Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries

1. Scaling and Universality near the Superfluid Transition of 4He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun Department of Physics and iQUEST, University of California, Santa Barbara Feng-Chuan Liu and Yuan-Ming Liu Jet Propulsion Laboratory, Pasadena, CA Guenter Ahlers, UC Santa Barbara Supported by NASA Grant NAG8-1429 Gordon Conference on Gravitational Effects in Physico-Chemical Systems, Connecticut College, New London, CT, July 31 2003

2. The Superfluid Transition Tl(P): line of second-order phase transitions He-I He-II l

3. Bulk Thermodynamic Properties G.A., Phys. Rev. A 3, 696 (1971)

4. Bulk Thermodynamic Properties, LPE LPE: “Lambda Point Experiment”, Oct. 1992, USMP1 on Columbia Lipa et al., Phys. Rev. Lett. 76, 944 (1996); and Phys. Rev. B, in print.

5. HRT Typical high-resolution Thermometer (HRT) Resolution ~ 10-10 K at 2 K Lipa, Chui, many others

6. Bulk Transport Properties Thermal conductivity l diverges at Tl, depends on P W.-Y. Tam and G.A. Phys. Rev. B 32, 3519 (1985).

7. Finite Size Effects • Confinement introduces additional length L • x cannot grow without bound Static properties: Some Theory and Experiment Transport properties: Very little Theory or Experiment Nho and Manousakis, Phys. Rev. B 64, 144513 (2001) (Monte Carlo) Topler and Dohm, Physica B, in print (RG) Kahn+A. [PRL 74, 944 (1995)] measured thermal conductivity l near Tl at SVP in a 1-dim. geometry for one L.

8. Finite Size Effects • 1.) Need a wide range of L to test scaling. • 2.) Need, e.g., a range of pressures to test universality. • Assume the existence of a universal scaling • function F(L/ x) • Depends on geometry and boundary conditions, i.e. there are several Universality Classes

9. Q The Geometries Confining geometries generate NEW UNIVERSALITY CLASSES Radius L L 2-dimensional I Q 2-dimensional II 1-dimensional Q Characteristic Length Scale L

10. Silicon wafer geometries M.O. Kimball, K.P. Mooney, and F.M. Gasparini, preprint.

11. Microchannel plates Confinement Medium: Microchannel Plate Diameter 1 to 100 mm length 0.3 to 5 mm

12. Rectangular Microchannel Plates Hamamatsu, 5 X 50 mm X 2 mm

13. 2-d finite size Cp 57 mm: CHeX (Columbia, 1997). Lipa et al., J. Low Temp. Phys. 113, 849 (1998); Phys. Rev. Lett. 84, 4894 (2000). Others: Gasparini group [Mehta, Kimball, and Gasparini, J. Low Temp. Phys. 114, 467 (1999); Kimball, Mehta, and Gasparini, J. Low Phys. 121, 29 (2000)].

14. 2-d finite size Cp 4He Heat Capacity Scaling relation: (57/0.21)1/n = 4500 ! 57 mm data from the CHeX flight experiment, Lipa et al., PRL 84, 4894 (2000). 0.211 mm from Mehta and Gasparini, PRL 78, 2596 (1997). 2-dimensional

15. 2-d finite size Cp

16. 2-d finite size CpCHeX f_2 CHeX: Lipa et al., Phys. Rev. Lett. 84, 4894 (2000). RGT: Dohm group [Schmolke et al., Physica 165B&166B, 575 (1990); Mohr and Dohm, Proc. LT22 (2000)].

17. 1-d finite size Cp Need CHeX II (re-flight with Cylindrical geometry) 0.26 mm (Anopore) FC Monte Carlo All is not well !!! 8 mm (m-channel plate) X J. Lipa, M. Coleman, and D.A. Stricker, J. Low Temp. Phys. 124, 443 (2001).

18. 1-d finite size Cp Needed: CHeX reflight with cylindrical (D = 1) microchannel plates. Solid circles: T. Aouaroun + G.A., unpub., L = 1mm

19. Conclusion: D = 2: Scaling works remarkably well from just below the maximum of Cp up to large T. Further below the maximum there are problems. Surface specific heat agrees quantitatively with calculations above the transition, but is larger than the theory by a factor of 3 below the transition. D = 1: The surface specific heat agrees with the D = 2 measurements, i.e. it agrees with theory above and disagrees by a factor of 3 below the transition. Scaling seems to break down near the transition.

20. Finite-Size Thermal Conductivity D = 2 mm 105 / l ( s cm K / erg ) 106 t A. Kahn + G.A., Phys. Rev. Lett. 74, 944 (1995).

21. BEST Project • BESTBoundary Effects on the Superfluid Transition • Test dynamic finite-size scaling and universality using the thermal resistivity r of 4He near Tl • Scaling • Measure r as a function of L • Is there a scaling function for r? • Universality • Measure r as a function of pressure • Is the scaling function independent of P?

22. Scaling Function To derive scaling function, write the bulk conductivity as a power law and the finite size effect as a function of L/x : Scaling function F in terms of X

23. SVP Results Data at different lengths scale L = 1mm L = 2mm Does not scale ! D. Murphy, E. Genio, G.A., F. Liu, and Y. Liu, Phys. Rev. Lett. (2003).

24. P-dependence Data at different P have same scaling function F 28 bars 0.05 bars L = 1mm

25. F(0) vs. P Is F(0) “Universal” ? 2 %

26. Results F(0) is independent of P F(-4) is not Is not universal !

27. l (t = 0) L- x/n Topler and Dohm At Tl agreement with theory is excellent.

28. Gravity Effect

29. Gravity Effect

30. Conclusions Within experimental resolution, Data at different sizes and SVP scale above Tl but not below Data at different pressures have the same scaling function above Tl but not below At Tl agreement with theory is excellent. Measurements for larger L are needed to provide a more stringent test of the theory, but require micro-gravity.

31. Future Ground Projects 1.) Take data as function of P at different L 2.) Study region below Tl in more detail 3.) Measurements on rectangular geometry