HIGHER TAMM-DANCOFF APPROXIMATION

A particle number conserving microscopic method to treat correlations in fermionic systems

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)

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

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

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)

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 expansionof the particle-hole basis when diagonalizing some « suitably » defined residual interaction, on a vacuum of the Hartree-Fock type, associated with the total hamiltonian anda 1-body density including the 1-body effects of the correlations

Hence its name

HTDA standing forHIGHER TAMM-DANCOFF APPROXIMATION

THE FORMALISM AND ITS IMPLEMENTATION

Start from H = 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)

Then H = H(SM) + v(residual)with H(SM) = K + V - <0|v|0>and v(residual) = v – V + <0|v|0>so that <0|H(SM)|0> = <0|H|0>and <0|v(residual)|0> = 0

HFfor H

Building the many-body

basis

N,ZOdd/Even

Symmetries

|i> , ei , |0>

N,Z, Odd/Even, Symmetriesn-p coupling or notNature of v(resid.)

(pairing and/or RPA)

|n>

Residual Interaction

v(resid.)

Wick’s Theoremfrom<n|H|m>

to<ij|v(resid.)|kl>

v(resid.)sp symmetries,n-p or not, …

Computing<ij|v(resid.)|kl>

Building<n|H|m>

Lanczössolutions

|> ground state|x> 1st exc. states

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 <j||i> = < | a+i aj | >

where i and j labels orthonormal sp basis statesSuch a 1-body matrixcontains the 1-body effects of the correlations

One then defines the Hartree-Fock Hamiltonianassociated 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

TruncationsIn 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 pairing20-30 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 statescarry 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

Convergence

Only n-pairstates here

(64Ge, |TZ| = 1)

Convergence

Only n-pairstates here

(64Ge, T=0 and 1)

PHENOMENOLOGICAL RESIDUAL INTERACTIONS

Starting with v(residual) = v – V + <0|v|0>One makes a multipole expansion THEN, IN PRACTICE

high multipoles ~ forcelow 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)

Fit of the phenomenological force

As in BCS etc… fit some odd-even mass differencesYet 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 analysisand it should not near N=Z due to the derivative discontinuity

Theoretically now

Let |> be the vacuum for N neutrons|> be the first unoccupied state for |>|a> the corresponding last occupied state

Define E[N,] = <|H(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 | >

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 < tr|v(resid.)| tr>antisym.

To compute -

N-2 neutrons are described by |N-2>N-1 by |a> on top of |N-2/ a>N by |a> |atr> on top of |N-2/a>

yielding

- = 1/2 [E(N-2,N-2/ a) -E(N-2,N-2)] –1/4 < a atr|v(resid.)| a atr>antisym.

Fit of the phenomenological QQ forces

Extend the sp space (beyond what is needed for pairing)so as to generate sufficient 1p-1h many-body statesTo fulfill some sum rules associated with moments mk(Q)

typically k=1,3, up to ~70-80 %

Solve the secular equation to generate a HTDA spectrumto reach a sufficient convergence of some Ek energiestypically k=1,3 with Ek = (mk(Q) / mk-2(Q) )1/2

Quadrupole modeSIII40Ca

Comparing with giant resonance data gives a handle on the parameter

MAGIC NUCLEI

Pairing correlations do exist in e.g. 208PbThey are however too small to depopulate onlyby themselves the proton 3s1/2 state So as to account for the observed charge density pattern

Hartree-Fock calculations add a spurious bump to the 208Pb charge density at the center of the nucleus

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

Occupation probabilities

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

ISOMERIC STATES : THE 178Hf EXAMPLE

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 178Hf these sp statesare precisely calculated (with SIII)on both sides of the Fermi surfacefor BOTHneutrons and protons

ENERGIES (MeV)

State Calculated Experimental

0+ 0 08-(n) 1.14 1.15 state known to be mixed (n x p)8-(p) 1.376+(n) 1.40 1.55 state known to be mixed (n x p) 6+(p) 2.0616+ 2.45 2.45 delta force strength fitted for agreement15+ 2.84 unseen yet, proposed14- 2.86 2.57

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.

HIGH SPIN STATES

Routhian variational calculationswith HTDA trial wavefunctions

(H-J) = 0

for yrast SD bands in the Hg / Pb region

Dynamical moments of inertia for Hg isotopesalso Sum i u iv i = tr((1-))1/2

and evolution of the quadrupole moments with I

Dynamical moments of inertia for Hg isotopes

Kinematic moments of inertia

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)

Neutron-proton pairing correlationsin BCS and BCS LipkinNogami

(K. Sieja, A Baran results)

ISOSPIN MIXINGDespite the important Coulomb interaction

the isobaric invariance is weakly violated

So that |g.s.> ~ | T0 , TZ> + | T0 + 1, TZ>

with 2 << 2, 2 + 2 =1, TZ = (N-Z)/2, T0 =| TZ |

Thus 2 ~ [<g.s.|T2|g.s.> - T0(T0+1)] / 2 (T0+1)

Meaningless in BCS, HFB : spurious mixing of TZ components !!!

COMPARAISON OF VARIOUS ESTIMATES OF THE MIXING PARAMETER

Illustrative HTDA results ( force strength yet to be adjusted)

Bohr et al. : Hydrodyn. ModelHamamoto et al. : HF + RPAHTDA : |TZ|=1 pairing

Some conclusions :HF negligible|TZ|=1 pairing importantTZ=0 pairing reduce 2 Role of « RPA » ?

ISOSPIN MIXING AS A FUNCTION OF T0

2 maximum for N=Z

Decrease by a factor T0 + 1(Lane and Soper)

VIBRATIONAL CORRELATIONS

Preliminary resultsIsoscalar quadrupole mode ( = 2) onlyCalculations as specified above

(in particular for the strength of v(resid.))SIII interaction for HFDiscrete calculations

(ad hoc widening to yield smooth distributions)At present no pairing yet (i.e. only Q2Q2 force)

Transition matrix elementssquared

0 and 2 Excitations

Contributionsto the m1 moment

Contribution to the m1 moment

ENERGIES E3

WIDTHS 2

k = ((m k / (m k-2 )- (m k-1 / (m k-2)2)1/2

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 » configurationsas well as configuration mixing calculations are under study