大振幅集団運動の微視的理論 26 Oct. 2010@ 湯川記念館 Canonical-basis TDHFB を用いた線形応答計算 Contents Motivation for construction of 江幡 修一郎 A,B Cb-TDHFB theory Formalizations 中務 孝 B,C , 稲倉 恒法 B , Cb-TDHFB from TDHFB Linear Response with TD scheme 吉田 賢市 B , 橋本 幸男 A,C , Results Comparison with QRPA cal. in ISQ 矢花 一浩 A,B,C Heavy and systematic results in IVD Summary & Future Graduate School of Pure and Applied Sciences, University of Tsukuba A RIKEN Nishina Center Theoretical Nuclear Physics Laboratory B Center for Computational Sciences, University of Tsukuba C
Propaganda of Phys. Rev. C 82 , 034306
RIKEN press release from http://release.nikkei.co.jp/ Nikkei press release from http://release.nikkei.co.jp/
Introduction Density Functional Theory from http://unedf.org/
Introduction Odd-even mass staggering W.J.Swiatecki, Phys. Rev. 100 (1955) 937 Spontaneous Fission Half-lives Density Functional Theory From J.H.E.Mattauch, W.Thiele and A.H.Wapstra, Nucl. Phys. 67 (1965) 1 First 2 + state J.L.Wood et.al , Phys. Rep. 215 (1992) 101 Gap Energy 2 + 50 Sn Isotopes 0 + from http://unedf.org/
Introduction from http://www.rarf.riken.go.jp/newcontents/contents/facility/RIBF.html
Introduction Future plan of RI facility in the world from http://www.rarf.riken.go.jp/newcontents/contents/facility/RIBF.html
Introduction Construction of theoretical framework to calculate structure and response of from light nuclei to heavy ones systematically Requirements : 1, Applicable to heavy nuclei 2, No symmetry restriction for any deformed nuclei 3, Able to describe excitations and various dynamics of nuclei 4, Including effects of Pairing Correlation 3-Dimensional Cb-TDHFB + coordinate-space
Recipe for the Canonical-basis TDHFB (Cb-TDHFB) Ebata et al , PRC 82 , 034306 TDHFB : Density matrix : Pair tensor : Arbitrary complete set : Canonical basis Canonical-basis diagonalize Density matrix. In this Canonical-basis, the number of matrix elements compress to diagonal components. The computational cost of TDHFB may be reduced also in Canonical-basis representation !! : Time-dependent Canonical basis : Time-dependent Canonical single-particle basis This set is assumed to be orthonormal.
Recipe for the Cb-TDHFB Ebata et al , PRC 82 , 034306 1, Canonical-basis representation TDHFB : Pair of k -state (no restriction of time-reversal) : Occupation probability : Pair probability : Normal density : Pair tensor Inversion We can get the derivatives of ρ k (t) and κ k (t) with respect to time.
Recipe for the Cb-TDHFB Ebata et al , PRC 82 , 034306 TDHFB are identical to gap parameters of BCS approximations, in the case where pair potential is computed as ρ k (t) and κ k (t) We can get the time-dependent equation for ? with orthonormal canonical basis
Recipe for the Cb-TDHFB Ebata et al , PRC 82 , 034306 Can we describe the inversion for this part with the orthonormal canonical basis ? We can not invert this pairing potential, because the two-particle state do not span the whole space. 2, Assumption for Pairing potential … Pair potential is diagonal. We can invert the pairing potential. Properties of Cb-TDHFB Cb-TDHFB equations TDHF HF+BCS
Recipe for the Cb-TDHFB Ebata et al , PRC 82 , 034306 3, We adopt a schematic pairing functional: This pairing potential violate the gauge invariance related to the phase degree of freedom of canonical basis. Cb-TDHFB equations are invariant with respect to the phase of canonical basis. This schematic pairing potential violate We must choose the special gauge in this schematic pairing functional.
Recipe for the Cb-TDHFB Ebata et al , PRC 82 , 034306 TDHFB 1, Canonical-basis representation 2, Assumption for Pairing potential Cb-TDHFB 3, We adopt a schematic pairing functional. We must choose the special gauge. Feasible Cb-TDHFB
How to calculate Linear Response with TD scheme ? Instantaneous external field then the equations can be automaticaly linearised with respect to V ext and the density fluctuation. ^ Strength function S(E;F) Initial state of Real-Time cal. Ground state of HF or HF+BCS Strength function is calculated as ^ Fourier transformed time dependent expectation value of F.
Calculation setup Interaction : Skyrme force ( SkM* ) Pairing strength : Smoothed Pairing strength Even-even Nuclei : 18-32 Ne, 18-40 Mg, 24-46 Si, 28-50 S, 32-58 Ar, 144-154 Sm, 172 Yb ( 12-22 C, 14-28 O, 34-64 Ca) External field : Isovector Dipole, Isoscalar Quadrupole Cal. space ( 3D-Spherical box ): Relatively light nuclei case (A < 60), we use the box has radius 12 [fm] & mesh 0.8 [fm]. Relatively heavy nuclei case (A > 100), we use the box has radius 15 [fm] & mesh 1.0 [fm]. Pairing model space Energy cutoff
Can we describe deformed nuclei ? in Quadrupole mode β =0.00 16 O In Spherical case, we can not distinguish Q 20 and Q 22 mode. ∆ n = 0.0 [MeV] In Spherical case, ∆ p = 0.0 [MeV] β =0.39 24 Mg ∆ n = 0.0 [MeV] In Quadrupole deformed case(prolate), the GR of Q 20 is lower than one of Q 22 . ∆ p = 0.0 [MeV] In oblate case, K=0 - the relation between Q 20 and Q 22 is reversed. K=1 -
Comparison with QRPA ( IS Quadrupole ) for 34 Mg 34 Mg (a) Cb-TDHFB with (b) Full Cb-TDHFB fixed LS & Coulomb potentials (c) QRPA without residual LS & (d) QRPA (delta-pairing) Coulomb interaction (delta-pairing) C. Losa et al. PRC81, 064307 (2010)
Summary of isoscalar Quadrupole mode We can describe the properties of deformed nuclei for ISQ vibrations with Cb-TDHFB in 3D-coordinate space. In spherical case, we can not distinguish ISQ vibrations which are called β - and γ - vibration. In prolate (oblate) deformed case, main peak of Q 20 is lower (higher) than one of Q 22 . Comparison with deformed HFB+QRPA results for 34 Mg and 24 //// Mg /////// ////// The results of Cb-TDHFB well agree with other deformed QRPA cal. in isoscalar quadrupole modes, except for height of the lowest peak. ( caused by using a schematic pairing functional ? ) The results of ISQ vibration are more sensitive for the residual spin- orbit interaction than ones of IVD mode.
Can we describe deformed nuclei ? in E1 mode In Spherical case, 16 O the Giant Dipole Resonance(GDR) ∆ n = 0.0 [MeV] will be a concentrated peak. β =0.00 ∆ p = 0.0 [MeV] 24 Mg From Nucl.Phys. A251,479(1975) β =0.39 ∆ n = 0.0 [MeV] In Quadrupole deformed case, the GDR have two components, ∆ p = 0.0 [MeV] K=0 - , 1 - K=0 - K=1 - From J,IZV,67,656,2003
Example of E1 mode for other heavier nuclei with SkM* 40 Ca ∆ n = 0.0 [MeV] β =0.00 ∆ p = 0.0 [MeV] From Nucl.Phys. A227, 513 (1974) 90 Zr ∆ n = 0.0 [MeV] 208 Pb β =0.00 ∆ p = 1.9 [MeV] β =0.00 ∆ n = 0.0 [MeV] ∆ p = 0.0 [MeV] From Nucl.Phys. A175, 609 (1971) From Nucl.Phys. A159, 561 (1970)
The shape transitional region in Sm isotopes N = 88 N = 82 β =0.20 β =0.00 Preliminary ∆ n = 0.0 [MeV] ∆ n = 0.9 [MeV] ∆ p = 2.0 [MeV] ∆ p = 1.5 [MeV] N = 84 N = 90 β =0.00 β =0.29 ∆ n = 1.0 [MeV] ∆ n = 0.9 [MeV] ∆ p = 1.1 [MeV] ∆ p = 1.9 [MeV] N = 86 β =0.11 β =0.32 Preliminary ∆ n = 0.9 [MeV] ∆ n = 0.9 [MeV] ∆ p = 1.6 [MeV] ∆ p = 1.0 [MeV] From Nucl.Phys. A225, 171 (1974)
E1 strength functions 172 Yb ( for computational cost ) J. Terasaki and J. Engel PRC 82 , 034326 172 Yb Box Size : ρ = z ± =20[fm], b-spline ( Cylindrical ) β =0.34 Single-quasiparticle space ( g.s. HFB ) : ∆ n = 0.773 [MeV] 5348 states for neutron, ∆ p = 1.248 [MeV] 4648 states for proton Total time : 136,000 CPU hours ( with Kraken ; Super computer of ORNL ) S. Ebata using Cb-TDHFB 172 Yb ( based on PRC 82 , 034306 ) Box size : R=15[fm], mesh=1[fm] ( 3D-Spherical ) β =0.32 Canonical basis space ( g.s. HF+BCS ) : ∆ n = 0.757 [MeV] 146 states for neutron, ∆ p = 0.551 [MeV] 98 states for proton Total time : 300 CPU hours ( with ONE CPU ; Intel Core i7 3.0 GHz )
E1 strength functions for Ne isotopes
E1 strength functions for Mg isotopes
In order to investigate the appearance of Low-energy E1 strength ...
The ratio of the E1 strength in Low-energy region Low-energy region < 10 [MeV] For Ne, Mg isotopes The low-energy E1 strength appear from N =16 in the both cases. (They start to occupy the s 1/2 orbit.) 16 Deformed Spherical
The ratio of the E1 strength in Low-energy region Low-energy region < 10 [MeV] For Si, S, Ar isotopes The low-energy E1 strength appear dramatically from N =30 in the HF+RPA case. (They start to occupy the p 3/2 orbit.) 30
Pygmy resonance is pure soft-dipole mode ?
Pygmy resonance is pure soft-dipole mode ? 24 O 26 O N = 16 N = 18 26 Ne 28 Ne N = 16 N = 18
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