Triaxial Projected Configuration Mixing

1. Collective wave functions?2. Old results on Zr

3. Few results on 24Mg4. Many questions

First triaxial calculations:P. Bonche, H. Flocard, J. MeyerJ. Dobaczewski, J. Skalski

New developments:M. Bender

Configuration MixingStarting point: set of Wa wave functions |a >, non-orthogonal:

New set of wave functions:

The unknown fm( )a are solutions of the HW equation:

The f’s are non orthogonal, ill-behaved, ….

Change of basis, using the overlap matrix, defining its square root:

Very nice but not used directly!

First, diagonalisation of the overlap I:

And then

Last summation restricted to a limited number of eigenvaluesIt is this equation that is solved!

The collective wave function is

And the eigenstates of the Hamiltonian is:

Neither g nor f are the overlap

Meaning of oblate, prolate, triaxial …. after configuration mixing?

Projection of triaxial map:

Triaxial minimum?lost of the meaning of q after projection!no orthogonality of wave functions!

z=symmetry axis the maps for the otherorientations have nosimpleinterpretations

Q=125 fm2, = 16°g(mean-field configuration)

z= longest intermediate smallest axis

Spectra obtained after projection of the lowest configuration:three possible orientations

Same results AFTER K-mixing

Spectroscopic properties of the min configurationbefore and after K-mixingcompared to the Davidoff rotor model

Configuration mixing:

comparison between different bases:

1. purely prolate2. axial3. purely triaxial 4. triaxial + a few prolate configurations

We are not using a hamiltonian but a density functionalgeneralized for non-diagonal matrix elements

One must avoid pathologies:possible problems determined by projecting on N and Zwith 9 and 29 pointsTriaxial region close to the oblate axis.No oblate points mixed with triaxial points.

Small eigenvalues of the norm kernel indicate redundancy in a basis

small eigenvalues (10-2) = not much information

All the GCM calculations: axial (prolate+oblate) purely triaxial (35 keV lower than axial) triaxial + prolate (160 keV lower than triaxial)

Triaxial correlations described by configuration mixing of axial configurations!

Cut in the Q,g plane: and GCM calculations

increase of energy for excited statesdue to the correlations in the ground state!

Spectra in 3 bases

No vectors in common!

Very careful about language:

« the nucleus is triaxial after projection on J » !

Analysis of phenomenological models (clever but with the hands)

Sign of triaxiality or K-bands?