neutron matter, symmetry energy and neutron...
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Neutron matter, symmetry energy and neutronstars
Stefano Gandolfi
Los Alamos National Laboratory (LANL)
Elba XII Workshop - Electron-Nucleus Scattering XIIMarciana Marina, Isola d’Elba, June 25-29, 2012
Homogeneous neutron matter
Inhomogeneous neutron matter
W. Nazarewicz − UNEDF
Outline
• The model and the method
• Homogeneous neutron matter• Three-neutron force and the equation of state of neutron matter
• Symmetry energy
• Neutron star structure
• Inhomogeneous neutron matter: Skyrme vs ab-initio.• Energy
• Density and radii
• Conclusions
Nuclear HamiltonianModel: non-relativistic nucleons interacting with an effectivenucleon-nucleon force (NN) and three-nucleon interaction (TNI).
H = − ~2
2m
A∑i=1
∇2i +
∑i<j
vij +∑
i<j<k
Vijk
vij NN (Argonne AV8’) fitted on scattering data. Sum of operators:
vij =∑
Op=1,8ij vp(rij) , Op
ij = (1, ~σi · ~σj ,Sij ,~Lij · ~Sij)× (1, ~τi · ~τj)
Urbana–Illinois Vijk models processes like
π
π
∆
π
π
π
π∆
π
π
π
∆
π
∆
+ short-range correlations (spin/isospin independent).
Quantum Monte Carlo
Evolution of Schrodinger equation in imaginary time t:
Ψ(t) = e−(H−ET )tΨ(0)
In the limit of t →∞ it approaches to the lowest energy eigenstate (notorthogonal to Ψ(0)).
Propagation performed by
ψ(R, t) = 〈R|ψ(t)〉 =
∫dR ′G (R,R ′, t)ψ(R ′, 0)
For a given microscopic Hamiltonian, this method solves the ground–statewithin a systematic uncertainty of 1–2% in a non-perturbative way.
GFMC: spin/isospin states included in the variational wavefunction.
AFDMC: spin/isospin states are sampled.
Light nuclei spectrum computed with GFMC
-100
-90
-80
-70
-60
-50
-40
-30
-20
Ener
gy (M
eV)
AV18
AV18+UIX
AV18+IL7 Expt.
0+
4He0+2+
6He 1+3+2+1+
6Li3/2−1/2−7/2−5/2−5/2−7/2−
7Li
0+2+
8He 2+2+
2+1+
0+
3+1+
4+
8Li
1+
0+2+
4+2+1+3+4+
0+
8Be3/2−1/2+5/2−1/2−5/2+3/2+
7/2−
3/2−
7/2−5/2+7/2+
9Be 3+1+
2+
4+
1+
3+2+
3+
10B
3+
1+
2+
4+
1+
3+2+
0+
12C
Argonne v18with UIX or Illinois-7GFMC Calculations
1 June 2011
Carlson, Pieper, Wiringa, many papers
Three-body force in neutron matter
Tsang et al., arXiv:1204.0466
Neutron matter
Assumptions:
• The two-nucleon interaction reproduces well (elastic) pp and npscattering data up to high energies (Elab ∼ 600MeV) in all channels.
• The three-neutron force (T = 3/2) very weak in light nuclei, whileT = 1/2 is the dominant part (but zero in neutron matter).Difficult to study in light nuclei.
• In neutron matter the short-range repulsive part of three-body forceis the dominant term.
Symmetry energyNuclear matter EOS:
E (ρ, x) = ESNM(ρ) + E (2)sym(ρ)(1− 2x)2 + · · · , ρ = ρn + ρp , x =
ρpρ
0ρ
0 = 0.16 fm
-3
E0 = -16 MeV
symmetric nuclear matterpure neutron matter
Nuclear saturation
Symmetry energy
Neutron matter
We consider different forms of three-neutron interaction by only requiringa particular value of Esym at saturation.
0 0.1 0.2 0.3 0.4 0.5
Neutron Density (fm-3
)
0
20
40
60
80
100
120
En
erg
y p
er
Ne
utr
on
(M
eV
)
Esym
= 33.7 MeV
different 3N different 3N:
• V2π + αVR
• V2π + αV µR
(several µ)
• V2π + αVR
• V3π + αVR
• . . .
Neutron matter and symmetry energyWe then try to change the neutron matter energy at saturation:
0 0.1 0.2 0.3 0.4 0.5
Neutron Density (fm-3
)
0
20
40
60
80
100E
ne
rgy p
er
Ne
utr
on
(M
eV
)E
sym = 35.1 MeV (AV8’+UIX)
Esym
= 33.7 MeV
Esym
= 32 MeV
Esym
= 30.5 MeV (AV8’)
Gandolfi, Carlson, Reddy PRC (2012).
Neutron matter and symmetry energy
From the EOS, we can fit the symmetry energy around ρ0 using
Esym(ρ) = Esym +L
3
ρ− 0.16
0.16+ · · ·
30 31 32 33 34 35 36E
sym (MeV)
30
35
40
45
50
55
60
65
70
L (
Me
V)
Very weak dependence to the model of 3N force for a given Esym.
Neutron star structureEOS used to solve the TOV equations.
8 9 10 11 12 13 14 15 16R (km)
0
0.5
1
1.5
2
2.5
3
M (
Mso
lar)
AV8’ + modified TNI
Esym
= 33.7 MeV
M = 1.97(4) Msolar
M = 1.4 Msolar
different 3N
8 9 10 11 12 13 14 15 16R (km)
0
0.5
1
1.5
2
2.5
3
M (
Mso
lar)
Causality: R>2.9 (G
M/c2 )
ρ central=2ρ 0
ρ central=3ρ 0
error associated to Esym
35.1
33.7
32
Esym
= 30.5 MeV (NN)
1.4 Msolar
1.97(4) Msolar
Accurate measurement of Esym would put a constraint to the radius ofneutron stars, OR observation of M and R would constrain Esym!
M = 1.97Msolar recently observed – Nature (2010).
Neutron stars
Observations of the mass-radius relation of neutron stars are becomingavailable:
9 10 11 12 13 14 15
0.8
1
1.2
1.4
1.6
1.8
2
2.2
R (km)
)M
(M =Rphr
9 10 11 12 13 14 15
0.8
1
1.2
1.4
1.6
1.8
2
2.2
R (km)
)M
(M >>Rphr
Steiner, Lattimer, Brown, ApJ (2010)
Neutron star matter model
Neutron star matter model:
• ρ < ρt , ρt = 0.28 . . . 0.48 fm−3
EPNM = a
(ρ
ρ0
)α+ b
(ρ
ρ0
)βEsym = a + b + 16 , L = 3(aα + bβ)
Note:
a and α sensitive to the 2-neutron force.b (and β) is strongly related to 3-neutron force.
• ρ > ρt ,
Polytropes or Polytrope+Quark matter (Alford et al.)
Neutron star matter model
1 2 3 40
1
2
3
4
Fiducial
No corr. unc.
Quarks
Quarks, no corr. unc.
b (MeV)
Pro
bab
ilit
y d
istr
ibuti
on
2.1 2.2 2.3 2.4
1
2
3
4
Fiducial
No corr. unc.
Quarks
Quarks, no corr. unc.
β
Pro
bab
ilit
y d
istr
ibuti
on
8 9 10 11 12 13 14 15R (km)
0
0.5
1
1.5
2
2.5
M (
Mso
lar)
Esym
= 33.7 MeV
Esym
=32 MeV
Polytropes
Quark matter
Steiner, Gandolfi, PRL (2012)
New constraints to 3-neutron force (to combine with NN)!!
Neutron stars: symmetry energy
20 40 60 80 100
2
4
6
8
Fiducial
No corr. unc.
Quarks
Quarks, no corr. unc.
IAS
PDR
HIC
Masses
Pb (p,p)
L (MeV)
Pro
bab
ilit
y d
istr
ibu
tio
n
Steiner, Gandolfi, PRL (2012) Tsang et al., arXiv:1204.0466
Neutron drops
Now let’s study inhomogeneous neutron matter.
We confine neutrons by adding an external potential:
H = − ~2
2m
A∑i=1
∇2i +
∑i<j
vij +∑
i<j<k
Vijk +∑i
Vext(ri )
Vext is a Wood-Saxon or Harmonic well:
VWS = − V0
1 + exp[(r − R)/a]
VHO =1
2mω2r2
=⇒ different geometries and densities.
Neutron drops, harmonic oscillator well
External well: harmonic oscillator with ~ω=5, 10 MeV.
0.7
0.8
0.9
1
1.1E
/ ω
N4
/3
0 10 20 30 40 50 60N
0.7
0.8
0.9
1
1.1
E /
ωN
4/3
AFDMCGFMCSLY4SKM*SKPBSK17E
TFξ=0.5, 1
5 MeV
10 MeV
Skyrme systematically overbind neutron drops.
Neutron drops, harmonic oscillator well
Fixing Skyrme force:
0 10 20 30 40 50 60
N
0.7
0.8
0.9
1
1.1E
/ ω
N4
/3
AFDMC
SLY4
GFMC
closed-shell:gradient term
10 MeV
1p or 1h:spin-orbit
mid-shell:spin-orbitand pairing
The correction is very similar in all the Skyrme forces we considered.
Neutron drops, adjusted Skyrme forceNote: bulk term of Skyrme fit neutron matter.
We add the missing repulsion by adjusting the gradient term Gd [∇ρn]2,the pairing and spin-orbit terms.
0.7
0.8
0.9
1
1.1
E /
ωN
4/3
0 10 20 30 40 50 60N
0.7
0.8
0.9
1
1.1
E /
ωN
4/3
AFDMCGFMCSLY4SLY4-adj
5 MeV
10 MeV
Gandolfi, Carlson, Pieper, PRL (2011).
Neutron drops, adjusted Skyrme force
Neutrons in the Wood-Saxon well are also better reproduced by theadjusted SLY4.
8 10 12 14 16 18 20N
-14
-12
-10
-8
E/N
(M
eV)
AFDMCGFMCSLY4SLY4-adj
Gandolfi, Carlson, Pieper, PRL (2011).
Neutron drops: radiiCorrection to radii using the adjusted-SLY4.
2.0
2.5
3.0
3.5
4.0
4.5<
r2 >
1/2
(fm
)
GFMCSLY4SLY4-adj
8 10 12 14 16N
2.0
2.5
3.0
3.5
< r
2 >
1/2
(fm
)
10 MeV
5 MeVHO
WS
Gandolfi, Carlson, Pieper, PRL (2011).
Neutron drops: radial densityNeutron radial density:
0
0.1
0.2
0.3r2
ρ (f
m-1
)
ω = 5 MeV ω = 10 MeVSLY4-adj
SLY4
0 1 2 3 4 5 6r (fm)
0
0.1
0.2
0.3
0.4
0.5
r2ρ
(f
m-1
)
N=8
N=14
Gandolfi, Carlson, Pieper, PRL (2011).
Gradient term
Where is the gradient term important?
Just few examples:
• Medium large neutron-rich nuclei
• Phases in the crust of neutron stars
• Isospin-asymmetry energy of nuclear matter
Conclusions
• Effect of three–neutron forces to high-density neutron matter; thesystematic uncertainty due to 3N is relatively small.
• Esym strongly constrain L. Weak dependence to the model of 3N.
• Uncertainty of the radius of neutron stars mainly due Esym ratherthan 3N.
• Neutron star observations becoming competitive with terrestrialexperiments.
• Skyrme can be better constrained by ab-initio calculations.
Thanks for the attention