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Understanding the Rotating Mean Field: Nucleonic Motion and Pair Correlations

This overview explores the concept of the rotating mean field as applied to nucleonic motion. Nucleons, moving independently within a mean field produced by all nucleons, form pairs that influence angular momentum and energy states. Utilizing the Hartree-Hartree-Fock density functionals and the variational principle, we delve into the effects of spontaneous symmetry breaking and how rotational degrees of freedom manifest in the cranking model. The interaction between particle configurations and the implications of pair correlations on moments of inertia are critically highlighted.

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Understanding the Rotating Mean Field: Nucleonic Motion and Pair Correlations

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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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