Quantum Monte Quantum Monte CarloCarlo Simulations Simulations of Mixedof Mixed 33He/He/44He He

ClustersClusters

dario.bressanini@uninsubria.itdario.bressanini@uninsubria.ithttp://www.unico.it/http://www.unico.it/~~dariodario

Dario Dario BressaniniBressanini

Universita’ degli Studi dell’Insubria

Trento 2 Sep 2003Trento 2 Sep 2003

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HeliumHelium A helium atom is an A helium atom is an

elementary particle. A weakly elementary particle. A weakly interacting hard sphere.interacting hard sphere.

Interatomic potential is Interatomic potential is known more accurately than known more accurately than any other atom. any other atom.

Two isotopes: Two isotopes: • 33He (fermion: antisymmetric trial function, spin 1/2) He (fermion: antisymmetric trial function, spin 1/2) • 44He (boson: symmetric trial function, spin zero)He (boson: symmetric trial function, spin zero)• The interaction potential is the sameThe interaction potential is the same

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Helium ClustersHelium Clusters1.1. Small mass of helium atomSmall mass of helium atom

2.2. Very weak He-He interactionVery weak He-He interaction

0.02 Kcal/mol0.9 * 10-3 cm-1

0.4 * 10-8 hartree10-7 eV

Highly non-classical systems. No equilibrium structure.Highly non-classical systems. No equilibrium structure.ab-initio methods and normal mode ab-initio methods and normal mode analysisanalysis useless useless

SuperfluiditySuperfluidityHigh resolution High resolution spectroscopyspectroscopy

Low temperature Low temperature chemistrychemistry

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44HeHenn Cluster Clusterss Stability Stability

44HeHe33 bound. Efimov effect? bound. Efimov effect?

Liquid: stableLiquid: stable

44HeHe22 dimer existsdimer exists

44HeHenn

All clusters All clusters boundbound

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33HeHenn Cluster Clusterss Stability Stability

Even less is known for mixed clusters. Even less is known for mixed clusters. Is Is 33HeHemm

44HeHenn stable ? stable ?

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 < 35critically boundcritically bound

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33HeHe44HeHenn Cluster Clusterss Stability Stability

33HeHe44HeHedimer dimer unboundunbound

33HeHe44HeHe22

Trimer boundTrimer bound33HeHe44HeHenn

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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Evidence of Evidence of 33HeHe2244HeHe2 2 Kalinin, Kornilov Kalinin, Kornilov

and Toenniesand Toennies

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33HeHe3344HeHenn Cluster Clusterss StabilityStability

Adding a third fermionic helium, introduces a Adding a third fermionic helium, introduces a nodalnodal surfacesurface into the wave function that destabilizes the system into the wave function that destabilizes the system

What is the smallest What is the smallest 33HeHe3344HeHenn stable cluster ? stable cluster ?

For L=0 n = 9, for L=1 n = 4For L=0 n = 9, for L=1 n = 4

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Evidence of Evidence of 44HeHe4433HeHe3 3 Kalinin, Kornilov Kalinin, Kornilov

and Toenniesand Toennies

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Mixed He clusters: Mixed He clusters: 33HeHemm

44HeHenn 44HeHenn 33HeHemm 0 1 2 3 4 5 6 70 1 2 3 4 5 6 7 8 9 10 11 8 9 10 1100

11

22

33

44

55

3535

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

Bound L=0Bound L=0

UnboundUnbound

UnknownUnknown

MaybeMaybe

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

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

44HeHe22 L=1 S=1 L=1 S=1

33HeHe3344HeHe44 L=0 S=1/2 L=0 S=1/2

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

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33HeHemm44HeHe1010 m = 0,1,2,3 m = 0,1,2,3

L=0L=033HeHe d distribution with respect to the center of massistribution with respect to the center of mass

((33He-He-C.O.M.C.O.M.))

c.o.mc.o.m

5 10 15 20 25 300

0.002

0.004

0.006

0.008

0.01

33HeHe44HeHe1010

5 10 15 20 25 300

0.002

0.004

0.006

0.008

0.0133HeHe22

44HeHe1010

5 10 15 20 25 300

0.002

0.004

0.006

0.008

0.01

0.012

33HeHe3344HeHe1010

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33HeHemm44HeHe1010 m = 0,1,2,3 m = 0,1,2,3

L=0L=033HeHe- - 33HeHe d distributionistributionss ((33He- He- 33He)He)

5 10 15 20 25 30 35 400

0.0025

0.005

0.0075

0.01

0.0125

0.015

The The (tentative)(tentative) picture: two picture: two 33HeHe outside outside ((, , ) and one ) and one inside, pushed away inside, pushed away from the other from the other 33HeHe

44HeHe1010

outside outside inside inside

outside outside inside inside

outside outside outside outside on opposite sideson opposite sides

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33HeHe3344HeHe1010 L=0 L=0

Why ?Why ?

44HeHe1010

It is a It is a Nodal EffectNodal Effect..The wave function is zero if the two The wave function is zero if the two 33HeHe are at the same distance from the are at the same distance from the 33HeHe. For . For this reason the three atoms are not free to this reason the three atoms are not free to move on the surface of the cluster. One is move on the surface of the cluster. One is pushed inside to avoid the wave function pushed inside to avoid the wave function node. node.