Editing Lambda point (section)

{{Short description|Superfluid transition temperature of helium-4}}
[[File:Lambda transition.svg|thumb|250px|The plot of the specific heat capacity versus temperature.]]
The '''lambda point''' is the [[temperature]] at which normal fluid [[helium]] (helium I) makes the transition to [[superfluid]] state ([[helium II]]). At pressure of 1 [[atmosphere (unit)|atmosphere]], the transition occurs at approximately 2.17 [[Kelvin|K]]. The lowest pressure at which He-I and He-II can coexist is the vapor−He-I−He-II [[triple point]] at {{convert|2.1768|K|C}} and {{convert|5.0418|kPa|atm|abbr=on}}, which is the "saturated [[vapor pressure]]" at that temperature (pure helium gas in thermal equilibrium over the liquid surface, in a [[Hermetic seal|hermetic]] container).<ref name=Donnelly>{{cite journal| title=The Observed Properties of Liquid Helium at the Saturated Vapor Pressure | first1=Russell J.| last1=Donnelly| first2=Carlo F.| last2=Barenghi | journal=[[Journal of Physical and Chemical Reference Data]]| year=1998| volume=27| issue=6| pages=1217–1274| doi=10.1063/1.556028|bibcode = 1998JPCRD..27.1217D }}</ref> The highest pressure at which He-I and He-II can coexist is the [[Body-centered cubic|bcc]]−He-I−He-II triple point with a helium solid at {{convert|1.762|K|C}}, {{convert|29.725|atm|kPa|abbr=on}}.<ref name=Hoffer>{{cite journal| title=Thermodynamic properties of <sup>4</sup>He. II. The bcc phase and the P-T and VT phase diagrams below 2 K | first1=J. K.| last1=Hoffer| first2=W. R.| last2=Gardner| first3=C. G.| last3=Waterfield| first4=N. E.| last4=Phillips| journal=[[Journal of Low Temperature Physics]]| date=April 1976| volume=23| issue=1| pages=63–102| doi=10.1007/BF00117245|bibcode = 1976JLTP...23...63H | s2cid=120473493}}</ref>

The point's name derives from the graph (pictured) that results from plotting the [[specific heat capacity]] as a function of [[temperature]] (for a given pressure in the above range, in the example shown, at 1 atmosphere), which resembles the [[Greek language|Greek]] letter [[lambda]] <math>\lambda</math>. The specific heat capacity has a sharp peak as the temperature approaches the lambda point. The tip of the peak is so sharp that a critical exponent characterizing the divergence of the heat capacity can be measured precisely only in zero gravity, to provide a uniform density over a substantial volume of fluid.  Hence, the heat capacity was measured within 2&nbsp;nK below the transition in an experiment included in a [[Space Shuttle]] payload in 1992.<ref name=JPL>{{cite journal| title=Heat Capacity and Thermal Relaxation of Bulk Helium very near the Lambda Point | first1=J.A.| last1=Lipa| first2=D. R.| last2=Swanson| first3=J. A.| last3=Nissen| first4=T. C. P.| last4=Chui| first5=U. E.| last5=Israelsson| journal=[[Physical Review Letters]]| year=1996| volume=76| issue=6| pages=944–7| doi=10.1103/PhysRevLett.76.944|bibcode = 1996PhRvL..76..944L| pmid=10061591| hdl=2060/19950007794| s2cid=29876364| hdl-access=free}}</ref>{{unsolved|physics|Explain the discrepancy between the experimental and theoretical determinations of the heat capacity critical exponent {{math|''α''}} for the superfluid transition in helium-4.<ref name="Rychkov"/>
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Although the heat capacity has a peak, it does not tend towards [[infinity]] (contrary to what the graph may suggest), but has finite limiting values when approaching the transition from above and below.<ref name=JPL /> The behavior of the heat capacity near the peak is described by the formula <math>C\approx A_\pm t^{-\alpha}+B_\pm</math> where <math>t=|1-T/T_c|</math> is the reduced temperature, <math>T_c</math> is the Lambda point temperature, <math>A_\pm,B_\pm</math> are constants (different above and below the transition temperature), and {{math|''α''}} is the [[critical exponent]]: <math>\alpha=-0.0127(3)</math>.<ref name=JPL /><ref>{{Cite journal|last1=Lipa|first1=J. A.|last2=Nissen|first2=J. A.|last3=Stricker|first3=D. A.|last4=Swanson|first4=D. R.|last5=Chui|first5=T. C. P.|date=2003-11-14|title=Specific heat of liquid helium in zero gravity very near the lambda point|journal=Physical Review B|volume=68|issue=17|pages=174518|doi=10.1103/PhysRevB.68.174518|bibcode=2003PhRvB..68q4518L|arxiv=cond-mat/0310163|s2cid=55646571}}</ref> Since this exponent is negative for the superfluid transition, specific heat remains finite.<ref>For other phase transitions <math>\alpha</math> may be positive (e.g. <math>\alpha\approx+0.1</math> for [[Critical point (thermodynamics)|the liquid-vapor critical point]] which has [[Ising critical exponents]]). For those phase transitions specific heat does tend to infinity.</ref>

The quoted experimental value of {{math|''α''}} is in a significant disagreement<ref>{{Cite book|last=Vicari|first=Ettore|chapter=Critical phenomena and renormalization-group flow of multi-parameter Phi4 theories |date=2008-03-21|title=Proceedings of the XXV International Symposium on Lattice Field Theory — PoS(LATTICE 2007)|volume=42|language=en|location=Regensburg, Germany|publisher=Sissa Medialab|pages=023|doi=10.22323/1.042.0023|doi-access=free}}</ref><ref name="Rychkov">{{Cite journal|last=Rychkov|first=Slava|date=2020-01-31|title=Conformal bootstrap and the λ-point specific heat experimental anomaly|url=https://www.condmatjclub.org/?p=4037|journal=Journal Club for Condensed Matter Physics|language=en|doi=10.36471/JCCM_January_2020_02|doi-access=free}}</ref> with the most precise theoretical determinations<ref>{{Cite journal|last1=Campostrini|first1=Massimo|last2=Hasenbusch|first2=Martin|last3=Pelissetto|first3=Andrea|last4=Vicari|first4=Ettore|date=2006-10-06|title=Theoretical estimates of the critical exponents of the superfluid transition in $^{4}\mathrm{He}$ by lattice methods|journal=Physical Review B|volume=74|issue=14|pages=144506|doi=10.1103/PhysRevB.74.144506|arxiv=cond-mat/0605083|s2cid=118924734}}</ref><ref>{{Cite journal|last=Hasenbusch|first=Martin|date=2019-12-26|title=Monte Carlo study of an improved clock model in three dimensions|arxiv=1910.05916|journal=Physical Review B|volume=100|issue=22|pages=224517|doi=10.1103/PhysRevB.100.224517|issn=2469-9950|bibcode=2019PhRvB.100v4517H|s2cid=204509042}}</ref><ref>{{cite journal|last1=Chester|first1=Shai M.|last2=Landry|first2=Walter|last3=Liu|first3=Junyu|last4=Poland|first4=David|last5=Simmons-Duffin|first5=David|last6=Su|first6=Ning|last7=Vichi|first7=Alessandro|title=Carving out OPE space and precise O(2) model critical exponents|journal=Journal of High Energy Physics|year=2020|volume=2020|issue=6|page=142|doi=10.1007/JHEP06(2020)142|arxiv=1912.03324|bibcode=2020JHEP...06..142C|s2cid=208910721}}</ref> coming from high temperature expansion techniques, [[Monte Carlo method|Monte Carlo]] methods and the [[conformal bootstrap]].