Uncertainty Analysis and Order-by-Order Optimization of Chiral Nuclear Interactions
Abstract
Chiral effective field theory ( ΧEFT) provides a systematic approach to describe low-energy nuclear forces. Moreover, EFT is able to provide well-founded estimates of statistical and systematic uncertainties | although this unique advantage has not yet been fully exploited. We ll this gap by performing an optimization and statistical analysis of all the low-energy constants (LECs) up to next-to-next-to-leading order. Our optimization protocol corresponds to a simultaneous t to scattering and bound-state observables in the pion-nucleon, nucleon-nucleon, and few-nucleon sectors, thereby utilizing the full model capabilities of EFT. Finally, we study the effect on other observables by demonstrating forward-error-propagation methods that can easily be adopted by future works. We employ mathematical optimization and implement automatic differentiation to attain e cient and machine-precise first- and second-order derivatives of the objective function with respect to the LECs. This is also vital for the regression analysis. We use power-counting arguments to estimate the systematic uncertainty that is inherent to EFT and we construct chiral interactions at different orders with quantified uncertainties. Statistical error propagation is compared with Monte Carlo sampling showing that statistical errors are in general small compared to systematic ones. In conclusion, we find that a simultaneous t to different setsmore »
- Authors:
- Publication Date:
- Research Org.:
- Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States). Oak Ridge Leadership Computing Facility (OLCF)
- Sponsoring Org.:
- USDOE Office of Science (SC)
- OSTI Identifier:
- 1239437
- Alternate Identifier(s):
- OSTI ID: 1261377
- Grant/Contract Number:
- SC0008499; AC05-00OR22725; FP7/2007-2013; DEFG02-96ER40963
- Resource Type:
- Published Article
- Journal Name:
- Physical Review. X
- Additional Journal Information:
- Journal Name: Physical Review. X Journal Volume: 6 Journal Issue: 1; Journal ID: ISSN 2160-3308
- Publisher:
- American Physical Society
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 73 NUCLEAR PHYSICS AND RADIATION PHYSICS
Citation Formats
Carlsson, B. D., Ekström, A., Forssén, C., Strömberg, D. Fahlin, Jansen, G. R., Lilja, O., Lindby, M., Mattsson, B. A., and Wendt, K. A. Uncertainty Analysis and Order-by-Order Optimization of Chiral Nuclear Interactions. United States: N. p., 2016.
Web. doi:10.1103/PhysRevX.6.011019.
Carlsson, B. D., Ekström, A., Forssén, C., Strömberg, D. Fahlin, Jansen, G. R., Lilja, O., Lindby, M., Mattsson, B. A., & Wendt, K. A. Uncertainty Analysis and Order-by-Order Optimization of Chiral Nuclear Interactions. United States. https://doi.org/10.1103/PhysRevX.6.011019
Carlsson, B. D., Ekström, A., Forssén, C., Strömberg, D. Fahlin, Jansen, G. R., Lilja, O., Lindby, M., Mattsson, B. A., and Wendt, K. A. Wed .
"Uncertainty Analysis and Order-by-Order Optimization of Chiral Nuclear Interactions". United States. https://doi.org/10.1103/PhysRevX.6.011019.
@article{osti_1239437,
title = {Uncertainty Analysis and Order-by-Order Optimization of Chiral Nuclear Interactions},
author = {Carlsson, B. D. and Ekström, A. and Forssén, C. and Strömberg, D. Fahlin and Jansen, G. R. and Lilja, O. and Lindby, M. and Mattsson, B. A. and Wendt, K. A.},
abstractNote = {Chiral effective field theory ( ΧEFT) provides a systematic approach to describe low-energy nuclear forces. Moreover, EFT is able to provide well-founded estimates of statistical and systematic uncertainties | although this unique advantage has not yet been fully exploited. We ll this gap by performing an optimization and statistical analysis of all the low-energy constants (LECs) up to next-to-next-to-leading order. Our optimization protocol corresponds to a simultaneous t to scattering and bound-state observables in the pion-nucleon, nucleon-nucleon, and few-nucleon sectors, thereby utilizing the full model capabilities of EFT. Finally, we study the effect on other observables by demonstrating forward-error-propagation methods that can easily be adopted by future works. We employ mathematical optimization and implement automatic differentiation to attain e cient and machine-precise first- and second-order derivatives of the objective function with respect to the LECs. This is also vital for the regression analysis. We use power-counting arguments to estimate the systematic uncertainty that is inherent to EFT and we construct chiral interactions at different orders with quantified uncertainties. Statistical error propagation is compared with Monte Carlo sampling showing that statistical errors are in general small compared to systematic ones. In conclusion, we find that a simultaneous t to different sets of data is critical to (i) identify the optimal set of LECs, (ii) capture all relevant correlations, (iii) reduce the statistical uncertainty, and (iv) attain order-by-order convergence in EFT. Furthermore, certain systematic uncertainties in the few-nucleon sector are shown to get substantially magnified in the many-body sector; in particlar when varying the cutoff in the chiral potentials. The methodology and results presented in this Paper open a new frontier for uncertainty quantification in ab initio nuclear theory.},
doi = {10.1103/PhysRevX.6.011019},
journal = {Physical Review. X},
number = 1,
volume = 6,
place = {United States},
year = {Wed Feb 24 00:00:00 EST 2016},
month = {Wed Feb 24 00:00:00 EST 2016}
}
https://doi.org/10.1103/PhysRevX.6.011019
Web of Science
Works referenced in this record:
Structure of Nuclei with Two- Plus Three-Nucleon Interactions from Chiral Effective Field Theory
journal, July 2007
- Navrátil, P.; Gueorguiev, V. G.; Vary, J. P.
- Physical Review Letters, Vol. 99, Issue 4
Ab initio Gorkov-Green's function calculations of open-shell nuclei
journal, January 2013
- Somà, V.; Barbieri, C.; Duguet, T.
- Physical Review C, Vol. 87, Issue 1
Scattering formalism for nonidentical spinor particles
journal, January 1980
- La France, P.; Winternitz, P.
- Journal de Physique, Vol. 41, Issue 12
Neutron-proton effective range parameters and zero-energy shape dependence
journal, April 2006
- Hackenburg, R. W.
- Physical Review C, Vol. 73, Issue 4
Uncertainty Propagation in Nerve Impulses Through the Action Potential Mechanism
journal, January 2015
- Torres Valderrama, Aldemar; Witteveen, Jeroen; Navarro, Maria
- The Journal of Mathematical Neuroscience, Vol. 5, Issue 1
Ab initio no core shell model
journal, March 2013
- Barrett, Bruce R.; Navrátil, Petr; Vary, James P.
- Progress in Particle and Nuclear Physics, Vol. 69
Accurate nucleon-nucleon potential with charge-independence breaking
journal, January 1995
- Wiringa, R. B.; Stoks, V. G. J.; Schiavilla, R.
- Physical Review C, Vol. 51, Issue 1
Proton- scattering calculation: Elastic and charge-exchange reactions up to 30 MeV
journal, March 2015
- Deltuva, A.; Fonseca, A. C.
- Physical Review C, Vol. 91, Issue 3
In-medium similarity renormalization group with chiral two- plus three-nucleon interactions
journal, March 2013
- Hergert, H.; Bogner, S. K.; Binder, S.
- Physical Review C, Vol. 87, Issue 3
Phase shift analysis of 0 – 30 MeV pp scattering data
journal, July 1988
- Bergervoet, J. R.; van Campen, P. C.; van der Sanden, W. A.
- Physical Review C, Vol. 38, Issue 1
The limits of the nuclear landscape
journal, June 2012
- Erler, Jochen; Birge, Noah; Kortelainen, Markus
- Nature, Vol. 486, Issue 7404
Updated analysis of NN elastic scattering data to 1.6 GeV
journal, December 1994
- Arndt, Richard A.; Strakovsky, Igor I.; Workman, Ron L.
- Physical Review C, Vol. 50, Issue 6
Isoscalar Hamiltonians for light atomic nuclei
journal, August 1999
- Kamuntavičius, G. P.; Navrátil, P.; Barrett, B. R.
- Physical Review C, Vol. 60, Issue 4
Half-life and <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mi>f</mml:mi><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo stretchy="false">/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> value for the bare triton
journal, March 2005
- Akulov, Yu. A.; Mamyrin, B. A.
- Physics Letters B, Vol. 610, Issue 1-2
Colloquium : Three-body forces: From cold atoms to nuclei
journal, January 2013
- Hammer, Hans-Werner; Nogga, Andreas; Schwenk, Achim
- Reviews of Modern Physics, Vol. 85, Issue 1
Vacuum Polarization Effects in Proton-Proton Scattering
journal, December 1957
- Durand, Loyal
- Physical Review, Vol. 108, Issue 6
Optimized Chiral Nucleon-Nucleon Interaction at Next-to-Next-to-Leading Order
journal, May 2013
- Ekström, A.; Baardsen, G.; Forssén, C.
- Physical Review Letters, Vol. 110, Issue 19
Low-energy chiral two-pion exchange potential with statistical uncertainties
journal, May 2015
- Pérez, R. Navarro; Amaro, J. E.; Arriola, E. Ruiz
- Physical Review C, Vol. 91, Issue 5
Hydrogen-Deuterium Isotope Shift and the Structure of the Deuteron
journal, January 1998
- Huber, A.; Udem, Th.; Gross, B.
- Physical Review Letters, Vol. 80, Issue 3
Proton radius, Darwin-Foldy term and radiative corrections
journal, November 2010
- Jentschura, U. D.
- The European Physical Journal D, Vol. 61, Issue 1
Renormalization of one-pion exchange and power counting
journal, November 2005
- Nogga, A.; Timmermans, R. G. E.; Kolck, U. van
- Physical Review C, Vol. 72, Issue 5
Efficient calculation of chiral three-nucleon forces up to for ab initio studies
journal, April 2015
- Hebeler, K.; Krebs, H.; Epelbaum, E.
- Physical Review C, Vol. 91, Issue 4
Cosmic calibration: Constraints from the matter power spectrum and the cosmic microwave background
journal, October 2007
- Habib, Salman; Heitmann, Katrin; Higdon, David
- Physical Review D, Vol. 76, Issue 8
Bayesian uncertainty quantification and propagation in molecular dynamics simulations: A high performance computing framework
journal, October 2012
- Angelikopoulos, Panagiotis; Papadimitriou, Costas; Koumoutsakos, Petros
- The Journal of Chemical Physics, Vol. 137, Issue 14
Realistic models of pion-exchange three-nucleon interactions
journal, June 2001
- Pieper, Steven C.; Pandharipande, V. R.; Wiringa, R. B.
- Physical Review C, Vol. 64, Issue 1
Precision Nucleon-Nucleon Potential at Fifth Order in the Chiral Expansion
journal, September 2015
- Epelbaum, E.; Krebs, H.; Meißner, U. -G.
- Physical Review Letters, Vol. 115, Issue 12
Uncertainties of predictions from parton distribution functions. I. The Lagrange multiplier method
journal, December 2001
- Stump, D.; Pumplin, J.; Brock, R.
- Physical Review D, Vol. 65, Issue 1
Construction of high-quality NN potential models
journal, June 1994
- Stoks, V. G. J.; Klomp, R. A. M.; Terheggen, C. P. F.
- Physical Review C, Vol. 49, Issue 6
Parameterization dependence of -matrix poles and eigenphases from a fit to elastic scattering data
journal, September 2012
- Workman, R. L.; Arndt, R. A.; Briscoe, W. J.
- Physical Review C, Vol. 86, Issue 3
Few-nucleon forces and systems in chiral effective field theory
journal, October 2006
- Epelbaum, Evgeny
- Progress in Particle and Nuclear Physics, Vol. 57, Issue 2
Microscopic Calculation of the Inclusive Electron Scattering Structure Function in
journal, February 2000
- Mihaila, Bogdan; Heisenberg, Jochen H.
- Physical Review Letters, Vol. 84, Issue 7
High-Precision Determination of the Pion-Nucleon Term from Roy-Steiner Equations
journal, August 2015
- Hoferichter, Martin; Ruiz de Elvira, Jacobo; Kubis, Bastian
- Physical Review Letters, Vol. 115, Issue 9
Nucleon-nucleon potential and minimal relativity
journal, August 1969
- Brown, G. E.; Jackson, A. D.; Kuo, T. T. S.
- Nuclear Physics A, Vol. 133, Issue 3
Magnetic moment interaction in nucleon-nucleon phase-shift analyses
journal, October 1990
- Stoks, V. G. J.; de Swart, J. J.
- Physical Review C, Vol. 42, Issue 4
Accurate charge-dependent nucleon-nucleon potential at fourth order of chiral perturbation theory
journal, October 2003
- Entem, D. R.; Machleidt, R.
- Physical Review C, Vol. 68, Issue 4
Charge symmetry, quarks and mesons
journal, October 1990
- Miller, G. A.; Nefkens, B. M. K.; Šlaus, I.
- Physics Reports, Vol. 194, Issue 1-2
Q UANTUM M ONTE C ARLO C ALCULATIONS OF L IGHT N UCLEI
journal, December 2001
- Pieper, Steven C.; Wiringa, R. B.
- Annual Review of Nuclear and Particle Science, Vol. 51, Issue 1
Peripheral nucleon-nucleon scattering at fifth order of chiral perturbation theory
journal, January 2015
- Entem, D. R.; Kaiser, N.; Machleidt, R.
- Physical Review C, Vol. 91, Issue 1
Improved Coulomb Potential
journal, June 1983
- Austen, G. J. M.; de Swart, J. J.
- Physical Review Letters, Vol. 50, Issue 26
Coupled-cluster computations of atomic nuclei
journal, September 2014
- Hagen, G.; Papenbrock, T.; Hjorth-Jensen, M.
- Reports on Progress in Physics, Vol. 77, Issue 9
Table of experimental nuclear ground state charge radii: An update
journal, January 2013
- Angeli, I.; Marinova, K. P.
- Atomic Data and Nuclear Data Tables, Vol. 99, Issue 1
Effective chiral lagrangians for nucleon-pion interactions and nuclear forces
journal, September 1991
- Weinberg, Steven
- Nuclear Physics B, Vol. 363, Issue 1
Electromagnetic corrections to scattering
journal, February 1977
- Tromborg, B.; Waldenstrøm, S.; Øverbø, I.
- Physical Review D, Vol. 15, Issue 3
Atomic Mass and Nuclear Binding Energy for O-16 (Oxygen)
book, January 2009
- Sukhoruchkin, S. I.; Soroko, Z. N.
- Nuclei with Z = 1 - 54
Coulomb corrections to πN elastic scattering
journal, July 1973
- Bugg, D. V.
- Nuclear Physics B, Vol. 58, Issue 2
CODATA recommended values of the fundamental physical constants: 2010
journal, November 2012
- Mohr, Peter J.; Taylor, Barry N.; Newell, David B.
- Reviews of Modern Physics, Vol. 84, Issue 4
Electromagnetic corrections in hadron scattering, with application to πN → πN
text, January 1978
- Waldenstrøm, S.; Øverbø, I.; Tromborg, B.
- Birkhäuser
Statistical error analysis for phenomenological nucleon-nucleon potentials
journal, June 2014
- Navarro Pérez, R.; Amaro, J. E.; Ruiz Arriola, E.
- Physical Review C, Vol. 89, Issue 6
Electromagnetic contributions to the nucleon-nucleon interaction
journal, October 1977
- Gersten, A.
- Nuclear Physics A, Vol. 290, Issue 2
Determination of the Axial-Vector Weak Coupling Constant with Ultracold Neutrons
journal, October 2010
- Liu, J.; Mendenhall, M. P.; Holley, A. T.
- Physical Review Letters, Vol. 105, Issue 18
Error estimates of theoretical models: a guide
journal, May 2014
- Dobaczewski, J.; Nazarewicz, W.; Reinhard, P-G
- Journal of Physics G: Nuclear and Particle Physics, Vol. 41, Issue 7
The two-nucleon system at next-to-next-to-next-to-leading order
journal, January 2005
- Epelbaum, E.; Glöckle, W.; Meißner, Ulf-G.
- Nuclear Physics A, Vol. 747, Issue 2-4
Quantification of modelling uncertainties in a large ensemble of climate change simulations
journal, August 2004
- Murphy, James M.; Sexton, David M. H.; Barnett, David N.
- Nature, Vol. 430, Issue 7001
Partial-wave analysis of all nucleon-nucleon scattering data below 350 MeV
journal, August 1993
- Stoks, V. G. J.; Klomp, R. A. M.; Rentmeester, M. C. M.
- Physical Review C, Vol. 48, Issue 2
Nuclear forces from chiral lagrangians
journal, November 1990
- Weinberg, Steven
- Physics Letters B, Vol. 251, Issue 2
Chiral effective field theory and nuclear forces
journal, June 2011
- Machleidt, R.; Entem, D. R.
- Physics Reports, Vol. 503, Issue 1
Electromagnetic corrections to hadron-hadron scattering
journal, July 1974
- Tromborg, B.; Hamilton, J.
- Nuclear Physics B, Vol. 76, Issue 3
Theory of the Effective Range in Nuclear Scattering
journal, July 1949
- Bethe, H. A.
- Physical Review, Vol. 76, Issue 1
Few-nucleon systems in a translationally invariant harmonic oscillator basis
journal, March 2000
- Navrátil, P.; Kamuntavičius, G. P.; Barrett, B. R.
- Physical Review C, Vol. 61, Issue 4
Review of Particle Physics
journal, July 2012
- Beringer, J.; Arguin, J. -F.; Barnett, R. M.
- Physical Review D, Vol. 86, Issue 1
Statistical uncertainties of a chiral interaction at next-to-next-to leading order
journal, February 2015
- Ekström, A.; Carlsson, B. D.; Wendt, K. A.
- Journal of Physics G: Nuclear and Particle Physics, Vol. 42, Issue 3
Modern theory of nuclear forces
journal, December 2009
- Epelbaum, E.; Hammer, H. -W.; Meißner, Ulf-G.
- Reviews of Modern Physics, Vol. 81, Issue 4
Effective Field Theory for Lattice Nuclei
journal, February 2015
- Barnea, N.; Contessi, L.; Gazit, D.
- Physical Review Letters, Vol. 114, Issue 5
Chiral three-nucleon force at N LO: Longest-range contributions
journal, May 2012
- Krebs, H.; Gasparyan, A.; Epelbaum, E.
- Physical Review C, Vol. 85, Issue 5
Search for Resonance Structure in the Total Cross Section below 800 MeV
journal, July 1982
- Lisowski, P. W.; Shamu, R. E.; Auchampaugh, G. F.
- Physical Review Letters, Vol. 49, Issue 4
Bayesian methods for parameter estimation in effective field theories
journal, March 2009
- Schindler, M. R.; Phillips, D. R.
- Annals of Physics, Vol. 324, Issue 3
Momentum space calculations and helicity formalism in nuclear physics
journal, November 1971
- Erkelenz, K.; Alzetta, R.; Holinde, K.
- Nuclear Physics A, Vol. 176, Issue 2
Wigner Symmetry in the Limit of Large Scattering Lengths
journal, August 1999
- Mehen, Thomas; Stewart, Iain W.; Wise, Mark B.
- Physical Review Letters, Vol. 83, Issue 5
Nuclear energy density optimization
journal, August 2010
- Kortelainen, M.; Lesinski, T.; Moré, J.
- Physical Review C, Vol. 82, Issue 2
Lattice effective field theory for medium-mass nuclei
journal, May 2014
- Lähde, Timo A.; Epelbaum, Evgeny; Krebs, Hermann
- Physics Letters B, Vol. 732
Nucleon-nucleon potential from an effective chiral Lagrangian
journal, March 1994
- Ordóñez, C.; Ray, L.; van Kolck, U.
- Physical Review Letters, Vol. 72, Issue 13
Modern ab initio approaches and applications in few-nucleon physics with
journal, January 2013
- Leidemann, Winfried; Orlandini, Giuseppina
- Progress in Particle and Nuclear Physics, Vol. 68
Three-Nucleon Low-Energy Constants from the Consistency of Interactions and Currents in Chiral Effective Field Theory
journal, September 2009
- Gazit, Doron; Quaglioni, Sofia; Navrátil, Petr
- Physical Review Letters, Vol. 103, Issue 10
Three-nucleon forces from chiral effective field theory
journal, December 2002
- Epelbaum, E.; Nogga, A.; Glöckle, W.
- Physical Review C, Vol. 66, Issue 6
Determination of the deuteron mean square radius
journal, August 1986
- Klarsfeld, S.; Martorell, J.; Oteo, J. A.
- Nuclear Physics A, Vol. 456, Issue 3
Lattice simulations for few- and many-body systems
journal, July 2009
- Lee, Dean
- Progress in Particle and Nuclear Physics, Vol. 63, Issue 1
Accurate momentum-space method for scattering by nuclear and Coulomb potentials
journal, July 1974
- Vincent, C. M.; Phatak, S. C.
- Physical Review C, Vol. 10, Issue 1
Improved chiral nucleon-nucleon potential up to next-to-next-to-next-to-leading order
journal, May 2015
- Epelbaum, E.; Krebs, H.; Meißner, U. -G.
- The European Physical Journal A, Vol. 51, Issue 5
Error analysis of nuclear forces and effective interactions
journal, February 2015
- Pérez, R. Navarro; Amaro, J. E.; Arriola, E. Ruiz
- Journal of Physics G: Nuclear and Particle Physics, Vol. 42, Issue 3
Meaningful characterisation of perturbative theoretical uncertainties
journal, September 2011
- Cacciari, Matteo; Houdeau, Nicolas
- Journal of High Energy Physics, Vol. 2011, Issue 9
Formalism of nucleon-nucleon elastic scattering experiments
journal, January 1978
- Bystricky, J.; Lehar, F.; Winternitz, P.
- Journal de Physique, Vol. 39, Issue 1
High-precision, charge-dependent Bonn nucleon-nucleon potential
journal, January 2001
- Machleidt, R.
- Physical Review C, Vol. 63, Issue 2
Dimensional Power Counting in Nuclei
journal, December 1997
- Friar, J. L.
- Few-Body Systems, Vol. 22, Issue 4
Phase-Shift Analysis of 310-Mev Proton-Proton Scattering Experiments
journal, January 1957
- Stapp, H. P.; Ypsilantis, T. J.; Metropolis, N.
- Physical Review, Vol. 105, Issue 1
Accurate nuclear radii and binding energies from a chiral interaction
journal, May 2015
- Ekström, A.; Jansen, G. R.; Wendt, K. A.
- Physical Review C, Vol. 91, Issue 5
Nonparametric Dark Energy Reconstruction from Supernova Data
journal, December 2010
- Holsclaw, Tracy; Alam, Ujjaini; Sansó, Bruno
- Physical Review Letters, Vol. 105, Issue 24
Phenomenological Lagrangians
journal, April 1979
- Weinberg, Steven
- Physica A: Statistical Mechanics and its Applications, Vol. 96, Issue 1-2
Nuclear saturation and the smoothness of nucleon-nucleon potentials
journal, December 1970
- Haftel, Michael I.; Tabakin, Frank
- Nuclear Physics A, Vol. 158, Issue 1
Power Counting of Contact-Range Currents in Effective Field Theory
journal, February 2015
- Valderrama, M. Pavón; Phillips, D. R.
- Physical Review Letters, Vol. 114, Issue 8
Phenomenological high precision neutron–proton delta-shell potential
journal, July 2013
- Navarro-Pérez, R.; Amaro, J. E.; Ruiz-Arriola, E.
- Physics Letters B, Vol. 724, Issue 1-3
Coarse-grained potential analysis of neutron-proton and proton-proton scattering below the pion production threshold
journal, December 2013
- Pérez, R. Navarro; Amaro, J. E.; Arriola, E. Ruiz
- Physical Review C, Vol. 88, Issue 6
Coarse-grained NN potential with chiral two-pion exchange
journal, February 2014
- Navarro Pérez, R.; Amaro, J. E.; Arriola, E. Ruiz
- Physical Review C, Vol. 89, Issue 2
Nuclear sizes and the isotope shift
journal, December 1997
- Friar, J. L.; Martorell, J.; Sprung, D. W. L.
- Physical Review A, Vol. 56, Issue 6
Continuum within an Ab initio Framework
journal, July 2014
- Romero-Redondo, Carolina; Quaglioni, Sofia; Navrátil, Petr
- Physical Review Letters, Vol. 113, Issue 3