global global mass region (matter ) & (I) M3Y semi-microscopic int. Ref.: H. N., P. R. C68, ( 03) N. P. A722, 117c ( 03) Proc. of NENS03 (to be
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1 Gogny hard core spin-isospin RCNP (Mar , 2004) Collaborator: M. Sato (Chiba U, ) ( )
2 global global mass region (matter ) & (I) M3Y semi-microscopic int. Ref.: H. N., P. R. C68, ( 03) N. P. A722, 117c ( 03) Proc. of NENS03 (to be published) G-matrix M3Y e.g. saturation, LS-splitting OPEP central part keep cf. χpt (Skyrme, Gogny) spin & spin-isospin property ( g l, g l) SkM SLy5 D1S M3Y-P2 g g g g Spin-isospin property OPEP new magic numbers? cf. Otsuka et al.
3 g 0, g 1 in Skyrme int. & στ excitation (E GTR ) : M. Bender et. al., PRC65, reasonable value : g 0 1
4 S.p. levels of N = 32 nuclei Z-dep. of ε n : ε n (0f 5/2 ), ε n (1p 1/2 ) relative to ε n (1p 3/2 ) 5 4 D1S SLy5 M3Y-P2 f 5/2 ε n [MeV] 3 2 p 1/2 1 0 Exp. (f 5/2 ) Exp. (p 1/2 ) Z Z-dep. of ε n (0f 5/2 ) N = 32 magicity at Z 20? M3Y-P2 reproduce inversion of 0f 5/2 & 1p 1/2! (without core-pol.)
5 (II) Gogny phenomenological int. M. Sato & H. N., many talks (JPS meetings, etc.) parameter (e.g. D1S) unphysical true minimum n-matter collapse (ρ E/A ) E/A [MeV] pure neutron matter -20 sym. nucl. mat. finite nuclei ρ [fm 3 ] ρ p ρ n ( physical solution ) D1S Properties of sym. nucl. matter (& surface energy) Symmetry energy Pairing properties : SE part ρ-dependent repulsion hard core
6 spin-isospin property? spin-isospin property?
7 Interactions : v 12 = v C 12 + v LS 12 + v TN 12 + v DD 12 ; v C 12 = n (tse v12 LS = n (tlse n v12 TN = n (ttne n n P SE + t TE n P TE + t SO n P SO + t TO n P TO )fn C (r 12 ), P TE + t LSO n P TO )fn LS (r 12 ) L 12 (s 1 + s 2 ), P TE + t TNO n P TO )fn TN (r 12 ) r12s 2 12, v DD 12 = t DD (1 + x DD P σ )[ρ(r 1 )] α δ(r 12 ). Skyrme interaction f n (r 12 ) = δ(r 12 ), 2 δ(r 12 ) Gogny interaction f C n (r 12 ) = e (µ nr 12 ) 2, f LS n (r 12 ) = 2 δ(r 12 ) D1S the only parameter-set widely applied in recent calculations D1S M3Y-type interaction f n (r 12 ) = e µ nr 12 /µ n r 12 Nuclear matter properties Basic characters of eff. int. (central part) v C 12 + v DD 12 E E/A ; E = E(ρ, η t, η s, η st ) ρ = στ ρ τσ = ρ p + ρ p + ρ n + ρ n, η s = η t = η st = ( ) σρ τσ στ ( ) τρ τσ στ ( ) στρ τσ στ /ρ = ρ p ρ p + ρ n ρ n ρ /ρ = ρ p + ρ p ρ n ρ n, ρ /ρ = ρ p ρ p ρ n + ρ n ρ,.
8 Saturation η t E = η s E = η st E = 0 η t = η s = η st = 0 ρ E = 0 ρ 0 ( k F0 ), E 0 (= b vol ) sat. Landau parameters (around the saturation point) (spin-saturated) sym. nucl. matter v C 12 + v DD 12 N 0 [ fk1 k 2 + f k 1 k 2 (τ 1 τ 2 ) + g k1 k 2 (σ 1 σ 2 ) f k1 k 2 +g k 1 k 2 (σ 1 σ 2 ) (τ 1 τ 2 ) ] (N 0 : level density at the Fermi surface) around k 1 = k 2 = k F0, including exchange terms δ2 ( V /Ω) δn(k 1 )δn(k 2 ), etc. (n(k) : occ. prob. of s.p. states) f l = 1 1 d(cos θ 12) f k1 k 2 P l (cos θ 12 ), etc. (cos θ 12 ˆk 1 ˆk 2 ) M 0 M = f 1, a t 1 2 a s 1 2 a st 1 2 K 9 2 ρ 2E 2 E ηt 2 = k2 F0 sat. 6M0 2 E ηs 2 = k2 F0 sat. 6M0 2 E ηst 2 = k2 F0 sat. 6M0 = 3k2 F0 (1 + f 0 ), sat. M 0 (1 + f 0), ( symmetry engergy) (1 + g 0 ), (1 + g 0).
9 5 parameters vs. Gogny D1S parameter? ( int. range fix) Symmetric nuclear matter On the η s = η t = η st = 0 line Interaction parameters : E(ρ, η = 0) = n c n + n c n 3 4 (tse +t DD ρ 1+α n 3 4 (tse n + t TE n ) (tso n + 9t TO n ) ρ n ) + t TE n ) 1 4 (tso n + 9t TO W n (ρ) W n (ρ) : appropriate function of ρ Gogny 2 ranges for v C 12 5 parameters (including α) : Note : Skyrme, M3Y 2 range Saturation ρ 0 & E 0 ( v12 DD ) ρ fm 3 density dist. & radius E 0 16 MeV binding energy Effective mass (k-mass) M 0 ( f 1 ) of Z N finite nuclei M 0 0.7M s.p. level spacing for ε ε F 20 MeV Incompressibility K ( f 0 ) K 210 MeV (in non-rel. models) E x of GMR Surface energy b surf Note : v LS b surf 20 MeV mass, fission barrier (in Gogny D1S) D1S sym. nucl. matter
10 (Volume) symmetry energy a t ( f 0) a t 30 MeV binding energy of Z < N nuclei Note : f 0 v nn 1 4 (v pp + v nn + 2v pn ) v T =1 v T =0 (v SE + v TO ) (v TE + v SO ) Pairing properties v SE for ρ < ρ 0 region Matter (ε F ) peak & characterize Gogny D1S pairing property ( int. range?) D1S Neutron matter E(ρ, η t = 1) v SE + v TO? v SE ρ-dep. or v SE < 0 for low ρ v SE > 0 for high ρ ( hard core) D1S no ρ-dep. in v SE ( x DD = 1) Note : v TO 0 Spin-isospin flip g 0 g 0 1 E x of GTR D1S g fit Note : v SE v C + v DD
11 D1S ( ) v C + v DD parameter : 11 (int. range fix) E(ρ, η = 0) (sym. nucl. matter energy profile) 5 constraints a t 1 constraint (ε F ) profile 2 constraints ( peak ) D1S keep( ) Note : v SE ρ-dependence ( hard core ) x DD 1 Neutron matter ρ limit x DD < 1 Unphysical min. x DD 0.94 Additional constraint 1 E(ρ, η = 0) v SE v TE compensate? even odd reasonable cf. α ( (0s) 4 ) v SE + v TE Free parameters : x DD & 1 ( ) a1 x DD = 0.94 b1 x DD = 0.90 b2 x DD = 0.90, (ε F ) ( )
12 Behavior of v SE in nucl. matter D1S a1 b1 b2-200 E/A [MeV] kf [fm 1 ]
13 Neutron matter profile pure neutron matter E/A [MeV] D1S Friedman and Pandharipande Pairing gap profile ρ[fm -3] symmetric nuclear matter (ε ) f b2 D1S a1 b k [fm -1 f ]
14 v SE ρ-dep. ( hard core) g? parameter x DD g 0 g 1 g 0 g 1 D1S a b b M3Y-P Note : x DD v SE ρ-dep. (1) E(ρ, η = 0) (2) a t (3) pairing (4) ρ-dep. (or hard core) v SE high ρ ρ < ρ 0 pairing property v SE ρ-indep. (4) compensate ρ = ρ 0 v SE (1 3) v TE, v SO, v TO e.g. v TE :, v TO : ( a t ) at ρ = ρ 0 channel v SE v TE v TO v SO total 3 : 3 1 : 9 g 0 = g 0 =
15 Gogny int. v SE ρ-dependence (i.e. x DD ) 2 n-matter στ property v SE high ρ ( hard core) στ property ( correlate )
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