Upload
others
View
6
Download
0
Embed Size (px)
Citation preview
New Horizons for the No-Core Shell Model
Robert Roth
Robert Roth - TU Darmstadt - February 2018
No-Core Shell Model & Friends
2
No-Core Shell Model
In-Medium Similarity Renormalization Group
Many-Body Perturbation Theory
• solution of matrix eigenvalue problem in truncated many-body model space
• universality: all nuclei and all bound-state observables on the same footing
• but: limited by model-space convergence
• power-series expansion of energies and states
• simplicity: low-order contributions can be evaluated very easily and efficiently
• but: order-by-order convergence problematic
• decoupling ground-state from excitations through unitary transformation via flow equation
• efficiency: favorable scaling gives access to medium-mass nuclei
• but: limited to ground-state observables
Robert Roth - TU Darmstadt - February 2018
No-Core Shell Model & Friends
§ complementarity of advantages and limitations of the different methods
§ combine NCSM with other methods to overcome limitations
§ expand reach in terms of observables, particle number or model-space size
§ target: spectroscopy of fully open-shell medium-mass nuclei
3
No-Core Shell Model
In-Medium Similarity Renormalization Group
Many-Body Perturbation Theory
Robert Roth - TU Darmstadt - February 2018 4
Hybrid NCSM Methods
IM-SRG decoupling
MBPTbasis optimization
NCSMmany-body solution
NCSMreference state
MBPTconvergence booster
NCSMmany-body solution
NCSMmany-body solution
NAT-NCSM IM-NCSMNCSM-PTNCSM
ground state
NCSMstrength distribution
SF-NCSM
Natural-Orbital NCSM
Robert Roth - TU Darmstadt - February 2018
J. Müller, A. Tichai, K. Vobig, R. Roth, in prep.
6
Natural-Orbital NCSM
NCSMmany-body solution
MBPTbasis optimization
• construct HF basis in large single-particle space
• compute perturbative corrections to one-body density matrix up to second order
• determine natural orbitals from one-body density matrix and transform matrix elements
• NCSM calculation with natural-orbital basis
• use importance truncation for large spaces and heavier nuclei (optional)
• use normal-order two-body approximation to include 3N interactions (optional)
cf. work of Ch. Constantinou, M. A. Caprio, J. P. Vary, P. Maris on construction of natural-orbital basis from NCSM solutions
Robert Roth - TU Darmstadt - February 2018
Natural Orbitals from MBPT
§ perform constrained spherical Hartree-Fock calculation to obtain unperturbed single-particle basis and ground state
§ compute MBPT corrections to HF ground state up to second order
7
§ write density-matrix corrections in terms of single-particle summations, evaluation only takes minutes…
§ solve eigenvalue problem of one-body density matrix, eigenvectors define expansion coefficients of natural-orbital single-particle states
§ transform all input matrix elements to natural-orbital basis
J. Müller, A. Tichai, K. Vobig, R. Roth, in prep.
§ evaluate one-body density matrix with perturbed state up to second order
<latexit sha1_base64="HtZmTYnn+p0V8TSZAqs1erp+7tY=">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</latexit>
<latexit sha1_base64="nyH0PbSgZwW2g8jF8RFMJ4Stgxc=">AAACIXicbY/PSwJBHMVn7Zdtv7Y6dhmSwChkV4LqEAhBeOhg4KagJrPjqJM7O8vMKMqy/01/Qf9A907RqehUf0m7tlJqD2Z4fN77Hp7ju1Qq03zXUguLS8sr6VV9bX1jc8vY3rmVvC8wsTF3uag6SBKXesRWVLmk6guCmOOSitO7jPPKgAhJuVdWI580GOp4tE0xUhFqGtd10eV3QbZUPgybAb0P4QVMUPFqgo5gfgLN/C9MkGUlSG8aGTNnjgXnjZWYDEhUahpP9RbHfUY8hV0kZc0yfdUIkFAUuyTU631JfIR7qENqkfUQI7IRjFeH8CAiLdjmInqegmP69yJATMoRc6ImQ6orZ7MY/p8JMgjhFBpKl7ZIXD0exj9DWPBQjxdbs/vmjZ3Pneesm5NM4SyZngZ7YB9kgQVOQQEUQQnYAINH8AY+wZf2oD1rL9rrTzWlJTe7YEraxzfv1aFr</latexit>
<latexit sha1_base64="DtZUz6mqYjY7maLgbIySSpJn4u4=">AAACF3icbY/LSsNAGIUn9VbjLerSzWARKkpJiqAuhKIgXUZobaGpZTKd1qEzSZiZlpaQF/EJfAtdiSvBlfo2JjGLXjwww+E7518cN2BUKtP80XJLyyura/l1fWNza3vH2N27l/5QYFLHPvNF00WSMOqRuqKKkWYgCOIuIw13cJPkjRERkvpeTU0C0uao79EexUjFqGNcOwOiQseW9CEs2rXjKIJXMGXV2wiewOnYStJZVI5RxyiYJTMVXDRWZgogk90xXpyuj4eceAozJGXLMgPVDpFQFDMS6c5QkgDhAeqTVmw9xIlsh+nWCB7FpAt7voifp2BKpy9CxKWccDducqQe5XyWwP8zQUYRnEFjyWiXJNXTcfJzhIUf6Xq82Jrft2jq5dJlybo7K1Qusul5cAAOQRFY4BxUQBXYoA4weAYf4At8a0/aq/amvf9Vc1p2sw9mpH3+AsdEnl8=</latexit>
ℏΩ [MeV]
E[MeV
]
14 16 18 20 22 24 26 28
-170
-165
-160
-155
-150
-145
-140
Natural Orbitals
NN
-onl
y, α
=0.
08 fm
4 , e
max
=12
ℏΩ [MeV]
E[MeV
]
14 16 18 20 22 24 26 28
-170
-165
-160
-155
-150
-145
-140
Robert Roth - TU Darmstadt - February 2018
J. Müller, A. Tichai, K. Vobig, R. Roth, in prep.
8
NCSM Convergence: Energies
§ MBPT natural-orbital basis eliminates frequency dependenceand accelerates convergence of NCSM
Harmonic Oscillator
16ONmax=2
12
4
6
Hartree-Fock
ℏΩ [MeV]
E[MeV
]
14 16 18 20 22 24 26 28
-170
-165
-160
-155
-150
-145
-140
ℏΩ [MeV]
E[MeV
]
12 14 16 18 20 22 24 26 28
-130
-120
-110
-100
-90
ℏΩ [MeV]
E[MeV
]
12 14 16 18 20 22 24 26 28
-130
-120
-110
-100
-90
Robert Roth - TU Darmstadt - February 2018
J. Müller, A. Tichai, K. Vobig, R. Roth, in prep.
9
NCSM Convergence: Energies
Natural Orbitals
NN
+3N
(400
), α
=0.
08 fm
4 , e
max
=12
§ MBPT natural-orbital basis eliminates frequency dependenceand accelerates convergence of NCSM
Harmonic Oscillator
Nmax=2
10
4
6
Hartree-Fock
ℏΩ [MeV]
E[MeV
]
12 14 16 18 20 22 24 26 28
-130
-120
-110
-100
-90
8
16O
ℏΩ [MeV]
E[MeV
]
12 14 16 18 20 22 24 26 28
-130
-120
-110
-100
-90
ℏΩ [MeV]
E[MeV
]
12 14 16 18 20 22 24 26 28
-130
-120
-110
-100
-90
Robert Roth - TU Darmstadt - February 2018
J. Müller, A. Tichai, K. Vobig, R. Roth, in prep.
10
NCSM Convergence: Energies
Natural Orbitals
NN
+3N
(400
), α
=0.
08 fm
4 , e
max
=12
§ MBPT natural-orbital basis eliminates frequency dependenceand accelerates convergence of NCSM
Harmonic Oscillator
Nmax=2
10
4
6
Hartree-Fock
ℏΩ [MeV]
E[MeV
]
12 14 16 18 20 22 24 26 28
-130
-120
-110
-100
-90
8
16O ●◆ ... explicit 3N○◇△... NO2B
ℏΩ [MeV]
Rp,rms
[fm]
12 14 16 18 20 22 24 26 28
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
ℏΩ [MeV]
Rp,rms
[fm]
12 14 16 18 20 22 24 26 28
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
Robert Roth - TU Darmstadt - February 2018
J. Müller, A. Tichai, K. Vobig, R. Roth, in prep.
11
NCSM Convergence: Radii
Natural Orbitals
NN
+3N
(400
), α
=0.
08 fm
4 , e
max
=12
§ MBPT natural-orbital basis eliminates frequency dependenceand accelerates convergence of NCSM
Harmonic Oscillator
Nmax=2
10
4
6
Hartree-Fock
ℏΩ [MeV]
Rp,rms
[fm]
12 14 16 18 20 22 24 26 28
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
16O ●◆ ... explicit 3N○◇△... NO2B
ℏΩ [MeV]
Rp,rms
[fm]
12 14 16 18 20 22 24 26 28
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
ℏΩ [MeV]
Rp,rms
[fm]
12 14 16 18 20 22 24 26 28
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
Robert Roth - TU Darmstadt - February 2018
J. Müller, A. Tichai, K. Vobig, R. Roth, in prep.
12
NCSM Convergence: Radii
Natural Orbitals
NN
+3N
(400
), α
=0.
08 fm
4 , e
max
=12
§ MBPT natural-orbital basis eliminates frequency dependenceand accelerates convergence of NCSM
Harmonic Oscillator
Nmax=2
10
4
6
Hartree-Fock
ℏΩ [MeV]
Rp,rms
[fm]
12 14 16 18 20 22 24 26 28
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
16O ●◆ ... explicit 3N○◇△... NO2B
Nmax
E x(1+)
[MeV
]
2 4 6 8 100
2
4
6
8
10
Nmax
E[MeV
]
2 4 6 8 10
-100
-95
-90
-85
-80
Robert Roth - TU Darmstadt - February 2018
J. Müller, A. Tichai, K. Vobig, R. Roth, in prep.
13
NCSM Convergence: Spectroscopy
NN+3N(500), α=0.08 fm4, emax=12
Nmax
E x(2+)
[MeV
]
2 4 6 8 100
2
4
6
8
10
Nmax
Q(2+)
[efm2]
2 4 6 8 100
1
2
3
4
5
6
Nmax
B(E2,2+
→0+
)[e2fm4]
2 4 6 8 100
1
2
3
4
5
6
Nmax
B(M1,1+
→0+
)[μN2]
2 4 6 8 100
0.01
0.02
12C
ground-state energy
2+ excitation energy 1+ excitation
energy
2+ quadrupolemoment
E2 transitionstrength
M1 transitionstrength
ħΩ
⦁ 16 MeV◆ 20 MeV
24 MeV
Robert Roth - TU Darmstadt - February 2018
J. Müller, A. Tichai, K. Vobig, R. Roth, in prep.
14
Oxygen Isotopes
Nmax
E[MeV
]
2 4 6 8 10 12 14-170
-160
-150
-140
-130
-120
-110
-100
-90
-80
14O
15O
16O
...
24O
§ excellent convergence with natural-orbital basis for all oxygen isotopes
chiral NN+3N Λ3N=400 MeV α=0.08 fm4
ħΩ=20 MeV emax=12
Robert Roth - TU Darmstadt - February 2018
J. Müller, A. Tichai, K. Vobig, R. Roth, in prep.
15
Oxygen Isotopes
§ excellent convergence with natural-orbital basis for all oxygen isotopes
§ very good agreement with experimental systematics and dripline
§ NO2B instead of explicit 3N causes ~1% overbinding
A
E[MeV
]
14 16 18 20 22 24 26
-170
-160
-150
-140
-130
-120
-110
-100
-90
AO
chiral NN+3N Λ3N=400 MeV α=0.08 fm4
ħΩ=20 MeV emax=12
⦁ NAT-NCSM, explicit 3N
⦁ NAT-NCSM, NO2B
Perturbatively Improved NCSM
• multi-configurational MBPT at second order for individual unperturbed states
• capture couplings in huge model-space through perturbative corrections
Robert Roth - TU Darmstadt - February 2018
Perturbatively Improved NCSM
17
• eigenstates from NCSM at small Nmax as unperturbed states
• access to all open-shell nuclei and systematically improvable
NCSMmany-body solution
MBPTconvergence booster
Tichai, Gebrerufael, Vobig, Roth; arXiv:1703.05664
Robert Roth - TU Darmstadt - February 2018
Tichai, Gebrerufael, Vobig, Roth; arXiv:1703.05664
Multi-Configurational Perturbation Theory
§ prior NCSM calculation: reference or unperturbed state is superposition of Slater determinants from reference space
18
§ define partitioning and unperturbed Hamiltonian
§ evaluate second-order correction to the energy at many-body level
§ reformulation in terms of single-particle summations gives access to very large model spaces
|�refi =X
�2Mref
C� |��i
E(2) = �X
� /2Mref
|h�� |H |�refi|2
�� � �ref
H0 = �ref |�refih�ref| +X
� /2Mref
�� |��ih�� |
Robert Roth - TU Darmstadt - February 2018
Tichai, Gebrerufael, Vobig, Roth; arXiv:1703.05664
19
Oxygen Isotopes
A
E[MeV
]
14 16 18 20 22 24 26
-170
-160
-150
-140
-130
-120
-110
-100
-90
AO
chiral NN+3N Λ3N=400 MeV α=0.08 fm4
ħΩ=20 MeV emax=12
HF basis Nmax=2 reference
Computational CostNAT-NCSM, expl. 3N 20000 CPU-hNAT-NCSM, NO2B 16000 CPU-hNCSM-PT 1000 CPU-h
§ excellent agreement with direct NCSM
§ all isotopes are accessible due simple m-scheme implementation
NCSM-PT
⦁ NAT-NCSM, explicit 3N
⦁ NAT-NCSM, NO2B
Robert Roth - TU Darmstadt - February 2018
Tichai, et al.; in prep.
20
Oxygen Isotopes: Excited 2+ States
A
E*(2+)
[MeV
]
14 16 18 20 22 24 260
2
4
6
8
10
AO§ all methods can treat
excited states natively
§ example: first 2+ states in even oxygen isotopes
§ excellent agreement among methods except for closed (sub-)shells 22O, 24O…
chiral NN+3N Λ3N=400 MeV α=0.08 fm4
ħΩ=20 MeV emax=12
HF basis Nmax=2 reference
NCSM-PT
⦁ NAT-NCSM, explicit 3N
⦁ NAT-NCSM, NO2B
Robert Roth - TU Darmstadt - February 2018
Tichai, Gebrerufael, Vobig, Roth; arXiv:1703.05664
21
Exploring sd-Shell PhenomenaE
[MeV
]
-170
-160
-150
-140
-130
-120
A
E[MeV
]
16 18 20 22 24 26 28 30 32-200-190-180-170-160-150-140-130-120
w/o chiral 3Nwith chiral 3N
§ exploring sd-shell phenomena, e.g., oxygen anomaly
§ low computational cost enables surveys with different interactions
AO
AF
NCSM-PT chiral NN+3N Λ3N=400 MeV α=0.08 fm4
ħΩ=20 MeV emax=12 HF basis Nmax=2 reference
In-Medium NCSM
Robert Roth - TU Darmstadt - February 2018
In-Medium NCSM
23
• use in-medium evolved Hamiltonian for a subsequent NCSM calculation
• access to ground and excited states and full suite of observables
• IM-SRG evolution of multi-reference normal-ordered Hamiltonian (and other operators)
• decoupling of particle-hole excitations, i.e., pre-diagonalization in many-body space
• ground-state from NCSM at small Nmax as reference state for multi-reference IM-SRG
• access to all open-shell nuclei and systematically improvable
NCSMreference state
IM-SRG decoupling
NCSMmany-body solution
talk by Klaus Vobig
on Thursday
Robert Roth - TU Darmstadt - February 2018
Gebrerufael, Vobig, Hergert, Roth; PRL 118, 152503 (2017)
24
Oxygen Isotopes
A
E[MeV
]
14 16 18 20 22 24 26
-170
-160
-150
-140
-130
-120
-110
-100
-90
AO
chiral NN+3N Λ3N=400 MeV α=0.08 fm4
ħΩ=20 MeV emax=12
HF basis Nmax=0 reference
§ excellent agreement with direct NCSM
§ IM-SRG evolution limited to J=0 reference states and thus even-mass isotopes
◆ IM-NCSM
⦁ NAT-NCSM, explicit 3N
⦁ NAT-NCSM, NO2B
Robert Roth - TU Darmstadt - February 2018
Vobig, Gebrerufael, Roth; in prep.
25
Oxygen Isotopes
A
E[MeV
]
14 16 18 20 22 24 26
-170
-160
-150
-140
-130
-120
-110
-100
-90
AO§ excellent agreement with
direct NCSM
§ IM-SRG evolution limited to J=0 reference states and thus even-mass isotopes
§ odd-mass nuclei via simple particle attachment or removal in final NCSM run
chiral NN+3N Λ3N=400 MeV α=0.08 fm4
ħΩ=20 MeV emax=12
HF basis Nmax=0 reference
IM-NCSM attached▼ IM-NCSM removed
◆ IM-NCSM
⦁ NAT-NCSM, explicit 3N
⦁ NAT-NCSM, NO2B
Strength-Function NCSM
• prepare pivot vector by applying transition operator to ground-state vector
• use simplistic Lanczos iterations to generate strength distribution
Robert Roth - TU Darmstadt - February 2018
Strength-Function NCSM
27
• regular NCSM calculation for ground state for a range of Nmax truncations
• access to all open-shell nuclei
NCSMground state
NCSMstrength distribution
Stumpf, Wolfgruber, Roth; arXiv:1709.06840
Stumpf, Wolfgruber, Roth; arXiv:1709.06840
Robert Roth - TU Darmstadt - February 2018
Strength-Function NCSM
§ perform NCSM calculation for ground state
§ prepare pivot vector with transition operator
§ perform Lanczos algorithm with Hamiltonian: obtain eigenvectors as superposition of Lanczos vectors
28
10�3
10�2
10�1
100
101100 LV
10�3
10�2
10�1
100
101200 LV
10�3
10�2
10�1
100
101400 LV
20 40 60 80 100E� [MeV]
10�3
10�2
10�1
100
101800 LV
R(E2)[e2fm
4MeV�1]
16O
, N
N-o
nly,
α=
0.08
fm
4 , H
F ba
sis,
ħΩ
=20
MeV
, e
max
=12
, N
max
=4
<latexit sha1_base64="oW0TF0NjWJs//zcw2N/Rz6anF9U=">AAAB23icbU9NS8NAFHypX7V+VT16CRbBg5RERPFWEMFjBWMrbS2b7WtdupsNu5vSEnLzJN7Em/4V/4f/xqTmYFsH3mOYmQdv/JAzbRzn2yosLa+srhXXSxubW9s75d29ey0jRdGjkkvV9IlGzgL0DDMcm6FCInyODX94lfmNESrNZHBnJiF2BBkErM8oMan00B6iia+7TtItV5yqM4W9SNycVCBHvVv+avckjQQGhnKidct1QtOJiTKMckxK7UhjSOiQDLCV0oAI1J14+nBiH6VKz+5LlU5g7Kn69yImQuuJ8NOkIOZJz3uZ+L+ncJTYM9JYc9bDLHoyzrYgVMmklBZ25+stEu+0ell1b88qtfO8eREO4BCOwYULqMEN1MEDCgLe4AM+rUfr2XqxXn+jBSu/2YcZWO8/wVWIeQ==</latexit>
<latexit sha1_base64="Djjp+TfxbGmANDWlMy0Hr/Cn5s0=">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</latexit>
<latexit sha1_base64="Yv2Onlskjjad94u8XZcpN6AzNZY=">AAAB23icbU9NS8NAFHxbv2r9qnr0EiyCBymJB8VbQQSPFYyttLVstq916W427G5LS8jNk3gTb/pX/B/+G5Oag60OvMcwMw/eBJHgxrruFyksLa+srhXXSxubW9s75d29O6NGmqHPlFC6GVCDgofoW24FNiONVAYCG8HwMvMbY9SGq/DWTiPsSDoIeZ8zalPpvj1EG191w6RbrrhVdwbnL/FyUoEc9W75s91TbCQxtExQY1qeG9lOTLXlTGBSao8MRpQN6QBbKQ2pRNOJZw8nzlGq9Jy+0umE1pmpvy9iKo2ZyiBNSmofzaKXif97GseJMydNjOA9zKInk2xLyrRKSmlhb7HeX+KfVi+q3o1bqZ3lzYtwAIdwDB6cQw2uoQ4+MJDwCu/wQR7IE3kmLz/RAslv9mEO5O0bGUSIsw==</latexit>
<latexit sha1_base64="BnCauYEobTkz8WhFQuvcT6/rUEE=">AAACBnicbY9NSsNAHMUn9avWr6hLN4NFqCAlKaKIFApF0F0FYwtNGybTaR06k4TMpLSEOYA38BauxF1xp1fwNiaxC9v6YIYf773/4rkBo0IaxreWW1ldW9/Ibxa2tnd29/T9g0fhRyEmFvaZH7ZcJAijHrEklYy0gpAg7jLSdIf1NG+OSCio7z3ISUA6HA082qcYycRy9Io9JDK+cTwFq9AWEXdiWjVVN75T9jWsO7Qbl7zTlLPiyKHK0YtG2cgEl8GcQRHM1HD0qd3zccSJJzFDQrRNI5CdGIWSYkZUwY4ECRAeogFpJ+ghTkQnzrYpeJI4Pdj3w+R5Embu34sYcSEm3E2aHMknsZil5v9ZSEYKzlljwWiPpNWzcfpzhENfFZLB5uK8ZbAq5auyeX9erF3MlufBETgGJWCCS1ADt6ABLIDBC5iCT/ClPWuv2pv2/lvNabObQzAn7eMHxy2Y4Q==</latexit>
§ first coefficient provides transition matrix element
§ construct discrete strength distribution
<latexit sha1_base64="56tvaD39UFWkH09G4hOtIg7i5oQ=">AAACJnicbU/bSsMwGE7nac5T1UtvikNQkNGKKCKCMASvdIpzgp0lTf/NsPRAko1JyAP5Bl75Cl6JeCPe6WPY1l6o84OEL9+B8PkJo0La9qtRGhufmJwqT1dmZufmF8zFpUsR9zmBJolZzK98LIDRCJqSSgZXCQcc+gxafq+e+a0BcEHj6ELeJdAOcTeiHUqwTCXPPK97zo1ajza0dWC5Psc9kGrgOVodeVGuhVjeEszUiXb38xenQ0hdWys3TtSp9pTL0g8DrL9Lnlm1a3YOa5Q4BamiAg3PfHSDmPRDiCRhWIhrx05kW2EuKWGgK25fQIJJD3fhOqURDkG0Vb5dW2upElidmKcnklau/mwoHApxF/ppMlsi/nqZ+L/HYaCtX9JQMBpAFt0cZneICY91JR3s/J03Sppbtb2ac7ZdPdwplpfRClpF68hBu+gQHaMGaiKCHtAb+kCfxr3xZDwbL9/RklF0ltEvGO9feP6mxA==</latexit>
<latexit sha1_base64="9AwvB4qzHMOKPUS2cPvZ7eEUDUw=">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</latexit>
Stumpf, Wolfgruber, Roth; arXiv:1709.06840
Robert Roth - TU Darmstadt - February 2018
Strength-Function NCSM
§ works with simplest Lanczos algorithm (no reorthogonalization, Lanczos vectors discarded)
§ same computational reach as regular NCSM
§ no ad-hoc truncations, convergence in Nmax and Lanczos iterations can be demonstrated explicitly
§ full convergence of individual transitions in the relevant energy regime after ~800 iterations
§ full access to fine structure of giant resonances
§ full access to below-threshold features
29
10�3
10�2
10�1
100
101100 LV
10�3
10�2
10�1
100
101200 LV
10�3
10�2
10�1
100
101400 LV
20 40 60 80 100E� [MeV]
10�3
10�2
10�1
100
101800 LV
R(E2)[e2fm
4MeV�1]
16O
, N
N-o
nly,
α=
0.08
fm
4 , H
F ba
sis,
ħΩ
=20
MeV
, e
max
=12
, N
max
=4ab initio approach to
strength distributions with many advantages
Robert Roth - TU Darmstadt - February 2018
Stumpf, Wolfgruber, Roth; arXiv:1709.06840
30
Discrete Strength Distribution
10�2
10�1
10016O
10�2
10�1
100
10�2
10�1
100
101
10�2
10�1
10018O
10�2
10�1
100
10�2
10�1
100
101
10�2
10�1
10020O
10�2
10�1
100
10�2
10�1
100
101
10�2
10�1
10022O
10�2
10�1
100
10�2
10�1
100
101
0 10 20 30 40 50 60E� [MeV]
10�2
10�1
10024O
0 10 20 30 40 50 60E� [MeV]
10�2
10�1
100
0 10 20 30 40 50 60E� [MeV]
10�2
10�1
100
101
R(E0)[e2fm
4MeV�1]
R(E1)[e2fm
2MeV�1]
R(E2)[e2fm
4MeV�1]
isoscalar E0 isovector E1 isoscalar E2
chiral
NN
+3N
(400
), α=
0.08
fm
4 ,
HF
basi
s, ħΩ
=20
MeV
, e
max
=12
, N
max
=8/
9, 1
000L
V• discrete strength distribution, even above particle-emission threshold, since we work with bound-state basis
• escape width: cannot be described, expected to be a few 100 keV in resonance region
• spreading width: fully included through configuration mixing, dominant in resonance region
• description of escape width needs continuum basis (Gamow)
• however, typical experimental resolution for unstable isotopes ~1 MeV, i.e., escape width is not resolved
• traditional approach: fold discrete strength distribution with Lorentzian of ~1 MeV width
0.20.40.60.81.0 16O
0.2
0.4
0.6
2
468
10
0.2
0.4
0.6
0.8 18O
0.1
0.2
0.3
2
4
6
8
0.5
1.0
1.5 20O
0.1
0.2
0.3
2468
10
0.5
1.0
1.5
2.0 22O
0.2
0.4
0.6
5
10
15
0 10 20 30 40 50 60E� [MeV]
0.5
1.0
1.5
2.0 24O
0 10 20 30 40 50 60E� [MeV]
0.2
0.4
0.6
0 10 20 30 40 50 60E� [MeV]
5
10
15
R(E0)[e2fm
4MeV�1]
R(E1)[e2fm
2MeV�1]
R(E2)[e2fm
4MeV�1]
Robert Roth - TU Darmstadt - February 2018
Stumpf, Wolfgruber, Roth; arXiv:1709.06840
31
Strength Distribution
chiral
NN
+3N
(400
) &
N2L
O-S
AT,
α=
0.08
fm
4 ,
HF
basi
s, ħΩ
=20
MeV
, e
max
=12
, N
max
=8/
9, 1
MeV
sm
earing
isoscalar E0 isovector E1 isoscalar E2
Robert Roth - TU Darmstadt - February 2018
Stumpf, Wolfgruber, Roth; arXiv:1709.06840
Comparison with RPA and SRPA
§ collective excitations traditionally described in RPA or SRPA
§ RPA (1p1h) cannot describe fragmentation, therefore, go to SRPA (2p2h)
§ NCSM shows much more fine structure than SRPA and resolves notorious problem with pathological SRPA energy-shifts
32
10�2
100
16O
10�2
100
10�2
100
10�2
100
10�2
100
10�2
100
10 20 30 40 50 60E� [MeV]
10�2
100
10 20 30 40 50 60E� [MeV]
10�2
100
10 20 30 40 50 60E� [MeV]
10�2
100
R(E0)[e2fm
4MeV�1]
R(E1)[e2fm
2MeV�1]
R(E2)[e2fm
4MeV�1]
isoscalar E0 isovector E1 isoscalar E2
NC
SM
RP
AS
RP
A
chiral
NN
+3N
(400
), α=
0.08
fm
4 ,
HF
basi
s, ħΩ
=20
MeV
, e
max
=12
, N
max
=8/
9
Robert Roth - TU Darmstadt - February 2018
Conclusions
§ hybrids built on the NCSM enable comprehensive access to ground and excited states of arbitrary open-shell nuclei
§mass reach: A≲30 if large Nmax is needed: NAT-NCSM, SF-NCSMA≲70 if small Nmax is sufficient: IM-NCSM, NCSM-PT
§more hybrids: NCSM with Continuum, HORSE,...
33
IM-SRG decoupling
MBPTbasis optimization
NCSMmany-body solution
NCSMreference state
MBPTconvergence booster
NCSMmany-body solution
NCSMmany-body solution
NAT-NCSM IM-NCSMNCSM-PTNCSM
ground state
NCSMstrength distribution
SF-NCSM
Robert Roth - TU Darmstadt - February 2018
� S. Alexa, D. Derr, M. Dupper, A. Geißel, S. Graf, T. Hüther, J. Kaltwasser, M. Knöll, L. Mertes, J. Müller, S. Schulz, C. Stumpf, T. Tilger, K. Vobig, R. WirthTechnische Universität Darmstadt
� A. Tichai CEA Saclay
� P. Navrátil TRIUMF, Vancouver
� H. HergertNSCL / Michigan State University
� J. Vary, P. MarisIowa State University
� E. Epelbaum, H. Krebs & the LENPIC CollaborationUniversität Bochum, …
Epilogue
§ thanks to my group and my collaborators
34
Epilogue
■ thanks to my group & my collaborators
● S. Binder, J. Braun, A. Calci, S. Fischer,E. Gebrerufael, H. Spiess, J. Langhammer, S. Schulz,C. Stumpf, A. Tichai, R. Trippel, R. Wirth, K. VobigInstitut für Kernphysik, TU Darmstadt
● P. NavrátilTRIUMF Vancouver, Canada
● J. Vary, P. MarisIowa State University, USA
● S. Quaglioni, G. HupinLLNL Livermore, USA
● P. PiecuchMichigan State University, USA
● H. HergertOhio State University, USA
● P. PapakonstantinouIBS/RISP, Korea
● C. ForssénChalmers University, Sweden
● H. Feldmeier, T. NeffGSI Helmholtzzentrum
JUROPA LOEWE-CSC EDISON
COMPUTING TIME
46