Nuclear structure III: Nuclear and neutron matter. National Nuclear Physics Summer School Massachusetts Institute of Technology (MIT) July 18-29, 2016
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1 Nuclear structure III: Nuclear and neutron matter Stefano Gandolfi Los Alamos National Laboratory (LANL) National Nuclear Physics Summer School Massachusetts Institute of Technology (MIT) July 18-29, 2016 Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 1 / 29
2 Fundamental questions in nuclear physics Physics of nuclei: How do nucleons interact? How are nuclei formed? How can their properties be so different for different A? What s the nature of closed shell numbers, and what s their evolution for neutron rich nuclei? What is the equation of state of dense matter? Can we describe simultaneously 2, 3, and many-body nuclei? Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 2 / 29
3 Uniform nuclear and neutron matter What is nuclear matter? Easy, an infinite system of nucleons! Infinite systems Symmetric nuclear matter: equal protons and neutrons Pure neutron matter: only neutrons W/o Coulomb: homogeneous Nuclear matter saturates (heavy nuclei, bulk ) Neutron matter positive pressure Properties of infinite matter important to constrain energy density functionals Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 3 / 29
4 Neutron Matter Low-density neutron matter and Cold atoms Low-density means ρ << ρ 0 Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 4 / 29
5 Ultracold Fermi atoms Dilute regime mainly s-wave interaction T fraction of T F T 0 Experimentally tunable interaction Crossover from weakly interacting Fermions (paired) to weakly repulsive Bosons (molecules) Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 5 / 29
6 Ultracold Fermi atoms Dilute regime mainly s-wave interaction T fraction of T F T 0 Experimentally tunable interaction Crossover from weakly interacting Fermions (paired) to weakly repulsive Bosons (molecules) Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 5 / 29
7 Ultracold Fermi atoms Dilute regime mainly s-wave interaction T fraction of T F T 0 Experimentally tunable interaction Crossover from weakly interacting Fermions (paired) to weakly repulsive Bosons (molecules) Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 5 / 29
8 Ultracold Fermi atoms Dilute regime mainly s-wave interaction T fraction of T F T 0 Experimentally tunable interaction Crossover from weakly interacting Fermions (paired) to weakly repulsive Bosons (molecules) Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 5 / 29
9 Ultracold Fermi atoms Dilute regime mainly s-wave interaction T fraction of T F T 0 Experimentally tunable interaction Crossover from weakly interacting Fermions (paired) to weakly repulsive Bosons (molecules) Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 5 / 29
10 Scattering length Two-body system with attractive interaction: a < 0 a = a > 0 u(r) u(r) u(r) v(r) v(r) v(r) No bound states 0 r Bound state with E b =0 0 r E b 0 r 2 2m a 2 Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 6 / 29
11 Very Low Density Neutron Matter: cold atoms Low density neutron matter unitary limit: r eff r 0 a, r eff = 0, a = Only one scale: E = ξe FG NN scattering length is large and negative, a = 18.5 fm NN effective range is small, r eff = 2.7 fm Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 7 / 29
12 Very Low Density Neutron Matter: cold atoms Low density neutron matter unitary limit: r eff r 0 a, r eff = 0, a = Only one scale: E = ξe FG Why neutron matter and cold atoms are so similar? NN scattering length is large and negative, a = 18.5 fm NN effective range is small, r eff = 2.7 fm Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 7 / 29
13 Very Low Density Neutron Matter: cold atoms Low density neutron matter unitary limit: r eff r 0 a, r eff = 0, a = Only one scale: E = ξe FG Why neutron matter and cold atoms are so similar? NN scattering length is large and negative, a = 18.5 fm NN effective range is small, r eff = 2.7 fm Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 7 / 29
14 Very Low Density Neutron Matter: cold atoms Low density neutron matter unitary limit: r eff r 0 a, r eff = 0, a = Only one scale: E = ξe FG Why neutron matter and cold atoms are so similar? NN scattering length is large and negative, a = 18.5 fm NN effective range is small, r eff = 2.7 fm BCS crossover BEC 1 Low density neutron matter a k F Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 7 / 29
15 Unitary Fermi gas Exact calculation of ξ using AFMC: ε k (2) ξ ε k (4) 0.34 N = 38 N = 48 N = 66 ε k (h) ρ 1/3 = αn 1/3 /L ξ = 0.372(5) Carlson, Gandolfi, Schmidt, Zhang, PRA 84, (2011) ξ = 0.376(5) Ku, Sommer, Cheuk, Zwierlein, Science 335, 563 (2012) Validation of Quantum Monte Carlo calculations Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 8 / 29
16 Fermi gas and neutron matter E / E FG k F [fm -1 ] Lee-Yang QMC s-wave QMC AV4 Cold Atoms k F a Carlson, Gandolfi, Gezerlis, PTEP 01A209 (2012). Ultracold atoms very useful for nuclear physics! Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 9 / 29
17 BCS and Fermi superfluidity BCS (Bardeen-Cooper-Schrieffer) pairs: an arbitrarily small attraction between Fermions can cause a paired state of particles and the system has a lower energy than the normal gas. Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 10 / 29
18 BCS and Fermi superfluidity BCS (Bardeen-Cooper-Schrieffer) pairs: an arbitrarily small attraction between Fermions can cause a paired state of particles and the system has a lower energy than the normal gas. The (condensate) pairs may then form a superfluid: The pairing gap is basically the energy needed to break a pair, and then excite the system to its normal energy. Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 10 / 29
19 Pairing gap and neutron stars The pairing gap is fundamental for the cooling of neutron stars. Neutron star crust made of nuclei arranged on a lattice surrounded by a gas of neutrons. Specific heat suppressed by superfluidity (similarly to the superconducting mechanism). Cooling dependent to the pairing gap! Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 11 / 29
20 Pairing gap and neutron stars The pairing gap is fundamental for the cooling of neutron stars. Neutron star crust made of nuclei arranged on a lattice surrounded by a gas of neutrons. Specific heat suppressed by superfluidity (similarly to the superconducting mechanism). Cooling dependent to the pairing gap! D. Page (2012) Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 11 / 29
21 Dilute neutron matter The pairing gap is the energy cost to excite one particle from a BCS (collective) state. Pairing gap of low density neutron matter vs cold atoms: k F [fm -1 ] / E F BCS-atoms QMC Unitarity BCS-neutrons Neutron Matter Cold Atoms k F a Gezerlis, Carlson (2008) Cold atoms results confirmed by experiments! Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 12 / 29
22 Neutron Matter Dense neutron matter Dense means ρ (0.5 few times)ρ 0 Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 13 / 29
23 Fermi gas (1/2) Non interacting two-components Fermi gas (non-relativistic): where k F = (3π 2 ρ) 1/3. 2 E N (k F ) = 3 5 2m k2 F = E FG (k F ), Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 14 / 29
24 Fermi gas (1/2) Non interacting two-components Fermi gas (non-relativistic): 2 E N (k F ) = 3 5 2m k2 F = E FG (k F ), where k F = (3π 2 ρ) 1/3. For a system made of neutrons and protons, define: Useful relations: ρ = ρ n + ρ p, ρ p = 1 α ρ, 2 α = ρ n ρ p ρ n + ρ p, ρ n = 1 + α ρ, 2 Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 14 / 29
25 Fermi gas (1/2) Non interacting two-components Fermi gas (non-relativistic): 2 E N (k F ) = 3 5 2m k2 F = E FG (k F ), where k F = (3π 2 ρ) 1/3. For a system made of neutrons and protons, define: Useful relations: ρ = ρ n + ρ p, ρ p = 1 α ρ, 2 The nuclear matter energy is given by: E A (ρ, x) = N E ( ) k (n) F + Z A N A = f (α) E FG (k F ). E ( Z k (p) F ) = 3 5 α = ρ n ρ p ρ n + ρ p, ρ n = 1 + α ρ, 2 2 2m k2 F 1 [(1 + α) 5/3 + (1 α) 5/3] 2 Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 14 / 29
26 Fermi gas (2/2) For small asymmetries, α 0, the function f (α) can be expanded f (α) = α α4 +..., And thus the equation of state is given by: E A (ρ, x) = 3 5 E FG (ρ)+ 5 9 E FG (ρ)α 2 + = E SNM +α 2 S(ρ)+α 4 S 4 (ρ)+..., where E SNM is the energy of symmetric nuclear matter (α = 0) and S(ρ) is the symmetry energy given by 2 S(ρ) = 1 2 α 2 [E(ρ, x)] α=0 E PNM(ρ) E SNM (ρ), and E PNM is the energy of pure neutron matter (α = 1). Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 15 / 29
27 Symmetry energy Around density ρ 0 nuclear matter saturates, thus and we can expand as E SNM (ρ) ρ = 0 ρ=ρ0 ( ) 2 ( ) 3 ρ ρ0 ρ ρ0 E SNM = E 0 + α + β +..., ρ 0 Pure neutron matter instead does not saturate, thus also linear power in ρ is fine. ρ 0 Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 16 / 29
28 Symmetry energy Around density ρ 0 nuclear matter saturates, thus and we can expand as E SNM (ρ) ρ = 0 ρ=ρ0 ( ) 2 ( ) 3 ρ ρ0 ρ ρ0 E SNM = E 0 + α + β +..., ρ 0 Pure neutron matter instead does not saturate, thus also linear power in ρ is fine. Then, around ρ 0 we can expand: ρ 0 E sym = S 0 + L 3 ρ ρ 0 ρ 0 + K sym 18 ( ) 2 ρ ρ0 +..., where L is the slope of the symmetry energy, and K sym is the symmetry compressibility. ρ 0 Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 16 / 29
29 Neutron matter equation of state Why do we care? EOS of neutron matter gives the symmetry energy and its slope. Assume that NN is very good - fit scattering data with very high precision. Three-neutron force (T = 3/2) very weak in light nuclei, while T = 1/2 is the dominant part. No direct T = 3/2 experiments. Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 17 / 29
30 Neutron matter equation of state Why do we care? EOS of neutron matter gives the symmetry energy and its slope. Assume that NN is very good - fit scattering data with very high precision. Three-neutron force (T = 3/2) very weak in light nuclei, while T = 1/2 is the dominant part. No direct T = 3/2 experiments. Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 17 / 29
31 Neutron matter equation of state Why do we care? EOS of neutron matter gives the symmetry energy and its slope. Assume that NN is very good - fit scattering data with very high precision. Three-neutron force (T = 3/2) very weak in light nuclei, while T = 1/2 is the dominant part. No direct T = 3/2 experiments. Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 17 / 29
32 Neutron matter equation of state Why do we care? EOS of neutron matter gives the symmetry energy and its slope. Assume that NN is very good - fit scattering data with very high precision. Three-neutron force (T = 3/2) very weak in light nuclei, while T = 1/2 is the dominant part. No direct T = 3/2 experiments. Pure neutron systems are the probe! Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 17 / 29
33 Neutron matter equation of state Why do we care? EOS of neutron matter gives the symmetry energy and its slope. Assume that NN is very good - fit scattering data with very high precision. Three-neutron force (T = 3/2) very weak in light nuclei, while T = 1/2 is the dominant part. No direct T = 3/2 experiments. Pure neutron systems are the probe! Why to study symmetry energy? Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 17 / 29
34 Neutron matter equation of state Why do we care? EOS of neutron matter gives the symmetry energy and its slope. Assume that NN is very good - fit scattering data with very high precision. Three-neutron force (T = 3/2) very weak in light nuclei, while T = 1/2 is the dominant part. No direct T = 3/2 experiments. Pure neutron systems are the probe! Why to study symmetry energy? Theory Esym, L Neutron stars Experiments Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 17 / 29
35 Neutron matter at nuclear densities At nuclear densities neutron matter cannot be modeled as in the dilute regime. Nucleon-nucleon (and three-nucleon) interaction become very important. Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 18 / 29
36 Neutron matter at nuclear densities At nuclear densities neutron matter cannot be modeled as in the dilute regime. Nucleon-nucleon (and three-nucleon) interaction become very important. Let s start from Hamiltonians used for nuclei: Energy (MeV) He 6 He Li 7/2 5/2 5/2 7/2 1/2 3/2 7 Li Argonne v 18 with UIX or Illinois-7 GFMC Calculations 1 June He Li Be /2 + 5/ / / / / / / /2 1/2 + 3/ Be B AV18 AV18 +UIX AV18 +IL7 Expt. 12 C 0 + Carlson, et al., RMP (2015) Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 18 / 29
37 Scattering data and neutron matter How much can we trust the nucleon-nucleon interactions? In a scattering event with energy E lab two nucleons have k E lab m/2, k F that correspond to k F ρ (E lab m/2) 3/2 2π 2. E lab =150 MeV corresponds to about 0.12 fm 3. E lab =350 MeV to 0.44 fm 3. Argonne potentials useful to study dense matter above ρ 0 =0.16 fm 3, other (soft) interactions not clear Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 19 / 29
38 What is the Symmetry energy? symmetric nuclear matter pure neutron matter Symmetry energy E 0 = -16 MeV 0 ρ 0 = 0.16 fm -3 Nuclear saturation Assumption from experiments: E SNM (ρ 0 ) = 16MeV, ρ 0 = 0.16fm 3, E sym = E PNM (ρ 0 ) + 16 At ρ 0 we access E sym by studying PNM Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 20 / 29
39 Neutron matter Equation of state of neutron matter using the AV8 +UIX Hamiltonian. 120 Energy per Neutron (MeV) exp. E sym AV8 +UIX AV Neutron Density (fm -3 ) Incidentally these can be considered as extremes with respect to the measured E sym. Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 21 / 29
40 Neutron matter Three-neutron interaction uncertainty: 120 Energy per Neutron (MeV) E sym = 33.7 MeV different 3N Neutron Density (fm -3 ) Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 22 / 29
41 Neutron matter Main sources of uncertainties: Experimental: E sym Theoretical: form of three-neutron interaction not totally understood Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 23 / 29
42 Neutron matter Main sources of uncertainties: Experimental: E sym Theoretical: form of three-neutron interaction not totally understood Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 23 / 29
43 Neutron matter Main sources of uncertainties: Experimental: E sym Theoretical: form of three-neutron interaction not totally understood Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 23 / 29
44 Neutron matter Main sources of uncertainties: Experimental: E sym Theoretical: form of three-neutron interaction not totally understood Which one dominates? Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 23 / 29
45 Neutron matter Equation of state of neutron matter using Argonne forces: Energy per Neutron (MeV) E sym = 35.1 MeV (AV8 +UIX) E sym = 33.7 MeV E sym = 32 MeV E sym = 30.5 MeV (AV8 ) Neutron Density (fm -3 ) Gandolfi, Carlson, Reddy, PRC (2012) Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 24 / 29
46 Symmetry energy Many experimental efforts to measure E sym (or S 0 ) and its slope L: Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 25 / 29
47 Neutron matter and symmetry energy From the EOS, we can fit the symmetry energy around ρ 0 using E sym (ρ) = E sym + L 3 ρ AV8 +UIX L (MeV) E sym =32.0 MeV E sym =33.7 MeV 35 AV E sym (MeV) Gandolfi et al., EPJ (2014) Tsang et al., PRC (2012) Very weak dependence to the model of 3N force for a given E sym. Knowing E sym or L useful to constrain 3N! (within this model...) Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 26 / 29
48 Neutron matter and the puzzle of the three-body force Energy (MeV) He 6 1 He + 6 Li Li 7/2 5/2 5/2 7/2 1/2 3/2 Argonne v 18 with UIX or Illinois-7 GFMC Calculations 1 June 2011 AV He Li AV18 +UIX Be AV18 +IL7 7/2 + 5/2 + 7/2 7/2 3/2 3/2 + 5/2 + 1/2 5/2 1/2 + 3/2 9 Be B Expt C 0 + Energy per Neutron [MeV] AV8 +UIX AV8 +IL7 AV ρ [fm -3 ] Note: AV8 +UIX and (almost) AV8 are stiff enough to support observed neutron stars, but AV8 +IL7 too soft. How to reconcile with nuclei??? Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 27 / 29
49 Neutron matter at N2LO EOS of pure neutron matter at N2LO, R 0 =1.0 fm. E/A (MeV) N 2 LO (D2, E1) N 2 LO (D2, EP) N 2 LO (D2, Eτ) n (fm 3 ) Energy per Neutron (MeV) AV8 +UIX AV Neutron Density (fm -3 ) Lynn, et al., PRL (2016). Note: the above (but not all) chiral Hamiltonian able to describe A=3,4,5 nuclei and neutron matter reasonably. Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 28 / 29
50 Summary of this lecture: Low-density neutron matter and cold atoms Superfluidity Dense matter: free Fermi gas Dense matter with interaction Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 29 / 29
51 Summary of this lecture: Low-density neutron matter and cold atoms Superfluidity Dense matter: free Fermi gas Dense matter with interaction Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 29 / 29
52 Summary of this lecture: Low-density neutron matter and cold atoms Superfluidity Dense matter: free Fermi gas Dense matter with interaction Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 29 / 29
53 Summary of this lecture: Low-density neutron matter and cold atoms Superfluidity Dense matter: free Fermi gas Dense matter with interaction Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 29 / 29
54 Summary of this lecture: Low-density neutron matter and cold atoms Superfluidity Dense matter: free Fermi gas Dense matter with interaction Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 29 / 29
55 Summary of this lecture: Low-density neutron matter and cold atoms Superfluidity Dense matter: free Fermi gas Dense matter with interaction End for today... Stefano Gandolfi (LANL) - stefano@lanl.gov Nuclear and neutron matter 29 / 29
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