A multi-dimensional constrained relativistic mean eld model

Transcription

1 A multi-dimensional constrained relativistic mean eld model and its application for hypernuclei and heavy nuclei * Bing-Nan Lu * En-Guang Zhao Shan-Gui Zhou Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 11 KLFTP-BLTP Joint Workshop on Nuclear Physics September, 6-8, 11 Beijing

2 Motivation Self-consistent symmetries are used to simplify the mean eld calculations. Dierent problems are treated with dierent SCS. spherical; axial symmetric; reection symmetric;... Problem with dierent SCS: ssion Solving these problems by P. Moller et al., Nature 409, 785 (01) Symmetry unrestricted calculation available for non-relativistic models Ajustable SCS

3 Relativistic mean eld models Lagrangian L = ψ(iγ µ µ mb)ψ ( µσ µ σ m 2 σ σ 2 ) + ( 1 4 Ωµν Ω µν m ωω 2 ) +( 1 4 R µν Rµν m ρ ρ 2 ) 1 4 F µν F µν g σ σ ψψ g ω ω µ ψγ µ ψ g ρ ρ µ ψγ µ τψ ea µ ψγ µ ψ 1 3 g 2σ g 3σ 4 Parameters: Meson-exchange + non-linear terms : NL3*, PK1... Point-coupling + non-linear terms : PC-F1, PC-PK1... Meson-exchange + density dependent : DD-ME2... Point-coupling + density dependent : DD-PC1...

4 Axially deformed basis The triaxial deformed potentials and wave functions are expanded on the axially deformed basis: ) ( ) = ψ i = ( ψ L i ψ S i L α f α i α L S β g β i β S (1) α T = Cα T n z,n r,m,s = Cα T φ nz (z)rn m 1 r (r) exp(imθ) χ s, T = L,S (2) 2π The basis wave functions are eigenfunctions of the Schroedinger equation: ( h2 2M M ( ) ω 2 r r 2 + ω 2 z z2) n z,n r,m,s = E α n z,n r,m,s (3) P. Ring, Y. K. Gambhir and G. A. Lalazissis, Comp. Phys. Comm. 105(1997) 77

5 Expansion of the potentials Any scalar function V (z,r,θ) can be expanded as: V (z,r,θ) = V 1 µ(z,r) exp(iµθ) (4) µ= 2π If V is not only real but also symmetric under P x and P y, So if we dene a set of basis functions: V µ = V µ = V µ, V 2n+1 = 0 (5) β 0 (θ) = 1 2π, β n (θ) = 1 π cos(2nθ) (6) for expanding the azimuth angle part of the densities and potentials: 1 V (z,r,θ) = V n (z,r)β n (θ) = V 0 (z,r) + n=0 2π V n (z,r) 1 cos(2nθ) (7) n=1 π where all the expansion coecients V n are real.

6 Numerical check E MF (MeV) N F =18 N F = (Ground state) basis

7 Outline

8 RMF model for the Λ hypernuclei The Lagranigan density for hypernuclei: where L = L 0 + L Λ (8) L Λ = ψ Λ ( iγ µ µ M Λ g σλ σ g ωλ γ µ ω µ ) ψλ + f ωλλ 4m Λ ψ Λ σ µν Ω µν ψ Λ (9) Ω µν = µ ω ν ν ω µ is the ω eld tensor. Parameter m Λ g σλ g ωλ f ωλλ N-N interaction PK1-Y MeV 80g σ 0.6g ω g ωλ PK1 NLSH-A MeV 0.621g σ 0.667g ω g ωλ NLSH C.Y. Song, J.M. Yao, H.F. Lv, and J. Meng, Intl. Jour. Mod. Phys. E 19, 2538 (10) M. T. Win and K. Hagino, Phys. Rev. C 78, (08)

9 Carbon isotopes: w/ & w/o Λ sin C C C C C 15 C 19 C 23 C cos 12 C 16 C C C 17 C 21 C

10 Outline

11 ssion barrier T h Outer barrier: β βs in γ E (M e V ) T h βc o s γ Inner barrier: γ β E (M e V ) β 2 0

12 ssion barrier Pu PC-PK1 w/ w/o 3 w/, w/ 3 w/o w/o 3 w/o, w/

13 ssion barrier 30 a 240 Pu b w/ 240 Pu w/o

14 ssion barrier = 0.25 = 0.30 = 0.35 = = 0.45 = 0 = 5 = = 0.65 = 0.70 = 0.75 = = 0.85 = 0.90 =

15 Application: heavy nuclei ssion barrier ssion barrier = 0 = 5 = 1.10 = = 1. = 1.25 = 1.30 = = 1.40 = 1.45 = 1.50 = = 1.60 = 1.65 = A multi-dimensional constrained rela 22 * Bing-Nan Lu * En-Guang Zhao Shan-Gui Zhou -0.1

16 ssion barrier 8 6 NL3* NL-Z2 PC-PK DD-ME2 DD-PC U

17 ssion barrier Th U Pu Cm Cf B [inner] (MeV) Th U Pu Cm Cf B [outer] (MeV) w/o w/ exp PC-PK1 Z, A

18 Application: heavy nuclei ssion barrier ssion barrier Cm 246 Cm Cm Cm Cm * Bing-Nan Lu * En-Guang Zhao Shan-Gui Zhou A multi-dimensional constrained rela

19 It is better to treat all self-consistent symmetry in a single framework. The inuence of additional Λ hyperon on nuclear shape is investigated for light nuclei. 12 C, 22 C and 28 Si may be the candidate for drastic shape change eect. Most of the potential energy surfaces are soften by Λ. The ssion barriers are calculated for the actinide nuclei. 3-dimensional constrained calculation including β 22 for 240 Pu is performed. The outer barriers are lowered by triaxial deformation up to 1 MeV. By allowing β 22 deformation, most of the barriers are reproduced. Dierent ssion path in Cm isotopes may inuence the calculated barrier heights. Thank you