SLIDE 1 Unified description of neutron-star interiors
Nicolas Chamel
Institute of Astronomy and Astrophysics Université Libre de Bruxelles, Belgium
in collaboration with
- S. Goriely (ULB), J. M. Pearson (UMontréal), A. F
. Fantina (ULB, GANIL) P . Haensel & J. L. Zdunik (CAMK)
- Y. D. Mutafchieva & Zh. Stoyanov (INRNE)
- A. Potekhin (Ioffe)
CAMK, Warsaw, 28 March 2018
SLIDE 2 Prelude
Haensel, Potekhin, Yakovlev, “Neutron Stars - 1. Equation of State and Structure” (Springer, 2007)
The interior of a neutron star exhibits very different phases (gas, liquid, solid, superfluid, etc.)
- ver a very wide range of densities
with possibly exotic particles (hyperons, quarks) in the inner core.
Blaschke&Chamel, contribution to the White Book of the COST Action MP1304, arXiv:1803.01836
SLIDE 3 Main motivation
Ad hoc matching of different models can lead to significant errors on the neutron-star structure & dynamics.
12 13 14 15
R (km)
0.5 1.0 1.5 2.0 2.5
M (M ⊙)
1.0 1.5 2.0 2.5
M (M ⊙)
0.5 1.0 1.5 2.0 2.5 3.0
lcr (km)
Unified nm =0.01 nm =nc nm =nt nm =n0 nm =0.5n0 −n0 nm =0.1n0 −nt
Fortin et al., Phys.Rev.C94, 035804 (2016)
Combining different microscopic inputs lead to multiple interpretations of astrophysical phenomena (degeneracy). This calls for a unified description of neutron-star interiors.
SLIDE 4 Outline
1
Internal constitution of a neutron star
⊲ Main assumptions on dense stellar matter ⊲ Constraints from laboratory experiments ⊲ Predictions from the nuclear-energy density theory ⊲ Phase transitions in the inner core ?
2
Description of specific neutron-star classes
⊲ Highly-magnetized neutron stars ⊲ Accreting neutron stars
3
Conclusions & perspectives
SLIDE 5
Neutron-star surface
The surface of a neutron star is expected to be made of iron, the end product of stellar nucleosynthesis (identification of broad Fe K emission lines from accretion disk around neutron stars in LMXB).
Stixrude, Phys.Rev.Lett. 108, 055505 (2012)
Compressed iron can be studied with nuclear explosions and laser-driven shock-wave experiments... But at pressures corresponding to about 0.1 0.1 0.1 mm below the surface (for a star with a mass of 1.4M⊙ and a radius of 12 km) ! Ab initio calculations predict various structural phase transitions.
SLIDE 6
Crystal Coulomb plasma
At a density ρeip ≈ 2 × 104 g cm−3 (about 22 cm below the surface), the interatomic spacing becomes comparable with the atomic radius.
Ruderman, Scientific American 224, 24 (1971)
At densities ρ ≫ ρeip, atoms are crushed into a dense plasma of nuclei and free electrons. Nuclei become more neutron rich with increasing pressure.
SLIDE 7 Description of the outer crust of a neutron star
Main assumptions:
cold “catalyzed” matter (full thermodynamic equilibrium)
Harrison, Wakano and Wheeler, Onzième Conseil de Physique Solvay (Stoops, Brussels, Belgium, 1958) pp 124-146
the crust is stratified into pure layers made of nuclei A
ZX
electrons are ∼ uniformly distributed and are highly degenerate T < TF ≈ 5.93 × 109(γr − 1) K γr ≡
r ,
xr ≡ pF mec ≈ 1.00884 ρ6Z A 1/3 nuclei are arranged on a perfect body-centered cubic lattice T < Tm ≈ 1.3 × 105Z 2 ρ6 A 1/3 K ρ6 ≡ ρ/106 g cm−3
Pearson et al.,Phys.Rev.C83, 065810 (2011) Chamel&Fantina,Phys.Rev.D93,063001 (2016)
SLIDE 8 Experimental “determination” of the outer crust
The composition of the crust is completely determined by experimental atomic masses down to about 200m for a 1.4M⊙ neutron star with a 10 km radius The physics governing the structure of atomic nuclei (magicity) leaves its imprint
Due to β equilibrium and electric charge neutrality, Z is more tightly constrained than N: only a few layers with Z = 28.
Kreim, Hempel, Lunney, Schaffner-Bielich, Int.J.M.Spec.349-350,63(2013)
SLIDE 9 Plumbing neutron stars to new depths
Precision mass measurements of
82Zn by the ISOLTRAP
collaboration at CERN’s ISOLDE radioactive-beam facility in 2013 allowed to "drill" deeper.
Wolf et al.,Phys.Rev.Lett.110,041101(2013)
The composition is very sensitive to uncertainties in nuclear masses. Errors of a few keV/c2 can change the results. Deeper in the star, recourse must be made to theoretical models.
SLIDE 10
Theoretical challenge
Models of dense matter required to compute all the necessary inputs to astrophysical simulations of neutron stars should be: versatile: applicable to compute all properties under various conditions thermodynamically consistent: avoid spurious instabilities as microscopic as possible: make reliable extrapolations numerically tractable: systematic calculations over a wide range of temperatures, pressures, composition, magnetic field The nuclear energy density functional theory is the most suitable (quantum) approach. Nucleons are treated as independent quasiparticles in a self-consistent potential field (Hartree-Fock-Bogolyubov method).
Dobaczewski&Nazarewicz, in ”50 years of Nuclear BCS” (World Scientific Publishing, 2013), pp.40-60; Chamel,Goriely,Pearson, ibid., pp.284-296
In principle, this theory describes the many-body system exactly provided the exact functional is known (Hohenberg-Kohn theorem)...
SLIDE 11 Nuclear-matter uncertainties
Because the exact functional is unknown, phenomenological functionals are employed. How to quantify nuclear-matter uncertainties ? The energy per nucleon of nuclear matter at T = 0 around saturation density n0 and for asymmetry η = (nn − np)/n, is usually written as e(n, η) = e0(n) + S(n)η2 + o
where e0(n) = av + Kv 18ǫ2 − K ′ 162 ǫ3 + o
with ǫ = (n − n0)/n0 S(n) = J + L 3 ǫ + Ksym 18 ǫ2 + o
is the symmetry energy The lack of knowledge is embedded in av, Kv, K ′, etc. In order to make meaningful comparisons, energy-density functionals corresponding to different values of these parameters should be fitted using the same protocole.
SLIDE 12 Brussels-Montreal Skyrme functionals (BSk)
For application to extreme astrophysical environments, functionals should reproduce global properties of both finite nuclei and infinite homogeneous nuclear matter. Experimental data/constraints: nuclear masses (rms ∼ 0.5 − 0.6 MeV/c2) nuclear charge radii (rms ∼ 0.03 fm) symmetry energy 29 ≤ J ≤ 32 MeV incompressibility Kv = 240 ± 10 MeV Many-body calculations using realistic interactions: equation of state of pure neutron matter
1S0 pairing gaps in nuclear matter
effective masses in nuclear matter stability against spin and spin-isospin fluctuations
Chamel et al., Acta Phys. Pol. B46, 349(2015)
SLIDE 13
Neutron-matter stiffness
BSk19, BSk20 and BSk21 were fitted to realistic neutron-matter equations of state with different of degrees of stiffness:
Goriely, Chamel, Pearson, Phys. Rev. C 82, 035804 (2010).
SLIDE 14
Neutron-matter constraints at low densities
All three functionals are consistent with ab initio calculations at densities relevant for the inner crust and outer core of neutron stars:
SLIDE 15
Symmetry-energy constraint
The functionals BSk22-26 were also fitted to realistic neutron-matter equations of state but with different values for J = 29 − 32 MeV:
Goriely, Chamel, Pearson, Phys.Rev.C 88, 024308 (2013).
SLIDE 16 Theoretical predictions of the outer crust
0m
HFB-19 HFB-21
a t
inner crust core
crust
200m 300m
62 64 66 86Kr 84Se 82Ge
Ni
124Mo
Sr
56Fe 80Zn 79Cu 122Zr 120 122 124
Ni
100m
78 80 121Y 62 64 66 86Kr 84Se 82Ge 80Ni 126Ru 124Mo 122 124
Sr
56Fe 80Zn 78Ni
Zr
120 122 124 126
Ni
82Zn
N≈82 N≈50 10km
Predictions from HFB-21 of
Composition dominated by nuclei with N close to magic number N = 82. Sr isotopes are the most abundant nuclides (∼ 40%).
- nly 0.04% in Earth’s crust
Pearson et al.,Phys.Rev.C83,065810(2011) Wolf et al.,Phys.Rev.Lett.110,041101(2013)
SLIDE 17 Role of the symmetry energy
The composition of the outer crust is only slightly influenced by the density dependence of the symmetry energy S(n). The proton fraction varies roughly as Yp = Z A ∼ 1 2 − (12π2(c)3P)1/4 8S
10
10
n [fm
0.3 0.35 0.4 Yp HFB-22 HFB-24 HFB-25 HFB-26
Pearson et al., Eur. Phys. J.A50,43(2014); Pearson et al. in prep.
SLIDE 18 Equation of state of the outer crust
The pressure, determined by electrons, is almost independent of the
- composition. Analytical fits: http://www.ioffe.ru/astro/NSG/BSk/
SLIDE 19 Stratification and equation of state
Transitions between adjacent crustal layers are accompanied by density discontinuities.
n P nmax
1
nmin
2
P
1 2
1 2 1+2 + Mixed solid phases cannot exist in a neutron star crust because P has to increase strictly monotonically with ¯ n (hydrostatic equilibrium).
SLIDE 20
Compounds in neutron-star crusts?
Multinary ionic compounds made of nuclei with charges {Zi} might exist in the crust of a neutron star.
Dyson, Ann. Phys.63, 1 (1971); Witten, ApJ 188, 615 (1974)
Necessary conditions: stability against weak and strong nuclear processes.
Jog&Smith, ApJ 253, 839(1982).
stability against the separation into pure (bcc) phases: R({Zi/Zj}) ≡ C Cbcc f({Zi}) ¯ Z Z 5/3 > 1 where f({Zi}) is the dimensionless lattice structure function of the compound and C the corresponding structure constant.
Chamel & Fantina, Phys. Rev. C94, 065802 (2016).
Stellar vs terrestrial compounds: (i) they are made of nuclei; (ii) electrons form an essentially uniform relativistic Fermi gas.
SLIDE 21 Substitutional compounds in neutron-star crusts
Compounds with CsCl structure are present at interfaces if Z1 = Z2.
n P nmax
1
nmin
2
nmin
1+2 nmax 1+2
P
1 1+2
P
2 1+2
P
1 2
1 2 1+2 But they only exist over an extremely small range of pressures.
Chamel&Fantina, Phys. Rev. C94, 065802 (2016).
SLIDE 22 Neutron-star crust and nuclear masses
The composition of the outer crust is completely determined by nuclear masses M′(A, Z). Essentially exact analytical expressions valid for any degree of relativity of the electron gas and including electrostatic correction:
Chamel&Fantina,Phys.Rev.C94,065802(2016)
In the limit of ultrarelativistic electron Fermi gas: P1→2 ≈ (µ1→2
e
)4 12π2(c)3 , ¯ nmax
1
≈ A1 Z1 (µ1→2
e
)3 3π2(c)3 , ¯ nmin
2
≈ A2 Z2 Z1 A1 ¯ nmax
1
µ1→2
e
≡ M′(A2, Z2)c2 A2 − M′(A1, Z1)c2 A1 Z1 A1 − Z2 A2 −1 + mec2 Since ¯ nmin
2
> ¯ nmax
1
in hydrostatic equilibrium, nuclei become more neutron rich (Z2/A2 < Z1/A1) and less bound with increasing depth.
SLIDE 23
Description of the inner crust of a neutron star
At densities ∼ 4.4 × 1011 g cm−3, neutrons drip out of nuclei thus marking the transition to the inner crust.
Negele&Vautherin,Nucl.Phys.A207,298(1973)
The neutron-saturated clusters owe their stability to the presence of a highly degenerate surrounding neutron liquid. Unbound neutrons are expected to be superfluid at T ≤ Tc by forming Cooper pairs analogously to electrons in conventional superconductors. The conditions prevailing in the inner crust of a neutron star cannot be reproduced in terrestrial laboratories.
SLIDE 24
Fast numerical implementation of HFB equations
We use the Extended Thomas-Fermi+Strutinsky Integral (ETFSI) approach with the same functional as in the outer crust: semiclassical expansion in powers of 2: the energy becomes a functional of nn(r r r) and np(r r r) and their gradients only. proton shell effects are added perturbatively (neutron shell effects are much smaller and therefore neglected). In order to further speed-up the calculations, clusters are supposed to be spherical (no pastas) and nn(r r r), np(r r r) are parametrized.
Pearson,Chamel,Pastore,Goriely,Phys.Rev.C91, 018801 (2015). Pearson,Chamel,Goriely,Ducoin,Phys.Rev.C85,065803(2012). Onsi,Dutta,Chatri,Goriely,Chamel,Pearson, Phys.Rev.C77,065805 (2008).
Advantages of the ETFSI method:
very fast approximation to the full HFB equations avoids the pitfalls related to continuum states
SLIDE 25 Structure of the inner crust of neutron star
With increasing density, clusters keep ∼ the same size but are more and more dilute, and dissolve at ¯ n ∼ 0.07 − 0.09 fm−3.
5 10 15 20 25 30 35 40 45 50 r [fm] 0.04 0.08 0.12 0.04 0.08 0.12 0.04 0.08 0.12 0.04 0.08 0.12 0.04 0.08 0.12 0.04 0.08 0.12 n = 0.0749994 n = 0.0608667 n = 0.0427486 n = 0.0210866 n = 0.0051307 nn(r), np(r) [fm
n = 0.00027
The crust-core transition is found to be very smooth.
SLIDE 26 Proton shell effects in stellar environments
The ordinary nuclear shell structure is altered in dense matter: Z = 28, 82 disappear, while 40, 58, 92 appear (quenched spin-orbit).
Energy per nucleon obtained with BSk24:
10 20 30 40 50 60 70 80 90 100 110 Z 6.956 6.958 6.96 6.962 6.964 6.966 6.968 6.97 e [MeV] n=0.0480922 fm
SLIDE 27
Role of shell effects and symmetry energy
The composition of the inner crust is strongly influenced by proton shell effects and the symmetry energy:
Terrestrial abundances: Zirconium (Z = 40): 0.02% Cerium (Z = 58): 0.007%
SLIDE 28 Symmetry energy and proton fraction
The proton fraction Yp of the inner crust is governed by the density dependence of the symmetry energy S(n): the lower S the lower Yp. Analytical fits: http://www.ioffe.ru/astro/NSG/BSk/
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
n [fm
2 4 6 8 10 12 14 16 18 20 22 24 26
S(n) [MeV] BSk22 BSk24 BSk25 BSk26
Pearson et al. in prep.
SLIDE 29 Equation of state of the inner crust
The pressure in the inner crust is related to the slope L of the symmetry energy P ∼ L 3 n2
n
n0 Analytical fits: http://www.ioffe.ru/astro/NSG/BSk/
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
n [fm
2 4 6 8 10 12 14 16 18 20 22 24 26
S(n) [MeV] BSk22 BSk24 BSk25 BSk26
Pearson et al., in prep.
SLIDE 30
Unified equations of state of neutron stars
The same functionals used in the crust can be also used in the core (n, p, e−, µ−) thus providing a unified and thermodynamically consistent description of neutron stars. Analytical fits: http://www.ioffe.ru/astro/NSG/BSk/
SLIDE 31 Adiabatic index
Unified equations of state can hardly be parametrized by polytropes!
10 11 12 13 14 15 16 log10 ρ [g cm
1 2 3 Γ BSk19 BSk20 BSk21 SLy FPS
Potekhin, Fantina, Chamel, Pearson, Goriely, A&A 560, A48 (2013)
SLIDE 32 Symmetry energy and proton fraction
The proton fraction Yp of the core is governed by the density dependence of the symmetry energy S(n): the lower S the lower Yp. Analytical fits: http://www.ioffe.ru/astro/NSG/BSk/
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
n [fm
100 200 300 400 500 600 700 800 900
S(n) [MeV] BSk22 BSk24 BSk25 BSk26
The direct Urca process is allowed in all models but BSk19, BSk20 and BSk26.
SLIDE 33
Gross properties of nonrotating neutron stars
Potekhin,Fantina,Chamel,Pearson,Goriely,A&A 560, A48 (2013) Fantina,Chamel,Pearson,Goriely,A&A 559, A128 (2013)
BSk19 is too soft: it is ruled out by observations of 2M⊙ neutron stars.
SLIDE 34
Gross properties of nonrotating neutron stars
The radius of a 1.4M⊙ neutron star is predicted to lie between 11.8 and 13.1 km: this difference is potentially observable.
SLIDE 35 Exotic neutron star cores
The inner core of a neutron star may contain hyperons, meson condensates, or even deconfined quarks:
Hydrogen/He atmosphere
R ~ 10 km n,p,e, µ neutron star with pion condensate quark−hybrid star hyperon star g/cm 3 10 11 g/cm 3 10 6 g/cm 3 10 14 Fe
− π
K−
s u e r c n d c t g p
i u
p r
s
color−superconducting strange quark matter (u,d,s quarks)
CFL−K + CFL−K0 CFL−
π
2SC+s 2SC n,p,e, µ
quarks u,d,s
crust N+e H traditional neutron star strange star N+e+n Σ,Λ,Ξ,∆ n superfluid nucleon star
CFL
CFL
2SC
picture from F . Weber
SLIDE 36
Phase transition in neutron-star cores
Given the current uncertainties, we consider a 1st order transition to an exotic phase with the stiffest possible equation of state:
causality constraint vs ≤ c quarks vs ≤ c/ √ 3
Chamel, Fantina, Pearson, Goriely,A&A553, A22 (2013)
SLIDE 37
Exotica and maximum neutron-star mass
causality constraint vs ≤ c quarks vs ≤ c/ √ 3
“Soft” nucleonic matter (BSk19) as suggested by heavy-ion collisions are not necessarily ruled out by observations. Neutron stars with M > 2M⊙ would be hardly compatible with the presence of free (noninteracting) quarks in their core.
Chamel, Fantina, Pearson, Goriely, A&A553, A22 (2013)
SLIDE 38
Highly-magnetized neutron stars
Some neutron stars are endowed with extremely high surface magnetic fields ∼ 1014 − 1015 G, as inferred from spin-down and spectroscopic studies. According to simulations, the internal field could reach 1018 G. Very high magnetic fields are thought to be at the origin of giant flares observed in Soft Gamma-ray Repeaters.
SLIDE 39 Role of a high magnetic field on dense matter?
At the surface of neutron stars B 2 × 1015 G. The electron motion perpendicular to B B B is quantised into Landau orbitals with a characteristic scale am = a0
- Bat/B, where a0 is the Bohr radius
For B ≫ Bat = m2
ee3c/3 ≃ 2.35 × 109 G, atoms are expected to
adopt a very elongated shape along B B B and to form linear chains
Ruderman, PRL27, 1306 (1971); Medin&Lai, Phys.Rev. A74, 062508 (2006)
The attractive interaction between these chains could lead to a transition into a condensed phase with a surface density ρs ∼ 560AZ −3/5(B/1012 G)6/5 g cm−3 In deeper regions of the crust, matter is very stiff ρ ≈ ρs
P0
P0 ≃ 1.45×1020(B/1012 G)7/5 Z A 2 dyn cm−2
Lai, Rev.Mod.Phys.73, 629 (2001); Chamel et al., Phys.Rev.C86, 055804 (2012)
SLIDE 40
The intriguing case of RX J1856.5-3754
X-ray observations with Chandra
Turolla et al., ApJ 603, 265 (2004) van Adelsberg et al., ApJ 628, 902 (2005) Trümper (2005), astro-ph/0502457
Recent review: Potekhin et al., Space Sci. Rev. 191, 171 (2015) The thermal X-ray emission is best fitted by a black body spectrum: evidence for a condensed surface? The presence of high B B B has found additional support from recent optical polarimetry measurements.
Mignani et al., MNRAS 465, 492 (2017)
SLIDE 41 Composition of highly-magnetized crust
The composition changes with B, but not the structure (bcc).
Kozhberov, Astrophys. Space Sci.361, 256 (2016)
Equilibrium nuclides for HFB-24 and B⋆ ≡ B/(4.4 × 1013 G): Nuclide B⋆
58Fe(-)
9
66Ni(-)
67
88Sr(+)
859
126Ru(+)
1031
80Ni(-)
1075
128Pd(+)
1445
78Ni(-)
1610
79Cu(-)
1617
64Ni(-)
1668
130Cd(+)
1697
132Sn(+)
1989
100 200 300 400 500 600 700 800
B/Brel
0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75
n [10
56Fe 64Ni 62Ni 86Kr 84Se 82Ge 80Zn 79Cu 78Ni 80Ni 124Mo 122Zr 121Y 124Sr 120Sr 122Sr
Chamel et al., Prog. Theor. Chem. & Phys. (Springer, 2017), pp 181-191.
SLIDE 42 Quantum oscillations
The neutron-drip density exhibits typical quantum oscillations. Example using HFB-24:
500 1000 1500 2000 B/Bcrit 1.8 2 2.2 2.4 2.6 2.8 3 3.2 n [10
HFB-22 HFB-23 HFB-24 HFB-25 HFB-26 HFB-27* DZ
Universal oscillations: ¯ nmin
drip
¯ ndrip(B⋆ = 0) ≈ 3 4 ¯ nmax
drip
¯ ndrip(B⋆ = 0) ≈ 35 + 13 √ 13 72 In the strongly quantising regime, ¯ ndrip ≈ A Z µdrip
e
mec2 B⋆ 2π2λ3
e
3CαZ 2/3 B⋆ 2π2 1/3 mec2 µdrip
e
2/3
Chamel et al.,Phys.Rev.C91, 065801(2015). Chamel et al.,J.Phys.:Conf.Ser.724, 012034 (2016).
SLIDE 43 Quantum oscillations
The neutron-drip density exhibits typical quantum oscillations. Example using HFB-24:
500 1000 1500 2000 B/Bcrit 1.8 2 2.2 2.4 2.6 2.8 3 3.2 n [10
HFB-22 HFB-23 HFB-24 HFB-25 HFB-26 HFB-27* DZ
Universal oscillations: ¯ nmin
drip
¯ ndrip(B⋆ = 0) ≈ 3 4 ¯ nmax
drip
¯ ndrip(B⋆ = 0) ≈ 35 + 13 √ 13 72 In the strongly quantising regime, ¯ ndrip ≈ A Z µdrip
e
mec2 B⋆ 2π2λ3
e
3CαZ 2/3 B⋆ 2π2 1/3 mec2 µdrip
e
2/3
Chamel et al.,Phys.Rev.C91, 065801(2015). Chamel et al.,J.Phys.:Conf.Ser.724, 012034 (2016).
SLIDE 44
Accretion and X-ray binaries
The composition of the surface layers may be changed by the fallback of material from the supernova explosion, the accretion of matter from a stellar companion. The accretion of matter triggers explosive thermonuclear reactions giving rise to X-ray bursts. Ashes are further processed as they sink inside the star, releasing heat.
SLIDE 45
Accreted neutron star crusts
The original crust is buried in the core and replaced by accreted material with very different properties.
Composition and crustal heating for ashes made of 56Fe:
Results are very sensitive to shell effects (magic number Z = 14)
Fantina,Zdunik,Chamel,Pearson,Haensel,Goriely, in prep.
SLIDE 46 Compounds in accreted crusts
Various compounds can form from ashes of X-ray bursts:
CsCl NaCl Z1 Z2 Z1 Z2 BaTiO3 Z1 Z2 Z3
Rocksalt: AgNe. Cesium chloride: AgCa, AgTi, AgCr, AgFe, AgCo, AgNi, AgZn, AgGe, AgAs, AgSe, AgKr, KrCa, KrTi, KrCr, KrFe, KrCo, KrNi, KrZn, KrGe, KrAs, KrSe, SeCa, SeTi, SeCr, SeFe, SeCo, SeNi, SeZn, SeGe, SeAs, AsCa, AsTi, AsCr, AsFe, AsCo, AsNi, AsZn, AsGe, GeCa, GeTi, GeCr, GeFe, GeCo, GeNi, GeZn, ZnCa, ZnTi, ZnCr, ZnFe, ZnCo, ZnNi, NiCa, NiTi, NiCr, NiFe, NiCo, CoCa, CoTi, CoCr, CoFe, FeCa, FeTi, FeCr, CrCa, CrTi, TiCa. Perovskite: AgNeO3.
Chamel, J. Phys. Conf.S.932, 012039 (2017)
SLIDE 47 Conclusions & Perspectives
We have developed a set of unified equations of state for neutron stars using accurately calibrated nuclear-energy density functionals varying the neutron-matter stiffness (BSk19-21) & symmetry energy (BSk22-26). Tables:
http://vizier.cfa.harvard.edu/viz-bin/VizieR?-source=J/A+A/559/A128
Analytical fits: (including composition & local nucleon distributions)
http://www.ioffe.ru/astro/NSG/BSk/
Unified equations of state for accreted neutron stars and magnetars are under development.
Perspectives:
Consistent calculations of superfluid properties, Extension to finite temperatures (neutron-star mergers), Allowance for nuclear “pasta” mantle (if any) beneath the crust, Inclusion of exotica in the core (neutron stars with M 1.4M⊙).
