Microscopic description of fission process within the Nuclear Density Functional Theory

Microscopic description of fission process within the Nuclear Density Functional Theory

Witold Nazarewicz

Topical FUSTIPEN Meeting on "Theory of Nuclear Fission"

GANIL, Caen, January 4-6, 2012

elongation necking

split

N,Z

N₂,Z₂ N₁,Z₁

N=N₁+N₂ Z=Z₁+Z₂

www.phys.utk.edu/witek/fission/fission.html Funded 2003

Fellow: DOE NNSA SSGF

UTK/ORNL Team

Witold Nazarewicz

PhD students: Jordan McDonnell, Nikola Nikolov

Postdocs: Jochen Erler, Junchen Pei Visitors: Andrzej Staszczak, Andrzej Baran, Javid Sheikh, Janusz Skalski, Michal Warda

+ Jacek Dobaczewski, Arthur Kerman Nicolas Schunck, Patrick Talou (NEUP)

UNEDF

Powerful phenomenology exists…

… but no satisfactory microscopic understanding of:

• Barriers

• Fission half-lives and mass/energy splits

• Fission dynamics

• Cross sections

• …

What is needed?

• Expertise in nuclear theory

• Collaborative efforts focused on programmatic needs

• Microscopic many-body techniques

• Computational algorithms

• Hardware

Optimized Functionals Optimized Functionals

Numerical Techniques Numerical Techniques

Large-scale DFT Large-scale DFT

Confrontation with experiment; predictions Confrontation with experiment; predictions

Collective dynamics Collective dynamics

Input

Forces, parameters

Many-body

dynamics Open channels

fission

… garbage in, garbage out…

… garbage in, garbage out…

Physics of fission is demanding

Physics of fission is demanding

Mean-Field Theory Density Functional Theory ⇒

• mean-field one-body densities ⇒

• zero-range local densities ⇒

• finite-range gradient terms ⇒

• particle-hole and pairing channels

• Has been extremely successful.

A broken-symmetry generalized product state does surprisingly good job for nuclei.

Nuclear DFT

• two fermi liquids

• self-bound

• superfluid

Degrees of freedom: nucleonic densities

NN+NNN interactions

NN+NNN interactions

Density Matrix Expansion Density Matrix

Expansion Input

Energy Density Functional Energy Density

Functional

Observables Observables

• Direct comparison with experiment

• Pseudo-data for reactions and astrophysics

Density dependent interactions Density dependent

interactions

Fit-observables

• experiment

• pseudo data Fit-observables

• experiment

• pseudo data

Optimization Optimization

DFT variational principle HF, HFB (self-consistency)

Symmetry breaking DFT variational principle HF, HFB (self-consistency)

Symmetry breaking

Symmetry restoration Multi-reference DFT (GCM) Time dependent DFT (TDHFB)

Symmetry restoration Multi-reference DFT (GCM) Time dependent DFT (TDHFB)

Nuclear Density Functional Theory and Extensions

• two fermi liquids

• self-bound

• superfluid (ph and pp channels)

• self-consistent mean-fields

• broken-symmetry generalized product states

Technology to calculate observables

Global properties Spectroscopy

DFT Solvers Functional form Functional optimization Estimation of theoretical errors

INPUT

Example: Large Scale Mass Table Calculations

 5,000 even-even nuclei, 250,000 HFB runs, 9,060 processors – about 2 CPU hours

 Full mass table: 20,000 nuclei, 12M configurations — full JAGUAR HFB+LN mass table, HFBTHO: embarrassingly parallel problem

Augmented Lagrangian Method for Fission A. Staszczak et al., Eur. Phys. J. A 46, 85 (2010)

The quadratic penalty method (QPM), frequently used in constrained DFT calculations for fission, often fails to deliver the requested average value of the collective operator with an acceptable accuracy. In the Augmented Lagrangian Method (ALM), a linear constraint is applied together with a quadratic

penalty function, and this guarantees that the constrained calculations properly converge. The ALM has been implemented in UNEDF DFT solvers HFODD and HFBTHO solvers

A comparison between the ALM (black squares) and the standard QPM (open squares) for the constrained self-consistent convergence scheme. The DFT calculations were carried out for the total energy surface of ²⁵²Fm in a two-dimensional plane of elongation, Q₂₀, and reflection-asymmetry, Q₃₀. Although QPM often fails to produce a solution at the required values of constrained variables on a rectangular grid, ALM performs very well in all cases.

Two dimensional constrained calculations in a (Q₂₀,Q₃₀) plane for ²⁵²Fm performed with HFODD using the ALM. Compared with the standard quadratic penalty method, one can see interesting physics in the region which was inaccessible by the latter approach, namely the appearance of the second (fusion) valley at large values of Q₃₀ separated from the spontaneous fission valley by a steep ridge.

fission isomer bandhead

Surface symmetry energy and fission of neutron-rich nuclei Surface symmetry energy and fission of neutron-rich nuclei

N. Nikolov et al., Phys. Rev. C 83, 034305 (2011)

fissility parameter

Adiabatic fusion barriers from self-consistent calculations

J. Skalski, Phys. Rev. C76, 044603 (2007)

Center of mass!!!

Center-of-mass correction neglected

Data on fission isomers bandheads included

UNEDF1 functional: focus on heavy nuclei and fission

UNEDF1 functional: focus on heavy nuclei and fission

Overall quality similar as UNEDF0

big improvement for

fission

inner inner

outer

outer

Collective potential V(q)

Universal nuclear energy density functional is yet to be developed

Choice of collective parameters

How to define a barrier?

How to connect valleys?

Dynamical corrections going beyond mean field important

• Center of mass

• Rotational and vibrational (zero-point quantum

correction)

• Particle number

Adiabatic Approaches to Fission

WKB:

multidimensional space of collective parameters collective inertia

(mass parameter)

Several collective coordinates The action has to be minimized

A. Baran et al., Nucl. Phys. A361, 83 (1981)

a

b

a

b

Generalized HFB density matrix is expanded around the quasi-stationary HFB solution up to quadratic terms in the collective momentum

HFB Hamiltonian

Kinetic energy

ATDHFB equation

ATDHFB equation in quasiparticle basis Mass parameter

ATDHFB equation in quasiparticle basis

Cranking approximation:

time-odd interaction matrix is neglected

NEXT:

Examples

Microscopic description of complex nuclear decay: Multimodal fission

A. Staszczak, A.Baran, J. Dobaczewski, W.N., Phys. Rev. C 80, 014309 (2009)

Calculated fission half lives of even-even fermium isotopes, with 242 ≤ A ≤ 260, compared with experimental data.

Summary of fission pathway results obtained in this study. Nuclei around

252Cf are predicted to fission along the asymmetric path aEF; those around

262No along the symmetric pathway sCF. These two regions are separated by the bimodal symmetric fission (sCF + sEF) around ²⁵⁸Fm. In a number of the Rf, Sg, and Hs nuclei, all three fission modes are likely (aEF + sCF + sEF; trimodal fission). In some cases, labeled by two-tone shading with one tone dominant, calculations predict coexistence of two decay scenarios with a preference for one. Typical nuclear shapes corresponding to the calculated nucleon densities are marked.

Two-dimensional total energy surface for ²⁵⁸Fm in the plane of two collective coordinates: elongation, Q₂₀, and necking, Q₄₀.

Two-dimensional total energy surface for ²⁵⁸Fm in the plane of two collective coordinates:

elongation, Q₂₀, and reflection- asymmetry, Q₃₀. Dashed lines show the fission pathways:

symmetric compact fragments (sCFs) and asymmetric elongated fragments (aEFs).

Nuclear shapes are shown as three-dimensional images that correspond to calculated nucleon densities.

Microscopic description of spontaneous fission Microscopic description of spontaneous fission

A. Staszczak et al. Phys. Rev. C 80, 014309 (2009)

- 1 3 6 8 - 1 3 7 6

- 1 3 8 0 - 1 3 6 0 - 1 3 5 6

- 1 3 9 0 - 1 3 9 4

- 1 3 9 6

- 1 3 9 0 - 1 4 0 8

- 1 3 9 4 - 1 3 9 6 - 1 3 8 0

- 1 3 8 0 - 1 3 9 0

- 1 3 7 6 - 1 3 6 8

- 1 3 7 6 - 1 3 6 0

- 1 3 9 6

- 1 3 6 8

- 1 3 9 4

- 5 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0

0 2 0 4 0 6 0 8 0 1 0 0

1 8 0

H g ( S k M * )

Q u a d r u p o l e m o m e n t Q _{2 0} ( b ) Octupole moment Q 30 (b3/2 )

- 1 4 1 0 - 1 4 0 2 - 1 3 9 4 - 1 3 8 6 - 1 3 7 8 - 1 3 7 0 - 1 3 6 2 - 1 3 5 4

- 1 5 2 8

- 1 5 3 0

- 1 5 2 4

- 1 5 1 0

- 1 5 0 4

- 1 5 0 0

- 1 5 3 2

- 1 5 3 2 - 1 5 3 4

- 1 5 3 6

- 1 5 3 4

- 1 5 4 2

- 1 5 3 6

- 1 5 4 2 - 1 5 3 0

- 1 5 2 8

- 1 5 3 0 - 1 5 3 0

- 1 5 2 4 - 1 5 2 4

- 1 5 2 4 - 1 5 2 4

- 1 5 2 4 - 1 5 4 2

- 1 5 3 6 - 1 5 2 8 - 1 5 1 0

- 5 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0

0 2 0 4 0 6 0 8 0 1 0 0

1 9 8

H g ( S k M * )

Q u a d r u p o l e m o m e n t Q _{2 0} ( b ) Octupole moment Q 30 (b3/2 )

- 1 5 7 0 - 1 5 6 0 - 1 5 5 0 - 1 5 4 0 - 1 5 3 0 - 1 5 2 0 - 1 5 1 0 - 1 5 0 0 - 1 4 9 0

s E F a E F

Gogny (M. Warda) SkM* (A. Staszczak)

- 5 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 - 1 5 7 0

- 1 5 6 0 - 1 5 5 0 - 1 5 4 0 - 1 5 3 0 - 1 5 2 0 - 1 5 1 0

a sy m

. fu s io n s y m

. fus ion s E F

a E F

1 9 8

H g

Total energy Etot (MeV)

Q u a d r u p o l e m o m e n t Q _{2 0} ( b )

- 5 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 - 1 4 2 0

- 1 4 1 0 - 1 4 0 0 - 1 3 9 0 - 1 3 8 0 - 1 3 7 0 - 1 3 6 0

s E F 1 a E F

s C F

1 8 0

H g

Total energy Etot (MeV)

Q u a d r u p o l e m o m e n t Q _{2 0} ( b )

s E F

s ym . fus ion

0 10 20

Energy [MeV]

S=0 S(kT=1) S(kT=2) S(kT=3) S(kT=4)

0 100 200 300

Q₂₀ [b]

0 10 20

Energy [MeV]

180Hg

sEF

aEF

SkM*

(J. Sheikh)

Fission barriers of compound nuclei

• J. Pei, J. Sheikh, WN, A. Kerman, Phys. Rev. Lett. 102, 192501 (2009)

• J. Sheikh, WN, J. Pei, Phys. Rev. C 80, 011302(R) (2009)

25 50 75 0

5 10

100 150 200

-30 -20 -10 0

1.5 2.0

Q

(b)

Q

( b

) Q

( b)

0

240 Pu

1.5 2.0

1.0 0

Systematic Study of Fission Barriers of Excited Superheavy Nuclei J. Sheikh, WN, J. Pei, Phys. Rev. C 80, 011302(R) (2009)

Focus on:

• Mirror asymmetry and triaxiality at high

temperatures

• Systematic analysis of barrier damping

J.D. McDonnell et al.

A highlight in 2011 SSAA Magazine!

Potential energy surfaces are calculated for the spontaneous fission of light

actinides with (finite-temperature) HF+BCS theory. The fission path favors

more symmetric scission configurations as excitation energy increases.

Example: Fission theory

Two DOE-sponsored workshops on exascale computing for nuclear physics, one common conclusion:

Two DOE-sponsored workshops on exascale computing for nuclear physics, one common conclusion:

The microscopic description of nuclear fission is one of four Priority Research Directions.

The microscopic description of nuclear fission is one of four Priority Research Directions.

“Understanding nuclear fission at a more comprehensive level is a critical problem in national security with important

implications in nuclear materials detection and nuclear energy, as well as enhancing the quantitative understanding of nuclear weapons”

“The ultimate outcome of the nuclear

fission project is a treatment of many-body dynamics that will have wide impacts in nuclear physics and beyond. The

computational framework developed in the context of fission will be applied to the

variety of phenomena associated with the large amplitude collective motion in nuclei and nuclear matter, molecules,

nanostructures and solids”

Slide 35

Scientific Grand Challenges in National Security:

the Role of Computing at the Extreme Scale

October 6-8, 2009 · Washington, D.C.

Priority Research Direction: Fission Theory

Scientific and computational challenges Summary of research direction

Expected Scientific and Computational Outcomes Potential impact on Nuclear Physics Problems that arise in NNSA/ASC?

Scientific: Tunneling and time-dependent dynamics of complex heavy nuclei.

Computational: Implementation of highly nonlinear, nonlocal, multi-dimensional system of time-dependent integro-differential equations.

Develop efficient solvers for the time-dependent (deterministic and stochastic) nuclear superfluid density functional theory (DFT). Solve the multi-dimensional optimization problem for collective action. Develop algorithms to compute bounce trajectories in imaginary time.

Uncertainty quantification for complex nuclear processes.

Scientific: Microscopic description, based on realistic internucleon interactions, of nuclear fission process.

Explanation of tunneling motion of a many-body system.

Computational: New fast multiresolution methods for the nuclear DFT. Implementation of new Importance Sampling Techniques for the time-dependent stochastic evolution equations.

Validate and predict fission cross sections, fragment mass and energy distributions.

Provide microscopic guidance (extrapolation capability) for phenomenological models of fission.

Microscopic input to materials damage simulations.

Neutron and photon spectra for stockpile simulation validation and nonproliferation.

• Quest for the universal interaction/functional

The major challenge for low-energy nuclear theory

• Many-dimensional problem

Tunneling of the complex system

Coupling between collective and single-particle Time dependence on different scales

• All intrinsic symmetries broken

Large elongations, necking, mass asymmetry, triaxiality, time reversal (odd, odd-odd systems),…

• Correlations important

Pairing makes the LACM more adiabatic. Quantum corrections impact dynamics.

Pre-Conclusions

Fission is a fundamental many-body phenomenon that possess the ultimate challenge for theory

Understanding crucial for many areas:

• Nuclear structure and reactions (superheavies)

• Astrophysics (n-rich fission and fusion, neutrino- induced fission)

• Numerous applications (energy, AFC, Stockpile Stewardship…)

• …

The light in the end of the tunnel: coupling between modern microscopic many-body theory and high-

performance computing

Conclusions