Scaling and Universality near the Superfluid Transition of 4 He 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 Supported by NASA Grant NAG Guenter Ahlers, UC Santa Barbara Gordon Conference on Gravitational Effects in Physico-Chemical Systems, Connecticut College, New London, CT, July

The Superfluid Transition T (P): line of second-order phase transitions He-II He-I

t1t1 t4t4 t2t2 t2t2 t3t3 t4t4 Q 5 mm t3t3 I II t i = ( T i - T 4,  T

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

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.

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

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

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

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/  ) Depends on geometry and boundary conditions, i.e. there are several Universality Classes

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

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

Microchannel plates Confinement Medium: Microchannel Plate Diameter 1 to 100  m length 0.3 to 5 mm

Rectangular Microchannel Plates Hamamatsu, 5 X 50  m X 2 mm

2-d finite size C p 57  m: 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)].

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

2-d finite size C p

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

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

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

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.

Finite-Size Thermal Conductivity 10 6 t 10 5 / ( s cm K / erg ) D = 2  m A. Kahn + G.A., Phys. Rev. Lett. 74, 944 (1995).

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

Scaling Function To derive scaling function, write the bulk conductivity as a power law and the finite size effect as a function of L  Scaling function F in terms of X

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

P-dependence Data at different P have same scaling function F L = 1  m 0.05 bars 28 bars

F(0) vs. P Is F(0) “Universal” ? 2 %

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

L - x/ Topler and Dohm At T agreement with theory is excellent.  (t = 0)

Gravity Effect

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

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

Detailed Scaling Function