Highly inhomogeneous 4He systems

from Density Functional calculations Francesco Ancilotto

Physics Department “G. Galilei” - University of Padova

andINFM-Democritos National Simulation Center (Trieste)

Barcelona, Jul 16-20 (2007)RPMBT14

Acknoledgements:

• Manuel Barranco, Marti Pi, Ricardo Mayol, Manuel Barranco, Marti Pi, Ricardo Mayol,

Alberto HernandoAlberto Hernando (University of Barcelona) (University of Barcelona)

• Frederic CaupinFrederic Caupin (Ecole Normale Superieure - (Ecole Normale Superieure -

Paris)Paris)

• Flavio Toigo, Stefano PaoliniFlavio Toigo, Stefano Paolini (University of (University of Padova & INFM-DEMOCRITOS )Padova & INFM-DEMOCRITOS )

•

Phase Diagram of 4He

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.

4He the “” transition line

Liquid helium: the prototypic quantum fluid

Both isotopes (fermionic 3He and bosonic 4He) become superfluids

below a critical temperature.

It is generally believed that thesuperfluid transition is connected to Bose-Einstein condensation

(Grebenev, Toennies, Vilesov, Science 279, 2083 (1998)).

Superfluid Superfluid 44He nanodroplets: a unique He nanodroplets: a unique nanomatrix nanomatrix for molecules and molecular complexes for molecules and molecular complexes

Why 4He nanodroplets ?

cold, liquid, inert

Tracing the onset of superfluidity:

~ 60 4He atoms suffice !

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).

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) superfluidT=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)

4He nanodroplets are superfluid

Re

lati

ve

De

ple

tio

n

[%]

• Pure 3He droplets

•T=0.15K •Broad peak

Experiment: (Toennies et al. Science, 1998)

Wave Number Change [ cm -1]

Re

lati

ve

De

ple

tio

n

[%]

• Pure 4He droplets

• T=0.38K

• free rotor spectrum (with increased inertia)

How small can a superfluid droplet be?

NHe4 ~ 60 !

The Density Functional Theory for fluids

The grand-potential is a functional of the fluid density r:

r is minimum at = 0 (equilibrium density)

is fixed by the normalization condition (r) dr = N

Vext r is the external potential (e.g., the fluid-substrate

potential: Vext Vrep(z) - C3/z3 )

(i)

(ii)

r E r - r Vext r dr

The phenomenological Density Functional theory for liquid 4He

(Dupont-Roc et al., 1990; Dalfovo et al, 1995; F.A. et al, 2000)

A few (temperature-dependent) parameters are fitted to reproduce experimental values of uniform liquid 4He

The non-local term Ecnl accounts for correlation effects

due to the short-range part of the He-He interaction

Calc.Expt.

(F.A., F. Faccin and F. Toigo, PRB 2000)

4He film on a Li surface

(L.Szybisz, M.Boninsegni JLTP 2004)

Phenomenological DFT for 4He: a quite accurate description of inhomogeneous (large!) systems

DFT: a valid alternative to “exact” (but more computationally demanding) methods (like, e.g., Path-Integral Monte Carlo)

4He: the liquid-vapor surface tension

The 4He/Cs contact angle: DFT vs. experiment

4He droplet at T = 0 (3000 atoms)

cos svsl

lv

DFT

(F.A., A. Sartori and F. Toigo, PRB 1998)

Liquid 4He wets any surface, with the exception of heavyalkali (Cs, Rb) surfaces

Cs surface

Alkali-doped 4He clusters

Rb atom

NHe=300

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 nanodroplets may

provide useful informations on surface excitations of the droplet

* F.A. et al., Z. Phys. B 1995; F. Stienkemeier et al., Z. Phys. D 1996

Ground state

Excited state(relaxed 4He)

Excited state (frozen 4He)

Excitation and decay of an alkali atom on the surface of 4He

Cs atom

Liquid 4He

Cs excitation

Cs de-excitation

A ring structure of 4He “atoms” spontaneously form around the Cs atom

4 He rela

xation

4He film on a Li surface

(L.Szybisz, M.Boninsegni JLTP 2004)

Phenomenological DFT for 4He: a quite accurate description of inhomogeneous (large!) systems...but not TOO inhomogeneous!

Be+ ion in a 4He cluster

■■■ PIMCDFT

Thanks to S.Paolini (UniPd)

A modified DF has been proposed which accurately describes highly inhomogeneous 4He systems (F.A., M.Barranco, F.Caupin, R.Mayol, M.Pi, PRB 72

(2005)

Solid 4He from DFT calculations

hcp crystal structure: two views *

* surfaces of constant density = L

At high pressure a low-energy solid-like phase is found !

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

L s

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 !)

Equation of States for 4He: DFT vs. experiment

liquid

solid

lines: DFT calculationsdots: experiments

From a “double-tangent” construction one can derive the freezing pressure

P = - (E/N)/(1/)

(and also the liq/sol densities)

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 )

1/S 1/L

The liquid-solid interface at T=0

Results from DFT calculations: a stable liquid/solid interface can be obtained (at the freezing pressure):

We find ~ 0.1 10-3 N/m

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)

solidliquid

Results in quantitative agreement with VMC calculations(Pederiva, Ferrante, Fantoni, Reatto (1994)

Positive ions immersed in liquid 4He

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 Be+@He70

DFT

■■■ PIMC

Experiments: Na+ and Mg+ ions-doped 4He nanodroplets(P.Class, S.O.Mende and F.Stienkemeier, Rev. Sci. Instr. 2003)

Positive ions doped 4He clusters

Persistence of a rigid structure in the 1st shell (from

ground-state PIMC calculations)

Be+@He70 Mg+@He70 Ca+@He70

S. Paolini, F. Ancilotto and F. Toigo, J. Chem. Phys. (2007)

VMC calculations (Rossi, Galli, Reatto, Phys. Rev. B (2004) :

• positive ions solvated in 4He: Na+, K+, Cs+, Be+, Mg+

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

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

M

(K. Lehman, PRL 2002)

Good agreement with VMC results: classical hydrodynamics works quite well for a quantum fluid!

Effective mass of positive ions in bulk liquid 4He from DFT calculations

Alkali ions: the 1st solvation layer must be treated as a fixed mass moving rigidly with the ion

(i) 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

Density Functional study of Scolium

●

The electron is (temporarily) prevented from neutralization by the helium-electron repulsion( energy barrier ~ 1 eV )

G.Scoles, unpublishedGolov and Sekatski, 1991

Rydberg atom 4He cluster

Scolium lifetime must be long enough to allow for its detection Possible detection of Scolium in recent experiments

(M.Drabbels, to be published)

e-

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

4He freezing pressure

Electrostriction should favor solidification of larger clusters

e-

He+

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)

ion-doped He cluster

Scolium

(1) (2)

(1) Cheng, Cole & Cohen (1994)

(2) Bellert & Beckenbridge (2002)

Squeezing a 4He cluster with a Rydberg electron

The electron pressure forces ~10 additional 4He atoms in

the second solvation shellAngular density distribution of 4He atoms in the 2nd shell

Ion-doped4He cluster Scolium

Dynamical criteria are necessary to determine the character (solid or liquid) of Scolium

Conclusions

Density Functional Theory (in its more recent

implementation) provides a quite accurate description

of extended & highly inhomogeneous 4He 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

Liquid Helium solidifies at T=0 only at pressures P>25 atm (in a hcp crystal structure)

4He

Solid helium

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)

Supersolid 4He discovered recently!

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….

Vortices in 4He clusters from DFT calculations

Rotating cluster

Equilibrium configuration by minimizing

E´ []= E [] - z (r) Lz (r) dr

Fixed cluster in a rotating frame

4He field velocity v = (ħ/MHe) v = 0

a rotational motion of the fluid can be obtained only through the nucleation of vortex lines

NHe=300

A quantized vortex spontaneously develops v · dl = (2ħ/MHe)

^

Vortices in Ca-doped 4He clusters from DFT calculations

(b) and (c): Stable states of Ca in a 4He

nanodroplet (with and without vortex):

The excitation/emission spectra of Ca should be very different in the two cases, allowing in principle to detect vortices in 4He nanodroplets

A Ca atom on the surface is drawn inthe interior of the He nanodroplet when a vortex is present

(F.A., M.Barranco and M.Pi, PRL 2003)

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 )