Opacidade de neutrinos e a estrutura da fase pasta · Opacidade de neutrinos e a estrutura da fase...

Opacidade de neutrinos e aestrutura da fase pasta

Debora Peres Menezes - Universidade Federal de Santa Catarina

Florianopolis - Brazil - PoA - julho de 2011

Building the Pasta Phase

The solid crust insulates thermally the neutron star cold surface from its

hot liquid interior and because of this fact, it is of vital importance in the

understanding of neutron star cooling and evolution.

Does the existence of the pasta phase on the crust influence neu-

trino opacities in stellar matter?

What is the pasta phase probably present in the crust of NS?

• It is the result of a frustrated system. At low densities a competition

between the strong and the electromagnetic interactions takes place

leading to a frustrated system.

• The individual interaction energies among the three types of pairing

cannot be minimised simultaneously: the system is frustrated (P.

Schiffer, Nature 420 (2002) 35).

• Normally the short and large distance scales related to the nuclear

and Coulomb interactions are well separated so that nucleons bind

into nuclei but at densities of the order of 1013 − 1014 g/cm3 these

length scales are comparable.

• A variety of complex structures exist: droplet (meatball, 3D), rod

(spaghetti, 2D), lazagna (slab, 1D), penne (tube, 2D), Swiss cheese

(bubble, 3D).

• The pasta phase is the ground state configuration if its free energy is

lower than the corresponding homogeneous phase.

Different models - npe matter

Variations of quantum hadrodynamics (non-linear Walecka model)

L =∑

i=p,n

Li + Le+Lσ+Lω+Lρ+Lδ+Lγ, (1)

with fixed couplings (coupling constants):

L(ψ, gsφ, gvVµ, gρ~b

µ, eAµ, κφ3, λφ4, ξ(VµVµ)2) (2)

or with density dependent couplings (non-linear terms are not present):

L(ψ,Γsφ,ΓvVµ,Γρ~b

µ, eAµ) (3)

Γi(ρ) = Γi(ρ0)hi(x), x = ρ/ρ0 (4)

These parameters are fitted to give especific nuclear matter properties

NL3: G. A. Lalazissis, J. Konig and P. Ring, Phys. Rev. C 55, 540 (1997)

TM1: K. Sumiyoshi, H. Kuwabara, H. Toki, Nucl. Phys. A 581, 725 (1995)

NLδ: B. Liu, V. Greco, V. Baran, M. Colonna and M. Di Toro, Phys. Rev. C 65,

045201 (2002)

GM1,GM3: N. K. Glendenning, Compact Stars, Springer-Verlag, New-York, 2000

TW: S. Typel and H. H. Wolter, Nucl. Phys. A656, 331 (1999)

DDME1: T. Niksic, D. Vretenar, P. Finelli and P. Ring, Phys. Rev. C 66, 024306

(2002)

DDHδ: T. Gaitanos, M. Di Toro, S. Typel, V. Baran, C. Fuchs,V. Greco and H. H.

Wolter, Nucl. Phys. A 732, 24 (2004); S.S. Avancini, L. Brito, D. P. Menezes and C.

Providencia, Phys. Rev. C 70, 015203 (2004)

GDFM: P. Gogelein, E.N.E. van Dalen, C. Fuchs and H. Muther, Phys. Rev. C 77,

025802 (2008)

Two methods are used: TF and CP.

The Thomas-Fermi Approximation

The densities are obtained by minimising the functional:

Ω =

∫

VWZ

d3x ǫTF [ρp(~x), ρn(~x), ρe(~x)] −∑

i=n,p,e

µi

∫

VWZ

d3rρi(~x) .

with adequate boundary conditions.

Numerical technique: All mesonic fields are expanded in a HO basis

for 1D, 2D or 3D and the differential equations become matrix equations

solved self-consitently.

Wigner-Seitz Cell Geometry

TF: Inside the WS cell npe matter is considered locally homogeneous and

described by a degenerate Fermi gas:

Ni =∫

d3x ρi(~x) , i = p, n, e

ETF =∫

d3x ǫ(ρp(~x), ρn(~x), φ(~x), V0(~x), b0(~x), δ0(~x)) + ǫCOUL(ρp(~x), ρe(~x))

• Body centered cubic crystal lattice (bcc) - spherical structures (3D)

- ρi(r, θ, φ) = ρi(r)

• Hexagonal lattice - cylindrical structures (2D) - ρi(ρ, θ, z) = ρi(ρ)

• Unidimensional lattice - slab structure - ρi(x, y, z) = ρi(z)

K. Oyamatsu, Nucl. Phys. A561 (1993) 431.

We fix ρB and calculate the energies related to the 3 Wigner-Seitz cell

geometries. The lower one is the preferential pasta structure. It is then

compared with the energy of a homogeneous system.

Coexisting Phases Method / MFA

Nuclear matter

ρ, Yp = ρp/ρ (global proton fraction) are fixed; the pasta structures arebuilt with different geometrical forms in a background nucleon gas.

Gibbs’ conditions :

P I(νIp , νIn,M

∗nI,M∗

pI) = P II(νIIp , ν

IIn ,M

∗nII ,M∗

pII), (5)

µIi = µIIi , i = p, n (6)

m2sφ

I0 = Γsρ

Is, m2

sφII0 = Γsρ

IIs , (7)

m2δ δI3 = Γδρ

Is3, m2

δ δII3 = Γδρ

IIs3, (8)

fρIp + (1 − f)ρIIp = ρp = Ypρ, (9)

I,II = labels of the phases, f = volume fraction of phase I:

f =ρ− ρII

ρI − ρII(10)

P = P I + Pe (11)

E = fEI + (1 − f)EII + Ee + Esurf + ECoul, Esurf = 2ECoul (12)

ECoul =2α

42/3(e2πΦ)1/3

(

σD(ρIp − ρIIp ))2/3

, (13)

α = f for droplets/rods, α = 1− f for bubbles/tubes; σ = surface energy

coefficient; D = dimension of the system.

Φ =

(

2−Df1−2/D

D−2 + f

)

1D+2, D = 1,3;

f−1−ln(f)D+2 , D = 2.

(14)

σ is AN IMPORTANT QUANTITY, which we parametrize.

Stellar matter

µp = µn − µe + µν, µe = µµ (15)

ρp = ρe + ρµ (16)

YL =ρe + ρν + ρµ

ρB(17)

The Gibbs’ conditions:

µIn = µIIn , µIe = µIIe , µIν = µIIν , (18)

f(

ρIp − ρIe − ρIµ)

+ (1 − f)(

ρIIp − ρIIe − ρIIµ)

= 0 (19)

0.00 0.04 0.08 0.120

20

40

60

80

E/A

-M (M

eV) a)

(fm-3)0.00 0.04 0.08 0.120

20

40

60

80

E/A

-M (M

eV) b)

(fm-3)

0.00 0.04 0.08 0.12 0.16

0

40

80 c)

(fm-3)

F/A

-M (M

eV)

0.00 0.04 0.08 0.12 0.16

0

40

80

(fm-3)F/

A-M

(MeV

)

d)

npe matter (CP method) with NL3 and a) T=0, Yp = 0.5, b) T=0 Yp = 0.3; c) T=5MeV, Yp = 0.5, d) T=5 MeV, Yp = 0.3. The different structures of the pasta arerepresented by different colors and the dashed line stands for homogenous matter.

0.00

0.02

0.04

0.06

0.08

0.10

0.12

T=0 T=5MeV

(fm-3

)

T=10MeV

Comparison of the phase diagrams, NL3, Yp = 0.3, CP method. From bottom to topthe colours represent droplets, rods, slabs and the homogeneous phase for T = 0 MeV,and homogeneous phase, droplets, rods, slabs and the homogeneous phase for T = 5MeV and T = 10 MeV.

0.00

0.02

0.04

0.06

0.08

0.10

0.12b)

(fm-3

)

TM1 TM1NL3TF CP

NL3CPTF

Phase diagrams, T=0, obtained both with CP and TF approximation for NL3 and TM1with Yp = 0.3. From bottom to top the colours represent droplets, rods, slabs, tubes,bubbles and the homogeneous phase, whenever present.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16ρ

(fm

−3)

NL3 NL3 TW DDH GDFM

(a)δ δ

Hom.MatterTubes

Slabs

Rods

Droplets

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

ρ(f

m−3

)

NL3 NL3 TW DDH GDFM

Droplets

Rods

Slabs

Tubes

Bubbles

MatterHom.

δδ

Phase diagrams at T = 0 MeV for Yp = 0.5 obtained with a) the coexisting phasesmethod (CP) and b) obtained with the TF method for several models.

Hom.BubblesTubesSlabsRodsDroplets

T= 0 MeV

eq-βa)


T= 0 MeV

Yp = 0.5


T = 5 MeV


T = 7 MeV

0.00 0.02 0.04 0.06 0.08 0.10 0.12

ρ ( fm-3 )


T = 8 MeV

b)


T = 0 MeV

Yp = 0.3


T = 5 MeV


T = 7 MeV

0.00 0.02 0.04 0.06 0.08 0.10 0.12

ρ ( fm-3 )


T = 8 MeV

c)

Pasta phases: comparison between NL3 (dashed line) and TW (full line). The thicklines - Thomas-Fermi (TF) calculation and the thin red lines - coexisting-phases (CP)calculation.

Comments on the pasta phase

• While the final EOS obtained with the different methods do not vary much, theinternal structure varies considerably.

• The TF approximation was performed to test the much simpler CP calculation. Itwas shown that the success of the CP calculation depends on the parametrisationof the surface energy.

• The pasta phase shrinks with the increase of the temperature and the homogeneousmatter can be the preferential phase also at very low densities depending on thetemperature and the proton fraction.

Neutrino cross sections

To calculate neutrino opacities and mean free paths we consider neutral current scat-tering reactions:

νe + n→ νe + n, (20)

νe + p → νe + p, (21)

and charged current absorption reactions:

νe + n→ e− + p. (22)

νe + p→ e+ + n. (23)

for which we need cross sections σn, σp and σa (huge expressions...)

The total neutrino mean free path in dense matter is given by

λν =1

ρnσn + ρpσp + ρBσa. (24)

Rosseland neutrino mean free paths are related with diffusion coefficients Dj by

λkν =Dk

∫ ∞

0dǫν ǫkνfν(1 − fν)

, (25)

where

Dk =

∫ ∞

0

dǫν ǫkνλνfν(1 − fν). k = 2,3,4 (26)

0.01 0.02 0.03 0.04 0.05456789

101112131415

(fm-3)

D2 (10

5 MeV

3 m)

Rods Droplets Homogeneous matter

T=3 MeVYL=0.2NL3

0.01 0.02 0.03 0.04 0.05 0.06 0.0723456789

101112131415

D2 (10

5 MeV

3 m)

Slabs Rods Droplets Homogeneous matter

T=3 MeVYL=0.4NL3

(fm-3)

0.01 0.02 0.03 0.04 0.05456789

101112131415

D2 (10

5 MeV

3 m)

(fm-3)


T=5 MeVYL=0.2NL3

0.01 0.02 0.03 0.04 0.05 0.06 0.07456789

101112131415

(fm-3)

D2 (10

5 MeV

3 m)

T=5 MeVYL=0.4NL3

Rods Droplets Slabs Homogeneous matter

Diffusion coefficient D2 as function of baryon density for different temperature andlepton fraction values for homogeneous matter and pasta phase.

0.01 0.02 0.03 0.04 0.050.0

1.5

3.0

4.5

6.0

7.5

9.0

(fm-3)

D3 (10

7 MeV

4 m)


T=3 MeVYL=0.2NL3

0.01 0.02 0.03 0.04 0.05 0.06 0.070

2

4

6

8

10

12

14


T=3 MeVYL=0.4NL3

D3 (10

7 MeV

4 m)

(fm-3)

0.01 0.02 0.03 0.04 0.050

1

2

3

4

5

6

7

8

9

D3 (10

7 MeV

4 m)

(fm-3)


T=5 MeVYL=0.2NL3

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080123456789

101112131415

(fm-3)D

3 (10

7 MeV

4 m)

T=5 MeVYL=0.4NL3



0.01 0.02 0.03 0.04 0.050.0

1.5

3.0

4.5

6.0

7.5

9.0

(fm-3)

D4

(109

MeV

5 m

)


T=3 MeVYL=0.2NL3

0.01 0.02 0.03 0.04 0.05 0.06 0.0702468

101214161820

T=3 MeVYL=0.4NL3


(fm-3)

D4

(109

MeV

5 m

)

0.01 0.02 0.03 0.04 0.050

1

2

3

4

5

6

7

8

9

D4

(109

MeV

5 m

)

(fm-3)


T=5 MeVYL=0.2NL3

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.0802468

101214161820

D4

(109

MeV

5 m

)

(fm-3)


T=5 MeVYL=0.4NL3


Conclusions

• Just NL3 obtained with CP; other parametrizations may give slightly different resultsfor the diffusion coefficients.

• The differences in the diffusion coefficients due to the existence of the pasta phasehave consequences in neutrino opacities.

• Neutrino opacities are used as input to the solution of the transport equations in theequilibrium diffusion approximation, which simulates the Kelvin-Helmholtz phase ofthe protoneutron stars.

• The existence of the pasta phase in the crust of protoneutron stars will probablyinfluence their cooling mechanism.

Collaborators

M. D. AlloyPh.D. student - Universidade Federal de Santa Catarina

S.S. Avancini

Universidade Federal de Santa Catarina

C. ProvidenciaUniversidade de Coimbra - Portugal

Thank you for your attention

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