Developments of the canonical-basis time-dependent Hartree-Fock-Bogoliubov theory 187 th RIBF Nuclear Physics Seminar S.Ebata Shuichiro EBATA Meme Madia Laboratory, Hokkaido University, Nuclear Reaction Data Centre, Faculty of Science, Hokkaido University (JCPRG) th RIBF Nuclear Physics Hall Phys. Rev. C82 (2010) , Phys. Rev. C90 (2014) , JPS Conf. Proc. 1, (2014), etc. Based on

Contents 187 th RIBF Nuclear Physics Seminar S.Ebata  Motivation to construct Cb-TDHFB & Several mean-field models  Derivation of Cb-TDHFB Eqs.  Application of Cb-TDHFB  Systematic study of E1 modes (Linear response cal.)  Fusion reaction with Pairing (Non-linear response cal.)  Summary & Perspective

Motivation 187 th RIBF Nuclear Physics Seminar S.Ebata Nucleus is a core of atom, which is composed nucleons. AtomNucleus electron ~ [m] ~ few [m] Nucleon Neutron Proton Finite quantum many-body system Nucleus has magic number Neutron number  B = Binding Energy(Exp.) – Empirical formula Nucleus makes an averaged “potential”. → Mean field → Shell structure

187 th RIBF Nuclear Physics Seminar S.Ebata Single-particle excitation Alpha-particle excitation Collective excitation Etc. Motivation In Nuclear system, many degree of freedom appear, while depending on a number of nucleons and energy of the system. To understand the complexity of nuclear system, we should know many excited modes of many nuclei, systematically.

187 th RIBF Nuclear Physics Seminar S.Ebata Motivation

187 th RIBF Nuclear Physics Seminar S.Ebata Motivation “Systematic” → stable, unstable, spherical, deformed... The theoretical method should satisfy some requirements, to describe the character of many nuclei systematically. Canonical-basis TDHFB (Cb-TDHFB) in 3-Dimensional coordinate-space Requirements : 4, is applicable to heavy nucleus with reasonable numerical cost 1, has No spatial symmetry restriction to describe any deformation 3, is able to describe excitation and various dynamics of nucleus 2, includes Pairing correlation (for heavy & open shell nuclei) Generally... To understand the complexity of nuclear system, we should know many excited modes of many nuclei, systematically.

Several mean-field models 187 th RIBF Nuclear Physics Seminar S.Ebata Static, Ground state Dynamics, Excited state With No Pairing Hartree-Fock(HF) Time-dependent HF (TDHF), RPA With Pairing (BCS) HF+BCS TDHF+BCS Cb-TDHFB With Pairing Hartree-Fock- Bogoliubov (HFB) TDHFB, QRPA Method Interaction (non-relativistic) *(Q)RPA: Quasi-particle Random Phase Approximation ← Small amplitude limit of TDHF(B) Contact (  ) - type Skyrme Finite rangeGogny Base Dimension Cal. costDemerit Func. Exp. (HO,Gauss,etc.) LightDilute density X Mesh Heavy Heavy Cal. cost X Cal. costShape 1D Spherical 2D Quadrupole 3D No restriction Light Heavy *Another key word: Self-consistency *Relativistic Mean Field (RMF); Method : Hartree, Hartee-Bogoliubov

Derivation of Cb-TDHFB 187 th RIBF Nuclear Physics Seminar S.Ebata TDHFB 1, Canonical-basis representation 2, Assumption for Pairing potential Cb-TDHFB S. Ebata, T. Nakatsukasa, T. Inakura, K. Yoshida, Y.Hashimoto and K.Yabana, Phys. Rev. C82 (2010)

: Generalized quasi-particle state : Canonical basis : BCS quasi-particle state HF vs. HF+BCS vs. HFB *One body density matrix is diagonalized in Canonical basis. HF HFBHF+BCS Pairing correlation Generalize : Generalized density matrix : Generalized Hamiltonian TDHFB N : nucleon # : Pair potential N' : canonical basis # M : basis # N = N' N < N' Dimension

Recipe for the Canonical-basis TDHFB (Cb-TDHFB) Ebata et al, Phys. Rev. C82, : 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. : Time-dependent Canonical basis The computational cost of TDHFB can be reduced in Canonical-basis representation !? : Time-dependent Canonical single-particle basis This set is assumed to be orthonormal. TDHFB Eq.

Note: HFB state can be represented in Canonical basis at each time... HFB at t BCS at t Time HFB at t' BCS at t'...

187 th RIBF Nuclear Physics Seminar S.Ebata 1, Canonical-basis representation : Occupation probability : Pair probability : Density matrix : Pair tensor Recipe for the Cb-TDHFB PRC82, : Pair of k-state (no restriction of time-reversal) Inversion We can obtain the derivatives of  k (t) and  k (t) with respect to time. TDHFB Eq.

Recipe for the Cb-TDHFB TDHFB Eq. We can get the time-dependent equation for with orthonormal canonical basis ?  k (t) and  k (t) are identical to gap parameters of BCS approximations, in the case where pair potential is computed as PRC82,

Recipe for the Cb-TDHFB Can we describe the inversion for this part with the orthonormal canonical basis ? 2, Assumption for Pairing potential … Pair potential is diagonal. We can not invert this pairing potential, because the two-particle state do not span the whole space. Cb-TDHFB equations Properties of Cb-TDHFB TDHF HF+BCS We can invert the pairing potential. PRC82,

3, If we adopt a schematic pairing functional: When we apply Cb-TDHFB to... This pairing potential violates 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. PRC82,

187 th RIBF Nuclear Physics Seminar S.Ebata TDHFB 1, Canonical-basis representation 2, Assumption for Pairing potential 3, We adopt a schematic pairing functional. We choose the special gauge. Cb-TDHFB Recipe for the Cb-TDHFB For applications of Cb-TDHFB Essential points of derivation Essential points of application PRC82,

Calculate HF or HF+BCS ground state Adding a instantaneous external field to ground state Calculate the time-evolution with TDHF or Cb-TDHFB Strength function S(E;F) is gotten as Fourier transformed TD-. : Smoothing parameter one-body operator Linear response calculation w/ TD scheme 187 th RIBF Nuclear Physics Seminar S.Ebata

We need to confirm the dependence on the amplitude k, and the operator. Linear response calculation w/ TD scheme 187 th RIBF Nuclear Physics Seminar S.Ebata

Example: Photo-absorption cross section of 172 Yb 187 th RIBF Nuclear Physics Seminar S.Ebata  =0.32  n = 0.76 [MeV]  p  = 0.55 [MeV] 3D Cb-TDHFB K=0 - K=1 - Box size : R=15[fm], mesh=1[fm] (3D-Spherical) Canonical-basis space (HF+BCS g.s.) : 146 states for neutron, 98 states for proton Total cal. cost : 300 CPU hours (with a Single processor; Intel Core i7 3.0 GHz)  = 1.0 [MeV] Experimental data: A.M.Goryachev and G.N.Zalesnyy Vopr. Teor. Yad. Fiz. 5, 42 (1976). Neutron Proton Cb-TDHFB cal. can reproduce the photo-absorption cross section of 172 Yb. Heavy nucleus Deformed nucleus Including pairing

172 Yb  =0.32  p  = 0.55 [MeV] S. Ebata using Cb-TDHFB ( based on PRC82, ) Box size : R=15[fm], mesh=1[fm] (3D-Spherical) Canonical-basis space (HF+BCS g.s.) : 146 states for neutron, 98 states for proton Total cal. cost : 300 CPU hours (with a Single processor; Intel Core i7 3.0 GHz)  n  = 0.76 [MeV] J. Terasaki and J. Engel PRC82,  =0.34 Box Size :  = z ± =20[fm], b-spline (Cylindrical) Single-quasiparticle space (HFB g.s.) : 5348 states for neutron, 4648 states for proton Total cal. cost : 100,000 CPU hours (with Kraken; Super computer of ORNL) = 0.77 [MeV] = 1.25 [MeV] Cb-TDHFB can reproduce the similar results of QRPA with very small numerical cost. 172 Yb 187 th RIBF Nuclear Physics Seminar S.Ebata Example: Numerical cost (Cb-TDHFB vs. QRPA)

Systematic study for E1 modes (Pygmy dipole mode) 187 th RIBF Nuclear Physics Seminar S.Ebata Strength Excitation Energy Experiments Soft dipole mode ? GDR ? PDR 26 Ne : J. Gibelin et al., PRL101, Ni : O. Wieland et al., PRL102, , 132 Sn : P. Adrich et al., PRL95, Ce : R.-D. Herzberg et al., PLB 390, Ba : R.-D. Herzberg et al., PRC60, Ba, 140 Ce, 144 Sm : A. Zilgas et al., PLB542, Pb : N. Ryezayeva et al., PRL89, Pb : J. Enders et al., NPA724, Pb : A. Tamii et al., PRL107, What is a nature of PDR ?  Collective mode or not  Characteristic mode of unstable nucleus  Light and Heavy mass region  For deformed nuclei … → UNCLEAR !! P. Adrich et al., PRL95, EXP. 187 th RIBF Nuclear Physics Seminar S.Ebata

187 th RIBF Nuclear Physics Seminar S.Ebata Systematic study for E1 modes S. Ebata, T. Nakatsukasa and T. Inakura, Phys. Rev. C90 (2014) We have prepared many wave functions of even-even nuclei (over 850), and had investigated the E1 modes for Z = 6 – 50, systematically.

187 th RIBF Nuclear Physics Seminar S.Ebata To quantify the low-lying E1 strength systematically... J. Gibelin, et al., Phys. Rev Lett. 101, (2008) 26 Ne on a 208 Pb target We use the ratio to analyze the low-lying E1 strength for all calculated nuclei. “Low-energy” ex.) The ratio of low-lying E1 strength in Total E1 strength (sum rule).  n =7.17 PRC90,

187 th RIBF Nuclear Physics Seminar S.Ebata N-# dependence of PDR (6 < Z < 20) HF+RPA vs. Cb-TDHFB: The low-lying E1 strength increases smoothly around N=28 in Si, S, and Ar isotopes, due to the fractional occupation of orbitals caused by the Pairing correlation. Neutron-# vs. PDR fraction ratio: The low-lying E1 strength is sensitive to shell structure. PRC90,

187 th RIBF Nuclear Physics Seminar S.Ebata N-# dependence of PDR (Z > 28) N=82 N=50 deformed Neutron-# vs. PDR fraction ratio: The shell structure dependence of PDR remains also in heavy mass region. How a relation between PDR and N-skin is ? Deformation vs. PDR PRC90,

Characteristic N-# dependence of PDR around N=82 N=82 Isotope Rewrite Isotone Something... However, the heavy isotopes have NOT only simple s.p.-excitation structure but also another one. In heavy isotopes also, the occupation single-particle state near the Fermi level is important for the emergence of PDR. f 7/2 PRC90,

 n =5.06  n =3.90  n =3.60  n =1.57  n =1.23 *Change smoothing function Lorentzian to W(t) at t=0 T : total real-time step The peaks which are appeared from N=84 are almost same as those obtained in unperturbed calculation (dotted lines). These peaks have very small coherency. Low-lying E1 strength of Zr isotopes around N=82 In 6~8 MeV, the strengths are smaller than unperturbed results. There is possibility that these peaks have some collectivities. PRC90,

187 th RIBF Nuclear Physics Seminar S.Ebata Character of PDR in each region Single-particle state Other mode (Collective mode?) Superposition Stable Unstable Light Heavy × Possibility O, Ne, Mg Solid line: Cb-TDHFB Dashed line: HF+RPA N=82

187 th RIBF Nuclear Physics Seminar S.Ebata Piekarewicz PRC (2006) T. Inakura, et al., PRC (R) (2011) RMF+RPA HF+RPA Linear relation between PDR and N-skin-thickness ?

187 th RIBF Nuclear Physics Seminar S.Ebata Linear relation between PDR and N-skin-thickness ? N=50 N=82 PRC90,

187 th RIBF Nuclear Physics Seminar S.Ebata PDR/GDR [%] 132 Sn 110 Sn 120 Sn Comparison ours with RMF+RPA results Piekarewicz cal. is RMF+RPA w/o pairing, in HO basis. PDR : E1 strength (5-10 MeV) GDR : E1 strength (10-25 MeV) diff. of R rms [fm]

187 th RIBF Nuclear Physics Seminar S.Ebata Linear relation between PDR and N-skin-thickness ? SkI3 Skyrme parameter is determined to reproduce the ordering of single-particle states of 208 Pb that obtained by RMF Cal. NPA584, 467(1994), P.-G. Reinharrd, H.Flocard ← Behavior of PDR is changed significantly, especially from N=50 to 82. The correlation between PDR and N-skin depends on mass region. PRC90,

187 th RIBF Nuclear Physics Seminar S.Ebata Deformation vs. PDR K=0 and 1 components of PDR strength in deformed nuclei D. Pena, E. Khan and P. Ring PRC (2009) 50 Sn isotope

187 th RIBF Nuclear Physics Seminar S.Ebata deformed z x y K=0 PDR ratio becomes large in prolate nuclei. ⇔ N-skin corresponds to it? K=0 and 1 components of PDR strength in deformed nuclei PRC90,

187 th RIBF Nuclear Physics Seminar S.Ebata N-skin thickness and deformation ( for Sr, Zr, Pd and Cd isotopes ) z x,y N-skin does not correspond to PDR ratio.

187 th RIBF Nuclear Physics Seminar S.Ebata Deformation neutron and proton ( for Sr, Zr, Pd and Cd isotopes )

187 th RIBF Nuclear Physics Seminar S.Ebata Summarize systematic study for E1 modes (PDR) PDR fraction ratio → Neutron number dependence characterized by shell structure appears over the wide mass region. → The PDR on nuclei with N=82 has multi-structure. (Single-particle and Collective) N-skin vs. PDR → The linear correlation between PDR and N-Skin appears, which depends on mass region. → The slope becomes gradually decrease as the proton number increases (Z > 40). → This relation has interaction dependence and shell structure dependence. PDR in deformed nuclei vs. N-skin (Sr, Zr, Pd, Cd with N=60-70) → In deformed nuclei, a K=0 dominance f PDR (K=0) > f PDR (K=1) (prolate) appears. → However, the dominance does not correspond to a direction dependence of N-skin. A correlation between PDR and N-skin is not so strong, but in much neutron-rich nuclei, the linear correlation appears clearly. PRC90,

187 th RIBF Nuclear Physics Seminar S.Ebata H.Flocard, S.E.Koonin and M.S.Weiss Phys. Rev. C17 (1978) 1682 Simulation of heavy ion collision using TDHF

187 th RIBF Nuclear Physics Seminar S.Ebata What kind of pairing effect is expected in low-energy Heavy ion collision ?  Level crossing  Energy Dissipation  Neck formation  Odd-even effects for spontaneous fission half-lives ?  Nucleon transfer reaction  Pair transfer  Nuclear Josephson effect  Fusion or Fission cross section W.J.Swiatecki, Phys. Rev. 100 (1955) 937. Spontaneous Fission Half-lives

187 th RIBF Nuclear Physics Seminar S.Ebata Constant G & Contact pairing Q 20 mode of 34 Mg with result of QRPA (C.Losa) For extension to heavy ion collision using Cb-TDHFB … Contact pairing V (NPA517(1990)275, Kringer, Bonche, Flocard, Quentin and Weiss) : spin-singlet zero-range interaction Phys.Rev.C81,064307

187 th RIBF Nuclear Physics Seminar S.Ebata Incident Energy : [MeV] ( E cm = 9.0 – 10 [MeV], V Coul. ~ 9 MeV ) Setup for collision Effective Interaction : Skyrme force (SkM*), Contact pairing Projectile : 22 O, Target : 22 O (HF g.s. has also spherical shape) Calculation space ( 3D meshed box ): Length of box for ( x, y, z ) is 36, 20, 40[fm] meshed by 1.0 [fm] Impact parameter : [fm] # of canonical-basis for HF+BCS g.s. ; (N, Z) = (32 (16+16), 16 (8+8) ) Average of gap energy ; S.E. and T. Nakatsukasa, JPS Conf. Proc. 1 (2014)

187 th RIBF Nuclear Physics Seminar S.Ebata Simulation of 22 O + 22 O collision with b = 3.0 [fm] and E cm =10 [MeV] Cb-TDHFBTDHF Time-evolution of Neutron density distribution JPS Conf. Proc. 1,

187 th RIBF Nuclear Physics Seminar S.Ebata Projectile : 52 Ca, Target : 52 Ca In both methods, the g.s. is spherical shape. E cm = 51.5 MeV (V Coul. ~ 49 MeV) Effective Interaction : SkM*, Contact pairing Impact parameter : [fm]  H ~ 189 mb →  B ~ 181 mb be small about 5% Cb-TDHFB TDHF To reproduce  n of 52 Ca In a little bit heavier system Point: increase of pair number

187 th RIBF Nuclear Physics Seminar S.Ebata Projectile : 52 Ca, Target : 52 Ca In both methods, the g.s. is spherical shape. E cm = 51.5 MeV (V Coul. ~ 49 MeV) Effective Interaction : SkM*, Contact pairing Impact parameter : [fm]  H ~ 189 mb →  B ~ 181 mb be small about 5% Cb-TDHFB TDHF To reproduce  n of 52 Ca In a little bit heavier system Point: increase of pair number

187 th RIBF Nuclear Physics Seminar S.Ebata Projectile : 22 O, Target : 52 Ca E cm = 25 MeV (V Coul. ~ 20.8 MeV) Effective Interaction : SkM*, Contact pairing In both methods, the g.s. is spherical shape. Impact parameter : [fm] Cb-TDHFB TDHF Average of strength in 22 O and 52 Ca  H ~ 528 mb →  B ~ 503 mb be small about 5% Case of different nuclei collision Point: Difference of chemical potential

187 th RIBF Nuclear Physics Seminar S.Ebata Cb-TDHFB TDHF  H ~ 528 mb →  B ~ 503 mb be small about 5% Projectile : 22 O, Target : 52 Ca E cm = 25 MeV (V Coul. ~ 20.8 MeV) Effective Interaction : SkM*, Contact pairing In both methods, the g.s. is spherical shape. Impact parameter : [fm] Average of strength in 22 O and 52 Ca Case of different nuclei collision Point: Difference of chemical potential

187 th RIBF Nuclear Physics Seminar S.Ebata Coulomb barrier for 96 Zr+ 124 Sn (F.D.) HF g.s. of 124 Sn has Oblate shape.  =-0.1 V FD =225 MeV 96 Zr 124 Sn V FD =216 MeV 96 Zr 124 Sn Projectile : 96 Zr, Target : 124 Sn HF+BCS : Both ground states are spherical shape. HF : 96 Zr ; spherical, 124 Sn ; oblate shape. E cm = 228 MeV ( V coul. =216, 225 MeV ) Effective Interaction : SLy4d, Contact pairing Very strong in order to check the effect Neglect proton pairing Is the pairing effect visible in much heavier system ?

187 th RIBF Nuclear Physics Seminar S.Ebata TDHF (z || Z) Cb-TDHFB TDHF (x,y || Z) Is the pairing effect visible in much heavier system ? Z Z z z

187 th RIBF Nuclear Physics Seminar S.Ebata TDHF (z || Z) Cb-TDHFB TDHF (x,y || Z) Time up to scission < !! Is the pairing effect visible in much heavier system ?

187 th RIBF Nuclear Physics Seminar S.Ebata We apply Cb-TDHFB to large amplitude collective phenomena such as collision, with a contact pairing functional. 22 O+ 22 O, 52 Ca+ 52 Ca, 22 O+ 52 Ca: Pairing effects in fusion reaction, have a repulsive aspects. 96 Zr+ 124 Sn: The pairing effects gets the time from a contact to scission to be short. Simulation of the collision phenomena with Cb-TDHFB Summarize fusion reaction with pairing correlation JPS Conf. Proc. 1,

Summary 187 th RIBF Nuclear Physics Seminar S.Ebata To study the properties of nucleus without restrictions of mass region, we suggest the Cb-TDHFB method in 3D-coordinate representation. Linear response calculation using Cb-TDHFB, can be compared with QRPA (TDHFB) results for IVD, ISQ modes. Due to the small computational cost, we can apply the method to systematic study of the excited states. PRC82, JPS Conf. Proc. 1, PRC90, Simulation of the collision phenomena with Cb-TDHFB PDR N-# dependence characterized by shell structure appears over the wide mass region. The PDR on nuclei with N=82 has multi-structure. (Single-particle and Collective) The linear relation has interaction dependence and shell structure dependence. Pairing effects in fusion reaction, have a repulsive aspects.

Perspective (data base) 187 th RIBF Nuclear Physics Seminar S.Ebata

Perspective (structure) 187 th RIBF Nuclear Physics Seminar S.Ebata  Extent to other modes (E0, E2, E3, M1, etc.)  Extent to Uranium isotopes  Ground state (deformation, skin, etc.)  Excited states (E1: PDR, GDR, Polarizability, etc.)  Reaction  Input data format (EoS, radioactive therapy, etc.)

Summary & Perspective 187 th RIBF Nuclear Physics Seminar S.Ebata To study the properties of nucleus without restrictions of mass region, we suggest the Cb-TDHFB method in 3D-coordinate representation. PRC82, JPS Conf. Proc. 1, PRC90, PDR N-# dependence characterized by shell structure appears over the wide mass region. The PDR on nuclei with N=82 has multi-structure. (Single-particle and Collective) The linear relation has interaction dependence and shell structure dependence. Pairing effects in fusion reaction, have a repulsive aspects. Perspective  Extent to other modes (E0, E2, E3, M1, etc.), up to Uranium.  Construct theoretical data base (G. & Ex. States, Reaction, EoS) Thank you !