Nuclear Structure Theory II

Transcription

1 uclear Structure Theory II The uclear Many-body Problem Alexander Volya Florida State University

2 Physics of light nuclei 1 H 4 Li 3 He 2 H Be 5 Li 4 He 3 B H Be 6 Li 5 He 4 B H Be 7 Li 6 He 5 B H B 9 Be 8 Li 7 He B 10 Be 9 Li 8 He B 11 Be 10 Li B 12 Be 11 Li 10 He B 13 Be B 14 Be B B hart of light nuclei colored by lifetimes 26 τ 1/2 < 0.1 as 0.1 as < τ 1/2 < 1 ms 1 ms < τ 1/2 < 0.1 s 0.1 s < τ 1/2 < 3 s 3 s < τ 1/2 < 2 min 2 min < τ 1/2 < 1 hour 1 day < τ 1/2 < 1 y 1 y < τ 1/2 < 10 Gy τ 1/2 > 1 Gy 2

3 Physics of deuteron, A=2 ucleon-nucleon interaction 3

4 Physics of deuteron, A=2 Symmetry, isospin and two-nucleon states otation: What are deuteron s quantum numbers? 4

5 Physics of deuteron, A=2 Symmetry, isospin and two-nucleon states Argonne V4 potential Prediction for tritium ( 3 H) is 7.65 MeV binding 5

6 Level scheme of light nuclei 6

7 Halo nuclei, 11 Be Extended radius Halo density 7

8 lustering in light nuclei ê ê He He 3 + n 3 5 Li p 2 6 He 4 + n + n Be 4 + lustering physics Be n + n e 10 8

9 hart of Isotopes PSS: July 9-11,

10 uclear many-body problem configuration interaction, the shell model Many-body configurations and Hamiltonian Example study Binding energy, shell evolution and monopole Pairing interaction Multipole-multipole interaction, emergence of deformation and rotations Statistical approach and random matrix theory

11 The uclear Shell Model Many-body Hamiltonian d 5/2 Mean field and residual interactions Residual interactions Residual, depends on mean-fied Depends on truncation of space and basis ot exactly two-body

12 Many-body formalism Many-body state reation and annihilation operators

13 Shell model interactions Two-body Hamiltonian in the particle-particle channel: where o-core shell model bare interaction Renormalized interactions improve convergence Traditional shell model Simple potential interactions Renormalized bare interactions to include core and core excitations Phenomenological interactions determined from fits.

14 Typical shell model study Shell model codes: ushell: Antoine: Redstick: osmo: (click osmo) For demonstration we use osmo code. Anatomy of a shell model study 1. Identify system, valence space, limitations on many-body states to study (cosmoxml) 2. reate a list of many-body states, typically fixed J z projection T z, and parity (Xsysmbs) 3. reate many-body Hamiltonian (XHH+JJ) 4. Diagonalize many-body Hamiltonian using exact, lanczos, davidson or other method (texactev, davidson_file). 5. Database eigenstates states and determine their spins (XSHLJT) 6. Define other operators and compute various properties verlaps and spectroscopic factors (XSHLSF) Electromagnetic transition rates (XSHLEMB)

15 The simple model Single-j level Ω=2j+1 single-particle orbitals: m=-j, j-1, j umber of nucleons : 0 Ω umber of many-body states: Ω!/((!(Ω-)!) Many-body states classified by rotational symmetry: (J,M) Dynamics Rotational invariance and two-body interactions particle-particle pair operator P L M =(a a) L M particle-hole pair operator M K κ =(a a ) K κ Hamiltonian Dynamics is fully determined by j+1/2 parameters V L

16 48 a =28 Isotones, data 49 Sc 9.6 MeV To the lowest order binding energy is proportional to the number of nucleons. 50 Ti 51 V 52 r 53 Mn 54 Fe 55 o

17 =28 Isotones, data 1-particle J=j=7/2 2 particles J=0,1,2,..7 but Pauli principle J=0,2,4,6 3 particles Total states 56=8!/(5!3!) J=15/2,11/2,9/2,7/2,5/2,3/2 3-particles on j=7/2, 28 m>0 states m 1 m 2 m 3 M 7/2 5/2 3/2 15/2 1 7/2 5/2 1/2 13/2 1 7/2 5/2-1/2 11/2 7/2 3/2 1/2 11/2 2 7/2 5/2-3/2 9/2 7/2 3/2-1/2 9/2 5/2 3/2 1/2 9/2 3 7/2 5/2-5/2 7/2 7/2 3/2-3/2 7/2 7/2 1/2-1/2 7/2 5/2 3/2-1/2 7/2 4 7/2 5/2-7/2 5/2 7/2 3/2-5/2 5/2 7/2 1/2-3/2 5/2 5/2 3/2-3/2 5/2 5/2 1/2-1/2 5/2 5 7/2 3/2-7/2 3/2 7/2 1/2-5/2 3/2 7/2-1/2-3/2 3/2 5/2 3/2-5/2 3/2 5/2 1/2-3/2 3/2 3/2 1/2-1/2 3/2 6 7/2 1/2-7/2 1/2 5/2 3/2-7/2 1/2 5/2 1/2-5/2 1/2 5/2-1/2-3/2 1/2 5/2-3/2-1/2 1/2 3/2 1/2-3/2 1/2 6

18 Monopole term 48 a 49 Sc 50 Ti 51 V 52 r 53 Mn 54 Fe 55 o 56 i onsider a constant component shift Shift term counts number of pairs What is needed to fix closed shell 56 i? Fully occupied shell 56 i Monopole term Matching binding across the shell gives

19 =28 isotones Monopole term Prediction S p =7.374 MeV in 56 i The single-particle energy changes from 9.6 MeV to 7.3 MeV Woods-Saxon prediction 9.9 MeV to 7.2 MeV

20 Energy: Shell evolutions and monopole term Effective single-particle energies Tensor nucleon-nucleon interaction off-diagonal monopole term g 9/2 f 5/2 f 7/2 proton neutron Example from From tsuka, GXPF1 interaction

21 Proton energy in f7/2 shell

22 =28 Best fit verall spectrum, ordering is well reproduced 31 state only 5 parameters There are discrepancies, p-h symmetry, seniority 48 a 49 Sc 55 o 56 i 53 Mn 50 Ti 51 V 54 Fe 52 r

23 Two-body interaction pairing and potential model Phenomenological residual interaction Short range interaction contributes most to pairing matrix element V 0

24 Pairing interaction in f7/2 shell nuclei

25 Pairing interaction in nuclei

26 Pairing interaction in nuclei

27 Literature G. E. Brown and A. D. Jackson, The nucleon-nucleon interaction (orth-holland Pub. o.; distributors for the U.S.A. and anada, American Elsevier Pub. o., Amsterdam; ew York, 1976) R.B.Wiringa, R.A.Smith and T.L.Ainsworth, Phys. Rev. 29, 1207 (1984). V. Zelevinsky, Quantum physics (Wiley-VH, Weinheim, 2011) D. Dean et al. Progress in Particle and uclear Physics 53 (2004) Broglia, Zelevinsky (eds), Fifty years of nuclear BS, Pairing in Finite Systems. (World Scientific, 2013) Rowe, uclear ollective Motion Models and Theory, (World Scientific, 2010)