ionization of copper oxide

ionization of copper oxide

International Journal of Mass Spectrometry and Ion Processes, 102 (1990) 115-132 115 Elsevier Science Publishers B.V.. Amsterdam COLLISION-INDUCED ...

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International Journal of Mass Spectrometry and Ion Processes, 102 (1990) 115-132

115

Elsevier Science Publishers B.V.. Amsterdam

COLLISION-INDUCED DISSOCIATION OF POSITIVE AND NEGATIVE COPPER OXIDE CLUSTER IONS GENERATED BY DIRECT LASER DESORPTION/IONIZATION OF COPPER OXIDE

JAMES R. GORD,* RAYMOND

J. BEMISH and BEN S. FREISER**

Herbert C. Brown Laboratory qf Chemistry, Purdue University, West Lqfayette, IN 47907 (U.S.A.)

(First received 21 November 1989; in final form 9 March 1990)

ABSTRACT Positive and negative cluster ions Cu,O:‘- (positive ions: n = 1-14, m = O-8; negative ions: n = l-21, m = O-12) have been generated by direct laser desorption/ionization from a pellet of copper oxide and studied using Fourier transform ion cyclotron resonance mass spectrometry (FT-ICR-MS). Energy-resolved collision-induced dissociation studies of these ions demonstrate interesting trends that may have implications for copper oxide cluster ion structures and stabilities. INTRODUCTION

This special issue of International Journal of Mass Spectrometry and Ion Processes is a testament to the scientific community’s ever-growing interest in the chemistry and physics of cluster species. As efforts devoted to cluster research continue to increase, we come ever closer to joining the regimes of small polynuclear species, which are amenable to study through gas-phase techniques and high-level theoretical methods, and condensed phases. Surely, bridging this gap is a primary motivator for the continued growth in cluster studies. In addition, a host of associated fields stand to benefit from the wealth of information being uncovered during the characterization of cluster species. Heterogeneous catalysis, atmospheric and interstellar chemistry, the chemistry of flames, aggregation phenomena, corrosion chemistry, and microelectronics are all integrally tied to a thorough understanding of the size-dependent properties of clusters. Several recent review articles document the depth and breadth of cluster studies currently under way [l-4]. *American Chemical Society Analytical Division Fellow. Present address: Joint Institute for Laboratory Astrophysics, University of Colorado and National Institute of Standards and Technology, Boulder, CO 80309, U.S.A. **TO whom correspondence should be addressed. 0168-1176/90/$03.50

0 1990 Elsevier Science Publishers B.V.

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The intense interest in this area has fueled the application of a host of experimental methodologies to the formation of clusters and the elucidation of their properties. Clusters and cluster ions have been successfully generated over a range of sizes using supersonic expansion sources, often coupled with laser desorption [5-91. A cold-cathode d.c. discharge source has been employed to prepare transition-metal anion clusters [ 10,111. Powerful spectroscopic techniques, including photoionization, photodissociation, and electron photodetachment, have been used to study clusters formed using these sources. In addition, the spectroscopy of clusters has been studied through matrix isolation methods [12]. Sputtering using fast ion and atom beams has also been a very successful approach to the generation of metal cluster species. The reactivity and collision-induced dissociation of these species are often studied using conventional mass spectrometers, guided ion beam apparatus, and flow/drift tubes [ 13-251. Clustering ion-molecule reactions of volatile metal carbonyl compounds have also been employed to form metal cluster ions [26-291. Finally, direct laser desorption/ionization (LDI) has also been demonstrated as a viable approach to the formation of cluster ions [30-321. Fourier transform ion cyclotron resonance mass spectrometry (FT-ICRMS) has emerged as an ideal approach to the study of cluster ions, and several of the sources mentioned above have been coupled with FT-ICR-MS instrumentation. Size- and composition-dependent studies of cluster ions are facilitated by the multistage mass spectrometric capability of FT-ICR-MS in addition to the technique’s ultrahigh mass resolution, large mass range, and ion trapping capability. Smalley and co-workers have combined the advantages of supersonic beam cluster ion generation and FT-ICR-MS detection in an instrument which has been used to study large clusters of niobium, silicon, and carbon [33-361. A similar source is currently under construction in our laboratory. Generation of metal clusters by direct laser desorption/ionization is particularly suited to FT-ICR-MS. Moini and Eyler have used LDI of pure foils of copper, silver, and gold to prepare small positive and negative cluster ions of these species (n = l-6 in the case of Au,+) [37,38]. Au; (n = l-3) were generated by Weil and Wilkins through LDI of Au0 [39]. Freiser and coworkers prepared Ag,’ (n = l-l 1) and Zn: by LDI of the respective oxides [40] as well as small noble metal cluster ions by LDI of “activated” pure metal surfaces [41]. Cassady and McElvaney used LDI of cobalt oxides and molybdenum oxides to form pure metal and metal oxide clusters [42]. In addition, Irion and Selinger have employed a SIMS sputtering source coupled to an FT-ICR-MS to form and study CUT (n = 1-41) [43]. In this paper, we extend our original work involving the direct LDI of AgO and ZnO to include the formation and study of positive and negative copper oxide cluster ions prepared by LDI of CuO. Extensive energy-resolved collision-induced dissociation studies of these species, especially the negative ion

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clusters, have uncovered some interesting characteristics of these clusters with implications for ion structure and stability. These data are discussed in terms of qualitative electronic shell models and compared and contrasted with collision-induced dissociation of other transition-metal oxide cluster ions as well as electron affinity measurements for size-selected copper clusters. EXPERIMENTAL

All experiments were performed using a Nicolet FTMS-2000 Fourier transform ion cyclotron resonance mass spectrometer. The details of FT-ICR-MS operation and experimental methodologies have been previously described 144-481. Although the FTMS-2000 is typically equipped with dual cubic trapping cells, our instrument is currently configured with a single rectangular trapping cell measuring 4.8cm x 4.8cm x 9.6cm. The cell is situated in the bore of a 3T superconducting magnet with the long dimension of the cell parallel to the magnetic field lines. Electrostatic trapping potentials were maintained at + 2 V for the study of positively and negatively charged cluster ions, respectively. Ion signals were collected and digitized in the mediumresolution mode, using 1 MHz detection bandwidth and 16 K, 32 K, or 64 K data points. These conditions proved to be sufficient to resolve isotopic progressions in all but the most massive cluster ions observed. Copper oxide pellets were prepared from commercially available powder (Aldrich) using a Parr pellet press. Our previous experience with metal oxide pellets formed in this manner has shown them to be quite delicate, so the copper oxide pellets were sintered for a week at 900°C, producing a more tenacious pellet. These pellets were affixed to the end of a solids probe using double-sided sticky tape and introduced to the vacuum chamber to a point just outside the cell. Base pressures with the solids probe in place were approximately lo-* Torr. Laser desorption/ionization was achieved using a Quanta-Ray Nd:YAG laser [49]. The fundamental output (A = 1064nm) of the laser was focussed using a 1 m focal length lens and passed through a quartz window on the external ion source affixed to the analyzer end of our FTMS-2000. Focussing was required to guide the laser through the 4 mm diameter holes in the center of the trapping plates of the single cell. The focussed output produced a spot size approximately 1 mm in diameter at the surface of the copper oxide pellet, and the laser was Q-switched to produce a 2.5 ns pulse duration. Laser power was varied from approximately 10 to 1OOmJ per pulse. Throughout these experiments, argon was maintained in the cell at a static pressure of 2 x 10e6Torr as measured with an uncalibrated Bayard-Alpert ionization gauge. The argon served to cool kinetically excited cluster ions

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c>

‘.1 .

I

MASS IN A.M.U. Fig. 1. Positive copper oxide cluster ion mass spectrum generated by direct laser desorption/ ionization of a CuO pellet.

collisionally [50] and acted as the collision gas for collision-induced dissociation experiments. Ion isolation using swept double resonance ejection techniques and collision-induced dissociation were carried out as previously described [51,52]. The spread in ion kinetic energy in the collision-induced dissociation experiment is dependent on the total average kinetic energy and is approximately 35% at 1 eV, 10% at lOeV, and 5% at 30eV [53]. Furthermore, under the conditions used for collision-induced dissociation, multiple collisions with the target gas occur. The maximum center-of-mass energies calculated for single-collision conditions in several of the figures, therefore, represent only a relative measure of the internal energy deposited into the ions. RESULTS

Positive cluster ions A positive cluster ion mass spectrum produced by direct LDI of a copper oxide pellet is displayed in Fig. I. Significant intensities of Cu, 0,’ are observed up to Cu,,O,+ . Mass assignments were made based on exact mass

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I

0

I

I

,,,I,,

2

I

4

6

No. Copper

8

I

10

I

I

12

I

I

14

Atoms

Fig. 2. Number of oxygen atoms vs. number of copper atoms in the major peaks observed in the C&O,+ mass spectrum.

measurements and observed isotopic distributions for a given cluster ion signal, although, under the conditions employed in the collection of Fig. 1, individual isotope peaks were not resolved for n > 10. All of the positive cluster ions observed belong to the oxygen-deficient class, Cu,OL (n > m). The intensities of the positive ion clusters decrease pseudo-exponentially with increasing size with a fairly significant discontinuous reduction in intensity for cluster ions larger than Cu, O+ . In addition, there is an even-odd alternation in intensity early in the cluster ion series with small Cu, 0; and Cu,O,+ intensities relative to Cu, 0: and Cu,O: . Figure 2 is a plot of the number of oxygen atoms vs. the number of copper atoms in the major positive cluster ions observed. The slope of this plot ( N 0.6)points out once again the extent of oxygen deficiency in these cluster ions. Due to the rapid drop-off of cluster ion intensity with increasing size, only the smallest positive cluster ions could be effectively isolated and studied by collision-induced dissociation (CID). The ionic products from the dissociation of Cu: , Cu, O+ , Cu: , Cu, O+ , and Cu, 0: are indicated in Table 1. Furthermore, the relative abundance of these products as a function of center-of-mass collision energy is depicted in Figs. 3 and 4 for the CID of Cut and Cu,O+, respectively. Negative cluster ions The LDI negative cluster ion mass spectrum displayed in Fig. 5 exhibits a number of interesting differences from the positive cluster ion mass spectrum.

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TABLE 1 Product ions from collision-induced Parent ion

dissociation of CU,~O,~,

Product ions (n,m)

130 cll:

TO

291

371

X

cu*o+

X

cll:

X

cujo+

X

X X

cu,o:

X

Negative cluster ions were observed up to Cu,, O;, and signals were sufficient to resolve isotopic structure for all of the cluster ions with n < 16 under appropriate digitization conditions. Most of the negative cluster ions belong to the oxygen-deficient category; however, some of the smaller cluster ions, including Cu, 0; and Cu,O; , have equivalent numbers of copper and oxygen atoms. The intensities of the negative cluster ions follow a significantly different pattern from that of the positive cluster ions. Although they are not displayed in Fig. 5, the signals for negative ions with masses below 200 a.m.u. are very small. Even the Ct.- signal is vanishingly small. The cluster ions appear with reasonable intensity beginning with Cu,O; and increase to a maximum intensity at Cu,O; , after which the signals drop off with increasing mass. No even-odd alternation is observed. A plot of the number of oxygen

“.”

0

4

8

Ecm (Max,

Fig. 3. Energy-resolved collision-induced

12

16

eV)

dissociation of Cue

.

121

0.01-

0

4



8



II:~,,, (Max.

Fig. 4. Energy-resolved collision-induced





17







16

CL’)

dissociation of Cu, O+ .

atoms vs. the number of copper atoms in the major cluster ions observed (Fig. 6) is characterized by a slope of N 0.5. The small intensity signals to each side of the abundant cluster ions are actually quite complex. For example, Fig. 7 displays an isolation of the ions contained in the low-intensity signal just to the left of Cu,O; . Although the expected distribution of isotopes is distorted, perhaps due to the proximity of the swept double-resonance ejection pulses used to isolate the ions in this region, the data strongly suggest the presence of Cu,O; and Cu,O-. The other peaks in the region cannot be adequately explained, although the locations of peaks associated with other possible cluster ions are indicated on the figure. Products from the CID of Cu,O; , Cu,O; , Cu,O; , Cu,O; , Cu,O; , Cu,O; , Cu, 0;) Cu, 0;) Cu,,O; , and Cu,, 0; are summarized in Table 2. Energy-resolved breakdown curves for the dissociation of Cu,O;, Cu,O;, cu,o, ) CU,O,) and Cu,O; are included in Figs. 8-12, respectively. Only the major product ions observed are included in these figures. DISCUSSION

Clusters of the noble metals, particularly copper, have been studied extensively [ 11,21,37,40,43,54-641. Given the wealth of information accumulated on the properties of pure metal clusters and their ions, the next logical step involves the investigation of adatom effects on cluster behavior. Indeed, studies of the characteristics of metal-adatom cluster species, especially for

122

JJ 6

ado

MASS

IN

1060

A.M.U.

1260

1400

1E IO

Fig. 5. Negative copper oxide cluster ion mass spectrum generated by direct laser desorption/ ionization of a CuO pellet.

the case in which the adatom is oxygen, have become very popular of late [16,17,19,20,24,25,40,42,43]. Our goal in this research was to determine how oxygen affects the structure and stability of pure copper clusters and compare and contrast oxygen’s effect in this case with its effects as studied in other cluster systems. The characteristics of the desorption event itself are of interest. During their study of carbon cluster ion formation by direct LDI of graphite pellets, McElvaney et al. determined that the size of the cluster ions generated was dependent upon the laser irradiation time at a particular location on the pellet [32]. The laser formed a small channel in the sample after extensive irradiation, and this channel served to constrict the laser-generated plasma, enhancing clustering reactions and shifting the observed ion distribution to highermass cluster ions. Although a similar channel was generated in the CuO pellets used in this study, the negative cluster ion distribution was independent of channel formation, and the positive cluster ion distribution actually showed decreased cluster ion intensities after channel formation. This may suggest that gas-phase clustering reactions play a less significant role in the formation of copper oxide cluster ions than in the case of carbon cluster ion generation.

123

0

~.‘,‘,‘.‘.“‘.‘.‘.‘,i 0

2

4

6

8

10

No. Copper

12

14

16

18

20

Atoms

Fig. 6. Number of oxygen atoms vs. number of copper atoms in the major peaks observed in the Cu,O, mass spectrum.

F

cumomn

m

c

8 7 9

E F

9 8

3 7 0 4 1 5

A B

D e

E 1

h

A 555

560

565

!

Fig. 7. Isolation of CU~O; from mass 555 to mass 600.

1

6 01 D

124

X

x x x x x x x x x x x

x

xxx

X

xxxxxxx xxxxxxxx

X

xxx

X

x

125

Fig. 8. Energy-resolved

collision-induced

dissociation of Cu,O;

.

Alternatively, annealing or other modification of the pellet surface induced by the laser may discriminate against positive cluster ion generation. Our earlier studies with Ag,t and Zn,+ showed similar behavior in that a relatively “fresh” metal oxide surface was required for efficient positive cluster ion formation.

Fig. 9. Energy-resolved

collision-induced

dissociation of Cu,O;

.

126

Fig. 10. Energy-resolved collision-induced

dissociation of C&O;.

Our understanding of this phenomenon is currently very limited, and additional studies of direct LDI of metal oxide pellets are currently under way to further characterize the effect. The copper oxide clusters generated in this study are exceptional in another

Fig. 11. Energy-resolved collision-induced

dissociation of Cy 0; .

127

0.01~

I

0

@/

-4”4 ’

4



8

Fig. 12. Energy-resolved collision-induced

12

16

20

dissociation of Cu,O; .

sense. Matsuda and co-workers have examined the intensity distributions for positive and negative cluster ions of copper, silver, and gold in the size range n = l-250 as generated by SIMS of the appropriate metal foils [57,58]. In each case, the cluster ion intensities decreased pseudo-exponentially with increasing cluster size and were characterized by two types of variation imposed on this intensity decay. The first was a low/high alternation in cluster ion intensity based on whether the cluster ion in question contained an even or odd number of atoms, respectively. The second involved the discontinuous reduction in cluster ion intensity which was observed at particular values of n. The even/odd intensity alternation has also been noted by Lineberger and co-workers [l l] and Smalley and co-workers [54,55] in their studies of Cu; photoelectron/photodetachment spectroscopy and by Irion and Selinger in their study of Cu,’ produced by SIMS [43]. The observation of this even/odd alternation in cluster ionization potentials and electron affinities is explained in terms of the electronic structure of the clusters. Even-n clusters are expected to have paired electrons in their highest occupied molecular orbitals, whereas odd-n clusters will have singly occupied HOMOs, resulting in higher ionization potentials for the even-n clusters. Furthermore, the LUMO of the even-n clusters is expected to be antibonding while the singly occupied HOMO of the odd-n clusters is expected to be non-bonding, so the electron affinities of the even-n clusters should be lower. In this study, the addition of oxygen to copper clusters resulted in positive ion clusters with some evidence for even/odd alternation, but no such

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behavior was observed in the case of negative ion clusters. This suggests that the oxygen adatoms have a significant effect on the nature of the clusters’ molecular orbitals. In support of the even/odd alternation in positive copper oxide cluster ions, Irion and Selinger observed only odd-n ions when Cu,’ species were accelerated into 0, resulting in cluster dissociation and oxygen addition. For instance, they observed the formation of Cu,O: and Cu,O+, and these ions are present in relatively high abundance in our positive-ion spectra. Another effect of the oxygen adatoms involves the discontinuities in cluster ion intensity observed at “magic numbers” of copper atoms. In Cu,’ spectra, the cluster ion intensity drops discontinuously at it = 3, 9, 19, 21, 35,41, etc., and, in Cu; spectra, this drop occurs at n = 1, 7, 17, 19, 33, 39, 57, etc. This behavior has been explained based on the “jellium” or one-electron shell model, which represents the electronic structure of the metal cluster using a single spherically symmetric potential well [65]. Each copper atom is assumed to donate a single s-type electron to the self-consistently calculated energy levels of this potential well, and a cluster stoichiometry which corresponds to the closing of a shell of this potential well results in a particularly stable cluster or “magic number” cluster. This model has also been applied to more complex cluster stoichiometries. Jarrold and Bower used the model to explain magic number behavior in Al,Oz CID cross-sections by assuming that each aluminum atom donates three electrons to the well and that each oxygen atom ties up two electrons through bonds to aluminum [ 161.Thus, the magic number behavior in Al, 0: species is shifted relative to the magic numbers in bare Al,+ by the number of additional aluminum atoms required to make up for the presence of the oxygen adatoms. The positive cluster ions generated in this study show a discontinuous reduction in intensity after Cu, O+ ; however, this observation cannot be rationalized in terms of the jellium model unless it is assumed that the oxygen adatom does not tie up any of the electrons donated by the copper atoms. If this were the case, then a second discontinuity would be expected after Cu, O,+; however, none is observed. Furthermore, no discontinuous behavior is observed in the negative cluster ion spectra at all. These results suggest that the electronic structures of copper oxide cluster ions may be ill-suited for the jellium model; however, careful cluster stability measurements, such as those described by Jarrold and Bower for Al,O,+, might uncover magic number effects in these copper species. The energy-resolved CID of copper oxide cluster ions shows some interesting trends. At the most basic level, the CID of Cu: is consistent with the Stevenson-Audier rule in that Cu+ is observed to be a more abundant product ion than Cu: , given their respective ionization potentials of 7.7 eV [66] and 7.9eV [55]. Furthermore, Cu,O+ dissociates just as previously described by Irion and Selinger to produce Cu+ and CuZO+.

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More meaningful conclusions concerning cluster ion characteristics can be drawn from the wealth of negative cluster ion CID data accumulated. In general, the oxygen-deficient negative cluster ions dissociate by loss of one or more copper atoms to produce an oxygen-equivalent species. These dissociate further by loss of multiple CuO units until Cu,O; is reached, and this ion dissociates to form Cu, 0; , CuO; , and CuO- . Similar CID results have been reported for Co,OL , in which Campana and co-workers used a Coulomb plus Born-Mayer pair-potential model to postulate cluster ion structures consistent with the observed CID [20]. Oxygen-equivalent clusters were predicted to be highly symmetric cage, ring, or ladder structures. Alternatively, oxygendeficient clusters were predicted to be unsymmetrical and characterized by strained structures with protruding metal atoms which could be easily removed to achieve oxygen-equivalent cluster ions. Similar structural characteristics are probably associated with the copper oxide cluster ions generated in this study. Finally, the very fact that the negative cluster ions undergo CID at all is of interest. In their studies of Cu; species, Smalley and co-workers observed exclusive electron photodetachment upon cluster ion photoactivation. They rationalized this behavior by simply assuming that the lowest energy process available to the pure copper cluster ions is detachment of the electron, i.e. there are no low-lying dissociation channels. However, we observed significant Cu,O; dissociation. This can be explained in two ways. First, it is reasonable to assume that addition of an adatom with high electronegativity, such as oxygen, to a pure metal cluster will produce a significant increase in the electron affinity of the resulting species. Thus, the energy required for electron photodetachment would increase, and dissociative reaction pathways could be observed. This simple explanation is based solely on the cluster ions’ relative stabilities toward dissociation and electron photodetachment. Alternatively, a dynamical explanation could be invoked. During photoactivation, energy is deposited in the cluster ion by promotion of an electron, so photoexcitation is dynamically suited for electron detachment. Low-energy CID, however, is believed to proceed by excitation of vibrational modes. This is the type of activation that would be most likely to result in cluster ion dissociation. In future studies of negative copper oxide cluster ions we intend to use ion-molecule bracketing techniques and wavelength-resolved electron photodetachment to determine the electron affinities of these species and investigate the possibility of dynamical effects. CONCLUSIONS

Oxygen adatoms have significant and interesting effects on the properties of copper cluster ions. Dramatic changes in the cluster ion intensity

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distributions relative to the pure metal cluster ions are observed, particularly in the case of negative copper oxide cluster ions. CID results indicative of extensive copper-oxygen bonding point toward cluster ion structures with oxygen distributed throughout the species rather than simply adsorbed to the surface of a pure metal cluster ion. Finally, the addition of oxygen to negative copper cluster ions opens up many dissociative pathways which are at low energies relative to electron detachment, perhaps due to increased cluster electron aftinities. ACKNOWLEDGEMENTS

Acknowledgement is made to the Division of Chemical Sciences in the Office of Basic Energy Sciences in the United States Department of Energy (DE-FGOZ87ER13766) for supporting this research and to the National Science Foundation (CHE-8612234) for continued support of FT-ICR-MS instrumentation. J.R.G. gratefully acknowledges the American Chemical Society’s Analytical Chemistry Division and the Tennessee Eastman Company for fellowship support. Finally, the authors would like to thank Extrel Corporation for the single cell assembly currently on loan to our research group. REFERENCES 1 M.A. Duncan and D.H. Rouvray, Sci. Am., 261 (1989) 110. 2 S.W. Buckner and B.S. Freiser, in D.H. Russell (Ed.), Gas Phase Inorganic Chemistry, Plenum Press, New York, 1988, p. 279. 3 J.E. Campana, Mass Spectrom. Rev., 6 (1987) 395. 4 M.D. Morse, Chem. Rev., 86 (1986) 1049. 5 Z. Fu and M.D. Morse, J. Chem. Phys., 90 (1989) 3417. 6 J.J. Breen, K. Kilgore, W.B. Tzeng, S. Wei, R.G. Keesee and A.W. Castleman, Jr., J. Chem. Phys., 90 (1989) 11. 7 M.E. Geusic, R.R. Freeman and M.A. Duncan, J. Chem. Phys., 89 (1988) 223. 8 R.J. St. Pierre, E.L. Chronister, L. Song and M.A. El-Sayed, J. Phys. Chem., 91 (1987) 4648. 9 R.E. Smalley, Laser Chem., 2 (1983) 167. 10 K.M. Ervin, J. Ho and W.C. Lineberger, J. Chem. Phys., 89 (1988) 4514. 11 D.G. Leopold, J. Ho and W.C. Lineberger, J. Chem. Phys., 86 (1987) 1715. 12 D.M. Kolb, in A.J. Barnes, A. Muller, 0. Thomas and R. Gaufres (Eds.), Matrix Isolation Spectroscopy, Reidel, Dordrecht, 1981. 13 S.K. Loh, L. Lian and P.B. Armentrout, J. Am. Chem. Sot., 111 (1989) 3167. 14 M.F. Jarrold, J.E. Bower and K. Creegan, J. Chem. Phys., 90 (1989) 3615. 15 M.F. Jarrold and J.E. Bower, J. Phys. Chem., 92 (1988) 5702. 16 M.F. Jarrold and J.E. Bower, J. Chem. Phys., 87 (1987) 1610. 17 M.F. Jarrold and J.E. Bower, J. Chem. Phys., 85 (1986) 5373. 18 M.R. Zakin, D.M. Cox and A. Kaldor, J. Phys. Chem., 91 (1987) 5224. 19 M.R. Zakin, D.M. Cox and A. Kaldor, J. Chem. Phys., 87 (1987) 5046.

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