Electrons in Liquid Helium

Humphrey J. MARIS

Department of Physics, Brown University, Providence, RI 02912, U.S.A.

(Received June 13, 2008; accepted July 15, 2008; published November 10, 2008)

An electron injected into liquid helium forces open a small cavity that is free of helium atoms. Thisobject is referred to as an electron bubble, and has been studied experimentally and theoretically formany years. At first sight, it would appear that because helium atoms have such a simple electronicstructure and are so chemically inert, it should be very easy to understand the properties of these electronbubbles. However, it turns out that while for some properties theory and experiment are in excellentquantitative agreement, there are other experiments for which there is currently no understanding at all.

KEYWORDS: superfluids, helium, vortices, bubbles, ionsDOI: 10.1143/JPSJ.77.111008

1. Introduction

In this short review, we summarize experimental andtheoretical studies of the properties of electrons in liquidhelium. Electrons can be introduced into liquid helium bymeans of a radioactive source, by field-emission from asharp tip, or by photoemission from the wall of a container.An electron entering helium loses kinetic energy byionization and excitation of helium atoms (provided itsenergy is high enough) and by the production of elementaryexcitations of the liquid (rotons and phonons). After theelectron has lost its kinetic energy it forces open a cavity inthe liquid and becomes trapped in this cavity. This trappingoccurs because the energy of the ‘‘bubble’’ state is lowerthan the energy that the electron would have if it weremoving through uniform bulk liquid. We discuss the detailedcalculation of the energy of these electron bubbles in thenext section.

There are several reasons the study of these electronbubbles is of interest. The electron bubble is a unique systemwith no analogue. The bubbles form nanostructures of welldefined geometry (almost perfectly spherical below 1 K).These nanostructures have the property that, unlike otherquantum dots, when the electron is excited to a higherenergy state there is a large change in the bubble size andshape. Thus it is interesting to study what happens whenlight is absorbed by an electron bubble. When the velocityof the bubble through the liquid exceeds a critical value,quantized vortices are produced, and the bubble can becometrapped on a vortex.

2. Electron Bubble in the Ground State

We first consider why it is energetically favorable for anelectron to become trapped in a bubble. The bubble state wasfirst proposed by Careri et al.1) following an earlier proposalby Ferrell2) regarding the structure of positronium in liquidhelium An electron entering helium has to overcome apotential barrier of height U0 of approximately 1 eV.3) Thisenergy can be compared with the energy Ebubble that theelectron has when in the bubble state. This energy is thesum of the zero-point energy EZP of the confined electron,the surface energy of the bubble ES, and the work EV doneagainst the liquid pressure in creating the cavity. Thus

Ebubble ¼ EZP þ ES þ EV ¼ EZP þ �Aþ PV ; ð1Þ

where � is the surface tension of helium, A is the surfacearea, P is the applied pressure, and V is the volume of thebubble. Hence, for a spherical bubble of radius R the energyis

Ebubble ¼h2

8mR2þ 4�R2�þ

4

3�R3P; ð2Þ

where m is the mass of the electron. In the absence of anyapplied pressure, the radius corresponding to the minimumenergy is

Rmin ¼h2

32�m�

� �1=4

; ð3Þ

and the bubble energy for this radius is

Emin ¼ h2��

m

� �1=2

: ð4Þ

The surface tension of helium at low temperatures is0.375 erg cm�2.4) Hence the radius Rmin is 18.9 A and theenergy is 0.21 eV. Thus, an electron in the bubble statehas an energy considerably less than the energy U0 of anelectron moving through bulk liquid.

The calculation of the energy that has just been givenis based on several simplifying assumptions. It has beenassumed that the surface energy of the helium is simplyproportional to the area of the surface. Clearly, for a bubbleof radius 19 A there may be significant corrections to thesurface energy arising from the curvature of the surface.In the calculation of the zero-point energy of the electron, ithas been assumed that the wave function of the electrongoes to zero at the bubble wall. The penetration of thewave function into the wall gives a decrease in the electronenergy. The degree to which this penetration takes placedepends on the potential UðrÞ experienced by the electron atthe helium surface (r is the distance from the center of thebubble). This potential depends on the density profile �ðrÞ ofthe helium in the vicinity of the bubble surface. The densityprofile �ðzÞ of the planar liquid–vapor interface for heliumhas been measured,5) and the width of the interface wasfound to be approximately 7 A. The density profile �ðrÞfor the electron bubble differs from �ðzÞ because of the

SPECIAL TOPICSJournal of the Physical Society of Japan

Vol. 77, No. 11, November, 2008, 111008

#2008 The Physical Society of Japan

New Frontiers of Quantum Fluids and Solids

111008-1

curvature of the surface and because the interaction betweenthe electron and the helium further modifies the heliumprofile. Thus, it is necessary to find a self consistent solu-tion for ðrÞ and �ðrÞ. This problem has been tackled in anumber of papers,6–11) mostly by using some form of densityfunctional theory for the helium. As an example, we show inFig. 1 the results of the recent calculation by Pi et al.10)

A further correction to the energy comes from polar-ization effects. The electron inside the bubble polarizes thesurrounding helium and this lowers the total energy of thesystem. The simplest approximation is to take the electron asfixed at the center of the bubble; this then leads to apolarization energy of

Epol ¼ �ð"� 1Þe2

2R; ð5Þ

where " is the dielectric constant of the helium. In a morecorrect calculation, one can allow for the fact that theelectron is not localized at the center of the bubble buthas a probability 2 dV of being within some volume dV .Evaluation of the polarization energy averaged over allpossible positions of the electron gives a value of Epol that isincreased by a factor of 1.345 relative to the value given ineq. (5). For R ¼ 19 A, this gives an energy of 0.028 eV, a14% decrease in the total energy of the bubble. There is alsoa slight decrease in the equilibrium radius.

At a finite temperature, the shape of the bubble is modifieddue to thermal fluctuations.12) The normal modes of aspherical bubble can be classified by the usual quantumnumbers l and m. Each mode has a frequency !l (dependentonly upon l) and the amplitude has a Gaussian probabilitydistribution. Representative shapes are shown in Fig. 2. Onecan see that at a temperature of 1 K or below, the fluctuationsin shape are small. The bubble shape is also modified if thebubble is moving through the liquid. The flow of the liquidaround the bubble gives a Bernoulli pressure at the bubblesurface which changes the equilibrium shape.13) There isonly one calculation of this effect so far. In this calculationthe liquid was treated as incompressible and the effects ofthe normal fluid component were ignored. As the velocity of

the bubble increased, the shape of the bubble was found tochange from spherical to become an oblate spheroid. Abovea critical velocity of 46 m s�1, no stable shape for the bubblecould be found, and so it is not clear what happens to anelectron that is accelerated to a velocity above this value. Itwould be very interesting to repeat this calculation allowingfor the compressibility of the liquid. It is remarkable that thisvelocity above which no stable shape can be found is veryclose to the velocity at which vortices are nucleated by themoving bubble.

There is no simple way to directly measure the energyof an electron bubble. However, an interesting test of thetheory of the bubble energy can be made by an investigationof the stability of the bubble at different pressures. Considerthe simplest expression for the bubble energy as givenby eq. (2). In Fig. 3, we show the energy of a bubble as afunction of its radius for several different pressures. One cansee that, as expected, applying a positive pressure makes theequilibrium size of a bubble smaller and a negative pressuremakes it larger. But if the pressure is more negative than acritical value Pc, there is no value of the radius at which theenergy is a minimum and so beyond this point the bubblebegins to grow very rapidly, i.e., ‘‘explodes’’. This effect wasfirst predicted by Akulichev and Boguslavskii14) and wasobserved experimentally by Classen et al.8) A focusingultrasonic transducer was used to generate a high amplitudepressure oscillation within a small volume of liquid helium.If there is an electron bubble within this volume, this bubblewill explode and quickly grow to a size large enough to bedetected by light scattering. The critical pressure Pc is ap-

| |

| |

r (Å)

0

3 x 10-4

2 x 10-4

1 x 10-4

0

ψ2

(Å-3

)

ψ 2 ρ

HE

LIUM

DE

NS

ITY

(g cm-3)

0.15

0.1

0.05

010 20 30

Fig. 1. Plot of the square of the electron wave function ðrÞ and the

helium density �ðrÞ as a function of the distance r from the center of the

bubble, as calculated in ref. 10.

1 K 2 K 3 K

Fig. 2. Examples of typical shapes of an electron bubble at 1, 2, and 3 K.

The procedure for calculating these shapes is described in ref. 12.

0

2000

4000

6000

8000

0 10 20 30 40 50

RADIUS (Å)

EN

ER

GY

(K

)

P=1

P=-2

P=-1

P=0

Fig. 3. The energy of an electron bubble as a function of radius. The

different curves are labeled by the pressure in bar.

J. Phys. Soc. Jpn., Vol. 77, No. 11 SPECIAL TOPICS H. J. MARIS

111008-2

proximately �1:9 bars at low temperatures and changes withtemperature primarily because of the variation in the surfacetension. A comparison of the experimentally measuredcritical pressure8,15) with theory10) is shown in Fig. 4.

This technique in which a sound pulse is used to explodean electron bubble can be used to make a movie showingthe motion of a single electron.16) This is technically moredifficult than the experiment just mentioned because it isnecessary to generate a sound wave that has an amplitudesufficient to exceed the critical pressure not just in a smallfocal region, but over a macroscopic volume (several cm3).Sound pulses are applied at a repetition rate of 32 per secondthereby causing the electron bubble to explode and becomevisible every 31 ms. The helium is illuminated with lightfrom a flash lamp synchronized to the application of thesound. As an example of the sort of images that can beobtained, Fig. 5 shows a recording of the motion of anelectron made in this way.

3. Excited States and Optical Transitions

To further test the theory described in the previoussection, one can measure the optical absorption of thebubbles. For an electron confined in a spherical box, electric-dipole transitions from the ground state can occur to energylevels with angular momentum l ¼ 1. If the potential isinfinite outside the box, these levels have electron energy of

Enl ¼h�

2�2nl

2mR2;

where �nl is the n-th zero of the spherical Bessel functionjlðxÞ. Thus, optical transitions occur at photon energies of

"n ¼h�

2

2mR2½�2

n1 � �2�: ð6Þ

Numerical values are �11 ¼ 4:493 and �21 ¼ 7:725. Thenusing a radius of 19 A appropriate to liquid with zero appliedpressure, the energies for the lowest two optical transitionsare found to be 0.11 and 0.53 eV. Of course, these simpleestimates do not take into account the corrections discussedin the previous section, such as the penetration of the wavefunction into the liquid, the width of the helium wall, etc.A detailed calculation allowing for these effects has beenperformed by Grau et al.17) using a density-functionalmethod and the results obtained over a range of pressureare shown in Fig. 6.

The measurement of the optical absorption is verychallenging. It is hard to have a high density of electronbubbles because even in the absence of an applied electricfield, the space charge drives the electrons to the walls ofthe cell. The first measurements18–21) did not measure theabsorption directly. It was discovered that when electronbubbles were illuminated with light of the correct wave-length to excite the electron, there was a change in themobility of the bubbles. It was proposed that this change inmobility comes about because before illumination theelectron bubbles are trapped on quantized vortices, and thatthe excitation of the electrons enables them to escape.18,19,22)

We discuss this mechanism in more detail in §6. The firstexperiments in which the optical absorption was observedby direct measurement were performed by Grimes andAdams23) and by Pereversev and Parshin.24) The data ofGrimes and Adams21,23) for the transition energy as afunction of pressure are included in Fig. 6. It can be seenthat the agreement between experiment and theory isexcellent thereby confirming the basic assumptions aboutthe physics of the bubble state.

In the optical absorption experiments, it is also possibleto measure the width of the absorption line. One contributionto the width comes from the finite lifetime of the excitedstate; however, as we will show later, this contribution isnegligible. The main contribution comes about because theshape of the bubble fluctuates and for each shape there is aslightly different photon energy required to make the opticaltransition. As mentioned earlier, a bubble has a spectrumof normal modes for shape oscillations; the amplitude ofthese modes is non-zero due to both zero-point and thermalfluctuations. The theory of the line width due to shapefluctuations was first considered by Fomin25) who gave onlyan order of magnitude estimate of the line width, specifi-

TEMPERATURE (K)

0 1 2 3 4 5

-1

-2

0

1

PR

ES

SU

RE

(ba

rs)

Fig. 4. Critical pressure at which electron bubbles explode. The solid

circles are experimental results (ref. 8) and the solid curve is the theory

from Pi et al. (ref. 10). The data points inside the circle are for electron

bubbles that are attached to quantized vortices.

1 cm

Fig. 5. A still from a movie recording the position of a single electron

moving in liquid helium at 2.4 K. The dark circular area in the center of

the picture is the interior of the helium cell as viewed through a window

in the cryostat. The position of the electron is recorded every 31 ms. This

was recorded by Guo and Jin (unpublished).

J. Phys. Soc. Jpn., Vol. 77, No. 11 SPECIAL TOPICS H. J. MARIS

111008-3

cally 1012 s�1 (¼0:0007 eV), about 30 times smaller than theexperimental result. Fowler and Dexter26) gave a morecareful analysis but not including quantitatively the effect ofall normal modes of vibration. They obtained a width of the1S to 1P transition of 0.013 eV, still significantly less thanthe measured line width. A much more detailed theory of theeffects of the different modes was then worked out by Marisand Guo.12) The line width arises from the mode with l ¼ 0

(the ‘‘breathing mode’’) and also the 5-fold degenerate l ¼ 2

modes. A displacement of the bubble surface correspondingto the mode with l ¼ 0 shifts each of the three 1P levelsequally and by itself would lead to a Gaussian line shape.Displacements of the bubble surface due to the l ¼ 2

vibrational modes, on the other hand, result in a splitting ofthe 1P electronic levels and lead to a non-Gaussian lineshape. The final results of the calculation12) are in excellentagreement with the line-shape measurements of Grimes andAdams.21,23) The fluctuations in bubble shape should alsogive a significant absorption cross section for transitionsfrom the 1S state to states with D symmetry.27)

Once the electron has made a transition to an excited state,the shape of the electron bubble undergoes a large changebefore reaching a shape of mechanical equilibrium. Thepressure exerted on the bubble wall by the electron is

Pel ¼h�

2

2mjr j2: ð7Þ

For mechanical equilibrium this pressure has to be balancedby the sum of the pressure P that is externally applied to theliquid and the force due to surface tension. Hence

h�2

2mjr j2 ¼ Pþ �ð�1 þ �2Þ; ð8Þ

where �1 and �2 are the principal curvatures. Thus, if thewave function changes, the curvature of the surface also hasto change. For example, for P states the wave functionvanishes everywhere in the z ¼ 0 plane, and so the electronexerts no pressure on the wall in this plane. Thus, if theliquid pressure is zero, the sum of the curvatures �1 and �2

must be zero. Calculated bubble shapes for some of the lowenergy excited states are shown in Fig. 7.28) Note that thesedifferent states are labeled with reference to the quantumnumbers of the electron in a spherical bubble. Thus, forexample, the 1P state is the state that results by starting witha spherical bubble containing an electron with the wavefunction

/sinð�r=RÞð�r=RÞ2

�cosð�r=RÞ�r=R

� �cos �;

and then adiabatically adjusting the bubble surface and wavefunction so as to lower the total energy, i.e., the sum of theenergy of the electron and the surface energy.

Investigations of these equilibrium shapes reveal anumber of interesting effects. Because of the complexityof the problem, most of the calculations have used thesimplest model, i.e., the bubble surface has been taken ashaving a sharp interface and the penetration of the electronwave function into the helium has been ignored. Some of theresults obtained are as follows:

a) The discussion of the 1P state just given refers to astarting state 110, i.e., a state with quantum numbers

PRESSURE (bars)

(a)

EN

ER

GY

(eV

)

0.1

0.14

0.18

0.22

0 205 10 15 25

1S to 1P

25

PRESSURE (bars)

EN

ER

GY

(eV

)

0

(b)

0.4

0.6

0.8

1

1S to 2P

5 10 15 20

Fig. 6. Photon energy required to excited an electron from the ground 1S

state to the 1P and 2P states. Experimental results are from ref. 21

(crosses), ref. 23 (diamonds), and ref. 20 (open circles). The solid line

shows the results of the Orsay–Paris finite-range density functional

calculation reported in ref. 17.

20 Å

1D2P1P

1S 2S

Fig. 7. Calculate shapes of bubbles containing electrons in different

quantum states. These shapes are for zero pressure. The details of the

calculations are described in ref. 28.

J. Phys. Soc. Jpn., Vol. 77, No. 11 SPECIAL TOPICS H. J. MARIS

111008-4

n ¼ 1 (no radial nodes), l ¼ 1, and m ¼ 0. Of course,for a spherical bubble there are also states withm ¼ �1. For these states the electron pressure Pel stillhas axial symmetry and so at first sight one mightexpect that from this initial state the electron bubblecould relax to an equilibrium shape with axialsymmetry. However, it turns out that such shapes areunstable against small non-axially symmetric pertur-bations. One finds that once the possibility of suchperturbations is allowed for, the bubble evolves into ashape that is identical to the 1P bubble shown in Fig. 7but with the symmetry axis rotated.29) This has beeninvestigated for the states with n ¼ 1 and l ¼ 1 butmay possibly be a general result.

b) As the pressure is raised, the ‘‘waist’’ of the 1P bubble(see Fig. 7) decreases. By the time a pressure of 10bars is reached, the radius of the waist has decreasedto a few times the interatomic spacing, and so the useof the simple model with a sharp interface becomesquestionable. Is the 1P bubble stable under theseconditions? To investigate this it is necessary to use areliable density functional for the helium. Lehtovaaraand Eloranta11) have used a DF method to calculate theproperties of the 1P bubble at zero pressure. It wouldbe very interesting to extend this type of calculation tohigh pressures.

c) For the 2P bubble, a different type of instability occurswhen the pressure is increased to 1.53 bar.30) Thisinstability occurs because the bubble shape becomessuch that the 2P electron wave function is degeneratewith the 1F state. Once the wave function hastransitioned to 1F the waist of the bubble shrinks tozero and the bubble splits into two bubbles. The wavefunction inside each of these bubbles is of the 1P form.

d) For the starting S states 200 and 300, there is again asurprise. The shape shown in Fig. 7 for the 2S state isbased on the assumption that the bubble has sphericalsymmetry. However, Grinfeld and Kojima31) showedthat unless the pressure in the liquid is below �1:23 barthe 2S bubble is unstable against perturbations that lackaxial symmetry. Since the bubble is unstable against auniform radial expansion when the pressure is below�1:33 bar, this means that the bubble is spherical overonly a very small pressure range. In the pressure rangeimmediately above �1:23 bar, the bubble was shownto have tetrahedral symmetry (see Fig. 8). Above apressure of �0:65 bar, the tetrahedral 2S bubblebecomes unstable and relaxes to the 1D state. The 3Sbubble has been studied Maris and Guo.32) and has aneven more complex behavior.

It would be very interesting to investigate all of theseeffects using a density functional approach. In addition, itshould be noted that these calculations just described allamount to searches for equilibrium states. By this we meanthat an initial shape is assumed for the bubble (usuallyspherical), and then an investigation is made to see if a stateof lower energy can be found by make a small variation inthe shape. This process is repeated until the energy can nolonger be lowered. Thus, the procedure is purely a numericalmethod for finding the minimum energy; it is not a methodthat considers the dynamics of how a bubble actually

evolves after optical excitation. We describe what is knownabout the bubble dynamics is §6.

When the bubble changes shape, the excess energy isdissipated either by transfer to the thermal excitations in theliquid or by the radiation of sound waves. Because of thelarge change in shape of the bubble after optical excitation,the energy of the photon emitted when the bubble returns tothe ground state is significantly less than the energy of thephoton that excited the bubble. The lifetime and energy ofthe emitted photon has been calculated for some of theexcited states. For the radiative decay of the 1P state, Fowlerand Dexter26) used a highly simplified model and estimatedthe lifetime � to be 13 ms. More recent detailed calculationshave obtained values of � of 45 ms30) and 60 ms,11) andemitted photon energies of 0.042 eV30) and 0.035 eV.11) Themeasured lifetime of the 1P state is 50 ns,33) about 1000times less than the time for radiative decay. This indicatesthat the relaxation of the 1P state is dominated by some typeof non-radiative decay process, but the nature of this decayprocess is not known. The 2P state can radiatively decay tothe 1S, 2S, and 1D states and the results of a calculation30) ofthe decay rates is shown in Fig. 9. Note that according to theFranck–Condon principle34) these transitions between elec-tronic states should be considered to take place while theshape of the bubble remains constant.

There are a number of experiments that could beperformed to investigate the excited states. A measurementof the photon energies of the emitted light would provide a

50 Å

Fig. 8. Calculated shape of the 2S electron bubble at a pressure of �0:7

bar from ref. 32.

2P

2S0.197 eV

1S

1D

0.173 eV

m=±1 0.137 eVm=0 0.140 eV

m=±2

54 μs

219 μs

2.6 μs

0.047 eV

0.237 eV

Fig. 9. Calculated decay modes of the 2P electron bubble at zero pressure

(ref. 30). Note that the five 1D states are split into three different energy

levels because the initial 2P bubble is not spherical.

J. Phys. Soc. Jpn., Vol. 77, No. 11 SPECIAL TOPICS H. J. MARIS

111008-5

valuable test of the theory. This would require the detectionof rather weak radiation in the far infrared but should bepossible. A measurement of the mobility of bubbles (seenext section) in excited states could give information aboutthe bubble size but appears to be impossible to performbecause of the short lifetime. One interesting way to test thetheory is by means of cavitation experiments of the typedescribed at the end of §2. As already mentioned, the 1Sbubble becomes unstable and explodes at a critical pressureof �1:9 bar. Electron bubbles in an excited state willexplode at a negative pressure of smaller magnitude, and thecalculated values35) are listed in Table I. Thus, if a heliumcell containing electron bubbles is illuminated with light ofthe appropriate wavelength, the threshold negative pressurefor cavitation should be reduced. This effect has beendetected36) and the measured reduction in the magnitude ofthe negative pressure is in agreement with theory.

4. Exotic Ions

In this and the following sections, we turn to consider anumber of experiments that have given results that are notunderstood. The first experiments that we discuss are ionmobility measurements. To measure the mobility, electronsare first introduced into the liquid at the top of theexperimental cell. They are prevented from moving intothe main part of the cell by means of a negative potentialapplied to a grid. The voltage on this grid is then switchedto allow a pulse of ions to pass into a drift region wherethey are subject to a uniform and known electric field. Thetime that it takes the ions to reach a collector electrodeis measured and from this time, the mobility of the ionsis determined. Schwarz37) has measured the mobility ofelectron bubbles over a wide temperature range using thistime-of-flight method The theory of the mobility has beenstudied by Barrera and Baym38) and by Bowley,39) and goodagreement between theory and experiment was achieved.However, in another experiment performed in 1969, Doakeand Gribbon40) detected negatively-charged ions that had amobility substantially higher than that of the normal electronbubble negative ion. The ions were produced using an �source in the liquid. These ‘‘fast ions’’ were next seen inanother time-of-flight experiment by Ihas and Sanders (IS) in1971.41) They showed that the fast ion could be produced byeither an � or � source, or by an electrical discharge in thehelium vapor above the liquid. At 1 K, the mobility of thefast ions was about 7 times higher than the mobility of thenormal ions. In addition, they reported the existence of twoadditional negative carriers, referred to as ‘‘exotic ions’’, thathad a mobility larger than the mobility n of the normalnegative ion, but less than the mobility f of the fast ion. In

a paper the following year, IS reported on further experi-ments42,43) in which 13 carriers with different mobilitieswere detected. In these later experiments the ions weregenerated by creating an electrical discharge in the vaporabove the surface of the liquid. Measurements could bemade only in the temperature range 0.96 to 1.1 K, where thedensity of the helium vapor was such that a discharge couldbe produced. The strength of the signal associated with eachcarrier depended in a complicated way on the magnitude ofthe voltage used to produce the discharge, on the geometryof the electrodes, and on the location of the liquid–vaporinterface. Figure 10 shows the mobilities 2, 4, and 8 ofthe three exotic ions that gave the strongest signals, togetherwith the mobility of the normal bubble and the fast ion. TheIS experiments are described in detail in the thesis of Ihas.42)

More recently, these ions have been studied by Eden andMcClintock (EM).44–46) They, too, detected the fast ion andseveral of the exotic ions with different mobilities. In theirmeasurements, the mobility in large electric fields wasstudied, whereas IS had investigated the mobility in lowfields. EM showed that, like the normal ion, the exotic ionsnucleate vortices when their velocity reaches a critical valuevc. This critical velocity was found to be larger for the exoticions of higher mobility. The ‘‘fast ion’’ did not nucleatevortices. In another study, Williams et al.47) investigatedhow the production of the exotic ions was affected by thecharacteristics of the electrical discharge.

For ions moving in superfluid helium at a temperaturearound 1 K, the mobility is limited by scattering by rotons,and so the mobility should vary with temperature approx-imately as expð�=kTÞ, where � is the roton energy gap. Onecan see from Fig. 10 that log is indeed proportional toT�1. The slope of the plot of log vs T�1 is close to, butslightly larger than �. At 1 K the roton mean free path is

Table I. Calculated values of the critical pressure at which electron

bubbles explode. The calculation is described in ref. 35.

Quantum stateCritical pressure

(bar)

1S �1:89

2S �1:33

1P �1:63

2P �1:22

1D �1:49

0.92

T-1 (K-1)

1.08

MO

BIL

ITY

(cm

2 V

-1 s

-1)

40

10

20

4

2

T (K)

N

F

3

10

5

0.96 1.0

1.04 1.0

Fig. 10. Mobility of the fast ion (F), exotic ions, and normal electron

bubble as a function of temperature. Data are taken from ref. 42. The

numbers assigned to the different exotic ions are the same as in Table II.

J. Phys. Soc. Jpn., Vol. 77, No. 11 SPECIAL TOPICS H. J. MARIS

111008-6

larger than the diameter of the ion, and so the drag exertedon the ion by the roton gas is roughly proportional to thecross-sectional area of the ion, i.e., proportional to the squareof the radius R. The normal electron bubble has a radius ofaround 19 A, and since the measured mobility of the exoticions is as much as 4 times the mobility of the normal ion, theradius of the exotic ions must lie in the range between about10 and 19 A. From the measurements of the mobility, theradius of the ions can be estimated as listed in Table II.48)

The measurements by Eden and McClintock45) show thatthe critical velocity vc for the nucleation of vortex ringsby exotic ions is larger than for normal bubbles, and thatamongst the different exotic ions, vc increases as themobility increases. Theory predicts that the critical velocityof an ion should increase with increasing radius of the ion.49)

Thus, the measurements of the critical velocity also indicatethat the exotic ions are smaller than normal electron bubbles.

So far it has not been possible to find any model that canexplain the nature of these objects. If they were bubblescontaining an electron in an excited state, the bubble wouldbe larger and so the mobility would be lower. In addition,the lifetime of the excited states is much less than the time�d it takes for the exotic ions to drift the length of theexperimental chamber.50) If the exotic ions were freeelectrons, i.e., electrons not trapped in a bubble, they shouldhave a much larger mobility than that measured experimen-tally. In addition, a free electron is expected to form a bubblein a very short time. Helium negative ions do exist butalso have a lifetime much less than �d.41) Sanders and Ihas51)

have considered the possibility that the exotic ions could bebubbles containing two or more electrons. Such bubbleswould be larger than normal electron bubbles, and so onewould expect that the critical velocity for vortex nucleationwould be smaller than for normal bubbles, in contrast to theexperimental findings.52) In addition, calculations show thata bubble containing two electrons is unstable against fissioninto two bubbles each containing a single electron.11,30,53)

Finally, one could fall back on the idea that the ions aresome sort of impurity ion, but this too appears to beuntenable. One would have to suppose that because of theelectrical discharge in the vapor above the liquid, atoms aresputtered off of the electrodes that produce the discharge and

off the walls of the experiment cell. The amplitudes of thesignals due to the 13 exotic ions that are detected are roughlycomparable, i.e., lie in a range spanning about a decade.Thus, the sputtered material would have to be composed of13 different elements in roughly equal amounts. This seemsvery implausible. In addition, there are not many elementsthat when introduced into liquid helium as negative ions willform bubbles in the required size range. For the normalelectron bubble one can consider that the bubble is preventedfrom collapsing by the outward pressure exerted on the wallby the wave function of the electron. Now consider, forexample, a negative ion in which the last electron has abinding energy of Eb when the ion is in vacuum. Theelectron wave function will fall off with distance asexpð�r

ffiffiffiffiffiffiffiffiffiffiffi2mEb

p=h� Þ. If, for example, Eb ¼ 1 eV this means

that the wave function will fall off as expð�0:5rÞ where r ismeasured in Angstroms. Thus, at a distance r of 6 A, forexample, the wave function has already decreased to a verysmall value. Hence, when such an ion is placed in liquidhelium the outermost electron cannot exert enough pressureto maintain a bubble with a radius of 10 to 15 A. To explainthe existence of a bubble in this size range, it is necessary toconsider impurity atoms in which the last electron has a veryweak binding, or even atoms which do not form boundnegative ions in vacuum. This point has been made veryclearly in a paper by Grigorev and Dyugaev.54) They showthat to have a bubble radius larger than 12 A, the electronaffinity in vacuum has to be smaller than about 0.15 eV.There are a very limited number of elements that meetthis condition and it is hard to believe that 13 of theseelements could be present as impurities in the electrodes orcell walls.

It is important to note that the exotic ions have been seenonly when the electrons are introduced through the use of adischarge in the vapor above the helium; this is not anecessary condition for the production of the fast ion.40) Onecould conjecture that some unknown charged complex isformed in the ionized vapor. Another possibility is that theintense light emitted from the discharge has some effect onthe electron bubbles once they have been formed in theliquid.

5. Ions and Vortices

The idea that vorticity in a superfluid should be quantizedwas first put forward by Onsager,55) and the concept ofdiscrete quantized vortex lines was proposed by Feynman.56)

The quantization of circulation was demonstrated in anelegant experiment by Vinen.57) In 1964, Rayfield andReif58) made the remarkable discovery that under someconditions an electron will move through superfluid heliumat a velocity that decreases as the applied electric fieldincreases. It was found that at a critical velocity a fastmoving electron bubble nucleates a vortex ring and becomesattached to it. The work done on the electron by the electricfield causes the vortex ring to increase in diameter andbecause of this increase the velocity of the ring decreases. Acomprehensive review of theoretical and experimental workon vortices has been presented in a book by Donnelly.59)

The theory of the nucleation of vorticity by a movingelectron bubble has been developed by Muirhead, Vinen,and Donnelly.49) Their result for the critical velocity was

Table II. Radius of the fast ion (F) and some of the exotic ions estimated

from their measured mobility as reported in ref. 42. This table is based on

the assumption that the radius of the normal ion (N) is 19 A.

IonRadius

(A)

F 7.3

1 8.9

2 9.3

3 9.6

4 11.9

5 12.3

6 13.0

7 13.6

8 14.2

9 15.6

10 16.8

N 19

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close to the value found experimentally.60) It is generallyaccepted that the binding of an electron bubble to a vortexline comes about because when the bubble is located with itscenter on the core of the vortex line, it displaces superfluidthat has a high kinetic energy. On this basis, the bindingenergy can be shown to be61)

V0 ¼2��sh�

2R

m21þ

a2

R2

� �1=2

sinh�1 R

a

� �� 1

" #; ð9Þ

where �s is the superfluid density, R is the radius of thebubble, and a is the ‘‘healing length’’ estimated to be 1.46 A.One can attempt to experimentally determine V0 from ananalysis of the time � it takes for bubbles to escape afterbeing trapped on vortices. This time is expected to vary withtemperature as

��1 ¼ �0 expð�V0=kTÞ; ð10Þ

where �0 is the attempt frequency for escape. Because �varies very rapidly with temperature, experimental data62,63)

can be obtained only over a very narrow temperature range;at saturated vapor pressure this range is from about 1.6 to1.7 K. These data have been analyzed in different ways.Douglass62) took �0 to be independent of temperature, andfound that the temperature-dependence of � was best fit withV0 ¼ 140 K. Pratt and Zimmerman obtained a similar valuewhen they analyzed the temperature-dependence of theirdata.63) Parks and Donnelly,61) on the other hand, developeda theory of the prefactor and then chose V0 to give themeasured magnitude of � in the center of the temperaturerange over which measurements were made. This gave abinding energy of about 45 K. For this to be consistent witheq. (9) the size of the bubble would have to be significantlyless that the value of around 19 A that is indicated by theoptical absorption measurements. Several authors have triedto resolve this inconsistency. Pi et al.64) have performed adensity functional calculation using a functional scheme thathas proven to be very accurate for the calculation of otherproperties of helium.65) They find values for V0 of 104.5 K atT ¼ 0 and 97.4 at 1.6 K. Padmore66) has suggested thatthe fluctuations in the position of the vortex line caused bythe presence of the bubble will modify the temperature-dependence of the escape rate. This idea has been developedfurther in a series of papers by McCauley and Onsager,67)

but the overall situation remains unclear.It appears likely that these difficulties are related to

another problem. In §2 we described the ultrasonic experi-ments in which it was possible to measure the negativepressure Pc at which an electron bubble becomes unstableand explodes. In these experiments it is possible to alsomeasure the explosion pressure Pvort

c for electrons that aretrapped on vortices. The magnitude of Pvort

c was found to beabout 13% less than the magnitude of Pc.

8,68) Qualitatively,this difference is not surprising. The circulation of liquidaround the core of the vortex gives a Bernoulli pressurewhich makes the pressure at the bubble lower than thepressure that is applied to the bulk liquid. However, thecalculated difference between Pvort

c and Pc is only 4%. Thisdifference has been obtained both from a simple theoryalong the same lines that leads to the result eq. (9) for thebubble energy,8) and also by a detailed density functionaltheory.64) Furthermore, it is curious that while theory appears

to give too large a value for the binding energy, it gives toosmall a shift in Pc. It would have seemed more likely that ifthere is something wrong with the general theoreticalapproach, the theory would either overestimate or under-estimate the magnitude of both effects. Finally, we note thatthe theory of both effects are perhaps based on someunrealistic assumptions. In the calculation of the bindingenergy,61,64) the calculated quantity V0 has been the differ-ence in energy between two configurations. The firstconfiguration can be considered to be a straight line vortexof length L running between two parallel plates a distance L

apart, together with an electron bubble at a large distancefrom the vortex. The second configuration is with the vortexstill running between the plates and the bubble centered onthe vortex line. Thus, in the calculation it is assumed thatwhen the bubble becomes attached to the vortex line thelength of vortex line that is present decreases by approx-imately the diameter of the bubble. It would be morerealistic to consider a vortex ring but then it is not obviousby how much the length of vortex line will change when thebubble becomes attached. The same issue arises in thecalculation of the change in the critical pressure.

The ultrasonic method has also revealed that there isanother type of electron bubble (unidentified object UO).68)

This is found to explode at a critical pressure PUOc of

magnitude significantly smaller than the magnitude of Pvortc ,

i.e., jPcj > jPvortc j > jPUO

c j. Since the UO explodes at asmaller pressure, it appears likely that it is larger thanthe normal electron bubble. The UO has only been seenwhen the liquid contains a high density of vortices at lowtemperatures and so it is probable that the UO is an electronbubble attached in some way to a vortex, or possibly to morethan one vortex. Two possible origins of the UO objectshave been considered. The magnitude of the explosionpressure would be reduced if a bubble could be attached to avortex line with two quanta of circulation. However, suchvortices of higher circulation are believed to be unstableagainst decay into two normal vortices. A second possibilityis that two vortices could be attached to a single electronbubble. This, again, would give a smaller explosion pressure.This has been investigated by Pi et al.64) who find that thechange in the explosion pressure is less than what is requiredto explain the experimental data.

Finally, we mention that the trapping of electrons providesa means to obtain information about the geometry of vortexlines. Packard and collaborators69) were able to photographthe places where vortex lines in rotating helium reached thefree surface of the liquid. More recently, in the movies madeas described in §2, one can see some electrons that move onsnake-like paths through the liquid.16) It appears that theseelectrons are attached to quantized vortices and are slidingalong the vortices as they move through the liquid. Currentefforts are directed towards using this technique to obtaindetailed images of the geometry of vortices.70)

6. Dynamics of the Electron Bubble

In this review, we have so far mostly focused on the staticproperties of the electron bubbles such as the equilibriumshapes of bubbles in different quantum states. We now turnto consider situations in which the shape of the bubblechanges rapidly with time. As a first example, consider the

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111008-8

‘‘solvation process’’, i.e., the initial formation of the bubblestate. This has been discussed by Rosenblitt and Jortner.71)

They consider an initial bubble of radius �3 A and find thatthe time �expand for the bubble to reach full size is 3.9 ps. Asthe bubble grows, sound waves are radiated away into thesurrounding liquid. In principle, the growth of the bubblecould be measured by means of ultrafast optical techniquesbut this has not yet been done.72) The time �expand canbe estimated from measurements in which electrons areinjected into liquid helium from a metal surface, and theseestimates73) are in reasonable agreement with the theoreticalvalue. If the electron is excited with light of sufficiently highphoton energy, it will be ejected from the bubble into a freestate, and the bubble will then collapse. Calculations give atime of 20 ps for this to happen.71) These calculations ofRosenblitt and Jortner treat the helium as a classical fluidwithout dissipation and the bubble is taken to have a sharpsurface. It would be interesting to extend these calculationsby using density functional methods and a first step in thisdirection has been taken by Eloranta and Apkarian.74)

A more complicated dynamical problem arises when theelectron is excited not out of the bubble but into the 1P or 2Pstates. As already mentioned, according to the Franck–Condon principle one should consider that the electronictransition occurs quickly, and the bubble begins to changeshape after this transition has taken place. After excitationthe electron exerts on the wall of the bubble a pressure asgiven by eq. (7), and the bubble evolves into a non-sphericalshape. A full calculation of the evolution of the bubble shapeshould include allowance for the damping of the motiondue to the normal fluid viscosity of the helium, the radiationof sound waves into the liquid, and the use of a densityfunctional to correctly treat the density variation near thesurface of the bubble. A calculation including all of theseeffects has not been done. In Fig. 11, we show the results ofa calculation75) for a highly simplified model in which theliquid is treated as incompressible and inviscid. The electronis excited to the 1P state at time zero and the liquid moves asa result of the surface tension forces and the outward pres-sure exerted by the electron. After about 11 ps, the shape ofthe bubble is close to the equilibrium shape for the 1P bubble

as shown in Fig. 7. However, at this instant the liquid at thepoles is flowing away from the bubble center and the liquidat the waist is flowing inwards. As a result of this inertia, thebubble shape continues to change and at approximately19.5 ps the bubble breaks into two pieces of equal size. Thetwo pieces separate with a substantial velocity.

Now consider some of the limitations of this calculation.In the calculation leading to the results shown in Fig. 11, ithas been assumed that the wave function has odd parity andthat the bubble shape has even parity. As a consequence,each of the two baby bubbles that are produced has exactlythe same size and contains electron wave function of equalamplitude. At first sight, one might argue that this isextremely unlikely to be the final state if a bubble in liquidhelium at finite temperature is excited by light. Because ofthermal fluctuations, the starting bubble will inevitably besomewhat distorted from spherical. Therefore as the point offission is approached, it would seem that the electron wavefunction will simply move into whichever part of the bubblehappens to be slightly larger and the other part will collapse.It turns out, however, that because of the peculiarities ofquantum mechanics exactly the opposite happens.28) Thisis because the system is in an excited state and so theamplitude of the wave function is larger in the smallerbubble. This effect thus tends to drive the system towardsa division of the bubble into two equal parts. Whether infact this effect is large enough to cause a finite fraction ofthe wave function to end up in each baby bubble is notestablished. Elser,76) for example, has argued that beforecomplete division of the bubble takes place the wavefunction of the electron will cease to deform adiabaticallyas the bubble shape develops and that, as a result, all thewave function will end up in one of the parts. This part willexpand to become a conventional ground state bubble andthe other part containing no wave function will collapse.

Clearly, if the energy loss due to the radiation of soundand the viscosity of the liquid is above a critical value,fission will not occur, and the bubble will relax to theequilibrium 1P shape. The loss due to sound radiation shouldnot have a marked dependence on temperature and so onecan certainly expect that the sum of these two energy lossmechanisms will increase with increasing temperature.There should also be a pressure effect. As already noted indiscussing equilibrium shapes, the waist of a 1P bubbledecreases markedly as the pressure is increased. Thus, forhigh pressures a time evolution that gives even a smallovershoot of the equilibrium 1P shape will lead to fissionand so fission should occur up to higher temperatures,perhaps even above the lambda transition. Thus, one cananticipate that fission will occur only in the region ofthe P–T plane as shown schematically in Fig. 12. Belowthe ‘‘fission line’’ 1P bubbles should be produced, whereasabove the line fission should occur and so there will be no 1Pbubbles resulting from optical illumination. A quantitativecalculation of the boundary line in the P–T plane has not yetbeen reported.

The behavior indicated qualitatively in Fig. 12 has beendetected using the ultrasonic technique77) and providesstrong support for the theoretical ideas just described. If nopressure is applied to the liquid, 1P bubbles are producedeven down to a temperature of 1.3 K. At a pressure of 1 bar,

t =0 ps

t =10 ps

t =19.5 ps

Fig. 11. Results of a calculation of the shape of an electron bubble after

excitation from the 1S state to the 1P state. The details of the calculation

are described in the text.

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1P bubbles are seen only above a temperature of approx-imately 1.5 K. Under the pressure–temperature conditionssuch that no 1P bubbles are produced by optical excitation, itappears that instead other objects are produced that require alarger negative pressure to explode. The nature of theseobjects is not understood at present.

Although this experiment provides strong evidence thatfission of the bubble does occur, it does not determine whathappens to the wave function of the electron. Jackiw et al.78)

and Rae and Vinen79) argue that even if some part of thewave function of the electron ends up in each of the babybubbles, the final state of the system will always be onebubble containing all of the wave function of the electron,and that this bubble would be no different from an ordinaryelectron bubble. If this is really what happens and it happensin a very short time (e.g., a time of the order of the 10 ps),then under conditions of pressure and temperature such thatfission of the bubble does occur, there should be no net effectof light on the electron bubbles, i.e., the bubbles should allreturn to their original state after the light is absorbed.However, this is not consistent with what is seen in theultrasonic experiment just mentioned.77)

Is there a connection between the fission process and theincrease in mobility that is seen when bubbles in helium areilluminated with light of the right wavelength to exciteoptical transitions? Northby and Sanders18) noted that themobility enhancement was observed only for fields greaterthan about 1 kV cm�1 at 1.3 K, and suggested that it ‘‘mayinvolve the interaction of the ion with turbulence orviscosity’’. A field of 1 kV cm�1 at this temperature shouldgive a velocity80) of around 6 m s�1, much less than thecritical velocity at which an electron bubble nucleatesvortices, so it is not clear why the mobility enhancementoccurs only above this field strength. But they also noted thatthe enhancement became unobservable above about 1.7 K.This certainly supports the idea that vortices are involvedsince it is known that at this temperature the trapping timefor electron bubbles on vortices becomes very short. Zipfeland Sanders19,20) studied the mobility enhancement as afunction of pressure. They found that the temperature atwhich the signal disappeared decreased as the pressure wasraised, becoming 1.34 K when the pressure reached 16 bar.This provides further support for the vortex interpretationbecause, from the results of Pratt and Zimmerman,63) the

escape time at zero pressure and 1.7 K (the condition atwhich the Northby–Sanders signal disappeared) is very closeto the escape time at 16 bar and 1.34 K. These escape timesare roughly 0.3 s. Grimes and Adams21) also attributed themobility enhancement that they saw to the escape of electronbubbles from vortices as a result of illumination, and againfound that there was a strong correlation between thetemperature at which the signal disappeared and the temper-ature at which the escape time dropped to a critical value.It is interesting to note, however, that Grimes and Adamscould detect no enhancement at pressures below 1 bar,whereas Northby and Sanders, and Zipfel and Sanders, diddetect a signal. It is possible that this is because Grimes andAdams were looking at the 1S! 1P transition whereas theother experiments studied the 1S! 2P transition. Elser81)

proposed that the signal in the Grimes and Adams experi-ment came about because after excitation to the 1P state thebubble undergoes fission, and the baby bubble that does notcontain the electron collapses releasing heat. He proposedthat if the pressure is below 1 bar, fission will not occur andso the amount of heat released will be reduced and notsufficient to enable the surviving bubble to escape from thevortex. The assumption that fission does not occur below apressure of 1 bar is consistent with the ultrasonic experi-ments already mentioned.77) Moreover, one would expectthat for the 1S! 2P transition the amount of heat releasedshould be larger, whether or not fission occurs, and so anexcited bubble can escape even at zero pressure.

An alternative possibility is that the bubble escapes fromthe vortex ‘‘mechanically’’. If fission occurs after the lightis absorbed, the two baby bubbles will be ejected with asubstantial velocity in opposing directions. Whether or notwe accept the idea that one of them immediately collapses(whatever ‘‘immediately’’ means!), clearly the survivingbubble (or bubbles if each baby bubble contains a fractionof the wave function) has a chance of escaping from thepotential binding it to the vortex just due to its high velocity.Since the ultrasonic experiments indicate that fission doesnot occur below 1 bar, one would not expect to see electronbubbles escaping from vortices in this pressure range. This isin agreement with experimental observations.

The initial experiments that we have been discussing didnot provide an estimate of the probability that an electronbubble will escape from a vortex after absorbing a photon.This probability has now been measured82) and found to be�10�4; surprisingly small. This measurement was made at1 K and a pressure of 1 bar. It would be very interesting tovary the temperature and the pressure and to check if thissmall probability is in fact consistent with the results of themobility enhancement experiments.

The final question to discuss is the possible relationbetween fission and the exotic ions. The exotic ions aresmaller than normal electron bubbles; could they be bubblescontaining a fraction of the wave function of the electron?28)

As already noted, Jackiw et al.78) and Rae and Vinen79) haveargued against this possibility. For this to be an explanationof the exotic ions there would need to be a mechanism bywhich bubbles containing at least 13 different fractions of theelectron probability (integral of j j2) are produced. In theexotic ion experiments, the electrical discharge in the vaporabove the liquid surface produces intense light over a broad

FISSION

NO 1P BUBBLES

1P BUBBLES PRODUCED

TEMPERATURE

PR

ES

SU

RE

Fig. 12. Qualitative plot of the regions of the P–T plane in which fission

occurs and in which 1P bubbles are produced when light is absorbed.

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spectrum. This light will be absorbed by electron bubbles inthe liquid and will cause different types of fission processesand will excite electrons out of the bubbles. It would bevery valuable to perform more experimental studies of theexotic ions to investigate their properties more carefully.For example, the results reported so far indicate that therelative strength of the signals due to the different ions varysignificantly with the position of the liquid level and thecharacter of the light emitted in the electrical discharge. Whatis the correlation between the spectrum of the emitted lightand the appearance of different ions? Is there a correlationbetween the intensities of different ions, e.g., when ionnumber 3 gives a strong signal is ion number 8 strong too,thereby implying some relation between their origins? Thistype of information cannot be extracted from the existingdata and might give a clue to the origin of the ions.

7. Summary

We can summarize the current understanding of electronbubbles in liquid helium as follows:

a) The basic structure of these objects has been confirmedthrough accurate measurements of the photon energiesrequired to cause optical absorption. The measuredenergies are in excellent agreement with detaileddensity functional calculations, and the shape of theabsorption lines is well understood. The negativepressure at which the 1S ground state bubble becomesunstable has been measured and is in good agreementwith theory.

b) The equilibrium shapes of electron bubbles in excitedstates have been calculated based on highly simplifiedmodels. It would be worthwhile to perform moredetailed density-functional calculations of the proper-ties of these objects and to detect the light that isemitted when they relax back to the ground state. Theexcited states undergo non-radiative decay by someunknown mechanism.

c) There are significant difficulties in understanding theinteraction of electron bubbles with vortices. Thecalculated binding energy is not in agreement withexperiment and the effect of vortices on the explosionpressure is not understood.

d) The exotic ions and the fast ion have been detected inseveral experiments by different groups. These ions aresmaller than the normal electron bubble. There is stillno understanding of the nature of these objects. This isremarkable when one considers that one is dealing herewith electrons in a very pure liquid composed of atomswith no chemical properties. In addition to the exoticand fast ions, another object has been detected that islarger than the normal electron bubble. It is clear thatmore experiments are needed to study these fascinatingobjects.

Acknowledgments

The author wishes to thank A. Ghosh, W. Guo, M. Hirsch,D. Jin, D. Konstantinov, and W. Wei for their contributionsto this research. He thanks S. Balibar, F. Caupin, and G. M.Seidel for helpful discussions. This work was supportedin part by the United States National Science Foundationthrough Grant DMR-0605355.

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48) This table is based on the radius estimate made by Ihas in his thesis.

Ihas took the radius of the normal electron bubble to be 16.1 A. We

have multiplied Ihas’ estimates of the radii of each exotic ion by a

factor of 19/16.1 to allow for the fact that elsewhere in this review we

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have taken the radius of the normal electron bubble to be 19 A. Note

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system used in Ihas’ thesis.

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75) H. J. Maris: unpublished.

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81) See ref. 26 of ref. 21.

82) D. Konstantinov, M. Hirsch, and H. J. Maris: AIP Conf. Proc. 850

(2006) 163.

Humphrey Maris was born in England in 1939.

He obtained his B. Sc. (1960) and Ph. D. (1963)

degrees from Imperial College, London. He was a

research associate at Case Institute from 1963 to

1965. Since 1965 he has been a professor of physics

at Brown University. He has worked in many areas

of physics, including low temperature physics and

liquid helium, ultrafast optics, neutrino detection,

ultrasonics and the development of new techniques

for semiconductor metrology.

J. Phys. Soc. Jpn., Vol. 77, No. 11 SPECIAL TOPICS H. J. MARIS

111008-12