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 ( ) -
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
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.
.
Top Related