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Discontinuous field-velocity relation for vortex rings in superfluid solutions
Volume 36A, number 4
PHYSICS LETTERS
13 September 1971
DISCONTINUOUS FIELD-VELOCITY VORTEX RINGS IN SUPERFLUID
RELATION SOLUTIONS
FOR *
M. KUCHNIR, J. B. KETTERSON and P. R. ROACH Argonne National Laboratory, Argonne, Illinois 60439, USA Received 2 August 1971.
3He-4He solutions The45simultaneous existence ofastwo different observed in dilute below mK can be interpreted evidence forring the velocities existence of rings with different quanta of circulation or a resonance in the ring—3He scattering cross section.
New data (fig. 1) involving large and small positively charged vortex rings in dilute 3He-4He solutions allow two different interpretations for the observed** discontinuity and double valued-
I
0
ness of the steady state drift velocity below 45 mK.
Io6ppni 3He
--
\.
-I
\ ~.
550m1( 370mK 85mK
-.
.
:--
..
•
I-
>2 -3
40-
U
550 mK
-
~O6 ppm 3He
0
U
~ 30
E(V/cm)
-
37OmK >-
Fig.
-
85mK
2.
Straight line fit of the high field (big ring) data according to eq. 3a.
0
~ 20
-
> I-
relationship between the electric field,
-
~,
and
ln(V/~). From the coefficients of the straight
o 10
00
line obtained by fitting the large ring data (see fig. 2) we obtain the drag coefficient[2], a, and
(
I9mK 18mK \ NEGATIVEJ ION / 10
20
30
40
50
ELECTRIC FIELD (V/cm)
Fig. 1. Steady state terminal velocity of charge car— riers as function of electric field. Note the discontinu— ity and simultaneous existence of two positively charged vortex rings at 19 mK and 8.4 V/cm.
the core radius, a, assuming rings with one quanturn of circulation (N= 1). For the temperatures of 19, 85, 370 and 550 mK the values obtained for a are: 7.1 ± 0.2, 6.4 ± 0.2 and 6.9 ± 0.2 and 6.7 ± 0.2 eV/cm while those for a are 15 ± 2, ii ± 2, 4.5 ± 1 and 2.8 ± 0.5 A respectively. We observe essentially no temperature dependence
The classical formula for the ring velocity, V, combined with the assumption of a viscous force proportional to the ring radius, R, yields a linear
of a as expected [1,3], but an anomalously large core radius is obtained at low tem~erature, mdicating, perhaps, condensation of He on the core of the vortex ring. The 15 A value of a and the condition R > a implies V ~ 10 rn/see; that is,
Work performed under the auspices of the United
extrapolation 1 to small rings of data obtained from big rings predicts a collapse of the ring for
*
States Atomic Energy Commission.
**
Kuchnir et al. Ti] reported previous measurements on a 1.70 ppm 3He—4He solution,
~ This extrapolation might be questioned with regard to the validity of the classical equations for small rings. 287
Volume 36A, number 4
1000
106I9m1< ppm
—
PHYSICS LETTERS
tion of the data is possible as it was recently pointed out [4]. If ~a/~V (1 - e~/a)1a/V for a limited range of velocities the resultant V versus
~
3He-4He
.
• • 00
—
0
•
00 •
£ A
•
00
•
~
•~
lO
-
I.28A
‘~
•~
/ I
_______
0 I
2
15.0 A
“~
303A 3
6.OA 4
6
7
aIeV/cml
Fig. 3. Drag coefficient, a, derived from velocity data and eq. (3b) for several values of the core radius, a, as a function of ring radius R. The dashed lines correspond to the discontinuity gap. a velocity close to the value at the end point of the high field regime. Under this condition a more energetic ring (higher circulation) might become stable and be responsible for the data points between the bare ion curve (data left of the maximum) and the discontinuity. Since this new ring would also be quite small, values for a and a cannot be meaningfully extracted from the data. If a depends on R (or V) for big rings in such a way as to still yield the observed straight line In the in V/~ versus ~ plot and yet presents a resonance for small rings a second interpreta-
288
curve will be triple valued in this range. One solution (with dV/d~ > 0) will correspond unstable rings and the other two would correspond to rings with the same circulation but with different velocities and this may be the origin of the observed discontinuity. Such a behavior could be caused by a resonance of the ring- 3He scattering cross section e.g., when the de Brogue wavelength, ~c, of the 3He(11 A at 19 mK) becomes of the order of R. In fig. 3 we show the a(R) that the 19 mK data would yield for several values of a. Large a (15A) implies collapse of the N = 1 ring at the discontinuity and cannot account for
I 5
13 September 1971
the high velocity branch (which would then correspond to N= 2); intermediate a (6 A) cannot fully account for the high velocity rings; small a (3.03 A or 1.28 A) can account for all the data but implies an unexpected dependence of a on R for large rings. Some problems remain. The first interpretation of the observed discontinuity requires an anomalously large core radius while the second requires discarding the drag per unit length argument for large rings. Both interpretations neglect the possibility of the core radius depending on the ring velocity.
References
[1] M.Kuchnir, J. B. Ketterson and P. R. Roach, Phys. Rev. Letters 26 (1971) 879. [2] Defined as in G. W. Rayfield and F. Reif, Phys. Rev. [3) 136 W.J.(1.964) Titus,A1164. Phys. Rev. A2 (1970) 206. [4] C. J. Goebel, private communication.
PHYSICS LETTERS
13 September 1971
DISCONTINUOUS FIELD-VELOCITY VORTEX RINGS IN SUPERFLUID
RELATION SOLUTIONS
FOR *
M. KUCHNIR, J. B. KETTERSON and P. R. ROACH Argonne National Laboratory, Argonne, Illinois 60439, USA Received 2 August 1971.
3He-4He solutions The45simultaneous existence ofastwo different observed in dilute below mK can be interpreted evidence forring the velocities existence of rings with different quanta of circulation or a resonance in the ring—3He scattering cross section.
New data (fig. 1) involving large and small positively charged vortex rings in dilute 3He-4He solutions allow two different interpretations for the observed** discontinuity and double valued-
I
0
ness of the steady state drift velocity below 45 mK.
Io6ppni 3He
--
\.
-I
\ ~.
550m1( 370mK 85mK
-.
.
:--
..
•
I-
>2 -3
40-
U
550 mK
-
~O6 ppm 3He
0
U
~ 30
E(V/cm)
-
37OmK >-
Fig.
-
85mK
2.
Straight line fit of the high field (big ring) data according to eq. 3a.
0
~ 20
-
> I-
relationship between the electric field,
-
~,
and
ln(V/~). From the coefficients of the straight
o 10
00
line obtained by fitting the large ring data (see fig. 2) we obtain the drag coefficient[2], a, and
(
I9mK 18mK \ NEGATIVEJ ION / 10
20
30
40
50
ELECTRIC FIELD (V/cm)
Fig. 1. Steady state terminal velocity of charge car— riers as function of electric field. Note the discontinu— ity and simultaneous existence of two positively charged vortex rings at 19 mK and 8.4 V/cm.
the core radius, a, assuming rings with one quanturn of circulation (N= 1). For the temperatures of 19, 85, 370 and 550 mK the values obtained for a are: 7.1 ± 0.2, 6.4 ± 0.2 and 6.9 ± 0.2 and 6.7 ± 0.2 eV/cm while those for a are 15 ± 2, ii ± 2, 4.5 ± 1 and 2.8 ± 0.5 A respectively. We observe essentially no temperature dependence
The classical formula for the ring velocity, V, combined with the assumption of a viscous force proportional to the ring radius, R, yields a linear
of a as expected [1,3], but an anomalously large core radius is obtained at low tem~erature, mdicating, perhaps, condensation of He on the core of the vortex ring. The 15 A value of a and the condition R > a implies V ~ 10 rn/see; that is,
Work performed under the auspices of the United
extrapolation 1 to small rings of data obtained from big rings predicts a collapse of the ring for
*
States Atomic Energy Commission.
**
Kuchnir et al. Ti] reported previous measurements on a 1.70 ppm 3He—4He solution,
~ This extrapolation might be questioned with regard to the validity of the classical equations for small rings. 287
Volume 36A, number 4
1000
106I9m1< ppm
—
PHYSICS LETTERS
tion of the data is possible as it was recently pointed out [4]. If ~a/~V (1 - e~/a)1a/V for a limited range of velocities the resultant V versus
~
3He-4He
.
• • 00
—
0
•
00 •
£ A
•
00
•
~
•~
lO
-
I.28A
‘~
•~
/ I
_______
0 I
2
15.0 A
“~
303A 3
6.OA 4
6
7
aIeV/cml
Fig. 3. Drag coefficient, a, derived from velocity data and eq. (3b) for several values of the core radius, a, as a function of ring radius R. The dashed lines correspond to the discontinuity gap. a velocity close to the value at the end point of the high field regime. Under this condition a more energetic ring (higher circulation) might become stable and be responsible for the data points between the bare ion curve (data left of the maximum) and the discontinuity. Since this new ring would also be quite small, values for a and a cannot be meaningfully extracted from the data. If a depends on R (or V) for big rings in such a way as to still yield the observed straight line In the in V/~ versus ~ plot and yet presents a resonance for small rings a second interpreta-
288
curve will be triple valued in this range. One solution (with dV/d~ > 0) will correspond unstable rings and the other two would correspond to rings with the same circulation but with different velocities and this may be the origin of the observed discontinuity. Such a behavior could be caused by a resonance of the ring- 3He scattering cross section e.g., when the de Brogue wavelength, ~c, of the 3He(11 A at 19 mK) becomes of the order of R. In fig. 3 we show the a(R) that the 19 mK data would yield for several values of a. Large a (15A) implies collapse of the N = 1 ring at the discontinuity and cannot account for
I 5
13 September 1971
the high velocity branch (which would then correspond to N= 2); intermediate a (6 A) cannot fully account for the high velocity rings; small a (3.03 A or 1.28 A) can account for all the data but implies an unexpected dependence of a on R for large rings. Some problems remain. The first interpretation of the observed discontinuity requires an anomalously large core radius while the second requires discarding the drag per unit length argument for large rings. Both interpretations neglect the possibility of the core radius depending on the ring velocity.
References
[1] M.Kuchnir, J. B. Ketterson and P. R. Roach, Phys. Rev. Letters 26 (1971) 879. [2] Defined as in G. W. Rayfield and F. Reif, Phys. Rev. [3) 136 W.J.(1.964) Titus,A1164. Phys. Rev. A2 (1970) 206. [4] C. J. Goebel, private communication.