Nuclear collective motions in a self-consistent Landau-Vlasov approach

Physics A476 (1988) 189-212 Nonh-Holland, Amsterdam

Nuclear

NUCLEAR

COLLECTIVE MOTIONS IN A SELF-CONSISTENT LANDAU-VLASOV APPROACH G.F.

Dipartimento

di Fisicu, Uniuersitd

BURG10 and M. DI TORO

di Caiania

and INFNSezione Italy

di Catania

57, Corso Iialia-95129

Cataniu,

Received 28 April 1987 (Revised 3 August 1987) Abstract: A semiclassical theory of nuclear collective motions based on the Vlasov equation is extended

to take into account self-consistency effects provided by the residual interaction. Using separable forces a correlated response function is derived which has several striking similarities with fully quantum results. Explicit spin-orbit terms are also included through a clear WKB correspondence with the quantum response. Finite temperature effects are analysed showing little change in the strength distribution. Through a relaxation time approach two-body collision terms are introduced. This is essential to account for the damping widths of isoscalar quad~pole and octupoie giant resonances. However a quite important interplay between self-consistent (Landau damping) and collisional damping is also revealed.

1. ~ntr~uctjon

A phase space description of nuclear many analyses in the last decade I-‘). The

main

motivations

are related

collective

motions

to the fact that

has been

the quantum

the subject random

of

phase

(RPA), although quite successful in nuclear physics, is complicated to work with and not very transparent in the results. In particular it is very difficult to answer some important questions emerging also from recent experimental data: (if Structure of collective modes built on excited states (temperature and angular momentum effects); (ii) Interplay between one-body dissipation (Landau damping) and explicit twobody collisions to account for the damping width of giant resonances of different multipolarities; (iii) Relation between collective motions and the structure of the residual interaction. The Vlasov equation represents the natural semiclassical limit of a time-dependentHartree-Fock (TDHF) theory6) and therefore its linearized version should be the right starting point for a phase space approach to RPA. Previous attempts along this line ‘*475)were limited only to the study of solutions in the scaling approximation, which were reproducing some bulk features of giant resonances. approximation

0375~9474/88/%03.50 @ Elsevier Science Publishers (Nosh-HolIand Physics ~blishing Division) January

1988

B.V.

190

G.F. Burgio, M. di Toro / Nuclear collective motions

In ref. ‘), in the following called (I), an approach to construct a general solution has been introduced in the case without self-consistency, i.e. neglecting the effect of the residual interaction. Since the results were very promising, we have extended the method with the introduction of correlations, which are actually needed to get collective states. The main result is that the nuclear response for small amplitude oscillations in the Vlasov theory shows several striking similarities with fully quantum calculations and quite good agreement with experiments. Moreover we have the nice features of an extremely reduced numerical effort joined to a much clearer interpretation: complicated nuclear spectra can be explained just in terms of a few simple classical quantities. The residual interaction, introduced through separable forces, strongly enhances the collectivity of the response. However, for high multipolarities (La 4) we see a persistence of the strength fragmentation, which means that the Landau damping is quite important for such modes, also in agreement with some indications of fully quantum RPA results. The introduction of an excited reference state, on top of which we consider the collective motions, at temperature T # 0, is quite straightforward in the Vlasov approach. While the uncorrelated response shows a larger fragmentation, we still see a strong collectivity after the introduction of residual interactions, with a slight increase of the low energy strength, as expected from the smoothing of the reference distribution function. Explicit particle-particle collisions are considered through a relaxation time approach which leads to an intrinsic two-body collisional width. However, the final result of our self-consistent analysis shows that we end up with much larger widths for the strength distribution in giant resonance regions. This seems to indicate the existence of a quite complicated interplay between one-body and two-body dissipations. A systematic analysis of the isoscalar quadrupole and octupole widths shows a quite good agreement with experiments, provided that a dependence upon multipolarity is supposed for the collisional time. In sect. 2 we remind the most significant results of (I) needed for the extension of the method. In sect. 3 we analyse the self-consistency problem using separable interactions with different form factors. Sect. 4 is devoted to the semiclassical correspondence with quantum RPA, used to take into account spin-orbit terms in the mean field. In sect. 5 we consider temperature effects and in sect. 6 we extend the theory to the inclusion of collision terms. Finally some conclusions are drawn in sect. 7. 2. Reminder of the theory In (I) an almost analytic method is derived to solve the nuclear Vlasov equation for small amplitude oscillations, in the case without particle-hole correlations. Before

191

G. F. Burgio, M. di Toro / Nuclear collective motions

extending

the approach,

it is important

to review the general

formalism

and some

of the main results. The Vlasov equation

for a system

under

an external

driving

field p( t)Q(r)

can

be written:

(2.1) with h( r, p, t} = ( p2/2m) + W(r, p, t), where W( r, p, t) is the self-consistent For small amplitude variations of the distribution function:

potential.

(2.2)

f(r, p, r) =_Mr, p’) + g(r, p, t) , we can expand

eq. (2.1) to the first order

which is the semi-classical

correspondent

in g. In this way we develop

a theory

of RPA. We get: (2.3)

where 6W is the self-consistent interaction u( r, r’):

variation

6W(r, t) = and 6p transition Since

d3r’ u(r, r’)Sp(r’,

field due to the residual

t)

(2.4)

density.

G(r, it is clear that eq. (2.3) is actually Of course

of the mean

the structure

r) =

d3pg(r,

(2.5)

p, t)

a complicated

integro-differential

of the equilibrium

solution

equation.

f0 is quite

represents the reference nuclear state on top of which we study collective Since {ho, fo} = 0 we can choose: fo(r, P’) = F[Mr, which implies

the equation, -iwg(r,p,

In a semiclassical picture nucleus (ground state):

for the Fourier

we choose

of g(r, p, t):

transform

SW+PQl.

a Thomas-Fermi

F(ho)= 40(EF(zTh)3

f0

excitations.

(2.6)

p2)1

w) ={ho, gl-F’(E){ho,

important.

approximation

(2.7) for a cold

ho)

and therefore from eq. (2.7) we see that we should expect to have contributions only from particle orbits at the Fermi energy, since F’(E) - S(& - E).

G.F. Burgio, M. di Toro / Nuclear collective motions

192

From this discussion it is also clear how straightforward it will be an extension to hot nuclei, i.e. collective oscillations about excited states. We must consider in (2.6) a general T f 0 Fermi distribution and therefore more orbits will participate in the collective motion, as discussed in detail in sect. 5. From the structure of eq. (2.7) we see that if we introduce some change of variables to a new set which includes constants of motion of ho, we can reduce the number of independent variables. Assuming a spherical symmetry for the reference state, in (I) the transformation: (2.9)

(r, P) --, (C A, r, a, P, Y)

has been suggested, where E, energy, A, angular momentum, r, radius and (cy, p, A), Euler angles. Apart from E and A, also (Yand p are constants of motion since the plane of the orbit does not change and cos /3 = (AZ/A). y represents the angle variable on the orbit plane, with 9 = A/ mr’. The y-dependence can be easily accounted for by using a multipole expansion:

g(E, A, r, a, P, Y, w) = C G~N(E,A, r, ~)(Dhd~, P, Y))*

(2.10)

LMN

with DhN rotation matrices, and eq. (2.7) is reduced to a set of radial equations for the functions G$N( E, A, r, u). The general response function D(r, r’, w) defined by: (2.11)

d3r’ D(r, r’, u)Q(r’)

6p(r, w) =p(w) I

can be expressed, for the L-multipolarity, in terms of the GLN solutions [see eq. (5.20) of (I)]. Therefore from the knowledge of the GbN we can construct the strength function associated with any operator of multipolarity L and radial dependence Q(r): rr2 dr’ Q*(r)DL(r,

r’, o)Q(r’)

= -i

(2.12)

Im n,(o)

where 17,(o) is the polarization propagator. In (I), solutions without correlations, i.e. without self-consistent variation SW of the mean field, are explicitly worked out. The corresponding response function has the structure:

with: d,LN(r ry =g( ,

-20;(N)),

y,,&,

;#CoS

$f(;

r)3 ‘OS $“;;)r’)l

.

(2.14)

G.F. Burg@ M. di Toro / Nuclear collective motions

193

Poles are in the positions: (2.15) where n, N are integer numbers -cc < n < co and -L c N s L, (( - ) N = ( - )L), with:

radial period and

angular “period” for each orbit (A, E) where we can find particles for a given mass, Fermi energy and unperturbed mean field. The quantity: (2.16) is the radial velocity and (r, , r2) are the classical turning points of the orbit (A, E). The phase s, (IV, I) is given by: s,(N, r) = W,(N)T(r) - W(r),

(2.17)

where:

are respectively the time elapsed and the angle spanned to reach the position r on the orbit (A, E). The radial dependence of the semiclassical D-function, eq. (2.14), shows singularities in r, r’ at the positions of classical turning points. This is at variance with the quantum case 8S9)and implies some difficulties in using a zero-range residual interaction, as shown in the following. Finally from eq. (2.12), (2.13) an uncorrelated strength function can be easily obtained as:

al

dEF’(E)

dh Aw,(N)TJYLN(f~,TT,~)l*)Q(n, N)~*~(oJ-w,(N)),

(2.18)

194

G.F. Burgio, M. di Toro / Nuclear collective motions

where the residues

at the poles are mainly

given by: (2.19)

In (I) uncorrelated strength functions for isoscalar monopole, quadrupole and octupole modes in 40Ca and *“Pb are shown, derived from a Woods-Saxon unperturbed field with standard parameters. The similarity to quantum p-h strength distributions is quite impressive. We remark that we have solved the problem within a fully classical dynamics, the only quantum effects being the Pauli principle included in the static distribution through eq. (2.8). Indeed the Pauli principle seems to be determinant in order to have a collective behaviour in the nuclear many-body response. The eigenfrequencies, eq. (2.15), and the residues, eq. (2.19), are provided by each orbit (A, E) with an almost uniform distribution. The Pauli effect, through the structure of F’(E) peaked at the Fermi energy, selects only contributions from orbits around EF leading to a collective structure of the response. This is a hint towards an understanding of the damping of collective modes as a partial relaxation of Pauli blocking which will allow two-body collisions with a breaking of the collective flow, as discussed in sect. 6.

3. Correlated results: A semiclassical The semiclassical response equation [for L-multipolarity, DL(r, r’, w) = D;(r,

r’, o)+

function eq. (2.11) see sect. 5 of (I)]:

J J x2 dx

y* dy Di(r,

RPA

satisfies

x, w)u,(x,

a RPA-type

y)D,(y,

integral

r’, w)

(3.1)

This is the starting point to introduce self-consistency effects due to the action of the residual interaction. From the structure of the singularities of Di, eq. (2.14), it is clear that a zero-range residual interaction, e.g. of Skyrme type, will imply uncurable singularities in the kernel of eq. (3.1). This is a drawback but also quite interesting because it shows at very microscopic level some intrinsic limits of a semiclassical

description

of a many-body

quantum

dynamics.

Moreover

it seems to

suggest, in order to improve the approximation, the use of some technique to overcome the classical turning points singularities. On the other hand the use of separable interactions, mostly active on the nuclear some consistency surface, has been proven to be quite realistic 6,10), provided requirements between the variation of the mean field and the local density are satisfied. Indeed nuclei are nearly incompressible drops with a potential field determined by the local density. For a general separable interaction, with any form factor, the polarization propagator (eq. (2.12)) can be algebraically constructed just from the knowledge of D”( r, r’, 0).

G.F. Burgio,

M. di Tore / Nuclear

195

collectiue motions

Let us define: uL(r, r’) = K(L)F(r)F(r’) and the general

(3.2)

integral: y* dy A(x)D,(x,

x2 dx 5

y, w)B(y)

= (ADLB) .

(3.3)

I

From eq. (3.2) we get (QLDLOL)=(QLDOLQL)+K(O(QLDOLF)(FDLQL) with

(FDLQL) = (FDO,Qd + K(L)(FD:F)(FDLQL) and therefore

a polarization

propagator:

n,(w) = (QLDLQL) =

IT;(w)[l

-K(L)(FDOLF)]+K(L)(QDOLF)’ ,

1 -K(L)(FDO,F)

(3.4)

where we have also made use of the separable structure, eq. (2.14), of the uncorrelated response function D”,( r, r’, w). From eq. (3.4) we get the new poles 0, through the dispersion relations:

dx Jx2 J

-&=(FD:F)= with a strength

function

Y* dy Hx)D’i(x,

y, w)F(y)

given by the residues:

~~(QLD’$> “(‘,) In the particular

(3.5)

= (d/dw)(FDO,F)

case of a form factor

relations of multipole-multipole The consistency condition

,sn,

’

(3.6)

F(r) = QL(r) = rL we get the dispersion

residual interactions. for incompressible displacements

L # 0 (isoscalar

modes): t$ = #Y,,(i)

SP = V(PV4),

(3.7)

and:

d$p leads to a factorized

structure

of the residual

(3.8) interaction:

uL(r, r’) = fc(L)rL-laVar Y=,(i)r

+‘&;Y,,(P)

(3.9)

with 1 K(L)=

J

r

r2L

avap ar’sdr’.

(3.10)

196

G.F. Burgh,

As a particular

case for harmonic

multipole-multipole consistency

M. di Toro / Nuclear collective motions

condition

residual

oscillator

interaction

field

V =fmw$’

with the well known

we recover

Bohr-Mottelson

the (BM)

lo): 1

(3.11)

2L+ 1 1: p(r)r2L dr

In our calculations we start, as in (I), from a Woods-Saxon field with standard parameters ( V, = -50 MeV, diffuseness a = 0.65 fm, r, = 1.25 fm). Therefore we consider two residual interactions: (i) Multipole-multipole type with BM condition Saxon form for the local density; (ii) A form:

(3.11) extended

to a Woods-

(3.12) with K(L)

(3.13)

=

\/l+(Z,R)2 and K,, fixed from the condition of zero energy for the spurious l- isoscalar mode. This allows to overcome the incompressible displacements ansatz, eq. (3.7), which is in contrast with the very general transition densities we are going to derive from the Vlasov dynamics. The Ldependence of the coupling constant K(L), eq. (3.13), comes from a finite range ((Y- 1 fm) angular structure of the interaction. Interaction of this type has been used with success to study surface response in nuclei in a fully RPA analysis of semi-infinite Fermi liquids ‘I). In figs. l-8 we report the fraction corresponds

to each eigenfrequency

of the energy-weighted for isoscalar

sum rule (EWSR)

2+, 3-, 4+, 5- modes.in

which

40Ca and

208Pb. In each figure we plot the response without correlations (a), with a multipolemultipole residual interaction with BM consistency (b), and finally with a residual interaction of the type eq. (3.12), (c). In table 1 we list the values of the EWSR fractions corresponding to the more collective states. Our results deserve some comments: 1) The energy-weighted sum rule for isoscalar one-body operators is very well conserved going from the uncorrelated to the correlated response. 2) The residual interaction strongly enhances the collectivity of the modes. This effect is less important for higher multipoles, starting from L = 4: also in the Vlosov theory the strength of multipole operators becomes gradually more fragmented with increasing L [ref. “)I. 3) We clearly see also low-lying collective states, which are usually missing, or hardly seen, in the fluid dynamical approaches 3,4) where one is forced to reduce the complexity of the time-dependent distribution function in phase-space, especially

G.F. Burgh,

M. di Toro / Nuclear

197

collective motions

d)

0.8 0.4

00 i,-

oo,.. 0.0

I 100

I 200

II

i 30.0

400

EnergyCMeV)

Fig. 1. Fraction

(a) no correlations; of EWSR for isoscalar 2+ modes in %a: residual interaction; (c) dV/dr form factor.

0.8 -

I

I

I

1

(b) multipole-multipole

’

a) -

EnergyNeW

Fig. 2. Same as fig. 1 for 3-

with respect to the distortions of the Fermi surface. for low-lying collective states is large and strongly of the mode, in good agreement with experiments. 4) Our results are strikingly similar to that of quantum RPA calculations with Skyrme-Gogny concentration of the strength for high multipoles states. This could be related to two points:

modes. The fragmentation of the strength increasing with the multipolarity complicated fully self-consistent forces 13). We have some larger particularly for the high energy

198

G.F. Burgio, M. di Toro / Nuclear collective motions I

I

I

I

’

a)_

0.4-

0.2-

0.0.

%. I

0Il.‘.. 1

1

0.4-

cl_

0.20.0

s1. I c 1,. tn.* ,, 1 0 ’ I 0.0 10.0 20.0 30.0 LO.0 500 EnergyCMeV)

Fig. 3. Same as fig. 1 for 4’ modes.

Note the change

,119

I

r

”

in the vertical

scale.

I

a)-

0.40.2-

1

0.0. ,.“.I...

5 JL1.. h ”

” CL

0.402-

I

o.o_L.L

00

Im1.L I 1‘1j

20.0

LOO

600

Energy(MeV)

Fig. 4. Same as fig. 1 for 5- modes.

Note the change

in the vertical

(i) In our calculations we do not have escape widths orbits are bound. This effect could imply a large spreading (ii) In the quantum calculations, also without continuum a truncation in the reference single-particle basis which high multipoles in the high energy region. 5) The separable interaction with a d V/dr form factor the multipole-multipole. Indeed it implies larger energy

scale.

since all the contributing of high energy strengths; coupling, there is always implies less strength for is in general stronger shifts and strength

than con-

G.F. Burgio,

hf. di Tore / Nuclear collective motions

0.8

04 00 0.8

r 5 2

II

a)

b)

04 00 0.8

5

199

c)

0.4:

o.oi..., 00

100

200

300

40.0

Energy(MeV)

Fig. 5. Fraction of EWSR for isoscalar 2+ modes in “‘Pb. . (a) no correlations; (b) multipole-multipole residual interaction; (c) dV/dr form factor.

L&LA 00

10.0

200

30.0

400

EnergyCMeV)

Fig. 6. Same as fig. 5 for 3- modes.

centrations. It is also clear that the dV/dr force gives results which are much closer to fully self-consistent calculations. Probably in the multipole-multipole interaction there are too many ansatz’ to get the right consistency (harmonic oscillator field, incompressible displacements). In conclusion of this section we would like to stress how reliable seems to be a semi-classical theory of nuclear excitations based on the Vlasov equation. We

200

G.F. Burgio, M. di Toro / Nuclear collective motions I

7

I

0.4-

a)_

0.2-

b)_

0.L -

.C 6 6 0.25 0.0

I. I ,I

L ..UY c,

OL0.20.0 .II~‘I&luL.L 00

10.0

. ..I. I ..I. 20.0 300 40.0

Energy(MeV)

Fig. 7. Same as fig. 5 for 4+ modes.

I

Note the change

scale.

I

I

4

04

:‘,.,,, 0.0

in the vertical

100

200

300

40.0

Energy(MeV)

Fig. 8. Same as fig. 5 for 5- modes.

Note the change

in the vertical

scale.

understand several features of nuclear spectra just in terms of few simple classical quantities, performing very reduced numerical calculations (our semiclassical RPA code requires no more than about 200 statements). One could argue that explicit quantum corrections to the Vlasov equation, of h2 type 436), might change the structure of the response. That seems not to be true. Apart from the good agreement with fully quantum calculations, in ref. r4) a quantitative evaluation of these effects is performed following an extended fluid dynamical

G.F. Burgio, M. di Toro / Nuclear collective motions

201

TABLE 1 Energies and EWSR fractions of the most collective states in %a “sPb evaluated with a multipole-multipole residual interaction. brackets the corresponding values using a d V/dr form factor.) A” %a

2+ 3-

4+

s-

“sPb

2+ 3-

4+

5-

procedure

E [MeV] 21.11 8.83 11.28 34.27 35.42 37.3 21.11 21.43 23.98 25.2 47.97 49.4 11.86 32.42 59.84 62.76

‘). For the octupole

%EWSR

(17.68) (4.82) (32.27)

(16.66)

(45.89) (10.94) (29.29) (59.07)

13.86 (11.34) 7.51 (4.13) 13.94 22.27 (20.28) 10.13 (7.09) 15.31 (14.57) 34.4 (28.52) 7.64 (6.78) 17.8 (17.11) 22.87 38.02 (36.74)

to solve the new phase space equation

of the interaction

and (In

88 (95) 5.8 (25.57) 6.5 28 (60.25) 25 17 7.19 (42) 5.98 8.43 5.39 16 (34) 31 2.35 (3) 30.77 (32.52) 11.86 (10) 20 84 (80) 8.24 (5.28) 11 66 (50) 2 (6.18) 11 (5) 48 (41) 2.76 (6) 6 (9) 9.38 36.55 (31)

with a macroscopic

parametrization

case, which is the lowest multipole

response

that could be affected, the corrections are of the order of 1% for the low-lying resonance energy and are completely negligible for the high energy mode. This conclusion seems to be at variance with some recent results on dynamical quantum effects for octupole states obtained by Kohl et al. 27). However the model hamiltonian used by these authors, harmonic oscillator mean field plus octupoleoctupole residual interaction, induces an overestimation of such corrections for the low-lying octupole mode. Indeed the absence of surface terms implies an exactly zero frequency for the low-lying motion in the semiclassical approximation [see eq. 6.444 of ref. ‘“)I. The use of more realistic mean fields with surface effects naturally leads to some low energy octupole strength, as shown here and in ref. 14), which is only slightly influenced by quantum corrections. From this discussion we

202

G.F. Burgio, M. di Toro / Nuclear collective motions

learn that one should use with some caution nuclear models since impo~ant physical effects can be masked by quantum corrections, with some renormalization of the coupling constants for the multipole-multipole residual interaction.

4. Inclusion of spin-orbit coupling There is a complete correspondence between the structure of the Vlasov theory and a quantum RPA. Some points are already discussed in (I), but a thorough analysis, based on a WKB Iimit of the RPA response, can be found in ref. “). In this section, we show how to use such semiclassical correspondence to evaluate typical quantum effects, like the spin-orbit coupling. In the Vlasov response without spin-orbit, the uncorrelated frequencies are: w,(N)=

nwo(A)+Nw,(h),

(2.15)

where w,, = 2rr/ T is the frequency of radial motion and wy is the precession frequency in the orbit plane, for each occupied orbit (A, E = I$). The corresponding strengths (residues) are given by the integrals:

I

an, N)=f ‘2

dr,

I1

-CL&r’) v(r’)

cos [sn( N, r‘)]

(2.19)

with

The uncorrelated quantum particle-hole response function shows poles in the p-h energies with strengths given by the p-h matrix element (n,l,lQjn&,). The WKB limit of this quantity becomes ‘s*‘6): (4-l)

with gb(Ah,AI, r)=[(np-nh)WO+(1~-Ib)Wy17(r)-(E,-I,)y(r)

(4.2)

while the p-h energies assume the form:

E,-E,=(n,-nh)hwo+(t,-I,)hw,.

(4.3)

Therefore the WKB limit of the quantum response naturally leads to the Vlasov response provided the correspondence: np-n,,@h,

I,-l,,++N

(4.4)

is introduced. As discussed in detail in ref. 15),we can use the WKB matrix elements and energy differences to evaluate spin-or1 ; effects in the Vlasov theory.

G.F. Burgio, M. di Toro / Nuclear collective motions

If the single-particle

hamiltonian

contains

a spin-orbit

v,, = -a(r)h21* the WKB limit of the p-h matrix

element

(n,l,j,lQ(r)ln,l,jh)=f

with the modified

203

term:

s

(4.5)

becomes:

rz~r$f-$cos I r1

4(An,Al,

Aj, r)

(4.6)

phase:

(4.7) where

r

6(r)=

dr’ --a(r)),

,, dr’)

A = 2S(r,),

respectively analogous to I, y(r) and T, K Correspondingly in the Vlasov theory we should

expect to see new frequencies

at: (4.8)

and residues

given by

cos[SAN, M Q(%NM)=~ Irr--$Q(r)

r)l

(4.9)

with s.(N,M,r)=(nwo+N[o,+2*~]-M~A}~(r)-Ny(r)-[2N-M]16(r), (4.10) where M assumes

the values: M=N-l,N,N,N+l.

To evaluate

the effect we have chosen

(4.11)

lo):

v,, = -0.1 hR, * (I * s) ) hRo = 41 A-“3

MeV .

(4.12)

Fig. 9 shows the Vlasov strength distribution for isoscalar quadrupole excitations in 4oCa. The uncorrelated response is distributed over more levels since now we have more eigenfrequencies, eq. (4.8). The introduction of residual interaction, of

204

G.F. Burgio, M. di Toro / Nuclear collective motions

00

00

I

I

I II.L 11

10.0

20.0

30.0

10.0

Energy (MeV)

Fig. 9. Fraction

of EWSR for isoscalar 2+ modes in @Ca with spin-orbit multipole-multipole residual interaction.

term: (a) no correlations;

(b)

quadrupole-quadrupole type with BM consistency, strongly reduces the fragmentation to a final response not much different from the case, fig. 1, without spin-orbit. In a sense this further justifies the comparison, made in the previous section, with fully microscopic quantum results.

5. Temperature effects on collective

motions

The reliability of the nuclear Vlasov response induces to use this extremely simplified theory in a wider context. In this section we will analyse the structure of collective modes built on excited states, which are characterized by some temperature. Many data of this kind are now available from fusion and deep inelastic heavy ion reactions i7). As already pointed out in sect. 2, the introduction theory is straightforward: it corresponds in changing eq. (2.8), for the reference

state in a general 4

‘(“P2)=(27rh)3

of temperature in the Vlasov the step distribution function,

T # 0 Fermi

distribution:

1 (5.1)

l+exp[(E-E,)/T]’

The energy derivative which enters in the strength distribution (see eq. (2.18)) now is not a delta function at E = EF, but a smooth curve peaked at EF with a width of the order of T. This means that we will have uncorrelated eigenfrequencies not only from orbits at the Fermi energy (for a given angular momentum) but also at neighbouring energies with strengths suitably weighted by the factor: W(E)

exp [(E - &)I =L T {1+exp[(E-E,)/T]}2’

Tl

The net effect will be an enhanced fragmentation of the uncorrelated distribution. This indeed is the main feature observed in the uncorrelated

(5.2) strength response

G.F. Burgh,

I

9

M. di Toro / Nuclear collective motions

I

’

!

0.8-

_

a a)--

Tz3MeV

205

b)

__

Tz3MeV

0.b

‘,I’-

.c b

0.0

1

t iiiL

0.8-

T=GMeV

1

1

1 18

I --

I

IL.,

/

I

T=6MeV I

ot00

00

I 100

I

I

200

30.0

00

100

I a,, “.. I 200

I

300

LOO

Ener~(MeV)

Fig. 10. Fraction of EWSR for isoscalar 2+ modes in 4oCa at T= 3, 6 MeV: (a) no correlations; (b) multipole-multipole residual interaction.

0.4

F-’ ’ -r T=3MeV

0.21 It: 57 it

0.0

2

OL-

1 J4I

I

c I I.ll

‘.m.b.

1

1 k.

T:GMeV

T=GMeV 02 0.0 ~~.~. 00 100

t 200

llll’kl~ 30.0

L h*.. 0.0

100

t 200

’

i 300

400

EnergyiMeV)

Fig. 11. Same as fig. 10 for 3-

0.0

10.0

200

modes.

30.0

Note the change in the vertical scale.

0.0

10.0

20.0

300

400

Energy(MeV)

Fig. 12. Fraction of EWSR for isoscalar 2+ modes in “‘Pb at T = 3, 6 MeV: (a) no correlations; (b) multipole-multipole residual interaction.

G.F. Burgio, M. di Toro / Nuclear

206

collective motions

calculated for quadrupole and octupole modes in 40Ca and *‘*Pb at temperatures T = 3, 6 MeV (see figs. lo-13a)). We have assumed the Fermi energy unchanged, which is quite reasonable in the limit: exp(-E,/T)Sl. For each angular momentum A we have chosen the contributions, suitable weighted, from EF-4T to EF+4T at intervals AE = T However when we switch on the residual interaction, either of multipole-multipole or dV/dr type, we observe a strong collectivity effect and a severe reduction of the fragmentation, see figs. lo-13b) and fig. 14 (dV/dr form factor). This result is indeed in quite nice agreement with fully quantum RPA analysis with realistic forces 18,19).We also see an increase of the low-energy strength for quadrupole transitions (see in particular fig. 14), as expected from a smoothing of the reference distribution function and a partial relaxation of Pauli effects. r’

0.4-

1’

t

1

I

a) --

T= 3MeV

c

/

7

I

’

b)_

T=3MeV

0.2f F

E 5

0.0

I.. 1 I I_

o,~_

T =GMeV

__

::.,I 0.0

100

I._

I J. I I

20.0

0.0

300

I .LL_l_l

T=GMeV

10.0

200

30.0

LOO

Energy(MeV)

Fig. 13. Same as fig. 12 for 3- modes.

Note the change

in the vertical

scale.

Energy(MeV)

Fig. 14. Fraction

of EWSR

for isoscalar

2+ modes in 40Ca at T=3.6 interaction.

MeV with a dV/dr

residual

G.F. Burgio, M. di Toro / Nuclear collective motions

In the analysis

we have kept fixed the coupling

multipole residual ansatz is probably

parameters

of the multipole-

interaction, without any explicit temperature dependence. This justified at low temperatures for isoscalar modes *‘). At higher

T-values the problem

deserves

at T # 0, of the best coupling invariance,

207

we have found

further investigations. constant

~~

However

as a result of a search,

(eq. 3.13) which restores

a small temperature

dependence,

the translational

about

10% increase

for T = 6 MeV.

6. Collision

terms and widths of giant resonances

In sect. 3, we have seen how the fragmentation of the strength distribution is extremely reduced for low multipolarities, L = 2,3, once we consider the residual interaction. This means that for these isoscalar modes the Landau damping (onebody dissipation) alone is not adequate to reproduce the width of the collective states, in agreement with the other analyses “). The Vlasov approach seems to be very convenient to study the interplay between one-body and two-body dissipation, since we can naturally extend it to include collision collision

terms, reaching integral 22).

the Landau

kinetic

equation

with a Uehling-Uhlenbeck

A simplified way to take into account particle-particle collisions is the relaxation time method where one term is added to the Vlasov equation, which describes the drive of the distribution towards a new equilibrium due to the effect of collisions, and it is ruled by a characteristic collisional time. This approach has been shown to be quite reliable to describe collision effects in medium energy heavy ion dynamics 23) and can be easily included in our solution of the linearized equation. The Vlasov equation

with collision

terms has the structure:

(6.1) where T is the collisional time and f$ is the new equilibrated distribution If we start from a cold system, fg will represent some Fermi distribution

function. at finite

temperature. The choice off: is also related to the condition that mass, momentum, kinetic energy conservations are not violated by the collision term. The linearized eq. (2.2) now assumes the form:

(6.2) where we are neglecting the term (fo-fg)/7 since the excitation energies involved can imply only relatively low temperatures. It can be immediately shown that the presence of a linear term in g leads to complex eigenfrequencies (see the structure of the eqs. (4.6), (4.7) and (4.8) of (I))

G.F. Burgio, M. di Toro / Nuclear collective motions

208

and complex

residues.

The new uncorrelated

poles are at:

;.(N)=n~+N~_f=w,(N)-l

(6.3)

7

with residues: &n,N)=A+iB

(6.4)

with 12 dr,

QLM(4 cos [s,(

____

N, r’)] cash

v(r))

r1 ‘2 dr, QLM (4

sin [s,(N,

7

,

(6.5)

r’)] sinh

We can easily check that for 7 + 03, no collisions, (A+ Q(n, N), eq. (2.19), and B+O). The free polarization propagator is now:

and therefore

r(r)

-

v(r’)

r1

=

11

we recover the previous

a~(~)+ @O,(o) we get an uncorrelated

s:(w) =

strength

-$3:(w)=g

In the case with collision

distribution: dE F’(E)

dh A(YL,&+)]*TL

(A’+

B2)m/7

T (w-o,(N))*+l/T*’

terms we have constructed

the self-consistent

using a multipole-multipole residual interaction. The form of the correlated gator, eq. (3.4), for that particular separable interaction is reduced to:

n,(w) = and therefore a correlated

results

introducing strength

l-

of n”,(w),

propa-

(6.8)

K(L)I;I;(W) structure

response

eq. (6.6), we finally get

distribution: 1

%(@) = -;

the complex

C(m)

(6.7)

SOL(w) Irn nL(w) =(I -K(Q(Y”L(W))‘+K(Q’&jF(U).

(6.9)

Therefore using eqs. (6.6), (6.7) and (6.9), we can easily construct all the responses once the relaxation time T is fixed. Finally from the behaviour of SL(w) in the giant resonance region we can evaluate the total damping width of the collective mode. In fig. 15 we show the two strength distributions, without and with correlations, for the isoscalar quadrupole in 40Ca obtained just choosing a relaxation 7 such that r = h/r = 1 MeV. We clearly see that the final width in the giant resonance region

G.F. Burgio, M. di Toro / Nuclear collective motions

209

Energy(MeV)

Fig. 15. Fraction

of EWSR for isoscaiar 2+ modes in ‘?Za with a collisional width correlations; (b) multipole-multipole residual interaction.

h/r

= 1 MeV: (a) no

is much higher, of the order 3-4 MeV. This seems to indicate that the final damping of the collective state is actually due to an interplay between two-body and one-body dissipation, which enhances the anti-coherence effect of the collisions. The main problem is the choice of the relaxation time for a finite cold Fermi system, in particular for giant modes. Two effects are contributing: (i) The finiteness of the nucleus imposes a smearing out of the Fermi momentum distribution on the nuclear surface 24). The Pauli blocking effect is very much relaxed on the surface allowing collisions to take place also at very low energy 25*26). (ii) Collective distortions in the momentum space are also increasing the probability of two-body collisions. Indeed let us consider quadrupole-type distortions, which are quite well suited to reproduce the momentum collective behaviour of L = 2,3 isoscalar giant resonances [scaling approximation, see refs. ‘-“)I. Nucleons sitting in the prolate part of the distribution can collide since the final momenta are not completely forbidden by the Pauli blocking, as in the case of spherical distributions. This effect, which is L-dependent

through

the amplitude

of the oscillation,

also more important on the nuclear surface where the Fermi momentum and the relative weight of the oscillation increases. From the previous points we can assume for h/7 a mass dependence to the relative weight of the surface region in a nucleus: .

h/7 = r,( L)Ap”3

will be

decreases

24)

proportional

(6.10)

For quadrupole we have chosen r,, in order to reproduce the experimental width in “‘Pb and then we have kept it fixed along the mass range. The results for some spherical nuclei are reported in table 2. The agreement with experiments 26) (spherical nuclei) is very good. From our results we can get a mass dependence, for isoscalar quadrupoles, of the type: r

2+

= 14 . 01 A-o.3’

,

(6.11)

G.F. Burgio, M. di Toro / Nuclear collective motions

210

TABLE

2

Collisional times and quadrupole, octupole final widths (in MeV) for some spherical nuclei. (In brackets the octupole widths obtained with a L(L+ 1) dependence of h/7.) Nucleus

ii/T [MeV]

l-z+

r,-

“‘Pb ‘++Sm “‘Cd ‘%a

0.65 0.734 0.803 1.126

2.5 2.94 3.14 4.29

3.17 (6.9) 3.45 (7.85) 4 (9.8) 5.05 (10.8)

i.e. an almost A-“3 behaviour

which is quite close to the estimate

of Nix and Sierk 26):

r,+ = 19.16 A-1’3 , obtained amplitude

using a macroscopic collective oscillation

(6.12)

surface friction force which is damping a small of an incompressible irrotational fluid of nucleons.

Both one- and two-body dissipations are mixed in this surface term. The strength of the friction force is fixed just to reproduce the average A-dependence of the giant quadrupole widths. In doing so the authors are probably overestimating the real dissipation because the variations of the experimental widths are largely due to ground state deformations. A reduction of the strength to fit the Adependence for spherical nuclei only should bring the result eq. (6.12) to a behaviour very close to our semiclassical microscopic value. In table 2 we report also the widths for octupole modes obtained using the same collisional width r, without any dependence on the multipolarity of the collective motion. The octupole than experiment.

widths are slightly higher than the quadrupole

but much lower

If we assume that the surface variations are affecting the collisional term, we should introduce a L(L+ 1) dependence in the r,. In doing so we finally obtain values (in bracket in table 2) which are in extremely good agreement with the present data for spherical nuclei. The results

are quite interesting

but to put this section

in a right perspective

we

must be aware of the problems related to the relaxation time approximation eq. (6.1) which is partially violating the local particle, momentum and energy density conservation. Some further studies should be done considering other terms in the collision integral which are restoring the conservation laws. Some preliminary results obtained in a fluid dynamical approach seem to indicate that the corresponding effects on final spreading widths are negligible 28). 7. Conclusions The main result reached with this article is that the nuclear Vlasov theory represents a very reliable tool for quantitative predictions in the study of nuclear collective

G.F. Burgio,

motions.

This is important

parency

of the interpretation,

fancier

problems,

collisions.

M. di Toro / Nuclear collective motions

not only for the numerical but also because

like the effects of temperature

Here we would

211

simplicity

and for the trans-

it seems to be quite easy to study or the damping

like to stress again the main points

due to two-body

of our semiclassical

analysis: (i) Apart from the Pauli principle, included in the static distribution function, a fully classical dynamics is able to reproduce detailed properties of nuclear excitations. (ii) The self-consistency provided by the residual interaction plays a fundamental role to built collective states, i.e. the response depends crucially on residual interaction. (iii) A clear correspondence with a fully quantum theory can be used to introduce quantum corrections, ranging from spin-orbit terms to exchange effects. (iv) Temperature is not affecting the final response too much, at least keeping unchanged the residual interaction. An increase of the low-energy strength is observed. (v) Two-body collisions are essential to reproduce the widths of isoscalar giant resonance of low multipolarities (L = 2,3). However the self-consistency implies a quite complicated interplay between Landau damping (one-body) and collisional damping. A quite interesting problem will be to follow this interplay at finite temperature. In any case the collisional time must be dependent on the multipolarity of the collective motion. Finally we would like to stress that this approach is completely general and suitable to be used to study collective motions of any finite Fermi system (electrons in atoms or molecules, quarks in bags,. . . ) just changing the properties of the mean field (single-particle

orbits)

and the residual

interaction.

We warmly thank Dr. D.M. Brink (Oxford) and Dr. A. Dellafiore (Florence) for continuous support and useful suggestions. We are grateful to Dr. G.F. Bertsch (Cyclotron Lab., MSU) and to Dr. I.N. Mikhailov (JINR, Dubna) for several enlightening

discussions.

References 1) G.F. Bertsch, in Nuclear physics with heavy ions and mesons, Les Houches 1977, eds. R. Balian, M. Rho and G. Ripka (North-Holland, Amsterdam, 1978) 2) J.R. Nix and A.J. Sierk, Phys. Rev. C21 (1980) 396 3) G. Eckart, G. Holzwarth and J.P. da Providencia, Nucl. Phys. A364 (1981) 1 4) M. Di Toro, in Winter College on fundamental nuclear physics, eds. K. Dietrich, M. Di Toro and H.J. Mang (World Scientific, Singapore, 1985), Vol. 1, p. 451 5) E.B. Balbutzev, I.N. Mikhailov and Z. Vaishvila, Nucl. Phys. A457 (1966) 222 and refs. therein 6) P. Ring and P. Schuck, The nuclear many body problem, (Springer, Berlin, 1980) 7) D.M. Brink, A. Dellafiore and M. Di Toro, Nucl. Phys. A456 (1986) 205 8) Z.E. Saperstein, S.A. Fayans and V.A. Khodel, Kurchatov Institute Report IAE-2580, MOCKBA 1976

212

G.F. Burgio, M. di Toro / Nuclear collective motions

9) V.A. Khodel and Z.E. Saperstein, Phys. Reports 92 (1982) 183 10) A. Bohr and B.R. Mottelson, Nuclear structure (W.A. Benjamin, Reading, 1975) 11) H. Esbensen and G.F. Bertsch, Ann. of Phys. 157 (1984) 255; Phys. Rev. Lett. 52 (1984) 2257; Phys. Lett. 161B (1985) 248 12) J. Speth and A. Van der Woude, Rep. Progr. Phys. 44 (1981) 46 13) T.S. Dumitrescu, C.H. Dasso, F.E. Serr and T. Suzuki, J. of Phys. G 12 (1986) 349 and T.S. Dumitrescu, Ph.D. thesis. TUM Garching 1986 14) E.B. Balbutzev and M. Di Toro, Explicit quantum effects in nuclear collective motion, preprint JINR-Dubna P4-86-551 (1986) J. of Phys. G, submitted 15) A. Dellafiore and F. Matera, Nucl. Phys. A460 (1986) 245 16) A.B. Migdal, Qualitative methods in quantum theory (Benjamin, NY, 1962) 17) K.A. Snover, in The reponse of nuclei under extreme conditions, E. Majorana School, Erice 1986, ed. R.A. Broglia (Plenum, London, 1987) in press, and refs. therein 18) H. Sagawa and G.F. Bertsch, Phys. Lett. 146B (1984) 138 19) P.F. Bortignon in Erice 1986, same Proceedings as ref. “) 20) M.I. Baznat, A.V. Ignatyuk and N.1. Pyatov, Sov. J. Nucl. Phys. 30 (1979) 493 21) C. Fiolhais, Ann. of Phys. 171 (1986) 186 and refs. therein 22) L. Landau, Sov. Phys. JETP 3 (1957) 920; 5 (1957) 101 23) H.S. Kohler, Nucl. Phys. 440 (1985) 165 and refs. therein 24) M. Durand, V.S. Ramamurthy and P. Schuck, Phys. Lett. 113B (1982) 116 25) R.W. Hasse and P. Schuck, Nucl. Phys., A438 (1985) 157 26) J.R. Nix and A.J. Sierk, Proc. Int. School-Seminar on heavy ion physics Dubna 1986, ed. G.N. Flerov and Y.T. Oganessian, in press 27) H. Kohl, P. Schuck and S. Stringari, Nucl. Phys. A459 (1986) 265 28) I.N. Mikhailov, private communication, Dubna 1987