IONS AND VORTICES IN He4

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IONS AND VORTICES IN He4

G. Gamota

To cite this version:

G. Gamota. IONS AND VORTICES IN He4. Journal de Physique Colloques, 1970, 31 (C3), pp.C3- 39-C3-54. �10.1051/jphyscol:1970304�. �jpa-00213847�

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JOURNAL DE PHYSIQUE Colloque C 3, supplkment au no 10, Tome 31, Octobre 1970, page C 3 - 39

IONS AND VORTICES IN He4

by G. GAMOTA

Bell Telephone Laboratories, Incorporated Murray Hill, New Jersey

1. Introduction. - The fascinating properties of ions and vortices in cold helium have taken many new directions since the first experiment was performed more than twelve years ago. At first, mobility studies of ions were made to understand the superfluid nature of HeII. Several years later ions became useful in understanding the nature of rotating helium, and played a major role in the discovery of quantized vortex lines and rings. At that time a very stimulating event occured in science. A beautiful classical theory that had lain dormant for almost a century again came into vogue, and included predictions that were verified in a quantum world of wave functions and order parameters. The hydrodynamic theory is that of classical vortices for a perfect fluid which is most closely represented by HeII. From the classical world we jump again back to quantum mechanics as we discuss the recent discovery of the neutral particle.

These neutrals, which are thought to be excited states of the helium atom, were detected with ions and are produced by the high energy particles emitted from the radioactive source.

This review is divided into five parts. In section 2 the ion experiments are presented and the results are discussed in the framework of current thoughts regarding the structure of ions in helium. The experiments are all carried out under circumstances where the fluid is irrotational and the ions move with velocities far below critical. Reference is also made to related work in cold helium gas, solid helium crystals, as well as film flow experiments.

Section 3 introduces quantized circulation and rotating liquid helium. In many experiments ions again are used to help clarify the picture of quantized vortex lines threading rotating superfluid helium.

Quantized circulation and vortex rings are discussed in section 4. Ions played an important role in the discovery and production of these smoke rings in HeII. The unusual motion of these rings in a quantum system is remarkable, yet it is understood in terms of very old theories derived before the dawn of modern physics.

In section 5 the remaining loose ends are discussed.

Here we Iook at the interaction of ions and vortices, the creation process of vortex rings and the related

critical velocities, and briefly at the new particle referred to as the neutral.

Section 6 contains concluding remarks and a brief paragraph on the state of the art covering some unanswered problems and a few experiments that are presently underway. Reference is also made to the similarity between vortex lines in rotating HeII and rotating neutron stars (pulsars).

2. Ions. - Historically, it is fair to say that ions were first used as microscopic probes to study the superfluid nature of HeII. It soon became apparent, however, that the ion in helium itself was a very interesting entity and thus much of the research turned towards the goal of understanding the ion structure itself. In this section the study of ions in low temperature helium will be discussed. Low temperature helium is specifically emphasized rather than HeII since many of the properties to be covered are also true for dense gaseous helium and probably solid helium crystals.

2. 1. SOURCE OF IONS. - In the early work on ions some form of radioactive source was used, such as Po210 or tritiated titanium foil. The former produces a particles of the order of 5 MeV and the later P particles in the keV range. Photoemission was used in both liquid and gaseous helium experiments. Much later came the cold cathode emitters (AI-A1,-0,-Au), which produce only low energy electrons, and field emission. The high energy radioactive particles produce short tracks of ionized helium and electrons (He+ and e-). Most of the ion pairs recombine, but in the presence of an electric field it is possible to produce a sizable current (up to A is easily obtainable). The polarity of the current is determined by the sign of the electric field.

2. 2. MOBILITY EXPERIMENTS. - Most of the early work consisted of measurements of the mobility of ions in helium. This included research by Williams, Careri and his co-workers, Reif and Meyer, and Sanders and his co-workers [I]. These experiments, while not only shedding light on the properties of HeII, also paved the way for understanding the nature

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1970304

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C 3 - 4 0 G. GAMOTA of the ions themselves. Using Reif and Meyer's [2]

results, for example, which are shown in figure 1, we

FIG. 1. - The zero field mobility po on a logarithmic scale as a function of the reciprocal absolute temperature T-1 (Reif and

Meyer [2]).

see the zero-field mobility po as a function of inverse temperature. We note that the mobility for positive and negative ions is nearly the same, within n factor of two between - 0.8 OK and 2.0 OK. Furthermore p0 varies as exp[A1!kT] where A' was found to be close to the roton energy gap. This fact made it abundantly clear, that in determining the mobility, the dominant scattering process in this temperature range is by rotons. At lower temperatures po increases more slowly and seems to go as some power law, suggesting phonon scattering. Further information about the nature of the ion was obtained by the pressure dependence of the mobilities 131, as shown in figure 2. The negative ion p i first increases with increasing pressure up to about 12 atm., and then starts to decrease following very closely the value of p:. To explain this data it was argued that as pressure is increased, the density of rotons increases thereby resulting in a decrease of mobility which explains the behavior of p:. The fact that p i first increased suggested that unlike the positive ion, the negative ion is easily compressible up to a certain radius, and the reduction in size more than makes up for the increase in roton density.

From these pioneering experiments it was obvious that the ions were not simply a free electron and a He+ atom but that their nature was rather complex, their effective mass was large compared to the helium atom, and the positive and negative ions seemed inherently different in structure. Various proposals were made for their structure, but the present picture for the positive ion was given by Atkins [4], while

ATMOSPHERES

FIG. 2. -Pressure dependence of the mobilities at various temperatures. The mobility p is measured in cm2 volt-1 s-1, the pressure in atmospheres. The solid curves refer to the positive ion, the dashed curves to the negative ion (Meyer and

Reif [3]).

Kuper [5], Levine and Sanders [6], the Chicago group [7-101, and Fowler and Dexter [ l l ] described the negative ion.

2.3. POSITIVE ION SNOWBALL. - Atkins [4] proposed that the positive ion as it moves through a polarizable fluid distorts the density of the liquid.

Near the ions thus an increase in density is expected due to the strong electric field. Using classical ther- modynamics and assuming the external pressure on the liquid to be the saturated vapor pressure, Atkins obtained the result shown in figure 3. The disconti-

0.1

0 2 4 6 8 10 12 14

DISTANCE FROM ION 1

FIG. 3. - Variation in density of liquid near a localized point charge (Atkins [4]).

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IONS AND VORTICES IN He4 C 3 - 4 1

nuity in density marks the distance where the pressure has risen to the solidification point. This brought to Atkins' mind the idea that the ion might be a small solid helium snowball. He also estimated the effective mass, M*, to be about 40 m (m = mass of a helium atom) and the ion radius to be - ⁶^A. ^{At about}

the same time but working independently, Careri, Scaramuzzi, and Thomson [12] proposed various structures for the ions, and later Careri, Fasoli, and Gaeta 1131 reported results of an experiment that was performed to select the proper model for ions in liquid helium. Their conclusion was that the positive ion is a cluster of polarized atoms around one charge, and the negative ion is a large cage where the electron is self-trapped.

The negative ion and its structure will be further discussed later in this section, but for now let us concentrate on the positive ion. About this time it was pertinent to ask what was the effective mass of these carriers. Several attempts to measure the effective mass were made with cyclotron resonance, but evidently most such experiments failed due to the appearance of a new phenomena, the run-away, which we will deal with later. One effort was made to determine the effective mass, and although it was not as direct as cyclotron resonance, it was successful.

Dahm and Sanders [I41 made a measurement of the relaxation time z for the ions. The relationship between M* and z can be obtained from an equation of motion of an ion in the presence of collisions and a dc electric field E,

For steady state, dv/dt = 0 and we have

where vd is the average drift velocity. If vd is linearly proportional to E it becomes useful to introduce the concept of mobility,

Dahm and Sanders measured ^z in a microwave experiment and combined it with known mobilities to obtain values for M*. Their results show that for the positive ion, M* increases approximately from 40 m up to about twice this value near TA, as seen in figure 4. The data for the negative ion are not as quantitative but show that M* 2 100m.

Schwarz and Stark [15] have recently reported very accurate measurements of ^,u: from 1 OK down to 0.4 OK. In covering such a temperature range, they were able to study the interactions of ions with phonons, rotons, and He3 impurities. They used theory developed by Baym, Barrera, and Pethick 1161 that

FIG. 4. - Positive and negative effective masses versus temperature. The solid lines on the left represent the theoretical hydrodynamic mass for radii of 5.2, 5.7 and 6.2 8. The solid line above 1.9 ^{O K}is the theoretical effective mass in the viscous regime for radius of 5.7 A. The numbers on top are upper limits of the experimental errors for the negative ions ( ~ a h m and

Sanders [14]).

interprets mobility in terms of microscopic scattering processes. Using this model they obtained not only information on the scatterers (e. g., an independent measure of roton energy gap A which is in very good agreement with the neutron scattering results) [17], but also calculated an effective radius of the ion to be 5.0 f 0.1 A. Mobility measurements in He1 have also now been extended up to the solidification point by Keshishev, Kovdrya, Mezhov-Deglin, and Shal' nikov [18]. Agreement is found with Atkins' model.

A considerable amount of information about the ions was obtained in rotating liquid helium experiments. These will be discussed in section 5, along with ion critical velocity measurements.

2.4. NEGATIVE ION BUBBLE. - The negative ion is by far the more interesting of the two species.

Almost from the earliest mobility experiments it was clear that it was not just a bare electron but a rather complicated structure. On the basis of their experiments, as mentioned previously, Careri, Fasoli, and Gaeta 1131 suggested that the electron is in a cage, self-trapped by a shell of polarized atoms. Several other structures were proposed for the negative ion but only the cage model, the so-called bubble model remained as more experimental data were collected.

It should be noted that a similar structure was proposed several years earlier by Ferrell [19] to explain the life times of positronium in liquid helium, a topic which we will not cover except to refer to a recent theoretical paper by Hernandez and Sang-il Choi [20]

and references therein. On the theoretical side, Kuper [5] calculated various properties of this bubble and in the most recent paper by the Chicago group, Springett, Cohen, and Jortner [lo] gave a detailed account of its structure, while Fowler and Dexter [ l l ] looked at its optical properties.

Some of the strongest early evidence for the

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C 3 - 4 2 G. GAMOTA

bubble came not from liquid helium but from cold dense helium gas experiments. In measuring the mobility of photoelectrons in the gas, Levine and Sanders 161 saw an abrupt drop in mobility as the density was increased. This is shown in figure 5.

FIG. 5. - Mobility versus pressure at three temperatures. The solid curves have no theoretical significance (Levine and San-

ders [6]).

They interpreted such a large change in mobility (4 orders of magnitude) to mean that the electron while in the high mobility range is essentially free and scatters off helium atoms. When the density is high enough however, it falls into a self-entrapped cage and thus becomes a large hydrodynamic entity with a much lower mobility. The bubble model was also supported by many experiments on ions in rotating helium which will be discussed in section 5.

The reason for the bubble forming is quantum mechanical in nature. Due to the Pauli exclusion principle, the short-range interaction between an electron and a neutral helium atom is repulsive, while the weak long range polarization force is attractive, so one can visualize the electron as being trapped in a three dimensional square well [21]. That such a cavity is energetically favorable can be seen from a crude calculation which gives an equilibrium radius for the bubble. We take the total free energy of the system to be

where the first term on the right represents the kinetic energy of an electron confined to an infinitely deep square well of radius R, the second term is due to the surface tension of the fluid, and the last is the work done in creating a cavity ; me is the mass of the elec- tron, R is the radius of the bubble, y the surface

tension is taken to be y = 0.37 erg/cm2, and P the pressure is taken to be the saturated vapor pressure.

By minimizing E with respect to R, the equilibrium radius and energy are found, Re -- 19 A and E, -- 0.2 eV respectively. The significance of this energy is that the barrier that helium poses to an electron is - 1 eV as calculated by Burdick 1221 and measured by Sommer [23] and later by Woolf and Rayfield [24]. Thus falling into a cavity, the electron goes into a lower energy state, and has an extremely short life time as a free electron before it forms a bubble. Onn and Silver [25] studied the life times of electrons injected into helium from a cold cathode emitter and found them to be approximately 10-l2 s.

Accurate measurements of mobility at low temperatures were done by Schwarz and Stark [26]. The results are in extremely good agreement with theory proposed by Baym, Barrera, and Pethick [27], who treat the problem of mobility of an electron bubble in the phonon-limited temperature region

(0.3 OK < T < 0.5 OK) .

Mobility measurements below 0.3 OK were done by Neeper and Meyer [28].

One can carry the bubble model further and find that it should be possible to optically excite the electron from the well. As a result, Northby and San- ders 1291, and later Zipfel and Sanders [30], performed a series of experiments on photo-ejecting electrons from bubbles. While the results of the experiments support the validity of the bubble model, the interpre- tation of the data has recently been questioned by Miyakawa and Dexter [31]. Their arguments, however, strengthen rather than weaken the case for the bubble structure. Miyakawa and Dexter [32] also calculate the stability of electronic bubbles in liquid neon and hydrogen, and conclude that bubbles can exist in hydrogen and possibly in neon at higher temperatures.

There has also been some suggestions that the negative ion structure changes with different concen- trations of ~ eFor example, Rayfield [33] measured ~ . a reduction in the critical velocity for vortex ring creation by negative ions when small concentration of He3 were added. To account for this change, Dahm [34] presented a model which shows that He3 atoms condense on the surface of bubbles in liquid helium.

2.5. SOLID HELIUM AND FILM FLOW. - In C O ~ C ~ U -

ding this section, let us briefly mention the work on ions in solid helium and film flow. Up to now the only solid helium experiments performed were those by Shal'nikov and his co-workers [35,36]. They measured I-Vcharacteristics of a simple diode cell and found very interesting results. On the basis of their data, Cohen and Jortner [37] have recently proposed an electron in a bubble in solid helium and estimated the pressure of bubble collapse at 2 4 kbars. Related to this topic is a suggestion by Nosanow and Titus [38]

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IONS AND VORTICES IN He4 C 3 - 4 3

that under certain circumstances, screw dislocations in solid helium behave in a similar manner to vortex rings, and ions could be bound to them.

Ions were recently employed to study films. Mara- viglia [39] observed that ions can be successfully injected into helium film, and Bianconi and Maravi- glia [40] used this method to study the oscillations of two helium baths. They observed the ion current in the film modulated with a frequency twice that of the level oscillations ; a surprising and unexplai- ned result.

3. Quantized vorticity and rotating helium. -

3.1. THEORY. - Presented in this section will be a review of the work that has been reported on vortex lines in rotating helium. Early rotating helium experiments will not be included since they are already quite adequately covered [41-441, nor will I touch upon the extensive but still speculative subject of vortices in orifices and capillaries.

We will start by introducing quantized circulation and vortex lines [45] following the ideas laid down by Onsager [46], and later Feynman [47]. First, let us consider the Onsager-Feynman argument for quantized circulation K. Suppose we have a bucket of helium at rest and for sake of simplicity take it to be at 0 O K . We represent the system by the ground state wave function. For a moving system in analogy with wave mechanics, Feynman writes

where the sum is over all the atoms and S(R) is a slowly varying function on the scale of atomic spa- cings. Using eq. (5) we calculate the current density and from it obtain

Equation (6) implies that

curl v = 0 , (7)

and seems to agree in principle with Landau's [48]

assumption that the superfluid part of the HeII can exhibit only irrotational flow. For such a fluid, circulation which is defined as

must be zero since by Stokes' law

$ ^v.dl= ^$fcurl v.ds. (9) By taking the continuity equation, and assuming an incompressible fluid, we derive that div v = 0, and together with curl v = 0, we are led to conclude that v = 0 everywhere. The above statements are in

fact true but only for a fluid contained in a simply connected region (i. e., a region where any line integral contour drawn can always be shrunk to zero).

Whenever we have a multiply connected region we can have a solution with v # 0. A multiply connected region is one in which all contours cannot be shrunk to zero, e. g., a wire or a singularity like a vortex line parallel to the axis of a bucket and touching the bottom. Let us now calculate the circulation around such a circuit using eq. (6) and (8).

As we go once around the path and return back to the starting point, the wave function $J~,,, must be single valued but S can be like a phase factor changing by a multiple of 2 TC ; as a result of this,

More recently, Ginzburg and Pitaevskii [49] suggested that quantization of ^IC in He11 follows from a complex single valued order parameter, having a multiple-valued phase.

The energy argument of why we should expect vortex lines in rotating helium is as follows ; again we follow Feynman [47]. As helium is rotated with an angular velocity o, the superfluid does not move at first (i. e., v = 0 because we have a simply connected region), until it is energetically favorable for a quantized vortex line to appear parallel to the axis of rotation. The higher the o, the more vortex lines appear.

For the lowest energy state, the lines are distributed uniformly throughout the fluid and the number of lines per unit area, no, is obtained by equating the total circulation in the fluid to the circulation of a classical fluid that is rotating like a rigid body,

IC, = (nR2) no K = I v.dl = _I:'~ R . R d0=2 nwR2 ⁽¹²⁾ We picture schematically the behavior of rotating HeII in figure 6.

For a recent discussion of the hydrodynamics of rotating liquid helium refer to Andronikashvili and Mamaladze's review article [50] and references therein. Also see Fetter [51, 521, and Stauffer and Fet- ter [53] for a discussion on the distribution of vortices in a rotating vessel.

3.2. EXPERIMENTS. ^-Shortly after Feynman's theoretical paper, Vinen 1541 experimentally verified the quantization of IC ; later, but with more precise data, Whitmore and Zimmermann [55] added their findings.

The experimental apparatus in both experiments

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G . GAMOTA

He II at T = O K CLASSICAL FLUID

( a ) ( b ) ( c )

FIG. 6. - Pictorial representation of rotating fluids. (a) Vessel full of superfluid rotating at subcritical angular velocity o (simply connected region), curl v = 0 and v = 0 everywhere.

(b) At high w, many vortex lines are formed in the superfluid (multiply connected region), curl v = 0 everywhere except for the space occupied by the vortex line filaments, v $: 0. (c) Clas- sical fluid with velocity v ⁼w x r. In (b) and (c) meniscus is of

the same parabolic form.

consisted of a bucket of HeII and a fine wire in the middle parallel to the axis of rotation. The method involved measuring circulation by means of its influence on the transverse vibrations of the wire. Whitmore and Zimmermann extended Vinen's measurements by detecting the direction of circulation, studying the temperature dependence (1.2 OK to 1.9 OK), and changing the diameter of the wire. Some typical results showing that ^IChas long periods of stability at levels n = 0, 1,2, and 3 are shown as a histogram in figure 7.

RUN E - 6

GF

0 I 2 3 0 I 2 3

APPARENT CIRCULATION IN UNITS OF h/m

FIG. 7. - Time in stable levels versus apparent circulation for runs E-6 and E-7. The unshaded columns represent stable circulation observed during the last hour of each run (Whitmore and

Zimmerman [55]).

They also noted that ^IC,even though stable at one of the levels, tended to smoothly change or even reverse direction. Experiments measuring quantization of ^IC for vortex rings will be discussed in the next section.

Experimental verification of vortex lines threading rotating HeII was at first rather indirect. For instance, in the earliest experiments Hall and Vinen [56] observed anisotropy of second sound in rotating HeII.

Careri, McCormick, and Scaramuzzi [57] measured the voltage-current characteristics of ions passing through rotating HeII, and also observed an anisotropy. The reason for the anisotropy in both of these experiments was explained in terms of the interaction of vortex lines with second sound and ions. That a vortex line should interact and trap negative ions was first conjectured by Onsager and referred to in Care- ri's (et al.) paper [57].

The ion trapping problem and related experiments will be further discussed in section 5. Presently, let us describe two recent experiments on the detection of individual lines. The first is due to Hess and Fair- bank [58] and the second to Packard and Sanders 1591.

Hess and Fairbank measured the angular momentum of a very small rotating vessel for a very low o in contrast to an earlier high density line experiment of Reppy and Lane 1601. Feynman [47] predicted that as we rotate a bucket of helium, the superfluid would remain at rest until such time that

where R is the radius of the vessel and a. the vortex core taken to be - ¹^A.At higher o, the state of the lowest free energy has one vortex in the middle and the superfluid rotating. The rotating superfluid will then carry angular momentum Lo

where N is the total number of helium atoms and p,/p the ratio of the superfluid to bulk fluid density.

FIG. 8. - Angular momentum Lof the superfluid versus angular velocity w of the rotor, after helium is cooled in rotation.

Lo = Nhp,/p and wo = f i l r n ~ 2 . Each point is the average for all runs at that angular velocity (Hess and Fairbank [58]).

2 0

15

10

-1 O

\ -I

5

I I I I I 1 I

EQUILIBRIUM ANGULAR MOMENTUM /' --- ^CLASSICAL

VORTEX MODEL

/

- /'

/' -

/' / '

- /'

,/' ^-

i P

/ ^' -

/' -

-

/ '

- -

/ ' - - P

,/' --

0 -

-2- ---_

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IONS AND VORTICES IN He4

As o increased one should see a series of steps in L as new vortex lines are formed. Hess [61] theoretically predicted these steps and compared them to his experimental data. The results are shown in figure 8. Quite clearly the data follow the vortex model prediction, but unfortunately the steps are not resolved due to lack of experimental points and scatter.

Packard and Sanders [59] used the ion-trapping technique and were able to clearly see the appearance of the first few vortex lines as o was increased from zero. The actual rotating vessel was made from four resistors glued end to end with a 1 mm hole drilled down the axis. The use of conducting material facili- tated easier application of homogeneous axial electric fields. The idea behind the experiment is as follows : Electrons are injected into a rotating cylindrical vessel for a certain length of time. If there are vortex lines present, the electrons will become trapped, other- wise they will diffuse out to the walls. Once charged, an axial electric field pushes the trapped electrons to a collector, where they are counted. The number of electrons reaching the collector is proportional to the number of lines so one expects a staircase-like current output as a function of w. In figure 9 such data is shown, obviously indicating the formation of at least the first five vortex lines.

FIG. 10. -TWO vortex lines with opposite circulation & ic, a core radius ao, and separated a distance 2 R. The net velocity field at any point in space is the vector sum of the velocities from each filament. Due to the velocity field of the other, both vortices will remain together and travel to the right with velocity

u = ~ / 4 nR.

FIG. 9. - Collected charge versus time. Data taken during an acceleration. Time is measured from the beginning of rotation

(Packard and Sanders [59]).

ANGULAR VELOCITY. w (rad/s) of the vortex, which we assume to be in solid body

- ^1.5 ^1.75 ²^D

4. Vortex rings in HeII. - Unlike quantized vortex lines in He11 which can be observed up to T,, vortex rings do not exist for a long enough time to do useful experiments unless the temperature T is well below 1 ^O^K (possible vortex ring creation above 1 ^{O K}will be discussed in section 5). Vortex rings appearing in superfluid helium were mentioned in Feynman's paper [47], where he stated that vortex lines instead of ending on surfaces can bend to form a toroidal shape resulting in the familiar smoke ring velocity pattern. The rings of course also have IC quantized, but unlike a single free line they always have to move. To see this, consider a cross section of two lines of opposite circulation as in figure 10. The velocity field is shown schematically by the arrows around the core

2 4.0

-

where R is the radius of the vortex ring and a, is the core radius. Other classical expressions for a vortex ring in an incompressible, inviscid fluid are [64]

I I I

where 8 is the energy and p is the dynamical impulse.

Many experiments have been described in the litera- ture where vortex rings were conjectured to be res- ponsible for low critical velocities in various geome- tries [65], but no direct evidence for their existence was given until the experiment of Rayfield and Reif [66].

They undertook the problem of understanding the runaway phenomena or hot ions that we will discuss in section 5. With a velocity spectrometer they rotation. A rigorous derivation from classical hydrodynamics [62] shows that a vortex filament must move with the local fluid velocity that is produced by all other sources. As a consequence, the top vortex filament must move with the velocity produced from the bottom filament, and vice versa. Therefore, the pair moves as a unit with velocity v = 1c/4 nR. For a ring, we have to carefully integrate the contributions from all parts of the filament, and obtain [63]

- 9

5 ^3.0

a I- 2 o

= ^2.0 P W

5 ^I.0-

LL I-

t: i;l ₁₅₀ ₁₇₀ ₁₉₀ ₂₁₀ ₂₃₀

- $NOISE -

.- .... ... -.... -... .:-:... ... . . ..

-

,-, ... ... . . . . . .' ^.A';

-...

..,. ^.

..-... .-. . ^-

*.'. ...

-... A -... ^r,...?. ...L ^I..I... I I I

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C 3 - 4 6 G. GAMOTA

measured the velocity of these hot ions as a function of energy. The experiment was performed at 0.3 OK

to insure negligible roton density. Their results were quite fascinating. The hot ions not only appeared to behave as charged free particles in a vacuum (i. e., lost no appreciable energy in traversing field free regions), but also more surprisingly, slowed down as they gained energy. Furthermore, both a positive ion and electron in a bubble produced the same results as shown in figure 11. Using eqs (16) and (17) Ray-

RADIUS ( P )

2 .O 4 . 0 6.0

I I I I I I

0 P O S I T I V E CHARGES A NEGATIVE CHARGES

FIG. 12. - Cut off energy versus opening size. The points are experimental. The curve is theoretical, calculated from Eq. (17) using the cut off condition R = + ¹(Gamota and Sanders [68]).

FIG. 11. - Relation between the velocity u and energy E of a vortex ring. The points are experimental data for positive and negative ions. The curve is theoretical with K = (him) and

ao = 1.2 A (Rayfield and Reif [66]).

field and Reif solved for u = U(E) and obtained the solid line in the figure. The best fit to the data gave them K, = (1.000 f 0.003) x cm2/s and a, = 1.28 A. The value of ^K,is within experimental error equal to a singly quantized circulation,

Their conclusion was that the hot ions were ions somehow attached to single quantized vortex rings.

They also made a measurement of the energy loss of a vortex ring and found it to be temperature dependent following the temperature dependence of the density of phonons, rotons, and He3 impurity [67].

The classical hydrodynamic equations show that the radius of the vortex ring would be energy dependent.

Gamota and Sanders [68] conducted an experiment that measured directly this size. The experiment consisted of measuring the transmission of a beam of charged vortices through meshes of various size.

Whenever the ring grew to a size where it could no longer pass through the openings, they found a cutoff in the current. Their results are shown in figure 12,

where the cutoff energy is plotted as a function of opening size. The solid line is the best fit to eq. (17) with two adjustable parameters, rc, and a,. The best fit produced K, = 1.002 + 0.030 cm2/s and a, = 0.9 f 0.5 A. This again is in very good agreement with the notion that singly quantized vortex rings exist in HeII. In another experiment, Gamota and Sanders [69] measured the scattering cross section for vortex- vortex interaction and found the interaction to be very short range and dependent on the size of vortices, in agreement with classicaI theory. In a related experiment, Schwarz [70] measured the cross section of the interaction between a beam of vortex rings and an array of vortex lines. He also gives an approximate calculation for such an interaction based on the for- malism in J. J. Thomson's treatise [71]. Meyer [72]

studied the motion of very small rings in a transverse magnetic field and found good agreement with theory.

On the theoretical side, Amit and Gross [73], and in a series of articles Fetter [74-771, have analyzed the problem of vortices from a quantum mechanical point of view and found that whenever the ratios of the radius of the ring to the core size is large (Ria, + ^I),

the classical equations are applicable. What is strange, however, is that experimental results indicate that the equations hold very well for small radii even in the presence of a large ion, which certainly should cause a large perturbation.

All of the vortex ring experiments discussed so far have used the charge both to create and detect vortex rings. Three methods have recently been reported that might lead to an eventual neutral vortex detector.

One, which will be discussed more fully in section 5, utilizes ions to be attracted to rings, and the other

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IONS AND VORTICES IN He4 C 3 - 4 7

two make use of the energy and dynamical impulse of the ring.

Trela and Fairbank [78] measured the energy loss that occurs with the flow of superfluid helium through an orifice at critical velocity and found that the energy loss agrees with Feynman's [47] idea about the formation of quantized vortex rings whose size is that of the orifice. In the discussion of the ac Josephson effect experiments performed by Richards and Ander- son [79] and repeated by Khorana, Chandrasekhar, and Douglass [SO], one also finds reference to the passage of quantized vortices through orifices. It is not clear, however, that these vortices are, in fact, vortex rings as suggested by Zimmermann [81] or vortex lines as conjectured by Anderson [82].

In a separate experiment Gamota and Barmatz [83]

measured the momentum transfer of a vortex ring impinging on a diaphragm. To insure that there are, in fact, vortex rings present in the experimental cham- ber, these authors produced them with the help of ions, and then focussed them upon the diaphragm that was also part of a capacitor. A deflection of the diaphragm produced a capacitance change that was recorded. The amount of charge was also measured, and by assuming that each ring had one charge, a measure of relative impulse as a function of energy was found to agree with theory, eq. (18). A typical curve is shown in figure 13. The experiment was

E( ELECTRON VOLTS )

FIG. 13. - Plot of relative impulse versus energy. The solid line is the theoretical prediction, using a0 = 1.3 d and

IC = 1.0 x 10-3 cm2/s .

The experimental values are normalized to the impulse Po at 200 eV energy (Gamota and Barmatz [83]).

also repeated as a function of temperature and it was shown that even though the amount of current stayed approximately constant, the capacitance change vanished rapidly when a temperature was reached

where the roton density was sufficient to cause large energy losses of a traveling vortex ring.

In concluding this section, I would like to describe a current experiment that is in its initial stage [84] : The behavior of a pulsed beam of vortex rings. The purpose of the experiment is to study the cooperative effects of a beam of charged vortex rings. If only one sign of charge is present we have a sort of ion- vortex cloud propagating through the liquid ; if both signs of charges are present, we might expect plasma-vortex effects. For now, only some ion cloud results will be presented since the plasma behavior is still under study. The apparatus used for this purpose consists of a radioactive source, three grids, and a collector. Two of the grids are spaced close to the source while the third acts as an electrical shield and is close to the collector. A voltage is pulsed between the source and the first grid while a dc voltage is applied between the next pair of grids. A long field- free region exists in the middle of the apparatus.

Typical data are shown in figure 14, where received

FIG. 14. - Current output as a function of received pulse width.

The label 1-4 indicates the transmitted pulse width (Gamota [84]).

signals (beam shapes) at the collector are plotted as a function of the width of the transmitted pulse. If we take the ratio of half widths of each curve and plot it against the width of the voltage pulse we get a curve as shown in figure 15a. Figure 15b shows the total charge Q = Zdt for each pulse as a function of pulse duration. S :

Qualitatively, the data are understood in terms of vortex dynamics. A charged vortex in front of the charged ensemble will feel both a Coulomb force as well as a hydrodynamic force. These forces tend to

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C 3 - 4 8 G . GAMOTA

TRANSMITTED PULSE WIDTH WT ( I O - ~ S )

FIG. 15. - (a) The ratio of the received pulse half width w~ to the transmitted pulse width ^{W T}versus WT. The triangles are for the 5 eV vortex rings and circles are for 8 eV vortex rings.

(b) The total charge Q as a function of W T defined as the time integral of the instantaneous current which is measured by a

fast electrometer.

enlarge the frontal rings, slowing them down and thus compressing the beam. A similar reason accounts for the compression from the rear. In a two dimensional approximation, a wider pulse will create a stronger electric field and will cause more compression, so one expects the decrease in ratio of the widths shown in figure 15a. As the two dimensional approximation starts to fail, when the width is comparable to the beam cross section, one would expect some penetration layer of pulse compression and in the middle section of the beam an equilibrium value of charge density.

This is seen quite clearly for curves with W , > 35 ms.

5. Ions and vortices. - In this next to last section 1 want to cover the various experiments and theories

that have been presented on the interaction between ions and vortices, ion critical velocities and the crea- tion of vortex rings, and the neutrals.

5 . 1 INTERACTION BETWEEN IONS AND VORTICES. -

As mentioned previously, the first experiment to detect an interaction between negative ions and vortex lines was by Careri, McCormick and Scaramuzzi 1571.

The binding energies of ions to vortices was first estimated by Douglass [85] and Cade 1861. Douglass measured the lifetime of trapped negative ions on vortex lines around 1.5 OK, while Cade measured the lifetime of trapped positive ions on vortex rings near 0.5 OK. Their estimates for the binding energy were

- 100 OK and 10 OK for the negative and positive ions respectively. Lord Kelvin (Sir William Thornson) [87]

first suggested, before the turn of the century, that a foreign body is attracted to the vortex core. With regard to ions in helium, Parks and Donnelly [88]

proposed, more recently, the following argument that gives a similar result. They assumed that the introduction of a foreign body into the superfluid velocity field of a vortex reduces the energy of the system by an amount equal to the kinetic energy of the superfluid replaced by the foreign body. In the case where the ion is far from the vortex core, the reduction in energy, A U, is

where ~ / 2 7tr is the velocity field of a vortex line with circulation K, r is the distance from the line to the ion, and R the ion radius. Differentiating (19) we get

which yields the l / r 3 force law predicted by Lord Kelvin.

The lowest energy state of the ion-vortex line system is reached when the ion is coaxially centered on a line (a bead on a string). The binding energy as before is

In eq. (19) we assumed p, to be constant ; here we cannot because p, goes to zero as r + 0. Using an equation by Fetter [74], Parks and Donnelly wrote for P,,

r2

= P S ( ~ ) ---

r2 + a: (22)

and obtained

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IONS AND VORTICES IN He4 C 3 - 4 9

The binding energies that this model yields are

- 50 OK and - 20 OK for negative and positive ions, respectively. The exact value of the binding energy depends, of course, on the size of the ion.

Let us now look at the analysis of the life time experiments of Douglass [85], Cade [86], and Pratt and Zimmermann [89]. We follow Parks and Don- nelly [88] who make use of Chandrasekhar's Brownian particle motion approach [go]. They consider the ion as a Brownian particle trapped in a vortex potential U(r). If an electric field E in the x direction is also present, the potential becomes

This potential well with E = 0 is shown graphically in figure 16a. The vortex core is along the z direction.

As E is turned on and increased, the well becomes

FIG. 16. - (a) Potential well in the z-x plane of a negative ion in a vortex field as a function of ion distance from the vortex filament (Parks and Donnelly [88]). (b) Three dimensional view of same potential well but with electric field E applied in the x direction. As E increases the well becomes more tipped

and 0 increases.

more and more tipped in the z - x plane (8 increases), so that the well depth AU decreases, and the ion can more easily escape (Fig. 16b). The probability that N ions remain after time t is given by

where No is the total number of ions and p is the escape probability per unit time. To solve for p, Parks and Donnelly use the Fokker-Planck equation to find the distribution function of a Brownian particle in a potential well ; by doing this they derive

where AU is the well depth. The proportionality constant A depends on the details of the shape of the potential well [91]. Good quantitative agreement was found between theory and experiments, and it became clear why no binding of the positive ion can be

will bind only up to T -- 1.7OK. This comes about because

N = N o exp ( ^-^{A exp}[=kg] l) ^, ⁽²⁷⁾

so a difference of a factor of 2 in AU makes a fantasti- cally large change in N.

Tanner [92] at saturated vapor pressure and Springett [93] at higher pressures measured the cross section of capture of negative ions by vortex lines.

Again using Parks and Donnelly's approach, a measured value of the pressure dependence of the negative ion radius was obtained. This is shown in figure 17. Similar curves, although the absolute value

FIG. 17. - Electron bubble radius as a function of applied pressure. The points are experimental and the solid curve is

theoretical (Springett [93]).

at P = 0 differed slightly, were obtained by Pratt and Zimmermann [89], and also by Zipfel [94].

Ions were also used to study the properties of the vortex cores. Douglass 1951, as well as Glaberson,

FIG. 18. -Temperature dependence of the transit time of negative ions over a distance L = 4.93 cm and at the vapor pressure, P = 0. The upper curve is the trapped-ion time ; the lower curve. the free-ion time. The solid line is theoretical

expected unless T 5 0.65 OK and the negative ion (~laberson, Strayer and Donnelly [96]).

4