4. The rotating mean field. The mean field concept A nucleon moves in the mean field generated by all nucleons. The mean field is a functional of the.

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Presentation transcript:

4. The rotating mean field

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

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

Spontaneous symmetry breaking Symmetry operation S

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

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  about the z axis. In the laboratory frame it corresponds to a uniformly rotating mean field state

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.

D D p h hp Pair potential

Can calculate molecule Comparison with experiment ok. Very different from

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

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

Experimental single particle routhians

Slope =

experiment Cranked shell model

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

band E band EAB bandcrossing band A band B

Rotational alignment

Energy small Energy large torque

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

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

[0] [A] [AB] backbending [B]

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

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.