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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 ?
9
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
10
3D harmonic oscillator levels3D harmonic oscillator levels
1s
1p
2s 1d
11
Rigid rotor levelsRigid rotor levels
l = 0
l = 1
l = 2
12
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)
13
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)
14
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
16
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
17
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
)
18
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 ____________
19
33HeHe3344HeHem m : energies: energies
l l = 1= 1 ____________
l l = 0= 0 ____________
20
Evidence of Evidence of 33HeHe3 3 44HeHe44
Kalinin, Kornilov and ToenniesKalinin, Kornilov and Toennies
21
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
22
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
24
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
28
33HeHe44HeHe9 9 : L = 2 state: L = 2 state
29
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
31
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
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