Cluster misti Cluster misti 33He/He/44HeHe : :

stati e strutturestati e strutture

Gabriele.Morosi@uninsubria.itGabriele.Morosi@uninsubria.ithttp://scienze-como.uninsubria.it/http://scienze-como.uninsubria.it/morosimorosi

Dario Dario BressaniniBressaniniGabriele Gabriele MorosiMorosi

Dario Dario BressaniniBressaniniGabriele Gabriele MorosiMorosi

Universita’ degli Studi dell’Insubria Universita’ degli Studi dell’Insubria

2

44HeHenn Cluster Clusterss Stability Stability

Liquid: stableLiquid: stable

44HeHe22 dimer existsdimer exists

44HeHenn

All clusters All clusters boundbound

3

33HeHemm Cluster Clusterss Stability Stability

What is the smallest What is the smallest 33HeHemm stable cluster ? stable cluster ?

Liquid: stableLiquid: stable

33HeHe22

dimer dimer unboundunbound

33HeHemm

m = ?m = ? 20 < 20 < mm < 35 < 35

critically boundcritically bound

4

33HeHenn / 4 / 4HeHemm Cluster Clusterss StabilityStability

Even less is known for mixed clusters. Even less is known for mixed clusters.

Is Is 33HeHemm44HeHenn stable ? stable ?

5

The SimulationsThe Simulations

Potential = sum of two-body TTYPotential = sum of two-body TTY

ji

ijHeHe rVV )()(R

Both VMC and DMC simulationsBoth VMC and DMC simulations

6

33HeHe44HeHenn Cluster Clusterss Stability Stability

33HeHe44HeHedimer dimer unboundunbound

33HeHe44HeHe22

Trimer boundTrimer bound

33HeHe44HeHenn

All clusters All clusters up boundup bound

44HeHe3 3 E = E = -0.08784(7)-0.08784(7) cm cm-1-1

33HeHe44HeHe22 E = E = -0.00984(5)-0.00984(5) cm cm-1-1

Bonding Bonding interactioninteractionNon-bonding Non-bonding interactioninteraction

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33HeHe2244HeHenn Cluster Clusterss StabilityStability

Now put two Now put two 33HeHe. Singlet state. . Singlet state. is positive everywhere is positive everywhere

33HeHe2244HeHenn

All clusters All clusters up boundup bound

33HeHe2244HeHe

Trimer Trimer unboundunbound

33HeHe2244HeHe22

Tetramer boundTetramer bound5 out of 6 unbound pairs5 out of 6 unbound pairs

44HeHe4 4 E = E = -0.-0.33888866((11)) cm cm-1-1

33HeHe44HeHe33 E = E = -0.-0.20622062((11)) cm cm-1-1

33HeHe2244HeHe22 E = E = -0.-0.071071((11)) cm cm-1-1

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Why excited states ?Why excited states ?

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N=19

N=3

0 10 20r (bohr)

0.000

0.005

0.010

0.015

(r)

(

boh

r-3)

3He4He2N=19

N=3

0 10 20r (bohr)

0.000

0.005

0.010

0.015

(r)

(

boh

r-3)

3He4He2

Distribution Functions in Distribution Functions in 33HeHe44HeHenn

0 10 20 30r (bohr)

0.000

0.005

0.010

0.015

(r)

(

boh

r-3) 3He4He2

N=19

N=3

0 10 20 30r (bohr)

0.000

0.005

0.010

0.015

(r)

(

boh

r-3) 3He4He2

N=19

N=3

((44He-c.o.m.)He-c.o.m.) ((33He-c.o.m.)He-c.o.m.)

c.o.m. = center of massc.o.m. = center of mass

Similar to pure Similar to pure clustersclusters

Fermion is pushed Fermion is pushed awayaway

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3D harmonic oscillator levels3D harmonic oscillator levels

1s

1p

2s 1d

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Rigid rotor levelsRigid rotor levels

l = 0

l = 1

l = 2

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33HeHe44HeHe1414

ll = 2 -12.304(5) = 2 -12.304(5)

E = 0.353(6)E = 0.353(6)

ll = 1 -12.657(3) = 1 -12.657(3)

E = 0.270(4)E = 0.270(4)

ll = 0 -12.927(3) = 0 -12.927(3)

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33HeHemm44HeHenn

m=2m=2

m=3m=3

m=4m=4

m=8m=8

R.Guardiola, J.Navarro, Few-Body Syst. Suppl. R.Guardiola, J.Navarro, Few-Body Syst. Suppl. 1414, 223 (2003), 223 (2003)

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Rigid rotor levelsRigid rotor levels

A wealth of possible statesA wealth of possible states

• Aufbau principleAufbau principle

• Pauli exclusion principlePauli exclusion principle

• Hund’s ruleHund’s rule

l = 0

l = 1

l = 2

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33HeHemm 4 4HeHenn Cluster Clusterss

ENERGETICSENERGETICS

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33HeHe44HeHen n : energies: energies

2 4 6 8 10 12

n

-10

-8

-6

-4

-2

0

E

(cm

-1)

L = 0

L = 1

L = 2

n = n = 55

n = n = 99

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33HeHe44HeHen n : excitation energies: excitation energies

L=2L=2 ____________

L=1L=1 ____________

L=0L=0 ____________

4 8 12 16 20 24

n

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

E (

cm-1

)

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33HeHe2244HeHen n : energies: energies

Energy relative to Energy relative to 44HeHenn energy energy

0 4 8 12 16 20n

-2.5

-2

-1.5

-1

-0.5

0

E (

cm

-1)

L S 0 0 s2

1 1 sp

1 0 sp

l l = 0= 0 ____________

l l = 1= 1 ____________

l l = 0= 0 ______ ______

l l = 1= 1 ____________

l l = 0= 0 ____________

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33HeHe3344HeHem m : energies: energies

l l = 1= 1 ____________

l l = 0= 0 ____________

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Evidence of Evidence of 33HeHe3 3 44HeHe44

Kalinin, Kornilov and ToenniesKalinin, Kornilov and Toennies

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33HeHemm44HeHen n stability chartstability chart

44HeHenn 33HeHemm 0 1 2 3 4 5 6 70 1 2 3 4 5 6 7 8 9 10 11 8 9 10 11

00

11

22

33

44

55

3535

Bound L=0Bound L=0

UnboundUnbound

UnknownUnknown

L=1 S=1/2L=1 S=1/2

L=1 S=1L=1 S=1

GuardiolaGuardiolaNavarroNavarro

33HeHe3344HeHe44 L=1 S=1/2 L=1 S=1/233HeHe22

44HeHe44 L=1 S=1 L=1 S=1

33HeHe2244HeHe22 L=0 S=0 L=0 S=0 33HeHe33

44HeHe88 L=0 S=1/2 L=0 S=1/2

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33HeHenn / 4 / 4HeHemm Cluster Clusterss

STRUCTURESTRUCTURE

23

Average valuesAverage values

d

dOO

0*0

0*0

ˆˆ

d

dOO

T

T

*0

*0

ˆˆ

DMC samplesDMC samples T*

0

0*0 must be must be

sampledsampled

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33HeHe3344HeHe2222

33He-c.o.m. distributionHe-c.o.m. distribution

0 10 20 30

r(3He-com)

0

0.002

0.004

0.006

0.008

0.01

g(r

)

VMC

DMC

)()()()( RRRR FFBFBBold L

25

33HeHe3344HeHe2222

33He-c.o.m. distributionHe-c.o.m. distribution

0 10 20 30

r(3He-com)

0

0.004

0.008

0.012

g(r

)

VMC

DMC

)( FcomFoldnew R

26

33HeHe44HeHenn

33He – com rdfHe – com rdf

27

33HeHe44HeHe7 7 : L = 1 state: L = 1 state

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33HeHe44HeHe9 9 : L = 2 state: L = 2 state

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33HeHe2244HeHe1717

3He – c.o.m. - 3He cosine distribution

c.o.mc.o.m

L=0 (s2)

L=1 (sp) singlet

triplet-1 -0 .5 0 0.5 1

co s

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33HeHe3344HeHe22 22 L = 1L = 1

Distribution with respect to the center of massDistribution with respect to the center of mass

c.o.mc.o.m

(He-c.o.m.)(He-c.o.m.)

32

3He – c.o.m. - 3He cosine distribution

c.o.mc.o.m

((ss)-)-(s)(s)

33HeHe3344HeHe22 22 L L

= 1= 1

33

Dario BressaniniDario Bressanini

Silvia TarascoSilvia Tarasco

Matteo BardinMatteo Bardin

Sara MarelliSara Marelli