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

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Presentation transcript:

How do nuclei rotate? 3. 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 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 auxillary 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

Can calculate molecule Comparison with experiment Very different from

The QQ-model

Mean field solution

Intrinsic frame Principal axes

Transition probabilities

Symmetries Broken by m.f. rotational bands

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

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

  /2) (-,1/2) (-,-1/2)

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

Experimental single particle routhians

experiment Cranked shell model

Rotational alignment

Energy small Energy large torque

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

Slope =

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.

The pairing+QQ model

Mean field approximation (CHFB) particle hole amplitudes

Configurations (bands)

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

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

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

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.

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.