# Lambda point – Wikipedia

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The **lambda point** is the temperature at which normal fluid helium (helium I) makes the transition to superfluid helium II (approximately 2.17 K at 1 atmosphere). The lowest pressure at which He-I and He-II can coexist is the vapor−He-I−He-II triple point at 2.1768 K (−270.9732 °C) and 5.0418 kPa (0.049759 atm), which is the “saturated vapor pressure” at that temperature (pure helium gas in thermal equilibrium over the liquid surface, in a hermetic container).^{[1]} The highest pressure at which He-I and He-II can coexist is the bcc−He-I−He-II triple point with a helium solid at 1.762 K (−271.388 °C), 29.725 atm (3,011.9 kPa).^{[2]}

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 letter lambda

${displaystyle lambda }$. 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 nK below the transition in an experiment included in a Space Shuttle payload in 1992.^{[3]}

Unsolved problem in physics:

Explain the discrepancy between the experimental and theoretical determinations of the heat capacity critical exponent *α* for the superfluid transition in helium-4.^{[4]}

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.^{[3]} The behavior of the heat capacity near the peak is described by the formula

where

${displaystyle t=|1-T/T_{c}|}$is the reduced temperature,

${displaystyle T_{c}}$is the Lambda point temperature,

${displaystyle A_{pm },B_{pm }}$ are constants (different above and below the transition temperature), and *α* is the critical exponent:

.^{[3]}^{[5]} Since this exponent is negative for the superfluid transition, specific heat remains finite.^{[6]}

The quoted experimental value of *α* is in a significant disagreement^{[7]}^{[4]} with the most precise theoretical determinations^{[8]}^{[9]}^{[10]} coming from high temperature expansion techniques, Monte Carlo methods and the conformal bootstrap.

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## References[edit]

**^**Donnelly, Russell J.; Barenghi, Carlo F. (1998). “The Observed Properties of Liquid Helium at the Saturated Vapor Pressure”.*Journal of Physical and Chemical Reference Data*.**27**(6): 1217–1274. Bibcode:1998JPCRD..27.1217D. doi:10.1063/1.556028.**^**Hoffer, J. K.; Gardner, W. R.; Waterfield, C. G.; Phillips, N. E. (April 1976). “Thermodynamic properties of^{4}He. II. The bcc phase and the P-T and VT phase diagrams below 2 K”.*Journal of Low Temperature Physics*.**23**(1): 63–102. Bibcode:1976JLTP…23…63H. doi:10.1007/BF00117245. S2CID 120473493.- ^
^{a}^{b}^{c}Lipa, J.A.; Swanson, D. R.; Nissen, J. A.; Chui, T. C. P.; Israelsson, U. E. (1996). “Heat Capacity and Thermal Relaxation of Bulk Helium very near the Lambda Point”.*Physical Review Letters*.**76**(6): 944–7. Bibcode:1996PhRvL..76..944L. doi:10.1103/PhysRevLett.76.944. hdl:2060/19950007794. PMID 10061591. S2CID 29876364. - ^
^{a}^{b}Rychkov, Slava (2020-01-31). “Conformal bootstrap and the λ-point specific heat experimental anomaly”.*Journal Club for Condensed Matter Physics*. doi:10.36471/JCCM_January_2020_02. **^**Lipa, J. A.; Nissen, J. A.; Stricker, D. A.; Swanson, D. R.; Chui, T. C. P. (2003-11-14). “Specific heat of liquid helium in zero gravity very near the lambda point”.*Physical Review B*.**68**(17): 174518. arXiv:cond-mat/0310163. Bibcode:2003PhRvB..68q4518L. doi:10.1103/PhysRevB.68.174518. S2CID 55646571.**^**For other phase transitions**^**Vicari, Ettore (2008-03-21). “Critical phenomena and renormalization-group flow of multi-parameter Phi4 theories”.*Proceedings of the XXV International Symposium on Lattice Field Theory — PoS(LATTICE 2007)*. Regensburg, Germany: Sissa Medialab.**42**: 023. doi:10.22323/1.042.0023.**^**Campostrini, Massimo; Hasenbusch, Martin; Pelissetto, Andrea; Vicari, Ettore (2006-10-06). “Theoretical estimates of the critical exponents of the superfluid transition in $^{4}mathrm{He}$ by lattice methods”.*Physical Review B*.**74**(14): 144506. arXiv:cond-mat/0605083. doi:10.1103/PhysRevB.74.144506. S2CID 118924734.**^**Hasenbusch, Martin (2019-12-26). “Monte Carlo study of an improved clock model in three dimensions”.*Physical Review B*.**100**(22): 224517. arXiv:1910.05916. Bibcode:2019PhRvB.100v4517H. doi:10.1103/PhysRevB.100.224517. ISSN 2469-9950. S2CID 204509042.**^**Chester, Shai M.; Landry, Walter; Liu, Junyu; Poland, David; Simmons-Duffin, David; Su, Ning; Vichi, Alessandro (2020). “Carving out OPE space and precise O(2) model critical exponents”.*Journal of High Energy Physics*.**2020**(6): 142. arXiv:1912.03324. Bibcode:2020JHEP…06..142C. doi:10.1007/JHEP06(2020)142. S2CID 208910721.

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