Deformed-jellium model for the fission of multiply charged simple metal clusters

A deformed-jellium model is used to calculate the fission barrier height of positive doubly charged sodium clusters within an extended Thomas-Fermi approximation. The fissioning cluster is continuously deformed from the parent configuration until it splits into two fragments. Although the shape of the fission barrier obviously depends on the parametrization of the fission path, we have found that remarkably, the maximum of the barrier corresponds to a configuration in which the emerging fragments are already formed and rather well apart. The implication of this finding in the calculation of critical numbers for fission is illustrated in the case of multiply charged Na clusters.

The fissioning cluster is con- tinuously deformed from the parent configuration until it splits into two fragments.Although the shape of the fission barrier obviously depends on the parametrization of the fission path, we have found that remarkably, the maximum of the barrier corresponds to a configuration in which the emerging fragments are already formed and rather well apart.The implication of this finding in the calculation of critical numbers for fission is illustrated in the case of multiply charged Na clusters.
The electrostatic repulsion in isolated positive multiply charged metal clusters may lead them to dissociate into aggregates with smaller sizes and charges.As a conse- quence of this phenomenon, multiply charged metal clus- ters MN~+ are usually observed in mass spectra only if the number N of atoms exceeds a critical size N that depends on the metal species M and on the charge state q.The critical numbers for multiply charged alkali-metal clusters up to q = 7 have been recently determined.'   The existence of these critical values can be explained by the competition between two main &agmentation mech- anisms for excited multiply charged clusters, evapora- tion of a neutral monomer and fission into two charged fragments, with preferential emission of &agments with a "magic" number of electrons (such as Lis+, Nas+, Ks+, or K9+) (Ref.5) due to electronic shell effects.Hot large clusters lose their excitation energy mainly evaporating neutral atoms, whereas for small clusters the preferred decay channel is asymmetric fission.Both &agmentation processes compete for cluster sizes around N"and under certain experimental conditions some MN+ clusters have also been detected below N~.
Old theoretical studies of cluster stability were based on pure energetic criteria which only involved the en- ergies of the initial and final states.
But the experi- mental findings indicate that cluster fission is a barrier- controlled process, and have prompted the interest of the- oreticians to obtain these barriers.
For large clusters monomer evaporation is favored over fission because the barrier against fission is larger than the heat of evapora- tion of the monomer.On the other hand, small clusters may undergo fission because the barrier height is, in this case, smaller than the heat of evaporation.Consequently, a fundamental understanding of the critical number N, requires the calculation of the fission barrier for the pro- cess M' '++M+ Np p ) and the same can be said for N, *, the critical number for spontaneous fission (N, is the size for which the fission barrier becomes zero; it is smaller than N, ).
Microscopic descriptions of the dynamics of the fis- sion process, based on local spin-density functional calculations for the electronic structure in conjunction with molecular dynamics simulations, have been performed for small doubly charged alkali-metal clusters.
These calcu- lations clearly show the inQuence of electronic shell efFects on the fission energetics and barriers heights (predom- inance of an asymmetric fission channel, double-hump fission-barrier shapes, etc).Whether or not these results can be generalized to large systems is difficult to ascer- tain, since microscopic calculations are not feasible for large multiply charged clusters.A systematic investiga- tion of &agmentation processes for clusters as large as Na44s + (Refs. 2 and 3) calls at present for much sim- pler, yet reliable methods to tackle the problem.One of the simplest methods is the liquid.drop model, in which the metallic character of the fissioning system is taken into account by explicitly concentrating the net charge on the cluster surface.Shell effects have been included in this model by applying the shell correction Inethod of Strutinsky (originally developed in nuclear physics r) to study symmetric and asymmetric fission of doubly charged silver clusters and of highly charged alkali- metal clusters.However, this type of calculation misses 0163-1829/95/51(3)/1897(5)/$06.00 51 1897 1995 The American Physical Society the sizable spillout of the electronic distribution beyond the cluster edge.
The aim of the present work is to address the study of the fragmentation of multiply charged simple metal clusters within a different approach, namely, the den- sity functional theory using a self-consistent deformed- jellium (DJ) model.One of the merits of this model is that the electronic spillout is self-consistently treated.

Previous work '
lies on the assumption that the fragments are already preformed at the early stages of the Gssion process, that is, before surpassing the barrier maximum.This means that to obtain the fission barrier height F one starts with a cluster of N -q valence elec- trons moving in the mean Geld created by two tangent jellium pieces with Np and p atoms, respectively, and the cluster &agmentation path is obtained by increasing the distance between those two jellium pieces, which were taken as spheres.We then get F as the difference be- tween the energy at the maximum of the barrier and that of the spherical parent cluster M~~+.Here, we shall substantiate the above crucial assumption of the two-jellium- sphere (TJS) model and, consequently, offer a sound and easy way to calculate the fission-barrier height for sim- ple metal clusters of (almost) arbitrarily large size and charge state.
For this purpose we have described the fission path by a series of deformed-jellium shapes connecting the initial configuration of the parent cluster with the final one corresponding to two separated.fragments.More pre- cisely, the positive background of the fissioning cluster is modeled by axially symmetric jellium shapes corresponding to two spheres smoothly joined by a portion of a third quadratic surface of revolution. 2 This family of shapes is characterized by the values of three parameters: the asymmetry 4, the distance parameter p which is proportional to the separation s between the emerging fragments, and the "deck" parameter A which takes into account the neck deformation.In this DJ model, the jellium self-energy and the jellium potential acting on the electrons, which are both analytical functions in the TJS model, have to be calculated numerically (see Ref. 24 for details).Given a cluster configuration defined by a set of values of the jellium parameters (b, , p, A), the density of the valence electrons is self-consistently calculated by minimizing the total energy of the system for this fixed jellium background configuration.To obtain the barrier corresponding to a fission channel we compare the re- sults following different fragmentation pathways, defined by the relations between the parameters A and p (A is fixed by the size of the final fragments).As a general criterion we start with a sphere (A = 1 -A, p = K) and follow the line corresponding to a cone capped with spheres (A = 1 -4 /p) up to a configuration where the neck starts to form.Next, we have chosen the line of fastest variation of the neck, which corresponds to the relation (A -1) -p2 = const, up to A = 0. From the theoretical results of Ref. 16 this fission path seems to be the one giving the minimum barrier height (see Figs. 2  and 4 below).Anyway, other parametrizations leading to cluster shapes much more compact (Fig. 1) or elongated (Fig. 3) will be compared.
As a Grst application of the DJ model, we have carried out some extended Thomas-Fermi (ETF) calcula- tions for the symmetric fission of Na42 + using the energy density functional of Ref. 20. Due to shell effects, sym- metric fission is predicted to be the preferred &agmenta- tion channel for this cluster.
The ETF approximation treats exchange and correlation effects in a local density approximation.
On the other hand, it includes the von Weizsacker quantum correction to the kinetic energy in addition to the local Thomas Fermi term.
We have used an effective value P=0.5 [in Hartree atomic units (a.u.)] for the coefficient of the von Weizsacker correction. 2 The semiclassical ETF method cannot ac- count for shell effects, which should be included in any proper study of small alkali-metal clusters in order to predict the most probable fission channel among all the possible p values [see (1)].However, for a fixed decay channel, the ETF method gives the correct value of the fusion barrier height B,i.e., the barrier height referred to the energy of the fragments at infinite separation.This has been explicitly shown in the case considered here, namely Na42 +: 2 Na2i+, by performing a fully self- consistent Kohn-Sham (KS) calculation within the TJS model, and it supports the use of the ETF approxima- tion to calculate B for that reaction.The fission-barrier height can then be obtained &om the relationship where LHy is the heat of fission AHy = 2E(Na2i+) -E(Na42 +), that is, the difference between the energy of the &agments at infinite separation and that of the spherical parent cluster.
Figure 1 displays the evolution of the total energy as a function of &agment separation when the fission path is described by means of intersecting jellium spheres (A = 1 -p).Plotted at the bottom of the figure are snap- shots of the positive background along the fission path at the corresponding s values.The dotted vertical line at s = 22 a.u.indicates the scission separation (touching jellium spheres in this case), i.e. , the starting point in the TJS model.For s ) 22 a.u. the jellium configura- tions correspond to two separated spheres (A = 0, p ) 1) and we recover the results of the previous TJS model.The 1/s curve (dashed line) is the classical Coulomb re- pulsion between the &agments as point charges and the horizontal line in the right side corresponds to the energy of the final state.The maximum of the barrier occurs at s 30 a.u. and the Gssion-barrier height has a value of 3.17 eV (0.85 eV for the fusion barrier height).
Figure 2 shows the barrier obtained for a different pathway parametrization in which the Gssioning cluster forms a neck at s 20 a.u.The scission distance is larger than in the previous case, but we obtain the same val- ues as before for the barrier height and for the position of the maximum.In the fission path corresponding to Fig. 3, the neck variation is slower than in the previous cases.This leads to a larger barrier height and scission occurs for a separation greater than the position of the maximum of the barrier.Within our model, the value of the fission-barrier height corresponds to the lowest result obtained (Figs. 1 or 2).
The calculations shown in Figs. 1 -3 constitute the key result of this study, which can be summarized as follows: within the jellium model, the saddle configuration corre- sponds to two disconnected, spherical in the present case, jellium pieces tied up by the electronic cloud.We would like to stress that this holds even in the study of sym- metric fission, in which two large clusters are emerging.
The conclusion would not be modified if a KS instead of a ETF calculation is carried out, since the maximum of FIG. 3. Same as Fig. 2 for a slower variation of the neck.the barrier occurs when the two jellium pieces are well apart, and humps and other structures originated by shell efFects are found at much shorter distances.' As a final comment, the chief importance of self-consistently incor- porating the electronic spillout to describe such extended saddle configurations should not be overlooked.
As a second example, we have also considered the asymmetric fission Na24 + ~Na2i+ + Na3+, whose ETF fission barrier is shown in Fig. 4. The solid line has been obtained using the decaying path corresponding to the jellium configurations schematically shown at the bottom of this figure.Scission occurs at 823 a.u.The dashed line corresponds to the intersecting-jellium-sphere fission path, as in Ref. 26 (similar to that of Fig. 1 although in the present case the jellium spheres have difFerent sizes), with the scission point at s 17 a.u.(the jellium config- uration shown at the top).Also for this asymmetric case The solid line corresponds to the fission pathway described by the jellium configurations schematically shown at the bottom, whose scission point occurs at 8 we verify that the simple TJS model yields the same fis- sion barrier height as the DJ model, F = 1.05 eV, with the position of the maximum at s 24 a.u.(the fu- sion barrier height B is, in this case, equal to 1.03 eV).
We conclude again that the maximum of the fission bar- rier corresponds to a configuration in which the emerging singly charged fragments are already formed and slightly tied up by the electron cloud.Thus, if the goal is the cal- culation of the barrier height and not the detailed shape of the barrier itself, one can obtain the correct results for the barrier height from the simpler model of two perfect jellium spheres with increasing separation between their centers (starting at the touching configuration).
The TJS model has already been used within the ETF method to calculate the critical value for the competi- tion between neutral monomer evaporation and the most asymmetric fission in Na~+ clusters, which turned out to be % = 36, in qualitative agreement with the exper- imental result N = 27.Consideration of other fission channels, like Na3+ emission, leads to a decrease of the critical number.The predicted critical size for sponta- neous fission of Na~+ clusters is ¹ 10.Another in- teresting result obtained for the fission of Na~+ is that the height of the ETF fission barrier can be estimated from the classical Coulomb repulsion between the two fragments, considered as point charges, and separated by a distance such that their electron densities begin to overlap.
Since for large clusters shell effects tend to play a di- minishing role, the semiclassical extended Thomas-Fermi model is expected to exhibit the main trends of excited medium-size or large cluster fragmentation.This is in- deed the case for highly charged alkali-metal clusters M~+, for which it is reasonable to assume that charged trimer emission is the predominant fission channel.The comparison between ETF and pure Coulomb barriers further suggests a simple approximation for the fission- barrier height: (5) where r, is the Wigner-Seitz radius of the metal and D3 is an effective distance that serves to take care of the large electron spillout beyond the jellium edge.We have verified.that the experimental critical sizes for Na~~+ clusters with q ) 2 are well reproduced by simple spher- ical extended Thomas-Fermi calculations in which one compares the heat of monomer evaporation with the fis- sion barrier obtained from Eq. ( 5) using an effective value D3 -18 a.u.(see Table I).Note that Ds gives, on average, the separation between the TJS contact config- uration and the point at which the pure Coulomb bar- TABLE I. Critical numbers for the competition between neutral monoxner evaporation and charged trimer emission in Na~~+ clusters, as obtained using Eq. ( 5) with D3 = 18 a.u., in
Although more complicated, the model presented in this paper can be extended to study multi&agmentation of clusters with q ) 3. However, we did not consider this possibility since no evidence for multi&agmentation exists in the literature.
We stress again that the ETF energy functional is un- able to account for electronic shell effects, which deter- mine in many cases the nature of the preferred fission channels, and a full Kohn-Sham calculation of the fission barriers for the different possible channels is desirable (this calculation has already been performed for selected channels in the TJS model i).Nevertheless, with ref- erence to Eq. (3), giving the fission-barrier height as a sum of two terms, those shell effects have a strong inHu- ence on the values of the heat of fission AHf (final state effect) but a relatively minor effect on B .This fact suggests the calculation of F by combining the ETF fusion barriers with the heats of fission obtained from Kohn-Sham calculations, which are easy to perform for the undeformed parent and for fragments infinitely sep- arated.This should provide a fair approximation to F In summary, in this paper we have calculated the barriers for fission of multiply charged clusters using a deformed-jelliurn model.As the main result we have found that the saddle configuration corresponds to two disconnected jellium pieces tied. up by the electronic cloud.
FIG.1.ETF Gssion barrier for Na42 +,' 2 Na2q+ ob- tained when the 6ssion pathway is described by intersecting jellium spheres up to the scission configuration (dotted line) and by two separated jellium spheres from this point on.The dashed line represents the classical Coulomb barrier and the horizontal line at the right side corresponds to the energy of the final state.
FIG.2.Same as Fig.1using elongated jellium configura- tions and imposing the fastest variation of the neck.
comparison with experimental values(Ref.3).Last column corresponds to the absolute critical sizes for spontaneous fission in the same approximation.