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4. The rotating mean field

4. The rotating mean field. 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.

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4. The rotating mean field

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  1. 4. 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 (axial) 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 auxiliary 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. 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.

  8. Pair potential D D p h p h

  9. Can calculate Very different from molecule Comparison with experiment ok.

  10. 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.

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

  12. Experimental single particle routhians

  13. Slope =

  14. Cranked shell model experiment

  15. Double dimensional occupation numbers. Different from standard Fermion occupation numbers! Pairing taken into account

  16. band EAB band E band A band B bandcrossing

  17. Rotational alignment

  18. Energy large Energy small torque

  19. 1 3 Rotational aligned 1 3 Deformation aligned “alignment of the orbital”

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

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

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

  23. Summary • The mean field may spontaneously break symmetries. • The non-spherical mean field defines orientation and the rotational degrees of freedom. • 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.

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