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Momentum distributions and pair distribution functions Stefano Gandolfi Los Alamos National Laboratory (LANL) Quantitative challenges in EMC and SRC Research and Data-Mining Massachusetts Institute of Technology, December 2-5, 2016


  1. Momentum distributions and pair distribution functions Stefano Gandolfi Los Alamos National Laboratory (LANL) Quantitative challenges in EMC and SRC Research and Data-Mining Massachusetts Institute of Technology, December 2-5, 2016 www.computingnuclei.org Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 1 / 17

  2. Nucleon-nucleon correlations Illustration of back-to-back proton-neutron pairs in Jefferson Lab Experiment, Subedi et al. , Science (2008) Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 2 / 17

  3. Nucleon-nucleon correlations Ratio of np to pp pairs in light (upper panel) and heavy (lower panel) nuclei, Subedi et al. , Science (2008), Hen et al. , Science (2014) Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 3 / 17

  4. Nucleon-nucleon correlations Overarching questions: What is the origin of nucleon-nucleon correlations? Are nucleons more correlated at low- or high-momenta? Can we extract information from momentum distributions? Can we extract information from pair correlation functions? Which are observables and data related to those correlations? In this talk I am not giving answers , but showing what we can calculate! One-body momentum distributions Two-body momentum distributions Contact parameter in cold atoms but not in nuclei Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 4 / 17

  5. Nuclear Hamiltonian Most common models: non-relativistic nucleons interacting with an effective nucleon-nucleon force (NN) and three-nucleon interaction (TNI). A H = − � 2 � ∇ 2 � � i + v ij + V ijk 2 m i =1 i < j i < j < k v ij NN, Argonne, chiral EFT, CD-Bonn, etc. V ijk TNI Urbana, Illinois, chiral EFT, etc. Many-body methods: GFMC, AFDMC, CC, SCGF, ... Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 5 / 17

  6. Momentum distribution - 4 He Preliminary! AFDMC calculations: 5 AV6’ AV7’ 100 4 2 LO (1.0 fm) N number of particles 2 LO (1.2 fm) N 3 3 ) n(k) (fm 1 2 AV6’ 4 He 2 LO (1.2 fm) N 1 4 He 0.01 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 -1 ) -1 ) K M (fm k (fm Useful to “check” the contact(s)? Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 6 / 17

  7. Universality Preliminary! By rescaling the momentum distributions of 4 He and 16 O, at large momenta they seem universal: 100 4 He, AV6’ 4 He, N 2 LO (1.2 fm) 16 O, AV6’ 16 O, N 2 LO (1.2 fm) 3 ) 1 n(k) (fm 0.01 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 -1 ) k (fm Again, useful to “check” the contact(s)? Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 7 / 17

  8. Momentum distribution - 40 Ca Preliminary! AFDMC calculations using AV6’ compared to Coupled-Cluster using N2LO SAT : 40 35 2 LO SAT N 2 AV6’ 30 10 number of particles 25 3 ) 0 n(k) (fm 10 20 15 AV6’ -2 10 40 Ca 2 LO SAT 10 N 40 Ca 5 -4 10 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 -1 ) -1 ) K M (fm k (fm CC calculations provided by G. Hagen (ORNL). Implications??? Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 8 / 17

  9. Symmetric nuclear matter Self Consistent Green’s Function calculations of nuclear matter: Rios, Polls, Dickhoff, PRC (2014) Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 9 / 17

  10. Two-body momentum distributions in light nuclei VMC calculations using AV18+UIX: 12 C 10 B 10 5 8 Be 10 5 10 3 6 Li 10 5 10 3 4 He 10 1 10 5 10 3 10 1 ρ pN (q,Q=0) (fm 3 ) 10 5 10 -1 10 3 0 1 2 3 4 5 10 1 10 -1 10 3 0 1 2 3 4 5 10 1 10 -1 0 1 2 3 4 5 10 1 10 -1 0 1 2 3 4 5 10 -1 0 1 2 3 4 5 q (fm -1 ) Carlson, et al. , RMP (2015). Blue symbols are T=0 pairs, and red symbols are T=1 pairs. Strong dominance of T=0 pairs! Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 10 / 17

  11. Two-body momentum distributions of 16 O Preliminary! AFDMC calculations using different NN interactions (Q integrated): 1000 10000 T=0 (np), AV6’ T=0 (np), AV6’ T=1 (pp), AV6’ 100 T=1 (pp), AV6’ 1000 2 LO (1.2 fm) 2 LO (1.2 fm) T=0 (np), N T=0 (np), N 100 10 2 LO (1.2 fm) 2 LO (1.2 fm) T=1 (pp), N T=1 (pp), N -1 ) -1 ) n 2 (k) (fm n 2 (k) (fm 10 1 0.1 1 0.01 0.1 4 He 16 O 0.001 0.01 0.0001 0.001 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.5 1.5 2 2.5 3 3.5 4 4.5 -1 ) -1 ) k (fm k (fm 3 36 32 T=0 (np), AV6’ 28 T=1 (pp), AV6’ number of pairs number of pairs 2 24 2 LO (1.2 fm) T=0 (np), N 2 LO (1.2 fm) 20 T=1 (pp), N T=0 (np), AV6’ 16 T=1 (pp), AV6’ 1 12 2 LO (1.2 fm) T=0 (np), N 2 LO (1.2 fm) 8 T=1 (pp), N 4 He 16 O 4 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -1 ) -1 ) K M (fm K M (fm Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 11 / 17

  12. Pair distribution functions Preliminary! AFDMC calculations using different NN interactions: 4 He, AV6’ 0.25 4 He, N2LO (1.0 fm) 40 Ca, AV6’ 0.2 40 Ca, N2LO (1.0 fm) -3 ) g(r) (fm 0.15 0.1 0.05 0 0 1 2 3 4 6 7 8 5 r (fm) Very similar at the origin, but more statistics needed. For the last time ... useful to “check” the contact(s)? Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 12 / 17

  13. Contact parameter Tan universal relations: Tan’s contact E ζ 5 ν C = 6 = ξ − k F a − 3( k F a ) 2 + . . . , 5 πζ E FG Nk F 10 π ζ k 4 k 4 8 2 C F F N ( k ) → k 4 = 3 π 2 Nk F k 4 g ↑↓ ( r ) → 9 π 20 ζ ( k F r ) − 2 = 3 C ( k F r ) − 2 8 Nk F S ↑↓ ( k ) → 3 π 10 ζ k F k = 1 C k F 4 Nk F k Shina Tan, Ann. Phys. (2008). Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 13 / 17

  14. Contact parameter 0.6 0.41 1 / k F a = 0 VMC 0.4 0.5 DMC 0.5 E / E FG ext. 4 n(k/k F ) 0.39 0.4 E / E FG 0.4 0.38 0 0.015 0.03 0.045 0.3 r e / r 0 (k/k F ) 0.3 0.2 0.2 0.1 0 0.1 0 0.5 1 1.5 2 2.5 3 3.5 4 -0.2 -0.1 0 0.1 0.2 k / k F 1 / k F a Momentum distribution Equation of state 1 200 1.5 2 g ↑↓ (k F r) 1 0.5 150 0.5 (k F r) (1) (k F r) g ↑↓ (k F r) 0 0 0 0.4 0.8 1.2 100 k F r ρ VMC VMC -0.5 DMC DMC 50 ext. ext. f(k F r)=a+b/(k F r) 2 f(k F r)=1 - c k F r -1 0 1 2 3 4 5 6 7 0 0 0.2 0.4 0.6 0.8 1 k F r k F r One-body density matrix Pair distribution function C / Nk F = 3 . 39(1), Gandolfi, Schmidt, Carlson, PRA 83, 041601 (2011). Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 14 / 17

  15. Contact parameter 20 0 -2 15 ξ fit -4 QMC C / Nk F -6 10 -2 -1 0 1 2 1 / a k F 5 QMC exp exp, new Enss. et al. 0 -1.5 -1 -0.5 0 0.5 1 1.5 1 / a k F Hoinka, Lingham, Fenech, Hu, Vale, Drut, Gandolfi, PRL 110, 055305 (2013). Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 15 / 17

  16. Contact parameter in nuclei? Preliminary! 25 4 He, AV6’ 4 He, N 2 LO (1.2 fm) 20 16 O, AV6’ -1 ) 16 O, N 2 LO (1.2 fm) 15 4 n(k) (fm 40 Ca, AV6’ 40 Ca, N 2 LO SAT 10 k 5 0 0 2 4 6 8 -1 ) k (fm Some flat region for harder interactions that is “maybe” universal for different A (but not using different interactions). Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 16 / 17

  17. Summary Thank you! Discussion ... Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 17 / 17

  18. Extra slides Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 1 / 27

  19. Nuclear Hamiltonian Model: non-relativistic nucleons interacting with an effective nucleon-nucleon force (NN) and three-nucleon interaction (TNI). A H = − � 2 � � � ∇ 2 i + v ij + V ijk 2 m i =1 i < j i < j < k v ij NN fitted on scattering data. Sum of operators: σ j , S ij ,� L ij · � � O p =1 , 8 v p ( r ij ) , O p v ij = ij = (1 ,� σ i · � S ij ) × (1 ,� τ i · � τ j ) ij Argonne AV8’. Local chiral forces up to N 2 LO has the similar spin/isospin operatorial structure of AV8’ - Gezerlis, Tews, et al. PRL (2013), PRC (2014) Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 2 / 27

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