Skyrme Hartree-Fock Methods and the Saclay-Lyon Forces

Skyrme Hartree-Fock Methods and the Saclay-Lyon Forces K. Bennaceur, IPNL UCB Lyon-1 / CEA ESNT The SLy force, quick overview Beyond the standard parameterization: new terms... Extended density dependence Tensor force... and new constraints Effective masses Instabilities

Zoology of the effective interactions Effective interactions Main ingredient for HF calculations (and beyond: HFB, RPA, GCM,...) Different flavors: Zero range (Skyrme forces), finite range (Gogny force), relativistic (effective lagrangians),... Different recipes: Density dependent term, spin-orbit term, isospin properties,... Very easy to construct a Skyrme force!... at most 10 Gogny forces about 20 relativistic lagrangians more than 100 Skyrme forces!

Effective interactions in mean field calculations H = T + V realistic NN interaction many body problem E = X i G = V V Q e G ki 2 2m + 1 X ij G E=ei +e 2 j ij ij<f e i = k2 i 2m + X ij G E=ei +e j ij j<f G matrix (Bethe-Goldstone equation) Effective phenomenological interaction V eff = V Skyrme, V Gogny,... Effective microscopic hamiltonian and mean field approximation Φ HF, δ Φ H eff Φ Φ Φ = 0,...

First benchmark: the symmetric infinite nuclear matter Infinite medium, no surface: N = Z with ρ = A V No Coulomb interaction = cste, I = N Z A = 0 No pairing correlations Binding energy liquid drop model E A = a v + a I I 2 + a s A 1/3 + a Is I 2 A 1/3 + a c Z 2 A 4/3 +... P. Möller, J.R. Nix, W.D. Myers and W.J. Swiatecki, 1995. E/A = 16.0 ± 0.2 MeV Density in the center of eavy nuclei electron scattering ρ 0 = 0.16 ± 0.002 fm 3 1654 known masses from 16 O to 263 106 J.B. Bellicard et al., 1981; H. de Vries, 1987.

BHF Calculations Coester lines E/A [MeV] ρ [fm 3 ] G = V V Q e G at lowest order misses the saturation point!

The standard Skyrme forces V (r 1,r 2 ) = t 0 (1 + x 0 ˆPσ )δ(r) central + 1 2 t 1(1 + x 1 ˆP σ ) [ P 2 δ(r) + δ(r)p 2] non local + t 2 (1 + x 2 ˆPσ )P δ(r)p non local + t 3 (1 + x 3 ˆP σ )ρ α (r)δ(r) density dependent + iw 0 σ [P δ(r)p] spin-orbit Beiner, Flocard, Giai, Quentin, 1975: SIII Krivine, Treiner, Bohigas, 1980: SkM Bartel et al., 1982: SkM* Giai, Sagawa, 1981: SGII Dobaczewski et al., 1984: SkP Rayet, Arnould, Tondeur, Paulus, 1982: RATP Chabanat et al., 1995: SLyxx

Density dependence 600 compression modulus (MeV) 500 400 300 200 Skyrme Gogny linear lagrangians non linear lagrangians Brink Boeker finite range 100 0.12 0.14 0.16 0.18 0.2 0.22 equilibrium density (fm** 3) Compression modulus K known to be 220 or 240 MeV

(local) Density functional Φ H Φ = H(r) dr H = K + H 0 + H 3 + H eff + H fin + H so + H sg + H coul H 0 = 1 4 t 0 ˆ(2 + x0 )ρ 2 (2x 0 + 1) `ρ 2 n + ρ 2 p H 3 = 1 24 t 3ρ α ˆ(2 + x 3 )ρ 2 (2x 3 + 1) `ρ 2 n + ρ 2 p H eff = 1 8 [t 1(2 + x 1 ) + t 2 (2 + x 2 )] τρ + 1 8 [t 2(2x 2 + 1) t 1 (2x 1 + 1)](τ n ρ n + τ p ρ p ) H fin = 1 32 [3t 1(2 + x 1 ) t 2 (2 + x 2 )]( ρ) 2 1 32 [3t 1(2x 1 + 1) + t 2 (2x 2 + 1)] ˆ( ρ n ) 2 + ( ρ p ) 2 H so = 1 2 W 0 [J ρ + J n ρ n + J p ρ p ] H sg = 1 16 (t 1x 1 + t 2 x 2 )J 2 + 1 16 (t 1 t 2 ) ˆJ 2 n + J 2 p ρ q = X i<f,σ J q = ϕ i (rσq) 2, τ q = X X i<f,σ ϕ i (rσq) 2, i<f,σσ ϕ i (rσ q) σ σ σ ϕ i (rσq)

(local) Density functional Correct description of the binding energies and radii of nuclei (Vautherin and Brink, 1972). Simple mathematical form of the functional Tractable calculations in r representation Simple form of the 1-body HF field Relation with realistic NN interactions (Negele and Vautherin, 1972-75).

Cooking recipe E A V Skyrme 10 parameters ρ 0 Saturation point of symmetric INM ρ 0, E/A ρ e 0 Compression modulus K = 9ρ 2 0 d 2 E dρ 2 A (ρ) ρ=ρ0 Giant breathing mode E0;T=0 K = 220 ± 20 MeV Isoscalar effective mass m m «1 s = 1 + 1 8 m 2 [3t 1 + t 2 (5 + 4x 2 )] ρ 0 (J.P. Blaizot) Giant mode E2;T=0 m m = 0.8 ± 0.1

Cooking recipe 1.0 0.9 T6 α = 1/6 α = 1/3 α = 1/2 α = 1 m*/m 0.8 SkM* SIII 0.7 RATP 0.6 150 200 250 300 350 400 K (MeV) K m m and α 1 6 to 1 3

Cooking recipe: asymmetric INM Y p = ρ p ρ E A Yp=0 Yp=0.5 E A = a v + a I I 2 + a s A 1/3 + a Is I 2 A 1/3 +... symmetry energy: a I = 1 d 2 E 2 dρ 2 A (ρ) I=0 a I 32 MeV (P.G. Reinhard, B.A. Li) ρ isovector effective mass: m 1 m v = 1 + m 4 2 [t 1 (2 + x 1 ) + t 2 (2 + x 2 )] ρ 0 = 1 + κ v κ v TRK enhancement factor for the m 1 sum rule κ v 0.4 to 0.5 neutron matter EOS (Wiringa et al., 1988; Akmal et al., 1998)

0.6 Ca Sn 0.4 r n - r p [fm] 0.2 0.0-0.2 a I = 38 37 36 35 34 33 32 31 30 29 28 r n r p (fm) 0.5 0.4 0.3 0.2 Hoffmann, 1980 Starodubsky, 1994 Krasznahorkay, 1994 Clark, 2003 Batty, 1989 HFB SLyxx -5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 N - Z 0.1 0 20 24 28 32 36 40 a (MeV) I

Cooking recipe: Surface energy The atomic nucleus is not a infinite system... especially near the surface. 30 1.0 E_surf (MeV) 25 20 SLy4 SIII SkM* peau de neutrons (fm) 0.8 0.6 0.4 0.2 SLy4 SIII SkM* 15 0.0 0.1 0.2 0.3 0.4 0.5 I = (N Z)/A 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 I = (N Z)/A

Cooking recipe: Ferromagnetic neutron matter Landau parameters 5 4 < x 2 1 ρ f (fm 3 ) 0.75 0.6 0.45 0.3 0.15 10 5 10 4 10 3 10 2 10 1 Y p SLy230a SLy230b SGII SkM* SIII D1S SKX

Cooking recipe: Summary ρ 0 E A K m m t i,x i,α a I neutron matter κ v t i,x i surface energy, Landau parameter t i,x i spherical magic nuclei: t i,x i,w 0 16 O, 40 48 Ca, 56 Ni, 132 Sn, 208 Pb (binding energies, charge radii, s.p.e.)

SLy force or forces? One force... with various realizations! Force J 2 terms 2-body c.m. Spin-orbit SLy4 no no standard SLy5 yes no standard SLy6 no yes standard SLy7 yes yes standard SLy10 yes yes extended Standard S.O. H so = W 0 2 2 4J ρ + X q 3 J q ρ q 5 Extended S.O. H so = W 1 2 J ρ + W 2 2 X J q ρ q q

The parameters can not reabsorb everything! 1.0 Isotopic shift in lead isotopes 0.5 SLy4 SLy5 r 2 (fm 2 ) 0.0-0.5-1.0 1.0 0.5 SLy6 SLy7 r 2 (fm 2 ) 0.0-0.5-1.0 110 115 120 125 130 135 N 110 115 120 125 130 135 N

Not so bad results E = E calc E exp 2.0 0.0-2.0 Z=50 16.0 14.0 12.0 SLy4 SkM* N=82 E (MeV) -4.0-6.0-8.0 SLy4 SkM* E (MeV) 10.0 8.0 6.0 4.0-10.0 2.0-12.0 50 55 60 65 70 75 80 85 90 N 0.0 50 55 60 65 70 Z E. Chabanat et al., Nucl. Phys. A 627, 710; ibid. 635, 231; ibid. 643, 441.

And not so good results p [MeV] n [MeV] 2 0-2 -4-6 -8-10 -12 0-2 -4-6 -8-10 -12 208 Pb 13/2 + 7/2-9/2-1/2 + 3/2 + 11/2-5/2 + 5/2 + 15/2-11/2 + 9/2 + 1/2-5/2-3/2-13/2 + 7/2-5/2 + Expt. Expt. FY FY SkP SkP SkM * SkM * SLy6 SLy6 SLy7 SLy7 SkI1 126 SkI1 82 SkI4 SkI4 SkI3 SkI3 NL3 NL3 NL-Z NL-Z 92 NL-Z2 NL-Z2 NL-VT1 NL-VT1 3p3/2-2f5/2-1i13/2 + 2f7/2-1h9/2-3s1/2 + 2d3/2 + 1h11/2-2d5/2 + 4s1/2 + 2g7/2 + 3d5/2 + 1i11/2 + 2g9/2 + 3p1/2-3p3/2-1i13/2 + 2f5/2-2f7/2 - Poor spectroscopic properties Shape description?... Beyond mean field correlations?... See M. Bender, et al., Rev. Mod. Phys. 75, 2003

We are far from the end of the story... Isospin evolution - a I is not enough - contraints on the neutron EOS seems to improve the results Surface properties - a s, a I s, fission barriers - crucial for SD and HD phenomena - isospin and surface properties should be adjusted together Dangerous simplifications: J 2 terms, 2-body c.m.,... Coulomb? Density dependence at the mean field level... and beyond - mean field (B. Cochet et al., Int. J. Mod. Phys. E13, 187, Nucl.Phys. A731, 34) - beyond (T. Duguet, P. Bonche, Phys. Rev. C67, 054308) Tensor terms (Fl. Stancu et al., Phys. Lett. 68B, 2, 108)

A simple extention: generalized density dependence Attemps made in the past: (with surprises along the way) V (r 1,r 2 ) = (1)... + 1 6 t 3(1 + x 3 P σ )ˆρ q1 (r 1 ) + ρ q2 (r 2 ) α = (2)... + h 1 2 t 4(1 + x 4 P σ ) k 2`ρ i q1 (r 1 ) + ρ q2 (r 2 ) α + cc. +... +... (1) e.g. M. Farine, J.M. Pearson, F. Tondeur, NPA 615 (1997) 137 (2) K.F. Liu, G.E. Brown, NPA 265 (1976) 385 General density dependence: V expansion in k F ρ 1/3 E/A = + V (r 1,r 2 ) = t 00 (1 + x 00 P σ )δ(r) h 3 80 3π 2 2 + t 01 (1 + x 01 P σ ) ρ 1/3 (R) δ(r) + t 02 (1 + x 02 P σ ) ρ 2/3 (R) δ(r)!... + t 03 (1 + x 03 P σ ) ρ(r) δ(r) + non local terms(ρ?) + spin-orbit term(ρ?) 2/3 Θs (m ) + 3 8 t 02i ρ 5/3 +... and m m (K )

Generalized density dependence E/A [ MeV ] 400 350 300 250 200 150 ρ 1/3 + ρ E/A = 16.0 MeV ρ 0 = 0.16 fm 3 K = 230 MeV fit of neutron matter 100 400 50 350 0 300-50 0.0 0.2 0.4 0.6 0.8 250 1.0 1.2 Density [ fm -3 ] E/A [ MeV ] 200 150 ρ 1/3 + ρ 2/3 m m remains free symmetric nuclear matter pure neutron matter symmetric nuclear matter A. Akmal et al., Phys. Rev. C 58 (1998) 1804 pure neutron matter 100 50 0 180 160-50 0.0 0.2 0.4 0.6 0.8 140 1.0 1.2 Density [ fm -3 ] 120 E/A [ MeV ] 220 200 100 80 60 40 20 0-20 ρ 2/3 + ρ -40 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Density [ fm -3 ]

What for? Improvement and interesting results... (B. Cochet et al., Int. J. Mod. Phys. E13, 187, Nucl.Phys. A731, 34) But it allows to vary freely the effective mass! Spectroscopic properties should be improved with m /m closer to 1 Neutrons and protons have different effective masses in asymmetric matter this degree of freedom was never studied in the past... With SLy forces: m = m n m p m I=1 < 0

m from (D)BHF calculations Phenomenological approach (reaction): B.-A. Li, PRC (2004) 064602. BHF calculations: I. Bombaci et al., PRC 44 (1991) 1892, PRC 60 (1999) 024605. DBHF calculations: F. Hofmann et al., PRC 64 (2001) 034314, E.N.E Van Dalen et al., PRL 95 (2005) 022302. m > 0 wins the election! with m 015 to 0.2

SLy type forces with different effective mass splittings SLy force m depends on both κ v and m /m m /m can not be varied without changing K Use of a Skyrme force with two density dependent terms ρ 1/3 + ρ 2/3 instead of ρ 1/6 x 2 can be different of -1 (Cf. B. Cochet et al. NPA A731, 34) m /m and κ v can be freely varied But: Spin instabilities must be kept under control Surprise 1!

Spin instabilities 0.45 Neutron matter f [fm -3 ] 0.40 0.35 Symmetric matter m* m = 0.7 0.30 m* m = 0.9 m* m = 0.8 0.25 0.0 0.1 0.2 0.3 0.4 0.5 0.6 v

Single particle energies in 132 Sn 10 0 neutrons = 0.25 = 0.60 protons = 0.25 = 0.60 s.p.e. [MeV] -10-20 -30-40 -50

Single particle energies in 170 Sn 10 0 neutrons = 0.25 = 0.60 protons = 0.25 = 0.60 s.p.e. [MeV] -10-20 -30-40 -50

Asymmetric matter in nuclei: 100 Sn 1.0 0.8 0.6 0.4 0.2 0.0-0.2-0.4-0.6-0.8-1.0 10 x m * I 0 2 4 6 8 10 r [fm] m 0 0 100 Sn

Asymmetric matter in nuclei: 120 Sn 1.0 0.8 0.6 0.4 0.2 0.0-0.2-0.4-0.6-0.8-1.0 10 x m * 0 2 4 6 8 10 r [fm] 0 m 0.03 I 120 Sn

Asymmetric matter in nuclei: 170 Sn 1.0 0.8 0.6 0.4 0.2 0.0-0.2-0.4-0.6-0.8-1.0 10 x m * 0 2 4 6 8 10 r [fm] m 0.06 I 0 170 Sn

(local) Density functional Φ H Φ = H(r) dr H = K + H 0 + H 3 + H eff + H fin + H so + H sg + H coul H 0 = 1 4 t 0 ˆ(2 + x0 )ρ 2 (2x 0 + 1) `ρ 2 n + ρ 2 p H 3 = 1 24 t 3ρ α ˆ(2 + x 3 )ρ 2 (2x 3 + 1) `ρ 2 n + ρ 2 p H eff = 1 8 [t 1(2 + x 1 ) + t 2 (2 + x 2 )] τρ + 1 8 [t 2(2x 2 + 1) t 1 (2x 1 + 1)](τ n ρ n + τ p ρ p ) H fin = 1 32 [3t 1(2 + x 1 ) t 2 (2 + x 2 )]( ρ) 2 ρ 0 = ρ n + ρ p ρ, 1 32 [3t 1(2x 1 + 1) + t 2 (2x 2 + 1)] ˆ( ρ n ) 2 + ( ρ p ) 2 H so Z = 1 2 W 0 [J Z ρ + J n ρ n + J p ρ p ] h H sg = 1 16 (t 1x 1 + t 2 x 2 )J 2 + 1 16 (t 1 t 2 ) ˆJ i dr H 2 n + J 2 fin = dr C ρ 0 ρ 0 ρ 0 + C ρ 1 ρ 1 ρ 1 p C ρ 1 = 3t 1(2x 1 + 1) + t 2 (2x 2 + 1) 32 ρ 1 = ρ n ρ p

Along the way... Surprise 2: Skyrme functionnal: H = Isospin instabilities With several forces... the nuclei seem to explode! dr {... + C ρ C ρ 1 = 3t 1 32 } 1 (ρ n ρ p ) (ρ n ρ p ) +... ( x 1 + 1 ) + t ( 2 x 2 + 1 ) 2 32 2 Large and positive C ρ 1 favors densities ρ n and ρ p with opposite curvatures C ρ 1 = C ρ 1 (t 1, x 1, t 2, x 2 ) correlated with m /m and κ v Empirically: C ρ 1 > 30 to 35 unstable nuclei

Isospin instabilities 10-2 [fm - 3 ] 0.10 0.05 1 / 3 + 2 / 3 ( 1 = -38,4 ) 10-4 0.00 0 2 4 6 8 r [fm] Force 4 ( 1 = -15,7 ) E / E 10-6 10-8 10-10 [fm - 3 ] 0.10 0.05 0.00 0 2 4 6 8 r [fm] [fm - 3 ] 0.10 0.05 0.00 0 2 4 6 8 r [fm] 0 50 100 150 200 250 300 350 Itérations

Isospin instabilities 0.00-5.00-10.00-15.00 C 1-20.00-25.00 m* m = 0.9 m* m = 0.7-30.00 m* m = 0.8-35.00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 v

m : Masses, gaps, position of drip lines,...

Better criteria to characterize the instabilities? Several instabilities pollute the Skyrme forces... Landau parameters are not always sufficiant to controle it... Most of the time we do not see it because we do not look for it... Force SLy4 SIII SKM* SkP C ρ 1 15.7 17.0 17.1 33.0???

Linear response Several instabilities often experienced with the Skyrme forces Ferromagnetic instabilities spin: polarization n, p spin-isospin: polarization n, p Isospin instabilities: neutrons-protons separation Response of the system to a perturbation described by: Θ ss a = 1 a, Q (α) = a eiq r a Θ (α) a, Θ vs a = σ a, The response fonctions are defined by Θ sv a = τ a, Θ vv a = σ a τ a (Cf. C. Garcia Recio et al., Ann. of Phys. 214 (1992) 293 340) χ (α) (ω,q) = 1 Ω ( n Q (α) 0 2 n ) 1 ω E n0 + iη 1 ω + E n0 iη

Linear response 0.64 0.64 ρc [fm 3 ] ρc [fm 3 ] 0.48 0.32 0.16 0 0.12 0.08 0.04 scalar-isovector f3 f4 f5 SLY5 scalar-isoscalar vector-isovector vector-isoscalar 0.48 0.32 0.16 0 0.64 0.48 0.32 0.16 ρc [fm 3 ] ρc [fm 3 ] 0 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0 q [fm -1 ] q [fm -1 ]

What about existing forces? 1.12 0.96 scalar-isovector ρ c [fm -3 ] 0.8 0.64 0.48 0.32 0.16 0 SLY4 SLY5 SIII SKM* SKP 0 0.5 1 1.5 2 2.5 3 3.5 4 q [fm -1 ]

What about existing forces? 1.12 0.96 scalar-isovector ρ c [fm -3 ] 0.8 0.64 0.48 0.32 0.16 0 SLY4 SLY5 SIII SKM* SKP 0 0.5 1 1.5 2 2.5 3 3.5 4 q [fm -1 ] J. Terasaki, J. Engel in nucl-th/0603062: For reasons we do not understand, we can not obtain solutions of the HFB equation with SkP for Ca and Ni isotopes.

Tensor force Tensor force: V = 1 2 T [ (k σ 1 )(k σ 2 )δ 1 3 k 2 (σ 1 σ 2 )δ + h.c. ] +U [ (k σ 1 )δ(σ 2 k) 1 3 (k δk)(σ 1 σ 2 ) ] In the functional: Terms s T, s s, ( s) 2 Terms J 2 : (J (1) ) 2 = 1 2 J2

Tensor force: Preliminary results 0 2d 3/2 3s 1/2 1g 7/2 2d 5/2-5 1h 11/2 3s 1/2 2d 3/2-10 1g 9/2 SLTa 2d 5/2 1g 7/2 2p 1/2 2p 3/2 1f 5/2-15 1g 9/2 2p 1/2-20 1f 7/2 114 Sn 116 Sn 118 Sn 120 Sn 122 Sn 124 Sn 126 Sn 128 Sn 130 Sn 132 Sn 2p 3/2

Back to the roots: INM eq. of state E/A S,T=1,0, SNM [MeV] 20 10 0-10 -20-30 f3 f4 f5 SLY5 Baldo 5 0-5 -10 E/A S,T=1,1, SNM [MeV] Surprise 3! E/A S,T=0,0, SNM [MeV] 10 5 0-5 -10-15 -20-25 0 0.08 0.16 0.24 0.32 0 0.08 0.16 0.24 0.32 0.4-5 -10-15 -20-25 -30 E/A S,T=0,1, SNM [MeV] ρ [fm -3 ] ρ [fm -3 ] A non local density dependent term can act in the (0,0) and (1,1) channels... 1 but not the one used by Farine et al.: 2 t 4(1 + x 4 ˆP σ ) ˆP 2 ρ(r)δ(r) + cc. need for 1 2 t 4(1 + x 4 ˆPσ )P ρ(r)δ(r)p?

People M. Bender CEA - ESNT K. Bennaceur IPNL - UCB Lyon 1 / CEA - ESNT P. Bonche CEA - SPhT B. Cochet IPNL - UCB Lyon 1 T. Duguet NSCL - MSU T. Lesinski IPNL - UCB Lyon 1 J. Meyer IPNL - UCB Lyon 1