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COLLECTIVE EXCITATIONS IN NUCLEAR HYDRODYNAMICS
W. Nawrocka
To cite this version:
W. Nawrocka. COLLECTIVE EXCITATIONS IN NUCLEAR HYDRODYNAMICS. Journal de
Physique Colloques, 1987, 48 (C2), pp.C2-75-C2-78. �10.1051/jphyscol:1987212�. �jpa-00226477�
JOURNAL DE PHYSIQUE
Colloque C2, suppl6rnent au n o 6, Tome 48, j u i n 1987
COLLECTIVE EXCITATIONS IN NUCLEAR HYDRODYNAMICS
W. NAWROCKA
Institute of Theoretical Physics, University of Wroclaw, ul. Cybulskiego 3 6 , PL-50-205 WrocZaw, Poland
Abstract - In the frame of extended Thomas-Fermi approximation dynamical deformation of the Fermi surface is taken into account. The monopole collective excitation frequencies are estimated with this correction.
Following the well known procedure [ l ] we can perform the transition from the quantum mechanical variational principle
to the collective variational principle 2
6 ( dt(s(:,t);(:,t) - H)
L
1
assuming that the wave function has the form [ 2 ]
9 = exp [i / d : s ( ; , t ) 0 ^ ( : ) I @(o(g,t))
where $ is the real determinant function,
and collective hamiltonian
where the collective potential energy
By introduction of a suitable assumption about the state equation E=E(p) the problem (2) can be solved. The state equation of the finite system have to depend on the bo- undary condition on the nuclear surface and on the corrections from the deformation of the Fermi surface. The bulk part of E ( p ) can be written as
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987212
C2-76 JOURNAL DE PHYSIQUE
where
T - k i n e t i c e n e r g y d e n s i t y e - p o t e n t i a l e n e r g y d e n s i t y .
P o t
K i n e t i c enerRy d e n s i t y can be e x p r e s s e d by t h e Wigner d i s t r i b u t i o n f u n c t i o n
Now, we assume t h a t t h e Fermi s u r f a c e d e f o r m a t i o n w i l l b e t a k e n i n t o a c c o u n t up t o t h e q u a d r u p o l e t e r m ( L 2 2 ) . I n t h i s c a s e [ 3 J [ 4 1
- +
where f i s t h e Wigner f u n c t i o n i n t h e s p h e r i c a l Fermi s u r f a c e c a s e , S i s t h e d i s p l a - cement f i e l d .
+ +
D e n s i t y p ( r , t ) t o t h e s e c o n d o r d e r i n S h a s t h e form
In t h e same a p p r o x i m a t i o n
- - - -
P , T , E V a r e c a l c u l a t e d w i t h f .
T r e a t i n g s and S a s v a r i a b l e we o b t a i n from -+ ( 2 ) t h e f o l l o w i n g e q u a t i o n s
where
and
i s t h e a d i a b a t i c c o m p r e s s i b i l i t y .
(In the sharp surface case p= p where p is the central density in the ground sta- te).
The equation (11) is the continuity equation, because p(l)=p-p in the linear in S + approximation is equal
and velocity field - v s .
'a-: a o
Let us limit the considerations to the monopole density vibrations of the nucleus with sharp surface.
In this case
p(r,t) = p (1 + n(r,t))O(R(t) - r) , (16)
where p - equilibrium density in the centre of the nucleus. In the equilibrium peq = p0O(R - r) , where R - nuclear radius in the ground state.
eq e q (17)
Now
in the linear in approximation.
Functions n and q are dependent:
what follows from the particle number conservation.
For the monopole vibrations the velocity field If is parallel to r and similarly the + displacement field $ is parallel to r. Therefore from (1 1),(12) -+ we have
Compressibility K ' differs from the adiabatic compressibility K:
This difference is due to the dependence of E (p) on the quadrupole Fermi surface de- formation. v
Now, we add the boundary condition on the nuclear surface [5]
JOURNAL DE PHYSIQUE
In our simple case
and
From (20) we can obtain
where a(t) - periodic function with frequency
r j l (kr)
( t ) = a t j (kr)O(Req-r) -F
01
0
R - r)1 ,
eq . eq and the displacement field
The boundary condition (22) can be rewritten in the form
where y = kR
eq '
Solution of the equation (28) and dispersion relation (26) provide the spectrum oE the monopole vibrations.
In comparison with the StaticFermi surface case we have two effects:
1" Compressibility renormalization K + K ' .
2" Renormalization of the C ,C coefficients in the boundary condition equation
0
