Shell Model Monte Carlo Methods and their Applications to Nuclear Level Densities

Transcription

1 Shell Model Monte Carlo Methods and their Applications to Nuclear Level Densities H. Nakada (Chiba U., RIKEN (Feb. 14, 2013) Contents : I. Introduction II. Shell model Monte Carlo methods III. Brief survey of nuclear level densities IV. SMMC level densities (in medium-mass nuclei) V. SMMC in rare-earth nuclei VI. Summary & future prospect

2 Collaborators : Y. Alhassid (Yale U., U.S.A.) C. Özen (Kadir Has U., Turkey) S. Liu (Yale U., U.S.A.) L. Fang (Yale U., U.S.A.) E. Akyüz (Istanbul U., Turkey) G.F. Bertsch (U. Washington, U.S.A.) ( quit)

3 I. Introduction Nuclear structure global nature (e.g. saturation, binding energies, shell structure) MF (or EDF) approaches 2-body correlations often important for quantitative description GCM, Interacting shell model (or CI), etc. Advantages & disadvantages of ISM A full quantum theory ( collective + non-collective d.o.f) good accuracy! up to (relatively) high excitation energies cf. GCM D model space renormalization, energy limit rapid increase of computational intensiveness application & development of QMC technique (SMMC) Ref. : G.H. Lang et al., P.R.C 48, 1518 ( 93)

4 Level scheme of 56 Fe : M1 distribution in 56 Fe : Ref. : H.N. et al., N.P.A 571, 467 ( 94) Ref. : T. Shizuma et al., P.R.C 87, ( 13)

5 Advantages & disadvantages of SMMC A implementable in large model space exact quantum mechanical method able to cover up to high excitation energies including statistical properties ( finite T ) D no explicit wave functions difficult to obtain detailed spectroscopic information statistical errors sign problem

6 Summed B(GT + ) : (comparison with diag.) B(GT + ) distribution in 54 Fe, 59,55 Co : Ref. : S.E. Koonin et al., Phys. Rep. 278, 1 ( 97)

7 (One of) the most successful applications of SMMC so far nuclear level densities H.N. & Y. Alhassid, P.R.L. 79, 2939 ( 97) W.E. Ormand, P.R.C 56, R1678 ( 97) H.N. & Y. Alhassid, P.L.B 436, 231 ( 98) K. Langanke, P.L.B 438, 235 ( 98) Y. Alhassid, S. Liu & H.N., P.R.L. 83, 4265 ( 99) Y. Alhassid, G.F. Bertsch, S. Liu & H.N., P.R.L. 84, 4313 ( 00) J.A. White, S.E. Koonin & D.J. Dean, P.R.C 61, ( 00) Y. Alhassid, S. Liu & H.N., P.R.L. 99, ( 07) C. Özen et al., P.R.C 75, ( 07) H.N. & Y. Alhassid, P.R.C 78, (R) ( 08) A. Mukherjee & Y. Alhassid, P.R.L. 109, ( 12) + deformation within ISM ( certain spectroscopic information) Y. Alhassid, L. Fang & H.N., P.R.L. 101, ( 08) C. Özen, Y. Alhassid & H.N., P.R.L. 110, ( 13)

8 II. Shell model Monte Carlo methods SMMC evaluate O β = Tr(Oe βh )/Z(β) (e.g. E(β) = H β ) H : shell model Hamiltonian ( body) e βh auxiliary-fields (σ α (τ)) path integral rep. 2-body int. Pandya transform. e βh : Suzuki-Trotter decomp. Hubbard-Stratonovich transform. j 3 j 4 π, ρ, etc α κ α σ α s α α j 1 J, T j 2

9 e βh = ( e βh) n t ; HS transform. : Tr(Oe βh ) e βh j exp( β κ α 2 ˆρ2 α) D[σ] G(σ) Tr(OU σ ) [ ] exp( βϵ j ˆNj ) α [ ] exp( β κ α 2 ˆρ2 α) + O ( ( β) 2) [ ] dσ α exp β( κ α 2 σ2 α + s α κ α σ α ˆρ α ) s α = ±1 (if κ α < 0) or ±i (if κ α > 0) (AF path integral) G(σ) = exp( β κ α 2 σ2 α), U σ = exp( βh σ ) ; h σ = j ϵ j ˆNj + α s α κ α σ α ˆρ α Tr gc (U σ ) = det(1 + U σ ) σ α : auxiliary field U σ : s.p. matrix for U σ calculable only via the s.p. matrices mean field cf. Mean-field approx. σ α = ˆρ α static mean-field (no fluctuation, no τ-dependence) [ ] Static-path approx. exp( β κ α 2 ˆρ2 α) dσ α exp β( κ α 2 σ2 α + s α κ α σ α ˆρ α ) σ α static auxiliary-field with fluctuation

10 MC sampling σ k = {σ α (τ)} k O β 1 O σk (with stat. error) N k k random walk of {σα (τ)} k Metropolis algorithm (& amendment) detailed balance cf. N k 4000 in practical SMMC calculations advantage : easier to handle large model space CPU time Ms.p. 3 n t, suitable for parallel computing finite-t method suitable for statistical properties (but not for distinguishing discrete levels) conservation laws projections Z & N proj. canonical ens. (Tr gc Tr) π, J (J z ), isospin proj. E 0 = lim β H β evaluated also by SMMC (stabilization method required for heavy nuclei) disadvantage : sign problem propagator Tr(e βh(σ k) ) : not necessarily positive-definite dominant part of nuclear int. sign good!

11 III. Brief survey of nuclear level densities Nuclear level densities ( partition fn.) one of the key inputs in calculations of low-energy nuclear reactions for A(a, b)b reaction σ (a,b) de f T (a) J i π i (E i ) T (b) J f π f (E f ) ρ Jf π f (E f ) J f π f Hauser-Feshbach formula T (a/b) Jπ (E) : transmission coefficient from compound state ρ Jπ (E) : level density application to nuclear-astrophysics e.g. s- & r-processes (n, γ) vs. β-decay σ (n,γ) (a = n, b = γ) E i = E f + E γ S n ( E f S n ) rp-process (p, γ) vs. β-decay

12 Experimental methods to measure nuclear level densities 1. Direct counting of levels lowest-lying states or light nuclei (E x 2 3 MeV) 2. Level spacing among neutron resonances (ρ = D 1 ) small energy range around E x = S n 8 MeV, restricted to s-wave 3. Ericson fluctuation E x 20 MeV 4. Charged particle reactions ( reaction model) 5. Oslo method γ-ray matrix P (E x, E γ ) = C(E x ) T (γ) (E γ ) ρ(e x E γ ) ( Brink-Axel hypothesis) E x 3 7 MeV

13 Conventional approach to nuclear level densities Backshifted Bethe s formula ( Fermi-gas model) π [ ρ(e x ) = 12 a 1/4 (E x ) 5/4 exp 2 ] a(e x ) (for state density) fits well to experimental data (except at very low E x ), if the parameter a (& ) is adjusted ( : backshift pairing & shell effects) ρ J,π (E x ) = ρ(e x ) 2J πσ exp[ J(J + 1)/2σ 2] ; σ = I (E x )/a 3 ρ(e x ) = J,π (2J + 1) ρ J,π (E x ) However, 1) a = A/6 A/10 MeV 1, in contrast to the Fermi-gas prediction a A/15 2) a : nucleus-dependent (not only A-dependent) shell effects, etc.

14 fitted values of a : Ref. : Bohr & Mottelson, vol. 1 Note : 10% change in a change in ρ(e x ) by greater than factor 10! (for A 150, E x 8 MeV)

15 For better E x -dep. correction for low E x part Constant-T formula ( T -dep. of pairing) ρ(e x ) exp[(e x E 1 )/T 1 ] for E x < E M matching to BBF at E x = E M To get less A-dep. parameters nucl.-dep. corrections ( a E x -dep. : a(e x ) = ã 1 + δw 1 exp[ γ(e ) x )] E x δw : shell correction energy, γ = γ 1 A 1/3 Collective enhancement factor K vib (E x ), K rot (E x ) ( e.g. K rot (E x ) = max ([ A 5/3 1+ β ) ] ) 2 Ex a 1+exp( E x E c d c ) + 1, 1 can be harmonious with a A/15 MeV 1 (Fermi-gas value) many corrections & parameters introduced origin? estimate? (physics?) significant nucleus-dependence still remains e.g. for ã( : asymptotic value of a) & σ

16 Ref. : A.J. Koning et al., N.P.A 810, 13 ( 08) = It has been difficult to predict nuclear level densities to good accuracy

17 What is needed? (1) shell effects (2) collective 2-body correlations e.g. V = κ 2 ˆρ2 (ˆρ : 1-body op.) typically, κ : large collective 'noncollective' singleparticle 'collective' (Both coll. & non-coll. d.o.f. relevant!) full shell model Both (1) & (2) are fully taken into account within the model space large model space required SMMC (H.N. & Y. Alhassid)

18 IV. SMMC level densities (in medium-mass nuclei) Nuclear temperature (average) excitation energy } Fermi gas BBF grand-can. ens. finite-t formalism MF SMMC can. ens. statistical properties state density thermodynamics in finite systems partition fn. ρ(e) = Tr δ(e H) Z(β) = Tr(e βh ) = Laplace transform. de ρ(e)e βe (Tr : can. trace) saddle-point approx. ( smoothening for E, nuclear temp.) ρ(e) e S 2πβ 2 C ; S = βe + ln Z(β), β 2 C = de dβ (S : entropy, C : heat capacity) cf. grand-can. ρ gc (E) e S gc (2π)3 ( N p ) 2 ( N n ) 2 β 2 C gc nucl.-dep.?

19 Nuclei around Fe-Ni region setup model space : full pf + 0g 9/2 Hamiltonian : s.p. energy W-S pot. monopole pairing (SE channel) strength even-odd mass difference int. surface-peaked multipole (isoscalar, λ = 2, 3, 4) strength self-consistency + renorm. renorm. factor realistic int. applications { ρ(e x ) ρ π (E x ) ( π proj.) no adjustable parameters! sign good (for even-even nuclei) small stat. error comparison to BBF extension to higher E x (via connection with HF) ρ J (E x ), ρ Jπ (E x ) ( J proj.) ρ T (E x ) ( isospin proj.) comparison to spin cut-off model correction subject to Z N nuclei

20 State density ρ(e x ) of 56 Fe : Ref. : Y. Alhassid et al., P.R.C 68, ( 03) } SMMC for pf + g 9/2 connection of F (β) ρ(e) finite-t HF SMMC HF SMMC+HF Exp. W. Dilg et al. N.P.A 217, 269 { E x 10 MeV strong correlations inside pf + g 9/2 shell E x 25 MeV weak correlation, s.p. d.o.f. is essential

21 State density ρ(e x ) of A = 55 isobars : 10 5 Ref. : Y. Alhassid, S. Liu, H.N., P.R.L. 83, 4265 ( 99) ρ Mn 55 Fe 55 Co Many empirical formulas predict equal ρ(e x ) among odd-a isobars not true! ( exp. & micro. cal.) E x (MeV) Exp. : W. Dilg et al., N.P.A 217, 269 ( 73)

22 J-dep. level density ρ Jπ (E x ) of 56 Fe : 55 Fe 56 Fe Ref. : Y. Alhassid, S. Liu, H.N., P.R.L. 99, ( 07) 60 Co 0.04 E x =4.39MeV E x =5.6MeV E x =3.39MeV 0.02 SMMC results E x =7.06MeV E x =7.5MeV E x =6.88MeV spin-cutoff model with σ { fitted to SMMC from rigid-body I ρ J /ρ E x =11.95MeV E x =17.37MeV E x =11.6MeV E x =16.1MeV E x =11.89MeV E x =17.71MeV significant deviation from spin-cutoff model { at low E x & in even-even nuclei 0.02 influence of pairing J J J

23 π-dep. state density ρ π (E x ) : SMMC Exp ρ [MeV -1 ] ρ π=+ ρ π= exp. HFB SMMC Fe E x [MeV] Ref. : H.N. & Y. Alhassid, P.R.L. 79, 2939 ( 97) strong π-dep.? Ref. : Y. Kalmykov et al. P.R.L. 99, ( 07) exc. out of sd-shell plays a role C. Özen, E. Akyüz & H.N., work in progress

24 V. SMMC in rare-earth nuclei Questions rotational (i.e. well-deformed) nuclei described within spherical shell model? with how large model space? crossover transition from vibrational to rotational nuclei? how to identify vibrational / rotational characters in SMMC? nuclear shape & level densities? micro. understanding of K vib (E x ), K rot (E x ) (for level density)

25 Nuclides : 162 Dy, even-n Nd-Sm isotopes Setup model space : { Ref : Y. Alhassid, L. Fang & H.N., P.R.L. 101, ( 08) C. Özen, Y. Alhassid & H.N., P.R.L. 110, ( 13) p : (Z = shell) + 1f 7/2 n : 0h 11/2 + (N = shell) + 1g 9/2 expand def. WS solutions by sph. WS orbitals (0.1 < ˆn αj /(2j + 1) < 0.9) biggest SMMC calculations to date! Hamiltonian : s.p. energy W-S pot. + HF-type correction pp & nn monopole pairing (g 0 = γ g0 BCS ) strength even-odd mass difference + fit to I int. g (p + n) surface-peaked multipole (λ = 2, 3, 4) strength self-consistency + renorm. renorm. factor for λ = 2 (k 2 ) fit adjust. parameters smooth function of N

26 Technical developments for SMMC calculations 1) Proton-neutron formalism p & n occupying different shell 2) Stabilization of canonical propagator E g, I g needs calculations at β 10 MeV 1 for heavy well-deformed nuclei propagator U σ ( e βh ) ill-behaved at large β stabilization for canonical propagator ( n-proj.) 3) Assessment of discretization effects extrapolation to β = 0 test case : 22 Ne for 162 Dy β = 1/32 & 1/64 MeV 1 linear extrapolation for β 3.25 MeV 1 average for β > 3.25 MeV 1 (β = 0.5, 0.75, 1 MeV 1 )

27 Rotational character? 162 Dy E x 1 MeV ( T 0.2 MeV) g.s. band only E(T ) E 0 + T, J 2 T 2I g T cf. J 2 T e E x(2 + )/T for vib. SMMC : E (MeV) <J 2 > 10 rot. exp T (MeV) SMMC I g = 35.8 ± 1.5 MeV 1 vs. I g = 37.2 (exp.)

28 Crossover from vibrational to rotational nuclei? even-n Nd-Sm isotopes { /( 30 e J 2 E x (2 + )/T 1 e ) E x(2 + )/T 2 (vib.) T 6 T / (at low T ) vib. / rot. E x (2 + ) (rot.) Sm Sm Nd <J 2 > T Sm 154 Sm Nd 148 Nd 10 <J 2 > T T (MeV) exp exp. (dis.) SMMC exp. discrete levels + BBF Nd 152 Nd T (MeV)

29 crossover from vib. to rot. well reproduced! (within shell model) 148 Sm : E x (2 + ) = ± MeV vs MeV (exp.) 154 Sm : ± MeV MeV fit to J 2 T 144 Nd : ± MeV MeV 150 Nd : ± MeV MeV 152 Nd : ± MeV MeV

30 SMMC state densities 162 Dy : (vs. exp. & finite-t HFB) ρ (MeV -1 ) E M E (MeV) β (MeV -1 ) E X (MeV) excellent agreement with exp. exp. count. n-res. Oslo HFB SMMC almost equal slope at high E x, but factor 10 2 enhancement from finite-t HFB collective rotation

31 Sm : ρ(e x ) (MeV -1 ) Sm 150 Sm p n n p 152 Sm 154 Sm p n p n E x (MeV) count. n-res. BBF HFB SMMC exp. pairing / shape trans. in HFB

32 Collective enhancement K(E x ) := ρ SMMC (E x ) / ρ HFB (E x ) K vib (E x ), K rot (E x ) (?) n p n p K Sm 150 Sm p p n n 152 Sm 154 Sm E x (MeV) decay of K losing collectivity { pairing rotation

33 VI. Summary & future prospect Summary SMMC methods reviewed briefly projections isolated finite system Applications to level densities successful rotational properties & crossover O.K. collective enhancement obtained microscopically Future prospect Systematic calculations! connection of different model spaces! powerful & massive CPUs (+ man-power)? simplification based on physics understanding? RIPL-4 hopefully contain SMMC level densities (by SNP2008) Quantitative derivation of collective enhancement factor?