Density functional theory of spontaneous fission life-times

Transcription

1 Density functional theory of spontaneous fission life-times Jhilam Sadhukhan University of Tennessee, Knoxville & Oak Ridge National Laboratory

2 Fission N,Z Microscopic understanding elongation necking split N=N 1 +N 2 Z=Z 1 +Z 2 Experimental results N 1,Z 1 N 2,Z 2

3 Quality Input ph channel: Skyrme functionals SkM* parametrization pp channel: mixed pairing interaction Fission: our strategy Stability of the heaviest nuclei, r-process, advanced fuel cycle Large-scale Simulations on Leadership-class Computers Dynamics Collective potential and inertia in 2D (Q 20,Q 22 ) for 264 Fm ( 264 Fm undergoes symmetric fission) Numerical Techniques Symmetry unrestricted DFT solver: HFODD (v2.49t) N. Schunck et al. Comp. Phys. Comm. 183, 166 (2012) HFB for potential ATDHFB for cranking inertia Action minimization technique Confrontation with experiment; predictions PRC 80, (2009) Spontaneous fission pathways & T 1/2 3

4 Plan Hetree-Fock-Bogoliubov (HFB) method Potential Energy Surface (PES) Adiabetic Time Dependent HFB formalism Collective Inertia ( ) Action minimization techniques Dynamic Programing Method (DPM) Ritz Method (RM) PES DPM RM SF path & T 1/2 Results: Spontaneous Fission (SF) paths and Half-lives (T 1/2 ) 4

5 HFB equation HFB formalism P. Ring and P. Schuck, The Nuclear Many-Body Problem (Springer-Verlag, Berlin, 1980) Generalized density Single particle Routhian Pairing potential Non-linear eigenvalue equation Chemical potential Diagonal matrix of quasiparticle energies Particle density Pairing density Paring form factor Calculation can be constraint at a particular value of quadrupole moment <Q> Converged solutions Á 0 & º 0 are achieved after solving HFB equation iteratively 5

6 Calculated potential energy Skyrme functional with SkM * parametrization optimized for fission barrier J. Bartel et al. Nucl. Phys. A 386, 79 (1982) J. Dobaczewski et al. Eur. Phys. J. A, 15, 21 (2002) 264 Fm Adjusted to reproduce the n & p pairing gaps of 252 Fm A. Staszczak et al. Phys. Rev. C 87, (2013) Vibrational Zero Point Energy (GOA) A. Staszczak et al. Nucl. Phys. A 504, 589 (1989) 6

7 Introducing dynamics :- ATDHFB formalism TDHFB talk by J. Dobaczewski Adiabatic approximation :- (dynamics is quasi-stationary) Expansion in powers of collective momentum χ time-even odd ATDHFB M. Baranger M. Veneroni, Ann. Phys. 114, 123 (1978) even Collective Inertia:- J. Dobaczewski J. Skalski, Nucl. Phys. A 369, 123 (1981) Comparing with the classical expression of KE A. Baran et al. Phys. Rev. C 84, (2011) 7

8 After a few steps:- Calculated Inertia A. Baran et al. Phys. Rev. C 84, (2011) Unit = /MeVb 2 Derivatives of densities can be calculated using Lagrange three-point formula (need to know time-odd densities to calculate E 1 ) Cranking Approximation:- more simplified way (perturbative approximation) :- Derivatives calculated approximately in terms of collective variables Widely used to calculate fission lifetimes 8

9 Understanding calculated Inertia Invariant under rotation in 2D variations almost disappeared Sharp variations variations similar to MC variations similar to MC Unit = /MeVb2 Large fluctuations of mass parameters are manifestations of crossings of single-particle levels near the Fermi energy 9

10 Numerical test s describes the path on 2D surface M Step 1: Step 2: Densities are rotated by Proper Eulerian angles Step 3: 10

11 Spontaneous fission half-life A. Baran, Phys. Lett. B 76, 8 (1978) n is the number of assaults on the fission barrier per unit time s -1 Penetration probability (WKB) Action integral along the fission path L(s) Q 22 s 1 L(s) s 2 Q 20 Most probable fission path = Minimum action path 11

12 Action minimization techniques A. Baran et al. Nucl. Phys. A 361, 83 (1981) Ritz method (RM) Two numerical methods Dynamic programing method (DPM) Ritz method (RM):- y s 2 y 2 Q 22 y 1 s 1 L(s) x 1 Q 20 x 2 x For the present calculation a 1,a 2 and a 3 are sufficient 12

13 Action minimization techniques A. Baran et al. Nucl. Phys. A 361, 83 (1981) Dynamic programing method (DPM) :- Select s 1 & s 2 Q 22 s 1 Surface between s 1 & s 2 is divided into 2D mesh S is calculated between s 1 & each point in 1 st column S is calculated for each point s 2 in 2 nd column with all points in 1 st column & Minimum action path Q 20 is retained Repeated for all points in column 2:- minimum action paths up to column 2 Repeated for all columns Finally we get the minimum action path between s 1 & s 2 13

14 Results(existing) R. A. Gherghescu et. al. Nucl. Phys. A 651, 237 (1999) Macroscopic-microscopic calculation 14

15 Results(existing) Microscopic HFB calculation With Perturbative-cranking inertia 15

16 Results(present calculation) E 0 = 1.0 MeV Static path (minimum potential path) Dynamic path with cont. M (DPM) Dynamic path with M C (DPM & RM) Dynamic path with M Cp (DPM & RM) Dynamical effects due to action minimization is not very prominent With M C :- dynamics is favoring triaxial saddle, similar to static path With M Cp :- Strong dynamical effects, triaxiality becomes unimportant 16

17 Results(present calculation) Orders of magnitude difference in T 1/2 calculated with M C and M Cp 17

18 Summary & conclusion Spontaneous fission lifetimes have been studied within a dynamic approach based on the minimization of the collective action in a two-dimensional collective space of elongation and triaxiality. A strong dynamical effect has been predicted. Although it offsets the static reduction of the inner barrier by triaxiality when the approximate perturbative cranking inertia is used, the strong effect of triaxiality is observed with the more appropriate non-perturbative cranking inertia. A more detailed study of dynamical effects due to triaxial and refection asymmetric degrees of freedom is in progress. 18

19 Collaborators: W. Nazarewicz J. Dobaczewski A. Baran K. Mazurek J. A. Sheikh 19