Neutron Radii and the Neutron Equation of State

Alex Brown PREX Aug-17-2008

Neutron Radii and the Neutron Equation of State


Skx-s20(5) Skyrme energy density functional

Skx-s15 Skx-s20 Skx-s25

0.15 0.20 0.25 fm for the 208Pb neutron skin

Neutron skin = S = ΔRnp = Rn – Rp where Rn is the rms radius for neutrons and Rp is the rms radius for protons


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, Skx, Skxc BAB, Phys. Rev. C58, 220 (1998).Displacement Energies with the Skyrme Hartree-Fock Method, BAB, W. A. Richter and R. Lindsay, Phys. Lett. B483, 49 (2000).Neutron Radii in Nuclei and the Neutron Equation of State, BAB, Phys. Rev. Lett. 85, 5296 (2000). S. Typel and BAB, Phys. Rev. C64, 027302 (2001).Charge Densities with the Skyrme Hartree-Fock Method, W. A. Richter and BAB, Phys. Rev. C67, 034317 (2003).Tensor interaction contributions to single-particle energies, BAB, T. Duguet, T. Otsuka, D. Abe and T. Suzuki, Phys. Rev. C 74, 061303, (2006).Neutron Skin Deduced from Antiprotonic Atom Data, BAB, G. Shen, G. C. Hillhouse, J. Meng and A. Trzcinska, Phys. Rev. C76, 034305 (2007).

Skx family of Skyrme functionals

Skx, Skx-ce

Skx-csb

Skx-ta, Skx-tb

Skx-s15, Skx-s20, Skx-s25


Skyrme interaction

(σ = α)


Skyrme energy density functional

Nuclear matter is this without the surface terms1

Nuclear matter (N=Z) depends on the t’sSymmetry energy and neutron matter also depends on the x’s


Skyrme single-particle wave equation

Effective mass m*(r)/m


Skyrme potential


Focus on properties of spherical nuclei in a spherical potential model – fast

but limited to properties of a few key nuclei

208Pb

132Sn

100Sn


Skx-s15


Skx-s20


Skx-s25


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.


1998 - Skx - fit to these data

Fitted parameters:

t0 t1 t2 t3 x0 x1 x2 x3 W (spin orbit term)

t0s (isospin symmetry breaking)

Vary α (power of the density dependence) by hand minimum at α = 0.5 (K=270 nuclear matter incompressibility)

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

x3 depends on neutron EOS

x1 and x2 not determined


Skx – single-particle energies

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 (Dobascewski, Duguet)


Skx Skyrme single-particle energies - implies that (m*/m)=1.00


Skx Skyrme single-particle energies


1998 - Neutron EOS and neutron skin -- x3

How can we constrain the neutron equation of state?

• Friedman-Pandharipanda neutron EOS - Phys. Rev. C33, 335 (1986)


Nuclear charge densities


Neutron density for 208Pb

Shows the shell layers

(Dashed line is the proton density)


Diffuseness of the charge density iscorrelated with nuclear matter

incompressibility

Best fit to charge density requiresK=200-230 MeV

Skx-s20(5) takes α = 1/6Which gives K=200

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

Ratios of charge densities (Skm*)


0.25

0.20

0.15

S (fm)

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

Ratios of neutron densities (Skm*)


0.25

0.20

0.15

S (fm)

K=200 MeV for nuclear matter incompressibility α = 1/6

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

Skx for charge density diffuseness and neutron skin


Assumption about neutron matter effective mass (m*/m)=1.00 used as a fit constraint


-0.5 to 0.5 -1.0 to 1.0


Ab-initio low-density value A. Gezerlis and J. Carlson, PRC77, 032801 (2008) also important to get low-density part right (Andrew Steiner…)


BE(132Sn)-BE(100Sn)(MeV)

277278282283284291296299

Exp = 278(1)


K=270 α =1/2S=0.25

K=200α = 1/6S=0.20

So next step would be to introduce two α valuesOne for nuclear matter and another for the symmetry potential


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.


122ZrS BE

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


S (fm) = 0.12 0.16


• Participating Institutions and Co-Investigators:Ames National Laboratory - SosonkinaANL - Pieper, Wiringa, Lusk, Moré, NorrisLawrence Berkeley National Laboratory - Ng, Yang

LLNL - Escher, Navratil, Ormand, ThompsonLos Alamos National Laboratory - Carlson, KawanoORNL - Arbanas, Dean, Nazarewicz, Fann, Roche, SheltonCentral Michigan University - HoroiIowa State University - VaryMichigan State University - Brown, BognerUniversity of North Carolina at Chapel Hill - EngelOhio State University - FurnstahlSan Diego State University - JohnsonUniversity of Tennessee - Bertulani, PapenbrockUniversity of Washington - Bertsch, Bulgac

• Funding Partners: Office of Science, Advanced Scientific Computing Research, and National Nuclear Security Agency

SciDAC -Building a universal energy density functional


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


• Mihai Horoi

Thomas Duguet

• Werner Richter

Taka Otsuka

D. Abe

T. Suzuki

• Funding from the NSF

Collaborations


Skx Skyrme Interaction


Displacement energy requires a new parameter


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


Rms charge radii


114Sn to 115Sb proton spectroscopic factors


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


A α β δ γ ε η θ κ λ μ ν π ρ σ τ υ φ χ ψ ω

A Ώ Γ Δ Λ Π Σ Φ Ψ Ω

Neutron Radii and the Neutron Equation of State

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