Nuclear Superfluidity in the Crust of Neutron Stars

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Nuclear Superfluidity in the Crust of Neutron
Stars
Nicolae Sandulescu
Institute of Atomic Physics, Bucharest

• Introduction inner crust structure
• Superfluid properties HFB
• Collective excitations QRPA
• Specific heat and cooling
• Summary and conclusions

Nguyen Van Giai IPN-Orsay Elias Khan
IPN-Orsay
R.J.Liotta KTH-Stockholm

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E. Khan, N. Sandulescu Nguyen Van Giai
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Inner Crust Matter
0.001r0
r0
0.5 r0
Crystal lattice structures
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Elementary cells
Wigner-Seitz cell
Elementary cell
Lattice
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Inner crust equation of state
F Enuc Ecoul Eelec TS - mnN-mpZ
Free energy
Enuc Evol rEsurf r, rEls r, r Epair
r,k
dF0, b-equilibrium
(N,Z,Rcell)
Microscopic calculations
mean field type HFB
I) Inner crust structure N/Z
II) Pairing properties D(r,T,w)
III) Collective excitations QRPA
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Crust Structure Negele Vautherin
Enuc Evol r Esurf r, r Els(p) Els(n)
Epair r,k
0
0

• HF calculations T0

• Wigner Seitz approximation

• Boundary conditions at rRws

jl0 leven djl/dr0 lodd
Main result cells with Z 40,50,32
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Inner Crust Structure
J.W. Negele, D. Vautherin, NPA207 (1973) 298
r0/100
500Zr40
r0/42.9
950Sn50
r0/7.84
1800Sn50
r0/3.37
1500Zr40
r0/2.03
982Ge32
Role of pairing correlations ?
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Finite-Temperature HFB
EnucEvol rEsurf r, rEls(p)Els(n) Epair
r,k
DT(r) Vpair tT(r)
fi(1eEi/kT)-1
N.S, Phys.Rev.C70 (2004) 025801
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Collective Modes in the Inner Crust

• Collective modes in a non-uniform condensate
• coherence length z hvF /pDF
• distance between clusters L
• ( a) L gtgt z the case of uniform
  condensate
• (b) L lt z needs for microscopic
  calculations !
  

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Time-dependent HFB
EnucEvol rEsurf r, rEls(p)Els(n) Epair
r,k
Linear response
Densities fluctuations

ph
pp
hh
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QRPA response
Residual interaction
E.Khan, N. S.,M.Grasso, Nguyen Van Giai, Phys.
Rev. C66 (2002)024309
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Pairing correlations
Enuc ESkyrme Epair r,k
Epair r,k
Pairing in uniform neutron matter
?
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Pairing effective interactions
kF lt 0.9
Vbare
Gogny force
Vpair V01-h(r/r0)ad(r-r)
a0.45 h0.7
I) V0-430
Dmax 3 MeV
Vtotal
II) V0-330
Dmax 1 MeV
Mtotal
Ecut-off60 MeV
C.Shen,U.Lombardo,P.Schuck,W.Zuo,N.S,
Phys.Rev.C67(2003)
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Wigner-Seitz cells
J.W. Negele, D. Vautherin, NPA207 (1973) 298
r0/100
500Zr40
r0/42.9
950Sn50
r0/7.84
1800Sn50
r0/3.37
1500Zr40
r0/2.03
982Ge32
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Density in the Wigner-Seitz Cells
N.S ,Nguyen Van Giai,R.J.Liotta,
Phys.Rev.C69(2004)045802
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Pairing Field in the Wigner-Seitz Cells
N.S, Phys.Rev.C70 (2004) 025801
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Pairing Field in the Wigner-Seitz Cells
N.S, Phys.Rev.C70 (2004) 025801
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Specific Heat of Inner Crust Matter
C(t)V CV (n) CV (e) CV(protonslattice)

• normal phase

CV (n D0) gt CV (e)
D
CV(n) CV(nD0) e-D/kT

• suprafluid phase

• FT-HFB approx

SHFB Si (1fi)ln(1fi)-filnfi
fi(1eEi/kT)-1
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Specific Heat in the FT-HFB Approach
N.S, Phys.Rev.C70 (2004) 025801
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Specific Heat in the FT-HFB Approach
N.S, Phys.Rev.C70 (2004) 025801
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Collective Modes in the Inner Crust

• Non-uniform condensate
• coherence length z hvF /pDF
• distance between clusters L
• ( a) L gtgt z the case of uniform
  condensate
• (b) L lt z QRPA
  calculations !
  

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Supergiant resonances
L1
Z50 N1750
L0
Effect on specific heat ?
E.Khan,N.S,Nguyen Van Giai, Phys.Rev.C, in press
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Specific heat of collective modes
Z50 N1750
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Cooling time of Neutron Stars
cooling time
conductivity
diffusion
Lattimer et al, ApJ425(1994)802
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Mass-Radius Constraints from Cooling Time
Lattimer et al, ApJ425(1994)802
1.15 M0 lt M lt1.5 M0 tw 10 years Superfluidity
9 km lt R lt 11.5 km No Superfluidity 6.8 km lt
R lt 8.5 km
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Summary and Conclusions

• Self- consistent treatment of inner crust matter
  
• - FT-HFB and QRPA with the same
  energy functional
• - pairing effects on inner crust
  structure ?
• Nuclear superfluidity in the inner crust
• - temperature is essential in the outer
  part of the crust and for weak pairing
• - nuclear clusters can either suppress or
  enhance the pairing correlations
• Specific heat of baryonic matter
• - sensitive to the presence of the
  clusters temperature
• - very strong dependence on the screening
  of pairing force !
• Collective modes in the inner crust
• - low-lying supergiant resonances
• - important contribution to the
  specific heat !

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Nguyen Van Giai IPN - Orsay
Elias Khan IPN - Orsay
R. J. Liotta KTH - Stockholm
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Density in the Wigner-Seitz Cells
N.S ,Nguyen Van Giai,R.J.Liotta,
Phys.Rev.C69(2004)045802
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Mean Field in the Wigner-Seitz Cells
N.S ,Nguyen Van Giai,R.J.Liotta,
Phys.Rev.C69(2004)045802
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Specific Heat of Collective Modes

• specific heat

• entropy

Scoll Sn (1gn)ln(1gn)-gnlngn
- Sij (1gij)ln(1gij)-gijlngij
gnexp(Wn/kT)-1-1
(Wn QRPA spectrum)
(Ei - HFB spectrum )
gijexp((EiEj)/kT)-1-1
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Density profiles of WS cells
Size ( R ) and coherence length (z)
1800Sn R28 fm z3 fm 982Ge R14 fm
z10 fm
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Pairing field in the WS cells