J . Phys. Chem. 1991, 95,6421-6429
6421
FEATURE ARTICLE Shell Structure of Clusters T.P.Martin,* T.Bergmann, H.Giihlich, and T.Lange Max-Planck-Institut fiir Festkorperforschung, Heisenbergstrasse 1 , 7000 Stuttgart 80, FRG (Received: April 8, 1991; In Final Form: May 29, 1991)
The properties of bulk metal are so different from those of a metal atom that it is sometimes hard to imagine how they might be related. However, a connection can be established by studying a succession of clusters containing 2, 3, 4, 5, ... atoms. If the properties of the solid w m found to evolve gradually and continuously with increasing cluster size, this type of investigation would not really be of much interest. But nature has presented us with a different situation. The properties of clusters change not continuously but often periodically with cluster size. This is due to the formation of shells: shells of electrons and shells of atoms.
1. Introduction The story of shell structure in metal clusters has an earlier parallel in nuclear physics. Many years ago, it was noticed that atomic nuclei containing 8,20,50,82, or 126 protons or neutrons have very long lifetimes. It was a challenge for the nuclear physicists in the 1940s to explain these so-called magic numbers. Since physicists tend to see most objects as perfectly round, it should come as no surprise that they assumed atomic nuclei are spherically symmetric. Under this assumption they had to solve only a radial Schriidinger equation:
where 1 is the angular momentum quantum number and V(r) is the radial dependence of the potential in which the nucleons move. They assumed further that the potential could be described by a simple potential well. Some confusion can arise because nuclear physicists and atomic physicists use slightly different definitions for the principal quantum number n: n(atomic) = n(nuc1ear) 1 (2) Throughout this paper we will use the principal quantum number from nuclear physics, Le., n denotes the number of extrema in the radial wave function. Eigenstates of the radial SchrMinger equation are often called subshells. The subshells of the infinite spherical potential well are shown ordered according to momentum in Figure 1. The lowest energy state is 1s and then come lp, Id, 2s, If, 2p, ..., etc. That is, with 2, 8, 18, 20, 34,40, 58, 68, 90, ... nucleons, subshells are completely filled and the corresponding nuclei could be expected to be exceptionally stable. However, these are not the observed magic numbers. In 1949 Goeppert-Mayer' and Haxel et a1.2 came up with a modified model that yielded the observed magic numbers. Their idea was that the spin-orbit interaction is unusually strong for nucleons. Subshells with high angular momentum split, and the states rearrange themselves into different groups. As we shall see, the original shell model, which the nuclear physicist had to discard, describes very nicely the electronic states of metal clusters.+'s
+
(1) Goeppert-Mayer. M. fhys. Reo. 1949, 75, 1969L. Jensen, J. H. D.; S u a , H. E. fhys. Rev. 1949,75, 1766L. (2) Haxel, 0.; (3) Knight, W. D.; Clemenger, K.; de Heer, W. A,; Saunders, W. A,; Chou, M. Y.; Cohen, M. L. fhys. Reo. Leu. 1984, 52, 2141.
0022-365419112095-6421S02.50/0
2. Subshells, Sbells, pnd Supersbells
If it can be assumed that the electrons in metal clusters move in a spherically symmetric potential, the problem is greatly simplified. Subshells for large values of angular momentum can contain hundreds of electrons having the same energy. The highest possible degeneracy assuming cubic symmetry is only 6. So under spherical symmetry the multitude of electronic states condenses into a few degenerate subshells. Each subshell is characterized by a pair of quantum numbers n and 1. Under certain circumstances the subshells themselves condense into a smaller number of highly degenerate shells. The reason for the formation of shells out of subshells requires more explanation. The concept of shells can be associated with a characteristic length. Every time the radius of a growing cluster increases by one unit of this characteristic length, a new shell is said to be added. The characteristic length for shells of atoms is approximately equal to the interatomic distance. The characteristic length for shells of electrons is related to the wavelength of an electron in the highest occupied energy level (Fermi energy). For the alkali metals these lengths differ by a factor of about 2. This concept is useful only because the characteristic lengths are, to a first (4) Kappcs, M. M.; Kunz, R.W.; Schumacher,E. Chem. fhys. Leu. 1982,
91, 413.
( 5 ) Katakuse, I.; Ichihara, I.; Fujita, Y.; Matsuo, T.; Sakurai, T.; Matsuda,
T. I n l . J . Mass Specrrom. Ion Processes 1985, 67, 229. (6) Brechignac, C.; Cahuzac, Ph.; Roux, J.-Ph. Chem. fhys. fetr. 1986, 127, 445. (7) Begemann, W.; Dreihofer, S.;Meiwes-Broer, K. H.;Lutz, H. 0. Z. fhvs. D 1986. 3. 183. -(8) Saunde&'W. A,; Clemenger, K.; de Heer, W. A,; Knight, W. D. fhys. Reo. B 1986, 32. 1366. (9) Bergmann, T.;Limberger, H.; Martin, T. P. fhys. Reo. Lerr. 1988.60,
..".. 1161
(IO) Martins, J. L.; Car, R.;Buttet, J. Surf. Sci. 1981, 106, 265.
(11) Ekardt, W. Ber. Bunsen-Ces. fhys. Chem. 1984, 88, 289. (12) Clemenger, K.fhys. Reo. B 1985, 32, 1359. (13) Ishii, Y.;Ohnishi, S.; Sugano, S. fhys. Reo. 1986, 833, 5271. (14) Bergmann, T.;Martin, T. P. J . Chem. fhys. 1989, 90, 2848. (15) Gijhlich,H.; Lange, T.; Bergmann, T.; Martin, T. P. fhys. Rev. Lett. 1990,65, 748; Z . fhys. D 1991, 19, 117. (16) Martin, T.P.; Bergmann, T.; Gahlich, H.; Lange, T. Chem. fhys. Lett. 1990, 172,209; Z . fhys. D 1991, 19, 25. (17) Bjc"lm, S.;Borggreen, J.; Echt. 0.;Hansen, K.; Pederson. J. Rasmussen, H. D. fhys. Reo. Leu. 1990, 65, 1627. (18) Penson, J. L.; Whetten, R. L.; Chcng. Hai-Ping; Berry, R.S.,to be published. Honea, E. C.; Homer,M. L.; Persson, J. L.; Whetten, R. L. Chem. fhys. Lerr. 1990, 171, 147.
0 1991 American Chemical Societv
Martin et al.
6422 The Journal of Physical Chemistry, Vol. 95, No. 17, 1991
n
5 '
748
/
2(21+11
/ / /
58
349 /
0
5
10
15
20
R \IZME/h
Figure 1. Degeneracy of states of the infinitely deep spherical well on a monentum scale. The total number of fermions needed to fill all states up to and including a given subshell is indicated above each bar.
Figure 2. States of the infinitely deep spherical well for very large values of 1. Notice the periodic bunching of states into shells. This periodic pattern is referred to as supershell structure.
approximation, independent of cluster size. The concept of shells can also be described in a different manner. An expansion of N,the total number of electrons, in terms of the shell index K will always have a leading term proportional to P.One power of K arises because we must sum over all shells up to K in order to obtain the total number of particles. One power of K arises because the number of subshells in a shell increases approximately linearly with shell index. Finally, the third power of K arises because the number of particles in the largest subshell also increases with shell index. Expressing this slightly more quantitatively, the total number of particles needed to fill all shells, k, up to and including K is
It is possible, however, to produce metal clusters containing such large numbers of electrons.
Shell structure is not necessarily an approximate and infrequent bunching of states as in the example of the spherical potential well, Figure 1. Clearly, almost none of the subshells occur exactly at the same energy for this potential. Shell structure can be the result of exactly overlapping states. Such degeneracies signal the presence of a symmetry higher than spherical symmetry. Subshells of hydrogen for which n + l have the same value have exactly the same energy. This additional degeneracy in the states of hydrogen is a result of the form of its potential, 1/ r , which bestows on hydrogen O(4) symmetry. Subshells of the spherical harmonic oscillator for which 2n + l have the same value also have exactly the same energy due to the form of the potential, 9,and the resulting symmetry, SU(3). For this reason it is said that these systems, hydrogen and oscillator, have quantum numbers n + 1 and 2n l that determine the energy. We have shown that 3n + l is an approximate energy quantum number for alkali-metal clusters.16 As the cluster increases in size, electron motion quantized in this way would finally be described as a closed triangular traje~t0ry.I~ The grouping of large subshells into shells is illustrated in Figure 2 for the spherical potential well. Here, it can again be seen that in certain energy or momentum regions the subshells bunch together. However, the states are so densely packed in this figure that the effect is perceived as an alternating light-dark pattern. That is, for the infinite potential well, bunching of states occurs periodically on the momentum scale. The periodic appearance of shell structure is referred to as supershell structure.20*21 Although supershell structure was predicted by nuclear physicists more than 15 years ago, it has never been observed in nuclei. The reason for this is very simple. The first supershell beat or interference occurs for a system containing 1500 fermions. There exist, of course, no nuclei containing so many protons and neutrons.
3. Experimental Section The technique we have used to study shell structure in metal clusters is photoionization time-of-flight (TOF) m a s spectrometry, Figure 3. The mass spectrometer has a mass range of 600000 amu and a mass resolution of up to 20000. The cluster source is a low-pressure, rare-gas, condensation cell. Sodium vapor was quenched in cold He gas having a pressure of about 1 mbar. Clusters condensed out of the quenched vapor were transported by the gas stream through a nozzle and through two chambers of intermediate pressure into a high-vacuum chamber. The size distribution of the clusters could be controlled by varying the oven-to-nozzle distance, the He gas pressure, and the oven temperature. The clusters were photoionized with a 1-pJ,2 X 1-mm, 15-11s dye laser pulse. The high-resolution mass spectra showed no evidence of cluster fragmentation or two-photon processes. Since phase space in the ion optics is anisotropically occupied at the moment of ionization, a quadrupole pair is used to focus the ions onto the detector. All ions in a volume of 1 mm3 that have less than 500 eV of kinetic energy at the moment of ionization are focused onto the detector.22 The reflector consists of two segments with highly homogeneous electric fields, separated by wire meshes. The first segment, which is twice completely traversed by the ions, is called the retarding field, and the other segment is called the reflecting field. This two-stage reflector allows a second-order time focusing of ions.23 Two channel plates in series are used to detect the ions. The secondary electrons are collected on a metal plate and conducted to the electronics. The following main design features of the instrument are necessary to achieve such a r e s ~ l u t i o n : ~ ~ (1) The ions are accelerated at right angles to the neutral cluster beam. If clusters are ionized by a laser pulse from the gas phase, there will always be a distribution of initial potential energies. The reflector is used to compensate for these. If the neutral beam is parallel to the acceleration direction, there is also an initial distribution of kinetic energies or velocity components parallel to the acceleration direction. If the reflector is used to compensate for the initial potential energy, it cannot also compensate for the kinetic energies. (2) A long (29 cm) retarding field segment is used in the reflector. In the vicinity of the wire meshes at the end of the two reflector segments the electric field is not perfectly homogeneous. This causes a slight deflection of ions passing through them and thus a small time error. By use of a long retarding field segment, the field in the vicinity of the wire meshes is lowered, and the deflection of ions passing through them is reduced.
(19) Balian, R.; Bloch, C. Ann. Phys. 1971, 69, 76. (20) Bohr, A.; Mottelson, B. R. Nuclear Structure; Benjamin: London,
(22) Bergmann, T.; Gtihlich, H.; Martin, T. P.; Schaber, H.; Malegiannakis, G. Rev. Sci. Instrum. 1990, 61, 2585. (23) Mamyrin, B. A.; Katataer, V. I.; Shmikk, D. V.; Zagulin, V. A. Sou.
(3)
where L(k) is the highest angular momentum subshell in shell
k.
+
1975.
(21) Nishioka, H.; Hansen, K.; Mottelson, B. R. Phys. Rev. B 1990, 42, 9377.
Phys. JETP 1973.37.45. (24) Bergmann, T.; Martin, T. P.; Schaber, H. Rev. Sci Instrum. 1990, 61, 2592.
The Journal of Physical Chemistry, Vol. 95, No. 17. 1991 6423
Feature Article
TO PUMP
TO PUMP
TO PUMP
Figure 3. Apparatus for the production, photoionization, and time-of-flight mass analysis of metal clusters. 2 0 0 ~ ~ ~ " " ' 1' "" "
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Figure 5. Mass spectrum of (Na)+, clusters ionized with high-intensity, 2.53-eV light. The clusters are fragmented by the ionizing laser. Fragments having closed-shell electronic configurationsare particularly stable.
Figure 4. Mass spectrum of Cs-0 clusters. Notice that the exact composition can be determined on an expanded mass scale.
(3) A reflector design is used that guarantees very stable and homogeneous electric fields. To achieve a good rigidity and to reduce temperature sensitivity, we have used as the main construction element of the reflector a 60-cm-long glass tube with a diameter of 15 cm. This glass tube holds all the field-defining elements. Since field errors as small as l P can cause significant time-of-flight errors, great care has to be taken to achieve the necessary field homogeneity. The mass spectra that will be displayed in this paper cover a large range of masses. For this reason it will not be possible to distinguish the individual mass peaks. For example, at the top of Figure 4 we have reproduced a mass spectrum of Cs-0 clusters that appears to be nothing more than a black smudge. How do we know how many oxygen atoms the clusters contain? This can be seen by graphically expanding the scale by a factor of 100, Figure 4. Because of the high resolution of our mass spectrometer, we are quite certain about the composition of the clusters examined.
4. Observation of Electronic Shell Structure Knight et al.3 first reported electronic shell structure in sodium clusters in 1984. Electronic shell structure can be demonstrated experimentally in several ways: as an abrupt decrease in the ionization energy with increasing cluster size, and as an abrupt increase or an abrupt decrease in the intensity of peaks in mass spectra. The first type of experiment can be easily understood. Electrons in newly opened shells are less tightly bound, Le., have lower ionization energies. However, considerable experimental effort is required to measure the ionization energy of even a single cluster. A complete photoionization spectrum must be obtained, and very often an appropriate source of tunable light is simply not available. It is much easier to observe shell closings in photoionization, TOF mass spectra. However, depending upon the intensity and wavelength of the ionizing laser pulse, the new shell is announced by either an increase or a decrease in mass peak height. For high laser intensities, multiple-photon processes cause the mass spectra to be less wavelength sensitive and also cause considerable fragmentation of large clusters. The resulting mass spectrum reflects the stability of cluster ion fragments. Clusters with newly opened shells are less stable and are weakly represented
6424 The Journal of Physical Chemistry, Vol. 95, No. 17, 19'91
Martin et al. 700 d
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Figure 6. Mass spectra of (Na), clusters obtained by using ionizing light near the ionization threshold (top) and well above the ionization threshold (bottom). In both cases the neutral cluster beam was heated with 2.54 and 2.41-eV laser light. I
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Figure 7. Mass spectra of CS,,+~(SO~) clusters with decreasing photon energy of the ionizing laser from 2.53 (top) to 2.33 eV (bottom). The values of n at the steps in the mass spectra have been indicated.
in the mass spectra. Notice in Figure 5 that as each new shell is opened there is a sharp step downward in the mass spectrum. Remember that cluster ions containing 9, 21,41, 59, ... sodium atoms contain the magic number (8,20,40, 58, ...) of electrons. For low laser fluence and wavelengths near the ionization threshold, the mass spectra have a completely different character. As each new shell is opened, there is a sharp step upward in the mass spectra, Figure 6 (top). Open-shell clusters have low ionization thresholds that fall below the energy of the incident photons, whereas closed shell clusters remain unionized. Finally, for low laser fluence and wavelengths well above the ionintion threshold, it is possible to observe the neutral distribution of cluster sizes. If the source conditions are appropriately chosen, this distribution can peak at sizes corresponding to closed electronic shells, Figure 6 (bottom). Cluster intensities can sometimes be increased by a factor of 10 by using a seed to nucleate the cluster growth. For example, by adding less than 0.02% SO, to the He cooling gas, Cs2S02 molecules form that apparently promote further cluster growth. Mass spectra of Cs,,+,(S02) clusters obtainedls by using four different dye-laser photon energies are shown in Figure 7. Although it is not possible to distinguish the individual mass peaks in this condensed plot, it is evident that the spectra are charac-
Figwe 8. Expanded mass spectra of CS,,+~(SO~), clusters for an ionizing photon energy of 2.48 eV. The lines connect mass peaks of clusters containing the same number z of SO, molecules. Notice that the steps for clusters containing (SO,),and (SO,),are shifted by two Cs atoms.
terized by steps. For example, a sharp increase in the mass-peak intensity occurs between n = 92 and 93. This can be more clearly seen if the mass scale is expanded by a factor of 50 (Figure 8). Notice also that the step occurs at the same value of n for clusters containing both one and two SOz molecules. In addition to the steps for n = 58 and 92 in Figure 7, there are broad minima in the 2.53-eV spectrum at about 140 and 200 Cs masses. These broad features become sharp if the ionizing photon energy is decreased to 2.43 eV. By successively decreasing the photon energy, steps can be observed for the magic numbers n = 58,92, 138, 198 f 2,263 f 5,341 f 5, and 557 f 5.1s*'7 However, the steps become less well defined with increasing mass. We have studied the mass spectra of not only Cs,+,(S02) but also CS,+~(SO,),, CS,,~, and CS,~O> They all show steplike features for the same values of n. First, we would like to offer a qualitative explanation for these results and then support this explanation with detailed calculation. Each cesmium atom contributes one delocalized electron that can move freely within the cluster. Each oxygen atom and each SO, molecule bonds with two of these electrons. Therefore, a cluster with composition Cs,,+,(S02), for example, can be said to have n delocalized electrons. The potential in which the electrons move is nearly spherically symmetric, so that the states are characterized by a well-defined angular momentum. Therefore, the delocalized electrons occupy subshells of constant angular momentum that in turn condense into shells. When one of these shells is fully populated with electrons, the ionization energy is high and the clusters will not appear in mass spectra obtained by using sufficiently low ionizing photon energy. In other experimentsgthe closing of small subshells of angular momentum was shown to be accompanied by a sharp step in the ionization energy for Cs-0 clusters having certain sizes, namely, for Cs,+2rOrwith n = 8, 18, 20, 34, 58, and 92. The closing at n = 40 seen in all other alkali-metal clusters could not be observed either in the experiments or in the calculations. The steps were observed for clusters containing from one to seven oxygen atoms. 5. Density Functional Calculation
Self-consistent calculations have been carried out applying the density functional approach to the spherical jellium model.lO,ll We used an exchange correlation term of the GunnarssonLundqvist form and a jellium density rs = 5.75 corresponding to the bulk value of cesium. This model implies two improvements over the hard-sphere model discussed earlier. First, electronelectron interaction is included. Second, the jellium is regarded to be a more realistic simplification of the positive ion background ion is taken into account only by than the hard sphere. The 0,omitting the cesium electrons presumably bound to oxygen. The calculations were performed on CsW We found that if a homogeneousjellium was used, the grouping of subshells was rather similar to the results of the infinite spherical (25) Lange, T.; GBhlich, H.; Bergmann, T.; Martin, T. P.2.P h p . D 1991, 19, 113.
The Journal of Physical Chemistry, Vol. 95, No. 17, 1991 6425
Feature Article Main quantum number
- 4.0
5
n 138 1
440
3 . 1 1 ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' 1100 0 Number of Na Atoms Figure 10. Ionization potentials calculated as a function of n for (Na), clusters. A positive background charge distribution slightly concentrated in the central region has been used. Notice the similar behavior of the ionization energies of the chemical elements (inset).
562
-2.0
Figure 9. Self-consistent, one-electron states of a 600-electron cesium cluster calculated by using a modified spherical jellium background.
potential well. However, a nonuniform jellium yielded a shell structure in better accordance to experimental results. We found that the subshells groups fairly well into the observed shells only if the background charge distribution is slightly concentrated in the central region. This was achieved, for example, by adding a weak Gaussian (0.5%total charge density, half-width of 6 au) charge distribution to the uniform distribution (width 48 au). Figure 9 shows the ordering of subshells obtained from this potential. This leads to the rather surprising result that the Cs' cores seem to have higher density in the neighborhood of the center, perhaps due to the existence of the 02- ion. All attempts to lower the positive charge density in the central region led to an incorrect ordering of states. This first calculation addressed the problem of the grouping of low-lying energy levels in one large Csm cluster. However, in the experiment the magic numbers were found by a rough examination of ionization potentials of the whole distribution of cluster sizes. A more direct way to explain magic numbers is to look for steps in the ionization potential curve of Cs-0 clusters. for Therefore, we calculated the ionization potentials of CS,+~O n I600 and of (Na), for n I1100 using the same local-density scheme described above, Figure 10. Starting from a known closed-shell configuration for n = 18, electrons were successively added. Three test configurations were calculated for each cluster size testing the opening of new subshells. The configuration with minimum total energy was choosen for the calculation of the ionization potential. We found that the lower magic numbers n = 34,40, 58, and 92 were well reproduced. For higher n distinct steps in the ionization potential were observed for n = 138, 196,268, 338,440, 562,704,854, and 1012. Magic number clusters exhibit unusually high ionization energies for the same reason that rare-gas atoms do: they possess a closed-shell electronic configuration, Figure 10. In this sense the metallic clusters behave like giant atoms. 6. Shells of Atoms
One might think that the definition of a shell of atoms is straightforward-one layer of atoms arranged on the surface of a core such that the newly formed, larger unit has the same (overall) outer symmetry as the core itself. However, as one begins to construct examples, it becomes quickly clear that this definition might lead to confusion. For example, consider a cluster composed of atoms placed at the sites of a simple cubic lattice and having
F i e 11.
Closed-shell 147-atomcuboctahedron. Notice that the atoms
in the square faces are not close-packed.
the overall outer shape of a cube. The first such cube that can be formed around a central atom contains 27 atoms, 3 atoms on a side; the next, 125 atoms, i.e., 5 atoms on a side, etc. But what happened to the 64-atom cube with 4 atoms on a side? It has no central atom. That is, this simple example might be considered to describe two distinct shell sequences, one set of shells possessing a central atom, the other set having a central 8-atom cube. One way of getting around this difficulty is to combine the two sets into a single set. Each successive member of this combined set is obtained by adding atoms to only three of the six faces of the preceding member. We will see that it is useful to designate a set of such shells as irregular shells in order to distinguish them from, for example, the regular shells of an icosahedron where atoms must be added to all faces in order to complete the next shell. 6.1. Shells Obtained from Close-Packed Spheres. A limited number of 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. For example, the cuboctahedron, an octahedron truncated by a cube, Figure l l , has 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 relatively unfavorable energetically and is a good candidate to accept the first atoms of a new shell. The hexagonal close-packed crystal is usually distinguished from a face-centered crystal by the ordering of the close-packed layers. The atoms in every other layer lie exactly above one another and one speaks of an ababa... layer sequence. For the fcc close-packed crystal this sequence is abcabca.... It is convenient to use the same notion for clusters. The cuboctahedron obeys the sequence a b . . . , Figure 12. In this sense the cuboctahedron can be said to have fcc structure, i.e., it can be cut out of an fcc crystal. However,
6426 The Journal of Physical Chemistry, Vol. 95, No. 17, 1991
Martin et al.
Figure 14. Smaller geometric figure found in a bcc lattice is outlined with heavy lines. This figure can be found in shells of increasing size.
Figure 12. The 147-atom cuboctahedron composed of seven close-packed layers having the stacking sequence characteristic of an fcc lattice. Figure 15. Closed-shell 55 atom icosahedron and a portion of the next shell.
Figure 13. A 3 X 3 X 3 atom rhombohedral portion of a bcc lattice. It can be viewed as a cube distorted along the body diagonal.
if the bottom three layers of Figure 11 are rotated 60°, the sequence changes from abcabca to abcacba. The result is a highly symmetric cluster, closely related to the cuboctahedron; however, it does not have fcc structure. 6.2. Shell Structures Related to the Bcc Lattice. In a previous 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 irregular shells but is unrealistic in that elemental matter does not usually condense into a simple cubic structure. However, if such a cube is squeezed along a body diagonal, the cube deforms into a rhombohedron that can be cut out of a bcc lattice, Figure 13. The bcc rhombohedron represents a set of irregular shells containing, of course, the same number of atoms as simple cubic shells. The bcc lattice contains also a set of regular shells. The first member in this set is shown in Figure 14. The atoms of such clusters are contained within 12 rhombic faces. 6.3. Shell Structures with 5-Fold Symmetry. Until now we have discussed shell structure in clusters of close-packed atoms or of
atoms on crystal lattice sites. Clusters in the form of icosahedra or decahedra are neither close-packed nor are they small pieces cut of a crystal. A 5-fold symmetry axis is not consistent with the crystalline requirement of translational symmetry. Icosahedra form a set of regular shells around a central atom, Figure 15. Nature has played a strange trick on us here. The number of atoms needed to complete icosahedral shells is exactly that needed to complete cuboctahedral shells. For this reason, the experimental observation of magic numbers corresponding to shell closings is not sufficient to allow us to distinguish between noncrystalline icosahedra and fcc cuboctahedra. Decahedra represent a set of irregular shells. The shells possess alternately a central atom and a central seven-atom decahedron and are formed by placing a large overlapping "umbrella" on top of the previous member of the set, Figure 16.
7. Observation of Shells and Subshells of Atoms Both calculationsand experiments2633indicate 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 fcc crystals. (26) Hoare, M. R. Adu. Chem. Phys. 1979,40,49. Sattler, K.; Recknagel, E. Phys. Rev. Lett. 1981,47, 1121. (27) Echt, 0.; (28) Farges, J.; de Feraudy, M. F.; Raoult, B.; Torchet, G. J. Chem. Phys. 1986,84, 349 1. (29) Northby, J. A. J. Chem. Phys. 1987,86,6166. Hams, I. A., Kidwell, R. S.; Northby, J. A. Phys. Rev. Lett. 1984, 53, 2390. (30) Lethbridge, P. G.; Stace, A. J. J . Chem. Phys. 1989, 91, 7685. (31) Miehle, W.; Kandler, 0.; Leisner, T.; Echt, 0.J. Chem. Phys. 1989, 91, 5940. (32) The subject of icosahedral shells of atoms in inert-gas clusters has a
long and interesting history. For a recent starting point into this extensive literature see: Proceedings of Faraday Symposium on Large Gas Phase Clusters. J . Chem. Soc., Faraday Trans. 1990,86. (33) Schriver, K. E.; Hahn, M. Y.; Persson, J. L.; LaVilla, M. E.; Whetten, R. L. J . Phys. Chem. 1989, 93, 2869.
The Journal of Physical Chemistry, Vol. 95, No. 17, 1991 6421
Feature Article
I
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Figure 18. Ratio spectrum (slightly smoothed mass spectrum divided by highly smoothed envelop spectrum) for Mg clusters. The filled mass peaks correspond to completely filled icosahedral shells.
W Figure 16. Decahedra form a set of irregular shells. Successively larger shells are formed by adding an umbrella-shaped partial layer. Shell of Atoms 10 12 1
,
1
14 1
1
16 I
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peaks, corresponding to completely filled icosahedral shells, are nearly equally spaced on this n1I3scale. The four mass peaks observed between shell closings indicate highly stable partial shells.
0
10000
20000
n
Figure 17. Averaged mass spectra of (Na), clusters photoionized with 2.99- and 2.93-eV light. Well-defined minima occur at values of n
correspondingto the total number of atoms in close-pcked cuboctahedra and nearly close-packed icosahedra (listed at top). Precrystalline structures have also been observed for metallic materials in condensed units large enough to yield sharp electron diffraction patterns.*% 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 by using the newly developed technique of high-resolution electron micro~copy.~' Additional evidence exists for icosahedral symmetry in metal clusters. Calculations predict that very small alkaline-earth-metal clusters prefer noncrystalline structures.'&42 The pattern of NH, (34) Shechtman, D.; Blech, 1.; Gratias; D.; Chan, J. W.Phys. Reu. Lerf. 1984,53. 1951. (35) Janot, C.; Dubois, J.-M. J. Non-Crysr. Solids 1988. 106, 193. (36) Hall, B. D.; FlOeli, M.; Monot, R.; Borel, J.-P. Z . Phys. D 1989, 12, 97.
(37) SCC: Proceedings: 4rh Inrl. Meerfngon Small Particles and Inorganic Clusrers; Chapon, C., Gillet. M.F., Henry, C. R., Eds.;Springer: Berlin 1989. (38) Blaisten-Barojas. E.;Khanna, S . N. Phys. Reo. Lcrr. 1988,61, 1477. (39) Rao, B. K.; Khanna, S.N.; Meng, J.; Jena, P. Z . Phys. D, in press. (40) Pacchioni, G.; Pewestorf. W.;Koutecky, J . Chem. Phys. 1984, 83,
201 * (41) Bauschlicher. C. W.;Bagus, P. S.;Cox, B. N. J. Chem. Phys. 1982, 77, 4032. (42) Koutecky, J.; Fantucci, P. Chem. Reu. 1986, 86, 539.
and H 2 0 binding energies with Co and Ni clusters has been interpreted as indicating icosahedral symmetry in metal clusters containing from 50 to 150 Mass spectra of Ba and Ba-O clusters seem to indicate an icosahedral growth sequence in the size range from 13 to 35 atoms4-' Recently, we observed a slow modulation in mass spectra of Na clusters, Figure 17, which we interpreted as evidence for the existence of shell structures, Le., a highly symmetric, onionlike cluster structure.16 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 cusplike 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 ico~ a h e d r a . ~However, * on the basis of such observations, it is not possible to conclude that the clusters have icosahedral symmetry because icosahedral shells and fcc cuboctahedral shells contain exactly the same number of atoms. Figure 18 shows a mass spectrum of pure Mg clusters containing up to 3000 atoms.49 The choice of ionizing photon energy and laser intensity is important. We have used 50 mJ/cm2 of 308 nm radiation per 10-ns pulse. With such high intensities massive fragmentation of the clusters is to be expected. For this reason we believe that strong peaks in the mass spectrum indicate cluster ion fragments with high stability. Even though we are using a (43) Klots, T. D.; Winter, B. J.; Parks, E. K.;Riley, S.J. J . Chem. Phys. 1990. 92, 21 IO. (44) Winter, B. J.; Klots, T. D.; Parks, E. K.; Riley, S.J. Z . Phys. D, in
press.
(45) Rayane, D.: Melinon, P.; Cabaud, B.; Hoareau, A,; Tribollet, B.; Broyer, M. Phys. Rev. A 1989, 39, 6056. (46) Whetten, R. L.,private communications. (47) Martin, T. P.; Bergmann, T. J . Chem. Phys. 1990, 90, 6664. (48) Mackay, A. L. Acta Crystollogr. 1962, IS, 916. (49) Martin, T. P.; Bergmann,T.; GBhlich, H.; Lang, T. Chem. Phys. Lerr.
1991, 176, 343.
6428 The Journal of Physical Chemistry, Vol. 95, No. 17, 1991
Martin et al.
TABLE I: Total Number of Atoms Needed To Comdete Structural Sbeb
K 1
2 3 4 5 6 7 8 9 10 11 12
13 14 15 16 17 18 19 20
a
b
C
d
e
f
g
h
i
13 55 147 309 56 1 923 1415 2057 2869 3871 5083 6525 8217 10179 12431 14993 17885 21 127 24739 28741
15 65 175 369 671 1 IO5 1695 2465 3439 4641 6095 7825 9855 12209 14911 17985 21455 25345 29679 34481
19 85 23 1 489 891 1469 2255 3281 4579 6181 81 I9 10425 13131 16269 19871 23969 28595 33781 39559 45961
7 23 54 105 181 287 428 609 835 1111 1442 1833 2289 2815 3416 4097 4863 5719 6670 7721
4 IO 20 35 56 84 120 165 220 286 364 455 560 680 816 969 1140 1330 1540 1771
6 19 44 85 146 23 1 344 489 670 891 1156 1469 1834 2255 2736 3281 3894 4579 5340 6181
5 14 30 55 91 140 204 285 385 506 650 819 1015 1240 1496 1785 2109 2470 2870
38 201 586 1289 2406 4033 6266 9201 12934 17561 23 178 29881 37766 46929 57466 69473 83046 98281 1 15274 134121
16 68 180 375 676 1 IO6 1688 2445 3400 4576 5996 7683 9660 11950 14576 17561 20928 24700 28900 33551
3311
i 8 27 64 125 216 343 512 729 lo00
1331 1728 2197 2744 3375 4096 4913 5832 6859 8000 9261
Olcosahedra and cuboctahedra. * Bcc. COctahedra(ca). dDecahedra (without restrictions). CTetrahedra.'Octahedra (without restrictions). #Square pyramid. *Truncated octahedra (hex faces). 'Truncated tetrahedra (hex faces). 'Simple cubic and bcc rhombohedra. high laser intensity, the signal is weak, about one cluster per laser shot. Over 200000 shots at 50 Hz were required to obtain this spectrum. Because of the low signal, some averaging is necessary to bring out the spectral features. First, an average is made over 500 time channels. This plus the fact that magnesium has three natural isotopes (79% %Mg, 10% ZSMg,11% =Mg) limits the mass resolution. In a next step the spectrum is averaged over 5000 time channels. The resulting curve contains no structure but is merely an envelope of the original data. Finally, we form the ratio of the slightly averaged spectrum to the strongly averaged (envelope) spectrum. The resulting ratio spectra are shown in Figures 18 and 19. The strongest mass peaks correspond to the number of atoms in closed shells having either icosahedral or cuboctahedral symmetry. The Mackay icosahedra can be constructed from nearly close-packed spheres. These structures might be called noncrystalline since they possess a point group that is not consistent with translational symmetry. Cuboctahedra, on the other hand, can be constructed from close-packed spheres. In fact, cuboctahedra containing an arbitrary number of shells can be cut out of a fcc crystal. The main sequence of strong mass peaks does not allow us to distinguish between these two structures. We have to look elsewhere for decisve experimental data. We believe these data are contained in the weaker mass peaks between shell closings. Figure 19 shows a portion of the previous mass spectrum but now plotted against n1/3.Notice that the four main peaks are equally spaced. This is a characteristic common to all types of shell structure. The reason for this can be seen in the following way. Every time the radius (proportional, of course, to n1I3)of a growing cluster increases by one unit of a characteristic length, a new shell is said to be added. Notice in Figure 19 that the spectral features between complete shells repeat exactly within the statistical accuracy of the experiment. In particular, we will focus our attention on the repetitive peak structure labeled 1,2,3,and 4 and suggest below that this structure corresponds to partial icosahedral shells. The clusters most probably grow by adding shells of atoms to a rigid core. The number of atoms contained in a growth shell is dependent on the preferred coordination and local symmetry of the atoms and on the overall symmetry of the shell. If we assume that the magnesium atoms are close-packed, or nearly so, and that the outer form is that of an cuboctahedron or an icosahedron, then the total number of atoms Nk in a cluster containing K shells of atoms is48
Nk = (lob? 4- l5Kz
4-
11K
3)/3
(4)
Clusters constructed of complete shells can be expected to be highly stable. For inert-gas clusters both experiments and cal-
Figure 20. The dots represent the atoms of the 7th shell of an icosahedron projected onto a plane. The bottom 76 atoms are not shown. The umbrella-shapedstructures are identical, each containing 76 atoms. We suggest that the umbrellas represent highly stable partial shells of magnesium atoms.
culations indicate that partial icosahedral shells of atoms also show enhanced 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 (8 triangular and 6 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 or atoms on these faces alone or in combination with other cuboctahedral faces could be found that matched the observed subshell magic numbers. Next, we turned to the icosahedron for which subshell structure had already been studied.2e31 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 sequence, observed in inert-gas clusters, although close, seems to deviate significantly from the observed magic numbers for Mg clusters. Therefore, we would like to suggest an alternative sequence. Assume that the atoms in the new shell take immediately their final positions. In Figure 20 the positions of the atoms in the
The Journal of Physical Chemistry, Vol. 95, No. 17, 1991 6429
Feature Article
small clusters (n I1500) the pattern appears to be due to the filling of electronic shells. For large clusters the shell seems to be composed of atoms. Why might one expect a transition from electronic shell structure to shells of atoms? For very small clusters the atoms are highly mobile. There is no difficulty for the atoms to arrange themselves into a spherelike conformation if this is demanded by the closing of an electronic shell. At a size corresponding to about 1500 atoms under our experimental conditions, the clusters become rigid. Thereafter, each newly added atom condenses onto the surface and remains there. Further growth takes place by the accumulation of shells of atoms.
Figure 21. Mass spectra of (Na), clusters photoionized with 3.02-eV photons. Two sequences of structures are observed at equally spaced
intervalson the shell sequence.
scalean electronicshell sequence and a structural
TABLE 11: Total Number of Electrons Needed To ComDlete Shells shell a b C shell C 1 2 2 2 21 2520 2 10 8 2872 8 22 20 3256 3 28 18 23 40 3672 4 60 34 24 70 4122 58 25 5 110 4608 6 182 112 90 26 5130 168 7 280 132 27 5690 240 8 408 186 28 6290 330 9 570 252 29 440 10 770 6930 332 30 428 31 11 1012 7612 572 8338 728 12 1300 540 32 9108 910 670 33 13 1638 14 2030 9924 1120 820 34 15 2480 10788 1360 990 35 16 2992 11700 1632 1182 36 17 3570 12662 1938 1398 37 18 4218 13676 2280 1638 38 19 4940 14742 2660 1904 39 20 5740 3080 15862 2198 40
* Hydrogen. Oscillator. Pseudo-quantum number. seventh shell have been projected onto a plane in the manner of NorthbydBWe suggest that umbrella-shaped intermediate groups have enhanced stability. Each of the umbrellas contains 76 atoms and each has the same shape (although they appear distorted in the projection shown in Figure 20). Only 51 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.
8. Transition from Shells of Electrons to Sbells of Atoms Recently two types of shell structure have been observed in the same mass spectrum of large sodium clusters, Figure 21. For
Appendix The expression for the total number of atoms in complete icosahedral shells is often quoted in the literature. We were surprised to find no similar expressions for other geometries. In this Appendix we include not only formulas but a short tabulation (Tables I and 11) of the number of particles in geometric and also a few electronic shells. Icosahedra and cuboctahedra, Figure 15: N(K) = % ( l o p 1 5 p 11K + 3)
+
+
Bcc, Figure 14:
+ 6K2 + 4K + 1
N(K) = 4 p Octahedra (central atom):
+ 14K + 3)
+
N(K) = y3(16P 2 4 p
Decahedra (without restrictions), Figure 16:
+ l5K2 + 16K + 6)
N(K) = 1/(5K3 Tetrahedra:
Octahedra (without restrictions): N(K) = % ( 2 p+ 6K2
+ 7K + 3)
Square pyramid: N(K) = g ( 2 p
+ 9K2 + 13K + 6)
Truncated octahedron (hex faces): N(K)=16P+lSfl+6K+l Truncated tetrahedron (hex faces): N(K) = g ( 2 3 p + 4 2 p
+ 25K + 6)
Simple cubic and bcc, Figure 13: N(K) = p
+ 3K2 + 3K + 1
Hydrogen: N(K) = K ( 2 p
+ 3K2 + K )
Harmonic oscillator: N(K) = X(p
+ 3P + 2K)
Repistry NO. CS,7440-46-2; 0,. 7782-44-7; 0, 17778-80-2; SO29 7446-59-%Na, 7440-23-5.
