The critical current Jc of superfluid 3He-A in 0.8-μm-diam channels has been measured by the observation of the pressure difference along the channels versus the mass current. During warming Jc was found to decrease by about 30% at TBA(cyl) and by another 30% at TBA; TBA(cyl) is the reduced B→A transition temperature in the narrow flow channels, with TBA(cyl)TBA=0.92 at 27.4 bars. Above TBA a second dissipative mechanism was observed at lower currents. These features are believed to be associated with the ends of the channels.Peer reviewe

Manninen, M. T.

Pekola, Jukka P.

Sharma, R. G.

Tagirov, M. S.

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This is an electronic reprint of the original article.
This reprint may differ from the original in pagination and typographic detail.
Author(s): Manninen, M. T. & Pekola, Jukka & Sharma, R. G. & Tagirov, M. S.
Title: Critical current of 3He-A in narrow channels
Year: 1982
Version: Final published version
Please cite the original version:
Manninen, M. T. & Pekola, Jukka & Sharma, R. G. & Tagirov, M. S. 1982. Critical current
of 3He-A in narrow channels. Physical Review B. Volume 26, Issue 9. 5233-5236. ISSN
1098-0121 (printed). DOI: 10.1103/physrevb.26.5233.
Rights: © 1982 American Physical Society (APS). http://www.aps.org/
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duplication or sale of all or part of any of the repository collections is not permitted, except that material may
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PHYSICAL REVIEW B VOLUME 26, NUMBER 9 1 NOVEMBER 1982
Critical current of 3He-A in narrow channels
M. T. Manninen, J. P. Pekola, R. G. Sharma, * and M. S. Tagirov
Low Temperature Laboratory, Helsinki University of Technology, SF-02150 Espoo 15, Finland
(Received 23 April 1982; revised manuscript received 5 August 1982)
The critical current J, of superfluid He-A in 0.8-pm-diam channels has been measured by the
observation of the pressure difference along the channels versus the mass current. During
warming J, was found to decrease by about 30'lo at T~z(cyl) and by another 30'lo at T~z,
T~q(cyl) is the reduced B A transition temperature in the narrow flow channels, with
T~~(cyl) I T~z =0.92 at 27.4 bars. Above T~z a second dissipative mechanism was observed at
lower currents. These features are believed to be associated with the ends of the channels.
The A phase of superfluid 'He, because of its an-
isotropic nature, is expected to possess unusual flow
properties. For example, the critical velocity v, in a
narrow flow channel should depend strongly on the
texture formed by the l vector. ' Measurements of
e, in channels with diameter large compared to the
coherence lengh, (0=0.076 pm at P = 0, have yield-
ed values between 0.1 and 0.8 mm/s. ' ' Even in a
channel as small as 18 p,m v, was not more than 2
mm/s. All these values are less than either the ob-
served v, in the 8 phase under similar conditions or
the predicted depairing critical velocity. '8 The
mechanisms responsible for the dissipation at these
low velocities are not well understood.
In this work we have measured the critical mass
current J, of 3He-A through 0.8-p,m-diam and 10-
p, m-long channels. Such a geometry should be suit-
able for observing the depairing current, because oth-
er dissipative mechanisms, like vortex movement,
become ineffective when the channel diameter ap-
proaches the coherence length. The I vector should
also be effectively fixed by the boundary condition at
the walls.
Our results show a number of new features not ob-
served in experiments with larger flow channels. In
particular, suppression of the 8 A transition in the
narrow channels permits us to measure J, with either
A- or 8-phase liquid outside the channels. The ob-
served behavior of J, at the 8 A phase transition
temperatures, Tsz in bulk liquid and Tsq (cyl) in the
channels, suggests that the ends of the channels have
a profound effect on the flow. Since we measure the
pressure difference as a function of the current, we
also obtain quantitative information about dissipation
above J,.
The experimental cell (cf. Fig. I in Ref. 9) consists
of two 'He compartments, coupled through a super-
leak, which was made of Nuclepore' filter; the total
cross-sectional area of the flow channels is 0.015
mm . An aluminized Mylar diaphragm forms a flexi-
ble wall between the two compartments and was
used, with capacitor plates on both sides, to induce a
flow through the channels and to detect the pressure
difference hP. Temperatures were measured with
two CLMN (cerium magnesium nitrate diluted to 3'/0
molar solution by the corresponding lanthanum salt)
thermometers, one on each side of the cell and both
calibrated against tabulated values of T, versus pres-
sure. " T, and T~~ were detected as a change in the
slope and as a plateau region in the temperature
versus time curve, respectively. All our measure-
ments were performed in zero external magnetic
field.
In our flow experiments a drive voltage U was ap-
plied on one side of the capacitor, such that the force
on the Mylar diaphragm, proportional to U, varied
linearly with time. Simultaneously, the response of
the capacitor hC on the other side was monitored.
The mass current was varied by employing different
sweep rates of U . An average pressure difference
AP,„was determined by integrating the instantaneous
hP, obtained from the recorded U' and b C as func-
tions of time, and an averaged supercurrent J, was
found from the slope of the AC versus time curve.
The uncertainty in converting the measured quanti-
ties to pressure and current is about 30'lo. The criti-
cal current was determined from a plot of AP,„vs J,
by extrapolating J, to AP,„=O. More details about
our experimental techniques have been reported else-
where. 9
Considerable changes in the phase diagram of 'He
are anticipated when the fluid is confined in a re-
stricted geometry, ' '" e.g. , in a channel with its ra-
dius R comparable to (0. In our channels, with
R /g, =23 at the highest pressure P = 27.4 bars, we
expect a 0.7'lo reduction in T, at most. " The 8
transition, however, should be suppressed much
more owing to the different surface energies in the A
and 8 phases. ' In our geometry, with relatively long
channels and free liquid on both sides, we found that
J, depends on whether the bulk liquid is in the A or
in the 8 phase. This is clearly seen from Fig. 1,
26 5233 1982 The American Physical Society
5234 BRIEF REPORTS 26
7 ~
CO
Ol
E
0)
Ol
u 5
l-
I
T
Vl
O
~ gO P
/ fp
h es+~~»
jt
+11
,
T„,TB„(cyl
I I
0. 1 0.2
I
0.3 0.4
20 — ~ o
el N
cs p
t 0 0
%10
~CO
Cl b
0 ' I
0
I
0.05
(kg/m's)S
1
C7
CO
CO
O O
O CK
0.10
1-T/Tc
FIG. 1. The normalized average current J„at P =27,4
bars corresponding to hP,„=—16.7 p, bar, plotted as a func-
tion of the reduced temperature. The B A transition in
bulk liquid (at T&&) and inside the flow channels fat
T&z (cyl)] are indicated by arrows.
where the average current J„,normalized with the
expected temperature dependence (1 —T/T, ) 3 ~ for
the depairing current, has been plotted as a function
of reduced temperature. J„,produced by a voltage
step U = 100 V applied on the driving capacitor, is
about 30% larger than the extrapolated J„but it
shows the general behavior of the critical current.
Three distinct regimes are evident from Fig. 1. Far
below T, the bulk liquid as well as He in the channels
are in the 8 phase. Upon warming the 8 3 transi-
tion occurs first in the channels, at 1 —Ts„(cyl)/
T, =0.18. The observed broadening of this transi-
tion is probably caused by the distribution of the
channel diameters; we found that 80% of the holes
have a diameter between 0.8 + 0.2 p, m. Finally,
the B A transition in the bulk liquid is seen as a
drop of J„ to a lower value at 1 —Tsq/T, =0.109.
Temperature hysteresis due to supercooling is charac-
teristic to both transitions.
Qualitatively similar behavior was found at P =22.7
bars, with 1 —Tsq (cyl)/T, = 0.11 and 1 —Tsq/T, = 0.021.
The observed reductions in the 8 A transition
temperatures are comparable in magnitude to those
reported by Ahonen, Krusius, and Paalanen' in a
4-p,m parallel-plate geometry; Saunders et al. ' have
found 1 —Ts~ (cyl)/T, ) 0.5 in 2-p, m-diam channels.
The reduction of T» in our experiments may be in-
fluenced by the relatively short length of our chan-
nels, compared with earlier static measurements, and
by the B liquid flowing into the channels.
Figure 2 illustrates the dependence of AP„on J„
measured at P = 24.6 bars. Below T~& the behavior
is similar to that observed in the B phase at lower
pressures. Above T~~, however, the onset of dissi-
pation is markedly different. AP,„increases first with
FIG, 2. The average pressure difference AP,„during the
flow at P =24.6 bars, plotted as a function of the mass
current J, ; the reduced temperatures for each set of points
are shown. 1 —Tsz/T, =0.064; because of supercoollns, the
points at 1 —T/T, =0.0643 have been measured, however,
with the bulk liquid outside the channe1s in the A phase.
The straight lines illustrate extrapolations to AP«=0 for
determining J,.
103
10
10
0.0 1 0.02
1-T/Tc
L
+a+4+
0.05 0.1
FIG. 3. The slope dhP, „/dJ, vs temperature for P =24.6
bars. Circles represent the steeper slope in Fig. 2; filled cir-
cles correspond to T & Tz&. The smaller slope in Fig. 2 is
drawn with crosses. The solid lines have the slope of
(1—T/T )3~~
a relatively small slope and then faster with a slope
similar to that observed below T~~. We attribute
these two distinct slopes to two different dissipative
mechanisms which will be discussed below. Two crit-
ical currents are thus obtained from these two slopes
by extrapolating the values of AP,„to zero.
The values of dhP„/dJ, are shown in Fig. 3 as
functions of temperature. The steeper slopes taken
above and below T~q lie on the same straight line
(upper curve), suggesting that this slope in the A-
phase region is due to the same dissipative mechan-
ism which operates below T». With large enough
BRIEF REPORTS 5235
hP, the current saturates to a value which is about
40% larger than the extrapolated J,.
The extrapolated critical current J, is shown as a
function of temperature in Fig. 4 at three different
pressures. The uppermost points, joined by a curve,
correspond to Ts& (cyl) ( T ( Ts&, in this region J,
is approximately proportional to (1 —T/T, )'~' . The
depairing current of the bulk A phase in the Ginz-
burg-Landau region is given by 1.86(IC~/ACscs)
x (pksT, /pq)(1 —T/T, )' '. ' Using the measured
specific-heat jump" this 'equals 13.8(l —T/T, )' '
kg/m s at P = 27.4 bars, which is about four times
larger than our experimental J,. Above T» the mag-
nitude of J, is about 30% smaller (the middle set of
points in Fig. 4) than below Ts„and the temperature
dependence is slightly weaker.
The extrapolated current corresponding to the
smaller slope in Fig. 2, as shown in Fig. 4, displays a
similar behavior. Near T, both currents tend to bend
down owing to a reduction in T,; they extrapolate to
zero at T/T, =0.994, in agreement with the theoreti-
cally expected decrease in T, without flow. ' The
change in J, at T~& and the existence of the second
critical current must be caused by effects associated
with the ends of the channels because the phase tran-
sition occurs only in the bulk liquid. The small value
of J, below T~& may be due to the A-8 interface
present in this region. Figure 4 indicates that above
Tsq(cyl) J, is independent of pressure within the
scatter of our data; below Tsq(cyl) J, changes by 10%
between 22.7 and 27.4 bars.
The effect of the ends of a cylindrical channel on
the superflow has been considered by Thuneberg
and Kurkijarvi. In particular, they show that the
critical current through a long channel containing a
disgyration texture vanishes, irrespective of the direc-
tion of the I vectors at the two ends of the channel.
With R =0.4 p, m, a disgyration is energetically the
most favorable texture. " Further, in this case an ap-
plied pressure induces a current, which is determined
by friction between the normal component and the I
texture. ' This current is given by J, = (m/k')'
x (pL ff/p)b P, where p, is the Cross-Anderson
viscosity coefficient' and L,ff =2R is the effective
length of the end zones. After inserting typical
values we obtain d hP/dJ, = 0.4(1 —T/T, ) ' ' m/s
which is close to our result in Fig. 3 (lower curve).
The nonzero J, can be qualitatively understood as
arising from flow channels of noncylindrical sym-
metry, i.e., from channels having two or more holes
touching each other. This model requires that the ra-
0.3
0.1
Ol
0.03
%at
0.01
~8
0.003
I
pO ~
0
~ 0
~
~
A
0.01 0.02 005
1-T/T
C
0.10 0.20
FIG. 4. The extrapolated critical current J, as a function
of temperature. Diamonds correspond to P =22.7 bars, cir-
cles to 24.6 bars, and triangles to 27.4 bars. Filled symbols
represent extrapolations of the smaller slope above T&z.
The solid line has the slope of (1 —T/T, )
tio of the critical currents corresponding to the
steeper and smaller slopes is constant, depending
only on the distribution of channels; this is not in-
consistent with our data.
In conclusion, the critical mass current of 3He-A in
narrow channels differs significantly from that mea-
sured in a more open geometry. At present there ex-
ists no theoretical calculations for the depairing
current in a restricted geometry; in the bulk liquid
the predicted value exceeds our measured J, by a fac-
tor of 4. The results show that J, and the nature of
dissipation depend on the phase, A or 8, of the bulk
liquid outside the flow channels. This suggests that
dissipation is restricted to the end zones of the chan-
nels.
ACKNOWLEDGMENTS
We are indebted to S. A. Nenonen for help during
the measurements and wish to acknowledge useful
discussions with O. V. Lounasmaa, H. Smith, and E.
V. Thuneberg. This work was financially supported
by the Academy of Finland and by the Emil Aaltonen
Foundation.
5236 BRIEF REPORTS 26
Permanent address: National Physical Laboratory, New
Delhi, India.
~Permanent address: Physics Department, Kazan State
University, Kazan 420008, USSR.
'V. Ambegaokar, P. G. deGennes, and D. Rainer, Phys.
Rev. A 9, 2676 (1974).
E. V. Thuneberg and J; Kurkijirvi, Phys. Lett. 86A, 35
(1981); (unpublished).
3M. A. Paalanen and D. D. Osheroff, Phys. Rev. Lett, 45,
362 (1980).
4E. B. Flint, R. M. Mueller, and E. D. Adams, J. Low
Temp. Phys. 33, 43 (1978).
5R. T. Johnson, R. L'. Kleinberg, R. A. Webb, and J. C.
Wheatley, J. Low Temp. Phys. 18, 501 (1975).
J. M. Parpia and J. D. Reppy, Phys. Rev. Lett. 43, 1332
(1979).
H. Kleinert, J. Low Temp. Phys. 39, 451 (1980); D.
Vollhardt, K. Maki, and N. Schopohl, J. Low Temp. Phys.
39, 79 (1980).
sA. L. Fetter, itt Quantum Statistics and the Many Body Prob
lems, edited by S. B. Trickey, %. P. Kirk, and J. W. Dufty
(Plenum, New York, 1976), p. 127.
9M. T. Manninen and J. P. Pekola, Phys. Rev. Lett. 48, 812
(1982); 48, 1369 (E} (1982); M. T. Manninen, Ph. D.
thesis (Helsinki University of Technology, 1982) (unpub-
lished).
' Nuclepore Corporation, Pleasanton, Calif. 94566.
~'T. A. Alvesalo, T. Haavasoja, M, T, Manninen, and A. T.
Soinne, Phys, Rev. Lett. 44, 1076 (1980).
' G. Barton and M. A, Moore, J. Low Temp. Phys. 21, 489
(1975).
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Temp. Phys. 33, 577 (1978).
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~ A. I. Ahonen, M. Krusius, and M. A. Paalanen, J. Low
Temp. Phys. 25, 421 (1976}.
J. Saunders, D. S. Betts, D. F. Brewer, S. J. Swithenby,
snd W. S. Truscott, Phys. Rev. Lett. 40, 1278 (1978).
' L. J. Bucholtz and A. L. Fetter, Phys. Lett. 58A, 93
(1976}.
M. C. Cross and P. W. Anderson, in Proceedings of the
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Amsterdam, 1975), Vol. 1, p. 29.


English

Critical current of 3He-A in narrow channels

Aaltodoc Publication Archive

This is an electronic reprint of the original article.This reprint may differ from the original in pagination and typographic detail.Author(s): Manninen, M. T. & Pekola, Jukka & Sharma, R. G. & Tagirov, M. S.Title: Critical current of 3He-A in narrow channelsYear: 1982Version: Final published versionPlease cite the original version:Manninen, M. T. & Pekola, Jukka & Sharma, R. G. & Tagirov, M. S. 1982. Critical currentof 3He-A in narrow channels. Physical Review B. Volume 26, Issue 9. 5233-5236. ISSN1098-0121 (printed). DOI: 10.1103/physrevb.26.5233.Rights: © 1982 American Physical Society (APS). http://www.aps.org/All material supplied via Aaltodoc is protected by copyright and other intellectual property rights, andduplication or sale of all or part of any of the repository collections is not permitted, except that material maybe duplicated by you for your research use or educational purposes in electronic or print form. You mustobtain permission for any other use. Electronic or print copies may not be offered, whether for sale orotherwise to anyone who is not an authorised user.Powered by TCPDF (www.tcpdf.org)PHYSICAL REVIEW B VOLUME 26, NUMBER 9 1 NOVEMBER 1982Critical current of 3He-A in narrow channelsM. T. Manninen, J. P. Pekola, R. G. Sharma, * and M. S. TagirovLow Temperature Laboratory, Helsinki University of Technology, SF-02150 Espoo 15, Finland(Received 23 April 1982; revised manuscript received 5 August 1982)The critical current J, of superfluid He-A in 0.8-pm-diam channels has been measured by theobservation of the pressure difference along the channels versus the mass current. Duringwarming J, was found to decrease by about 30'lo at T~z(cyl) and by another 30'lo at T~z,T~q(cyl) is the reduced B A transition temperature in the narrow flow channels, withT~~(cyl) I T~z =0.92 at 27.4 bars. Above T~z a second dissipative mechanism was observed atlower currents. These features are believed to be associated with the ends of the channels.The A phase of superfluid 'He, because of its an-isotropic nature, is expected to possess unusual flowproperties. For example, the critical velocity v, in anarrow flow channel should depend strongly on thetexture formed by the l vector. ' Measurements ofe, in channels with diameter large compared to thecoherence lengh, (0=0.076 pm at P = 0, have yield-ed values between 0.1 and 0.8 mm/s. ' ' Even in achannel as small as 18 p,m v, was not more than 2mm/s. All these values are less than either the ob-served v, in the 8 phase under similar conditions orthe predicted depairing critical velocity. '8 Themechanisms responsible for the dissipation at theselow velocities are not well understood.In this work we have measured the critical masscurrent J, of 3He-A through 0.8-p,m-diam and 10-p, m-long channels. Such a geometry should be suit-able for observing the depairing current, because oth-er dissipative mechanisms, like vortex movement,become ineffective when the channel diameter ap-proaches the coherence length. The I vector shouldalso be effectively fixed by the boundary condition atthe walls.Our results show a number of new features not ob-served in experiments with larger flow channels. Inparticular, suppression of the 8 A transition in thenarrow channels permits us to measure J, with eitherA- or 8-phase liquid outside the channels. The ob-served behavior of J, at the 8 A phase transitiontemperatures, Tsz in bulk liquid and Tsq (cyl) in thechannels, suggests that the ends of the channels havea profound effect on the flow. Since we measure thepressure difference as a function of the current, wealso obtain quantitative information about dissipationabove J,.The experimental cell (cf. Fig. I in Ref. 9) consistsof two 'He compartments, coupled through a super-leak, which was made of Nuclepore' filter; the totalcross-sectional area of the flow channels is 0.015mm . An aluminized Mylar diaphragm forms a flexi-ble wall between the two compartments and wasused, with capacitor plates on both sides, to induce aflow through the channels and to detect the pressuredifference hP. Temperatures were measured withtwo CLMN (cerium magnesium nitrate diluted to 3'/0molar solution by the corresponding lanthanum salt)thermometers, one on each side of the cell and bothcalibrated against tabulated values of T, versus pres-sure. " T, and T~~ were detected as a change in theslope and as a plateau region in the temperatureversus time curve, respectively. All our measure-ments were performed in zero external magneticfield.In our flow experiments a drive voltage U was ap-plied on one side of the capacitor, such that the forceon the Mylar diaphragm, proportional to U, variedlinearly with time. Simultaneously, the response ofthe capacitor hC on the other side was monitored.The mass current was varied by employing differentsweep rates of U . An average pressure differenceAP,„was determined by integrating the instantaneoushP, obtained from the recorded U' and b C as func-tions of time, and an averaged supercurrent J, wasfound from the slope of the AC versus time curve.The uncertainty in converting the measured quanti-ties to pressure and current is about 30'lo. The criti-cal current was determined from a plot of AP,„vs J,by extrapolating J, to AP,„=O. More details aboutour experimental techniques have been reported else-where. 9Considerable changes in the phase diagram of 'Heare anticipated when the fluid is confined in a re-stricted geometry, ' '" e.g. , in a channel with its ra-dius R comparable to (0. In our channels, withR /g, =23 at the highest pressure P = 27.4 bars, weexpect a 0.7'lo reduction in T, at most. " The 8transition, however, should be suppressed muchmore owing to the different surface energies in the Aand 8 phases. ' In our geometry, with relatively longchannels and free liquid on both sides, we found thatJ, depends on whether the bulk liquid is in the A orin the 8 phase. This is clearly seen from Fig. 1,26 5233 1982 The American Physical Society5234 BRIEF REPORTS 267 ~COOlE0)Olu 5l-ITVlO~ gO P/ fph es+~~»jt+11,T„,TB„(cylI I0. 1 0.2I0.3 0.420 — ~ oel Ncs pt 0 0%10~COCl b0 ' I0I0.05(kg/m's)S1C7COCOO OO CK0.101-T/TcFIG. 1. The normalized average current J„at P =27,4bars corresponding to hP,„=—16.7 p, bar, plotted as a func-tion of the reduced temperature. The B A transition inbulk liquid (at T&&) and inside the flow channels fatT&z (cyl)] are indicated by arrows.where the average current J„,normalized with theexpected temperature dependence (1 —T/T, ) 3 ~ forthe depairing current, has been plotted as a functionof reduced temperature. J„,produced by a voltagestep U = 100 V applied on the driving capacitor, isabout 30% larger than the extrapolated J„but itshows the general behavior of the critical current.Three distinct regimes are evident from Fig. 1. Farbelow T, the bulk liquid as well as He in the channelsare in the 8 phase. Upon warming the 8 3 transi-tion occurs first in the channels, at 1 —Ts„(cyl)/T, =0.18. The observed broadening of this transi-tion is probably caused by the distribution of thechannel diameters; we found that 80% of the holeshave a diameter between 0.8 + 0.2 p, m. Finally,the B A transition in the bulk liquid is seen as adrop of J„ to a lower value at 1 —Tsq/T, =0.109.Temperature hysteresis due to supercooling is charac-teristic to both transitions.Qualitatively similar behavior was found at P =22.7bars, with 1 —Tsq (cyl)/T, = 0.11 and 1 —Tsq/T, = 0.021.The observed reductions in the 8 A transitiontemperatures are comparable in magnitude to thosereported by Ahonen, Krusius, and Paalanen' in a4-p,m parallel-plate geometry; Saunders et al. ' havefound 1 —Ts~ (cyl)/T, ) 0.5 in 2-p, m-diam channels.The reduction of T» in our experiments may be in-fluenced by the relatively short length of our chan-nels, compared with earlier static measurements, andby the B liquid flowing into the channels.Figure 2 illustrates the dependence of AP„on J„measured at P = 24.6 bars. Below T~& the behavioris similar to that observed in the B phase at lowerpressures. Above T~~, however, the onset of dissi-pation is markedly different. AP,„increases first withFIG, 2. The average pressure difference AP,„during theflow at P =24.6 bars, plotted as a function of the masscurrent J, ; the reduced temperatures for each set of pointsare shown. 1 —Tsz/T, =0.064; because of supercoollns, thepoints at 1 —T/T, =0.0643 have been measured, however,with the bulk liquid outside the channe1s in the A phase.The straight lines illustrate extrapolations to AP«=0 fordetermining J,.10310100.0 1 0.021-T/TcL+a+4+0.05 0.1FIG. 3. The slope dhP, „/dJ, vs temperature for P =24.6bars. Circles represent the steeper slope in Fig. 2; filled cir-cles correspond to T & Tz&. The smaller slope in Fig. 2 isdrawn with crosses. The solid lines have the slope of(1—T/T )3~~a relatively small slope and then faster with a slopesimilar to that observed below T~~. We attributethese two distinct slopes to two different dissipativemechanisms which will be discussed below. Two crit-ical currents are thus obtained from these two slopesby extrapolating the values of AP,„to zero.The values of dhP„/dJ, are shown in Fig. 3 asfunctions of temperature. The steeper slopes takenabove and below T~q lie on the same straight line(upper curve), suggesting that this slope in the A-phase region is due to the same dissipative mechan-ism which operates below T». With large enoughBRIEF REPORTS 5235hP, the current saturates to a value which is about40% larger than the extrapolated J,.The extrapolated critical current J, is shown as afunction of temperature in Fig. 4 at three differentpressures. The uppermost points, joined by a curve,correspond to Ts& (cyl) ( T ( Ts&, in this region J,is approximately proportional to (1 —T/T, )'~' . Thedepairing current of the bulk A phase in the Ginz-burg-Landau region is given by 1.86(IC~/ACscs)x (pksT, /pq)(1 —T/T, )' '. ' Using the measuredspecific-heat jump" this 'equals 13.8(l —T/T, )' 'kg/m s at P = 27.4 bars, which is about four timeslarger than our experimental J,. Above T» the mag-nitude of J, is about 30% smaller (the middle set ofpoints in Fig. 4) than below Ts„and the temperaturedependence is slightly weaker.The extrapolated current corresponding to thesmaller slope in Fig. 2, as shown in Fig. 4, displays asimilar behavior. Near T, both currents tend to benddown owing to a reduction in T,; they extrapolate tozero at T/T, =0.994, in agreement with the theoreti-cally expected decrease in T, without flow. ' Thechange in J, at T~& and the existence of the secondcritical current must be caused by effects associatedwith the ends of the channels because the phase tran-sition occurs only in the bulk liquid. The small valueof J, below T~& may be due to the A-8 interfacepresent in this region. Figure 4 indicates that aboveTsq(cyl) J, is independent of pressure within thescatter of our data; below Tsq(cyl) J, changes by 10%between 22.7 and 27.4 bars.The effect of the ends of a cylindrical channel onthe superflow has been considered by Thunebergand Kurkijarvi. In particular, they show that thecritical current through a long channel containing adisgyration texture vanishes, irrespective of the direc-tion of the I vectors at the two ends of the channel.With R =0.4 p, m, a disgyration is energetically themost favorable texture. " Further, in this case an ap-plied pressure induces a current, which is determinedby friction between the normal component and the Itexture. ' This current is given by J, = (m/k')'x (pL ff/p)b P, where p, is the Cross-Andersonviscosity coefficient' and L,ff =2R is the effectivelength of the end zones. After inserting typicalvalues we obtain d hP/dJ, = 0.4(1 —T/T, ) ' ' m/swhich is close to our result in Fig. 3 (lower curve).The nonzero J, can be qualitatively understood asarising from flow channels of noncylindrical sym-metry, i.e., from channels having two or more holestouching each other. This model requires that the ra-0.30.1Ol0.03%at0.01~80.003IpO ~0~ 0~~A0.01 0.02 0051-T/TC0.10 0.20FIG. 4. The extrapolated critical current J, as a functionof temperature. Diamonds correspond to P =22.7 bars, cir-cles to 24.6 bars, and triangles to 27.4 bars. Filled symbolsrepresent extrapolations of the smaller slope above T&z.The solid line has the slope of (1 —T/T, )tio of the critical currents corresponding to thesteeper and smaller slopes is constant, dependingonly on the distribution of channels; this is not in-consistent with our data.In conclusion, the critical mass current of 3He-A innarrow channels differs significantly from that mea-sured in a more open geometry. At present there ex-ists no theoretical calculations for the depairingcurrent in a restricted geometry; in the bulk liquidthe predicted value exceeds our measured J, by a fac-tor of 4. The results show that J, and the nature ofdissipation depend on the phase, A or 8, of the bulkliquid outside the flow channels. This suggests thatdissipation is restricted to the end zones of the chan-nels.ACKNOWLEDGMENTSWe are indebted to S. A. Nenonen for help duringthe measurements and wish to acknowledge usefuldiscussions with O. V. Lounasmaa, H. Smith, and E.V. Thuneberg. This work was financially supportedby the Academy of Finland and by the Emil AaltonenFoundation.5236 BRIEF REPORTS 26Permanent address: National Physical Laboratory, NewDelhi, India.~Permanent address: Physics Department, Kazan StateUniversity, Kazan 420008, USSR.'V. Ambegaokar, P. G. deGennes, and D. Rainer, Phys.Rev. A 9, 2676 (1974).E. V. Thuneberg and J; Kurkijirvi, Phys. Lett. 86A, 35(1981); (unpublished).3M. A. Paalanen and D. D. Osheroff, Phys. Rev. Lett, 45,362 (1980).4E. B. Flint, R. M. Mueller, and E. D. Adams, J. LowTemp. Phys. 33, 43 (1978).5R. T. Johnson, R. L'. Kleinberg, R. A. Webb, and J. C.Wheatley, J. Low Temp. Phys. 18, 501 (1975).J. M. Parpia and J. D. Reppy, Phys. Rev. Lett. 43, 1332(1979).H. Kleinert, J. Low Temp. Phys. 39, 451 (1980); D.Vollhardt, K. Maki, and N. Schopohl, J. Low Temp. Phys.39, 79 (1980).sA. L. Fetter, itt Quantum Statistics and the Many Body Problems, edited by S. B. Trickey, %. P. Kirk, and J. W. Dufty(Plenum, New York, 1976), p. 127.9M. T. Manninen and J. P. Pekola, Phys. Rev. Lett. 48, 812(1982); 48, 1369 (E} (1982); M. T. Manninen, Ph. D.thesis (Helsinki University of Technology, 1982) (unpub-lished).' Nuclepore Corporation, Pleasanton, Calif. 94566.~'T. A. Alvesalo, T. Haavasoja, M, T, Manninen, and A. T.Soinne, Phys, Rev. Lett. 44, 1076 (1980).' G. Barton and M. A, Moore, J. Low Temp. Phys. 21, 489(1975).'3L. H. Kjadman, J. KurkijRrvi, and D. Rainer, J. LowTemp. Phys. 33, 577 (1978).' I. A. Privorotskii, Phys. Rev. B 12, 4825 (1975).~ A. I. Ahonen, M. Krusius, and M. A. Paalanen, J. LowTemp. Phys. 25, 421 (1976}.J. Saunders, D. S. Betts, D. F. Brewer, S. J. Swithenby,snd W. S. Truscott, Phys. Rev. Lett. 40, 1278 (1978).' L. J. Bucholtz and A. L. Fetter, Phys. Lett. 58A, 93(1976}.M. C. Cross and P. W. Anderson, in Proceedings of theFourteenth International Conference on I.ow Temperature Phy-sics, edited by M. Krusius and M. Vuorio (North-Holland,Amsterdam, 1975), Vol. 1, p. 29.

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Critical current of 3He-A in narrow channels

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