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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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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.
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) …….
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 w 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.
Pair potential D D p h p h
Can calculate Very different from molecule Comparison with experiment ok.
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.
The cranked shell model Many nuclei have a relatively stable shape. Each configuration of particles corresponds to a band.
Cranked shell model experiment
Double dimensional occupation numbers. Different from standard Fermion occupation numbers! Pairing taken into account
band EAB band E band A band B bandcrossing
Energy large Energy small torque
1 3 Rotational aligned 1 3 Deformation aligned “alignment of the orbital”
Double dimensional occupation numbers. Different from standard Fermion occupation numbers!
backbending [AB] [AB] [A] [B] [0]
The backbending effect s-band [AB] ground band [0]
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.