U. N&I-IERa, S. BJORNHOLMb, S. FRAUENDORF”, F. GARCIASd, C...

FISSION OF METAL CLUSTERS

U. N&I-IERa, S. BJORNHOLMb, S. FRAUENDORF”, F. GARCIASd, C. GUET”

” Max-Planck Institut fur Festkiirperforschung, Heisenbergstrasse I, D-70569 Stuttgart, Germany b The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100, Copenhagen 0,

Denmark ’ Institut fur Kern und Hadronenphysik, Forschungszentrum Rossendorfl Postfach 510 I 19,

D-01 314 Dresden, Germany d Departament de Fisica, Universitat de les Illes Balears, E-07071 Palma de Mallorca, Spain

e Departement de Recherche Fondamentale sur la Matidre Condensbe. Service des Ions, Atomes et Agregats, CEA, Grenoble, I7 Rue des Martyrs. F-38054 Grenoble, Cedex 9, France

AMSTERDAM - LAUSANNE - NEW YORK - OXFORD - SHANNON - TOKYO

PHYSICS REPORTS

EJSEVIER Physics Reports 285 (1997) 245-320

Fission of metal clusters

U. Nghera,‘, S. Bjnrrnholm by*, S. Frauendorf c, F. Garciasd, C. Guete a Max-Planck Institut fir Festkiirperforschung, Heisenbergstrasse I, D-70569 Stuttgart, Germany

b The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark ’ Institut fur Kern und Hadronenphysik, Forschungszentrum Rossendorf; Postfach 510 119,

D-01314 Dresden, Germany ‘Departement de Fisica, Universitat de les Illes Balears, E-07071 Palma de Mallorca, Spain

“Departement de Recherche Fondamentale sur la Mat&e Condensee. Service des Ions, Atomes et Agregats, CEA, Grenoble, 11 Rue des Martyrs. F-38054 Grenoble Cedex 9, France

Received May 1996; editor: J. Eichler

Contents

1. Introduction 1.1. Surface and Coulomb forces

1.2. Metals versus nuclear matter 1.3. Experimental and theoretical status

1.4. Main findings

1.5. Outline 2. Fission of metal droplets

2.1. Rayleigh model 2.2. Q-values 2.3. Interacting-sphere models: Polarization

effects 2.4. Choice of model description 2.5. Size dependence of fission barriers 2.6. Multiply charged clusters 2.7. Influence of the model parameters on the

barrier heights 2.8. Conclusions

3. Density functional calculations of fission barriers

248 248 250

251 254

255

255 255 258

259

264

265 265

266 267

268

3.1. The spherical jellium model and the extended Thomas-Fermi approximation

3.2. Calculation of the fission barrier

3.3. The two-sphere-jellium model 3.4. Chemical bond model for the outer barrier

3.5. Shell effects 4. Experimental evidence

4. I, Review of experiments 4.2. Analysis and interpretation of experiments

4.3. Summing up 5. Future perspectives

5.1. Towards fissility X = 1, the nuclear

connection

269 270 271 274

276

278 278 290

301 303

5.2. Beyond fissility X = 1 6. Conclusions Appendix A. The classical image charge model Appendix B. Experimental bulk properties of

monovalent metals at different temperatures References

303 307 311 313

315 318

* Corresponding author. ’ Present address: Siemens AG, Otto-Hahn-Ring 6, D-81739 Mtinchen, Germany.

0370-1573/97/$32.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved PZI SO370-1573(96)00040-3

U. Niiher et al. I Physics Reports 285 (1997) 245-320 247

Abstract

Experimental results on the fission of doubly and other multiply charged metal clusters are reviewed and examined in the light of a simple model, where the fission barrier is approximated as two charged spheres in near-contact, at a mutual distance given by the balance between Coulomb repulsion and attractive polarization effects. The barriers are estimated for different mass and charge splits and are compared with the activation energy for the competing evaporation process.

From the mode1 as well as in experiment one finds a strong preference for singly charged trimers (with two electrons) in the “fission” channel, but also fragments with the higher magic electron numbers 8 and 20 may occur with enhanced abundance. In addition, there is a pronounced odd-even effect. In most of the experiments that have been carried out so far, fission occurs as the termination of a chain of evaporations of neutral atoms. This limits the observations to a range where the surface energy dominates over the Coulomb energy of the fissioning cluster, explaining the tendency for asymmetric fission and justifying the two-sphere barrier approximation. Conditions favoring symmetric fission and other fission modes specific to highly charged metal drops are discussed, and experimental approaches are suggested.

PACS: 36.40.-c

Keywords: Fission; Metal clusters; Statistical decay; Polarization; Shell effects

248 U. Nliher et al. I Physics Reports 285 (1997) 245-320

1. Introduction

I. I. Surface and Coulomb forces

Fission is a process where the analogy between nuclei and metal clusters immediately suggests itself. In both cases, a spherical droplet of sufficiently high charge will become unstable towards the division into two or more fragments. This can be described with a high degree of success by picturing the nucleus or the cluster as a classical charged drop, deforming under volume conservation through elongated shapes that ultimately develop one or more constrictions and snap into separate droplets. Initially, the short-range cohesive forces in the surface successfully resist deformation, but if it happens anyway because of thermal agitation, the long-range electrostatic force will get the upper hand at some point and drive the system apart. Thus the competition between short-range attractive forces and long-range Coulomb repulsion leads to the existence of a barrier, separating bound states from fissioning configurations, cf. Fig. 1. The relative weight of repulsive and cohesive forces is expressed through the fissility parameter [I]

When the repulsive electrostatic energy equals twice the cohesive surface energy, i.e. when the fissility parameter equals 1, the barrier is 0. For smaller values of X, fission from internally excited nuclei or clusters can be observed as long as the barrier remains comparable to or smaller than the activation energy of other decay modes, such as evaporation of neutral particles [2].

When the instability towards fission is described in terms of a classical liquid drop [3] there is a considerable analogy between nuclei and metal clusters, although nuclei are uniformly charged droplets while metal clusters are surface-charged objects. The analogy can be extended further by

Fig. 1. Fission barriers of uniformly charged classical drops. The spherical equilibrium shape is at a local minimum, but the interplay of short-range cohesive forces and the long-range electrostatic repulsion allows the system to separate after passing a maximum in deformation energy. For high fissilities, where the electrostatic energy dominates, the barrier towards symmetric divisions forms a saddle which is stable against asymmetric distortions. The shape of the saddle configuration varies from a single sphere to an elongated dumbbell as the fissility decreases.


including the effects of shell structure, which is associated with the quantized motion of the constituent delocalized nucleons or electrons in a mean field confined by the surface of the droplet.

In nuclear physics a separation of the total energy of the charged quantum droplet, expressed as a function of its size and shape, into a classical (liquid drop, LD) part and a quanta1 (shell) correction has proved extremely useful [4,5]:

G,,(N, Z, shape) = &D(N, 5 shape) + ~sheil(~~ Z shape) . (1.2)

The parameters of ELo(N,Z, shape) are smooth functions of neutron and proton numbers (N,Z) and express the average- or bulk-properties of the nuclear medium. By contrast, EShell(N,Z, shape) expresses the oscillations due to the quantized motion of the particles in the nuclear mean field. This term is usually obtained from a calculation of the single-particle eigenvalues by subsequently forming the sum of eigenvalues corresponding to the filled orbitals and extracting the oscillating part, [4]. A similar procedure suggests itself for metal clusters in molten form. Whether this general description is applicable also to small clusters, or to solid clusters of large sizes, where the ionic structure plays a role, is an open question.

A major problem is to describe the sequence of shapes that the charged quantum drop undergoes during fission. In particular, the barrier-, or saddle-shape together with its energy are both important. The energy, because it determines the fission decay rate in competition with evaporation processes, and the shape, because the shell energy correction cannot be determined unless the barrier shape is known. Generally speaking, the electrostatic energy favors a symmetric saddle shape which may lead to fission into two equal halves, while the surface energy favors a saddle shape composed of an infinitesimally small sphere in contact with a large sphere, with highly asymmetric binary fission as the result. In real nuclei or clusters, this classical prediction of extreme asymmetry is modified by the quantization of charge and mass in the direction of predicting a small, but nevertheless finite fragment size. The resulting asymmetric saddle shape remains close to two spheres in near-contact [3].

When surface and Coulomb forces act together, the result naturally depends on their relative strength. For a sufficiently high Coulomb energy, with X-values approaching 1, there is a distinct fission channel favoring symmetric fission relative to binary divisions that are somewhat less symmetric. Nevertheless, the symmetric fission always finds itself in competition with highly asymmetric “fission” channels. In nuclei this is seen as coexisting alpha particle emission and fission from heavy compound nuclei. At lower Coulomb energies, symmetric fission will have the highest barrier of all binary divisions. In this case, symmetric fission will cease to exist as a distinct decay channel. Fig. 2 illustrates the situation. Calculations of the barriers for uniformly charged drops (i.e. nuclei) show that the transition from Coulomb-dominated, distinctly symmetric fission (larger X-values) to surface-dominated, asymmetric fission (low X-values) occurs at fissility

x = 0.39 ) (1.3)

the so-called Businaro-Gallone point [3]. Light nuclei lie below this point, and here the compound nucleus decay through charged particle emission is dominated by alpha (and proton) decay with slightly heavier particles in weak competition. This decay pattern continues with the more heavy nuclei that lie above the BusinarwGallone point. Only for the very heaviest nuclei has the barrier against symmetric fission become low enough to allow this mode to compete effectively with proton and alpha decay, i.e. for X 2 0.65.

250 U. Ndher et al. IPhysics Reports 285 (1997) 245-320

x d 0.39

CHARGE and MA!% ASYMMETRY

x -00.8

CHARGE ond MASS ASYMMETRY

Fig. 2. The barrier height against nuclear fission as a function of charge (and mass) asymmetry for a fixed initial size. When fission is dominated by the energy of surface deformations (low X-values), asymmetric fission will have the lowest

barrier and dominate fission decay, left hand figure. For large X-values, the electrostatic energy dominates and the barrier towards division into two equal charges, i.e. symmetric fission, becomes small, right-hand figure. For metallic droplets, the region where this happens is expected to be even more narrowly confined to X-values close to one.

1.2. Metals versus nuclear matter

The fissility parameter can also be expressed as

x = (Z2/N)l(Z2/N),,i, >

where (Z2/N)Ctit, corresponding to X = 1, is

(1.4)

(Z2/N)crit = 1 &q&c/e2 (1.5)

for a metal. Here u,, is the Wigner-Seitz radius, e is the elementary charge, and G is the surface tension (both r,, and 0 are temperature-dependent quantities). For the nuclear medium (Z2/N),,, M 50, while for metals a typical value of (Z2/N)crit is 0.50, i.e. one hundredth. This is a reflection of the extraordinarily strong cohesion and surface tension that one finds in nuclei by virtue of the strong force, while in metals the cohesion is of electromagnetic origin like the Coulomb repulsion. For practical work this means that the discreteness of the cluster charge is felt very strongly.

The different nature of the two media, nuclei and metals, makes the study of how metal clusters undergo fission especially interesting.

Metal clusters can be neutral, nuclei cannot. As a result, nuclei with more than 300 nucleons are unstable and do not exist. The cohesive energy of the nuclear medium has furthermore a distinct maximum for equal neutron and proton numbers with narrow margins for deviations. Within the allowable margin, the Coulomb energy nevertheless favors a small neutron excess in heavy nuclei. The so-called symmetry energy ensures at the same time that the local density of neutron and proton matter remains essentially constant under all shape changes; polarization effects are negligible. In the process of separation, two nuclear fragments are therefore repelled as essentially two point charges, located at the respective centers of mass, while the mutual attraction, being limited by the range of the nuclear forces, has the character of a proximity force between surfaces. By contrast, the


charge-to-mass ratio of a metal cluster is a much more free parameter. It may even change sign. In practice, it is limited only by the available experimental tools. Also the local charge-to-mass ratio will vary, since the excess charge tends to be located at the surface [6]. As a result of this, the production of two fragments with radically different charge-to-mass ratios is readily possible. In addition, the electronic charge is easily polarized under the influence of the charge of a nearby fragment, leading to long-range attractive interactions of the image charge type. This will significantly modify the primary Coulomb repulsion between the centers-of-mass and consequently the barrier height.

Most work so far has been done with doubly positive cluster ions. In this case the fission process must necessarily be symmetric with respect to charge. But again, as opposed to nuclei, it still leaves open the mass asymmetry as a virtually independent parameter. More generally, we thus have two asymmetry parameters in binary cluster fission, one for charge and one for mass. The position of the Businaro-Gallone point, below which the saddle shape becomes instable towards asymmetric distortions, is also expected to be different for metals. The high mobility of the surface charge will push the Businaro-Gallone point to considerably higher X-values. For even higher X-values - in excess of unity - the tendency for direct fission decay into a large number of small fragments may, furthermore, be more pronounced in the case of metals than it is for nuclei [7]. The similarity that remains after all is the nature of a Fermi liquid with delocalized fermions that is common to both nuclei and (molten) metal clusters. One therefore anticipates that modulations of the liquid drop trends due to shell effects will play similar roles in the two cases.

1.3. Experimental and theoretical status

Historically, the study of cluster fission began, not with metals, but with van der Waal’s clusters and other media bound by dipole forces [8]. The basic observation was that doubly charged species, recognized as peaks with half-integer values of the charge-to-mass ratio in time-of-flight mass spectra, did not occur below a lower bound, loosely denoted the appearance size, N,, or more precisely, the effective appearance size, Niff. Prior to time-of-flight mass analysis, a neutral beam of clusters was multiply ionized and inevitably heated in the same process. Before entering the acceleration stage, the heating results in thermally activated emission of neutral fragments. In this way the size of the clusters is reduced without affecting the charge. At some point, however, fission, i.e. emission of charged fragments, enters into competition with the evaporation of neutrals, resulting in the disappearance of small clusters with multiple charge. In the present report, the real appearance size, N,, will be identified as the size where thermally activated emission of neutral and charged fragments, respectively, are equally probable.

There are many interesting observations in connection with the fission of van der Waals and other polar-bound clusters, e.g. Ar, Kr, Xe; or COz,02,N2, CO [9, lo]. On the other hand, they are not directly relevant to the fission of metal clusters. The first group are clusters built from basically neutral elements and held together by static or dynamic dipole forces. The introduction of one or more electric monopoles in the form of a net charge modifies the conditions of binding drastically and complicates the description of the fission process. Similar complications are not anticipated with metal clusters, because they are built of a large number of charged species (ions and electrons), where a few charges more or less is unlikely to cause any radical change in binding.

252 U. Niiher et al. I Physics Reports 285 (1997) 245-320

Most experiments performed so far with simple metal clusters are dealing with sizes close to the appearance size N,. An early study of silver clusters, produced by sputtering [ 111, presents the so far most complete picture of the size distribution of singly charged fragments from doubly charged parents containing from 12 to 22 atoms. There is evidence of a preference for fission products with an even number of electrons (i.e. odd mass) as opposed to those with an odd number (i.e. even mass). Among the odd-mass fission fragments, those corresponding to magic electron numbers 2 or 8 have a certain prominence, but other products are also present. (More recent investigations [ 12-141 performed with an ion trap are at variance with this, indicating a strong preference for charged trimer emission).

In a study of doubly charged gold clusters produced by a liquid metal ion source [ 15, 161 a tendency for even-electron clusters to fission rather than evaporate neutral atoms, and vice versa for odd-electron species, is very clear. Superposed on this odd-even staggering is a strong increase in the fission-to-evaporation rate as the cluster size is reduced from Au&+ to Au:.+. Finally, emission of a charged trimer is found to be the dominant fission channel, although not the only one. These results are analyzed [ 15, 161 in terms of the liquid drop model for symmetric fission. See also

WI* There are several studies of the alkali metals. Asymmetric fission dominates, and the fission-

evaporation competition is described in terms of the threshold energy for evaporation relative to the estimated fission barrier height, including experimental, independently determined shell energy corrections [ 17-191. The model used in doing so is that of two spheres in contact [20]. It is sketched in Fig. 3. With an estimate of the ground-to-ground state energy difference of initial and final states Qr, the fission barrier is calculated with the help of Coulomb’s law:

B+- = B, - (E, - Ef ) = B, - Qf , (1.6)

where B, = ZlZ2e2/(R, + R2 + s,). Here, s, is a parameter that accounts for the mutual polarization of the fragments. This model also finds support in a detailed and very comprehensive study of all the alkali metals, which involves charges in excess of seven units and sizes up to 500 atoms [21, 221. Here, 18 different effective appearance sizes, spanning the entire size space and charge range, are found to have closely similar (Z2/N,“ff)-values and fissility parameters, with X lying in the interval 0.38-0.28. The authors of [21,22] examine the fission-evaporation competition with the help of the two-sphere model of Fig. 3 using a purely classical liquid drop description without shell corrections, but taking into account, however, the quantization of charge, the weak coupling between mass and charge asymmetry, and the special polarizability of the charge distribution in metals. More recently, the Stuttgart group has performed similar experiments with clusters of the alkaline earth metals [23]. Much of this report will be devoted to a closer examination of the above and related models, including the role that shell structure in the initial and final states has in shaping the details.

There are a few experiments where fission of clusters with X-values close to unity and even in excess of this critical value are examined. In such experiments, there is no competition from evaporation, instead they rely on techniques where the charge is suddenly increased. For example, singly charged potassium clusters in the size range N = 5-12 have been made doubly charged with the aid of a strong pulse from a UV-laser; and it has been possible to observe how they fission [24]. Another method is to multiply ionize a metal cluster by charge transfer in a peripheral collision with a highly charged atomic ion [25]. By this method it is possible to reach very high charge states

U Niiher et al. IPhysics Reports 285 (1997) 245-320 253

V(R) !ENERG’f Sod/e

I- 4

I i

Two Fragments

Liquid drop Coulomb barrier

0 N6 (N,+j~)~~‘~2

Fig. 3. Model of (asymmetric) fission for low X-values. The saddle configuration can be approximated by two spheres a certain distance apart. When this barrier Br(N) is comparable to the activation energy for emission of neutral particles DI (N)‘, shown to the left, fission will compete with evaporation. The fission barrier height is determined by the energy release in the process (i.e. Qr-value) and the electrostatic energy B, needed to reach the saddle from the outside, see

Eq. (1.6), and insert to the upper right. The energy release Qr can be estimated by using the bulk liquid drop model (full-drawn horizontal lines). This energy can be corrected by adding (negative) shell correction energies to the initial

and final states, &h&(N) and A&&Nr , Nz), respectively. The outer Coulomb barrier is then shifted, and the dashed lines show the resulting barrier. Analogous considerations apply to the activation energy, DIG.

within a collision time of a few femtoseconds [26]. When X is close to unity, it is anticipated [5] that shell structure in the initial and final states - and also for shapes in the vicinity of the barrier - will influence the fission process in interesting ways. This is what happens in nuclei, where the existence of an island of superheavy elements (2 ~114) depends on shell stabilization of the nuclear ground states against fission. It is also well known that the fragment distribution from uranium fission deviates from the expected symmetry due to quantum size effects. The fission barrier may also split into two with an isomeric energy minimum in between [27]. Regions of size and charge where one can expect situations similar to this for metal clusters will be discussed.

The experimental studies have developed hand in hand with theoretical investigations. Some of the latter address the changes necessary in the liquid drop description, when going from a uniformly charged (nuclear) liquid to a charged metal droplet [7,28]. Others focus on the shell structure for shapes relevant to fission. This requires a minimum of three shape variables: an elongation parameter,

254 U. N&her et al. I Physics Reports 285 (1997) 245-320

a neck parameter, and (mass) asymmetry. There is no universally applicable parameter set of this kind. For small X-values, two spheres merging into one, possibly with a hyperboloid of revolution connecting the spheres, will be adequate [29,30]. For intermediate X-values, a parametrization based on Cassini ovaloids, with the Lemmiskat as a separatrix, seems realistic for a description of the saddle shapes [7,3 11; and for X-values close to unity, the parametrization in terms of the cylindrical variables (B, C, y), has proved to be effective [5]. This latter model has been used in several studies [32-341. The same is true of the two-sphere model [35,36], which no doubt is the one most adequate for comparison with experiments involving fission-evaporation competition (i.e. low X-values).

Finally, a judiciously adjusted molecular dynamics model has been applied to the fission of small alkali clusters with large X-values [24,37].

1.4. Main jindings

Our main result is that the picture originally proposed in Refs. [ 17-191 and sketched in Fig. 3 is an adequate starting point for the description of observations that involve the competition between evaporation and fission. The important quantities are the threshold for evaporation D,(N)Z typically ~1 eV, relative to the fission barrier height &, which again depends on the fusion barrier B, and the energy release in fission, Qr, according to Eq. (1.6). With the exception of D,(N)’ these quantities depend sensitively on the charge and mass asymmetry in the fission process and thus have to be evaluated for different charge and mass splits in terms of a liquid drop (bulk) part and a shell correction. Ideally, the comparison between model predictions and experiment, requires a definition of all these quantities to a precision distinctly better than k,T or typically much better than 0.03 eV. This goal turns out to be unreachable at present. Neither a purely theoretical approach based on local density models nor estimates based on experimental bulk parameters, or on independent experiments with clusters, allow such a high precision. The model predicts, on the other hand, a number of systematic trends that are less sensitive to the model parameters. Thus the emission of a singly charged trimer is predicted to be the preferred liquid drop fission channel, and this is confirmed experimentally. The metallic character of the clusters as expressed by the strong mutual polarization of the two nascent fragments and the ensuing reduction of the barrier height is also born out by the experiment. It appears though that the polarization calculated from the classical image charge model of two spheres with diffuse surfaces somewhat overestimates the actual polarization. The influence of shell structure and odd-even effects are also clearly seen. The extra stability of the magic fragments Mel and Me& is at least in one case, Li&+, sufficient to overcome the liquid drop bias against nearly symmetric mass splits [38]. Attempts to quantify the observations in terms of the influence of the shell energy on the Qr-values and thereby on the fission barrier height have, on the other hand, met with limited success. It appears that the saddle configuration suggested in the two-sphere model, Fig. 3, is too simplistic, and that the shell stabilization in the final state is not fully developed at the saddle, especially for near-symmetric mass splits. A new generation of experiments with much improved precision is required to settle these questions. An effort in this direction should be weighed against the interest in studying the fission of highly charged clusters with fissility near or even beyond unity. An examination of the latter scenario points to a number of interesting possibilities for future experiments.


1.5. Outline

The next section will examine a purely classical model of two charged spheres with either sharp surface profiles or with diffuse surfaces, as a representation of fission for low X-values. Special attention will be devoted to their mutual polarization interactions, and to the sensitivity of the numerical results to the precise values chosen for the bulk surface tension, the electrostatic energy and for the diffuseness. Symmetric fission, in the framework of the classical, sharp surface, liquid drop description, is also examined and compared to the two-sphere model.

In Section 3, this classical model is compared to results of density functional calculations of the barrier heights and shapes, both in the Thomas-Fermi approximation and in the Kohn-Sham formalism. Similarities and differences will be discussed, among them the effect of shell structure.

Section 4 reviews the experimental data. An attempt will be made to separate shell oscillations from the average trends. Special emphasis will be placed on identifying areas of virtual agreement as distinct from areas where there are clear discrepancies between model and observation.

Section 5 presents speculations about a new area of cluster fission where the fissility is almost equal to 1. Here, special attention will be paid to regions of (iV,Z)-space where one should expect symmetric fission and where the barrier heights and the fragment mass distributions might be influenced by the existence of closed shells in the initial charged cluster or in the final fragments, under the assumption of a close analogy to the fission of heavy nuclei.

The speculations are taken a step further by discussing the physics of metal droplets with charges in excess of what is required to reach fissility 1, together with methods for producing such clusters.

Section 6 presents the conclusions.

2. Fission of metal droplets

2.1. Rayleigh model

The classical liquid droplet model is at the base of all approaches to the fission processes in nuclear as well as cluster physics. This model was conceived at the end of the last century by Rayleigh [l], and its ideas are still essential for understanding fission, as has been clearly demonstrated in nuclear physics.

Of course, metal clusters are different from nuclei because they should be treated as conducting droplets and not as homogeneously charged objects. The absence of the mass-to-charge coupling (as for neutrons and protons) leads to polarizability effects that have a strong influence on the fission of metal clusters. In addition, mass and charge asymmetry become independent variables.

There are two different approaches to the description of the fission barrier height and the shape of a charged droplet at the saddle point. One possibility is to start with the parent cluster (which we assume spherical) and study the change in total energy during the deformation. This is a good method as long as the barrier towards fission is low, i.e. the X-value is close to unity, because then the barrier shape is not far from spherical, cf. Fig. 1.

On the other hand, there is the approach from the direction of the fragments. This uses the knowledge of the @-value, i.e. (Ei - Er) in Eq. (1.6), and calculates the barrier by integrating the repulsive forces between the charged fragments, Fig. 3. This is a realistic method if the X-value is


low and the barrier shape corresponds to two fragments that have lost contact. For thermally induced, highly asymmetric fission in competition with evaporation of neutral particles, this condition may well be fulfilled.

The Rayleigh model refers to the former case, X 21 1. Rayleigh considered the two major forces in a charged, incompressible droplet: the Coulomb force and the surface tension. The Coulomb force tries to deform the droplet, which leads to fission. The surface tension keeps the droplet spherical. Rayleigh introduced a curvature-independent surface tension G, and defined the surface energy

Es = ods (2.1)

and the Coulomb energy of a conducting droplet

(2.2)

where ps is the surface charge density. For an isolated sphere, ps is homogeneous, but for a deformed droplet Eq. (2.2) has to be minimized under the constraint that the total charge is conserved [28]. This turns out to be the same as the condition of constant potential along the metal surface.

An expansion of the total energy as a function of appropriate deformation parameters leads to the classical result that for EC/Es > 2 the droplet becomes unstable, e.g. it spontaneously deforms and undergoes fission. This holds for both homogeneously charged droplets (e.g. nuclei) and metal droplets (e.g. clusters). The application of this result to metal clusters gives a critical particle size, which depends on the surface tension and the Wigner-Seitz radius. For sodium clusters Z2/N,h, N 0.4 is obtained (see Eq. (1.5) and Table 4). For doubly charged clusters this corresponds to a critical size of about ten atoms. This is the absolute lowest limit for stability. The appearance size, N, or Ni’, tends to be at least twice as large.

For fissilities smaller than unity, the shape at the top of the fission barrier deviates strongly from that of a sphere, see Fig. 1. Barrier shapes are illustrated in Fig. 4 [3]. Here, only the symmetrical case is considered. In Fig. 5 the variation of the symmetric barrier height with fissility parameter is shown. For X = 1 (barrierless fission) the droplet is unstable with respect to any small quadrupole deformation. For such small quadrupole deformations, it turns out that the change in Coulomb energy is the same for both surface- and volume-charged droplets, and the barrier height for 1 > X > 0.6 is described quite well in both cases by the Bohr-Wheeler formula [2]

B&Y) = $Es . (1 -X)3 ,

viz. Fig. 5. See, however, Ref. [7].

(2.3)

At the other extreme, X = 0, there is no charge and the saddle configuration is that of two touching spheres. We are here assuming the surfaces to be infinitely sharp. The barrier is equal to the liquid drop Q-value for a symmetric mass split, or Br(X = 0) = E,(2’13 - 1) = Es x 0.2599. With increasing charge, the energy of a “saddle” configuration made of two spheres in contact decreases as a linear function of X. For two unpolarized metal spheres, the slope parameter is 0.1102, as explained in Ref. [ 161. This is illustrated with the thin line in Fig. 5. Later we shall return to the remaining features of this figure, where the effects of metallic polarization and deformed saddle shapes are considered.

U. Ntiher et al. IPhysics Reports 285 (1997) 245-320 257

x = 0.8

x=06

x =0.9

x=07

Fig. 4. Symmetric saddle-point shapes for uniformly charged droplets (nuclei) for various values of X. For X = 1, the saddle point shape is a sphere. With increasing barrier height the saddle shape deforms more and more and approaches two spheres in contact for X = 0.

SYMMETRIC FISSION OF A METALUC DROP

_ Deformed saddle’

. (Interpolation) \

0 ,‘.‘.‘,‘,‘.‘,‘,

0 0.2 0.4 0.6 0.6 1

FISSILITY (X)

Fig. 5. The connection between barrier height Br and fissility for symmetric fission of surface charged droplets. The barrier height is expressed in units of the surface energy. The Bohr-Wheeler estimate, presumed to be valid for large X-values, is shown as a thin curve. The two-sphere approximation for unpolarized spheres is the thin, downsloping line. The fat line takes metallic polarization into account assuming a sharp metal surface; and the additional effects of the deformations is described by the dashed-dotted curve.

258 U. Nliher et al. IPhysics Reports 285 (1997) 245-320

It should be remembered that for a volume-charged droplet the symmetric barrier becomes unstable towards asymmetry for all values of X below 0.39, cf. Fig. 2; and for a surface-charged droplet this limit will lie even higher. At low fissility, one therefore expects asymmetric fission. In this case the inclusion of metallic polarizability leads to the surprising result that the fragments loose contact on their way to fission before they reach the barrier; cf. Fig. 3. For such situations the fission barrier is given by the difference between the height of the outer barrier and the Q-value for fission (Eq. (1.6)). We will first discuss the Q-values.

2.2. Q-values

The description of individual neutral clusters as liquid droplets is nothing else than the expansion of the total energy E,,,(N) as a function of size N in the form

&0,(N) = a&J - a,N2j3 + &hell(N) , (2.4)

where a, and a, = 4rr&~ are the coefficients for the volume and surface contribution, respectively. Eshell is the shell correction, e.g. the deviation due to the quantized motion of the N valence electrons from the smooth trend set out by Eq. (2.4).

For a charged cluster, further terms depending on the charge Z have to be added. The energy necessary to remove an electron from the neutral cluster is

E,(Z = 1) = IV, + ce2/(rw,N”3), (2.5)

where Wb is the bulk work function and, classically, the coefficient c should equal 0.5. Photoionization experiments are at variance with this and indicate a c-value of about 0.4 [ 17,391. This deviation is of pure quanta1 origin as has been theoretically confirmed through density functional theory [40].

The total energy needed to bring a neutral cluster [ 19,411 to charge Z is

&(N,Z) = ZW, + (Z(c - l) + {Z2)[e2/(rw,N’ ‘)] . (2.6)

This term has to be subtracted from the cohesive energy of the neutral cluster (Eq. (2.4)). Note that since the volume and the charge of the fissioning clusters are conserved, Wb and a, have no influence on the energy change during the fission process. What remains, the surface energy

E,(N) = a,N2/3 (2.7)

and the size-dependent part of the Coulomb energy EC, of the spherical cluster A&:+. This can be written as

Ed(N,Z) = E,(N) + [e2/(2rwsN”3)](Z2 + (2c - 1)Z). P-8)

If the droplet undergoes fission into two charged fragments, the surface energy will increase and the Coulomb energy will decrease. If the size of the fragments is xN and (1 - a)N and their charges PZ and ( 1 - /?)Z, the change in total energy (Q-value) can be written as

- Qf = E,(~N,PZ) + &((I - a)N,(l - p)z) - &(N,z). (2.9)

For a doubly charged cluster (/l = i) the Q-value is shown as a function of fragment size in Figs. 6(a) and (b) for X = 0.3 and X = 1, respectively. Due to the small number of charges

U. Ntiher et al. I Physics Reports 285 (1997) 245-320 259

1 , I

_ x=0.3 -r[ 1

0 0.2 0.4 0.6 0.8 1

(a) MASS ASYMMETRY, a

Coulomb Energy -1 i Lo__ lmiLm . ..A

0 0.2 0.4 0.6 0.8 1

b) MASS ASYMMETRY, a

Fig. 6. The et-value (solid line in the middle) of charged metal droplets for X = 0.3 (a) and X = 1 (b), respectively - in both cases for symmetric charge divisions. The upper line corresponds to the cost in surface energy while the lower line gives the gain of Coulomb energy. For low fissility the relative influence of the surface energy increases. This gives a trend in Qf-value favoring mass asymmetric fission.

involved in the fission of clusters there are only a few discrete values for p. To speak about fission at all, one fragment will keep at least one charge. With metal clusters, asymmetry therefore primarily refers to the mass asymmetry CY.

2.3. Interacting-sphere models: Polarization efects

The experimentally observed fission of metal clusters occurs at a fissility parameter of X near 0.3 in all those cases where the fission is thermally activated and has to compete with evaporation [2 1, 221. The barrier heights are comparable to the neutral monomer separation energy, typically about 1 eV. In the following, we will concentrate on models which are capable of describing such low-fissility processes.

If the fissility is small and the Q-value is known, the height of the fission barrier is obtained by considering two interacting spheres. As a first approximation, one may describe the saddle shape as two tangent spheres having the size of the fragments, and calculate their electrostatic energy as if they were point charges. This gives for the height of the fission barrier (cf. Eq. (1.6))

B; 21 Z2e2P(1 - B)

rw,iv’3(a’~3 + (1 - c()‘/3) _ e

f (2.10)

The upper line of Fig. 7 shows the barrier height for two touching unpolarized spheres as a function of symmetry. In preparing the figure we chose Naif, which is the experimentally observed appearance size for doubly charged sodium clusters. The fissility of this cluster is 0.35. The barrier height is nearly independent of the fragment size, except for very asymmetric mass splits.

Metal clusters are highly polarizable. Therefore a realistic description will have to include polarization effects. Various methods to account for these effects have been considered. In the next section a microscopic Thomas-Fermi description will be given. Here we will take a purely classical approach and begin by imagining a point charge in front of a charged sphere with a sharp surface. As known from textbook physics the problem of finding the electric field (and the electrostatic

U. Ntiher et al. I Physics Reports 285 (1997) 245-320

0 3 6 9 12 15 16 21 24 27

FRAGMENT SIZE

Fig. 7. The fission barrier Br of NazT (X-value 0.35) as a function of fragment size based on the parameters of Table 4, in

the point charge model (upper line), and in the image charge model with 6R = 0.7 8, (lower line). In both cases c = 0.4. Polarization effects reduce the barrier by 0.2-0.4 eV and favor asymmetric fission. (For comparison: the symmetric barrier according to Fig. 5 and Eq. (2.23) lies at 0.94 eV).

energy) of a point charge q1 in front of an isolated conducting sphere of radius R can be solved easily by using the image charge method.

The point charge q1 induces an image charge on the conducting sphere. The magnitude, qimage, of the image charge depends on the distance of the point charge from the center of the sphere, s, and the radius of the sphere, R. It is

Climage = -(R/s)ql . (2.11)

The position of that image charge is at R2/s. Since the total charge on the sphere is conserved, a second charge of equal magnitude and opposite sign has to be added and placed at the center of the sphere. When the point charge on the outside moves to the surface of the sphere, the same happens to the image charge on the inside. This gives an attractive interaction. The image charges thus simulate the effect of polarization of real charges on the surface of the conducting sphere. Together with the point charge, the image charges create a field outside the sphere, which is identical to the field of the point charge and the induced surface charges; in particular, the potential on the surface of the conducting sphere is everywhere the same. The net force acting on the point charge is

F(s) = q2qJs’ + q;R/s3 - q:Rlb(s - (R2/s))21 >

where q2 is the net charge on the sphere. This yields a total interaction energy E(s):

(2.12)

E(s) = Jrn F(x)dx = y - 2s2(;iT R2) . ,5

(2.13)

If the assumption that one of the fragments is a point charge is relaxed, an expansion in terms of higher-order image charges is still possible. Now, there are pairs of image charges on both fragments. Additionally, every such pair of image charges will induce another one on the opposite fragment. The result is an infinite sum of image charges with alternating signs on every fragment. The expansion converges very rapidly. Even for symmetric fission, ten image charges on each fragment are sufficient. This is described in more detail in Appendix A.

1.6

1.4

2 1.2

6'

5 5 0.0

0.6

0.4 5

6R=O

R,+ R2 \

-

\

.---

/ L

10

5R=O. 7 ii

15 20 5 10

CENTER DISTANCE ( 8, )

U. Niiher et al. /Physics Reports 285 ( ‘1997) 245-320

1.6

6R=0

R,+R 2

\

-

\ \

/

(a) (b) CENTER DISTANCE (A )

Fig. 8. The electrostatic energy E(s) of two spherical, charged fragments as a function of distances for asymmetric (a) and symmetric fission (b), respectively. The upper line corresponds to the Coulomb energy of the spheres without any polarization effects. The line in the middle takes image charges, i.e. polarizability, into account with 6R = 0. The lowest line furthermore takes surface diffuseness into account with 6R = 0.7 A. The vertical lines give the distance for touching spheres with 6R = 0 and 0.7 A, respectively. For asymmetric fission, the barrier maximum B, lies beyond the touching

configuration.

The barrier heights calculated using the full image charge model are shown by the lower curve of Fig. 7. The difference between the two curves corresponds to the energy gain due to the polarizability of the fragments. As one sees, there is a significant lowering of the barrier; in addition, polarization favors mass-asymmetric fission.

So far we have mainly considered metal spheres with a sharp surface. In the following, we will look more closely at the surface diffuseness, as expressed through the spill-out parameter 6R, see Eqs. (2.19) and (2.20). In Figs. 8(a) and (b) the potential energy of two spherical, charged fragments of Na$ is plotted as a function of distance. The upper curves correspond to the point charge model while the two lower ones include polarization effects, the middle one assuming 6R = 0, the lowest 6R = 0.7 A. The vertical lines indicate the distance between the centers of touching spheres with and without diffuseness, respectively. While for the fragment Nal (Fig. 8(a)) there is a maximum outside the touching sphere radius in both cases, there is none for NaTj (Fig. 8(b)). (For that case we generally assume the touching configuration with 6R = 0.7 A to be the barrier maximum. ) For Na,f the lowest barrier maximum appears at a distance where the spherical fragments are already separated by about 2A and the energy is lowered from 1.55 to l.O4eV, i.e. by 33%. This is shown in Fig. 9. Without surface diffuseness (6R = 0), the Nat-hairier is placed only 0.8 A beyond touching of the spheres. Its energy is 1.16 eV, or 25% less than the reference energy, 1.55 eV. See also Fig. 10. From this numerical exercise we conclude that, thanks to the polarization, the two-sphere model seems to be justified as a description of the most probable fission of clusters with X-values below 0.4 (and p = 0.5). The polarization drives the system towards asymmetry, and this in turn pushes the fragments away from direct contact at the barrier maximum. Conversely, the two-sphere model appears less adequate for the description of the competing symmetric-fission mode. Here, deformations may also come into play, cf. Figs. 4 and 5.

As an alternative to the classical image charge method with sharp surfaces, which gives the same result for any conducting medium, one may work out the interaction energy of two polarizable


2+ l+ it Na --> Na + Na

27 24 3 IO 12 14 16 18 20

CENTER-OF-MASS D1STANCE.s ( a)

22 24

Fig. 9. The saddle-point shape of asymmetric fission of Nai;: Two spheres which have already lost contact. The distance between the (sharp) surfaces is 2.1 A. Thin lines indicate surface diffuseness with 6R = 0.7 A.

Fig. 10. The interaction energy (outer barrier, E(s) or B,(s)) between Nat and Na& as a function of the distance s of their centers. The dots denote the Coulomb energy of two point charges. The short dashes include the dipole polarization for the classical metal sphere with a sharp surface. The long dashes show a similar calculation that takes into account an additional dipole polarization, due to spill-out with 6R = 0.8 A. The dashed-dotted line is based on the classical image charge treatment with 6R = 0. The full drawn line is also based on the image charge treatment, but it is including additional polarization due to spill-out with 6R = 0.8 A.

spheres through a multipole expansion. This is a standard method in molecular physics for describing the interaction of molecules and molecular ions [42].

The first of these polarization, or induction, terms describes the attraction of the molecular charge ql by the dipole it induces in the other molecule, and vice versa. Introducing the polarization CC

P=xE, (2.14)

where E is the electrical field acting on the molecule and P the induced dipole, the monopole- monopole and monopole-dipole interaction energy, at the center distance s is

E(s) = 4142;s - @,q: + x2q:)/(2s4). (2.15)

The maximum of this function is the outer barrier. It lies at

SC = M~lqzlYl t ~2ql/q2)li.'3 (2.16)

and the height of the outer barrier is given by

& = %,q2/(4s,). (2.17)

This is a quite general result for any polarizable material. If we assume both fragments to be ideal metallic spheres with sharp surfaces, then the static dipole polarizability will be

r = R3, R = Y,,N’~~ (2.18)

In this case, Eq. (2.16) shows that the position S, of a charge symmetric fission barrier lies at

s, = [2(R: + R2)] . 3 “3 It is located outside the sum R, + R2 of the two fragment radii, provided the


ratio of the larger over the smaller radius, RJR*, is larger than 3.73, that is, when the heavy-to-light mass ratio is larger than 52. This condition is practically never met in the fission of metal clusters.

Therefore, a simplified dipole approximation of this kind appears inadequate. It has at least two shortcomings. One is that small metal clusters are more easily polarized than indicated by Eq. (2.18). An enhanced polarizability M will increase the barrier distance S, according to Eq. (2.16) and lower the outer barrier (Eq. (2.17)). It is known that due to surface diffuseness, or equivalently the spill-out 6R of the electron charge density beyond the surface, static dipole polarizabilities of small neutral alkali metal clusters substantially exceed the classical value of Eq. (2.18) [43,39]. This can be taken into account by introducing an effective radius

R e =r N’13+gR WS 3

and an effective value for the static dipole polarizability

(2.19)

r = R; . (2.20)

There is no direct measurement of these quantities for charged clusters, but they can be inferred from beam deflection experiments on neutral clusters [43] and from the measured plasmon frequencies of charged clusters [39,44,45]. For Na clusters, a value of 6R = 0.7 A will be used in our numerical estimates.

The other shortcoming is that the dipole induction term is just the first one in a multipole series. Higher multipoles become important at small distances when the inducing electric field is inhomogeneous. Here, the classical image charge model is superior because it presents the result of a complete multipole expansion in closed form. The difference between the two models is readily illustrated by returning to the case of a point charge (CI~ = 0) in front of a charged sphere (a, = R3 ). In this case the dipole approximation (Eq. (2.15)) predicts the interaction energy:

Edi@ = q,qJs - q;R3/(2s4) . s (2.21)

Comparing this expression with Eq. (2.13), one sees that the polarization effect is much stronger in the image charge model at distances where s is only slightly larger than R. (On the other hand, expanding the denominator of Eq. (2.13) in powers of R2/s2 the dipole approximation (Eq. (2.2 1)) is recovered at the first order).

It is desirable to combine the advantages of the classical image charge model with the freedom of the induction model to consider different polarizabilities, as due for example to the electron spill-out. The solution that immediately suggests itself is to extend the image charge model by simply inserting values of R, according to Eq. (2.19) instead of the original R-value. To get an impression of what effect this may have, we rewrite Eq. (2.13):

E(x) = y (f. - gx2(f_ l)) ) x

(2.22)

where x = s/R. This function has its maximum at a definite value of s/R. For q1 = q2, it is equal to (1 + &)/2 (the golden ratio!), and Es is exactly equal to one-half of the prefactor q2/R. If we replace R by R + 6R, only the prefactor will change and the barrier maximum will lowered by a factor R/(R + 6R), a result that will carry over to the more general case of two spheres interacting via image charges, see also Figs. 8(a) and (b).


To summarize the foregoing discussion we show in Fig. 10 how the interaction energy E(s), or the outer barrier B,(s) between Na& and Nal is influenced by the various polarization effects. Whereas the point charge interaction at the contact distance amounts to 1.55 eV in this example, the image charge calculation with the inclusion of spill-out gives a maximum lying 2 A beyond contact at an energy of 1 .O eV (i.e. a reduction in energy by 35%).

To round off this discussion, we should also mention a term that is not included in the image charge model but emphasized in molecular physics, namely the so-called “dispersion term” [42]. It describes the dynamic interaction between the ground state dipole fluctuations in the two clusters and is responsible for the van der Waals attraction between neutral molecules. (Such a term is also missing in the microscopic local density approach discussed in Section 3.) We have estimated this term using experimentally known plasmon frequencies, and found it to be negligible.

Another influence that has been disregarded so far arises because some clusters are deformed. This leads to a change of the interaction energy at the barrier. Our estimates of the modifications of both the field and the polarizabilities based on realistic deformation parameters [46,74] indicate that these effects are at most of the order of 5% of the outer barrier height. Deformations will therefore

not be considered further.

2.4. Choice of model description

On the basis of the foregoing discussion we consider the classical liquid drop model, including the classical image charge model for two interacting spheres, as the most instructive starting point for the comparison with experimental data. Since we are dealing with objects of atomic dimensions we include two particularly pertinent quantum effects namely (i) the surface diffuseness, or spill- out, in the form of a phenomenological effective radius, rwsN113 + 6R, and (ii) the deviation of the c-parameter in the electrostatic energy expression from the classical value of one-half (we will use c = 0.4 throughout). The correction (i) acts on the outer barrier B, and hence on the total kinetic energy release, while (ii) modifies the @values.

Both theoretical and experimental evidence indicates that strongly asymmetric fission is the dominant decay channel in situations where fission competes with evaporation. Nevertheless, for small, doubly charged clusters the symmetric barrier lies close in energy, cf. e.g. Fig. 7. When appropriate we will therefore include estimates of the symmetric barrier. As mentioned in connection with the discussion of Fig. 5 there are well-defined expressions, not only for the barrier height in the limit of X close to unity, but also for X close to zero if one assumes the two touching spheres to be unpolarizable. Assuming a metallic dipole polarizability of a = R3, i.e. sharp surfaces, Saunders derives an increase in the linear reduction of the barrier height [ 161. This is the fat straight line in Fig. 5. Polarization increases the slope from 0.1102 to 0.1983, a value that lies close to our own estimates. To take into account that the symmetric barrier shapes will deviate more and more from two spheres in contact as X increases, Swiatecki [47] has suggested an interpolation procedure in the form of a third-order polynomial that is fitted at X = 0 to the straight line, and at X = 0.6 to the Bohr-Wheeler expression (Eq. (2.3)). This leads to [16]

B “f’” = &(0.2599 - 0.1983X - 0.5369X2 + 0.4575X3), (2.23)

shown as the fat dashed-dotted line in Fig. 5. It is no better and no worse than a curve drawn with free hand for the same purpose. The difference between the fat line and the fat curve describes the


Touching spheres

cn 2 ii

0 5 10 15 20 25 30 35 40 0 50 100 150 200 250 300

PARENT SIZE, N FRAGMENT SIZE

Fig. 1 I. The increase of the fission barrier Br(N) of doubly charged sodium clusters with size. The thin upsloping curve gives the predictions of the sharp-surface liquid drop mode1 for symmetric fission, Eq. (2.23). The results of the two-sphere model without (triangles) and with (circles) polarizability 6R = 0.7A are included. For comparison, the threshold for evaporation, D,(N)++, calculated from Eq. (4.11) is also shown (thin line).

Fig. 12. The fission channels for NaG,, as a function of mass asymmetry for different charge splits fi = i, I, 2, & 2. The most asymmetric fission channel, fi = i, is the one with the lowest barrier.

effect of deformations at the symmetric barrier. As one sees this effect is comparable to the effect of dipole polarization.

2.5. Size dependence of jission barriers

Choosing NaiT as an example, Fig. 11 compares fission barriers calculated in three different ways with the activation energy &(N) ++ for monomer evaporation. This is given as

d(E,,, - E,)/dN = D,(N)++ = a, - $a,N-‘i3 - dE,/dN , (2.24)

where E,, refers to Eq. (2.4) (without shell correction energy) and EC refers to Eq. (2.6). The top curve (triangles) is just the Coulomb energy at touching distance, R(24) + R(3), of two point charges minus the liquid drop Q-value. It serves as a reference. The black circles represent barriers towards trimer emission calculated with the image charge model including a spill-out of both fragments of 6R = 0.7 A. The thin line describes the symmetric barrier for a drop- with a sharp surface according to Eq. (2.23) with X = 9.36/N, where 9.36 is the critical size for doubly charged sodium, cf. Table 4, which again is based on Appendix B. As one sees, polarization and diffuseness lowers the asymmetric barrier quite significantly. With diminishing size, or increasing fissility, the symmetric barrier becomes the lower of the two. Where D,(N) ++ becomes equal to the (lowest) barrier one expects equal competition between evaporation and fission, i.e. this intersection defines the appearance size N,.

2.6. Multiply charged clusters

If the parent cluster is more than triply charged, there are several ways to split the initial charges onto the fragments. In Fig. 12 the fission barrier of Nat& with X = 0.3 is calculated in the


0-, __.-

1 10 100

FRAGMENT SIZE

Fig. 13. The fission of Na$, as in Fig. 12, here shown in a semi-logarithmic plot. The loss of a singly charged sodium dimer or trimer ion is the most probable exit channel.

Fig. 14. The saddle configuration for asymmetric fission of Na$ Surface diffuseness corresponding to 6R = &?‘A is

indicated.

two-sphere model with image charges, using Rz for the polarizability (Eq. (2.19)) and c = 0.4. The possible values for the splitting of the charge are ,!3 = i, I, 5, %, and 5. Each curve corresponds to a different splitting. One sees that the charge asymmetric fission has a much smaller barrier than the symmetric fission. This is shown more clearly in the semi-logarithmic plot, Fig. 13. The lowest barrier corresponds to the loss of a singly charged dimer or trimer.

This calculation is the clue for understanding why the appearance sizes seem to scale with the liquid drop parameter Z2/N, when 2 and N are increased [ 10,21,22]. It is due to a more or less accidental cancellation of two opposing effects. For a given value of Z2/N, an increase in 2 allows for more and more charge asymmetric saddles, resulting in a lowering of the effective barrier height compared to the symmetric barrier, as one sees by comparing Fig. 7 with Figs. 12 and 13. This decrease is compensated by the increase in N-value proportional to 2 2, because the height of the symmetric

saddle, which we use as reference point, is proportional to the surface energy of the sphere, i.e. it will increase as N2/3. The relevant parameters for most cluster materials, together with the discretization of the charge onto small, integer numbers, conspire to make the two effects cancel (and to match the essentially constant threshold energy for evaporation) for values of Z2/N that are nearly constant.

In calculations of this type one furthermore finds that the size of the singly charged fragment is nearly independent of the size (and total charge) of the decaying cluster and always close to 3.

As already mentioned, the two-sphere model is a particularly good approximation when the fission process is strongly asymmetric. For a small singly charged fragment, the saddle point is always found to be beyond the contact point in the region of X-values considered here. The configuration of Nat& at the saddle point is shown in Fig. 14.

2.7. Infruence of the model parameters on the barrier heights

The droplet model contains only few parameters. For quantitative predictions, the sensitivity of the model to slight changes in these parameters is extremely important. The parameters used in the model are the surface tension, a,, the Wigner-Seitz radius r,,, the spill-out 6R, and the prefactor of the ionization energy c. Unless otherwise stated, the calculations in this section are made using the


2+ I+ l+ Na --> Na + Na

27 24 3

0.7 ' /.,I ,/,,/ 0' i I1

0 3 6 9 12 15 18 21 24 27 0.2 0.4 0.6 0.8 1 1.2

FRAGMENT SIZE SURFACE PARAMETER, s

Fig. 15. Sensitivity to the prefactor in the Coulomb energy, Eq. (2.5). The fission of Na$ for c = 0.5 (classical value) and c = 0.4 (quantum theoretical and experimental value). The barrier decreases by 0.2 eV, asymmetric fission is favored.

Fig. 16. The sensitivity of the trimer emission barrier against changes in the surface tension coefficient a, for Na$. The

barrier increases with surface tension almost proportionally.

parameter set a, = I .04 eV, a, = 0.72 eV, r,, = 2.15 A, 6R = 0.7 A, and c = 0.4, cf. Table 4. If the value c = 0.5 is used instead of c = 0.4 (see discussion in connection with Eq. (2.5)), the fission barrier increases and the trend towards asymmetric fission is less strong. This is demonstrated for the fission of NGT in Fig. 15. The trimer barrier increases by approximately 0.2 eV.

Even more critical is the influence of the surface tension. It is very difficult to obtain precise data from experiments with clusters. Instead we choose to base the comparisons of model predictions with experiment on the explicit, and in fact more interesting, assumption that bulk values for a, (as well as for a, and rws) remain valid down to the smallest cluster sizes. To get an idea of the sensitivity, the barrier height for the trimer decay of NaiT is shown as a function of the surface tension parameter a, in Fig. 16. A large surface tension will increase the fission barrier. It also favors asymmetric fission. The same trend is observed if a curvature-dependent surface tension is introduced, but in that case the trend will decrease with increasing drop size.

In Fig. 17 we illustrate the sensitivity of the fission barrier to surface diffuseness, comparing the case of 6R = 0 to 6R = 0.7 A for the fission of Naz:, see also Figs. 8 and 10.

The separation energies D(N) are particularly sensitive to the choice of volume energy parameter, a,. In fission it is assumed that the volume energy does not change and the barriers are independent of a,. How well this holds in practice is less clear.

Figs. 15-17 demonstrate how strongly the model predictions depend on the choice of parameters. An error of 20% in the surface energy has a same effect as a change in c from 0.5 to 0.4. Even if one assumes, as we do, that the bulk values for rws,a,, and a, remain valid for small clusters, it is necessary that they be accurately known, and furthermore that they refer to the temperature of the experiment, see Section 4.2.1 and Appendix B.

2.8. Conclusions

The classical liquid droplet model can be used to understand basic features of cluster fission. Two approaches are possible in describing the fission process: For high fissility, X close to 1, a slight


0.7' 1 1

0 3 6 9 12 15 16 21 24 27

FRAGMENT SIZE

Fig. 17. The sensitivity of the fission barrier of Na&+ to surface diffuseness 6R. The parameter c is set equal to 0.4 in both cases.

deformation of a spherical droplet will bring it over the saddle point. For low fissility and large asymmetry it is sufficient to describe the clusters in terms of two interacting spheres. Generally, low fissility favors asymmetric fission thereby adding credibility to the two-sphere model and resulting in dependable estimates of the height of the barrier.

The introduction of polarizability to the two-sphere model using the image charge method results in an attractive interaction between the charged fragments at close distances. This lowers the fission barrier significantly, especially the asymmetric one, and results in a saddle-point configuration where the two fragments have already lost contact.

The classical model can be made more realistic by taking into account surface diffuseness and the quantum correction c to the Coulomb energy.

In the following section the present results will be discussed and compared to calculations where quantum effects are treated explicitly, both with respect to polarizability and as regards shell structure.

3. Density functional calculations of fission barriers

A microscopic description of the dynamics of the fission process, based on local spin-density functional calculations for the electronic structure in conjunction with molecular dynamics simulations, has been performed for small Na: and Kc clusters (N < 12) [37]. That calculation has shown the possible existence of double-hump barriers, as in nuclear fission, and the influence of quantum effects in determining the predominant fragmentation channel. Whether or not these results can be directly generalized to large systems is difficult to assess, since molecular dynamics simulations become very difficult for large clusters. To get an overview of fragmentation processes for large clusters it may therefore be justified to try simpler methods. One of these methods is the macroscopic two-sphere droplet model discussed in the Section 2. Another approach is the density functional theory using the jellium model. Here we review recent work done within this framework [35,36,49-5 11. See also Ref. [48] where the effect of ionic structure is taken into account.

U Niiher et al. IPhysics Reports 285 (1997) 245-320 269

3.1. The spherical jellium model and the extended Thomas-Fermi approximation

An important part of the ground state energy of alkali-metal clusters is determined by the motion of the valence electrons. In a first approximation, details of the positive charge configurations seem to be of minor importance. This has led to the introduction of the jellium model [52] with the fundamental assumption that any ionic structure can be neglected. The jellium model, which has indeed enjoyed considerable success with regard to ground state electronic properties and optical response of simple metal clusters, may also be used to get insight into the potential energy landscape and to estimate fission barriers.

In the spherical jellium model [52] one assumes the alkali-metal cations to form a spherical positive charge distribution with a sharp surface at a radius R = rw,N1/3, where N is the number of ions in the cluster and r,, the Wigner-Seitz radius of the bulk metal. The jellium density is uniform, equal to the bulk density of the alkali metal, n + = 3/(4w3 ). The jellium radius R is assumed to

remain unchanged when the cluster is ionized. More generl; shapes of the jellium distributions are also considered. The treatment of the electron system is described in the following.

The valence electron density n(r) is self-consistently calculated in the presence of the external potential Vi(u), generated by the jellium, by minimizing the total electron energy within the framework of the density functional theory [53]. If shell structure is disregarded, the extended Thomas-Fermi (ETF) version of the energy functional is appropriate [35]:

E[n] = $ (37t2)2’3 I

ns/3(,)d,. + ! 8 I

(on(r))2dr + 1 n(r) 2 JJ n(v) n(r’) &’ &

Ir - r’l

3 3 li3 +

s V,(r) n(r)dr - - -

0 s 4 rt n(r)4i3 dr -

s

0.44 n(r)

7.8 + (3/(47cn(r)))“3

dr (3.1)

(Hartree atomic units (a.u.) are used throughout this section unless explicitly stated, i.e. h = m = e2 = 1, length unit a0 = 0.53 A, energy unit 1 Hat-tree = 27.2 eV). The first two terms in Eq. (3.1) correspond to the electron kinetic energy, which is the sum of the local Thomas-Fermi term and the von Weizsacker gradient correction. This Weizsacker term is the first of a series expansion. In order to make further terms redundant, the coefficient /I is treated as an adjustable parameter. (The symbol /? is used here in another meaning than in Section 2). The third term is the classical Coulomb interaction (Hartree part) and the fourth one corresponds to the electron-jellium interaction. The last two terms are, respectively, the exchange and correlation energies taken in the local density approximation.

To find n(r) one must solve the Euler-Lagrange equation associated with the energy functional in Eq. (3.1):

WInllWr) = p , (3.2)

where ,U is the electron chemical potential. Adding the jellium self-energy to E[n] gives the total energy of the system:

E cluster = E[nl + Ejellium . (3.3)

The fall-off of the electronic density is almost completely determined by the gradient correction in the kinetic energy timctional. In the variational equation (3.2) the Weizslcker term leads to an

270 U. Ntiher et al. I Physics Reports 285 (1997) 245-320

asymptotic fall-off with the correct exponential form. However, the surface diffuseness and the surface tension obtained depend quite strongly of the coefficient p. When fl decreases, the cluster surface becomes sharper. In the calculations discussed below an effective value p = i based on an empirical estimate, is adopted. In particular, using this effective value the extended Thomas-Fermi approach reproduces on average the Kohn-Sham total energies and electron radii of spherical clusters [54]. Moreover, using /3 = i the threshold energy for monomer evaporation (% 1.1 eV) is in agreement with the experimental cohesive energy in the bulk metal.

The ETF total energy of a charged spherical Na:+ cluster can be parametrized according to Eqs. (2.4)-(2.7) as

&,,(N,Z) = a,N + a,N2i3 + W,Z + a,[2Z (c - i) + Z2]N-“3 . (3.4)

Using the energy functional equation (3.1) with p = i the best fit gives a, % -2.2 eV, a, M 1.1 eV, w, = 3.0eV, a, E 3.0 eV and c M 0.37. An a,-value of -2.2 eV is not in conflict with the generally quoted bulk cohesive energy of - 1.1 eV. In jellium calculations the atom itself is endowed with a non-zero binding energy al (= a, - a, = - 1.1 eV). The net work required to remove an atom from bulk jellium is therefore a, - al =: 1.1 eV in agreement with experiment and with the conventional definition of the bulk cohesive energy per atom. Note on the other hand that the effective surface energy a, is significantly larger than the bulk value of Table 4.3 (0.72 eV) and that the Coulomb energy a, is smaller than e2/2 r,, = 3.5 eV derived from the bulk density. Consequently, the ETF model predicts a critical ratio (Z2/N),rit = 2a,/a, - 0.73, which is much larger than the value 0.43, expected from bulk parameters. One should bear these differences in mind when comparing fission barriers, or Q-values (D,(N) and Qf) obtained within the ETF model with the phenomenological models discussed in the previous section.

3.2. Calculation of the &ion barrier

To obtain the fission barrier, a set of deformed axially symmetric jellium configurations representing the fragments at various separations is combined with the extended Thomas-Fermi calculation, providing the total cluster energy E(s) as a function of fragment separation s. The maximum of this curve is the fission barrier Bf . The simplest choice of a fragmenting jellium distribution consists of two spheres with their centers separated by the distance s [35, 36, 49, 501. For large fragment separation, E(s) is given by the Coulomb repulsion of two point charges. At intermediate separations E(s) is modified by the mutual polarization of the charged fragments, in analogy with the classical image charge effect. At still smaller distance the overlap between the electron wave functions of the two fragments becomes important, in analogy with the forming of a chemical bond. Within the jellium model and for low fissility (X 2 0.3) clusters, the maximum of the calculated barrier is always found at separations larger than that corresponding to touching of the two jellium spheres (which is the most compact configuration considered in the two-sphere-jellium model).

An improved description of the barrier requires the solution of the extended Thomas-Fermi problem for a sequence of jellium shapes continuously connecting the initial configuration of the parent cluster with the final one of two separated fragments. This problem has been studied in Refs.[51,54] where the positive background of the fissioning cluster is modeled by two spheres smoothly joined by a portion of a quadratic surface of revolution [29,30]. For nuclei this is an adequate parametrization when fission is dominated by the surface energy, i.e. for low and medium X-values. This family

U. Niiher et al. IPhysics Reports 285 (1997) 245-320 211

of shapes is characterized by three parameters: the asymmetry A, the elongation parameter p, which is proportional to the separation s between the emerging fragments, and the “neck” parameter L.

In order to obtain the barrier corresponding to a certain fission channel defined by the mass ratio of the fragments, one has, in principle, to study the three-dimensional energy surface E( A, p, II). In a simplified (and computer-time saving) version one assumes that on the path to fission the parameter A is constant and equal to its value in the exit channel. As an example, Fig. 18 displays some equal- energy contour lines of E( A = 0, p, 2) for symmetric fission, Nai,f + 2Na,f,. The fission barrier is situated near the line 3, = 0. A further simplification consists of restricting the estimates to a limited number of one-dimensional fission paths. The dashed curve to the left corresponds to the fission path described by means of intersecting jellium spheres (A = 1 - p) up to p = 1, or s = 11.7 A (the contact point), and of two separated jellium spheres (3. = 0, p > 1) afterwards. The dashed line to the right corresponds to the following path: One starts with a sphere (L = 1 - A, p = A = 0) and follows the line corresponding to a cylinder capped at each end with spheres (A = 1 -- A*/p) up to a configuration near p = 1 where a concave neck tends to form. Next the line of fastest variation of the neck is chosen, which corresponds to the relation (A - 1)’ - p* = constant, up to ;1 = 0 (i.e. two separated spheres). These two pathways are different, but we obtain identical values for the barrier heights (B, = 0.9 eV, Bf = 3.2 eV) and nearly identical center-of-mass distance at the maximum. However, the path with two intersecting spheres (left) seems to lead somewhat more efficiently over the saddle. For the third fission path shown (dashed-dotted line) the neck variation is chosen to be slower than in the previous cases and scission occurs for a separation greater than the position of the maximum of the barrier. This path has a higher barrier.

From Fig. 18 we conclude that a reasonable estimate of the barrier can be obtained by studying the one-dimensional paths indicated by the dashed lines. Fig. 19 thus displays E(s) for the symmetric fission of Nail (with X = 0.13). The solid line gives the barrier obtained for a pathway parametrization in which the fissioning cluster begins to form a concave neck at s = 10.6 A (dashed line to the right in Fig. 18). Plotted at the bottom of the figure are snapshots of the jellium background shapes along this fission path. The dashed line corresponds to the pathway along intersecting and separated jellium spheres. The l/s curve (dotted line) gives the classical Coulomb repulsion between the fragments as point charges and the horizontal line on the right side corresponds to the energy of the final state. In this case, the Q-value for fission is -2.3 eV. This value is much lower than the liquid drop estimate of about -0.8 eV and results from the higher surface energy coefficient a, and lower Coulomb coefficient a, of the Thomas-Fermi model. As a consequence, the symmetric fission barrier is almost twice higher in this model (compare Figs. 11 and 19). In addition, it places the barrier maximum at s = 15.9 A, or 4.2 A beyond the two spheres in contact (S = 11.7 A).

3.3. The two-sphere-jellium model

The results presented in Figs. 18 and 19 confirm that in the example chosen the maximum of the barrier corresponds to two disconnected jellium spheres tied up by the polarization forces due to the electronic cloud. We would like to stress that in the present approach this holds even in the case of symmetric fission, provided X is very small. This conclusion would not be modified by inclusion of shell effects, since the maximum of the barrier occurs when the two jellium pieces are well apart, and humps and other structures originated by non-spherical shell effects should only be expected at much shorter distances [37]. This feature of metal clusters is quite at variance with


h

0 0.5 1 1.5 2

P

Naz,+ + NazlC

,C------ / / / / / / J __ / I

/ /

/

,’

0 ooc3 00

t, 0 5 10

s @)

Fig. 18. Equal-energy contour lines for the dissociation Na$ + 2Na:, obtained in the extended Thomas-Fermi ap-

proximation. From the left to the right, solid curves correspond to E = -71.3, -70.7, -70.2, -69.6, -69.1, -68.8, -68.7 and -68.5 eV, respectively. The two dashed lines represent the fission pathways compared in Fig. 19, whereas the dashed-dotted curve is a different jellium parametrization, much more elongated.

Fig. 19. Extended Thomas-Fermi fission barrier for Nai; + 2Na;,. The solid line corresponds to the fission pathway described by the jellium configurations schematically shown at the bottom, whose scission point occurs at s = 14.8A. The dashed line is the result for intersecting and separated jellium spheres, which touch at a distance of 11.5 A. We have used rws = 2.12 A.

the standard situation in nuclear physics where X is close to unity (see, however, Section 4.2.4). The importance of incorporating the electronic spill-out and the mutual polarization in describing the saddle configurations should, on the other hand, be stressed when comparing with the nuclear case.

As a more realistic example of the extended Thomas-Fermi calculation for doubly charged sodium clusters within the two-jellium-sphere model, Fig. 20 shows the fission barrier obtained for the asymmetric process Na$ + Na& + Nal, which yields Br = 1.17 eV, to be compared to the value D, (27)++ = 1.09 eV. Fig. 21 displays the valence electron density in the same case for a separation .s = 12.4 A, close to the maximum configuration. The distance between the two jellium surfaces is about 3 A, whereas it amounts to about 2 A in the classical model including polarization forces and spill-out (see Figs. 9 and lo), thus indicating once more that the present ETF model yields a somewhat larger electron spill-out.

The ETF two-jellium-sphere model has been used to calculate the appearance size determined by the competition between evaporation and trimer fission in Nat clusters [36]. The appearance size turns out to be A$ = 25, in good agreement with the experimental result N,“” = 27, Table 2 and [18,22]. In Fig. 22 we compare results of the Thomas-Fermi calculations with similar results from the classical model described in Section 2. Since not only the fission barrier height but also the outer barrier B, is accessible to experiment (being equal to the total fragment kinetic energy) we plot separately the components, B, and Qr, that combine to give Bf according to Eq. (1.6). Both models predict a lowering of B, relative to the point-charge interaction energy (dotted line). As was to be expected this influence of diffuseness and polarization is stronger in the Thomas-Fermi


Na,,+ + Na,+ -39.0

-40.5

1 -41.0 - -::T Bf

Dl(N)++

--____------ _

-41.5 4 8 10 12 14 16 1E

s (A)

Fig. 20. Extended Thomas-Fermi fission barrier (solid line) for charged trimer emission from Nag:. The open circle on

the left side of the barrier indicates the touching point of the two jellium spheres representing the fragments. Dashed line gives the energy of the parent cluster (assumed spherical), whereas dotted curve corresponds to the classical Coulomb repulsion between the fragments as point charges. The short horizontal line to the right gives the final state energy of the fragments at infinite separation. D1 (N) ‘+ indicates the threshold for evaporation of a neutral atom.

Fig. 21. Three-dimensional view of the extended Thomas-Fermi electron density in the case Na$ -+ Na&+Nal in the two-jellium-sphere model, for a fragment separation s = 12.4 A.

picture. The &values are also lower, this time because of the larger surface energy coefficient a, and smaller Coulomb coefficient a,. The end result in this case is a fission barrier, Bf = B, - Qf, that has almost the same value in the two models. Measurements of fission barriers or appearance sizes are therefore unsuited if one wants to discriminate between these models. High-precision measurements of the total kinetic energy release in the process, on the other hand, would be able to do so.

U. Niiher et al. I Physics Reports 285 (1997) 245-320

0 10 20 30 40

N

-614

16 16 20 22 24 28

s (A)

Fig. 22. Fusion barrier height B, (upper curves) and Qr-value for fission (lower curves) as a function of cluster size in the case of charged trimer emission from doubly charged sodium clusters. Solid lines correspond to the extended Thomas-Fermi calculation and the dashed curves give the liquid droplet model predictions within the image charge model with SR = 0 and c = 0.5. Dotted line corresponds to the point charge Coulomb repulsion between the fragments at contact.

Fig. 23. Extended Thomas-Fermi fission barrier (solid line) for the charged trimer emission from Nat:,, cf. Fig. 20 for explanations of the symbols.

3.4. Chemical bond model jbr the outer barrier

Fig. 23 displays the extended Thomas-Fermi fission barrier and Q-values corresponding to the reaction Na:& + Na& + Na;. Parent and fragments are treated as spherical clusters and the fission pathway is described by the two-sphere-jellium model. This example demonstrates that the two- sphere model can be used for all metallic clusters with low X values, irrespective of size. This general validity has led the authors of Ref. [49] to invoke an analogy to chemical bonding, where the energy as a function of separation between the reaction partners is the balance between classical Coulomb interactions and an effective attraction, which sets in when the electron densities begin to overlap, and expresses the same physics that was discussed in Section 2.3. A semi-empirical model is proposed [55, 561 to calculate the fission barrier height for large, highly charged simple-metal clusters, making use of a bonding potential V(S). This potential is akin to the description of the long-range part of the affinity for forming a chemical bond. The electronic clouds of the interacting clusters mutually polarize, giving rise to a bonding contribution, which is effective even if the two positive jellium pieces do not overlap. The incipient bonding is responsible for the lowering of the fusion barrier B,(S) below the point-charge Coulombic repulsion.

A number of calculated fusion barriers for Na v clusters (obtained within the two-sphere-jellium model) have been fitted by a simple expression. For charged trimer emission, the best agreement has been found using the following expression for the bonding potential:

V(s) = -I+, Ii0 eC’(s-Ho), s > R. , (3.5)

U N&her et al. I Physics Reports 285 (1997) 245-320 275

where & = 0.008/r,,, R0 = rWS[(N - 3)‘i3 + 31/3] and y = 0.2. The bonding potential has to be added to the Coulomb repulsion between the fragments considered as point charges to obtain the fusion barrier B,(s):

B,(s) = V(s) + (2 - 1)/s . (3.6)

For medium-size and large charged sodium clusters the agreement with the more detailed calculation is remarkable [55]. An ansatz similar to Eq. (3.5) has been proposed in [57] for the symmetric fusion of neutral sodium clusters. Using Eqs. (3.5) and (3.6) the maximum of the fusion barrier reads

B, = &t&n) = HZ - lY&nl[1 - lh%l)l >

where the position of the maximum S, is obtained as the solution of

s* ep7’m = (2 - l)e-YRo/(y &)Ro) . In

(3.7)

(3.8)

Eq. (3.7) may be applied to simple-metal clusters to obtain the barrier height B, for the fusion process MF_<‘)+ + A43f + Mi+ Thus, also the appearance sizes for highly charged alkali-metal .

clusters can be obtained. As an example, Table 1 displays the appearance sizes predicted by the semi-empirical model [55] for charged cesium clusters and compares them with experimental results from Tables 2 and 3.

To conclude, we may say that the bonding potential is a convenient way to parametrize the mutual polarization of the separated fragments as well as the overlap of the electron densities that extend quite far out beyond the jellium edges.

Table 1 Appearance sizes Na for Cs? clusters obtained within the semiempirical model presented in Section 3.3, as well as the predicted critical numbers N,,, for barrierless fission. We have used r,, = 2.98 8, and a, = 0.66 eV. The experimental results for the appearance sizes have been taken from [22] and Tables 2 and 3

Z 2 3 4 5 6 7

N,(model) 23 57 105 165 236 319 Mexp.) 18+ I _ 84 * 5 190 * 10 _

N,‘“(exp.) 19* 1 49* 1 94* 1 1551l12 230 f 5 325 + 10 X&model) 11 27 51 80 114 155

Table 2 Effective appearance sizes for fission, N,“s. From Ref. [22]

z=2 z=3 z=4 z=5 Z=6 z-7

Lithium 25f 1 Sodium 27& 1 63 * 1 123 f2 206 * 4 310 + 10 445 + 10 Potassium 2oIt 1 55 * 1 llOf5 Rubidium 19* 1 54 * 1 108f3 Cesium 19f 1 49 It 1 94f 1 155 f2 230 i 5 325 i 10

276 U. Niiher et al. IPhysics Reports 285 (1997) 245-320

Table 3 Appearance sizes for fission of charged cesium clusters, N,. For details, see Section 4.1.4.3

Z N, alkeT N$ N$ - N,

2 18f I 0.4 f 0.1 19 1 4 84 + 5 0.10 i 0.03 94 10 6 190 i IO 0.06 i 0.02 230 40

3.5. Shell efects

The quantities, D,(N)’ and Qf are very sensitive to shell effects. These cause some clusters to be more tightly bound (because they have closed electronic shells) than others, and may also give rise to deformed shapes [45, 46, 58611. The equilibrium deformations of parent and fragments can be determined selfconsistently by minimization of the total energy. Here we shall discuss the role played by shell effects in influencing the fission barrier height. In the preceding sections we have used the jellium model and the local density approximation in combination with the semiclassical Thomas-Fermi method, an approach which cannot account for shell structure effects. To include them one has to perform a Kohn-Sham calculation of the fission barrier, cf. Ref. [50].

In the Kohn-Sham approximation the kinetic energy is treated exactly, and the single-particle wave functions & are obtained by solving the equations

[-iv2 + Veff(Y)] $4 = &id4 > (3.9)

with the same effective potential

Jf&(Y) = V,(r) + Vtl(J,) + Vxc(f.> (3.10)

as in Eq. (3.1) i.e. the sum of the electrostatic potential of the jellium background VI(v), the Hartree potential of the electronic cloud V”(Y), and the exchange-correlation potential V,,(v) treated as usual in the local density approximation. The electron density is then evaluated as

(3.11)

where the sum is extended over the occupied orbitals. Finally, the total energy of the system is obtained from a functional similar to that given in Eq. (3.1) with the substitution of the kinetic energy part by the exact single-particle kinetic energy

(3.12)

Fig. 24 shows the Kohn-Sham fission barrier corresponding to the process Nai,’ + Nail + Na:. Notice that the fission products, Na& and Na,f, are magic clusters while the parent Nai$ has two electrons in the lg-shell. The calculation of the barrier (continuous curve) used the two-jellium- sphere parametrization. The dashed-dotted line with energy -81.09 eV on the left side corresponds to the energy of Nat: treated simply as a spherical cluster, whereas the horizontal dashed line


-80.0 Nail+ + NE+*

-82.0 - 10 15 20

s (4

Fig. 24. Kohr-Sham fission barrier (solid line) for Na$ + Na&+Na:.

at -81.63 eV indicates its energy when ellipsoidal deformations are considered. For this case we obtain Bf = 0.7eV and D1(44)2f = 1.1 eV. The value of D1(44)2f is obtained by also treating Nail as a non-spherical cluster. The charged trimer emission is predicted to be the most favorable dissociation channel for Nas4f. Indeed, for this decay channel shell effects confer the daughter clusters a special stability and Qf becomes positive. A similar effect occurs in the symmetric fission of Nai,f . There is an important decrease in the fission barrier height for Nat; + 2Nal, when shell effects are included, from Bf(TF) = 3.2 eV to Br(KS) = 0.4 eV. This is essentially due to the increase of Qf, which is negative in the extended Thomas-Fermi approximation and positive in the Kohn-Sham approach, while the fusion barrier height B, has the same value in both calculations (B, = 0.9eV). These Kohn- Sham calculations predict that fission into fragments with magic numbers of electrons can become dominant in the competition with evaporation because the calculated fission barriers are lower than the D(N)-values for evaporation by several units of kBT (sz 0.04 eV). As a consequence, evaporation chains should terminate at Na&+ or Naz2’ contrary to what is observed [21,22], see also Fig. 3 1. The two-sphere Kohn-Sham model thus appears to exaggerate the shell structure effects somewhat.

Fig. 25 displays the Kohn-Sham single-particle energies for Nai,f -+ 2Na,f, when the fission path is described by the two-sphere-jellium model. One can see how the single-particle levels evolve from the spherical configuration of the parent cluster to the spherical configuration of the identical fragments. The single-particle levels of Na$ are represented by open dots on the right side of the figure. The maximum of the fission barrier is located at s M 15.9 8, [49] where the electron levels have already approached their values in the fragments. As a consequence, the shell effects should be the same as for the fragments at infinite distance. As one sees from Fig. 25 this conclusion depends rather crucially on the large center-of-mass separation at the saddle. If this were shifted to within 1 A from touching, i.e. to 12.7 A, the shell effects would be substantially reduced in magnitude.

The present study of the fission barriers in the Kohn-Sham approach gives an enhanced insight into the working hypothesis used throughout this review: The shell effects at the top of the barrier are assumed to be the same as the ones in the final fission fragments and the outer barrier height B, therefore becomes independent of shell structure effects. These enter only through their influence on the Q-values.

278 I/. Niiher et al. I Physics Reports 285 (1997) 245-320

/ Na2,’ + Na2,+

t~~l~~~~l~~~,l,i,,,l,~,,l,,, 0 5 10 15 20

s (4

Pld

Fig. 25. Evolution of the Kohn-Sham single-particle levels for Na:; + 2Nai,. Dashed-dotted lines represent empty levels. Dashed vertical line indicates the touching sphere separation.

For a comparison of experimental and theoretically calculated shell energies, including those at

the saddle, see Sections 4.2.2 and 4.2.4.

4. Experimental evidence

In this section the experimental results will be presented and discussed. We will start by reviewing the most important experiments and the way they are analyzed, Section 4.1. This is to be followed, Section 4.2, by a closer examination of the combined evidence in the light of the concepts developed in Sections 2 and 3. The aim is to assess the entire body of data for consistency in terms of the underlying model assumptions. One aspect of particular interest is the separation of average, or bulk, trends from quantum size effects in the form of shell energy corrections.

4.1. Review of experiments

4. I. I. Silver Katakuse et al. [ 1 l] have produced doubly (and singly) charged silver clusters in vacuum through

a sputtering process by directing a Xe + beam from an ion gun onto a silver foil. The products are analyzed in a two-sector, electric and magnetic, mass spectrometer. Only those clusters which have a definite energy and mass per charge unit are recorded in each run. If that cluster does not change its charge and mass from the onset of acceleration till it has passed through the spectrometer, it requires a definite acceleration voltage V. to enable it to reach the detector. This is so, also for doubly charged clusters. If such a cluster breaks into two equal parts after acceleration but before analysis, the two fragments will also be recorded. A lighter doubly charged cluster with energy 2F$ may fission asymmetrically to produce a singly charged cluster with the same mass as above, but its velocity and energy will now be too high. The lighter the parent cluster is, the greater the resulting excess energy. It turns out that one can measure these asymmetric fission events with very high

L! Ntiher et al. I Physics Reports 285 (1997) 245-320 279

Cm~kmfI 2 3 4 5 6 Cm~mmll23456769

FRAGMENT SIZE FRAGMENT SIZE

Fig. 26. Fragment distributions from fission of doubly charged silver clusters (parent) produced by sputtering and analyzed in a high-resolution electric and magnetic mass spectrometer. From Ref. [l 11.

sensitivity by detuning the acceleration voltage. By scanning the voItage from OS& to V, one can identify and measure the intensity of doubly charged parent clusters that fission into the preselected singly charged product. In a consecutive run one may then scan for another fission product, and so on. By reshuffling the data it is finally possible to construct the full fragment mass spectrum for each parent cluster - or rather the heavier half section; the lighter half section then follows from an assumption of mass conservation. Fig. 26 shows the results for even parent clusters; it remains until today the most detailed mass distributions that have appeared in the literature. One sees an odd-even staggering with two odd-mass, even-electron products being the more likely. In addition, there is a clear preference for products (observed or inferred) with magic numbers 3+ and 9+, but also peaks at (7+), 1 l+, (13+), and 15+ are present in high intensity from the fission of the heaviest parents. It is particularly noticeable that trimer emission appears strongly suppressed relative to heavier products from Agi: and Agi..

Sputtering is a violent process. There is no doubt that the clusters are produced hot and tend to evaporate or fission thermally, not only before but also after the acceleration, albeit here more weakly. The results should therefore be understood in terms of thermally activated fission with the different final fragment channels competing with each other and with evaporation.

280 U. Ndher et al. I Physics Reports 285 (1997) 245-320

These pioneering experiments suffer from the rather indirect way the fragment distributions in Fig. 26 are obtained. It is therefore interesting that the Mainz group has succeeded quite recently in observing the fission of similar silver clusters suspended in a Penning trap [ 12-141. They determine the appearance size for silver to be N, = 16. At the same time their more direct observation methods point to a dominance of fission into the charged trimer at variance with the Osaka results [ 111. This will be discussed further in Section 4.2.4.

4. I .2. Gold Doubly charged clusters of gold have been produced by Saunders [ 15, 161 using a liquid-metal

ion source. From a sharp tungsten tip, wetted with molten gold, clusters are extracted in vacuum by the action of a high electric field - a process that, incidentally, may be closely similar to the fission of freely suspended metal clusters [62]. The clusters produced in this way are relatively cold, with little tendency for evaporation or thermally activated fission.

In order to observe fission, the doubly charged clusters are brought to collide with the atoms of dilute, stationary krypton gas at relative center-of-mass energies from 2 to 10 eV. The conversion of part of the collision energy into heat raises the temperature and provokes fission. With the aid of three quadrupole mass spectrometers in series, doubly charged clusters are first size-selected, then heated, and finally analyzed with respect to the ensuing fission- and evaporation products - in each sector, respectively. In this way the fission-evaporation competition and the fragment size distributions from selected parent clusters with sizes from 12 to 18 are observed for several values of the collision energy.

Trimer emission dominates the mass distributions, especially at low collision energies, but there is also what appears to be evidence of fission into a pentamer and a heavier partner, as well as into the magic A$-fragment. No preference for near-symmetric fission is observed. The results on fission-evaporation competition is shown in Fig. 27. As one sees, the ratio of decay rates, kf/ke, is systematically higher for parents with even mass. Even more conspicuous is the steep rise in kf/k, with decreasing parent size. The appearance size is found to be N, = 14 (Fig. 27), a result that agrees with more recent measurements [ 121.

Although the observed mass distributions are manifestedly dominated by asymmetric fission, Saunders discusses the results in terms of a pure liquid drop model for fission across a symmetric barrier with no shell effects, i.e. according to Fig. 5 and Eq. (2.23). In order to avoid conflict with the observed asymmetry in the final fragment distribution he speculates that the asymmetry might develop on the way from the (symmetric) saddle configuration towards scission. The critical size for doubly charged gold clusters is found to be 3.6 according to Eq. (1 S). Hence the fissility of At.@- AU&’ is in the range X = 0.2-0.3 and the corresponding saddle shapes (Fig. 4) highly constricted; a situation that renders the above speculation rather improbable.

Despite these objections Saunders’ analysis serves an important purpose by bringing up the question of how much the deformable liquid drop description of symmetric fission and the asymmetric two-sphere model, which disregards the possibility of strong deformations at the saddle, differ from one another in concrete cases, cf. e.g. Fig. 11.

In Fig. 28 the height of the symmetric barrier is shown for doubly charged gold clusters as a function of N, where it is compared to the liquid drop evaporation threshold, Di (N), obtained by differentiation of Eq. (2.4) disregarding shell effects (and charge), see Ref. [16].


Id I I

0.20 0.24 0.28 0.32 0.36

FISSILITY. X

ID- 14 I6 18 20 22 24 26 28

CLUSTER SIZE, N

Fig. 27. Branching ratios, i.e. the yields of singly charged fission fragments relative to evaporation products (neutral atoms) for doubly charged gold clusters produced in a liquid-metal ion source and heated by gas collisions. From Ref.

[161.

Fig. 28. Calculated fission barrier height Br for symmetric fission and the approximate evaporation threshold III(N) for

doubly charged gold clusters according to Ref. [ 161.

Knowing the difference in activation energy for the two competing processes, B:‘“(N) - D(N) from Fig. 28, the fission-evaporation competition can now be analyzed in terms of the statistical expression

b/k, = (sf/&)exp[-(B~y”(N) - D(N))/(kBT)l . (4.1)

From the energies plotted in Fig. 28 and the general slope of the data in Fig. 27 one derives a value of kBT = 0.11 eV, or a temperature of about 1300 K. It is not far from what one should expect, if the competing decay processes take place within the experimentally allotted time of a few microseconds, see also Table 4. In the same spirit, the odd-even staggering in the kf/k,-values, amounting to a factor of five on average (Fig. 27), can be converted to an odd-even staggering in the energy difference, (&(N) -D(N)++), using Eq. (4.1) and the above temperature. The result is an energy difference of 0.2 eV, which is to be ascribed to an odd-even staggering in the evaporation and hence in the D(N)++-values. (One does not expect the fission barrier to exhibit odd-even energy fluctuations, because the transition state at the saddle is most likely to have the same odd-even character as the initial state.)

The appearance size N, should lie near N = 19 according to Fig. 28 if one assumes gr/gk to be near unity. This is in disagreement with the value N, = 14 derived from Fig. 27. The resulting discrepancy amounts to 1 eV in D,(N) - Bf(N). Since the asymmetric barrier should be even lower the discrepancy is even worse.

Nevertheless, important aspects of the experiments on gold fission appears to be describable within a framework of pure liquid drop fission across a symmetric barrier - without taking shell effects or mass asymmetry into account. This goes counter to the observed asymmetry and to the arguments put forward in Sections l-3. We will therefore return to the question in Sections 4.2.1 and 4.2.4.

282 U. Nliher et al. /Physics Reports 285 (1997) 245-320

Table 4

Bulk values of cohesive energy a,, surface energy coefficient a,, Wigner-Seitz radius r,,, and spill-out 6R of the molten metal at (or near) temperatures knT = a,/29

Element kBT (ev) a, (eV) as (eV) rws (A) 6R 6% ZZ

i > 77 N, (2 = 2)

crit

Silver Ag 0.106 2.66 1.98 1.66 0.50 0.911 4.38 Gold Au 0.117 3.32 2.42 1.65 0.50 1.11 3.61 Lithium Li 0.053 1.56 0.92 1.77 0.50 0.453 8.83

Sodium Na 0.036 1.04 0.72 2.15 0.70 0.427 9.36 Potassium K 0.030 0.88 0.61 2.60 0.85 0.450 8.90 Cesium cs 0.028 0.74 0.51 3.06 1.00 0.432 9.26

Note: With the exception of the &R-values, this table is based on the more detailed tabulations of Appendix B. For sodium, a SR = 0.78, is estimated on the basis of static polarization experiments [39, 441 and plasmon frequencies [45]. For the remainder of the alkali metals 6R is simply scaled according to their Wigner- Seitz radii r,,. The values adopted for gold and silver are no more than educated guesses. As discussed in

Appendix B there is some ambiguity about the values of kn7’. Here we quote knT = a,/29 (Weisskopf), but the alternative value ks T = a,/21 (Kassel) cannot be excluded.

4.1.3. Doubly charged alkali clusters Alkali-metal clusters produced either in a neat, supersonic expansion source or in a gas aggre-

gation source have been studied with the aid of a tandem time-of-flight mass spectrometer system by Brechignac and co-workers. Initially, the evaporation of neutral atoms and dimers from size- selected, singly charged cluster cations was studied [63-651. Later, the competing decay of doubly charged species through evaporation or fission [ 18, 19,381 was investigated for sodium, as well as for potassium and lithium. A recent review can be found in Ref. [66].

Thanks to these studies, the statistical transition state theory for the evaporation has been put on a semiquantitative footing; and the activation energies for neutral monomer and dimer decay have been measured, together with the temperatures occurring in the decay processes. Based on the very plausible assumption that the neutral species are evaporated with thermal kinetic energies only and that statistical equilibrium is reached in the transition state, the activation energies are identified as the thermodynamic separation energies, D(N)+. Further details of their method are given in Appendix B. The resulting tables of D(N)+-values show both an odd-even staggering and shell structure in the form of steps at the magic electron numbers 2, 8, 20 (and 40). Total cohesive energies E(N)+ are subsequently obtained as a sum of the D(N)+-values from 2 to N. With the less plausible assumption that measured vertical-ionization thresholds, ZP(N), for cold neutral clusters can be used to obtain the cohesive energy E(N) of the neutral clusters in a Born cycle, cohesive energies of the neutral clusters are derived. These are analyzed in terms of the liquid drop expression (2.4) with the result that the measurements appear to agree with the zero-temperature bulk values for a, and a,; or at least that they are consistent with these values.

Evaporation and fission from doubly charged clusters is now examined in a similar way, exploit- ing the results already obtained. As an example, we reproduce measurements from the fission of potassium clusters K2+2+, K2+4+ and K&+, respectively, in Fig. 29. A number of new assumptions are made in order to analyze this and similar results. They are the ones expressed e.g. in Fig. 3, as well as in Sections 2 and 3, see also Ref. [ 191. Thus:


K;,(K;) K ++ 21

K;,( K;)

K;(K,*)

?

TIME OF FLIGHT

Fig. 29. Mass spectra of doubly charged potassium clusters (contaminated with singly charged species of identical mass-to-charge ratio) with their heavy fission products (left) and evaporation residues (right), obtained by tandem time-of-flight mass spectrometty. The time available for fission and evaporation decay is a few microseconds. From Ref. [19].

a

(i) The two-sphere model is a valid approximation. (ii) Polarization reduces the external barrier B, and hence the fission barrier Bf.

(iii) Shell corrections at the saddle are identical to those in the final fragments. (iv) The ionization energy of a singly charged cluster with IV, electrons is larger than that of neutral one with the same number of electrons by the amount

(4.2)

With these assumptions and extensive use of Born-cycles, i.e. energy conservation, a large body of measurements on fission mass distributions and (kr/k,)-values are examined. Compared with the model presented in Section 2 and in Appendix A, the main difference is the treatment of polarization in terms of an effective barrier distance, s, as shown in Fig. 3. This leads similarly to estimates


0 5 IO 15 20 25

FRAGMENT SIZE, P

Fig. 30. Estimated fission barriers as a function of mass asymmetry in the fission of Klz and K:: clusters, respectively, compared to the estimated activation energy for evaporation, Dl(N)+‘. The barrier heights are derived in a two-sphere liquid drop model taking into account polarization and correcting for shell structure in the initial and final states. The shell correction energies are obtained in independent experiments, where all the atomic separation energies III(N)’ are determined through measurements of monomer and dimer evaporation rates. From Ref. [19].

of the outer barrier B, as a function of mass split to which the @-values from the Born-cycles are added. Two examples are shown in Fig. 30. From Fig. 29 one sees that evaporation of neutral atoms and fission in the form of trimer emission compete on a more or less equal footing. This is not inconsistent with the activation energies for evaporation and fission shown in Fig. 30, if one allows for uncertainties of the order of 0.05 eV. On the other hand, the fragment distributions found experimentally are very strongly dominated by trimer emission (and its complement), Fig. 29, while the prediction in Fig. 30 should allow the magic fragment K; and its complement to compete with equal intensity. This may look like a serious discrepancy, but if one remembers that the experimental arrangement is somewhat less sensitive to near-symmetric mass splits [ 191 and in addition that the relevant temperature is about 0.04 eV, one realizes that an underestimation of the barrier for Kz-decay by an amount of the order of 0.12 eV or less suffice to explain the disagreement.

These findings nevertheless raise doubt about the validity of the assumptions (i)-(iv), in particular whether assumptions (i) and (iii) remain valid for near-symmetric fission or whether they are only valid for the highly asymmetric trimer decay. It is the closed shell with eight electrons in the final fragment that lowers the appropriate barrier. If this shell is perturbed at the saddle, the resulting barrier will be increased and the fission through KG-channel will be suppressed. The only case where one sees unambiguous evidence of fission into the eight-electron fragment is in the fission of L& Ref. [38]. In that case there is an enhanced effect of magicity since the fission goes into 9


the Li; + Li$ channel with closed shells in both fragments. We will discuss this important point further in Section 4.2.4.

The competition between (trimer) fission and evaporation has been analyzed [38] for the case of lithium fission in the range N = 24-32, and for potassium with N = 20-28. Using Eq. (4.1) with temperatures kBT = 0.07 eV for lithium and 0.04 eV for potassium, i.e. values resulting from the Kassel formula, see [67] and Appendix B, a value of (Br - D) is derived for each cluster. The resulting average change in (Br - 0) with N across these size ranges is about

d(Br - D)/dN M dBr/dN x 0.040 % 0.01 eV/atom

for lithium, and

(4.3)

dBr/dN M 0.020 f 0.007 eV/atom (4.4)

for potassium. Since sodium is intermediate between lithium and potassium it presumably has a slope parameter of about 0.03 eV/atom. The corresponding value calculated with the two-sphere liquid drop model can be extracted from Fig. 11 in the case of sodium. One indeed finds a slope of 0.03 eV/atom, while the Thomas-Fermi model gives about 0.05 eV/atom.

All in all, the experiments with doubly charged alkali metal clusters have contributed decisively to our present picture of how fission proceeds in small metal clusters. Both the two-sphere droplet model with polarization and the emphasis on realistic shell corrections finds strong support through this work. What remains is to understand why the (near) symmetric fission channels show up so weakly. These studies also serve to demonstrate the need for high accuracy and for separate measurements of fragment kinetic energies [69,70], in order for more detailed quantitative tests to become possible.

4.1.4. Multiply charged alkali-metal clusters

4.1.4.1. Appearance sizes. The Stuttgart group has used a multistep photoionization technique to study the decay of multiply charged alkali clusters [2 1,221. The cluster source was a low-pressure inert gas condensation cell, which produces a continuous beam of very large neutral clusters (up to several thousand atoms). These clusters were ionized by the pulse of a strong UV laser (6.4 or 7.9 eV). It was shown experimentally [22] that a photon energy of approximately

hv= W, + (2 + c)[e2/(rwsN’i3)] (4.5)

is needed to extract an additional electron from an already Z-fold charged cluster (see also Eq. (2.5)). Therefore, there is a size-dependent limit for obtaining multiply charged cluster, which is due to the photon energy of the ionizing laser and not to the stability of the clusters. Essentially, all multiply charged clusters that can be directly ionized with a UV laser pulse turn out to be stable towards fission. To obtain fission a second laser pulse with photons of energies near the plasmon resonance of the alkali clusters was used for additional heating. It was fired a few nanoseconds after the UV laser. The multiply charged alkali clusters heated by this laser pulse evaporate neutral atoms. In the process they shrink in size without losing charge until they reach a critical size where they begin to undergo fission.

In a high-resolution time-of-flight mass spectrometer, multiply charged clusters are easily recognized since the corresponding mass peaks fall between those of the singly charged clusters. A mass

286 U. Nliher et al. I Physics Reports 285 (1997) 245-320

SIZE/CHARGE, N/Z

Fig. 3 1. Above a mass spectrum of multiply charged sodium clusters. The stepwise darkening of the spectrum indicates new sets of mass peaks of increasing charge. Below, an expanded view of the interval of N/Z-values. From Ref. [21].

spectrum, or better said, an N/Z spectrum for sodium clusters is shown in Fig. 31. The highest mass peaks belong to singly charged sodium clusters. The peaks which occur exactly halfway between the Na$ peaks are mainly due to Nar with an odd number of atoms. New sets of peaks appear in the spectrum at various critical values of N/Z. This is perceived as a stepwise darkening of the mass spectrum. In the lower part of Fig. 31 the threshold region for the appearance of Na: (in black) can be seen on an expanded scale. By careful examination of the fine structure in the mass spectra, it is possible to determine the critical size for appearance of hot Nap clusters with 2 5 Z 5 7. The experimental - or effective - appearance size, Niff, is defined as the minimum size of the Z-fold charged clusters that can be identified in the spectrum. In the following section, we will give a more quantitative definition of the appearance size.

The values of Niff directly determined from the spectra of sodium clusters are plotted on a double logarithmic scale in Fig. 32. They fall on a straight line with a slope of 2.3 with magnitudes that correspond to fissilities around 0.3.

The appearance sizes for all other alkali metals have been determined in similar experiments. The results are shown in Table 2 (Section 3.4). For the monoisotopic elements cesium and sodium, Nt’ could be observed up to Z = 7. For lithium, potassium and rubidium two or more stable isotopes complicated the spectra, so that appearance sizes could only be identified unambiguously up to Z = 4.

What these experiments show is that fission begins to compete with evaporation at fissilities that lie in a narrow interval of X-values (0.25 5 X < 0.35) irrespective of charge state 2 5 Z 5 7 and irrespective of which alkali metal one considers.

One may wonder why the appearance sizes lie so close to a line of constant Z’/N. For a given droplet shape at the fission saddle, the liquid drop barrier heights will scale with the surface energy,


NUMBER OF ATOMS, N

Fig. 32. The experimentally observed appearance sizes N,” of multiply charged sodium clusters. The lines correspond to fissilities X N 1(Z2/N = 0.4) and X N 0.3 (Z*/N = 0.13).

i.e. with N2!3, whereas the activation energy for the competing evaporation process is essentially independent of N. This problem is discussed in Section 2.6 where reasons for the observed Z2/N scaling in terms of the accidental cancellation of two opposing effects are given.

Cesium clusters have the smallest appearance sizes. At first glance this is surprising, because cesium has the lowest surface tension of all the alkali metals. But it also has the lowest density. If we transform appearance sizes into radii according to the relation

R, = (N,“’ )1’3rws , (4.6)

one finds that cesium clusters in fact have the largest critical radii at the appearance point. This is shown in Fig. 33.

More recently, the Stuttgart group has carried out an analogous, equally complete series of measurements with the earth alkali metals, Mg, Ca, Sr and Ba [23]. When the results are plotted in a diagram like Fig. 32 one finds quite surprisingly that the appearance sizes lie at the considerably higher X-values, 0.50-0.73; in one case (Mgl+) even at X= 1.0.

4.1.4.2. Evidence for asymmetric trimer decay. The decay paths in terms of the mass and charge of the fission products are not directly observed in the experiments described above. Neverthe- less, there is strong evidence that the process is dominated by the loss of small singly charged fragments.

Fig. 34(a) shows a typical spectrum of multiply charged cesium clusters plotted on an N/Z scale. In Fig. 34(b) the same spectrum is plotted as a function of N. On this scale the groups of differently charged clusters separate into almost non-overlapping domains, each domain containing clusters of a differently charged state. There is a simple explanation to this behavior. To the left, each domain is limited by the stability of the Z-fold charged clusters. To the right, it is limited by the stability of clusters with charge Z + 1. This means that every group of clusters is charged as highly as possible. It perfectly fits to the scenario proposed earlier in this section. The highly charged clusters

288 U Niiher et al. I Physics Reports 285 (1997) 245-320

CLUSTER RADIUS [i]

Fig. 33. Radii of multiply charged alkali clusters at the appearance point. Since the surface tension of cesium is the lowest among the alkali metals these clusters are expected to have the largest radii, as indeed they have.

:

SIZE /CHARGE, N/Z

100 x-3

CLUSTER SIZE, N

Fig. 34. (a) Mass spectrum of multiply charged cesium clusters. The scale is N/Z. (b) The same spectrum, now on an N scale. The groups of differently charged clusters separate into almost non overlapping domains.


shrink by evaporation of neutrals till they have reached a fissility where they undergo fission. Rather than breaking apart into fragments of similar size and charge they lose only a small singly charged fragment, for example a trimer. This reduces their fissility and so the following steps are again evaporation processes until the next critical size is reached. The resulting cascade of decaying clusters leads to a spectrum as seen in Fig. 34(b). If the clusters broke apart in the middle, the spectrum would tend to look quite different with overlapping domains of differently charged states.

4.1.4.3. Systematics of fission-evaporation competition. In hot clusters that have been allowed to evaporate in vacuum for times of the order of microseconds, the temperature will typically be such that D(N)/knT is about 20-30 (the Gspann factor, see also Table 4 and Appendix B) [71]. If the activation energies for fission and evaporation are equal within knT the two processes will compete on an equal footing.

For small doubly charged clusters the increase of the fission barrier with cluster size is relatively steep, and it is only within a short size range that one can observe both fission and evaporation, see Figs. 27 and 29. The situation is rather different for large, highly charged clusters. Here, the barrier height changes less steeply and therefore fission and evaporation can occur in parallel over a wide range of N-values.

In the experiments [21, 221 the directly measured quantities are the cluster abundances, P(N). The competition between fission and monomer evaporation takes place under nearly steady state conditions where for each size gain equals loss, therefore (see Fig. 35)

dP(N)/dt=k,(N + l)P(N + 1) - (k,(N) + k&V))P(N)=O. (4.7)

Under the simplifying assumption of size independent evaporation rates, i.e. k,(N) = k,(N + l), it follows that the branching ratio between fission and evaporation can be written

~fwYkw)=(~(~ + WyN)) - 1 . (4.8)

For the purpose of the present analysis the appearance size of a multiply charged cluster, N,, will be defined as the size where the fission barrier equals the separation energy. Using this definition the size-dependent fission barrier &(N) can be expanded around N,:

Br(N) r” D -t (dBr(N,)/dN)(N - N,) =D + a(N - N,) , (4.9)

with the gradient of the fission barrier d&(N,)/dN=a. If an Arrhenius expression is used to describe the decay rates this equation leads to (see Eq. (4.1))

ln(kf(N)lk,(N))=ln([P(N + 1 )/P(N)] - 1 )=ln(gf/&> - [a/(kBT)l(N - %) . (4.10)

As one can see, this ratio can be derived directly from the experiment, e.g. Figs. 34(a) and (b). The result is shown in Fig. 36 for the decay of fourfold charged cesium clusters. Under the somewhat arbitrary assumption that gf is exactly equal to ge a regression analysis will give the appearance size of the charged clusters in addition to the a/kBT-value. For fourfold charged cesium clusters the appearance size is found to be N, = 84 i 5. This value deviates from the effective appearance size, N,“’ = 94 found in the previous paragraph by looking at the experimental spectrum. Results are summarized in Table 3 (Section 3.4), and it is found that N,“” - N, increases with charge.


Evaporatlan, h, (Nl cst’ vaparatm, k, (Ntl) cs I*

Ntl

I.?-I)’

CSN-X

Fig. 35. Competition between fission and evaporation along an evaporative decay chain.

a0 90 100 110 120

NUMBER OF ATOMS, N

Fig. 36. The branching ratio, k,/kf of fourfold charged cesium clusters as a function of size. The extrapolation to ln(k,/kr) = 0 determines the true appearance size, N,.

The interpretation is the following: Due to the slow increase of the fission barrier with size, the competition between fission and evaporation sets in for large, highly charged clusters already at cluster sizes well above the appearance size. Although the branching ratio favors evaporation in this size range, at each evaporation step a measurable amount of clusters is irreversibly lost through fission. This cumulative depletion is sufficient to lower the signal of the highly charged clusters almost to zero long before the true appearance size is reached.

4.2. Analysis and interpretation of experiments

Since most of the experiments involve fission-evaporation competition or competition between different fission channels, a quantitative description of the results would require a precision in the definition of the energy of fission barriers and evaporation thresholds better than ksT, otherwise rates and branching ratios, which always contain a Boltzman factor, will have uncertainties larger than a factor e (= 2.72). This implies precisions better than about 0.03 eV for the alkali metals and better than about 0.10 eV for gold and silver. With present day techniques such a goal is unattainable in any individual reaction. The best one can hope for is to identify trends and to examine how the relative variations in energy depend on quantities like size N or charge Z.

In the interpretation of the trends it is furthermore desirable to make a clear distinction between the average bulk effects including polarization, on one side, and contributions due to shell structure,


on the other. Otherwise, it will not be possible to extrapolate with any degree of reliability to larger clusters and to higher fissilities. Thus the partition of the energy into a bulk part and a shell part, Eq. (1.2), is central - also to the interpretation of experimental data. In the following, we will base the analysis on a carefully chosen set of liquid drop parameters, which will be taken from tabulated properties of the liquid metals in bulk form at temperatures as close as possible to the temperatures occurring in the experiments, see Appendix B. As a second step we will derive shell correction energies from evaporation experiments [64,65] with cluster cations with sizes up to N = 42. These will be compared with calculated shell energies for various assumptions about deformations. Based on the understanding of the influence of polarization on the fusion barrier, fission-evaporation results will then be discussed in terms of the model, Fig. 3, assuming trimer fission to be the dominant fission channel. At the end, the observed fragment mass distributions will be analyzed in a similar way. When appropriate, the alternative model description in terms of symmetric fission will be brought in for comparison.

4.2.1. Liquid drop j&ion barriers and evaporation thresholds The important bulk parameters are the cohesive energy per particle, a,, the surface energy c or the

surface coefficient a, =4x&o, and the Wigner-Seitz radius r,,. From these quantities the average monomer separation energy is given (Eqs. (2.8) and (2.24)) by

Q(N)+Z=a, - + aSN-“3 + i eyz2 + (2c - 1)Z)

r,, N413 ’ (4.11)

The &value follows from Eq. (2.9). The fission barrier is given by Eq. (2.10) with an additional correction based on numerical estimates of the polarization in terms of image charges with inclusion of the spill-out parameter 6R, according to Appendix A. The three quantities, a,, a, and r,, vary with temperature. In Appendix B the bulk properties of interest are tabulated for different temperatures. Based on the tables we present our best choice for the various parameters in Table 4 (Section 4.1.2).

With this choice of parameters we may now compare the liquid drop predictions with relevant experimental results. Since at this stage shell effects are neglected we shall begin by examining the size dependence of the fission*vaporation competition as expressed by the quantity d(& - D)/dN of Eqs. (4.3) and (4.4), or by the coefficient a in Eq. (4.9).

Fig. 37(a) shows the predictions for doubly charged silver clusters. One notices that the competing decay channels (charged trimer and neutral monomer emission) have equal activation energies for n = 22.5. Assuming the ratio of prefactors, gf/ge, Eq. (4.1), to be unity this also defines the appearance size N,. The experiments [ 121 give a value of N, = 16. Thus the model either overestimates the value of D*(N)++ or underestimates the trimer barrier by 0.5 eV. We will return to this problem below in Section 4.2.3.

Fig. 37(b) presents the predictions for doubly charged gold clusters. Here an estimate of the height of the barrier for symmetric fission according to Eq. (2.23) is included. Again one notices a predicted appearance size N, = 24 for trimer decay compared to an experimental value of N, M 14, see Fig. 27 and [ 16, 121. According to the model, symmetric fission will be suppressed, having a much higher barrier. The upsloping curve marked Bf in Fig. 28 should in principle be identical to the thin upsloping line in Fig. 37(b), and similarly for the two curves marked &(N) or D,(N)++ in the respective figures. The differences are primarily due to a different choice of model parameters, a,, a, and r,,.


0 L__ I

5 10 15 20 25 30 35 40

(a) PARENT SIZE, N

AU++

J 5 10 15 20 25 30

(b) PARENT SIZE, N

Fig. 37. (a) Model calculations of the increase in the barrier against trimer fission (triangles) and the threshold for evaporation, D,(N)’ ’ , for doubly charged silver ions (Eq. (4.11), with parameters from Table 4). (b) The same for doubly charged gold ions. The thin upsloping curve is the prediction of the liquid drop model for symmetric fission, Eq. (2.23).

In discussing the gold measurements, Section 4.1.2, we found a cluster temperature of 1300 K based on the value of d(& - D)/dN from Fig. 28 and the change in branching ratio with size, Fig. 27. If we now do the same using the less steep curve in Fig. 37(b) describing trimer emission, we find a temperature of 700K, i.e. one-half the previously estimated value. If the temperature is really so low it is difficult to see how there can be any thermally activated monomer evaporation at all in the experiment unless the true separation energy D1(N)++ is considerably lower than the estimate shown in Fig. 37(b). A relatively early result by the Mainz group seemed to corroborate this speculation [72], while more recently, the monomer separation energy (of Ai&‘) is found to agree reasonably well with the calculated value in Fig. 37(b) [ 131. The discrepancy therefore persists. It will be interesting to see if in future experiments one will find a slope as steep as the one shown in Fig. 27. If the real slope had indeed just half the value indicated in Fig. 27 the discrepancy would disappear.

For the alkali metals there is less ambiguity about the trend in the separation energies D(N)+, while for the relevant temperatures there exist two suggested sets of values, differing by as much as 30-40%, see Appendix B. Figs. 38(a) and (b) present model calculations for Liz+ and Kz+, respectively. From these one derives value of d(& - D)/dN = 0.035 eV/atom for lithium, which compares favorably with the experimental value 0.04 f 0.01 quoted previously, Eq. (4.3). Similarly, for potassium the model value of 0.025 eV/atom agrees with the experimental one: 0.020 f 0.007 eV/atom, Eq. (4.4). It should be noticed, though, that these comparisons, including also the situation with sodium, are based on the higher (Kassel) value of the two possible temperatures. If one chooses the lower (Weisskopf) value, the agreement is less satisfactory.

We conclude this discussion by examining the situation with cesium. Fig. 39 presents calculations for three different charge states. The slopes derived from this figure can be compared to the experimental values of a/kBT give already in Table 3. According to Eqs. (4.9) and (4.10)

d(Bf - D)/dN M a = (a/ksT)kBT. (4.12)

U. Niiher et al. IPhysics Reports 285 (1997) 24.5-320 293

2.5 ) I 1.5 I

E OS Li++ ++

K

0 1 ,.,/,,. ,, ,. 0 '~,'1',11'1'111"11'1~'11,1',1, "',,',',",- 15 20 25 30 35 40 45 50 5 10 15 20 25 30 35 40 45 50

(a) PARENT SIZE, N (b) PARENT SIZE, N

Fig. 38. The same as Fig. 37(a): (a) for doubly charged lithium ions; (b) for potassium ions.

1

0 100 200 300 400

PARENT SIZE, N

Fig. 39. Calculated trimer fission barriers for cesium clusters in three different charge states (triangles). Thin lines are the corresponding thresholds for monomer evaporation, Eq. (4.11). Based on the parameters of Table 4.

The results of the comparison are presented in Table 5. The model calculation is clearly able to describe the experimental trend realistically, but once again the best agreement is obtained with the higher of the two possible temperatures, i.e. Kassel.

Thus in these comparisons, instead of being able to check how accurate the assumptions about the bulk liquid drop parameters are, we are reminded of the difficulties in obtaining unambiguous estimates of the experimental cluster temperatures. If one turns the question around and assumes the bulk parameters to be essentially correct, the observations just reviewed can be seen as an indication that the higher of the two temperature estimates is the more realistic one - as opposed to the choice made in preparing Table 4. The effect of 30-40% differences in temperature on the bulk parameter values remains rather slight, however, (about l%), as one sees from the table in Appendix B .

Further comparisons with experiment will require insight into the magnitude of the shell corrections.


Table 5 Measured and calculated values of the parameter a for cesium, describing trimer fission-evaporation competition as a function of size - with the two different estimates of the temperature. Measured values of a/kar from Table 3

Z Experiment Model

a/W kBT (ev) a (eV/atom) a (eV/atom)

2 0.4 i 0.10 0.028 0.01 I * 0.003 0.022 0.039 0.016 ZIG 0.004

4 0.10 f 0.03 0.028 0.003 i 0.00 1 0.0056 0.039 0.004 + 0.001

6 0.06 It 0.02 0.028 0.00 I 7 f 0.0006 0.0025 0.039 0.0023 * 0.0008

4.2.2. Experimental shell corrections In the simple two-sphere model, shell corrections at the transition state (saddle) are identical

to those of the final fragments. As discussed already in Section 4.1.3 it is therefore possible to use independent experiments on evaporation from the size-selected fragment cations to derive shell corrections at the saddle [19,38,63-651. The directly measured quantities to be used for this purpose are the activation energies for monomer evaporation from singly charged cluster cations, D(N)+. Forming the sum

E(N)+ = CD(i)‘, (4.13) 2

one obtains an expression for the total binding energy of the cluster Na$ (using sodium as an example). The purely experimental quantity E(N)+ contains both the smooth, bulk-like contributions and the shell effects, including odd-even staggering. This can be written as

E(N)+ = E(E)+ + E(N)&.,, . (4.14)

In order to separate the oscillating contribution from the smooth term E(i))+, we now consider the Born cycle

IP(1

(N-1)

and obtain

! IP(N >

Na, +Na++e E(N)’

___f Nai + e

(4.15)

E(N)+ = E(N) + ZP( 1) - ZP(N) = a:N - afN2!3 + lP( 1) - IP(N) + E(N),+,,,, , (4.16)

U. Nliher et al. I Physics Reports 285 (1997) 245-320 295

where IP( 1) is the atomic ionization energy and P(N) the smoothly varying adiabatic ionization energy of neutral cluster NaN, while E(N)$,,, is the shell- and odd-even energy of the cluster cations that we are looking for. For the adiabatic ionization energy one can adopt the smooth expression

(Eq. (25)),

P(N) = w, + ce2/(r,,N”3) ) (4.17)

with c = 0.4. One now obtains

E(N XX,, = E(N)+ - E(fi)+ (4.18)

as the difference between the purely experimental quantity, E(N)+, and the smooth quantity E(i)+ = a:N - a:NZi3 + IP( 1) -P(N). Here the parameters u: and u6 should be varied and chosen in such a way that the shell oscillations are isolated in the best possible way, which means that on the average the shell energies of deformed, mid-shell clusters should be close to zero. (In principle W,, r,, and c could also be varied, but this introduces an unwarranted redundancy into the problem.) The adopted values of a: and uk should of course be close to the bulk values, a, and a,; but they can only be strictly equal if all systematic errors in the measurement of the D(N)+-values can be excluded. Thus a 5% error in the absolute magnitude of D(N) translates directly into a 5% systematic error in a:, etc. The real interest focuses on the relative variations in D(N)+ as a function of N, and in that respect the measurements are more likely to be free of systematic errors.

Fig. 40(a) shows an example of shell correction energies derived from the evaporation measurement, Ref. [63], with the parameter set a: = 0.97 eV and CZ~ = 0.67 eV. The sensitivity to the choice of parameters is illustrated through Fig. 40(b), where the volume energy has been changed by 1% i.e. setting Q: = 0.96 eV and having ai unchanged. (An increase in CZ~ by 5% leaving a, unchanged would have produced essentially the same result.) We consider the plot in Fig. 40(b) as the more satisfactory representation of the shell energies, because the experimental values for non-magic (deformed) clusters come closer to the zero line, representing the bulk. One notices the dips at magic numbers 2, 8 and 20, and the relatively strong attenuation of the odd-even difference with increasing size,

Leaving the odd-even staggering aside, we replot the shell corrections for even-electron clusters in Fig. 41. Here it is compared to several theoretical estimates [46, 731. One sees how strongly the calculation based on a purely spherical binding field exaggerates the shell effect. Allowing for axially symmetric deformations of all multipoles from P2 to P6 brings the calculation into much better agreement with experiment, although the shell amplitudes still seem to be overestimated by a factor close to two. The last calculation, which relaxes all a priori symmetry requirements (ultimate jellium, [73]), comes very close to the experimental amplitudes. In this calculation the density and the Wigner-Seitz radius is determined self-consistently, and the value (rws = 2.21 A) is quite close to that of sodium (rws = 2.15 A). The agreement is truly remarkable, because the calculation - being a variant of the jellium model - completely disregards the role of the discretized positive charges on the ions. It indicates that the ions are under complete slavery of the metallic electron gas - at least at elevated temperatures and as far as the energetics is concerned.

The experimental shell energies determined in this way and shown in Fig. 40(b) are listed in Table 6. They may now serve as realistic corrections to the bulk, liquid drop energies derived in the previous sections.

296 U Niiher et al. IPhysics Reports 285 (1997) 245-320

Sodium 1

a even no, of electrons

A odd no. of electrons

(a)

0 IO 20 30 40

CLUSTER CATION SIZE, N

0 even no. of electrons

A odd I-O of electrons

1 I I I I 1 0 IO 20 30 40

(b) CLUSTER CATION SIZE. N

Fig. 40. (a) Experimental shell correction energies E(N)&, for singly charged sodium clusters according to Eqs. (4.15)- (4.18) using measured separation energies D(N)+ from Ref. [63] and smooth parameters as given in Table 4, except that the bulk binding energy has been set equal to av = 0.97eV. (b) Alternative plot of the experimental shell corrections. The only difference from (a) is that the volume energy coefficient is lowered by l%, i.e. to 0.96eV.

U. NZiher et al. IPhysics Reports 285 (1997) 245-320 297

1.8

t

Even sodium 0 talc spheres Clusters

I --a-- talc mal deformaions p.0 . . 14

t

: : -+- talc. ultm jellium 0

0 IO 20 30 40

ELECTRON NUMBER, Ne

Fig. 41. Experimental shell correction energies E(N)&,, for singly charged, even-electron clusters as given in Table 6,

compared with three different theoretical estimates [46,73,74] valid at zero temperature. For these relatively small clusters temperature is not expected to attenuate the shell oscillations. The theoretical energies are normalized to the experimental

value for Ne = 20. For a discussion, see text.

Table 6 lists similar results from measurements of the separation energies in lithium cation clusters [65]. The shell dip at N = 9 is somewhat less pronounced than for sodium, but more surprisingly, there is very little indication of a shell dip at N = 3 (magic number 2) and the shell correction energies for Na and Li are of opposite sign at this N-value. The odd-even staggering, on the other hand, is more or less the same in the two cases.

4.2.3. Appearance sizes and the fusion barrier, polarization efSects and fragment kinetic energies In Section 4.2.1 we examined the slopes of the curves in Figs. 37-39 in relation to trends in

the competition between fission and evaporation. Here, we shall look at the intersection, where Bf equals D,(N) ++ The experimental location of this point is much less sensitive to uncertainties in .

temperature. Neglecting to begin with any deviation of the ratio of prefactors (gr/gk) (Eq. (4.1)) from unity, the intersection defines the liquid drop appearance size, N,(LD). A glance at the figures shows that N,(LD) is systematically larger than the measured values, N;s, given in Table 2. Taking into account that the trimer has a closed two-electron shell, cf. Fig. 41, will tend to lower the predicted barrier further and increase the discrepancy. In the following we shall examine the situation a little closer.

The relevant data are gathered in Table 7. The difference between observed and calculated liquid drop appearance sizes is converted to an energy discrepancy from the calculated values of d(Bf - D)/dYV (or directly from the plots, Figs. 11 and 37-39). This discrepancy AE is

AE = (Br(exp.) - B(LD)) - (DT+(exp.) - DT+(LD)). (4.19)


Table 6 Experimental shell correction energies E(N)& for sodium and lithium, according to Eq. (4.16). For sodium, the smooth

energy E(a)’ is based on the following parameter: a{ = 0.96 eV, at = 0.667 eV corresponding to a bulk surface tension

cr = 190 dyn/cm or 0.190 J/m’, Wigner-Seitz radius (molten sodium) rwS = 2.15 8. The atomic ionization potential is IP( 1) = 5.139 eV, and the bulk work function W oo is 2.75 eV. Experimental separation energies D(N)+ are from Refs. [63, 671. For lithium, the corresponding parameters are: a: = 1.39eV, u: = a, = 0.92eV (X 0.38 J/m*), rwS = 1.77 8, IP( 1) = 5.39 eV, W, = 2.38 eV. Experimental separation energies. are from [65]

Cluster

ion size

N Na

Exp. shell energy

E(N)&, (eV)

Li

Cluster ion size

N Na

Exp. shell energy

E(N),f,,,, (eV)

Li

0.029 0.305

-0.227 0.078

0.020 0.218

-0.127 -0.107 -0.610

0.227 0.444 0.21 I 0.588

0.105 0.042

-0.131 -0.004 -0.457

20 -0.250 -0.240

21 -0.366 -0.403

22 -0.170 -0.256

23 -0.005 -0.279

24 0.053 -0.202

25 0.016 -0.215

26 0.013 -0.158

27 0.015 -0.131

28 0.009 -0.084

29 -0.037 -0.077

10 -0.296 -0.140 30 -0.05 1 -0.040 11 -0.280 -0.283 31 -0.114 -0.013 12 -0.163 -0.166 32 -0.106 -0.014

13 -0.265 -0.209 33 -0.176 -0.001

14 -0.184 -0.132 34 -0.246 0.028

15 -0.171 -0.195 35 -0.324 -0.005

16 -0.056 -0.108 36 -0.292 0.012

17 -0.070 -0.161 37 -0.001

18 -0.120 -0.074 38 -0.014

19 -0.212 -0.257 39 -0.027

40 -0.010

41 -0.033

42 0.034

The last parenthesis can be evaluated at the relevant sizes by comparing model values of D,(N)+ with experiments [63-651, and thus the first term Br(exp.) - &(LD) can be isolated (except for cesium). One sees from the last line in the table that &(LD) is underestimated by some 0.1-0.2 eV.

A shell stabilization of the charged trimer will aggravate the problem. On the other hand, one should also take into account that the prefactor gr/gk might differ from unity in a direction that diminishes the shell effect. Estimates of this aspect leads to a correction to the (shell) energy of approximately 0.1 eV [74]. For sodium this means that the liquid drop model calculation seems to underestimate the barrier by about 0.2 eV. Examining Fig. 10 one notices that a calculation of the outer barrier B, that neglects polarization effects beyond the dipole term actually leads to an increase in the estimated barrier height of 0.15 eV. There is no experimental evidence for polarizabilities of multipole orders higher than the dipole and it is perhaps not entirely unrealistic to imagine that a

U. Mher et al. I Physics Reports 285 (1997) 245-320 299

Table I

Appearance sizes I% and trimer barriers Bf. The liquid drop appearance sizes N&LD) are taken from Figs. 11, 37, 38 and 39. The experimental values, N,(exp.), are estimated from the results presented in Table 2, taken into account the (more direct) measurements of Ref. [38] for lithium and potassium. Cesium values from Table 3. For the remaining entries, see text

Metal ion Li ‘+ Na2+ K2+ cs2+ cs4+ cs6+

N&D) 39 32 31 31 130 218 N,(exp.) 26 25 23 18 84 190 N&D) - N,(exp.) 13 7 8 13 46 88 AE (eV) 0.37 0.19 0.21 0.30 0.26 0.22 D(exp.)+ + - D(LD)++ (eV) -0.20 -0.06 -0.07 Wexp.) - Bf(LD) (eV) 0.17 0.13 0.14 -

quantum gas of electrons is less polarizable with regard to the higher multipole distortions than the free charges in the classical metal assumed in the image charge model. In the classical metal there is no electron kinetic energy and hence no electron pressure. Such a hypothesis would explain the discrepancy. (As one sees from Fig. 10 disregarding surface diffuseness would also “explain” it, but that is certainly not a realistic proposition).

These considerations are nothing but speculation inspired by the calculations presented in Fig. 10. They are not corroborated by the Thomas-Fermi theory outlined in Section 3. On the other hand, if true it leads to a simple prediction, which may also be read from Fig. 10: For trimers emitted in the fission of Ns~2~ ++ the total kinetic energy release should be of 1.15 eV and not 1.00 eV. More generally, the kinetic energies should amount to 0.75 times the Coulomb energy calculated for point charges at the contact distance, RI + R2, rather than 0.65 times this energy. Experimental values of the total kinetic energy release have been obtained for K2f5’ (0.93 eV) and for Li&’ (1.19 eV) [69,70]. This amounts to 0.72 and 0.64 times the calculated Coulomb energy B,, respectively. The results cannot be seen as a confirmation of the above speculations. On the other hand, the experimental uncertainties are not negligible. Experiments with sufficient accuracy to distinguish firmly between the two possibilities would be interesting.

4.2.4. Mass distributions, shell efsects In the foregoing sections we saw that the question of shell structure

as long as the analysis is restricted to the competition between charged effects remains undecided, trimer emission and evap-

oration. The observed preference for trimer emission is explained on the basis of the liquid drop model alone, and possible enhancements of the emission rate due to the closed shell with No = 2 remain ambiguous. What is not ambiguous, on the other hand, is the odd-even staggering with clear preferences for small even-electron fragments, N = 3,5, as opposed to odd-electron fragments N = 2,4,6.

Just because our smooth liquid drop model favors trimer emission, the appearance of fragments with magic electron numbers No = 8 or 20 is a clear sign of shell structure actually playing a role at the barrier. Therefore the positive identification of fragments with No = 8 in the fission of Ag&‘,

Ag,+dt, and Ag,:f, Fig. 26 and with No = 8 in combination with No = 20 in the fission of Liz,, Section 4.1.3 and [38], becomes particularly interesting and significant. In the following we shall


make use of the quantitative estimates of shell energies in the final fragments, which were obtained in Section 4.2.2 and listed in Table 6. These will be combined with liquid drop model calculations of the barrier height as a function of mass asymmetry, shown for example in Fig. 7. An analysis of this kind is already presented in Fig. 30 with reference to the fission of K&+ and K2+6f. Here, however, there is no evidence for other than trimer fission and the situation with the fission of potassium is therefore inconclusive.

Fig. 42 shows the prediction of the two-sphere model for the fission of Na,f,+ with and without shell corrections. The barrier is

Br = Br (LD) + Br (shell) , (4.20)

with

&(shell) = &M(N)+ i- &hell(N - 27)+ - &h&6)+ (4.21)

where the last term is taken to represent the unknown shell energy in Na&‘, which has the same number of free electrons as Na&. The liquid drop barrier &(LD) is identical to the one calculated in Fig. 7 taking diffuseness and image charges into account. As discussed in the previous section, this barrier seems to underestimate the actual liquid drop barrier by some 0.1 to 0.2 eV. Taking this into account, the competition between evaporation and trimer fission seems correctly described. What is clearly incorrect, on the other hand, is the predicted barrier for fission into the pair (N,,Nz) = (9,18) of 0.2 eV; because experimentally there is no evidence whatsoever for fission into this channel [ 171.

A similar conflicting situation arises in the case of LI,, ‘++-fission. The calculated barriers are shown in Fig. 43. Experimentally, there is evidence for approximately equal fission rates into the trimer and the doubly magic (Nl, A$) = (9,21) channels [38], whereas the calculation predicts the complete dominance of the double magic channel since its barrier is lower by as much as 0.7eV.

We conclude from these two examples that the model hypothesis of unaltered shell structure at the saddle (compared to the fragments in their final equilibrium states) has to be abandoned when fission becomes near-symmetric.

With reference to Fig. 25 and the discussion in connection with this figure it is evident that the validity of the original model hypothesis is strongly dependent on the actual center-of-mass separation of the incipient fragments at the saddle. In the foregoing section we already discussed this aspect in connection with the possible suppression of polarization from multipoles higher than the dipole polarization; because - as Fig. 10 shows - with dipole polarization alone the center-of-mass distance at the saddle becomes smaller, approaching the touching configuration. (Fig. 10 relates to trimer fission, one expects an even stronger effect for near-symmetric fission, see Fig. 8).

The experimental appearance in the fission of Li&+ of fragments heavier than the trimer shows that the final-state shell structure cannot be completely washed out at the saddle. Examining Figs. 42 and 43 a little closer, one can make an estimate of the degree of attenuation of the shell structure. The attenuation has to be at least 50%, perhaps as much as 70%. Qualitatively, this is compatible with the calculation, Fig. 25, provided the center of mass distance at the saddle is less than one Angstrom from the distance of touching spheres.

There remains the results [l l] for the fission of silver, Section 4.1.1. As one sees from Fig. 26 fission into Agz and into Agg’ seems to compete on an equal footing, with Agg’ becoming the dominant channel for the heavier clusters. Here, we have no independent estimates of the shell energy

U. Ntiher et al. IPhysics Reports 285 (1997) 245-320 301

No;; -NrJ + t No’ 27-p P

I 0 6 12 18 24

FRAGMENT SIZE, P

0.6

0.4 I I 0 5 IO 15 20 25 30

FRAGMENT SIZE, P

Fig. 42. The fission barrier height of Na$ as a function of fragment size like in Fig. 30. The thin line is the model

calculation without shell effects. It is identical to the lower curve in Fig. 7. The fat line with large filled circles represents the barrier after correction with the experimental shell energies of Table 6.

Fig. 43. The same as Fig. 42, here for the fission of Lifl. Model parameters are taken from Table 4 and shell energies

from Table 6.

of the final fragments. The predictions of the pure liquid drop model are shown in Figs. 44(a) and (b). Again, without the influence of shell structure trimer emission is expected to be the only significant decay channel, as actually found by the Mainz group [ 121. To explain the alleged occurrence of the fragment Agl in the decay of Ag&+ one needs to invoke an extra shell energy correction at the

saddle equal to 0.2 eV in favor of Agl. For the decay of Ag,, ++ the necessary shell energy has to

be at least 1.0 eV. The global scale of energies in silver is 2-3 times that of sodium, so this energy has to be scaled down to about 0.3-0.4eV if one wants to relate the estimate to the prediction of Fig. 42 for example. A residual shell energy of 0.3-0.4eV is not far from one-half of the full shell energy predicted in Fig. 42. Therefore the experimental result for Ag,‘;’ does not contradict the conclusion reached above about 50-70% attenuation of shell structure effects at the saddle. On the other hand, it remains a puzzle that the residual shell energy that has to be invoked in the case of Ag:,’ is so much smaller. Hopefully, future experiments will resolve the puzzle.

4.3. Summing up

We have reviewed the main body of experiments on the fission of simple monovalent metals performed before 1995 and a few more recent ones. They all involve the competition between decay through evaporation of neutral atoms and decay by fission, or the competition between different


2.4

D,(N)++

0 3 6 9 12

(a) FRAGMENT SIZE

3.4

3.2

z 2.4 0 z 2.2

E 2

I.8 I 0 5 10 15 20

(b) FRAGMENT SIZE

Fig. 44. (a) Calculated fission barrier of Agf,’ as a function of fragment size according to the smooth, liquid droplet model with parameters from Table 4. The threshold for monomer evaporation D,(N) ‘+ is given for comparison. (b) The

same for Ag$.

mass splits in the fission channel. In general, the fissilities are rather low, X M 0.3, and the mass- asymmetric emission of charged trimers dominates. Odd-even effects are prominent and at least in one case fragments with magic electron numbers N, = 8 or 20 compete successfully with N, = 2 trimer emission. Statistical decay laws are assumed to govern the processes and are expressed rather crudely by equations of the Arrhenius type in various forms. Based on this, an extrapolation method is presented in Section 4.1.4.3 that allows a precise definition of the size N,, where the evaporation rate equals the fission rate.

For comparison with the experimental results we have introduced a reference model in Section 2 that describes the clusters as surface-charged liquid droplets with a diffuse surface. The density as well as the specific volume and surface energy of the cluster are assumed equal to those of the bulk material at the same temperature. From the comparisons we learn:

(i) that the threshold energies for evaporation from singly charged clusters oscillate relative to the model predictions in a way that reflect strong odd-even differences for small clusters and also a systematic extra binding of magic clusters with N, = 8 and 20;

(ii) that the magnitude, E,h,a(N), of the extra binding is in accord with theoretical jellium model calculations, provided these allow for a complete relaxation of the cluster shape in terms of non- spherical and non-axial deformations (Section 4.2.2).

In order to describe fission, the model is extended by assuming a saddle shape composed of the two fragments next to each other at a distance given by the balance between Coulomb repulsion and attraction due to metallic polarization, and with shell stabilization energies identical to those found in the separated fragments. From this we further learn:

(iii) that the evolution in the relative rates for evaporation versus fission as a function of size is sensitive to the assumed cluster temperatures, and that uncertainties in assumed temperature stand in the way for a closer test of the model parameters, Section 4.2.1;

(iv) that the observed appearance sizes are systematically somewhat smaller than predicted by the model. Future measurements of the kinetic energy released in fission may decide whether this is due to a mutual polarizability that is lower than anticipated by the model, implying suppression of polarization terms higher than the dipole term, Section 4.2.3;


(v) that the strikingly small appearance sizes observed with the divalent metals Mg-Ba do not appear to be explainable within the framework developed for the monovalent metals;

(vi) shell effects are present in near-symmetric saddle configurations, but they are attenuated by a factor of OS-O.3 compared to the effect in the separated fragments. This may be related to the reduced polarizability conjectured in (iv); Section 4.2.4.

5. Future perspectives

In the foregoing sections we have made status of the experimental situation as it is known at present. Before concluding this review it may be appropriate to briefly take a look at the future and present some speculations about the prospective outcome of future investigations in the field of metal cluster fission.

The first section to follow leans heavily on possible analogies between nuclear and cluster fission, whereas the second and final section takes inspiration more directly from the original formulation of the fission problem by Lord Rayleigh [ 11, and from Taylor’s later experiments with bulk materials [62].

5.1. Towards jssility X = 1, the nuclear connection

In this section we will ask what new physics will emerge if it should become possible to work with larger clusters and with charge states that will bring the fissility close to unity. All experience with nuclear fission corresponds to this situation. It is an interesting question to which extent charged metal droplets will behave analogously, and to which extent one will encounter completely new and different effects. The experimental possibilities for reaching this new regime will be discussed in Sections 5.2.2 and 5.2.3. An indication of how to proceed is given by the experiment of Brechignac et al. [24], where a singly charged K$ cluster is photoionized a second time to form K&+ with an X-value of 0.94.

As we have seen in the foregoing sections, experiments with hot clusters of fixed charge tend to be limited by the competition between fission and evaporation. The binding conditions in simple monovalent metals are such that this competition becomes acute already for values of X E 0.3. The result is a strong favoring of charged trimer emission. In nuclear physics terms, this is similar to the evaporation of an alpha particle from the compound nucleus rather than to fission in its true sense.

5.1. I. Fission of the nuclear quantum droplet It should be realized that the character of atomic nuclei as droplets of a quantum liquid - as

opposed to being small samples of crystalline matter - is made particularly transparent through the fission process. A major rearrangement of the nuclear many-body system can be described with a high degree of success by picturing the nucleus as a classical charged drop. The competition between short-range attractive forces and long-range Coulomb repulsion leads to the existence of a barrier, separating bound states from fissioning configurations as shown in Fig. 1.

The more the Coulomb forces dominate, the lower barrier, the more compact the saddle shape, and the stronger the preference for divisions into two equal fragments. This is brought out in detailed theoretical studies of uniformly charged classical drops [3] and it is confirmed through an impressive


body of experimental observations [27,75]. In this type of fission the saddle shape is a highly non- trivial configuration that cannot be accessed in any other way.

The quanta1 aspects of the nuclear fluid show up in the details. The barrier height is influenced by shell structure in the initial state, while shell structure in the final states may give rise to pronounced deviations from the predicted symmetric fragment distribution, as seen in the fission of uranium for example [75]. In addition, shell effects may modify the saddle configuration. When this is close to an ellipsoid with a ratio of axes 2 : 1, a metastable energy minimum may develop, giving the barrier a double-humped profile [4,5,27] with a shape isomeric state in the minimum in-between. Finally, the flow of the nuclear quantum droplet, as it is driven from the saddle towards complete separation into two fragments is highly dissipative, in a way that is specific to liquid where the particle mean free path exceeds the linear dimensions of the flowing object. A long mean free path and the existence of a mean field with shell structure are indeed two sides of the same coin.

5.1.2. Fission of metallic droplets Like nuclei, metal drops are stabilized shapewise by surface tension, and they are also likely to

have a characteristic density and hence volume, which will be preserved under shape changes. As a result they will fission if given a sufficiently high charge. The saddle configuration will presumably be reflection-symmetric and stable towards asymmetric distortions for X-values sufficiently close to unity, [ 11, but whether or not this will result in the subsequent formation of two equal fragments is not so evident [6,7]. For lack of better knowledge we shall assume this here. Finally, like in nuclei, the average liquid drop behavior will be significantly modified by (electronic) quantum size effects. These similarities should give scope for interesting experiments. But the perspective becomes even larger, when one considers the differences between nuclei and metal clusters.

First of all, metal clusters will be crystalline, or at least solid, at sufficiently low temperatures. There is no parallel to that in nuclei. For metal clusters there may therefore be two distinctly different regimes for the descent from saddle to scission, with solid clusters exhibiting a very slow, creeping motion as opposed to a faster process for the liquid phase. In addition, the uniform charge distribution in nuclei and the strong preference for equal neutron and proton densities severely restrict the range of occurrence of stable, or metastable nuclei, see Fig. 45. In metals there is no restriction of this kind. The excess charge Z and the size N are virtually free parameters, and the field for fission studies therefore becomes much wider with metals. This is illustrated in Fig. 46. Here one also sees how the approach towards more fissile species through “evaporative shrinking” [2 1,221 fails to bring the charged cluster into the interesting region of fissilities close to one, because charged trimer evaporation takes over around X = 0.3, resulting in a reduction of the fissility. It follows that all fission barriers in the interesting domain of X < 1 are significantly lower than the evaporation thresholds. For alkali metals this means significantly lower than 1 eV. Being able to suddenly increase the charge by several units and detecting the fission decay right away is the challenge that has to be met in order to explore the interesting domain. The question of heating during the charging process is an important aspect in this connection, see Section 5.2. In the following, we will discuss how shell structure might influence the energetics of fission.

5.1.3. Injhence of shell structure For X close to unity, the liquid drop fission barrier becomes small compared to the energy

scale of the electronic shell structure. If X exceeds 0.8, Eq. (2.3) predicts a liquid drop barrier,


N

i

f$ 100 100

5 y 50 50

Y vc7porafion of mJfr0ns

20 20

IO 20 50 100 200 300

NUCLEAR MASS, A

Fig. 45. Double-logarithmic plot of the nuclear isotope chart. The dashed line delineates the relatively narrow region of stable and radioactive nuclei. Its extension toward large sizes is limited by the zero fission barrier line, X = 1. If excited, or heated, nuclei will evaporate neutrons, but the nuclear symmetry force, which prefers equal numbers of protons and neutrons, causes the neutron threshold to increase, eventually allowing for the evaporation of charged particles (protons and alpha particles) to compete. This is indicated by oblique hatching to the left of the line marked km/k, = 1. Only for the heaviest nuclei, symmetric fission enters as an alternative deexcitation process, see line marked kf/k, = 1 and the vertical hatching.

Br( LD) < O.OO~U,N~‘~, or less than 0.14 eV for Nazoo. Shell structure in the initial states will strongly modify such low liquid drop barriers. Fig. 47 shows the (free) shell energies for Na clusters for three different temperatures. They are calculated in Refs. [46, 761 for the canonical ensemble of the valence electrons, minimizing the total free energy with respect to the cluster shape. The steep dips at the magic numbers 58, 92, 138, 198 and 260 reflect the shell gaps in the electronic levels of these spherical clusters. The extra binding due to the shell closure amounts to about 1 eV for zero temperature and remains larger than 0.5 eV for kBT = 0.06 eV. The plateaus between the magic numbers correspond to deformed clusters. For these the shell energy is less than 0.1 eV.

The dominance of the spherical shell energies over the the liquid drop energy will influence fission in a decisive way. For kBT of the order of 0.01-0.03 eV all deformed clusters should fission rapidly, because the fission barrier is roughly given by the liquid drop estimate, which is less than 0.2 eV as indicated above. In contrast to this, the magic clusters are expected to survive symmetric fission, because the spherical shape is highly stable. In order to fission symmetrically the cluster must deform, i.e. thermal agitation must overcome a barrier equivalent to the energy difference between the bottom of the dips and the plateaus in Fig. 47. This stabilization due to shell structure is analogous to the predicted stabilization of superheavy nuclei around 2 = 114 [77]. The search for these elements has been successful in recent experiments, which are approaching 2 = 114 [78]. One indeed finds the barrier against fission to be exclusively due to shell structure effects [79]. On the other hand, the emission of a singly charged trimer from the shell-stabilized cluster is not hindered by the shell closure cf. Fig. 39. The main effect of shell structure is therefore its influence on the decay mode, and not on the lifetime.


I I !“‘I I 11’11 50 - -50

Unstable

N 20-

I I L ,<II I LICII IO 20 50 100 200 500 1000

CLUSTER SIZE, N

Fig. 46. The wide open field of metal cluster fission, plotted in the same way as Fig. 45 with the exception that the charges on the ordinate scale have been divided by 10. The upper left part corresponds to clusters without fission barrier. There is a broad band of clusters below the X = 1 line, extending ad injinitum to the right, where thermally activated fission should be observable. The nearly parallel line labeled “trimeri evaporation” marks the limit for observation of fission from hot clusters in competition with the evaporation of neutral atoms. Above this line only relatively cold clusters will be stable for finite amounts of time.

5 -I = 0 06 e\'

-13 I

50 100 150 200 250 300

CLUSTER SIZE. N

Fig. 47. Shell contribution to the free energy for Na clusters. The full drawn lines are the zero lines of shell energies, representing the bulk properties. For each cluster the free energy is minimized with respect to the shape of the cluster (quadrupole and hexadecapole deformation). From Refs. [46,76].

As seen from Fig. 46, one may always come close to the line X = 1 by charging the cluster by a sufficient amount. Although the discreteness of the charge imposes some restrictions on the X-values, by choosing the right charge Z one may always reach X > 0.8, where the spherical shell energy is the main stabilization mechanism. Let us take the shell closure at the electron number N, = 138 in

U. N&her et al. IPhysics Reports 285 (1997) 245-320 307

Na clusters as an example. To reach a fissility close to unity, a charge of Z = 8 is needed. The liquid drop fission barrier &(LD) is less than 0.2 eV. For Z = 9 there will be no liquid drop barrier. Fig. 47 shows that the extra binding due to the spherial shell closure is considerably larger than 0.2 eV for 130 < N, < 150. Hence, one expects that NayTi9+, . . . , Nayli9+ will not fission readily across the symmetric or near-symmetric barrier configuration, whereas the lighter and heavier clusters will. Above mass 198 (N, = 190) one expects again stability due to the next spherical shell closure at N, = 198. The rapid symmetric fission will cease already at somewhat lower mass, because the shell energy slowly increases already for the deformed clusters towards the end of the shell (cf. Fig. 47). A similar pattern is expected for all the shell closures at N, =20,40,58,92,138,198,258 and maybe 336. For still larger masses the shell stabilization becomes rather weak due to the thermal fluctuations, as is seen from Fig. 47.

The shell energies will also influence the mass distributions of the fragments in binary fission. One expects that at least one fragment will have a size corresponding to a magic electron number. The energy gain is so substantial that it may already be felt at the saddle, thereby selecting the channel. The other fragment can readily accommodate the rest of the initial electrons because there is no special preference with non-magic clusters. The analogous effect is well known from fission of heavy nuclei where asymmetric mass distributions are observed, always with one fragment taking a nearly magic number [7_5]. In the case of nuclei the shell energies and the liquid drop fission barriers are comparable. Hence, for clusters one expects this kind of shell effect to become even more prominent.

Returning to the example of NaF, one expects that the clusters above mass 160 will decay in an asymmetric way with respect to mass: NaF + Na$ + Na:_,,. This is caused by the dramatic gain in shell energy for the magic fragment, which is much larger than the loss in the liquid drop energy for an asymmetric mass split. Approaching the next shell closure at N = 200 the shell structure of the fragments favors a symmetric mass split. For the next lower shell 92 < N, < 138 the shell structure favors the decay of the deformed clusters into two relatively symmetric fragments Na:? + Na$ + NaE,, but the asymmetric channel Nat + Nait + Nak,, could compete favorably.

The discussion assumes the existence of electronic shell as one knows them from molten clusters. If the clusters are cold and solid the electronic shell structure may be rather different or even be completely absent. If so, a dramatic change of cluster fission with temperature can be expected also for this reason.

5.2. Beyond fissility X = 1

Thus far we have been concerned with the fission process of metal clusters characterized by a fissility parameter X below the critical value (X < 1). This situation corresponds to the existence of a barrier against fission. In such a case, where only thermally activated fission can occur within typical experimental time scales, the topology of the potential energy hypersurface rules to a large extent the ultimate dissociation channels. For X > 1, the charged cluster may find a path towards fission without having to overcome any barrier. One expects that the dynamics of the charged conducting cluster will now play a major role and govern the fragmentation process. In order to investigate this fragmentation dynamics and whether small metal clusters with not too many units of charge do indeed


behave the same way as large conducting droplets, one would need to prepare clusters of a hundred to about a thousand atoms in relatively high charge states. How can this be done in practice? After recalling some stability criteria for charged droplets, we shall discuss various experimental ways to multi-ionize metal clusters and we shall especially focus on a recently developed method that allows one to obtain highly charged clusters without heating them too much at the same time [25].

5.2.1. Highly charged metal clusters and the Rayleigh model Lord Rayleigh was the first to quantitatively examine the conditions for stability of a liquid

conducting drop charged with electricity [ 11. Following him let us first consider a neutral droplet of surface tension cr and assume incompressible, non-viscous, h-rotational flow. Under the action of capillary forces, oscillations can develop and cause the surface of the drop to deviate from the spherical equilibrium shape of radius R ,+ For small amplitude oscillations it is convenient to parametrize the surface in terms of an expansion in spherical harmonics:

R(t 4) = Rx 1 + c ~LMYLM(& 4) I > (5.1) LM

where the radius parameter R, ensures volume conservation. The eigenfrequencies of capillary oscillations of the spherical drop are given by

co; = [o/(pR;)]L(L - l)(L + 2) , (5.2)

where p is the specific density. Note that all modes are stable and characterized by a (2L + 1)-fold degeneracy. The smallest possible frequency of oscillations of the drop corresponds to L = 2. Let us now add some electrical charge that, for a conducting drop, will distribute itself across the surface. Under the action of the destabilizing Coulomb field the eigenfrequencies of oscillations of the drop of fissility parameter X are modified by an X-dependent factor:

co; = [o/(pR;)]L(L - l)(L + 2)[1 - 4X@ + 2)] . (5.3)

One sees that all modes up to Lhlgh = 4X - 2 become unstable. If the charge is distributed uniformly throughout the entire volume, as it is the case for protons in atomic nuclei, the instability occurs

instead for multipoles up to a value of Lhigh = ci-” 10X + G - i. For X sufficiently close to unity both

cases are identical with only the quadrupole mode being unstable. Quite different behaviors occur for X % 1, where the conducting sphere develops instabilities for higher multipoles much more easily than the nucleus. Whereas it is impossible to prepare nuclei with such high fissilities, it is actually possible with metal clusters. It is therefore relevant to notice that not only does the highest unstable multipole grow linearly with X, but also the most unstable mode (having the largest modulus of oL) grows as a linear function of X. As an example, a Na,,, ‘*+ cluster with X N 3 shows instabilities up to Lhigh = 10 with the largest one at L = 7. As a dramatic consequence of the occurrence of this kind of unstable ripples, Lord Rayleigh suggested that “the liquid is thrown out in fine jets” [l]. A recent theoretical study [7] is most suggestive in this respect, see also [62]. It is therefore tempting to extend the analogy to the highly charged metal cluster by predicting that tiny droplets


are rapidly emitted from its surface. These tiny droplets (monomer, dimer, trimers, . . .> would carry one unit of charge. In this light what would occur for X N 1 remains particularly intriguing, since here both Q-value considerations, as discussed in Section 2, and the dynamics of the liquid drop seem to favor symmetric fission, or at least a symmetric saddle configuration.

5.2.2. Experimental means for producing highly charged metal clusters Sizes and electrical charges of atomic clusters can be varied independently. A neutral cluster is

prepared by some conventional means, and then multiply ionized by some other method. In simple metal clusters there is a neat separation of core electrons from the conduction electrons. Not being aware of actual experimental results, we shall refrain from discussing ionization methods that would take advantage of the electronic (Auger-type) cascades following a deep core hole excitation, induced for example by X-ray absorption. On the other hand, focusing on methods for removing valence electrons only, one may contemplate the possibility of creating cold, highly charged clusters in this way. Before discussing a collisional method which may open this possibility, a few comments on the alternative method of multi-UV photon absorption ought to be made. This method is described in Section 4.1.4. It follows from Eq. (4.5) that the highest charge state to which a NaN cluster can be brought by using a 7.9 eV high-fluency laser does not exceed Z = 4 for N = 200 and Z = 8 for N = 1000. Such clusters have quite low fissilities. The Stuttgart group showed that it was actually possible to shrink the size without changing the charge by heating with an additional laser and thereby provoking evaporation [21,22]. Nevertheless the highest available charge remains quite limited, and it is not possible to study the fission of large and cold clusters with high fissility by this method.

An alternative method, just recently developed [25], exploits the large amount of potential energy that is stored in a multiply charged atomic ion. When such an ion approaches an alkali-metal cluster in a low-energy peripheral collision, conduction electrons are inevitably transferred from the cluster to the projectile ion. This transfer occurs at quite large distances. The electrostatic potential seen by an electron outside the cluster is composed from the contributions from the charged ion and from the image charges that the latter induces in the cluster. There are also (smaller) contributions from the ionized cluster charge, and from the image charges induced by the electron itself. As the ion passes by, the top of this potential can be lowered enough to completely eliminate the barrier that otherwise prevents the electrons near the Fermi surface in the cluster from escaping. The critical distance d, at which the barrier disappears is roughly given by

(5.4)

where Q is the ion charge and IP the ionization potential of the cluster, Eq. (2.5). As an illustrative example we show in Fig. 48 the potential experienced by an electron from a Nazoo cluster in the presence of an ion of charge 5. For a relative distance of 53A, which is about four times the cluster radius, the maximum of the barrier is equal to the negative of the ionization potential of NazoO, and lies at about 20 A. Note that “hollow atoms” are likely to be formed in such peripheral collisions, as it has already been observed when highly charged atomic ions interact with surfaces [80]. Since only electrons near the Fermi level can escape, one expects that the resulting highly charged cluster remains weakly excited. The whole multiple ionization process takes place on a very short time scale of a few femtoseconds, during which the constituent ions in the cluster remain fixed.


2 -- IP

: F Na,,,’ Ifi

i -10 -

5 F Impact parameter = 53 A s

Fig. 48. Potential barrier for an electron located on the radial axis between a Nazoo cluster and an atomic ion of charge 5 at a distance 53 A. The part just around and inside the cluster radius, R, = 12.6& is only schematic. The ionization potential of an isolated Na2oo cluster is indicated by IP.

In a real experiment these peripheral collisions occur together with more central collisions. The latter lead to very high electronic excitations [Sl] that eventually turn into violent vibrational motion and thermal disintegration of the cluster. The possibility of strong multiple excitations of the electronic plasmon modes in the cluster provoked by the swift passing of the atomic ion may also imply a certain degree of heating in periferal encounters.

5.2.3. Highly ionized sodium clusters from collisions with multiply charged ions First results from a cross-beam experiment in which H+, 05+, and Ars+ ions of velocity around

the Fermi velocity of the electrons in the cluster were colliding with free sodium clusters of a few hundred atoms are available [25, 261. They show that the measured appearance sizes for clusters with charges from 3 up to 6 are systematically lower when the ionization has been induced by a multiply charged ion than in proton-induced ionization. Moreover, the proton data agree with the appearance sizes obtained by the high-fluency laser method [21]. The authors invoke the temperature effect to explain the observed differences [25]. A fraction of the multiply charged clusters produced in (peripheral) collisions with highly charged ions have an internal temperature, which is low enough to preclude evaporation within the time available in the experiment. These multiply charged clusters survive and their decay by fission can be observed although their fission barrier is lower than the activation energy D, (N)Z for monomer evaporation (N 1 eV). To go further and be able to determine the size dependence of the fission barrier for a given value of the cluster charge it will be necessary to have a better control of the initial state of the highly charged clusters produced in the experiment. The main aim remains of course to be able to produce very high and well-defined charge states and to see how such highly instable metal clusters rid themselves of their charge excess. In order to fulfill these expectations, coincidence techniques are required. These are presently being developed and one may expect that such quantities as the multiplicity of fragments and their size distribution in individual fragmentation events will soon become available, see [26]. This will open new perspectives in the field of metal cluster fission.


6. Conclusions

Studies of fission of multiply charged metal clusters have greatly widened the insight into cluster fragmentation processes, otherwise limited to the evaporation of neutral atoms or dimer molecules. All evidence supports a description in terms of competing thermally activated processes involving a transition state and an activation energy for each decay channel. With the aid of somewhat simplistic transition-state models of the Arrhenius type one arrives at threshold energies, or activation barriers, from observations of the decay into the various channels. The experimental result can take the form of an appearance size, where fission and evaporation of neutrals are equally probable, or it can be a determination of the relative probabilities for observing fission products of various sizes and charge states. In practice, the fission barriers are measured relative to the evaporation thresholds for neutral atoms, while these are known or assumed to be known from other sources.

There results a considerable body of data on the heights and therefore the relative importance of different fission channels. In addition to ordinary experimental uncertainties in these heights there are systematic indeterminacies stemming from the simplifications involved in the choice of transition state model and other simplifications. Nevertheless, an analysis of systematic trends in the data and comparisons with theoretical models has proved to be instructive.

The existence of odd-even staggering in the data and the preference for fragments with 2 or 8 free electrons immediately indicate that electronic quantum effects and shell structure play important roles. These effects are superposed on general systematic trends that fit into a liquid drop picture of the fissioning cluster. A theoretical treatment in the framework of local density jellium theory reproduces these trends rather well, both with respect to the liquid drop properties described in the extended Thomas-Fermi approximation, and as regards shell structure obtained in the Kohn-Sham approach. The existence of a pronounced surface diffuseness and a strong electric polarizability are features that naturally emerge from this quantum theoretical description.

Another model of the liquid drop features is based on a classical picture, where it is specifically assumed that the density as well as the volume and surface energy terms are identical in the bulk and in the clusters, provided that they refer to the same temperature. Supplementing this picture with the empirically known surface diffuseness and an equally well-known quantum correction to the classical electrostatic charging energy implies a model that is both simple and well defined. It will not describe shell- and odd-even effects, however. Here one can introduce the additional hypothesis that shell effects in the transition state (i.e. at the saddle) are identical to those found experimentally in the separated fragments, and more generally, that the saddle energy can be computed as the maximum in the repulsive potential between the two unperturbed fragments. An important feature of this potential is the classical image charge interaction between the fragments, i.e. the mutual polarizability. This may produce a maximum in the potential before the surfaces touch.

When this picture is confronted with the entire body of experimental data one finds agreement on the whole, except for the divalent metals Mg-Ba. For the monovalent metals the general trends are reproduced in a way that strongly supports the model in its main features, and there are no alarming discrepancies. In particular, the predicted predominance of charged trimer emission is confirmed. It is also quite striking that parameters taken from the properties of bulk matter appear to be valid down to the smallest cluster sizes. On two points, however, small but systematic differences between model predictions and direct observations makes it possible to discern an influence of physical effects that are neglected in the simple model. One is a systematic overestimation of the mutual polarization. The

312 U Ntiher et al. I Physics Reports 28.5 (1997) 245-320

image charge treatment is equivalent to a molecular polarization model that takes all multipoles into account with the weighting factors appropriate to a classical metallic medium. In reality, confining the polarization to the dipole term gives a somewhat better fit to experiment. Perhaps this signals that the electron gas is less polarizable than a classical metallic droplet (with diffuse surface), except for its dipole polarizability.

The other effect may be related to this. It is the systematic suppression of fission into the channels with a magic number of 8, and sometimes also 20 electrons, compared to model predictions. For this type of fission it is therefore not correct to assume that the shell structure of the final fragments is fully developed already at the saddle, it rather appears to be strongly attenuated here.

This observation accentuates the interest in new experiments performed with large clusters and at higher fissilities, where fission through a more compact symmetric saddle configuration is predicted. By analogy to nuclear fission one may here expect specific shell structure effect for ellipsoidal saddle shapes. As opposed to nuclei, clusters with fissility near one can in principle be produced within a broad range of mass values. Most of these will fission instantaneously because the height of the liquid drop barrier is negligible; but the ones that hold a magic number of electrons will be stabilized by their spherical shell configurations in analogy to the island of stable superheavy elements in nuclear physics. The fission of large clusters of near-critical fissility may reveal many other interesting properties of solid and liquid clusters besides those anticipated by the analogy to nuclear fission.

It was Lord Rayleigh, who in 1882 posed the problem of what happens to a charged liquid drop. Will it be stable, will it divide into two or perhaps three fragments or will it sprout out a stream of tiny charged droplets until the charge is reduced to a reasonable value? He had a conducting liquid in mind, but it was through studies of nuclear fission that most of the answers were produced so far.

Important as this episode is, it leaves essentially all questions related to fission of supercriti- cal drops unanswered. It now appears possible to advance into this region through charge transfer experiments using crossed beams of metal clusters and highly charged atomic ions. Perhaps we shall therefore soon have more complete answers to Lord Rayleigh’s problem.

Acknowledgements

We thank J.A. Alonso, M. Barranco, J.M. Lopez, A. Mafianes, J. Nkmeth and R. Lombard for their contribution to the work described in Section 3 as well as V.V. Pashkevich for contributing to Section 5, where we have also received decisive inspiration from conversations and correspon- dence with W.J. Swiatecki. This work has been partially supported by the DGICYT (Spain), grant PB92-002 1 -CO2-02, and by EU contracts, Science SC 1 -CT9 l-0654, Science SC 1 -CT92-0770, HCM - Network ERBCHRXCT 940612, and furthermore by grants from the Danish Natural Science Research Council. Francesca Garcias, Claude Guet, Stefan Frauendorf and Ulrich Ngher acknowl- edge the Niels Bohr Institute for its kind hospitality and for financial support. During the preparation of the manuscript we have enjoyed most valuable advise and received information about unpublished results from several colleagues. We wish to express our special thanks to our referee as well as to P. Frijbrich, M. Heinebrodt, H. Ito, I. Katakuse, S. Kriickeberg, L. Schweikhard and R. Vandenbosch.


Appendix A. The classical image charge model

The image charge expansion is a useful tool for describing the interaction energy of two charged metal spheres. It is a purely classical description. As known from textbook physics the problem of finding the electrical field (and the electrostatic energy) of a point charge q at a distance s from the center of a conducting sphere of radius R can be solved easily. The interaction between the point charge and the conduction sphere is described by introducing two image charges, one at the center of the sphere and the other at position R2/s (see Fig. A. 1). The size of the induced charges is +qR/s (at the center) and -qR/ s, respectively. The image charges simulate the effect of polarization in the form of induced real charges on the surface of the conducting sphere. Together with the point charge they create a field outside the sphere, which is identical to the field of the point charge and the induced surface charges; in particular, the potential on the surface of the conducting sphere is everywhere the same. In this way the image charges describe the attractive interaction between the conducting sphere and the point charge. At s = R the total energy diverges. If the sphere is not neutral but has an additional surplus charge Q, the interaction of Q with the point charge q

is uninfluenced by the induced image charges. Thus, the total force between a point charge and a charged conducting sphere can be written as

F(s) = Qq/s2 + q2R/s3 - q2R/[s(s - (R2/s))2] , (A.1)

which is the sum of a monopole-monopole and a monopole-dipole interaction. This yields for the total energy E as a function of d, the distance of the centers of the spheres,

J d

E(d) = F(s)& = ii!?! _ q2R3 00 d 2d2(d2 - R2) ’

(A-2)

The extension of the problem to that of two charged, conduction spheres is straightforward. The exact solution uses an infinte number of image charges of rapidly decreasing size (see Fig. A. 1). First, an image charge is induced on each sphere. Furthermore, each image charge creates another one on the opposite sphere. This procedure results in an infinite number of image charges with alternating sign and rapidly decreasing magnitude. In the following description Q and q are the total net charges on each sphere (equal to the initial charges), and Qo and q. are the charges in the center of each. R and r are the radii of the spheres. The centers of the spheres are located at So and so, with SO 5 SO + R + r. The sizes of the induced image charges, Qi and qi, are

qn+l = -Qnd@o - &I >

Qnt~ = -qnR/(s, - So) .

The positions, si and 5’i, of the induced charges are given by

Sn+l = so - r2/(so - S,) ,

S n+t = So + R’/(sn - So) . (A-4)

(A.3)

For even-numbered n Eq. (A.3) can be expressed as

qn = Pn . 90 ,


s

Fig. Al. (a) The interaction of a point charge q with an isolated conducting sphere can be described by two image charges, one at the center and the other at a distance R2/s from the center. (b) The interaction of two charged metal

spheres can be described by an infinite number of image charges, q,, and Qll, on each sphere.

en = Ll . Qo ,

and for odd-numbered n, respectively,

Qn = I,, * qo . (A-5)

p and L can be extracted easily from Eqs. (A.3) and (A.4). The conservation of total charge on each sphere specifies the magnitude of the charge in the center:

i=l

This is an alternating sum which converges rapidly. It yields

(A.61

Yo=Y-~(I”uYo+~2i-leo,, 1=l

U. N&her et al. I Physics Reports 285 (1997) 245-320 315

or

Qt 1 + C:,PZ;) - 4 CEl A2i-1 Q” = 1+ c:, (P 21 + l2i) + CE, Cj”=, (P2iA2j - PZi-112j--l> ’

q(1 + CjZ, i2i) - Q Cr, ~2r-I q” = 1 + c:, (p 21 + A2i) + CFl C,Z1(P2ii/2j - PZi-lA2j-1) ’

(A-7)

which can be solved easily if the infinite sum is extended only to a maximum value, i,,,. This summation converges very fast. For two spheres of equal size and charge an accuracy better than 0.1% is obtained with i,,, = 10 . When size and position of the image charges are known, the total energy of the system can be calculated by varying the position of one sphere from cc to So + d:

E(d)=/:-li z;sdro. (A.8)

Together with Eqs. (A.3), (A.4) and (A.7) the total energy of the system can be expressed as an exclusive function of the input parameters, Q,q, R,Y, and the distance of the center of the spheres, d = so - So.

Appendix B. Experimental bulk properties of monovalent metals at different temperatures

The comparison between experimental results and model calculations in this review is based on the hypothesis - right or wrong - that the cohesive energy, surface energy and density of a cluster of a given internal temperature are equal to those found experimentally for the bulk form at the same temperature. Clusters that cool by evaporation in vacuum during times of the order of microseconds will have a characteristic internal temperature, which is almost independent of the temperature of the initially hot starting condition. Leygnier [67] has estimated this temperature for sodium and potassium clusters in the size range 5 < N < 40 using either the so-called Kassel formula or the Weisskopf formula to describe the measured evaporation rates. The evaporation rate k(N, E) reads in the Kassel formulation [24, 671

k,(N,E) = vg(1 - D(N)/E)3N-7 . (B.1)

Here v is the Debye frequency, g is the number of surface atoms, M (4)N2’3, D(N) the separation energy, either for monomer or for dimer evaporation, and E is the excitation energy, M kaT(3N - 6). For not too small clusters this reduces to the Arrhenius expression

kx(N, T) = 4vN2/3e-D(N)lkBT , U3.2)

The exponent D(N)/ksT is often called the Gspann factor. The analogous, so-called Weisskopf expression for the decay rate reads according to Refs. [24,67]

k,(N, E) = Sv3[(3N - 7)/E]hm( 1 - D(N)/E)3N-8 . U3.3)

The quantity S is the geometrical cross-section of the cluster, xr& N213, and m is the mass of the evaporated atom or dimer. This formula also reduces to an Arrhenius-type expression, but the prefactor to the exponential, or to the paranthesis, is significantly larger.

316

Table 8 Bulk properties of metals

U. Niiher et al. I Physics Reports 285 (1997) 245-320

Temperature Cohesive energy Surface energy Wigner-Seitz radius crit gr Z = 2)

(K) (eV) H,,,,(KJ/mol) a, (eV) (T (J/m* 1 as (eV) p (kg/m3 1 rws (8)

Silver

Abs zero

Room Melting,s Melting,1

Boiling Critical

Gold

Abs zero Room Melting,s Melting,1

Boiling Critical

Lithium

Abs zero Room Melting,s Melting,1

Boiling 1615 0.139 147 1.39 0.24 0.67 Critical 3200 0.276 0 0 0 0

0 0 298 0.025

1235 0.106 1235 0.106

2435 0.210

0 0 298 0.025

1337 0.115 1337 0.115 1360 0.117

3130 0.270

0 0 298 0.025

454 0.039 454 0.039

625 0.053

286 2.94

280 2.90 267 2.66

256 2.44 0.70 1.67 0 0 0 0

368 3.81 350 3.60 345 3.45 332 3.33 332 3.32

322 3.12 0 0

161 1.64 161 1.63 157 1.60 155 1.56

-

0.91 1.98 9340 1.66 0.91 4.38

8100 1.74 0.80 4.94

1.13 2.42

1.13 2.42

19320 1.59 18500 1.61 17320 I .65 17300 1.65

0 0

0.40 0.95 516 0.38 0.92 500

400

1.60

171

1.75 1.77

1.90

1.11 3.61 1.11 3.61

0.46 8.66 0.45 8.33

0.35 11.40

The Orsay group [24,67] measures the decay rates k(E,N) for monomer and dimer evaporation for a series of consecutive cluster sizes, obtaining thereby two sets of experimentally determined values of the energy ratios D,(N)/kaT and &(N)/kaT from either the Kassel formula (B.2) or the equivalent Weisskopf formula, respectively. By further requiring energy conservation

D,(N) + DI(N - 1) = D&V) + E(2)) (B.4)

where E(2) is the known dimer binding energy, it becomes possible to determine absolute energies, D,(N), D,(N) and kBT. The resulting D(N) values are practically identical in the two formulations but, as a result of the different prefactors, the temperatures come out differently. For sodium clusters the group finds T ~580 K according to Kassel’s formula and T ~4420 K from Weisskopf’s formula.

Table 8 (Continued.)


Sodium

Temperature Cohesive energy Surface energy Wigner-Seitz radius crit gr Z = 2)

(K) (ev) &,(KJ/mol) a, (eV) 0 (J/m2) as (ev) P (kg/m3 1 rws (8)

Abs zero Melting,s Melting,1

Boiling Critical

0 0 107 1.11 371 0.032 107 1.08 371 0.032 104 1.04 420 0.036 104 1.04 580 0.050 103 1.02

1156 0.100 97 0.91 2500 0.215 0 0

Abs zero Melting,s Melting,1

Boiling Critical

Cesium

Abs zero Melting,s Melting,]

Boiling Critical

0 0 90 0.93 336 0.029 90 0.90 336 0.029 88 0.88 350 0.030 88 0.88 480 0.041 86 0.85

1033 0.089 79 0.73 2200 0.190 0 0

0 0 78 0.80 301 0.026 77 0.077 301 0.026 75 0.75 327 0.028 74 0.74 452 0.039 73 0.72 942 0.081 68 0.62

2050 0.176 0 0

0.20 0.73 0.20 0.72 0.18 0.68 0.12 0.50 0 0

0.110 0.61 0.109 0.61 0.100 0.57 0.065 0.41 0 0

0.070 0.51 0.069 0.51 0.063 0.48 0.038 0.32 0 0

1013 2.08 968 2.11 927 2.14 915 2.15 877 2.18 740 2.31

910 2.57 860 2.62 824 2.66 820 2.66 792 2.69 669 2.85

1997 2.98 -

1850 3.05 1830 3.06 1760 3.11 1510 3.26

0.43 9.24

0.43 9.36

0.41 9.65 0.31 12.86

0.45 8.88

0.45 8.91

0.43 9.38 0.33 12.19

0.43 9.24

0.43 9.26

0.41 9.72

0.29 13.90

Both values are listed in the first column of Table 8 and similarly for potassium (480 K and 350 K, respectively). The corresponding values of kBT are listed in the second column.

In order to be specific we have taken the temperatures determined from the Weisskopf formula as the basis for our model calculations. To obtain estimates for the remaining metals, we note that the ratio a,/kBT is equal to 29 for both potassium and sodium, and we simply take the cluster temperature relevant to fission-evaporation experiments to be kBT = a,/29 for all the metals; see Table 8. The choice of the Weisskopf expression may actually be the less realistic of the two options. It follows that the temperature itself is subject to a considerable uncertainty. The other model parameters are much less affected. In order to show this the values of a,, a, and r,, are given in the table for a large span of temperatures. In this way, the dilatation and the increase in Wigner-Seitz radius at melting are also illustrated. For all metals, the temperature relevant to the experiments lie above the melting temperature. When known, values of the critical temperature are also tabulated. This is to emphasize that at sufficiently high temperature the surface energy approaches zero and the notion of a Wigner-Seitz radius loses its meaning.


The cohesive energies a, are calculated from the quantity (AH,,, - kBT), remembering that AH = AE + PAV and that PAV E PV,,, = kBT for evaporation from a dense liquid phase to a dilute ideal gas phase. The values of AHevap are (mostly) taken from Ref. [82] or from Ref. [83]. They are corrected for the presence of (a few percent) dimers in the gas phase above the bulk metal, and thus refer exclusively to the evaporation of single atoms. The units for the AH and AE-values are converted according to: 1 eV/atom = 96.48 KJ/mol = 23.06 Kcal/mol.

Surface energies c are from [83]. The unit J/m2 is converted to a,-values as follows: a,(eV) =

0.78430 (J/m2)r& (A2). Densities are again taken from [83] and converted to r,,-values: r,, (A) = 7.345(MW/p (kg/m3))‘13,

where MW is the molecular weight. A few r,,-values are from [84]. Finally (Z2/N)crit = 0.2778a,r,,,, (A).

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