Structure and Properties
of Neutron-Star Crusts with
Brussels-Montreal Functionals

Nicolas ChamelInstitute of Astronomy and Astrophysics
Université Libre de Bruxelles, Belgium
in collaboration with:J. M. Pearson, A. F. Fantina, S. Goriely, P. Haensel, J. L. Zdunik,
T. Delsate, N. Gürlebeck, Y. D. Mutafchieva, Zh. Stoyanov

Hirschegg, 16 - 20 January 2017

Prelude
Although the crust of a neutronstar represents about ∼ 1% of themass and ∼ 10% of the radius, it isrelated to various phenomena:
pulsar sudden spin-ups,
X-ray (super)bursts,
thermal relaxation intransiently accreting stars,
quasiperiodic oscillations insoft gamma-ray repeaters
r-process nucleosynthesis inneutron-star mergers
mountains and gravitational
wave emission

Plumbing neutron-star crusts

Chamel&Haensel, Living Reviews in Relativity 11 (2008), 10
http://relativity.livingreviews.org/Articles/lrr-2008-10/

The nuclear energy density functional theory provides a consistentand numerically tractable treatment of all these different phases.

Outline

1

Nuclear energy density functionals for astrophysics. nuclear energy-density functional theory. Brussels-Montreal functionals

2

Applications to neutron-star crusts. composition and equation of state. crystallography. role of a high magnetic fieldNuclear energy density functional theory in a nut shell
The energy E of a nuclear system (q = n, p for neutrons, protons) isexpressed as a (universal) functional ofnq (rr , σ; r 0 , σ 0 ) =< Ψ|cq (rr 0 σ 0 )† cq (rr σ)|Ψ >eq (rr , σ; r 0 , σ 0 ) = −σ 0 < Ψ|cq (rr 0 − σ 0 )cq (rr σ)|Ψ >,nwhere cq (rr σ)† and cq (rr σ) are the creation and destruction operatorsfor nucleon q at position r with spin σ = ±1.
In turn, these matrices are expressed in terms of independent(q)(q)
quasiparticle wavefunctions ϕ1k (rr ) and ϕ2k (rr ) as
X (q)
(q)
nq (rr , σ; r 0 , σ 0 ) =ϕ2k (rr , σ)ϕ2k (rr 0 , σ 0 )∗eq (rr , σ; r 0 , σ 0 ) = −
n

X
k (q)

(q)

k (q)
(q)

ϕ2k (rr , σ)ϕ1k (rr 0 , σ 0 )∗ = −

X

(q)

(q)

ϕ1k (rr , σ)ϕ2k (rr 0 , σ 0 )∗ .

k

The exact ground-state energy is obtained by minimizing theeq (rr , σ; r 0 , σ 0 )] under the constraint of fixed
functional E[nq (rr , σ; r 0 , σ 0 ), n(q)(q)
nucleon numbers (and completeness relations on ϕ1k (rr ) and ϕ2k (rr )).

Hartree-Fock-Bogoliubov equations
Constrained variations of the nuclear energy functional yield theself-consistent Hartree-Fock Bogoliubov (HFB) equations
!
!
(q)
0
0
XZ
eq (rr , σ; r 0 , σ 0 )ψ1k (rr 0 , σ 0 )h
3 0 hq (rr , σ; r , σ )
d r e
(q)
hq (rr , σ; r 0 , σ 0 ) −hq (rr , σ; r 0 , σ 0 )ψ2k (rr 0 , σ 0 )
σ0
!
!
(q)
(q)
Ek + µq0ψ1k (rr , σ)
=
,
(q)
(q)
0
Ek − µqψ2k (rr , σ)where µq are the chemical potentials, and the non-local fields aredefined byhq (rr , σ; r 0 , σ 0 ) =δE,δnq (rr , σ; r 0 , σ 0 )eq (rr , σ; r 0 , σ 0 ) =hδE.
eq (rr , σ; r 0 , σ 0 )δnDuguet, Lecture Notes in Physics 879 (Springer-Verlag, 2014), p. 293
Dobaczewski & Nazarewicz, in ”50 years of Nuclear BCS” (World Scientific Publishing,2013), pp.40-60

Problem: we do not know what the exact functional is... We have thus
to rely on phenomenological functionals.

Phenomenological nuclear energy density functionals
For simplicity, the functional is generally written as
Z hieq (rr ), . . . d3r
E = E nq (rr ), ∇ nq (rr ), τq (rr ), J q (rr ), n
where (σ σσ0 denotes the Pauli spin matrices)XX
eq (rr ) =eq (rr , σ; r , σ)nq (rr ) =nq (rr , σ; r , σ), n
n
σ=±1
σ=±1
X Z
0
3 0
∇ · ∇0 nq (rr , σ; r 0 , σ)τq (rr ) =d r δ(rr − r )∇σ=±1 ZX∇nq (rr , σ; r 0 , σ 0 ) × σ σ0 σJ q (rr ) = −id3r 0 δ(rr − r 0 )∇σ,σ 0 =±1

Such functionals can be constructed from Skyrme type zero-range
effective interactions in the “mean-field” approximation.
Remark: fitting directly the energy functional E to experimental and/or
microscopic nuclear data may lead to self-interaction errors.
Chamel, Phys. Rev. C 82, 061307(R) (2010).

Phenomenological corrections for atomic nuclei
For atomic nuclei, we add the following corrections to the HFB energy:
Wigner energy(
EW = VW exp

N −Z
−λ
A

!2 )

(
0
+ VW
|N − Z | exp

−

A
A0

!2 )

0
∼ 1 MeV, λ ∼ 300 MeV, A0 ∼ 20
VW ∼ −2 MeV, VW

rotational and vibrational spurious collective energy
n
o
crank
Ecoll = Erotb tanh(c|β2 |) + d|β2 | exp{−l(|β2 | − β20 )2 }This latter correction was shown to be in good agreement withcalculations using 5D collective Hamiltonian.
Goriely, Chamel, Pearson, Phys.Rev.C82,035804(2010).

In this way, these collective effects do not contaminate theparameters (≤ 20) of the functional.

Brussels-Montreal Skyrme functionals (BSk)
Experimental data:
all atomic masses with Z , N ≥ 8 from the Atomic Mass
Evaluation (root-mean square deviation: 0.5-0.6 MeV)
nuclear charge radii
symmetry energy 29 ≤ J ≤ 32 MeV
incompressibility Kv = 240 ± 10 MeV (ISGMR)Colò et al., Phys.Rev.C70, 024307 (2004).

N-body calculations using realistic forces:
equation of state of pure neutron matter1

S0 pairing gaps in nuclear matter
effective masses in nuclear matterstability against spin and spin-isospin fluctuations
Chamel et al., Acta Phys. Pol. B46, 349(2015)

Nonlocal and relativistic functionals have been also developed:
Goriely et al., Eur. Phys. J. A 52, 202 (2016).
Pena Arteaga, Goriely, Chamel, Eur. Phys. J. A 52, 320 (2016)

Brussels-Montreal Skyrme functionals. fit to realistic 1 S0 pairing gaps (no self-energy) (BSk16-17)
Chamel, Goriely, Pearson, Nucl.Phys.A812,72 (2008)
Goriely, Chamel, Pearson, PRL102,152503 (2009).
Chamel, Phys. Rev. C 82, 014313 (2010).

. removal of spurious spin-isospin instabilities (BSk18)
Chamel, Goriely, Pearson, Phys.Rev.C80,065804(2009)
Chamel & Goriely, Phys. Rev. C 82, 045804 (2010).

. fit to realistic neutron-matter equations of state (BSk19-21)
Goriely, Chamel, Pearson, Phys.Rev.C82,035804(2010). fit to different symmetry energies (BSk22-26)
Goriely, Chamel, Pearson, Phys.Rev.C88,024308(2013). optimal fit of the 2012 AME - rms 0.512 MeV (BSk27*)
Goriely, Chamel, Pearson, Phys.Rev.C88,061302(R)(2013). generalized spin-orbit coupling (BSk28-29)
Goriely, Nucl.Phys.A933,68(2015).

. fit to realistic 1 S0 pairing gaps with self-energy (BSk30-32)
Goriely, Chamel, Pearson, Phys.Rev. C93,034337(2016).

Description of the outer crust of a neutron star
Main assumptions:matter is in full thermodynamic equilibrium
the crust is stratified into pure layers made of nuclei AZ X
atoms are fully pressure ionized ρ  10AZ g cm−3
electrons are uniformly distributed and are highly degenerate
nuclei are arranged on a perfect body-centered cubic lattice ρ 1/3
6
T < Tm ≈ 1.3 × 105 Z 2K
ρ6 ≡ ρ/106 g cm−3
A
The only microscopic inputs are nuclear masses. We have madeuse of the experimental data from the Atomic Mass Evaluation
complemented with our HFB mass tables available at
http://www.astro.ulb.ac.be/bruslib/
Pearson,Goriely,Chamel,Phys.Rev.C83,065810(2011)

Electron polarization effects are included using the expressions given
in Chamel & Fantina,Phys.Rev.D93, 063001 (2016)

Composition of the outer crust of a neutron star
The composition of the crust is completely determined byexperimental nuclear masses down to about 200m for a 1.4M
neutron star with a 10 km radiusPearson,Goriely,Chamel,Phys.Rev.C83,065810(2011)
Kreim, Hempel, Lunney, Schaffner-Bielich, Int.J.M.Spec.349-350,63(2013)
Wolf et al.,PRL 110,041101(2013)

Composition of the outer crust of a neutron star
Role of the symmetry energy
HFB-22-25 were fitted to different values of the symmetry energy
coefficient at saturation, from J = 29 MeV (HFB-25) to J = 32 MeV(HFB-22).
HFB-25(29)
78 Ni
126 Ru124 Mo122 Zr
121 Y
120 Sr122 Sr-0.0005
HFB-22
HFB-250.0004-3

HFB-24
(30)
78 Ni
80 Ni
124 Mo122 Zr
121 Y
120 Sr122 Sr
124 Sr-P [MeV fm ]HFB-22
(32)
79 Cu
82 Zn
78 Ni
80 Ni
124 Mo122 Zr
121 Y
122 Sr
124 Sr
128 Sr0.00030.00020.000110-5-3

n [fm ]

10

-4

Composition of the outer crust of a neutron star
Role of the spin-orbit couplingi
σ i + σ j ) · p ij × δ(rr ij ) p ij
W0 (σ
~2
i
σ i + σ j ) · p ij × (nqi + nqj )ν δ(rr ij ) p ij
HFB-28: vijso → vijso + 2 W1 (σ
~


X
1
HFB-29: Eso =J · ∇ n + (1 + yw )Jq · ∇ nq
2
q
HFB-24: vijso =HFB-2478 Ni
80 Ni
124 Mo122 Zr
121 Y
120 Sr122 Sr
124 Sr0.00050.0004

HFB-28
HFB-29-3

HFB-2979 Cu
78 Ni
126 Ru124 Mo122 Zr
121 Y
120 Sr122 Sr
124 SrP [MeV fm ]HFB-2879 Cu
78 Ni
128 Pd80 Ni
126 Ru124 Mo122 Zr
120 Sr122 Sr
124 Sr0.00030.00020.00010-5
10-3n [fm ]

10

-4

Composition of the outer crust of a neutron star
Role of nuclear pairing
HFB-27∗ is based on an empirical pairing functional.
HFB-29 (HFB-30) was fitted to EBHF 1 S0 pairing gaps including
medium polarization effects without (with) self-energy effects.
0.0005

HFB-3078 Ni
80 Ni
126 Ru124 Mo122 Zr
121 Y
120 Sr122 Sr
124 Sr0.0004

HFB-29
HFB-30
HFB-27-3

HFB-2979 Cu
78 Ni
126 Ru124 Mo122 Zr
121 Y
120 Sr122 Sr
124 SrP [MeV fm ]HFB-27∗78 Ni
126 Ru124 Mo122 Zr
120 Sr122 Sr
124 Sr0.00030.00020.00010-5
10-3n [fm ]

10

-4

Catalyzed vs accreted crusts
The composition of accreted crusts can be substantially different:
matter is not in full thermodynamical equilibrium !
Using the HFB-27∗ nuclear mass model and considering 56 Fe ashes:
130
120
110
100
A
90
80
70
N
60
50
40
30
Z
20
10
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4-4-3
n [10 fm ]Chamel,Fantina,Zdunik,Haensel, Nuclear Theory 34, pp.126-131 (Heron
Press, Sofia, 2015)

Stratification and equation of state
So far, we have assumed pure layers made of only one kind of nuclei

P
2
+

P1 2

1+2
1

nmax
1

nmin
2

"

#
n̄2min − n̄1max
A2 Z1
Cbcc α2/3
2/3≈
1+Z1 − Z2−1n̄1maxZ2 A2(3π 2 )1/3
with Cbcc = −1.444231 and α = e2 /~c.

nStratification and equation of state
So far, we have assumed pure layers made of only one kind of nuclei

P
2
+

P1 2

1+2
1

nmax
1

nmin
2

n

n̄2min − n̄1max
Z2
Z1
>0⇒
<
: the denser, the more neutron rich.
n̄1max
A2
A1.

Stratification and equation of state
So far, we have assumed pure layers made of only one kind of nuclei

P
2
+

P1 2

1+2
1

nmax
1

nmin
2

n

Mixed solid phases cannot exist in a neutron star crust because P has toincrease strictly monotonically with n̄.
.

Compounds in neutron-star crusts?
The structure could be determined using molecular dynamics
simulations. However this would be extremely costly because thecomposition must be also optimized.
Multinary compounds made of nuclei with charges {Zi } could exist in
the crust of a neutron star provided
they are stable against the separation into pure (bcc) phases:
R({Zi /Zj }) ≡CZ̄f ({Zi })
>1
Cbcc
Z 5/3

where f ({Zi }) is the dimensionless lattice structure function of
the compound and C the corresponding structure constant.
Chamel & Fantina, Phys. Rev. C 94, 065802 (2016).

This condition is independent of the stellar environment and can thus
be easily tested for any given compound structure and composition !
they are stable against weak and strong nuclear processes.

Binary compounds in neutron-star crusts?
We have investigated the formation of various compounds:

fcc1

fcc2

sc1

sc2

Terrestrial examples:
fcc1: rocksalt (NaCl), oxydes, carbonitrides
fcc2: fluorite (CaF2 )sc1: cesium chloride (CsCl), β-brass (CuZn)
sc2: auricupride (AuCu3 )Stellar compounds differ in two fundamental ways from theirterrestrial counterparts: (i) they are made of nuclei; (ii) electrons form
an essentially uniform relativistic Fermi gas.

Binary compounds in neutron-star crusts?
Oher cubic compounds with same structure as perovskites:

p1

p2

p3

Non-cubic compounds (e.g. tungsten carbide):hcp

Binary compounds in neutron-star crusts?
Some compounds are unstable against phase separation for any
charge ratio q = Z2 /Z1 and can thus be ruled out:
1.004

1.001
sc1
sc2
fcc1
fcc2

1.003
1.002

1.00051

1.001

0.9995

1

0.999

0.999

0.9985

0.998
0

0.5

1
q

1.5

2

0.998
0

hcp
p1
p2
p3

0.5

1
q

Chamel & Fantina, Phys. Rev. C 94, 065802 (2016).

The most likely compounds are those with CsCl structure.

1.5

2Substitutional compounds in neutron-star crusts
Compounds with CsCl structure are present at interfaces if Z1 6= Z2 .

P
2

P1+2 2
P1 2
P1 1+2

1+2
1
max
nmin
nmax
1+2 n1+2
1

max
min
n̄1+2
− n̄1+2
3Cbcc α≈maxmin
(3π 2 )1/3n̄2 − n̄1

nmin
2

Z 5/3
f̃ (Z1 , Z2 ) −  Z̄
 1
Z̄ A1
Z̄ A2
1−
1−
ĀZ1
ĀZ2

n

Ternary compounds in neutron-star crusts?
We have also considered ternary compounds with cubic perovskite
structure such as BaTiO3 :Such compounds do not exist in catalyzed crusts and but could beformed in accreting neutron stars.
Chamel, in preparation

Neutron-drip transition: general considerations
Nuclei are actually stable against neutron emission but are
unstable against electron captures accompanied by neutron emissionA−∆NA−
Z X + ∆Ze →Z −∆Z Y + ∆N n + ∆Z νenonaccreting neutron stars
All kinds of reactions are allowed: the ground state is reached for∆Z = Z and ∆N = A
HFB-19
HFB-20
HFB-21outer crust

drip lineρdrip (g cm−3 )Pdrip (dyn cm−2 )126 Sr (0.73)

121 Sr (-0.62)

4.40 × 1011

126 Sr (0.48)

121 Sr (-0.71)

124 Sr (0.83)

121 Sr (-0.33)

4.39 × 1011
4.30 × 1011

7.91 × 1029
7.89 × 1029
7.84 × 1029

accreting neutron stars
Multiple electron captures are very unlikely: ∆Z = 1 (∆N ≥ 1)
HFB-21

ρdrip (g cm−3 )2.83 − 5.84 × 1011

Pdrip (dyn cm−2 )4.79 − 12.3 × 1029

ρdrip and Pdrip can be expressed by simple analytical formulas.
Chamel, Fantina, Zdunik, Haensel, Phys. Rev. C91,055803(2015).

Impact of a strong magnetic field on the crust?

~ (along let’s say the z-axis), the electron
In a strong magnetic field B
motion perpendicular to the field is quantized:
Landau-Rabi levels
Rabi, Z.Phys.49, 507 (1928).

qeν = c 2 pz2 + me2 c 4 (1 + 2νB? )where ν = 0, 1, ... and B? = B/Bc
m2 c3
with Bc = e ' 4.4 × 1013 G.
~eMaximum number of occupied Landau levels for HFB-21:

B?
νmax

1500
1

1000
2

500
3

Only ν = 0 is filled for ρ < 2.07 × 106

100
14

50
28

10
137

1
1365

 
A
3/2
B? g cm−3 .
Z

Landau quantization can change the properties of the crust.

Equation of state of the outer crust of magnetars
Matter in a magnetar is much more incompressible and lessneutron-rich than in a weakly magnetised neutron star.
-3

0.5

10

■

0.48

-4

10

-3

P [MeV fm ]

0.46

0
10
100
1000

-5

10

0.44
■

0.42

-6

ye

10

-7

0.4
0.38

10

■

0.36

-8

10

0.34

-9

0.32

10

0.3
0

-10

10

-9

10

-8

10

-7

10

10

-6

-5

10

1

2

3

4
-4

-4

5

6

7

-3

P [10 MeV fm ]10

-3n [fm ]P ≈ P0

n
−1
ns

2



s
π 2 λ3e me c 2 P 
1
ye ≈
1−
2
4B? J 2

Chamel et al.,Phys.Rev.C86, 055804(2012).

8

Composition of the outer crust of a magnetar
The magnetic field changes the composition:
Equilibrium nuclides for HFB-24 and B? ≡ B/(4.4 × 1013 G):
2.75

124

Sr

2.5
122

Sr

2.25
120

2

Sr

121

-3

1.75

Y

122

1.5

Zr

-4

B?
9
67
859
1031
1075
1445
1610
1617
1668
1697
1989

n [10 fm ]Nuclide
58
Fe(-)
66
Ni(-)
88
Sr(+)
126
Ru(+)
80
Ni(-)
128
Pd(+)
78
Ni(-)
79
Cu(-)
64
Ni(-)
130
Cd(+)
132
Sn(+)

1.25
124

1
0.75

80
79

0.5

Ni
78

Cu 80

82

0.25
0
100

300

400

500

B/BrelChamel et al., to appear in QSCP series, Springer (2017).

600

700

800

Ni

Zn

Ge 84
Se
Kr 64
Ni
Ni 56
Fe

86
62

200

Mo

Neutron-drip transition in magnetars
3.2

These oscillations are almost
universal:

3

min
ndrip
3
≈
ndrip (B? = 0)
4

2.6

-4

-3

n [10 fm ]

2.8

2.4
2.2
2
1.8
0

HFB-22
HFB-23
HFB-24
HFB-25
HFB-26
HFB-27*DZ
500√
35 + 13 13≈
ndrip (B? = 0)
72
max
ndrip

1000
B/Bcrit

1500

2000

In the strongly quantizing regime,
1/3 
2/3 
A µdripB?
4B?
me c 2e
2/3
ndrip ≈
1 − CαZZ me c 2 2π 2 λ3e
3
2π 2
µdrip
e
Chamel et al.,Phys.Rev.C91, 065801(2015).
Chamel et al.,J.Phys.:Conf.Ser.724, 012034 (2016).

Neutron-drip transition in magnetars
3.2

These oscillations are almost
universal:

3

min
ndrip
3
≈
ndrip (B? = 0)
4

2.6

-4

-3

n [10 fm ]

2.8

2.4
2.2
2
1.8
0

HFB-22
HFB-23
HFB-24
HFB-25
HFB-26
HFB-27*DZ

500√
35 + 13 13≈
ndrip (B? = 0)
72
max
ndrip

1000
B/Bcrit

1500

2000

In the strongly quantizing regime,
1/3 
2/3 
A µdripB?
4B?
me c 2e
2/3
ndrip ≈
1 − CαZZ me c 2 2π 2 λ3e
3
2π 2
µdrip
e
Chamel et al.,Phys.Rev.C91, 065801(2015).
Chamel et al.,J.Phys.:Conf.Ser.724, 012034 (2016).

Neutron-drip transition: role of the symmetry energy

3.5
-4

-3

ndrip [10 fm ]3

B* = 0B* = 500
B* = 1000
B* = 1500
B* = 2000-4-3

ndrip [10 fm ]The lack of knowledge of the symmetry energy translates into
uncertainties in the neutron-drip density:

A = 723 A = 76
A = 56
A = 682.5
A = 105
A= 98
2

A = 66
A = 60-4-3

ndrip [10 fm ]

4

2
35

40

45

50
55
L [MeV]

60

65

70

A = 64

3.5
A = 1043
2.5
2

A = 106
A = 103

1.5
30

35

40

45

50
55
L [MeV]

60

65

In accreted crusts, the neutron-drip transition may be more sensitive
to nuclear-structure effects than the symmetry energy.
Fantina et al.,Phys.Rev.C93,015801(2016).

70

Description of neutron star crust beyond neutron drip
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 becomesa functional of nq (rr ) and their gradients only.
proton shell effects are added perturbatively (neutron shelleffects are much smaller and therefore neglected).
In order to further speed-up the calculations, clusters are supposed to
be spherical (no pastas) and nq (rr ) 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 difficulties related to boundary conditions
Chamel et al.,Phys.Rev.C75(2007),055806.

Structure of nonaccreting neutron star crusts

BSk27*
BSk25
BSk24
BSk22

n̄cc (fm−3 )0.09190.08560.08080.0716

Pcc (MeV fm−3 )
0.439
0.211
0.268
0.291

-3

The crust-core transition predictedby the ETFSI method agrees very
well with the instability analysis ofhomogeneous nuclear matter.

nn(r), np(r) [fm ]With increasing density, the clusters keep essentially the same size
but become more and more dilute.
0.12
0.08
0.04
0
0.12
0.08
0.04
0
0.12
0.08
0.04
0
0.12
0.08
0.04
0
0.12
0.08
0.04
0
0.12
0.08
0.04
0
0

BSk21

n = 0.0003
n = 0.005

n = 0.02

n = 0.04

n = 0.06

n = 0.08
5

10

15

20

25

r [fm]

Chamel et al., Acta Phys. Pol.46,349(2015).
Pearson,Chamel,Goriely,Ducoin,Phys.Rev.C85,065803(2012).

The crust-core transition is found to be very smooth.

30

35

40

45

Role of proton shell effects on the composition of theinner crust of a neutron star
The ordinary nuclear shell structure seems to be preserved apartfrom Z = 40 (quenched spin-orbit?).
The energy differences between different configurations become
very small as the density increases!
SLy4

BSk19

-1.4
-3

n=0.00025 fm

-1.5
-1.6
-1.7
-1.8
8.016

n=0.06 fm

-1.3
-1.4
-1.5
-1.6
-1.7
-1.8

-3

n=0.00026 fm-3

n=0.06 fm

7.61

8.014

-3

7.605

8.012
8.01

7.6

8.008
8.006

7.595

20

30

Z

40

50

20

ETF: dashed lines - ETFSI: solid lines

30

40
Z

50

Role of proton pairing on the composition of the innercrust of a neutron star
Proton shell effects are washed out due to pairing.
-1.4
-1.5

n=0.00026 fm-3

Example with BSk21.

with pairingwithout pairing

-1.6
-1.7

At low densities, Z = 42 isenergetically favored overZ = 40, but by less than
5 × 10−4 MeV per nucleon.

-1.8
-1.9
7.308
-3

n=0.055 fm

7.306
7.304

A large range of values ofZ could thus be present ina real neutron-star crust.

7.302
7.3
7.298
7.296

20

30

Z

40

50

60

Pearson,Chamel,Pastore,Goriely,Phys.Rev.C91, 018801 (2015).

Due to proton pairing, the inner crust of a neutron star is expected tocontain many impurities.

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 thermodynamicallyconsistent description of neutron stars.

Tables of the full equations of state for HFB-19, HFB-20, and
HFB-21:
http://vizier.cfa.harvard.edu/viz-bin/VizieR?-source=J/A+A/559/A128

Fantina, Chamel, Pearson, Goriely, A&A 559, A128 (2013)

Analytical representations of the full equations of state (fortran
subroutines):
http://www.ioffe.ru/astro/NSG/BSk/

Potekhin, Fantina, Chamel, Pearson, Goriely, A&A 560, A48 (2013)

Equations of state for our latest functionals will appear soon.

Nuclear uncertainties in the mass-radius
Mass-radius relation of nonrotating and nonaccreting neutron stars:
2.5
BSk22
BSk24
BSk25
BSk26

M [solar masses]

2
1.5
1
0.5
0
9

10

11

12

13 14
R [km]

15

16

17

The radius of a 1.4M neutron star is predicted to lie between 11.8
and 13 km.
Delsate et al., Phys. Rev. D 94, 023008 (2016)

Conclusions
We have developed accurately calibrated nuclear energy
density functionals fitted to essentially all nuclear mass data as
well as to microscopic calculations.
These functionals provide a unified and consistent descriptionof neutron-star crusts.
The equation of state of the outer crust is fairly well known,
but its composition depends on the nuclear structure of very
exotic nuclei (e.g. spin-orbit coupling, pairing).
The constitution of the inner crust is much more uncertain
due to the tiny energy differences between different
configurations (nuclear pastas?)
Magnetars may have different crusts.
Systematic studies of crustal properties for both nonaccreted andaccreted neutron stars are under way.