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Low Energy Theory of the Neutron Star Crust and its Observable
Implications. Sanjay Reddy
Los Alamos National Laboratory
Collaborators: Aguilera, Cirigliano, Chamel, Cumming, Page, Pethick, Pons &
Sharma
INT @ 20. The future of nuclear physics and its intersections. July 1-2, 2010Tuesday, July 6, 2010
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nuclei
neutronsuperfluid
Neutron Star Crust:
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New Phenomena in Neutron Stars
• Crustal heating and subsequent thermal relaxation in accreting neutron stars.
• Possible excitation of shear modes in the solid crusts of magnetars during giant flares.
- a window into the thermal and mechanical properties of the crust.
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Simulations by Dany Page
Transient Accretion• Nuclear reactions heat the crust during accretion.
• Crust relaxes during quiescence. Haensel & Zdunik 1990, Brown, Bildsten, Rutledge (1998)
Shternin & Yakovlev (2007), Brown & Cumming (2009)
crust
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Simulations by Dany Page
Transient Accretion• Nuclear reactions heat the crust during accretion.
• Crust relaxes during quiescence. Haensel & Zdunik 1990, Brown, Bildsten, Rutledge (1998)
Shternin & Yakovlev (2007), Brown & Cumming (2009)
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Simulations by Dany Page
Transient Accretion• Nuclear reactions heat the crust during accretion.
• Crust relaxes during quiescence. Haensel & Zdunik 1990, Brown, Bildsten, Rutledge (1998)
Shternin & Yakovlev (2007), Brown & Cumming (2009)
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More than one source !Cackett et al. 2006 Cackett et al. 2008
4.4 yr6.6 yrMXB 1659-29 KS 1731-260
!Cool = 305± 50 days!Cool = 465± 25 days
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Connecting to Crust Microphysics
!Cool !CV
"(!R)2
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Connecting to Crust Microphysics
!Cool !CV
"(!R)2
Crustal Specific Heat
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Connecting to Crust Microphysics
!Cool !CV
"(!R)2
Crustal Specific Heat
Thermal Conductivity
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Connecting to Crust Microphysics
!Cool !CV
"(!R)2
Crust ThicknessCrustal Specific Heat
Thermal Conductivity
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Explosions on Magnetars: Giant Flares
SGRs exhibit powerful outburst ~ 1046 ergs/s
http://www.physics.mcgill.ca/~pulsar/magnetar/main.html
Anomalous X-Ray Pulsars (10)Soft Gamma Repeaters (8)
Inferred to have surface fields of the order of 1015 Gauss.
SGR 0525-66 : (1979)SGR 1806-20 (1979/1986/2004)* SGR 1900+14 (1979/1986/1998)SGR 1627-41 (1998)
SGR 1806-20: 2004 Flare
Hurley et al. (2005)
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QPOs are likely to be shear modes in the solid crustDuncan (1998), Strohmayer, Watts (2006)
!n=0,l=2 ! 2 ctR
!n=1 ! " ctR
!R
R
Similar frequencies observed in 2 sources.
SGR 1806 2004 Giant Flare
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Shear mode
velocity
QPOs are likely to be shear modes in the solid crustDuncan (1998), Strohmayer, Watts (2006)
!n=0,l=2 ! 2 ctR
!n=1 ! " ctR
!R
R
Similar frequencies observed in 2 sources.
SGR 1806 2004 Giant Flare
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Microscopic Structure of the Crust
Negele & Vautherin (1973)
2 aneutrons
protons
Out
er C
rust
Inner Crustdripped
superfluid neutrons
Baym Pethick & Sutherland (1971)Tuesday, July 6, 2010
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Microscopic Structure of the Crust
Negele & Vautherin (1973)
2 aneutrons
protons
Out
er C
rust
Inner Crustdripped
superfluid neutrons
Baym Pethick & Sutherland (1971)
Theory of electr
ons and phonon
s
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Separation of Scales•Protons cluster (pairing + shell gaps) •Proton clusters form a Coulomb lattice. •Neutrons pair to form a superfluid.
!Debye !c
a! 0.45 !plasma
Ener
gy (
Tem
pera
ture
)
Longitudinal and Transverse Lattice
Phonons
!plasma =
!4"# Z2 nI
A mn
Nuclei (protons)
Superfluid Phonons
! ! EFn exp!
"1N(0) Vnn
"
Neutrons
Sing
le p
artic
le
exci
tatio
nsC
olle
ctiv
e ex
cita
tions
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Superfluid Critical Temperature
Pairing gap is difficult to calculate in strong coupling.No small expansion parameter kF a > 1.
Cold atom experiments at kF a =∞ validate QMC. Polarization measurements in imbalanced systems are exponentially sensitive to the gap.
Tc ! 0.57 !
At kF a =∞: ! =!EF
= 0.45± 0.05
Gezerlis & Carlson (2009)
Carlson & Reddy (2010)
(?)
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Superfluid Critical Temperature
Pairing gap is difficult to calculate in strong coupling.No small expansion parameter kF a > 1.
Cold atom experiments at kF a =∞ validate QMC. Polarization measurements in imbalanced systems are exponentially sensitive to the gap.
Tc ! 0.57 !
At kF a =∞: ! =!EF
= 0.45± 0.05
Gezerlis & Carlson (2009)
Carlson & Reddy (2010)
(?)
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Relevant Temperature Scales in the Crust
Electro
ns TF
Ion TPlasma
Ions TDeBroglie
Ion TMelt (Γ=200)
TUmklapp
InnerCrust
OuterCrust
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Relevant Temperature Scales in the Crust
Electro
ns TF
Ion TPlasma
Ions TDeBroglie
Ion TMelt (Γ=200)
TUmklapp
InnerCrust
OuterCrust
Acc
retin
g an
d M
agne
tized
Neu
tron
Sta
rs
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Relevant Temperature Scales in the Crust
Electro
ns TF
Ion TPlasma
Ions TDeBroglie
Ion TMelt (Γ=200)
TUmklapp
InnerCrust
OuterCrust
Tc
Acc
retin
g an
d M
agne
tized
Neu
tron
Sta
rs
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1011 1012 1013 1014
ρ (g/cm3)
0.01
0.1
Spee
d (in
uni
ts o
f c=3
x1010
cm
/s)
Longitudinal Lattice PhononsTransverse Lattice PhononsSuperfluid Phonons
Low Energy Excitations
1011 1012 1013 1014
ρ (g/cm3)
0.01
0.1
Spee
d (in
uni
ts o
f c=3
x1010
cm
/s)
Longitudinal Lattice PhononsTransverse Lattice PhononsSuperfluid Phonons
vs
cl
ct
!lPh(q) = cl q
!tPh(q) = ct q
!sPh(q) = vs q
!electron = q
How are these low energy modes coupled ? Tuesday, July 6, 2010
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Low Energy Effective Field Theory
Neutron superfluid: Goldstone excitation is the phase of the condensate.
Proton (clusters) move collectively on lattice sites. Displacement is a good coordinate.
neutrons
protons
neutrons
protons
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Low Energy Effective Field Theory
!i(x, y, z)
Neutron superfluid: Goldstone excitation is the phase of the condensate.
Proton (clusters) move collectively on lattice sites. Displacement is a good coordinate.
neutrons
protons
neutrons
protons
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Low Energy Effective Field Theory
!i(x, y, z)
Neutron superfluid: Goldstone excitation is the phase of the condensate.
Proton (clusters) move collectively on lattice sites. Displacement is a good coordinate.
neutrons
protons
neutrons
protons
“coarse-grain”
Collective coordinates:
Vector Field: Scalar Field:
!i(r, t)!(r, t)
!!!(r)!"(r)" = |!| exp (#2i ")
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Inner Crust EFT
Lp =12npm !t"i !t"i !
12
K !i"i!i"i !14
µs "ij "ij + · · ·
Compressibility
K ! !2E
!np!np
Shear Modulus
µs !Z2 e2
a4
!ij = "i!j + "j!i !23#ij "k!k
Protons:
Neutrons:
Son & Wingate (2006)
!!!(r)!"(r)" = |!| exp (#2i ")
! = µn t
Ground-state
Ln = P (µn)Tuesday, July 6, 2010
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Inner Crust EFT
Lp =12npm !t"i !t"i !
12
K !i"i!i"i !14
µs "ij "ij + · · ·
Compressibility
K ! !2E
!np!np
Shear Modulus
µs !Z2 e2
a4
!ij = "i!j + "j!i !23#ij "k!k
Protons:
Neutrons:
Son & Wingate (2006)
!!!(r)!"(r)" = |!| exp (#2i ")
! = µn t
Ground-state
! !
Fluctuations (Superfluid Phonons)
Ln = P (µn)Tuesday, July 6, 2010
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Coupling Neutrons and Protons. (or the superfluid and the lattice)
Gibbs-Duhem Relation: !µn = Enn !nn + Enp!np
!µn = !"t#! Enpnp"i$i
Velocities and current-current coupling :
Ln = P (µn) +!P
!µn"µn +
12
!2P
!µn!µn"µ2
n + · · ·
!µn = ! ("i#)2
2m+
12$ m (%vn ! %vp)2
!vn ="i#
m!vp = m "t$i
density-density interaction
current-current interaction
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Coupling Neutrons and Protons. (or the superfluid and the lattice)
Gibbs-Duhem Relation: !µn = Enn !nn + Enp!np
!µn = !"t#! Enpnp"i$i
Velocities and current-current coupling :
Ln = P (µn) +!P
!µn"µn +
12
!2P
!µn!µn"µ2
n + · · ·
!µn = ! ("i#)2
2m+
12$ m (%vn ! %vp)2
!vn ="i#
m!vp = m "t$i
density-density interaction
current-current interaction
Ln = P (µn) + nn !µn +12"n! µ2
n + · · ·
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The Coupled SystemLn+p =
12(!t")2 ! 1
2v2
s (!i")2 +12(!t#i)2 !
12(c2
l ! g2) (!i#i)2
v2s =
nf
m!nc2l =
K + 4µs/3m(np + nb)
nb = ! nnBound neutrons:Free neutrons: nf = nn (1! !){Entrainment: protons
drag neutrons.
Velocities :
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The Coupled SystemLn+p =
12(!t")2 ! 1
2v2
s (!i")2 +12(!t#i)2 !
12(c2
l ! g2) (!i#i)2
v2s =
nf
m!nc2l =
K + 4µs/3m(np + nb)
nb = ! nnBound neutrons:Free neutrons: nf = nn (1! !){Entrainment: protons
drag neutrons.
Velocities :
+ g !t" !i#i + $̃ !i" !t#i
Longitudinal lattice phonons and superfluid phonons are coupled:
g = np Enp
!!n
m(np + nb)"̃ =
!nb vs"(np + nb)nf
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The Coupled SystemLn+p =
12(!t")2 ! 1
2v2
s (!i")2 +12(!t#i)2 !
12(c2
l ! g2) (!i#i)2
v2s =
nf
m!nc2l =
K + 4µs/3m(np + nb)
nb = ! nnBound neutrons:Free neutrons: nf = nn (1! !){Entrainment: protons
drag neutrons.
Velocities :
+ g !t" !i#i + $̃ !i" !t#i
Longitudinal lattice phonons and superfluid phonons are coupled:
g = np Enp
!!n
m(np + nb)"̃ =
!nb vs"(np + nb)nf
Lt =12(!t"i)2 !
12c2t (!i"j + !j"i)2
Transverse lattice phonons:
c2t =
µs
m(np + nb)⇒
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List of low energy constants
Thermodynamic Derivatives: Enn =
!2E
!nn!nnEpp =
!2E
!np!npEnp =
!2E
!nn!np
cl ct vs g !̃
K µs Enn Enp nb
g ! 10!3 " 10!2
Coupling between superfluid and lattice: !̃ ! 10!3 " 10!2
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Two Applications
• Superfluid heat conduction.
• Effects of mixing and entrainment on the sound speeds.
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Mixing and DissipationMixing of sound modes
Dissipation of lPh leads to dissipation of sPh
1011 1012 1013 1014
ρ (g/cm3)
0.01
0.1
Spee
d (in
uni
ts o
f c=3
x1010
cm
/s)
Longitudinal Lattice PhononSuperfluid PhononTransverse Lattice Phonons
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Mixing and DissipationMixing of sound modes
Dissipation of lPh leads to dissipation of sPh
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Mixing and DissipationMixing of sound modes
Dissipation of lPh leads to dissipation of sPh
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Mixing and DissipationMixing of sound modes
Dissipation of lPh leads to dissipation of sPh
sPh mean free path
lPh mean free path
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g̃mix = g + !̃ " =cl
vs
!lPh = cl "lPh
Mixing and DissipationMixing of sound modes
Dissipation of lPh leads to dissipation of sPh
sPh mean free path
lPh mean free path
Tuesday, July 6, 2010
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g̃mix = g + !̃ " =cl
vs
!lPh = cl "lPh
Mixing and DissipationMixing of sound modes
Dissipation of lPh leads to dissipation of sPh
sPh mean free path
lPh mean free path
Away from resonance
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Thermal Conductivity
107 K
108 K
109 K
neutron-drip
! =1
3Cv ! v ! "
Typically electrons dominate heat conduction.Processes:•Electron-phonon•Electron-impurity
Flowers & Itoh (1976)Uripin & Yakovlev (1980)
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Thermal Conductivity
107 K
108 K
109 K
109 K
108 K
107 K
For T>108 K superfluid phonons play a role.
neutron-drip
! =1
3Cv ! v ! "
Typically electrons dominate heat conduction.Processes:•Electron-phonon•Electron-impurity
Flowers & Itoh (1976)Uripin & Yakovlev (1980)
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Conductivity in Magnetized Neutron StarsMagnetic field suppresses transverse conduction
= Collision time
=Gyrofrequency
Canuto and Ventura (1977)Uripin & Yakovlev (1980)
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Conductivity in Magnetized Neutron StarsMagnetic field suppresses transverse conduction
= Collision time
=Gyrofrequency
Canuto and Ventura (1977)Uripin & Yakovlev (1980)
Aguilera et al. (2009)
1013 G
1013 G1014 G
1014 G
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Shear Mode Velocity:
Lowest energy modes dominate the specific heat.
c2t =
µs
m(np + nb)
Entrainment effects
µs !Z2e2
a4i
nb = ! nn
CV ! 4!2
15
T 3
c3t(T < TDebye)
Specific Heat
Chamel, Pethick, Reddy (in prep)Tuesday, July 6, 2010
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Specific Heat at 1012 g/cm3
electronssh
ear m
ode
long
itudin
al m
ode
supe
rfluid
mod
e
107 108 109
T (K)
0.0001
0.001
0.01
0.1
1C
v (in
units
of n
i) Normal Neutrons !
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Specific Heat at 1012 g/cm3
electronssh
ear m
ode
long
itudin
al m
ode
supe
rfluid
mod
e
107 108 109
T (K)
0.0001
0.001
0.01
0.1
1C
v (in
units
of n
i) Normal Neutrons !
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Specific Heat at 1012 g/cm3
electronssh
ear m
ode
long
itudin
al m
ode
supe
rfluid
mod
e
107 108 109
T (K)
0.0001
0.001
0.01
0.1
1C
v (in
units
of n
i) Normal Neutrons !
CV ! CnormalV exp ("!
T)
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Shear Mode Velocity & Magnetar QPOs
Duncan (1998)Watts & Strohmayer (2006)
!n=0,l=2 ! 2 ctR
!n=1 ! " ctR
!R
R
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Conclusions
!Cool !CV
"(!R)2
Astronomy (Observations) Astrophysics
( Theory)
Nuclear Physics& Many-body
Theory
Insights about the nature of matter at extreme density
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Back up slides
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MixingMixing leads to oscillations
1011 1012 1013 1014
ρ (g/cm3)
0.01
0.1
Spee
d (in
uni
ts o
f c=3
x1010
cm
/s)
Longitudinal Lattice PhononSuperfluid PhononTransverse Lattice Phonons
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Dissipative Processes
Electrons lPhs sPhs
Electron-phononprocesses
Impurity (Rayleigh)scattering
Multi electron and phonon processes
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Dissipative Processes
Electrons lPhs sPhs
Electron-phononprocesses
Impurity (Rayleigh)scattering
Multi electron and phonon processes
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Coupling to Electrons - Landau Damping
Electron particle-hole excitations damp collective modesand vice-versa.
Lel!lph =1
fel!ph!i"i #†
e #e
Lel!sPh !g̃mix
v2s(1" !2)
1fel!ph
"t# $†e $e
Induced coupling between superfluid phonons and electrons:
g̃mix = g + !̃ " =cl
vswhere
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Heating in an Accreting Crust
Haensel & Zdunik (2003,2008) Gupta et al. (2007 & 2009)
Non-equilibrium reactions:Electron capture: Neutron transfers:Pycno-nuclear fusion:
e! + [A,Z] ! [A,Z " 1] + !e + Heat
n + [A,Z] ! [A+ 1, Z] + Heat
Tuesday, July 6, 2010