Volume 177, number 3,4
PHYSICS LETTERS B
18 September 1986
SURFACE T E N S I O N AND N U C L E A R L O W - L Y I N G V I B R A T I O N S IN A F L U I D - D Y N A M I C A L A P P R O A C H ~ L.P. B R I T O
Departamentode F~tsica,Universidadede Coimbra,P-3000Coimbra,Portugal Received 24 April 1986, revised manuscript received 3 July 1986 As a development of a previous variational approach to the Vlasov dynamics of small amplitude nuclear oscillations, a phenomenological surface energy is considered. This is shown to yield a finite energy to the low-lying states, while keeping unchanged the main features of the nuclear fluid. In recent years a great deal of work has been undertaken in order to obtain a semiclassical model which provides simultaneously low-lying and high-lying nuclear vibrational states as solutions of the same equations [1-3]. Extensive microscopic calculations [4,5] indicate that the low-lying states are characterized by rotational flow, a property which is absent from the low-lying surface vibrations of the standard liquid-drop model [6]. In order to make a distinction, as far as possible, between classical and quantal features of the nuclear collective motion, the Vlasov equation is an appropriate tool, since it is the classical limit of the T D H F equation, which arises when a W i g n e r - K i r k w o o d expansion is performed and h corrections are ignored. In previous papers [7,8] we have proposed a variational formulation of nuclear fluid dynamics, as described by the Vlasov equation, which goes beyond the scaling approximation [9,10], by including distorsions of the local Fermi sphere in the time-even distribution function
fE=fO +
(fo, P}+½{(fo, P), P) +---
/
= 0(X-h0(r, p ) - W(,, t)
* Work supported by Instituto Nacional de Investigac~lo Cientifica, Lisbon, Portugal.
where P is the generator of the time-even distorsions, ho(r, p) is the single-particle hamiltonian, corresponding to the equilibrium distribution function fo(r, p), and W(r, t) and X~l~(r, t) are variational fields. The dynamical deformations of the distribution function are introduced by means of a generator Q with non-local components [11]:
f=fE+(fE,
Q}+½((fE, Q},Q},
(2)
with
Q=ep(r, t ) + ½Y'.p~pl/p,a(r, t), a#
(3)
where ~(r, t) and ~ a ( r , t) are further variational fields. We have emphasized that the occurrence in this model of zero frequency modes, for some multipolarities, contributing in an appreciable way to the sum rule $1, was related to the absence of surface tension. In the present letter we wish to investigate the effect of the surface tension on such states. For this purpose, we add to the energy a surface energy term which only modifies the boundary conditions, keeping unchanged the equations of motion in the interior of the nucleus. A similar idea has already been developed both for isoscalar [1] and isovector [2] modes, in the framework of different approaches. The surface energy is written as Esurf = o 9~) dS,
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(4) 251
Volume 177, number 3,4
PHYSICS LETTERS B
where the surface tension coefficient, o, is to be c o n s i d e r e d as a p h e n o m e n o l o g i c a l p a r a m e t e r a n d d S is the element of the nuclear surface d S = R2[1 +
(1/R2)(OR/~O)2+ [1/R 2 sin 2 0 )
of the nuclear d r o p l e t m o d e l [12]. T h e force p a r a m e t e r s have been a d j u s t e d in o r d e r to r e p r o d u c e the following s a t u r a t i o n p r o p e r t i e s of n u c l e a r matter: k F = 1.26 fm -1
X ( 0 R / 0 c p ) 2] 1/2sin 0 dO dcp
E/A
- - R 2 [ 1 + ½ ( V R ) 2] s i n 0 d 0 d c p , where ~7 is the g r a d i e n t operator, acting only on the a n g u l a r variables a n d R = R 0 + 8R. F o l l o w i n g closely the contents of our previous p a p e r [8], the s e m M a s s i c a l lagrangian, i n the h a r m o n i c a p p r o x i m a t i o n , reads now as L(2)= f d 3 r [ _ (pa _ ½}oX~)(~, + ~ _ ~ 2 ~ ) 1
18 September 1986
(P0 = 0.135 f m - 3 ) ,
= - 13.8 MeV.
In table 1 we c o m p a r e the excitation energies a n d the percentages of the energy-weighted sum rule ( E W S R ) o b t a i n e d in the present calculation Table 1 For the states indicated, energies and percentages of the EWSR are given. The excitation operator rtYto has been used. Experimental results are given also. a(MeV/m -2)
J'~
E (MeV)
% of EWSR
1.017
2+
3.34 11.27 2.51 8.13 18.19 23.25 24.49 27.99 3.64 11.88 23.88 26.88 29.69 34.1
31.15 66.15 34.5 2.75 x 10-1 54.9 3.34 x 10-1 1.68 6.8 34.63 2.05 41.15 1.59 5.58 11.14
2.96 11.00 0 8.05 17.68 22.75 23.89 26.82 0 11.75 22.96 26.58 28.61 33.03
32.26 63.11 36.7 3 x l 0 -1 50.36 6.95 x 10-1 3.31 6.97 37.2 1.96 34.98 5.2 x 10- 2 10.38 10.77 b) 15 c) 70 33 b) 60 +20 c) 15 b)
3+
-EEP,,
X.•]-
T[*, *.,]
+
½l(l + 1 ) ] ~
-(a/R2)[1 x
4+
d~(SR, h)
h),
(5)
where the field p l ( r , t) replaces the field W(r, t) a n d E[pa, X,¢] a n d T [ ~ , ~,p] denote, respectively, the p o t e n t i a l a n d kinetic energy functionals. O n e obtains, t h r o u g h variation, the s a m e equations of m o t i o n for the fields in the b u l k of the nucleus, as in ref. [8]. T h e b o u n d a r y c o n d i t i o n s are also the same, except for the case of + ~, P-Fz "q ~
[r=Ro = 0,
which is r e p l a c e d b y i -2" 6 + r~pFeo,~,~ + (O/PoR2 )[1
X(SR.h)lr=no=O.
252
0
2+ 3-
(6)
4+
+ ½1(l+ 1)] (7)
A s a consequence of (7) the surface t e r m does c o n t r i b u t e to the n o r m a l i z a t i o n conditions. W e have w o r k e d o u t n u m e r i c a l results for 2°8pb, using a S k y r m e - t y p e effective interaction, with zero-range two- a n d t h r e e - b o d y parts. F o r the surface tension we have taken the value o = 1.017 M e V fm -2, derived from the surface energy term
(8)
experiment
2+ 34+
") Ref. [81. b) from ref. [13]. c) from ref. [14].
4.09 10.9+0.3 2.61 17.5+0.8 4.32
Volume 177, number 3,4
PHYSICS LETTERS B
with the previous results [8]. We observe that the inclusion of the surface tension does not affect substantially the location of the giant resonances, but plays a crucial role for the low-lying surface vibrations. Indeed, the first 3 - and 4 + states gain energy due to the surface tension. On the other side, the distribution of strength is not very much affected when the surface tension is taken into account. The agreement of the energies of the low-lying modes with the experimental results [15] is remarkably good. For instance, we obtain for the well established low-lying 3 - state the energy 2.51 MeV which is quite close to the value 2.61 MeV reported in several experiments [13,15]. Since in the present model the energies of the giant resonances are not significantly affected by the surface tension, their dependence on the number of particles A is still given by an A -t/3 law. However, the energies of the low-lying modes will tend to zero faster than A -1/3 being well fitted by an A -t/2 law. Although we have not derived an analytic expression in terms of the multipolarity l, the energies obtained for the low-lying 3 - follow closely the curve 36.2 A - 1/2 MeV. The surface tension takes into account quantum effects not included in the Vlasov equation and which play an important role for the low-lying states. The application of the Vlasov dynamics to semi-infinite Fermi liquids [16,17] has shown that a mode with zero energy appears, which is strongly damped. The finite energy of the low-lying surface vibrations arises only as a consequence of the finiteness of the nuclear system. A short comment concerning the monopole modes is in order. The presence of surface tension leads to the increase of the incompressibility of nuclear matter. Therefore, the excitation energy of the first monopole compression mode is still higher than the result obtained before (18.11 MeV). This fact, which is essentially a deficiency of the use of Skyrme-type forces, may be overcome by replacing the three-body part of the interaction by a two-body term with non-integer powers ( a ) of the density, p. In table 2 we compare the results for the monopole modesl obtained with these different assumptions.
18 September 1986
Table 2 Excitation energies of the monopole modes for different effective interactions and different values of o. Interaction
o (MeV fm- 2) . 0 1.017
two- and three-body &forces ( a = 1)
18.11 a) 20.03 a) 29.72 a) 38.95 a)
21.01 21.77 32.00 42.02
two-body forces including non-integer powers of P ( a = 1/6)
13.67 16.57 24.62 32.23
16.62 18.99 28.19 36.91
a) Ref. [8]. The main features of the nuclear fluid are not strongly modified when the surface tension is considered. The behaviour of the transition currents, corresponding to the low-lying 3 - state, illustrates this fact (see fig. 1). The transition current is defined as
j(r, t ) - [ d3-----~pP f(r, p, t), - J 2~.3
(9)
and the different functions represented in fig. 1 are defined according to J = J + ( r ) Yt,t+ 1,0 + J - Yt,t-1,0,
~7"j=d(r)Yi,o,
grxj=it(r)YL,.o.
(10)
The fluid exhibits significant rotational components, which are typical of the low-lying vibrations [4,5,18]. The main effect of the surface tension is reflected in the function d(r) which does not vanish anymore. We conclude that our nuclear fluid-dynamical model allows for a complete description of the most important vibrational states of heavy nuclei in the sense that both low-lying modes and giant vibrations emerge as solutions of the same variational problem. The inclusion of a surface tension on such a model does not affect sensitively the main characteristics of the nuclear fluid but leads to the disappearance of zero-frequency modes corresponding to a free surface. It is remarkable that the Vlasov dynamics may correctly account for the nuclear response over a wide range of energies. 253
Volume 177, number 3,4
PHYSICS LETTERS B
18 September 1986
b
a
40-
I~=a-
A=2oa
/
/
E=O
/
1~=3-
A=208
E=2.51 MeV
20
/
//
// O
%%
%,~%
~2
ii
/
----...2
.
.
.
.
.
.
.
/ x
--
/
t(r)
t(r)
. .
/ x
I --20
~ j ÷ ( r ) --4
o,IVm]
xx
/
. . . . .
~
.
j(rl
--40
-
--
jwIr)
.j2r) ..........
d(r)(×lOI
Fig. 1. (a) The functions j+ (r), j_ (r) and t(r), defined in eq. (10), are plotted (arbitrary units), for the zero-frequency 3- state. (b). The radial functions of the current, according to eq. (10), are plotted (arbitrary units), for the low-lying 3- state, corresponding to a surface tension o = 1.017 MeV fro-2. T h e a u t h o r is v e r y g r a t e f u l to P r o f e s s o r J. d a P r o v i d S n c i a for his c o n t i n u o u s o r i e n t a t i o n d u r i n g t h e work. V a l u a b l e s u g g e s t i o n s f r o m P r o f e s s o r J. M a r t o r e l l a n d u s e f u l c o m m e n t s f r o m P r o f e s s o r C. F i o l h a i s are a c k n o w l e d g e d also.
References [1] C.Y. Wong and N. Azziz, Phys. Rev. C 24 (1981) 2290. [2] S. Stringari, Ann. Phys. (NY) 151 (1983) 35. [3] J.P. da Providencia and G. Holzwarth, Nucl. Phys. A 439 (1985) 477. [4] F.E. Serr et al., Nucl. Phys. A. 404 (1983) 359. [5] A. Schuh et al., Nucl. Phys. A. 412 (1984) 34. [6] A. Bohr and B. Mottelson, Nuclear structure, Vol. 2 (Benjamin, New York, 1975).
254
[7] LP. Brito and C. da ProvidSncia, Phys. Lett. B 143 (1984) 36. [8] L.P. Brito and C. da ProvidSncia, Phys. Rev. C 32 (1985) 2049. [9] J.P. da ProvidSnca and G. Holzwarth, Nucl. Phys. A 398 (1983) 59. [10] S. Stringari, Nucl. Phys. A 279 (1977) 454. [11] S. Stringari, Nucl. Phys. A 325 (1979) 199. [12] W.D. Myers and W.J. Swiatecki, Ann. Phys. (NY) 84 (1974) 186. [13] M.N. Harakeh et al., Nucl. Phys. A 327 (1979) 373. [14] H.P. Morsch et al., Phys. Rev. Lett. 45 (1980) 337. [151 C.D. Djalali et al., Nucl. Phys. A 380 (1982) 42. [16] Y.B. Ivanov, Nucl. Phys. A 365 (1981) 301. 117] V.I. Abrosimov and J. Randrup, Nucl. Phys. A 449 (1986) 446. [18] T.S. Dumitrescu et al., Nordita preprint (1985).