1. "How to create a 1qp state and what does it mean?
Odd nuclei have been routinely studied by mean-field methods (called HFB in the 90’s)
In particular in conjunction with the self consistent cranking approximation
Today first beyond-mean-field calculations on a mean-field basis
How is such a basis constructed, what happens along a path (shape constraint)
Fission?
How many 1qp states can be constructed?
Odd Nuclei ECT* 09/2017
CANHP 2015

2. P. Bonche (CEA) H. Flocard (IN2P3) M. Bender (IPNL) Contributors:
Collaboration
P. Bonche (CEA) H. Flocard (IN2P3) M. Bender (IPNL)
Contributors:
J. Terasaki, W. Ryssens, B. Avez, B. Bally, S. Cwiok, W. Nazarewicz,
R. Janssens, B. Gall, C. Rigollet…
Odd Nuclei ECT*

3. Description of an odd nucleus (i) First step: vacuum or “false vacuum”
Constraint on N and Z but no qp excitation.
Put the Fermi energy at the right place.
(ii) Second step: several 1qp states are calculated self-consistently
for each set of (q)
|a (q) > = ba |HFB (q)>
where a labels the qp states.
Odd Nuclei ECT*
CANHP 2015

4. What does it imply such 1qp excitation?
Change in the density matrix or the pairing tensor
(rB,a ) m,n = (V*VT )mn + Um a U* n a – V* m a Vn a
(kB,a ) m,n = (V*UT )mn + Um a V* n a – V* m a Un a
exchange of U and V in the columns corresponding to the index a
Equal filling approximation: one takes into account half the contributions of a
and of its time-reversed partner
In this case, no TR breaking, all time odd terms of the EDF vanish
Usually, these terms do not change the spectrum by less than 100 keV
No cranking for EFA.
Odd Nuclei ECT*

5. Odd Nuclei ECT*

6. Odd Nuclei ECT*

7. Choice of the qp?
Odd Nuclei ECT*

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9. Super heavy nuclei
Odd Nuclei ECT*

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14. Nobel Symp 2016

15. axial q2 +octupole +cranking
222 Th
axial q octupole cranking
Odd Nuclei ECT*

16. 251 Md projection on N, Z, J of 1qp states
States created at the min axial deformation
Odd nuclei ECT*

17. For 25Mg and 46Mg , use of the same forces in the mean-field and pairing channels SLyMR0 and SLyMR1
Coulomb direct and exchange calculated exactly
Odd Nuclei ECT*

18. Points in the triaxial plane every 40 fm2 (around 30 points)
25Mg
Size of the basis:
Points in the triaxial plane every 40 fm2 (around 30 points)
604 1qp states of positive parity
222 1qp states of negative parity
Selection on energy between all these states, finally 100 and 60
states selected for projection and configuration mixing.
After K-mixing, each of these 1 qp states can generate several states
for each J-value.
Final dimension of the bases: 226 for 5/2+
149 for 3/2+
106 for 3/2-
- Accuracy around 20 keV
Odd Nuclei ECT*

19. Odd Nuclei ECT*

20. Adding 2qp excitations
Construction of a few 2qp excitations without (q1,q2) constraints for 46Ca
At the end, 3 are relevant (energy low enough):
J= energy
1/2- +3/ K= lowest
1/2- +7/ K= lowest
5/2- +7/ K= lowest
Odd Nuclei ECT*

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23. Some final comments
Many applications of mean-field methods to odd nuclei.
The meaning of 1qp excitations not always obvious
Fission path? Breaking of symmetries? Spectra?
BMF is probably the best tool to answer these questions
Odd Nuclei ECT*