Evidence for three-nucleon interaction in isotope shifts of Z = magic - PowerPoint PPT Presentation

Evidence for three-nucleon interaction in isotope shifts of Z = magic nuclei H. Nakada ( Chiba U., Japan ) @ Tsukuba; May 11, 2016 Contents : I. Introduction II. Mean-field approaches with semi-realistic interaction III. Incorporating 3 N LS interaction IV. 3 N LS interaction & isotope shifts V. Summary H.N. & T. Inakura, P.R.C 91, 021302(R) (’15) H.N., P.R.C 92, 044307 (’15)

I. Introduction Shell structure ( → magic number) — fundamental concept for nuclear structure astrophysical importance  waiting point in s - & r -processes   constraint on EoS ← subtracting shell effects  clusters in n -star inner crust ( ↔ e.g. QPO)  ⋆ Z - & N -dep. ! (“shell evolution”) ⋆ central + ℓs potential ℓs pot. ↔ ℓs splitting ( → magic #’s in Z, N > 20 ) · · · origin ? ◦ 2 N LS int. — insufficient ( ⇒ strong LS int. used in phenomenology) ◦ tensor int. (+ α ) → Z - & N -dep. (1st-order effects) contribution to overall strength (2nd-order effects) ? · · · small ✬ ✩ comprehension of its origin → correct prediction of shell structure ✫ ✪

⋆ 3 N int. ( χ EFT) → ρ -dep. LS int. Ref. : M. Kohno, P.R.C 86, 061301(R) stronger LS int. at higher ρ ( → stronger ℓs pot.) — complementary to 2 N LS int ! ⇒ experimental evidence independent of ℓs splitting ? ( → good reliability)

∆ ⟨ r 2 ⟩ p ( A Pb ) := ⟨ r 2 ⟩ p ( A Pb ) − ⟨ r 2 ⟩ p ( 208 Pb ) Isotope shifts in Pb nuclei exp. ⇒ kink at N = 126 !  electron scatt.   X-ray freq. difference ( µ − atom, Kα , OIS)   Ref. : M.M. Sharma et al. , P.R.L. 74, 3744 reproduced by RMF, but not by Skyrme EDF up to ’95 → dep. on isospin content of LS int. ( → extension of Skyrme EDF) · · · but cannot be a complete solution !

kink in ∆ ⟨ r 2 ⟩ p ( A Pb ) at N = 126 ← − n 0 i 11 / 2 occupation ⇑  larger ⟨ r 2 ⟩ p - n attraction    than neighboring orbits    N < 126 — unocc.  N > 126 — sizable occ. prob.     ( ∵ pairing)  Ref. : P.-G. Reinhard & H. Flocard, N.P.A 584, 467 ‘understanding’ in ’90’s · · · ε n (0 i 11 / 2 ) − ε n (1 g 9 / 2 ) is a key ↔ isospin content of LS int. However, ε n (0 i 11 / 2 ) ≈ ε n (1 g 9 / 2 ) required ! ( ↔ equal occ. prob.) on the contrary · · · 11 / 2 + ✻ 0 . 78 MeV ❄ 9 / 2 + 209 Pb

II. Mean-field approaches with semi-realistic interaction “Semi-realistic” nucleonic interaction ← − microscopic 2 N (+ 3 N ) int. ↑ phenomenological modification { saturation properties ℓs splitting (?) ⇒ MF (HF, HFB) & RPA calculations nuclear reactions · · · future project

p 2 ∑ ∑ i Effective Hamiltonian H = H N + V C − H c . m . ; H N = 2 M + v ij ˆ i i<j (rotational & translational inv.) v (C) v (LS) v (TN) v (C ρ ) v (LS ρ ) ( ) v ij = ˆ ˆ ij + ˆ + ˆ + ˆ +ˆ ; ij ij ij ij v (C) ∑ t (SE) P SE + t (TE) P TE + t (SO) P SO + t (TO) f (C) ( ) ˆ = P TO n ( r ij ) , n n n n ij n   ( 1 − P σ ) ( 1 + P τ ) ( 1 + P σ ) ( 1 − P τ ) P SE := , P TE := , 2 2 2 2     ( 1 − P σ ) ( 1 − P τ ) ( 1 + P σ ) ( 1 + P τ )   P SO := , P TO :=   2 2 2 2 v (LS) ∑ t (LSE) P TE + t (LSO) f (LS) ( ) ˆ = P TO ( r ij ) L ij · ( s i + s j ) , n n n ij n v (TN) ∑ t (TNE) P TE + t (TNO) f (TN) ( r ij ) r 2 ( ) ˆ = P TO ij S ij ij n n n n v (C ρ ) C (SE) [ ρ ( r i )] P SE + C (TE) [ ρ ( r i )] P TE ( ) ˆ = δ ( r ij ) ; ij r ij ) − σ i · σ j , f n ( r ) = e − µ n r /µ n r L ij := r ij × p ij , S ij := 3( σ i · ˆ r ij )( σ j · ˆ ρ ρ α (Y) ( Y = SE or TE ; α (SE) = 1 , α (TE) = 1 / 3 ) C (Y) [ ρ ] = t (Y)

M3Y int. · · · Yukawa function → fit to G -matrix (at ρ ≈ ρ 0 / 3 ) v (C) v (C) • OPEP → longest part of ˆ ( ≡ ˆ OPEP ) ij • popular in reaction problems v (C ρ ) • no saturation (without modification) → add ˆ ij ‘ M3Y-P n ’ { v (C) by ˆ v (C ρ ) replace short-range part of ˆ • modifying M3Y-Paris v (LS) enhance ˆ ( ↔ ℓs splitting) v (C) • keeping ˆ OPEP v (TN) from M3Y-Paris in M3Y-P5 to P7 • no change for ˆ (— realistic tensor force) · · · leading order of chiral dynamics basic formulae : H.N., P.R.C 68, 014316 (’03) M3Y-P6, P7 : H.N., P.R.C 87, 014336 (’13) ⇒ nuclear matter & finite nuclei

Numerical methods for finite nuclei — Gaussian expansion method • spherical HF · · · H.N. & M. Sato, N.P.A 699, 511 (’02); 714, 696 (’03) • spherical HFB · · · H.N., N.P.A 764, 117 (’06); 801, 169 (’08) • axial HF & HFB · · · H.N., N.P.A 808, 47 (’08) • spherical RPA · · · H.N., K. Mizuyama, M. Yamagami & M. Matsuo, N.P.A 828, 283 (’09) ✬ ✩ Advantages of the method (i) ability to describe ε -dep. exponential/oscillatory asymptotics (ii) tractability of various 2-body interactions (iii) basis parameters insensitive to nuclide (iv) exact treatment of Coulomb & c.m. Hamiltonian ✫ ✪

⋆ Energies & radii of doubly magic nuclei : SLy5 D1S M3Y-P6 M3Y-P7 CCSD Exp. 16 O − E 128 . 6 129 . 5 126 . 3 125 . 9 107 . 5 127 . 6 √ ⟨ r 2 ⟩ 2 . 59 2 . 61 2 . 59 2 . 57 — 2 . 61 40 Ca − E 344 . 3 344 . 6 335 . 9 334 . 3 308 . 8 342 . 1 √ ⟨ r 2 ⟩ 3 . 29 3 . 37 3 . 37 3 . 35 — 3 . 47 48 Ca − E 416 . 0 416 . 8 413 . 8 414 . 9 355 . 2 416 . 0 √ ⟨ r 2 ⟩ 3 . 44 3 . 51 3 . 51 3 . 49 — 3 . 57 90 Zr − E 782 . 4 785 . 9 781 . 1 780 . 8 — 783 . 9 √ ⟨ r 2 ⟩ 4 . 22 4 . 24 4 . 23 4 . 22 — 4 . 32 208 Pb − E 1635 . 2 1639 . 0 1634 . 5 1635 . 5 — 1636 . 4 √ ⟨ r 2 ⟩ 5 . 52 5 . 51 5 . 53 5 . 51 — 5 . 49 CCSD · · · G. Hagen et al. , P.R.L. 101, 092502 (’08) (chiral N 3 LO without 3NF)

⋆ Separation energies of proton- / neutron-magic nuclei : · · · important in astrophysics S n ( Z, N ) := E ( Z, N − 1) − E ( Z, N ) S 2 n ( Z, N ) := E ( Z, N − 2) − E ( Z, N ) = S n ( Z, N ) + S n ( Z, N − 1) ( Z, N − 1) := E ( Z, N − 1) − 1 ∆ mass [ ] E ( Z, N − 2) + E ( Z, N ) n 2 = 1 [ ] S n ( Z, N ) − S n ( Z, N − 1) (for N − 1 = odd ) 2 ↔ pairing ✻ ( Z, N − 2) S 2 n ✻ ✻ ( Z, N − 1) ∆ mass n S n ( Z, N ) ( S p , S 2 p , ∆ mass → obtained analogously) p ∆ mass for Z = 50 & ∆ mass for N = 82 · · · fitted n p

!"#$% &!"#$'( !"#$%& '!"#$%( ! ! ! ! # $ %&'()* # $ %&'()* $ % &'()*+ $ % &'()*+ "# "# " " "! "! ! ! # # " " ! ! 1 / " , 2 1 / ," ,! +" +! 2" 2! !" "1 "0 "- ! 1 0 - "0 "- ! 1 0 - 2! - - / / )!"#$'% *!"#$+( )!"#$*& +!"#$(% ! ! ! ! # $ %&'()* # $ %&'()* $ % &'()*+ $ % &'()*+ "# "# " " "! "! ! ! # # " " ! ! +" 2" !" 1" 1" 0" /" ." 2! 2# 1! 1# #! #! ## 0! 0# .! - - / / ,!"#$%' ,!"#$-%. ! ! # $ %&'()* $ % &'()*+ "# " "! ! # " ! "" "! " ! ," ,! +" .! .# -! -# ,! ,# - / • : M3Y-P7 △ : D1M × : Exp.

⋆ N -dependence of single-proton energies below Z = 20 — 1 s 1 / 2 - 0 d 3 / 2 inversion ∆ ε p = ε p (1 s 1 / 2 ) − ε p (0 d 3 / 2 ) Ref : M. Grasso et al. , P.R.C 76, 044319 (’07) (Exp. : average weighted by spectroscopic factor)

Case of semi-realistic int. (M3Y-P5 ′ ) " # De % &'()*+ $ ⇓ v (TN) ⇒ # &/#0 &/#( " &(!1 2,3 &45%6 ! "$ ", !$ !, -$ -, ,$ . v (TN) → correct N -dep. of ∆ ε p (in N = 20 – 28 ) ! realistic ˆ H.N., K. Sugiura & J. Margueron, P.R.C 87, 067305 (’13)

⋆ Magic numbers 126 120 Z magic N magic M3Y-P6 submagic (0.5) submagic (0.5) (0.8) (0.8) 92 Z=82 N=184 64 164 58 Z=50 N=126 40 Z Z=28 N=82 Z=20 N N=50 14 Z=8 28 H.N. & K. Sugiura, P.T.E.P. 2014, 033D02

III. Incorporating 3 N LS interaction Semi-realistic M3Y-P6 int. · · · reasonable shell structure ⇒ yardstick v (LS ρ ) ) Ref. : M. Kohno, P.R.C 86, 061301(R) 3 N LS int. ↔ ρ -dep. LS int. ( ˆ ⇓ v (LS) M3Y-P6 — ˆ M3Y × 2 . 2 (equate ε n (0 i 11 / 2 ) − ε n (0 i 13 / 2 ) at 208 Pb) vs. v (LS) v (LS ρ ) M3Y-P6a — ˆ M3Y + ˆ v (LS ρ ) = 2 i D [ ρ ( R ij )] p ij × δ ( r ij ) p ij · ( s i + s j ) ; ρ ( r ) ( ) D [ ρ ( r )] = − w 1 ≈ − w 1 ρ ( r ) 1 + d 1 ρ ( r ) d 1 = 1 . 0 fm 3 (prefix), w 1 ( > 0) : fitted