second sound

Normal sound waves are fluctuations in the displacement and density of molecules in a substance;{{cite book |last1=Feynman |first1=Richard |title=Feynman Lectures on Physics |date=4 October 2011 |publisher=Basic Books |isbn=978-0465024933}}{{cite web |last1=Feynman |title=Sound. The wave equation |url=https://feynmanlectures.caltech.edu/I_47.html |website=feynmanlectures.caltech.edu |publisher=Caltech |access-date=20 July 2021}}

second sound waves are fluctuations in the density of quasiparticle thermal excitations (rotons and phonons{{cite book

| title = Transport Phenomena

| last1 = Smith | first1 = Henrik

| last2 = Jensen |first2 = H. Hojgaard

| year = 1989

| publisher = Oxford University Press

| isbn = 0-19-851985-0

| url = https://books.google.com/books?id=xPt9AAAAIAAJ

| chapter = Section 4.3: Second Sound

}}). Second sound can be observed in any system in which most phonon-phonon collisions conserve momentum, like superfluids{{cite journal

| title = Second Sound: Waves of Entropy and Temperature

| last = Srinivasan

| first = R

| journal = Resonance

| volume = 3

|date=March 1999

| pages = 16–24

| doi = 10.1007/BF02838720

| s2cid = 123957486

| url = http://www.ias.ac.in/resonance/Mar1999/pdf/Mar1999p16-24.pdf

}} and in some dielectric crystals{{cite journal

| title = Second Sound: The Role of Elastic Waves

| last = Srinivasan

| first = R

| journal = Resonance

| volume = 4

|date=June 1999

| pages = 15–19

| url = http://www.ias.ac.in/resonance/June1999/pdf/June1999p15-19.pdf

| doi=10.1007/bf02834631

| s2cid = 124849291

}}{{Cite journal | last1 = Prohofsky | first1 = E. | last2 = Krumhansl | first2 = J. | doi = 10.1103/PhysRev.133.A1403 | title = Second-Sound Propagation in Dielectric Solids | journal = Physical Review | volume = 133 | issue = 5A | pages = A1403 | year = 1964 |bibcode = 1964PhRv..133.1403P }}{{Cite journal | last1 = Chester | first1 = M. | title = Second Sound in Solids | doi = 10.1103/PhysRev.131.2013 | journal = Physical Review | volume = 131 | issue = 5 | pages = 2013–2015 | year = 1963 |bibcode = 1963PhRv..131.2013C }} 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.{{Cite book |last1=Ashcroft |first1=Neil W. |url=https://books.google.com/books?id=oXIfAQAAMAAJ |title=Solid State Physics |last2=Mermin |first2=N. David |date=1976 |publisher=Holt, Rinehart and Winston |isbn=978-0-03-083993-1 |language=en}}

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

:$\backslash frac\{1\}\{\backslash tau\_\{\backslash rm\; u\}\}\; \backslash ll\; \backslash omega\backslash ll\; \backslash frac\{1\}\{\backslash tau\_N\}$

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 phenomenon which makes it very different from usual heat diffusion.

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).{{cite report

| title = Superfluid helium as a technical coolant

| last = Lebrun

| first = Phillipe

| date = July 17, 1997

| page = 4

| type = LHC-Project-Report-125

| publisher = CERN

| url = http://cdsweb.cern.ch/record/330851/files/lhc-project-report-125.pdf

}} Second sound can be observed either as pulses or in a resonant cavity.{{Cite journal | last1 = Van Der Boog | first1 = A. G. M. | last2 = Husson | first2 = L. P. J. | last3 = Disatnik | first3 = Y. | last4 = Kramers | first4 = H. C. | title = Experimental results on the velocity of second sound and the viscosity in dilute 3He-4He mixtures | doi = 10.1016/0378-4363(81)90176-5 | journal = Physica B+C | volume = 104 | issue = 3 | pages = 303–315 | year = 1981 |bibcode = 1981PhyBC.104..303V }}

The speed of second sound is close to zero near the lambda point, increasing to approximately 20 m/s around 1.8 K,{{Cite journal | last1 = Wang | first1 = R. T. | last2 = Wagner | first2 = W. T. | last3 = Donnelly | first3 = R. J. | year = 1987 | title = Precision second-sound velocity measurements in helium II | journal = Journal of Low Temperature Physics | volume = 68 | issue = 5–6 | pages = 409–417 | bibcode = 1987JLTP...68..409W | doi = 10.1007/BF00682305| s2cid = 120789592 }} about ten times slower than normal sound waves.{{Cite journal | last1 = Lane | first1 = C. | last2 = Fairbank | first2 = H. | last3 = Fairbank | first3 = W. | doi = 10.1103/PhysRev.71.600 | title = Second Sound in Liquid Helium II | journal = Physical Review | volume = 71 | issue = 9 | pages = 600–605 | year = 1947 |bibcode = 1947PhRv...71..600L }}

At temperatures below 1 K, the speed of second sound in helium II increases as the temperature decreases.{{Cite journal | last1 = De Klerk | first1 = D. | last2 = Hudson | first2 = R. | last3 = Pellam | first3 = J. | title = Second Sound Propagation below 1K | doi = 10.1103/PhysRev.93.28 | journal = Physical Review | volume = 93 | pages = 28–37 | year = 1954 | issue = 1 |bibcode = 1954PhRv...93...28D }}

Second sound is also observed in superfluid helium-3 below its lambda point 2.5 mK.{{Cite journal | last1 = Lu | first1 = S. | last2 = Kojima | first2 = H. | doi = 10.1103/PhysRevLett.55.1677 | title = Observation of Second Sound in Superfluid ^{3}He-B | journal = Physical Review Letters | volume = 55 | issue = 16 | pages = 1677–1680 | year = 1985 | pmid = 10031890|bibcode = 1985PhRvL..55.1677L }}

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

$c\_2\; =\; \backslash left(\backslash frac\{TS^2\}\{C\}\backslash ,\backslash frac\{\backslash rho\_s\}\{\backslash rho\_n\}\backslash right)^\{1/2\}$

where $T$ is the temperature, $S$ is the entropy, $C$ is the specific heat, $\backslash rho\_s$ is the superfluid density and $\backslash rho\_n$ is the normal fluid density. As $T\backslash rightarrow\; 0$, $c\_2=c/\backslash sqrt\{3\}$, where $c=(\backslash partial\; p/\backslash partial\; \backslash rho)\_S\backslash approx\; (\backslash partial\; p/\backslash partial\; \backslash rho)\_T$ is the ordinary (or first) sound speed.

Second sound has been observed in solid ⁴He and ³He,{{Cite journal | last1 = Ackerman | first1 = C. | last2 = Bertman | first2 = B. | last3 = Fairbank | first3 = H. | last4 = Guyer | first4 = R. | title = Second Sound in Solid Helium | doi = 10.1103/PhysRevLett.16.789 | journal = Physical Review Letters | volume = 16 | issue = 18 | pages = 789–791 | year = 1966 |bibcode = 1966PhRvL..16..789A }}{{Cite journal | last1 = Ackerman | first1 = C. | last2 = Overton | first2 = W. | doi = 10.1103/PhysRevLett.22.764 | title = Second Sound in Solid Helium-3 | journal = Physical Review Letters | volume = 22 | issue = 15 | pages = 764–766 | year = 1969 |bibcode = 1969PhRvL..22..764A }}

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,{{Cite journal | last1 = Narayanamurti | first1 = V. | last2 = Dynes | first2 = R. | doi = 10.1103/PhysRevLett.28.1461 | title = Observation of Second Sound in Bismuth | journal = Physical Review Letters | volume = 28 | issue = 22 | pages = 1461–1465 | year = 1972 |bibcode = 1972PhRvL..28.1461N }}

or solid sodium fluoride (NaF) around 10 to 20 K.{{Cite journal | last1 = Jackson | first1 = H. | last2 = Walker | first2 = C. | last3 = McNelly | first3 = T. | title = Second Sound in NaF | doi = 10.1103/PhysRevLett.25.26 | journal = Physical Review Letters | volume = 25 | pages = 26–28 | year = 1970 | issue = 1 |bibcode = 1970PhRvL..25...26J }} In 2021 this effect was observed in a BKT superfluid{{cite journal |vauthors=Christodoulou P, Gałka M, Dogra N et al |date=10 June 2021 |title=Observation of first and second sound in a BKT superfluid |url=https://www.nature.com/articles/s41586-021-03537-9 |journal=Nature |volume=594 |issue=7862 |pages=191–194 |arxiv=2008.06044 |bibcode=2021Natur.594..191C |doi=10.1038/s41586-021-03537-9 |pmid=34108696 |s2cid=235394222}} as well as in a germanium semiconductor{{Cite journal |last1=Beardo |first1=Albert |last2=López-Suárez |first2=Miquel |last3=Pérez |first3=Luis Alberto |last4=Sendra |first4=Lluc |last5=Alonso |first5=Maria Isabel |last6=Melis |first6=Claudio |last7=Bafaluy |first7=Javier |last8=Camacho |first8=Juan |last9=Colombo |first9=Luciano |last10=Rurali |first10=Riccardo |last11=Alvarez |first11=Francesc Xavier |last12=Reparaz |first12=Sebastian |date=2021-06-01 |title=Observation of second sound in a rapidly varying temperature field in Ge |journal=Science Advances |language=en |volume=7 |issue=27 |pages=eabg4677 |arxiv=2007.05487 |bibcode=2021SciA....7.4677B |doi=10.1126/sciadv.abg4677 |issn=2375-2548 |pmc=8245038 |pmid=34193427}}{{Cite web |date=2021-07-18 |title='Second sound' appears in germanium |url=https://physicsworld.com/a/second-sound-appears-in-germanium/ |access-date=2021-07-20 |website=Physics World |language=en-GB}}

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.{{Cite journal | last1 = Huberman | first1 = S. | last2 = Duncan | first2 = R.A. | doi = 10.1126/science.aav3548 | title = Observation of second sound in graphite at temperatures above 100 K | journal = Science | year = 2019 | volume = 364 | issue = 6438 | pages = 375–379 | pmid = 30872535 | arxiv = 1901.09160 | bibcode = 2019Sci...364..375H | s2cid = 78091609 }} 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.{{Cite book | last1 = Pitre | first1 = L. | chapter = The Comparison between a Second-Sound Thermometer and a Melting-Curve Thermometer from 0.8 K Down to 20 mK | doi = 10.1063/1.1627108 | title = AIP Conference Proceedings | volume = 684 | pages = 101–106 | year = 2003 }}

Oscillating superleak transducers (OST){{Cite journal | last1 = Sherlock | first1 = R. A. | title = Oscillating Superleak Second Sound Transducers | doi = 10.1063/1.1684354 | journal = Review of Scientific Instruments | volume = 41 | issue = 11 | pages = 1603–1609 | year = 1970 |bibcode = 1970RScI...41.1603S | doi-access = free }} use second sound to locate defects in superconducting accelerator cavities.{{cite journal

| journal = ILC Newsline

| url = http://newsline.linearcollider.org/2011/04/21/the-sound-of-accelerator-cavities/

| last = Hesla

| first = Leah

| title = The sound of accelerator cavities

| date = 21 April 2011

| access-date = 26 October 2012

}}{{Cite journal | last1 = Quadt | first1 = A. | last2 = Schröder | first2 = B. | last3 = Uhrmacher | first3 = M. | last4 = Weingarten | first4 = J. | last5 = Willenberg | first5 = B. | last6 = Vennekate | first6 = H. | doi = 10.1103/PhysRevSTAB.15.031001 | title = Response of an oscillating superleak transducer to a pointlike heat source | journal = Physical Review Special Topics: Accelerators and Beams | volume = 15 | issue = 3 | year = 2012 | page = 031001 |arxiv = 1111.5520 |bibcode = 2012PhRvS..15c1001Q | s2cid = 118996515 }}

Researchers made significant advances in directly observing second sound in distinct quantum fluids. At the Massachusetts Institute of Technology (MIT), physicists visualized second sound in a unitary Fermi gas of ultracold lithium-6 atoms by tracking temperature-dependent resonant frequencies, enabling the first direct imaging of heat waves in such a dilute system.{{Cite journal |last1=Yu |first1=S. |last2=Borchert |first2=M. |last3=Nishida |first3=Y. |last4=Zwierlein |first4=M. |title=Second sound in a unitary Fermi gas |journal=Science |volume=384 |issue=6693 |pages=1064–1069 |year=2024 |doi=10.1126/science.adg3430 |pmid=38679494}} Furthermore, scientists at Université Grenoble Alpes developed a micromachined heater-thermometer system that enabled direct detection of second sound in superfluid helium-4, further validating the wave-like heat propagation in bosonic quantum fluids under controlled cryogenic conditions.{{Cite journal |title=Kolmogorov Cascade as the Governing Mechanism for Intervortex Spacing in Quantum Turbulence |last1=Bret |first1=Clément |display-authors=et al. |year=2025 |journal=arxiv |eprint=2504.21416 }}

• 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, [http://www.inf.ethz.ch/personal/fcellier/MS/piram_ms.pdf "Numerical investigation of second sound in liquid helium,"] Dipl.-Ing. Dissertation (1991). Retrieved on April 15, 2007.

Category:Quantum mechanics

Category:Thermodynamics

Category:Superfluidity

Category:Lev Landau