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) - [email protected] Nuclear and neutron matter 1 / 29 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) - [email protected] Nuclear and neutron matter 2 / 29 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) - [email protected] Nuclear and neutron matter 3 / 29 Neutron Matter Low-density neutron matter and Cold atoms \Low-density" means ρ << ρ0 Stefano Gandolfi (LANL) - [email protected] Nuclear and neutron matter 4 / 29 Ultracold Fermi atoms Dilute regime mainly s-wave interaction ! T fraction of TF T 0 ! ∼ Experimentally tunable interaction Crossover from weakly interacting Fermions (paired) to weakly repulsive Bosons (molecules) Stefano Gandolfi (LANL) - [email protected] Nuclear and neutron matter 5 / 29 Ultracold Fermi atoms Dilute regime mainly s-wave interaction ! T fraction of TF T 0 ! ∼ Experimentally tunable interaction Crossover from weakly interacting Fermions (paired) to weakly repulsive Bosons (molecules) Stefano Gandolfi (LANL) - [email protected] Nuclear and neutron matter 5 / 29 Ultracold Fermi atoms Dilute regime mainly s-wave interaction ! T fraction of TF T 0 ! ∼ Experimentally tunable interaction Crossover from weakly interacting Fermions (paired) to weakly repulsive Bosons (molecules) Stefano Gandolfi (LANL) - [email protected] Nuclear and neutron matter 5 / 29 Ultracold Fermi atoms Dilute regime mainly s-wave interaction ! T fraction of TF T 0 ! ∼ Experimentally tunable interaction Crossover from weakly interacting Fermions (paired) to weakly repulsive Bosons (molecules) Stefano Gandolfi (LANL) - [email protected] Nuclear and neutron matter 5 / 29 Ultracold Fermi atoms Dilute regime mainly s-wave interaction ! T fraction of TF T 0 ! ∼ Experimentally tunable interaction Crossover from weakly interacting Fermions (paired) to weakly repulsive Bosons (molecules) Stefano Gandolfi (LANL) - [email protected] Nuclear and neutron matter 5 / 29 Scattering length Two-body system with attractive interaction: a = ∞ a < 0 a > 0 u(r) u(r) u(r) 0 0 0 v(r) v(r) v(r) 0 0 0 r r r 2 ~ No bound states Bound state with E =0 Eb 2 b ∼ 2m a Stefano Gandolfi (LANL) - [email protected] Nuclear and neutron matter 6 / 29 Why neutron matter and cold atoms are so similar? BCS crossover BEC 1 a k Low density neutron matter F Very Low Density Neutron Matter: cold atoms Low density neutron matter unitary limit: ! reff r0 a ; reff = 0 ; a = j j j j 1 Only one scale: E = ξEFG ! NN scattering length is large and negative, a = 18:5 fm − NN effective range is small, reff = 2:7 fm Stefano Gandolfi (LANL) - [email protected] Nuclear and neutron matter 7 / 29 BCS crossover BEC 1 a k Low density neutron matter F Very Low Density Neutron Matter: cold atoms Low density neutron matter unitary limit: ! reff r0 a ; reff = 0 ; a = j j j j 1 Only one scale: E = ξEFG ! 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, reff = 2:7 fm Stefano Gandolfi (LANL) - [email protected] Nuclear and neutron matter 7 / 29 BCS crossover BEC 1 a k Low density neutron matter F Very Low Density Neutron Matter: cold atoms Low density neutron matter unitary limit: ! reff r0 a ; reff = 0 ; a = j j j j 1 Only one scale: E = ξEFG ! 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, reff = 2:7 fm Stefano Gandolfi (LANL) - [email protected] Nuclear and neutron matter 7 / 29 BCS crossover BEC 1 a k Low density neutron matter F Very Low Density Neutron Matter: cold atoms Low density neutron matter unitary limit: ! reff r0 a ; reff = 0 ; a = j j j j 1 Only one scale: E = ξEFG ! 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, reff = 2:7 fm Stefano Gandolfi (LANL) - [email protected] Nuclear and neutron matter 7 / 29 Unitary Fermi gas Exact calculation of ξ using AFMC: 0.42 (2) ε 0.40 k 0.38 (4) ξ εk 0.36 N = 38 0.34 N = 48 N = 66 (h) εk 0.32 0 0.1 0.2 0.3 0.4 1/3 1/3 ρ = αN /L ξ = 0:372(5) Carlson, Gandolfi, Schmidt, Zhang, PRA 84, 061602 (2011) ξ = 0:376(5) Ku, Sommer, Cheuk, Zwierlein, Science 335, 563 (2012) Validation of Quantum Monte Carlo calculations Stefano Gandolfi (LANL) - [email protected] Nuclear and neutron matter 8 / 29 Fermi gas and neutron matter -1 kF [fm ] 0 0.1 0.2 0.3 0.4 0.5 1 Lee-Yang QMC s-wave 0.9 QMC AV4 Cold Atoms 0.8 FG 0.7 E / 0.6 0.5 0.4 0 2 4 6 8 10 - kF a Carlson, Gandolfi, Gezerlis, PTEP 01A209 (2012). Ultracold atoms very useful for nuclear physics! Stefano Gandolfi (LANL) - [email protected] Nuclear and neutron matter 9 / 29 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. 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) - [email protected] Nuclear and neutron matter 10 / 29 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) - [email protected] Nuclear and neutron matter 10 / 29 D. Page (2012) 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) - [email protected] Nuclear and neutron matter 11 / 29 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) - [email protected] Nuclear and neutron matter 11 / 29 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: -1 kF [fm ] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.6 BCS-atoms QMC Unitarity 0.5 F 0.4 / E BCS-neutrons ∆ 0.3 0.2 Neutron Matter 0.1 Cold Atoms 0 0 2 4 6 8 10 12 - kF a Gezerlis, Carlson (2008) Cold atoms results confirmed by experiments! Stefano Gandolfi (LANL) - [email protected] Nuclear and neutron matter 12 / 29 Neutron Matter Dense neutron matter \Dense" means ρ (0:5 few times)ρ0 ∼ − Stefano Gandolfi (LANL) - [email protected] Nuclear and neutron matter 13 / 29 For a system made of neutrons and protons, define: ρn ρp ρ = ρn + ρp ; α = − ; ρn + ρp Useful relations: 1 α 1 + α ρ = − ρ ; ρ = ρ ; p 2 n 2 The nuclear matter energy is given by: E N E Z E 3 2 1 h i (ρ, x) = k(n) + k(p) = ~ k2 (1 + α)5=3 + (1 α)5=3 A A N F A Z F 5 2m F 2 − = f (α) EFG (kF ) : Fermi gas (1/2) Non interacting two-components Fermi gas (non-relativistic): E 3 2 (k ) = ~ k2 = E (k ) ; N F 5 2m F FG F 2 1=3 where kF = (3π ρ) . Stefano Gandolfi (LANL) - [email protected] Nuclear and neutron matter 14 / 29 The nuclear matter energy is given by: E N E Z E 3 2 1 h i (ρ, x) = k(n) + k(p) = ~ k2 (1 + α)5=3 + (1 α)5=3 A A N F A Z F 5 2m F 2 − = f (α) EFG (kF ) : Fermi gas (1/2) Non interacting two-components Fermi gas (non-relativistic): E 3 2 (k ) = ~ k2 = E (k ) ; N F 5 2m F FG F 2 1=3 where kF = (3π ρ) . For a system made of neutrons and protons, define: ρn ρp ρ = ρn + ρp ; α = − ; ρn + ρp Useful relations: 1 α 1 + α ρ = − ρ ; ρ = ρ ; p 2 n 2 Stefano Gandolfi (LANL) - [email protected] Nuclear and neutron matter 14 / 29 Fermi gas (1/2) Non interacting two-components Fermi gas (non-relativistic): E 3 2 (k ) = ~ k2 = E (k ) ; N F 5 2m F FG F 2 1=3 where kF = (3π ρ) .