Lecture 5: Helium Droplets Grebenev, Toennies & Vilesov Science 279, 2083 (1998)

Lecture 5: Helium Droplets

Grebenev, Toennies & VilesovScience 279, 2083 (1998)

Helium Droplets

T0 ≤ 35 KP0 ≥ 20 bar

Droplets are cooledby evaporation to=0.38 K (4He),=0.15 K (3He)

Brink and Stringari,Z. Phys. D 15, 257 (1990)

Some Microscopic Manifestations of Superfluidity

1. Free Rotations of Molecules

2. The Roton Gap (Phonon Wing)

3. Anomalously Small Moments of Inertia

How many atoms are needed for superfluidity?

How will this number depend on the observed property?

2002-03-01-T3a-Ka

Low temp.nozzle

Scatteringchamber

Photon absorptionand

Evaporation

Ionizer

Massspectrometer

Mirror

La

ser

be

am

T0

0

-

-

v

v

20 K

20 bar

d=5 mm

P

Apparatus for Laser Depletion Spectoscopy

Mass.Spect.Signal

Laser Frequency n

none IR photon evaporates

4

DN ~ ~~ ~h

7.2K-7%

400 atoms

For an N=6000 He dropletthis leads to a 7%signal depletion

+

Laser Depletion Spectroscopy

Sharp spectral features indicate that the molecule rotates without friction

The closer spacing of the lines indicates a factor 2.7 largermoment of inertia

Is this a new microscopic manifestation of superfluidity?

OCS

Since IR absorption lines are so sharp, what about electronic transitions?

The Phase Diagram and Phonons in Liquid 4He

The experimental sideband reflects the DOS of Elementary Excitations

rotational lines

Large 4He Clusters: 100< N< 5000

Small 4He Clusters: N< 100

Two Methods Used to Produce Mixed 4He/3He Droplets

Aggregation of 4He Atoms Around an OCS Molecule Inside a 3He Droplet

3He

OCS surrounded by a cage of 4He

IR Spectra of OCS in 3He Droplets

with Increasing Numbers of 4He

Atoms

~ 60 He atoms are needed to restore free rotations:

Number needed for superfluidity? Grebenev Toennies and Vilesov Science, 279, 2083 (1998)

Wavenumber [cm-1]

Rel

ativ

e D

eple

tion

[%]

The Appearance of a Phonon Wing Heralds the Opening up of the Roton Gap

Pörtner, Toennies and Vilesov, in preparation

According to this Criterium 90 4He Atoms are needed for Superfluidity!

maxon

roton

rotons: in 4He only

maxons: in both 4He and 3He

SEARCH FOR SUPERFLUIDITY INPARA- H ( pH ) CLUSTERS2 2

(Ginzburg and Sobyanin, JETP Lett. , 242 (1972))15

pH has no total nuclear spin, I = 0at T = 0 all molecules are in j = 0

pH are spinless Bosons like He indistinguishable

The superfluid transition temperature is given by

T = n 3.31 g Mk

c

T = 6.0 K c

22/3

2/3B

for pH g = 12

but H solidifies at

T = 13.8 K !m

2

2

2

T = 1.4 K

For ortho - H (oH ), I = 1 and j = 1, g = 9. 2 2

c

Para-Hydrogen Has Long Been A Candidate for Superfluidity

Bose condensed

Non-condensed

The reduced coordinationIn small droplets favorssuperfluid response

Decrease in the moment of inertia indicatessuperfluidity

5.

24 3

2001-06-13-t2-kus

4.

3.

2.

1.OCS in largemixed droplet

Capture of firstH molecule2

Capture of secondH molecule2

H molecule movesfreely in liq. He andbinds at OCS replacinga He atom

24

After many H capturesOCS is surrounded by rings of H

2

2

H2

Aggregation of p-H2 molecules around an OCS molecule inside a

mixed 4He/3He droplet

(5-6 H2)

(3-4 H2)

(5-6 H2)

Average Moments of Inertia

Ia Ib Ic

840 1590 1590

55 1590 1590

880 2500 2500

This is the first evidencefor superfluidity of p-H2

In 1959 Migdal applied BCS theory (1957) to explain superfluidity in nuclei

end of lecture 5

Lecture 6: Helium clusters

he-he pot

The large zero point energy also affects the dimer

The large zero-point energy makes liquid Helium the most tenuous of all liquids

About 10 years ago it was not known whether the He dimer had a bound state

The diffraction angle is inversely prop. to N

Low temp.cluster source

T0

0

-

-

v

v

40 K

1 barP

Non - destructive Diffraction Grating “Mass Spectrometer”

Previous: Na atoms, Pritchard et al (1988); He*, Mlynek et al (1991)

m m 5 5slit slit

80 cm

Mass spectrometerdetector

m 20~~slit +

detect

He atoms at mass 4 4

2003-01-24-T1-Ka

J

He clusters at mass 8 4

Can discriminate against atoms with mass spectrometer set at mass 8 and larger from J. P. Toennies

Electron Microscope Picture of the SiNx Transmission Gratings

Courtesy of Prof. H. Smith and Dr. Tim Savas, M. I. T.

2002-07-24-T2-WK

He Atom Diffraction Pattern for 300 K Beam

22 22

n=

15 15

8 8

5 5

105

10 4

10 3

102

10 1

Mas

s 4

Io

n S

ign

al [

cts/

sec]

-12 -8 -4 0 4 8 12

Deflection Angle [mrad]

T = 294 K0

P = 140 bar0

= 0.56 A°

-1 1

Bragg: A°0.561000 A°

nd

= (n=1) = 0.56 10-3 rad..

= 150 radmDJ

from J. P. Toennies

At Low Source Temperatures New Diffraction Peaks Appear

Measure Size of Dimer from Cross Sectionon Scattering from Grating Bars

<R>2

s0 seff- :

s0 seff

He (1s)2

He<R>

Break-up reduces effective slit width

Hegerfeldt and Köhler, PRL 84 (2000)

2003-07-10-T1-Schr.

from J. P. Toennies

n=-1

n=-2

n=+1

n=+2

2003-07-17-T1-Schr.

n=0

Single Slit Diffraction is Envelope of Grating Diffraction

Single slit:

Grating:

p-

p+

p

p

Matter Waves: Feynmann: Lecture Notes in Physics

Dps-~

eff

Slit function

from J. P. Toennies

0 500 1000 1500 200056

57

58

59

60

61

62

63

64

Effe

ctiv

e S

l it W

idth

s

[nm

]e

ff

Particle Velocity v [m/s]

Effective Slit Widths vs Particle VelocityHe Atom versus He Dimer

Scattering length a = 2 <R> = 97 A

C =0.12 meV nm33

He

He2

Grisenti, Schöllkopf, Toennies Hegerfeldt, Köhler and StollPhys. Rev. Lett. 85 2284 (2000)

=2.5nm

SeffD

oo

V (particle-wall) = 33C

X-

<R> = 52.0 +

Eb -~4m 2

2

<R>

=1.2 10 K-3.1 10-3 K

104 A°

=1.1 10-3 K

0.4 A

Grisenti; Schöllkopf, Toennies, Hegerfeldt, Köhler and Stoll, Phys. Rev. Lett. 85 2284 (2000)

Since <R> is much greater than Rout the dimeris a classically forbidden molecule

<R>

The 4He dimer: the world‘s weakest bound and largest ground state molecule

A frail GIANT!

from J. P. Toennies

He

He

He

2

23

+

He, He ,He

Cluster beam

Kr

l

3

n

Cluster Size Resolved Integral Cross Sections

0 2.0 4.0 6.0 8.0 10.0103

104

Pea

k A

rea

[arb

. uni

ts]

He4

He3

He2

12.0

Pressure Krypton Gas [10 mbar]-5

He

7 10 4.

2003-06-26-T1-Schr.

I=I exp (- n l)s. .o

See Monday poster No 172

of He Clusters in Scattering from Kr Atoms

A.Kalinin, O. Kornilov, L. Rusin, J. P. Toennies, and G. Vladimirov, Phys. Rev. Lett. 93, 163402 (2004)

To Further Study the Dimer it is Interestingto Scatter from an Object Smaller than the Dimer: An Atom!

The Kr atom can pass through the middle of the molecule without its being affected

The dimer is nearly invisible:

magic!

from J. P. Toennies

end of lecture 6

2003-06-23-T2a-Schr.

Cluster Magic Numbers

Geometrical Electronic

Metal clusters

Fermi Level

Do liquid He clusters have magic numbers?

R. Melzer and J.G. Zabolitzky say No!

Ar55 C60

J. Phys. A: Math. Gen. 17 L565 (1984)

Cluster Magic Numbers

Det

achm

ent E

nerg

y [K

]

2004-08-16-T1-Schr.

Ground State Energies of He Clusters

Guardiola and Navarro, priv. comm.

Monte Carlo Calculations: Diffusion

0

1

2

3

4

5

0

0

10

10

20

20

30

30

40

40

50

50-150

-100

-50

0

Binding Energies

Bin

ding

Ene

rgy

E

[K]

b

Atom DetachmentEnergies

m = EN

DD

Recent highly accurate diffusion Monte Carlo (T=0) calculationrules out existence of magic numbers due to stabilities:

R. Guardiola,O. Kornilov, J. Navarro and J. P. Toennies, J. Chem Phys, 2006

Cluster Number Size N

He

He

2

3

4 5678

He

N=

Deflection Angle [mrad]

-4 -3 -2 -1 00

5

10

15

20

He

Sig

nal [

cts/

sec]

+

T =6.7 KoP =1.5 bar

=4.0 Alo

Magic Numbers in He Clusters: He4 4N

Angular resolution 20 10 rad.-6DJ .

x0.03

2003-08-11-T1a-Schr.

Searching for Large 4He Clusters: 4HeN

He2+

from J. P. Toennies

Magic Numbers in Large 4He Clusters

10-4

10-3

10-2

10-1

100

G(N

)

Cluster Size Distributions G(N), N < 100

0 10 20 30 40 50 60 70 80 90 100Cluster Number Size N

0

1

2

3

4

5

Ge

xp(N

) / G

fit(N

)P0 = 1.33 bar

1.28 bar

1.22 bar

1.16 bar

1.10 bar

P0 = 1.33 bar

1.28 bar

1.22 bar

1.16 bar

1.10 bar

Brühl et al. Phys. Rev. Lett. 92 185301-1 (2004)

42

23

13,149,10

2004-01-21-T6-Schr.

T =6.7 K0

G (N) = I J( ) N

G (N) = I J J( ) ddN

-2

J N-1

26

2003-06-26-T1-Fu

C lus tergro wth

Evapo ra tive Co oling

d= 5 mm

Clusters Reach Final Sizes in Early,“ Hot “ Stage of Expansion

Growth reaction

Equilibrium constant

Abrupt changes in equilibrium constants areknown to affect size distributions

He + He HeN-1

N-1 1

N

NNK =

X

X

X X

S g j e-E j /kT

j

Where are partition functionsX

The K have sharp peaks whenever the N cluster has a new excited state. Then both Ξ and K will increase.

But for the N+1 cluster both Ξ will be about the same and K will fall back.

To explain Magic numbers recall that clusters

are formed in early „hot“ stages of the expansion

fro

m J

. P

. T

oe

nn

ies

0n 0)( ,01 ndRkj

)()()0(

2

)12(

)(

,,02

,

2

,

ndndnd

nd

RIRkjR

n

dS

Rd

P

)/(

,)()(

22

0

22

dB

xd

TRMk

dxxxjeRI

Single-particle excitation theory of evaporation and cluster stability

Magic numbers!

evaporation probability

200 /2 MVk

2006

Thermalization via evaporation (DFT)

Binding energy per atom

Barranco et al (2006)

Atomic radial distributions

3Hen

4Hen

Barranco et al (2006)

Barranco et al (2006)

one-particle states

3He in 4Hen

Barranco et al (2006)

4He / 3He phase separation

Barranco et al (2006)

Stable 4He + 3He mixed clusters

Barranco et al (2006)

Electron bubbles in 4He droplets

R 1.7 nm

0.48 dyn/cm

E 0.26 eV

322

22

3

44

2PRR

RmE

e