Forces for extensions of mean-field PhD Thesis Marlène Assié Denis Lacroix (LPC Caen), Jean-Antoine Scarpaci (IPN Orsay)  Extensions of mean-field ?  Why a new force ? and which forces ?  Some results  Perspectives

Beyond the mean-field static :HF dynamical : TDHF mean-field Skyrme Forces LOW ENERGY INTERMEDIATE ENERGY E/A=5MeV/uE/A=150MeV/u Boltzmann term which force ? collisions important + mean-field Extended TDHF Bonche et al PRC 13 (1976) 1226 Skyrme Forces

Zero range force versus finite range force Time Dependent Density Matrix :  function used : v=v 0  3 (r 1 –r 2 ) Extended TDHF : = C (| ) cut off  k~range With a finite range force : range M. Tohyama Prog. Theor. Phys. 94(1995) S.Wang, W. Cassing, Nucl. Phys. A 652 (1999) D.Lacroix, S. Ayik, Ph. Chomaz, Prog. in Part. Nucl. Phys. 52(2004)

The collision term Forces in the collision term might be fitted to the nucleon- nucleon cross-section In the semi-classical limit p1 p2 p4 p3 loss gain k k’ interaction dependence

Li & Machleidt Cross Section for various Skyrme forces and the Gogny force Li & Machleidt Phys. Rev. C48 (1993); Phys. Rev. C 49 (1994) Total Cross Section (mb) Energy (MeV) Li & Machleidt parametrization

Li & Machleidt SGII SkM SkM* SkP SLy4 SLy7 Cross Section for various Skyrme forces and the Gogny force Total Cross Section (mb) Energy (MeV) Yildirim et al, Eur. Phys. J. A 10 (2001) ; Yilmaz et al, Phys. Lett. B 472 (2000)

Cross section with Skyrme forces Yildrim et al, Eur. Phys. J. A 10 (2001) ; Yilmaz et al, Phys. Lett. B 472 (2000) Cross section calculated with the Skyrme force :

Li & Machleidt SGII SkM SkM* SkP SLy4 SLy7 Gogny Cross Section for various Skyrme forces and the Gogny force Total Cross Section (mb) energy (MeV) None of these forces reproduce nucleon- nucleon in medium cross-section new force

Fitted force Force of finite-range and separable  finite-range  already used for pairing correlations Duguet PRC 69 (2004)  separable : simpler numerically energy (MeV)  Gaussian (G) Total cross section (mb) Fit with Gaussian = -655 MeV fm3  ² = fm2  0 =0,18 fm -3 Li & Machleidt fit (G)  Gaussian +constant (G+C) Fit with Gaussian + c = -137 MeV fm3  1 ²= 1.54 fm2 c=1.34  0 =0,18 fm -3 energy (MeV)

Density dependence Li & Machleidt parameterization E=50MeV Parameterization of Li & Machleidt density (fm-3) Total cross section (mb) Li & Machleidt Phys. Rev. C48 (1993); Phys. Rev. C 49 (1994)

Density dependence Li & Machleidt parameterization Gaussian Gaussian + constant E=50MeV Density dependence (DDG) similar as Skyrme introduced to reproduce a density dependence Parameterization of Li & Machleidt density (fm -3 ) Total cross section (mb) Li & Machleidt Phys. Rev. C48,1702; Phys. Rev. C 49, 566

Fit with the density dependent function Fit method : 2 dimensions fit on the energy and the density with 4 parameters E=50 MeV E=100 MeV density (fm -3 ) energy (MeV)  =0.05 fm -3  =0.15 fm -3 Total cross section (mb) Li & Machleidt fit with the density dependent force (DDG)

Fit with a sum of Gaussian and density dependence Fit method : 2 dimensions fit on the density and the energy with 5 parameters energy (MeV) Total cross-section (mb)  =0.05 fm -3  =0.15 fm -3 Li & Machleidt fit with the density dependent force (DDGC) density (fm -3 ) E=100 MeV E=50 MeV

Summary 2 forces which reproduce density and energy dependence of the nucleon-nucleon cross section particle-hole channel Skyrme force particle-particle channel hole-hole channel pairing force Case of pairing Case of 2 body correlations particle-hole channel Skyrme force 2 particle- 2 hole channel separable finite-range force should use the same force independent

Comparison with the Skyrme force when k  0 and  =0 Skyrme forcesGaussian (G)Gaussian+ constant (GC) Gaussian density dependent Gaussian+ constant density dependent Up to 10 parameters2 parameters3 parameters4 parameters5 parameters t0=-2931 to MeV.fm MeV.fm MeV.fm MeV.fm MeV.fm 3 t1=235 to 970 MeV.fm MeV.fm MeV.fm MeV.fm MeV.fm 5 t2=-556 to 107 MeV.fm 5 Neglected t3=8000 to MeV.fm (3+3  ) MeV.fm 3+3  MeV.fm 3+3   =1/6 to Skyrme Force Gaussian t 0  t 1  -2  ² Gaussian density dependent t 0  t 1  -2  ² t3  c3

Skyrme Gaussian density Gaussian + c + density Li & Machleidt t0=-2572 MeV.fm3 t0= MeV.fm3 t0=-2645 MeV.fm3 t1= 1896 MeV.fm3 t1= 1038 MeV.fm3 t1= 410 MeV.fm3

Perspectives Time Dependent Density Matrix (TDDM) W. Cassing, S. J. Wang, Z. Phys. A, 328 (1987)  truncation of the BBGKY hierarchy  takes into account all the two body correlations (extension of HF) Skyrme force separable finite- range force Perspectives : study of correlations between two neutrons in borromean nuclei Experiment planned for July 2006 at GANIL

Skyrme Gaussian Gaussian + c Li & Machleidt

Skyrme Gaussian density Gaussian + c + densite Gaussian Gaussian + c Li & Machleidt Comparison of slopes and values in zero for all fits

About the Skyrme force REALISTIC FINITE-RANGE FORCE In momentum space In a nucleus  short-range expansion ZERO-RANGE FORCE