Experiments on the twisted vortex state in superfluid 3He-B
V.B. Eltsov, R. de Graaf, R. Hanninen, M. Krusius, R.E. Solntsev
aa r X i v : . [ c ond - m a t . s o f t ] J un Journal of Low Temperature Physics manuscript No. (will be inserted by the editor)
V.B. Eltsov · R. de Graaf · R. H¨anninen · M. Krusius · R.E. Solntsev
Experiments on the twisted vortex state insuperfluid He-B
Keywords superfluid He, quantized vortices, vortex dynamics, NMR
Abstract
We have performed measurements and numerical simulations on a bun-dle of vortex lines which is expanding along a rotating column of initially vortex-free He-B. Expanding vortices form a propagating front: Within the front the su-perfluid is involved in rotation and behind the front the twisted vortex state forms,which eventually relaxes to the equilibrium vortex state. We have measured themagnitude of the twist and its relaxation rate as function of temperature above0 . T c . We also demonstrate that the integrity of the propagating vortex front re-sults from axial superfluid flow, induced by the twist.PACS numbers: 67.57.Fg, 47.32.y, 67.40.Vs Since the pioneering works by Feynman, and by Hall and Vinen a rotating su-perfluid has been associated with an array of rectilinear vortex lines, stretchedalong the axis of rotation. This lowest energy state is known as the equilibriumvortex state. Different kinds of collective excitations of vortex arrays have beendiscussed over the years [1], including analogs of Kelvin waves on bundles ofvortices [2, 3] and Tkachenko waves [4, 5]. Recently a new state of vortex matter,the twisted vortex state, was experimentally identified and theoretically describedin the B phase of superfluid He [6]. In this state vortices have helical configura-tion circling around the axis of rotation. Remarkably, certain such configurations,uniform in the direction of the rotation axis, do not decay, although the energy ofthis state is higher than that of the equilibrium vortex array. The reason is that theforce, acting on vortices, is directed along the vortex cores everywhere.
Low Temperature Laboratory, Helsinki University of Technology, P.O.Box 2200, 02015 HUT,FinlandTel.: +358-9-4512973Fax: +358-9-4512969E-mail: [email protected].fi z , cm z , cm z , cm v s , cm/s v s , cm/s v s , cm/s v s (cid:30) . : R / (cid:0) v s z . : R / Fig. 1 (Color online) Numerical simulations of vortex dynamics demonstrate the twisted vor-tex state during the expansion of the vortex bundle along the rotating column. The subsequentrelaxation of the twisted state proceeds from the top and bottom sample boundaries. The vor-tex configuration and the z -dependencies of the superfluid velocity components at given radialdistance are shown at three points in time. The sample radius is 1.5 mm, length 40 mm, angularvelocity 1 rad/s and temperature 0 . T c . The twisted vortex state is created when a bundle of vortex lines expands alongan initially vortex-free rotating superfluid column, Fig. 1. Those segments of thevortex lines, which terminate on the side wall of the sample cylinder, propagatetowards the vortex-free part and simultaneously precess around the central axisunder the action of the Magnus and mutual friction forces. Such two-componentmotion leads to a vortex bundle which is helically twisted. Expansion along thecolumn becomes slower with decreasing temperature as mutual friction decreases.Thus the spiral, created by the motion of the vortex end on the side wall, becomestighter. One may expect that the resulting twist, characterized by the wave vector Q of the vortex helix in the bundle, becomes stronger as the temperature is reduced.We have measured the magnitude of the twist in the range 0.3 – 0.8 T c and observedthis expected behavior only at T > . T c . Below 0 . T c the magnitude of thetwist decreases again.In the real sample, which is not infinitely long, the twist cannot be completelyuniform: At the top and bottom ends of the sample the vortices are perpendicularto the wall and the twist disappears there. The twist in the bulk unwinds when thevortex ends slide over the end plates of the sample cylinder. The model in Ref. [6]predicts that the relaxation of the twist becomes faster with decreasing temperatureand distance to the wall. We have experimentally confirmed both properties andestablished reasonable agreement with the model.We also examine in this report the role of the twist-induced superflow in thepropagation of the vortex front, which separates the vortex-free superfluid from r , cm0.000.050.10 v s (cid:30) , c m / s r , cm-0.06-0.04-0.020.000.02 v s z , c m / s Fig. 2
Radial dependencies of the superfluid velocity from the simulation snapshot in the centerpanel of Fig. 1. Circles are the simulation results at z = . Q = . R − . At r > . the twisted vortex bundle. At T & . T c the thickness of the vortex front in-creases while it propagates. At lower temperatures the twist-induced axial super-flow pushes vortices at the rear of the front forward. Eventually they catch up withthe vortices in the head of the front and the front propagates in a thin steady-stateconfiguration. The essential features of the vortex front and the twisted state can be displayedby means of numerical calculations of vortex dynamics in a rotating cylinder. Thesimulation technique accounts fully for inter-vortex interaction and for the effectof solid walls [7]. In the initial state at t = v s f shows an almost linear transition from the non-rotatingstate v s f = v s f ≈ W r within the region of the vortexfront. This shear flow within the front is created by vortices which terminate on thecylindrical wall perpendicular to the axis of rotation. At the temperature of 0 . T c the thickness of the vortex front grows with time. Simulations at 0 . T c on the otherhand demonstrate a thin time-invariant front [6]. We discuss this difference below.The appearance of the twist is reflected in the axial superflow at the velocity v s z , which is along the vortex expansion direction close to the cylindrical boundaryand in the opposite direction close to the axis. As shown in Fig. 2, at constantheight z the r -dependencies of v s f and v s z are reasonably well described by themodel suggested in Ref. [6]: v s f ( r ) = ( W + Qv ) r + Q r , v s z ( r ) = v − Q W r + Q r , (1)where v = ( W / Q )[ Q R / ln ( + Q R ) − ] and R is the sample radius. The wavevector Q of the twist has its maximum value close to the rear end of the front and c m c m c m . c m . c m ? He−A He−Binjectedvorticesbottom platewith ori(cid:2)ce top NMR pick−up H D : f D :
03 MHzbottom NMR pick−up H D : f D :
965 MHzbottom NMR pick−up H D : f D :
682 MHztop NMR pick−up H D : f D :
961 MHz setup 1 setup 2
Fig. 3 (Color online) Experimental setup. decreases to zero at the bottom and top ends of the sample. The axial superflowaffects the NMR spectrum of He-B and this allows us to observe the twistedvortex state in the experiment.
The experimental techniques are similar to those in the recent studies of non-equilibrium vortex dynamics in He [8]. The sample of He-B at 29 bar pressureis contained in a cylindrical smooth-walled container with dimensions shown inFig. 3. The sample is split in two independent B-phase volumes by an A phaselayer, stabilized with applied magnetic field. In each B-phase volume an indepen-dent NMR spectrometer is used to monitor the vortex configuration. Two arrange-ments for pick-up coils have been used. They are labelled throughout this report as“setup 1” and “setup 2”. These arrangements differ in the placement of the pick-upcoils with respect to the upper and lower ends of the container, in Larmor frequen-cies, in the design of pick-up coils, and in the field homogeneity. The last propertyis mostly determined by the field distortion from the superconducting wire in thepick-up coils. For solenoidal coils in setup 1 it is D H / H ≈ · − , while the coilsin setup 2 distort the field more and D H / H ≈ . · − .The initial vortex-free state is prepared by thermal cycling of the sample totemperatures above 0 . T c , where one waits at W = W = . W = .
03 rad/s above the critical velocity of the AB interface instability W cAB [10], after which N M R a b s o r p ti on , m V exp(- t / t Lar ) h tw t CF Larmor peakcounterflow peak1.21.3 W , r a d / s W cAB Fig. 4 (Color online) Examples of NMR records from a vortex injection experiment and defi-nition of the parameters t CF , t Lar and h tw . The signal traces are recorded with the bottom spec-trometer in setup 2 at T = . T c . W is kept constant, Fig. 4. In the instability event about 10 vortices are injectedinto the B-phase close to the AB interface. At T < . T c a turbulent burst immedi-ately follows and generates almost the equilibrium number of vortices [11]. Thesevortices then propagate towards the pick-up coil. The instability velocity W cAB iscontrolled by temperature and magnetic field profile and varies between 1.1 and1.5 rad/s. A modification of this injection technique has been used at T > . T c .Here the sample initially rotates at W > v n f − v s f .It is at maximum in the initial vortex-free state with v s f =
0. When v s f increasesthe absorption decreases rapidly and drops to zero when v s f ∼ ( / ) v n f [13].Absorption in the Larmor peak is mostly sensitive to the axial flow v s z . It is zeroin the vortex-free state while for a vortex cluster it has some finite value, whichincreases monotonously with increasing v s z [13].An example of NMR measurement is presented in Fig. 4. To record signals atboth Larmor and counterflow peaks with the same spectrometer the experimentis repeated twice in identical conditions. A rapid drop in the counterflow signalis seen which corresponds to the passage of the vortex front through the coil. Wecharacterize this drop with time t CF . The Larmor peak grows first to a maximumvalue h tw , which corresponds to the maximum twist. Then the signal relaxes ex-ponentially with the time constant t Lar to a value characteristic for an equilibrium
T/T c h t w / h ( W = ) setup 1setup 2simul. 0.00.51.01.5 Q R Fig. 5 (Color online) Strength of the twist as a function of temperature. The measurements areperformed using the bottom spectrometer in both setups of Fig. 3. The ratio of the maximumamplitude of the Larmor peak in the twisted state to the amplitude of the Larmor peak in thenonrotating sample is plotted on the left axis. The maximum value of the twist wave vector Q ,obtained in simulations, is plotted on the right axis. The solid curve shows the fit QR = . ( − a ′ ) / a . The broken curve shows the minimum magnitude of the twist for which propagation ofthe vortex front in a thin steady-state configuration is possible. vortex cluster. The propagation velocity V f of the vortices at the head of the frontcan be determined from their flight time between the injection moment, where W ( t ) = W cAB , and the arrival to the edge of the peak-up coil, as monitored withthe counterflow peak. We do not discuss V f in detail in this report. The temperature dependence of the magnitude of the twist is presented in Fig. 5.From the experiment the raw data are plotted: The maximum height of the Larmorpeak h tw (Fig. 4), normalized to the height of the Larmor peak at W = . T c . The same behavior of themagnitude of the twist is confirmed in the numerical simulations: The maximumvalue of the twist wave vector behind the front, as determined from a fit of thevelocity profiles to Eq. (1), also peaks at 0 . T c .The initial growth of the twist with decreasing temperature is expected. Fromthe equations of motion of a single vortex which ends on the cylindrical wall, theestimate for the expansion velocity along z is v L z = a ( T ) [ v n f ( R ) − v s f ( R )] , (2) T/T c t L a r W setup 2 bottom, b = 1.5 cmsetup 1 bottom, b = 1.0 cmsetup 2 top, b = 0.5 cm Fig. 6 (Color online) Relaxation time of the twist versus temperature. Solid lines are fit to Eq. (3)with C = where a is a mutual friction coefficient [14]. As the normal fluid velocity atthe side wall v n f ( R ) = W R and the superfluid velocity induced by a single vor-tex can be neglected in our conditions, Eq. (2) gives v L z ≈ aW R . For the az-imuthal velocity of the vortex end in the rotating frame a similar estimation gives v L f ≈ ( − a ′ ) W R , where a ′ is another mutual friction coefficient [14]. Thusthe trajectory of the vortex end on the side wall is a spiral with wave vector Q = v L f / ( Rv L z ) ≈ ( − a ′ ) / ( R a ) . This value can be used as a rough estimationof the Q vector in the twisted vortex state. As one sees from Fig. 5, the simulationresults at T & . T c indeed follow this dependence.For the decreasing twist at T < . T c a couple of reasons can be suggested.First, the twist can relax through reconnections between vortices in the bundle,which become more prominent with decreasing temperature. Second, relaxation ofthe twist proceeds in diffusive manner within the twisted cluster. The source of thetwist is at the vortex front, while the sink is at the end plate of the cylinder, wherethe twist vanishes because of the boundary conditions. The effective diffusioncoefficient increases as the temperature decreases [6]. The faster diffusion limitsthe maximum twist in the finite-size sample at low temperatures.The relaxation of the twisted state is observed as a decay in the amplitude ofthe Larmor peak. The measured time constant t Lar of the exponential decay ispresented in Fig. 6 for three different positions of the detector coil with respect tothe end plate of the sample cylinder. It is clear that the relaxation indeed becomesfaster with decreasing temperature. It is also evident that the relaxation of thetwist proceeds faster as the observation point moves closer to the end plate of thesample cylinder. Both of these features are well described by the model presentedin Ref. [6]. According to this model the relaxation time of the twist is t ≈ C W (cid:18) bR (cid:19) a [( − a ′ ) + a ] , (3) where C ∼ b is the distance along z to the end plate, where the twist van-ishes. The fit to this model using C as the only fitting parameter demonstratesreasonable agreement with the measurements in Fig. 6. The experimental pointsshow somewhat faster temperature dependence than the model. The discrepancymay result from shortcomings of both the model and the experiment. The modelis constructed for the case of weak twist ( QR ≪ b itself.When Eq. (2) is applied to vortices within the vortex front, the following prob-lem arises: At the head of the front the superfluid component is almost at rest, v s f ≪ v n f , and the expansion velocity of vortices is V f ≈ aW R . Behind the frontthe density of vortices is close to the equilibrium and v s f ≈ v n f . Thus vortices atthe tail of the front are barely able to expand and the thickness of the front shouldrapidly increase in time. On the other hand simulations show that vortices at thetail of the front do expand, Fig. 1. Moreover, at sufficiently low temperatures theexpansion velocity of these vortices reaches the velocity of the foremost vortices,so that the front propagates in a thin steady-state configuration [6].The explanation is that the vortex state behind the front is not the equilib-rium state, but the twisted vortex state. Taking into account the axial superflow,induced by the twist, Eq. (2) should be modified as v L z = a [ v n f ( R ) − v s f ( R )] +( − a ′ ) v s z ( R ) . Given that v s z ( R ) is in the direction of the front propagation and v s f ( R ) < v n f ( R ) in the twisted state, the expansion velocity V t of the vortices inthe tail of the front is enhanced. This velocity can be estimated taking v s z ( R ) and v s f ( R ) from Eq. (1): V t = aW R (cid:20) + − a ′ a Q R (cid:21) (cid:20) − Q R ( + Q R ) log ( + Q R ) (cid:21) (4)This velocity has a maximum as a function of the Q vector. If a / ( − a ′ ) > . T > . T c [14]), the maximum value of V t is less than the velocity of theforemost vortices V f ≈ aW R . In these conditions the thickness of the front in-creases while it propagates. When T < . T c a wide range of Q values exists forwhich formally V t > V f . The minimum possible value of Q is shown in Fig. 5 bythe broken curve. In these conditions the vortex front propagates in a steady-state“thin” configuration. These simple considerations become inapplicable, however,when vortices interact strongly within the thin layer. To determine, say, the sta-ble front thickness and the magnitude of the twist in the regime below 0 . T c adifferent approach would be needed.The change in the front propagation at T ≈ . T c is observed in both exper-iment and simulations. In the experiment the decay time of the counterflow peak t CF in Fig. 4 can be used to extract the front thickness. The decay of the coun-terflow peak starts when the head of the vortex front arrives at that edge of thedetector coil, which is closer to the injection point. The counterflow signal van-ishes when the last part of the front, which still possesses enough counterflow togenerate the NMR response, leaves the far edge of the detector coil. The product t CF V f has the dimension of length and can be called the apparent thickness of thefront. When the real thickness of the front grows with time its apparent thicknessdepends on the distance of the observation point from the injection point and onthe rate with which the real thickness increases. When the front remains thin thevalue of its apparent thickness equals the height of the pick-up coil d coil = T/T c V f t C F , c m Fig. 7
Apparent thickness of the vortex front as function of temperature. The solid line is theprediction of the model in Eq. (5).
The measurements of the apparent thickness of the front are presented inFig. 7. The scatter at higher temperatures has two sources. Partially it is due tothe uncertainty in the determination of V f from flight times which are not longcompared to the measuring resolution. Another contribution is the variation inthe decay profile of the counterflow signal which may display weak oscillationsaround a roughly linear decrease. At T > . T c we have t CF V f > d coil and theapparent thickness decreases with temperature. Finally at T < . T c the frontbecomes thin compared to the height of the pick-up coil. Assuming that at themoment of injection the front is infinitely thin we can write t CF = d + d coil V ∗ t − dV f , (5)where d is the distance from the injection point (i.e. position of the AB interface)to the nearest edge of the pick-up coil and V ∗ t is the expansion velocity at theposition in the front where the NMR signal from the counterflow vanishes. Giventhat the latter condition roughly corresponds to v s f ∼ ( / ) v n f we take V ∗ t = ( V t + V f ) / V t < V f and simply V ∗ t = V f otherwise. Using V t from Eq. (4) and the simpleestimates QR = ( − a ′ ) / a and V f = aW R , we get from Eq. (5) the solid line inFig. 7, which is in reasonable agreement with the experiment.In the simulations the thickness of the front grows with time at higher tempera-tures. This process slows down as the twist increases with decreasing temperature(Fig. 5). Finally at T ≈ . T c the twist reaches the value which is enough, ac-cording to Eq. (4), to support a thin front configuration. At lower temperatures thethickness of the front indeed becomes time-independent and roughly equal to theradius of the sample. Simultaneously the twist behind the front starts to drop, ashas been discussed above. It remains, however, within the limits where the twist-induced superflow is able to keep the front thin. We have studied the formation and relaxation of the twisted vortex state in He-B in the temperature range between 0 . T c and 0 . T c . At higher temperatures T & . T c the twist behind the propagating vortex front grows with decreasingtemperature as Q (cid:181) ( − a ′ ) / a . Here the thickness of the front increases while itpropagates along the rotating column. The axial superflow, induced by the twistedstate, boosts the expansion velocity of vortices in the tail of the front. This en-hancement increases as the twist grows with decreasing temperature. Finally at T ≈ . T c vortices in the tail of the front are able to catch up with vortices in thehead. At lower temperatures the front propagates in a thin steady-state configura-tion, while the twist starts to decrease.The relaxation of the twist in a sample of finite length is not related to thefront. It proceeds from the walls which limit the length of the sample along therotation axis. The relaxation speeds up with decreasing temperature, unlike manyother processes in vortex dynamics.While the understanding of the front propagation and of the formation of thetwisted vortex state in the high-temperature regime T & . T c is quite good, inthe low-temperature regime of a thin front the theoretical understanding is lacking.Especially interesting would be to consider the role of vortex reconnections andturbulence, both of which should increase with decreasing temperature. References
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