Atomic structure and segregation in alkali-metal heteroclusters

The ground-state atomic and electronic distributions in Na Cs„clusters with composition m =n and m =2n have been calculated by minimizing the total cluster energy using the density-functional formalism. The approximation is made by replacing the total external potential of the ions by its spherical average around the cluster center during the iterative process of solving the Kohn-Sham equations for each geometry tested. In the size range studied here (up to 90 atoms per cluster), the cluster is composed of well-separated homoatomic Na and Cs shells, the external one always being a Cs shell. We have also found that the cohesive energy goes rapidly to the bulk limit. An analysis of the geometries shows strong cluster reconstruction with increasing size. By comparing the geometry of pure Na„with that of the Na„core in Na„Cs„ for clusters formed by only an inner Na layer and an outer Cs layer, we have observed that the Na„core adopts a geometry different in most cases from that of the free Na„cluster, and such that the number of faces of the polyhedron formed by the Na„core is as close as possible to the number of external Cs atoms, in order to accomodate these Cs atoms on top of the faces of the polyhedron.

stratification and segregation effects will dominate the en- tire cluster structure.Some preliminary results presented by the authors have shown that the cluster Na&pCS&p is formed by an inner Na shell surrounded by a Cs shell.
Using the Car-Parrinello method, Ballone et al. ' have found segregation of potassium to the surface of a Na&OK&o cluster.
Relevant questions in this area are the following: How does the segregation effect depend on cluster size?Evidently, segregation is a surface effect.But what happens when one looks at the whole cluster?Is there random mixing of Na and Cs? Or, rather, do ordering or full phase separation occur?Bulk Na-Cs alloys are miscible in the liquid state." In the solid state a single intermetal- lic compound, of composition Na2Cs, forms at low tem- perature, but solid solutions do not exist.This indicates the importance of the temperature for mixing.In sum- mary, it should be interesting to know if the miscibility properties change in microalloys with fewer than 100 atoms.In this work we study the atomic arrangements of Na Cs"heteroclusters with m = n and 2n, and with a to- tal number of atoms up to X = n +m =90, paying partic- ular attention to questions like mixing, ordering, segrega- tion, and relative stability of the clusters as a function of size and magic numbers.

II. MODEL
In this section we give a brief sketch of the computational method used to obtain the ground-state geometry, and the corresponding energy, of the clusters.
First of all, the atoms are placed at random positions, and the valence-electron distribution is self-consistently calculated by minimization of the total energy, according to the Kohn-Sham version of density-functional theory'   using the local-density approximation for exchange and correlation effects." At this moment, the net forces at the ionic sites are nonzero, and then the ions are allowed 1990 The American Physical Society to move a small distance in the force directions.The electron distribution is computed for this new atomic arrangement, and the whole cycle is iterated until a minimum of the total energy with respect to all ionic coordinates is obtained, that is, until all the forces vanish.
As the only guarantee for reaching the absolute minimum, and not a relative one, the whole process is repeated for a sufhcient number of random initial configurations of the atoms (in practice, this number has been taken to be larger than 40).
Empty-core model potentials' are used to describe the pseudopotentials of the individual ions.The empty-core radii, r, (Na) =1.74 a.u. and r, (Cs) =2.74 a.u., lead to the experimental ionization potentials in the free-atom limit for the same density-functional method.
An essential approximation of the method is that the total ionic potential due to the ions at positions R, , Vl(r)=g v(~r -R, ~), is substituted by its spherical average around the cluster center (center of ionic charge), VI"(r), in the process of integrating the Kohn-Sham equations.For this reason we call this method the spherically averaged pseudopo- tential (SAPS) method.' However, the exact ion-ion Coulombic repulsion is evaluated in our calculations, which makes the total energy a function dependent on the precise location of the atoms.
Additional details concerning the density-functional formalism used here can be consulted in Ref. 15, where applications to different pure-metal clusters have been done.

A. Layering, segregation, and growth
Let us consider first the main features of the atomic distribution in Na Cs"clusters with m = n and 2n and a tota1 number of atoms N 90.A common characteristic for the two concentrations and all sizes is the formation of homoatomic shells, that is, separated shells of Na and Cs atoms, respectively, with the most external shell al- ways occupied by Cs atoms.The most salient features concerning the formation and evolution of these shells with cluster size can be observed in Fig. 1.The radial distribution of atoms (with respect to the center of ionic charge of the cluster) has been plotted for a few represen- tative cases (N=8, 20, 34, 40, 58, and 90 for Na"Cs"and %=9, 21, 39, and 90 for Naz"Cs").Clusters up to about %=42 have two homoatomic layers (for both concentra- tions), the inner one formed by the Na atoms and the outer one by the Cs atoms.The definition of the Na layer is more ambiguous than that of the Cs layer, this one be- ing appreciably thinner.For Na"Cs" it is perhaps more realistic to describe the distribution of the Na atoms by a well-defined shell only in the restricted size range N ~26.
Instead, from N=28 up to %=44 there are a few inter- mediate Na atoms halfway between the internal (Na) and external (Cs) shells (see, for instance, NazoCszo in Fig. 1).
For Na2"Cs"one observes an appreciable broadening of the whole Na shell with respect to Na"Cs".This difference is due to the different concentration, and it suggests that the stoichiometry NazCs develops ordering more easily than the NaCs stoichiometry, which is in agreement with the situation observed in the bulk phase." In summary, we can look at the process of clus- ter growth as the successive formation of inner homoa- tomic shells, at least in the size range N & 100.
Besides the existence of homoatomic shells, we ob- serve, for most clusters with N & 10, a strong tendency to find one single atom at the cluster center.For Na"Cs" the central atom is Na up to N=26, whereas at larger sizes the nature of the central atom is anticorrelated with that of the innermost shell; that is, the central atom is Cs when the innermost shell is formed by Na atoms and vice versa.For Naz"Cs" the central atom is Na with very few exceptions.
The valence-electron density is also plotted in Fig. 1, in units of the average bulk electron density p,", defined as p, "=l/Q.,", where Q"=[mQ(Na)+nQ(Cs)]/N, Q(Na) and Q(Cs) being the volumes per atom in bulk Na and Cs, respectively.
The (average) radii of the shells and their widths (mea- sured by the standard deviations) are given in Fig. 2 for Na"Cs".The width of the surface (Cs) shell never exceeds 1 a.u.In contrast, the dispersion in the Na shell is larger, about 2 a.u. in some cases, and it would increase further if the intermediate Na atoms in the size range N =28 44 (these have b-een omitted from the figure) were ascribed to this shell.The radius of the surface shell, which is a measure of the cluster size, is a linear function of N'   with a slope approximately equal to r, * [r, *=(30, ."/4 r) ' 7].Radii for Naz"Cs"are not shown in the figure, but it is interesting to note that the radius of the external Cs shell is very similar for the two concen- trations at each size, considering the large atomic size difference, between Na and Cs.
The distribution of Cs atoms in Na"Cs" is given in Fig. 3.The majority of Cs atoms is in the surface shell.First, the number of atoms in the surface shell increases steeply with n, but later this number grows at a lower rate due to the formation of the inner Cs shell.
Previous calculations' ' for homoatomic Na"and Cs" clusters using the SAPS method have also produced a structure formed by atomic layers.In the case of sodium clusters, there is one single layer for n (7, one layer plus a central atom for 7 ~n ~19 (except for Nas), ' and two layers for 20~n ~50.The situation for Cs" is very simi- lar' to the small difference that Cs8 is not an exception and that the configuration of two layers starts at n=19.
The geometries of homoatomic clusters (to be discussed in more detail below) are rather similar to those obtained by Manninen, ' minimizing the electrostatic interactions of point positive ions in a spherical and homogeneous negative background.This trend is, however, very different from that for Na"Cs".In heteroclusters, the difference between the core radii of the Na and Cs atoms is responsible for (a) the formation of distinct homoatom- ic layers, which we interpret as a precursor of the order- has proposed a way of interpreting the abundance spectrum and magic numbers of alkali-metal clusters based on geometrical considerations.Compared with our results, there is, however, an essential difference in the pattern of cluster growth, both for homoatomic and heteroatomic clusters.In the model proposed by Anagnostatos, the clusters grow outwards; that is, the external layers form over the previously formed inner lay- ers.In contrast, our results indicate a strong reconstruc- tion of the whole cluster as it grows.In the size range covered by our calculations, additional inner shells grow when the external shell is suSciently filled.We may in- terpret this effect as a tendency to minimize the surface energy of the cluster.A particularly evident manifesta- tion of this effect can be seen in Fig. 3, where, in the range N=42 -52, the population of the surface shell remains nearly constant, while the additional Cs atoms give rise to an internal Cs layer.
Summarizing the features exposed so far, we identify three main conclusions: (a) First of all, in the small size regime considered in this paper, the geometries bear no relation to those of bulk crystals, since the surface effect is very strong and dominates the cluster reconstruction as N grows.(b) Cs atoms strongly prefer the outer part of the cluster.This is again a surface-controlled effect, since Cs segregation lowers the surface energy of the cluster.
(c) However, when the cluster is large enough, Cs atoms begin to form a new shell in the inner part of the cluster, and for Na60CS30 we have found a sequence of layers Na- Cs-Na-Cs.We interpret this as a precursor of the tenden- cy to superlattice formation in the bulk solid alloy, due mostly to atomic size difference.
At this point it is fair to point out that in the process of calculating the equilibrium geometry of Na-Cs clusters (also of pure Na"or Cs"), we find a large number of rela- tive minima with energies close to that of the ground state.This is a manifestation of the "soft" nature of clus- ters formed by alkali metals.This softness accounts for the strong reconstruction with increasing size, and it also suggests that very large cluster sizes are needed to exhibit the crystallographic structure of the bulk solid.This con- trasts with the situation for clusters with ionic bonding.
In a study of Cs"O, we have found that the coordination around the oxygen impurity is rather stable as a function of cluster size, and furthermore, this coordination is simi- lar to that in the bulk oxide.'

B. Stability and magic numbers
Sharp variations in the relative stability of the clusters as a function of size give rise to the so-called magic num- bers, which appear as prominent features (maxima or pronounced steps) in the abundance spectrum.
Figure 4 gives the total energy per atom, E(X)/X, of Na"Cs"as a function of the number of atoms in the clus- ter.E(N)/N shows pronounced minima for %=8, 20, 34, 40, and 58.This is a consequence of electronic shell- closing effects.Clusters with 8, 20, 34, 40, and 58 valence electrons have the closed-shell electronic configura- tions ls 1p, ls lp 2s ld', ls lp 2s ld' 1 f ',   ls lp 2s ld'olf' 2p and ls lp 2s2ldiolf i~2p61g18 respectively.We then predict that these clusters should be prominent in the abundance spectra.The general trend in the energy curve is similar to that obtained for homoatomic alkali-metal clusters both in the jellium and SAPS models, and the above numbers, X =8, 20, 34, 40, and 58, are, in fact, experimental magic numbers of pure alkali-metal clusters' and mixed alkali-metal cluster.
We then conclude that the magic numbers of pure and mixed alkali-metal clusters depend only on the number of electrons in the cluster.However, the geometry and atomic distribution are sensitive to the nature of the atoms in the cluster and to their relative proportion.
The convergence of the energy of the cluster to the bulk limit can be studied by calculating the cohesive en- ergy per atom.For the clusters with composition Na Cs", the cohesive energy per atom can be written E"h(Na Cs") = [mE(Na)+nE(Cs) 1 m+n -E(Na Cs")], where E(Na) and E(Cs) are free-atom energies calculated by the same model.The cohesive energies are plotted in CQ where E""(Na) and E"h(Cs) are the cohesive energies of the pure metals, and AH is the heat of formation of the alloy with respect to the pure metals.AH is negative, since the compound exists, but its precise value has not yet been determined.The only theoretical calculation for this compound gives the wrong sign for bH, but at least suggests that the absolute value of AH is small com- pared to the cohesive energies of the pure metals Na and Cs), which is the usual situation for metallic alloys.Ad- ditional evidence for a small ~bH ~comes from the experimental fact that the number of intermetallic compounds in this binary system is only one (Na2Cs).Consequently, we can approximate E"h(NazCs alloy) = -, '[2E, , h(Na)+E"h(Cs)] . ( 4 Using experimental values for the cohesive energies of the pure metals, Eq. ( 4) gives E"h(Na2Cs alloy)=0.037 a.u.
On the other hand, the cohesive energy of the cluster Na26Cs» is already equal to 0.031 a.u./atom.From the evolution of E"h with cluster size in Fig. 5, we can con- clude that the bulk limit of E, ,h is approached rather rapidly.
One could argue that a better idea of the convergence to the bulk limit can be obtained by using in Eq. ( 4) theoretical predictions for E,"h(Na) and E"h(Cs) (ob- tained under the same assumptions used in our cluster calculations) instead of the experimental values.Never- theless, it is well known that density-functional theory and the pseudopotential approximation work extremely well for the simple alkali metals and the theoretical pre- dictions for the cohesive energy are in very good agree- ment with experiment.For instance, an all-electron den- sity functional calculation by Moruzzi et al. gives E""(Na)=1. 116eV/atom, in nearly perfect agreement with the experimental value of 1.113 eV/atom.Calcu- lations using the empty-core pseudopotential ' also give very good results for the binding energies of Na and Cs.Actually, we estimate that the error from using ex- perimental pure-metal cohesive energies instead of calcu- lated ones is not larger than the error from neglecting hH in Eq. ( 3).
In Fig. 6 we give the energy balance of the reaction Na2, +Cs"~Na, "Cs".
This reaction is exothermic; that is, the mixed cluster is stable with respect to the pure fragments, but the magni- tude of the heat of the reaction, if measured per atom, de-  creases rapidly with increasing size.In the case n~t his energy balance becomes the energy of formation hH of the bulk NazCs compound starting from the pure met- als, but at small n the formation of the Naz"Cs"clusters does not proceed, evidently, through reaction ( 5).The heat of the reaction Na"+Cs"~Na"Cs" is also included in Fig. 6.
ues' for the ground state of Na"Cs" is given in Fig. 7.
One of the noticeable features of this figure is the behav- ior of the 1d and 2s levels.In the size range N 26 these two levels are close in energy, although the 2s is below the 1d state.Then the levels cross and the 1d state be- comes lower in energy for N~28.For comparison we notice that the ld state is always below the 2s in the jelli- um model of pure alkaline-metal clusters.
The peculiar behavior observed here for the mixed clusters is correlated with the nature of the atom that in most cases exists at the cluster center.This central atom is Na for N ~26, Cs between N=28 and 46, and Na again for N )46.A Na atom provides a more attractive potential than a Cs atom, giving rise to an enhanced probability for s-type electrons in the central region.For small clusters (X ~26) this effect is strong enough to place the 2s level below the 1d level.Inversion between the 1d and 2s lev- els has also been found in the case of divalent impurities in alkali-metal clusters.'  This inversion is responsible for the appearance of a new magic number in those clus- ters, corresponding to ten electrons.Also, the 2p and 1g states are closer here than in the jellium model for pure clusters.
This effect is also present in recent jellium-on-jellium calculations for Na clusters coated by Cs.Finally, the oscillations of the 2s eigenvalue in the region %=50 -58 are correlated with the number of Cs atoms in the inner shell of the cluster.

C. Na"geometries
It is the purpose of the next section (Sec.III D) to dis- cuss the geometries of small Na Cs"clusters formed by an inner Na shell and and outer Cs shell.In particular, we wish to show the way the external Cs layer modifies the geometrical arrangement of the internal Na content with respect to the geometry of the pure Na"counterpart.
As a prior step, we first discuss the geometries of the pure Na"clusters, comparing these with the results of other calculations which avoid the spherical approxima- tion for the total ionic potential.Because of this approxi- mation, our resulting geometries for very small clusters are more spherical than those obtained in more accurate calculations, which are restricted, however, to date, to very small clusters.Ab initio configuration-interaction (CI), ' extended-Hiickel,   and density-functional pseudo- potential results are available for the equilibrium geometries of Na"up to n=9.These are, as ours, static calculations.Our stable geometries for some clusters are  but with an atom in its center in the case of Na7.Na6 is a pentagonal pyramid and Na7 a pentagonal bipyramid in more accurate calculations.'  We have found that the pentagonal bypyramid is the first excited local minimum of Na7 in the SAPS model, being almost degenerate with the centered octahedron (the difFerence in energy is only 0.004 eV/atom, whereas the cohesive energy per atom is 0.65 eV).Our geometry for Nas is a square antiprism, in agreement with the density functional results of Man-  ninen and co-workers '  The equilibrium geometries of Na, o and Na&~a re very regular.Na, o is a centered tricapped trigonal prism (D3I, ), and Na» is a centered square antiprism bicapped at the square faces (C~,, ).Na, z is a distorted centered icosahedron with a vertex lost and Na» a perfect centered icosahedron.A configuration-interaction calculation by Pacchioni and Koutecky' suggested that the equilibrium geometry of Li» is a centered icosahedron.A more complete dynami- cal study has been performed by Ballone et al. ' for Na2O at T=0 and 200 K.The equilibrium geometry at T=200 K (in an average sense) resembles our static geometry (see Fig. 9) in that there are two atoms inside a more external arrangement of the remaining 18 atoms.
In summary, we have found that the equilibrium geometries predicted by the SAPS model for small Na clusters are often identical to those of more accurate cal- culations and, in other cases, coincide with the geometries of low-lying isomers.Since, first of all, the SAPS model is expected to become better for larger clus- ters (which are more spherical than the smaller ones) and, on the other hand, the energy differences between the ground-state equilibrium geometry and those of low-lying isomers are rather small, we can have some confidence in our predictions for "large" mixed clusters.A few representative clusters (corresponding to n =6, 7, 8, 9, 12, and 13) have been chosen as illustrative exam- ples of the small size range.Figure 8 shows, for each n, the geometry of pure Na"(left-hand part), that of the Na core (denoted Na"*) in the Na"Cs"heterocluster (middle column), and the whole Na"Cs"geometry (right-hand column).The pure Na"geometries have already been discussed in Sec.III C. Na6 is a nonregular centered tri- gonal bipyramid (D3h).This is a polyhedron with six faces, to which the six outer Cs atoms of Na6Cs6 can be easily coordinated.Each Cs atom is on top of one of the faces, and the six Cs atoms then form a trigonal prism surrounding the Na6 core.Na7 brings out the central atom of Na7 on top of a face of the octahedron.The seven Cs atoms are then on top of the remaining seven faces of the octahedron.The square antiprism Nas reconstructs in such a way that Nas is a centered octahedron with one vertex split into two symmetrical ones and C2"symmetry, as the way to provide the eight faces needed to host the eight Cs atoms.The Na9 structure (centered square antiprism) remains the same in forming the heterocluster, and the nine Cs atoms become coordinated to nine of the ten faces, leaving one triangular face empty.Note, however, that the interatomic distances in the Na9 core have increased with respect to those in the free Na9 cluster.The structures of the Na&o and Na~, cores are different from those of the pure Na clusters, the trend governing the changes being the tendency to form a polyhedron with a number of faces closer to the number of Cs atoms.The Na» core in Na»Cs» has D3h symme- try; it is formed by one Na atom at the center of a po- lyhedron with 12 faces.Each of the 12 Cs atoms is coor- dinated to one of the 12 faces of the Na polyhedron, preserving the D31, symmetry.Finally, Na, 3 is a little dis- torted with respect to Na&3, but it remains as an icosaedral arrangement, covered by 13 Cs atoms in the following way: Four Cs atoms (atoms E) are on top of edges of the Na» (Ih ) core, and each one is coordinated to four Na atoms.The remaining nine Cs atoms are on top of nine triangular faces (atoms T), whereas three faces rest empty.
Interatomic distances are also given for Na"and Na"* in Fig. 8.When only one distance is reported, this means that the polyhedron is regular or that an average (d ) of the polyhedron edges is made.Averaged values for the edges of the external polyhedra joining Cs and Na atoms are between 8.1 and 8.3 a.u.for the clusters given in Fig. 8.
From the analysis of the cluster topologies, we con- clude that the geometry of the Na"' core in Na"Cs" is different from that of free Na", and that the mechanism of reconstruction is the formation of a polyhedron with a number of faces consistent with the number of external Cs atoms.Then these Cs atoms can sit on top of those faces.This result indeed suggests that specially stable clusters will be obtained by starting with the equilibrium polyhedron for Na"and capping each face with a Cs atom.Note, however, that the geometrical stabilization effects are expected to be very small in pure and mixed alkali-metal clusters and will be unobservable in the usual mass spectra.By looking again at Fig. 4, we note that the stability peaks of Na"Cs"clusters (%=8, 20, 34, 40, and 58) correspond to electronic shell-closing effects.
FIG.2.Average radii of Cs (+ and o) and Na (+) shells and their widths (measured by the standard deviation) in Na"Cs"vs N' '.N =2n is the number of atoms in the cluster.
FIG.3.Number of atoms in the inner (I) and surface (S) Cs shells of Na"Cs"as a function of size.Note that these clusters have an even number of atoms.

Fig. 5
Fig.5for m =n and 2n.As mentioned above, the only bulk solid alloy is the ordered compound NazCs.The FIG.5.Cohesive energy (per atom) of Na"Cs"and Na2"Cs" FIG.6.Energy balance for the reactions Na2"+ Cs" FIG. 7.One-electron Kohn-Sham eigenvalues as a function of cluster size for Na"Cs".The outermost occupied level is indicated by a circle.
eV/atom above the stable planar form.The geometry for both Na6 and Na7 is the regular octahedron, FIG.8.Geometries of pure Na"(left-hand column), Na" part in Na"Cs"(central column), and whole Na"Cs"(right-hand column), for n=6, 7, 8, 9, 12, and 13.Open and solid circles represent Na and Cs atoms, respectively.Distances are all in a.u.
Larger clus- ters with 46~N ~60 develop a Cs layer in the inner re- gion.Then their configuration is that of a Na region with a large dispersion of radial distances bound on the inside by one Cs shell and on the outside by another, more pop- ulated, Cs shell.It is noticeable that the cluster size N at which the inner Cs shell appears is about the same for the two concentrations studied here.One could have expected some difference due to concentration because theCs- and Arvati et a/.' This occurs