Reaction mechanism of fusion-fission process in superheavy mass region

Transcription

1 Reaction mechanism of fusion-fission process in superheavy mass region Y. Aritomo 1,, K. Hagino 3, K. Nishio 1, and S. Chiba 1 1 Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Japan Flerov Laboratory of Nuclear Reactions, Dubna, Russia Department of Physics, Tohoku University, Sendai, Japan TAN11 the 4 th International conference on the Chemistry and Physics of the Transactinide Ekements Sochi, Russia, 5-11 September, 011

2 1. Model Coupled-channels method + Dynamical Langevin calculation Trajectory analysis Langevin equation Two center shell model. Results 36 S+ 38 U and 30 Si+ 38 U Capture Cross-section Fusion Cross-section Mass distribution of Fission fragments 3. Mechanism of Dynamical process Potential energy surface on scission line Analysis of trajectory behavior Analysis of probability distribution 4. Summary

3 Zcn= Si + 38 U 36 S + 38 U Zcn=108 Exp. by K. Nishio et al. at JAEA Fusion-fission process Quasi-fission process

4 1. Model Estimation of cross sections. 1-. Dynamical Equation

5 Estimation of cross sections Capture Cross Section Coupled-channel method Fusion Cross Section CN Formation probability P CN Dynamical calculation Langevin eq. Axial symmetric configuration deformation effect by assuming the angle dependent touching distance

6 Nuclear shape two-center parametrization ( z,, ) (Maruhn and Greiner, Z. Phys. 51(197) 431) qz (,, ) δ=0 (δ1=δ) Trajectory which enters into the spherical region = fusion trajectory

7 Potential Energy ( 1) V ( q,, T ) VDM ( q) VSH ( q, T ) Iq ( ) V ( q) E ( q) E ( q) DM S C V q T E q T 0 SH (, ) shell ( ) ( ) T : nuclear temperature E * =at a : level density parameter Toke and Swiatecki E S : Generalized surface energy (finite range effect) E C : Coulomb repulsion for diffused surface E 0 shell : Shell correction energy at T=0 I : Moment of inertia for rigid body Φ(T) : Temperature dependent factor ( T ) exp E 0 MeV d at E d

8 Multi-dimensional 多次元ランジュバン方程式 Langevin Equation dq dt dp dt i i m 1 ij V q i p j 1 m p p m p g R ( t) 1 1 jk j k ij qi Friction Random force dissipation fluctuation jk k ij j Newton equation R k i( t) 0, Ri ( t1) Rj ( t) ij ( t1 t g ik g jk T ij ) : white noise (Markovian process) q i : deformation coordinate (nuclear shape) two-center parametrization ( z,, ) (Maruhn and Greiner, Z. Phys. 51(197) 431) p i : momentum m ij : Hydrodynamical mass (inertia mass) γ ij : Wall and Window (one-body) dissipation (friction ) E int E * m p p V ( q) 1 1 ij i j E int :intrinsic energy, E * : excitation energy

9 . Results -1. Capture and Fusion Cross sections Mass distribution of Fission Fragments 36 S+ 38 U and 30 Si+ 38 U -. Touching probability CC method Perform a trajectory calculation starting from the touching distance between target and projectile to the end each process.

10 Fusion box and Sample trajectory 36 S+ 38 U z z-α δ=0 z-δ α=0 DQF (Deep Quasi-fission process) within A CN / ±0 E*= 40 MeV Trajectory does not enter L = 0 the fusion box θ = 0 Y. Aritomo and M. Ohta, Nucl. Phys. A 744, 3 (004) z QF QF FF FF? FF DQF

11 Results 36 S+ 38 U MDFF and Cross sections FF process Exp. by K. Nishio et al. at JAEA

12 Exp. by K. Nishio et al. at JAEA Results 30 Si+ 38 U MDFF and Cross sections

13 3. Mechanism of Dynamical process MDFF at Low incident energy 30 Si+ 38 U 36 S+ 38 U 134u 90u 178u 74u 137u 00u Clarification of the mechanism of Fusion-fission process

14 (a) 1-dim Potential energy on scission line 30 Si+ 38 U QF process QF process FF process FF process 36 S+ 38 U α = 0 Nuclear shape at scission point QF -process FF-process

15 (b) Trajectory Analysis on Potential Energy Surface 30 Si+ 38 U Extract the trajectories with A ~ 175 which correspond to the peak of MDFF E* = 35.5 MeV L=0, θ=0 A ~ S+ 38 U E* = 39.5 MeV L=0, θ=0

16 Whole Dynamical Process using ALL trajectories The whole coordinate space is divided into boxes B Whether trajectory enters into a box or not. If Yes mark the box z Δz = 0.10 Δδ =0.06 Δα =0.06 ~40,000 points

17 (c) Trajectory Analysis Probability Distribution 30 Si+ 38 U 36 S+ 38 U E* = 35.5 MeV L=0, θ=0 E* = 39.5 MeV L=0, θ=0

18 Probability distribution on the z-δ plane 39.5 MeV

19 Time evolution of probability distribution z-a plane z-δ plane QF via mono-nucleus t < 30 x < δ < 0.5 FF and DQF t > 50 x < δ < 0.

20 Time evolution of probability distribution E * =39.5 MeV z-a plane E * =39.5 MeV z-δ plane QF t < 0 x < δ < 0. FF and DQF t < 30 x10-1 0< δ < 0.4

21 30 Si + 38 U 36 S + 38 U 134u 90u 178u 74u 137u 00u QF via mono-nucleus t < 30 x 10-1 sec 0. < δ < 0.5 (peak 0.4) FF and DQF t > 50 x 10-1 sec -0. < δ < 0. (peak 0) QF t < 10 x 10-1 sec 0 < δ < 0. (peak 0) FF and DQF t < 30 x 10-1 sec 0 < δ < 0.4 (peak 0.)

22 4. Summary 3. Summary 1. In order to analyze the fusion-fission process in superheavy mass region, we apply the Couple channels method + Langevin calculation.. Incident energy dependence of mass distribution of fission fragments (MDFF) is reproduced in reaction 36 S+ 38 U and 30 Si+ 38 U. 3. The shape of the MDFF is analyzed using (a) 1-dim potential energy surface on the scission line (b) sample trajectory on the potential energy surface (c) probability distribution 4. The relation between the touching point and the ridge line is very important to decide the process fusion hindrance 5. Understanding the dynamics of QF and FF processes will be established more realistic model which can predict the opportunity to form wider range of SHE isotope.

23 計算処方の概観 Model: Outlook of calculation methods Time-evolution of nuclear shape in fusion-fission process 1. Potential energy surface. Trajectory described by equations

24 neck parameter ( e ) Time dependent adiabatic fusion-fission potential 4 Th QF r/r 0 FF V. Zagrebaev, A. Karpov, Y. Aritomo, M. Naumenko and W. Greiner, Phys. Part. Nucl. 38 (007) 469 Time-dependent weight function

25 Two Center Shell Model Hˆ V ( r) VLS( rps) V ( rp) L m 0 z1z r1r z z1 z1z 1 c1z d1z r1 1 g1z r z1 z0 1 V ( rz) m 0 zz 1 cz dz r 1 gz r 0 z z z r z z z r ε = 1.0 V 0 Neck parameter is the ratio of smoothed potential height to the original one where two harmonic oscillator potential cross each other ( a ) ( b ) ( c) V V e= E/E 0 V E 0 z z z1 z 0 z z z E z r r r z 1=z =0 0.0 a 1 a b 1 a 1 z 1 z a a 1 z 1 z a b 1 b 1 b b b z J. Maruhn and W. Greiner, Z. Phys, 197 z max z min 1 r=0.75 R (1+ ) r 0 /3 r

26 Results 34 S+ 38 U MDFF and Cross sections

27 What we can obtain under the conditions Phenomenalism Dynamical Model based on Fluctuation-dissipation theory (Langevin eq, Fokker-Plank eq, etc) Classical trajectory analysis We can obtain. Fission, Synthesis of SHE Mass and TKE distribution of fission fragments A CN : 00~300 Neutron multiplicity Charge distribution Cross section (capture, mass symmetric fission, fusion) Angle of ejected particle, Kinetic energy loss ( two body) Conditions Nuclear shape parameter Potential energy surface (LDM, shell correction energy, LS force) Transport coefficients (friction, inertia mass) Liner Response Theory Dynamical equation (memory effect, Einstein relation)

28 Evaporation residue cross section Total fission Fusion cross-section 34 S + 38 U K.Nishio et al., Phys.Rev.C, 8, (010).

29 Estimation of cross sections under Bass barrier region Ecm=170 E* =58. Ecm=164 E* =5. Taking into account the contributions of all configurations V Bass V Bass Ecm=158 E* =46. Ecm=15 E* =40. Touching probability CC method + After touching Langevin calculation Ecm=148 E* =36. Aritomo, Hagino and Nishio

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31 M.A.Nagarajan, A.B. Balantekin, N. Takigawa, PRC 34, 894 (1986)

32 Trajectory Analysis on Potential Energy Surface 36 S+ 38 U E* = 39.5 MeV L=0, θ=0

33 Key Words Phenomenalism ( 現象論 ) 現象論的理論というのは 現象を説明するためにある仮定のもとに理論を展開するにあたり この仮定が必ずしも厳密に実在認識に相当しなくとも これによって導き出された結果が現象における諸関係をよく説明し得るならば それで一応満足するが如き理論を言う 学現象論 竹山説三著 電磁気

34 Zcn= Si + 38 U 36 S + 38 U Zcn= S + 38 U 74 u 137 u 00 u Exp. by K. Nishio et al. 30 Si + 38 U 90 u 134 u 178 u

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36 Probability distribution 30 Si+ 38 U on the z-a plane δ = 0. ε = 1.0 E* = 35.5 MeV L=0, θ=0 δ = 0. ε = 0.35

37 Probability distribution 36 S+ 38 U on the z-a plane E* = 39.5 MeV L=0, θ=0 δ = 0. ε = 1.0

38 Cross section V Bass Exp. by K. Nishio et al., Phys. Rev. C, 77 (008)

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40 Diabatic and Adiabatic Potential Energy V. Zagrebaev, A. Karpov, Y. Aritomo, M. Naumenko and W. Greiner, Phys. Part. Nucl. 38 (007) 469 Time-dependent weight function G. F. Bertsch, 1978; W. Cassing, W. Nörenberg, A. Diaz-Torres, 004; A. Diaz-Torres and W. Scheid, 005.

41 Group of Theoretical and Computational Physics, FLNR, Russia V.I. Zagrebaev A.V. Karpov A.S. Denikin M.A. Naumenko E.A. Cherepanov Y. Aritomo Frankfurt Institute for Advanced Studies, Germany W. Greiner Universite Libre de Bruxelles, Belgium F. Hanappe Japan Atomic Energy Agency, Japan K. Nishio

42 System of coupled Langevin type Equations of Motion

43 The problem of mass exchange

44 V.I. Zagrebaev and W. Greiner, J. Phys. G. 31 (005) 85

45 t > 10-1 s with mass asymmetry at the entrance channel The trajectory has already felt First, the system feels the diabatic potential the adiabatic potential 9.4 % 7.6 %

46 Langevin type equation Before touching nucleon transfer mij : Hydrodynamical mass (mono-nucleus region), Reduced mass (separated region) γij: Wall and Window (one-body) dissipation

47 neck parameter ( e ) Time dependent adiabatic fusion-fission potential 4 Th r/r 0 V. Zagrebaev, A. Karpov, Y. Aritomo, M. Naumenko and W. Greiner, Phys. Part. Nucl. 38 (007) 469 Time-dependent weight function

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49 Transport coefficients (inertia mass and friction) Inertia Mass (Hydrodynamical mass) Total kinetic energy of system 1 1 T r v d r m ( q) q q 3 m ij i j Werner-Wheeler approximation z r v 0 v re ze r P r A ( z; q) q i P P( z; q) z B ( z; q) q i For an incompressible fluid the total (convective) time derivative of any fluid volume must vanish i i Incompressible fluid Axially symmetric shape K.T.R. Davies, A.J. Sierk, R. Nix, PRC 13 (1976) 385 z r max mij rm P z Ai Aj P Ai Aj dz min max A ( z; q) P ( z ; q) dz i P ( z; q) qi 1 A z q P z q dz z i( ; ) ( ; ) P ( z; q) q zmin i 1 Ai B i( z; q) P z z z P(z;q) z

50 Transport coefficients (inertia mass and friction) Friction (Two body friction) Incompressible fluid Rayleigh dissipation function 1 1 F r d r q q q 3 ( ) ij ( ) i j Total kinetic energy of system 1 1 T r v d r m ( q) q q 3 m ij i j ( r) v ( v ) Euler-Lagrange equation v d L L F dt q q q Constant two-body viscosity coefficient i i i zmax ij 3 z i j i j 1 max A ( z; q) P ( z ; q) dz P ( z; q) qi 1 A z q P z q dz z i( ; ) ( ; ) P ( z; q) q zmin i 1 Ai B i( z; q) P z K.T.R. Davies, A.J. Sierk, R. Nix, PRC 13 (1976) 385 i min 1 P A A P A A dz 8 Two body friction z z

51 Transport coefficients (inertia mass and friction) Friction (One body friction) Rayleigh dissipation function 1 de 1 F ( q) q q dt ij i j Incompressible fluid constant two-body viscosity coefficient 1 1 F r d r q q q 3 ( ) ij ( ) i j Loss of energy to particles inside the mean filed at the rate y de r dt Sv n ds z Wall formula r r (, ) v n r S rs n v nˆ 1 t z i S S qz mass density of nucleus average nucleon speed relative normal velocity of the wall 1 r S rs qirs rs rs qi z 1 ij Euler-Lagrange equation d L L F dt q q q i i i rv z r r 1 r qi q j 4 z max S S S dz r z S min One body friction (Wall formula) A.J. Sierk, R. Nix, PRC 1 (1980) 98 1

52 (b) Trajectory Analysis on Potential Energy Surface z-a plane 30 Si+ 38 U E* = 35.5 MeV L=0, θ=0 36 S+ 38 U E* = 39.5 MeV L=0, θ=0 δ=0., ε=1.0 δ=0., ε=1.0 δ=0., ε=0.35

53 (a) 1-dim Potential energy on scission line neck parameter ( e ) 30 Si+ 38 U 36 S+ 38 U QF r/r 0 FF