Correlated Momentum distributions in nuclear systems

Correlated Momentum
distributions in nuclear systems
A. Rios (UM), A. Polls (UB), W. Dickhoff (WU)
ƒMotivation
ƒSingle particle spectral functions at zero and finite Temperature
ƒSingle-particle properties
ƒEntropy and Free energy
ƒThermodynamical properties. Liquid-gas coexistence.
ƒConclusions and perspectives
PRC71 (2005) 014313, PRC69(2004)054305, PRC72(2005)024316,PRC74 (2006)
054317, PRC73 (2006)024305,PRC78(2008)044314, PRC79(2009)025802
The equation of state (EOS) of asymmetric matter (different number of
protons and neutrons) is a necessary ingredient in the description of
astrophysical environments such as supernova explosions or the structure
of neutron stars.
Recent data concerning nuclei far from the stability valley has revitalized the
interest for asymmetric systems.
Heavy ion collisions.
The EOS of asymmetric nuclear matter in a wide range of densities and
temperatures is one of the challenging open problems in nuclear physics.
It is a very old many-body problem and still has not been fully solved. We
have two problems : the NN interaction and the many-body problem.
Here we will assume that the NN interaction is realistic and we will
concentrate in the many-body problem.
NN correlations and single particle properties
The microscopic study of the single particle properties in nuclear systems
requires a rigorous treatment of the nucleon-nucleon (NN) correlations.
ƒ Strong short range repulsion and tensor components, in realistic
interactions to fit NN scattering data îîImportant modifications of
the nuclear wave function.
ƒ Simple Hartree-Fock for nuclear matter at the empirical saturation
density using such realistic NN interactions provides positive energies
rather than the empirical -16 MeV per nucleon.
ƒThe effects of correlations appear also in the single-particle
properties:
ƒPartial occupation of the single particle states which would be fully
occupied in a mean field description and a wide distribution in energy
of the single-particle strength. Evidencies from (e,e’p) experiments.
The Single particle propagator a good tool to study single particle properties
Not necessary to know all the details of the system ( the full many-body
wave function) but just what happens when we add or remove a particle
to the system.
It gives access to all single particle properties as :
ƒ momentum distributions
ƒ self-energy ( Optical potential)
ƒ effective masses
ƒ spectral functions
Also permits to calculate the expectation value of a very special twobody operator: the Hamiltonian in the ground state.
Recently, enormous progress has been achieved in the calculation of
the single-particle propagator:
Self-consistent Green’s function (SCGF) and Correlated Basis Function
(CBF).
Single particle propagator
Zero temperature
Heisenberg picture
T is the time ordering operator
Finite temperature
The trace is to be taken over all energy eigenstates and all particle
number eigenstates of the many-body system
9Z is the grand partition function
Lehmann representation + Spectral functions
FT+ clossureîLehmann representation
The summation runs over all energy eigenstates and all particle number eigenstates
The spectral function
with
where
and
Momentum distribution
T=0 MeV
Finite T
therefore
Is the Fermi function
Spectral functions at zero tempearture
F
Free system
r
î Interactions î Correlated system
Spectral functions at finite Temperature
Free system
î Interactions î
Correlated system
Tails extend to the high energy
range.
Quasi-particle peak shifting with
density.
Peaks broaden with density.
Dyson equation
How to calculate the self-energy
The self-energy accounts for the interactions of a particle with the particles
in the medium.
We consider the irreducible self-energy. The repetitions of this
block are generated by the Dyson equation.
The first contribution corresponds to a generalized HF, weighted with n(k)
The second term contains the renormalized interaction, which is calculated in
the ladder approximation by propagating particles and holes. The ladder is the
minimum approximation that makes sense to treat short-range correlations.
It is a complex quantity, one calculates its imaginary part and after the real part
is calculated by dispersion relation.
The interaction in the medium
How to calculate the energy
Koltun sum-rule
The BHF approach î
is the BHF quasi-particle energy
Does not include propagation of holes
Momentum distributions for symmetric nuclear matter
At T= 5 MeV , for FFG k<kF 86 per cent of the particles! and 73 per cent at
T=10 MeV. In the correlated case, at T=5 MeV, for FFG k< kF 75 per cent and
66 per cent at T= 10 MeV.
-3
1
ρ=0.16 fm , T=10 MeV
Av18
FFG
0.8
Av18
FFG
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0.5
1
1.5
Momentum, k/kF
2
0
0.5
1
1.5
Momentum, k/kF
2
0
Momentum distribution, n(k)
Momentum distribution, n(k)
-3
ρ=0.16 fm , T=5 MeV
Density dependence of the occupation of lowest momentum state at
T=5 MeV .
Symmetric, T=5 MeV
Neutron, T=5 MeV
1
1
0.95
0.8
0.9
0.7
Av18
CDBonn
FFG
0.6
0
0.08
0.16
0.24
-3
Density, ρ [fm ]
Av18
CDBonn
FFG
0.32 0
0.08
0.16
0.24
-3
Density, ρ [fm ]
0.85
0.32
n(k=0)
n(k=0)
0.9
Momentum distributions of symmetric and neutron matter at T=5 MeV
Argonne v18
T=5 MeV
1
Symmetric
matter
-3
ρ=0.08 fm
-3
ρ=0.16 fm
-3
ρ=0.24 fm
-3
ρ=0.32 fm
0.4
0.2
n(k)
-5
10
-6
0
1
10
Neutron
matter
0.8
-4
10
n(k)
-3
10
0.6
-2
10
-3
0.6
10
0.4
10
0.2
10
0
600 800 1000 1200 1400 10
Momentum, k [MeV]
-4
-5
-6
0
1
0.5
1.5
Momentum, k/kF
2
n(k)
n(k)
0.8
-2
10
Relation of the momentum distribution and the derivative of the real part
time ordered self-energy at the quasi-particle energy.
The self-energy down has contributions from 2p1h self-energy diagrams
The self-energy up has contributions from 2h1p self-energy diagrams
Argonne v18
1
-3
ρ=0.16 fm , T=5 MeV
n(k)
n↑(k)
n↓(k)
0.8
0.6
0.4
10
10
-2
-3
0.2
0
10
-4
-0.2
-0.4
0
0.5
1
1.5
Momentum, k/kF
2 200 400 600 800 1000 1200 10
Momentum, k [MeV]
-5
Momentum distribution, n(k)
Momentum distribution, n(k)
Momentum distributions obtained from the derivatives of the self-energy
Different components of the imaginary and real parts of the self-energy
-3
Re Σ(0,ω) [MeV]
Im Σ(0,ω) [MeV]
CDBonn, ρ=0.16 fm , T=5 MeV
20
0
-20
-40
ΣR
Σ↑
Σ↓
20
0
-20
ΣR
Σ↑
Σ↓
-40
-60
-1000 -500
0
500 1000 1500 2000
ω−µ [MeV]
CDBonn
0
CDBonn
0
-20
-20
-40
-40
-60
-60
0
Argonne v18
Argonne v18
0
-40
-40
-80
-80
-3
ρ=0.08 fm
-3
ρ=0.16 fm
-3
ρ=0.24 fm
-3
ρ=0.32 fm
-120
-160
-500
0
500 1000 1500 2000 2500 -500
ω−µ [MeV]
0
500 1000 1500 2000 2500
ω−µ [MeV]
-120
-160
ImΣ↓(0,ω) [MeV]
Neutron matter, T=5 MeV
ImΣ↓(0,ω) [MeV]
ImΣ↓(0,ω) [MeV]
ImΣ↓(0,ω) [MeV]
Symmetric matter, T=5 MeV
The circles represent the position of the quasi-particle energy
CDBonn
CDBonn
-20
-20
-40
-40
-3
-60
-60
ρ=0.08 fm
-3
ρ=0.16 fm
-3
ρ=0.24 fm
-3
ρ=0.32 fm
-80
-80
-100
ReΣ↓(0,ω) [MeV]
0
ReΣ↓(0,ω) [MeV]
0
Neutron matter, T=5 MeV
-100
0
Argonne v18
Argonne v18
0
-40
-40
-80
-80
-120
-120
-160
-160
-200
-200
-200
0
200 400 600 800
ω−µ [MeV]
-200
0
200 400 600 800
ω−µ [MeV]
ReΣ↓(0,ω) [MeV]
ReΣ↓(0,ω) [MeV]
Symmetric matter, T=5 MeV
CDBonn
Symmetric
matter
0
Symmetric
matter
-0.1
-0.1
-0.2
-0.2
∂ωReΣ↓(0,ω)
0
0
Neutron
matter
Neutron
matter
-0.05
∂ωReΣ↓(0,ω)
∂ωReΣ↓(0,ω)
0
∂ωReΣ↓(0,ω)
Argonne v18
-0.05
-3
ρ=0.08 fm
-3
ρ=0.16 fm
-3
ρ=0.24 fm
-3
ρ=0.32 fm
-0.1
-150 -125 -100 -75
-50
ω−µ [MeV]
-25
0
-125 -100 -75
-0.1
-50
ω−µ [MeV]
-25
0
Proton and neutron momentum distributions a=0.2, r=0.16 fm-3
9The BHF n(k) do not contain
correlation effects and very
similar to a normal thermal
Fermi distribution.
9The SCGF n(k) contain
thermal and correlation
effects.
9Depletion at low momenta
and larger occupation than the
BHF n(k) at larger momenta.
9The proton depletion is
larger than the neutron
depletion. Relevant for (e,e’p).
Dependence of n(k=0) on the asymmetry
-3
1
nν(k=0)
0.95
ρ=0.16 fm , T=5 MeV
Neutrons
0.9
0.85
0.8
0.75
0.7
0
Protons
Av18
CDBONN
FFG
0.2
0.4
0.6
0.8
Asymmetry, β
1
Free Fermi Gas
β=0.0
β=0.2
β=0.4
β=0.6
β=0.8
1
0.8
0.6
0.4
0.2
Argonne v18
β=0.0
β=0.2
β=0.4
β=0.6
β=0.8
Neutrons
Protons
Neutrons
Protons
1
0.8
0.6
0.4
0.2
0
0
0 100 200 300 400 500 0 100 200 300 400 500
Momentum, k [MeV]
Momentum, k [MeV]
Momentum distribution, n(k)
Momentum distribution, n(k)
Neutron and proton momentum distributions for different asymmetries
K=0 MeV proton spectral function for different asymmetries
9a→ 1, kFp→ 0 MeV, the
quasi-particle peak gets
narrower and higher.
9The spectral function at
positive energies is larger
with increasing asymmetry.
9 Tails extend to the highenergy range.
9 Peak broadens with
density
Density and temperature dependence of the spectral function for neutron matter
Spectral function, T=5 MeV
Argonne V18
Argonne V18
k=0
k=0
-2
-2
10
10
-3
-3
10
10
-4
-4
10
10
-5
-5
10
-6
10
-1
k=kF
10
-2
k=kF
10
-3
-3
10
-4
-4
10
10
-5
-5
10
10
-6
-6
10
-2
10
-3
10
-4
10
-2
10
10
-1
-1
10
10
-3
ρ=0.04 fm
-3
ρ=0.08 fm
-3
ρ=0.16 fm
-3
ρ=0.24 fm
-3
ρ=0.32 fm
k=2kF
k=2kF
T=5 MeV
T=10 MeV
T=15 MeV
T=20 MeV
-1
10
-2
10
-3
10
-4
10
-5
-5
10
10
-6
10
-500
-250
0
250
ω−µ [MeV]
500
-500
-250
0
250
ω−µ [MeV]
500
-1
-6
10
-1
-1
-1
(2π) A(k,ω) [MeV ]
10
10
-1
10
(2π) A(k,ω) [MeV ]
-1
10
-3
Spectral function, ρ=0.16 fm
Partial derivative
0
-40
-60
-0.1
-80
-0.2
-100
Neutrons
-120
Neutrons
-0.3
Neutrons
-140
-40
0
Σ↓(k,ω) [MeV]
0
α=0.5
-60
-20
α=0.4
α=0.3
-40
-80
α=0.2
-60
α=0.1
-100
-80
-100
-120
Protons
Protons
-120
-140
-200 -100 0 100 200 300
-500 0 500 1000 1500 2000
ω−µτ [MeV]
ω−µτ [MeV]
∂ω ReΣ↓(k,ω)
Σ↓(k,ω) [MeV]
0
-20
-40
-60
-80
-100
-120
Real part
-0.1
-0.2
-0.3
Protons
-80
-40
-20
-60
ω−µτ [MeV]
0
∂ω ReΣ↓(k,ω)
Imaginary part
n(k=0) for nuclear and neutron matter,
-3
nn(k=0)-np(k=0)
0.25
0.2
0.15
0.1
ρ=0.16 fm , T=5 MeV
CDBONN
Reid93
Av18
Av8’
Av6’
Av4’
FFG
0.05
0
0
0.2
0.4
0.6
0.8
Asymmetry, β
1
Real part of the on-shell self-energy for neutron matter
-3
ρ=0.08 fm
ρ=0.16 fm
-3
-3
ρ=0.24 fm
0
0
Re Σ(k,ε(k)) [MeV]
Re Σ(k,ε(k)) [MeV]
Argonne V18
SCGF
-20
-20
BHF
SCGF
SCGF
-40
-40
T=5 MeV
T=20 MeV
BHF
-60
-60
BHF
0
1
2
3
-1
k [fm ]
40
1
2
3
-1
k [fm ]
40
1
2
3
-1
k [fm ]
4
1
1
T=5 MeV
0.8
T=5 MeV
0.8
0.6
0.6
-3
ρ=0.04 fm
-3
ρ=0.08 fm
-3
ρ=0.16 fm
-3
ρ=0.24 fm
-3
ρ=0.32 fm
0.4
0.2
0
1
ρ=0.16 fm
0.8
0.4
0.2
-3
ρ=0.16 fm
0
1
-3
0.8
0.6
0.6
T=20 MeV
T=16 MeV
T=12 MeV
T=8 MeV
T=4 MeV
0.4
0.2
0
0
0.5
1
k/kF
1.5
2 0
0.5
0.4
0.2
1
k/kF
1.5
2
0
Momentum distribution Momentum distribution
Momentum distribution Momentum distribution
Argonne V18
frhsfj
CDBONN
Occupation of the lowest momentum state as a function of density for
neutron matter.
1
0.95
0.95
0.9
0.9
0.85
0.85
T=5 MeV
T=10 MeV
T=15 MeV
T=20 MeV
T=5 MeV, FFG
0.8
0.75
0.7
0
0.08
0.16
-3
ρ [fm ]
0.24
0.32 0
0.8
0.75
0.08
0.16
-3
ρ [fm ]
0.24
0.32
n(k=0)
1
n(k=0)
Argonne V18
CDBONN
Summary
™The calculation and use of the single particle green function is suitable and
it is easily extended to finite T. Temperature helps to avoid the “np” pairing
instability.
™The propagation of holes and the use of the spectral functions in the
intermediate states of the G-matrix produces repulsion. The effects increase
with density.
™Important interplay between thermal and dynamical correlation effects.
™For a given temperature and decreasing density, the system approaches
the calssical limit and the depletion of n(k) increases.
™For larger densities, closer to the degenerate limit, dynamical correlations
play a more important role and n(0) decreases with increasing density.
™For a given density when the asymmetry increases, the neutrons get more
degenerate and the protons loss degeneracy. The depletion of the protons is
larger and has important thermal effects.
™Three-body forces should not change the qualitative behavior.