ANNALS
OF PHYSICS
151, 35-70 (1983)
Fluid-Dynamical Description of Nuclear Collective Excitations S. STRINGARI Dipartimento
di Fisica,
Universitd
di Trento,
38050
Povo,
Italy
Received December 28. 1982
Fluid-dynamic models are derived in the framework of the time-dependent Hartree-Fock theory. The generalized scaling (GS) and the hydrodynamic (HD) models are discussed in detail and their applicability to nuclear giant resonances is explicitly investigated. The response function of the system (dynamic polarizability) and its connections to sum rules are analyzed in the framework of the fluid-dynamic approach. It is shown that the GS model correctly reproduces the high-frequency limit of the response function, while the HD model is more suited to investigate its static limit. In terms of sum rules, this means that the two models are expected to reproduce the cubic energy-weighted and inverse energy-weighted sum rules, respectively. Arguments based on sum rules and on the form of the boundary conditions suggest that the HD model provides a useful tool to investigate isoscalar compression excitations and, in general, isovector modes, while it dramati’cally fails in the description of divergency free motions, where the GS model results are much more successful. The role of surface effects on the nuclear motion is also discussed. It is shown that the effect of the surface symmetry energy term on isovector excitations can be taken into account by means of a suitable choice of the boundary conditions.
1. INTR~DDCT~ON Considerable interest has developed in recent years in approaches to the nuclear motion based on the dynamics of fluids (see, for example, [ 1 I). This kind of approach provides the interesting possibility of describing general features of collective states in a systematic way, using a limited set of parameters which characterize global properties of nuclei (volume and possibly surface properties). The idea of treating the nucleus as a fluid has a long history, dating back to the pioneering works on the liquid-drop model (1937-1939) [2-4], where the fission mode and the surface vibrations were first discussed in a systematic way. The hydrodynamic model was later (1950) successfully used to describe higher energy collective states (giant resonances) and, in particular, the dipole giant resonance [5]. More recently (1974-75) it has been suggested [6] that nuclei can vibrate as an elastic medium. The elastic vibrations are characterized by the presence in the stress tensor of nondiagonal components, which are absent in the hydrodynamic picture. The presence of these components can be related microscopically to a typical quantum effect exhibited by Fermi systems, i.e., to the deformations of the Fermi 35 OOO3-4916183 $7.50 All
Copyright 0 1983 by Academic Press, Inc. rights of reproduction in any form reserved.
36
S. STRINGARI
sphere in the momentum space during the nuclear motion, and permits description of a new class of phenomena, the most famous of which is the giant quadrupole resonance. It is the aim of this work to present in a compact way the main features of elastic and hydrodynamic vibrations in nuclei. We shall start from a microscopic point of view with the aim of getting a connection between microscopic models of nuclear dynamics, such as the time-dependent Hartree-Fock (TDHF) theory, and macroscopic approaches. The starting point is a generalized scaling transformation applied to the nuclear ground state from which one can derive what is called the generalized scaling (GS) model (Sections 2 and 3). The hydrodynamic (HD) model is obtained simply by ignoring in the GS derivation the terms associated with the distortions of the Fermi sphere. Since the GS model has been mainly developed for the isoscalar modes, particular attention will be devoted to the isovector case. When possible, we avoid a detailed analysis of the solutions of the equations of motion. Rather, we prefer to discuss the main features of the collective motion in terms of sum rules (Section 4). The main part of the work is concerned with the case of large systems in which surface effects are ignored. The inclusion of these effects is, of course, of crucial interest in actual nuclei, and in Section 5 we include them, using a method alternative to the standard description of finite systems. In Section 6 we present a (necessarily) brief description of isoscalar and isovector collective modes and discuss, in particular detail, the case of the isovector dipole excitation, for which interesting features are predicted by the GS model. The role of surface effects on this mode is also discussed. In Section 7 we draw some conclusions on the use of fluiddynamic models in the description of giant resonances and we comment on some of the problems which remain to be explored in this area.
2. FLUID-DYNAMIC DESCRIPTION OF NUCLEAR MOTION: THE GENERALIZED SCALING MODEL Collective oscillations in nuclei were first described as vibrations of an elastic medium by Bertsch in 1974-1975 [6]. Similar approaches to the nuclear motion were later developed in [7-91. To derive the equations of motion, one looks for solutions of the time-dependent Hartree-Fock (TDHF) equations in the space of Slater determinants 1w) built up with single-particle wavefunctions of the type: (2.1) where t,up is the single-particle wavefunction filling the Hartree-Fock (HF) ground state IO), ((r, t) is interpreted as a velocity potential and u is the displacement field. ti is equal to 1 for isoscalar modes, while for isovector modes it is equal to the third component of the isospin operator rf (i.e., 1 for neutrons and -1 for protons). The set of single-particle wavefunctions defined by Eq. (2.1) is orthonormalized only to first
NUCLEAR
37
FLUID-DYNAMICS
order in u. A set orthonormalized to all orders (and identical to Eq. (2.1) to first order) can be generated by means of the explicitly unitary transformation [7] I : Iw>=e
im ri5(Ii,t)tjeiC1/2)
Zi(U(ri.l)’
fi+ Pi’U(rt,t))ti
(2.2)
10)
applied to the ground state IO). In Eq. (2.2) pi is the momentum operator acting on the ith particle. If IO) is a Slater determinant, the state 1w) is also a Slater determinant. The exponential containing u defines a time-even unitary transformation which changes the nuclear density of the state IO). Special choices for this transformation have been extensively used in the past in order to investigate the restoring force parameter for various isoscalar and isovector excitations [l&14]. The term in r introduces time-odd components in the wavefunction without modifying its density and is a simplified form of the most general unitary transformation introduced by Baranger and Veneroni I15 1, 1~) = eiX / 0), where x is hermitian and time-even. In the present work, we discuss the fluid-dynamic equations of motion in the framework of coordinate space. One can also derive equivalent approaches based on the formalism of the distribution function [ 16, 17 I. (For a recent review of various semiclassical approaches to nuclear dynamics see 118, 19 I.) The form of the transition density associated with transformation (2.2) is easily expressed in terms of the displacement field u: to first order in u one has:
PJ~, t) = (~11
I
J(r - ri) ti / v/) - (0 I x 6(r - ri) ti (0) = V(u(r, t) p&>>,
(2.3)
i
where pO(r) is the density of the ground state IO). Conversely the form of the velocity Iield, which depends only on the time-odd component of the wavefunction, is directly written in terms of the local velocity potential <: v(rt) = -&
(WI T&F
I
@(r - ri) Pi + h-C+) tiI v/>=
Kb t>
(2.4)
and results to be automatically irrotational [20]. The condition of irrotationality implicitly contained in Eq. (2.2) suggests that the method is well suited to investigate giant resonances exhausting most of the excitation strength. In fact, it has been shown [ 211 that irrotational flow for the nuclear motion is microscopically obtained if one assumes that a single collective state completely exhausts the sum rules (doorway hypothesis [22]). The predictions of the irrotational Bohr-Tassie (231 and Werntz-oberall [24] models for the transition density and for the velocity field are easily obtained by setting u = const V( with < = r’.yl, and < = r2 for multipole and monopole deformations, respectively. The equations of motion can be derived by using the variational principle applied to the action integral
(2.5)
6Z= 0 I Some changeswith
respect
to 171 have been introduced
in the notation.
38
S. STRINGARI
with
z= Jt’df(y/ ig-HIV). to
(2.6)
The method reproduces [25] the TDHF equations if 1w) is the most general Slater determinant. Here we are consequently deriving an approximate solution of the TDHF theory. Furthermore we will study small oscillations around the ground-state configuration. Due to the explicit connections between the TDHF theory and the random phase approximation (RPA) [26, 121, the method should then be regarded as an approximation to the RPA. By keeping only terms quadratic in < and u and neglecting total derivatives with respect to the time, which do not affect the equations of motion, we find: (2.7) and (WI HI v> = (01 H 10) + E(u) + K(r).
(2.8)
In Eq. (2.8) we have introduced the collective potential energy E(u) = (01 e=-
i(1/2)~i(u.pi+h.c.)ti*ei(l/2)~i(u~pi+h.c.)t,
: (ol[t~
(u . pi
IO)- (0‘I HI01
+ kc.) ti,
H, f F7 j (u * pi + kc.) ti [ I
11
(2.9)
IO>
and the collective kinetic energy K: (2.10)
The terms linear in u and < in Eqs. (2.9) and (2.10) vanish due to a well-known property of the Hartree-Fock ground state IO) ([26, p.1431). Assuming that the nuclear potential commutes with the isoscalar operator Zi <(rit) and that it contributes to the commutator with the isovector operator JJi <(rit) t: through a density dependent term, one can very simply evaluate K as: K = irn
I
(isoscalar),
v2p, dv
K = fm [ v”( 1 + k&J)
(2.11) p,, dv
(isovector),
NUCLEAR
39
FLUID-DYNAMICS
where k&J is related to the nonlocality terms of the force (in the case of Skyrme forces one has k, = (m/2)(t, + t,)p, [27, lo]). Equation (2.11) shows that the collective kinetic energy depends on the scalar potential < only through the velocity field v = V<. This suggests the use of the classical expression (2.11) for the collective kinetic energy, also, when the velocity field contains rotational components, and cannot be written as the gradient of a scalar field. Of course, with such an assumption one is considering also states which are not reproduced by the original ansatz (2.2), and which cannot be explicitly identified as Slater determinants. With this assumption, the equations of motion can be derived by performing the variation (25) with respect to the displacement and velocity fields u and v: 61 -=o 6u
’
-= 61 6v
0
(2.12)
.
For the isoscalar and isovector modes we find
6Es mtp,, --= &I
(2.13)
li=-v, and
6E, -
6u
= mip,,
i = -v(l
(2.14) t k,@,)),
respectively. By looking for solutions u(rt) = u(r) u(t) oscillating with frequency w. one finally finds the following linear equations, called the equations of the generalized scaling (GS) model [6, 7,9] : (2.15) for the isoscalar
case, and
1 aEu W) I+ k&d
mw’u(r)
~dr)
for the isovector case. It is also interesting to derive the continuity equation for the isoscalar isovector densities. Using Eqs. (2.3), (2.13), and (2.14) one fmds @i = p,“) g t V(vp,) = 0
(2.16)
and
(2.17)
40 in the isoscalar
S. STRINGARI
case @ = p, + p,), and
Jg + V(v( I + k,.@,)PO)= 0 in the isovector one @i = pr -p,). Equation (2.18) clearly shows that the exchange terms of the nuclear force modify the isovector current with respect to the usual convection term p0 v. Before concluding this section we want to comment on some theoretical aspects of the present approach. We have started from a restricted set of Slater determinants (see Eqs. (2.1) (2.2)), and consequently the natural theory with which to make the comparison is the TDHF theory. In deriving the equations of motion (2.15) and (2.16) we have, however, made the further assumption that expression (2.11) for the collective kinetic energy holds also for velocity fields containing rotational components. This assumption, needed to carry out the complete variation 61/6v, has the advantage of yielding a description of the nuclear motion including rotational components, but makes the comparison of the approach with microscopic theories (TDHF, RPA) less clear. The existence of a set of Slater determinants for which the collective kinetic energy has the form (2.11) of a classical fluid irrespective of the nature of the velocity field should be investigated in order to make this comparison in a rigorous way. This problem is of crucial relevance for the analysis of the rotational motion [28-291 and in general of low-lying collective states [ 30-3 1 1, but can have implications for giant resonances also. For example, it has recently been shown [32-331 that the flow patterns of the isoscalar quadrupole resonance as predicted by microscopic RPA calculations are not fully reproduced by the GS model. As concerns the validity of Eq. (2.11) see also the discussion on the propagation of transverse zero sound in infinite systems at the end of Section 4. The equations of motion (2.15) (2.16) can also be derived using the fluid-dynamic formulation of [9], which has been extensively used to investigate various collective excitations in nuclei [34, 351. As concerns the comparison with the TDHF theory. this method exhibits difficulties similar to the ones discussed above. In fact, except for the case of irrotational flow, it does not make explicit use of single-particle states (Slater determinants). Other methods to derive equations of fluid-dynamic type have been developed by introducing suitable reduction procedures or truncation schemes in the solution of the microscopic equations of motion [ 16, 36, 37 1. However, these approaches are not of variational type because the approximation is made in the equations of motion, i.e., after variation. Finally, we recall that alternative procedures, based on a variational approach in the framework of the TDHF theory, have been recently proposed to derive equations of fluid-dynamic type. In one of these methods [32], one abandons the description of rotational components by imposing explicitly a constraint of irrotationality in the derivation of the equations of motion, In practice, one writes u = Vv and carries out the variation (2.5) with respect to n and c. A second and more ambitious method
41
NUCLEAR FLUID-DYNAMICS
1381 introduces rotational components in the wave function by replacing the local operator c in Eq. (2.2) with a more general nonlocal operator2 x = r + (l/2) Cnb(PaP4$a!3 + h.c.1. In the following sections we will discuss the solutions of Eqs. (2.15), (2.16) in the limit of large systems and develop the formalism for the isoscalar and isovector modes.
3. ELASTIC AND HYDRODYNAMIC COLLECTIVE
The energy derived minant)
VIBRATIONS
equations of motion (2.15) and (2.16) functional E(u) is given. Explicit and by assuming that the expectation value can be written as a local functional of
E=(y/lWv)= I(&+i
MODELS FOR IN NUCLEI
can in principle be solved once the rather compact forms for E(u) are of H on a general state (Slater deterthe type3
r + v@) + uNL@)@r - J2) - vE”‘J;
~,~,@)p:
+ grad. terms
dv,
depending on a few l-body densities: p =p, + pP, p, =p, -ppr J = J, + J, and J, = J, - J,. The above densities are defined by:
(3.1) r = 5, + so,
(3.2)
(9 indicates neurons or protons) The term u ““@) in Eq. (3.1) gives nonlocality effects due to isoscalar currents and is built up to ensure the local galilean invariance; the term VT”“@) reflects the possible dependence of the force on isovector currents; the term containing u,,,@) is responsible for the nuclear symmetry potential. ’ The importance of the presence of non-local terms of this type in the time-odd part of the nuclear deformation is suggested, for example, by the analysis of low-lying vibrations in the framework of the schematic model [39]. The same analysis, however, suggests that also the (time-even) scaling deformation of Eq. (2.2) should be replaced by a more general transformation. ’ Terms like ~,(t, - rP) have not been considered in Eq. (3.1) because their effect on the equations of motion can be taken into account by means of a simple renormalization of the symmetry energy potential us”, .
42
S. STRINGARI
The assumption (3.1) of a local energy functional has important consequences on the nature of the equations of motion (2.15) and (2.16), which then reduce to linear differential equations. The limit of large systems is particularly simple and the isoscalar case has already been discussed by several authors [6,34,40,41]. We present here a unified derivation for isoscalar and isovector deformations which clarilies the differences and analogies between the two cases. To evaluate E(u), one needs the changes 6p, 6p, and 6r associated with the unitary transformation (2.2) (the current terms do not enter in the evaluation of E(u)). Up to terms qudratic in u we have:
SP= V,(~,Po)+ swkw,Po>> dp, =o 6r = V,(u,r,)
+ ght,
+ fV,(Vu)
+ 3uG5(~,~0>) + g&l,
Vur,)
v,p,
+ iw,
+ v,%)*
- $I/( V,(Vu>
(3.3)
to
50 + bV,(Vu)
v/p>
po
V,(Vu)
p.
for the isoscalar case, and
6z = ~V,(u,V,(u,z,))
+ @,u,
+ ~v&,vur,)
+ v,u,)*
- $4,V,(Vu)
50
70 + &((Vu)
(3.4)
for the isovector case. In Eqs. (3.3) and (3.4), to = r, + rP is the kinetic energy density of nuclear matter, which can be written in terms of the nuclear density using the Thomas-Fermi relation to = api13,
(3.5)
where a = (3/5)(37~‘/2)“‘. In deriving 6r we have neglected, for the sake of simplicity, some terms whose contribution is negligible in the limit of large systems. Furthermore, we have neglected contributions proportional to N- Z. It is worth noting that if one assumes the Thomas-Fermi relation (3.5) as a functional relation for the kinetic energy density, one finds a different result for the variations of the kinetic energy density: GrTF = &, - fV,(Vu) + $
v,p,
- ;(v,u,
zo - :qvu>
V,(Vu)
+ v,u,y
to
po
(3.6)
for the isoscalar case, and GrTF
=
67”
- #Qd,
+ v,u,)*
To + $(vu)*
To - $,(Vu)
V,(Vu)p,
(3.7)
NUCLEAR
43
FLUID-DYNAMICS
for the isovector one. In Eqs. (3.6) and (3.7), 6r, and 6r, are the variations of the kinetic energy density given by Eqs. (3.3) and (3.4). The differences are due to the distortions, absent in the Thomas-Fermi approximation, produced by transformation (2.2) on the Fermi sphere [42]. Using the changes (3.3) and (3.4), we get the asymptotic expressions:
a,(.)=~~~j(r(V,,,+OiUlr)*-~~vu)*)Pod~ kl
+
&x
j
(vu)”
PO d”
+
&
j
v,(vu>
‘ktvu>
PO d”
(3.8)
and
E,,(u) = +F j (; (‘k”, + v,u,)2-4 (\7”)2)PO dv + ;
L,
j (Vu)’
PO dv
1
(3.9)
-jvk(VU)~k(%)~odV+E~r‘(U)
+ 8m*
for the isoscalar and isovector collective potential energies, respectively. In deriving Eqs. (3.8) and (3.9) we have made use of the nuclear matter saturation condition: (3.10) with
7@)
given by Eq. (3.5). Furthermore,
we have introduced the Fermi energy (3.11)
the effective mass defined by the relation -
1
2m*
= -& (1 + 2mp,uNL@,)),
(3.12)
the nuclear matter incompressibility x=9pz
p2 v@)+i7@) 8P c 3P c P
2m
p
+
VNL@)
74)
Y
(3.13)
and the nuclear matter symmetry energy (3.14)
44
S. STRINGARI
The contribution Errf (u) entering in the isovector collective energy (3.9) depends on the value of the radial component of the displacement field u at the surface and can not be ignored, as will be discussed later. If one neglects the term 1/8m* J” V,(Vu) V,(Vu)p, dv, Eq. (3.8) has the form of the potential energy of an elastic medium [43 1: (3.15) with the Lame’s elastic constants
given by
Ji = Cdx- I$,) PO
(3.16)
and
P = ho.
(3.17)
We note that neglecting the term 1/8m* J’ V,(Vu) V,(Vu)p,dv usually gives a good approximation for large systems except for processes associated with a large momentum transfer. Assuming, for example, u = q cos qr, it is easy to verify that the contribution to E(u) coming from this term is the dominant one when q + co, but is negligible when q -+ 0. In the following, this contribution will be ignored. The equations of motion for the isoscalar case can now be derived very easily. Using Eq. (2.15) and integrating Eq. (3.15) by parts, one gets: (~+p)V,Vu+pv*u,+mp,w*u,=0 inside the nucleus, with the boundary
condition
(3.18) at the surface [43 1:
Zgu+nX(VXu)
= 0,
(3.19)
r=R
which directly arises when one integrates Eq. (3.15) by parts. In Eq. (3.19), R is the nuclear radius and n = r/r is the unit vector in the radial direction. Equation (3.18) admits both compressional and shear solutions characterized by the following dispersion laws: (j&z-
m*x(F+y”8 j
(compressional)
and
(3.20) (32 22, m5’
(shear),
respectively. Equations (3.18) and (3.19) have been used by several authors 411 to study isoscalar multipole excitations in nuclei.
[34, 40,
NUCLEAR
45
FLUID-DYNAMICS
Also, the isovector potential energy (3.9) has the form of the potential (3.15) of an elastic medium, with the Lame’s constants given by
energy
A = L, - +rJ PO (3.21)
and 2
Pu= IEFPO.
However, in this case an additional surface term E:,“‘f(u) enters in the potential energy E(u). For large systems, this term has a particularly simple form: E,surf_ -+
v,~,,,@) - 3)
(u - VP)’ dv
(3.22)
and, except for special choices of u, gives most of the contribution to the isovector potential energy E,,(u) (Eq. (3.9); in the case of Goldhaber-Teller deformations u = Vz only this term would contribute to E,(u). The presence of the surface term (3.22), which physically gives the nuclear resistence against the formation of a neutron (or proton) skin, strongly affects the solutions of the equations of motion. In fact, in the limit of large systems,the solutions with the lowest frequency (following a mass dependence of the type o = const A - ‘13) are found when the surface contribution (3.22) vanishes. This requirement is equivalent to imposing the vanishing of the radial component of the displacement field at the surface:
u * rllrER =o.
(3.23)
Of course, the boundary condition (3.23) holds only in the limit A + co. In finite nuclei, the presence of surface symmetry energy effects is responsible for important modifications of condition (3.23) which will be discussed in Section 5. Equation (3.23) is equivalent to the well-known condition of Steinwedel and Jensen [5], which requires that the neutrons oscillate against the protons with the surface fixed. Equation (3.23) has to be taken into account in the determination of the equations of motion (2.16). This can be done, for example, by adding the constraining term - ~,,,,a(@) u . n to the potential energy (3.15). E(@) plays the role of a Lagrangian multiplier and lsurf is the integral on the surface. Free variation with respect to u then yields the new equations of motion. In the interior of the nucleus one finds: (A + p) v,vu
+ pv2u, + muPO02Ul= 0,
(3.24)
which is similar to Eq. (3.18) for the isoscalar case, except for the presence of an isovector effective mass (k&j,) = 2mp, $Ypo))
46
S. STRINGARI
and for the value of A, which is related to the symmetry energy (Eq. (3.21)) rather than to the incompressibility (Eq. (3.16)). C onversely, the boundary conditions at the surface are significantly changed with respect to the isoscalar case. In fact, they are given [35] by Eq. (3.23) and by the following equation: 2~“+“X(vX”)-2$(“*n)n r=R
=0.
(3.26)
In contrast to the isoscalar case, in which the boundary condition (3.19) is given by the vanishing of the force at the surface of the nucleus, the isovector case only requires the tangential component of the force to vanish, the additional condition being given by the vanishing of the radial component of the velocity field at the surface.4 These results show that the nature of isovector vibrations is very different with respect to the one of isoscalar vibrations and that one is not physically authorized to use the same boundary conditions in the two cases. The dispersion laws predicted by Eq. (3.24) for the isovector compressional and shear waves are: 8 vol + E&F
(compressional) i
(3.27)
and w2=4-~ 2 2
(shear),
m, 5 F
respectively. The equations of standard hydrodynamics can easily be derived in the framework of the present formalism by setting the term proportional to E, in Eqs. (3.8) and (3.9) equal to zero. From a microscopic point of view, this corresponds to imposing a condition of local equilibrium during the nuclear motion which ensures that the Fermi sphere preserves its sphericity. This assumption is equivalent to using the Thomas-Fermi relation (3.5) in order to derive the changes of the kinetic energy density (Eqs. (3.6) and (3.7)) in the evaluation of the potential energy E(u). The elastic and hydrodynamic potential energies coincide only for very special choices of u, which make the term in .sFin Eqs. (3.8) and (3.9) vanish. It is not difficult to show that this happens if and only if the following conditions for u are satisfied:
vxux=vyuy=vzuz VxUy+VyUx=VyUz+VzUy=V*Ux+VxUr=O. ’ Equations (3.23) isovector oscillation.
and (3.26)
differ
from
the boundary
conditions
(3.28)
used in 144, 451 to describe
the
47
NUCLEAR FLUID-DYNAMICS
The rigid translation (u = Vz) corresponding to a Goldhaber-Teller deformation in the isovector case, the scaling monopole deformation (u = r), and the nuclear rotation (u = Q x r), are some of the few cases satisfying Eq. (3.28). Another interesting case that arises for the description of the isovector dipole will be discussed in Section 6. The hydrodynamic equations of motion read:
+,vu +mw2u, =0
(3.29)
vu IrER = 0
(3.30)
with
for the isoscalar case, and b,,,V,Vu + m,.o’u,
=0
(3.31)
with (3.32)
u.nl,,,=O
for the isovector case. The dispersion laws become:
co;,,=--9* x
(3.33)
m9
and (3.34) respectively. The solutions of Eqs. (3.29t(3.32) are extensively discussed in 146). As follows from Eqs. (3.29) and (3.31), in the HD limit the velocity field results to be irrotational. However, Eqs. (3.29) and (3.31) admit solutions u, occurring at w = 0, containing rotational components. These motions usually have nonvanishing frequencies in the elastic case and can contribute to sum rules, as will be discussed later (see Section 6).
4. SUM RULES IN THE ELASTIC AND HYDRODYNAMIC
MODELS
The sum rule approach provides a useful microscopic tool to study the main features of collective states (excitation energy, total strength, etc.). Since sum rules are often associated with important global properties of the system, it is interesting to investigate if different macroscopic models, accounting for a classical description of the nuclear motion, are able to reproduce them in a correct way. Such an
48
S. STRINGARI
investigation can either provide microscopic support to macroscopic models, or reveal their drawback in a very explicit way. In practice, only few sum rules can easily be evaluated for a given excitation operator. In the following, the discussion will be limited to these sum rules, the knowledge of which clearly does not exhaust the investigation of all the dynamic properties of the system. In order to evaluate sum rules in the framework of the fluid dynamical models previously described, we find it convenient to study the dynamic polarizability, also known as the linear response function. Let us suppose that the nucleus interacts with an external oscillating field through the interaction --EF cos wt, where E is the strength of the field and F is a onebody excitation operator depending on spatial and possibly spin or isospin coordinates. The dynamic polarizability a(w) is defined as: a(w)=lim
(WIFIy/)-(oIFIo) E+O
& cos
wt
(4.1)
3
where 1w) is the solution of the Schrodinger equation in the presence of the interaction term. a(w) is related to the excitation strength of the operator F through the relation (4.2)
which follows from the use of the first-order perturbation theory. Equation (4.2) holds also in the TDHF-RPA scheme [12]. In this case a(w) is evaluated by solving the time dependent Hertree-Fock equations, while (01 F/n) and o,, - w. are matrix elements and excitation energies of the random phase approximation. By evaluating a(w) we get information on sum rules. This follows by the expansion of Eq. (4.2) in thelimitsw+Oandw-+co: a(w),,o=
2(X-,
+ w2Sp3 + da*)
(4.3 1
and
a(w),+,= -$ (S,+ A,+... o2
))
(4.4)
where (4.5)
are moments of the strength of order k relative to the excitation operator F In order to evaluate a(w) we have to add the term Iiot =
to the action (2.5).
” dt &(I//[ F ( I/) cos wt I to
(4.6)
NUCLEAR
49
FLUID-DYNAMICS
We first consider the case of local operators F of the type F = If(q)
In this case, the contribution
ti.
(4.6) can be written
-t1 Iint = EJ df I
dvf(r)
(4.7) as
p(r, t) cos wt
(4.8)
to
in the isoscalar case, and Iint = E ” dt dvf(r)p,(r, !’to I
t) cos wt
in the isovector one. Using Eqs. (3.3) and (3.4), which give the changes of p and p, in terms of the displacement field u, the equations of motion (3.18) and (3.24) contain an additional term, and setting u(r, t) = u(r) cos wt, become (m,, = m in the isoscalar case) = 0.
(4.10)
field u(r) is found, the dynamic polarizability
(4.1) is deter-
(A + p) V,(Vu) + ,uV*q + mt,pod~, Once the displacement mined as:
a(w) = _ i VW
Another interesting case is when happens, for example, if one chooses:
. 44 PO(r) dv E
the external
F = s f(uext(ri)
- woVtf
(4.11)
field is velocity
dependent,
(4.12)
- pi + kc.) ti.
In this case, the additional term (4.6) in the action gets a contribution presence of the velocity field v(r, t) in the nucleus: Iint = E i” dt S d u mu,,,(r) dto
. v(r, t) pO(r) cos of.
Setting v(r, t) = V(t) v(r) and u(r, t) = U(t) u(r), the equations interior of the nucleus become (m, = m in the isoscalar case) (A + p) V,Vu + P’u,
+ mupow2ut - wpOm&4ext>t
as
from the
(4.13)
of motion
= 0.
in the
(4.14)
50
S. STRINGARI
Once more, the solution u(r) can be used to evaluate a(w) through:
where v = ml,/m(cvext - wu). Of course, the hydrodynamic equations for o(o) are simply obtained by setting p = 0 and identifying A with l/9 xp,, and b,,, p,, in Eqs. (4.10) and (4.14) (and in the relative boundary conditions), respectively. Explicit expressions for sum rules will be given for special choices of F in Section 6. Here we will discuss some general results when F is of the form (4.7). The most famous sum rule is the energy weighted one:
S,=‘(W,--Wg)l(OlFI~)l*=t(O/[F~
[KFIIIO)
(4.15)
that can be evaluated microscopically in terms of a double commutator involving the nuclear hamiltonian. In most of cases, the nuclear potential commutes with isoscalar operators F, while it gives a contribution to the commutator if F is isovector. Using, for example, Shyme-type forces, we have [27, lo] sRPA 1
=+-J
F’fl*~&
(4.16)
for the isoscalar case, and sRPA = -,jnjlV~l*(l+kl.@o))~odtl 1
for the isovector case. It is immediate to verify that either the elastic or the hydrodynamic models reproduce correctly the S, sum rule. This can be verified by taking the limit w -+ co of Eq. (4.10), which gives for the scaling deformation field (m,, = m in the isoscalar case)
This field can be used to evaluate a(w) (Eq. (4.11)) in the limit o + co. By comparison with Eq. (4.4), one then recovers results (4.16) and (4.17). Unlike the S, case, the other sum rules are more model dependent. Let us first consider the cubic energy weighted sum rule S,. One can prove that in the random phase approximation (RPA) S, is related to the energy change associated with a generalized scaling deformation of the type of Eq. (2.2), applied to the Hartree-Fock ground state. More explictly one has (m, = m in the isoscalar case): RPA
s3 =$(VIm”=o
NUCLEAR
FLUID-DYNAMICS
51
This result, with IV) = exp(v[H, FIRPA)IO) = exp(iv$ Ci(l/m, Vf. pi + h.c.) tJ0). holding for isoscalar [47] as well as for isovector [ 121 excitations, suggests that the solutions of the elasticity equations, being derived by a generalized scaling ansatz, correctly reproduce this sum rule. One can then write:
Conversely, the HD model in general does not reproduce the sum rule SyPA correctly. This is due to the fact that the evaluation of SyPA is usually associated with a deformation of the Fermi sphere in the momentum space. The most evident case in which the HD model dramatically fails in reproducing S, is when one considers excitation operators f yielding divergency free scaling deformations for Vf, as happens for the quadrupole operator f = r2ylm. In this case, most of the contribution to ,ppA comes from the shear term in Eq. (3.15). Since the term proportional to E, in Eqs. (3.8) and (3.9) is definite positive, one can conclude that in general SF” > sy.
(4.21)
The situation is quite different for the S-, sum rule which is related to the static polarizability, as follows from Eq. (4.3). In fact, the HD picture is more suited than the GS one to reproduce this sum rule. This follows from a well-known result, derived in the framework of the Landau theory of Fermi systems [48], according to which the sphericity of the Fermi sphere is preserved in the presence of an external static field, and consequently the linear response function is correctly given by the HD model in the static limit (more precisely, the validity of the HD picture is ensured as long as the momentum transfer q and the frequency w of the field satisfy the condition k, s q 4 w/v,, where v, = k,/m * is the Fermi velocity). In [49] the polarizability sum rule has been explicitly evaluated in the framework of the hydrodynamic model for the isoscalar monopole excitation. For higher multipolarities the existence of surface vibrations of the nucleus can not be ignored in the evaluation of s-1 for isoscalar excitations. In fact, these vibrations occurr at low energy and, except for special choices of the excitation operator F, they exhaust the sum rule S-, in the limit of large systems. In order to utilize the polarizability sum rule for a description of compression modes, one consequently has to require the isoscalar excitation operator F not to excite the surface vibrations. In the limit of large A this requirement is satisfied by excitation operators F vanishing at the surface [ 501: f(r=R,8,ql)=O.
(4.22)
Requirement (4.22) is not demanded in principle for monopole excitations because surface vibrations of monopole type cannot exist. In order to derive a general expression for the polarizability sum rule, it is. however, convenient, without being restrictive (it suffices to substract the constant f(R) from the operator f(r)), to impose condition (4.22) also to the monopole case.
52
S. STRINGARI
The polarizability sum rule is easily evaluated 01= 032 = k/9) PO):
by solving
Eqs. (4.10) for w = 0
(4.23) which, by virtue of Eq. (4.22), automatically In conclusion, one finds 150) SRPA -1
satisfies the boundary condition
= + ii/‘(r)
PO(r) dv.
(3.30).
(4.24)
In the isovector case, one proceeds in an analogous way. However, in this case low-lying states of surface type do not appear and consequently condition (4.22) is no longer requested. Equation (4.10) gives @ = 0, A= bvD, p,-,) :
where the term (f) = l/A Ifp, dv has been included in order to ensure the existence of a solution u satisfying the boundary condition (3.28). One can then write: sRPA -1
=
+
j+
i
(f-
U))’
~0
dv.
(4.26)
VOI
It is interesting to note that, in contrast to the sum rules S, and S,, the polarizability sum rule (4.26) is not affected by the isovector effective mass m, (Eq. (3.25)). Of course Eqs. (4.24) and (4.26) hold only in the limit of large systems. General expressions of the type of Eqs. (4.24) and (4.26) are not available in the GS model, though this sum rule can be evaluated case by case and compared with the HD predictions. In general, the following inequality holds: SGS -1, < SHD -I’
(4.27)
Rather general results in the framework of the HD and GS models can also be derived for the sum rule S _ 3. To evaluate S _ ) one expands the solution of Eq. (4.10) around w = 0 (m, = m in the ioscalar case): u = &(U, + u2u* + .*. ),
(4.28)
where u. and u1 satisfy the equations L(uo) = -vfpo
(4.29)
Uu,)
(4.30)
and = %UoPo,
NUCLEAR
respectively and L is the linear differential approximation) L(u) = g
Evaluation
53
FLUID-DYNAMICS
operator defined by (,u = 0 in the HD
= -(A + p) V(Vu) - ‘uv2u.
(4.3 1)
of a(w) then yields:
a(w)=-~u,V~,-w?ju,V~p,,+... . Using Eq. (4.30) together with the hermiticity Eq. (4.32) with Eq. (4.3), one finally finds:
property
= f i m,,uip, dv.
s-,
(4.32)
of L, and comparing
(4.33)
This result represents a clear example in which the sum rule S- 3 can be directly evaluated by using the solution of a constrained calculation [ 5 1, 3 11. To conclude this section, we discuss the case in which an excitation operator of the type F = Ci &q’ 6 ti is applied to an infinite system. In this case, the evaluation of sum rules is particularly simple and gives the following expressions, holding for low q (see 1521 for a recent discussion of sum rules in infinite systems): s,RPA 2&f
(4.34) (4.35)
S RPA -I=--
9 1 2 x
A
(4.36)
for the isoscalar case, and 1
sRPA 1
=--
sRPA 3
=++$A
S RPA -1
=&L
q2
A
2 m,,
I,’ VOI
(4.38) (4.39)
for the isovector case. We note that the compressional frequency (3.20) of the GS model is simply related to the ratio ml while that of the HD model, i.e., the frequency of first sound, is given by dm. Physically, this follows from the fact
54
S. STRINGARI
that in both models the plane-wave operator excites a single collective state exhausting the whole strength. In terms of the Laudau parameters we can write: (4.40) and (4.4 1) for the isoscalar
case, while for the isovector
case one has (4.42)
and (4.43) The coefficients F,, F, , Fh, and F; are related to the physical quantities entering in Eqs. (4.34)-(4.39) through the relations: X=9%
(f+fF,)
m*=m (1 +fFj) (4.44) h,,,=s
(f+fF;)
We note that Eqs. (4.34)-(4.39) and hence (4.4.0)-(4.43) have been evaluated starting from energy functionals of the type of Eq. (3.1). The Landau parameters F,(F;) associated with these functionals vanish for I > 1 and consequently the above results are correct only within this approximation (actually one can show that the sum rule S, is affected also by the Landau parameter F,(F;)). It is. interesting to compare the mean excitation energies (4.40~(4.43), defined in terms of the RPA sum rules, with the energy of the collective solution of the RPA equations of infinite systems in the long wavelength limit. In this limit, the RPA equations of motion coincide with the Landau theory equations [49, 53 1 and are particularly simple to investigate (for a recent review of the Landau theory, see 1541). Assuming that the Landau parameters F,(F;) vanish for 1 > 1, the dispersion relation for the longitudinal collective solution, known as zero sound, can be
NUCLEAR
FLUID-DYNAMICS
5.5
explicitly written as (in the isovector case one has to replace F, and I;, with F; and F; , respectively): $ln--
s+l S-l
1 F, + s*F,/(l + +F,) *
l=
(4.45)
The dimensionless quantity s=w
(4.46)
P,
is the ratio of the phase velocity of the wave to the Fermi velocity (v, = k,/m *). In Fig. 1 we plot the values of s given by Eq. (4.45) together with the prediction of Eqs. (4.40) and (4.4 1) as a function of the interaction strength F,, for two different values of the nonlocality term F, (in the nuclear case F, N 0.1). The figure shows that
-2
-1
FIG. l(a, b). The square of the reduced velocity different values of F,. The full line corresponds dotdashed line to S,/S ~, (first sound).
0
1
as a function of the interaction strength F,, for two to zero sound, the dashed line to SJS, and the
56
S. STRINGARI
(4.47) and that the ratio S,/S, very closely reproduces the square of the zero sound energy when F0 ti 1 (see also [36]). This result can be formally obtained expanding the solutions of Eq. (4.45) for F0 9 1 [54,55 J. One gets:
s,=s&+o ($ ) ) 2
0
where sds = d(3/5 + 1/3Fo)(l +F1/3) is the reduced velocity obtained from the collective frequency dm. Commenting on the role of the nonlocality term F,, we note that it dlects the condition for the existence of undamped zero sound. In fact, by looking at Eq. (4.45) one finds that this condition can be written as
When F, > j (as illustrated in Fig. l(b)), it is possible to have a solution of the equation for zero sound also if the usual condition F, > -1, ensuring the stability of first sound, is not satisfied. This example shows how critical can be the difference between the deformations responsible for zero and first sound. Using the GS model it is possible to investigate also the transverse sound. In this case, one can show [35] that the elastic description of the collective motion, as given by the GS model, is not as successfulas for the longitudinal case. It is interesting to notice that the main reason for this is due to the fact that the classical expression for the collective kinetic energy (2.11) is not adequate to describe the propagation of transverse sound. As for the longitudinal case, also for the transverse one. the GS model reproduces correctly two sum rules. However, one of these sum rules, the one associatedwith the collective mass parameter (2.11) takes an important contribution not only from the collective state, but also from the single-particle-type excitations occurring in the low-energy part of the excitation spectrum [561. This explains why the GS frequency results to be lower than the zero sound one (a similar situation occurs, in the longitudinal case for the hydrodynamic sum rule S_ ,). The elastic behavior of normal Fermi fluids and its implications on liquid 3He have been recently investigated in 155, 571.
5. SURFACE EFFECTS ON THE NUCLEAR MOTION In the preceding sections we have derived the equations of motion of the GS and HD models in the limit of large systems, i.e., neglecting surface contributions to the nuclear potential energy. The presence of the surface has been, however, taken into
NUCLEAR
57
FLUID-DYNAMICS
account to some extent through the imposition of boundary conditions to the solutions of the equations of motion. Complete calculations of nuclear oscillations, including finite-size effects, have been recently carried out in the framework of the GS model 134, 351. This approach should be compared with complete RPA calculations, which, however, include additional distortions of the Fermi sphere with higher multipolarity, as well as pure quantum effects like the spin orbit effect. In this section we try to include tine-size effects using an alternative approach. The idea is to add a surface energy term to the volume expression (3.15) which modifies the boundary conditions at the surface, without affecting the equations of motion in the interior of the nucleus. Such an approach, already used to study the effect of surface tension on isoscalar excitations [53,41], gives rise to a contribution to the stress at the surface which modifies the solutions of the equations of motion. In the following, we will focus on the isovector case. In Section 3 we have seen that a surface term (Eq. (3.22)) depending on the radial component of the displacement field, arises naturally in the isovector collective potential. However, such a term cannot be easily worked out in finite nuclei. For this reason we look for a more phenomenological approach in which the potential energy functional is written as:
(5.1) which generalizes Eq. (3.15) by including a surface contribution (58 ]. This surface term gives the nuclear stiffness against isovector polarizations at the surface, and its form is suggested by the comparison with the liquid droplet model (LDM) 1591, where the quantity iporo Q I surfr2 gives the surface contribution to the symmetry energy (7 = l/r,(R,(O~) -R,(&)) is the neutron-proton skin in units of r. and can be connected to the displacement field II of isovector deformations through the relation t = (2/r,) u . n). The comparison with the LDM provides a physical interpretation of the coefficient Q, which is related to the surface symmetry term entering in the semiempirical mass formula: 1 (N-Z)2 E sym = 2 A
b
b surfA -I/.? + ,.. b >(>I
VOI 1 -
(5.2)
through the relation (J = ib.,,,) b surf -=-b YOI Let us discuss
how the equations
9 J 4
of motion
(5.3!
Q' are modified
by the inclusion
of this
58
S. STRINGARI
surface term. By inserting Eq. (5.1) in the equations of motion (2.16) following equation in the interior of the nucleus:
one finds the
(A + p) V,(Vu) + pv2u, + mvpoW2Ul = 0, and the following
boundary
A(Vu)n+p
condition
Z~u+n~Vxu
(5.4)
at the surface:
+$pQ(u.n)n
r-R
0
= 0.
(5.5)
By comparing Eqs. (5.4) and (5.5) with Eqs. (3.24), (3.23), and (3.26), one notes that the presence of the surface term in the potential energy (5.1) does not change the equations of motion in the interior of the nucleus, nor the boundary condition (3.26) for the transverse component of the stress. Instead, the surface effects modify the condition (3.23), which now becomes: mu+2p-+
++FQm 0
r:A
= 0.
(5.6)
It is worth noting thst in the absence of surface symmetry energy (Q = co), the usual condition u . nlrZR = 0 is recovered. Equation (5.5) provides a straightforward way to include surface effects in the solutions of the equations of motion for isovector excitations. In the hydrodynmic limit, the new equations of motion generalize the classical Steinwedel-Jensen [S] model for isovector oscillations by modifying the boundary condition at the surface, which becomes: (5.7) The surface polarizability
effects are particularly easy to study if one evaluates the static in the HD model. In this case, one has to solve the equation
b,,,V . (Vu)= EVA-
(5.8)
with the condition (5.7). For excitation operators f with multipolarity ff 0 the solution is given by Vu = c/b,,,f which can be inserted in Eq. (4.11). Using Eqs. (5.7) and (5.3), we find the following expression for the inverse-energy weighted sum rule.
Equation (5.9) generalizes result (4.26), including the contributions to the polarizability sum rule given by the surface symmetry energy term. This formula was first derived for the dipole mode in [60], where a hydrodynamic approach to isovector modes was developed in finite systems. The same result has been also derived in the framework of the liquid droplet model [61].
59
NUCLEAR FLUID-DYNAMICS
6. SYSTEMATICS
OF GIANT
RESONANCES
In this section we discuss the physical results emerging from the solutions of the equations of motion; for the isoscalar case these results have been extensively investigated in recent years, so the discussion will be limited to some particular aspects. For the isovector case we will illustrate with particular emphasisthe dipole mode, the investigation of which cannot be considered exhausted despite the extensive work performed in recent years. In particular, the role of surface effects demands further theoretical work. A. Isoscalar Modes For each multipolarity 1, one can solve the equations of motion of the GS or HD models. While the hydrodynamic model presented here describesonly compressional (longitudinal) motions, the GS approach admits also transverse displacement fields of shear type. For a given multipolarity, the transverse and the longitudinal components are coupled by the boundary conditions (3.19). However, it may happen that the longitudinal (or transverse) components are dominant for some collective excitations. For example, it has been shown [40] that the first quadrupole state in the GS approach is essentially dominated by transverse components, its energy being almost independent on the value of the nuclear incompressibility. The compression modes are of particular interest for low multipolarities (I = 0 and 1= I), where irrotational divergency free motions are not possible (for I= 1 this motion would correspond to the translational motion). Conversely, the excitations dominated by the transverse components are expected to be the most interesting for multipolarities 12 2, since in this case the compression modes occur at higher energies. We shall try to derive an explicit estimate of the energy of compressionand shear modesin terms of sum rules. For compression modes, the HD approach can be profitably used to derive a mean excitation energy using the polarizability sum rule as ]50] : ! CL) compr
=
As,
h’s_,
.
By choosing the following multipolar excitation operators which statisfy condition (4.22) f = r’y,,(r2 -R’), one easily evaluates the sum rules S, and S-i (11yJ2 6 PR 471 m 21$5
s, =4-L
(6.2) d.C’= 1):
21+2
108 A 1 ‘-‘=4n~(21+3)(21+5)(~+7)~
(6.3) Z/+4
’
(6.4)
60
S. STRINGARI
from which one gets
Though surface contributions to the compressional frequencies are important [62-641, the relative location of energies with different multipolarities is expected to depend less critically on these effects. For example, one finds: &I
compr -pir= com!Jr
.-
15 -7
(6.6)
in rather good agreement with the experimental situation in “‘Pb, where wl’o - 13.5 MeV, and where some indications of isoscalar dipole strength have been recently found at w”’ - 21.5 MeV [65]. Prediction (6.6) agrees reasonably well with recent RPA calculations carried out using Skyrme forces [66] and turns out to be very close to the ratio between the energies of the first dipole and monopole excited states of the HD model [46] (cL$~‘,/w~~~~= 1.43). Incidentally, we note that prediction (6.5) for the monopole mode (I = 0) differs from the well-known expression w = &m [67, 68, 47, 121 by a factor of m. Since the latter expression corresponds, in terms of sum rules, to the ratio w = m, on concludes [49, 621 that for a large system the excitation strength of the operatorf= r2 is significantly fragmented. If one studies excitations dominated by shear terms using a sum rule approach, one should avoid to use operators f exciting compressional states. In this case, it is convenient to evaluate the excitation energy through the ratio:
0
=
(6.7)
f = r’ylm.
(6.8)
shear
using the choice
In fact, the scaling field u = (l/m) V’ which permits Eq. (4.20), is in this case divergency free. One finds:
to evaluate S, through
s, +Ls-/p2 s, =$---$(I-
1)(2f + 1)R21P4
NUCLEAR
FLUID-DYNAMICS
61
and hence w shear
=
(6.11)
Result (6.11) coincides with the expression found in 1691, where the restoring force and the collective mass of isoscalar giant resonances are determined imposing scaling deformations and assuming incompressible, irrotational flow. One should, however, remark that result (6.11) is valid only if one uses Skyrme type forces (Eq. (3.1)). An additional contribution, which cannot be expressed in terms of the Fermi energy, enters in the evaluation of the restoring force associated with a scaling deformation (S,) if one uses finite range forces. The presence of this additional contribution is related to the possibility that the Landau coefficient F, of the force is different from zero, and has been investigated for the quadrupole case in (70-72). In the quadrupole case (1= 2), Eq. (6.11) results rather succesfully in reproducing the general trend of experimental data (wz+ ‘v 654 ‘j3 MeV). This case represents a beautiful example in which the distortions of the Fermi sphere, taken into account in the GS approach yielding Eq. (6.11) and totally absent in the HD one, are crucial to determining the collective frequency. A careful analysis 1401 of the lowest quadrupole excitation in the GS model reveals the presence of important rotational components in the velocity field. These components tend to lower the energy of the giant quadrupole state by -15 % with respect to predictions (6.11). One should, however, be careful before carrying any conclusion about this effect. In fact, it has been recently shown [32-331 that the flow patterns given by the GS model for the quadrupole state differ from the predictions of microscopic RPA calculations, and this suggests that the GS model is not fully adequate to describe the fine details of this state, though it reproduces the sum rules (6.9) and (6.10) exactly. It is interesting to note that the irrotational fluid-dynamic model of [32], which also reproduces the sum rules (6.9) and (6.10), is in better agreement with the RPA calculations. As concerns the relative location of the compressional and shear modes, we observe that the former are always higher than the latter. For example, in the quadrupole case one has ~~~~~~~~~~~~~= 2 (for a recent theoretical investigation of the 2’ compressional mode, see [ 73 1). Isovector
Modes
The classification of collective states in the isovector case proceeds in a rather different way as compared to the isoscalar case, due to the different boundary conditions satisfied by the solutions of the equations of motion. In particular, one expects that compression type modes will still be present, but, unlike the isoscalar case, one cannot use irrotational and divergency-free displacement fields to classify the collective oscillation. In fact, in this case the contribution to the potential energy would become critically large at the surface. This situation occurs, for example, in the evaluation of the sum rule S, in the dipole case or, equivalently, in the evaluation of
62
S. STRINGARI
the restoring force due to instructive in the study of In order to classify the convenient to evaluate the
a Goldhaber-Teller deformation [74], and is particularly surface effects [ 12, 14, 741. isovector compressional modes (polarization modes), it is energy
using the HD model for calculating S-i. Differently from the isoscalar we were forced to satisfy condition (4.22), we here can choose:
(6.13)
(t > 0).
f = r’Yim
case, where
We find: s, E-$2$
lR2’-2 1
se,
1 --R2’
L-31
(6.15)
477 2 b,,, 21+ 3 and ‘b @CT) -4 -$1(2/f t,
(6.16)
(I > 01,
where m,, is given by Eq. (3.25). For I= 1 the formula gives the well-known Migdal’s result [ 751. Results (6.16) can be easily generalized by including the surface correction in S ~, (Eq. (5.9)). One finds 1761:
W(T) --4
$‘+(21+ L’
3)
1 1 f 5(2Zt 3) (b,,,Jb,,,)
A -I”
’
(6.17)
If one looks at the solution of the GS equations, the situation looks, however, more complicate. In fact, the coupling between the transverse and longitudinal components in the solutions of the equations of motion, gives rise to a fragmentation of the strength, especially pronounced in the dipole case [35]. The Dipole Case The HD model for the isovector dipole mode was successfully used by Steinwedel and Jensen (SJ) [5] to extract the law og = 79.4- ‘I3 MeV. which reproduces fairly well the peak energy of the dipole giant resonance in the region of heavy nuclei. The SJ result is obtained by solving the equations of motion (3.3 1) for I= 1 : u = W,(qr)yl,).
(6.18)
NUCLEAR
63
FLUID-DYNAMICS
The boundary condition (3.32) then gives qR = 2.08 for the lowest solution, which exhausts almost the whole strength of the dipole operator F = Ciry,,r3. Inserting this value of q in the dispersion law (3.27) and choosing b,,, = 50 MeV, and m,, = m, one gets the SJ result. The success of the HD prediction has strongly encouraged in the last years the use of classical models in the description of collective modes. However, one should remark that more sophisticated calculations of the dipole strength give a worse description of the experimental situation. In [27] it was first pointed out that the RPA strength for the dipole mode with typical Skyrme forces has two distinct peaks whose origin cannot be attributed to the fragmentation due to the single-particle effects, and which are not exhibited by experiments. This anomalous behaviour of the RPA strength has been recently investigated in a systematic way using Skyrme-type forces 1771. As discussed in [34] the GS model also predicts a strong fragmentation of the dipole strength. It is interesting to understand the origin of this fragmentation and compare the GS description with the HD one. As already discussed, the HD equations of motion automatically define irrotational solutions. They, however, admit divergency-free solutions containing rotational components occurring at zero energy. These solutions, which do not carry any strength of the dipole operator, can be written as: (6.19)
u = aV(v,m) +PV x W,(9r)JJ,,).
where L = -ir x V, and a relation between a, /?, and q is fixed by the boundary condition u . nlrER = 0. The energy associated with the displacement field (6.19) is removed from zero in the GS model due to the presence of the shear term in the potential energy. Furthermore, the coupling to the compressional fields (6.18) gives rise to the fragmentation of the dipole strength. It is also interesting to discuss the dipole mode in terms of sum rules. We first consider the sum rule S-, for which an explicit expression was derived in Section 4 using the HD model (f= l/22):
sHD--~A~ -1
The displacement field associated the solution of the equation
with the constraint
of irrotationality
“=Ei
20 L,
-
40
(6.20)
bv,,’
with the HD constraining
and the condition
z ( v (I)(r’-3rR2)+3+z(r2-R’)).
calculation
is given by
u . nlrZR = 0. One finds:
64
S. STRINGARI
Result (6.22) can be used to evaluate the sum rule S-, pff=-
(see Eq. (4.33)):
1 anz,.jg. 175 VOI
(6.23)
Finally, we notice that the transition density (2.3) associated with the displacement field (6.22) coincides, as expected, with the Migdal deformation [75] : dp, = const zpO.
(6.24)
Also, the GS Eqs. (4.10) can be explicitly solved. The resulting solution for o = 0, satisfying the boundary conditions (3.23) and (3.26) is given by
“=&C-v
YOI
~~)(r3+rR2)+;z(r2-R2).
Though Eq. (6.25) differs from the HD solution (6.21), the divergency and hence the transition density associated with Eq. (6.25) coincide with the HD predictions (6.21) and (6.24). As a consequence, one has SC’: = sy
.
(6.26)
Result (6.26) reflects also the fact that the field u of Eq. (6.25) satisfies the conditions (3.28), which imply the vanishing of the shear contribution to the potential energy. The above results show that inserting the field u of Eq. (6.25) in Eq. (2.2), one can explicitly construct a unitary transformation acting on the nuclear wavefunction yielding the Migdal deformation (6.24) for the isovector density and the Migdal value for the static polarizability. This transformation significantly differs from the Goldhaber-Teller one 1781, which is obtained choosing u = const Vz, and yields 6p, = const Vlp. The difference between the two fields (6.22) and (6.25) is reflected in the sum rule S _ 3, for which one find the following GS prediction 2 p =d?l~,g+“:. 315 VOI -3
(6.27)
This result is independent on the value of sF, and indicates that in the limit E, -+ 0 the lowest GS solution, which tends to a divergency-free solution of the type of Eq. (6.19) and whose energy goes to zero as &, carries a dipole strength ](O ( xi l/22,‘; 1n)] * proportional to e,3’2. In the limit eF+ 0 this mode does not contribute to S-i, but affects S _ 3. Until now we have ignored the role of the finite size of the nucleus, which demands a rather accurate analysis. Here we discuss only some general features of the problem. The inclusion of surface effects, as suggested in Section 5, yields the new boundary condition (5.6) for the radial component of the stress. The new term tends to lower the value of q, thereby lowering the energy of the dipole states and yielding a
NUCLEAR
65
FLUID-DYNAMICS
shape between the Steinwedel-Jensen (SJ) and the Goldhaber-Teller (GT) models for the transition density [79]. However, we note that, differently from the suggestions of [80], the solutions of Eq. (5.4) are not linear combinations of SJ and GT deformations. The predictions of the HD model, for which the dispersion law is given by Eq. (3.34) and the boundary condition by Eq. (5.7), are particularly illustrative. For example, assuming A = 208, the lowest solution gives qR = 2.08, 1.89, 1.74, and 1.61 for b,,,r/b,,, = 0, 1,2, and 3, respectively. Figure 2 shows the A-dependence of the dipole energy as predicted by the HD model using bsurr/bvo, = 2. The lowering of the dipole state due to the inclusion of surface effects is also reflected in the mean excitation energy o = v’m. In fact, Eq. (6.17) gives (1= 1): 1 1 + ‘,(b,,,f/b,,,)
A -“3 ’
(6.28)
Equation (6.28), which generalizes the Migdal result of Eq. (6.16), interpolates between the A - l/6 behavior of the Goldhaber-Teller model and the A -W behaviour of the Steinwedel-Jensen model in agreement with the systematics of the experimental dipole energies [ 8 11. Also the GS solutions are lowered by the inclusion of the surface term. Furthermore, the finite-size effects might be responsible for a redistribution of the strength among the first two states, as suggested in [34]. These effects could be investigated more systematically by solving, for example, Eqs. (5.4) and (5.5) for different values of b,,,f/b,,, . Comparison with full RPA calculations of the dipole strength might provide a useful test of the GS model.
ISOVECTOR
DIPOLE
RESONANCE
:~~~~~
20
60
FIG. 2. Systematics of the isovector giant prediction of the hydrodynamic model including Experimental data are from [Sl 1.
100 NUCLEAR
140 MASS
I80
220
resonance excitation energy. The full surface effects (b,,,(m/m,) = 71 MeV,
line gives the b,,,,/b,,, = 2).
66
S. STRINGARI
7.
CONCLUSIONS
One of the aims of this work was to illustrate the properties of some macroscopic models of fluid-dynamic type currently employed in nuclear physics. Though these models admit a completely classical interpretation, they can be derived starting from a microscopic point of view in which explicit use of a quantum energy functional is made. The microscopic theory considered in the present work is the random phase approximation or, equivalently, the linearized time-dependent Hartree-Fock theory. Several approximations and assumptions are explicitly introduced in the derivation of the macroscopic models. In the hydrodynamic case, the main assumption was the use of the Thomas-Fermi approximation or, equivalently, that the sphericity of the Fermi sphere in the momentum space is preserved during the nuclear motion. Physically, this assumption is justified only when the frequency of the oscillation is smaller than the typical frequencies of the collisions between particles which tend to bring everywhere a state of local equilibrium. In principle, this means that at temperature T = 0 the hydrodynamic model should not be employed to describe giant resonances in nuclei or the propagation of zero sound in infinite systems. However, if one studies the linear nuclear response to an external static field, one is authorized to use the HD picture, since in this case the sphericity of the Fermi sphere is preserved. As is well known, the linear static response a of the system is related to the polarizability sum rule S-i = 1/2a = C,, I(01 F 1n)l*/(w,, - wO). When S-i is dominated by a collective state (giant resonance), its evaluation is particularly useful, and this is expected to happen when F is an isoscalar operator exciting only compression modes, or when F is of isovector nature. A different situation occurs for the generalized scaling model in which the nuclear motion is described in terms of the classical equations of elasticity. In this case, the relevant assumption is a scaling deformation applied to the wave function (see Eq. (2.1) or (2.2)). This assumption is known to reproduce correctly the nuclear response to an external oscillating field when o -+ co. In terms of sum rules, one can show that the generalized scaling model correctly reproduces the sum rule S, = Cn(% - wJ3 l(lFl a29 evaluated in the random phase approximation. The GS model results are particularly successful in describing divergency-free motions of isoscalar nature where the HD picture is totally inadequate. In the present work, we have also emphasized the role of surface effects. In order to take into account these effects in an explicit way, we have modified the boundary conditions at the surface with respect to the usual form holding in the limit of large systems, without affecting the equations of motion in the interior of the nucleus. In the isovector case, we have shown that the new boundary conditions produce a lowering of the excitation energies with respect to the Steinwedel-Jensen predictions. Surface effects considerably modify also the polarizability sum rule, for which compact expressions can be found in the present approach. A classification of irrotational collective motions in nuclei has been finally proposed in Section 6. In particular, an important difference between the isoscalar
67
NUCLEAR FLUID-DYNAMICS
and the isovector case has been pointed out. While for isoscalar excitations one is naturally led to distinguish between compression and divergency-free oscillations (characterized by the incompressibility modulus and by the Fermi energy parameter, respectively), for the isovector excitations, only compression-type motions arise naturally in the framework of the present approach. Divergency-free oscillations are in fact hindered by the boundary conditions u . nlrzR = 0. Of course, the inclusion of finite-size effects can partially modify the above classification. We finally conclude by recalling some problems connected to the present work which remain to be explored: (a) The formalism has been here presented only for spin-independent excitations. Of course, one could derive equations of motion also for spin and spin-isospin excitations in analogy with the derivation of isospin modes. However, a realistic description of these modes should take into account the role of the spin-orbit force in an explicit way. A fluid-dynamic description of spin modes, including the spin-orbit effect, has not yet been formulated. (b) We have already commented in the text on the difficulties encounterd in order to derive the GS model starting from a variational principle in the framework of the TDHF theory. We stress again that the validity of the classical expression (2.11) for the collective kinetic energy of the nuclear system should be investigated in the case of states involving rotational velocity fields. Clearly, a systematic investigation of flow patterns and a detailed comparison with microscopic calculations (see 132-33 1) can provide a unique test on the quality of the predictions of fluid-dynamic models. (c) During the present work we have always ignored possible effects arising from the presence of a neutron excess in the nucleus (r,, = (N- Z)/2 # 0). In N # Z nuclei, new physical features arise that should be investigated. First, we expect a coupling between isoscalar and isovector modes. Second, the excitations in the T, = To f 1 nuclei are split with respect to the excitations in the T, = To nucleus, due to the role of symmetry energy. It is clear that a fluid-dynamic description of excitations in the isospin channels would be of great interest. (d) Finally, we have not discussed damping mechanisms occuring in giant resonances. These mechanisms have been rather extensively investigated in recent years and have also been considered in the framework of fluid-dynamic descriptions [69,82, and references therein]. However, the connections between microscopic and macroscopic approaches to nuclear damping cannot be considered sufficiently understood, and deserve further theoretical work.
ACKNOWLEDGMENTS I am particularly indebted to G. F. Bertsch for stimulating discussions discussions with 0. Bohigas and G. Holzwarth are also acknowledged.
and suggestions.
Helpful
68
S. STRINGARI
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