Post on 30-Jul-2018
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...
International School of Nuclear Physics: nuclei in the laboratory and in the cosmos, ERICE, Sept. 17th, 2014
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∑
International School of Nuclear Physics: nuclei in the laboratory and in the cosmos, ERICE, Sept. 17th, 2014
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.
International School of Nuclear Physics: nuclei in the laboratory and in the cosmos, ERICE, Sept. 17th, 2014
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)
International School of Nuclear Physics: nuclei in the laboratory and in the cosmos, ERICE, Sept. 17th, 2014
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
International School of Nuclear Physics: nuclei in the laboratory and in the cosmos, ERICE, Sept. 17th, 2014
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
International School of Nuclear Physics: nuclei in the laboratory and in the cosmos, ERICE, Sept. 17th, 2014
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
International School of Nuclear Physics: nuclei in the laboratory and in the cosmos, ERICE, Sept. 17th, 2014
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 :
International School of Nuclear Physics: nuclei in the laboratory and in the cosmos, ERICE, Sept. 17th, 2014
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
International School of Nuclear Physics: nuclei in the laboratory and in the cosmos, ERICE, Sept. 17th, 2014
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)
International School of Nuclear Physics: nuclei in the laboratory and in the cosmos, ERICE, Sept. 17th, 2014
Surprising features of superfluidity
Atmosphere
Outer Crust
Inner Crust
Outer Core
Inner Core
Transition outer / inner crust
Fε
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
J. Margueron 32 International School of Nuclear Physics: nuclei in the laboratory and in the cosmos, ERICE, Sept. 17th, 2014
The crust of neutron stars
Neu
tron
drip
SPIRAL2
Nuclear matter Lattice of nuclei
Negele & Vautherin NPA 207 (1973)
International School of Nuclear Physics: nuclei in the laboratory and in the cosmos, ERICE, Sept. 17th, 2014
A systematic study based on 8 isotopes with 28<Z<50
Suppression or persistence of pairing upon overflowing neutrons.
J.M. & Khan, PRC 2012
Fε
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.
J.M. & Khan, PRC 2012
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
International School of Nuclear Physics: nuclei in the laboratory and in the cosmos, ERICE, Sept. 17th, 2014
BCS/BEC condensate of Cooper pairs
J. Margueron 38 International School of Nuclear Physics: nuclei in the laboratory and in the cosmos, ERICE, Sept. 17th, 2014
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