UNEDF UNEDF ProjectProject: : TowardsTowards a a UniversalUniversalNuclearNuclear EnergyEnergy DensityDensity FunctionalFunctional

Atomic nucleus

Piotr MagierskiWarsaw University of Technology/University of Washington

protons

neutrons

Nuclear LandscapeNuclear Landscape

82

50

28

28

50

82

2082

28

20

Stablenuclei

known nuclei

neutron stars

126

r-process

rp-process

superheavynuclei

neutron

drip line

Terraincognita

proton

drip l

ine

What are the basic degrees of freedom of a nuclear system?

It depends on the energy scale we are interested?

gg

g

u d_

g

g

ggg

g

gg Quarks and gluons

QCD energy scale: 1000MeV

p n Baryons and mesonsEnergy scale: 100MeV

NucleonsEnergy scale: 10MeV

Collective degrees offreedom: 0.1-1MeV

Nucleon-nucleon (N-N) interaction is an effective interaction

3central spin tensor spin orbit bodyV V V V V V = + + + +N-N force can be determined (except for the three-body term)

from the proton-proton and proton-neutron scattering experiments withsome guidance coming from QCD (symmetries).

Results of solvingSchroedinger eq.with N-N potential.

Blue only two-bodyterms included.Red two-bodyand three-body terms.Green experiment.

3-body interaction is important!

(S. Pieper, ANL)

Configuration Interaction (CI)For heavier nuclei one may solve (stationary) Schroedinger equation

in a limited many-body basis (configuration space).

Excitation energies of the lowest2+ states. Size of the basis:~109 Size of the basis grows rapidly

with the number of protons and neutrons.

Honma, Otsuka et al., PRC69, 034335 (2004)

Can we calculate the wave function for medium and heavy nuclei?

The radius of a nucleus of mass number A (number of nucleons)is of the order of 1/ 3

0 0, 1.2R r A r fm= In order to make a reliable calculation of the wave function wehave to consider a volume at least of the order of

3 30( 2 ) 8V R r A =

How many points inside the volume V do we need?From the Fermi gas model:

1 / 32 33/ , 0.16 -nuclear saturation density

2F Fp k fm = =

2 F

FFermi wavelengthk

=

Fmax 2

Fx = Maximumdistance between points

maxx

Therefore values of the wavefunction have to be known at least in ( )

30

3

8( )3

max

Fk rV A Ax

= points

But the wave function depends on A variables (disregarding spin):

1 2( , , . . . , )Ar r rTo store the wave function we need to store AA complex numbers.

Not possible now andnever will be!!!For A=100 it means 10

200 complex numbers

Nuclear wave function contains too much informationInstead of wave function one may use a density distribution:

3 3 22 2( , ) ( ) ... | ( , ,..., ) |A Ar r r d r d r r r r = =

TheoremTheorem ((HohenbergHohenberg & & KohnKohn):):The energy of the nondegenerate ground state of

the Fermi system is uniquely determined by its density distribution.

It is sufficient to search for the density functional: [ ( )]E rThe ground state energy is obtained through the requirementthat the functional reaches the minimum value for the ground

state density distribution.

From W. Nazarewicz talk

Building blocks of Nuclear Energy Density Functional

0

r ( )= 0

r ,

r ( )= r ; r ( )

isoscalar (T=0) density 0 = n + p( )

1

r ( )= 1

r ,

r ( )= r ; r ( )

isovector (T=1) density 1 = n p( )

s 1

r ( )= r ; r '( ) ' ' isovector spin density

s 0

r ( )= r ; r '( ) ' ' isoscalar spin density

j T

r ( )= i

2

'

( )T r , r '( ) r '= r

J T

r ( )= i

2

'

( ) s T r , r '( ) r '= r

T

r ( )=

'T

r ,

r '( ) r '= r

T T

r ( )=

'

s T

r ,

r '( ) r '= r

current density

spin-current tensor density

kinetic density

kinetic spin density

Local densitiesand current

In nuclear systems we have to generalize the density functional taking into account also spin and isospin.

HT

r ( )=CTT2 +CTssT2 +CTTT +CTs

s T

s T

+CT TT jT

2( )+CTT s T

T T

J T2( )+CTJ T

J T +

s T

j T( )[ ]

Example: SkyrmeFunctional

Etot =

2

2m0 + H0

r ( )+ H1

r ( )

d3r Total ground-state energy

Pairing field has still to be added...

www.unedf.org

5-year project(Second year just passed)Director: George Bertsch,University of Washington

Goals

Constraints on the densityfunctional from ab-initioapproaches

Equation of state of dilute neutron matter:

Pairing gap in diluteneutron matter:

Ab-initio calculations in medium mass nuclei:

Coupled Cluster (CC) method and Configuration Interaction (CI)method has to agree with each other up to 1% error (in bindingenergy) for the following nuclei: 8He, 16O, 40Ca.

Ab-inition calculations should agree withDensity Functional Theory with respect to:-one body density matrix,-binding energy,-energy as a function of various external fields: monopole, quadrupole, etc.-energy as a function of isospin degree of freedom,-energy as a function of general density perturbations.

Density Functional Theory (DFT)And Applications

Present status:-There is entire ZOO of parametrizations of nuclearDFT.-The role of various terms in DFT is still not wellunderstood.-DFT works well for differences.-Dependence of DFT on pairing fields is poorly known.-Present accuracy of nuclear mass determination isof the order of 700 keV (above 2500 nuclei werecalculated).

Examples

S. Cwiok, P.H. Heenen, W. NazarewiczNature, 433, 705 (2005)

Stoitsov et al., PRL 98, 132502 (2007)

UTK/ORNL (Nazarewicz, Schunck, Stoitsov)MSU (Brown), UW (Bertsch),Texas Commerce (Bertulani)ANL (Mor, Sarich)Warsaw, Jyvskyl (Dobaczewski)

UTK/ORNL (Nazarewicz, Schunck, Stoitsov)UW (Bulgac)ANL (Mor, Norris, Sarich)ORNL (Fann, Shelton, Roche)Warsaw (Dobaczewski, Magierski)UTK (Pei)

UTK/ORNL (Nazarewicz)ANL (Mor, Norris, Sarich)Bruyeres (Goutte)Lublin (Baran, Staszczak)

UNEDF PhysicsUNEDF CS/AMUNEDF Foreign CollaboratorOutside UNEDF

W. Nazarewicz talk

Construction of 3-D DFT SolversThe important part of the project is to develop numerical codes whichsolve with controllable accuracy the stationary nuclear many-bodyproblem using the energy density functional theory.The following requirements have to be met:-No symmetry limitations: any nuclear shape has to be tractable withthe same accuracy.-No assumed time-reversal invariance: odd and even nuclear systemsare treated on the same footing.

-Coordinate representation (DVR basis or adaptive basis): The wavefunction and nuclear densities are represented on the spatial lattice.-Parallelization: The code is supposed to run on the largestcontemporary computer clusters and therefore the computational taskhas to be efficiently split into many processors.

ComputationalComputational issuesissues ((solvedsolved thanksthanks to to PhysicsPhysics--ComputerComputerScienceScience partnershippartnership).).Optimization techniques for nuclear structure DFT codesSolving large-scale systems of nonlinear equationsEvaluation of performance and scalability in DFT calculationsEvaluation of derivative-free methods for noisy, nonlinear problems3-D adaptive multi-resolution method for atomic nuclei (Madness)

Example:Example: ComputationalComputational ComplexityComplexity ofofLarge Scale Mass Table CalculationsLarge Scale Mass Table Calculations

M. Stoitsov; HFB+LN mass table, HFBTHO

EvenEven--Even NucleiEven NucleiThe The SkMSkM* mass table contains 2525 even* mass table contains 2525 even--even nucleieven nucleiA single processor calculates each nucleus 3 times (A single processor calculates each nucleus 3 times (prolateprolate, oblate, , oblate, spherical) and records all nuclear characteristics and candidatespherical) and records all nuclear characteristics and candidates for s for blocked calculations in the neighborsblocked calculations in the neighborsUsing 2,525 processors Using 2,525 processors -- about 4 CPU hours (1 CPU about 4 CPU hours (1 CPU hour/configuration)hour/configuration)

9,210 nuclei9,210 nuclei599,265 configurations599,265 configurationsUsing 3,000 processors Using 3,000 processors -- about 25 CPU hoursabout 25 CPU hours

All NucleiAll Nuclei

Number of processors > number of nuclei!

Jaguar Cray XT4 at ORNL

Dynamic extension of DensityFunctional Theory

See also Ela Ganiolus talk later today

(present status)

Main challenges

Descipription of spontaneousfission process:

Description of shapecoexistence:

What makes us believe we can make a breakthrough?

Solid microscopic foundationlink to ab-initio approacheslimits obeyed (e.g., unitary regime)

Unique opportunities provided by coupling to CS/AM Comprehensive phenomenology probing crucial parts of the

functionaldifferent observables probing different physics

Stringent optimization protocol providing not only the coupling constants but also their uncertainties (theoretical errors)

Unprecedented international effort

Conclusion: we can deliver a well theoretically founded EnergyDensity Functional, of spectroscopic quality, for structure and reactions, based on as much as possible ab initio input at this point in time.

UNEDF Project: Towards a Universal Nuclear Energy Density FunctionalGoals

/ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputCondition () /PDFXRegistryName (http://www.color.org) /PDFXTrapped /Unknown

/Description >>> setdistillerparams> setpagedevice