Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In...
-
Upload
lisa-jennings -
Category
Documents
-
view
215 -
download
2
Transcript of Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In...
![Page 1: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/1.jpg)
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
![Page 2: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/2.jpg)
The Superfluid Transition
T(P): line of second-order phase transitions
He-II
He-I
![Page 3: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/3.jpg)
Bulk Thermodynamic Properties
G.A., Phys. Rev. A 3, 696 (1971)
![Page 4: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/4.jpg)
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.
![Page 5: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/5.jpg)
HRT
Typical high-resolutionThermometer (HRT)
Resolution ~ 10-10 K at 2 K
Lipa, Chui, many others
![Page 6: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/6.jpg)
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).
![Page 7: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/7.jpg)
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
![Page 8: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/8.jpg)
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
![Page 9: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/9.jpg)
The Geometries
Q
LRadius L
Q
Confining geometries generateNEW UNIVERSALITY CLASSES
1-dimensional
2-dimensional IIQ
2-dimensional I
Characteristic Length Scale L
![Page 10: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/10.jpg)
Silicon wafer geometries
M.O. Kimball, K.P. Mooney, and F.M. Gasparini, preprint.
![Page 11: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/11.jpg)
Microchannel plates
Confinement Medium: Microchannel Plate
Diameter 1 to 100 m length 0.3 to 5 mm
![Page 12: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/12.jpg)
Rectangular Microchannel Plates
Hamamatsu, 5 X 50 m X 2 mm
![Page 13: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/13.jpg)
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)].
![Page 14: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/14.jpg)
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 !
![Page 15: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/15.jpg)
2-d finite size Cp
![Page 16: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/16.jpg)
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).
![Page 17: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/17.jpg)
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)
![Page 18: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/18.jpg)
1-d finite size Cp
Solid circles: T. Aouaroun + G.A., unpub., L = 1m
Needed: CHeX reflightwith cylindrical (D = 1)microchannel plates.
![Page 19: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/19.jpg)
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.
![Page 20: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/20.jpg)
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).
![Page 21: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/21.jpg)
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?
![Page 22: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/22.jpg)
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
![Page 23: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/23.jpg)
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
![Page 24: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/24.jpg)
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
![Page 25: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/25.jpg)
F(0) vs. P
Is F(0)“Universal” ?
2 %
![Page 26: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/26.jpg)
Results
F(0) is independent of P
F(-4) is notIs not universal !
![Page 27: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/27.jpg)
L- x/
Topler and Dohm
At T agreement with theory is excellent.
(t = 0)
![Page 28: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/28.jpg)
Gravity Effect
![Page 29: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/29.jpg)
Gravity Effect
![Page 30: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/30.jpg)
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.
![Page 31: Scaling and Universality near the Superfluid Transition of 4 He in Restricted Geometries In collaboration with Edgar Genio, Daniel Murphy, and Tahar Aouaroun.](https://reader035.fdocuments.us/reader035/viewer/2022081516/5697bf9c1a28abf838c930a6/html5/thumbnails/31.jpg)
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