Highly inhomogeneous 4 He systems from Density Functional calculations

1. Highly inhomogeneous 4He systemsfrom Density Functional calculations Francesco Ancilotto Physics Department “G. Galilei” - University of Padova and INFM-Democritos National Simulation Center (Trieste) RPMBT14 Barcelona, Jul 16-20 (2007)

2. Acknoledgements: • Manuel Barranco, Marti Pi, Ricardo Mayol, Alberto Hernando (University of Barcelona) • Frederic Caupin (Ecole Normale Superieure - Paris) • Flavio Toigo, Stefano Paolini (University of Padova & INFM-DEMOCRITOS )

3. Phase Diagram of 4He Liquid helium: the prototypic quantum fluid • Helium is the only element that remains a liquid at T=0. • The reasons for that are (a) large zero-point oscillations of atoms and • (b) the binding forces between the atoms are very weak. • Both isotopes (fermionic 3He and bosonic 4He) become superfluids below a critical temperature. the “” transition line 4He It is generally believed that the superfluid transition is connected to Bose-Einstein condensation

4. Superfluid 4He nanodroplets: a unique nanomatrix for molecules and molecular complexes Why 4He nanodroplets ? cold, liquid, inert Tracing the onset of superfluidity: ~ 60 4He atoms suffice ! (Grebenev, Toennies, Vilesov, Science279, 2083 (1998)). a nearly ideal “spectroscopic matrix” for studying atomic/molecular species at ultra-low temperature (which may be unstable or weakly interacting in the gas phase).

5. Experiments onHelium droplets • Adiabatic expansion cools helium to below the critical point, forming droplets. • Droplets then cool by evaporation to: T=0.38 K, (4He) superfluid T=0.15 K, (3He) normal fluid • The droplets are sent through a scattering chamber to pick up impurities, and are detected either with a mass spectrometer with electron-impact ionizer or a bolometer. Toennies and Vilesov, Ann. Rev. Phys. Chem. 49, 1 (1998)

6. Pure 3He droplets • T=0.15K • Broad peak Relative Depletion [%] Relative Depletion [%] • Pure 4Hedroplets • T=0.38K • free rotor spectrum • (with increased • inertia) Wave Number Change [ cm -1] 4He nanodroplets are superfluid Experiment: (Toennies et al. Science, 1998) How small can a superfluid droplet be? NHe4~ 60 !

7.  is fixed by the normalization condition  (r) dr = N (i) Vext r is the external potential (e.g., the fluid-substrate potential: Vext  Vrep(z) - C3/z3 ) (ii) The Density Functional Theory for fluids The grand-potential  is a functional of the fluid densityr: •  r  E r -     r Vext r dr •  r is minimumat  = 0 (equilibrium density)

8. The phenomenological Density Functional theory for liquid 4He (Dupont-Roc et al., 1990; Dalfovo et al, 1995; F.A. et al, 2000) • The non-local term Ecnlaccounts for correlation effects due to the short-range part of the He-He interaction • A few (temperature-dependent) parameters are fitted to reproduce experimental values of uniformliquid 4He

9. Phenomenological DFT for 4He: a quite accurate description of inhomogeneous (large!) systems 4He: the liquid-vapor surface tension 4He film on a Li surface Expt. Calc.  (L.Szybisz, M.Boninsegni JLTP 2004) (F.A., F. Faccin and F. Toigo, PRB 2000) DFT: a valid alternative to “exact” (but more computationally demanding) methods (like, e.g., Path-Integral Monte Carlo)

10. DFT svsl cos  lv (F.A., A. Sartori and F. Toigo, PRB 1998) The 4He/Cs contact angle: DFT vs. experiment Liquid 4He wets any surface, with the exception of heavy alkali (Cs, Rb) surfaces 4He droplet at T = 0 (3000 atoms)  Cs surface

11. Alkali-doped 4He clusters • Most impurities are stable inside a 4He cluster • Exceptions: alkali atoms*(and some alkaline-earth) bind to the surface of the cluster • Such “dimple” states are weakly • bound (Eb~10 K) • Spectroscopic studies of alkali- doped 4He nanodropletsmay provide useful informations on surface excitations of the droplet Rb atom NHe=300 *F.A. et al., Z. Phys. B 1995; F. Stienkemeier et al., Z. Phys. D 1996

12. Excitation and decay of an alkali atom on the surface of 4He Excited  state (frozen 4He) Ground state Cs atom Cs excitation Liquid 4He 4He relaxation Excited  state (relaxed 4He) Cs de-excitation A ring structure of 4He “atoms” spontaneously form around the Cs atom

13. Be+ ion in a 4He cluster DFT ■■■ PIMC Phenomenological DFT for 4He: a quite accurate description of inhomogeneous (large!) systems... but not TOO inhomogeneous! 4He film on a Li surface (L.Szybisz, M.Boninsegni JLTP 2004) Thanks to S.Paolini (UniPd) A modified DF has been proposed which accurately describes highly inhomogeneous4He systems (F.A., M.Barranco, F.Caupin, R.Mayol, M.Pi, PRB 72 (2005)

14. Solid 4He from DFT calculations At high pressure a low-energy solid-like phase is found ! hcp crystal structure: two views * *surfaces of constant density = L

15. L s Solution: add a penalty term in the energy functional which inhibit excessive (unphysical!) pile-up of density Problem: 4He “atoms” are too localized: the resulting solid is too dense compared with experiments A density-dependent “ceiling” potential term in the effective potential of the 4He Schrodinger equation It has no effect in the region of “physical” densities (OK !)

16. Equation of States for 4He: DFT vs. experiment lines: DFT calculations dots: experiments liquid solid From a “double-tangent” construction one can derive the freezing pressure P = - (E/N)/(1/) (and also the liq/sol densities) 1/S 1/L P = 25.8 atm (expt.: 25 atm ) S =0.0294 Å-3 (expt.: 0.0286 Å-3) L =0.0263 Å-3 (expt.: 0.0260 Å-3 )

17. The liquid-solid interface at T=0 • Helium is among the few substances in which the liq-sol • surface tension  can be measured:  =0.17 10-3 N/m • (O.A. Andreeva and K.O. Keshishev, 1991) • Results from DFT calculations: a stable liquid/solid interface can be obtained (at the freezing pressure): liquid solid • We find  ~ 0.1 10-3 N/m Results in quantitative agreement with VMC calculations (Pederiva, Ferrante, Fantoni, Reatto (1994)

18. Be+@He70 DFT ■■■ PIMC Positive ions immersed in liquid 4He Experiments: Na+ and Mg+ ions-doped 4He nanodroplets (P.Class, S.O.Mende and F.Stienkemeier, Rev. Sci. Instr. 2003) • The ion polarizes the He atoms: a region of increased density develops due to electrostriction • A solid-like cage (“snowball”) may develop around the ion (whose structure depends on the ion-He interaction potential

19. Be+@He70 Mg+@He70 Ca+@He70 Positive ions doped 4He clusters • VMC calculations(Rossi, Galli, Reatto, Phys. Rev. B (2004) : • positive ions solvated in 4He: Na+, K+, Cs+, Be+, Mg+ Persistence ofarigid structurein the1st shell (from ground-state PIMC calculations) S. Paolini, F. Ancilotto and F. Toigo, J. Chem. Phys. (2007)

20. Alkali & alkali-earth ions in bulk liquid 4He DFT radial 4He density around the ion The potential well depth decreases with increasing the ion atomic number

21. M Positive ions moving in liquid 4He: the hydrodynamical mass Effective mass due to the displaced fluid: M* = M + (1/2) mliquid • Hydrodynamical (irrotational) flow of 4He around a • spherical solute can be computed using the equilibrium • 4He density profile around the ion (K. Lehman, PRL 2002) • Good agreement with VMC results:classical • hydrodynamics works quite well for a quantum fluid!

22. Effective mass of positive ions in bulk liquid 4He from DFT calculations • Compute the equilibrium density profile (r)using DFT • (ii) Use the hydrodynamical approximation to compute the • effective mass from (r) • We get semi-quantitative agreement with VMC calculations • only for alkaline-earth ions • Alkali ions: the 1st solvation layer must be treated as a fixed mass moving rigidly with the ion

23. ● Rydberg atom 4He cluster Density Functional study of Scolium G.Scoles, unpublished Golov and Sekatski, 1991 e- The electron is (temporarily) prevented from neutralization by the helium-electron repulsion ( energy barrier ~ 1 eV ) • Scolium lifetime must be long enough to allow for its detection • Possible detection of Scolium in recent experiments (M.Drabbels, to be published)

24. 4He freezing pressure e- + He Possible solidification of Scolium • The outer electron exerts an additional electrostatic pressure, which may cause the 4He solidification • A simple model (no electrostriction • effects) predicts that sufficiently • small 4He clusters may turn solid • Electrostriction should favor solidification of larger clusters

25. ion-doped He cluster Scolium Squeezing a 4He cluster with a Rydberg electron • DFT calculations of a Scolium made of Be+ doped 4He nanodroplets enveloped by an electron • (F.A., M.Pi, M.Barranco, R.Mayol, K.Lehmann, submitted to JPC) (1) (2) • Cheng, Cole & • Cohen (1994) (2) Bellert & Beckenbridge (2002)

26. Squeezing a 4He cluster with a Rydberg electron • The electron pressure forces ~10 additional 4He atoms in the second solvation shell Angular density distribution of 4He atoms in the 2nd shell Ion-doped 4He cluster Scolium • Dynamical criteria are necessary to determine the • character (solid or liquid) of Scolium

27. Conclusions • Density Functional Theory (in its more recent implementation) provides a quite accurate description of extended & highly inhomogeneous4He systems like, e.g.: • the solid phase & the liquid-solid interface • Positive ions in liquid 4He • “Scolium”: an electron circulating around a positively charged 4He nanodroplet

28. Solid helium Liquid Helium solidifies at T=0 only at pressures P>25 atm (in a hcp crystal structure) Supersolid 4He discovered recently! A solid like no other: cold, solid helium flows like a liquid “…. abrupt drop in the rotational inertia of the confined solid below a certain critical temperature. The most likely interpretation of the inertia drop is entry into the supersolid phase. If confirmed, our results show that all three states of matter—gas, liquid and solid—can undergo Bose–Einstein condensation…..” (E.S.Kim and M.Chan, Nature 2004) 4He

29. Vortices in 4He clusters from DFT calculations • Possible mechanisms for producing vortices in 4He droplets : adiabatic expansion, cavitation,… (J.D.Close et al., JLTP 1998) • Open issues: existence? stability? dynamics? • Detection of vortices in 4He clusters from spectroscopic • measurements of atomic impurities trapped by the • vortex line • Ideal candidates: atoms which are (i) barely stable on the surface of a pure clusters but are (ii) trapped inside the cluster when a vortex is present….

30. v· dl = (2ħ/MHe) Vortices in 4He clusters from DFT calculations • 4He field velocity v = (ħ/MHe) v = 0 • a rotational motion of the fluid can be obtained only • through the nucleation of vortex lines Fixed cluster in a rotating frame • Rotating cluster • Equilibrium configuration by minimizing • E´ []= E [] - z  (r) Lz (r) dr NHe=300 ^ • A quantizedvortex spontaneously develops

31. Vortices in Ca-doped 4He clusters from DFT calculations • (b) and (c): Stable states of Ca in a 4He nanodroplet (with and without vortex): • A Ca atom on the surface is drawn in • the interior of the He nanodroplet when • a vortex is present • The excitation/emission spectra of Ca • should be very different in the two cases, • allowing in principleto detect vortices in • 4He nanodroplets (F.A., M.Barranco and M.Pi, PRL 2003)

32. Helium-solvated ions: solid-like vs liquid-like structures “snowballs” or “bubbles” ? The paradox of ion mobilities in liquid 4He: (Glaberson et al., 1975) Alkali ions:  decreases as Z increases Alkali-earth ions: increases as Z increases ! (in the simplest model one would expect ~ R-1 ~ Z-1 )