How do nuclei rotate?

1. How do nuclei rotate? 3. The rotating mean field

2. The mean field is a functional of the single particle states determined by an averaging procedure. The mean field concept A nucleon moves in the mean field generated by all nucleons. The nucleons move independently.

3. Total energy is a minimized (stationary) with respect to the single particle states. Start from the two-body Hamiltonian effective interaction Use the variational principle Calculation of the mean field: Hartree Hartree-Fock density functionals Micro-Macro (Strutinsky method) …….

4. Spontaneous symmetry breaking Symmetry operation S

5. Deformed mean field solutions Measures orientation. Rotational degree of freedom and rotational bands. Microscopic approach to the Unified Model. 5/32

6. Cranking model Seek a mean field solution carrying finite angular momentum. Use the variational principle with the auxillary condition The state |> is the stationary mean field solution in the frame that rotates uniformly with the angular velocity w about the z axis. In the laboratory frame it corresponds to a uniformly rotating mean field state

7. Can calculate Very different from molecule Comparison with experiment

8. The QQ-model

9. Mean field solution

10. Intrinsic frame Principal axes

11. Transition probabilities

12. Broken by m.f. rotational bands Symmetries

13. Principal Axis Cranking PAC solutions Tilted Axis Cranking TAC or planar tilted solutions Chiral or aplanar solutions Doubling of states

14. The cranked shell model Many nuclei have a relatively stable shape. Each configuration of particles corresponds to a band.

15. (-,-1/2) (-,1/2) (+,-1/2) (+,1/2)

16. (+,1/2) (-,1/2) (+,-1/2) (-,1/2) (+,1/2)

17. Experimental single particle routhians

18. Cranked shell model experiment

19. Rotational alignment

20. Energy large Energy small torque

21. 1 1 3 3 Deformation aligned Rotational aligned “alignment of the orbital”

22. Slope =

23. Pair correlations

24. Pair correlations Nucleons like to form pairs carrying zero angular momentum. Like electrons form Cooper pairs in a superconductor. Pair correlations reduce the angular momentum.

25. The pairing+QQ model

26. particle hole amplitudes Mean field approximation (CHFB)

27. Configurations (bands)

28. Double dimensional occupation numbers. Different from standard Fermion occupation numbers!

29. backbending [AB] [AB] [A] [B] [0]

30. The backbending effect s-band [AB] ground band [0]

31. Moments of inertia at low spin are well reproduced by cranking calculations including pair correlations. rigid irrotational Non-local superfluidity: size of the Cooper pairs larger than size of the nucleus.

32. Summary • The pairing+QQ model leads to a simple version of mean field theory. • The mean field may spontaneously break symmetries. • The non-spherical mean field defines orientation and the rotational degrees of freedom. • There are various discrete symmetries types of the mean field. • The rotating mean field (cranking model) describes the response of the nucleonic motion to rotation. • The inertial forces align the angular momentum of the orbits with the rotational axis. • The bands are classified as single particle configurations in the rotating mean field. The cranked shell model (fixed shape) is a very handy tool. • At moderate spin one must take into account pair correlations. The bands are classified as quasiparticle configurations. • Band crossings (backbends) are well accounted for.