Cranked-RPA description of wobbling motion in triaially deformed nuclei Takuya Shoji and Yoshifumi R. Shimizu Department of Physics, Graduate School of Sciences, Kyushu University, Fukuoka 81-8581,Japan aially-sym. def. triaial def.
Wobbling motion (triaially( deformed) Rotor Model I I H rot = + + J J J tilting - 軸 y I z y z I E Bohr Mottelson vol.ii E (ΔI=±1) I ( I 1) 1 E( I, n) = + + ω wob n + 1 J I 1 I J J ω wob = 1 1 J Jy Jz E (ΔI=-) -phonon ecited band 1-phonon ecited band yrast band (vacuum of phonon) Body-fied frame I I +
Evidence of wobbling motion 163 Lu 71 Evidence of wobbling motion S. W. Ødegård et al., Phys. Rev. Lett. 85 (001) 5866 D. R. Jensen et al., Phys. Rev. Lett. 89 (00) 14503 I I Rotor Model B(E)in : large E B(E)out TSD3 I I 1 B(E)out : large TSD TSD1 B(E)in I I +
Triaial deformation in the rotor model Triaial deformation γ Three moments of inertia J, Jy, Jz B(E)out (ΔI= -1) γ > 0 TSD1 TSD ΔI= +1 ΔI= -1 J>J>J y z Two intrinsic Q-moments Q0 ( z y ) a Q ( y ) a tan γ = Q Q 0 in the Lund convention irrotational moments of inertia ( J J )( J J ) ω wob y z
Microscopic study of wobbling motion Cranked mean field + Random Phase Approimation (RPA) E ecited band h' = h ω J def rot ω rot ω n yrast band H ' = h' + V int = ω n n [ H ', X ] X n total angular momentum
Microscopic Calculation Rotor Model Picture (Interpretation) Microscopic wobbling theory E. R. Marshalek, Nucl. Phys. A331 (1979) 49 H = hsph. + P+ QQ Realistic Calculation M. Matsuzaki, Nucl. Phys. A509 (1990) 69 Y. R. Shimizu, M. Matsuzaki, Nucl. Phys. A588 (1995) 559 H = h + V Recent Calculation for Lu, Hf nuclei M. Matsuzaki, Y. R. Shimizu, K. Matsuyanagi, Phys. Rev. C 65,041303(R) def res symmetry (J) restoring int. general def. Phys. Rev. C 69,03435(004) using Nilsson potential as a mean field
Uniformly-Rotating otating frame -ais J Marshalek Nucl. Phys. A331 (1979) 49 Principal-Ais frame -ais J ω PA-conditions PA Q z = PA Q y = 0 0 Shape fluctuation UR h t = hdef ωrot J () κ Q () t Q κ Q () t Q z z z y y y Quadrupole tensor 5 Q r δ A ij = [ i j ij ] a 16π a = 1 Angular momentum fluctuation PA h ( t) = hdef ω( t) J moment of inertia J () t =J ω () t ( n) ( n) ( n) yz, yz, yz, J I J= J / ωrot I/ ωrot I J J ωn = 1 1 ( n) ( n) J Jy Jz
J Mean-field ( Nilsson potential is too large due to l term Woods-Saon potential good parametrization field (Woods-Saon potential) ρ Density distribution ρ W.S.:R.Wyss ( r) 08 8 Pb 15 66 Dy ρ r [fm] 08 8 Pb No ls +ls H.O. +ls +ls +l Skyrme:S3 r [fm]
0.4 0.3 0. 0.1 0 163 Shape of nuclei at I=53/ Cranked Nilsson Strutinsky i ( π, α ) = ( ++, 1/) proton 13/ γ = 60 ND γ ~ 0 TSD 71 Lu γ = 0 Lund convention -0.1-0. γ = 60 0.1 0. 0.3 0.4 o ε cos( γ + 30 ) β = 0.4 γ =18 o β 4 = 0.034
Δ ( ω ) Paring gaps (monopole pairing) 1 ω rot Δ 0 1 ω 0 = Δ 0 ω 0 ω rot rot Δ ( ω ) = 0 Δ 0 for ω for ω < ω rot 0 ω rot 0 Δ [MeV] Self-consistent cal. Δ [MeV] Δ 0 Δ 0 Self-consistent cal. Used in Cal. ω 0 collapse! ω [MeV] rot ω [MeV] rot
Separable force 1 Vres = κ kf RPA interaction ( minimal( minimal ) k= y, z k [ ] κ,, k = hdef ijk ijk F = h ij k [ ] def, k Broken rotational symmetry [ h, J ] 0 def k Restore broken symmetry in RPA order [ hdef + Vres, J k ] RPA = 0 No adjustable parameters Bohr-Mottelson vol.ii C. G. Andersson et al., Nucl. Phys. A361 (1981) 147 RPA equation of motion is determined by a given mean field
163 71 Lu Moment of inertia CAL. EXP. CAL. (p+rotor) ω rot [MeV] nice agreement with eperimental data CAL. (p+rotor) I. Hamamoto,G. B. Hagemann, Phys. Rev. C 67, 014319 (003) EXP. D. R. Jensen et al., Nucl. Phys. A 703 (00) 3 J ω = rot J ω rot E = I γ = 0 o (R) J = 48 (R) Jy = 45 (R) Jz = 17
163 71 Lu Ecitation energy of RPA-phonon CAL. (p+rotor) EXP. CAL. (RPA) disappear ωwob ω J rot 1 J = 1 Jy Jz ω rot [MeV] Wobbling-like RPA solution eists, even if parameters are changed in a reasonable range. CAL. (p+rotor) EXP. I. Hamamoto,G. B. Hagemann, Phys. Rev. C 67, 014319 (003) D. R. Jensen et al., Nucl. Phys. A 703 (00) 3
163 71 Lu B(E) ratio CAL. (p+rotor) EXP. IN OUT CAL. (RPA) Etremely collective transitions ω rot [MeV] ~ 50 Weisskopf units CAL. (p+rotor) EXP. I. Hamamoto,G. B. Hagemann, Phys. Rev. C 67, 014319 (003) A. Görgen et al., Phys. Rev. C 69, 031301(R) (004)
163 71 Lu Three moments of inertia J J y J J = J ωrot (wob) J (wob) yz, yz, = (wob) ωyz, J z J is the largest ω rot [MeV] ( J J )( J J ) ω wob y z
Conclusion We have studied nuclear wobbling motions by using Cranked mean field plus RPA on microscopic view point. Woods-Saon mean field and symmetry restoring RPA interaction Wobbling-like RPA solutions eist. The ecitation energy is small compared with the eperimental data (Effects of particle-rotation coupling?) The B(E) ratio of the in-band and out-of-band is small compared with the eperimental data (Definitions of?) γ Microscopic justification of the rotor model
Effects of particle-rotation rotation coupling Particle plus Rotor Model Hamamoto-Hagemann, Phys. Rev. C 67,014319 H rot ( I j) I I I = + + = + + + J J J J J J J J y Iz y Iz I j j y z y z I J J j E( I + 1, n = 1) E( I, n = 0) = 1 1 + J J Jz J j 13 j J J -1 70 [MeV ] 93 [kev] y ω rot [MeV]
163 71 Lu BE ( ) / BE ( ) out Definitions of in γ EXP. γ dens γ in the rotor model ω [MeV] rot Q 3 y 1 dens = = tan Q0 z y if J = J, y z γ pot BE ( ) / BE ( ) out in + o tan ( γ 30 )
Conclusion We studied wobbling motion by using Cranked mean field plus RPA on microscopic view point. Woods-Saon mean field and symmetry restoring RPA interaction Wobbling-like RPA solutions eist. The ecitation energy is small compared with the eperimental data (Effects of particle-rotation coupling?) The B(E) ratio of the in-band and out-band is small compared with eperiments (Definitions of?) γ Microscopic justification of the rotor model
Density distribution ρ ( r) of 08 ρ ( r ) ρ ( r) 8 Pb W.S.:R.Wyss W.S.:Universal r [fm] ρ ( r) ρ ( r) r [fm] Skyrme:SKM* Gogny:D1S r [fm] r [fm]
163 71 Lu Ecitation energy of RPA-phonon EXP. ω rot [MeV] CAL. (RPA) EXP. D. R. Jensen et al., Nucl. Phys. A 703 (00) 3
15 66 Dy Woods-Saon parameter set R.Wyss Universal
16 70 Yb Quasi-particle energies β = 0.4 γ =18 o β 4 = 0.034 Δ= 0.50 [MeV] λ = 8.8 [MeV] N λ = 3.6 [MeV] P α = + 1/ α = 1/ PROTON NEUTRON ω rot [MeV] ω rot [MeV]
163 71 Lu In-band transitions CAL. EXP. ω rot [MeV] EXP. A. Görgen et al., Phys. Rev. C 69, 031301(R) (004)
Woods-Saon potential 0.1 0.5 0.9 S 0 Vw.s. S r Vw.s.
LS potential Coulomb potential Pairing potential