I) Pair condensation in nuclear systems - general properties - BCS, Gorkov and Bogoliubov equations II) Linear response with pairing correlations and continuum

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


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


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)


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


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


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 ?


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

Documents

I) Pair condensation in nuclear systems - general properties - BCS, Gorkov and Bogoliubov equations II) Linear response with pairing correlations and continuum