Periodically-driven quantum systems: Effective Hamiltonians and ...
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![Page 1: Alex Brown UNEDF Feb-22-2008 Strategies for extracting optimal effective Hamiltonians for CI and Skyrme EDF applications.](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ede5503460f94bee8a5/html5/thumbnails/1.jpg)
Alex Brown UNEDF Feb-22-2008
Strategies for extracting optimal effective Hamiltonians for CI and Skyrme EDF applications
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Alex Brown UNEDF Feb-22-2008
sd
pf
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Alex Brown UNEDF Feb-22-2008
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
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Alex Brown UNEDF Feb-22-2008
3 spe 63 tbme for the sd-shell
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Alex Brown UNEDF Feb-22-2008
Starting Hamiltonian Renormalized NN
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Alex Brown UNEDF Feb-22-2008
Sigma_th = 100 keV
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Alex Brown UNEDF Feb-22-2008
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Alex Brown UNEDF Feb-22-2008
About 5 iterations needed
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Alex Brown UNEDF Feb-22-2008
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Alex Brown UNEDF Feb-22-2008
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Alex Brown UNEDF Feb-22-2008
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Alex Brown UNEDF Feb-22-2008
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Alex Brown UNEDF Feb-22-2008
rms for the 608 levels
rms for the tbme
USDA30
USDB56
Linear combinations of two-body matrix elements
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Alex Brown UNEDF Feb-22-2008
USDA 170 keV rms
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Alex Brown UNEDF Feb-22-2008
USDB 137 keV rms
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Alex Brown UNEDF Feb-22-2008
USDA ground state energy differences
MeV
theory underbound
oxygen beyond N=16 all unbound
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Alex Brown UNEDF Feb-22-2008
USDA 170 keV rms for 608 levels290 keV rms for tbme (4.1% of largest)
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Alex Brown UNEDF Feb-22-2008
USDB 137 keV rms for 608 levels376 keV rms for tbme
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Alex Brown UNEDF Feb-22-2008
USD 150 keV rms for 380 levels450 keV rms for tbme
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Alex Brown UNEDF Feb-22-2008
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.
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Alex Brown UNEDF Feb-22-2008
TBME depend on the target nucleus and model spaceComparison of 24O (with proton p1/2) and 22C (without p1/2)
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Alex Brown UNEDF Feb-22-2008
USD G
Effective spe for the oxygen isotopes
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Alex Brown UNEDF Feb-22-2008
A tour of the sd shell on the web
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Alex Brown UNEDF Feb-22-2008
Positive parity states for 26Al
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Alex Brown UNEDF Feb-22-2008
Positive parity states for 26Mg
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Alex Brown UNEDF Feb-22-2008
gsp = 5.586 gs
n = -3.826gl
p = 1 gln = 0
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Alex Brown UNEDF Feb-22-2008
gsp = 5.586 gs
n = -3.826gl
p = 1 gln = 0
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Alex Brown UNEDF Feb-22-2008
gsp = 5.586 gs
n = -3.826gl
p = 1 gln = 0
gsp = 5.586 gs
n = -3.826gl
p = 1 gln = 0
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Alex Brown UNEDF Feb-22-2008
gsp = 5.127 gs
n = -3.543gl
p = 1.147 gln = -0.090
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Alex Brown UNEDF Feb-22-2008
gsp = 5.586 gs
n = -3.826gl
p = 1 gln = 0
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Alex Brown UNEDF Feb-22-2008
gsp = 5.127 gs
n = -3.543gl
p = 1.147 gln = -0.090
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Alex Brown UNEDF Feb-22-2008
sd
pf
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Alex Brown UNEDF Feb-22-2008
jj44 means f5/2, p3/2, p1/2, g 9/2 orbits for protons and neutrons
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Alex Brown UNEDF Feb-22-2008
USDA 170 keV rms for 608 levels290 keV rms for tbme (4.1% of largest)
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Alex Brown UNEDF Feb-22-2008
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
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Alex Brown UNEDF Feb-22-2008
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, 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).
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Alex Brown UNEDF Feb-22-2008
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.
Constraint to FP curve for the neutron EOS
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Alex Brown UNEDF Feb-22-2008
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
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Alex Brown UNEDF Feb-22-2008
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 UNEDF Feb-22-2008
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)
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Alex Brown UNEDF Feb-22-2008
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Alex Brown UNEDF Feb-22-2008
Displacement energy requires a new parameter
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Alex Brown UNEDF Feb-22-2008
Rms charge radii
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Alex Brown UNEDF Feb-22-2008
Skx Skyrme Interaction
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Alex Brown UNEDF Feb-22-2008
Skx Skyrme Interaction
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Alex Brown UNEDF Feb-22-2008
Skx Skyrme Interaction
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Alex Brown UNEDF Feb-22-2008
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
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Alex Brown UNEDF Feb-22-2008
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Alex Brown UNEDF Feb-22-2008
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 UNEDF Feb-22-2008
Skx – fit to single-particle energies
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Alex Brown UNEDF Feb-22-2008
Skx with G matrix tensorCHI jumps up from 0.6 to 1.5 due to spe
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Alex Brown UNEDF Feb-22-2008
normal spin-orbit
tensor terms
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Alex Brown UNEDF Feb-22-2008
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Alex Brown UNEDF Feb-22-2008
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
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Alex Brown UNEDF Feb-22-2008
122ZrS BE
(fm) (MeV)0.15 -928.60.20 –931.30.25 –934.2
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Alex Brown UNEDF Feb-22-2008
S (fm) = 0.12 0.16
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Alex Brown UNEDF Feb-22-2008
Neutron matter effective mass can constrain x1 and x2
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Alex Brown UNEDF Feb-22-2008
Phys. Rev. C 76, 034305 (2007).
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Alex Brown UNEDF Feb-22-2008
28O
34Si42Si
48Ca
24O
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Alex Brown UNEDF Feb-22-2008
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
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Alex Brown UNEDF Feb-22-2008
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
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Alex Brown UNEDF Feb-22-2008
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)..
N=16
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Alex Brown UNEDF Feb-22-2008
In 28O the d3/2 is bound by 0.2 MeV
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Alex Brown UNEDF Feb-22-2008
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)..
N=20
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N=20
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N=28
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N=28
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Alex Brown UNEDF Feb-22-2008
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Alex Brown UNEDF Feb-22-2008
114Sn to 115Sb proton spectroscopic factors
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Alex Brown UNEDF Feb-22-2008
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
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Alex Brown UNEDF Feb-22-2008
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).
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Alex Brown UNEDF Feb-22-2008
ep=1 en=0
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Alex Brown UNEDF Feb-22-2008
ep=1.37 en=0.45
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ep=1.37 en=0.45ep=1.10 en=0.68
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ep=1 en=0
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Alex Brown UNEDF Feb-22-2008
ep=1.37 en=0.45
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Alex Brown UNEDF Feb-22-2008
|ga/gv|=1.26
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Alex Brown UNEDF Feb-22-2008
|ga/gv|=0.97
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Alex Brown UNEDF Feb-22-2008
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 UNEDF Feb-22-2008
USDB ground state energy differences
MeV
theory underbound
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Alex Brown UNEDF Feb-22-2008
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Alex Brown UNEDF Feb-22-2008
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Alex Brown UNEDF Feb-22-2008
PRL98, 102502 (2007)RIKEN
PRL99 1125012 (2007)NSCLTheory has 10 eV width
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Alex Brown UNEDF Feb-22-2008
• Mihai Horoi
Thomas Duguet
• Werner Richter
Taka Otsuka
D. Abe
T. Suzuki
• Funding from the NSF
Collaborations
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Alex Brown UNEDF Feb-22-2008
Monopole interactions
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Alex Brown UNEDF Feb-22-2008
Monopole interaction changes