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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