RPA and QRPA calculations with Gaussian expansion method

Transcription

1 RPA and QRPA calculations with Gaussian expansion method H. Nakada (Chiba U., DCEN11 Symposium (YITP, Sep. 6, 11) Contents : I. Introduction II. Test of GEM for MF calculations III. Test of GEM for RPA (& QRPA) calculations IV. Several topics V. Summary & future prospect

2 I. Introduction Exotic structure of unstable nuclei presence of halo, skin Z,N-dep. of shell structure & magic number influence of continuum, resonance asymptotics of wave functions effective NN int. coupling to continuum new methods of numerical calculations desired adaptable to unstable nuclei (as well as stable nuclei) MF theory s.p. states { 1st approx. good basis for extensive study in nuclear structure calculations new methods of MF cal. : PTG basis, WS basis, GEM, GF method, etc.

3 Excitations of unstable nuclei influence of halo, skin, exotic shell structure,? reliable MF may be crucial! RPA (& QRPA) well connected to MF theory (self-consistency) in principle, no additional parameters! suitable to describe global characters of excitations (in mass table & ω) well adapted to collective excitations spurious modes well separated (if numerical method is good) lacks coupling to p-h d.o.f. ( keep in mind!) MF theory EDF approach finite-range int. non-local EDF RPA (& QRPA) (small amp. limit of) TDEDF approach

4 II. Test of GEM for MF calculations Gaussian expansion method (GEM) E. Hiyama et al., Prog. Part. Nucl. Phys. 51, 3 ( 3) Applications of GEM to MF calculations spherical HF H.N. & M. Sato, N.P.A 699, 511 ( ); 714, 696 ( 3) spherical HFB H.N., N.P.A 764, 117 ( 6); 81, 169 ( 8) axial HF & HFB H.N., N.P.A 88, 47 ( 8) Advantages of the method (i) ability to describe ε-dep. exponential/oscillatory asymptotics (ii) tractability of various -body interactions (iii) basis parameters insensitive to nuclide (iv) exact treatment of Coulomb & c.m. Hamiltonian

5 Outline of the method basis : ϕ νljm (r) = R νlj (r) [Y (l) (ˆr)χ σ ] (j) m ; R νlj (r) = N νlj r l exp( νr ) ν complex ν = ν r + iν } { i Re[R νlj (r)] r l exp( ν r r cos(νi r ) ) Im[R νlj (r)] sin(ν i r ) GEM vs. HO bases (s 1/ ) 1 h.o. bases GEM Real- & complex-range GEM bases (s 1/, ν i /ν r = π/) 1.1 E=-8MeV.1 rr s1/.1 E=-1MeV rr s1/ r [fm] r [fm] approximate wave-function asymptotics (ε-dep., exp. & osc.)

6 -body int. matrix elements Fourier transform. various interactions central, LS, tensor channels function form of r delta, Gauss, Yukawa, etc. c.m., Coulomb (including exchange terms) solve HF/HFB eq. as generalized eigenvalue problem iteration (alternatively, gradient method may be applied)

7 Wave-function asymptotics in HFB for large r U i (r) { cos(pi+ r + θ i+ )/r for λ + ε i > ( p i+ M(λ + ε i ) ) exp( η i r)/r for λ + ε i < ( η i M( λ ε i ) ), V i (r) exp( η i+ r)/r for λ ε i < ( η i+ = M( λ + ε i ) ) λ(< ) : chem. pot., ε i (> ) : q.p. energy exponential & oscillatory asymptotics! continuum discrete levels continuum /////////////////////////////// ////////////////////////////// ε λ + λ (λ < )

8 ns 1/ levels in 6 O (Gogny-D1S) 1 neutron s 1/ oscillatory asymptotics? Fourier transform ru(r), rv(r) r [fm] G(k) [fm] x1-3 x k [fm -1 ] 1.5 k [fm -1 ] p 1/..5 d 5/ s 1/ k [fm -1 ] ε-dep. exponential & oscillatory asymptotics well described by GEM

9 Single basis-set? various doubly magic nuclei (HF with Gogny-D1S) B.E. (MeV) Nuclide HO GEM 16 O O Ca Ca Zr Pb r [fm -3 ] r [fm] ( 16 O, 4 O, 4 Ca, 48 Ca, 9 Zr & 8 Pb) basis : R νlj (r) = N νlj r l exp( νr ), ν = ν r + iν i : ν ν r = ν b k i = (k =, 1,, 5), ν i = ± π ν r (k =, 1, ) ; ν = (.4 fm), b = 1.5 #(basis)=1, irrespective of (l, j) store int. matrix elements ( 4 GB), & use them for all calculations!

10 Deformation with spherical basis? Contour plot of ρ τ (r) : good for normal deformation (HFB with Gogny-D1S) Mg 14 Mg 14 Mg z [fm] r p r p r p z [fm] r n r n r n x [fm] x [fm] x [fm] peanut-shape neutron halo in 4 Mg!

11 III. Test of GEM for RPA (& QRPA) calculations Comparison of methods for contact force MF Woods-Saxon pot. Ref. : H.N. et al., Nucl. Phys. A 88, 83 ( 9) residual int. contact force Shlomo & Bertsch, N.P.A 43, 57 ( 75) ˆv res = f [ t (1 + x P σ ) δ(r 1 r ) t 3(1 + x 3 P σ ) ρ(r 1 ) δ(r 1 r ) ] sample nuclei 4,48,6 Ca f ω s = (spurious c.m. state) Methods to be compared 1. GEM (basis-set adopted in HF & HFB cal.). quasi-ho basis (N osc 11) 3. r-space with box boundary (h =. fm, r max = fm, ε cut 5 MeV) 4. continuum RPA (h =. fm) exact 1, (, 3). MF cal. solve RPA eq.

12 Strength function : O (λ,τ=) = i r λ i Y (λ) (ˆr i ) S g (w) [fm 6 /MeV] 16x Ca (l=3, t=) Cont. Box quasi-ho GEM x1 3 S g (w) [fm 6 /MeV] S g (w) [fm 6 /MeV] x Ca 1 6 Ca ω & B(λ) for discrete states ω, B(λ) & σ for continuum } well described by GEM (not so well by quasi-ho, particularly for 6 Ca) w [MeV] 5 3

13 Transition density : r r tr,n (r) [fm ] Ca (for peaks at ω = 5 7 MeV) Cont. quasi-ho GEM 5 1 r [fm] 15 The GEM results are in fair agreement with the exact ones The quasi-ho results have significant discrepancy for low-energy excitation near the drip line difficulty in describing r 1 fm components

14 HF + RPA (by GEM with Gogny-D1S) Ĥ = ˆK + ˆV N + ˆV C Ĥc.m. ˆK = p i M i ˆV N : effective NN int. (finite-range!) ˆV C : Coulomb int. (including exchange terms exactly) Ĥ c.m. : c.m. Hamiltonian (1- + -body terms) Spurious c.m. state ω s =? nuclide ωs 4 Ca Ca Ca well separated from physical mode (also confirmed by B(E1))

15 EWSR ω α α O (λ,τ=) = 1 [O(λ,τ=), [H, O (λ,τ=) ]] α = λ(λ + 1) 4π 4π λ 1 [ 1 (1 1 ) ri λ M A i 1 A i j ri λ 1 Y (λ 1) (ˆr i ) rj λ 1 Y (λ 1) (ˆr j ) (last term -body terms of H c.m. ) ] R (λ,τ) = α ω α α O (λ,τ) / 1 [O(λ,τ), [H, O (λ,τ) ]] = 1? nuclide R (λ=,τ=) R (λ=3,τ=) 4 Ca Ca Ca

16 HFB + QRPA (by GEM with Gogny-D1S) ε cut : cut-off energy for unperturbed q.p. states convergence? HF + RPA (for reference) w%&'()*! " # $ # 3 " "$ 56 + ",! -.$$ # + ",! -.$$$ De /1 %&'()* HFB + QRPA ω s (MeV ) : ε cut (MeV) 44 Ca 1 11 Sn w%&'()*! " # $ + 3 # 4. 3 " 56 + ",! -.$$ # + ",! -.$$$ De /1 %&'()*

17 IV. Several topics Application with semi-realistic interaction M3Y-Pn M3Y int. + phenomenological modification (mainly, saturation & ls splitting) Ref. : H.N., P.R.C 68, ( 3) M3Y-P5 & P5 full inclusion of v (C) OPEP (central part of OPEP) & v (TN) (G-matrix-based tensor force) Effects of v (C) OPEP & v(tn) on shell structure Ref. : H.N., P.R.C 78, 5431 ( 8); 8, 99(E) ( 1) P.R.C 81, 731 ( 1); 8, 993(E) ( 1) see Refs. Ref. : H.N., P.R.C 81, 513(R) ( 1) E.P.J.A 4, 565 ( 9) in AIP Proc (to be published)

18 Tensor-force effects on M1 transition in 8 Pb (HF + RPA) 1p-1h excitation p : (h 11/ ) 1 (h 9/ ), n : (i 13/ ) 1 (i 11/ ) n ˆV res IV = c p + c 1 n = c IS + c 1 IV p (c 1 > c > ) HF approx. (unperturbed) RPA IS = c 1 p + c n = c 1 IS + c IV (c 1.c ) or (c 1, c ) ˆV res IV dominance B(M1; ( IS )) sensitive to ˆV res

19 M3Y-P5 ˆv ˆv (TN) ˆv Exp. IS E x (MeV) B(M1) (µ N ) IV E x (MeV) (Ēx) (9.9) (9.6) B(M1) (µ N ) or 18. ( ˆT (M1) including corrections from p-h & MEC) role of tensor force reconfirmed Ref. : T. Shizuma et al., P.R.C 78, 6133(R) ( 8) ( { ) IS weak effects of -body correlations? IV strong

20 Low-energy excitations of doubly-magic nuclei 78 Ni erosion of Z = 8 magicity due to v (TN)? Ref. : T. Otsuka et al., P.R.L. 95, 35 ( 5) MF cal. with semi-realistic int. (incl. v (TN) ) magicity preserved will be resolved by near-future experiments! a key exp. E x ( + 1 ) HF + RPA prediction (with Z = 8 magicity) reference of experiments D1S D1M M3Y-P5 E x ( + 1 ) (MeV) B(E; ) (e fm 4 ) Ref. : H.N., P.R.C 81, 513(R) ( 1)

21 B(E) in Sn isotopes ( preliminary!) &'($) *+ $,- ".!!! "!!! #!!! $!!! %!!! + Shell Model (e eff n = 1.) D1S M3Y-P5! %!! %! %%! %% / %$! B(E; ) in N 7 well adjusted to exp. by e eff n = 1. e in shell model Ref. : A. Banu et al., P.R.C 7, 6135(R) ( 5) well reproduced in HFB + QRPA with no adjustable parameters %$ %#!

22 B(E; ) in N 6 A) underestimate in shell model (with e eff n = 1. e) B) overestimate in HFB + QRPA with M3Y-P5 C) severe overestimate in HFB + QRPA with D1S (& SLy4? Ref. : J. Terasaki et al., P.R.C 78, ( 8)) A) vs. B) & B) vs. C)

23 B) QRPA with M3Y-P5 vs. C) QRPA with D1S problem of D1S at N 6 close to deformation because of strong ng 7/ n1d 3/ excitation shell structure! 9 Zr 1 Sn 11 Sn n1d 3/ ng 7/ n1d 5/ v (TN) 9 Zr 1 Sn 11 Sn D1S M3Y-P5 (3/) Zr 15 Sn 17 Sn 19 Sn 111 Sn (7/) + (5/) + Exp.

24 A) Shell model vs. B) QRPA with M3Y-P5 shell model (on top of 1 Sn core) only neutrons in valence shell coupling to GQR (core pol.) e eff n constant e eff n hypothesis true? e eff n E strength T p (E) T n (E) / HFB + QRPA valence (norm within valence) %& '()%& ' *+, (%#' *+, -!" #!$ #!" "!$ "!" #"" ##" # " #/" #." A-dep. of e eff n? (but overshooting)

25 to be considered q.p. states canonical bases (in defining valence orbits) Ref. : J. Dobaczewski et al., P.R.C 53, 89 ( 96) backward amp. on/off

26 IV. Summary Gaussian expansion method (GEM) has successfully been applied to MF & RPA calculations. Advantages : (i) ability to describe ε-dep. exponential/oscillatory asymptotics (ii) tractability of various -body interactions (iii) basis parameters insensitive to nuclide (iv) exact treatment of Coulomb & c.m. Hamiltonian Several results have been shown : Tensor-force effects on M1 transition in 8 Pb Low-energy excitation of doubly-magic nuclei (e.g. + 1 in 78 Ni) B(E) in Sn isotopes (tensor-force effect, A-dep. effective charge?)

27 Contributors : HF : M. Sato (Chiba U. until 4) RPA : K. Mizuyama (Milan) M. Yamagami (Aizu) M. Matsuo (Niigata) 8 Pb M1 : T. Shizuma. (JAEA) Sn E : Special thanks to K. Matsuyanagi (for giving key information)