# Lambda point – Wikipedia

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 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

${displaystyle Capprox A_{pm }t^{-alpha }+B_{pm }}$

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:

${displaystyle alpha =-0.0127(3)}$

.[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.

## References

1. ^ 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.
2. ^ Hoffer, J. K.; Gardner, W. R.; Waterfield, C. G.; Phillips, N. E. (April 1976). “Thermodynamic properties of 4He. 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.
3. ^ 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.
4. ^ 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.
5. ^ 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.
6. ^ For other phase transitions
${displaystyle alpha }$

may be negative (e.g.

${displaystyle alpha approx +0.1}$

for the liquid-vapor critical point which has Ising critical exponents). For those phase transitions specific heat does tend to infinity.

7. ^ 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.
8. ^ 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.
9. ^ 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.
10. ^ 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.