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Quantized vortices at a moving A-B interface in superfluid3He
PflYSICA
Physica B 194-196 (1994) 779-780 North-Holland
Quantized vortices at a moving A-B interface in superfluid 3He E.V. T h u n e b e r g , / J . Parts, Y. Kondo*, J.S. Korhonen t, and M. Krusius Low T e m p e r a t u r e Laboratory, Helsinki University of Technology, Otakaari 3 A, 02150 Espoo, Finland We have studied the B~-A phase transitions in a rotating container. The initial state contains the equilibrium number of vortices, and after the A-B interface has traversed the container, the vortex state was studied using NMR. We find that the A-phase state resulting from a B--~A transition contains the equilibrium number ~f continuous vortices. Possible explanations are discussed. (a)
The A-B transition of superfiuid 3He allows a unique possibility to study the continuity of topological objects through a phase interface. We will limit the discussion in this paper to the transition from the B to the A phase; the opposite A--+B transition will be discussed in detail elsewhere. The situation is depicted in Fig. 1. In 3He-B, the vortices resemble those in superfluid 4He: they have a singular core of the size of the coherence length (~ ~ 10nm) and a single quanturn of circulation (~ = f v s . dr = h/2m) [1]. In 3He-A, the prevailing vortex type is continuous. Its vorticity is distributed over a much larger area (~D ~ 10pm) in the form of a smooth distribution of the anisotropic order parameter. The continuous vortices are bound to pairs and thus have two quanta of circulation. We may first ask, do the B-phase vortices penetrate through the interface when the A phase grows at the expense of the B phase, or are the A-phase vortices generated independently of the B-phase vortices? Secondly, if the penetration takes place, do the vortices in the A phase remain singular, which is predicted to be the minimum-energy state at low rotation velocities [2], or do they transform to continuous vortices by forming pairs? Our N M R cell is a cylindrical chamber with radius R = 3.5 m m and height L = 7 mm, rotated around its s y m m e t r y axis (Fig. 2). Its only connection to the rest of the 3He volume is via an orifice in the center of b o t t o m plate. The critical velocities for the nucleation of vortices in the N M R cell are on the order of 2.5 r a d / s in the B
A
pha: ~e
Figure 1. Three possible fates of a singular vortex (lines) in front of a growing A phase: (a) the vortex is pushed ahead in front of the interface, (b) it penetrates through as a singular vortex and (c) it finds a pair and forms a continuous vortex (shaded tube) in the A phase.
phase and 0.1 r a d / s in the A phase. Our experiments were done at varying rotation speeds from 0.4 r a d / s to 1.2 r a d / s at 29.3 bar pressure. Both the number and the type of the vortices can be inferred from peaks in the absorption spectrum of continuous-wave N M R [1]. Before the B - ~ A transition, the B phase was prepared to contain the equilibrium number of vorrices: N = 2rR2~/t~, where 9t is the angular velocity of the container. In the B phase, the number of vortices can be determined with a resolution of ~ 0.01 rad/sec, which corresponds to approximately 10 vortices [3]. Then the transition was allowed to take place at constant ~. After the transition was finished, the N M R ab-
*Present address: Physics Department, University of Bayreuth, 8580 Bayreuth, Germany. tPresent address: Physics Department, University of Kyoto, Kyoto, Japan.
0921-4526/94/$07.00 © 1994 - Elsevier Science 0921-4526(93)E0969-N
SSDI
B.V.
All
rights reserved
780
coil cell
nel
VlR thermometer
sintered heat~ex~hanger and nuclear cooling stage
Figure 2. The experimental cell.
sorption spectrum in the A phase was analyzed. C o n t r a r y to our expectation, it was identical to the one obtained after starting rotation within the A phase: The frequency shift of the satellite peak from the bulk absorption frequency indicated continuous vortices, and the amplitude of the peak corresponded to the equilibrium number of vortices within a resolution +50 vortices. No evidence of singular vortices was seen. According to theory, singular vortices are distinguished by a much smaller frequency shift [4], but this lacks experimental verification because the only evidence of t h e m comes from measurements of ion mobility [1]. An identical N M R spectrum in 3He-A was obtained even in the ease that the initial B phase was in a metastable state containing no vortices. For the interpretation of these results, it is imp o r t a n t t h a t the phase of the order parameter is continuous across the A-B interface, which means t h a t vortices cannot terminate on the interface. The presence of continuous vortices in 3He-A implies t h a t either (i) B-phase vortices transform to continuous ones in penetrating through the interface, or (ii) the A-phase vortices are nucleated independently of the interface. Both alternatives contain puzzles t h a t remain unsolved here. W h a t is the mechanism t h a t in the first alternative (i) makes the vortices to come so close each other t h a t they can form the pair, which is necessary for the formation of a continuous vortex? Moreover, singular vortices should be the absolute en-
ergy minimum at ft < 1 r a d / s in the A phase [2]. The latter possibility (ii) is reasonable because the measured critical velocity for the A phase is low, near the resolution threshold. It is more difficult to prove t h a t a sufficiently strong force exist for pushing the B-phase vortices in front of the interface. Such a force may arise from two mechanisms. Firstly, the superfluid densities in the two phases are different: p~ in the A phase is anisotropie with components Pit parallel and p z perpendicular to the orbital anisotropy vector 1, whereas it is isotropic = P B in the B phase. Weak-coupling theory gives Ptl = (3/5)pB and pz = (6/5)pB near the transition temperature. Because of the kinetic energy density ps(vs - v n ) 2 / 2 , the B-phase vortex is repelled from the interface if 1 is parallel to the line, but it is attracted for perpendicular 1. The magnetic field ( H = 14.2 or 28.4 mT, parallel to the axis of the cylinder) favors the latter case in our cell. It is therefore unlikely t h a t this mechanism could prohibit the vortex penetration through the interface. Secondly, a force might arise from the interaction of the vortex core with the interface. A quantitative estimate of the latter force requires a complicated numerical calculation. Thus we can only conclude t h a t the most plausible explanation of the experimental results would be that the core interaction is strong enough to prevent vortex penetration until vortices are nucleated in the A phase. We thank G. Volovik for useful discussions. REFERENCES 1.
2. 3.
4.
P. Hakonen, O.V. Lounasmaa, and J. Simola, Physica B 160 (1989) 1; M.M. Salomaa and G.E. Votovik, Rev. Mod. Phys. 59 (1987) 533. A.L. Fetter, ,J. Low Temp. Phys. 67 (1987) 145. Y. Kondo, J.S. Korhonen, fJ. Parts, M. Krusius, O.V. Lounasmaa, and A.D. Gongadze, Physica B 178 (1992) 90. H.K. SeppSJii and G.E. Volovik, J. Low Temp. Phys. 51 (1983) 179; V.Z. Vulovic, D.L. Stein, and A.L. Fetter, Phys. Rev. B 29 (1984) 6090.
Physica B 194-196 (1994) 779-780 North-Holland
Quantized vortices at a moving A-B interface in superfluid 3He E.V. T h u n e b e r g , / J . Parts, Y. Kondo*, J.S. Korhonen t, and M. Krusius Low T e m p e r a t u r e Laboratory, Helsinki University of Technology, Otakaari 3 A, 02150 Espoo, Finland We have studied the B~-A phase transitions in a rotating container. The initial state contains the equilibrium number of vortices, and after the A-B interface has traversed the container, the vortex state was studied using NMR. We find that the A-phase state resulting from a B--~A transition contains the equilibrium number ~f continuous vortices. Possible explanations are discussed. (a)
The A-B transition of superfiuid 3He allows a unique possibility to study the continuity of topological objects through a phase interface. We will limit the discussion in this paper to the transition from the B to the A phase; the opposite A--+B transition will be discussed in detail elsewhere. The situation is depicted in Fig. 1. In 3He-B, the vortices resemble those in superfluid 4He: they have a singular core of the size of the coherence length (~ ~ 10nm) and a single quanturn of circulation (~ = f v s . dr = h/2m) [1]. In 3He-A, the prevailing vortex type is continuous. Its vorticity is distributed over a much larger area (~D ~ 10pm) in the form of a smooth distribution of the anisotropic order parameter. The continuous vortices are bound to pairs and thus have two quanta of circulation. We may first ask, do the B-phase vortices penetrate through the interface when the A phase grows at the expense of the B phase, or are the A-phase vortices generated independently of the B-phase vortices? Secondly, if the penetration takes place, do the vortices in the A phase remain singular, which is predicted to be the minimum-energy state at low rotation velocities [2], or do they transform to continuous vortices by forming pairs? Our N M R cell is a cylindrical chamber with radius R = 3.5 m m and height L = 7 mm, rotated around its s y m m e t r y axis (Fig. 2). Its only connection to the rest of the 3He volume is via an orifice in the center of b o t t o m plate. The critical velocities for the nucleation of vortices in the N M R cell are on the order of 2.5 r a d / s in the B
A
pha: ~e
Figure 1. Three possible fates of a singular vortex (lines) in front of a growing A phase: (a) the vortex is pushed ahead in front of the interface, (b) it penetrates through as a singular vortex and (c) it finds a pair and forms a continuous vortex (shaded tube) in the A phase.
phase and 0.1 r a d / s in the A phase. Our experiments were done at varying rotation speeds from 0.4 r a d / s to 1.2 r a d / s at 29.3 bar pressure. Both the number and the type of the vortices can be inferred from peaks in the absorption spectrum of continuous-wave N M R [1]. Before the B - ~ A transition, the B phase was prepared to contain the equilibrium number of vorrices: N = 2rR2~/t~, where 9t is the angular velocity of the container. In the B phase, the number of vortices can be determined with a resolution of ~ 0.01 rad/sec, which corresponds to approximately 10 vortices [3]. Then the transition was allowed to take place at constant ~. After the transition was finished, the N M R ab-
*Present address: Physics Department, University of Bayreuth, 8580 Bayreuth, Germany. tPresent address: Physics Department, University of Kyoto, Kyoto, Japan.
0921-4526/94/$07.00 © 1994 - Elsevier Science 0921-4526(93)E0969-N
SSDI
B.V.
All
rights reserved
780
coil cell
nel
VlR thermometer
sintered heat~ex~hanger and nuclear cooling stage
Figure 2. The experimental cell.
sorption spectrum in the A phase was analyzed. C o n t r a r y to our expectation, it was identical to the one obtained after starting rotation within the A phase: The frequency shift of the satellite peak from the bulk absorption frequency indicated continuous vortices, and the amplitude of the peak corresponded to the equilibrium number of vortices within a resolution +50 vortices. No evidence of singular vortices was seen. According to theory, singular vortices are distinguished by a much smaller frequency shift [4], but this lacks experimental verification because the only evidence of t h e m comes from measurements of ion mobility [1]. An identical N M R spectrum in 3He-A was obtained even in the ease that the initial B phase was in a metastable state containing no vortices. For the interpretation of these results, it is imp o r t a n t t h a t the phase of the order parameter is continuous across the A-B interface, which means t h a t vortices cannot terminate on the interface. The presence of continuous vortices in 3He-A implies t h a t either (i) B-phase vortices transform to continuous ones in penetrating through the interface, or (ii) the A-phase vortices are nucleated independently of the interface. Both alternatives contain puzzles t h a t remain unsolved here. W h a t is the mechanism t h a t in the first alternative (i) makes the vortices to come so close each other t h a t they can form the pair, which is necessary for the formation of a continuous vortex? Moreover, singular vortices should be the absolute en-
ergy minimum at ft < 1 r a d / s in the A phase [2]. The latter possibility (ii) is reasonable because the measured critical velocity for the A phase is low, near the resolution threshold. It is more difficult to prove t h a t a sufficiently strong force exist for pushing the B-phase vortices in front of the interface. Such a force may arise from two mechanisms. Firstly, the superfluid densities in the two phases are different: p~ in the A phase is anisotropie with components Pit parallel and p z perpendicular to the orbital anisotropy vector 1, whereas it is isotropic = P B in the B phase. Weak-coupling theory gives Ptl = (3/5)pB and pz = (6/5)pB near the transition temperature. Because of the kinetic energy density ps(vs - v n ) 2 / 2 , the B-phase vortex is repelled from the interface if 1 is parallel to the line, but it is attracted for perpendicular 1. The magnetic field ( H = 14.2 or 28.4 mT, parallel to the axis of the cylinder) favors the latter case in our cell. It is therefore unlikely t h a t this mechanism could prohibit the vortex penetration through the interface. Secondly, a force might arise from the interaction of the vortex core with the interface. A quantitative estimate of the latter force requires a complicated numerical calculation. Thus we can only conclude t h a t the most plausible explanation of the experimental results would be that the core interaction is strong enough to prevent vortex penetration until vortices are nucleated in the A phase. We thank G. Volovik for useful discussions. REFERENCES 1.
2. 3.
4.
P. Hakonen, O.V. Lounasmaa, and J. Simola, Physica B 160 (1989) 1; M.M. Salomaa and G.E. Votovik, Rev. Mod. Phys. 59 (1987) 533. A.L. Fetter, ,J. Low Temp. Phys. 67 (1987) 145. Y. Kondo, J.S. Korhonen, fJ. Parts, M. Krusius, O.V. Lounasmaa, and A.D. Gongadze, Physica B 178 (1992) 90. H.K. SeppSJii and G.E. Volovik, J. Low Temp. Phys. 51 (1983) 179; V.Z. Vulovic, D.L. Stein, and A.L. Fetter, Phys. Rev. B 29 (1984) 6090.