Equation of state for the dense matter of neutron stars ...sfb.aei.mpg.de/Meetings/School04/Talks/muether04.pdf · for the dense matter of neutron stars and supernovae ... • White

Equation of state of Nuclear Matter, Golm 2004 Folie 1

Equation of state for the dense matter of neutron

stars and supernovae

SFB/Tr7 Summer School

Albert Einstein Institut

20.-25.9. 2004

Herbert Müther


Normal matter versus nuclear matter

Distance between atoms:

10-10 m

Radius of nucleus:

3*10-15 m

10 km

0.3 m

Compress matter to nuclear matter:

Some densities:

• Earth 5.5 g/cm3

• White dwarf ~106 g/cm3

• Nuclear matter 2.7 1014 g/cm3

Compress matter to nuclear matter:

Some densities:

• Earth 5.5 g/cm3

• White dwarf ~106 g/cm3

• Nuclear matter 2.7 1014 g/cm3


Gravitation to compress matter:

r

Balance of forces:

[ ]

2

22

)()(

)()(

)()(

)()(

)(

0

r

rmGr

dr

d

d

dP

r

rrPrP

rrPrPfFr

rmGrfr

r

rmGmF

FF

press

grav

pressgrav

ρρρ

ρ

−=�

∆∆+−

∆+−∆=

∆∆−=∆−=

=+

Mass inside the shell of radius r: m(r)

)(4)(

)(4)()(

2

2

rrdr

rmd

rrrrmrrm

ρπ

ρπ

=

∆=−∆+

So we need the

equation of state: P(ρ)

Simple EoS: ideal gas

TPTNVP ρ==


Non-relativistic degenerate Fermi gas

• single particle energiesm

kii 2

2

=ε

Energy and density for infinite system of Fermions:

( ) ( ) dkkV

kdxdFk

i

=�

0

23

333 2

42

2

2

ππ

π

�

�

Density:

Energy:

Pressure 35

5

20

22

2

32

0

22

1

25

1

2

1

3

11

ρ

ππ

ππρ

mdV

dEP

m

kdkk

m

k

V

E

kdkkV

N

F

k

F

k

F

F

∝−=

==

===

Note T=0Note T=0

Fermi momentum: kF


massesrest including

)()()(

if

E

kEkEkE

nepnFn

eFe

pFp

e

>+

+→+ ν

Density g/cm3

101

106

1015

„ Normal Matter “ stabilized by valence e-

White dwar f stabilized by degenerate e-gas

its cheaper to buy a neutron:

Homog. neutron matter

Nuclear Pasta

Phase transition to quark gluon plasma ?

(„ filling factor“ of nuclear matter : ~1/3 …1/24)


Its not neutrons only:decomposition of neutron star matter

n

Σ∆


Outline of this lecture

• strong interaction– realistic NN interaction

– meson exchange

– many-body forces

• many-body theories– hole-line expansion

– selfconsistent Green function

• relativsitic aspects

• superfluidity

• interaction with neutrinos

• phase transitions– pion condensation

– kaon condensation

– quark matter


Few references:

• R. Machleidt, Adv. Nucl. Phys.19 (1989) 185

• H. Heiselberg, M. Hjorth-Jensen, Phys. Rep. 328 (2000) 237

• H. Müther, A.Polls, Prog. Part. & Nucl. Phys. 45 (2000) 243

•A. Akmal, V.R. Pandharipande, D.G. Ravenhall, Phys. Rev. C58 (1998) 1804

• M. Prakash et al. Phys. Rep. 280 (1997) 1

• H. Riffert, H. Müther, H. Herold, H. Ruder: Matter at High Densities in Matter at High Densities in AstrophysicsAstrophysics Springer Tracts in M.P. 133(1996)

• D.Blaschke, N.K.Glendenning, A.Sedrakian:Physics of Neutron Stars InteriorsPhysics of Neutron Stars InteriorsSpringer (2001)


Basic Concept of microscopic nuclear structure study:Basic Concept of microscopic nuclear structure study:

Realistic NN interaction:

• model of the interaction

• adjust parameter to describe NN data

Solve Many-Body problem at normal densities:

• describe saturation of nuclear matter

E/A = -16 MeV, ρ0=0.16 fm-3, kF=1.36 fm-1

• and finite nuclei

without any adjustable parameterwithout any adjustable parameter

Predict properties of matter at densities above ρ0


Relativistic

Meson-Exchange Model π,ρ,σ,...qq

quark- versus meson-exch.

k -k

k’ -k’

q = k - k’

Ingredients:

1) Dirac spinor of nucleons

2) meson nucleon vertices

λλλ ��

��

��

��

+

+=mE

km

mEku

k

k 21

2),(

�

vector2

4

arpseudoscal 4

mesonscalar 45

+=Γ

=Γ

=Γ

υµυµ σγπ

γπ

π

qim

fg

gi

g

vvv

psps

ss

3) meson propagator

mesons for vector

mesonsscalar for 1

2220

2

2220

µµ

µ

υµµυ

−−

+−=

−−=

qq

qqg

P

qqP

s

s

��

[ ] [ ])()'()()'(),'( kukuPkukukkV αααα Γ−Γ−=


Solve NN Problem

nonrel.: Lippmann-Schwinger eq.:

TiHE

VVT

VT

ViHE

η

η

+−+=

Ψ=Φ

Ψ+−

+Φ=Ψ

0

0

1

1

explicitly:

relat: Bethe-Salpeter eq.:

• propagate + spinors only Blankb.

• fix k0 Sugar

• define:

• propagate + spinors only Blankb.

• fix k0 Sugar

• define:


Typical One-Boson-Exchange potential:

• good fit of NN data with only a few parameter

• ππππ : dominates long range, tensor force

• σσσσ :::: medium range attraction

• ωωωω : short range repulsion

• ρ ρ ρ ρ : short range tensor force

• other mesons for fine tuning

however: there is no σ σ σ σ in the particle data booklet !?!

σσσσππππ

∆∆∆∆


Local versus nonlocal InteractionInteraction depends on relative coordinate r but not on center of mass R:

<r ‘ |V|r>

Transfomation to momentum space:

<k‘ |V|k> = dr dr ‘ <k‘ |r ‘><r ‘ |V|r><r |k>

= dr dr ‘ eikr e -ik‘ r‘<r ‘ |V|r>

I f the interaction is local (<r ‘ |V|r>= V(r ) δδδδ(r -r ‘ ))

<k‘ |V|k> = dr e i(k-k‘ )r V(r)

the representation in momentum space depends on q=k-k‘ only;

Otherwise it also depends on k+k‘

Nonlocality corrsponds to momentum dependence:

OBE is non-local: CD Bonn, Nijm1


Ener gy of the deuter on:

one number or iginat ing fr om differ ent sour ces

-2.22 = T S +T D + V SS + V DD + 2 V SD

T S T D V SS V DD 2 V SD

CDBo 9.79 5.69 -4.77 1.34 -14.27V18 11.31 8.57 -3.96 0.77 -18.94Nij m1 9.66 7.91 -1.35 2.37 -20.82Nij m2 12.11 8.10 -5.40 0.59 -17.83Reid 12.61 9.53 -0.47 4.55 -28.45

Local and non-local NN interactions

)()( 13

13 DSDeuteron Ψ+Ψ=Ψ


Wavefunctions of deuteron

from various interactions

Local interactions

V18, Nijm2

stiff

Nonlocal interactions

CdBonn, Nijm1

soft

D-state probability:

a measure of the tensor force

r


Hartree-Fock:

first step in many-body theories

Basis of all EoS

Variational approach: find Slaterdeterminant which yields minimal E

( )hhfh

Fh

iji

Fi

tE

hjVihjhVihjUi

jUijtijhi

i

ε

δε

+=

−=

=+=

��

�

��

�

=Φ ∏

<

<

<

2

1

A

U

= +

j

i

h

• nucleons move independently in the mean field U

• determine single-particle wave functions i self-consistent

• trivial for nuclear matter


Energy of nuclear matter in HF approx.

EHF TCDBo 4.64 23.01V18 30.34 23.01Nijm1 12.08 23.01Reid 176.2 23.01

� Correlations are necessary to obtain binding energy� Local interactions are stiffer than non-local

r

V,Φ,Ψ

Potential

uncorrel wave

correl wave

Correlations:

• <V> more attractive

• <T> larger


Various ways to account for correlation effects

• hole line expansion, Brueckner theory

determine effective operator, such that

• Coupled cluster method, (Exp S)

determine exact wave function in form

• self-consistent evaluation of Green function

e.g.

• Variational approach (Monte Carlo techniques)

Φ=Φ

Ψ=Φ

EH

P

eff

( ) Φ=Ψ Sexp

ΨΨ= + )0()()´,,( ´kk ataTtkkG

0=ΨΨ

ΨΨ Hδ


Hole line expansion, Brueckner th: Our aim: solveΨ=Ψ EH

1) define appropriate model wave function Φ

iiiiH

VHUVH

UVUTVTH

ϕϕεϕ of Slaterdet. :

)(

)()(

0

0120

1212

Φ=+=−+=

−++=+=

2) Determine Heff mit ΦΦ= effHE

Linked

eff

effeff

effeff

VHE

QVV

VHE

QVVV

VHH

��

�

��

�

−+=

−+=

+=

00

mod

0

mod

0

Linked Cluster

theorem

Note: - order expansion according to number of hole lines

- Veff is a many-body operator

- choice of U1 influences convergence


Brueckner Hartree Fock for nuclear matter

Cm

k

ijwGijt

wGKkhw

KkQVkdVwG

i

jFj

iii

+≈

>+=<+=−

+=

<

*

2

3

2

|)(|

)(),(

),()(

εεε

�

�

Bethe Goldstone equation

corresponds to LS eq. in medium

Pauli quenching - dispersive qu.

G less attractive than T

Tht

VVT1212

1

−+=

Problems:

self-consistent solution

how to choose particle spectrum

(conventional vs continous)

convergence (3 hole lines contr.?)


Particle spectrum and convergence:

Solve 3-body problem in medium:

Bethe Fadeev eq.

Hartree-Fock: 30 MeV

BHF: -10 or -17 MeV

BHF +3: -15 MeV

H.Q.Song et al: Phys.Rev.Lett. 81 (1998) 1584

using V14


Ener gy of nuclear matter

EH F T V ππππ ECDBo 4.64 36.23 -22.30 -17.11V 18 30.34 47.07 -40.35 -15.85Nij m1 12.08 39.26 -28.98 -15.82Reid 176.2 49.04 -27.33 -12.47

� Cor r elat ions ar e necessar y to obtain binding ener gy� Pionic (tensor ) cor r elat ions ar e ver y impor tant� Cor r elat ions a f inger pr int of the inter act ion

Dependence on Interaction:Soft nonlocal interactions yield more binding and softer EoS

Employ Hellmann-Feynmantheorem to determineexpectation values:

1

0

|||

||)(

)(

=∂∂>=Ψ∆Ψ<

>Ψ>=Ψ∆+=

λλ

λλλ

λ

λλλ

EV

EH

VHH


Saturation point of nuclear matter:

thethe CoesterCoester bandband

CD Bonn

V18

We dont get energy and density


Saturation in finite nuclei: 16O

Include effects of long rangecorrelations: generalized ring


Relativistic effects: Dirac BHF

π,ρ,σ,...

k -k

k’ -k’

q = k - k’λλλ �

��

�

��

��

+

+=mE

km

mEku

k

k 21

2),(

�

Ingredients of meson exchange:

1) Dirac spinor of nucleons

2) meson nucleon vertices

[ ] vector4

meson scalar 4µγπ

π

vv

ss

g

g

=Γ

=Γ

µµγγvectscalar VVV +•= 11

σ attraction

ω repulsion

µµγ Σ+Σ•=Σ sHartree 1

Should we not consider

• the relativistic structure of Σ

• change of Dirac spinors in medium


Note:

0)0(

*

Σ−Σ+==Σ+=

s

s

mk

mm

ε

• effective mass small

• no binding in DHF

• correlation effects in

∆G reduce Σs and Σ0

• binding in DBHF

• large cancellation


Effective masses decrease with density

enhancement of small Dirac component


DBHF reproduces saturation point of nuclear matter.....

and improves for finite nuclei


How reliable is the EoS ????

♣ with respect to the many-body approach

♣ with respect to the NN interaction

♣ with respect to relativistic dynamic and/or 3N forces


with respect to the many body approach

Both calculations use V18 interaction


Energy density for β-stable nucleon matter

with respect to the NN interaction


with respect to relativistic effects or 3 N forces

Energy density pure neutron matter


Single-particle Greens function:

Def:

ΨΨ−Θ

−ΨΨ−Θ=−+

+

)(´)()´(

´)()(´)(´),(

tatatt

tatattttkig

kk

kkparticle propagation

hole propagation

Fourier Transformation:

( ) ( )∞−

∞

−++

+−+

−−

�

−−−

ΨΨ+

+−−

ΨΨ=

Fh

F

p

kA

kA

i

kSd

i

kSd

iEE

a

iEE

akg

ηωωωω

ηωωω

ω

ηωηωω

δ δ

δ

γ γ

γ

´

´,´

´

´,´

)()(),(

21

21

Spectral function Sh(k, ω ):

probability for the removal of

particle with momentum k and

energy ω


),(),(),(),(),( 00 ωδβωγδωαγωαβωαβγδ

gggg Σ+=

ηωωω

iktkg

k ±Σ+−=

)),((

1),(

infinite matter

Solve Dyson equation

Calculate Greens function

Evaluate self-energy Σ(k,ω)

self-consistency required


),(),(),( 1212 ωωω kkk phhp ∆Σ+Σ=Σ

hole-line expansion:

Ignore ∆Σ∆Σ∆Σ∆Σ2h1p

i.e.imaginary par t at

ω<µω<µω<µω<µ

The nucleon self-energy


Self-energy on-shell: Σ(k,ω=tk+Σ)

PhD thesis of

Tobias Frick


Examples for Spectral functions S(k,ωωωω) The effect of self-consistency:

QPGF: quasipar ticle approx. in evaluating ΣΣΣΣ

SCGF: self-consistent

∞

=µ

ωω-

)S(k, )( dkn

Spectral function and

momentum distribution:


Spectral function and momentum distribution

Correlations lead to nucleons with high momenta

How can we observe these momenta?

Correlations lead to partial occupanciesfor momenta k below kF :

Compare with effects of finite T


Explore momentum distribution in (e,e‘ )p ?

Exp: Blomqvist et al. Mainz

Nukleons with large momenta

• only at large „missing energies“

• ω << εF

• large excitation energy in residual nucleus

High momenta butsmall nissing enegies: to ground-state of A-1


Spectral function at high momenta and missing energies:

Exper imental data:

D. Rohe and I . Sick

Theory:

Local density approx.

We need:

Better description of spectral function at low energies


with respect to strange matter

we have to calculate selfenergies for Σ and Λ

G

Σ,Λwe have to solve generalizedBethe Goldstone eq.

(M. Hjorth-Jensen)

e + n Σ + ν

n

Σ∆


But „correlations“ lead to abundancies of excited baryonsalready at normal densities:

ππππ

∆∆∆∆

Mutual polar izationsof baryons in themedium


Propagation, creation and absorption of Neutrinos

q,ω

ν

ν

n

q,ω

ν

e

n

p

Neutral current: mean free path of Neutrino

Charged current: p+e n+ν

n p+e+ ν

Cross section propor tional to L indhard function:

„ density“ of excitations with momentum q and energy ωωωω

p

URCA


Example: mean free path of Neutrinos:

Results of mean field (BHF)

(black, Jerome Margueron)

Importance of RPA:


Effects of partial occupations:

Effects are veryimportant for chargedcurrents:

Pauli effect supressesURCA


Between Nuclei and Nuclear Matter : Pasta

x

dens

ity

n

p

Thomas Fermi Approximation

versus

BHF calculation in a Wigner Seitz Box


Typical density profile

Energy versus density

(in various WS cells)


Shell Effects:

Enhance proton abundance Reduce pairing gap


Conclusions

• nuclear part of the EoSrather well established– interaction model ?

– many-body forces ?

– relativistic effects ?

• up to around 5 ρ0

• inhomogenous matter– nuclear pasta

– shell effects

• problems for EoS at larger densities– strangeness

– excited baryons

– quark gluon plasma

• more information:– neutrino production etc.

– pairing properties

– …

Documents

Equation of state for the dense matter of neutron stars ...sfb.aei.mpg.de/Meetings/School04/Talks/muether04.pdf · for the dense matter of neutron stars and supernovae ... • White