Clusters in Low-Density Nuclear Matter

Clusters in Low-Density Nuclear Matter Excellence Cluster Universe, Technische Universität München GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt Helmholtz International Summer School: Nuclear Theory and Astrophysical Applications Bogoliubov Laboratory of Theoretical Physics Joint Institute for Nuclear Research Dubna, Russia July 25 - August 2, 2011

Motivation: The Life and Death of Stars when the fuel for nuclear fusion reactions is consumed: last phases in the life of a massive star (8M sun M star 30M sun ) core-collapse supernova neutron star or black hole X-ray: NASA/CXC/J.Hester (ASU) Optical: NASA/ESA/J.Hester & A.Loll (ASU) Infrared: NASA/JPL-Caltech/R.Gehrz (Univ. Minn.) NASA/ESA/R.Sankrit & W.Blair (Johns Hopkins Univ.) Clusters in Low-Density Nuclear Matter - 2

Motivation: The Life and Death of Stars when the fuel for nuclear fusion reactions is consumed: last phases in the life of a massive star (8M sun M star 30M sun ) core-collapse supernova neutron star or black hole essential ingredient in astrophysical model calculations: equation of state (EoS) of dense matter dynamical evolution of supernova static properties of neutron star conditions for nucleosynthesis energetics, chemical composition, transport properties,... X-ray: NASA/CXC/J.Hester (ASU) Optical: NASA/ESA/J.Hester & A.Loll (ASU) Infrared: NASA/JPL-Caltech/R.Gehrz (Univ. Minn.) NASA/ESA/R.Sankrit & W.Blair (Johns Hopkins Univ.) Clusters in Low-Density Nuclear Matter - 2

Motivation: EoS - Thermodynamical Conditions Relevant Parameters: Baryon density, log 10 (ρ [g/cm 3 ]) densities: 10 9 / sat 10 with nuclear saturation density sat 2.5 10 14 g/cm 3 (n sat = sat /m n 0.15 fm 3 ) temperatures: 0 MeV k B T 50 MeV (ˆ= 5.8 10 11 K) Temperature, T [MeV] 10 2 10 1 10 0 6 7 8 9 10 11 12 13 14 15 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 electron fraction: 0 Y e 0.6 10 1 10 9 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 1 10 0 Baryon density, n [fm 3 ] B Y e 0.1 0.05 T. Fischer, GSI Darmstadt Clusters in Low-Density Nuclear Matter - 3

Motivation: EoS - Thermodynamical Conditions Most Relevant Particles: (at finite temperatures and not too high densities) 10 2 Baryon density, log 10 (ρ [g/cm 3 ]) 6 7 8 9 10 11 12 13 14 15 0.5 neutrons, protons nuclei electrons, (muons) (charge neutrality! Y e Y p ) Temperature, T [MeV] 10 1 10 0 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 10 1 10 9 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 1 10 0 Baryon density, n [fm 3 ] B Y e T. Fischer, GSI Darmstadt Clusters in Low-Density Nuclear Matter - 3

Motivation: EoS - Thermodynamical Conditions Most Relevant Particles: (at finite temperatures and not too high densities) neutrons, protons nuclei electrons, (muons) (charge neutrality! Y e Y p ) Timescale of Reactions Timescale of System Evolution thermodynamical equilibrium construction of equation of state here: consider system of nucleons and (light) nuclei Temperature, T [MeV] 10 2 10 1 10 0 10 1 Baryon density, log 10 (ρ [g/cm 3 ]) 6 7 8 9 10 11 12 13 14 15 10 9 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 1 10 0 Baryon density, n [fm 3 ] B T. Fischer, GSI Darmstadt Y e 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 Clusters in Low-Density Nuclear Matter - 3

Questions How does the composition of matter change with density, temperature and neutron-proton asymmetry? How do interactions between particles modify the thermodynamical properties of matter? How do the properties of nuclei change in the medium? Clusters in Low-Density Nuclear Matter - 4

Outline Theory Different Theoretical Concepts Overview of Methods/Conventions Thermodynamical Relations Nuclear Statistical Equilibrium Virial Equation of State ( more details, examples in seminar) Generalized Beth-Uhlenbeck Approach Generalized Relativistic Density Functional Applications Formation and Dissolution of Clusters Thermodynamical Properties Symmetry Energy of Nuclear Matter Clusters in Low-Density Nuclear Matter - 5

Different Theoretical Concepts Composition of Nuclear Matter depends strongly on density, temperature and neutron-proton asymmetry affects thermodynamical properties Theoretical Models: different points of view Clusters in Low-Density Nuclear Matter - 6

Different Theoretical Concepts Composition of Nuclear Matter depends strongly on density, temperature and neutron-proton asymmetry affects thermodynamical properties Theoretical Models: different points of view chemical picture mixture of different nuclear species and nucleons in chemical equilibrium properties of constituents independent of medium interaction between particles? dissolution of nuclei at high densities? n n p t p n d n p n p p α p n n p n p Clusters in Low-Density Nuclear Matter - 6

Different Theoretical Concepts Composition of Nuclear Matter depends strongly on density, temperature and neutron-proton asymmetry affects thermodynamical properties Theoretical Models: different points of view chemical picture mixture of different nuclear species and nucleons in chemical equilibrium properties of constituents independent of medium interaction between particles? dissolution of nuclei at high densities? n n p t p n d n p n p p α p n n p n p physical picture interaction between nucleons correlations formation of bound states/resonances treatment of two-, three-,... many-body correlations? choice of interaction? n n n p n n p n n p p p n p p n p p n p n n p n p Clusters in Low-Density Nuclear Matter - 6

Overview of Methods/Conventions Improving the description step by step: Clusters in Low-Density Nuclear Matter - 7

Overview of Methods/Conventions Improving the description step by step: most simple approach, ideal mixture of independent particles, no interaction Nuclear Statistical Equilibrium Clusters in Low-Density Nuclear Matter - 7

Overview of Methods/Conventions Improving the description step by step: most simple approach, ideal mixture of independent particles, no interaction Nuclear Statistical Equilibrium low-density limit, with interactions/correlations Virial Equation of State Clusters in Low-Density Nuclear Matter - 7

Overview of Methods/Conventions Improving the description step by step: most simple approach, ideal mixture of independent particles, no interaction Nuclear Statistical Equilibrium low-density limit, with interactions/correlations Virial Equation of State consider medium effects with increasing density Generalized Beth-Uhlenbeck Approach Clusters in Low-Density Nuclear Matter - 7

Overview of Methods/Conventions Improving the description step by step: most simple approach, ideal mixture of independent particles, no interaction Nuclear Statistical Equilibrium low-density limit, with interactions/correlations Virial Equation of State consider medium effects with increasing density Generalized Beth-Uhlenbeck Approach connecting to densities around nuclear saturation Generalized Relativistic Density Functional Clusters in Low-Density Nuclear Matter - 7

Overview of Methods/Conventions Improving the description step by step: most simple approach, ideal mixture of independent particles, no interaction Nuclear Statistical Equilibrium low-density limit, with interactions/correlations Virial Equation of State consider medium effects with increasing density Generalized Beth-Uhlenbeck Approach connecting to densities around nuclear saturation Generalized Relativistic Density Functional system of units such that = c = k B = 1 Clusters in Low-Density Nuclear Matter - 7

Thermodynamical Relations Grand Canonical Ensemble particles i with chemical potentials µ i at Temperature T in Volume V natural variables: µ i,t, V Clusters in Low-Density Nuclear Matter - 8

Thermodynamical Relations Grand Canonical Ensemble particles i with chemical potentials µ i at Temperature T in Volume V natural variables: µ i,t, V thermodynamical potential: grand canonical potential Ω = Ω(µ i, T,V ) = pv contains all information of the system Clusters in Low-Density Nuclear Matter - 8

Thermodynamical Relations Grand Canonical Ensemble particles i with chemical potentials µ i at Temperature T in Volume V natural variables: µ i,t, V thermodynamical potential: grand canonical potential Ω = Ω(µ i, T,V ) = pv contains all information of the system equations of state: n i (µ i, T,V ) = N i V = 1 V S(µ i,t, V ) = Ω T µi,v Ω µ i T,V entropy particle number densities p(µ i, T, V ) = Ω V µi,v = Ω V pressure Clusters in Low-Density Nuclear Matter - 8

Thermodynamical Relations Grand Canonical Ensemble particles i with chemical potentials µ i at Temperature T in Volume V natural variables: µ i,t, V thermodynamical potential: grand canonical potential Ω = Ω(µ i, T,V ) = pv contains all information of the system equations of state: n i (µ i, T,V ) = N i V = 1 V S(µ i,t, V ) = Ω T µi,v Ω µ i T,V entropy particle number densities p(µ i, T, V ) = Ω V µi,v = Ω V pressure connection to microphysics: grand canonical partition function [ Z(µ i,t, V ) = traceexp β (Ĥ i µ ˆN )] i i Ω = T ln Z with Hamilton operator Ĥ, particle number operators ˆN i and β = 1/T Clusters in Low-Density Nuclear Matter - 8

Nuclear Statistical Equilibrium (NSE) most simple approach ideal mixture of nucleons (p,n) and nuclei X in chemical equilibrium Zp + Nn A ZX N Z µ p + N µ n = µ X or Zµ p + Nµ n = µ X B X with relativistic chemical potentials µ i = µ i + m i and binding energy B X of nucleus X with mass m X Clusters in Low-Density Nuclear Matter - 9

Nuclear Statistical Equilibrium (NSE) most simple approach ideal mixture of nucleons (p,n) and nuclei X in chemical equilibrium Zp + Nn A ZX N Z µ p + N µ n = µ X or Zµ p + Nµ n = µ X B X with relativistic chemical potentials µ i = µ i + m i and binding energy B X of nucleus X with mass m X independent particles without mutual interaction factorization Z = i Z i Clusters in Low-Density Nuclear Matter - 9

Nuclear Statistical Equilibrium (NSE) most simple approach ideal mixture of nucleons (p,n) and nuclei X in chemical equilibrium Zp + Nn A ZX N Z µ p + N µ n = µ X or Zµ p + Nµ n = µ X B X with relativistic chemical potentials µ i = µ i + m i and binding energy B X of nucleus X with mass m X independent particles without mutual interaction factorization Z = i Z i Maxwell-Boltzmann statistics 1 Z i = N i! QN i i z N i i = exp(q i z i ) with fugacities z i = exp N i =0 and single-particle canonical partition functions Q i = traceexp with single-particle Hamilton operator Ĥ i ( µi T ) ( ) βĥi Clusters in Low-Density Nuclear Matter - 9

Nuclear Statistical Equilibrium (NSE) nonrelativistic kinematics: Ĥ i = ˆp 2 i /(2m i) E i = p 2 i /(2m i) Q i = g ( i d 3 r (2π) 3 i d 3 p i exp E ) i V = g i T λ 3 i with degeneracy factor g i = (2J i + 1) (= 2 for nucleons) and thermal wavelength λ i = 2π m i T Clusters in Low-Density Nuclear Matter - 10

Nuclear Statistical Equilibrium (NSE) nonrelativistic kinematics: Ĥ i = ˆp 2 i /(2m i) E i = p 2 i /(2m i) Q i = g ( i d 3 r (2π) 3 i d 3 p i exp E ) i V = g i T λ 3 i with degeneracy factor g i = (2J i + 1) (= 2 for nucleons) and thermal wavelength λ i = 2π m i T Z = ( ) V exp g i λ 3 z i and Ω = T ln Z = TV i i i g i λ 3 i exp ( µi T ) Clusters in Low-Density Nuclear Matter - 10

Nuclear Statistical Equilibrium (NSE) nonrelativistic kinematics: Ĥ i = ˆp 2 i /(2m i) E i = p 2 i /(2m i) Q i = g ( i d 3 r (2π) 3 i d 3 p i exp E ) i V = g i T λ 3 i with degeneracy factor g i = (2J i + 1) (= 2 for nucleons) and thermal wavelength λ i = 2π m i T Z = ( ) V exp g i λ 3 z i and Ω = T ln Z = TV i i i including excited states (x) of nuclei g i g i (T) = (2J gs i (2J gs i + 1) + + 1) + x de i(e)exp (2J x + 1)exp ( E T ) ( E ) x T g i λ 3 i exp with level density i(e) ( µi T ) Clusters in Low-Density Nuclear Matter - 10

Nuclear Statistical Equilibrium (NSE) equations of state n i (µ i, T,V ) = 1 V Ω µ i = g i T,V λ 3 i exp ( µi T ) particle number densities Ω = TV i n i Clusters in Low-Density Nuclear Matter - 11

Nuclear Statistical Equilibrium (NSE) equations of state n i (µ i, T,V ) = 1 V Ω µ i = g i T,V λ 3 i exp ( µi T ) particle number densities Ω = TV i n i S(µ i,t, V ) = Ω T µi,v = 5 Ω 2T V T µ i n i entropy i Clusters in Low-Density Nuclear Matter - 11

Nuclear Statistical Equilibrium (NSE) equations of state n i (µ i, T,V ) = 1 V Ω µ i = g i T,V λ 3 i exp ( µi T ) particle number densities Ω = TV i n i S(µ i,t, V ) = Ω T p(µ i,t, V ) = Ω V µi,v µi,v = 5 Ω 2T V T = Ω V pressure µ i n i entropy i pv = NT with N = i N i Clusters in Low-Density Nuclear Matter - 11

Nuclear Statistical Equilibrium (NSE) equations of state n i (µ i, T,V ) = 1 V Ω µ i = g i T,V λ 3 i exp ( µi T ) particle number densities Ω = TV i n i S(µ i,t, V ) = Ω T p(µ i,t, V ) = Ω V µi,v µi,v = 5 Ω 2T V T = Ω V pressure µ i n i entropy i pv = NT with N = i N i E = TS pv + i µ i N i = 3 NT internal energy 2 mixture of ideal gases Clusters in Low-Density Nuclear Matter - 11

Nuclear Statistical Equilibrium (NSE) example ideal mixture of neutrons, protons and deuterons (d = 2 H) p + n d µ p + µ n = µ d B d with deuteron binding energy B d = 2.225 MeV Clusters in Low-Density Nuclear Matter - 12

Nuclear Statistical Equilibrium (NSE) example ideal mixture of neutrons, protons and deuterons (d = 2 H) p + n d µ p + µ n = µ d B d with deuteron binding energy B d = 2.225 MeV particle number densities n n,p = 2 exp ( µ n,p ) λ 3 n,p T n d = 3 λ 3 d law of mass action n d = 3 ( )3 2πmd 2 ( ) Bd exp n n n p 4 m n m p T T exp ( µ d ) T Clusters in Low-Density Nuclear Matter - 12

Nuclear Statistical Equilibrium (NSE) example ideal mixture of neutrons, protons and deuterons (d = 2 H) p + n d µ p + µ n = µ d B d with deuteron binding energy B d = 2.225 MeV particle number densities n n,p = 2 exp ( µ n,p ) λ 3 n,p T n d = 3 λ 3 d law of mass action n d = 3 ( )3 2πmd 2 ( ) Bd exp n n n p 4 m n m p T T exp ( µ d ) T symmetric nuclear matter (n n = n p ) total nucleon density n = n n + n p + 2n d deuteron fraction X d = 2n d n deuteron fraction 2n d /n 1.0 0.8 0.6 0.4 0.2 T = 1 MeV T = 2 MeV T = 5 MeV T = 10 MeV T = 20 MeV Y p = 0.5 0.0 10-6 10-5 10-4 10-3 10-2 10-1 10 0 nucleon density n [fm -3 ] Clusters in Low-Density Nuclear Matter - 12

Virial Equation of State (VEoS) consider two-, (three-,... many-) body correlations! expansion of grand canonical partition function in powers of fugacities z i = exp ( µ i ) T Z(µ i,t, V ) = 1 + Q i z i + 1 Q ij z i z j + 1 Q ijk z i z j z k +... 2 6 i ij ijk Clusters in Low-Density Nuclear Matter - 13

Virial Equation of State (VEoS) consider two-, (three-,... many-) body correlations! expansion of grand canonical partition function in powers of fugacities z i = exp ( µ i ) T Z(µ i,t, V ) = 1 + Q i z i + 1 Q ij z i z j + 1 Q ijk z i z j z k +... 2 6 i ij with one-, two-, three-,... many-body canonical partition functions ) ) Q i = trace 1 exp ( βĥ i Q ij = trace 2 exp ( βĥ ij ( Q ijk = trace 3 exp βĥijk ) ijk and Hamiltonians Ĥi, Ĥ ij, Ĥ ijk,... Clusters in Low-Density Nuclear Matter - 13

Virial Equation of State (VEoS) consider two-, (three-,... many-) body correlations! expansion of grand canonical partition function in powers of fugacities z i = exp ( µ i ) T Z(µ i,t, V ) = 1 + Q i z i + 1 Q ij z i z j + 1 Q ijk z i z j z k +... 2 6 i ij with one-, two-, three-,... many-body canonical partition functions ) ) Q i = trace 1 exp ( βĥ i Q ij = trace 2 exp ( βĥ ij ( Q ijk = trace 3 exp βĥijk expansion valid only for z i 1 ) ijk and Hamiltonians Ĥi, Ĥ ij, Ĥ ijk,... Clusters in Low-Density Nuclear Matter - 13

Virial Equation of State (VEoS) consider two-, (three-,... many-) body correlations! expansion of grand canonical partition function in powers of fugacities z i = exp ( µ i ) T Z(µ i,t, V ) = 1 + Q i z i + 1 Q ij z i z j + 1 Q ijk z i z j z k +... 2 6 i ij with one-, two-, three-,... many-body canonical partition functions ) ) Q i = trace 1 exp ( βĥ i Q ij = trace 2 exp ( βĥ ij ( Q ijk = trace 3 exp βĥijk expansion valid only for z i 1 for independent particles without interaction: Ĥ ij = Ĥ i + Ĥ j, Ĥ ijk = Ĥ i + Ĥ j + Ĥ k,... ) ijk and Hamiltonians Ĥi, Ĥ ij, Ĥ ijk,... factorization Q ij = Q i Q j Q ijk = Q i Q j Q k... Clusters in Low-Density Nuclear Matter - 13

Virial Equation of State (VEoS) use ln(1 + x) = x 1 2 x2 + 1 3 x3... ln Z = Q i z i + 1 2 i (Q ij Q i Q j )z i z j +... ij Clusters in Low-Density Nuclear Matter - 14

Virial Equation of State (VEoS) use ln(1 + x) = x 1 2 x2 + 1 3 x3... ln Z = i Q i z i + 1 2 (Q ij Q i Q j )z i z j +... ij introduce (dimensionless) cluster (virial) coefficients b i = g i b ij = λ 3/2 i λ 3/2 j (Q ij Q i Q j ) /(2V ) b ijk = λ i λ j λ k (Q ijk Q i Q jk Q j Q ik Q k Q ij + 2Q i Q j Q k ) /(6V ) Clusters in Low-Density Nuclear Matter - 14

Virial Equation of State (VEoS) use ln(1 + x) = x 1 2 x2 + 1 3 x3... ln Z = i Q i z i + 1 2 (Q ij Q i Q j )z i z j +... ij introduce (dimensionless) cluster (virial) coefficients b i = g i b ij = λ 3/2 i λ 3/2 j (Q ij Q i Q j ) /(2V ) b ijk = λ i λ j λ k (Q ijk Q i Q jk Q j Q ik Q k Q ij + 2Q i Q j Q k ) /(6V ) grand canonical potential Ω = T ln Z = TV i b i z i λ 3 i + ij b ij z i z j λ 3/2 i λ 3/2 j + ijk z i z j z k b ijk +... λ i λ j λ k Clusters in Low-Density Nuclear Matter - 14

Virial Equation of State (VEoS) use ln(1 + x) = x 1 2 x2 + 1 3 x3... ln Z = i Q i z i + 1 2 (Q ij Q i Q j )z i z j +... ij introduce (dimensionless) cluster (virial) coefficients b i = g i b ij = λ 3/2 i λ 3/2 j (Q ij Q i Q j ) /(2V ) b ijk = λ i λ j λ k (Q ijk Q i Q jk Q j Q ik Q k Q ij + 2Q i Q j Q k ) /(6V ) grand canonical potential Ω = T ln Z = TV i b i z i λ 3 i + ij b ij z i z j λ 3/2 i λ 3/2 j + ijk z i z j z k b ijk +... λ i λ j λ k b ij (T), b ijk (T) encode effects of two- and three-body correlations no correlations independent particles b ij = 0, b ijk = 0,... Clusters in Low-Density Nuclear Matter - 14

Virial Equation of State (VEoS) particle number densities n i = 1 Ω z i V µ i = b i T,V λ 3 i + 2 j b ij z i z j + 3 λ 3/2 i λ 3/2 j jk contributions from free particles and correlated particles b ijk z i z j z k λ i λ j λ k +... Clusters in Low-Density Nuclear Matter - 15

Virial Equation of State (VEoS) particle number densities n i = 1 Ω z i V µ i = b i T,V λ 3 i + 2 j b ij z i z j + 3 λ 3/2 i λ 3/2 j jk contributions from free particles and correlated particles virial equation of state pv NT = Ω i n iv T b ijk z i z j z k λ i λ j λ k +... = 1 + ij a ij (λ 3 in i ) 1/2 (λ 3 jn j ) 1/2 + ijk a ijk (λ 3 in i ) 2/3 (λ 3 jn j ) 2/3 (λ 3 kn k ) 2/3 +... with virial coefficients a ij, total particle number N = i N i by eliminating fugacities Clusters in Low-Density Nuclear Matter - 15

Virial Equation of State (VEoS) particle number densities n i = 1 Ω z i V µ i = b i T,V λ 3 i + 2 j b ij z i z j + 3 λ 3/2 i λ 3/2 j jk contributions from free particles and correlated particles virial equation of state pv NT = Ω i n iv T b ijk z i z j z k λ i λ j λ k +... = 1 + ij a ij (λ 3 in i ) 1/2 (λ 3 jn j ) 1/2 + ijk a ijk (λ 3 in i ) 2/3 (λ 3 jn j ) 2/3 (λ 3 kn k ) 2/3 +... with virial coefficients a ij, total particle number N = i N i by eliminating fugacities final task: determine cluster (virial) coefficients! b ij simple! b ijk,... difficult Clusters in Low-Density Nuclear Matter - 15

Virial Equation of State (VEoS) determination of second cluster (virial) coefficient interaction between two-particles independent of c.m. momentum transformation to c.m. and relative coordinates Clusters in Low-Density Nuclear Matter - 16

Virial Equation of State (VEoS) determination of second cluster (virial) coefficient interaction between two-particles independent of c.m. momentum transformation to c.m. and relative coordinates R ij = 1 M ij (m i r i + m j r j ) P ij = p i + p j r ij = r i r j p ij = µ ij ( pi m i p j m j ) with total mass M ij = m i + m j and reduced mass µ ij = m i m j /M ij Clusters in Low-Density Nuclear Matter - 16

Virial Equation of State (VEoS) determination of second cluster (virial) coefficient interaction between two-particles independent of c.m. momentum transformation to c.m. and relative coordinates R ij = 1 M ij (m i r i + m j r j ) P ij = p i + p j r ij = r i r j p ij = µ ij ( pi m i p j m j ) with total mass M ij = m i + m j and reduced mass µ ij = m i m j /M ij second cluster (virial) coefficient in classical mechanics b ij = λ 3/2 i λ 3/2 j (Q ij Q i Q j ) /(2V ) = 1 2 g ij λ 3/2 i λ 3/2 j d 3 r ij {exp[ βv ij (r ij )] 1} with two-body Hamiltonian H ij = p2 i 2m i + p2 j 2m j + V ij (r ij ) = P 2 ij 2M ij + p2 ij 2µ ij + V ij (r ij ) Clusters in Low-Density Nuclear Matter - 16

Virial Equation of State (VEoS) second cluster (virial) coefficient in quantum mechanics (G. E. Beth and E. Uhlenbeck Physica 3 (1936) 729, Physica 4 (1937) 915) Clusters in Low-Density Nuclear Matter - 17

Virial Equation of State (VEoS) second cluster (virial) coefficient in quantum mechanics (G. E. Beth and E. Uhlenbeck Physica 3 (1936) 729, Physica 4 (1937) 915) ) b ij (T) = 1 + δ ij 2 ( ) 3/2 mi + m j mi m j with two-body density of states de D ij (E) exp ( E T D ij (E) = k g (ij) k δ(e E (ij) k ) + l g (ij) l π dδ (ij) l de with contributions of bound states at energies E (ij) k < 0 and scattering states with phase shifts δ (ij) l (E) Clusters in Low-Density Nuclear Matter - 17

Virial Equation of State (VEoS) second cluster (virial) coefficient in quantum mechanics (G. E. Beth and E. Uhlenbeck Physica 3 (1936) 729, Physica 4 (1937) 915) ) b ij (T) = 1 + δ ij 2 ( ) 3/2 mi + m j mi m j with two-body density of states de D ij (E) exp ( E T D ij (E) = k g (ij) k δ(e E (ij) k ) + l g (ij) l π dδ (ij) l de with contributions of bound states at energies E (ij) k < 0 and scattering states with phase shifts δ (ij) l (E) experimental bound state energies/phase shifts available low-density behaviour of EoS established model-independently (see e.g. C. J. Horowitz, A. Schwenk, Nucl. Phys. A 776 (2006) 55) Clusters in Low-Density Nuclear Matter - 17

Virial Equation of State (VEoS) second cluster (virial) coefficient in quantum mechanics (G. E. Beth and E. Uhlenbeck Physica 3 (1936) 729, Physica 4 (1937) 915) ) b ij (T) = 1 + δ ij 2 ( ) 3/2 mi + m j mi m j with two-body density of states de D ij (E) exp ( E T D ij (E) = k g (ij) k δ(e E (ij) k ) + l g (ij) l π dδ (ij) l de with contributions of bound states at energies E (ij) k < 0 and scattering states with phase shifts δ (ij) l (E) experimental bound state energies/phase shifts available low-density behaviour of EoS established model-independently (see e.g. C. J. Horowitz, A. Schwenk, Nucl. Phys. A 776 (2006) 55) limitation: n i λ 3 i 1 (very) low densities Clusters in Low-Density Nuclear Matter - 17

Virial Equation of State (VEoS) possible topics for seminar: explicit calculation of second cluster (virial) coefficient b ij (T) with effective-range expansion for phase shifts in two-nucleon system neutron matter and unitary limit relativistic corrections effects of Fermi-Dirac/Bose-Einstein statistics Clusters in Low-Density Nuclear Matter - 18

Generalized Beth-Uhlenbeck Approach extension to higher densities consider effects of the medium (M. Schmidt, G. Röpke, H. Schulz, Ann. Phys. (N.Y.) 202 (1990) 57) Clusters in Low-Density Nuclear Matter - 19

Generalized Beth-Uhlenbeck Approach extension to higher densities consider effects of the medium (M. Schmidt, G. Röpke, H. Schulz, Ann. Phys. (N.Y.) 202 (1990) 57) use quantum statistical (QS) approach with thermodynamic Green s functions Clusters in Low-Density Nuclear Matter - 19

Generalized Beth-Uhlenbeck Approach extension to higher densities consider effects of the medium (M. Schmidt, G. Röpke, H. Schulz, Ann. Phys. (N.Y.) 202 (1990) 57) use quantum statistical (QS) approach with thermodynamic Green s functions derive equation of state of interacting many-body system from single-particle Green s function Clusters in Low-Density Nuclear Matter - 19

Generalized Beth-Uhlenbeck Approach extension to higher densities consider effects of the medium (M. Schmidt, G. Röpke, H. Schulz, Ann. Phys. (N.Y.) 202 (1990) 57) use quantum statistical (QS) approach with thermodynamic Green s functions derive equation of state of interacting many-body system from single-particle Green s function Green s function of noninteracting single-particle state j ( p j,σ j, τ j ) G 0 (j,z) = [z E(j)] 1 with E(j) = p 2 j /(2m j) Clusters in Low-Density Nuclear Matter - 19

Generalized Beth-Uhlenbeck Approach extension to higher densities consider effects of the medium (M. Schmidt, G. Röpke, H. Schulz, Ann. Phys. (N.Y.) 202 (1990) 57) use quantum statistical (QS) approach with thermodynamic Green s functions derive equation of state of interacting many-body system from single-particle Green s function Green s function of noninteracting single-particle state j ( p j,σ j, τ j ) G 0 (j,z) = [z E(j)] 1 with E(j) = p 2 j /(2m j) Green s function G(j, z) of interacting single-particle state j G(j, z) = G 0 (j, z) + G 0 (j,z)σ(j, z)g(j, z) (Dyson equation) with self-energy Σ(j, z) (contains information on the interaction) Clusters in Low-Density Nuclear Matter - 19

Generalized Beth-Uhlenbeck Approach extension to higher densities consider effects of the medium (M. Schmidt, G. Röpke, H. Schulz, Ann. Phys. (N.Y.) 202 (1990) 57) use quantum statistical (QS) approach with thermodynamic Green s functions derive equation of state of interacting many-body system from single-particle Green s function Green s function of noninteracting single-particle state j ( p j,σ j, τ j ) G 0 (j,z) = [z E(j)] 1 with E(j) = p 2 j /(2m j) Green s function G(j, z) of interacting single-particle state j G(j, z) = G 0 (j, z) + G 0 (j,z)σ(j, z)g(j, z) (Dyson equation) with self-energy Σ(j, z) (contains information on the interaction) consider spectral function A(j,E) = i[g(j, E + i0) G(j, E i0)] probability distribution in E, p replaces dispersion relation E j = E j ( p j ) of particle j in non-interacting system Clusters in Low-Density Nuclear Matter - 19

Generalized Beth-Uhlenbeck Approach total particle density equation of state n(µ,t, V ) = de 2π f +(E)A(j, E) j with Fermi distribution function f + (E) = {exp[β (E µ)] + 1} 1 Clusters in Low-Density Nuclear Matter - 20

Generalized Beth-Uhlenbeck Approach total particle density equation of state n(µ,t, V ) = de 2π f +(E)A(j, E) j with Fermi distribution function f + (E) = {exp[β (E µ)] + 1} 1 formal solution of Dyson equation G(j, z) = [z E(j) Σ(j, z)] 1 with self-energy Σ = Σ R + iσ I A(j,E) = 2Σ I (j,e i0) [E E(j) Σ R (j,e)] 2 +[Σ I (j,e i0)] 2 Clusters in Low-Density Nuclear Matter - 20

Generalized Beth-Uhlenbeck Approach total particle density equation of state n(µ,t, V ) = de 2π f +(E)A(j, E) j with Fermi distribution function f + (E) = {exp[β (E µ)] + 1} 1 formal solution of Dyson equation G(j, z) = [z E(j) Σ(j, z)] 1 with self-energy Σ = Σ R + iσ I A(j,E) = 2Σ I (j,e i0) [E E(j) Σ R (j,e)] 2 +[Σ I (j,e i0)] 2 expansion of A(j, E) for small Σ I = ε with ε = πδ(x) ε d x 2 +ε 2 dx P 1 x +... A(j,E) = 2πδ [E E(j) Σ R (j, E)] quasiparticle contribution 2Σ I (j,e i0) d +... correlation contribution dx P 1 x x=e E(j) ΣR (j,e) Clusters in Low-Density Nuclear Matter - 20

Generalized Beth-Uhlenbeck Approach total particle density equation of state n(µ,t, V ) = de 2π f +(E)A(j, E) j with Fermi distribution function f + (E) = {exp[β (E µ)] + 1} 1 formal solution of Dyson equation G(j, z) = [z E(j) Σ(j, z)] 1 with self-energy Σ = Σ R + iσ I A(j,E) = 2Σ I (j,e i0) [E E(j) Σ R (j,e)] 2 +[Σ I (j,e i0)] 2 expansion of A(j, E) for small Σ I = ε with ε = πδ(x) ε d x 2 +ε 2 dx P 1 x +... A(j,E) = 2πδ [E E(j) Σ R (j, E)] quasiparticle contribution 2Σ I (j,e i0) d +... correlation contribution dx P 1 x x=e E(j) ΣR (j,e) define quasparticle energy e(j): solution of e(j) = E(j) + Σ R (j, e(j)) Clusters in Low-Density Nuclear Matter - 20

Generalized Beth-Uhlenbeck Approach total particle density n(µ,t, V ) = j de 2π f +(E)A(j, E) = n free + 2n corr Clusters in Low-Density Nuclear Matter - 21

Generalized Beth-Uhlenbeck Approach total particle density n(µ,t, V ) = j de 2π f +(E)A(j, E) = n free + 2n corr with n free = j f +[e(j)] density of free quasiparticles with medium dependent self-energies Clusters in Low-Density Nuclear Matter - 21

Generalized Beth-Uhlenbeck Approach total particle density n(µ,t, V ) = j de 2π f +(E)A(j, E) = n free + 2n corr with n free = j f +[e(j)] density of free quasiparticles with medium dependent self-energies n corr correlation contribution from two-body states Clusters in Low-Density Nuclear Matter - 21

Generalized Beth-Uhlenbeck Approach total particle density n(µ,t, V ) = j de 2π f +(E)A(j, E) = n free + 2n corr with n free = j f +[e(j)] density of free quasiparticles with medium dependent self-energies n corr correlation contribution from two-body states properties of two-body states depend on c.m. momentum P Clusters in Low-Density Nuclear Matter - 21

Generalized Beth-Uhlenbeck Approach total particle density n(µ,t, V ) = j de 2π f +(E)A(j, E) = n free + 2n corr with n free = j f +[e(j)] density of free quasiparticles with medium dependent self-energies n corr correlation contribution from two-body states properties of two-body states depend on c.m. momentum P no bound two-body states for P < P Mott Mott momentum (Pauli principle!) Clusters in Low-Density Nuclear Matter - 21

Generalized Beth-Uhlenbeck Approach total particle density n(µ,t, V ) = j de 2π f +(E)A(j, E) = n free + 2n corr with n free = j f +[e(j)] density of free quasiparticles with medium dependent self-energies n corr correlation contribution from two-body states properties of two-body states depend on c.m. momentum P no bound two-body states for P < P Mott Mott momentum (Pauli principle!) binding energies B k = B k (P,µ,T) and two-body scattering phase shifts δ l = δ l (P,µ, T) depend on P and medium properties (µ,t) Clusters in Low-Density Nuclear Matter - 21

Generalized Beth-Uhlenbeck Approach total particle density n(µ,t, V ) = j de 2π f +(E)A(j, E) = n free + 2n corr with n free = j f +[e(j)] density of free quasiparticles with medium dependent self-energies n corr correlation contribution from two-body states properties of two-body states depend on c.m. momentum P no bound two-body states for P < P Mott Mott momentum (Pauli principle!) binding energies B k = B k (P,µ,T) and two-body scattering phase shifts δ l = δ l (P,µ, T) depend on P and medium properties (µ,t) define continuum edge E cont = E cont (P,µ, T) energy of scattering state with zero relative momentum Clusters in Low-Density Nuclear Matter - 21

Generalized Beth-Uhlenbeck Approach correlation density n corr = k + l g (2) k g (2) l f (E cont B k ) contribution of bound states P,P>P Mott de π 2 dδ l sin2 δ l de f (E cont + E) continuum contribution P with Bose-Einstein distribution function f (E) = {exp[β (E µ)] 1} 1 Clusters in Low-Density Nuclear Matter - 23

Generalized Beth-Uhlenbeck Approach correlation density n corr = k + l g (2) k g (2) l f (E cont B k ) contribution of bound states P,P>P Mott de π 2 dδ l sin2 δ l de f (E cont + E) continuum contribution P with Bose-Einstein distribution function f (E) = {exp[β (E µ)] 1} 1 comparison with standard second cluster (virial) coefficient explicit summation over c.m. momentum P Bose-Einstein statistics additional 2 sin 2 δ l factor medium-dependent binding energies B k and phase shifts δ l Clusters in Low-Density Nuclear Matter - 23

Generalized Beth-Uhlenbeck Approach important features of QS approach medium-dependent self-energies for free quasiparticles for states of correlated particles calculation e.g. in Skyrme-Hartree-Fock (SHF) or relativistic mean-field (RMF) models Clusters in Low-Density Nuclear Matter - 24

Generalized Beth-Uhlenbeck Approach important features of QS approach medium-dependent self-energies for free quasiparticles for states of correlated particles calculation e.g. in Skyrme-Hartree-Fock (SHF) or relativistic mean-field (RMF) models medium-dependent shift of binding energies main effect: Pauli principle blocking of states by medium! calculation perturbatively/variationally with (separable) realistic nucleon-nucleon potentials Clusters in Low-Density Nuclear Matter - 24

Generalized Beth-Uhlenbeck Approach important features of QS approach medium-dependent self-energies for free quasiparticles for states of correlated particles calculation e.g. in Skyrme-Hartree-Fock (SHF) or relativistic mean-field (RMF) models medium-dependent shift of binding energies main effect: Pauli principle blocking of states by medium! calculation perturbatively/variationally with (separable) realistic nucleon-nucleon potentials scattering phase shifts from in-medium T-matrix part of continuum strength moved to self-energies Clusters in Low-Density Nuclear Matter - 24

Generalized Beth-Uhlenbeck Approach important features of QS approach medium-dependent self-energies for free quasiparticles for states of correlated particles calculation e.g. in Skyrme-Hartree-Fock (SHF) or relativistic mean-field (RMF) models medium-dependent shift of binding energies main effect: Pauli principle blocking of states by medium! calculation perturbatively/variationally with (separable) realistic nucleon-nucleon potentials scattering phase shifts from in-medium T-matrix part of continuum strength moved to self-energies dissolution of clusters at high densities Mott effect Clusters in Low-Density Nuclear Matter - 24

Generalized Beth-Uhlenbeck Approach important features of QS approach medium-dependent self-energies for free quasiparticles for states of correlated particles calculation e.g. in Skyrme-Hartree-Fock (SHF) or relativistic mean-field (RMF) models medium-dependent shift of binding energies main effect: Pauli principle blocking of states by medium! calculation perturbatively/variationally with (separable) realistic nucleon-nucleon potentials scattering phase shifts from in-medium T-matrix part of continuum strength moved to self-energies dissolution of clusters at high densities Mott effect extensions possible to include three- and four-body correlations Clusters in Low-Density Nuclear Matter - 24

Generalized Beth-Uhlenbeck Approach shift of binding energies example: symmetric nuclear matter, nuclei at rest in medium in vacuum: experimental binding energies nuclei become unbound (B i < 0) with increasing density of medium binding energy B d [MeV] binding energy B h [MeV] 3 2 1 0 T = 0 MeV T = 5 MeV T = 10 MeV T = 15 MeV T = 20 MeV -1 0.00 0.01 0.02 10 8 6 4 2 0 2 H density n [fm -3 ] -2 0.00 0.01 0.02 density n [fm -3 ] binding energy B t [MeV] binding energy B α [MeV] 10 8 6 4 2 0-2 0.00 0.01 0.02 30 15 10 5 0 3 H density n [fm -3 ] 3 He 25 4 He 20-5 0.00 0.01 0.02 density n [fm -3 ] Clusters in Low-Density Nuclear Matter - 25

Generalized Beth-Uhlenbeck Approach shift of binding energies example: symmetric nuclear matter, nuclei at rest in medium in vacuum: experimental binding energies nuclei become unbound (B i < 0) with increasing density of medium parametrization of results used in generalized relativistic density functional binding energy B d [MeV] binding energy B h [MeV] 3 2 1 0 T = 0 MeV T = 5 MeV T = 10 MeV T = 15 MeV T = 20 MeV -1 0.00 0.01 0.02 10 8 6 4 2 0 2 H density n [fm -3 ] -2 0.00 0.01 0.02 density n [fm -3 ] binding energy B t [MeV] binding energy B α [MeV] 10 8 6 4 2 0-2 0.00 0.01 0.02 30 15 10 5 0 3 H density n [fm -3 ] 3 He 25 4 He 20-5 0.00 0.01 0.02 density n [fm -3 ] Clusters in Low-Density Nuclear Matter - 25

Generalized Relativistic Density Functional nuclear matter around saturation density successful methods: phenomenological mean-field approaches effective interactions, parameters fitted to properties of finite nuclei Clusters in Low-Density Nuclear Matter - 26

Generalized Relativistic Density Functional nuclear matter around saturation density successful methods: phenomenological mean-field approaches effective interactions, parameters fitted to properties of finite nuclei nonrelativistic Hartree-Fock calculations with, e.g., Skyrme/Gogny interaction Clusters in Low-Density Nuclear Matter - 26

Generalized Relativistic Density Functional nuclear matter around saturation density successful methods: phenomenological mean-field approaches effective interactions, parameters fitted to properties of finite nuclei nonrelativistic Hartree-Fock calculations with, e.g., Skyrme/Gogny interaction relativistic mean-field models with nonlinear meson self-interactions or density dependent meson-nucleon couplings Clusters in Low-Density Nuclear Matter - 26

Generalized Relativistic Density Functional nuclear matter around saturation density successful methods: phenomenological mean-field approaches effective interactions, parameters fitted to properties of finite nuclei nonrelativistic Hartree-Fock calculations with, e.g., Skyrme/Gogny interaction relativistic mean-field models with nonlinear meson self-interactions or density dependent meson-nucleon couplings problem: no correlations! idea: include two-, three-, four-body correlations as new degrees of freedom (clusters) with medium-dependent properties Clusters in Low-Density Nuclear Matter - 26

Generalized Relativistic Density Functional generalization of relativistic mean-field (RMF) models extended relativistic density functional with nucleons (ψ p, ψ n ), deuterons (ϕ µ d ), tritons (ψ t), helions (ψ h ), α-particles (ϕ α ), mesons (σ, ω µ, ρ µ ), electrons (ψ e ) and photons (A µ ) as degrees of freedom Clusters in Low-Density Nuclear Matter - 27

Generalized Relativistic Density Functional generalization of relativistic mean-field (RMF) models extended relativistic density functional with nucleons (ψ p, ψ n ), deuterons (ϕ µ d ), tritons (ψ t), helions (ψ h ), α-particles (ϕ α ), mesons (σ, ω µ, ρ µ ), electrons (ψ e ) and photons (A µ ) as degrees of freedom only minimal (linear) meson-nucleon couplings Clusters in Low-Density Nuclear Matter - 27

Generalized Relativistic Density Functional generalization of relativistic mean-field (RMF) models extended relativistic density functional with nucleons (ψ p, ψ n ), deuterons (ϕ µ d ), tritons (ψ t), helions (ψ h ), α-particles (ϕ α ), mesons (σ, ω µ, ρ µ ), electrons (ψ e ) and photons (A µ ) as degrees of freedom only minimal (linear) meson-nucleon couplings density-dependent meson-nucleon couplings Γ i functional form as suggested by Dirac-Brueckner calculations of nuclear matter more flexible approach than models with non-linear meson self-interactions Clusters in Low-Density Nuclear Matter - 27

Generalized Relativistic Density Functional generalization of relativistic mean-field (RMF) models extended relativistic density functional with nucleons (ψ p, ψ n ), deuterons (ϕ µ d ), tritons (ψ t), helions (ψ h ), α-particles (ϕ α ), mesons (σ, ω µ, ρ µ ), electrons (ψ e ) and photons (A µ ) as degrees of freedom only minimal (linear) meson-nucleon couplings density-dependent meson-nucleon couplings Γ i functional form as suggested by Dirac-Brueckner calculations of nuclear matter more flexible approach than models with non-linear meson self-interactions parameters: nucleon/meson masses, coupling strengths/density dependence fitted to properties of finite nuclei Clusters in Low-Density Nuclear Matter - 27

Generalized Relativistic Density Functional generalization of relativistic mean-field (RMF) models extended relativistic density functional with nucleons (ψ p, ψ n ), deuterons (ϕ µ d ), tritons (ψ t), helions (ψ h ), α-particles (ϕ α ), mesons (σ, ω µ, ρ µ ), electrons (ψ e ) and photons (A µ ) as degrees of freedom only minimal (linear) meson-nucleon couplings density-dependent meson-nucleon couplings Γ i functional form as suggested by Dirac-Brueckner calculations of nuclear matter more flexible approach than models with non-linear meson self-interactions parameters: nucleon/meson masses, coupling strengths/density dependence fitted to properties of finite nuclei cluster binding energies/effective resonance energies density dependence replaced by dependence on vector meson fields Clusters in Low-Density Nuclear Matter - 27

Generalized Relativistic Density Functional generalization of relativistic mean-field (RMF) models extended relativistic density functional with nucleons (ψ p, ψ n ), deuterons (ϕ µ d ), tritons (ψ t), helions (ψ h ), α-particles (ϕ α ), mesons (σ, ω µ, ρ µ ), electrons (ψ e ) and photons (A µ ) as degrees of freedom only minimal (linear) meson-nucleon couplings density-dependent meson-nucleon couplings Γ i functional form as suggested by Dirac-Brueckner calculations of nuclear matter more flexible approach than models with non-linear meson self-interactions parameters: nucleon/meson masses, coupling strengths/density dependence fitted to properties of finite nuclei cluster binding energies/effective resonance energies density dependence replaced by dependence on vector meson fields nucleon/cluster/meson/photon field equations, solved selfconsistently Clusters in Low-Density Nuclear Matter - 27

Generalized Relativistic Density Functional grand canonical thermodynamical potential Ω = pv = d 3 r ω g (T,µ i, σ,ω 0,ρ 0, A, σ, ω 0, ρ 0, A) with density functional ω g depending on temperature T, chemical potentials µ i, meson and photon fields σ, δ,ω 0, ρ 0,A fields equations with additional rearrangement contributions consistent derivation of thermodynamical quantities Clusters in Low-Density Nuclear Matter - 28

Generalized Relativistic Density Functional grand canonical thermodynamical potential Ω = pv = d 3 r ω g (T,µ i, σ,ω 0,ρ 0, A, σ, ω 0, ρ 0, A) with density functional ω g depending on temperature T, chemical potentials µ i, meson and photon fields σ, δ,ω 0, ρ 0,A fields equations with additional rearrangement contributions consistent derivation of thermodynamical quantities light clusters ( 2 H, 3 H, 3 He, 4 He), two-nucleon scattering correlations: explicitly included Clusters in Low-Density Nuclear Matter - 28

Generalized Relativistic Density Functional grand canonical thermodynamical potential Ω = pv = d 3 r ω g (T,µ i, σ,ω 0,ρ 0, A, σ, ω 0, ρ 0, A) with density functional ω g depending on temperature T, chemical potentials µ i, meson and photon fields σ, δ,ω 0, ρ 0,A fields equations with additional rearrangement contributions consistent derivation of thermodynamical quantities light clusters ( 2 H, 3 H, 3 He, 4 He), two-nucleon scattering correlations: explicitly included heavy clusters: (not considered here) Thomas-Fermi approximation in spherical Wigner-Seitz cells Clusters in Low-Density Nuclear Matter - 28

Generalized Relativistic Density Functional grand canonical thermodynamical potential Ω = pv = d 3 r ω g (T,µ i, σ,ω 0,ρ 0, A, σ, ω 0, ρ 0, A) with density functional ω g depending on temperature T, chemical potentials µ i, meson and photon fields σ, δ,ω 0, ρ 0,A fields equations with additional rearrangement contributions consistent derivation of thermodynamical quantities light clusters ( 2 H, 3 H, 3 He, 4 He), two-nucleon scattering correlations: explicitly included heavy clusters: (not considered here) Thomas-Fermi approximation in spherical Wigner-Seitz cells low-density limit, finite temperature: only nucleons and light clusters reproduction of standard virial EoS (details: S. Typel et al., Phys. Rev. C 81 (2010) 015803) Clusters in Low-Density Nuclear Matter - 28

Generalized Relativistic Density Functional Relation and Differences of Models quantum statistical approach generalized relativistic density functional empirical nucleon-nucleon potential medium dependence of parametrization of binding energy shifts cluster binding energies phenomenological meson-nucleon interaction parametrization of nucleon scalar/vector nucleon self-energies self-energy and effective mass in nonrelativistic approximation no effect of cluster formation medium-dependent change of on nucleon mean fields cluster properties induces change of mean fields Clusters in Low-Density Nuclear Matter - 29

Formation and Dissolution of Clusters particle fractions X i = A i n i n b n b = i A in i generalized relativistic density functional 10 0 low densities: two-body correlation most important T = 10 MeV high densities: dissolution of clusters Mott effect particle fraction X i 10-1 10-2 Y p = 0.4 p n 2 H 3 H 3 He 4 He 10-3 10-5 10-4 10-3 10-2 10-1 (without heavy clusters) density n [fm -3 ] Clusters in Low-Density Nuclear Matter - 30

Formation and Dissolution of Clusters particle fractions X i = A i n i n b n b = i A in i low densities: two-body correlation most important high densities: dissolution of clusters Mott effect effect of NN continuum correlations dashed lines: without continuum solid lines: with continuum reduction of deuteron fraction, redistribution of other particles correct limits in grdf model generalized relativistic density functional particle fraction X i 10 0 10-1 10-2 10-3 T = 10 MeV Y p = 0.4 10-5 10-4 10-3 10-2 10-1 (without heavy clusters) density n [fm -3 ] p n 2 H 3 H 3 He 4 He Clusters in Low-Density Nuclear Matter - 30

Formation and Dissolution of Clusters fraction of free protons in symmetric nuclear matter, thin lines: NSE generalized RDF model vs. NSE QS approach vs. NSE 0.5 0.5 proton fraction X p 0.4 0.3 0.2 0.1 proton fraction X p 0.4 0.3 0.2 0.1 2 MeV 4 MeV 6 MeV 8 MeV 10 MeV 12 MeV 14 MeV 16 MeV 18 MeV 20 MeV 0.0 10-5 10-4 10-3 10-2 10-1 10 0 density n [fm -3 ] 0.0 10-5 10-4 10-3 10-2 10-1 10 0 density n [fm -3 ] Clusters in Low-Density Nuclear Matter - 31

Formation and Dissolution of Clusters fraction of clusters in symmetric nuclear matter, generalized RDF vs. NSE helion fraction X h deuteron fraction X d 10 0 10 0 10-1 10-2 10-3 10-4 10-5 10-5 10-4 10-3 10-2 10-1 10 0 10-1 10-2 10-3 10-4 10-5 10-5 10-4 10-3 10-2 10-1 10 0 density n [fm -3 ] density n [fm -3 ] 10 0 10 0 10-1 10-2 10-3 10-4 2 H 3 H triton fraction X t 3 He 4 He α-particle fraction X α 10-1 10-2 10-3 10-4 2 MeV 4 MeV 6 MeV 8 MeV 10 MeV 12 MeV 14 MeV 16 MeV 18 MeV 20 MeV 10-5 10-5 10-4 10-3 10-2 10-1 10 0 density n [fm -3 ] 10-5 10-5 10-4 10-3 10-2 10-1 10 0 density n [fm -3 ] Clusters in Low-Density Nuclear Matter - 32

Formation and Dissolution of Clusters fraction of clusters in symmetric nuclear matter, QS approach vs. NSE helion fraction X h deuteron fraction X d 10 0 10 0 10-1 10-2 10-3 10-4 10-5 10-5 10-4 10-3 10-2 10-1 10 0 10-1 10-2 10-3 10-4 10-5 10-5 10-4 10-3 10-2 10-1 10 0 density n [fm -3 ] density n [fm -3 ] 10 0 10 0 10-1 10-2 10-3 10-4 2 H 3 H triton fraction X t 3 He 4 He α-particle fraction X α 10-1 10-2 10-3 10-4 2 MeV 4 MeV 6 MeV 8 MeV 10 MeV 12 MeV 14 MeV 16 MeV 18 MeV 20 MeV 10-5 10-5 10-4 10-3 10-2 10-1 10 0 density n [fm -3 ] 10-5 10-5 10-4 10-3 10-2 10-1 10 0 density n [fm -3 ] Clusters in Low-Density Nuclear Matter - 33

Formation and Dissolution of Clusters alpha-particle fraction in symmetric nuclear matter α-particle fraction X α α-particle fraction X α 10 0 10-1 10-2 10-3 10-5 10-4 10-3 10-2 10-1 10 0 10-1 10-2 10 0 4 MeV 8 MeV 10-1 10-2 12 MeV 20 MeV 10-3 10-4 10-3 10-2 10-1 10 0 10-1 10-2 virial EoS NSE Shen et al. EoS generalized RDF model QS approach 10-3 10-3 10-2 10-1 density n [fm -3 ] 10-3 10-3 10-2 10-1 density n [fm -3 ] Clusters in Low-Density Nuclear Matter - 34

Thermodynamical Properties pressure p = Ω/V in symmetric nuclear matter, thin lines: NSE generalized RDF model vs. NSE 10 1 10 1 QS approach vs. NSE pressure p [MeV fm -3 ] 10 0 10-1 10-2 10-3 pressure p [MeV fm -3 ] 10 0 10-1 10-2 10-3 2 MeV 4 MeV 6 MeV 8 MeV 10 MeV 12 MeV 14 MeV 16 MeV 18 MeV 20 MeV 10-4 10-5 10-4 10-3 10-2 10-1 10 0 density n [fm -3 ] 10-4 10-5 10-4 10-3 10-2 10-1 10 0 density n [fm -3 ] Clusters in Low-Density Nuclear Matter - 35

Thermodynamical Properties p/n in symmetric nuclear matter, lim n 0 (p/n) = T (ideal gas) 40 generalized RDF vs. NSE 40 QS approach vs. NSE pressure/density p/n [MeV] 30 20 10 0 pressure/density p/n [MeV] 30 20 10 0 2 MeV 4 MeV 6 MeV 8 MeV 10 MeV 12 MeV 14 MeV 16 MeV 18 MeV 20 MeV -10 10-5 10-4 10-3 10-2 10-1 10 0 density n [fm -3 ] -10 10-5 10-4 10-3 10-2 10-1 10 0 density n [fm -3 ] Clusters in Low-Density Nuclear Matter - 36

Thermodynamical Properties internal energy per nucleon E A in symmetric nuclear matter, lim n 0 E A = 3 2 T +... generalized RDF model vs. NSE 40 40 QS approach vs. NSE internal energy per nucleon E A [MeV] 30 20 10 0-10 internal energy per nucleon E A [MeV] 30 20 10 0-10 2 MeV 4 MeV 6 MeV 8 MeV 10 MeV 12 MeV 14 MeV 16 MeV 18 MeV 20 MeV -20 10-5 10-4 10-3 10-2 10-1 10 0 density n [fm -3 ] -20 10-5 10-4 10-3 10-2 10-1 10 0 density n [fm -3 ] Clusters in Low-Density Nuclear Matter - 37

Thermodynamical Properties neutron matter comparison: different models and effects nonrelativistic ideal gas 15.4 T = 10 MeV internal energy per nucleon [MeV] 15.2 15.0 14.8 14.6 ideal gas 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 density n [fm -3 ] Clusters in Low-Density Nuclear Matter - 38

Thermodynamical Properties neutron matter comparison: different models and effects nonrelativistic ideal gas rel. kinematics + statistics relativistic Fermi gas internal energy per nucleon [MeV] 15.4 15.2 15.0 14.8 14.6 T = 10 MeV ideal gas relativistic Fermi gas 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 density n [fm -3 ] Clusters in Low-Density Nuclear Matter - 38

Thermodynamical Properties neutron matter comparison: different models and effects nonrelativistic ideal gas rel. kinematics + statistics relativistic Fermi gas two-body correlations virial EoS with relativistic correction internal energy per nucleon [MeV] 15.4 15.2 15.0 14.8 14.6 T = 10 MeV ideal gas relativistic Fermi gas relativistic virial EoS 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 density n [fm -3 ] Clusters in Low-Density Nuclear Matter - 38

Thermodynamical Properties neutron matter comparison: different models and effects nonrelativistic ideal gas rel. kinematics + statistics relativistic Fermi gas two-body correlations virial EoS with relativistic correction (not included in standard virial EoS) mean-field effects standard RMF model with density dependent couplings internal energy per nucleon [MeV] 15.4 15.2 15.0 14.8 14.6 T = 10 MeV ideal gas relativistic Fermi gas relativistic virial EoS standard RMF 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 density n [fm -3 ] Clusters in Low-Density Nuclear Matter - 38