Equation of state in the inner crust: discussion of the ... · Conclusions Equation of state...
Transcript of Equation of state in the inner crust: discussion of the ... · Conclusions Equation of state...
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Equation of state in the inner crust:discussion of the shell effects
“bare” calculations of the EoS Discussion of the shell effects (cluster / continuum) Equation of state in the inner crust with partial removal of continuum shell effects ...
Jérôme MARGUERON
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Neutron star at equilibrium : statistical, betastability.
ρ < 103 g/cm3 : 56Fe bcc lattice + cloud of electronsρ > 103 g/cm3 : electrons fully ionizedρ > 106 g/cm3 : electrons are relativisticρ > 107 g/cm3 : 56Fe → 84Se → ...ρ > 109 g/cm3 : e capture:
n rich exotic nuclei 82Ge → ...→ 118Kr
ρ > 4.3 1011 g/cm3 : neutron drip→ nuclear clusters (lattice) surrounded by a neutron gas
ρ > 2.4 1014 g/cm3 : nuclei dissolve→ homogeneous nuclear matter
Anatomy of a neutron star
zone11 0.001710 0.0024... ...1 0.5
density 0
microscopicdescription:
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Still some open theoretical questions:
● Deformation of the nuclear cluster
● Is the neutron gas correctly described ?● shell effects, ● band theory.
● How far the nuclear force is reliable ? ● low densities, ● large asymmetries
● Description of the pairing at low density● uniform matter: strong/weak, ● nonuniform matter
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Selfconsistent HartreeFockBogoliubov meanfield based on a Skyrme interaction
Skyrme interaction : zero range “in medium” nuclear interaction Fitted to reproduce the properties of nuclear matter and tuned on nuclei (here SLy4).
HartreeFockBogoliubov : meanfield model with treat the pairing correlations acting between all single particle states. well adapted for nonhomogeneous systems.
More details in N. Sandulescu's talk
Difficulties : but only numerical, is to treat large boxes (R>30 fm) > smooth cutoff of the pairing field @ 20 MeV
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For a sharp cutoff:
→ Could be extended to smooth cutoff with a Gaussian shape
Esbensen, Bertsch, Hencken, PRC 1997
Zero range pairing interactionv r1−r2=v0 [1−
0
]r1−r2
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Zero range pairing interaction
for a smooth cutoff of 20 MeV with a tail of 10 MeV
mimic the pairing gap inhomogeneous nuclear matter
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Selfconsistent meanfield calculations with pairing (HFB) for NegeleVautherin cells
Densities MeanField Potentials
From Ch. Monrozeau, PhD thesis 2007cf. talk of N. Sandulescu
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Weak and Strong pairing fields
cf. talk ofN. Sandulescu
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Binding energy in WS cell & EoS
EV , N , Z =
ZmpNmn
V
AV
B , Rbox E lattice
V
E electronsV
E latticeV
=−0.89593 Z 2 e2
r c
E electrons V
=∫dr1 dr 2 er1
e2
r12
[er 2 −
pr2 ]
For a given density (): ➔ minimisation of F/V ➔ stability: pe=n
B ,p ,n
e=dE /VdN e
=ℏ k FedE electrons/V
dNe
bbc lattice
given by HF or HFB and
Equation of state
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Mapping of (Z,N) cells @ constant density● For all the cells, the step is fixed dR=0.2 fm → dV
ρ is fixed, then A change by step dA ~ ρdV=ρ4πR2dR
● Define a windows (N,Z) where to scan:20 < Z < 60 with dZ=2 → dN=dAdZ
Z
NdN=dA2
dZ=2
A=cst
=NZ43R3
For a given ρ :
βstabilityzone #
0.126 z030.055 z040.036 z050.023 z060.009 z070.005 z080.003 z090.002 z100.001 z11
0
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HF: Binding energy (zone 05)
2.1906 2.1931 2.1944
preliminary results
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HF: B.E., proton fraction and radius17914 cells
preliminary results
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HF: (Z,N)preliminary results
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HFB: Binding energy (zone 05)
2.006
preliminary results
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HFB: B.E., proton fraction, radiuspreliminary results
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HFB: (Z,N)preliminary results
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How solid are the results ?
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Are there spurious fluctuations spoiling the research of the minimum
energy cell ?
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What about unbound neutrons ?
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Unbound neutrons : the problem of spurious shell effects
Band theoryFermi Gas
WS approximation:
Density of states (unbound neutrons)
Shell effects :
Integrated density of states (unbound neutrons)
N. Chamel et al., Phys. Rev. C 75, 055806 (2007)
~100keV
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Modelization of the continuumwith finite size boxes eigenstates
E
r
EF average level spacing
in the continuum :
~ℏ2
2mRbox2
if Rbox
Rbox
The distribution of unbound states can vary depending :● on the box radius :● on the boundary conditions.
bound states
unbound states
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Estimation of shell effects for unbound neutrons
Simulate homogeneous pure neutron matter in a cell:Z=0, only N → no clusterSet of cells with radius [10:50] fm @ fixed density
asymptoticallymatches homogeneousenergy calculatedwith Skyrme interaction(on plane wave basis).
Bhom.
Bbox−hom.Rbox ,
E
0.0230.0090.005
/0
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Smooth and residual shell effects
Smooth component:
The finite size effect is fitted with
89.05 0 0.1425
Rbox−2
Bhom. −Bbox−hom.Rbox ,
~300keV
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Smooth and residual shell effects
Smooth component:
The finite size effect is fitted with
89.05 0 0.1425
Rbox−2
Bhom. −Bbox−hom.Rbox ,
~300keV
after substraction
residual component:
~50keV
Irreducible error of the model~50keV
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Back to inhomogeneous matterunbound neutrons fills the boundary conditions
→ induces spurious shell effects in the continuum
B ,Rbox ≈Bbox−inhom. ,Rbox[Bhom. unbound −Bbox−hom. unbound ,Rbox ]
B ,Rbox ≈Bbox−inhom. ,Rbox 89.05 0 0.1425
Rbox−2
“Corrected” binding energy (exact in homogeneous matter) :
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Treatment of the spurious shell effects (HF)
preliminary results
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HF: B.E., proton fraction, radiuspreliminary results
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HF: (Z,N)preliminary results
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Conclusions● Equation of state obtained is based on an unique nuclear interaction (Skyrme)
● The properties of the inner crust are mainly given by unbound neutrons states and electrons.
then :→ nuclear clusters induce corrections to homogeneous matter properties→ It is then important to have the best description of homogeneous
neutron gas.
● Spurious shell effects could induce shifts of the energy up to 300 keV (depending of the radius)
We have proposed a method to partly remove spurious shell effectsand reduce the fluctuations to about 50 keV.
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Outlooks
● Perform the same analysis for system with● pairing● finite temperature
● The model provides the basic ingredients of macroscopic models ;● Link between gap in homogeneous matter / nonhom. ;● Application : cooling process (specific heat). cf N. Sandulecu● Explore the sensitivity of the macroscopic variables on :
● Nuclear interaction ;● Symmetry energy ;● Pairing interaction (SP/WP) ;