–315. doi:10.1016/0378-4363(81)90176-5. Bibcode: 1981PhyBC.104..303V.
11. ↑ Wang, R. T.; Wagner, W. T.; Donnelly, R. J. (1987). "Precision second-sound velocity measurements in helium II". Journal of Low Temperature Physics 68 (5–6): 409–417. doi:10.1007/BF00682305. Bibcode: 1987JLTP...68..409W.
12. ↑ Lane, C.; Fairbank, H.; Fairbank, W. (1947). "Second Sound in Liquid Helium II". Physical Review 71 (9): 600–605. doi:10.1103/PhysRev.71.600. Bibcode: 1947PhRv...71..600L.
13. ↑ De Klerk, D.; Hudson, R.; Pellam, J. (1954). "Second Sound Propagation below 1K". Physical Review 93 (1): 28–37. doi:10.1103/PhysRev.93.28. Bibcode: 1954PhRv...93...28D.
14. ↑ Lu, S.; Kojima, H. (1985). "Observation of Second Sound in Superfluid ^{3}He-B". Physical Review Letters 55 (16): 1677–1680. doi:10.1103/PhysRevLett.55.1677. PMID 10031890. Bibcode: 1985PhRvL..55.1677L.
15. ↑ Ackerman, C.; Bertman, B.; Fairbank, H.; Guyer, R. (1966). "Second Sound in Solid Helium". Physical Review Letters 16 (18): 789–791. doi:10.1103/PhysRevLett.16.789. Bibcode: 1966PhRvL..16..789A.
16. ↑ Ackerman, C.; Overton, W. (1969). "Second Sound in Solid Helium-3". Physical Review Letters 22 (15): 764–766. doi:10.1103/PhysRevLett.22.764. Bibcode: 1969PhRvL..22..764A.
17. ↑ Narayanamurti, V.; Dynes, R. (1972). "Observation of Second Sound in Bismuth". Physical Review Letters 28 (22): 1461–1465. doi:10.1103/PhysRevLett.28.1461. Bibcode: 1972PhRvL..28.1461N.
18. ↑ Jackson, H.; Walker, C.; McNelly, T. (1970). "Second Sound in NaF". Physical Review Letters 25 (1): 26–28. doi:10.1103/PhysRevLett.25.26. Bibcode: 1970PhRvL..25...26J.
19. ↑ Huberman, S.; Duncan, R.A. (2019). "Observation of second sound in graphite at temperatures above 100 K". Science 364 (6438): 375–379. doi:10.1126/science.aav3548. PMID 30872535. Bibcode: 2019Sci...364..375H.
20. ↑ "Observation of first and second sound in a BKT superfluid". Nature 594 (7862): 191–194. 10 June 2021. doi:10.1038/s41586-021-03537-9. PMID 34108696. Bibcode: 2021Natur.594..191C. https://www.nature.com/articles/s41586-021-03537-9.
21. ↑ Beardo, Albert; López-Suárez, Miquel; Pérez, Luis Alberto; Sendra, Lluc; Alonso, Maria Isabel; Melis, Claudio; Bafaluy, Javier; Camacho, Juan et al. (2021-06-01). "Observation of second sound in a rapidly varying temperature field in Ge" (in en). Science Advances 7 (27): eabg4677. doi:10.1126/sciadv.abg4677. ISSN 2375-2548. PMID 34193427. Bibcode: 2021SciA....7.4677B.
22. ↑ "'Second sound' appears in germanium" (in en-GB). 2021-07-18. https://physicsworld.com/a/second-sound-appears-in-germanium/.
23. ↑ Pitre, L. (2003). "The Comparison between a Second-Sound Thermometer and a Melting-Curve Thermometer from 0.8 K Down to 20 mK". AIP Conference Proceedings. 684. pp. 101–106. doi:10.1063/1.1627108.
24. ↑ Sherlock, R. A. (1970). "Oscillating Superleak Second Sound Transducers". Review of Scientific Instruments 41 (11): 1603–1609. doi:10.1063/1.1684354. Bibcode: 1970RScI...41.1603S.
25. ↑ Hesla, Leah (21 April 2011). "The sound of accelerator cavities". ILC Newsline. http://newsline.linearcollider.org/2011/04/21/the-sound-of-accelerator-cavities/. Retrieved 26 October 2012.
26. ↑ Quadt, A.; Schröder, B.; Uhrmacher, M.; Weingarten, J.; Willenberg, B.; Vennekate, H. (2012). "Response of an oscillating superleak transducer to a pointlike heat source". Physical Review Special Topics: Accelerators and Beams 15 (3): 031001. doi:10.1103/PhysRevSTAB.15.031001. Bibcode: 2012PhRvS..15c1001Q.

## Bibliography

* 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.

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