Vlasov eq. for finite Fermi systems

with pairing

Vlasov eq. for finite Fermi systems

with pairing

V.I.Abrosimov, D.M.Brink, A.Dellafiore, F.Matera

V.I.Abrosimov, D.M.Brink, A.Dellafiore, F.Matera

Kazimierz Dolny, September‘07

Outline: Outline:

Introduction

Semiclassical TDHFB eqs.

Extended Vlasov eq.

One-dimensional systems

Conclusions

Introduction

Semiclassical TDHFB eqs.

Extended Vlasov eq.

One-dimensional systems

Conclusions

.

Normal system (no pairing)Normal system (no pairing)

TDHF eq.

Normal Vlasov eq.

TDHF eq.

Normal Vlasov eq.

it [h,]

t(r,p,t) {h,}

.

Linear approx.Linear approx.

0( , , ) ( , ) ( , , )h t h h t r p r p r p

Linearized Vlasov eq.

t(r,p, t) {h0,} {h,0}

0h h

0( , , ) ( , ) ( , , )t t r p r p r p

Correlated sys.(with pairing)Correlated sys.(with pairing)

TDHFB eqs.

where

TDHFB eqs.

where

.

†( , ) | ( ) ( ) |

( , ) | ( ) ( ) |

r r r r

r r r r

,

( ) (

t

t

i h

i h h

Semiclassical TDHFB eqs.:

Semiclassical TDHFB eqs.:

.

( , , ) { , } 2 Im( )

( , , ) { , } Re{ , }

( , , ) 2( ) (2 1) { , }

t ev od

t od ev

t ev od

i t i h i

i t i h i

i t h i

r p

r p

r p

herehere

( )

1( , , ) ( , , ) ( , , )

2ev od t t t r p r p r p

from Wigner-transf. TDHFB eqs. where wehave kept only terms of first order in

from Wigner-transf. TDHFB eqs. where wehave kept only terms of first order in

Semiclassical TDHFB eqs.-Static limit Semiclassical TDHFB eqs.-Static limit (Bengtsson and Schuck (1980)) (Bengtsson and Schuck (1980))

A 4

(2)3drdp 0 (r,p)

where where

is related to by the energy gap eq.is related to by the energy gap eq.

the chemical potential is determined by the number of particles ,

the chemical potential is determined by the number of particles ,

0 ( , ) r p 0 ( , ) r p

0 (r,p) 1

21

h0 (r,p) E(r,p)

0 (r,p) 0 (r,p)

2E(r,p)(note different sign from B. S.!)

quasiparticle energy,quasiparticle energy,

Constant- approx.Constant- approx.

Approximate Approximate

0( , , ) ( , ) . t phenom pairing gap r p r p

then semiclass.TDHFB eqs.become

( , , ) { , } 2 Im( )

( , , ) { , }

( , , ) 2( ) (2 1)

t ev od

t od ev

t ev

i t i h i

i t i h

i t h

r p

r p

r p

h0 (r,p),

In static limit one finds the equilibrium solution:

In static limit one finds the equilibrium solution:

with the particle energy andwith the particle energy and

the quasiparticle energythe quasiparticle energy

Note that in the limit of , we have and .

Note that in the limit of , we have and .

0

0 ( ) 0 0 ( ) -

Equilibrium distributionsEquilibrium distributions

33MeV

= 1MeV

0

[MeV]

0

EXTENDED linearizedVlasov eq.

EXTENDED linearizedVlasov eq.

From semiclass. TDHFB eqs. follows From semiclass. TDHFB eqs. follows

t(r,p,t) {h0,} {h,0} 2

i (r,p,t)

t i (r,p,t) E 2 ( )

[(r,p, t) (r, p, t)]

d0

dh(r,p,t)

with

( , , ) ( , , ) ( , , ),r it t i t r p r p r p

(with constant-

approx.)

One-dimensional systems: zero-order solution

One-dimensional systems: zero-order solution

Zero-order approx.: Zero-order approx.: 2( , ) ( )exth V t t x r

˜ n n0 1( )

n 0

2

for n 0

˜ n ( ) 2E( )

for n 0

Note that eigenfrequencies of normal system:

Note that eigenfrequencies of normal system:

Eigenfrequencies:Eigenfrequencies:

where frequency with

where frequency with0

2( )

( )T

2

1

1( ) 2

( , )

x

xT dx

v x

0( ) ( )n n

Correlated zero-order propagator Correlated zero-order propagator A propagator is defined by A propagator is defined by

0 0 0 0 0 02 cos[ ( )] 1 cos[ ( )]( , , ) 2

( , ) ( , )n n n

d n n n x n xD x x d

d T v x i v x

gives

˜ M 1 M 1gives and

A(t) 0

24 4( , ) ( , ) ( , , )

2 2x xx dp x p dx D x x x

To ensure particle-number conservation and to eliminate the spurious strength, we expressdensity fluctuations in terms current density through continuity eq.:

To ensure particle-number conservation and to eliminate the spurious strength, we expressdensity fluctuations in terms current density through continuity eq.:

ThenThen

1 4( , ) ( , ).

2x

x x x

px dp x p

i m

Uncorr. vs. corr.Uncorr. vs. corr. Uncorrelated Eigenfrequencies

No gap No spreading

EWSR

Uncorrelated Eigenfrequencies

No gap No spreading

EWSR

Correlated Eigenfrequences

Gap Spreading

EWSR

Spurious strength is subtracted!

Correlated Eigenfrequences

Gap Spreading

EWSR

Spurious strength is subtracted!

n0

d0

d ( )

˜ n n0 1( )

n 0

2

for n 0

˜ n ( ) 2E( )

for n 0

20

3= smoother

2 ( )

d

d E

2

M 1 2

3 2 AL2

m

˜ M 1 M 1

Response functionResponse function

External field External field

2 24( ) ( , , )

2R dx dx x D x x x

2( , ) ( )extV t t x r

1( ) Im ( )S R

Small size:Small size: 0

0 10 MeV

dashed = uncorr.

full = correlated

= 0.1 MeV

[MeV]

S 0 ( )

Medium size:Medium size:

0 1 MeV

dashed = uncorr.

full = correlated

= 0.1 MeV

[MeV]

S 0 ( )

0

Large size:Large size:

S 0 ( )

0

0 0.2 MeV

dashed uncorr.

full correlated

0.1 MeV

Collective solutionCollective solution

Coll. propagator satisfies integral eq. Coll. propagator satisfies integral eq.

4( ) ( ) ( , , ) ( )

2R dx dx Q x D x x Q x

0 0( , , ) ( , , ) ( , , ) ( , ) ( , , )D x x D x x dy dzD x y v y z D z x Separable interactionSeparable interaction ( , ) ( ) ( )v y z Q y Q z

gives collective response function by gives collective response function by 0

0

( )( )

1 - ( )

RR

R

wherewhere

int( , ) ( , )exth V t V t r r

Collective response(medium size: )Collective response(medium size: )

0 1 MeV

dashed = uncorr.

full = correlated

= 0.1 MeV

S 0 ( )

0

Conclusions Conclusions Semiclassical TDHFB eqs. have been studied in a

simplified model, in which the pairing field is treated as a constant phenomenological parameter.

A simple prescription for restoring both global and local particle-number conservation is proposed.

We have shown in one-dimensional system that our model represents the main effects of pairing correlations.

It is of interest to extend the present method to three-dimensional systems. Work on this problem is in progress.

Semiclassical TDHFB eqs. have been studied in a simplified model, in which the pairing field is treated as a constant phenomenological parameter.

A simple prescription for restoring both global and local particle-number conservation is proposed.

We have shown in one-dimensional system that our model represents the main effects of pairing correlations.

It is of interest to extend the present method to three-dimensional systems. Work on this problem is in progress.

.