cluster misti 3 he/ 4 he : stati e strutture [email protected] dario bressanini...
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Cluster misti Cluster misti 33He/He/44HeHe : :
stati e strutturestati e strutture
[email protected]@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
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44HeHenn Cluster Clusterss Stability Stability
Liquid: stableLiquid: stable
44HeHe22 dimer existsdimer exists
44HeHenn
All clusters All clusters boundbound
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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
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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 ?
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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
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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
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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
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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
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33HeHe44HeHenn
33He – com rdfHe – com rdf
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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.)
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3He – c.o.m. - 3He cosine distribution
c.o.mc.o.m
((ss)-)-(s)(s)
33HeHe3344HeHe22 22 L L
= 1= 1
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Dario BressaniniDario Bressanini
Silvia TarascoSilvia Tarasco
Matteo BardinMatteo Bardin
Sara MarelliSara Marelli