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Universita’ degli Studi dell’Insubria
Cluster misti 3He/4He :
stati e strutture
Dario Bressanini
Gabriele Morosi
[email protected]
http://scienze-como.uninsubria.it/morosi
4He
n
Clusters Stability
4He
n
4He dimer exists
2
All clusters
bound
Liquid: stable
2
3He
m
Clusters Stability
3He
3He
2
dimer unbound
m
m = ? 20 < m < 35
critically bound
Liquid: stable
What is the smallest 3Hem stable cluster ?
3
3He / 4He
n
m
Clusters Stability
Even less is known for mixed clusters.
Is 3Hem4Hen stable ?
4
The Simulations
Potential = sum of two-body TTY
V (R ) VHe He (rij )
i j
Both VMC and DMC simulations
5
3He4He
n
Clusters Stability
Bonding interaction
Non-bonding interaction
3He4He
3He4He
2
dimer unbound
Trimer bound
3He4He
2
4He
3
3He4He
n
All clusters
up bound
E = -0.00984(5) cm-1
E = -0.08784(7) cm-1
6
3He 4He
2
n
Clusters Stability
Now put two 3He. Singlet state. Y is positive everywhere
3He 4He
2
3He 4He
2
2
Trimer unbound
Tetramer bound
5 out of 6 unbound pairs
4He
4
3He4He
3
3He 4He
2
2
3He 4He
2
n
All clusters
up bound
E = -0.3886(1) cm-1
E = -0.2062(1) cm-1
E = -0.071(1) cm-1
7
Why excited states ?
8
Distribution Functions in 3He4Hen
c.o.m. = center of mass
(4He-c.o.m.)
(3He-c.o.m.)
0.015
0.015
N=3
0.010
3He4He
N=19
(r) (bohr-3)
(r)
-3
(bohr )
N=19
2
0.005
0.000
0.010
3He4He
2
N=3
0.005
0.000
0
10
r (bohr)
Similar to pure clusters
20
0
10
20
30
r (bohr)
Fermion is pushed away
9
3D harmonic oscillator levels
2s
1d
1p
1s
10
Rigid rotor levels
l=2
l=1
l=0
11
3He4He
l=2
14
-12.304(5)
E = 0.353(6)
l=1
-12.657(3)
E = 0.270(4)
l=0
-12.927(3)
12
3He 4He
m
n
m=2
m=3
m=4
m=8
R.Guardiola, J.Navarro, Few-Body Syst. Suppl. 14, 223 (2003)
13
Rigid rotor levels
•
•
•
Aufbau principle
Pauli exclusion principle
Hund’s rule
A wealth of possible states
14
3He
m
4He
n
Clusters
ENERGETICS
15
3He4He
n
: energies
0
n=5
E (cm-1)
-2
n=9
-4
L=0
L=1
L=2
-6
-8
-10
2
4
6
8
10
12
n
16
3He4He
n
: excitation energies
0.7
0.6
L=2 ______
E (cm-1)
0.5
0.4
L=1 ______
0.3
L=0 ______
0.2
0.1
0
4
8
12
16
n
20
24
17
3He 4He
2
n
Energy relative to 4Hen energy
0
l = 1 ______
l = 0 ______
-0.5
L
S
0
1
1
0
1
0
s2
sp
sp
-1
l = 1 ______
l = 0 ______
E (cm-1)
: energies
-1.5
-2
l = 0 ______
-2.5
0
4
8
12
16
20
n
18
3He 4He
3
m
: energies
l = 1 ______
l = 0 ______
19
Evidence of 3He3 4He4
Kalinin, Kornilov and Toennies
20
3He 4He
m
n
4He
n
3He
m
0
1
2
3
4
5
6
stability chart
7
8
9 10 11
0
Bound L=0
Unbound
1
2
Unknown
3
L=1 S=1/2
4
L=1 S=1
5
Guardiola
Navarro
35
3He 4He
2
2
3He 4He
3
8
L=0 S=0
3He 4He
2
4
L=1 S=1
3He 4He
3
4
L=0 S=1/2
L=1 S=1/2
21
3He / 4He
n
m
Clusters
STRUCTURE
22
Average values
Oˆ
ˆ Y d
Y
O
0
*
0
Y Y0
*
0
Y Y d
*
0
0
Y YT
DMC samples
Oˆ
must be
sampled
*
0
ˆ Y d
Y
O
T
*
0
Y Y d
*
0
T
23
3He 4He
3
22
3He-c.o.m.
distribution
0.01
0.008
VMC
DMC
g(r)
0.006
0.004
0.002
0
0
10
20
30
r(3He-com)
Yold L(R) BB (R) BF (R) FF (R)
24
3He 4He
3
22
3He-c.o.m.
distribution
0.012
VMC
DMC
g(r)
0.008
0.004
0
0
10
20
30
r(3He-com)
Ynew Yold F com (R F )
25
3He4He
3He
n
– com rdf
26
3He4He
7:
L = 1 state
27
3He4He
9:
L = 2 state
28
3He 4He
2
17
3He –
c.o.m. - 3He cosine distribution
c.o.m
L=0 (s )
2
L=1 (sp) singlet
-1
-0.5
0
cos
0.5
1
triplet
29
3He 4He
3
22
L=1
Distribution with respect to the center of mass
(He-c.o.m.)
c.o.m
31
3He 4He
3
22
3He –
L=1
c.o.m. - 3He cosine distribution
aa
c.o.m
a(s)-b(s)
32
Dario Bressanini
Silvia Tarasco
Matteo Bardin
Sara Marelli
33