The Nuclear Equation of
State
Abhishek Mukherjee
University of Illinois at Urbana-Champaign
Work done with : Vijay Pandharipande, Gordon Baym, Geoff Ravenhall, Jaime Morales and Bob Wiringa
National Nuclear Physics Summer School 2007
Abhishek Mukherjee: Nuclear EOS,
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Outline
EOS of Hot Dense Matter
Variational Theory for Finite T EOS
The Hamiltonian
Results
Conclusions
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EOS of Hot Dense Matter
I
Nuclear Matter
Infinite Matter with Neutrons + Protons only
• Symmetric Nuclear Matter
Equal Number of Neutrons and Protons
• Pure Neutron Matter
Neutrons only
• Asymmetric Nuclear Matter
Unequal number of neutrons and protons
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EOS of Hot Dense Matter
Why is the EOS interesting/useful ?
I
Supernova evolution
I
Cooling of Neutron Stars
I
Heavy Ion Collisions
I
Novel Many Body Systems
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EOS of Hot Dense Matter
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Variational Theory for Finite T EOS
Challenges for a microscopic theory:
1. Include correlations beyond mean field
2. Good description of the excited states (for T 6= 0)
3. Calculate the energy, E(ρ, T ) accurately
4. Calculate the entropy, S(ρ, T ) accurately
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Variational Theory for Finite T EOS
Correlated Basis States
Landau Fermi Liquid Theory
Eigenstates of
Interacting System
one - one
←→
Fermi Gas
States (FGS)
Correlated Basis States (CBS)
|ΨI ) = |CBS) =
Abhishek Mukherjee: Nuclear EOS,
G |FGS]
[FGS |G † G| FGS]
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Variational Theory for Finite T EOS
Correlation Operator
G=S
Y
F(rij )
i<j
I
Generalization of Jastrow method of pair correlation
functions
I
Correlations are included in the functions F(rij )
(ΨI |H|ΨI )
(ΨI |H|ΨJ )
Abhishek Mukherjee: Nuclear EOS,
−→ VCS (FHNC/SOC)
−→ No such method
8
Variational Theory for Finite T EOS
Gibbs-Bogoliubov variational principle for T 6= 0
F ≤ FV ≡ Tr(ρV H) + T TrρV lnρV .
FV is an approximation for F
Analogy : Rayleigh-Ritz variational principle for T = 0
E0 ≤ (Φ0 | H |Φ0 )
e.g. the Akmal-Pandharipande-Ravenhall EOS Phys. Rev. C 58, 1804(1998)
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Variational Theory for Finite T EOS
What we want to do
Variational ‘Hamiltonian’
Hv |ΨI ) = EvI |ΨI ) =
X
T (k )nI (k )|ΨI )
k
Variational density matrix
I
ρv (T ) ∝ e−Hv /T = e−Ev /T |ΨI )(ΨI |
Sv (T ) = −
X
(n̄(k ) ln n̄(k ) + (1 − n̄(k )) ln(1 − n̄(k )))
k
X
1
I
e−Ev /T (ΨI |H|ΨI )
I
−Ev /T
Ie
I
Tr (ρv H) = P
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Variational Theory for Finite T EOS
The Orthogonality Problem
The Correlated basis states are not Orthonormal to each other
|ΨI )(CBS) −→ |ΘI i ≡ Orthonormal Correlated Basis States (OCBS)
X
Hv |ΘI i = EvI |ΘI i =
T (k )nI (k )|ΘI i
k
1
−EvI /T
Ie
Tr (ρv H) = P
X
I
e−Ev /T [(Ψi |H|Ψi ) + ∆EI ]
i
∆EI ( Orthogonality Corrections) −→ (ΨI |H|ΨJ ), non
diagonal matrix elements
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Variational Theory for Finite T EOS
Our Solution
I
Microcanonical ensemble instead of a canonical ensemble
ρMC
=
v
1 X
|ΘI ihΘI |
NM
I
M(T ) ≡ the microcanonical ensemble at temperature T
NM = # of elements in M
I
Mixed orthonormalization procedure (Lówdin +
Gram-Schmidt).
∆EI (Orthogonality Corrections) −→ 0
Details: A. Mukherjee and V.R. Pandharipande , Phys. Rev. C 75, 035802 (2007)
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The Hamiltonian
The Hamiltonian
H=
X
i
−
X
X
~2 ∇2 X
vij +
Vijk +
δvij .
+
2m
i<j
i<j<k
I
vij = NN potential (Argonne v 18)
I
Vijk = NNN potential (Urbana IX)
I
δvij = Relativistic Boost Corrections
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i<j
13
The Hamiltonian
Argonne v 18
Av 18 = Long Range Part + Intermediate Range Part + Short
Range Part
• Long Range Part ∼ One Pion
Exchange Potential
I
Intermediate Range Part : Motivated by Meson Exchange
Theory
I
Short Range Part : Phenomenological
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The Hamiltonian
Urbana IX and Relativistic Boost Corrections
Urbana IX
I
UIX = Two Pion Exchange + Phenomelogical Repulsion
I
Fits Binding Energy of Light nuclei + Binding of nuclear
matter at saturation
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The Hamiltonian
Urbana IX and Relativistic Boost Corrections
Urbana IX
I
UIX = Two Pion Exchange + Phenomelogical Repulsion
I
Fits Binding Energy of Light nuclei + Binding of nuclear
matter at saturation
Relativistic Boost Corrections
I
Leading order corrections due to Special Relativity
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Results
I
The (temperature and density dependent) eigenvalue
spectrum of HV is an additional ‘parameter’ to be varied.
X
T (k )n̄(k )
EvI =
k
Effective mass approximation
T (k ) =
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k2
+ U(ρ, T )
2m∗ (ρ, T )
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Results
I
Minimize the energy of the two body cluster at finite
temperature
I
Long range part of the wavefunctions constrained by the
model (uncorrelated) wavefunction
Euler-Lagrange equations → Correlation functions(F)
I
F has variational parameters : 2 correlation lengths
(dc , dt ), spin-isospin quenching parameter (α) and the
effective mass (m∗ )
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Results
100
Free Energy (MeV)
Symmetric Nuclear Matter
0
5
10
20
0
0.2
0.4
-3
Density (fm )
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Results
50
40
Pure Neutron Matter
Free Energy (MeV)
30
20
10
10
0
-10
-20
20
0
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0.1
0.2
-3
Density (fm )
0.3
0.4
19
Results
Symmetric Nuclear Matter
5 HDP
m*/m
1
5 LDP
10
20
0.5
0
0.1
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0.2
0.3
-3
Density (fm )
0.4
0.5
0.6
20
Results
1
Pure Neutron Matter
0.9
20
m*/m
0.8
10
0.7
0.6
0.5
0
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0.1
0.2
-3
Density (fm )
0.3
0.4
21
Results
Pion Condensation
I
πs0 condensation = Softening of the spin-isospin sound
mode (same quantum number as a pion)
I
Pions are absorbed in the NN interaction
I
Pionic modes ∼ Tensor interactions
Pion condensation =⇒ enhancement of the tensor
correlations
dt increases dramatically in the HDP
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Results
6
Tensor Correlation Length (d t / r0)
10 PNM HDP
5 SNM HDP
5
10 PNM LDP
4
10 SNM
20 SNM
3
5 SNM LDP
2
0
0.1
Abhishek Mukherjee: Nuclear EOS,
0.2
0.3
-3
Density (fm )
0.4
0.5
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Conclusions
I
The method of correlated basis functions which has in the
past been successfully applied to study the ground state of
dense strongly interacting systems has been extended to
study the thermodynamics and response at finite
temperature.
I
An EOS for nuclear matter at low and moderate
temperatures is being developed.
I
In future the influence of thermal pions at very high
temperature and pairing at low densities will be examined.
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Appendix
Correlated Basis States
Correlation operator
G=S
Y
Fij
i<j
Fij contains the same operators as the NN interaction.
Generalization of the Jastrow correlation functions.
Correlated Basis States (CBS)
G|ΦI i
|ΨI ) = p
hΦI |G † G|ΦI i
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Appendix
Orthogonalization
The CBS are not orthogonal to each other
|ΦI )
(CBS)
orthonormalization
−→
|ΘI i (OCBS)
OCBS = Orthonormalized Correlated Basis States
hΘI |H|ΘI i = (ΦI | H |ΦI ) + Orthogonality Corrections
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Appendix
Variational Hamiltonian


X
HV |ΘI {nI (k, σz )}i = 
nI (k, σz )V (k, σz ) |ΘI {nI (k, σz )}i ,
k,σz
= EIV |ΘI {nI (k, σz )}i
For present Calculations:
V =
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~2 k 2
+ constant
2m∗ (ρ, T )
27
Appendix
We define
ρV =
1 X
|ΘI ihΘI |
NM
I∈M
M = Microcaninical ensemble of the Variational Hamiltonian
HV
FV
=
1 X (ΨI |H| ΨI )
NM
I∈M
+ Orthogonality Corrections − TSV (T )
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Appendix
Vanishing Orthogonality corrections
We showed that Orthogonality Corrections →, in the
thermodynamic limit.
FV =
1 X
(ΨI |H| ΨI ) − TSV (T )
NM
I∈M
First Term : Variational Chain Summation = Fermi Hypernetted
Chain summation + Single Operator Chain summation
Second Term : Trivial
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