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Transcript of I) Pair condensation in nuclear systems - general properties - BCS, Gorkov and Bogoliubov equations...
I) Pair condensation in nuclear systems
- general properties - BCS, Gorkov and Bogoliubov equations
II) Linear response with pairing correlations and continuum coupling - collective excitations in neutron-rich nuclei - two-neutron transfer III) Nuclear superfluidity and thermal properties of neutron stars
PAIRING CORRELATIONS AND COLLECTIVE EXCITATIONS IN NEUTRON-RICH SYSTEMS
Nicolae SandulescuInstitute of Atomic Physics, Bucharest
CEA/DAM – Bruyeres le Chatel
Outline
Neutron-Rich Systems
Sites of nucleosynthesis processes
bound nuclei weakly bound nuclei
Neutron-Rich Systems
nuclei+ neutron gas neutron matter
F F
F
weakly bound nuclei
Pairing in Neutron-Rich Systems
nuclei+ neutron gas uniform matter
- binding (e.g., neutron skins and halos)
- quasiparticle excitations • Consequences : - giant glitches - cooling
• mainly neutron 1S0 superfluidity
• Consequences • Core : - neutron 3PF superfluidity - proton 1S0 superconductivity
• Crust: - neutron 1S0 superfluidity in non-uniform nuclear matter
FF
• One- Cooper- pair problem
• Condensate of pairs: a few properties
• The condensate within the BCS approach
• Non-uniform condensate: Bogoliubov equations
Pair Condensation in Nuclear Systems
Outline
Moment of inertia : Migdal
One of the first hints on the nuclear superfluidity in neutron stars
One- Cooper-pair problem (I)
Physical system: two fermions subjected to an atractive interaction and situated on top (k >kF) of a free gas of fermions
. .• Free states ( box of length L)rik
Liiekr 1
2/3),(
• Two-electrons with CM at rest, i.e., kkk 21
21)(),( 21
ikr
kk
ikr eekgrrF
)2()],(22
[ 212
2
1
2
FErrVmm
)()2()(2'
'' kgEVkg F
kkkkk
F
otherwise 0
if '' 3 cutFkkFkk L
GV
Fkk
kF
kgL
GC
CkgE
)(
)()22(
3
Fkk Fk EL
G
)(2
11
3
Fkk Fk EL
G
)(2
11
3
cutF
FE
dNG
F
)(2
)(1
cutF
FE
dGN
FF
)(2)(1
1
2
)(
2
FGN
cut
e
E
2 )(
2
FGNcuteE
1)( if FGN
One- Cooper-pair problem (2)
.)( of degeneracy the todue and
condition the todue isfact This problem.body - two with theein varianc
strength,n interactio smally arbitrarilan for exists 0ith solution w
FF Nkk
EA
.particles) ginteractin-non the to(relativepair theofenergy binding theis Thus
.22fact in is electrons two theof eigenvalue The
FF EW
. (k,-k)pair with for the maximum isenergy binding theThus . 2
v
: isenergy binding the, q momentum with themoving ispair theIf
F qq
One- Cooper-pair problem (3)
One-Cooper-pair in finite nuclei
Physical system: two nucleons extra a closed shell
The most exotic system: 11Li
9Li
nn
Condensate of Cooper pairs
I) Condensate of Bose-Einstein type: if the pairs are well-separated, i.e., d. In this case the system can be considered as formed by (bound) independent pairs, which are correlated only through the Pauli principle.
The type of the condensate depends on the ratio between the size of the Cooper pairs , and the average distance between pairs, d .
II) BCS-type condensate: if the pairs are strongly overlapping, i.e., >>d. For a typical superconductor, in the region of a given pair (of size one can find located the center of mass of about 106 pairs.
• The condensate cannot be considered anymore as formed by bound Cooper pairs. However, as in the one-Cooper-pair problem, in the condensate there is a strong corelation between the states (k,-k).
• The strong correlations among pairs are induced mainly by Paul principle. Thus in the lowest order one can consider that the dynamical interactions affect only the two nucleons which form the Cooper pair.
The structure of a condensate
Physical system: N fermions in the presence of an attractive force
Cooper pair instability >>> system of identical pairs
21)(),( 21ikr
k
ikr eekgrr k is not restricted by k>kF
),1)...(2,1)(,()....,(r ),...,( 1211 NNrrrArr NNNN • each pair is described by the same wave function
.).........2()1( )()...(),...( 2111
2/,1 ..2/11 rikrik
kkNNN eeAkgkgrr
N Slater determinant
|....)()...(|
2/2/11
2/...1
2/1 NN
N
kkkkkk
NN cccckgkg
k
kk
NN cckgSS )( with ; |)( | 2/
Note: S+ is not a boson operator, so the wave functionis not a Bose-Einstein condensate
Example: pair condensation in Sn isotopes
Sn isotopes : neutrons in the major shell N=50-82
N. Sandulescu et al, PRC55 (1997)2708
02/ )( ; |)( |
Jjj
jjN
N ccgSS2||)0(| NJSMZ
function wavemodel shell )0(| JSM
Superfluid Flow
),1)...(2,1)(,()....,(r ),...,( 12110 NNrrrArr NNNN
21)(),( 21ikrikr eekgrr • Center of mass at rest:
• Translation motion: if the system is Galilean invariant, one can simply shift the CM of each pair with a given amount in momentum space
02)(2
1
)(2
2111121
...),(....),(r),...,( N
qri
rrqi
NN
rrqi
NqN
N
jj
NN
eerrerArr
valid in general if q does not change much over a coherence length (London)
• Superfluid flow: arbitrary motion ( London & Feynman)
0)(
2
11),...,( N
jsi
NsN
N
jerr
0)vcurl(
);(v
s
s
sgrad
m
Note: a condensate corresponds to a metastable equilibrium since a change of it would involve a simultaneous transition of many pairs.
Pair condensate: BCS ansatz
|....)()...(|
2/2/11
2/...1
1 NN
N
kkkkkk
NN cccckgkg
|)1(|
kk
kk ccgCBCS
SeSBCS |)(
!
1 |
0
u
v
k
kkg
|)vu(| kk
kk
k
ccBCS
BCSNBCSN |ˆ| :condition
- the width of the distribution is
- the relative fluctuation is 1/
• Condensate with a given number of pairs
• BCS ansatz : distribution of pairs
(“coherent” distribution)
• Essential properties:
BCS equations
0|ˆˆ|v, BCSNHBCSkku
|)vu(| kk kkk
ccBCS
))(
1(2
1v
22
2k
kk
k
22)(2 lk
l
lklk V
• Gap for constant interaction: ]1
exp[2GNF
cut
• Condensation energy: ]2
exp[22
1 22
GNNNWW
FFcutFsn
• Condensation amplitude:k
kkkkkk E
uBCSccBCS2
v||
is the energy region where the pairing is important, i.e, v2 goes from 1 to 0
(Pippard)length coherence r 2
2
m
k
k
mk F
Fk
compare to the binding energy of Cooper’s pair !)
Pairing Hamiltonian
Exact solution(Richardson)
Projected BCS
|)vu(| kk kkk
ccBCSBCS solution
Quasiparticle excitations
),1)...(2,1)(,()....,(r ),...,( 1211 NNrrrArr NNNN
),1)(,1)...(2,1(),()....,(r ),...,( 112111
NNNerrrArr NmrikNNN
mN
• excited states: 1,2,…,broken pairs (2,4,…unpaired particles)
)',).(,1)...(2,1(),()....,(r ),...,( 123211 NNeerrrArr NnNm rikrik
NNNmnN
• ground state: all particles are paired
Even number of particles (N=2N’)
Odd number of particles (N+1=2N’+1)
• ground state: all particles paired, except one
• excited states: no pairs broken (one unpaired particle)
1,2,… broken pairs (3,5,…unpaired particles)
|)vu(~
| kk
mkk
mk
m ccc
Quasiparticle excitations in BCS
|)vu(~
|,
kk
nmkk
nmk
mn cccc• Two-quasiparticle excitations
• One-quasiparticle excitations
0~
| : mBCSNote
• Qusiparticle operators (Bogoliubov)
|| BCSmm ''},{ kkkk
0 | BCSm
BCSc
BCScu
mm
m
mm
m
|v
1~|
|1~
| mmmm ccu mv
mmmm ccu mv
• Independent q.p. representation: iii
igeff EEH ˆ
iii
iii
EH
EH
],[
],[
22)( iiiE || BCSmm • Quasiparticle energy:
BCS at finite temperature
Finite temperature: thermally excited states of 1,2,3,4,… quasiparticles
the average effect of their interaction is described by =(T)
the excitations of the system gas of independent quasiparticles
The probability of finding a quasiparticle k at temp. T:
,......| , | ' BCSBCS kkk
kkkf
thermal averagemmmm ccu mv
BCS equations: minimizing F=<H>-TS with respect to uk, vk and fk
)21()(2 22 l
lk
l
lklk fV
)/exp(1
1
TkEf
Bkk
))(
1(2
1v
22
2k
kk
k
])1([v 2
k
2k kkk fufN
Pairing in Uniform Nuclear Matter
22)(2 lk
l
lklk V
Pairing in Neutron Matter : BCS with bare forces
(U.Lombardo, H-J. Schulze, Lect.Notes Phys. 578 ,2001, 30 )
Pairing in Neutron Matter: beyond BCS
Gorkov equations ;0|)(|0 aaTiG 0|)(|0 aaTiF
)]'(')]['('[
)'()',(')(
''
''
'
kk
kkk
kk Vd
C.Shen, U.Lombardo,P.Schuck,W.Zuo, N.S, Phys.Rev.C67(2003)
Self- energy
C.Shen, U.Lombardo,P.Schuck,W.Zuo, N.S, Phys.Rev.C67(2003)
Screening of Pairing Force
Pairing in neutron matter: beyond BCS (II)
C.Shen, U.Lombardo,P.Schuck,W.Zuo, N.Sandulescu, Phys.Rev.C67(2003)
Pairing in uniform neutron matter within various aproximations
(U.Lombardo, H-J. Schulze, Lect.Notes Phys. 578 ,2001, 30 )
Pairing in non-uniform systems
How to form Cooper pairs in inhomogeneous systems ?
• One alternative: use time-reversed states in an “appropiate” basis (e.g. HF)
• More general : use intrinsic properties of the condensate ( in “field” picture)
coordinate representation:2
1 ; )(r, );( r force: V=-V0(r-r’)
)()'()'()(2
1)(ˆ)(ˆ
',0
rrrrdrVrTrdrH
)}2
1,()
2
1()()
2
1()
2
1()(])(ˆ[{ * rrrrrrrTdrH eff
HF)in (like timeaat particle aon only acting potential aby V replace 0
)()2
1()
2
1()( 00 rVrrVr
)}2
1,()
2
1()()
2
1()
2
1()(])(ˆ[{ * rrrrrrrTdrH eff
r) ,*(r)act as a ‘classical’ coherent field, through which the condensate drives the particles by absorbing a pair and then restoring it ( eventually at a different point).
)2
1,(ˆ)()
2
1,(ˆ)ˆ(],ˆ[)
2
1,(ˆ
rrrTHrt
i eff
)2
1,(ˆ)()
2
1,(ˆ)ˆ(],ˆ[)
2
1,(ˆ * rrrTHr
ti eff
(Heisenberg representation)
• Spectrum: canonical transformation
iii
igeff EEH ˆ
iieffi
iieffi
EH
EH
],[
],[
Bogoliubov Equations (I)
20 |)(|)( rVVr
)(*)()( 0 rVrUVr ii
Pairing density:
*iU )
2
1,()
2
1,()( iVrrr
Bogoliubov Equations (II)
consistency:
k
kk
k
k
V
UE
V
U
T
T *
Particle density :
| | )()()( 2 iVrrr
Bogoliubov equations at finite temperature
)21)(()( *iiiT frVrU ]|)(|)21(|)(|[ 22
iiiiT frUfrV
fi=(1+eEi/kT)-1
)()( 0 rVr TT )()( 0 rVr T
20 |)(|)( rVVr
)(*)()( 0 rVrUVr ii
• translation of pairs with a total momentum 2q:
• Pairing density: *iU- )
2
1,()
2
1,()( iVrrr
(r) describes the center of mass motion of the condensed pairs !
• a stationary state of a homogeneous system: r) =constant
• vortex type motion : iner ||)(
Bogoliubov Equations (III)
consistency:
k
kk
k
k
V
UE
V
U
T
T *
• a stationary state of an inhomogeneous system: r) =(r)
(“condensate” wave function)
),()(),(2
),()(),(2
2
2
rEVErEVm
rEUErEUm
),(
),(
),(
),(
)(ˆ)(
)()(ˆ** rEV
rEUE
rEV
rEU
rTr
rrT
Physical condition for bound nuclei (<0): at large distances (r)=0,r) =0
Quasiparticle spectrum in finite nuclei
r)exp(-k' r)V(E, ;)exp(),(
spectrum bound
krrEU
E
r)exp(-k' r)V(E, );cos(),(
spectrum continuous
krrEU
E
)( 22E
2
||
Bogoliubov equations in finite nuclei: continuum solutions
M.Grasso, N. S., Nguyen Van Giai, R.J.Liotta, Phys. Rev. C64 (2001)064321
Bound solutions: E<-Unbound solutions: E>-
Densities
0 - E. . .
Solution at r=0
22
,
|),(|||)( rEVdEVrEcut
Eii
i
**
,
),(),( )( rEVrEUdEVUrEcut
Eiii
i
Quasiparticle resonant states
r)exp(-k' r)V(E, );cos(),(
spectrum continuous
krrEU
E
Quasiparticle resonances
• originating from single-particle (HF) resonances
• originating from bound s.p. states with
Quasiparticle resonant states in oxygen isotopes
E.Khan, N. Sandulescu.,M.Grasso, Nguyen Van Giai, Phys. Rev. C66 (2002)024309
M.Grasso, N. Sandulescu, Nguyen Van Giai, R.J.Liotta, Phys. Rev. C64 (2001)064321
Quasiparticle resonances at drip line
HF
N. Sandulescu, P. Schuck, PRC 71 (2005) 054303
Pair Distribution in Finite Nuclei
• Problematics: two-quasiparticle excitations in finite systems
• Time-dependent HFB approach and the linear response • Applications: - collective excitations and pair transfer in exotic nuclei - collective modes in the inner crust of neutron stars
Linear response with pairing correlationsand continuum coupling
Outline
Two-quasiparticle states
two-quasiparticle resonances ?
• bound-bound
• bound-unbound
• unbound-unbound
)}2
1,()
2
1()()
2
1()
2
1()(])(ˆ[{ * rrrrrrrTdrHeff
Time-Dependent HFB Equations
Static fields:
)( )]([ rVr )( )]([ rVr
k
kk
k
k
V
UE
V
U
T
T **
Time-dependent fields
)}2
1,()
2
1(),()
2
1()
2
1(),(]),(ˆ[{ * rrtrrrtrtrTdrHeff
),( )],([ trVtr
),( )],([ trVtr
)(
)(
)(ˆ)(
)()(ˆ
)(
)(
** tV
tU
tTt
ttT
tV
tU
ti
),( )],([ ** trVtr
)( )]([ ** rVr
Linear Response Theory (I)
• Probe the system with a weak time-dependent external field
pair transfer
• The external field induces strong oscillations of the nuclear densities whenever the frequency is close to an eigenmode of the system
; .)( )()0( ccet ti
• The external field produces small changes which can be treated in the linear order
21
12
11
333231
232221
131211
)(
)(
)(
F
F
F
RRR
RRR
RRR
k
; .)( )()0( ccet ti ; .)( )()0( ccet ti
• Advantage: a method to derive the (Q)RPA equations for density-dependent forces
FR ˆˆˆ )( response function
density' transition' )(
Linear Response Theory (II)
21
12
11
333231
232221
131211
)(
)(
)(
F
F
F
RRR
RRR
RRR
k
• simple to implement for zero range forces
')'(ˆ)',(ˆ)(ˆ )( drrFrrRr
- one can treat systems with large number of particles
• no needs to diagonalize large matrices
• well-suited for the treament of continuum configurations
example: heavy nuclei, trapped atoms, neutron stars matter
important for exotic nuclei close to the driplines
Coordinate representation:
i
ii crr )()(
Bogoliubov Transformation
kkk
kk rVrUr )()()( *
kikKk
iki VUc *iiki
iikk cVcU
),(),(
),(),(
*
rVrV
rUrU
ii
ikk
ii
ikk
*
*
UV
VU
c
c W
*
*
UV
VU WAWAA ~
Configuration space
Coordinate representation
Densities
|)()(|)( rrr
|)()(|)( rrr
• Density matrix
ij
jiji rrr )()()( *
ij
jiji rrr )()()( *
ij
jiji rrr *** )()()(
• Densities: coordinate representation i
ii crr )()( kk
kkk rVrUr )()()( *
**''
''
1
||||
||||
llll
llll
cccc
ccccR
10
00
||||
||||~
''
''
llll
llllRWWR
- quasiparticle representation
|)()(|)( rrr
kljijiij ccccccH ijklv4
1ˆ
int'22
''11
'0 .).(
2
1ˆ HchHHHH kkkkkkkk
iii
igHFB EEH ˆ
*11*20
2011~
HH
HHH HFB
E
EH HFB 0
0~
***11*20
2011
W ˆh
h
HH
HHWH HFB
HFB equations in various representations
kikKk
iki VUc *
V
UE
V
U
h
h **
''''' v
qqqqqllqll
''''' v
2
1
qqqqqqllll k
Configuration representation
Quasiparticle representation
0]~
,~
[ RH HFB
Mapping between representations
10
00
~R
Transition Densities (I)
|)()(|0)()( rrr
kkk
kk rVrUr )()()( *
Time-dependent densities tiet |0|)(|
)(|)()(|)(),( trrttr
|)()(|0)()( rrrTransition densities
|)()(|0)()( rrr
Transition densities in qp representation:
||0)()(||0)()()( **)(ijijijj
iji rVrUrVrUr
||0)()(||0)()()( **)(ijjiijj
iji rVrVrUrUr
||0)()(||0)()()( **)(ijjiijj
iji rUrUrVrVr
ccerr ti .)()( )()0(
)(||)()(||)(
)(||)()(||)()(
~
''
''
tttt
tttttR
llll
llll
tiet |0|)(|
h.c. 0||0
||00
10
00)(
~
'
'
ti
ll
ll etR
.. ~~
)(~ '0 cheRRtR ti
Time-Dependent Density Matrix
Quasiparticle representation
Transition Densities (II)
Matrix notations:
Time-Dependent HFB
External field:
'
'' ' *'*
H
)]([)]([
)]([ )]([ )( **
tTtk
ttTtH
)(1)(
)()()( ** tt
tttR
TDHFB in quasiparticle representation
• both H0 and R0 are diagonal
• the matrix R’ is off-diagonal
• The solution of TDHFB equations in qp representation
• Quasiparticle representation: WAWAA ~
Transition Densities (III)
Variation of the HFB Hamiltonian (I)
• Energy functional
• HFB Hamiltonian
.. ]2
1
2
1[)0()( )()()( che
hhhhth ti
klkl
ijkl
kl
ijkl
kl kl
ijijij
][][
][ ][ **
hk
hH
• Linear perturbation
*ij* ,
ijijij kh
Variation of the HFB Hamiltonian (II)
H’ in coordinate representation
Residual interaction
ij
jiji rrr )()()( *
ij
jiji rrr )()()( *
ij
jiji rrr *** )()()(
2or 3 is then 3or 2 is
ever that whenmeans
i
ii crr )()(
Variation of HFB Hamiltonian (III)
Quasiparticle representation
Variation in coordinate representation
Transition Densities (III)
Linear response
Bethe-Salpeter equation (coordinate representation)
Residual interaction
Unperturbed Green function
QRPA response
E.Khan, N. Sandulescu.,M.Grasso, Nguyen Van Giai, Phys. Rev. C66 (2002)024309
Green function: spherical symmetry (I)
Spherical symmetry: quantum numbers for q.p. states : (E,l.j,m)
Green functions (after summation over spin and m)
Green function: spherical symmetry (II)
Unperturbed Green function:
Green Function
Excitation strength )(|||0|)( 2
FS
Strength Function
Energy-weighted sum rule
21 |||0| FS
)'(ˆ)',(ˆ)(ˆ'Im1
)( rFrrGrFdrdrS
)'(ˆ)',(ˆ)(ˆ'Im2
1 rFrrGrFdrdrS
)'(ˆ)',(ˆ)(ˆ')( rFrrGrFdrdrRF
Strength functions
• within the same nucleus:
•two-particle transfer :
)'(ˆ)',(ˆ)(ˆ'Im1
)( rFrrGrFdrdrS
• Excitations in dripline nuclei - oxigen isotopes : E.Khan, N.S, M.Grasso,Nguyen Van Giai Phys. Rev.C66, 024309 (2002)• Two- particle transfer: - oxigen isotopes: E.Khan, N.S,Nguyen Van Giai, M. Grasso
Phys. Rev.C69, 01430914 (2004)• Nuclear Astrophysics - neutron stars: E.Khan, N.S, Nguyen Van Giai Phys.Rev.C71 (2005) 042801R - r-processes: S.Goriely, E. Khan, M.Samyn,
Nucl.Phys. A739 (2004) 331-352
Applications of the continuum- QRPA response
Quadrupole excitations in oxygen isotopes
Isoscalar quadrupole strength for oxygen 22
Two-particle transfer
M.Herzog, R.J.Liotta, T.Vertse, Phys. Lett.B165, 35 (1985) L. Fortunato, W. Ortzen, H. Sofia, A.Vitturi, Eur. Phys.J. A14, 37 (2002
• pair transfer in unbound (resonant) states ?
• giant pair vibrations ?
Giant Pair Vibration (GPV)
Pair transfer in high energy shell ? (analogy with giant resonances)
• Calculations: IF high energy shells exist, the GPV mode could exist
• Experiment: even if the GPV mode may exist, most probably it is destroyed by the excitation of the other modes
What about pair transfer with exotic nuclei ?
Two-neutron transfer
E.Khan, N.Sandulescu,Nguyen Van Giai, M. Grasso Phys. Rev.C69, 01430914 (2004)
Two-neutron transfer: cross sections
E.Khan, N.Sandulescu,Nguyen Van Giai, M. Grasso Phys. Rev.C69, 01430914 (2004)
Giant pair vibration mode
L. Fortunato, W. Ortzen, H. Sofia, A.Vitturi, Eur. Phys.J. A14, 37 (2002
Nuclear Superfluidity and Thermal
Properties of Neutron Stars
• Superfluid properties of the inner crust • Specific heat and cooling time
Outline
Superfluidity in Neutron Stars
• Crust : - neutron 1S0 superfluidity
• Core : - neutron 3PF2 superfluidity
- proton 1S0 superconductivity
- “exotic” superfluidity
• Consequences : - giant glitches
- cooling
22)(2 lk
l
lklk V
Superfluidity and Giant Glitches
• Spin-up: P/P~10-6
• Recovery: ~ 1-3 months
• Energy : 1043 erg
• Scenario: vortex depinning ( Anderson & Itoh, 1975)
Epin ~ [ f
fV
Superfluidity and Cooling Time
(Yakovle
v et a
l)Non superfluid Superfluid
• Thermal photon emission• Neutrino pair bremsstrahlung
• URCA processes: direct : n > p +e +; p+e > n+
modified : n+n >n+ p +e + ; n+p+e >n+ n+
bound nuclei
Nuclei in the Crust of Neutron Stars
nuclei+ neutron gas
• neutron 1S0 superfluidity • neutron 1S0 superfluidity
Outer Crust Inner Crust
Inner crust: microscopic treatment
I) Inner crust structure: N/Z, Rws
II) Pairing properties : (r,T,)
electronspairingSkyrme EEEE
F=0, equilibrium
Self-consistent mean field calculations (HFB)
Inner Crust Structure J.W. Negele, D. Vautherin, NPA207 (1973) 298
Finite-Temperature HFB
Enuc= ESkyrme + Epair [ ,]
fi=(1+eEi/kT)-1
T(r) = Vpair T(r)
N.Sandulescu, Phys.Rev.C70 (2004) 025801where :
)21)(()()12(4
1 *iiiiT frVrUj
Efective Pairing Interactions
VbareGogny force
kF < 0.9
Vpair =V0[1-r-r’)
I) V0=-430
II) V0=-330
=0.7
max =3 MeV
max =1 MeV
C.Shen, U.Lombardo,P.Schuck,W.Zuo, N.S, Phys.Rev.C67(2003)
N.Sandulescu ,Nguyen Van Giai,R.J.Liotta, Phys.Rev.C69(2004)045802
Density in the Wigner-Seitz Cells
r
Pairing Field in the Wigner-Seitz Cells
N.Sandulescu, Phys.Rev.C70 (2004) 025801
Pairing Field in the Wigner-Seitz Cells
N.Sandulescu, Phys.Rev.C70 (2004) 025801
Superfluidity and Specific Heat of Crust Matter
C(t)V = CV (n) + CV (e) + CV(lattice)
• normal phase :
• suprafluid phase : CV(n) CV(n;=0) e-/kT
CV (n) > CV (e)
iE
• issues: - effect of nuclear clusters on superfluidity & specific heat ?
- screening effects of pairing interaction ?
- effect of collective modes ?
• Main effects on cooling time ?
N.Sandulescu, Phys.Rev.C70 (2004) 025801
Specific Heat in the FT-HFB Approach
bare fore
screening
)(electronsVC
Specific Heat Across the Inner Crust
T=100 keV
Thermal Diffusivity and Cooling Time
D
Rtdiff
2
RVC
D
][)](,[
1 2
dR
TDt
shell
cdiff
][RR
(Landau & Lifshitz, Fluid Mechanics)
;
Constant thermal diffusivity
Non-constant thermal diffusivity
given by Tolman –Oppenheimer –Volkov equation
Cooling Time
)/log( 0
T=100 keV
dif
fusi
on
tim
e (y
ears
)
Non-uniform condensate:
coherence length : vFF
distance between clusters: L
( a) L >> : ~ the case of uniform condensate
(b) L < : need of QRPA calculations !
Collective Modes in the Crust of Neutron Stars
Supergiant resonances in the crust of neutron stars
L=1
L=0
L=2
71% EWSR
QRPA
HFB
Effect on specific heat ?
Z=50 N=1750
E.Khan,N.Sandulescu,Nguyen Van Giai, Phys.Rev.C71 (2005) 042801R
Specific heat of collective modes
Z=50 N=1750
(N. Sandulescu et al, in preparation)
Mass-Radius Constraints from Cooling Time
tw= 10 years
No Superfluidity:
6.8 km < R < 8.5 km
Superfluidity:
9 km < R < 11.5 km
Lattimer et al, ApJ425(1994)802