Nonlinear dynamics of nuclei and metal clusters
( ) F. CALVAYRAC(
1)
, S. EL-GAMMAL(
2)
, C. KOHL(
2)
P.-G. REINHARD(
2)
and E. SURAUD(
1)(
)
( 1) Laboratoire de Physique Mol´eculaire et Chimie Quantique, Universit´e Paul Sabatier 118 route de Narbonne, F-31062 Toulouse Cedex, France
( 2
) Institut f¨ur Theoretische Physik, Universit¨at Erlangen Staudtstr. 7, D-91077 Erlangen, Germany
(ricevuto il 29 Luglio 1997; approvato il 15 Ottobre 1997)
Summary. — We discuss and compare dipole resonance modes in nuclei and metal
clus-ters. In both systems, we observe well-developed collectivity with harmonic oscillations which persist high up into the regime of multiple resonances. Deformations tend to in-duce spectral fragmentation. In clusters, these effects are hardly strong enough to destroy all collectivity, except for the extreme of a fissioning cluster. Nuclear surface modes, on the other hand, are found to interfere strongly with the giant resonances being able to dissolve them completely.
PACS 24.30.Cz – Giant resonances. PACS 24.60.Lz – Chaos in nuclear systems .
PACS 33.40 – Multiple resonances (including double and higher-order resonance pro-cesses, such as double nuclear magnetic resonance, electron double resonance, and mi-crowave optical double resonance).
PACS 36.40.Gk – Plasma and collective effects in clusters. PACS 01.30.Cc – Conference proceedings.
1. – Introduction
The appearance of collective motion as coherent superposition of many single-particle modes is one of the most interesting and much studied dynamical features of many-fermion systems [1]. Typical examples are the Mie plasmon excitations in metal clus-ters [2,3] or their nuclear analogue, the giant resonances [4]. In both cases, there is nowa-days growing interest in the regime of large-amplitude motion associated with the exci-tation of multiple resonances. Those are often accessed by Coulomb exciexci-tation through a
(
)Paper presented at the 174. WE-Heraeus-Seminar “New Ideas on Clustering in Nuclear and Atomic Physics”, Rauischholzhausen (Germany), 9-13 June 1997.
(
)Membre de l’Institut Universitaire de France.
1176 F. CALVAYRAC, S. EL-GAMMAL, C. KOHL, P.-G. REINHARDandE. SURAUD
fast and highly charged ion; for a nuclear example see [5] and for metal clusters [6]. Ex-citation by intense laser pulses is an alternative option in case of clusters [7]. In any case, the new experimental possibilities motivate a theoretical attack to collective motion in the nonlinear regime. It is the aim of this contribution to report on recent investigations on nonlinear effects in nuclear giant resonances and plasmon resonances of metal clusters. Thereby we will concentrate on the question to what extent the resonance modes survive as harmonic oscillations and under which conditions a destruction of collectivity sets on.
2. – Time-dependent density functional theory
The only practical tool for a microscopic description of the dynamics of dense many-fermion systems is the time-dependent local-density approximation (TDLDA) where all complicated correlations are hidden into an effective energy-density functional and the dyamical equations for independent single particle states are derived by variation there-from. Linearized TDLDA has been employed since long time to compute plasmon res-onances in metal clusters [8, 9] and its theoretical foundations in the sense of a time-dependent density-functional theory are just being clarfied [10]. A much similar proce-dure has been used even longer in nuclear physics for extensive studies of nuclear heavy-ion reactheavy-ions [11]. It was called there time-dependent Hartree-Fock, which is somehwat misleading because the actual calculations use the Skyrme energy functional which rep-resents an effective interaction just as in the case of electronic TDLDA.
The time dependence of the (effective) single-particle states in TDLDA is given by i@
t
= ^
h, where the mean-field Hamiltonian depends on the accumulated densities. In the nuclear case, we employh
^
= ^
h(
; ;J)
with density, kinetic-energy density , and spin-orbit currentJ and the parametrization as outlined in [12]. In the cluster case, we use^
h= ^
h(
" ;
#
)
with the spin-densities" ;
# and the actual functional form from [13]. For the numerical solution, the single-particle wavefunctions and the mean fields are represented on a grid in coordinate space. For the nuclear case we use the code as described in [11] with an axial grid and Peacemann-Rachford propagator. For the cluster case, we use a 3D grid and an interlaced exponential propagator for kinetic and potential energy [14].
3. – Spectral analysis
A most prominent observable emerging from the dynamics is the spectrum of emitted photons. It is related to the power spectrum of electric- dipole oscillations of the many-fermion system. We compute it by taking a protocol of the dipole momentD
(
t)
during the dynamical evolution. For this, the dipole momentum is evaluated in an analyzing box confined to the vicinity of the system. The box diameter isR+6fm
for nuclei andR+8
a0 for Na clusters whereRis the system radius in both cases. The restriction to the analyzing box avoids perturbation of the signal from emitted particles flowing through the box to the (absorbing) boundaries. The dipole signalD
(
t)
is Fourier transformed into the frequency domain yieldingD~
(
!)
. The dipole power spectrum is then simplyP(
!) =
jD~
(
!)
j2 . It is the appropriate observable in the nonlinear regime as it is proportional to the rate of emitting photons of a given frequency!. It is to be remarked that a simple Fourier transform can produce artefacts if the original signalD
(
t)
was not fully dampened out at the final time. Some filtering is needed in that case. For a more detailed description of the spectral analysis see [15].Fig. 1. – Dipole power spectra for various nuclei and excitation mechanisms with average excitation mechanisms as indicated. Lower panel:36
S excited by an initial dipole shift with small (dotted) and large (full line) amplitude. Upper panel: dotted line:44
S excited by an initial dipole shift, full line: 36
S excited by a head-on collision of12 C+
24 Ne.
For the excitation mechanism, we have in mind the Coulomb excitation through a highly charged, fast ion passing by. A fast ion excerts a very short pulse to the system which delivers an instantaneous collective dipole excitation. We thus simplify the initial stage by describing the excitation as an instantaneous initial shift of the centers of mass, protons vs. neutrons in case of nuclei and the electron cloud vs. ionic background in case of metal clusters. Moreover, the instantaneous kick excites all modes with equal spectral weight and is thus ideal for exploring the general spectral pattern of a system [15].
4. – Nuclear collective motion
As first example, we consider the nuclear dipole giant resonance. The lower panel of fig. 1 shows the dipole power spectrum of36
S for the case of small and large amplitudes with average excitation energies as indicated. The small-amplitude case represents the usual nuclear excitation spectrum as it is also known from nuclear RPA calculations. We see a pronounced peak corresponding to the dipole giant resonance along thez-axis. This resonance obviously survives the strong excitation associated with the large-amplitude
1178 F. CALVAYRAC, S. EL-GAMMAL, C. KOHL, P.-G. REINHARDandE. SURAUD
case. The average excitation energy corresponds to as much as 3.5 resonance modes stacked on top of each other. The giant resonance resembles thus much a harmonic os-cillator where coherent excitation of higher harmonics is easily possible. The enormous robustness of the dipole giant resonance is supported by experiments where at least a double resonance excitation has been confirmed [5]. At second glance, one sees some modifications of the resonance peak at higher energy, namely a slight broadening which is quite an expected effect because the underlying mean field undergoes stronger fluctua-tions.
Variations of the conditions, however, show that this collectivity and harmonicity of the nuclear resonance modes can easily be perturbed. The upper panel of figure 1 shows two examples, a different nucleus with large proton-neutron asymmetry and the same nucleus as before but excited via a strongly interacting collision. The spectrum from the collisional case (full line) is much more noisy almost filling all frequencies up the dipole resonance. This is the typical result of all former studies of heavy-ion dynamics which has led to the conclusion that nuclear collective motion is very anharmonic and tends to chaos if driven in the nonlinear regime [17]. To understand these seemingly contradictory results, one has to remind that there is another class of collective modes in nuclei associated with de-formations of the surface and appearing at low energies in the range of 1-2 MeV. These surface modes are generally very anharmonic. It is obvious that strong changes in the nuclear shape on the way to compound nucleus formation excite a large fraction of (an-harmonic) surface modes together with the dipole resonance which eventually results is this noisy spectrum. The example shows that a chaotic or harmonic pattern of a nuclear spectrum can depend sensitively on the excitation mechanism. Another “destructor” of collectivity is the charge asymmetry of the ground state. The dotted line shows the result from the excitation of44
S
by a dipole shift. This nucleus has much more neutrons than 36
S
. And accordingly, one sees a more diffuse resonance spectrum for44
S
. Collectivity is not as badly destroyed as in the case of collisional excitation. In any case, the best chance to observe clean multiple resonances exists for pure Coulomb excitation of nuclei deep in the valley of stability.
5. – Plasmon resonances in metal clusters
The Mie plasmon in metal clusters has much in common with the nuclear giant res-onance. It is a collective mode of the center of mass of electrons against the positively charged background. The other features, however, differ. There is no analogue for the charge asymmetry and there are no electronic surface modes comparable to nuclear sur-face modes (the analogy would be here the ionic modes which run, however, at a much slower velocity and are dynamically decoupled from the resonance modes). On the other hand, the shape of a cluster can be tuned independently of the electronic motion by exert-ing constraints on the ions. This freedom will povide us with the variants in the followexert-ing studies of the plasmon resonance in Na clusters.
The lower panel of fig. 2 shows the dipole power spectrum for
Na
9 +in the ground-state configuration which consists of one ion at the symmetry axis followed by two subsequent rings each covering four ions, called thus the 144 configuration [18]. The ions are described by local pseudopotentials for which we take the local part of [19]. The ground-state con-figuration has been reoptimized for the given pseudopotential (for details see [20]). One sees clearly one all-dominating Mie resonance which stays stable high up into the multi-plasmon regime (the average excitation energy here would allow simultaneous excitation of four plasmons). The Mie plasmon is thus also a collective and robust harmonic mode,
Fig. 2. – Dipole power spectra in the small and large amplitude regime for Na clusters with various shapes. Lower panel: Na
9 +
ground state configuration (144 = 1 ion + ring of 4 ions + ring of 4 ions). Middel panel:Na
18 ++ !Na 15 + +Na 3 +
at scission point. Upper panel:Na
8 attached to a NaCl surface, isomer with vertical extension.
1180 F. CALVAYRAC, S. EL-GAMMAL, C. KOHL, P.-G. REINHARDandE. SURAUD
similar to the giant resonance in nuclei. There is the same broadening of the resonance due to enhanced fluctuations of the mean field. But this time the resonance peak is slightly upshifted with increasing amplitude. This is due to electron emission which leaves the cluster in a higher charge state which, in turn, enhances the frequency of plasmons.
A dramatic destruction of collectivity was seen in the case of nuclear collisions associ-ated with violent changes in the nuclear shape. It is hard to find similar fuzzy shapes in the ground states of metal clusters. But extreme deformations develop in the transient stages of fissioning clusters [16] and these should be inspected for spectral fragmentation. The middle panel of fig. 2 shows the power spectrum at the scission point of the process
Na
18 ++ !Na
15 ++ Na
3 +. Its shape contains large deformations with high multipolarity, and the effect on the spectra is obvious. There is an enormous spectral fragmentation al-ready in the small-amplitude regime, and excitation to large amplitudes induces additional broadening yielding a very noisy spectrum.
The above example shows that fuzzy shapes produce fuzzy electronic spectra. This particular scission configuration is, however, not easily accesible experimentally and one should search for other circumstances which could produce extreme deformations. Such a situation could be provided by a cluster attached to a surface. We have investigated the case of Na clusters on NaCl surfaces which was first studied in [22]. Planar (i.e. one layer) clusters have the preferred configuration due to the strongly attractive interface potential. But vertically extended isomers exist for clusters which have a magic electron number (8, 20, ...) [23]. Those isomers develop substantial octupole deformation from the attachment to the surface. The upper panel of fig. 2 shows the dipole power spectra for
Na
8on a NaCl surface (for details of the modelling and structural optimization see [24]). The spectrum shows indeed more fragmentation than the free cluster (see lower panel). But the resonance is still fairly well concentrated in one energy region around the peak. The reason is that the deformation of a cluster on a surface is larger than usually in the ground state, but does not reach by far the curious shapes during fission. On the other hand, the example demonstrates that plasmons persist to be a useful analysing tool for clusters on surfaces.6. – Towards collisional broadening
All above considerations have been confined to the TDLDA. This means that line broadening effects from Landau damping and from the low-energy modes are properly accounted for. The effect of two-particle collisions on the collectivity (or dissipation) has yet to be explored. An efficient way to sample the corresponding collision term is provided by stochastic TDHF which consists in occasional jumps between TDHF trajectories [25]. Just recently, we have started to implement that scheme with a jump strategy borrowed from the semiclassical Boltzmann- ¨Uhling-Uhlenbeck (BUU) approach [26]. First applica-tion to the case of nuclear16
O+
16O
collisions gives promising results. Figure 3 shows the quadrupole momentum as a function of time. One sees that pure TDHF (dotted curve) produces long- living oscillations of the quadrupole momentum, whereas the collisional damping in stochastic TDHF damps the quadrupole motion. The comparison with the well established BUU calculations (dashed curve) shows that the damping has the appropriate rate. We mention in passing, that stochastic TDHF does more than that. It produces an ensemble of events which describes also the fluctuations appearing always in connection
Fig. 3. – Time evolution of quadrupole momentum and variance thereof for a head-on collision of 16
O+ 16
OatE=A=50MeV. Full curve from stochastic TDHF, dotted curve from TDHF, dashed curve from semiclassical BUU.
with dissipation. For example, stochastic TDHF provides a reasonable amount of fluctua-tions of the quadrupole momentum in the above test case [25]. With further development, stochastic TDHF (which should be called, in fact, stochastic TDLDA) will provide a closed description of the resonance modes including their collisional widths.
7. – Conclusions
Nuclei and metal clusters both display a prominent dipole resonance mode which con-sists in collective vibrations of the centers of mass with opposite charges against each other. It is the giant dipole resonance in nuclei and the Mie plasmon in clusters. We have investigated the robustness of these modes when driving the excitation into the regime of large amplitudes and nonlinear dynamics. It turns out that these modes survive as harmonic oscillations even at high excitation energies, at least when taking place on top of a well bound ground-state configuration. Spectral fragmentation is induced by defor-mations. These are hardly strong enough to destroy collectivity in case of metal clusters, except perhaps for the extreme shapes appearing at the transient scission configuration of a fissioning cluster. The situation is different in case of nuclei. The nuclear surface modes show strong nonlinear effects (anharmonic couplings, transition to chaos) and they can couple dynamically to the dipole resonance. This perturbing channel is excited typically in heavy-ion collisions at small impact parameters, and subsequently the spectra emerg-ing from this excitation mechanism are extremely noisy leavemerg-ing no trace of collectivity. Furthermore, we have seen that collectivity is also lost for nuclei close to the drip lines with their large charge asymmetry. Altogether, we see much similarity between clusters and nuclei what the resonance modes as such are concerned. But there are substantial differences what the shapes and the coupling to the low-energy modes (surface vibration
1182 F. CALVAYRAC, S. EL-GAMMAL, C. KOHL, P.-G. REINHARDandE. SURAUD
We thank IDRIS and CNUSC for providing access to their parallel computers. We thank B. FISCHERfor providing us with the sequential version of the FALR Coulomb solver. Finally the authors would like to thank Profs. M. BRACKand S. K ¨UMMEL for enlightening discussions concerning the isomerism in Na clusters. Institut Universitaire de France and French-German exchange program PROCOPE (number 95073) are also acknowledged for financial support.
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