Nuclear matter inspired Energy density functional for finite nuclei: the BCP EDF M. Baldo a, L.M. Robledo b, P. Schuck c, X. Vinyes d a Instituto Nazionale di Fisica Nucleare, Sezione di Catania, Catania, Italy b Universidad Autónoma de Madrid, Madrid, Spain c Institut de Physique Nucléaire, Orsay, France d Universitat de Barcelona, Barcelona, Spain August 7th, 2009 Workshop on The Lead Radius Experiment and Neutron Rich Matter...
Why a new EDF? Bare nucleon-nucleon Bare nucleon-nucleon interaction well known at long distances. At short distances the repulsive core is less known. Three body forces are more or less understood. Short range in-medium correlations Short range in-medium correlations (Pauli blocking) cancel out the repulsive core and yield a smooth effective in medium interaction Effective interactions Handling of short range correlations requires Brueckner-like methods which are extremely hard to implement in finite nuclei. The smooth effective in-medium interaction is replaced by phenomenological effective interactions like Skyrme, Gogny or RFM
Why a new EDF? (II) Non-relativistic Skyrme /Gogny Central part, spin-orbit, Coulomb and a phenomenological density dependent term (involving non-integer powers of the density) Skyrme: Zero range central part δ( r r ) + gradient terms Gogny: Finite range central part exp( ( r r ) 2 /µ 2 ) 10-15 params fitted to nuclear matter (E/A, k F, K,... ) at equilibrium and finite nuclei (mostly spherical at the valley of stability). Skyrme has 100 parametrizations, Gogny 3, RMF 10-15 Asymptotic freedom problem: predictions from different parametrizations scatter around when moving away from the valley of stability suggest other strategies
Recent strategies Use more information from symmetric and neutron EoS to constrain the parameters ρ < ρ 0 relevant at the surface of finite nuclei Considering neutron EoS should improve description of neutron rich nuclei Skyrme SLy, Gogny D1N, etc Used as a requirement for reasonable interactions (Chamel s talk) Not so many results up to now to make sure we are free from asymptotic freedom
Skyrme, Gogny, RMF Fixed central parts with 10 parameters Fitted to nuclear matter EoS obtained in realistic calculations (BHF + AV18, etc) Fit the EoS with a given function of ρ and use the LDA to translate to finite nuclei. Not so easy to reproduce the EoS in the whole range of relevant densities Proliferation of parametrizations Similar to DFT strategy to guess the unknown exchange terms In nuclear physics it was floating around in the past (Vautherin, Negele) and somehow implemented by Fayans in 2001
Our requirements Not demanding in terms of computer resources Free from non-integer powers of the density for a hassle free implementation of beyond mean field methods Mass table quality for binding energies and radii (for astrophysical applications!) Suited to accurately describe other properties like Quadrupole and octupole deformation Fission Moments of inertia Good behavior beyond mean field (correlations are very important)
Realistic EoS M. Baldo, C. Maieron, P. Schuck and X. Viñas, Nucl. Phys. A736 (2004) 241 Bethe-Brueckner + Converged hole line expansion AV18 + Three body forces (Carlson, Schiavilla, Pandharipande, Wiringa) Symmetric + Neutron EoS For other asymmetries β = (ρ n ρ p)/ρ a quadratic interpolation is used e = e nβ 2 + e s(1 β 2 )
Fitting the EoS The symmetric (s) and neutron (n) matter EoS are now fitted with polynomials P s and P n of the total density ρ j P 5 P s(ρ) = k=1 a(n) k x k x < 1 P s(ρ 0) + a 1(x 1) + a 2(x 1) 2 x > 1 with x = ρ/ρ 0 and ρ 0 = 0.17fm 3 P n(ρ) = Can be used up to ρ = 0.24 fm 3 5X k=1 b (n) k x k The interpolating polynomial for symmetric matter has been constrained to allow a minimum exactly at the energy E/A = 16MeV and Fermi momentum k F = 1.36fm 1, i.e. ρ 0 = 0.17 fm 3. Integer powers of the density (unlike expansions in k F )
Fitting the EoS, results
The BCP functional In the spirit of the LDA it is proposed to use the previous fit in finite nuclei just replacing the nuclear matter density ρ by the finite nuclei one ρ( r). This is the key ingredient defining the BCP (Barcelona, Catania, Paris) energy density functional (EDF). The energy of a finite nucleus is given by where E = T 0 + E int + E FR int + E s.o. + E C + E pair. Z Eint [ρ p, ρ n] = drˆp s(ρ)(1 β 2 ) + P n(ρ)β 2 ρ with ρ( r) = ρ n( r) + ρ p( r) and β( r) = ρ n( r) ρ p( r)/ρ( r) The other terms are the kinetic energy T 0, a surface term E FR int, the spin-orbit energy E s.o., the Coulomb term E C and finally the pairing energy E pair M.Baldo, P.Schuck and X. Viñas, Phys. Lett. B663 (2008) 390
Nuclear matter properties B/A(MeV) ρ 0 (fm 3 ) m/m K (MeV) -16.0 0.17 1.00 249. J (MeV) L(MeV) E sym(mev fm 3 ) 33.55 56.39-99.5
Remaining contributions to the EDF Phenomenological surface contribution Eint FR [ρ n, ρ p] = 1 X jz Z Z d rd r 2 ρ t( r)v t,t ( r r )ρ t ( r ) γ t,t t,t ff d rρ t( r)ρ t ( r) with v t,t (r) = V t,t e r 2 2 /r 0 and γt,t = R drv t,t (r) V n,n = V p,p = V L, V n,p = V p,n = V U and r 0 are free parameters to be fitted using finite nuclei data Coulomb Direct E H C = (1/2) R R drdr ρ p(r) r r 1 ρ p(r ) Exchange: EC ex = (3/4)(3/π) R 1/3 drρ p(r) 4/3 Spin-Orbit ˆv ij so = iw 0( σ i + σ j ) [ k δ( r i r j ) k]
Remaining contributions to the EDF Pairing Correlations Zero-range interaction, tailored to m=m,» «α v pp (n( r)) = v0 n( r) 1 η, n 0 = 2 2 3π k 3 2 F. L.N. Oliveira, E.K.U. Gross and W. Kohn, Phys. Rev. Lett. 60 (1988) 2430. E. Garrido, P. Sarriguren, E. Moya de Guerra, and P. Schuck, Phys. Rev. C 60, 064312 (1999) Parameters fitted to reproduce Gogny s pairing gap in nuclear matter n 0 Two-body center of mass correction Pocket formula based on HO M.N. Butler, D.W.L. Sprung and J.Martorell, Nucl. Phys. A422, 157 (1984).
Fitting procedure The free parameters, V L, V U, r 0 and W 0 are fitted to reproduce the binding energies of the spherical nuclei 16 O, 40 Ca, 48 Ca, 72 Ni, 90 Zr, 116 Sn, 124 Sn, 132 Sn, 204 Pb, 208 Pb, 214 Pb, and 210 Po the charge radii r c = rp 2 + 0.64 fm of the spherical nuclei 16 O, 40 Ca, 48 Ca, 90 Zr, 116 Sn, 124 Sn, 204 Pb, 208 Pb and 214 Pb. r 0 (fm) V L (MeV) V U (MeV) W 0 (MeV) BCP 1 1.05-93.520-60.577 113.829 BCP 2 1.25-33.700-32.483 110.812
Using 161 spherical nuclei BCP1 BCP2 D1S NL3 SLy4 rms E 1.775 2.057 2.414 3.582 1.711 MeV rms R 0.031 0.028 0.020 0.020 0.024 fm
Improving the fit Using an idea of Bertsch to improve locally any fit by using a linear approximation to the function to be minimized G.F. Bertsch et al Phys. Rev. C71 (2005) 054311 r 0 (fm) V L (MeV) V U (MeV) W 0 (MeV) BCP 1.05-64.562-91.255 114.927 BCP1 BCP rms E 1.775 1.599 rms R 0.031 0.033
Neutron drip line nuclei Noticeable differences at the tail of the densities
Neutron skin thickness r np = (1.01 ± 0.15)I + ( 0.4 ± 0.03) vs I = (N Z)/A Tzcińska et al Phys. Rev. Lett. 87 (2001) 082501
Typical example of deformation: 164 Er Same position maxima and minima, and even shoulders Spherical barrier higher in D1S Stronger pairing correlations in D1S
Dash: D1S, Full: BCP a s = 17.74 (BCP1), 18.20 (D1S) MeV
Dy isotopic chain BCP1 and BCP2 very similar Bigger E def in D1S Similar def parameter η (axis ratio) Reasonable δr (r - A 1/3 estimate) Reasonable S 2N
Single particle energies in 160 Dy Left protons, right neutrons m =1 in BCP m =0.7 in Gogny D1S Stretched D1S Reordering of level
Binding energies ( 154 Sm - 182 Hf) ( 230 Th - 250 Cf) With 161 spherical and 66 deformed nuclei rms E (MeV) BCP1 2.417 D1S 4.208 NLS3 3.062 Notice the large contribution of deformed nuclei, except for NL3
Fission Barrier of 240 Pu Similar BCP1 and BCP2 Lower barrier heights in BCP Higher collective masses Similar WKB half lives τ BCP1 = 1.2 10 28 s τ BCP2 = 1.1 10 27 s τ D1S = 1.5 10 26 s
Fission Barriers of Superheavy Elements
Octupole Deformations
Conclusions... BCP (1 or 2) works pretty well for finite nuclei properties rms s are a little too high but of the same quality as SLy4, D1S... but they are very poor for a mass table...
BCP 2009 There is a new version of the EDF called BCP 2009 where the symmetric and neutron matter fit has been redone to avoid wiggles in other quantities like K
Fit to spherical nuclei Blue circles: BCP1, Red crosses: BCP2009*, Black crosses: D1S BCP1 BCP2009 D1S NL3 rms E 1.775 1.133 2.414 3.582 MeV rms R 0.031 0.027 0.020 0.020 fm 161 spherical nuclei for rms E and 86 for rms R
Fit to deformed nuclei Work in progress... According to a suggestion by P. Ring, is better to fit deformed nuclei as they are more numerous and more mean field like (additional correlations are mostly static, not dynamic as in spherical nuclei... ) We have chosen 84 deformed nuclei in the rare earth, actinides and superheavy regions to run a deformed fit The binding energy is the mean field energy supplemented with the rotational energy correction and an estimation of the effect of the finite size of the basis. We start from the spherical fit parameters and then run a linear refit ala Bertsch on V U, V L and W SO for different choices of r 0 Using correlation analysis, we find that there is only one linearly independent parameter out of the three considered
Preliminary results Comparison with D1M (S. Goriely et al., PRL 102, 242501 (2009)) The rms E are impressive and encouraging and of the same quality as D1M (at a fraction of the computational cost)... a mass table is now possible...
what is next... Look at spherical nuclei Compute Mass table Explore properties of neutron rich nuclei Repeat calculations on quadrupole, fission, octupole, etc Include triaxiality and rotational bands in the check list Odd-A nuclei (in progress with spherical nuclei) Explore beyond mean field approaches like symmetry restorations Explore other pairing functionals...
Conclusions A new EDF based on a fit to realistic EoS and the LDA is postulated It contains essentially two free parameters (apart from the ones of the fit) Its local character makes it fast on the computer Nice results for finite nuclei comparable to those of the performant Gogny D1S The new BCP 2009 seems good enough as to produce a mass table Contrary to most of Skyme s, all Gogny s and most RMF s, its density dependence comes through integer powers of the density and therefore there is no hassle with beyond mean field approaches (no poles, branch-cuts, complex energies, etc) that could help improve rms E...
a single item to take away in the community of nuclear structure theorists there is still interest in producing improved mass tables for astrophysics applications... (there is still life after Skyrme, etc)