Cluster Magic Numbers

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

He2+

from J. P. Toennies

HeN

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

Bruehl et al Phys. Rev. Lett. 92 185301 (2004)

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



one-particle states

3He in 4Hen


l

4He / 3He phase separation


Stable 4He + 3He mixed clusters


Electron bubbles in 4He droplets

R 1.7 nm

0.48 dyn/cm

E 0.26 eV

322

22

3

44

2PRR

RmE

e

dynamics?

end of lecture 7

In quest of 4He supersolid

a work with J. Peter Toennies (MPI-DSO Göttingen), Franco Dalfovo (Uni Trento),

Robert Grisenti & Manuel Käsz (Uni Frankfurt), Pablo Nieto (Automoma Madrid)

History of a conjecture: BEC in a quantum solid ?

Vacancy diffusivity and solid 4He Poisson ratio

The Geyser effect in solid 4He vacuum expansion

Bernoulli flow of a nominal 4He solid

Suppression of flow anomalies by 1% 3He

4He vacuum expansion from low -T sources

Firenze 2005 - 1

History of a conjecture:

BEC in a quantum solid?

1969

Andreev $ Lifshitz

1970

Chester Leggett

1977

Greywall

2004

Kim & Chan

2004

Ceperley & Bernu

Firenze 2005 - 2

Kim & Chan

2004

measurements of non-classical rotational inertia

Firenze 2005 - 3

no trend ?

Kim

& C

han

Firenze 2005 - 4

Galli & Reatto

2001

(a) no ground state vacancies but only thermal vacancies

(b-d) ground state + thermal vacancies (for different vacancy formation energies)

what about injected (non-equilibrium) vacancies?

Firenze 2005 - 5

Vacuum expansion of solid 4He

)/(4 2dmkTSPu detdet

Firenze 2005 - 6

2/1/

200

)/2(

)/4(

s

s

P

dAuu

continuity

Bernoulli

Firenze 2005 - 7

4He phase diagram

Firenze 2005 - 8

The Geyser effect

Period vs. T at constant pressure

TTm 032.0 bar

35.0 bar

40.7 bar

Period versus P0 at constant temperature

3

2

2

1

)(0

mPP

Bernoulli

Firenze 2005 - 11

D 1)( min,/0 lsPPP

Ps/l information on dynamical processes inside solid 4He

DP information on Poisson ratio of solid 4He

Firenze 2005 - 12

Poisson ratio of solid 4He

Firenze 2005 - 13

Plastic flowmotion of dislocation

motion of vacancies dominant in solid He(high diffusivity!)

Polturak et al experiment (PRL 1998)

vacancy injection at s/l

interface + sweeping by

pressure gradient

Firenze 2005 - 14

PVa DF

kTD vvvv mm 0uFu

v

P

u

v

P

u

Vacancy drift

solid 4He p-type SC

Firenze 2005 - 15

DVa = V* - Va Va = 35.15 Å3 (atomic volume)

V* 0.45Va (vacancy isobaric formation

volume)

A0

As/l

L

Virtual volume to be filled by vacancies

in the time L/u0

u0

a

lsv

VX

AA

u

u

0

0/

0 2

/1

The vacancy mechanism

poise1016

8

0

/0

aav

lssolid VXV

AA

Dm

Firenze 2005 - 16

accumulation of vacancies up to a critical concentration Xc

drift + diffusion

diffusion

Pre

ssur

e

distance from s/l interface

0 L

COLLAPSE!

Geyser mechanism

vacancy bleaching &

resetting of initial conditions

Data on vacancy diffusivity and concentration in 4He

Transport theory

),(2

2txG

vvC

x

vu

x

vD

t

v

rvvv

PVC vvv2 Dm

),()()()()(),( 00 txuXxLxtXtxG s

0),(),( XtxXtxv

v

uvuvuu v

vvv

)(

vCvuvPVvu

vx

v

v

uvuvvu

vvvvvionlinearizat

vvv

')('

'')(

2Dm

vreffrr C211,

1

Generation function

surface generation velocity

Firenze 2005 - 18

]'4

'4

erfc[),(0

'4/)'(/'/21

2

t

tDxtuts

to

vvrr eetvD

dtutvD

xtvueXtxv

*erf

*

2erf

),0(),0(')(

/21*/

41

0

t

u

utee

tuX

tvutvDtj

vv

s

v

ttvv

vvosc

v

)/(* rvrv

FukTuD vvv /4/4 2

Solution for L

Excess vacancies

Current at the s/l interface (x = 0) due to excess vacancies

rvs uu /2

= surface depletion layer thickness

Firenze 2005 - 19

- the shape of the current depends on 2 parameters (, )

- the time scale implies another parameter (v)

- the ratio of the oscillation amplitude to the constant

background is measured by X0Vauv/u0 and is of the order

of a few percent (as seen in experiment)

fitting

reduced form:

1*//2/ vvsv uuty

]erferf2[)( 041 yye

euXtj y

y

vosc

Theory vs. experimentDv = 1.3·10-5 cm2/s

mv = 5.4·1010 s/g

uv = 2.0·10-3 cm/s

us = 2uv

s = 60 s

v = 13 s

* = 10.7 s

0 = 82 s

P0 = 31 bar T0 = 1.74 K

best fit with = 4 = 1.214

better fits are obtained with finite

L (one more parameter)

large means fast recombination

Period 0 vs. diffusivity

finite L approximate solution by Green’s function method

2

021

010 )*

(*

v

c

c

XX

XXerf Xc = critical concentration

v

vc

X

X

*1

*1)( 21

0

L

D

XX

v

c

)(0

0

2

LL

L

Firenze 2005 - 23

mm3.00

2

L

LvD

64.05.0

Firenze 2005 - 24

Anomalies below the ’

point!

a sharp transition in the flow regime at 1.58 K !

Effects of 3He

on the anomalies

from R. Richardson et al

Firenze 2005 - 27

small amounts of 3He remove the anomaly!

normal behaviour induced by less than

1% 3He !

normal behaviour induced by less than

1% 3He !

CONCLUSIONS

1. The geyser effect indicates (via Bernoulli’s law) an oscillation of the s/l (quasi-)equilibrium pressure at a given T: vacancy concentration appears to be the only system variable which can give such effect.

2. Below the ’ temperature flow anomalies are observed:

(a) The most dramatic one is the occurrence of a Bernoulli flow corresponding to pressures > Pm, at which 4He should be solid. (b) Below 1.58 K a sharp drop of the geyser period signals a dramatic change in the flow properties of solid 4He. These anomalies, suggesting superflow conditions, are attributed to injected excess vacancies, and agree with Galli and Reatto predictions for a vacancy-induced (Andreev-Lifshitz) supersolid phase.

3. A 3He concentration of 0.1% is shown to suppress the flow anomalies, suggesting a quantum nature of the superflow.

Miklos Gyulassy, 2004

„There is no end to this wonderful world of experimental discovery and mental constructions of reality as new facts become known. That is why physicists have more fun than most people“

end of lecture 8

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Cluster Magic Numbers