Detection of vortices in superfluid 4He in the T = 0limit using charged vortex rings

P. M. Walmsleya, A. A. Levchenkob, S. E. Maya, and A. I. Golova

aSchool of Physics and Astronomy, The University of Manchester,Manchester M13 9PL, UK

bInstitute of Solid State Physics, Russian Academy of Sciences,Chernogolovka 142432, Russia

We describe the experiment to study the dynamics of quantized vortex linesin superfluid 4He at temperatures down to 40 mK, i. e. without any normalcomponent. This was achieved by monitoring the trapping of small chargedvortex rings by an array of rectilinear vortex lines produced by rotating thecryostat. The design of the experimental cell is described. Our first obser-vations of the response of the superfluid in the T = 0 limit to starting andstopping rotation are presented.PACS numbers: 67.40.Vs, 67.40.Jg, 47.32.-y.

1. INTRODUCTION

Exactly 50 years ago, in line with the predictions of Onsager1 andFeynman2, experimental evidence for quantized vortices in superfluid 4Hewas obtained by Hall and Vinen3. They showed that the mutual friction,which appears between the superfluid and normal components when su-perfluid helium is rotated, is consistent with the presence of topologically-stable linear defects, each carrying one quantum of circulation, κ ≡ h

m4=

1 × 10−3 cm2s−1, where h is the Planck’s constant and m4 is the mass ofa 4He atom. This was soon followed by the discovery of Abrikosov vor-tices in type-II superconductors, and the research into the vortex state insuperfluids and superconductors has been extremely active to date. Thevortices in 4He were thoroughly studied both in the form of regular arrays ina rotating cryostat and irregular tangles in turbulent flow4. What initiallycame as a surprise is that many properties of the superfluid turbulence werevery similar to those of classical turbulence5. This was soon explained by

P. M. Walmsley et al.

the fact that most experiments have been performed at high temperaturesT > 1 K, where the very presence of viscous normal component (coupledto the superfluid by mutual friction) assures the quasi-classical behavior ofthe combined system. The question was therefore risen: what will be thedynamics of quantized vortices (and eventually of the turbulent tangle ofthese) at temperatures low enough so there is no any interference from thenormal component left, i. e. in the limit of zero temperature5?

For superfluid 4He, such temperatures, T < 0.3 K, are easy to achieve. Aharder question is: do we have a convenient detection technique for trackingthe vortices? So far, second sound attenuation was the detection techniqueof choice at T > 1 K, because of its great sensitivity to the presence ofvortices and convenience to use3,6. However, it relies on the presence of thenormal component and mutual friction, so must be ruled out.

The other technique to consider is trapping of injected ions by vortices7,4.The ions are microscopic probe particles that can be injected into helium,manipulated by electric field and detected electronically. They are attractedto vortex cores and can be trapped by them. Hence, by observing a reduc-tion or deflection of the ion current, one can learn about the presence anddynamics of vortices. Indeed, ions were first used to prove that quantizedvortices are discrete continuous defects: first observation of a vortex tanglein 19608, trapping of negative ions by a vortex array in a rotating bucket in19629, observation of entry of vortices one by one after starting rotation10;visualization of vortex arrays11, proof there always exist remanent vorticesin 4He 12, etc.

In 2000 Davis et al. reported an experiment at low temperatures13

in which they created turbulence in isotopically pure 4He 14 at pressurep = 15.5 bar by a vibrating grid and detected the presence of vortices by ob-serving a reduction of the current of negative ions propagating through thegrid. At temperatures below 70 mK the time required for the vortex tangleto decay was found to be of order 10 seconds, independent of temperature.This result should be interpreted as the observation of the decay of a local(inhomogeneous) turbulence in a superfluid without any normal component.However, many questions remained, one of them being: what was the actualdensity of vortex lines – as the cross-section for the interaction of ions withvortices at these low temperatures was never measured yet?

Hence, the aims of our experiment were two-fold. Firstly, we would likejust to check the very possibility of using ions to gain information aboutthe vortices at T < 0.3 K in a rotating cryostat where the density of vortexlines is known precisely. This would imply measuring the cross-section for iontrapping and studying the dynamics of growth and disappearance of a vortexarray upon starting and stopping rotation. Secondly, provided it works, to

Detection of vortices in 4He in T = 0 limit

use ions to study more subtle properties of vortex arrays or tangles, like therate of vortex motion and reconnections within the tangle, grid turbulence,etc. Our preliminary results are encouraging and they are reported in thispaper.

2. NEGATIVE IONS AND VORTICES IN HELIUM

Compared to positive ions (cluster ions of radius ∼ 7 A), negative ionsare easier to inject and are more strongly attracted to vortex lines; hence,only the latter will be considered here. An electron injected into liquid he-lium self-localizes in a spherical void (“bubble”) of radius some 18–12 A,at pressures p = 0–25 bar (see reviews7,4). This singly charged bubble is aclassical object of effective mass some 250–100 m4 (p = 0–25 bar) that canbe pulled by external electric fields and its motion detected as electric cur-rent. At T > 1 K their motion is characterized by a temperature-dependentmobility, while at lower temperatures they quickly reach the Landau veloc-ity for roton emission vL ≈ 50 m/s in the direction of the driving force.Moreover, at these velocities they normally create a vortex ring and get per-manently trapped by this ring16,17,4. Hence, the dynamics of the ion-ringcomplex is that of a quantized vortex ring (see Section 4), i.e. it is markedlydifferent from that of the free ion. In principle, it is possible to suppressthe process of vortex ring nucleation by using isotopically pure 4He at highpressure and low electric field15,17. As we will see, in our experiments withisotopically pure 4He 14 at pressure 15 bar and field of order 20 V/cm, atleast 70% of ions ended up nucleating and carrying rings. As the trappingcross-section for ion-ring complexes is much greater, this actually simplifiesthose experiments where existing vortex lines had to be charged by the ioncurrent.

The ion is attracted to the center of a vortex by the combined actionof the Bernoulli force and reduced condensation energy in the vortex core,that make a potential well of the depth ∼ 50 K 4 for the ion. Hence, thelifetime of the trapped ion at temperatures less than 1 K is virtually infinite– provided the applied electric field is much smaller than some 104 V/cm.Now the ion can only leave the volume either by sliding along the vortex lineto the wall (provided the line touches the wall) or by leaving the line andriding on a vortex ring after a reconnection event.

The cross-section (actually, “trapping diameter”) of a free ion movingperpendicular to a straight vortex was only measured above 0.8 K. Above1 K, it was found to be inversely proportional to the driving field and oforder 100 A in field 10 V/cm; but decreased substantially between 1 K and

P. M. Walmsley et al.

0.8 K 18. The decrease was related to the fact that the mean free path of ionsexceeded the characteristic width of the trapping potential, so it is harder tothermalize the ion that is moving at the terminal velocity of vL ≈ 50 m/s.What is important for our experiment is the fact that the cross-section of thecharged vortex rings is roughly equal to their diameter19, which in turn isnearly proportional to the ring energy acquired in the driving electric field.For energies 200–300 eV, the diameter is about 10 µm, making the trappingof ions a simple task. As a substantial fraction of ions in our experimentsappeared in the shape of charged rings (see below), this cross-section is therelevant one.

In equilibrium, rotating superfluid helium should host a stationary (inthe rotating frame) array of rectilinear vortex lines of density L = 2Ω/κ(“Feynman’s rule”). In reality, long relaxation times, pinning and agitationcan disturb this picture, especially at low temperatures when the stabilizingaction of mutual friction is absent. It is known that in the presence of heatflow or vibrations vortices become “kinky”24 and keep moving around20,and there are always pinned remanent vortices present even without anyrotation12.

3. CELL DESIGN

The experimental cell was designed to be as versatile as possible toallow a number of different types of experimental setup, such as: trappingof vortex lines moving either perpendicular to or along the vortex lines, aswell as at an angle; collecting ions arriving either along the vortex lines ormoving perpendicular to them; measurements of the ion mobility both alongthe vortex lines and perpendicular to them, etc. The size of the cell wasmade large to enhance the efficiency of ion trapping and the time resolutionof vortex dynamics as well as to ensure that the total number of vortexlines was much greater than unity even for very small angular velocities of∼ 0.01 rad/s. Hence, a cell of cubic geometry with sides of length l = 4.5 cmwas built. The six side plates (electrodes) that make up the cube can belabelled as “top”, “bottom”, “left”, “right”, and two “side” electrodes. Aphotograph and schematic drawing of the cell are shown in figures 1 and 2.

The top, bottom, left and right electrodes had circular grids in theircenters. All grids were made of square tungsten mesh with period 0.5 mmand wire diameter 0.020 mm, giving a geometrical transparency of 92%.The grids in the bottom, left and right plates had diameter 10 mm and wereelectrically connected to those plates. The grid in the top plate was different:it was stretched on a brass ring with inner (mesh) diameter of 13 mm and

Detection of vortices in 4He in T = 0 limit

Top flange

(o-ring surface)

Top electrode

Left electrode

and grid

Side electrode

Studding

Insulating

bushes

Top grid

Fig. 1. A view of three (out of total of six) electrode plates (top plate, leftplate, side plate) assembled on the cell flange.

outer diameter of 19 mm and located in a hole of diameter 21 mm in thecenter of the top plate. The grid was electrically isolated from the plate(allowing its use as a separate electrode/collector).

There were two injectors of electrons, one at the center of the bottomelectrode and another one at the center of the left electrode, positioned about1.5–2 mm behind the grids which separated them from the main volume ofthe cell. The injectors were field emission tips made of 0.1 mm diametertungsten wire that was electroetched in NaOH 21. The threshold for ionemission was ' -100 V and -210 V for the bottom and left injectors respec-tively.

There were also two collector plates that were placed 2.5 mm behindthe Frisch grids in the centers of the top and right plates. For experimentswhere the electric field configuration was kept steady during the measure-ments, the electrode plates and grids could be used as collectors themselves.The Frisch grids were necessary to minimize the capacitive pick-up for exper-

P. M. Walmsley et al.

Left tip

Bottom tipBottom electrode

with grid

Topgrid

Top collector

Right

collector

Right electrode

with grid

Left electrode

with grid

Top electrode Ω

Fig. 2. A side cross-section of the cell. The distance between oppositeelectrode plates is 4.5 cm. Thin vertical lines represent the array of rectilinearvortices when the cryostat rotates at an angular velocity Ω; for our typicalvalues of Ω = 0.5 rad/s, the spacing between vortices is about ten timessmaller than that shown. An example of “diverging” electric field lines isshown by dashed lines. It was calculated for the following potentials ofthe electrodes: left electrode at -90 V, side electrodes at -45 V, all otherelectrodes and collectors at 0. To inject ions, the left tip was usually keptat -305 V.

imental setups where sudden changes of electrical potentials were required,for example when the charge remaining trapped on rectilinear vortex lineshas to be promptly moved along these lines to the top collector. There weretwo conflicting requirements for the mesh size of the top grid then. In orderto minimize the pick-up, the mesh size should be much smaller then thedistance to the collector (2.5 mm). On the other hand, for the trapped ionssliding along vortices to reach the collector, the vortices have to terminateat the collector and not at the grid, meaning that the mesh size should bemuch greater than the typical equilibrium spacing between vortices at therequired angular velocity of rotation Ω. Our choice of grid mesh ensuredthat at least half of all vortices penetrated the grid at Ω ≥ 0.4 rad/s.

The electrodes were attached to a flange at the top of the cell using 3 mmdiameter brass studding and nuts, nylon bushes were used to electrically

Detection of vortices in 4He in T = 0 limit

Iright

Itop

Igrid

-90 V

-90 V

-320 V

0

0

0

240 eV

5 eV

Fig. 3. Calculated trajectories of charged vortex rings (solid lines for initialenergies H0 = 5, 15, 30, 60, 120, 240 eV) and electric field lines (dashedlines). The potentials of electrode are indicated; both side plates were keptat -45 V. In the experiment, electrons were injected from the left tip keptat -320 V. Bare ions were expected to follow the line of field (thick dashedline) and arrive in the center of top grid. Currents to the top grid, top plateand right plate were monitored and are shown in Fig. 4 versus temperature.

isolate the electrodes. A photograph of the partially assembled cell, withonly three of the electrode plates present, is shown in Fig. 1. The top flangehad eleven hermetic SMA coaxial feedthroughs for electrical connections andwas sealed to a cylindrical can (95 mm long and 70 mm inner diameter) usingan indium o-ring. The flange, can and all electrodes were made from brass.The electrodes were gold-plated and thoroughly degreased before the cell wasassembled. The total available volume in the whole cell for liquid helium was' 250 cm3. Commercial 4He gas was used in the first experimental run butsubsequent cooldowns have used isotopically pure 4He produced by Prof.McClintock’s group at Lancaster14.

The cell was mounted on a rotating cryostat22 and attached to the mix-ing chamber of the dilution refrigerator through four brass posts (each 48 mmlong, 9.5 mm o. d., 3 mm i. d.), which acted as a weak thermal link ensuring

P. M. Walmsley et al.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.0

0.2

0.4

0.6

0.8

1.0

I / I to

tal

T (K)

Top Grid Top Plate Right Plate

Fig. 4. Currents to the top grid, top plate and right plate in the experimentexplained in Fig. 3. p = 14.6 bar.

that the dilution refrigerator (Oxford 400) could be run continuously evenwhen the cell was at high temperatures. By applying power up to 5 mW toa heater mounted on the outer body of the cell it was possible to achievestable temperatures between 40 mK and 1.8 K, using a computer controlledPID algorithm. There were two separate fill lines from room temperature tothe cell, both were thoroughly thermally anchored to the 4.2 K flange and1 K pot, as well as having heat sinks at the still and 100 mK plate. Fromthe 1 K pot to the cell the lines had 0.3 mm i. d. and lengths of 1.5 m. Oneof these fill lines was used for filling the cell whilst the other enabled thepressure in the cell to be continuously monitored. It took about 30 hoursto condense helium into the cell while running the dilution refrigerator andsome 10 hours to cool the cell from 2 K to under 100 mK. Thermometry wasprovided via a factory calibrated (down to 50 mK) Ge resistor thermometer(Lake Shore GR-200A-30-0.05A, 20 kΩ at 45 mK) mounted on the top flangeof the cell.

To answer the question whether the very presence of ions affects thedynamics of the superfluid, let us compare the relevant energies. The totalkinetic energy of flow is ∼ ρl5Ω2/12; at our typical Ω ' 1 rad/s, it makes∼ 2×10−6 J. The highest possible space charge in field E = 20 V/cm wouldgain the electrostatic energy ∼ ε0E

2l3 = 3× 10−7 J. Hence, by keeping the

Detection of vortices in 4He in T = 0 limit

-500 0 500 1000 1500 2000 25000.0

0.5

1.0

1.5

2.0

2.5

3.0

Ω = 0.75 rad/sΩ = 0 Ω = 0

T = 175 mKp = 14.6 bar

I righ

t (pA

)

t (s)

Fig. 5. The absolute value of the ion current to the right plate versus timeduring starting (at t = 0, acceleration to Ω = 0.75 rad/s in 75 s) andstopping (deceleration from Ω = 0.75 rad/s to 0 between t = 1200 and 1275 s)rotation. The shaded areas indicate the regimes of uniform acceleration(left) and deceleration (right). The potentials of the electrodes are: left tip-305 V, left plate -90 V, side plates -45 V, right, top and bottom plates 0 V,producing the “diverging” field configuration as shown in Fig. 2.

space charge small and rotating sufficiently fast, the effect of the Coulombinteraction of trapped ions on the macroscopic flow can be made negligible.

4. DYNAMICS OF CHARGED RINGS

While charged vortex rings are more convenient than bare ions becauseof their much greater cross-section for charge trapping on vortex lines, theirdynamics is more peculiar, especially in a tilted electric field. Because ofhigh associated inertia of the flow, they never follow bent field lines – unlikebare ions. Hence, it is useful to know how to understand and predict theirtrajectories.

The energy H, velocity v and momentum P of a quantized vortex ringof radius R with a hollow core and no potential energy in the core are23,4

H =12ρsκ

2R(η − 2), (1)

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0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.2

0.4

0.6

0.8

1.0

I / I 0

Ω (rad/s)

p = 14.6 bar, E = 20 V/cm, T = 0.17 K I(Ω) / I(0) = 0.13 + 0.87 exp(- Ω / 0.33)

Fig. 6. Dependence of the current to the right electrode on the angularvelocity of rotation; The potentials of the electrodes are: left tip -305 V,left plate -90 V, side plates -45 V, right, top and bottom plates 0 V; the“diverging” field configuration as in Fig. 2. The line shows the fit of theexperimental points at 0 ≤ Ω ≤ 1 rad/s to (1− A) exp(−Ω/Ω∗) + A, whereΩ∗ = 0.33 rad/s and A = 0.13.

v =κ

4πR(η − 1), (2)

P = πρsκR2. (3)

where η = ln 8Ra ; the vortex core radius and the superfluid density (at p = 0,

T = 0) are a ≈ 0.8 A 4 and ρs = 0.145 g/cm3. If charged rings are producednear the field emission tip, they will enter the main volume with an initialenergy H0 ∼ 200 eV (corresponding to R ∼ 4 µm and v ∼ 2.5 cm/s). Aftertravelling to the collector subject to our typical driving voltage of some100 V the ring’s energy becomes H ∼ 300 eV (corresponding R ∼ 6 µm andv ∼ 1.6 cm/s). Increasing pressure from 0 to 25 bar will cause some 20%increase in density ρs and 30% increase in core radius a. For the same valueof the energy of a ring H, such pressure increase should make the radius Rdrop by 20%. As R À a, all these changes affect the value of η by only a fewpercent; hence, in what follows, we approximate η with a constant η ≈ 13.

Let us assume the electric field E = (Ex, Ey) does not have z−component

Detection of vortices in 4He in T = 0 limit

and the ring moves in plane xy. After using px = eEx, py = eEy, wherepx = pvx

v , py = pvy

v , v = (v2x + v2

y)1/2, we arrive at the equations for the

evolution of the components of the ring velocity,

vx =8πe

ρsκ3(η − 1)2((2v2

y − v2x)Ex − 3vxvyEy

)v, (4)

vy =8πe

ρsκ3(η − 1)2((2v2

x − v2y)Ey − 3vxvyEx

)v, (5)

These should be integrated for the particular field configuration E(x, y).Let us use these formulae in connection with an experiment. Suppose

that at t = 0 we injected a charged ring from a point x = y = 0 in thedirection y, normal to the injector plate. Hence, vx(0) = 0,

vy(0) = v(0) =κ3(η − 1)2ρs

8πH0(6)

and H0 ∼ 200 eV. The calculated trajectories for such rings with initialenergies H0 in the range between 5 and 240 eV are shown in Fig. 3 for theelectric field configuration E(x, y) corresponding to the shown arrangementof the potentials of different electrodes. The measured currents to the topgrid (effective catching diameter 20 mm, including its rim), top plate (aroundthe top grid) and right plate are shown in Fig. 4 as function of temperature.We can see that at T > 1.1 K, all ions behave as bare ions, following thebent field line from the injector in the center of the left plate to the topgrid. However, at T < 1.0 K only a third of injected ions followed the fieldline while another third arrived at the top plate but outside the central gridand another third actually arrived at the right plate in an almost ballisticmanner. These observations, combined with the calculated pattern shownin Fig. 3, imply that about a third of injected ions enter the main space ascharged rings of energy at least 30 eV, while another third appears as chargedrings too but with initial energies of some 20 eV or less. The remaining thirdof ions either never nucleated a ring at all or did it later on their way whilealready inside the main experimental volume.

5. FIRST RESULTS ON STARTING/STOPPING ROTATION

In Fig. 5 examples of transient currents after starting and stopping ro-tation of the cryostat are shown for isotopically pure 4He at p = 14.7 bar.After starting rotation, the cryostat accelerates to the angular velocity Ω =0.75 rad/s in 75 seconds. The collector current reaches a new much sup-pressed value some 500 s after starting rotation. The effect of rotation is

P. M. Walmsley et al.

reversible: after stopping the rotation (uniform deceleration within 75 s),the current returns to its original unsuppressed value after another 500 s.These transients were independent of temperatures down to 40 mK. How-ever, they looked quite differently at T > 1 K, where we do not expectcharged rings to be present at all. And no effect of rotation on the collectorcurrent was detected at T > 1.3 K, i. e. when the thermally activated escapefrom the trapping potential does not allow to keep the ion on a vortex forsufficiently long.

It is clear that the presence of vortex lines prevents some ions fromreaching the collector. The usual explanation is that ions trapped on vortexlines get quickly removed by sliding along the vortex lines up or down underthe vertical component of the diverging field (as in Fig. 2). In Fig. 6 we plotthe measured values of the collector current versus Ω. With our model oftrapping in mind, an exponential dependence of I/I0 on Ω is expected:

I = I0 exp(−Ω/Ω∗), (7)

where Ω∗ = κ/2lσ, σ is the trapping diameter, l = 4.5 cm is the distancebetween the injector and collector. While the data for 0 ≤ Ω ≤ 1 rad/slook pretty much like this formula, there are two problems. The minorone is that the current does not go to zero for higher Ω; this might be theconsequence of having a small fraction of bare ions running in parallel withcharged rings. As the trapping diameter of bare ions is expected to be muchsmaller than that of charged rings, their contribution should show little ofΩ-dependence. Bearing this in mind, the solid line in Fig. 6 models theexponentially decaying contribution from charged rings as in Eq. 7 on top ofa constant contribution from bare ions that makes some 13 % of total currentat Ω = 0. From the obtained value of Ω∗ = 0.33 rad/s, one can calculate thecharacteristic trapping diameter σ = 3.3 µm. Using the fact that ion-ringsstrongly interact with rectilinear vortex lines within their diameters19, thisimplies these are the rings of typical radius of R ∼ 1.5 µm and hence ofenergy H ∼ 60 eV.

The second problem that we face in Fig. 6 is that the current is slowlyincreasing above Ω = 2 rad/s. As yet, we do not have an explanation whatmight cause this effect. Obviously, more experiments are needed.

6. CONCLUSION

In their experiment on vortex detection, Davis et al.13 tried to eliminatevortex rings creation by using isotopically pure 4He, elevated pressure p =15.5 bar, and not too high electric field E = 50 V/cm. In our experiment,

Detection of vortices in 4He in T = 0 limit

we tried to simulate their conditions (isotopically pure 4He, p = 14.7 bar,E = 20 V/cm), but found that still a substantial fraction of ions arrives witha ring. This fact was actually beneficial as it helped to boost the ion-vortextrapping cross-section, hence sensitivity to the presence of vortices.

The observed transients of current show that, in T = 0 limit in a con-tainer of ∼ 5 cm width, it takes some 500 s for an array of rectilinear vortexlines to appear after starting rotation, or disappear after stopping rotation.The dynamics of a vortex tangle without mutual friction is complicatedas vortices are unstable and prone to reconnections and multiplication25.To understand how vortices multiply and move inside the cell, subject tomacroscopic flow, numerical simulations of the type shown in25 would bevery desirable. We defer a proper discussion of this subject until a furtherpublication.

Back in 1966, Schwarz and Donnelly wrote: “quantized vortex ringsare very sensitive “vortex-line detectors,” making them suitable probes for anumber of problems in quantum hydrodynamics”19. Indeed, we have shownthat charged rings can be used to learn about the existence and dynamics ofquantized vortex lines in superfluid 4He – even down to very low tempera-tures. In the future, we plan to employ negative ions to study the dynamicsof turbulence in superfluid 4He in the T = 0 limit.

ACKNOWLEDGMENTS

Discussions with Henry Hall, Joe Vinen and Peter McClintock as wellas help by Andrew Murray, Lev Levitin and Sio Lon Chan are acknowledged.The research is supported by EPSRC through grant GR/R94855.

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