J. Phys. Chem. 1987, 91, 3141-3149
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FEATURE ARTICLE Physics of Metal Clusters Marvin L. Cohen,* Department of Physics, University of California, and Materials and Molecular Research Division, Lawrence Berkeley Laboratory, Berkeley, California 94720
M. Y. Chou, Corporate Research Science Laboratories, Exxon Research and Engineering Company, Annandale, New Jersey 08801
W. D. Knight, and Walt A. de Heer Department of Physics, University of California, Berkeley, California 94720 (Received: October 13, 1986; In Final Form: February 20, 1987)
The theoretical and experimental basis for some current physical models are described and applied to explain the properties of clusters of a few prototype s-p metals. Emphasis is given to the successful use of models such as jellium which were originally developed for bulk solids. It is shown that the experimental data and the results of the jellium model point to a shell model description of metal clusters.
Introduction Metal clusters have attracted increased attention recently in both the physics and chemistry communities. Experimental techniques have become more sophisticated and flexible so that reproducible data can be used to distinguish between competing models and theories. On the theoretical side, physicists and chemists have been able to apply methods developed for other systems to explore the structural and electronic properties of clusters. Because clusters lie somewhere between molecules and surfaces in the hierarchy of atoms to molecules to solid surfaces to bulk solids, approaches and models to study them have come from both sides. In most cases, condensed matter theorists have applied solid-state models while quantum chemists have moved from calculations on complex molecules to clusters. Experimentalists have also witnessed a synthesis of approaches from physics and chemistry. The objectives of those working in the two sciences often differ considerably, and language problems have developed. Part of the motivation for the discussion here is to clarify some of the models used in the more physics-oriented approaches. A recent review by the authors' contains more detailed information and more references. The focus of this review is on clusters of simple metals (without conduction d-electrons). We will describe a method which treats each metal cluster as a structureless ionic jelly permeated by itinerant electrons. This jellium model has had considerable success in explaining many properties of bulk simple metals and the approximations involved are well understood. When the crystal structure is eliminated, the quantum energy levels for the valence electrons are determined by the confining region. For solids and clusters, the constraints of the region can, for example, be modeled with a macroscopic box or a microscopic spheroid, respectively. In both cases, the valence electrons are no longer identified with specific atoms but are part of a sea of indistinguishable electrons with each experiencing the effects of the positive background, the average potential of the other electrons, and the constraints of the confining region. (1) de Heer, W. A.; Knight, W. State Phys., in press.
D.;Chou, M. Y . ;Cohen, M. L. Solid
This model for a cluster has many features in common with modern models of nuclei. Unlike the model of an atom where the Coulomb interaction dictates that the principal and orbital angular momentum quantum numbers n and I obey the rule 1 < n, a spherical or spheroidal cluster model implies that the total resultant potential of the cluster is not Coulombic in form and I is not restricted. In fact, a lowest order approximation is to consider a square-well, harmonic oscillator, or hybrid potential (Figure 1) as is done for nuclei. To justify this model and explain its construction, we begin by examining the bulk solid. This is where the approximations and applications were first tested. Next, we make the extension to clusters and explore the main consequences of the model which include the existence of shell structure. The experimental results are then described followed by discussions of the comparisons between experiment and theory.
Theoretical Models Model of a Bulk Metal. The one-electron model has been very successful in describing a wide variety of solid-state properties. In this model, an electron in a solid is viewed as interacting with a periodic potential consiting of contributions from the ions and an average interaction from the other electrons. The total electronic energy of this system is then the sum of the energies of the individual electrons. To calculate the individual electron energies, it is necessary to provide a model for the crystal. Several successful approaches are available. Here we focus on the pseudopotential approach and later view jellium with its smeared out positive background as a special case of the pseudopotential model where the solid is assumed to consist of a periodic array of positive cores or ions and a sea of itinerant valence electrons. A core consists of a nucleus plus its core electrons, and it is considered to be the same in the solid as it is in an isolated atom. For example, sodium is described as a bcc array of cores immersed in a free-electron gas of valence electrons. The cores each contain a nucleus and the 10 core electrons [ ( 1 ~ ) ~ ( 2 ~ ) * ( 2 pEach ) ~ ] . sodium core has a net single positive charge, and the (3s)' valence electrons are no longer associated with a specific core but are shared among the cores.
0022-3654/87/2091-3141$01.50/00 1987 American Chemical Society
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Harmonic Oscillator
Intermediate
Square Well
Figure 1. Energy-level occupations for three-dimensional harmonic oscillator, intermediate, and square potential wells.
- % BOND LENGTH ~r &CORE
REGION
Cohen et al. Coulomb, exchange, and correlation interactions among the valence electrons. Each can be evaluated by using potentials which depend only on the charge density. This is the local density approximation (LDA) since the potentials based on this approximation are local in space; that is, the potential at position r for exchange and correlation is a functional of the electron density a t point r. These contributions to the energy will be computed explicitly later within the jellium approximation. Both the pseudopotential and LDA methods have become standard theoretical t00ls~-~ in the physics of solids and molecules. Together they provide a method for determining the structural and electronic properties of solids. The approach to structure determination involves the computation of the total energy of a solid for varying structural arrangements. The minimum energy structure (for a given volume fixed by the applied pressure) among likely candidates is the preferred choice. Unfortunately, not all structures can be tested, but considerable success has been achieved in predicting high-pressure structuresSaby using less than a dozen candidate structures. For a given structure, many structural and electronic properties can be determined t h e ~ r e t i c a l l y .These ~~~~ include lattice constants, bulk moduli, cohesive energies, phonon spectra, structural phase transitions, superconductivity, and other solid-state properties. The successes of this approach for extended systems motivate the use of similar schemes for clusters. The above model has been applied directly to clusters of atoms, but when these ab initio methods are used they become computationally complex for large clusters. Hence, as a first step, the jellium model can be used. This model has had wide application in solid-state physics for bulk and surface6 properties of solids. The approach is similar in principle for computing the electronic For the bulk, the solid is viewed in the energies for same way as it is in the pseudopotential model, but the discreteness of the cores is eliminated by smearing out the positive charge into a uniform structureless background “jelly”. In the jellium or homogeneous electron gas model, the positive background charge density epo+ is cancelled by a similar electronic contribution -epo to assure charge neutrality. Here the average density of the electrons is used Po = W Q
Figure 2. A schematic pseudopotential (solid line) showing the reduction of the attractive Coulombic ion potential (dashed line) in the core region.
The total energy of the system consists of the kinetic energies of the cores and electrons and the electrostatic potential energies arising from c o r m r e , electron-core, and electron-electron interactions. Since the core separations are large compared to their sizes, the interactions between the cores are Coulombic and can be evaluated by using standard Madelung summations.* In the approach described here, electron-core interactions are evaluated via a p~eudopotential,~ and electron-electron contributions are approximated by a local density approximation4described below. The electron-core pseudopotential contains both the attractive Coulomb interaction of the positive core and a repulsive potential which is constructed to reproduce the effect of the Pauli exclusion principle between the valence and core electrons. The latter term tends to force the valence electrons out of the core region where it cancels much of the attractive Coulomb potential. A schematic picture of a pseudopotential appears in Figure 2. Because of the cancellation in the core region, the effective potential is weak. In addition, there is electronic screening of the core potential which further reduces the interaction of each of the electrons with the cores. The last of the three potential energy terms arises from electron-electron interactions. This term is usually divided into
(1)
when N is the total number of elbctrons and Q is the volume. When constructing the total energy ET of this jellium system, the average electrostatic electronic energy is left out since this is cancelled by the Coulomb interaction of the uniform positive background with itself and the Coulomb interaction of the uniform positive background with the electrons. Now ET = NE, where E, is the ground-state energy of each individual electron. A parameter r, is used to characterize the density of the electron gas; r, is the radius per electron measured in Bohr radii a. 4
1
3 ~ ( r , a=~ )~ Po
The various energies of the electron gas can now be expressed in terms of r,. In rydbergs, the Fermi energy2EF = 3.6832/r: and the average kinetic energy per electron of an electron gas2 is EKE = 3/5EFor
The direct Coulomb interaction between electrons is omitted as described above, but an exchange interaction remains. Because of the Pauli principle, two electrons cannot have all their quantum numbers alike. Antiparallel spin electrons differ in their spin quantum number, but parallel spin electrons do not and hence avoid each other in an effort to satisfy Pauli’s rule. Because of (5) (a) Cohen, M. L. Science 1986,234,549. (b) Cohen, M. L. Phys. Scr. 1982, TI,5.
(2) Kittel, C. Introduction to Solid State Physics, 6th 4.;Wiley: New fiork, 1986. (3) Cohen, M. L.; Heine, V. Solid State Phys. 1970, 24, 37. (4) Kohn, W.; Sham, L. J. Phys. Pev. 1965, 140, A1333.
(6) Lang, N, D. Solid State Phys. 1973, 28, 225. (7) Martins, J. L.; Buttet, J.; Car, R. Phys. Rev. B 1985, 31, 1804. (8) Knight, W. D.; Clemenger, K.; de Heer, W. A.; Saunders, W. A.; Chou, M. Y.; Cohen, M. L. Phys. Rev. Lett. 1984,52,2141.
The Journal of Physical Chemistry, Vol. 91, No. 12, 1987 3143
Feature Article the charge on the electron, this repulsion changes the Coulomb energy of the system. Within the jellium model, the exchange energy
Ex = -0.9163/rS
(4)
Higher order correlations beyond exchange give rise to terms in power of r:, In (rS),r,, etc. The first terms in the correlation energy E, are E, = -0.094 + 0.0622 In (r,) (5)
So for bulk jellium E e = 2*2099 ---0‘9163 0.094 rsz r,
-
+ 0.0622 In (r,)
(6)
and this value is accurate as r, 0 which is the high-density limit. In the above model, the volume is assumed to become very large and surfaces are neglected. When surface effects are included: the electron gas is not terminated at the edge of the jellium but rather it spills out. This effect changes the potential, and the calculation must be done self-consistently; that is, the potential producing the electronic charge density depends on the spatial form of the density which in turn acts back on the potential. Self-consistent jellium models for surfaces give excellent insight into the surface properties of real metals. Since the approximations used are only appropriate for ground-state energies, excitations involving electron removal are not determined accurately. Work functions are within 10-20% of experimental values, but it is not clear whether the discrepancy arises from the elimination of crystal structure or from the approximations used for exchange and correlation in this “ground-state” model. In general, the jellium model for bulk solids is expected to be a good approximation when the electron-core interactions given by the pseudopotential are a small perturbation on the energy. For crystals like diamond, silicon, or NaCI, the pseudopotentials are relatively large and the charge density is not uniform. Covalent bonds pile up charge, and ionic potentials transfer electrons. For metals like N a and other alkalis, the pseudopotentials are small, and the charge density is quite uniform. Covalentlike contributions giving directional bonding do not appear. These systems are ideal cases for using the jellium and free-electron approaches and applications have become standard textbook examplesa2 Jellium Model for Clusters. The preferred approach for computing properties of clusters would be to use ab initio techniques like the total energy pseudopotential approach. However, since the structure of the cluster is unknown, the positions of the atoms must be determined by a minimum energy principle as described above for solids. This approach would require repeated calculations7 each time an atom is moved and hence can be done for only small clusters for a predetermined structure or for more simplified theoretical models. For clusters of silicon, the energy associated with the positions of the atoms is critical in determining the covalent bonds and the large contribution to the energy coming from the pile-up of charge. Hence, for this case, an accurate model of the electron-core potential is necessary. For rare-gas clusters, the interactions are weak, and there is little charge rearrangement. The total energy depends on packing; therefore, structure is again critical. However, in the case of simple metal clusters, one argues that structure is not critical and the pseudopotential is weak; hence, the use of a jellium model is appropriate. Another post hoc argument for the applicability of jellium for clusters is that the structural energy has a small variation as a function of cluster size. Since this is not the case for the electronic contribution, the latter is dominant in determining the relative stabilities of clusters. This aspect will be discussed later. Because of the simplicity of the jellium background potential, it can be applied to clusters of arbitrary size. The cluster density is fixed by using the r, value of the bulk solid. Cluster sizes for the model are determined by the number of atoms in the cluster. Hence, the jellium calculation for clusters has some of the features of the self-consistent surface calculation for solids described above. The effects of charge rearrangement have to be considered.
A simpler non-self-consistent approximation is to confine the electrons in a spherical box with infinite potential walls. This would not allow for the “spilling out” of the charge at the surface, but this approximation does illustrate the overall nature of a free-electron gas confined to a sphere. The energy eigenstates are also eigenstates of angular momentum, and shell structure imposed by the potential is evident (Figure 1 ) . A next possible approximation to account for the surface effects is to use a finite square well with rounded corners. This model also yields shell structure (Figure 1); its detailed properties depend on the parameters used to describe the potential. The approximate models give a reasonably accurate picture of the energy levels and the Occurrence of degenerate states arising from the spherical symmetry. When a fully self-consistent jellium model calculation is used, the results do not change much from the well models. Again, degeneracies and shells are apparent. When the total energy of the system is considered, the special stability associated with completed shells becomes manifest. The principal success of the jellium model for clusters is that it accounts for the steps in the total electronic energy as a function of the number of atoms N . These steps are a direct consequence of the spherical symmetry leading to degenerate electronic energy levels and shells which resemble the energy-level structures of nuclei and atoms. Stable alkali metal clusters are predicted and observed for N = 2, 8, 18, 20, 40, 5 8 , and 92, which represent the closed-shell spherical clusters. Further fine structure in the abundance spectra is predicted9 for the open-shell clusters by allowing for ellipsoidal deformations corresponding to observed subshell closing at, for example, N = 26, 30, 34, 36, 3 8 , etc. Analogous distortions were predicted and observed in nuclei according to the Nilsson model. Discussion of other refinements and limitations of the jellium model and the LDA will be presented in the section on theoretical results. Ellipsoidal Distortions. The Jahn-Teller theorem states that a molecule in a symmetrical configuration with both vibrational and non-Kramers electronic degeneracies is unstable against distortions which lower the energy. Analogous distortions occur for both atomic nuclei and.meta1 clusters. Detailed calculation9 has shown that most open-shell clusters which are allowed to distort at constant volume assume minimum energy ellipsoidal shapes. For x = y # z with respective major axes b and a, the distortion parameter is given by q=-
2(a - b) a+b
(7)
which can have values up to 7 = 0.5 for open-shell alkali clusters N < 100. A fully self-consistent treatment for ellipsoidal clusters has not yet appeared. However, the simplified models which are known to work for spherical clusters can be applied. A modified three-dimensional harmonic oscillator potential may be used to derive the energy levels for a spheroidal cluster. It may be inferred from Figure 1 that a “squared” harmonic oscillator potential closely resembles a “rounded“ square well, and the dynamical degeneracies inherent in the harmonic oscillator are similarly broken as when the Id level splits and drops below the 2s level in Figure 1 . It may also be seen that the spheroidal harmonic oscillator has two characteristic frequencies. For the prolate ellipsoid the curvature of the potential is the smaller and the frequency lower along the major axis. The difference frequency is a measure of the energy splitting which is characteristic of the distortion parameter 7. The scaling factor for the oscillator frequencies is the spherical frequency9 wo
-
EFN-‘~’
(8)
where EF is the Fermi energy, for example, 2.7 eV for sodium. The similarities among the several “well” models are clear enough, the utility of the intermediate rounded or squared well for dealing with spherical clusters is well established, and it should be applicable to both the spheroidal frequencies wXs)and w, which (9) Clemenger, K. Phys. Rev. B 1986, 86, 619; see also ref 66.
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Cohen et al.
characterize an open-shell cluster. If cluster distortions are constrained to occur at constant volume, the frequency w, is associated with the symmetry axis, the transverse frequency wxy is weighted doubly, and the frequencies are related by9 (9)
The main first-order effects will depend on symmetry, and these are observed, for example, as fine structure in the mass abundance spectra corresponding to increments of N + 2 (clusters 35-36 and 37-38) or N + 4 (clusters 27-30, 31-34). The first corresponds to the fact that the spin degeneracies persist in the spheroids. The second results from the axial symmetry. It is easily shown that the energy levels of the ellipsoidal model9 correspond with experimental abundance spectra and naturally include the results of spherical shell theory. In the ClemengerNilsson model9 the total electronic energy for a cluster is found by summing the individual electron energies, using the harmonic oscillator frequency wo as a scaling factor. The Nilsson Hamiltonian includes an anharmonic term which squares the well as discussed above and a spin-orbit coupling term which is needed to describe nucleon interactions but which is omitted for alkali clusters whose spin-orbit effects turn out to be smaller than any other energy splittings in the problem. The total electronic energy is calculated for all N electrons in a cluster and is a function of the distortion factor q since oscillator frequencies wxy and w, replace the wo of the spherical problem. When this total electronic energy is calculated as a function of 11, minima are found for both prolate and oblate ellipsoids. The deeper of these two minima determines the equilibrium value of q, identifies which shape is the more stable, andgives the equilibrium energy for the cluster. The minima for the closed-shell clusters occur as expected for 17 = 0. The equilibrium electronic energies for all open- and closed-shell clusters may be plotted as a function of N. The second differences8 A2(IV) then give the relative stabilities of the clusters and show peaks a t the spherical shell closing and lesser features at the spheroidal subshell closings. Although a totally self-consistent calculation has not been made, it appears that the configuration of the electron wave functions govern the predicted shapes. In the Clemenger-Nilsson model, even the small clusters appear to have symmetries which correlate closely with the results of detailed’ calculations. Thus, allowing for the three-dimensional extent of the wave functions, clusters 3 and 4 are “prolate”, 5, 6, and 7 are “oblate”, 8 is spherical as expected, and 13 is “prolate”. Martins et al. have described’ the situation in terms of the p-like orbitals for the incomplete shell for clusters N = 3-7. The shapes described above differ significantly from geometrical closepacking arguments. The important geometry here is the shape of the collective electron distribution as in the case of the self-consistent spherical jellium model. Both models assume delocalized electrons confined within potential boundaries determined by a uniform structureless positive background. The progression of energy levels is similar except for the spheroidal splittings. Essentially the same physical picture aca u n t s for both major (spherical) and minor (spheroidal) features in the abundance spectra.
Experimental Results At the present writing, a number of metal clusters and cluster ions have been studied; see the following reviews1.1*12and conference reports.I3-l5 Properties are known for alkali8-16and n ~ b i e ‘ ~metals, , ’ ~ some d i ~ a l e n t ’and ~ , ~trivalent2’-24metals, lead:’ Phillips, J. C. Chem. Rev. 1986, 86, 619. Whetten, R. L.; Cox, D. M.; Kaldor, A. Surf. Sci. 1985, 156, 8. (a) Gole, J. T.;Stwalley, W. C., to be published in Advances in Atomic and Molecular Physics. (b) Morse, M. D. Chem. Rev., in press. (13) J . Phys. Colloq. 1977, 38, C2. (14) Surf.Sci. 1981, 106. (15) Surf. Sci. 1985, 156. (16) Knight, W. D.; de Heer, W. A.; Clemenger, K.; Saunders, W. A. Solid State Commun. 1985, 53, 445. (17) Katakuse, I.; Ichihara, I.; Fujita, Y.; Matsuo, T.;Sakurai, T.; Matsuda, H. Int. J. Mass Spectrom. Ion Processes 1985, 67, 229.
II
s
20
Figure 3. Abundance spectrum for sodium: (a, top) continuous sweep covering clusters 2-63; (b, bottom) clusters 44-105. In addition to the major shell closings at 8, 20, 40, 58, and 92, significant minor features appear at 26, 30, and 34.
and a number of transition metals,lOslland semiconductor^.^^^^^ In addition a number of reports have described mixed clust e r ~ . ~ ~ Much , ~ ~ of~ the , ~experimental ~ , ~ ~ , work ~ ~ has dealt with the abundance spectra, which are related to the stabilities of clusters. Studies of properties have included electric29 and dipole moments, ionization potential^,^^ and chemical reactivity.’1,33,34The cluster state has unique features of its own
(18) Begemann, W.; Meiwes-Broer, K. H.; Lutz, H. 0. Presented at the Xth International Symposium on Molecular Beams, Cannes, France, June 1985. “Proceedings of the International Symposium on Metal Clusters, Heidelberg 1986”, Z . Phys. D 1986, 3, 109. (19) Katakuse, I.; Ichihara, T.; Fujita, Y.; Matsuo, T.; Matsuda, H. Int. J . Mass Spectrom. Ion Processes 1986, 69, 153. (20) (a) Brkhignac, C.; Broyer, M.; Cahuzac, Ph.; Delacretax, G.; Labastie, p.; Woste, L. Chem. Phys. Lett. 1985, 120, 559. (b) Brcchignac, c.; Cahuzac, Ph.; Roux, J. Chem. Phys. Lett. 1986, 127, 445. (21) Miihlbach, J.; Pfau, P.; Recknagel, E.; Sattler, K. Z . Phys. B 1982, 47, 233. (22) Devienne, F. Marcel; Roustan, Jean-Claude, Organic Mass Spectrom. 1982, 17, 173. (23) Martin, T. P. J . Chem. Phys. 1985, 83, 78. (24) Meiwes-Broer, K. H., private communication. (25) Bloomfield, L.; Freeman, R.; Brown, W. L. Phys. Rev. Lett. 1985.54, 2446. (26) Heath, J. R.; Liu, Yuan; OBrien, S. C.; Zhang, Qing-Ling.; Curl, R. F.; Tittel, F. K.; Smalley, R. E. J . Chem. Phys. 1985, 83, 5520. (27) Sattler, K. Surf. Sci. 1985, 156, 292. (28) Kappes, M.; Radi, P.; Schar, M.; Schumacher, E. Chem. Phys. Lett. 1985, 119, 11; J . Chem. Phys. 1986,84, 1863. (29) Knight, W. D.; Clemenger, K.; de Heer, W. A,; Saunders, W. A. Phys. Rev. B 1985, 31, 2539. (30) de Heer, W. A. Ph.D. Thesis, University of California, Berkeley, 1985. (31) Knight, W. D. Helv. Phys. Acta 1983, 56, 521. (32) Saunders, W. A.; Clemenger, K.; de Heer, W. A,; Knight, W . D. Phys. Rev. B 1985, 32, 1366. (33) Geusic, M. F.; Morse, M. D.; Smalley, R. E. J . Chem. Phys. 1985, 82. 590.
Th!eJournal of Physical Chemistry, Vol. 91, No. 12, 1987 3145
Feature Article
I
I I I
(b) Sodium
4
Y Figure 4. Abundance spectrum for potassium: (a, top) clusters 3-51; (b, bottom) clusters 50-100. Major shell closings are marked. Minor features are seen at 26, 30, and 34.
as well as features which relate it to molecules on the one hand and the bulk crystalline solids on the other. Questions such as “at which size does a metal cluster begin to act like a metal?” have stimulated several lines of work; for example, measurement of electric dipole polari~ability~~ of alkali metal clusters indicates that metallic screening is evident even in dimers. A study of inner-shell autoionization in mercury clustersZOaindicates that for n < 8 metallic character has not yet developed. The most prominent feature of the cluster state of metals is the electronic shell structure, which has been observed in a variety of pure and mixed metal clusters. The shell structure is reflected in the abundance spectra,8 ionization potential^,^^ electric dipole p ~ l a r i z a b i l i t y ,neutral ~~ cluster time of flight spectra,35 and photofragmentation spectra.36 In the rest of this experimental section, we will concentrate on a discussion of the abundance patterns (mass spectra) and the ionization potentials which give promising prospects for a future spectroscopy of clusters. Abundance Patterns. Shell structure was first recognized in the abundance patterns of sodium clusters. Features occurring at N = 8, 18, 20, 40, 58, and 92 identify a series of particularly stable clusters3’ (Figure 3). The same features have also been seen in potassium clusters 29 (Figure 4) and in potassium clusters containing single atom sodium impuritiesI6 (Figure 5 ) . Correspondingly a series of particularly stable positive cluster ions have been seen in secondary ion mass spectra (SIMS) spectra for the noble metals,”*’8 for example, for silver a t masses W = 9, 19, 21, 35, 41, 59, 93, 139, and 199. The members of this second series differ respectively from the first by one atom, and the stabilities correspond to clusters born as ions, with stabilities characterized by 8, 18, 20, ... electrons. Negative ions s h o ~ ~ (34) Richtsmeier, S.C.; Parks, E. K.; Liu, K.; Pobo,L. G.; Riley, S. J. J . Chem. Phys. 1985,82, 3659. (35) Clemenger, K., unpublished work. (36) de Heer, W. A,, unpublished work. (37) Although 18 and 34 appear as shell closings in spherical models, they are not prominent in the experimental spectra. Cluster 34 is calculated to be di~torted,~ and it appears in the abundance spectra as a definite but minor feature. Cluster 18 is now thought to be sphericalMbut is small in comparison with 20, which is thought to be exaggerated by evaporation effects. (38) Hortig, G.; Muller, M. Z . Phys. 1969, 221, 119. (39) Joyes, P.; Sudraud, P. Surf. Sci. 1985, 156, 451.
2 -
,. Number of atoms per cluster, N
Figure 5. (a) Abundance spectrum for potassium clusters with sodium impurities. The intermediate peaks represent mixed clusters containing one sodium impurity. Shell closings are seen for clusters containing 8, 20, and 40 electrons. At higher mass resolution the minor features at 26, 30, and 34 are clear in the mixed clusters. (b) Pure sodium abundance spectrum for comparison.
abundance patterns according to N- = 7, 17, 19, etc. also corresponding to 8, 18, 20, ... electrons. It was explained previously’ how the observed abundance patterns in a properly formed neutral cluster beam originate from relative cluster stabilities and local thermodynamic equilibrium among neighboring cluster sizes. If the equilibrium conditions are not met, distortions are observed in the abundance patterns. The existence of an energy gap following the filling of a spherical shell implies an abrupt drop in the relative stabilities and intensities of the clusters immediately following. Thus, steplike or edge discontinuities are expected and observed. The well-known abundance spectra for alkali metals now serve as a basis for comparison with other spectra related to electronic shell structure; for example, see ref 17. Enough is understood about the effects of experimental conditions that the spectra can be used to diagnose those conditions. Source Conditions and Cluster Formation. The quality of experimental results depends directly on the capacity and performance of the apparatus. We describe, for example, an apparatus’ which has proven to be exceptionally effective in producing relatively undistorted high-intensity alkali cluster abundance spectra. It employs a detector with high sensitivity and ~low , ~noise ~ permitting the detection of low abundances and important fine features in the spectra. The clusters are produced in a supersonic jet beam source. The cylindrical nozzle has a uniform bore of 0.076 mm, which is equal to half the length. Argon carrier gas pressures are between 1 and 10 atm. Typical nozzle temperatures are 100 K relative to the oven. The oven source temperature is adjusted to give -5% mixing ratio of metal vapor to carrier gas pressure. Final cluster temperatures are typically -300-400 K. Generally, in a seeded jet source, an inert gas is mixed with a low concentration of, for example, alkali metal vapor and the mixture is ejected.8*40s42-44a If the seed material can condense,
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3146 The Journal of Physical Chemistry, Vol. 91, No. 12, 1987
clusters will be formed in the expansion. This situation differs from the pure vapor expansion$I particularly because the carrier gas can serve as a heat bath for the seed. Collisions between seed particles facilitate the establishment of equilibrium among the cluster species in the nozzle channel.' The carrier gas mediates the equilibrium and absorbs most of the heat of formation which is converted into translational energy of the beam.36 Outside the nozzle, the densities and temperatures drop rapidly. Since the vapor density is much smaller than the carrier density, collisions among the clusters will cease before the cluster carrier collisions die out so that the cluster cooling continues after the cluster formation processes stop. This final cooling freezes the distribution. Significant cooling of the vibrational states of the sodium trimer in seeded molecular beams has been demonstrated44afor carrier pressures ranging from 1 to 8 atm where it is observed that spectral features which are broad at 1 atm carrier pressure become narrow a t 8 atm, and final temperatures are estimated to be around 20 K. The relative heights of peaks in the abundance spectra are in proportion to relative cluster stabilities reflecting the frozen equilibrium distribution of clusters. Numerical estimates' indicate a formation temperature which is consistent with the actual temperature in the nozzle channel. If there is significant additional growth by agglomeration or by addition of atoms to the clusters which have already formed, the abundance maxima will shift to larger numbers than predicted by cluster stabilities. This may occur in a diverging conical n ~ z z l e , * for * ~example, ~ ~ ~ ~ ~tending to shift the features at 8 and 20 to 9 and 21. On the other hand, if the expansion cooling is insufficient to stabilize the clusters which were formed in the nozzle channel, evaporation of atoms or dimers may shift the maxima to lower cluster numbers giving rise to maxima around 6,7 and ' ~ ~excessively ~ hot 18,19 instead of 8 and 20. This is ~ b s e r v e dfor nozzles or high mixing ratios. Since at a given temperature the evaporation rates and stabilities are inversely related, at intermediate temperatures evaporation will tend to enhance the high stability peak at N , at the expense of those clusters at N, 1 and N, 2 immediately above in the sequence. This accentuates the edges at 8, 20, and 40 in the Na and K abundance spectra. The effect of different carrier gases on the production of sodium clusters was studied43for He, Ne, Ar, Kr, and N2. The heavier noble gas carriers progressively increase production of larger clusters. The nozzle temperature and mixing ratio were high (35%) resulting in hot clusters. The effects of different gases on the production was explained in terms of varying efficiencies in removing the heat of condensation from the clusters. Although the temperatures of clusters produced in seeded beams have not been widely investigated,44ait is possible to estimate a final upper temperature limit based on cluster stabilities. Since the clusters in the apparatus being describedl survive the 2-m flight from source to detector, the evaporation rate must be less than lo3 atoms/s. An order of magnitude estimate of the evaporation rate can be made by assuming that, per unit area, the clusters evaporate at approximately the same rate as the bulk, which can be estimated from the vapor pressure. Using the bulk vapor pressure of sodium, it is found' that for Nazo the vibrational temperature must be less than 500 K.
+
+
(40) Knight, W. D.; Monot, R.; Dietz, E. R.; George, A. Phys. Rev. Lett. 1978, 40, 1324.
(41) Delacretaz, G.; GaniCre, J. D.; Monot, R.; Woste, L. Appl. Phys. B. 1982, 29, 55.
(42) Larsen, R. A.; Neoh, S. K.; Herschbach, D. R. Rev. Sci. Instrum. 1974. 45. 1511.
(43) Kappes, M. M.; Kunz, R. W.; Schumacher, E. Chem. Phys. Lett. 1982. 91. 413.
(44) (a) DelacrEtaz, G.; Wbste, L. Surf. Sci. 1985, 156, 770. (b) Hermann, A,; Hofman, M.; Leutwyler, s.;Woste, L.; Schumacher, E. Chem. Phys. Lett. 1982, 26, 2247. (c) Gole, J. L.;Green, G. J.; Preuss, D. R. J . Chem. Phys. Lett. 1982, 26, 2247. ( d ) Morse, M. D.; Hopkins, J. D.; langridge-Smith, P. R. R.; Smalley, R. J . Chem. Phys. 1983, 79, 5316. (45) Hagena, 0. f.; Obert, W. J . Chem. Phys. 1972, 56, 1793 (46) Brtchignac, C.; Cahuzac, Ph., private communications. (47) Daly, N. R. Rev. Sci. Instrum. 1960, 31, 264.
Cohen et al. I
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Figure 6. Ionization potentials for potassium clusters 3-100. Note the drops in IP a t major shell closings. Note also clear minor features at 26 and 30. Although the overall precision is 0.06 eV, comparative values for neighboring clusters are good to 0.01 eV.
Although the seeded cluster jet source is a simple and reliable method of producing a wide range of cluster sizes, its operation requires high vapor pressures, and hence it is unsuitable for high boiling point metals. Other cluster sources are described in ref 1. Photoionization. In the detector, clusters are photoionized and focused on the entrance of the on-axis quadrupole mass analyzer (QMA) which is designed to pass 50% of the selected masses. The range of the QMA is >4000 amu. The light intensity is monitored continuously during data collection and fed to the computer which provides a continuous plot of corrected and normalized signal against photon wavelength or energy as the light source is scanned by a computer-controlled stepping motor. A shutter is provided to turn the arc lamp on and off during laser irradiation in order to monitor the wavelength-independent intensity of the detected cluster. A light chopper occults the ionizing light at a 15-Hz rate for synchronous lock-in processing in order to eliminate non-light-related signals. The cluster beam intensities obtainable with the seeded source and sensitive detector yield signals of >20000 counts/s for sodium cluster N = 20 at detector noise levels of -1 Hz. The major sources of noise are statistical fluctuations in the cluster beam or light intensity. The detector lies approximately 2 m downstream from the source, and the cluster beam flux at the detector is orders of magnitude smaller than would be observed at the more commonly used shorter distances. This loss of intensity is more than compensated for by the large source production and the sensitivity of the detector. The original design, which was intended to provide adequate path length for deflection experiments, has special advantages, for example, in assessing the expansion cooling at the source and photofragmentation in the beam. The light sources are a xenon arc lamp (with optional filter or monochromator (2.5-5.0 eV at 12 nm resolution) or a tunable CW laser. Although the xenon lamp output is low compared to the HgXe lamp, it is superior to the latter because of the relatively flat spectral output, which permits precise location of spectral features without requiring elaborate deconvolutions. Ionization potentials are determined by linear e x t r a p ~ l a t i o n . ~ ~ ~ ~ ~ In recent years, considerable attention has been devoted to alkali cluster ionization potentials (IP). The most recent results for K clusters span a wide range from N = 3 to 101 (ref 32) (Figure 6) and exhibit several important features. Immediately following each spherical cluster, the IP drops abruptly. The IP jumps at the spherical shell closings N = 8, 18, 20, 40, 58, and 92 are consistent with the abundance spectra. Although the absolute error of the IP values may be comparable with the observed jumps, it is evident from Figure 6 that the IP variation from cluster t o
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(48) Brechignac, C.; Cahuzac, Ph. Chem. Phys. Lett. 1985, 117, 365.
The Journal of Physical Chemistry, Vol. 91, No. 12, 1987
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cluster can be determined to an accuracy of -0.01 The TABLE I: Electron Eigenvalues (in eV) for 13- and 15-Atom high accuracy permits the recognition of I P jumps at the larger Clustersa cluster closings at 40, 58, and 92 and also fine structure features Na13 NalS in the IP curve. state jellium cluster jellium cluster Instrumental effects at the detector can produce distortions in 1s -5.037 (2) -4.805 (2) -5.097 (2) -4.933 (2) the abundance spectra, although under common conditions the -4.104 (6) -3.930 (6) lp -3.985 (6) -3.675 (6) spectra are independent of the ionizing light spectrum. Hence, Id -2.718 (10) -2.498 (6) -2.916 (IO) -2.930 (4) cluster 9 (whose IP is below the I P for 8) appears to dominate -2.255 (4) -2.636 (6) 8 if laser photon energy is setzobbetween the respective thresholds. 2s -2.265 (2) -2.195 (2) -2.505 (2) -2.433 (2) Similarly two-photon ionizationZobproduces a strong odd-even alternation in the abundance spectrum which causes 9 to dominate 'Degeneracies are noted in parentheses. The electrons occupy the lowest energy levels. 8 and 10, and raises 21 relative to 20. Although photodissociation occurs at high laser f l ~ e n c e ,it~is~ not ~ , ~an~important factor in spectra produced by broadband arc lamp s o ~ r c e s . ~ ~ ~ ~ ~ Implications for Spectroscopy. The spectra of dimer^^'*^^-^' -14 and trimers44a4 of alkali and other metal clusters have been Na explored with considerable success. It was expected that the spectra of larger clusters would be complex and congested with vibrational and rotational effects and thermal broadening. Few data are yet available for larger clusters except for measurements of ionization potential. In addition to the recognition of IP jumps at the spherical closings, new patterns emerge in variations of the I P S for the nonspherical clusters. For example, variations of the I P are seen for the following sequences of four: 15-1 8, 23-26, I and 27-30, 41-44. Although the magnitude of these features is 58 82 smaller than the I P jumps at spherical clusters, they are seen consistently and reflect fine structure of the energy-level system -22 ' ' ' ' 0 10 20 30 40 60 BO 70 BO 80 100 of the clusters. They are also consistent with fourfold patterns Number of Atoms, N in the level structure derived from the Nilsson theory9 for Figure 7. Calculated total energy per atom in the spherical jellium model spheroidally distorted clusters. The case of N = 12 which is for sodium as a function of cluster size. ambiguous in spheroidal theory is resolved in a three-dimensional Clemenger-Nilsson analysisSoand is definitely predicted to be Hiickel molecular orbital (HMO) method; and LiN ( N I13)58 stable relative to its neighbors as is observed. by the generalized valence bond formalism (GVB). The results Furthermore, resolvable peaksMare observed near the thresholds for the equilibrium geometrical structures obtained from miniof certain clusters, that is, those following the spherical clusters mizing the total energies in the different calculations are usually N = 40, 58, and 92. The intensities of these peaks at N = 59, in agreement. For example, it is found in these studies that alkali 60,61, and 62 are linearly proportional to 1:2:3:4 which indicates metal clusters containing fewer than six atoms have planar that every electron at the beginning of a new shell contributes equilibrium structures. identically to the PIE curve. Similar behavior is observed at 93-96 There have also been some calculations of the electronic and indicated at 41-44. Single-particle levels begin to emerge properties for certain larger clusters with fixed high-symmetry even for the smallest clusters N = 3, 4, as pointed out in ref 32 structures; and results are available for Na,59Li,6O and NL61 It where one can also see evidence for partially resolved peaks at is instructive to compare these fixed-structure results with jellium well-defined energies which shift slightly as electrons are added results for the same size cluster. For example, if we focus on the successively to the shells. The shifts of these peaks with cluster results for Na,, and N a I 5clusters,59the most striking feature is electron number gives further evidence for fine structure needing the lack of directional bonding. Charge density plots59illustrate refined theoretical models for interpretation. that the Na valence electrons are not localized around the atoms Theoretical Results but spread out over the cluster. Even though the structures for As discussed previously, because of the complexities involved, Na,, and Na15are taken to be bcc and fcc, the electron charge a b initio theoretical studies of the structural and electronic appears delocalized in the manner expected for a nearly free properties of metal clusters have only been carried out for clusters electron-like system. containing small numbers of atoms. In some cases, only a few Another important comparison is the energy eigenvalues of the optimal atomic arrangements are considered. In particular, alkali Na,, and Nals fixed structures with the corresponding jellium metal clusters have been examined by using various theoretical values. The results appear in Table I which lists the eigenvalues approaches: Na, ( N = 2, ..., 8, and 13)7,55by the pseudopotential and degeneracies. The results for the two comparisons are remethod within the L D A LiN ( N I5)56by the self-consistent-field, markably similar. For the bcc and fcc structures, crystal field molecular-orbital method (SCF-LCAO-MO) within perturbative effects split the tenfold spherical degeneracy of the Id states. configuration interaction; MN ( N I 14, M = Li C S ) by ~ ~the Otherwise, the same sequence of degeneracies appear in the jellium and in the fixed-structure models. Even for the Id levels, the (49) Commonly quoted experimental accuracies**are not better than 0.1 splittings are small. Therefore, for self-consistent jellium calcueV which is inadequate to distinguish easily the largest of the IP jumps at 9, l a t i o n ~ , ~we ~ " expect ~ that the results will be similar both to a 18, and 20.
1.
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(50) Saunders, W. A. Ph.D. Thesis, University of California, Berkeley, 1986. (51) Feldman, D. L.; Lengel, R. K.; Zare, R. N. Chem. Phys. Lett. 1977, 52, 413. (52) Hermann, A.; Leutwyler, S.; Schumacher, E.; Woste, L. Chem. Phys. Lett. 1977, 52, 418. (53) Broyer, M.; Chevalyre, J.; Delacretax, G.; Fayet, P.; Woste, L. Chem. Phys. Lett. 1985, 114, 447. (54) Martins, J . L.; Buttet, J.; Car, R. Phys. Rev. Lett. 1984, 53, 655. (55) Martins, J. L.; Car, R.; Buttet, J. J . Chem. Phys. 1983, 78, 5646. (56) Rao, B. K.; Kharma, S . N.; Jena, P. Solid State Commun.1985,56, 731. Rao, B. K.; Jena, P. Phys. Rev. B 1985, 32, 2058. (57) Wang, Y.; George, T. F.; Lindsay, D. M.; Beri, A. C., to be published; Lindsay, D. M.; Wang, Y.; George, T. M., to be published.
(58) McAdon, M. H.; Goddard 111, W. A. J . Non-Cryst. Solids 1985, 75, 149. (59) Cleland, A. N.; Cohen, M. L. Solid Stale Commun.1985, 55, 35. (60) Redfern, F. R.; Chaney, R. C.; Rudolf, P. G. Phys. Rev. B 1985, 32, 5023. (61) Melius, C. F.; Upton, T. H.; Goddard 111, W. A. Solid State Commun. 1978, 28, 501. (62) Chou, M. Y.; Cleland, A.; Cohen, M. L. Solid State Commun. 1984, 52, 645. (63) Beck, D. E. Solid State Commun.1984, 49, 381. (64) Ekardt, W. Phys. Rev. B 1984, 29, 1558. (65) Chou, M. Y.; Cohen, M. L. Phys. Lett. 1986, 113A, 420.
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The Journal of Physical Chemistry, Vol. 91, No. 12, 1987 I
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rounded square well with depth approximately equal to the sum of the bulk Fermi energy plus the work function and to a full pseudopotential calculation if the pseudopotential is weak. Shell structure on the whole survives the differences in these models and dominates the total energy. The overall model and energylevel scheme resemble those of a system with properties somewhere between those of an atom and a solid. It is the discreteness of the energy levels that gives rise to discontinuous changes in the total energy as a function of the number of atoms N . The splittings are determined by the electron density of the jellium sphere or spheroid. Figure 7 displays the total energy per atom for a N a jellium sphere as a function of N . When the N value corresponds to a cluster with enough electrons to completely fill a level (to close a shell), the total energy per atom usually exhibits a minimum (Figure 7). The gaps in the energy spectrum give the minima and stability at the special values N = 8, 18, 20, 34, 40, 5 8 , and 92. Experimentally, if the clusters in the nozzle region are almost in thermal equilibrium and the density distribution is unchanged during the free expansion and ionization processes (Le., ignoring fragmentation and ionization cross section differences for different clusters), the observed abundances should be closely related to the second derivative of the total energy with respect to N'
where A2(N) 2 E ( N ) - E(N-1) - E ( N + l ) , and E ( N ) and IN are respectively the total electronic energy and the observed abundance intensity for the N-atom cluster. The calculated' A,(N) is shown in Figure 8 for Li, Na, and K where the peak positions a t N = 8, 18, 20, 40, 5 8 , 92 correlate excellently with the discontinuities in the mass spectra (Figures 3 and 4) for N a and K. The peak at 34 in Figure 8 has only a weak apparent counterpart in the experimental mass ~ p e c t r a .Experiment ~ and theory according to eq 10 are compred in Figure 9. The overall agreement between theory and experiment for the mass spectra supports the (66) Nilsson, S. G. K . Dan. Vidensk. Selsk. Mat. Fys. Medd. 1955, 2 9 Gustafson, C.; Lamm, I. L.; Nilsson, B.; Nilsson, S. G. Ark. Fys. 1967, 36, 613.
, , , , ( , , , , , m _ . , , ,
~
' " ~of the ' fjellium ' ~ approximation ' ~ " ~ and ~ 'the~assertion ~ ' that ' ' jellium ~ ' ~ validity gives a satisfactory physical picture for alkali metal clusters. The effects of ellipsoidal distortions were mentioned earlier. The application of Nilsson theoryg represents so far only a first approximation, but symmetry arguments and correspondence with experiment suggest that further extension of the jellium model to include a self-consistent treatment of the distortions would be a valuable theoretical advance. The criteria for deciding the range of applicability of the jellium model to clusters are fairly straightforward. Since the jellium model is based on the assumption that the valence electrons are delocalized and weakly interacting with the ionic potentials, it is applicable to systems in which this assumption is satisfied. For example, it is not expected to be a good approximation for clusters containing only a few atoms. In these cases, the atomic arrangements are important in determining the cluster properties. Furthermore, the smearing of the ionic cores is a better approximation for singly charged cores than for multiply charged cores because the latter have larger Madelung sums which scale as the square of the charge. With these points in mind, one does not expect that the jellium model would be highly accurate for predicting the properties of small A1 clusters. It is found67that the orbital structure of AIN ( N = 2-6) with equilibrium atomic configurations displays some differences compared with results65 for an analogous jellium model. Differences of this kind are expected and are not an indication that the jellium model is inappropriate for larger clusters. The alkali metal clusters are among the best candidates for applications of the jellium model. The results of a recent calculation6*support some of the statements made earlier. In this study, the variation of the structural part of the total energy of alkali metal clusters is shown to be smaller than the variation arising from the filling of discrete single-electron energy levels, especially for N > 10. Hence, on the basis of a variety of theoretical calculations, it is concluded that in free-electron-like clusters the origin of the discontinuities in the mass spectra is not related to the geometric packing pattern but to the electronic structure. The merit of the jellium model is that it gives a sound description of the observed shell structure and serves as a starting point for extensions and improvements to explain more detailed properties. In nonideal clusters, crystal field effects will split the highly degenerate levels found for clusters with perfect spherical symmetry. If the pseudopotential is weak, the splittings are small compared with the
(67) Upton, T.H . Phys. Rev. Lett. 1986, 56, 2168. (68) Manninen, M. Solid State Commun. 1986, 59, 281.
~
~
~
The Journal of Physical Chemistry, Vol. 91, No. 12, 1987 3149
Feature Article gaps between different shells as in the case of Na13and Na15.59 Instead of forming a shell composed of a single level for crystalline examples, the shells are sometimes composed of groups of leve l ~ , ~and ~@ these energy levels are filled in ascending order by electrons. The experimental finding that clusters with even numbers of electrons are, in general, more abundant than those with odd numbers of electrons indicates that there is a tendency to fill these energy levels with pairs of electrons of opposite spin, and the spin-polarization energy is smaller than the level splittings. In addition, the consideration of the crystal field effect is expected to decrease the discrepancies found between the observed ionization potentials and values calculated from the spherical jellium model. The calculated are larger than the experimental values by 15% and tend to overestimate the drop at shell closing numbers because of the neglect of effects such as cluster deformations. In the study of metal surfaces using the jellium approximation, it was found6 that including the effects of the lattice pseudopotentials as perturbations reduces the calculated work functions by 10% in better agreement with experiment. The same trend is expected for the cluster case. In the spherical jellium model, the large drops in the ionization potential occur only at major shell closing numbers. The inclusion of crystal field effects may give rise to some subshell structure and reduce the drop, but a t this point, no estimate for the size of the reduction is available. However, encouraging results have been found when the discontinuous variations of the ionization thresholds for K clusters are compared with the highest last occupied energy levels calculated with the spheroidally deformed one-particle potential discussed above. The shape of the variations as a function of cluster size agrees very well. One point worth emphasizing is that even when accurate pseudopotentials are used self-consistently the standard calculations are still based on the LDA, and in the case of calculations for atoms, this approximation often overestimates the ionization potentials by -0.1-0.3 eV.69 In addition, care must be exercised in studying excited-state
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(69) Gunnarsson, 0.J . Appl. Phys. 1978.49, 1399.
properties using the LDA since density functional theory is not designed to deal with excitation^.^^ The self-consistent jellium model has also been used to evaluate the static polarizability within the linear-response ~ c h e m e . ~ l - ~ ~ Theoretical values are typically too small by 15-20%.1 It has been suggested74that an approximate increase of 10% can be obtained by including the lattice structure with pseudopotentials. The size-dependence dynamic properties, e.g., photoabsorption and photoemission cross sections, are also e v a l ~ a t e d within ~~~~* the jellium framework. Sum rules, collective multipole excitation, and inelastic electron scattering for these metal clusters have also been s t ~ d i e d . ~ Specific ~ . ~ ~ predictions are available; further experimental work is necessary to test them. Recent photoabsorption experiment^^^ are consistent with the predicted split surface plasma resonance peak for spheroidal clusters and verify in detail cluster-to-cluster variations in absorption cross section. Although the theoretical picture will be improved by imposing self-consistency on the calculations, the simple modified spheroidal harmonic oscillator model successfully predicts a large number of experimentally observed facts.
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Acknowledgment. This work (M.L.C.) was supported by National Science Foundation Grant No. DMR8319024 and by the Director, Office of Energy Research, Office of Basic Energy Sciences, Materials Sciences Division of the US.Department of Energy, under Contract No. DE-AC03-76SF00098, and (W.D.K.) by the National Science Foundation Grant DMR8417823. (70) Schliiter, M.; Sham, L. J. Phys. Today 1982, 36. (71) Ekardt, W. Phys. Rev. Lett. 1984,52, 1925; Phys. Rev. B 1985,31, 6360. (72) Puska, M. J.; Niemenen, R. M.; Manninen, M. Phys. Rev. B 1985, 31, 3486. (73) Beck, D. E. Phys. Rev. B 1984, 30, 6935. (74) Ekardt, W.; Penzar, 2. Solid State Commun. 1986, 57, 661. (75) Bertsch, G.; Ekardt, W. Phys. Rev. B 1985, 32, 7659. (76) Ekardt, W. Phys. Rev. B 1985, 32, 1961; 1986, 33, 8803. (77) de Heer, W.; Selby, K.; Kresin, V.; Masui, J.; Knight, W. D. Bull. Am. Phys. SOC.1987, 32, 484.
