Alex Brown UNEDF Feb-22-2008 Strategies for extracting optimal effective Hamiltonians for CI and...

Alex Brown UNEDF Feb-22-2008

Strategies for extracting optimal effective Hamiltonians for CI and Skyrme EDF applications


sd

pf


77 gs BE and 530 excited states, 137 keV rmsB. A. Brown and W. A. Richter, Phys. Rev. C 74, 034315 (2006).

Number of data for each nucleus


3 spe 63 tbme for the sd-shell


Starting Hamiltonian Renormalized NN


Sigma_th = 100 keV


About 5 iterations needed


rms for the 608 levels

rms for the tbme

USDA30

USDB56

Linear combinations of two-body matrix elements


USDA 170 keV rms


USDB 137 keV rms


USDA ground state energy differences

MeV

theory underbound

oxygen beyond N=16 all unbound


USDA 170 keV rms for 608 levels290 keV rms for tbme (4.1% of largest)


USDB 137 keV rms for 608 levels376 keV rms for tbme


USD 150 keV rms for 380 levels450 keV rms for tbme


A few notes

• Need a realistic model for the starting and background Hamiltonians

• What do we use for undetermined linear combinations?

starting Hamiltonian or

Hamiltonian from previous iteration

Not obvious that the same (universal) Hamiltonian should apply to all sd-shell nuclei – probably a special case –

we now know that other situations (like O vs C) require an explicit change in the TBME due to changes coming from core-polarization of difference cores.


TBME depend on the target nucleus and model spaceComparison of 24O (with proton p1/2) and 22C (without p1/2)


USD G

Effective spe for the oxygen isotopes


A tour of the sd shell on the web


Positive parity states for 26Al


Positive parity states for 26Mg


gsp = 5.586 gs

n = -3.826gl

p = 1 gln = 0


gsp = 5.586 gs

n = -3.826gl

p = 1 gln = 0

gsp = 5.586 gs

n = -3.826gl

p = 1 gln = 0


gsp = 5.127 gs

n = -3.543gl

p = 1.147 gln = -0.090


gsp = 5.586 gs

n = -3.826gl

p = 1 gln = 0


gsp = 5.127 gs

n = -3.543gl

p = 1.147 gln = -0.090


sd

pf


jj44 means f5/2, p3/2, p1/2, g 9/2 orbits for protons and neutrons


USDA 170 keV rms for 608 levels290 keV rms for tbme (4.1% of largest)


Why do we need to modify the renormalized G matrix for USD

• Is the renormalization adequate• Difference between HO and finite well• Effective three-body terms• Real three-body interactions


Skyrme parameters based on fits to experimentaldata for properties of spherical nuclei, including single-particle energies, and nuclear matter.

A New Skyrme Interaction for Normal and Exotic Nuclei, B. A. Brown, Phys. Rev. C58, 220 (1998).Displacement Energies with the Skyrme Hartree-Fock Method, B. A. Brown, W. A. Richter and R. Lindsay, Phys. Lett. B483, 49 (2000).Neutron Radii in Nuclei and the Neutron Equation of State, B. A. Brown, Phys. Rev. Lett. 85, 5296 (2000).Charge Densities with the Skyrme Hartree-Fock Method, W. A. Richter and B. A. Brown, Phys. Rev. C67, 034317 (2003).Tensor interaction contributions to single-particle energies, B. A. Brown, T. Duguet, T. Otsuka, D. Abe and T. Suzuki, Phys. Rev. C 74, 061303, (2006).Neutron Skin Deduced from Antiprotonic Atom Data, B. A. Brown, G. Shen, G. C. Hillhouse, J. Meng and A. Trzcinska, Phys. Rev. C76, 034305 (2007).


Data for Skx

• BE for 16O, 24O, 34Si, 40Ca, 48Ca, 48Ni, 68Ni, 88Sr, 100Sn, 132Sn and 208Pb with “errors” ranging from 1.0 MeV for 16O to

0.5 MeV for 208Pb

• rms charge radii for 16O, 40Ca, 48Ca, 88Sr and 208Pb with “errors” ranging from

0.03 fm for 16O to 0.01 fm for 208Pb

• About 50 Single particle energies with “errors” ranging from 2.0 MeV for 16O to 0.5 MeV for 208Pb.

Constraint to FP curve for the neutron EOS


Skx - fit to these data

Fitted parameters:

t0 t1 t2 t3 x0 x1 x2 x3

W Wx (extra spin orbit term)

t0s (isospin symmetry breaking)

Vary α by hand (density dependence) minimum at α = 0.5 (K=270)

t0 t0s t1 t2 t3 x0 and W well determined from exp data

x3 constrained from neutron EOS

Wx x1 and x2 poorly determined


Skx - fit to all of these data

Fit done by 2p calculations for the values V and V+epsilon of the p parameters. Then using Bevington’s routine for a “fit to an arbitrary function”. After one fit, iterate until convergence – 20-50 iterations.

10 nuclei, 8 parameters, so each fit requires 2000-5000 spherical calculations.

Takes about 30 min on the laptop.

Goodness of fit characterized by CHI with best fit obtained for “Skx” with CHI=0.6


Skx - fit to all of these data

Single-particle states from the Skyrme potential of the close-shell nucleus (A) are associated with experimental values for the differences

-[BE(A) - BE(A-1)] or = -[BE(A+1)-BE(A)] based on the HF modelThe potential spe are typically within 200 keV of those calculated from the theoretical values for -[BE(A) - BE(A-1)] or = -[BE(A+1)-BE(A)]

No time-odd type interactions, but time-odd contribution to spe are typically not more than 200 keV (Thomas Duguet)


Displacement energy requires a new parameter


Rms charge radii


Skx Skyrme Interaction


Neutron EOS related to neutron skin -- x3

How can we constrain the neutron equation of state?

• We know the proton density from electron scattering

• The neutron skin is S = R_p – R_n where R are the rms radii


For Skxtbα t = -118, β t = 110

For Skxtaα t = 60, β t = 110

For Skxα t = 0, β t = 0


Skx – fit to single-particle energies


Skx with G matrix tensorCHI jumps up from 0.6 to 1.5 due to spe


normal spin-orbit

tensor terms


0.25

0.20

0.15

S (fm)

K=200 MeV for nuclear matter incompressibility

Phys. Rev. C 76, 034305 (2007).

Skx for charge density diffuseness and neutron skin


122ZrS BE

(fm) (MeV)0.15 -928.60.20 –931.30.25 –934.2


S (fm) = 0.12 0.16


Neutron matter effective mass can constrain x1 and x2


Phys. Rev. C 76, 034305 (2007).


28O

34Si42Si

48Ca

24O


Skxta/b: Skx with the inclusion of the rho+pi tensor in fits to spe, BE and radii,B. A. Brown, T. Duguet, T. Otsuka, D. Abe and T. Suzuki, Phys. Rev. C 74, 061303(R) (2006)..

Z=8



N=16


In 28O the d3/2 is bound by 0.2 MeV



N=20


N=28


114Sn to 115Sb proton spectroscopic factors


32Cl

33Ar

p

32Cl(p,gamma)33Ar

Rp-process path

Experiment needed to get energy of states in 33Ar to 5 keV accuracy. Theory needed to

get proton decay widths to ground and excited states of 32Cl and gamma widths for 33Ar

32P

33P

R. R. C. Clement et al., Phys. Rev. Lett. 92, 172502 (2004)H. Schatz, et al., Phys. Rev. C 72, 065804 (2005)

Role of excited state in other nuclei - Janina Grineviciute


Full pf space for 56Ni with GXPF1A Hamiltonian(order of one day computing time)

M. Horoi, B. A. Brown, T. Otsuka, M. Honma and T. Mizusaki, Phys. Rev. C 73, 061305(R) (2006).


ep=1 en=0


ep=1.37 en=0.45


ep=1.37 en=0.45ep=1.10 en=0.68


ep=1 en=0


ep=1.37 en=0.45


|ga/gv|=1.26


|ga/gv|=0.97


Nuclear Structure Theory - Confrontation and Convergence

• (AI) Ab initio methods with NN and NNN

• (CI) Shell model configuration interactions with effective single-particle and two-body matrix elements

• (DFT) Density functionals plus GCM…

My examples with Skyrme Hartree-Fock (Skx)

• Cluster models, group theoretical models …..

• Good – most “fundamental”

• Bad – only for light nuclei, need NNN parameters, “complicated wf”

• Good – applicable to more nuclei, 150 keV rms, “good wf”

• Bad – limited to specific mass regions and Ex, need effective spe and tbme for good results

• Good – applicable to all nuclei

• Bad – limited mainly to gs and yrast, 600 keV rms mass, need interaction parameters

• Good – simple understanding of special situations

• Bad – certain classes of states, need effective hamiltonian

Each of these has its own computational challenges


USDB ground state energy differences

MeV

theory underbound


PRL98, 102502 (2007)RIKEN

PRL99 1125012 (2007)NSCLTheory has 10 eV width


• Mihai Horoi

Thomas Duguet

• Werner Richter

Taka Otsuka

D. Abe

T. Suzuki

• Funding from the NSF

Collaborations


Monopole interactions


Monopole interaction changes

Alex Brown UNEDF Feb-22-2008 Strategies for extracting optimal effective Hamiltonians for CI and...

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