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SHELLS OF ATOMS

T.P. MARTIN

Max-Planck-Institut ftir Festkijrperforschung, Heisenbergstr. I, 70569 Stuttgart, Germany

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PHYSICS REPORTS

2s . __ -_ El ELSEVIER Physics Reports 273 (1996) 199-241

Shells of atoms

T.P. Martin

Max-Plan&Institut,ftir FestkSrperforschung, Heisenhergstr. 1, 70569 Stuttgart, Germany

Received November 1995; editor: E.W. McDaniel

Contents

1. Historical introduction 201 5. Test for shells 219 2. Classification of shells 205 6. Examples of shell structure 222

3. Example of an experiment ~ photoionization 7. The melting of clusters 234 of aluminum clusters 212 8. Concluding remarks 238

4. Experimental techniques for revealing shell structure 215

References 240

Abstract

It is interesting to consider how a solid evolves during the earliest stages of growth. The atoms reorganize into a completely new structure each time an atom is added when a cluster is very small. However, this cannot go on

indefinitely. Eventually, a preferred symmetry becomes frozen into the cluster. Further growth takes place by adding layers of atoms to this frozen core. One layer is sometimes referred to as a geometric shell of atoms. Shell structure is the subject of this article.

PACS: 36.40. -c; 61.46. +w; 61.48. fc

Keywords: Clusters; Fullerenes; Nanocrystals; Melting; Mass spectroscopy

0370-1573/96/$32.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved SSDI 0370-1573(95)00083-6

T.P. Martin /Physics Reports 273 (1996) 199-241 201

1. Historical introduction

The microscopic structure of condensed matter is very difficult to determine experimentally. An exception exists for crystalline materials which possess long-range translational periodicity. In this case, the development of structure-determining diffraction techniques triggered the explosive growth of what is now called solid state science. The study of condensed matter not having translational symmetry is greatly hindered by a lack of detailed knowledge of its structure. Clusters of atoms are no exception to this state of affairs. However, there is increasing evidence that clusters possess a periodicity that is not translational in character, but rather a kind of “shell periodicity”. This periodicity appears because the clusters grow by the accretion of shells or layers of atoms. It often turns out that concentric layers of atoms are added in such a way that the overall symmetry of the clusters is left unchanged. It will be shown that the shell periodicity of clusters allows, in some cases, the symmetry to be deduced.

Layered growth imposes certain restrictions on the outer symmetry or morphology of the cluster. The cluster could, for example, take the form of one of Plato’s five geometric bodies, the tetrahedron, the cube, the octahedron, the pentagonal dodecahedron, or the icosahedron. Later we will show examples where clusters do indeed take the form of platonic solids. In addition to these polyhedra known since antiquity there exist two modern geometries that are relevant to the discussion of clusters. The “modern” polyhedra were introduced by Johannes Kepler [l] in 1611.

The faces of Plato’s polyhedra are triangles, squares or pentagons. Kepler considered the possibility of constructing highly symmetric polyhedra having rhombic faces. He seems to have been inspired by observing the shape of pomegranate seeds. More will be said concerning that inspiration later. The two new polyhedra that Kepler discovered are the rhombic dodecahedron with 12 faces, Fig. 1, and the rhombic triacontahedron with 30 faces, Fig. 2. Notice the special relationship between the rhombic dodecahedra and the cube and the octahedron. A similar relationship can be seen to exist between the rhombic triacontahedron and Plato’s decahedron and icosahedron. In order to make contact with clusters of atoms here, we will merely state that rhombic dodecahedral shells of any size can be cut out of a body-centered cubic (bee) lattice and that a metallic layer grown epitaxially on Cso could form a rhombic triacontahedron. But specific examples will be discussed in detail later. Suffice it to say that these seven geometric forms, five platonic and two Keplereon, form the basis for a discussion of the outer symmetry of most clusters. However, there exists microscopic symmetry that must be discussed first, the arrangement of individual atoms.

A good starting point for the discussion of the arrangement of atoms is the packing of spheres. Once again, it all began with Kepler. Kepler described the arrangement of spheres which we now call face-centered cubic (fee). He mentioned in an off-handed way that this is the densest possible packing of spheres. Mathematicians have been trying to prove this assertion for the past 300 years [2,3]. Why is this so difficult to prove? It is clear that the densest packing of just four spheres is a tetrahedron, and it is clear how this densest packing ofjust four spheres is a tetrahedron, and it is clear how this densest packing should continue. Each newly added sphere should nestle into the hole made by three other spheres, forming a new tetrahedral arrangement. If you were able to continue indefinitely this building scheme and fill up all of space, you would construct a body having a density of 0.78 . . . and your name would go down in history, along with Plato, Archimedes

202 T.P. Martin 1 Physics Reports 273 (1996) 199-241

Fig. 1. Kepler’s rhombic dodecahedron and its relationship with a cube.

Fig. 2. Kepler’s rhombic triacontahedron and its relationship with a decahedron and an icosahedron.

and Kepler, because the highest sphere packing density that has ever been obtained is 0.74. The problem is that the tetrahedron is not a space-filling geometric figure. If you tried to follow the building principle mentioned above, you would eventually find yourself with a structure having a hole in the surface that was too small to accept a sphere and you would have no alternative but to cover it over, leaving a void in the solid, and thereby lowering its density. Although the tetrahedron is not space-filling, Kepler’s rhombic dodecahedron is. This brings us back to the pomegranate seeds with rhombic faces. Kepler envisioned that pomegranates grow in the following way: when the fruit is young the seeds are hard spheres packed in a face-centered cubic arrangement, having maximum density. As the fruit matures, the seeds expand, but the total volume of the fruit remains constant, constrained by the hard, outer shell. The seeds have no alternative but to expand into the voids between each seed, thus changing their shape from spherical to rhombic dodecahedral. The twelve flat rhombic faces are formed where the seed butts up against one of its twelve neighbors. In this way space is filled by tiny, rhombic dodecahedra fitting perfectly together, leaving no voids, Fig. 3.

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Fig. 3. Rhombic dodecahedra can be packed together to fill space. Packing at a threefold and a fourfold vertex is shown.

Fig. 4. Spheres in a close-packed overlayer can occupy either white sites (W) or black sites (B).

Clusters of certain materials consist of layers of close-packed atoms stacked one on the other. However, countless variations of this stacking arrangement can be envisioned. If one starts out with a single close-packed layer, denoted by (a), there are two ways of constructing the close- packed overlayer. The spheres in the overlayer can be placed in either the white triangular sites or

204 T.P. Martin / Physics Reports 273 (I 996) 199-241

Fig. 5. The 147-atom cuboctahedron is composed of seven close-packed layers having the stacking sequence characteristic of an fee lattice.

the black triangular sites, shown in Fig. 4. Now the double layer can be denoted as either (aw) or (ab), depending on which set of triangular sites are occupied by the overlayer. An underlayer can now be constructed, again with the option of occupying either white or black sites. Two possible arrangements of the triple layer can then be denoted as (bab) or as (baw). The former, repeated indefinitely, leads to hexagonal close-packing, the latter to cubic close-packing. Clearly, there is an infinite variety of both periodic and nonperiodic arrangements of layers. However, it should be remembered that each of these arrangements has perfect close-packing and leads to a density of 0.74. This notation can be applied also to clusters, as shown in Fig. 5. Here, seven layers have been


Fig. 6. 147-atom cuboctahedron. Notice that the atoms in the square faces are not close-packed.

stacked according to the sequence abw, abw, a. The repeating sequence abw indicates that this structure can be cut out of the cubic close-packed lattice. As seen in Fig. 6, this cluster has the form of a so-called cuboctahedron. Many symmetric clusters can be constructed from the close-packing of hard spheres, e.g., tetrahedra, octahedra and their truncated forms. The truncated forms can have triangular, square or hexagonal faces. The cuboctahedron is just one example of this set of geometric figures. It can be described as an octahedron truncated by a cube, and as shown in the figure it has six square faces and eight triangular faces. Notice that although this figure is constructed from close-packed layers, the square face reveals a surface that is not close-packed.

2. Classification of shells

An excellent example of shell structure is the set of nested Mackay [4] icosahedra. One can arrange 12 neighboring atoms around a central atom in several highly symmetric ways. One way is to put the atoms at the corners of an icosahedron, thus completing the first shell. This 13-atom core can be covered by a second layer of 42 atoms, forming again a perfect icosahedron, containing a total of 55 atoms. This larger core can now be covered by a third layer having icosahedral symmetry, as shown in Fig. 7. This example suggests a plausible definition for a shell - one complete layer of atoms arranged on the surface of a core having a well-defined symmetry. However, as one begins to construct examples it becomes quickly clear that this definition might lead to confusion. Consider another example, a cluster composed of atoms placed at the sites of a simple cubic lattice and having the overall outer shape of a cube. The first such cube that can be formed around a central atom contains 27 atoms, three atoms on a side; the next, 125 atoms, i.e., five atoms on a side, etc. But what happened to the 64-atom cube with four atoms on a side? By restricting ourselves to shells composed of complete layers of atoms we have missed a very important geometric figure. How can we get around this difficulty? The 64-atom cube has no

206 T.P. Martin /Physics Reports 273 (I 996) 199-241

Fig. 7. A closed-shell 55atom icosahedron and a portion of the next shell.

Fig. 8. A tetrahedron of the next larger size is obtained by adding only one triangular layer.

central atom, so one might say that the set of all cubes actually consists of two sets; one having at its center a single atom, and the other having at its center an eight-atom cube. However, such a division makes life unnecessarily complicated. It is better to use an alternative definition for a shell of atoms.

The 64-atom cube can be formed by covering only half the surface of the core with a new layer, i.e., by covering only three of the six square faces. An even more extreme example is the case of the tetrahedron, Fig. 8, for which it is necessary to add atoms to only one triangular face in order to obtain the next larger tetrahedron. We see that it is meaningful to define a shell as consisting of only enough atoms to regain the original symmetry. Therefore, adding one complete layer of atoms to a tetrahedron results in the addition of not one, but four shells.

Each set of shells is labelled by a geometry. It is useful to label each shell within a set by a shell index K. There exists an ambiguity in defining K, which can again lead to confusion. In some papers [S] the central atom is not considered a shell and is therefore labelled with K = 0. In other papers the central atom is given the shell index K = 1. In this report we will use the latter definition because it allows the index of a large shell to be determined at a glance. The shell index is simply the number of atoms along one edge of a regular face. For example, the total number of atoms in a cube having K complete shells is simply K3. The total number of atoms in a tetrahedron is

y1 = bK3 + 4 K2 + SK (tetrahedron) . (1)

Icosahedral clusters, if they continue to grow, will never become crystals. They possess 12 fivefold symmetry axes. This is a symmetry operation not consistent with the translational

T.P. Martin / Physics Reports 273 (1996) 199-241 201

Fig. 9. Five tetrahedra sharing an edge form almost, but not quite, a perfect decahedron.

Fig. 10. Successively larger decahedra are formed by adding an umbrella-shaped partial layer.

symmetry found in crystals. In spite of this, early electron microscope studies of gold clusters showed that they had icosahedral symmetry. Ino explained this observation by suggesting that the icosahedra were really 20 tetrahedra sharing a common vertex. Tetrahedra, in contrast to icosahedra, can be cut out of an fee crystal. So in this sense crystalline symmetry is somehow retained. The problem is that tetrahedra are not space-filling. Ino [6] proposed that there were small gaps in the regions separating the tetrahedra. For this reason, icosahedral particles are often referred to as multi-twinned particles, since they can be thought of as tetrahedra, each having twin boundaries with its neighbors. Alternatively, the tetrahedra might distort, as they are packed together, like Kepler’s pomegranate seeds, so that they fill out the entire volume of the icosahedron. In this case, the term multi-twinned particles loses its meaning. The atoms in the cluster can be said to have a true icosahedral arrangement. The total number of atoms in a Mackay icosahedron composed of K shells is

n = y K3 - 5K2 + 9 K - 1 (icosahedron) . (2)

Icosahedral packing arrangements have many favorable attributes. The icosahedron has the highest symmetry of all discrete point groups. Each of the 20 triangular faces of the icosahedron can be constructed with close-packed spheres. Then why is it that these clusters do not grow into pieces of bulk matter? The icosahedron is a highly strained structure. The interatomic distances between shells is smaller than those within a shell. For this reason, icosahedral packing is found only in relatively small clusters having a high surface-to-volume ratio.

The icosahedron is not the only noncrystalline structure having a fivefold symmetry axis. The decahedron also belongs to this class of geometric figures. The decahedron can also be constructed out of tetrahedra - almost. If five tetrahedra are packed together so that they share a common edge, the resulting figure is a decahedron with a gap between two connecting surfaces, Fig. 9. However, if the tetrahedra are allowed to deform, the result is a perfect decahedron with internal strains [7].

It is possible to arrange spheres in decahedra1 arrangements having a shell structure. However, unlike the icosahedron it is not necessary to cover the entire surface to obtain the next perfect geometric figure. The next large decahedron is formed by placing a large overlapping umbrella on top of the previous member of the set, Fig. 10. The members of the set of decahedra1 shells possess

208 T.P. Martin / Ph_vsics Reports 273 (1996) 199-241

Fig. 11. A 309-atom truncated decahedron with square faces.

alternately a central atom and a central seven-atom decahedron. The total number of atoms in a decahedron consisting of K shells is

y1 = $K3 + +K (decahedron) . (3)

A great advantage of the decahedron is that the atoms in its surface are close-packed. However, it has a very large surface area in addition to internal strain. For these reasons the decahedron is not expected to be energetically preferred at any cluster size. This is not the case for truncated forms of the decahedron.

The decahedron can be truncated [6] in order to make it more spherical, Fig. 11. However, as can be seen, this truncation exposes (100) faces which are not close-packed, and which have a correspondingly high surface energy. The degree of truncation can be varied in order to minimize total energy for given material [S]. If the (1 0 0) faces are square, the total number of atoms in a cluster is given by the expression

n = yK3 - 5K2 + YK - 1 (truncated decahedron) . (4)

In order to reduce the energy still further, Marks [9] has introduced another form of truncation which exposes reentrant (1 1 1) faces in addition to (1 0 0) faces.

A large number of symmetric clusters can be constructed from the fee close-packing of hard spheres. We have already mentioned the tetrahedron at the beginning of this section. An octahedron of any size can also be cut out of an fee crystal. The triangular faces of the octahedron consists of close-packed (1 1 1) planes which tend to minimize surface energy. Because of these properties, the octahedron is without internal strain and has a low surface energy. The total number of atoms in a cluster containing K complete octahedral shells is

y1= *K3 + 3K (octahedron). (5)

The octahedron is far from spherical; as a consequence, it has a large surface area. This deficiency can be improved through truncation.


Fig. 12. A 3 x 3 x 3 atom rhombohedral portion of a bee lattice. It can be viewed as a cube compressed along a body diagonal.

Truncated forms of the octahedron can have triangular, square, or hexagonal faces. For example, the cube-octahedron, an octahedron truncated by a cube, Fig. 6, can have six square faces and eight triangular faces. Although this figure is constructed from close-packed layers, the cut forming a square face reveals a surface that is not close-packed. Such a surface is energetically unfavourable. The great advantage of the cube-octahedron is that its total surface area is small. The total number of atoms in a cubed octahedron of this type is

n = yK3 - 5K2 + YK - 1 (cuboctahedron, triangular faces) . (6)

Alternatively, the octahedron can be truncated so that it exposes hexagonal (1 1 1) faces and square (1 0 0) faces, Fig. 12. Several examples have been given where a new shell can be formed with less-than-complete coverage of a surface by a new layer. This truncated octahedron with hexagonal faces can be regained only by adding more than one layer of atoms to the surface. The total number of atoms in the truncated cuboctahedron with hexagonal faces is given by the expression

n = 16 K3 - 33K2 + 24K - 6 (cuboctahedron, hexagonal faces) . (7)

There exists two shell structures related to the bee lattice. At the beginning of this section we used the example of a cube-shaped cluster cut out of a simple cubic lattice. This was convenient to illustrate the concept of shells formed from partial layers, but is unrealistic in that elemental matter does not usually condense into a simple cubic structure. However, if such a cube is squeezed along

T P. Martin / Physics Reports 2 73 (I 996) 199-241

Fig. 13. A 586-atom truncated octahedron with square and hexagonal faces.

Fig. 14. The fifth shell of Kepler’s rhombic dodecahedron.

a body diagonal the cube deforms into a rhombahedron that can be cut out of a bee lattice. It is difficult to represent this rhombic hexahedron so that it does not look like a cube, an attempt is made in Fig. 13, The bee rhombic hexahedron contains, of course, the same number of atoms as the undeformed cube,

n=K3 (h b’ h hd r om ic exa e ron ). (8)

The bee lattice contains also a set of shells based on Kepler’s rhombic dodecahedron. Fig. 14 shows a rhombic dodecahedron for K = 5. The total number of atoms in clusters with complete shells of this kind is

IZ = 4K3 - 6K2 + 4K - 1 (rhombic dodecahedron) . (9)

Shell structures can be found also in the hcp lattice. An example is shown in Fig. 15. Although the truncated hexagonal faces are close-packed, the other faces are less favorably constructed. Notice that the layers in these faces occur in pairs.

In the discussion above we have written a separate expression for the total number of atoms for each of the shell geometries. It is possible to summarize all of these formulas in a more general expression [lo]. However, this requires partitioning the surface of a cluster into what we will call a standard triangles. The standard triangle is made up of K rows of atoms. The first row contains one atom, the second row two atoms, the third row three and the Kth row K atoms. The total number of atoms in such a standard triangle is $K2 + &K. In most of the cases we have discussed until now the surface of the Kth shell can be partitioned into small standard triangles, all containing the same number of atoms. For example, the surface of the icosahedron contains 20

T.P. Martin 1 Physics Reports 273 (1996) 199-241 211

Fig. 15. The truncated hexagonal bipyramid is a portion of an hcp lattice.

standard triangles. Perhaps it is not quite so obvious that the surface of the cuboctahedron also contains 20 standard triangles. Remember, the cuboctahedron has eight triangular faces and six square faces; however, each of these square faces can be divided down the middle into two standard triangles. For clusters containing t standard triangles and requiring complete coverage of the surface to complete a shell, the total number of atoms can be written as

n=&t(2K3-3KZ+K)+2K-1. (10)

However, for cluster geometries that require coverage of only one-half of the surface to complete a shell, the total number of atoms is given by

n=&t(P-K)+K . (11)

Until now we have discussed shell structures that can be visualized as the packing of spheres. Recently, quite a different type of shell structure has been observed [ll, 121. These shells may consist of nested hyperfullerenes [13]. Many years ago, Goldberg [14] considered the set of polyhedra which have only trihedral vertices and only pentagonal and hexagonal faces. Con- strained by Euler’s rule, all of these polyhedra contain 12 pentagons andf hexagonal faces. The total number of vertices in these figures is 2f+ 20. However, we would like to consider only a subset of this family of polyhedra, those polyhedra having I or Ii, symmetry. The reason for considering this subset is that it contains those figures for which the pentagons have maximum separation. Goldberg showed that the number of vertices on these icosahedral polyhedra is 20(b2 + bc + c’) where b and c are integers. If we further restrict ourselves to considering Goldberg polyhedra with full I, symmetry, constraints are placed on the values of b and c, i.e., either b = c or bc = 0. Yoshida and Osawa [lS] have called these type-1 and type-II hyperfullerenes. The number of vertices in the type-1 fullerene shell is

I/ = 60K2, (12)

212 T.P. Martin /Physics Reports 273 (1996) 199-241

Fig. 16. The first three shells of type-1 hyperfullerenes contain 60, 240 and 640 atoms, respectively (see Ref. [lS]).

and for the type-II hyperfullerene

I/ = 20(K + 1)2 ) (13)

where K is an integer. Fig. 16 shows a cluster composed of the first three shells of type-1 hyperfullerencs. Type-I hyperfullerenes are of particular interest for several reasons. The intershell distance 3.5 A is very nearly equal to the interlayer distance 3.35 A found in crystalline graphite. This fact allows shells to form without strain. In addition, type-1 hyperfullerenes have closed electronic shells. It is not expected that the icosahedral symmetry will be lowered through a Jahn-Teller distortion. The total number of atoms in a cluster containing K type-1 shells is

n = 20K3 + 30K2 + 10K (type-I hyperfullerenes) . (14)

3. Example of an experiment - photoionization of aluminum clusters

If an apparatus is to be used to study the shell structure of clusters it must possess several features: (1) a cluster source capable of producing a broad distribution of large clusters; (2) a means of developing magic numbers in the mass spectrum; (3) the capability of mass selecting and detecting large clusters. The second feature will be treated at length in the next section of this report. This section will concentrate on cluster production, mass selection and detection.

The cluster source is a low-pressure, rare-gas, condensation cell. Metal vapor is produced in an oven which is resistively heated with a tantalum wire, Fig. 17. The temperature of the oven is


ACCELERATION REGION

CLUSTER CONDENSATION

CELL

PUMPING STAGE

TOF MASS SPECTROMETER

Fig. 17. Apparatus for the production, photoionization and time-of-flight mass analysis of clusters.

adjusted to produce a metal vapor having a partial pressure of approximately 0.1 mbar. The outer walls of the condensation cell is cooled with flowing liquid nitrogen. Thermal contact between the cold walls and the hot vapor is provided by a helium buffer gas at a pressure of 1 mbar. The metal vapor is quickly cooled through contact with the helium gas resulting in supersaturation and the formation of tiny metal droplets. As the clusters move away from the oven they stop growing in size, but their temperature continues to drop until they come into thermal equilibrium with the He gas having a temperature of approximately 100 K. The cold clusters are transported by the He gas stream through a nozzle about 3 mm in diameter and 3 cm in length into the first chamber of intermediate pressure. This chamber is pumped by a 1000 l/min roots blower. The clusters continue through a 3-mm diameter skimmer into a second chamber having an intermediate pressure of 6 x lo-’ mbar achieved with a turbomolecular pump. After passing through a second skimmer, the clusters arrive in an ionization chamber having a pressure of 1 x lo- 6 mbar. Here the clusters are ionized with a light pulse and deflected by means of an electric field into a time-of-flight mass spectrometer. Deflection of 90” is important if the mass spectrometer is to have a high mass

214 T.P. Martin 1 Physics Reports 273 (1996) 199-241

I (Al), I

600 700

Number of Atoms, n

800

Fig. 18. Mass spectrum of aluminum clusters obtained using threshold photoionization. Shell structure is revealed if the mass spectrum is smoothed with a spline function.

resolution. Forcing the ions to make a 90” turn, in order to enter the spectrometer, becomes increasingly difficult for large clusters having high kinetic energies perpendicular to the spectrometer. In fact, the kinetic energy or velocity of the neutral clusters sets an upper limit on the mass that can be analyzed in the spectrometer. After deflection and acceleration, the clusters are focussed with quadrupole optics onto the detector [ 161. All ions in a volume 2 mm x 2 mm x 0.2 mm and having a kinetic energy of less than 300 eV at the moment of ionization are focussed by the optics onto the detector. Before detection, however, the ions first pass through a reflector divided into two segments by a wire mesh. The reflector is used to compensate for the distribution of initial potential energies of clusters having the same mass. On emerging from the reflector, the clusters are post-accelerated giving them an additional energy from 8 keV. Finally they are detected with a pair of channel plates.

The method of processing mass spectra will be illustrated using aluminum clusters as an example. Aluminum is not an easy material to work with, it requires high temperatures for evaporation. It has the tendency to creep out of the crucibles and to coat the heating filament. It reacts with tantalum filaments causing them to burn out. For these reasons, the oven must be carefully designed and constructed. We used a BN crucible surrounded by a Ta-strip heater protected by an overhanging lip on the crucible. Fig. 18 shows a portion of a mass spectrum of aluminum clusters containing 600-800 atoms. Each peak in the spectrum corresponds to a pure singly ionized aluminum cluster. However, this representation of a mass spectrum is not appropriate for studying shell structure. The mass resolution is simply too high. Clearly, there are variations of the intensity of the mass peaks from one cluster size to the next; however, these variations are purely statistical in nature. Shell structure causes variations on a much larger mass scale. These large-scale variations can be seen much more clearly if a smoothing function is convoluted into the mass spectrum. Of course, the choice of the width of this smoothing function is very important. If it is chosen too broad, the shell structure itself will be filtered out. If the smoothing function is too narrow, the processed spectrum will be filtered out. If the smoothing function is too narrow, the processed spectrum will be unnecessarily noisy. The wavy line in Fig. 18 shows a processed spectrum obtained by convoluting a spline function having a width of five-hundred 64 ns time

T.P. Murtin / Physics Reports 273 (1996) 199-241 215

Number of Electrons

,,;I 6000 12000 18000 24000 _300+0 /’ “”

Number of Atoms

Number of Electrons

0 6000 12000 18000 24000 30000 15L I I / I ,

0 2000

: 1 ; ! : 1 4000 6000 8000 10000

Number of Atoms

Fig. 19. Large-scale mass spectrum of aluminum clusters.

Fig. 20. Difference mass spectrum of aluminum clusters obtained by subtracting a highly smoothed envelope mass spectrum from a slightly smoothed spectrum. Notice that the shell structure is not equally spaced on a number-of-atoms

scale.

channels with the original mass spectrum. The processed mass spectrum has been multiplied by a factor of 2 in order to lift it out of the original mass spectrum for display. The processed mass spectrum has four undulations. This is just the effect we are looking for, as can be seen in Fig. 19. Here the same mass spectra, processed and unprocessed, are displayed but now over a much larger mass scale for clusters containing l-10000 atoms. The spectrum can now be taken through one more step of processing. The original mass spectrum can be convoluted with a much broader smoothing function resulting in a smooth envelope curve. The envelope contains no shell oscillations. When this envelope is subtracted from the processed mass spectrum containing shell oscillations, the result is so-called difference spectrum, shown in Fig. 20. In some sense we have put the data through a frequency filter. The first stage of processing removes the high frequencies, the second stage of filtering, the very low frequencies. Needless to say, putting experimental data through such a filter must be done with great discretion.

4. Experimental techniques for revealing shell structure

Cluster growth demands nonequilibrium conditions. If growth were slow it might be possible to speak of a quasi-equilibrium where structures having a minimum free energy would be preferred. Unfortunately, cluster growth is usually not slow and the resulting size distribution is dictated by kinetic considerations. Magic numbers must be induced in the size distribution, subsequent to growth, using one of several experimental techniques [17, IS].

Shell structure can be revealed by heating clusters to a temperature at which they evaporate atoms. They can be heated, for example, with a laser. This might be either the same laser used for ionization or a second laser devoted solely to heating. For small clusters containing ten atoms the

216 T.P. Martin / Physics Reports 273 (1996) 199-241

absorption of just one photon may be sufficient to induce evaporation. However, large clusters containing thousands of atoms may have to absorb as many as one-hundred photons before the evaporation temperature is reached [19,20]. When we speak of multi-photon absorption in this section, it must be made clear that we are not speaking of the simultaneous absorption of photons involving a virtual state. The photons are absorbed one after the other with enough time between absorptions to allow the cluster to come into internal thermal equilibrium.

An important consideration in planning evaporation experiments is time scale. For example, one might want the temperature of the cluster to be high enough that it has a high probability of evaporation in a time much less than 10e8 s. This would allow several evaporation events to take place within the duration of the warming laser pulse. A heating experiment could be designed as follows: A laser wavelength is chosen for which the cluster has a high absorption cross-section. The laser Auence is chosen so that the cluster absorbs several photons during one laser pulse. After the first photon is absorbed and excites an electronic state, the cluster decays back down to the ground state almost immediately on the time scale we are speaking of. This decay is associated with the generation of heat. After establishing a new thermal equilibrium, the cluster absorbs a second photon. This process continues until the cluster reaches a temperature high enough that it evaporates an atom before the next photon arrives. The evaporation cools the cluster momentarily

_.i : L3.135 eV i : 5

7 ./ “‘.-3.154 eV -7,

i 1

2.6 2.8 3.0 3.2 3.4

Ionization Energy (eV)

Fig. 21. Photoionization efficiency spectra of (Na), clusters with values of n corresponding to the eighth and 16th shell. The asterisks and solid line correspond to a closed shell, while the circles and dashed line correspond to open-shell clusters. Notice that the ionization thresholds of closed-shell clusters lie at higher energies.

T.P. Martin /Physics Reports 273 (1996) 199~-241

- i 2 3.20

Wa),,

217

5 10 15 20 25 30 ” 1113)

Fig. 22. Ionization threshold energies of Na clusters. The oscillations occur at equally spaced intervals. Theoretical values of n for icosahedral (cuboctahedral) shell closings are indicated.

until the next absorption event. In this way a steady state is reached. The energy lost through evaporation is equal to the energy gained by absorption. Although the temperature (internal energy) of the cluster remains constant, the cluster shrinks in size through evaporation. It loses mass until it reaches a size for which it is particularly stable. At this size it resists further evaporation, at least momentarily, until the temperature rises to a higher value. However, this momentary hesitation of the evaporation process is sufficient to enhance the size distribution in clusters having high stability.

Not all materials are appropriate for this type of warming experiment. If the technique is to reveal structural magic numbers it is important that the clusters not be molten at the temperature at which they evaporate atoms on the lop8 s time scale. Rewording this in the terminology of materials science we could say that most materials melt before they boil. The technique described above requires a material which sublimes from the crystalline state.

Heating is not the only way to induce magic numbers in the size distribution of clusters. It is also possible to make use of the fact that the ionization threshold of geometrically perfect clusters is higher in energy than that of imperfect clusters [21,22]. This can be seen in Fig. 21 which contains two ionization efficiency curves. Plotted here is the ion signal in the mass spectrometer against the photon energy of the ionizing light. The asterisks represent the ions having a mass corresponding to the completion of exactly 16 icosahedral or cuboctahedral shells. The circles correspond to clusters having slightly smaller mass, that is, to clusters having an incomplete structural shell. The ionization energy can be extracted by fitting the data with an error function. The maximum of the first derivative of this function is then assumed to be the average ionization


Shell of Atoms 10 12 14 16 18 I I 1 I , I I I I 7

0 10000 20000

n

Fig. 23. Mass spectra of (Na), clusters photoionized with 2.99 and 2.93 eV photons. Minima occur at values of n corresponding to the icosahedral (cuboctahedral) shell closings listed at the top.

energy of an ensemble of clusters having this mass. But even without this fitting procedure, it is clear from Fig. 21 that clusters having a complete outer shell of atoms have a slightly higher ionization energy.

Why do closed-shell clusters have high ionization energies? Studies in surface science offer us a plausible explanation. The work function of a metal is lowered by the presence of steps on its surface. The degree of lowering is linearly proportional to the step density [23]. It is thought that the steps decrease the ionization energy by creating an electric dipole moment. Since steps can be expected to appear on the surface of clusters having incomplete shells, this effect could explain the modulation of the ionization energy. This modulation is seen clearly in Fig. 22 where ionization energy is plotted against the cluster size on an n1’3 scale. The maxima of the oscillatory curve correspond very well to the number of atoms necessary to complete icosahedral or cuboctahedral shells.

It is necessary to measure and analyze hundreds of photoionization efficiency curves in order to obtain the data shown in Fig. 22. Information concerning shells can be obtained in a much easier experiment. If the energy of the ionizing photons is chosen to be well above threshold, all clusters will be ionized with equal probability and the mass spectrum will be smooth. However, if a photon energy is chosen which lies within the band of oscillations, the mass spectrum should be highly structured. Clusters having complete shells of atoms will be hard to ionize because they have high ionization energies. For this reason such clusters will show up as minima in the mass spectra. This can be seen in Fig. 23.

T.P. Martin / Physics Reports 273 (1996) 199-241 219

5. Test for shells

Cluster mass spectra of almost any material show intensity anomalies, so-called magic numbers. A knowledge of one or two magic numbers is seldom sufficient to allow a unique guess as to the geometry of the cluster. A notable exception is with CeO. However, if the experimentalist observes

a series of magic numbers that obeys specific rules, it is often possible to uniquely determine the geometry of the clusters. The rules are necessary in order to establish the fact that a set of geometric shells has been observed.

There is a quick test available to the experimentalist to determine if a set of shells has indeed been observed. This test is based on the fact that in expressions for the total number of atoms, Eqs. (l)-(lo), the highest power in K, the shell index, is always K3. This, of course, must be the case since K is essentially the number of layers in the cluster, thus determining its radial dimension. The total number of atoms in the cluster will be proportional to its volume, which is in turn proportional to the third power of the radial dimension. Throughout this section we will make the approximation that powers of K smaller than 3 can be neglected. This permits us to write down some simple expressions which allow an immediate analysis of experimental data. It turns out that the expressions are quite useful for all but the smallest clusters.

The first consequence of the approximation is that if mass spectra are plotted against n1i3, magic numbers caused by shell structure will be equally spaced. This simple test for shell structure is so useful that the experimentalist would be well advised to develop the possibility of plotting data in

h . v=299eV

h-v=2.93eV

Fig. 24. Mass spectra of (Na), clusters photoionized with 2.99 and 2.93 eV light. Notice that the oscillations are equally spaced on an n113 scale.


0 experiment

+ icosahedra

0 2 4 6 I3 10 12 14 16 18 20 22 24 26 28

N1/3

Fig. 25. Shell index plotted against n ‘j3 for (Na), clusters. Two types of shells are observed. For small n, the observed electronic shell closings are compared with those predicted using a pseudo-quantum-number 3n + 1. For large n, the

observed shells of atoms are compared with those predicted assuming icosahedral (cuboctahedral) symmetry.

this way “on-line”. Strictly periodic oscillations in the data allow shell structure to be identified immediately in even quite noisy mass spectra. Mass spectra of sodium clusters obtained using two different photon energies near the ionization threshold are shown in Fig. 24. Both have been plotted on an nii3 scale, and in both cases the oscillations are equally spaced.

The existence of shells can be demonstrated in a different, although equivalent, manner. For threshold ionization mass spectra we could plot the values of n113 at minima in the spectra against

the shell index K. If the oscillations are equally spaced the points should, of course, fall on a straight line. Such a plot is shown for sodium clusters in Fig. 25. Notice that the points fall on two straight lines. It turns out that the data points for small clusters are caused by electronic shell structure [24-301 and those for large clusters by shells of atoms.

The concept of shells, whether shells of electrons or shells of atoms, is associated with a characteristic length. For shells of atoms the characteristic length is the thickness of a layer. For shells of electrons the characteristic length can be envisioned in the following way. Valence electrons can move freely within the cluster. The clusters are so big that the classical picture of an electron bouncing specularly off the sides of the cluster is not without meaning. The characteristic length then is the length of a closed electron trajectory within the cluster [31,32].

The presence of equally spaced magic numbers on an n1j3 scale allows the existence of shell structure to be established. However, even more detailed information can be extracted from such a plot. The size of the interval between the magic numbers, read directly from the mass spectra, will usually determine uniquely the geometry of the shells. It should be emphasized that this interval on an n1j3 scale is without dimensions and depends upon no physical properties such as density of material, size of atoms or character of bonding between the atoms. The size of the interval is determined by geometric considerations only. Nature has been particularly kind in allowing mass spectrometric data to lead so directly to a purely geometric result.


Icosahedrory Cube ~ Octahedron Tetrahedron 1-1

/i I ! ~

12 13 14 15 16 “‘A

Fig. 26. Segment of a difference mass spectrum of (Al), clusters. The oscillations occur at equal intervals, but this interval does not correspond to that expected for shells of atoms on regular polyhedra.

The interval between two magic numbers (shell oscillations) is just the cube root of the coefficient of the K3 term in Eqs. (lt-(lO), or more generally using Eq. (1 l),

d z (t~/6)“~, (15)

where t is the number of effective triangles and c is the coverage. That is, the number of effective triangles defining the shell can be determined from the experiment to be

t z6A3/c. (16)

Let us try out this expression using the shell data for sodium shown in Fig. 24. The interval between oscillations on the nri3 scale is quickly seen to be about 1.5 dimensionless units. Substituting this number into Eq. (16) and assuming that the coverage factor is 1, we find that the number of effective triangles in the sodium clusters is 20. That is, the sodium clusters could be either icosahedra or cuboctahedra. It is worth emphasizing once again that this is merely a method to quickly analyze data. The data can later be compared with the exact number of atoms necessary to fill icosahedra or cuboctahedra shells.

Applying this method of quick analysis to aluminum clusters [33-361 demonstrates a difficulty that might arise. A portion of a mass spectrum of aluminum clusters plotted on an n1’3 scale is shown in Fig. 26. Also shown are the intervals that could be expected between oscillations for icosahedra, cubes, octahedra and tetrahedra. Notice that the experimental oscillations are shorter than that expected of even the tetrahedron. Remember that for the tetrahedron it is necessary to complete only one additional triangular face in order to complete the shell. It is difficult to imagine how a shell could be completed with even fewer atoms, as the data suggest. This problem is resolved if one notices that exactly four oscillations are contained within the interval expected for one octahedral shell. Four is also the number of triangular faces that must be completed in order to construct an octahedral shell. It would appear that each of the oscillations in the aluminum data correspond not to the completion of a shell but rather to the coverage of one triangular face.


6. Examples of shell structure

Both calculations [37] and experiments [38%44] indicate that inert-gas clusters containing from 13 to 923 atoms have icosahedral symmetry. These might be referred to as precrystalline structures since the inert gases are known to condense into fee crystals. Precrystalline structures have also been observed for metallic materials in condensed units large enough to yield sharp electron diffraction patterns [45547]. These quasicrystals present a fascinating challenge to scientists to develop methods for describing a regular but nonperiodic state of bulk matter. Smaller icosahedral metal particles have been observed directly using the newly developed technique of high-resolution electron microscopy [48].

Additional evidence exists for icosahedral symmetry in metal clusters [49]. Calculations predict that very small alkaline earth clusters prefer noncrystalline structures [50-531. The pattern of NH, and Hz0 binding energies with Co and Ni clusters has been interpreted as indicating icosahedral symmetry in metal clusters containing from 50 to 150 atoms [54, 551. Mass spectra of Ba and Ba-0 clusters seem to indicate an icosahedral growth sequence in the size range from 13 to 35 atoms [56-581.

The slow modulation in mass spectra of Na clusters, Fig. 23, can be interpreted as evidence for the existence of shell structures, i.e., a highly symmetric, onion-like cluster structure [59]. The modulation appeared only if the energy of the ionizing photons was chosen to coincide with the ionization potential of the clusters and was found to be almost periodic when plotted on a cube root of mass scale. The cusp-like minima of the mass spectra pointed to characteristic masses or numbers of atoms. Within the accuracy of reading the minima, these magic numbers correspond to the number of atoms in complete Mackay icosahedra [60]. However, on the basis of such observations, it is not possible to conclude that the clusters have icosahedral symmetry because icosahedral shells and fee cuboctahedral shells contain exactly the same number of atoms. We have to look elsewhere for decisive experimental data. We believe these data are contained in weaker mass peaks between shell closings.

Fig. 27 shows a mass spectrum of calcium clusters containing up to 5000 atoms. The mass resolution is poor for several reasons. The clusters have been heated to encourage evaporation. Since loss of mass occurs in the acceleration region, the resolution suffers. Another factor contributing to our inability to separate cluster peaks is the fact that calcium possesses several relatively abundant natural isotopes including 97% 40Ca and 2.1% 44Ca. Since no additional information can be gained by using narrow time channels, the spectrum was averaged over one-hundred 64 ns time channels. An envelope function has been obtained by averaging over 3000 times channels. A convenient form for display is the difference of these averaged spectra shown in Fig. 27.

The difference spectrum for (Ca), is characterized by seven strong peaks. Although these peaks are not equally spaced on a scale linear in mass, they do occur at equal intervals on an nli3 scale. This signals the presence of shell structure, in this case geometric shells of atoms. Clusters having closed shells with icosahedral or cuboctahedral structure would be expected to contain 561, 923, 1415, 2057, 2869, 2869, 3869 and 5083 atoms. In general, the observed peaks in the mass spectra occur about 0.2% lower in mass. The reason for this small discrepancy is not fully understood. This main sequence of mass peaks gives strong evidence that as atoms evaporate from clusters, perfect geometric shells are successively revealed. Cluster having perfect symmetry are rather immune to


/“’ I”“I”“I”“I”“I”“,““,““,.,,.l

0 1000 2000 3000 4000 5000 Number of Atoms, n

Fig. 27. Difference spectrum (highly smoothed envelope spectrum, a, subtracted from a slightly smoothed spectrum, b, for Ca clusters. The strong peaks correspond to completely filled icosahedral shells of atoms.

further loss of mass. The number of atoms in cluster belonging to the main sequence indicate the geometry is either icosahedral or cuboctahedral.

Notice the weaker subshell structure occurring between shell closings. In order to better compare subshell structure, the mass spectrum in Fig. 2 has been cut into segments, each segment corresponding to the formation of one shell. Since the number of atoms in a shell is approximately proportional to the square of the shell index, the mass scale of each segment is adjusted so that complete shells lie above one another, Fig. 28. The same subshell structure is found in the mass spectra of strontium clusters, Fig. 29. When compared in this way, the similarity of the subshell structure from shell to shell is quite apparent. But what are these subshells?

For inert gas clusters both experiments and calculations indicate that partial icosahedral shells of atoms also show enhanced stability [41-431. For example, one might expect that completely covered facets of a cluster surface represent intermediate structures of high stability. Since the facet structure of the icosahedron (20 triangular faces) and the cuboctahedron (eight triangular and six square faces) are quite different, a determination of partial shell sizes should make it possible to distinguish between the two structures.

The square faces of the cuboctahedron would be likely candidates to accept the first atoms of a newly deposited layer because the atoms in these faces are not close-packed. However, no arrangement of atoms on these faces alone or in combination with other cuboctahedral faces could be found which matched the observed subshell magic numbers. Next, we turned to the icosahedron for which subshell structure had already been studied [41-431. The first atoms to form a new shell on an inert gas icosahedron apparently do not immediately take their final positions. This would force atoms on the border between two triangular faces to have contact with only two substrate atoms. Instead, the triangular faces are first filled with a close-packed layer. Only after the shell is more complete do the atoms rearrange into their final icosahedral positions. This shell-filling


2200 2600 Number of Atoms, n

Ca Atoms 300 400 500 600

700 800 Sr Atoms

Fig. 28. The same spectrum shown in Fig. 27 but cut into segments and each segment resealed so that they have the same length. Notice that the subshell structure repeats itself from shell to shell.

Fig. 29. Mass spectra showing subshells in the fourth shell of Ca and the fifth shell of Sr.

sequence, observed in inert-gas clusters, although close, seems to deviate significantly from the observed magic numbers for Ca and Sr clusters. Therefore, we would like to suggest an alternative sequence supported by a simple calculation.

First, we will try to explain the six strong, broad peaks shown in Fig. 29 for calcium and strontium clusters. Assume the atoms in the new shell take immediately their final positions and we merely count the numbers of atoms needed to cover combinations of triangular faces. In order to assist this counting, in Fig. 30 the positions of the atoms in the sixth shell have been projected onto a plane in the manner’of Northby [40]. We suggest that umbrella-shaped intermediate groups have enhanced stability. Each of the umbrellas contains 76 atoms and each has the same shape (although

T.P. Martin I Physics Reports 273 (1996) 199-241 225

Fig. 30. The dots represent the atoms of the seventh shell of an icosahedron projected onto a plane. The bottom 76 atoms are not shown. The umbrella-shaped structures are identical, each containing 76 atoms. We suggest that the umbrellas

represent highly stable partial shells of calcium atoms.

they appear distorted in the projection shown in Fig. 4). Only 5 1 additional atoms are necessary to complete the second umbrella because it shares atoms with the first. The third and fourth umbrellas overlap two others. Therefore, they require only 36 additional atoms for completion. This umbrella model is consistent with the observed subshell structure.

In order to explain the fine subshell structure occurring between completed umbrellas, it is necessary to go one small step further. The procedure is as follows. The outermost shell of a cluster is successively filled up with atoms. The already filled inner shells are considered to be rigid. Only icosahedral sites are allowed, in particular no fee substructure on the faces of the icosahedron is admitted. Although the atoms are added one by one, all atoms of the partly filled shell may change their sites when one new atom is added to the ensemble. Under these assumptions we tried to find the energy minimum of the cluster as a function of the total number of atoms.

The interactions of the atoms have been described by a Lennard-Jones potential, its minimum being the bonding length between neighboring shells. This is certainly not a very realistic model for the binding energy of a metal cluster, but as we will show, the agreement with the experiment is worthy of note. Using such a potential, nearest neighbor interactions dominate the total energy. Therefore, the binding energy of one atom could be (although we have not done this) roughly estimated by simply counting the occupied neighbor sites and adding a nearly constant energy term

226 T.P. Martin J Physics Reports 273 (1996) 199-241

as a substitute for the interactions with the rest of the cluster atoms. The most important factor determining the binding energy is the number of occupied nearest neighbor sites. This number can be as large as 9 for face atoms (6 nearest neighbors within the shell, 3 in the shell below), 8 for edge atoms (6 + 2) and 6 for corner atoms (5 + 1).

How will the atoms arrange as the shell is filled? The positions of the atoms will be determined by two, sometimes conflicting, tendencies. On the one hand, icosahedral corner and edge sites should be avoided because they provide less binding energy. On the other hand, the occupied surface area of the partial shell should have a minimum circumference without occupying corner or edge sites. A compromise has to be found.

The results of the calculation are summarized in Fig. 31. The fifth shell of an icosahedron is complete with 561 atoms and the sixth shell with 923 atoms. The mass interval shown in Fig. 31 represents, therefore, the growth of the sixth shell. The top curve shows the calculated maximum energy gained by the successive addition of one atom. The units of energy are left intentionally arbitrary in order to emphasize that the assumptions made in the calculation allow only a qualitative comparison with experiment. The 362 calculated energies (one for each atom added) take on one of the 15 values. Each value can be associated with a specific nearest-neighbor environment.

The calculations indicate that subshells are characterized not so much by particularly stable structures, as by particularly unstable islands. Notice that an atom at an edge site with only four nearest neighbors (two in the new shell and two in the substrate) is weakly bound. Such unstable conformations would be expected to be associated with the minima in the mass spectra. In fact, the correlation with the minima in the measured spectrum, also shown in Fig. 31 is very good. The minima are associated with the atomic arrangements shown at the bottom of Fig. 31. The dots in the pentagonal figures represent the positions of atoms in the sixth shell of an icosahedron projected onto a plane. For clarity, the atoms on the bottom of the icosahedron are not shown in this projection. The shaded areas cover those sites which are occupied by atoms. In all cases the structures consist of compact islands plus a single atom at an adjacent edge site (large black dot).

The growth of a shell seems to start by the successive coverage of 1,2,3 and 4 adjacent triangular faces. Then a reorganization of the atoms on the surface occurs prior to the completion of five triangular faces around a central vertex. This “umbrella” is a recurring motif. Four of the following minima can be described as overlapping umbrellas plus, of course, one atom.

The tendency for alkali halide clusters to grow as cuboids is well documented [62-721. Suppose a cluster has the shape of a rectangular solid with nl, ~1~ and 12~ atoms along each edge. It can be stoichiometric, (MX),, if the product ~1~ x n2 x n3 is an even integer. However, if nl, ~1~ and n3 are all odd numbers, the product will be odd. Such clusters must contain an extra cation or anion. This is just what is required of the positively charged clusters discussed in this section.

Fig. 32 shows a mass spectrum of (NaBr),Na+ clusters. As expected, the mass peaks corresponding to cuboid clusters with an odd number of atoms along each edge are particularly strong. However, the spectrum has another characteristic that is not so immediately obvious. Notice the broad minima that are not directly correlated with the sharp peaks. This series of minima is the most outstanding characteristic of the mass spectra of large alkali halide clusters as demonstrated in Fig. 33 for (NaI),Na+. The broad minima are equally spaced when plotted on a (mass)l13 scale. For this reason, one can assume that they reflect a shell or subshell structure. The intervals between minima correspond to the number of atoms necessary to cover one face of a cuboid. Subshell

T.P. Martin /Physics Reports 273 (1996) 199-241

Edge Atom 7 near. neigh.

Face Atom 6 near. neigh.





Edge 4 near.

Face 3 near.

Atom neigh.

Atom neigh.

1 ;/: ‘.‘.

. . . .

2

\ ado\ \ 960 \

Fig. 31. Subshells (stable islands) observed during the formation of the sixth shell (561 atoms to 923 atoms) of a (Ca),, cluster. The experimental mass spectrum (smooth curve) is compared with the calculated energy gained with each new atom. The addition of one atom after completion of subshells leads to characteristic minima in the energy and the mass spectrum. The arrangements of surface atoms at the minima are shown by the shaded areas in the figures at the bottom. The dots in the pentagons represent the atoms of the sixth shell of an icosahedron projected onto a plane.

structure shows up as a series of minima because a small group of atoms sitting on a smooth face is highly unstable with respect to evaporation. Although it is necessary to cover three faces to regain a perfect cube, there is no evidence for a superstructure with period 3 in the mass spectrum. This indicates that the faces are probably added or removed in a statistical manner.


0 5000 10000 15000 20000 Mass (amu)

0

c

(amu) 50 000 100 000 150 000

‘1 , . M&s- ’

40 ‘13

(amu)

Fig. 32. Mass spectrum of (NaBr),Na+ clusters. Perfect cuboids with this composition can be formed only if the number

of atoms along each edge is an odd number.

Fig. 33. Mass spectrum of Na(Na1): clusters. The interval between the oscillations corresponds to the number of atoms needed to cover one face of cuboid-shaped clusters.

We conclude from this data that the large alkali halide clusters are very likely imperfect cuboids, the number of atoms along each edge being only approximately equal. The clusters can probably be characterized as cubes with missing faces, edges and corners.

It is possible to cover the outside of a fullerene molecule with metal atoms [37-391. The experimental arrangement is very similar to that described earlier, with one exception. The cluster condensation cell now contains two ovens. One oven for the fuherne and one oven for the metal. By changing the temperatures of the two ovens, it is possible to control the composition of the clusters so that they have either high or low metal content. Fig. 34 is an example of a mass spectrum obtained if we put C& in one oven and calcium in the other oven. Notice that the first mass peak to stand out particularly strongly is x = 32. Presumably this peak is strong because this cluster is resistant to evaporation, that is, this cluster is particularly stable. Now the question arises, why is it stable? Perhaps the atoms are arranged in a highly symmetric manner. CbO has 12 pentagonal faces and 20 hexagonal faces, for a total of 32. If we place one metal atom in the center of each of the 32 faces, we would have a highly symmetric and presumably a highly stable cluster. This explanation immediately suggests a new experiment. Co has 12 pentagonal faces and 25 hexagonal faces for a total of 37. Therefore, we repeated the experiment using C 70 as the core molecule and found that x = 37 stands out as a magic number. This now begins to give one confidence that there is some truth in the geometric explanation for these magic numbers. It also seems to indicate that by putting alkaline earth metal atoms on fullerene molecules it is possible to “count” the number of


10000 mass [amu]

Fig. 34. Mass spectrum of C&Ca~ clusters. The labelled features appear to correspond to completion of each of the first

four layers of metal around CGO.

faces on these fullerene molecules by means of mass spectrometry. Thirty-two calcium atoms on CGO form an almost perfectly close-packed layer with very high icosahedral symmetry. This layer is shown at the far left side in Fig. 34. The black circles represent calcium atoms at the vertices of an icosahedron, the white circles are calcium atoms centered on the triangular faces of an icosahedron. Suppose we try to construct a second layer on top of the first layer using the same geometric scheme. This layer is also shown in Fig. 34. Once again, there are 12 black atoms at the vertices of the icosahedron. However, three atoms can be placed on each of the triangular faces, for a total of 72. If we add to this the 32 atoms in the first layer, we have a total of 104 metal atoms. With increasing metal coverage one sees that x = 104 also corresponds to a very stable cluster. It is worth pointing out here that the icosahedral shells that we are discussing now are not the Mackay icosahedra. Both types of icosahedral structures have metal atoms at the vertices. However, the Mackay icosahedra contain atoms on the edge of the triangular faces. The metal-coated CbO shells described here do not contain edge atoms.

Fig. 34 also shows what can be expected if we complete a third and a fourth shell. The third shell is formed by putting six atoms in each triangular face and the fourth shell by covering each face with ten atoms. The total number of atoms needed to complete the first, second, third and fourth shells are 32, 104, 236 and 448, respectively. The total number of atoms in a cluster composed of K layers around CGO is

M=~?K3+10K2+J$K. (171

Notice that the mass spectrum shows intensity anomalies at 32, 104,236 and 448, just as predicted by the geometric model. For this reason, we believe that calcium and barium atoms form icosahedral shells around C6,,.

230

0

T.P. Martin i Ph_vsics Reports 273 (1996) 199-241

n=55 I

(‘60) n+ 1

20000 40000 60000 80000

Mass (amu)

Fig. 35. Low resolution mass spectrum of (C,,,),, clusters. Notice the strong peaks for n = 13 an 55. Notice also that the mass peak corresponding to doubly ionized (Ch0)55 is unusually strong.

A low resolution mass spectrum of (C,,),, clusters can be seen in Fig. 35. Notice that mass peaks corresponding to N = 13 and 55 stand out strongly, where 13 and 55 are the numbers of spheres necessary to complete the first two shells in the sequence of the so-called Mackay icosahedra. Starting OLE with a central sphere, it is possible to arrange 12 additional spheres around it to form a perfect icosahedron. Using this 13 icosahedron as a core, one can arrange 42 spheres around that to form again a perfect icosahedron containing a total of 55 spheres. This sequence continues indefinitely, each time adding on a new layer of spheres and each time arriving at a larger perfect icosahedron. Unfortunately, on the basis of these two magic numbers 13 and 55, it is not possible to conclude that the geometry of the clusters we are observing is icosahedral, because there exist three distinctly different geometries with exactly the same magic number sequence. These are the icosahedra, cuboctahedra and truncated decahedra. Although the 55-sphere icosahedron may not look too different from the 55-sphere cuboctahedron, it is important to make the distinction. A cuboctahedron of any size, i.e., containing any number of shells, can be cut out of an fee close-packed crystal. ChO in the condensed form has fee structure. For this reason it would not be surprising to find that the clusters condense into cuboctahedral structures. The icosahedron cannot be constructed with hard spheres having dense packing. The icosahedron has fivefold symmetry axes, a symmetry not consistent with translational symmetry. In this sense the icosahedron is noncrystalline. Although these two structures, icosahedra and cuboctahedra, are so different, they do have the same magic numbers for shell completions.


Fig. 36. (Ch0)19 could be an octahedral arrangement of nearly spherical molecules or a “capped” icosahedron

The next magic number in Fig. 35 is IZ = 19. Starting out with the 13-sphere cuboctahedra, it is possible to place an additional sphere in the center of each of the square faces. After doing this, one obtains a perfect 19-sphere octahedron which might also be expected to have high stability, Fig. 36. On the other hand, starting out with a 13-sphere icosahedra, it is possible to put a six-sphere “hat” on top of one of the vertices to obtain again a highly stable 19-sphere configuration. So, unfortunately, the magic number 19 brings us no closer to the answer to our question.

We feel decisive data concerning cluster structure can be obtained by using a second laser to heat the clusters. As mentioned previously, clusters with complete geometric shells resist evaporation when heated. This explains why mass peaks corresponding to closed-shell clusters are so strong. However, if the temperature is high enough, even closed-shell clusters begin to evaporate. Evapor- ation can probably be envisioned to occur in the following way: molecules are first removed from vertex positions. These molecules are poorly bonded to the layer underneath because they have contact with only one molecule in this layer. After removing all of the vertex molecules of one facet, the edge molecules of that facet begin to evaporate. This is followed by molecules internal to the facet until finally a complete face of the regular geometric figure has evaporated. The result is a cluster with a high intermediate stability which also tends to resist further evaporation. In this way, faces of the regular geometric figure can be peeled off one after the other. The removal of each face will result in a cluster with high stability causing a strong peak in the mass spectrum. This sequence of strong peaks occurring between major shell closings is referred to as subshell structure. A cuboctahedron has eight triangular faces and six square faces. The icosahedron has 20 identical triangular faces. For this reason, by studying the subshell structure, it should be possible to make a clear distinction between the two geometric forms, However, life is not quite so simple.

The positions that molecules occupy in a perfect icosahedron will be called “icosahedral sites”. The assumption made in the previous paragraph is that the molecules of an incomplete surface layer will occupy icosahedral sites. This is not necessarily the case. The molecules in one triangular face of the icosahedron can be thought of as a portion of a flat, closed-packed layer of spheres. We know from introductory solid state physics that there are two ways of constructing a close-packed overlayer. Only one of these ways corresponds to icosahedral sites. In fact, the alternative set of sites are more favorable for low surface coverage. This is made plausible in Fig. 4 which shows one triangular face of a large icosahedron. There are two ways of constructing the next layer. The icosahedral sites have been labelled W, the alternative sites B. Notice that there are no alternative

232 T. P. Martin / Ph,vsics Reports 273 (I 996) 199-241

60000

Mass (amu)

Fig. 37. Difference mass spectrum of laser-warmed (C,,,), clusters.

sites on the edge of the triangle. If all of the alternative sites were occupied with molecules, the result would be a series of disconnected islands on the surface of the cluster. At high surface coverage, clearly, this is an unfavorable situation and the icosahedral sites will be preferred. It is necessary to occupy edge sites if the surface is to be complete, however, the edge sites are energetically unfavorable because molecules at these sites have contact with only two molecules in the substrate layer. However, for low coverage connectivity is less of a problem and all of the alternative sites offer the advantage of making good contact with three substrate molecules. Alternative sites should be favored for low surface coverage. Tcosahedral sites should be favored for high surface coverage [39,40].

Fig. 37 shows a mass spectrum of C 60 clusters that have been induced to evaporate molecules by means of heating. Notice the subshell magic numbers between the major shell closings at 15 and 55. However, particularly easy to interpret is the series of high mass peaks occurring after 55. Each time three molecules are added to the surface of the 55-molecule cluster, a particularly stable configuration is achieved. There are three, and only three, alternative sites on each of the triangular faces of the 55-molecule icosahedron, Fig. 38.

The subshell structure between 13 and 55 appears to be complicated. However, it is worthy of our attention because it offers a unique opportunity to study the formation of the shell from beginning to end. The sequence starts with a single isolated molecule sitting alone on the surface of a perfect 13-atom icosahedral core and ends with a single vacancy in an otherwise completed shell. Notice that the mass peak for n = 14 is the weakest of the entire series. It is certainly plausible that the first atom to start a new layer should be the most weakly bound and therefore easily evaporated. There is a surprising agreement between the subshell structure observed here and the


1000 2000 3000 4000

Mass (amu)

Fig. 38. All alternative sites are occupied on the surface of a 55molecule icosahedron.

Fig. 39. Mass spectrum of (ZnS)’ and (ZnS),,(ZnS,)’ clusters.

magic numbers reported in Ar mass spectra. In both cases the mass peaks corresponding to y1= 19, 23, 43, 46 and 49 stand out strongly [40-421.

It might seem impossible to produce ZnS clusters in the vapor phase. If ZnS is heated in a vacuum, the vapor consists of Zn atoms and Sz molecules. The diatomic ZnS molecule is not produced in appreciable quantities. However, the relative amounts of these two vapor components, Zn and Sz, are just right to form stoichiometric thin films of polycrystalline ZnS if the vapor is condensed onto a substrate. This seems to indicate that if the vapor were to be quenched, the clusters formed might well have composition (ZnS),.

Fig. 39 shows the mass spectrum of clusters obtained when the vapor above ZnS is quenched in He gas. The mass peaks appear in pairs. The peaks colored black are due to stoichiometric ZnS clusters, the white peaks are due to ZnS clusters containing one extra sulfur atom. It may seem surprising to find positively charged clusters containing one extra sulfur atom. It may seem surprising to find positively charged clusters that contain an extra anion, but sulfur is a very versatile element. It forms readily Si + polyanions and it is these that are responsible for the second series of peaks. We will concentrate on the stoichiometric ZnS clusters. Notice the strong peaks at

(ZnS)r 3, (ZnSL and (ZnS)3,. These clusters have resisted evaporation when heated with a laser and have, therefore, a high stability.

One plausible structure for (ZnS)34 is based on the atomic ordering in a zinc blend crystal. The zinc blend structure can be described as two interpenetrating fee lattices. One lattice contains the zinc atoms, the other sulfur atoms. A set of octahedral shells can be cut out of an fee lattice. Imagine cutting an octahedron out of the Zn sublattice of a zinc blend crystal. The associated sulfur atoms should of course be carried along. Although this procedure would yield a set of highly symmetric clusters, they would not have the desired composition (ZnS),. This deficiency is easily removed by stripping off one face of the octahedron. This procedure is shown in Fig. 40. Notice that the same scheme can be used to construct (ZnS) r3. In fact, the scheme can be used to construct a whole series of ZnS clusters. The number of ZnS molecules in each member of this series can be written as

y1= $(4K3 - 3K2 - K) )


Fig. 40. Removal of one face from an octahedral arrangement of Zn-S reveals (ZnS),, and (ZnS)34

where K is the number of Zn atoms along one edge of the original octahedron. The number of ZnS molecules in clusters corresponding to K = 3,4, $6 are 13, 34, 70, 125, respectively. There is no trace of enhancement in the mass peak corresponding to (ZnS)TO. However, (ZnSh 24 is represented by a strong peak, and more importantly, the mass peak corresponding to (ZnSh,, is very weak.

The first two magic numbers in the ZnS cluster mass spectrum, 13 and 34, seem to show promise of a complete set of shells, however, this series is broken after only two. The future will tell if a complete set of shell closings can be developed with different growth conditions.

7. The melting of clusters

Perhaps it was Lord Kelvin who first asked the question: “Does the melting temperature of a small particle depend on its size ?” [74]. More recent theoretical considerations [75-771, computer simulations [78-811 and experimental observations [82-851, support the conclusion that the melting temperature depends strongly on cluster size. One method to measure this size dependence is based on the disappearance of geometric shell structure in the mass spectra of clusters as they are heated. However, not all materials are suitable for this type of investigation. A primary requirement, of course is that cold clusters demonstrate a well-defined geometric shell structure. Na, Ca, Xe, Al, NaCl, Ni and related materials all pass this first requirement. However, Ca and Xe clusters do not really melt when heated: they sublime. So, a second requirement is that


the clusters do not lose appreciable mass by the evaporation of atoms as they approach the melting temperature. The third requirement is related to the manner in which the shell structure manifests itself in the experiments. Shell structure may appear in mass spectra because (1) neutral cluster formation favors closed shells or (2) the ionization cross-section favors open shells or (3) photofrag- mentation favors closed shells. Only mechanism (2) is suitable for studying the melting process. If the magic numbers already exist in the distribution of neutral clusters, they will not disappear just because the clusters are subsequently melted. Sodium meets all three requirements at easily attainable temperatures.

The cluster source is a low-pressure, inert-gas, condensation cell. Sodium vapor is quenched in cold (100K) He gas having a pressure of 1 mbar. Clusters condense out of the vapor and are transported by the He gas stream through a heated nozzle. The warmed clusters pass through two chambers of intermediate pressure into a high vacuum chamber. There the clusters are ionized with a 415 nm, 0.3 uJ, 2 x 2 mm, 15 ns dye laser pulse.

It is important in this investigation to have a reliable estimate of the heat transfer between the metal walls of the nozzle and the clusters. Unfortunately, we must carry out the experiment in a gas pressure range that falls between the two extremes covered by usual engineering formulas, i.e., between laminar flow and molecular transport. For this reason it was important to test the validity of these formulas with various nozzle designs.

If the wavelength of the ionizing light is so chosen that the photon energy corresponds to the ionization threshold of the sodium cluster, an oscillatory structure appears in the mass spectrum. In order to make these oscillations more convenient to analyze, we construct difference spectra. Difference mass spectra for sodium clusters obtained using 415 nm ionizing light are shown in Fig. 41. Each spectrum is marked with a corresponding measured nozzle temperature.

Consider first the low temperature spectrum obtained at T = - 80 “C. Seven oscillations are observed for clusters ranging in size from n = 2000 to M = 10 000 atoms. The minima correlate very well with the number of atoms necessary to complete icosahedral or cuboctahedral shells. These values have been indicated in Fig. 41 by vertical lines.

At low temperature the rather broad minima fall almost exactly at the number of atoms needed to fill geometric shells. These minima occur only for low intensities of the ionizing radiation ( < 0.05 ltJ/cm2) and only for threshold values of the ionizing radiation (410 nm < /1 < 420 nm). Nothing much happens to the spectrum as the nozzle temperature increases to T = 0°C. At T = 9’C the minima for small values of shell index (6 I K 2 9) have sharpened noticeably and the minima for the larger shells appear to be structured. At 18 “C the shells begin to disappear starting with the smallest. At the same time the minima for larger shells shifts to lower values of n. At 23 ‘C shells corresponding to K = 6,7,8 and 9 have disappeared. At 26” shells 10 and 11 disappear. Finally, all the shells have disappeared for nozzle temperatures above 34 “C.

Our best values of the temperatures at which successive geometric shell disappear from the mass spectra are shown in Fig. 42, where r is the radius of a sphere into which y1 sodium atoms can be packed assuming the filling factor of an icosahedron. It should be pointed out here that the radius scale in Ref. [86] is incorrect due to the use of an inappropriate Na atom diameter. Although there is no reason to expect that the relative changes in melting temperature for gold [83] and sodium should have the same size dependence, it would seem that they are not too different, even though the bulk melting temperatures are quite different: Tgold = 1337 K, Tsodium = 371 K.


-/ +34”c

I I I I I 2000 6000 10000

Number of Atoms, n

Fig. 41. Difference mass spectra of (Na), clusters using near-threshold photoionization. Minima are compared with geometric shell closings (vertical lines). The shell structure disappears as the clusters are warmed from 17 to 42 “C.

The classical thermodynamical treatment of the melting of small spherical particles expression of the form [75-771

~m/~bulk = 1 + constant. l/r .

Note that our data do not extrapolate well to the value 1 as Y goes to infinity, Fig. 42,

yields an

A long chain of assumptions has been made in interpreting the results of this experiment. It is assumed that the He gas remains in contact with the large sodium clusters long enough so that they too reach the nozzle temperature. The cluster surface topology is assumed to affect the ionization cross-section resulting in a size-dependent modulation of the mass spectrum for facetted structures, and a smoothly varying mass spectrum for droplets. In Fig. 42 we have normalized all temperatures


Number of Atoms 100000 10000 1000

1.0 I I I I I 1 I ,100 50 30 20 \

\ r A

0.9 - \

Y \ 2

e \ 2

0.8 -

To O \

0.7 I , \

0 0.02 0.04 0.06

l/r (W-l)

Fig. 42. Ratio of cluster melting temperature to bulk melting temperature for various cluster sizes. A straight line extrapolation to the bulk melting temperature is not apparent.

against the bulk melting temperature. This tacitly assumes that we are observing volume melting and not a surface melting and that the bulk melting temperature has something to do with the melting temperature of icosahedra. A few words must be spent discussing these assumptions.

For a nozzle with 3 mm diameter and for 1 mbar of He gas, all estimates indicate that thermal equilibrium should be established between gas and nozzle within the 100 l,~s it takes for the gas to pass through the nozzle. It is more difficult for the gas to transfer its heat to the large clusters. However, here again estimates indicate an equilibrium should exist. But suppose it does not. This would lower the cluster temperatures still further increasing the discrepancy between bulk and cluster melting temperatures. For this reason, we feel confident in concluding that the cluster melting temperatures lie at least 20% below the bulk melting temperature.

Crystalline sodium has bee structure. The sodium clusters could be either cuboctahedral (fee) or more likely icosahedral. The cluster structure must undergo a transition to the bulk bee form at some larger size than investigated here. This transition should be accompanied by a more steeply rising melting temperature curve in Fig. 42. The data are consistent with such a behavior.

As a cluster heats up it is plausible that disorder might first appear at the surface resulting in what could be called surface melting. It is possible that this disorder so effects the ionization energy that the oscillations in the mass spectra disappear at temperatures below volume melting. To demonstrate such an effect with a computer simulation would be a very demanding challenge.

238 TP. Martin / Physics Reports 273 (1996j 199-241

8. Concluding remarks

Under certain experimental conditions, a periodic structure can be induced in the mass spectra of clusters, Figs 43-46, For example, the clusters can be heated with a laser beam. In this way the mass spectrum reveals clusters resistant to the evaporation of atoms, i.e., clusters with large binding energies.

I 364 (Nal),Na+

Fig. 43. Each oscillation in the mass spectrum of NaI clusters corresponds to the completion of one square face on a cuboid structure.

lOdO

mass [amir]

Fig. 44. Each new complete icosahedral layer of metal atoms added to a C,, molecule results in a particularly stable structure.


n=891

Mass (amu)

Fig. 45. Each oscillation in the mass spectrum of (In), clusters corresponds to the completion of one triangular face on an octahedral structure.

5 100 L

5 g 50

0

8 0

: z = -50 .- n

0 1000 2000 3000 4000 5000

Number of Atoms, n

Fig. 46. The main peaks in the mass spectrum of Ca clusters correspond to the completion of icosahedral layers, the weaker peaks correspond to the completion of triangular faces.

Such mass spectra reveal that NaCl and aluminum clusters assume at a very early stage the structural properties of the bulk material. NaCl clusters, in fact all alkali halide clusters, tend to grow as tiny cubes with a local atomic arrangement almost identical to that found in a crystal, Fig. 43. Aluminum and indium clusters take the form of octahedra. Within these octahedra, the


atoms are close-packed just as they are in the bulk. However, with calcium and magnesium clusters nature had a surprise in store for us. They take on the form of icosahedra. The local atomic arrangement, although highly ordered, is not crystalline. In each of these three cases growth takes place by adding on layers or shells to regular geometric figures; cubes for NaCl, octahedra for Al and icosahedra for Ca. This mode of growth is so general that it might be regarded as a building principle for clusters.

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