1. UWC Beyond mean-field: present and future
Mean-field and Beyond for low-energy nuclear spectroscopy Present and Future
The long term goal
Possible alternative
The ingredients
Applications to even-even nuclei
Very recent extension to odd nuclei
Perspectives
17/01/2019
UWC Beyond mean-field: present and future
York

2. A few points on which I will stop to give more details
The Hartree Fock method: principle and realisations;
Pairing correlations
Breaking of symmetries, why, how and which?
Beyond mean-field: the different meanings
Correlations, what does it mean, can one measure experimentally correlations?
Nucleon-nucleon interaction: realistic, effective, energy density functional….
The code Ev8, solution of mean-field equations on a mesh
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UWC Beyond mean-field: present and future

3. The ultimate (unreachable?) goal
To set up a method for nuclear spectroscopy
To describe spectra and transition probabilities
Applicable to all nuclei, from the lightest to the heaviest
With a single phenomenological input, the modelling of the strong force
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UWC Beyond mean-field: present and future

4. The main alternative: the shell model
Successful in the description of many nuclei up to mass 100
Its range of applicability is extending
What we want to avoid:
Only valence particles (inert core, a few active shells)
Finely tuned interactions
Effective charges
Applicable up to the heaviest nuclei
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UWC Beyond mean-field: present and future

5. UWC Beyond mean-field: present and future
25Mg
3 lowest states are 0+
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UWC Beyond mean-field: present and future

6. Beyond mean-field methods
Starting point: mean-field wave functions: Hartree-Fock + pairing correlations
Effective nucleon-nucleon interaction (EDF or an interaction or relativistic)
Pairing correlations treated at the BCS or Bogoliubov level with the same or a different interaction
Constraints on the shape of the nucleus introduced in the mean-field equations. One determines the best HFB wave function for each shape
H -> H – lN N – lZ Z – l1q1 – l2q2
The wave function has the form:
Variational parameters: those of the basis describing the individual wave functions and the v’s
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UWC Beyond mean-field: present and future

7. The Shrödinger equation is equivalent to a wave function variational
principle: minimize < | H | Y> under the constraint <  | Y> =1:
One-body
Two-body
Hartree-Fock method:
the ground state wave function is a Slater determinant.

8. The Hartree-Fock method
Wave-function of a many particle system= Slater determinant
The particles interact trough a 2-body interaction v(r1-r2)
They are also confined by a central potential vext(r).
The total energy is:
direct
exchange

9. Minimize the energy with respect to fi
Minimize the energy with respect to fi * with a constraint on norm conservation:
One defines the one-body diagonal and non diagonal densities:

10. One rewrites the HF equations as a function of these densities:
where
The first line is easy: problem in a potential.
The second line is complicate: non local exchange term.
HF single particle energy:
2-body matrix element
between i and all other j
Total energy:
no double counting!

11. Mean-field Methods
Based on an “effective interaction” or a “density functional”
The (small number of) parameters of the effective interaction
are fixed by general considerations (no local adjustments)
Pairing correlations are included at the BCS or better HFB level
Full self-consistency
No restrictions to a few shells, mean-field equations are solved
as precisely as one wishes.
Spherical and deformed nuclei are treated on the same footing,
no “parametric deformation”

12. Deformation of the nucleus introduced by a Lagrange multiplier:
by varying l, one obtains solutions for different deformations
Missing ingredient: pairing correlations (superconductivity)
They can be introduced using the BCS theory:
single particle states are occupied with a probability v2 between 1 and 0
nucleons are grouped in pairs of opposite spin projections
The nuclear density becomes:
HF equations with modified densities
+ BCS equations to determine the occupations
Total wave function with only the right mean particle number!

17. Argonne Beyond mean-field: present and future
208Pb
180Hg
202Rn
170Hf
Mean-field energy curves (b2 proportional to Q)
Argonne Beyond mean-field: present and future
17/01/2019
York

18. UWC Beyond mean-field: present and future
17/01/2019
UWC Beyond mean-field: present and future

19. UWC Beyond mean-field: present and future
17/01/2019
UWC Beyond mean-field: present and future

20. Skyrme HFB
Deformation properties
of super-heavies

21. The shell model: a schematic view
All upper shells neglected
Active space:
Inactive core 40Ca

22. Active space
2-body matrix elements
between all pairs of states
Diagonalization of a big matrix in the active space
Starting point of the model: nucleons in a potential well.
Problem of the model: active space grows very quickly.
Respect always symmetries of the 2-body hamiltonian:
work in the laboratory frame of reference

23. Mean-field methods
Construct all the orbitals from the mean 2-body interaction of a particle with the others
All orbitals are active!
Caution: not true single particle levels
Break symmetries and work in a frame of reference intrinsic to the nucleus!

24. UWC Beyond mean-field: present and future
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UWC Beyond mean-field: present and future

25. for super heavy elements (only even-even)
Skyrme HFB
Qa for isotopic chains
for super heavy elements
(only even-even)
Cwiok, Heenen, Nazarewicz
Nature 2005

26. Argonne Beyond mean-field: present and future
Plus and minus of the mean-field approach:
Plus:
Starts from an effective interaction: generality
Can describe any kind of shapes (from ground state to fission)
Cranking (or qp excitations) well justified for deformed nuclei
Minus:
Valid only for energy (variational) and one-body operators
Breaking of symmetries (no direct determination of transitions)
Soft nuclei? Shape coexistence?
Effects of correlations beyond mean-field on masses?
17/01/2019
Argonne Beyond mean-field: present and future
York

27. UWC Beyond mean-field: present and future
Restoration of symmetry:
A deformed wave function breaks the rotational invariance:
Not an eigenstate of J2 and Jz.
Nuclear density
Any rotated density would also be a solution of
the HFB equations, with the same energy.
Symmetry restoration: mixing of all these degenerate
wave functions with appropriate weights to have
good J2 and Jz
17/01/2019
UWC Beyond mean-field: present and future

28. Projection on particle number
Projection operator:
To check that it extracts an eigenstate of particle number,
one applies it to |BCS> = Sn cn |n>
Effect on a BCS wave function:
vk replaced by eif vk
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UWC Beyond mean-field: present and future

29. UWC Beyond mean-field: present and future
Configuration mixing
Mixing of projected mean-field wave functions as a function of shape variable(s)
The coefficients f are determined by varying the energy:
They are solution of the Hill Wheeler and Griffin equation:
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UWC Beyond mean-field: present and future

30. UWC Beyond mean-field: present and future
17/01/2019
UWC Beyond mean-field: present and future

31. 16O: mean-field and projection

32. Single particle levels

33. Configuration mixing:
Energies
Wave functions

34. UWC Beyond mean-field: present and future
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UWC Beyond mean-field: present and future

35. Shape coexistence York April 2013
Mean field
Mean field projected on J=0
Bars in red: 0+ states obtained after configuration mixing
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Shape coexistence York April 2013
York

36. UWC Beyond mean-field: present and future
186Pb
exp cal.
conf. mix of mean field states (Skyrme intera.SLy6)
M. Bender et al. PRC 69 (2004), & privat com.
16+
E/MeV
E/MeV
(14+) 8+
pro
14+
pro
242 WU
(12+) 3 3
12+
(10+) 2+ 6+
β = 0.30(2)
β = 0.32
10+
(8+)
205 WU 2 8+ 2 4+
(6+) 8+
543 WU 6+
152 WU
246(42) WU
(4+) 6+ 2+
480 WU
438(98)WU
56 WU 4+
(2+) 1 4+ 1 0+
480(90)WU 2+
398 WU 0+ 2+
9.4 WU 0+ 0+
5.5 (15)WU
1.6 WU
obl
obl 0+ 0+
sph
sph
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UWC Beyond mean-field: present and future

37. Application to neutron deficient nuclei around Pb
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UWC Beyond mean-field: present and future
J. Yao, M. Bender and PHH, PRC (2013)

38. Systematic calculation of ground state properties “shell quenching”
M. Bender, G. Bertsch, PHH PRC (2008)
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UWC Beyond mean-field: present and future

39. UWC Beyond mean-field: present and future
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UWC Beyond mean-field: present and future

40. UWC Beyond mean-field: present and future
Quadrupole and octupole deformations
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UWC Beyond mean-field: present and future

41. UWC Beyond mean-field: present and future
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
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UWC Beyond mean-field: present and future

42. UWC Beyond mean-field: present and future
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UWC Beyond mean-field: present and future

43. UWC Beyond mean-field: present and future
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UWC Beyond mean-field: present and future

44. UWC Beyond mean-field: present and future
Very heavy calculations (but mainy gains are possible)
Crucial need to improve the interaction
Same on cranked states…
Unique interaction for all nuclei
Will be applicable to any nucleus
No core, positive and negative parity states described in a
single calculation
No effective charges.
Different “philosophy than the UNEDF project: some correlations must be
introduced explicitely
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UWC Beyond mean-field: present and future

45. UWC Beyond mean-field: present and future
First developments by Paul Bonche (Saclay), Hubert Flocard (Orsay)
and PHH (Brussels) with several collaborations (M. Weiss, J. Dobaczeswki, G. Bertsch…)
Actual developments by M. Bender (Bordeaux), T. Duguet (Saclay),
K. Bennaceur (Lyon) and several Ph D students B. Bally (Bordeaux),
W. Ryssens (Brussels), R. Jodon (Lyon) and post docs V. Hellemans, J. Yao, K. Washyiama, S. Baroni (Brussels) and B. Avez (Bordeaux)
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UWC Beyond mean-field: present and future

46. UWC Beyond mean-field: present and future
Fission barriers:
Influence of the surface tension
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UWC Beyond mean-field: present and future