Magic Numbers of Boson Clusters

Shell Effects – Erice 1

a) He cluster mass selection via diffraction

b) The magic 4He dimer

c) Magic numbers in larger 4He clusters? The Auger evaporation picture

Giorgio Benedek withJ. Peter Toennies (MPI-DSO, Göttingen)

Oleg Kornilov (UCB, Berkeley)Elena Spreafico (UNIMIB, Milano)

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

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

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

N = 4,5,6….

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

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

b) Magic numbers (or stability regions)

Classical noble gas (van der Waals) clusters:

- geometrical constraints only

- magic numbers = highest point symmetry

Quantum Bose clusters (4He)N are superfluid

- no apparent geometrical constraint

- no shell-closure argument

are there magic numbers or stability regions for boson

clusters?Shell Effects – Erice 2

4He clusters

T0= 6.7K

P0 ≥ 20bar

T= 0.37K

- formed in nozzle beam vacuum expansion

- stabilized through evaporative cooling

clusters are superfluid!

Shell Effects – Erice 3

Theory (QMC): no magic numbers

predicted for 4He clusters!- R. Melzer and J. G. Zabolitzky (1984)- M. Barranco, R. Guardiola, S. Hernàndez, R. Mayol, J. Navarro, and M. Pi. (2006)

Binding energy per atom vs. N:

a monotonous slope, with

no peaks nor regions of

larger stability!

Shell Effects – Erice 4

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

More 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

Diffraction experiments with neutral (4Ne)N

clusters show instead stability regions!

Shell Effects – Erice 5

Magic numbers, excitation levels, and other properties of small neutral

4He clusters

Rafael GuardiolaDepartamento de Física Atómica y Nuclear, Facultad de Fisica, Universidad de Valencia, 46100 Burjassot,

Spain

Oleg KornilovMax-Planck-Institut fur Dynamik und

Selbstorganisation, Bunsenstrasse 10, 37073 Gottingen, Germany

Jesús NavarroIFIC (CSIC-Universidad de Valencia), Apartado 22085,

46071 Valencia, Spain

J. Peter ToenniesMax-Planck-Institut fur Dynamik und

Selbstorganisation, Bunsenstrasse 10, 37073 Gottingen, Germany

R. Brühl, R. Guardiola, A. Kalinin, O. Kornilov, J. Navarro, T. Savas and J. P. Toennies,

Phys. Rev. Lett. 92, 185301 (2004)

Shell Effects – Erice 6

The size of 4He clusters

QMC (V. R. Pandharipande, J.G. Zabolitzky, S. C. Pieper, R. B.

Wiringa, and U. Helmbrecht, Phys. Rev. Lett. 50, 1676 (1973)

R(N) = (1.88Å) N 1/3 + (1.13 Å) / (N 1/3 1)

Shell Effects – Erice 7

0n

0)( ,01 ndRkj

Single-particle excitation theory of evaporation and cluster stability

Magic numbers!

200 /2 MVk

spherical box model

Shell Effects – Erice 8

Atomic radial distributions

3Hen

4Hen

Barranco et al (2006)

Fitting a spherical-box model (SBM) to QMC calculations

Condition: same number of quantum single-particle levels

this can be achieved with:

- a(N) = QMC average radius

- V0(N) = μB of bulk liquid

- a constant effective mass

m*

)(

)12(max

8

*2

222

Na

n

mm

m

B

m

)(

)12(

*81

2

222

Na

n

m

E

BB

n

m

mFrom:

Shell Effects – Erice 12

Bm

the linear fit of QMC shell

energies () for

(4He)70 rescaled

to the bulk liquid

μB gives

m*~ 3.2 m

constant)(

)12(max2

2

Na

n

QMC (Pandharipande et al 1988)

this m*/m value works well for all N since

Shell Effects – Erice 13

The Auger-evaporation mechanism

2/12/3

22

2

4)( E

mVEg

)'(),()(),(2

)',',( 121

2

11''2121 2121EEEgrEEEEW EEijEE 22 rrrr

v

exchange-symmetric two-atom wavefunction

2,1,)12()]([*8

)(),( 22

22

ilnNRm

NlnE iiii

m

6

60

6

60

60

6

21)(

r

R

r

R

R

Crv

3/4 30db = 40 Å3

C6 = 1.461 a.u.

d0 < r < R(N)

R(N) = cluster radius

6-12 Lennard-Jones potential

Integration volumeShell Effects – Erice 10

322

2

)()()(

nnn

nr-

attrep

r

Cr)( feA

rrr VVV

n

k

kx

n k

xexf

2

02 !

1)(

Tang-Toennies potential

Replaced by co-volume (excluded volume)

Shell Effects – Erice 11

)]'(1)[()()',,()( 221',,

221221

EnEnEnEEEEWdENP BBBEEE

)()()12()1(8

),(),( 0,2/12/

,121 LL

LLkEE YkrjLr

N2rr

)](2[*2 2121 NEEmk m

- Auger-evaporation probability

- Center-of-mass reference

total L = even

1/))(( ]1[)( TkNEB

BeEn m

m* = 3.2 4 a.u. μ() = 7.3 K

Shell Effects – Erice 14

- Cluster size distribution:

- Cluster kinetics in a supersonic beam

stationaryfission and coalescence neglected:cluster relative velocity very small

)(/1)( NPNn

' 42

2

2

)'(exp)'(

1)(*

N Ns

NNNn

sNNn

- Comparison to experiment:

N

1

Jacobian factor

Gaussian spread (s 0.002)

Ionisation efficiency

Shell Effects – Erice 15

Calculated 4He cluster size distribution at different temperatures

Shell Effects – Erice 16

Comparison to experiment I

Comparison to experiment II

Guardiola et al thermodynamic approach

Guardiola et al., JCP (2006)

HeN-1 + He ↔ HeN Formation-evaporation equilibrium:

Equilibrium constant:

ZN = partition function:

at each insertion of a new bound

state

Magic Numbers

SIF 2008 Genova - 14

In conclusion we have seen that…

Experimental evidence for the stability of the 4He dimer and the existence of magic numbers in 4He boson clusters

A kinetic theory based on the Auger evaporation mechanism for a spherical-box model qualitatively accounts for the experimental cluster size distributions

Substantial agreement with Guardiola et al thermodynamic approach: magic numbers related to the insertion of new bound states with increasing N

High-resolution grating diffraction experiments allow to study the stability of 4He clusters

Electron Microscope Picture of the SiNx Transmission Gratings

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

Lecture 2: Helium Droplets

Grebenev, Toennies & VilesovScience 279, 2083 (1998)

Helium Droplets

T0 ≤ 35 KP0 ≥ 20 bar

Droplets are cooledby evaporation to=0.37 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 experimental sideband reflects the DOS of Elementary Excitations

Roton gap:signature of superfluidity

rotational lines

stable for N > 30

(p + 1)(p + 2)(p + 3)/3

Magic number in fermionic 3He clusters (Barranco et al, 2006)

= 2, 8, 20, 40, 70, 112, 168, 240, 330, ...

Large 4He Clusters: 100< N< 5000

Small 4He Clusters: N< 100

Mixed 4He/3He Droplets: Two Production Methods

4He / 3He phase separation

Barranco et al (2006)

4HeN3He

Stable 4He + 3He mixed clusters

Barranco et al (2006)

1

2

3

4

1 3

0

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

Localized phonon in 3He at the impurity molecule

Space localization spectral localization!

The localized phonon (LP)

is much sharper than the

bulk phonon width!

)(),(!

1),( ,, ijiegieg

NRRrRr

]|)([1

),( o,

o,, ii iegegieg rRr

N

]exp[),(1

),( ,,, j j

egjiegNieg i

VRkRrRr

0,oo3

0,o

.]c.cˆ[1

ΨˆΨ

j

ii

i

iji

giiei V i

ieggeeg

eψψeRdV

e

kRk

k

r

r

DD

D

mm

E

E

),()exp()/()(,,,

2/13,, J

egeg mlegego

eg Yrr

)/exp()/()( 02/3

02/1

1 arZaZ eeis Rr

electron – collective excitation coupling

spatial decay of molecule electronic

wavefunctions

molecule

He atoms

4

2/33

01

)(

)(46)(

iqZ

aeq

ge

ge

e

ineleg

m

),()(4)(22 EqSqnE

qinelegm mn

)]/exp(1/[),(Im),( 1 kTEEqEqS

)],()(1/[),(),( 00 EqqEqEq v

iΓqEE

AEqph

)(),(

Inelastic part of dipolar matrix element:

Sideband absorption coefficient:

Dynamic form factor:

Response function:

non-interacting atoms

interatomic potential

0),(Re)(1 0 Eqq vCollective excitations: E = E(q)

100 ]/Re)[(Im E

100 ]/Reln[ E

)()2( 112

20

nnn uufun

ff m

D 01 EEEΛ m 2/))(/(Δ 2212

0

11111/11 )](1)[(Im

)1(

1)(

EΛEe

ES phphkTE

),(|,1)(2

11 EqqE phq

ph

)]cos(/)[sin(|0,1 11121

0 RqRqRqq /- N

“Shell” model for dynamics n

n +1

Barranco et al

2211)(

1)(

LPLP

LP

ΓEE

ΓES

111 )Re(

1

2

1 phmLP EE

111 )Im(

1 phLPΓ

particle-hole excitation spectrum collective excitation (phonon) spectrum

EEEΛ m 2/))(/(Δ 2212

0

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

para-Hydrogen

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