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

Geometries

In collaboration withEdgar Genio, Daniel Murphy, and Tahar Aouaroun

Department of Physics and iQUEST,University of California, Santa Barbara

Feng-Chuan Liu and Yuan-Ming LiuJet Propulsion Laboratory, Pasadena, CA

Supported by NASA Grant NAG8-1429

Guenter Ahlers, UC Santa Barbara

Gordon Conference on Gravitational Effects in Physico-Chemical Systems, Connecticut College, New London, CT, July 31 2003

The Superfluid Transition

T(P): line of second-order phase transitions

He-II

He-I

Bulk Thermodynamic Properties

G.A., Phys. Rev. A 3, 696 (1971)

Bulk Thermodynamic Properties, LPE

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

HRT

Typical high-resolutionThermometer (HRT)

Resolution ~ 10-10 K at 2 K

Lipa, Chui, many others

Bulk Transport Properties

2

3

4

567

0.001

2

3

10-5

10-4

10-3

10-2

10-1

100

t

SVP 28 bar

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, 144513 (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 severalUniversality Classes

The Geometries

Q

LRadius L

Q

Confining geometries generateNEW UNIVERSALITY CLASSES

1-dimensional

2-dimensional IIQ

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 Cp

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 Cp

Scaling relation:

€

FC = Lξ 0

( )−α /ν

CP ,0(t,L) − CP ,0(t0,∞)[ ]

t0 = ξ 0 /L( )1/ν

X = Lξ 0

( )1/ν

t

-30

-20

-10

0

10

40200-20

X

57 m 0.211 m

57 m data from the CHeX flight experiment, Lipa et al., PRL 84, 4894 (2000).0.211 m from Mehta and Gasparini, PRL 78, 2596 (1997).

4He Heat Capacity

2-dimensional

(57/0.21)1/ = 4500 !

2-d finite size Cp

2-d finite size Cp 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 Cp

€

FC = Lξ 0

( )−α /ν

CP ,0(t,L) − CP ,0(t0,∞)[ ]

t0 = ξ 0 /L( )1/ν

X = Lξ 0

( )1/ν

t

J. Lipa, M. Coleman, and D.A. Stricker, J. Low Temp. Phys. 124, 443 (2001).

8 m(channelplate)

0.26 m (Anopore)

All is notwell !!!

Monte CarloF C

X

Need CHeX II (re-flight withCylindrical geometry)

1-d finite size Cp

Solid circles: T. Aouaroun + G.A., unpub., L = 1m

Needed: CHeX reflightwith 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

106 t

10

5 /

( s

cm K

/ e

rg ) 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 4He near T

• ScalingMeasure as a function of LIs there a scaling function for ?

• UniversalityMeasure as a function of pressureIs the scaling function independent of P?

Scaling Function

€

(t,L) = ρ(t) ˜ F Lξ( ) = ρ 0t

x ˜ F Lξ( )

F = Lξ( )

x /ν ˜ F Lξ( )

€

F(X) = Lξ 0

( )x /ν

ρ(t,L) /ρ 0[ ]

X = Lξ( )

1/ν= t / t0

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

€

F = Lξ 0

( )x /ν

ρ (t,L) /ρ 0[ ]

X = Lξ 0

( )1/ν

t

Data at different lengths scale

Does not scale !

D. Murphy, E. Genio, G.A., F. Liu, and Y. Liu, Phys. Rev. Lett. (2003).

L = 1m

L = 2m

P-dependence

€

F = Lξ 0

( )x /ν

ρ (t,L) /ρ 0[ ]

X = Lξ 0

( )1/ν

t

Data at different P have same scaling function F

L = 1m

0.05 bars

28 bars

F(0) vs. P

Is F(0)“Universal” ?

2 %

Results

F(0) is independent of P

F(-4) is notIs not universal !

L- x/

Topler and Dohm

At T agreement with theory is excellent.

(t = 0)

Gravity Effect

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