Financial Bubbles, Real Estate bubbles, Derivative Bubbles, and
The photomobility of electron bubbles in liquid helium
2
Volume 44A, number 2 PHYSICS LETTERS 21 May 1973 THE PHOTOMOBILITY OF ELECTRON BUBBLES IN LIQUID HELIUM J.R. FLETCHER and R.M. BOWLEY Department of Physics, University of Nottingham, University Park, Nottingham, UK Received 4 April 1973 An explanation is given for the change in the drift properties of negative ions in liquid helium when they are exposed to near infra-red radiation. In an experiment studying the mobility of electrons Average rate of momentum loss in liquid helium, Northby and Sanders [1] have (1) reported that exposure to near infra-red radiation 2ir ~klM(k)I 2(n(k)+l)ö(h~—L~--E(k)÷hk.u) produces an increase in mobility. They explained this k result by assuming that the radiation ejects electrons from bubbles in the helium into continuum states of where hw is the energy of the photon, & is the differ- higher mobility. The purpose of this letter is to pro- ence in the energy of the ‘p’ and ‘s’ states, E(k) is the pose an alternative mechanism for the mobility in- energy of the phonon (roton) of momentum hk, crease. liku is the kinetic energy lost by the electron bubble, There are several difficulties with the Northby and and n(k) is the usual Bose distribution functions bf Sanders explanation, as pointed out in their paper. phonons The most serious is that photons having energies of n(k) = {exp(~3E(k))— l}_1 only 0.7 eV cause increased mobility, whereas the depth of the potential well is believed to be ‘-l eV. The spontaneous emission term in eq. (1) averages These photons are unable to eject electrons from to zero whereas the stimulated emission term gives the bubbles. to first order in u, When a photon is absorbed by an electron bubble the electron can make a transition from the ground +u2ir ~ 12 ~ ‘s’ state to a ‘p’ state without any change of bubble k ~ IM(k) dE(k) radius. After the electronic transition the bubble relaxes to a new equilibrium size and shape for the This rate of momentum loss is in the opposite direc- new electronic state, releasing energy in the process. tions to u(dn(k)/dE(k) is negative) so that the ion From the calculations of Fowler and Dexter [2] we is accelerated. One can obtain a similar expression can estimate the energy released for the ls ~ for the rate of momentum loss due to two phonon transition to be -‘0.2 eV. If this energy creates and multiphonon emission. phonons, they are radiated preferentially in the ~ It is easy to estimate an upper limit for the extra posite direction to the motion of the bubble due to momentum given to the electron bUbble for each stimulated emission by thermal phonons. There is photon absorbed. The maximum possible effect therefore a net loss of momentum by the bubble would occur if all the phonons were radiated in the in the opposite direction to the motion, and it must backward direction giving a momentum increase be accelerated. A similar acceleration occurs during of WIs where W is the energy released in relaxation fluorescence. and s the velocity of sound. Using the mobility For simplicity let us calculate the rate of momen- measured by Schwarz and Stark [3] this momentum turn loss by the ion due to one phonon radiation cause the electron bubble to move an extra distance when it drifts at velocity u. If the matrix element of 2 X i0~m at 0.5°K. for this process is M(k), then This upper limit would not be reached in practice 121
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Transcript of The photomobility of electron bubbles in liquid helium
Volume 44A, number 2 PHYSICS LETTERS 21 May 1973
THE PHOTOMOBILITY OF ELECTRON BUBBLES IN LIQUID HELIUM
J.R. FLETCHERandR.M. BOWLEYDepartmentof Physics, UniversityofNottingham,University Park, Nottingham,UK
Received4 April 1973
An explanation is given for the changein the drift properties of negativeions in liquid helium when they areexposedto near infra-red radiation.
In an experimentstudyingthe mobility of electrons Averagerateof momentumlossin liquid helium, Northby andSanders[1] have
(1)reportedthatexposureto near infra-redradiation 2ir �~klM(k)I2(n(k)+l)ö(h~—L~�--E(k)÷hk.u)producesan increasein mobility. Theyexplainedthis k
result by assumingthat the radiationejectselectronsfrom bubblesin the heliuminto continuumstatesof wherehw is the energy of the photon,& is the differ-highermobility. Thepurposeof this letteris to pro- encein the energyof the ‘p’ and ‘s’ states,E(k) is theposeanalternativemechanismfor themobility in- energyof the phonon(roton)of momentumhk,crease. liku is the kinetic energylost by theelectronbubble,
Thereareseveraldifficulties with the Northby and andn(k) is the usual Bosedistribution functionsbf
Sandersexplanation,aspointedoutin their paper. phononsThemostseriousis thatphotonshavingenergiesof n(k) = {exp(~3E(k))—l}_1only 0.7 eV causeincreasedmobility, whereasthedepthof thepotentialwell is believedto be ‘-l eV. Thespontaneousemissionterm ineq.(1) averagesThesephotonsare unableto ejectelectronsfrom to zerowhereasthe stimulatedemissiontermgivesthe bubbles. to first orderin u,
Whena photonis absorbedby an electronbubblethe electroncanmakea transitionfrom the ground +u2ir ~ 12 ~‘s’ stateto a ‘p’ statewithout any changeof bubble k ~ IM(k) dE(k)radius.After the electronictransitionthe bubblerelaxesto a newequilibrium size andshapefor the This rateof momentumloss is in the oppositedirec-new electronicstate,releasingenergyin the process. tions to u(dn(k)/dE(k)is negative)sothat the ionFrom the calculationsof Fowler andDexter [2] we is accelerated.Onecanobtaina similarexpressioncan estimatethe energyreleasedfor the ls ~ for the rateof momentumlossdue to two phonontransitionto be -‘0.2 eV. If this energycreates andmultiphononemission.phonons,theyare radiatedpreferentiallyin the ~ It is easyto estimatean upperlimit for theextrapositedirectionto the motionof thebubbledue to momentumgiven to theelectronbUbble for eachstimulatedemissionby thermalphonons.Thereis photonabsorbed.The maximumpossibleeffectthereforea net loss of momentumby the bubble would occurif all the phononswere radiatedin thein the oppositedirectionto themotion,andit must backwarddirectiongiving a momentumincreasebe accelerated.A similaraccelerationoccursduring of WIswhere W is the energyreleasedin relaxationfluorescence. ands thevelocity of sound.Usingthe mobility
For simplicity let us calculatethe rateof momen- measuredby SchwarzandStark [3] this momentumturn lossby the ion due to onephononradiation causetheelectronbubble tomove an extradistancewhen it drifts atvelocity u. If the matrix element of 2 X i0~m at 0.5°K.for thisprocessisM(k), then Thisupperlimit would notbe reachedin practice
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Volume 44A, number 2 PHYSICS LETTERS 21 M~1973
because the phononsare not a]! radiatedin the inteiisity ratio on the temperatureandthe driftbackwarddirection. Theangulardistribution of velocity of the bubble. The signal to intensity ratioemit tedphononsis a complicatedfunction of tern- probablyvariesslightly with the anglebetweentheperatureandvelocity. For low velocitiestheeffective planeof polarisationof thelight andthe velocity olnumber of phononsradiatedin the backwarddirection theelectronbubbles.The reasonis that the radiationis proportionalto u. of phononsis anisotropic.sincein the ‘p’ statethe
The number of electronbubblesacceleratedby cavity is not spherical,aswell asbeing anisotropicthe light is proportionalto theintensity of thelight dueto thedrift velocity of theelectronbubble.and the absorptioncrosssection.If all theelectron Measurementsof the fluorescencespectrumof thebubbleswhich absorb a photon areaccelerated excitedelectronbubblewould also be of interest.sufficiently to be detected,then the signal detectedby Northby andSanderswill be proportionalto thelight intensity. The signal to intensity ratio is then Referencesproportionalto the optical absorptionspectrurn
If theelectronbubbleswhichabsorba photon are Ill l.A. Northby andTM. Sanders.Jr..Phys.Rev. Lctt.
not all acceleratedsufficiently to be detected,then 18(1967)1184.
the signal to intensity ratio will not beproportional 121 W.t3. Fowlerand1).L. Dexter,Phys. Rev. 176 (1968)
to the optical absorptio:~spectrum,althoughit should [31 k.W.SchwarzandR.W. Stark. l’hys. Rev. Let). 21
reflect the structureof thatspectrum.This could he (1968)967.
theexplanationfor thedependenceof the signal to
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