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Alex Brown PREX Aug-17-2008
Neutron Radii and the Neutron Equation of State
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Alex Brown PREX Aug-17-2008
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
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Alex Brown PREX Aug-17-2008
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
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Alex Brown PREX Aug-17-2008
Skyrme interaction
(σ = α)
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Alex Brown PREX Aug-17-2008
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
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Alex Brown PREX Aug-17-2008
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Alex Brown PREX Aug-17-2008
Skyrme single-particle wave equation
Effective mass m*(r)/m
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Alex Brown PREX Aug-17-2008
Skyrme potential
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Alex Brown PREX Aug-17-2008
Focus on properties of spherical nuclei in a spherical potential model – fast
but limited to properties of a few key nuclei
208Pb
132Sn
100Sn
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Alex Brown PREX Aug-17-2008
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Alex Brown PREX Aug-17-2008
Skx-s15
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Alex Brown PREX Aug-17-2008
Skx-s20
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Alex Brown PREX Aug-17-2008
Skx-s25
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Alex Brown PREX Aug-17-2008
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.
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Alex Brown PREX Aug-17-2008
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
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Alex Brown PREX Aug-17-2008
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)
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Alex Brown PREX Aug-17-2008
Skx Skyrme single-particle energies - implies that (m*/m)=1.00
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Alex Brown PREX Aug-17-2008
Skx Skyrme single-particle energies
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Alex Brown PREX Aug-17-2008
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)
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Alex Brown PREX Aug-17-2008
Nuclear charge densities
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Alex Brown PREX Aug-17-2008
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Alex Brown PREX Aug-17-2008
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Alex Brown PREX Aug-17-2008
Neutron density for 208Pb
Shows the shell layers
(Dashed line is the proton density)
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Alex Brown PREX Aug-17-2008
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Alex Brown PREX Aug-17-2008
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*)
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Alex Brown PREX Aug-17-2008
0.25
0.20
0.15
S (fm)
Phys. Rev. C 76, 034305 (2007).
Ratios of neutron densities (Skm*)
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Alex Brown PREX Aug-17-2008
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
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Alex Brown PREX Aug-17-2008
Assumption about neutron matter effective mass (m*/m)=1.00 used as a fit constraint
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Alex Brown PREX Aug-17-2008
-0.5 to 0.5 -1.0 to 1.0
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Alex Brown PREX Aug-17-2008
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Alex Brown PREX Aug-17-2008
Ab-initio low-density value A. Gezerlis and J. Carlson, PRC77, 032801 (2008) also important to get low-density part right (Andrew Steiner…)
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Alex Brown PREX Aug-17-2008
BE(132Sn)-BE(100Sn)(MeV)
277278282283284291296299
Exp = 278(1)
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Alex Brown PREX Aug-17-2008
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
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Alex Brown PREX Aug-17-2008
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Alex Brown PREX Aug-17-2008
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.
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Alex Brown PREX Aug-17-2008
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Alex Brown PREX Aug-17-2008
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Alex Brown PREX Aug-17-2008
122ZrS BE
(fm) (MeV)0.15 -928.60.20 –931.30.25 –934.2
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Alex Brown PREX Aug-17-2008
S (fm) = 0.12 0.16
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Alex Brown PREX Aug-17-2008
• 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
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Alex Brown PREX Aug-17-2008
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
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Alex Brown PREX Aug-17-2008
• Mihai Horoi
Thomas Duguet
• Werner Richter
Taka Otsuka
D. Abe
T. Suzuki
• Funding from the NSF
Collaborations
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Alex Brown PREX Aug-17-2008
Skx Skyrme Interaction
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Alex Brown PREX Aug-17-2008
Displacement energy requires a new parameter
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Alex Brown PREX Aug-17-2008
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
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Alex Brown PREX Aug-17-2008
Rms charge radii
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Alex Brown PREX Aug-17-2008
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Alex Brown PREX Aug-17-2008
114Sn to 115Sb proton spectroscopic factors
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Alex Brown PREX Aug-17-2008
For Skxtbα t = -118, β t = 110
For Skxtaα t = 60, β t = 110
For Skxα t = 0, β t = 0
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Alex Brown PREX Aug-17-2008
Skx – fit to single-particle energies
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Alex Brown PREX Aug-17-2008
Skx with G matrix tensorCHI jumps up from 0.6 to 1.5 due to spe
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Alex Brown PREX Aug-17-2008
normal spin-orbit
tensor terms
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Alex Brown PREX Aug-17-2008
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