Infrared absorption and emission spectrum of electron bubbles attached to linear vortices in liquid 4 He

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1 Journal of Low Temperature Physics - QFS2009 manuscript No. (will be inserted by the editor) Infrared absorption and emission spectrum of electron bubbles attached to linear vortices in liquid 4 He D. Mateo A. Hernando M. Barranco M. Pi Received: date / Accepted: date Abstract Within finite-range density-functional theory, we have addressed the infrared absorption and emission spectrum of electron bubbles attached to linear vortices in liquid 4 He as a function of pressure. We have found that the presence of vortices affects very little the absorption spectrum, only causing a small shift in the 1s 2p peak. The energy of the lowest emission transition is also shown as a function of pressure for a vortex-free bubble and for a trapped bubble. In the emission energy the shift induced by the vortex line is proportionally bigger, especially when the waist around the electron probability density of the 1p state collapses, which happens at a pressure of 8 bar. Keywords Electron bubbles Infrared spectrum Vortices in liquid helium PACS dk E D- 1 Introduction Electron bubbles (e-bubbles) produced by electrons injected into liquid helium have been the subject of many experimental and theoretical investigations, see e.g. Refs. [1, 2] and references therein. In particular, spectroscopic studies on e-bubbles have drawn considerable attention [3 5], and the infrared absorption spectrum of the electron bubble in liquid 4 He has been analyzed in detail in a wide region of the pressuretemperature (P T ) plane from P = 0 to the solidification pressure, and up to temperatures above 4 K [6,7]. On the theory side, the density functional (DF) approach has been successfully used to obtain the 1s 1p and 1s 2p absorption energies as a function of pressure [8, 9], yielding values in good agreement with experiment [6, 7]. More recently, this formalism has been also applied to the description of one- and two-electron bubbles in superfluid 4 He, addressing in particular the de-excitation of 1p and 1d non-spherical solvation structures at zero pressure [10]. These calculations Supported in part by Grant No. FIS /FIS from DGI, Spain (FEDER). Departament E.C.M., Facultat de Física, Universitat de Barcelona. Avg. Diagonal 647, Barcelona, Spain

2 2 have further confirmed the suitability of DF theory to address inhomogeneous liquid helium, and have stressed the relevance of a correct description of the surface thickness of the electron bubble to achieve a quantitative description of the experimental data. In their first paper on the absorption spectrum of e-bubbles, Grimes and Adams [6] have used a photoconductive mechanism to detect the transitions which appeared to be associated with trapping of bubbles on vortices in the superfluid, and argued that in their experimental setup, roughly for P < 18 atm and T < 1.6 K, e-bubbles injected from field-emission tips spend a large fraction of their transit time through the cell trapped on vortex lines [7]. It is thus quite natural to address the influence of vortex lines on the absorption spectrum, and in particular, on the transition energies. Even within the simpler bubble model in which the surface thickness of the electron bubble is neglected [4, 5], to take properly into account the deformation caused by the vortex line in the e-bubble is not a trivial task [11]. Assuming that this deformation is small, it is possible to construct a deformed bubble model that allows to determine the structure of the e-bubble in the presence of a vortex line, and to obtain the shift it causes in the transition energies [7]. We have recently described e-bubbles attached to vortex lines within DF theory [12]. Our approach is fully variational and selfconsistent, and does not make any assumption about the shape and structure of the deformed bubble. It is based on the use of an accurate DF to describe the superfluid [13], and of an electron-helium (e-he) effective interaction [14]. These are the same ingredients we have employed to address e-bubble cavitation in liquid helium [15], as well as the infrared spectrum of electron bubbles [9]. In the present work, we extend our previous studies to investigate, within DF theory, the effect of linear vortices on the infrared absorption and emission spectrum of electron bubbles in liquid 4 He. 2 Method We have used the Orsay-Trento (OT) density functional [13] neglecting the dynamical terms that mimic backflow effects. The inclusion of these terms in three dimensional (3D) calculations is extremely cumbersome and very time consuming from the computational viewpoint. In recent applications of the OT functional to some dynamical problems, these terms have been also neglected [16,17], or have been found to contribute minimally to the studied process [8]. We use the electron-he interaction derived by Cheng et al. [14]. This allows us to write the energy of the electron-helium system as a functional of the electron wavefunction Φ(r) and the 4 He order parameter Ψ(r) = ρ(r) exp[ıs(r)], where ρ(r) is the particle density and v(r) = h S(r)/m He is the velocity field of the superfluid. Varying the functional with respect to Ψ and Φ one obtains a system of two coupled equations, an Euler-Lagrange equation for the liquid and a Schrödinger equation for the electron, that have to be solved selfconsistently. We refer the reader to Ref. [12] for a thorough discussion of how we have generated the vortex lines and the method we have used to solve the variational equations.

3 3 3 Results 3.1 Absorption spectrum ±3 ± ±3 ± Fig. 1 Lowest lying electron energy levels (ev) at P = 0 (left panel) and 20 atm (right panel) for a spherical e-bubble and for an e-bubble attached to a vortex line. We have obtained the electron spectrum in the mean field created by the superfluid using two different approaches. The simplest one invokes the Franck-Condon principle, and consists in solving the Schrödinger equation for the electron keeping frozen the helium configuration. This approximation is a priori well justified in view of the very different time scales for the helium and electron evolution, given their large mass difference. In this case, we have implemented a conventional Gram-Schmidt orthogonalization scheme to obtain a sufficient number of electron states to work out the dipole response. Alternatively, one may go beyond the Franck-Condon approximation and solve the coupled dynamical equations that result from the time-dependent DF approach for helium, and the time-dependent Schrödinger equation for the electron. Schematically, ı h t Ψ(r, t) = H HeΨ(r, t) ı h t Φ(r, t) = H eφ(r, t). (1) These equations are solved imposing as initial conditions the stationary solution of the superfluid, Ψ(r, t = 0) = Ψ 0 (r), and giving to the electron ground state wavefunction Φ 0 (r) a dipolar boost in the r direction, namely Φ(r, t = 0) = e ik r Φ 0 (r). If k is small enough, the Fourier analysis of the expectation value k r yields the dipole infrared response of the system [18]. In practice, we have solved Eqs. (1) using a fourth order Runge-Kutta method for the first time steps, and have used this information

4 4 to determine the solution for subsequents time values using Hamming s (predictormodifier-corrector) method [19]. This procedure is very robust and accurate, even for large amplitude motions [20]. As expected, we have found that both methods sensibly yield the same dipole spectrum, and we will only discuss the infrared spectrum obtained within the Franck-Condon approximation. Figure 1 displays the lowest lying electron energy levels for the e-bubble configurations at P = 0 and 20 atm considered in this work. In the spherical, vortex free case, they are labeled as nl by the principal quantum number n = 1, 2,... and the value of the orbital angular momentum l = 0 (s), 1 (p), 2 (d),... Each level is 2(2l + 1)-fold degenerate, the electron spin playing no role. In the presence of a vortex line, the energy levels are labeled by the principal quantum number and the projection m of the orbital angular momentum on the symmetry z-axis. The ±m states are degenerate. We have found that the states arising from a spherical (nl) level are ordered in increasing m values. With vortex Spherical With vortex Spherical S(ω) S(ω) ω (ev) ω (ev) Fig. 2 Left panel: dipole strength function (arbitrary scale) at P = 0 atm for a spherical e-bubble (solid line) and for an e-bubble attached to a vortex (dashed line). The inset displays the strength in logarithmic scale. Right panel: same as left panel for P = 20 atm. The dipole strength function S(ω) = n r 0 2 δ(ω ω n0 ) (2) n 0 is displayed in Fig. 2 at P = 0 and 20 atm, for both the spherical e-bubble and for that attached to a vortex line. To make the representation more realistic, the absorption peaks have been convoluted with a Lorentzian whose width has been taken from the experiment (about 1/6 of the 1s 1p excitation energy [7]). We recall that to reproduce the experimental width, we should have considered the coupling of the electron dipole excitation with the -quadrupole- shape fluctuations of the e-bubble [21]. At P = 0 atm, in the absence of vortices the 1s 1p transition is at ev and takes a 97.1 % of the oscillator strength. The 1s 2p transition is at ev and takes a 2.5 % of the oscillator strength [9]. At P = 20 atm, in the absence of vortices the 1s 1p transition is at ev and takes a 97.4 % of the oscillator strength, whereas the 1s 2p transition is at ev and takes a 2.4 % of the oscillator strength.

5 5 Figure 2 clearly shows that the effect of vortices on the absorption spectrum is very small, especially on the 1s 1p transition. More sizeable changes appear in the 1s 2p transition at low pressures. This larger effect is expectable as the 2p electron wavefunction penetrates more in the liquid than the 1p one, and it is more affected by any change in the structure of the bubble. Unfortunately, this transition carries little strength. We mention that the splitting of the spherical 1d and 1f states yields m = states that may be excited from the ground state by the dipole interaction, see Fig. 1. However, the strength of these transitions is so weak that it does not show up in the absorption spectrum, Fig Emission spectrum Z (Å) E (mev) Free e-bubble With vortex X (Å) P (bar) Fig. 3 Left panel: e-bubble configurations with and without a vortex line, relaxed around the 1p state for P = 0, 5, and 15 bar. The equidensity lines correspond to 0.1 to 0.9 ρ 0, where ρ 0 is the density of liquid helium at the given pressure. Right panel: emission energy of the 1p 1s transition as a function of pressure. Dashed line, deformed bubble attached to a vortex. The last published estimate of the radiative lifetime of the 1p state is about 60 µs [10]. We have obtained 56 µs, while the value obtained by Maris is 44 µs [22]. This time is much longer than the relaxation time of the liquid and, as a consequence, the e-bubble adapts its shape to the 1p electron probability density before decaying by photoemission to the deformed 1s state; therefore the bubble configuration at the time of emission can be obtained by minimizing the total energy of the system while keeping the electron in the first excited state. At P = 0, we have found that the 1p 1s emission energy is 36 mev, close to the 35 mev found in Ref. [10]. This agreement is an excellent test of the numerical method used by us and by these authors, on the one hand, and on the physical ingredients of both calculations on the other hand. We have carried out systematic calculations of the e-bubble equilibrium configuration relaxed around the 1p state as a function of P, and show some of them in Fig. 3. At pressures below 8 bar, the vortex line has little effect on the bubble configuration, and the presence of the vortex lowers the emission energy by a rather constant amount of 4 mev. Above 8 bar, the situation suddenly changes. The waist of the bubble collapses as a result of the pressure applied to the liquid (see e.g., the P = 15 bar configuration in Fig. 3). Interestingly, the presence of the vortex line prevents the waist from collapsing,

6 6 as it expels the superfluid off the vortex line. It can be seen that the emission energy of the free e-bubble drops from the P = 0 value by an order of magnitude. This is expectable, as the main difference between the 1s and 1p probability densities at these pressures appears in the waist region. If this region is inaccessible to the electron due to the presence of helium, these states become almost degenerate. When the bubble is trapped in a vortex line, a small waist whose extent is about the vortex core radius, 1Å, holds together the two baby bubbles around the 1p lobes. This is the reason why the emission energy smoothly depends on P at the point where the e-bubble starts to split. 4 Summary We have studied the absorption spectrum of an e-bubble attached to a vortex line as a function of pressure, and have found that it changes very little with respect to the free e-bubble spectrum. We have also studied the structure of the bubble relaxed around the 1p state as a function of P. At P = 0, we essentially reproduce the results obtained by previous authors [10,22]. We have found that the collapse of the 1p bubble at about 8 bar leads to the formation of two distinct baby bubbles. These bubbles may be held together by a tiny waist if the collapsing bubble is attached to a vortex line. Some of these results may change if the collapse of the excited bubble is dynamically addressed, even adiabatically [2]. Work along this line is currently in progress. References 1. M. Rosenblit and J. Jortner, J. Chem. Phys. 124, , (2006); ibid. 124, (2006). 2. H. J. Maris, J. Phys. Soc. of Jpn. 77, 1 (2008). 3. J. A. Northby and T. M. Sanders, Phys. Rev. Lett. 18, 1184 (1967). 4. W. B. Fowler and D. L. Dexter, Phys. Rev. 176, 337 (1968). 5. T. Miyakawa and D. L. Dexter, Phys. Rev. A 1, 513 (1970). 6. C. C. Grimes and G. Adams, Phys. Rev. B 41, 6366 (1990). 7. C. C. Grimes and G. Adams, Phys. Rev. B 45, 2305 (1992). 8. J. Eloranta and V. A. Apkarian, J. of Chem. Phys. 117, (2002). 9. V. Grau, M. Barranco, R. Mayol, and M. Pi, Phys. Rev. B 73, (2006). 10. L. Lehtovaara and J. Eloranta, J. Low. Temp. Phys. 148, 43 (2007). 11. B. DuVall and V. Celli, Phys. Rev. 180, 276 (1969). 12. M. Pi, R. Mayol, A. Hernando, M. Barranco, and F. Ancilotto, J. Chem. Phys. 126, (2007). 13. F. Dalfovo, A. Lastri, L. Pricaupenko, S. Stringari, and J. Treiner, Phys. Rev. B 52, 1193 (1995). 14. E. Cheng, M. W. Cole, and M. H. Cohen, Phys. Rev. B 50, 1136, (1994); Erratum ibid. 50, (1994). 15. M. Pi, M. Barranco, R. Mayol, and V. Grau, J. Low Temp. Phys. 139, 397 (2005). 16. F. Ancilotto, M. Barranco, and M. Pi, Phys. Rev. Lett. 91, (2003). 17. L. Lehtovaara, T. Kiljunen, and J. Eloranta, J. of Comp. Phys. 194, 78 (2004). 18. M. Pi, F. Ancilotto, E. Lipparini, and R. Mayol, Physica. E 24, 297 (2004). 19. A. Ralston and H.S. Wilf, Mathematical methods for digital computers (John Wiley and Sons, New York, 1960) 20. M. Barranco, R. Guardiola, S. Hernández, R. Mayol, and M. Pi, J. Low Temp. Phys. 142, 1 (2006). 21. H. J. Maris and W. Guo, J. Low Temp. Phys. 137, 491 (2004). 22. H. J. Maris, J. Low Temp. Phys. 132, 77 (2003).