1. HIGHER TAMM-DANCOFF APPROXIMATION A particle number conserving microscopic method to treat correlations in fermionic systems

2. Nathalie Pillet (CENBG Bordeaux, now CEA Bruyères le Châtel) Ha Thuy Long (CENBG Bordeaux, Vietnam Nat. U. Hanoï) Kamila Sieja (UMCS Lublin, CENBG Bordeaux, now GSI Darmstadt) Houda Naidja (CENBG Bordeaux, U. Setif) Tran Viet Nhan Hao (CENBG Bordeaux, Vietnam Nat. U. Hanoï) Johann Bartel (IPHC Strasbourg) Ludovic Bonneau (T-Div LANL Los Alamos, now CENBG Bordeaux) Hristo Lafchiev (INRNE Sofia) Jean Libert (IPN Orsay) Dan D. Strottman (T-Div LANL Los Alamos)

3. Introduction The formalism and its implementation Phenomenological residual interactions Low level density regimes Isomeric states High spin states Neutron-proton pairing correlations Isospin mixing Vibrational correlations Conclusions

4. INTRODUCTION Current theoretical descriptions of nuclear structure, imply de facto two steps possibly intertwinned - definition of a reference mean field - treatment of correlations (pairing, vibrational, long range) This is the case for - so-called shell model calculations - self-consistent approaches In the latter case, the often used Bogoliubov quasi-particle vacuum ansatz meets with a basic a priori problem

5. The general quasi particle (qp) Bogoliubov transformation mixes particle creation and anihilation operators. Consequently the particle number of the corresponding qp vacuum is not a good particle number state. Moreover, on more practical grounds, the relatively complicated structure of these qp operators makes it somewhat complicated to handle the further treatment of : - vibrational correlations (so-called QRPA) - configuration mixing (either for treating large amplitude collective motion or for restauring symmetries, e.g. intrinsic parity or rotation) (Balian-Brezin vs Löwdin handling of GCM kernels)

6. There is thus room for a comprehensive treatment able to implement all these correlations within an approach : unified particle-number conserving tractable yet retaining the successful phenomenological properties of Skyrme or Gogny effective interactions In a nutshell, one may perform a highly truncated expansion of the particle-hole basis when diagonalizing some « suitably » defined residual interaction, on a vacuum of the Hartree-Fock type, associated with the total hamiltonian and a 1-body density including the 1-body effects of the correlations Hence its name HTDA standing for HIGHER TAMM-DANCOFF APPROXIMATION

7. THE FORMALISM AND ITS IMPLEMENTATION Start fromH = K + v K 1-body : kinetic energy plus possibly constraints etc., v 2-body : phenomenological « effective » interaction Consider a « suitable » 1-body mean field V, namely in what follows a Hartree-Fock mean field associated with H for a Slater determinant |0> (vacuum) ThenH = H(SM) + v(residual) withH(SM) = K + V - andv(residual) = v – V + so that = and = 0

8. HF for H Building the many-body basis N,Z Odd/Even Symmetries |  i >, e i, |  0 > N,Z, Odd/Even, Symmetries n-p coupling or not Nature of v(resid.) (pairing and/or RPA) |  n > Residual Interaction v(resid.) Wick’s Theorem from to

9. v(resid.) sp symmetries, n-p or not, … Computing Building Lanczös solutions |  > ground state |  x > 1st exc. states

10. It is possible to perform HTDA calculations self-consistently ? Given a correlated solution |  >, one may evaluate the 1-body reduced density matrix  by  i,j = where i and j labels orthonormal sp basis states Such a 1-body  matrix contains the 1-body effects of the correlations One then defines the Hartree-Fock Hamiltonian associated with such a matrix , hence gets the vacuum |  0 > From which HTDA calculations are performed as above NOTE : this approach is self-consitent but not variational

11. Truncations In practice one should make two type of truncations Truncation of the sp space typically 6-8 MeV on each side of the Fermi surface, for pairing 20-30 MeV on each side of the Fermi surface, for « RPA » Truncation of the many body state space To describe reasonably pairing correlations, one may limit oneself to 1p-1h, 2p-2h (of which pair transfer states carry most of the probability of non-vacuum states) However including two pair transfer states enhances somehow the one pair transfer phenomenon In the large window (for « RPA ») only 1p-1h

12. Convergence Only n-pair states here ( 64 Ge, |T Z | = 1) Convergence Only n-pair states here ( 64 Ge, T=0 and 1)

13. PHENOMENOLOGICAL RESIDUAL INTERACTIONS Starting with v(residual) = v – V + One makes a multipole expansion THEN, IN PRACTICE high multipoles ~  force low multipoles ~ Sum   Q Q (  being a constant) Quite an old story… As a further approximation one may neglect some/all the low multipole terms The sp space yielding the many-body states is generally more restricted than the one used for computing the HF field then renormalisation of the multipole components thus lack of consistency between H(SM) and v(residual) therefore state (i.e.  ) – dependence of H !!!! « We are simply forced to simplify the force » (B.R. Mottelson)

14. Fit of the phenomenological  force As in BCS etc… fit some odd-even mass differences Yet here no pairing gaps available for the fit So we have to devise some specific procedure Restricting to a 3-point analysis (e.g. for N neutrons, N even) either centered :  0 = 1/2[E(N+1) + E(N-1)- 2 E(N)] or not :  +/- = 1/2 [2 E(N+/-1) - E(N+/-2)- E(N)] The latter are to be preferred when dealing near N=Z due to the Wigner term cusp It may be latter symmetrised :  =1/2 (  + +  - ) Of course this  is not the one obtained with a 5-point analysis and it should not near N=Z due to the derivative discontinuity

15. Theoretically now Let |  > be the vacuum for N neutrons |  > be the first  unoccupied state for |  > |a> the corresponding last occupied state Define E[N,  ] = expliciting the N-dependence of H We will consider four HTDA solutions |N-2>, |N> for H(N-2), (H(N) with N, N-2 neutrons |N-2/a> for H(N-2) with N-2 neutrons on a spectrum without |a> |N/  > for H(N) with N neutrons on a spectrum without |  >

16. To compute  + N neutrons are described by |N> N+1 by |  > on top of |N/  > N+2 by |  > |  tr > on top of |N/  > yielding  + = 1/2 [E(N,N/  ) -E(N,N)] –1/4 antisym. To compute  - N-2 neutrons are described by |N-2> N-1 by |a> on top of |N-2/ a> N by |a> |a tr > on top of |N-2/a> yielding  - = 1/2 [E(N-2,N-2/ a) -E(N-2,N-2)] –1/4 antisym.

17. Fit of the phenomenological Q Q forces Extend the sp space (beyond what is needed for pairing) so as to generate sufficient 1p-1h many-body states To fulfill some sum rules associated with moments m k (Q) typically k=1,3, up to ~70-80 % Solve the secular equation to generate a HTDA spectrum to reach a sufficient convergence of some E k energies typically k=1,3 with E k = (m k (Q) / m k-2 (Q) ) 1/2 Quadrupole mode SIII 40 Ca

18. Comparing with giant resonance data gives a handle on the   parameter

19. MAGIC NUCLEI Pairing correlations do exist in e.g. 208 Pb They are however too small to depopulate onlyby themselves the proton 3s 1/2 state So as to account for the observed charge density pattern Hartree-Fock calculations add a spurious bump to the 208 Pb charge density at the center of the nucleus

20. To flatten the density at r = 0, one needs a depopulation of ~ 20 % of the 3s1/2

21. Occupation probabilities Any « reasonable » pairing would not be able to depopulate the 3s1/2 by more than 7%

22. ISOMERIC STATES : THE 178 Hf EXAMPLE

23. In this nuclei (and around at higher E*) One may build two aligned qp structures for neutrons : 8 - = 7/2 - x 9/2 + for protons : 8 - = 7/2 + x 9/2 - In 178 Hf these sp states are precisely calculated (with SIII) on both sides of the Fermi surface for BOTH neutrons and protons

24. ENERGIES (MeV) StateCalculated Experimental 0 + 00 8 - (n)1.141.15state known to be mixed (n x p) 8 - (p)1.37 6 + (n)1.401.55 state known to be mixed (n x p) 6 + (p)2.06 16 + 2.452.45delta force strength fitted for agreement 15 + 2.84unseen yet, proposed 14 - 2.862.57 This type of agreement needs a reasonably good pairing description of the Pauli-blocked (V.G. Soloviev) isomeric states, for a pairing force shown to provide also good pairing properties for the g.s.

25. HIGH SPIN STATES Routhian variational calculations with HTDA trial wavefunctions  (H-  J) = 0 for yrast SD bands in the Hg / Pb region

26. Dynamical moments of inertia for Hg isotopes also Sum i u i v i = tr(  (1-  )) 1/2 and evolution of the quadrupole moments with I

27. Dynamical moments of inertia for Hg isotopes

28. Kinematic moments of inertia

29. Neutron-proton pairing correlations Correlation energies as functions of x (ratio of T=0/T=1 pairing interaction strengths) T=0 and T=1 correlations coexist (at variance with BCS or BCS/LN results)

30. Neutron-proton pairing correlations in BCS and BCS LipkinNogami (K. Sieja, A Baran results)

31. ISOSPIN MIXING Despite the important Coulomb interaction the isobaric invariance is weakly violated So that |g.s.> ~  | T 0, T Z > +  | T 0 + 1, T Z > with  2 <<  2,  2 +  2 =1, T Z = (N-Z)/2, T 0 =| T Z | Thus  2 ~ [ - T 0 (T 0 +1)] / 2 (T 0 +1) Meaningless in BCS, HFB : spurious mixing of T Z components !!!

32. COMPARAISON OF VARIOUS ESTIMATES OF THE MIXING PARAMETER Illustrative HTDA results (  force strength yet to be adjusted) Bohr et al. : Hydrodyn. Model Hamamoto et al. : HF + RPA HTDA : |T Z |=1 pairing Some conclusions : HF negligible |T Z |=1 pairing important T Z =0 pairing reduce  2 Role of « RPA » ?

33. ISOSPIN MIXING AS A FUNCTION OF T 0  2 maximum for N=Z Decrease by a factor T 0 + 1 (Lane and Soper)

34. VIBRATIONAL CORRELATIONS Preliminary results Isoscalar quadrupole mode ( = 2) only Calculations as specified above (in particular for the strength of v(resid.)) SIII interaction for HF Discrete calculations (ad hoc widening to yield smooth distributions) At present no pairing yet (i.e. only Q 2 Q 2 force)

35. Transition matrix elements squared 0 and 2   Excitations

36. Contributions to the m 1 moment

37. Contribution to the m 1 moment

38. ENERGIES E 3 WIDTHS  2  k = ((m k / (m k-2 )- (m k-1 / (m k-2 ) 2 ) 1/2

39. CONCLUSIONS This method seems rather promising and in some cases provides results not yet obtained (routhian, isospin mixing) The main drawback is the state dependence of the Hamiltonian More generally, its microscopic foundation should be worked out Currently, « QRPA », odd nuclei and more generally « qp » configurations as well as configuration mixing calculations are under study