Metal Ciuster-Rare Gas van der Waals Complexes - American

and Ni,Kr, species. These vdW species may be considered ex- amples of metal-rare gas physisorption on a microscopic scale. An important phenomenon whi...
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J. Phys. Chem. 1992, 96, 661 1-6616

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Metal Ciuster-Rare Gas van der Waals Complexes: Physisorption on a Microscopic Scale Mark B. Knickelbein* and Warren J. C. Menezes Chemistry Division, Argonne National Laboratory, Argonne, Illinois 60439 (Received: March 16, 1992; In Final Form: May 4, 1992)

Van der Waals complexes consisting of argon or krypton atoms bound to clusters of iron, cobalt, nickel, and niobium atoms have been generated by supersonic expansion from a liquid nitrogen-cooled laser vaporization source. Laser photoionization time-of-flight mass spectrometry reveals significant nonmonotonic variations in the efficiency of formation of these species as a function of metal cluster size. A correlation between these variations and the size-dependent reactivities of the bare metal clusters toward molecular hydrogen is observed. This correlation suggests that those metal cluster surface sites which readily support the binding of rare gas atoms are also highly reactive sites in direct cluster-hydrogen encounters. Alternatively, this correlation may imply that these cluster-hydrogen reactions occur by a two-step mechanism involving the production of a physisorbed M,--H, intermediate. Analysis of the post-threshold behavior in the photoionization efficiency spectra of niobium cluster-argon van der Waals species (Nbkr,) suggests that argon atoms bind with different efficiencies to each of the structural isomers of Nb,, Nb9, and Nbll.

Introduction The large, sizedependent variations observed in the chemical14 and physica11*12 properties of small, unsupported transition-metal clusters (M,) have prompted much speculation as to their source. The size dependence of cluster geometric structure is often sited as a primary source of the large variations observed in chemical r ~ a c t i v i t y . 'On ~ ~the ~ other hand, an ionization potential-reactivity anticorrelationI3-I5has been noted for reactions of several transition-metal clusters with small molecules such as hydrogen and nitrogen, implicating the importance of electronic structure, in this case as measured by variations in cluster HOMO energy. Clearly, electronic and geometric structures are intimately related and their individual effects difficult to distinguish in the analysis of metal cluster physiochemical properties. In this paper, we present several examples of transition-metal cluster-rare gas van der Waals (vdW) species M,R, with n > m, in which significant cluster-size-dependent variations in the extent of formation are observed. A preliminary account of this work, in which we discuss Nb,,Ar, and Fe,Kr species, has been published.16 In studies of iron cluster magnetic properties, Milani and deHeer" exploited the extent of Fe&, formation as a rough indicator of iron cluster temperature and also noted sizedependent variations in argon complex formation efficiency. In the present paper, we present a more quantitative analysis of the cluster size dependence for the formation of these species and extend our study to include Fe&, Co&, Corn,,,, Ni&,, and Ni,Kr, species. These vdW species may be considered examples of metal-rare gas physisorption on a microscopic scale. An important phenomenon which has emerged in the study of physisorption on high-index transition metal surfaces is the preferential adsorption of rare gas atoms onto surface defects (steps) rather than onto the topologically smooth terraces, the latter occurring with measurably lower adsorption energies.1*-20We argue by analogy that the variations in the formation of M,R, species with cluster size reflect changes in the relative depth of the M,-R potentials as the nature of the adsorption sites changes with cluster geometry. Because the metal cluster-rare gas van der Waals bond is relatively weak, the binding of rare gas atoms can be considered a nonperturbative probe primarily sensitive to cluster geometric structure. ExperiwnW Metbods

Because details of the molecular beam apparatus have been published previously,I0 only a brief description will be given here. Metal clusters are generated by laser vaporization of atoms from a cylindrical rod of the corresponding pure metal. Targets of 0022-3654/92/2096-6611%03.00/0

isotopically pure %Fe,58Ni,and W i were fabricated from material obtained from Oak Ridge National Laboratory. Metal clusters grow and cool via collisions with helium (15-25 Torr) within a liquid nitrogen-cooled cylindrical flow tube 7.6 cm X 0.32 cm (length X inner diameter). After residing in the flow tube approximately 1 ms, the clusters expand into vacuum through a 0.13-cm-diameter cylindrical orifice at the end of the flow tube which opens into a 15' (half-angle) conical nozzle, where there is further cooling in the supersonic expansion. Cluster-rare gas vdW species are generated by adding the corresponding rare gas (O.l-5% Kr, 3-15% Ar) to the helium flow stream and are only observed when the flow tube is cooled with liquid nitrogen. The expanding free jet is collimated by a skimmer, forming a molecular beam, which passes through a second differentially pumped vacuum stage and into a detection chamber. A photoionization laser intersects the molecular beam at 90° within the ion extraction region of a time-of-flight (TOF) mass spectrometer. The photoionization laser consists of either an ArF excimer laser (193 nm) or a frequency-doubled dye laser (200-250 nm). The mass-separated cluster photoions are detected by a microchannel plate, the output of which is digitized using an oscilloscope equipped with a histogramming memory. An averaged mass spectrum consisting of the sum of 1000-5000 individual TOF spectra is then transferred to a microcomputer for analysis. Photoionization efficiency (PIE) spectra are generated by recording TOF mass peak areas as a function of dye laser wavelength at 0.25-nm wavelength intervals and normalizing for ionization laser fluence. The fluence is kept below 200 pJ cm-2 to minimize multiphoton effects. Multiple wavelength scans are added to improve signal-to-noise ratios in the PIE spectra.

Results and Discussion A. Product Distributions. We observe that the propensity of metal clusters to form vdW complexes varies with the identity of the metal as Fen < Co, < Ni, < Nb, based on TOF spectra recorded under similar gas flow conditions. Time-of-flight spectra of Co,/Co&, and Wi,/60Ni,,Arm species are shown in Figures 1 and 2, respectively. An examination of the TOF mass spectra reveals that the extent to which rare gas atoms add to the metal clusters M, in the expansion varies nonmonotonically as a function of cluster size. Such variations can be observed in TOF spectra as modulation in the relative abundance of Co,,Ar, and Niar,,, species, seen as the smaller mass peaks interspersed among the bare cluster peaks in Figures 1 and 2. Similar behavior is displayed by Fenand Nb,.16 To quantify these variations, we have plotted 0 1992 American Chemical Society

Knickelbein and Menezes

6612 The Journal of Physical Chemistry, Vol. 96, No. 16, 1992

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ud time-of-fIight Figure 1. Photoionization time-of-flight mass spectrum of Conand Co,,Ar, recorded at X = 211.6 nm (5.85 eV). X,, = 0.10.

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ti me-of-f Iig - ht and 60NinArmrecorded at 211.6 nm (5.85 eV). X,, = 0.05. Because 2M(@"i) = 3M('"Ar), the @"in mass peaks are overlapped by those of @"iF2Ar3;60Ni,Ar peaks are overlapped by those of 60Ni,2Ar4, etc. Figure 2. Photoionization time-of-flight mass spectrum of "in

The Journal of Physical Chemistry, Vol. 96, No. 16, 1992 6613

Metal Cluster-Rare Gas van der Waals Complexes

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Figure 4. Formation fraction, Fdw(n),for Co,,Ar, and Conk, van der Waals species and relative reactivity of Contoward D2(ref 1). Uncertaintics in Fdw(n) are *lo%. Experimental conditions: Co,,Ar,, X, = 0.09; ConKrmr XK,= 0.04;X = 193 nm (6.4eV). Co, and Cosspecies possess ionization potentials greater than 6.4 eV.

the hctional mversion Fdw(n) of the metal clusters M, to M,,Rm vdW species, with Fvdw(n)defined as

where A,,,,, are the integrated TOF mass peak areas for the observed M a mspecies for a given n, with 1 I m, I 5 , depending on the extent of complex formation. The results for Fe,Rm (R = Ar, Kr), Co,,R,,, (R = Ar, Kr), Ni,Rm (R = Ar, Kr) m d Nb,,Ar,,, are shown in Figures 3-6, with the mole fraction of rare gas Ar (X,) or Kr (X,) in the carrier gas given in the captions. The estimated errors in the Fvdw(n)values are f10% based on repeated determinations under identical conditions. Also shown in Fiies 3-6 are the relative rate coefficientsfor reaction of these metal clusters with molecular hydrogen (vide infra). The analysis of product distributions above relies on the assumption that the intensities of the M a mTOF mass peaks reflect the distribution of neutral species prior to photoionization. However, differences in the ionization efficiencies of the various M a mspecies can cause the observed distributions of ions to differ from those of the neutrals. Although we have no information on the relative photoionization efficiencies of these species, we will assume that they are not significantly different. Indirect support for this assumption comes from photoionization spectroscopic

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Formation fraction, Fydw(n),for Nb,,Ar, van der Waals species and relative reactivity of Nb, toward D2(ref 4). Uncertainties in Fdw(n) are *lo%. Experimental conditions: XAr= 0.05, X = 193 nm (6.4eV). F i i 6.

studies of Nb,,Arm species (see section C). Unimolecular loss of rare gas atoms following photoionization may also lead to differences between the actual distribution of neutral Ma,,, species and that inferred from the TOF mass spectra,leading to apparent Fdwvalues which are lower than the actual values for the neutral species. It has been shownl0J1that, at the laser fluences employed in this study, multiphoton ionization/fragmentation of these transition-metal clusters is negligible compared to one-photon ionization; however, because the present photoionization experiments were conducted at energies ranging from 0.2 to 0.8 eV above the metal cluster ionization thresholds,lO.'l dissociative ionization is energetically possible even for the onephoton procas. Although the M,-Ar and M,-Kr binding energies are not known, first-order estimates from Ar and Kr physisorption onto extended transition-metal surfaces2' place them in the 0.03-0.10-eV range. indicate that Statistical unimolecular (RRKM) rate calc~lations~~ M,-R fission reactions for the metal clusters in the size range studied here occur faster than lo-* s even for excitation energies as low as 0.1 eV above the M,-R dissociation threshold. The predicted unimolecular lifetimes decrease rapidly with decreasing cluster size and with increasing excitation energy. From these estimates of unimolecular lifetime, we conclude that dissociative ionization leading to rare gas atom loss is essentially complete for all sizes on the time scale of ion extraction in the mass

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6614 The Journal of Physical Chemistry, Vol. 96, No. 16, 1992

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Figure 7. Photoionization efficiency spectra for Nb,Ar, species (m = 0-2). X, = 0.09 except for top panel where pure helium carrier gas was used. Vertical I P assignments are shown by arrows. The IP assignment obtained using helium carrier only (top panel) is reproduced in the second panel to show the ionization threshold shift with argon present in the expansion.

spectrometer (- lod s). The extent to which dissociative ionizaGon distorts the apparent product distribution (and hence Fdw(n)) will vary with the available energy above the dissociation threshold and is thus predicted to be largest for those clusters with low ionization potentials. We have measured Fdw(n) for Co,,Ar, at 211.6 nm (5.85 eV) and found it to be virtually identical to that measured at 6.4 eV shown in Figure 4, however. Furthermore, there is no obvious correlation between local extrema in Fydw(n) and the ionization potentials of the corresponding metal clusters, Mn.loJ1 We conclude that although the FvdW values may be influenced by dissociative ionization, the effects are not strongly size dependent and that Fvdw(n)provides a reasonably accurate measure of the size-dependent variations in the formation of the neutral rare gas van der Waals species. On the basis of general collision cross section arguments, we would expect a priori that, for metal clusters differing only in physical dimension, M,R, formation efficiency should increase monotonically as a function of size. We believe that the variations in Fydw(n)with cluster size n are due to differences in surface topology (Le., the relative "smoothness" or "roughness") of the metal clusters. Variations in cluster surface structure would in turn result in cluster sizedependentM,-R binding energies. Thus, those clusters which do not form vdW complexes easily may pcasess tightly packed geometric structures, whereas those clusters which form complexes with high probability may have surfaces with "steps", exposed atoms, or other imperfections to which a rare gas atom can more easily bind. While, in general, metal clusters smaller than 7-8 atoms form rare gas complexes with ~ comlower probability than larger clusters, Ni3-Sand C Oform plexes readily, as shown in Figures 3 and 4, respectively. A chemical probe study' has provided evidence that bare nickel clusters undergo structural changes beginning at 37, 41, and 49 atoms, in correspondence with the sudden changes in the relative populations of N i k r , species commencing at these same cluster

photon energy (eV) Figure 8. Photoionization efficiency spectra with vertical IP assignments

for NbsAr, species (m = 0-2).

sizes, as seen in Figure 2. While the exact nature of these structures is not known, they clearly must possess different Nin-Ar potentials. The selective binding of rare gas atoms to defects on macroscopic metal surfaces is now well established. Studies of rare gas adsorption on high-index surfaces of single-crystal ruthenium1* and palladiumIg indicate that xenon atoms bind with higher energies to the step defect sites than to terrace sites and thus bind selectively to step sites in the limit of low coverage. More recently, similar results have been obtained for xenon adsorption on stepped, single-crystal platinum surfaces.20 We believe that such studies of adsorption on single-crystal metal surfaces provide additional support for our assertion that the observed size dependence in the probability of these transition-metal clusters to form rare gas vdW species can be traced to variations in their geometric structures. B. Fvdw vs Hydrogen Reactivity. The relative rate coefficients kD for reactions of small clusters of iron? cobalt,' n i ~ k e l and ,~ niobium1atoms with molecular deuterium are plotted along with Fvdw(n)in Figures 3-6. While a correlation of D2reactivity and Fvdw(n) exists for Fen, Con,and Ni, clusters, Nb, species show no such relationship. Although this correlation is not perfect for Fe, Co,, and Ni, species (e.g., see C013-18 and NizSm), it is clear that, in general, those clusters which form rare gas complexes most efficiently also possess the highest reactivity, and vice versa. This correlation suggests that for these M, the surface sites which most readily support the binding of a rare gas atom may be the most reactive sites for a direct (i.e., concerted) reaction with hydrogen. Another possibility is that these M, + H2 reactions proceed through a short-lived intermediite M,-H2 van der Waals species, in which the hydrogen molecule is weakly bound prior to undergoing dissociative "chemisorption": M, + H2 Mn*.*H2 (2) Mn.*.H2 -+ M,HH

(3)

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The Journal of Physical Chemistry, Vol. 96, No. 16, 1992 6615

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Figure 9. Photoionization efficiency spectra with vertical IP assignments for NbllAr, species (m = 0-3).

In this twestep mechanism, the overall reaction rate is predicted to be proportional to the steady-state concentration of the M,--H2 intermediate, so that those clusters which readily form such intermediates would display higher reactivities than those which do not. We believe it is reasonable to expect that those clusters which most easily form rare gas complexes will also form M,-H2 van der Waals complexes with the greatest probability. Experimental support for this mechanism is emerging from temperature-dependent studies of the rates of reactions of Ni, with hyd r ~ g e n .Theoretical ~~ support for this mechanism is provided by the dynamical simulations of Jellinek and GUvenp of the Ni13+ D2 reaction, in which physisorbed precursors are found to play an important role.24 Moreover, these simulations indicate that the extent to which this two-step mechanism competes with the direct reaction is dependent on the isomeric form of Ni13,i.e., on cluster geometric structure. The observation that this correlation is not perfect for the fmt-row metals and is absent entirely for Nb, indicates that other factors such as cluster electronic structure cannot be neglected. The importance of cluster electronic structure has been inferred from an anticorrelation of Fen, V,, and Nb, cluster ionization potential (IP) with their corresponding hydrogen reactivity and has given rise to a cluster HOMO to hydrogen LUMO electron-transfer model.I3-l5 However, both the ‘geometric” and “electronic” models of cluster reactivity can be accommodated by assuming that those clusters with surface sites supporting enhanced rare gas atom binding efficiency (and displaying high reactivity toward hydrogen) also tend to have lower ionization potentials, in analogy with the lower work functions observed for

the stepped or corrugated faces of single-crystal metals as compared to smooth (low index) surfaces.25 Despite the desirability of distinguishing the effects of cluster geometric and electronic structures, however, they are not separable properties but rather are intimately intertwined. C. Niobium Cluster Isomers: PIE Spectra. Photoionization efficiency (PIE) spectra recorded for selected Nb&, species are shown in Figures 7-1 1. Spectra for NbgAr, and Nb2&, have been presented previously.16 The shift in Nb,,Ar, ionization potentials to lower energies with increasing m has been shownI6 to be in reasonable accord with a simple charge-induced dipole model which treat M,,Ar, complexes as classical metal spheres (MJsurrounded by polarizable point bodies (Ar).Here, we wish to focus on the shapes of the PIE spectra of selected Nb, and Nb,,Ar, species. Because the presence of a small number of adsorbed argon atoms can be reasonably expected to be a relatively weak perturbation the electronic structure of a polyatomic niobium cluster, we would expect that the spectroscopic properties of a N b k r , complex to be quite similar to that of the corresponding bare cluster, Nb,. As shown in Figures 8, 10, and 11 for NbsArm, Nb12Ar,, and Nb17Arm,respectively, aside from slight downward shifts in the ionization thresholds, the shapes of the near-threshold PIE spectra of clusters of a given n are qualitatively similar for m = 0-3. This behavior is typical for most clusters in the size range investigated, Nb7-NbZO,and suggests that the presence of a small number or argon atoms does not significantly affect the photoionization efficiencies of the Nb, species. A slight “turnover” is observed in the photoionization efficiencies of Nb,Ar,, Nb8Arm, and N b l & , , , starting at -0.2 eV above the ionization thresholds, and may indicate the onset of dissociative ionization. For Nbll we observe significant changes in the relative intensity of the threshold “steps” which characterize the PIE spectra of this species upon formation of the corresponding argon vdW complexes as shown in Figure 9. Similar behavior is observed in the PIE spectra of N b specieS.l6 In addition, we observe that the threshold behaviors in the PIE spectra of bare Nb,, Nb, (ref 16), and Nbll

6616 The Journal of Physical Chemistry, Vol. 96, No. 16, 1992

Knickelbein and Menezes observed in previous chemical kinetic indicating that these isomers have similar reactivities toward Hzand N,. Interestingly, NbI2,which has previously been shown" to be formed under these experimental conditions as (at least) two isomers with measurably different IPS, does not display the large changes in near-threshold photoionization behavior observed for Nb7Ar,, NbgAr,, and NbllAr,, suggesting that the isomers of Nblz form rare gas vdW adducts with similar propensity.

Conclusions

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The size-dependent variations in the formation efficiency of transition-metal cluster-rare gas van der Waals species reflect the differences in topology of the underlying metal cluster. The and hydrogen reactivity implies strong correlation between Fvdw that the dramatic variations in the rate coefficients kH2and kD, observed for transition-metalclusters are strongly tied to cluster geometric structure via their surface topology. The fact that the correlation is not perfect indicates, however, that geometry is only part of the story and that cluster electronic properties must also be considered. Acknowledgment. We thank Julius Jellinek, Eric Parks, and Steve Riley for informative discussions and for their continued interest in this work. This work is supported by the U S . Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, under Contract W-31-109-ENG-38.

References and Notes

photon energy (eV)

Figure 11. Photoionization efficiency spectra with vertical IP assignments for Nb17Armspecies (m = 0-2).

recorded with argon present in the carrier gas (Le., recorded simultaneouslywith Nb,,Arm species) are quite different from those recorded with helium only, a result not observed for other clusters in this size range. Interestingly, for Nb,, we observe an apparent increase in the ionization threshold with argon present in the carrier gas, as shown in Figure 7. We believe these effects can be traced to the presence of two (or more) structural isomers for these species. Gas-phase kinetics studies4,*using methods of metal cluster production similar to those employed here have produced results suggesting that Nb9, Nbll, and Nb12are formed in at least two isomeric forms. The PIE spectra (and ionization potentials) for the different isomers of a cluster will generally be different, as shown for Nb9 and Nb12 (ref 11). The PIE spectra of those niobium clusters produced as multiple isomers as well as the PIE spectra of their argon vdW adducts are thus composites, consisting of contributions from each isomer. However, because different isomers will in general form vdW complexes with different propensities, it can be expected that the ratio of isomers for a given cluster size n will change with m as each isomer binds argon atoms to a different extent. This effect is illustrated in the PIE spectra for NbllAr, ( m = 0-3), shown in Figure 9, in which the nearthreshold "step" is observed to change in relative intensity with increasing m, reflecting the changing ratios of Nbl isomers. In addition, it is observed that the PIE spectrum of bare Nbll recorded with pure helium a m e r gas possesses a different threshold shape than the spectrum recorded with argon present in the expansion. Similar changes in PIE spectral shapes are observed16 for N W , . An apparent increase in the ionization threshold for bare Nb7 recorded with argon present in the expansion (Figure 7) also points to the existence of two (or more) isomers for this species as well, a high IP form which is "inert" toward vdW complex formation and a low IP "reactive" form which readily forms Nb,Ar,. Thus, in the PIE spectrum of Nb, recorded with argon present in the expansion, the more "reactive" form was titrated away forming WAr,, leaving the "inert" isomer as the remaining bare cluster. Evidence for the existence of multiple isomers of Nb7 was not

(3) b l d o r , A.; Cox, D. M. J. Chem. Soc., Faraday Trans. 1990,86,2459. (4) Zakin, M. R.; Brickman, R. 0.;Cox, D. M.; Kaldor, A. J. Chem. Phys. 1988,88, 3555. (5) Hoffman 111, W. F.; Parks, E. K.; Nieman, G. C.; Pobo, L. G.; Riley, S . J. 2.Phys. D 1987, 7 , 83. (6) Hoffman 111, W. F.; Parks, E. K.; Riley, S.J. J. Chem. Phys. 1989, 90, 1526. (7) Parks, E. K.; Winter, B. J.; Klots, T. D.; Riley, S . J. J. Chem. Phys. 1991, 94, 1882. (8) (a) Hamrick, Y.; Taylor, S . ; Lemire, G . W.; Fu, Z.-W.; Shui, J.-C.; Morse, M. D. J. Chem. Phys. 1988,88,4095. (b) Hamrick, Y. M.; Morse, M. D. J. Phys. Chem. 1989, 93, 6494. (9) El-Sayed, M. A. J. Phys. Chem. 1991, 95, 3898. (10) (a) Knickelbein, M. B.;Yang, S.;Riley, S . J. J. Chem. Phys. 1990, 93, 94. (b) Yang, S.;Knickelbein, M. B. J. Chem. Phys. 1990, 93, 1533. (11) Knickelbein, M. B.; Yang, S . J. Chem. Phys. 1990, 93, 5760. (12) (a) Gantefir, G.; Gausa, M.; Meiwes-Brcer, K.-H.; Lutz, H. 0. Faraday Discuss. Chem. Soc. 1988,86, 197. (b) Seidl, M.; Meiwes-Broer, K.-H.; Brack, M. J. Chem. Phys. 1991, 95, 1295. (13) Whetten, R. L.; Cox, D. M.; Trevor, D. J.; Kaldor, A. Phys. Reo. Lett. 198s. 54. 1494. (14) Whetten, R. L.; Zakin, M. R.; Cox, D. M.; Trevor, D. J.; Kaldor, A. J. Chem. Phys. 1986,85, 1697. (1 5) Cox, D. M.;Whetten, R. L.; Zakin, M. R.; Trevor,D. J.; Reichmann, K. C.; Kaldor. A. In Advances in Loser Science: Stwallev. W.C., Lam. M.. as.; AIP Conference Proceedings No. 146; American institute of Physics: New York, 1986; Vol. I, p 527. (16) Knickelbein, M. B.; Menezes, W. J. C. Chem. Phys. Lett. 1991, 184, 433. (17) Milani, P.; de Heer, W. A. Phys. Reu. E 1991, 44, 8346. (18) Wandelt, K.; Hulse, J.; Kuppers, J. Surf. Sci. 1981, 104, 212. (19) Miranda, R.; Daiser, S.;Wandelt, K.; Ertl, G. Surf. Sci. 1983, 131, 61. (20) Siddiqui, H. R.;Chen, P. J.; Guo, X.;Yaks, J. T. J. Chem. Phys. 1990, 92, 7690. (21) Vidali, G.; Ihm, G.; Kim, H.-Y.; Cole, M. W. Sur/. Sci. Rep. 1991, 12, 133. (22) Menezes, W. J. C.; Knickelbein, M. B. To be published. (23) Parks, E. K.; Zhu, L.; Ho, J.; Riley, S. J. Unpublished results. (24). (a) Jellinek, J.; GIiveng, Z. B. In Mode Selective Chemistry; Jortner, J., Levine, R. D.,Pullman, JB., Eds.; Kluwer Academic Publishers: Dordrccht, Holland, 1991; pp 153-164. (b) Jellinek, J.; Giiveng, 2. B. In Physics and Chemistry of Finite Systems: From Clusters to Crystals; Jena, P., Khanna, S.N., Rao, 8 . K., Eds.; Kluwer Academic Publishers: Dordrecht, Holland, in press. (c) Jellinek, J.; Giiveng, Z. B. In Nuclear Physics Concepts in Atomic Cluster Physics (Lecture Notes in Physics); Lutz, H. O., Schmidt, R., Eds.; Springer-Verlag: Heidelberg, Germany, in press. (25) Smoluchowski, R. Phys. Rev. 1941, 60, 661.