Cluster misti 3 He/ 4 He : stati e strutture Dario Bressanini Gabriele Morosi Dario Bressanini Gabriele Morosi Universita’ degli Studi dell’Insubria

2 4 He n Clusters Stability Liquid: stable 4 He 2 dimer exists 4 He n All clusters bound

3 3 He m Clusters Stability What is the smallest 3 He m stable cluster ? What is the smallest 3 He m stable cluster ? Liquid: stable 3 He 2 dimer unbound 3 He m m = ? 20 < m < 35 critically bound

4 3 He n / 4 He m Clusters Stability Even less is known for mixed clusters. Even less is known for mixed clusters. Is 3 He m 4 He n stable ? Is 3 He m 4 He n stable ?

5 The Simulations Potential = sum of two-body TTY Potential = sum of two-body TTY Both VMC and DMC simulations Both VMC and DMC simulations

6 3 He 4 He n Clusters Stability 3 He 4 He dimer unbound 3 He 4 He 2 Trimer bound 3 He 4 He n All clusters up bound 4 He 3 E = (7) cm -1 3 He 4 He 2 E = (5) cm -1 Bonding interaction Non-bonding interaction

7 3 He 2 4 He n Clusters Stability Now put two 3 He. Singlet state.  is positive everywhere Now put two 3 He. Singlet state.  is positive everywhere 3 He 2 4 He n All clusters up bound 3 He 2 4 He Trimer unbound 3 He 2 4 He 2 Tetramer bound 5 out of 6 unbound pairs 4 He 4 E = (1) cm -1 3 He 4 He 3 E = (1) cm -1 3 He 2 4 He 2 E = (1) cm -1

8 Why excited states ?

9 Distribution Functions in 3 He 4 He n  ( 4 He-c.o.m.)  ( 3 He-c.o.m.) c.o.m. = center of mass Similar to pure clusters Fermion is pushed away

10 3D harmonic oscillator levels

11 Rigid rotor levels

12 3 He 4 He 14 l = (5) l = (5)  E = 0.353(6)  E = 0.353(6) l = (3) l = (3)  E = 0.270(4)  E = 0.270(4) l = (3) l = (3)

13 3 He m 4 He n m=2m=3m=4m=8 R.Guardiola, J.Navarro, Few-Body Syst. Suppl. 14, 223 (2003)

14 Rigid rotor levels A wealth of possible states A wealth of possible states Aufbau principle Aufbau principle Pauli exclusion principle Pauli exclusion principle Hund’s rule Hund’s rule

15 3 He m 4 He n Clusters ENERGETICS

16 3 He 4 He n : energies n = 5 n = 9

17 3 He 4 He n : excitation energies L=2 ______ L=1 ______ L=0 ______

18 3 He 2 4 He n : energies 3 He 2 4 He n : energies Energy relative to 4 He n energy Energy relative to 4 He n energy l = 0 ______ l = 1 ______ l = 0 ______ l = 1 ______ l = 0 ______

19 3 He 3 4 He m : energies 3 He 3 4 He m : energies l = 1 ______ l = 0 ______

20 Evidence of 3 He 3 4 He 4 Kalinin, Kornilov and Toennies

21 3 He m 4 He n stability chart 4 He n 4 He n 3 He m 3 He m Bound L=0 Unbound Unknown L=1 S=1/2 L=1 S=1 Guardiola Navarro 3 He 3 4 He 4 L=1 S=1/2 3 He 2 4 He 4 L=1 S=1 3 He 2 4 He 2 L=0 S=0 3 He 3 4 He 8 L=0 S=1/2

22 3 He n / 4 He m Clusters STRUCTURE

23 Average values DMC samples must be sampled

24 3 He 3 4 He 22 3 He-c.o.m. distribution 3 He-c.o.m. distribution

25 3 He 3 4 He 22 3 He-c.o.m. distribution 3 He-c.o.m. distribution

26 3 He 4 He n 3 He – com rdf

27 3 He 4 He 7 : L = 1 state

28 3 He 4 He 9 : L = 2 state

29 3 He 2 4 He 17 3 He – c.o.m. - 3 He cosine distribution c.o.m  L=0 (s 2 )  L=1 (sp) singlet  triplet

30 3 He 2 4 He 14 3 He – c.o.m. - 3 He angle distribution c.o.m  L=0 (s 2 )  L=1 (sp) singlet  triplet

31 3 He 3 4 He 22 L = 1 Distribution with respect to the center of mass c.o.m  (He-c.o.m.)

32 3 He – c.o.m. - 3 He cosine distribution c.o.m   ( s )-  (s) 3 He 3 4 He 22 L = 1

33 Dario Bressanini Silvia Tarasco Matteo Bardin Sara Marelli