Electrons in Superfluid Helium-4

Transcription

1 Eletrons in Superfluid Helium-4 D. Jin, W. Guo, W. Wei and H.J. Maris Department of Physis, Brown University, Providene, Rhode Island 091, USA Abstrat We review the urrent status of theory and experiment onerning eletrons in superfluid helium-4 with an emphasis on some of the unsolved problems. Although alulations of the bubble struture have been onfirmed in detail by measurements of the optial absorption spetrum, the energy of interation of a bubble with a vortex is not understood. We disuss the theory behind a proposed experiment that may shed light on the physial nature of the exoti ions. Keywords Superfluid helium, eletrons. PACS 67., 67.40Jg 1. INTRODUCTION In this paper we review some aspets of the behavior of eletrons in liquid helium that are of urrent interest. 1 An eletron injeted into helium fores open a spherial avity of radius approximately 19 Å. This radius R is suh that the total energy E is a minimum, the energy being the sum of the zero-point energy of the eletron, the surfae energy of the bubble, and the work done against the applied pressure P in forming the avity. Thus h 4π 3 E = + 4πR α + R P (1) 8 3 mr where α is the surfae tension and m is the eletron mass. A plot of the energy as given by Eq. 1 is shown in Fig. 1. This form for the energy is based on a number of simplifiations and approximations. 1 The penetration of the eletron wave funtion into the liquid is negleted (the eletron is treated as a partile in a box with rigid walls), the finite width of the liquid surfae is ignored, and the polarization of the liquid by the eletri field of the eletron is not taken into aount. At non-zero temperature there is also a ontribution to the energy from thermal flutuations. These negleted effets make a relatively small orretion to the energy; the eletron has to overome a barrier U 0 of about 1 ev to enter the helium and so the wave funtion of the eletron penetrates into the helium only a distane of around 1 Å, the 7 Å width of the liquid surfae is signifiantly less than the bubble diameter, and the polarization energy ontributes about 14% to the total energy of the bubble. 1 We will desribe alulations that allow for some of these effets later in this artile. Eletron bubbles were first studied through mobility measurements. 3 At finite temperatures the mobility of a bubble is limited by ollisions of phonons and rotons with the bubble. Exept at temperatures very lose to the lambda point, the mean free path of the exitations is larger than the size of the bubble and onsequently the drag on the moving bubble is proportional to the square of the

2 radius. Detailed theories of the mobility have been provided by Barrera and Baym 4 and by Bowley. 5 The agreement of these theories with the measured mobility provides strong evidene in support of the bubble model. More reently, a wider range of experimental probes for the study of the bubbles has been developed. These inlude optial 1,6,7,8 and ultrasoni studies. 9,10,11,1,13 Here, we disuss some of the results obtained, omment on some issues where there is a lak of theoretial understanding, and mention some possible diretions for future researh. As we will show, it is remarkable that although there are many experiments that give results in exellent and quantitative agreement, there are other experiments that give results that are extremely hard to understand. Fig. 1. Energy of an eletron bubble as a funtion of radius based on Eq. 1. The different urves are labeled by the pressure in bars.. BUBBLE ENERGETICS There is no obvious way to measure the total energy of the bubble. However, one way to test the theory of the bubble struture and energetis is through measurements of the optial absorption spetrum. The energy as given by Eq. 1 is the total energy of a bubble with the eletron in the lowest energy quantum state. When a photon is absorbed the eletron wave funtion hanges before the onfiguration of the helium an hange (Frank-Condon priniple 14 ). Thus the photon energy assoiated with an optial transition gives information about the size of the bubble when the eletron is in the ground state. Measurements have been made of the photon energies required to take the eletron from the ground 1S state to the 1P and to the P exited states; 6,7,8 these energies depend on the pressure. Careful alulations of the photon energies for these transitions have been performed by Grau et al. 15 These alulations allow for the effet of the penetration of the eletron wave funtion into the helium and use a density funtional sheme to find the variation of the density of the helium within the wall of the bubble. The agreement between theory and experiment is exellent for both transitions. The line shape has also been alulated and this too is in exellent agreement with experiment. 16 It is important to note that measurement of these optial transitions provides a more stringent test of theory than is provided by mobility measurements. The mobility is ertainly proportional to the inverse square of the radius but

3 beause of the finite width of the helium surfae it is not lear what is the effetive radius of the bubble for sattering of phonons and rotons. A seond test of the theory of the bubble struture and energetis an be made using ultrasonis. 9 One an see from Eq. 1 that the appliation of a positive pressure will make the radius of the bubble at whih the energy is a minimum smaller. Appliation of a negative pressure makes the equilibrium radius larger. At a ritial negative pressure P there is no longer a minimum in a plot of energy as a funtion of R. Beyond this pressure, the bubble is unstable and grows without limit ( explodes ). A negative pressure an be applied using a sound wave and the value of P determined. Aording to Eq. 1 the value should be /4 16 πmα P =. () 5 5h A more aurate alulation using the density funtional method has been performed by Pi et al. 18 The agreement between theory and experiment is exellent. From the two tests just desribed it would seem that the energetis of an eletron bubble are well understood. However, the same general type of theory that has just been desribed does not work when applied to two other problems. Both of these problems onern the interation of an eletron bubble with a quantized vortex line. It is well known that an eletron bubble is attrated to a quantized vortex line and an beome trapped on it. 19 The binding energy omes about beause as the bubble approahes the ore of the vortex it displaes superfluid that has a high veloity and hene a high kineti energy. Thus, when a bubble is plaed with its enter on the vortex line the energy of the system is lowered by an amount equal to the kineti energy of the liquid that has been displaed. There is also a smaller extra ontribution to the energy that omes about beause the shape of the bubble hanges when it approahes the vortex line. The binding energy has been alulated using the density funtional approah and the value obtained is K at T=0 and 97.4 K at 1.6 K. 0 This ompares with the experimental value whih is around 50 K. The reason for this large disrepany is unknown at present. MCauley and Onsager 1 have onsidered the effet of flutuations of the position of the vortex line on the binding energy but it seems unlikely that this effet an be large enough to explain the disrepany. A seond, and presumably related, problem arises from the ultrasoni experiments. 9 It is found that an eletron bubble that is attahed to a vortex line explodes at a negative pressure P vort that has a smaller magnitude than the explosion pressure P for an eletron moving freely in the liquid. A differene between these two quantities is to be expeted beause a bubble on a vortex line is subjet to not only the pressure applied to the liquid but also to an extra negative pressure due to the Bernoulli effet. However, the experimentally measured vort differene between P and P is 13% whereas the alulated differene is only 4%. 18 It therefore appears that there is something seriously wrong with the urrent understanding of the interation of vorties and bubbles. 3. BUBBLE DYNAMICS

4 The eletron bubbles have the interesting property that the wave funtion of the eletron is determined by the shape of the bubble in the helium and, at the same time, this shape is determined by the fore exerted by the eletron on the helium. For mehanial equilibrium of the surfae the ondition P m ψ = + ακ (3) must be satisfied at eah point. Here ψ is the eletron wave funtion, and κ is the sum of the prinipal urvatures of the bubble surfae. (Equation 3 is based on the same model as Eq. 1, i.e., the eletron wave funtion does not penetrate into the liquid and the helium is taken to have a sharp interfae.) If the eletron is in the ground state, the wave funtion has spherial symmetry and Eq. 3 leads to the same value for the bubble radius as does Eq. 1. If light is absorbed by the eletron ausing it to make a transition to an exited state, the wave funtion will hange and from Eq. 3 one an see that the bubble must hange size and shape. The equilibrium shapes for different quantum states have been alulated and show many interesting features. As an example, the alulated shapes of the 1P and P states for zero pressure are shown in Fig.. A harateristi feature of the P states is that the bubble is narrow at the waist. This is beause the wave funtion is zero over all of the z=0 plane and so the value of ψ is zero. Thus when the pressure is zero the sum of the prinipal urvatures in this plane must be zero. One of these urvatures is simply the reiproal of the radius of the waist of the bubble, i.e., the radius in the z = 0 plane, and so the other urvature has to be the negative of this. As the pressure is inreased, the waist of the bubble shrinks and by the time the pressure reahes 10 bars the radius of the waist is just a few Angstroms. To alulate the shape of the bubble in this regime it is learly neessary to use the density funtional method and to allow for the penetration of the eletron wave funtion into the helium. Preliminary alulations of this type have been performed by the Barelona group. 3 Fig.. Calulated shapes of the 1P and P eletron bubbles for zero pressure. The disussion just given onerns the equilibrium shape of the bubble. However, the dynamis of the bubble is perhaps of more interest. Aording to the Frank-Condon priniple, 14 one should onsider that when light is absorbed the wave funtion of the eletron hanges before there is any hange in the shape of the bubble. After the eletron wave funtion has hanged, there will be an imbalane in the fore ating on the surfae and motion will begin. When the bubble surfae starts to move, the motion will be damped by the radiation of sound away from the moving surfae and also by the sattering of thermal exitations that ollide with the surfae. If these damping mehanisms are suffiiently large, the bubble will smoothly relax to reah the equilibrium shape in whih Eq. 3 is

5 satisfied at all points on the surfae. If on the other hand the damping is small, the bubble will overshoot the equilibrium onfiguration. What happens next depends on the pressure. At zero pressure, density funtional alulations by the Barelona group find that the bubble goes past the equilibrium shape but then relaxes bak towards it. 3 At zero pressure this is predited to happen even if the temperature is very low so that there is no energy loss due to interation with thermal exitations. However, at pressures above a ritial value P s and at suffiiently low temperatures the bubble ontinues past the equilibrium shape, the waist of the bubble shrinks to zero, and the bubble splits into two. An aurate value for P s is not known at this point, but the simulations indiate that it lies between 0 and 5 bars. What happens after the bubble undergoes fission is not lear at the moment. It has been argued by Rae and Vinen 4 and by Jakiw et al. 5 that immediately after the bubble splits all of the wave funtion will be in one of the bubbles and that onsequently the other bubble will immediately ollapse. It is not lear what is meant by immediately. In Fig. 3 we show the results of omputer simulations performed to investigate this. In the simulation the wave funtion of the eletron is time-developed using Shrodinger s equation (SE) and a modified version of the Gross-Pitaevskii (GP) is used to model the motion of the helium. 6 The liquid pressure is 5 bars. The bubble starts with the eletron in the equilibrium 1S state. At t = 0 the wave funtion is hanged abruptly to the 1P state. In Fig. 3a we show the helium density at a series of later times as obtained by a time development of the SE and GP equations. The end result is two bubbles smaller than the original one and moving away from eah other. In Fig. 3b we show what happens if at the time t = 30 ps, the wave funtion in the left hand bubble is made zero and the wave funtion in the right hand bubble inreased so that the integral of ψ is unity inside that bubble. In this ase the left hand bubble ollapses and while so doing aelerates away to the left. Experiments to investigate the bubble dynamis have been performed using the ultrasoni tehnique. 11 The 1P bubbles explode at a different ritial pressure ( P = 1.63 bars) from the 1S bubbles ( P = 1.89 bars). This makes it possible to perform experiments in whih the number densities of 1S and 1P bubbles are determined. These experiments show that above a ritial temperature of around 1.5 K, 1P bubbles are produed when the liquid is illuminated with light of the appropriate wavelength to exite the 1S to 1P transition. This orresponds to the situation in whih the damping is suffiiently large that the bubbles relax to the equilibrium 1P shape and do not reah the fission point. When the temperature is lowered and the pressure is above about 1 bar the experiments show that no 1P bubbles are produed. This has to be beause fission has ourred. However, as developed so far the experiments annot determine what has happened to the bubbles after fission.

6 (a) t= (b) t= Fig. 3. (olor online) Plots of the density of liquid helium surrounding an eletron bubble. These are the results of numerial integration of the Shrodinger equation for the eletron wave funtion and the modified Gross- Pitaevski equation for the helium. The time is in units of ps and the pressure is 5 bars. In eah part of the figure the eletron is exited from the ground 1S state to the 1P state at time zero. In part (b) the time evolution is modified by abruptly hanging the wave funtion in the left hand part of the bubble to zero at a time of 30 ps.

7 4. Exoti Ions and Other Objets In several papers by different groups negatively harged objets that are different from the normal eletron bubbles have been observed. The physial nature of these ions has not been determined. Doake and Gribbon 7 performed a mobility experiment using an alpha partile soure in the liquid and deteted a fast ion in addition to the normal eletron bubble. This ion had a mobility about 7 times higher than the normal ion. Subsequently, Ihas and Sanders 8,9 used an eletrial disharge in the vapor above the liquid as an ion soure and in addition to the exoti ion were able to detet a series of 13 ions with mobility between that of the normal ion and the fast ion. They alled these objets exoti ions. These same objets have been studied in a series of papers by Eden, MClintok, and oworkers. 30 It has been found that the strength of the signals orresponding to the different exoti ions varies in a omplex way with the onditions of the eletrial disharge. Sine these ions have a higher mobility than the normal eletron bubble, they must be smaller than normal eletron bubbles. This assumption is supported by the observation that the exoti ions nuleate vorties at higher veloities than the normal ion; theory predits that the ritial veloity for vortex nuleation inreases as the size of the ion dereases. 31 Ihas 9 has alulated the radius from the mobility and found values that lie in the range between 7 and 15 Å. To date it has not been possible to find any model to explain the existene of the exoti ions. Bubbles ontaining an eletron in an exited state would be larger than a normal bubble. In addition, the lifetime of these states is muh less than the time τ d that it takes for an eletron to drift from one end of a mobility ell to the other. Helium negative ions also have a lifetime muh less than τ d. Bubbles ontaining two eletrons have also been onsidered; 9 these might have a higher mobility than normal bubbles, but alulations show that they are unstable against breakup into two single bubbles. 3 It is also unrealisti to try to explain the exoti ions in terms of negative ions of impurities. In the first plae, one would have to suppose that the liquid helium ontains 13 different impurities! In addition, in order for the bubbles formed by these impurity ions to have the size of exoti ions the impurities would have to have eletron affinities that lie in a narrow range. 1 As a more radial proposal, we have onsidered the possibility that the exoti ions are bubbles in whih some fration of the eletron wave funtion is trapped, in other words bubbles in whih the integral F of ψ over the volume of the bubble is less than unity. 33 This trapping might our, for example, when a normal bubble is exited from the ground state to a higher quantum state and the bubble then undergoes fission as already desribed. Although the observation of fast and exoti ions has been reported in several papers and by different groups, the only property of these ions that has been measured so far is the mobility. To determine the physial nature of these objets it is important to learn more about their struture. One possible way to investigate this is through measurements of the ritial pressure P at whih they explode. If the exoti ions are bubbles ontaining a fration of ψ, it is possible to alulate this pressure We have performed this alulation using the simplified

8 density funtional sheme desribed in ref. 9. We use the equation of state of helium at negative pressure as desribed in the appendix of ref. 34. We inlude in the energy a term λ ρ and hoose λ so that the model gives the orret value of the surfae energy of a free surfae of the liquid. We then solve the oupled differential equations for the eletron wave funtion and the helium density 1 f( ρ) f( ρ) ρ= + U0 ψ λ ρ ρ ρ= ρ 0 (4) h ψ + U0ρψ = Eelψ m where f ( ρ) is the energy per unit volume of helium of density ρ, ρ 0 is the density in the bulk liquid, U 0 is the energy of an eletron moving through uniform liquid of the density of helium under zero applied pressure, and Eel is the eletron energy. The numerial solution of these differential equations is straightforward and solutions an be found as a funtion of the parameter F and the density in the bulk liquid. In Fig. 4 we show how the radius of the bubble varies with the value of the quantity F. In this figure the radius of the bubble is taken to be the radius at whih the density is equal to half of the bulk liquid density. Then in Fig. 5, we show a plot of the explosion pressure P as a funtion of the radius. Thus, this alulation makes a predition for how the explosion pressure should vary with radius for the different exoti ions. Based on these results, the explosion pressure for the fast ion should be around -4 bars, roughly double the explosion pressure for the normal eletron bubble. A measurement of P for the fast and exoti ions appears feasible using the ultrasoni tehniques already developed and would provide a valuable test of the theory. In reent work it has been possible to use the ultrasoni tehnique to make movies showing the motion of individual eletrons. 35 It would be attrative to use this same tehnique to measure the pressure at whih the fast and exoti ions explode and at the same time reord the path that the ions take. Fig. 4. The radius of an eletron bubble as a funtion of the parameter F.

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