Superfluidity and thermal properties of the crust of...

International School of Nuclear Physics: nuclei in the laboratory and in the cosmos, ERICE, 2014

Superfluidity and thermal properties of the crust of Neutron Stars

Jérôme Margueron, IPN-Lyon, France.

Neutron drip

Transition Outer/inner crust

Fε

Equivalent piece of nuclear matter

Superfluidity in non-uniform systems, Cooling and crust thermalisation of neutron stars Surprising phenomenons in non-uniform systems

Pairing around and beyond the drip line

Global picture of a Neutron Star

J. Margueron 2 International School of Nuclear Physics: nuclei in the laboratory and in the cosmos, ERICE, Sept. 17th, 2014

Atmosphere

Outer Crust

Inner Crust

Outer Core

Inner Core

R ~ 10 - 12 km M ~ 1.2 - 2 Msun P ~ 1 ms - 10 s B ~1012 - 1016 G

Remnant of a core-collapse supernova (Baade & Zwickly, Phys. Rev. 1934)

J. Margueron 3

More detailed anatomy of a neutron star

ρ < 103 g/cm3 : 56Fe bcc lattice+cloud of electrons ρ > 103 g/cm3 : electrons fully ionized ρ > 106 g/cm3 : electrons are relativistic ρ > 107 g/cm3 : 56Fe → 84Se → ... ρ > 109 g/cm3 : e capture:

n rich exotic nuclei 82Ge → ...→ 118Kr

microscopic description: ρ > 4.3 1011 g/cm3 : neutron drip

→ nuclear clusters (lattice) surrounded by a neutron gas

ρ > 2.0 1014 g/cm3 : nuclei dissolve → homogeneous nuclear matter

Pasta phase

Composition of the crust

International School of Nuclear Physics: nuclei in the laboratory and in the cosmos, ERICE, Sept. 17th, 2014

We need a theoretical modeling able to bridge the transition from nuclei to uniform matter.

4

Self-consistent nuclear mean field models

Self-consistent mean-field theories and density functional theory

Properties of finite systems: masses, radii, pairing, evolution of the shells, deformation, collective modes, molecular states, …

General properties of matter: incompressibility, symmetry energy equation of state, …

↔ ↔

10 fm

Application to neutron stars and supernovae: Masses, radii, cooling, Glitches, neutrinos processes, …

20 km

Going towards very N rich nuclei

Fε

Fε

Fε

Dilute nuclei

Finite temperature, Beyond drip line.

Exotic nuclei

J. Margueron 5

A laboratory for modern physics ! Interdisciplinary field:

Neutron star is a laboratory to study matter under extreme conditions (density, temperature, ...)

Quantum Liquids: EF/T~100-1000 → Fermi-Dirac statistic

General Relativity: Rsh/R~0.1-0.2

Magnetic fields: B~1012-1016 G → origin, MHD ?

Special Relativity: vF/c~0.2-0.4

Deconfined Quarks matter: ρ > 1015 g/cm3

Superfluid, rotation, cold atomic gas, BEC/BCS crossover...


Superfluidity in uniform and non-uniform systems

Atmosphere

Outer Crust

Inner Crust

Outer Core

Inner Core

J. Margueron

BCS

Gezerlis, Carlson, Phys. Rev. C 81 (2010) Weak

(max at 1 MeV).

Pairing gap in neutron matter: comparisons of different approaches

Lombardo, Schulze, Lect. Notes Phys. 578 (2001)

What is the effect of strong/weak pairing on the cooling of neutron stars ?

BCS, BCS+polarisation, QMC, AFDMC, ...

Strong (max at 3 MeV),

7 International School of Nuclear Physics: nuclei in the laboratory and in the cosmos, ERICE, Sept. 17th, 2014

Superfluidity in non-uniform systems

J. Margueron 8

In non-uniform systems, Bogoliubov introduced field operators:

ψ↑(r) = un (r)bn↑+ vn (r)bn↓+

n∑

ψ↓(r) = un (r)bn↓ − vn (r)bn↑+

n∑

Generalized Bogoliubov-Valatin transformation:

ψ̂(r)

With a little bit of algebra:

h−µ Δ

Δ* −h+µ

#

$

%%

&

'

((

un (r)vn (r)

)

*

++

,

-

.

.= En

un (r)vn (r)

)

*

++

,

-

.

.

Quasi-particle wave function

Densities ρ(r) = (2Ja +1)va

2 (r)a∑

Mean field h[ρ(r)],Δ[ρ(r)]

! Self-consistent mean-field

κ (r) = (2Ja +1)ua (r)va (r)a∑


Hartree-Fock-Bogoliubov equation

J. Margueron 9

Application to semi-magic isotopes

J.M., Sagawa, Hagino, PRC 77 (2008)

BCS BCS++ BCS no isovector term

BCS with isovector term reproduce better the isotopic trend. BCS++ is too weak.


J. Margueron 10

BCS BCS++ BCS no IV term

Application to semi-magic isotones

Coulomb interaction disregarded. BCS with isovector term reproduce better the isotonic trend. BCS++ is too weak.

J.M., Sagawa, Hagino, PRC 77 (2008) International School of Nuclear Physics: nuclei in the laboratory and in the cosmos, ERICE, Sept. 17th, 2014

J. Margueron 11

Why BCS++ is so poor in nuclei ?

•  Polarisation is a surface effect : the phonons in uniform matter and in nuclei are very different. •  In uniform matter, the effective mass is peaked at the Fermi energy, while in Skyrme mean field, there is no surface peaked effective mass.

A surface peaked effective mass in the mean field model could increase the level density and enhance the pairing gap at the Fermi level.

➔

More in Duguet et al., PRC (2008)


J. Margueron 12

Mean field with surface peaked effective mass

Bender et al., Rev. Mod. Phys. 75 (2003)

Standard Skyrme energy density :

surface peaked effective mass

mean field compensation

Correction term : (isoscalar) Zalewski, Olbratowski,

Satula, PRC81 (2010)

Our approach : introduce a correction to the EDF such as to get : à a surface peaked effective mass (energy-independent) à a moderate effect on the mean field

Fantina et al., J. Phys. G 38 (2011)

Inspired from Ma & Wambach, NPA 1983


J. Margueron 13

Effective mass profile in 40Ca & 208Pb

standard Skyrme k-mass

Effective mass :

Fantina et al., J. Phys. G 38 (2011) International School of Nuclear Physics: nuclei in the laboratory and in the cosmos, ERICE, Sept. 17th, 2014

J. Margueron 14

Level density Density of states:

Number of states:

Fantina et al., J. Phys. G 38 (2011)

pairing properties

Pairing interaction: volume mixed surface

Pairing gaps


Thermal relaxation of the neutron star crust

Atmosphere

Outer Crust

Inner Crust

Outer Core

Inner Core

Ts Tc

J. Margueron 16

Cooling of Neutron stars

Several phases of cooling: •  Convective •  Conductive •  Radiative

Lattimer & Prakash, Phys. Rep. 442 (2007)

Time-scales: s 1-100 y >105 y

I: Crust thermalization epoch II: ν cooling epoch III: Photon epoch

low L No URCA

high L URCA


J. Margueron 17

➙ after ~1 year: Tcore << Tcrust ~0.5 MeV, ➙ next ~10-100 years: thermalization of the crust.

Crust thermalization

Ts

Lattimer et al, ApJ425 (1994) 802

Fast cooling (emission of n)

Atmosphere

Outer Crust

Inner Crust

Outer Core

Inner Core

Tc

core

crust

Fast cooling of the core (induced by URCA process) leave the crust at a temperature around 109-1010 K after few days.

with

Thermalization time :


1- Superfluidity and cooling of neutron stars

Log T = 9

Suppression of Cv in the superfluid phase

Fortin et al., PRC 88 (2010)

Increase the diffusivity in the crust (D=K/Cv)

Reduces the thermal relaxation time of the crust (τ=R^2/D)

Outer crust Inner crust Core

cooling of young neutron stars

Relaxation time of LMXRT

Neutron drip

J. Margueron

Temperature profiles in the crust

1st year

next 20 yrs

1.6 Mo Fast cooling

Fortin et al., PRC 88 065804 (2010)

Cooling code of D. Page.

19

Heat transport equation (Thorne 1977):

PBF processes

PBF processes


J. Margueron

Surface temperature

Hierarchy in the cooling time depending on the cooling scenario. For weak pairing: the presence of non-uniform matter makes the cooling faster. For strong pairing: almost no effect of non-uniform matter.

NC=No clusters

1.6 Mo Fast cooling

20

Fortin et al., PRC 88 065804 (2010)


Surprising features of superfluidity

Atmosphere

Outer Crust

Inner Crust

Outer Core

Inner Core

Transition outer / inner crust

Fε

BCS at finite temperature

T/Tc

pairing gap Specific heat

B C S

critical temperature

Neutrons specific heat in 500Zr

Pairing field profile at various temperatures:

Neutron specific heat:

Disappearance of superfluidity

N=460, Z=40

Fortin et al., PRC 88 065804 (2010)

Classical regime

in the neutron gas in the cluster

Pairing reentrance phenomenon in Sn at the drip

Temperature populates excited states: 1- kinetic energy cost induces a quenching of pairing, 2- in some cases, pairing occurs among thermally occupied excited states.

J.M. & Khan, PRC 2012

Pairing reentrance phenomenon

In nuclear matter: pairing in the T=0 (deuteron) channel Pairing in heated rotating nuclei

In spin-asymmetric cold atom gas

Superfluidity is destroyed by increasing the temperature… But a bit of temperature sometimes helps in restoring superfluidity !

Pairing reentrance in asymmetric systems:

Pairing in symmetric systems

Asymmetry detroys pairing

Temperature in asymmetric systems restore superfluidity

Sedrakian, Alm, Lombardo, PRC 55, R582 (1997) Dean, Langanke, Nam, and Nazarewicz, PRL105, 212504 (2010).

Castorina, Grasso, Oertel, Urban, Zappala, PRA 72, 025601 (2005) Chien, Chen, He, Levin, PRL 97, 090402 (2006)

Pairing reentrance in finite systems: In magic nuclei, the presence of low-energy resonances, populated at low temperature, can help superfluidity to appear.

J.M., Khan, PRC 2012

T

In higly polarized Liquid 3He, 4He Frossati, Bedell, Wiegers, Vermeulen, PRL 57 (1986)

A simple picture beyond the drip?

A

A

A

+ Almost independent:

unseparable

Depending on the shell structure of drip nucleus:

What is the interplay between the gas and the nuclei ?

Based on the presence or absence of resonant states

The limit of very dilute clusters

A

Rbox

Changing Rbox

At which density ρgas the gas can be neglected?

Interaction of a shallow gas with a nucleus

166Zr 166Zr

124Zr

Pastore, JM, Schuck, Vinas, PRC 2013

Decreasing the gas density (by increasing the volume)

Important role of resonant states Pastore, JM, Schuck, Vinas, PRC 2013

Fix Rbox, and decrease the total number of neutrons

Decreasing the gas density (by decreasing N)

Important role of resonant states

Pastore, JM, Schuck, Vinas, PRC 2013

weakly bound nuclei

Fε

FεFε

neutron star crust

Towards a better understanding of the neutron drip (line and -ing)

€

µ

Usual picture

€

µ

With continuum coupling

Conclusions Ø  The transition between the outer / inner crust offers a fascinating

playground to apply and test pairing theories.

Ø Since two superfluids overlap (gas+nucleus), surprising features occurs, mostly due to the resonant states.

Ø Neutron stars: Models for the crust including pairing shall be revised taking into account finite temperature in non-uniform nuclear clusters.

Ø Could resonant states be responsible for the entrainment phenomenon?

Ø  For nuclear experiments: role of continuum coupling going towards the drip line for 20<Z<40.

These non-trivial features of superfluidity are interesting for:

Pairing reentrance in other nuclei? Under investigation,

Similar effects in cold-atomic gas (Pastore, ArXiv 2014)

Correlations beyond HFB (QRPA, etc…)

Collaborators: A. Fantina (ULB Bruxelles), A. Pastore (ULB Bruxelles), N. Sandulescu (NIPNE Romania), N. Van Giai (IPNO), E. Khan (IPNO), P. Schuck (IPNO), X. Vinas (Barcelona), D. Page (UNAM Mexico), P. Pizzochero (Milan).

Outlooks

The crust of neutron stars

Neu

tron

drip

SPIRAL2

Nuclear matter Lattice of nuclei

Negele & Vautherin NPA 207 (1973)


A systematic study based on 8 isotopes with 28<Z<50

Suppression or persistence of pairing upon overflowing neutrons.


Fε

Weak dependence on the model

Group A2

Group A1

A systematic study based on 8 isotopes with 28<Z<50

Pairing is suppressed in the absence of occupied resonant states at the drip-line.


Fε

Spontaneous symmetry breaking

J. Margueron 37

Uniform matter Superfluid matter

Ground state: BCSHF

invariant under rotation: Symmetry U(1)

HF = exp(−iϕ ) HF BCS ≠ exp(−iϕ ) BCS

HF ck↑c−k↓ HF = 0 HF ck↑c−k↓ HF = ΔkNon-vanishing operator:

2 recent Nobel prices: Yoichiro Nambu

"for the discovery of the mechanism of spontaneous broken symmetry in subatomic physics" Makoto Kobayashi and Toshihide Maskawa "for the discovery of the origin of the broken symmetry which predicts the existence of at least three families of quarks in nature"

François Englert and Peter W. Higgs "for the theoretical discovery of a mechanism that contributes to our understanding of the origin of mass of subatomic particles, and which recently was confirmed through the discovery of the predicted fundamental particle, by the ATLAS and CMS experiments at CERN's Large Hadron Collider"

2008

2013


BCS/BEC condensate of Cooper pairs


a: scattering length, d: average distance. a>0 a<0

BEC (molecules) BCS (weak limit)

(Feshbach resonance)

N of atoms ~104-106 Densities ~1013-1014 atoms/cm3 Temperatures~10-100 nK

(6Li, 40K) Fermi gas (2 components) Interaction in the s-wave

ξ <<d ξ >>d

bound independent pairs Strong correlation between the states (k,-k)

(ultra-cold atomic gases)

! Low density neutron matter is close to the unitary limit (crossover)

CR

OSS

OVE

R

For neutrons: a ~ -18.5 fm

δεk ≈ Δ⇒ δk ≈ mΔ!2kF

⇒ δr ≈ !2kFmΔ

= ξ

Superfluidity and thermal properties of the crust of...

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