theor.jinr.rutheor.jinr.ru/~ntaa/11/files/lectures/typel_ntaa2011.pdf · motivation: the life and...

Clusters in Low-Density Nuclear Matter

Stefan TypelExcellence Cluster ‘Universe’, Technische Universitat MunchenGSI Helmholtzzentrum fur 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

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.)

when the fuel for nuclear fusion reactions is consumed:

• last phases in the life of a massive star(8Msun . Mstar . 30Msun)

⇒ core-collapse supernova

⇒ neutron star or black hole

Clusters in Low-Density Nuclear Matter - 2 Stefan Typel

Motivation: The Life and Death of Stars

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.)

when the fuel for nuclear fusion reactions is consumed:

• last phases in the life of a massive star(8Msun . Mstar . 30Msun)

⇒ core-collapse supernova

⇒ neutron star or black hole

• essential ingredient in astrophysicalmodel 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, . . .


Motivation: EoS - Thermodynamical Conditions

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

10−1

100

101

102

Baryon density, nB [fm−3]

Tem

pera

ture

, T [M

eV]

Ye

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.56 7 8 9 10 11 12 13 14 15

Baryon density, log10

(ρ [g/cm3])

T. Fischer, GSI Darmstadt

Relevant Parameters:

• densities:

10−9 . /sat . 10

with nuclear saturation densitysat ≈ 2.5 · 1014 g/cm3

(nsat = sat/mn ≈ 0.15 fm−3)

• temperatures:

0 MeV ≤ kBT . 50 MeV(= 5.8 · 1011 K)

• electron fraction:

0 ≤ Ye . 0.6



10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

10−1

100

101

102


Tem

pera

ture

, T [M

eV]

Ye

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.56 7 8 9 10 11 12 13 14 15


(ρ [g/cm3])


Most Relevant Particles:(at finite temperatures andnot too high densities)

• neutrons, protons

• nuclei

• electrons, (muons)(charge neutrality! Ye ≡ Yp)



10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

10−1

100

101

102


Tem

pera

ture

, T [M

eV]

Ye

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.56 7 8 9 10 11 12 13 14 15


(ρ [g/cm3])


Most Relevant Particles:(at finite temperatures andnot too high densities)

• neutrons, protons

• nuclei

• electrons, (muons)(charge neutrality! Ye ≡ Yp)

Timescale of Reactions≪

Timescale of System Evolution

⇒ thermodynamical equilibrium

⇒ construction of equation of state

here: consider system of nucleonsand (light) nuclei


Questions

• How does the composition of matter change withdensity, temperature and neutron-proton asymmetry ?

• How do interactions between particles modifythe thermodynamical properties of matter ?

• How do the properties of nuclei change in the medium ?


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


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



n

n

n

n

nn

n

np

p

p

p

p

p

p

p

d

t

α




• chemical picturemixture of different nuclear species and nucleonsin chemical equilibrium− properties of constituents independent of medium− interaction between particles ?− dissolution of nuclei at high densities ?



n

n

n

n

nn

n

np

p

p

p

p

p

p

p

d

t

α

n

n

n

n

nn

n

np

p

p

p

p

p

p

pp

pp

p

n n

nn

n




• chemical picturemixture of different nuclear species and nucleonsin chemical equilibrium− properties of constituents independent of medium− interaction between particles ?− dissolution of nuclei at high densities ?

• physical pictureinteraction between nucleons ⇒ correlations⇒ formation of bound states/resonances− treatment of two-, three-, . . . many-body correlations ?− choice of interaction ?


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







• connecting to densities around nuclear saturation⇒ Generalized Relativistic Density Functional

system of units such that ~ = c = kB = 1


Thermodynamical Relations

Grand Canonical Ensemble

• particles i with chemical potentials µi at Temperature T in Volume V

⇒ natural variables: µi, T, V






• thermodynamical potential: grand canonical potential Ω = Ω(µi, T, V ) = −pV⇒ contains all information of the system







• equations of state:

ni(µi, T, V ) =Ni

V= − 1

V

∂Ω

∂µi

∣

∣

∣

∣

T,V

particle number densities

S(µi, T, V ) = − ∂Ω

∂T

∣

∣

∣

∣

µi,V

entropy p(µi, T, V ) = − ∂Ω

∂V

∣

∣

∣

∣

µi,V

= −Ω

Vpressure







• equations of state:

ni(µi, T, V ) =Ni

V= − 1

V

∂Ω

∂µi

∣

∣

∣

∣

T,V


S(µi, T, V ) = − ∂Ω

∂T

∣

∣

∣

∣

µi,V

entropy p(µi, T, V ) = − ∂Ω

∂V

∣

∣

∣

∣

µi,V

= −Ω

Vpressure

• connection to microphysics: grand canonical partition function

Z(µi, T, V ) = trace exp[

−β(

H − ∑

i µiNi

)]

⇒ Ω = −T lnZ

with Hamilton operator H, particle number operators Ni and β = 1/T


Nuclear Statistical Equilibrium (NSE)

most simple approach

• ideal mixture of nucleons (p, n) and nuclei X in chemical equilibrium

Zp+Nn⇔ AZXN ⇒ Zµp +Nµn = µX or Zµp +Nµn = µX −BX

with relativistic chemical potentials µi = µi +mi andbinding energy BX of nucleus X with mass mX







• independent particles without mutual interaction

⇒ factorization Z =∏

i Zi







• independent particles without mutual interaction

⇒ factorization Z =∏

i Zi

• Maxwell-Boltzmann statistics

Zi =

∞∑

Ni=0

1

Ni!QNi

i zNii = exp (Qizi) with fugacities zi = exp

(µi

T

)

and single-particle canonical partition functions Qi = trace exp(

−βHi

)

with single-particle Hamilton operator Hi



• nonrelativistic kinematics: Hi = p2i/(2mi) ⇒ Ei = p2

i/(2mi)

⇒ Qi =gi

(2π)3

∫

d3ri

∫

d3pi exp

(

−Ei

T

)

= giV

λ3i

with degeneracy factor gi = (2Ji + 1) (= 2 for nucleons)

and thermal wavelength λi =√

2πmiT




i/(2mi)

⇒ Qi =gi

(2π)3

∫

d3ri

∫

d3pi exp

(

−Ei

T

)

= giV

λ3i



2πmiT

⇒ Z =∏

i

exp

(

giV

λ3i

zi

)

and Ω = −T lnZ = −TV∑

i

gi

λ3i

exp(µi

T

)




i/(2mi)

⇒ Qi =gi

(2π)3

∫

d3ri

∫

d3pi exp

(

−Ei

T

)

= giV

λ3i



2πmiT

⇒ Z =∏

i

exp

(

giV

λ3i

zi

)

and Ω = −T lnZ = −TV∑

i

gi

λ3i

exp(µi

T

)

• including excited states (x) of nuclei

gi → gi(T ) = (2Jgsi + 1) +

∑

x

(2Jx + 1) exp

(

−Ex

T

)

→ (2Jgsi + 1) +

∫

dE i(E) exp

(

−ET

)

with level density i(E)



• equations of state

ni(µi, T, V ) = − 1

V

∂Ω

∂µi

∣

∣

∣

∣

T,V

=gi

λ3i

exp(µi

T

)


⇒ Ω = −TV ∑

i ni




ni(µi, T, V ) = − 1

V

∂Ω

∂µi

∣

∣

∣

∣

T,V

=gi

λ3i

exp(µi

T

)


⇒ Ω = −TV ∑

i ni

S(µi, T, V ) = − ∂Ω

∂T

∣

∣

∣

∣

µi,V

= −5

2

Ω

T− V

T

∑

i

µini entropy




ni(µi, T, V ) = − 1

V

∂Ω

∂µi

∣

∣

∣

∣

T,V

=gi

λ3i

exp(µi

T

)


⇒ Ω = −TV ∑

i ni

S(µi, T, V ) = − ∂Ω

∂T

∣

∣

∣

∣

µi,V

= −5

2

Ω

T− V

T

∑

i

µini entropy

p(µi, T, V ) = − ∂Ω

∂V

∣

∣

∣

∣

µi,V

= −Ω

Vpressure

⇒ pV = NT with N =∑

iNi




ni(µi, T, V ) = − 1

V

∂Ω

∂µi

∣

∣

∣

∣

T,V

=gi

λ3i

exp(µi

T

)


⇒ Ω = −TV ∑

i ni

S(µi, T, V ) = − ∂Ω

∂T

∣

∣

∣

∣

µi,V

= −5

2

Ω

T− V

T

∑

i

µini entropy

p(µi, T, V ) = − ∂Ω

∂V

∣

∣

∣

∣

µi,V

= −Ω

Vpressure

⇒ pV = NT with N =∑

iNi

E = TS − pV +∑

i

µiNi =3

2NT internal energy

⇒ mixture of ideal gases



example

• ideal mixture of neutrons, protons and deuterons (d = 2H)

p+ n⇔ d ⇒ µp + µn = µd −Bd

with deuteron binding energy Bd = 2.225 MeV



example




• particle number densities

nn,p = 2λ3

n,pexp

(µn,p

T

)

nd = 3λ3

d

exp(µd

T

)

⇒ law of mass action

nd

nnnp=

3

4

(

2πmd

mnmpT

)32

exp

(

Bd

T

)



10-6

10-5

10-4

10-3

10-2

10-1

100

nucleon density n [fm-3

]

0.0

0.2

0.4

0.6

0.8

1.0

deut

eron

fra

ctio

n 2

n d/n

T = 1 MeVT = 2 MeVT = 5 MeVT = 10 MeVT = 20 MeV

Yp = 0.5

example





nn,p = 2λ3

n,pexp

(µn,p

T

)

nd = 3λ3

d

exp(µd

T

)

⇒ law of mass action

nd

nnnp=

3

4

(

2πmd

mnmpT

)32

exp

(

Bd

T

)

• symmetric nuclear matter (nn = np)

total nucleon density n = nn + np + 2nd

deuteron fraction Xd = 2ndn


Virial Equation of State (VEoS)

consider two-, (three-, . . . many-) body correlations !

• expansion of grand canonical partition function in powers of fugacities zi = exp(

µiT

)

Z(µi, T, V ) = 1 +∑

i

Qizi +1

2

∑

ij

Qijzizj +1

6

∑

ijk

Qijkzizjzk + . . .





µiT

)

Z(µi, T, V ) = 1 +∑

i

Qizi +1

2

∑

ij

Qijzizj +1

6

∑

ijk

Qijkzizjzk + . . .

with one-, two-, three-, . . . many-body canonical partition functions

Qi = trace1 exp(

−βHi

)

Qij = trace2 exp(

−βHij

)

Qijk = trace3 exp(

−βHijk

)

and Hamiltonians Hi, Hij, Hijk, . . .





µiT

)

Z(µi, T, V ) = 1 +∑

i

Qizi +1

2

∑

ij

Qijzizj +1

6

∑

ijk

Qijkzizjzk + . . .


Qi = trace1 exp(

−βHi

)

Qij = trace2 exp(

−βHij

)

Qijk = trace3 exp(

−βHijk

)


• expansion valid only for zi ≪ 1





µiT

)

Z(µi, T, V ) = 1 +∑

i

Qizi +1

2

∑

ij

Qijzizj +1

6

∑

ijk

Qijkzizjzk + . . .


Qi = trace1 exp(

−βHi

)

Qij = trace2 exp(

−βHij

)

Qijk = trace3 exp(

−βHijk

)


• expansion valid only for zi ≪ 1

• for independent particles without interaction:

Hij = Hi + Hj, Hijk = Hi + Hj + Hk, . . .

⇒ factorization Qij = QiQj Qijk = QiQjQk . . .



• use ln(1 + x) = x− 12x

2 + 13x

3 − . . .

lnZ =∑

i

Qizi +1

2

∑

ij

(Qij −QiQj) zizj + . . .



• use ln(1 + x) = x− 12x

2 + 13x

3 − . . .

lnZ =∑

i

Qizi +1

2

∑

ij


• introduce (dimensionless) cluster (virial) coefficients

bi = gi bij = λ3/2i λ

3/2j (Qij −QiQj) /(2V )

bijk = λiλjλk (Qijk −QiQjk −QjQik −QkQij + 2QiQjQk) /(6V )



• use ln(1 + x) = x− 12x

2 + 13x

3 − . . .

lnZ =∑

i

Qizi +1

2

∑

ij




3/2j (Qij −QiQj) /(2V )


⇒ grand canonical potential

Ω = −T lnZ = −TV

∑

i

bizi

λ3i

+∑

ij

bijzizj

λ3/2i λ

3/2j

+∑

ijk

bijkzizjzk

λiλjλk+ . . .



• use ln(1 + x) = x− 12x

2 + 13x

3 − . . .

lnZ =∑

i

Qizi +1

2

∑

ij




3/2j (Qij −QiQj) /(2V )


⇒ grand canonical potential

Ω = −T lnZ = −TV

∑

i

bizi

λ3i

+∑

ij

bijzizj

λ3/2i λ

3/2j

+∑

ijk

bijkzizjzk

λiλjλk+ . . .

• bij(T ), bijk(T ) encode effects of two- and three-body correlations

no correlations ⇒ independent particles ⇒ bij = 0, bijk = 0, . . .




ni = − 1

V

∂Ω

∂µi

∣

∣

∣

∣

T,V

= bizi

λ3i

+ 2∑

j

bijzizj

λ3/2i λ

3/2j

+ 3∑

jk

bijkzizjzk

λiλjλk+ . . .

contributions from free particles and correlated particles




ni = − 1

V

∂Ω

∂µi

∣

∣

∣

∣

T,V

= bizi

λ3i

+ 2∑

j

bijzizj

λ3/2i λ

3/2j

+ 3∑

jk

bijkzizjzk

λiλjλk+ . . .


• virial equation of state

pV

NT= − Ω

∑

i niV T

= 1 +∑

ij

aij(λ3ini)

1/2(λ3jnj)

1/2 +∑

ijk

aijk(λ3ini)

2/3(λ3jnj)

2/3(λ3knk)

2/3 + . . .

with virial coefficients aij, total particle number N =∑

iNi by eliminating fugacities




ni = − 1

V

∂Ω

∂µi

∣

∣

∣

∣

T,V

= bizi

λ3i

+ 2∑

j

bijzizj

λ3/2i λ

3/2j

+ 3∑

jk

bijkzizjzk

λiλjλk+ . . .


• virial equation of state

pV

NT= − Ω

∑

i niV T

= 1 +∑

ij

aij(λ3ini)

1/2(λ3jnj)

1/2 +∑

ijk

aijk(λ3ini)

2/3(λ3jnj)

2/3(λ3knk)

2/3 + . . .

with virial coefficients aij, total particle number N =∑

iNi by eliminating fugacities

• final task: determine cluster (virial) coefficients !

bij simple ! bijk, . . . difficult



determination of second cluster (virial) coefficient

• interaction between two-particles independent of c.m. momentum⇒ transformation to c.m. and relative coordinates





~Rij = 1Mij

(mi~ri +mj~rj) ~rij = ~ri − ~rj

~Pij = ~pi + ~pj ~pij = µij

(

~pimi

− ~pj

mj

)

with total mass Mij = mi +mj and reduced mass µij = mimj/Mij





~Rij = 1Mij

(mi~ri +mj~rj) ~rij = ~ri − ~rj

~Pij = ~pi + ~pj ~pij = µij

(

~pimi

− ~pj

mj

)

with total mass Mij = mi +mj and reduced mass µij = mimj/Mij

• second cluster (virial) coefficient in classical mechanics

bij = λ3/2i λ

3/2j (Qij −QiQj) /(2V )

=1

2

gij

λ3/2i λ

3/2j

∫

d3rij exp [−βVij(rij)] − 1

with two-body Hamiltonian Hij =p2

i2mi

+p2

j

2mj+ Vij(rij) =

P 2ij

2Mij+

p2ij

2µij+ Vij(rij)



• second cluster (virial) coefficient in quantum mechanics(G. E. Beth and E. Uhlenbeck Physica 3 (1936) 729, Physica 4 (1937) 915)




bij(T ) =1 + δij

2

(

mi +mj√mimj

)3/2 ∫

dE Dij(E) exp

(

−ET

)

with two-body density of states

Dij(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)




bij(T ) =1 + δij

2

(

mi +mj√mimj

)3/2 ∫

dE Dij(E) exp

(

−ET

)


Dij(E) =∑

k

g(ij)k δ(E − E

(ij)k ) +

∑

l

g(ij)l

π

dδ(ij)l

dE



• 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)




bij(T ) =1 + δij

2

(

mi +mj√mimj

)3/2 ∫

dE Dij(E) exp

(

−ET

)


Dij(E) =∑

k

g(ij)k δ(E − E

(ij)k ) +

∑

l

g(ij)l

π

dδ(ij)l

dE



• 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: niλ3i ≪ 1 ⇒ (very) low densities



possible topics for seminar:

• explicit calculation of second cluster (virial) coefficient bij(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


Generalized Beth-Uhlenbeck Approach

extension to higher densities ⇒ consider effects of the medium

(M. Schmidt, G. Ropke, 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 systemfrom single-particle Green’s function







• Green’s function of noninteracting single-particle state j ≡ (~pj, σj, τj)

G0(j, z) = [z −E(j)]−1 with E(j) = p2j/(2mj)









• Green’s function G(j, z) of interacting single-particle state j

G(j, z) = G0(j, z) +G0(j, z)Σ(j, z)G(j, z) (Dyson equation)

with self-energy Σ(j, z) (contains information on the interaction)









• Green’s function G(j, z) of interacting single-particle state j

G(j, z) = G0(j, z) +G0(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 Ej = Ej(~pj) of particle j in non-interacting system



• total particle density ⇒ equation of state

n(µ, T, V ) =∑

j

∫

dE

2πf+(E)A(j, E)

with Fermi distribution function f+(E) = exp [β (E − µ)] + 1−1




n(µ, T, V ) =∑

j

∫

dE

2πf+(E)A(j, E)


• formal solution of Dyson equation

G(j, z) = [z −E(j) − Σ(j, z)]−1 with self-energy Σ = ΣR + iΣI

⇒ A(j, E) = 2ΣI(j,E−i0)

[E−E(j)−ΣR(j,E)]2+[ΣI(j,E−i0)]2




n(µ, T, V ) =∑

j

∫

dE

2πf+(E)A(j, E)




⇒ A(j, E) = 2ΣI(j,E−i0)

[E−E(j)−ΣR(j,E)]2+[ΣI(j,E−i0)]2

• expansion of A(j, E) for small ΣI = ε with εx2+ε2 = πδ(x) − ε d

dxP 1x + . . .

⇒ A(j, E) = 2πδ [E −E(j) − ΣR(j, E)] quasiparticle contribution

−2ΣI(j, E − i0) ddxP 1

x

∣

∣

x=E−E(j)−ΣR(j,E)+ . . . correlation contribution




n(µ, T, V ) =∑

j

∫

dE

2πf+(E)A(j, E)




⇒ A(j, E) = 2ΣI(j,E−i0)

[E−E(j)−ΣR(j,E)]2+[ΣI(j,E−i0)]2

• expansion of A(j, E) for small ΣI = ε with εx2+ε2 = πδ(x) − ε d

dxP 1x + . . .

⇒ A(j, E) = 2πδ [E −E(j) − ΣR(j, E)] quasiparticle contribution

−2ΣI(j, E − i0) ddxP 1

x

∣

∣

x=E−E(j)−ΣR(j,E)+ . . . correlation contribution

⇒ define quasparticle energy e(j): solution of e(j) = E(j) + ΣR(j, e(j))



⇒ total particle density

n(µ, T, V ) =∑

j

∫

dE

2πf+(E)A(j, E) = nfree + 2ncorr




n(µ, T, V ) =∑

j

∫

dE

2πf+(E)A(j, E) = nfree + 2ncorr with

• nfree =∑

j f+[e(j)] density of free quasiparticles

with medium dependent self-energies




n(µ, T, V ) =∑

j

∫

dE


• nfree =∑



• ncorr correlation contribution from two-body states




n(µ, T, V ) =∑

j

∫

dE


• nfree =∑




properties of two-body states depend on c.m. momentum ~P




n(µ, T, V ) =∑

j

∫

dE


• nfree =∑





no bound two-body states for P < PMott Mott momentum (Pauli principle !)




n(µ, T, V ) =∑

j

∫

dE


• nfree =∑






binding energies Bk = Bk(P, µ, T ) and two-body scattering phase shifts

δl = δl(P,µ, T ) depend on ~P and medium properties (µ, T )




n(µ, T, V ) =∑

j

∫

dE


• nfree =∑






binding energies Bk = Bk(P, µ, T ) and two-body scattering phase shifts

δl = δl(P,µ, T ) depend on ~P and medium properties (µ, T )

define continuum edge Econt = Econt(P,µ, T ) ≡ energy of scattering statewith zero relative momentum



• correlation density

ncorr =∑

k

g(2)k

∑

~P ,P>PMott

f−(Econt −Bk) contribution of bound states

+∑

l

g(2)l

∑

~P

∫

dE

π2 sin2 δl

dδldE

f−(Econt +E) continuum contribution

with Bose-Einstein distribution function f−(E) = exp [β (E − µ)] − 1−1



• correlation density

ncorr =∑

k

g(2)k

∑

~P ,P>PMott

f−(Econt −Bk) contribution of bound states

+∑

l

g(2)l

∑

~P

∫

dE

π2 sin2 δl

dδldE

f−(Econt +E) continuum contribution

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 sin2 δl factor

medium-dependent binding energies Bk and phase shifts δl



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









• dissolution of clusters at high densities ⇒ Mott effect

• extensions possible to include three- and four-body correlations



0.00 0.01 0.02

density n [fm-3

]

-1

0

1

2

3

bind

ing

ener

gy

Bd

[MeV

]


0.00 0.01 0.02

density n [fm-3

]

-2

0

2

4

6

8

10

bind

ing

ener

gy

Bt

[MeV

]

0.00 0.01 0.02

density n [fm-3

]

-2

0

2

4

6

8

10

bind

ing

ener

gy

Bh

[MeV

]

0.00 0.01 0.02

density n [fm-3

]

-5

0

5

10

15

20

25

30

bind

ing

ener

gy

Bα

[MeV

]

2H

3H

3He

4He

shift of binding energies

• example: symmetric nuclear matter,nuclei at rest in medium

• in vacuum:experimental binding energies

• nuclei become unbound (Bi < 0)with increasing density of medium



0.00 0.01 0.02

density n [fm-3

]

-1

0

1

2

3

bind

ing

ener

gy

Bd

[MeV

]


0.00 0.01 0.02

density n [fm-3

]

-2

0

2

4

6

8

10

bind

ing

ener

gy

Bt

[MeV

]

0.00 0.01 0.02

density n [fm-3

]

-2

0

2

4

6

8

10

bind

ing

ener

gy

Bh

[MeV

]

0.00 0.01 0.02

density n [fm-3

]

-5

0

5

10

15

20

25

30

bind

ing

ener

gy

Bα

[MeV

]

2H

3H

3He

4He

shift of binding energies

• example: symmetric nuclear matter,nuclei at rest in medium

• in vacuum:experimental binding energies

• nuclei become unbound (Bi < 0)with increasing density of medium

• parametrization of results used ingeneralized relativistic densityfunctional


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 calculationswith, e.g., Skyrme/Gogny interaction







relativistic mean-field modelswith nonlinear meson self-interactionsor density dependent meson-nucleon couplings







relativistic mean-field modelswith nonlinear meson self-interactionsor 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



generalization of relativistic mean-field (RMF) models

• extended relativistic density functional

with nucleons (ψp, ψn), deuterons (ϕµd), tritons (ψt), helions (ψh), α-particles (ϕα),

mesons (σ, ωµ, ~ρµ), electrons (ψe) and photons (Aµ) as degrees of freedom







only minimal (linear) meson-nucleon couplings








density-dependent meson-nucleon couplings Γi

− 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













cluster binding energies/effective resonance energies

− density dependence replaced by dependence on vector meson fields

⇒ nucleon/cluster/meson/photon field equations, solved selfconsistently



• grand canonical thermodynamical potential

Ω = −pV =

∫

d3r ωg(T, µi, σ, ω0, ρ0, A, ~∇σ, ~∇ω0, ~∇ρ0, ~∇A)

with density functional ωg depending ontemperature T , chemical potentials µi, meson and photon fields σ, δ, ω0, ρ0, A

fields equations with additional rearrangement contributions consistent derivation of thermodynamical quantities




Ω = −pV =

∫




• light clusters (2H, 3H, 3He, 4He), two-nucleon scattering correlations: explicitly included




Ω = −pV =

∫





• heavy clusters: (not considered here) Thomas-Fermi approximation in spherical Wigner-Seitz cells




Ω = −pV =

∫





• 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)



Relation and Differences of Models

quantum statistical approach generalized relativistic density functional

empirical nucleon-nucleon potential⇓

medium dependence of ⇒ parametrization of binding energy shiftscluster binding energies

phenomenological meson-nucleon interaction⇓

parametrization of nucleon ⇐ scalar/vector nucleon self-energiesself-energy and effective massin nonrelativistic approximationno effect of cluster formation medium-dependent change ofon nucleon mean fields cluster properties induces

change of mean fields


Formation and Dissolution of Clusters

10-5

10-4

10-3

10-2

10-1

density n [fm-3

]

10-3

10-2

10-1

100

part

icle

fra

ctio

n X

i

pn2H

3H

3He

4He

T = 10 MeV

Yp = 0.4

generalized relativistic density functional

(without heavy clusters)

• particle fractions

Xi = Aininb

nb =∑

iAini

• low densities:two-body correlation most important

• high densities:dissolution of clusters⇒ Mott effect



10-5

10-4

10-3

10-2

10-1

density n [fm-3

]

10-3

10-2

10-1

100

part

icle

fra

ctio

n X

i

pn2H

3H

3He

4He

T = 10 MeV

Yp = 0.4

generalized relativistic density functional

(without heavy clusters)

• particle fractions

Xi = Aininb

nb =∑

iAini

• 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



10-5

10-4

10-3

10-2

10-1

100

density n [fm-3

]

0.0

0.1

0.2

0.3

0.4

0.5

prot

on f

ract

ion

Xp

generalized RDF model vs. NSE

10-5

10-4

10-3

10-2

10-1

100

density n [fm-3

]

0.0

0.1

0.2

0.3

0.4

0.5

prot

on f

ract

ion

Xp

QS approach vs. NSE

2 MeV4 MeV6 MeV8 MeV10 MeV12 MeV14 MeV16 MeV18 MeV20 MeV

fraction of free protons in symmetric nuclear matter, thin lines: NSE



10-5

10-4

10-3

10-2

10-1

100

density n [fm-3

]

10-5

10-4

10-3

10-2

10-1

100

deut

eron

fra

ctio

n X

d

10-5

10-4

10-3

10-2

10-1

100

density n [fm-3

]

10-5

10-4

10-3

10-2

10-1

100

trito

n fr

actio

n X

t

10-5

10-4

10-3

10-2

10-1

100

density n [fm-3

]

10-5

10-4

10-3

10-2

10-1

100

helio

n fr

actio

n X

h

10-5

10-4

10-3

10-2

10-1

100

density n [fm-3

]

10-5

10-4

10-3

10-2

10-1

100

α-pa

rtic

le f

ract

ion

Xα

2H

3H

3He

4He


fraction of clusters in symmetric nuclear matter, generalized RDF vs. NSE



10-5

10-4

10-3

10-2

10-1

100

density n [fm-3

]

10-5

10-4

10-3

10-2

10-1

100

deut

eron

fra

ctio

n X

d

10-5

10-4

10-3

10-2

10-1

100

density n [fm-3

]

10-5

10-4

10-3

10-2

10-1

100

trito

n fr

actio

n X

t

10-5

10-4

10-3

10-2

10-1

100

density n [fm-3

]

10-5

10-4

10-3

10-2

10-1

100

helio

n fr

actio

n X

h

10-5

10-4

10-3

10-2

10-1

100

density n [fm-3

]

10-5

10-4

10-3

10-2

10-1

100

α-pa

rtic

le f

ract

ion

Xα

2H

3H

3He

4He


fraction of clusters in symmetric nuclear matter, QS approach vs. NSE



10-5

10-4

10-3

10-2

10-110

-3

10-2

10-1

100

α-pa

rtic

le f

ract

ion

Xα

10-4

10-3

10-2

10-110

-3

10-2

10-1

100

10-3

10-2

10-1

density n [fm-3

]

10-3

10-2

10-1

100

α-pa

rtic

le f

ract

ion

Xα

10-3

10-2

10-1

density n [fm-3

]

10-3

10-2

10-1

100

4 MeV 8 MeV

12 MeV 20 MeV

virial EoSNSEShen et al. EoSgeneralized RDF modelQS approach

alpha-particle fraction in symmetric nuclear matter


Thermodynamical Properties

10-5

10-4

10-3

10-2

10-1

100

density n [fm-3

]

10-4

10-3

10-2

10-1

100

101

pres

sure

p

[M

eV f

m-3

]


10-5

10-4

10-3

10-2

10-1

100

density n [fm-3

]

10-4

10-3

10-2

10-1

100

101

pres

sure

p

[M

eV f

m-3

]

QS approach vs. NSE


pressure p = −Ω/V in symmetric nuclear matter, thin lines: NSE



10-5

10-4

10-3

10-2

10-1

100

density n [fm-3

]

-10

0

10

20

30

40

pres

sure

/den

sity

p/

n [

MeV

]

generalized RDF vs. NSE

10-5

10-4

10-3

10-2

10-1

100

density n [fm-3

]

-10

0

10

20

30

40

pres

sure

/den

sity

p/

n [

MeV

]

QS approach vs. NSE


p/n in symmetric nuclear matter, limn→0(p/n) = T (ideal gas)



10-5

10-4

10-3

10-2

10-1

100

density n [fm-3

]

-20

-10

0

10

20

30

40

inte

rnal

ene

rgy

per

nucl

eon

EA

[M

eV]


10-5

10-4

10-3

10-2

10-1

100

density n [fm-3

]

-20

-10

0

10

20

30

40

inte

rnal

ene

rgy

per

nucl

eon

EA

[M

eV]

QS approach vs. NSE


internal energy per nucleon EA in symmetric nuclear matter, limn→0 EA = 32T + . . .



0.0000 0.0002 0.0004 0.0006 0.0008 0.0010

density n [fm-3

]

14.6

14.8

15.0

15.2

15.4

inte

rnal

ene

rgy

per

nucl

eon

[M

eV]

ideal gas

T = 10 MeV

neutron matter

comparison: different models and effects

• nonrelativistic ideal gas



0.0000 0.0002 0.0004 0.0006 0.0008 0.0010

density n [fm-3

]

14.6

14.8

15.0

15.2

15.4

inte

rnal

ene

rgy

per

nucl

eon

[M

eV]

ideal gasrelativistic Fermi gas

T = 10 MeV

neutron matter


• nonrelativistic ideal gas⇓ rel. kinematics + statistics

• relativistic Fermi gas



0.0000 0.0002 0.0004 0.0006 0.0008 0.0010

density n [fm-3

]

14.6

14.8

15.0

15.2

15.4

inte

rnal

ene

rgy

per

nucl

eon

[M

eV]

ideal gasrelativistic Fermi gasrelativistic virial EoS

T = 10 MeV

neutron matter




⇓ two-body correlations• virial EoS with relativistic correction



0.0000 0.0002 0.0004 0.0006 0.0008 0.0010

density n [fm-3

]

14.6

14.8

15.0

15.2

15.4

inte

rnal

ene

rgy

per

nucl

eon

[M

eV]

ideal gasrelativistic Fermi gasrelativistic virial EoSstandard RMF

T = 10 MeV

neutron matter





(not included in standard virial EoS)

⇓ mean-field effects• standard RMF model with

density dependent couplings



0.0000 0.0002 0.0004 0.0006 0.0008 0.0010

density n [fm-3

]

14.6

14.8

15.0

15.2

15.4

inte

rnal

ene

rgy

per

nucl

eon

[M

eV]

ideal gasrelativistic Fermi gasrelativistic virial EoSstandard RMFgRDF

T = 10 MeV

neutron matter





(not included in standard virial EoS)

⇓ mean-field effects• standard RMF model with

density dependent couplings⇓ two-body correlations

• generalized relativistic densityfunctional (gRDF) with contributionsfrom nn scattering



• internal energy E and free energy F = E − TS = −pV +∑

i µiNi

• dependence on neutron-proton asymmetry β =(

ntotn − ntot

p

)

/(

ntotn + ntot

p

)

• constant density n = 10−3 fm−3



-1.0 -0.5 0.0 0.5 1.0asymmetry β

930

940

950

960

970

inte

rnal

ene

rgy

per

nucl

eon

E/A

[M

eV] T = 2 MeV

T = 4 MeVT = 6 MeVT = 8 MeVT = 10 MeVT = 12 MeVT = 14 MeV

-1.0 -0.5 0.0 0.5 1.0asymmetry β

860

880

900

920

940

free

ene

rgy

per

nucl

eon

F/A

[M

eV]

T = 2 MeVT = 4 MeVT = 6 MeVT = 8 MeVT = 10 MeVT = 12 MeVT = 14 MeV


i µiNi


ntotn − ntot

p

)

/(

ntotn + ntot

p

)

• constant density n = 10−3 fm−3, without (dashed lines) light clusters



-1.0 -0.5 0.0 0.5 1.0asymmetry β

930

940

950

960

970

inte

rnal

ene

rgy

per

nucl

eon

E/A

[M

eV] T = 2 MeV

T = 4 MeVT = 6 MeVT = 8 MeVT = 10 MeVT = 12 MeVT = 14 MeV

-1.0 -0.5 0.0 0.5 1.0asymmetry β

860

880

900

920

940

free

ene

rgy

per

nucl

eon

F/A

[M

eV]

T = 2 MeVT = 4 MeVT = 6 MeVT = 8 MeVT = 10 MeVT = 12 MeVT = 14 MeV


i µiNi


ntotn − ntot

p

)

/(

ntotn + ntot

p

)

• constant density n = 10−3 fm−3, with (solid lines) light clusters


Symmetry Energy

0 0.1 0.2 0.3 0.4 0.5density n [fm

-3]

0

20

40

60

80

100

sym

met

ry e

nerg

y E

s [M

eV]

NL1NL3NL-SHTM1TW99DD-ME1DD

D3C

T = 0 MeVwithout clusters

• general definition for zero temperature:

Es(n) =1

2

∂2

∂β2

E

A(n, β)

∣

∣

∣

∣

β=0

β =nn−np

nn+np

⇒ nuclear matter parameters

J = Es(nsat) L = 3n ddnEs

∣

∣

n=nsat


Symmetry Energy

40 60 80 100 120 140 160symmetry energy slope coefficient L [MeV]

0.15

0.20

0.25

0.30

0.35

neut

ron

skin

thic

knes

s r sk

in in

208 Pb

[f

m] NL1

NL3NL-SHTM1TW99DD-ME1DD

D3C

experiment

Starodubsky & Hintz, PRC 49 (1994) 2118


Es(n) =1

2

∂2

∂β2

E

A(n, β)

∣

∣

∣

∣

β=0

β =nn−np

nn+np



∣

∣

n=nsat

• correlation: neutron skin thickness⇔ slope of neutron matter EoS (⇔ L)B. A. Brown, Phys. Rev. Lett. 85 (2000) 5296,

S. Typel, B. A. Brown, Phys. Rev. C 64 (2001) 027302


Symmetry Energy

40 60 80 100 120 140 160symmetry energy slope coefficient L [MeV]

0.15

0.20

0.25

0.30

0.35

neut

ron

skin

thic

knes

s r sk

in in

208 Pb

[f

m] NL1

NL3NL-SHTM1TW99DD-ME1DD

D3C

experiment

Starodubsky & Hintz, PRC 49 (1994) 2118


Es(n) =1

2

∂2

∂β2

E

A(n, β)

∣

∣

∣

∣

β=0

β =nn−np

nn+np



∣

∣

n=nsat

• correlation: neutron skin thickness⇔ slope of neutron matter EoS (⇔ L)B. A. Brown, Phys. Rev. Lett. 85 (2000) 5296,

S. Typel, B. A. Brown, Phys. Rev. C 64 (2001) 027302

• with clusters and at finite temperatures: use finite differences

Es(n) = 12

[

EA(n, 1) − 2E

A(n, 0) + EA(n,−1)

]

distinguish free symmetry energy Fs and internal symmetry energy Es


Symmetry Energy

0.0 0.5 1.0 1.5density n/n

0

0.5

1.0

1.5

inte

rnal

sym

met

ry e

nerg

y E

sym

(n)/

Esy

m(n

0)

MDI, x = 1MDI, x = 0MDI, x = -1

T = 0 MeV

temperature T = 0 MeV

• mean-field models without clusters

e.g. model with momentum-dependentinteraction (MDI), parameter x controlsdensity dependence of Esym

(B. A. Li et al., Phys. Rep. 464 (2008) 113)


Symmetry Energy

0.0 0.5 1.0 1.5density n/n

0

0.5

1.0

1.5

inte

rnal

sym

met

ry e

nerg

y E

sym

(n)/

Esy

m(n

0)

MDI, x = 1MDI, x = 0MDI, x = -1

T = 0 MeV





⇒ low-density behaviour not correct


Symmetry Energy

0.0 0.5 1.0 1.5density n/n

0

0.5

1.0

1.5

inte

rnal

sym

met

ry e

nerg

y E

sym

(n)/

Esy

m(n

0)

RMF with clustersMDI, x = 1MDI, x = 0MDI, x = -1

T = 0 MeV





⇒ low-density behaviour not correct

• gRDF with clusters

⇒ increase of Esym at low densitiesdue to formation of clusters

⇒ finite symmetry energyin the limit n→ 0


Symmetry Energy

finite temperature

• experimental determination of symmetry energy heavy-ion collisions of 64Zn on 92Mo and 197Au at 35 A MeV

temperature, density, free symmetry energy derived as functions ofparameter vsurf (measures time when particles leave the source)(S. Kowalski et al., Phys. Rev. C 75 (2007) 014601)


Symmetry Energy

0 1 2 3 4 5V

surf [cm/ns]

0

5

10

15

20

free

sym

met

ry e

nerg

y F

sym

[M

eV]

experimentRMF without clustersQS with clusters

0 1 2 3 4 5V

surf [cm/ns]

0

5

10

15

20

inte

rnal

sym

met

ry e

nerg

y E

sym

[M

eV]


(b)(a)

finite temperature




Symmetry Energy

0 1 2 3 4 5V

surf [cm/ns]

0

5

10

15

20

free

sym

met

ry e

nerg

y F

sym

[M

eV]


0 1 2 3 4 5V

surf [cm/ns]

0

5

10

15

20

inte

rnal

sym

met

ry e

nerg

y E

sym

[M

eV]


(b)(a)

finite temperature



• symmetry energies in RMFcalculation without clustersare too small


Symmetry Energy

0 1 2 3 4 5V

surf [cm/ns]

0

5

10

15

20

free

sym

met

ry e

nerg

y F

sym

[M

eV]


0 1 2 3 4 5V

surf [cm/ns]

0

5

10

15

20

inte

rnal

sym

met

ry e

nerg

y E

sym

[M

eV]


(b)(a)

finite temperature



• symmetry energies in RMFcalculation without clustersare too small

• very good agreementwith QS calculationwith light clusters

(J. B. Natowitz et al., Phys. Rev.

Lett. 104 (2010) 202501)


Summary

• correlations in nuclear matter

change of composition⇒ e.g. formation and dissolution of clusters

modification of thermodynamical properties⇒ e.g. change of symmetry energy


Summary




• various theoretical approaches

nuclear statistical equilibrium virial equation of state generalized Beth-Uhlenbeck approach generalized relativistic density functional (gRDF)


Summary




• various theoretical approaches

nuclear statistical equilibrium virial equation of state generalized Beth-Uhlenbeck approach generalized relativistic density functional (gRDF)

• important features

quasiparticles with medium dependent properties bound state and continuum contributions to correlations dissolution of clusters ⇒ Mott effect correct limits at low and high densities


theor.jinr.rutheor.jinr.ru/~ntaa/11/files/lectures/typel_ntaa2011.pdf · motivation: the life and...

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