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NUCLEAR RESPONSE IN LANDAU VLASOV DYNAMICS
G. Burgio, M. Di Toro
To cite this version:
G. Burgio, M. Di Toro. NUCLEAR RESPONSE IN LANDAU VLASOV DYNAMICS. Journal de
Physique Colloques, 1987, 48 (C2), pp.C2-37-C2-43. �10.1051/jphyscol:1987205�. �jpa-00226470�
JOURNAL D E PHYSIQUE
Colloque C2, suppl6ment au n o 6 , Tome 48, juin 1987
G.F. BURG10 and M. D I TORO
Dipartimento di Fisica, Universitd di Catania, and Istitute Nazionale di Fisica Nucleare, Sezione d i Catania, Corso Italia 57 , I-95129 Catania, Italy
Be s u m 6
Nous pr6sentons une m6thode s6miclassique pour 6 t u d i e r les mouve- ments nucl6aire c o l l e c t i f s , d e v e l o p p g e B partir de l'equation de V l a s o v ,
avec tous les terms de autoconsistence. Nous avons inclus aussi les effects de la tem- perature et des collisions entre particules, qui sont assez importants pour expliquer les largeurs des r6sonances gcantes.
Abstract
We present a semiclassical method based on the Vlasov equation to study nuclear collective motions within a fully self-consestent approach. We include also temperatu- re effects and two-body collisions, which are quite important to account for the damp- ing widths of giant resonances.
The Vlasov equation represenra the natural semiclassical limit of a Time-Dependent- Hartree-Fock theory 1/ and therefore its linearized version is the right starting point for a phase space approach to RPA. Previous attempts along this line were limited to the study of solutions in the scaling approximation, able to reproduce only some bulk features of giant resonances. In ref. 2 / an approach to find a general solution of the linearized Vlasov equation has been worked out in the case without self-consistency, i.e. neglecting the effect of residual interactions. We extend here that procedure thmough the introduction of correlations, actually needed to enhance the collective behaviour of the response. The results show several striking similarities with fully quantum calculations and a pretty good agreement with experiments, but with the nice features of an extremely reduced numerical effort joined to a much clearer interpreta- tion: complicated nuclear spectra can be explained just in terms of few simple classi- cal quantities. All that induces to consider the nuclear Vlasov approach as a conveni- ent and reliable theory also in order to make quantitative predictions. Therefore we have introduced some temperature in the reference state in order to analyse the beha- viour of collective motions built on excited states. Finally we have studied the effect of two-body collisions, introduced through a relaxation time term, in competition with the Landau damping to account for the observed widths of giant resonances.
Small amplitude variations g(r,~,t) of the distribution function under an external driving field B(t)Q(z) satisfy the linearized Vlasov equation
ag/at {hO,g) + {&w + B Q , ~ J
(')presented by M. DI TORO
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987205
C2-38 JOURNAL DE PHYSIQUE
fo(r,p2) represents the distribution function of the reference nuclear state on top of which we study collective excitations, 6w is the selfconsistent variation of the mean field due to the residual interacti.on 6w = /d3r u(z,r')6p(r1,t).
Since 60 (r,t)= jd3Lg(r,p,t) it is clear that eq.(l) is a complicated integro- differential equation. However from (ho,Eo} =O we can choose fo=F(ho) which leads to the equation (Fourier transform in t)
Two points must be stressed: i) For a cold nucleus (ground state) a Thomas-Fermi approximation ~(h,)=(4/ (2~ti)~) 8 (EF-h,) implies that we will have contribu+.ions on- ly from particle orbits at the Ferm, energy, i..e. the Pauli principle is strongly enhancing the collective structure of the response; ii) Only Poisson brackets of ho are appearing and therefore if we can manage to change the variables (r,p) to a set with a large number of integrals of motion we will strongly reduce the-c~mplexity of
the problem.
For a spherical symmetric reference system the transformation (fl,~)+>(E,X,r,cr,B,Y) is very convenient 21, with E, energy,A, angular momentum, r, radius-.and (a, 13, Y) Euler . angles. Apart from E and 1, alsoaandBare constants of motion since the plane of the
orbit does not change and cos6-X7/h. - represents the .ankle variable on the orbit plane, with ?=X/mr2. The y-dependence can be easily accounted for using a rotation matrices expansion and finally eq. (2) reduces to a set of radial equations that can be exactly solved in the case without self-consistency 21, i.e. neglecting the 6w term. The uncor- related response function DL(r,rl ,w) ,defined by
06 p ( r , w ) = 6 ( u ) j d ' f l ' ~ 0 ( L , L 1 ,w)Q(<),
has the structure
Poles are in the positions un(N) = n2n/T + Nr/T , where n,N are integer numbers -W<n<m and -L 5N5
--- L, ((-1)~=(-1)~), with T=2 r1jr2 dr/v(r) radial period and
r=2r11r2 (~/mr') dr/v(r) angular "period" for each orbit (X,E) where we can find particles for given mass number, Fermi energy and umperturbed mean field wO(r).
v(r)aJ(2/tn) (E-wo(r)- h/2mr2) is the radial velocity and r1,r2 are the classical turning points of the orbit (X,E). The phases are defined by sn(N,r)=Un(N)T(r)-NY(r) where r(r)= rljr dr'/v(r') and y(r)= (X/mr'2)dr'/v(r') are respect,ivety the time elapsed and the angle spanned to reach the position r on the orbit (X,E).
The uncorrelated polarization operator rn =jr2dr jr12dr'~t(r)~f(r,r',w)Q(r') and the corresponding strength distribution L Sf(w) = - Imlle(w)/n can be easily constructed. The similarity to quantum particle-hole strength distributions is quite impressive 2/.
The correlated semiclassical response function satisfies a Dyson equation 21
To get a first evaluation of residual interaction effects we use separable forces, mostly active on the nuclear surface, which have been proven to be quite realistic
3 , 4 , 6 / : uL(r,rl)=k(L) F(r)F(rf). The separability condition leads to a simplified form
for the polarization lIL(w) 5 1 , from which we can get the new poles through dispersion relations and new corresponding strength functions. We use two different form factors:
i) F(r)=rL, in that case we get the dispersion relations of multipole-multipole
residual interactions, with k(L) given by a consistency condition 4 1 .
ii) F(r)=dV/dr, with k(~)=k~/J(l+(a~/~) ' ) . ko is fixed from the condition of zero energy for the spurious 1- isoscalar mode, and &lfm 6/.
In figures 1, 2, 3 we report the fraction of EWSR which corresponds to each eig- enfrequency for isoscalar 2 + , 3 - , 5 - modes in 4 0 ~ a and 208~b. The main effect of the introduction of residual interactions is to shift down the excit~tion energy and con- centrate the strength, closely to quantum calculations performed using an effective Skyrme-Gogny force 7/. Moreover our semiclassical distributions satisfie the Thouless theorem 3/ and reproduce the low lying collective states, which are very hardly seen in usual fluid-dynamical approaches. We still have some large strength concentration for high multipoles, particularly for the high energy state. This could be related to the fact that in our approach, with only bound orbits, we do not have escape widths.
The dV/dr form factor produces a residual interaction stronger than the multipole- multipole, with larger energy shifts and strength concentrations, with an overall better agreement with quantum results.
Fig. 1. EWSR fractlon for 2+ modes in 4 0 ~ a and ' O ' p b : a) no correlations, b) rL form factor, c) dV/dr form factor.
Another interesting chance given by the Vlasov approach is to study the nuclear
collective motions built on excited states, characterized by some temperature. In that
case the only change is the choice of a general T#O Fermi distribution for the referen-
ce state
JOURNAL DE PHYSIQUE
F i g . 2. Same as f i g . 1 f o r 3- modes i n 4 0 ~ a and B 0 8 ~ b .
Then we have contributions to the eigenfrequencies and to the strength functions not only from orbits at the Fermi energy, but also at neighbouringenergies. Thenet effect will be an enha~eLfragmentation of the uncorrelated strength distribution. However, like in the T=O case, the residual interaction prodkces a strong collectivity effect and a reduction of the fragmentation (see Fig.4). So the final strentgh distributions are not much affected by the temperature. We note two points: i) The coupling constants k(L) are kept fixed varying the temperature 8 1 ; ii) Using a dV/dr form factor, at high temperature we see an increase of the low energy strength for quadrupole transitions (see Fig.5), as expected from a smoothing of the reference ditribution function and a partial relaxation of Pauli effects. Moreover the coupling ko must be increased of about a 10% (at T=6 MeV).
Collective momentum distortiions are reducing the Pauli blocking allowing two-body collisions to take place. This effect is larger on the nuclear surface due to the de- crease of the Fermi Enrgy. If we include collision terms through a relaxation time approach, the Vlasov equation becomes
where 't: is the collisional time and
L,the new equilibrated distribution function.
The linearization of eq. (6) leads to complex frequencies 5 1 G,(N)=~,(N)-~/, and
to a very simple form of the uncorrelated strength function
Fig. 3 . Same as fig. 1 for 5- modes in 4 0 ~ a and 2 0 8 ~ b .
1
C
cn
C
z 2
Fig. 4.
0.4
EWSR fraction for 2' modes in 4Oca at T=3,6 MeV: a) no correlations, b) rL form factor
0.4
I l l ' 1 1 ' ' 1 '
- a> -
-
8
I
1I l l
I-
- a)-
0.2 0.0.
0.4
L
C
0 7
2 0.2-
L
iTi
0.0 0.4
Ener gy(MeV) E nergy(MeV)
-
0.0 0 00 20.0 . 40.0 60.0 ~ 0.0 2
-
-
- - 0.2
1.
