(5.1)Chapter 5
Agn+ Photodissociation
A Model of Metal Cluster Fragmentation
Introduction
RRKM Model
Implementation of the Model
Fragmentation Energies
Path Degeneracy
Oscillator Frequency Distributions
Conversion to Cannonical
Distributions
Preliminary Experimental Results and Comparisons
with the Model
Ag9+ Fragmentation
When a metal cluster is photoexcited with an energy below its ionization potential, a precisely known quantity of energy is added to the cluster which, in all excepting the smallest clusters, rapidly degenerates into thermal or vibrational energy, rather than being re–emitted as fluorescence or enduring as an excited state.93 94 If the cluster is then sufficiently excited above the binding energy, fragmentation occurs. The combined properties of low binding energy and rapid electronic energy transfer have been exploited in metal clusters to determine optical absorption properties in a technique referred to as photodissociation spectroscopy. This has been particularly successful for alkali metal clusters, e.g., sodium clusters95 and potassium clusters96, whose binding energies are quite low – typically less than 1eV. However for transition metal clusters whose binding energies are considerably higher – approximately 2–3eV for silver – absorption of a single visible photon does not necessarily induce dissociation. However of all the noble metals silver has the bulk cohesive energy and is therefore most likely to behave like a alkali metal in terms of fragmentation. Vibrational resolution has been obtained for transitional metal clusters as large as Cu
4+ ,97 however as the nuclearity increases it becomes more difficult to induce dissociation. Kinetic factors rapidly decrease the reaction rate and the experiment is often restricted to the microsecond time scale.In addition to kinetic effects, photofragmentation requires that the cluster posses a sufficiently large absorption cross-section at that wavelength for excitation. For silver clusters in particular, the process of photofragmentation depends upon both optical absorption properties and on energetic and kinetic parameters that govern the rate of fragmentation. Although some information is known about the absorption cross-sections of silver clusters,98 99 100 this study aims to observe in detail the parameters, such as temperature and nuclearity, that effect fragmentation rates and optical cross-sections.
For s1 metals the higher ionization potential of the atom over that of the cluster usually results in fragmentation products consisting of a small neutral fragment and a larger cation fragment. The most common fragmentation pathway involves evaporation of a neutral atom or a dimer. For alkali metal clusters this has been explicitly observed by re-ionizing neutral fragments,101 102 while for silver clusters it has been indirectly assessed from the remaining charged fragment and binding energy requirements. These processes have also been observed indirectly in Agn(AgI)m cluster fragmentation.103 In the case of high photon energies or multiphoton absorption additional pathways such as cluster fission may become important.
Although no direct probe of cluster temperature is available, in the experiment presented here the cluster ions are equilibrated with the carrier gas and thus have approximately the same temperature as the carrier gas.104 This is in contrast to many alkali metal studies that use a high temperature oven source, photoionization or electron impact and whose clusters have comparatively high levels of internal excitation and size dependent temperatures.105 This excitation has so far prevented vibrational resolution for all clusters larger than tetramers. Equilibration of the cluster ions with the carrier gas avoids the use of the "evaporative ensemble" model and the approximations required to estimate cluster temperature when photoionization is employed.106 107 (This method could, however, be used to help explain multiphoton absorption and multistep fragmentation that we have observed with these systems. Such a technique would be especially applicable to larger clusters where single photon fragmentation is too slow to observe.) Other researchers have investigated the photodissociation of silver clusters generated by sputtering,99103 a technique that produces 'hot' clusters and often restricts the study to odd N clusters due to their intensity in the molecular beam. The results of such experiments have been compared to silver clusters deposited into inert rare gas matrices whose temperatures are low and precisely known but are perturbed by matrix effects.108
The experiments presented here probe the optical absorption properties and fragmentation pathways of silver cluster ions with a lower internal temperatures than have been observed previously in the gas phase. However as preliminary work, this study has been restricted to determining size specific fragmentation pathways and rate estimates. Additional information is also available in a recent progress report. 109 In what follows, a statistical model for silver cluster photofragmentation is presented and then experimental results are described and compared to the properties of the model when this is possible. It should be noted that the main purpose of this chapter is to document research in progress and to guide future investigations rather than to present a completed project. Thus the description of the dissociation model described in the following section is somewhat pedagogical although the model itself constitutes original research.
The experimental analysis is often complicated by the requirement that two factors, photon cross-section and fragmentation rate, must combine to produce experimentally observable fragments. Complications can arise from low oscillator strength at or above the fragmentation threshold. Furthermore, fluence dependence measurements necessary to establish the number of photons absorbed are notoriously difficult. 93 110 Because of these challenges, an elucidation of the rates of cluster fragmentation can certainly be assisted by understanding a kinetic model. The theoretical investigation of the section that follows provides some guidelines as to how much energy is required to fragment a particular silver cluster on the experimental time frame, independent of optical properties, thereby simplifying experimental analysis. A comparison of the preliminary experimental results with this theoretical investigation clarifies the relationship between kinetic and energetic factors that govern fragmentation. Additionally, the model proposes the effects of cluster internal temperature, excitation energy, and fragmentation pathway that can be further investigated experimentally.
An important tool for the study of fragmentation processes of pulsed beams of cluster ions is the reflectron time-of-flight described in the experimental section. The apparatus described as Phase II in the experimental section was used for photodissociation. The source block and nozzle were held at room temperature for preliminary observations and later cooled with liquid nitrogen to observe temperature effects. Cluster ion excitation occurred within the reflectron TOF after the mass gate but before the reflecting fields. With this technique the masses of both parent and product are known. A packet of ions of the same mass are fragmented and mass separated daughter fragment ions are observed. The mass-gate technique used here removes all parent ions except those of interest and simplifies peak assignment. For low mass clusters (or when high resolution is not required) the mass gate can be disabled and all cluster ions can be fragmented simultaneously. If fragmentation is not extensive, an appropriate tuning of the reflecting fields separates parent and daughter ions for all species. However, for larger clusters with closely spaced peaks, each size cluster requires a specific mass gate timing. Mass spectra were collected at 15-20Hz with only alternate pulses irradiated by the laser so that a reference spectrum was also collected.
Fragmentation of the cluster releases some kinetic energy; however conservation of momentum requires that most of the velocity goes into the smaller fragment. As a result, for the cluster sizes observed here the reduction in detection efficiency of fragmentation products was negligible. Any fragmentation that occurs after passing through the reflecting fields was not observed. The time taken to pass through the reflecting fields varied with cluster size but was on the order of 20m s.
For the experiments described here, the dissociation laser was either an excimer laser at 308nm or a dye laser at 440nm. Experiments probing other wavelengths are in progress.
A Model of Metal Cluster Fragmentation
The established theories of unimolecular reaction dynamics can be used to understand which cluster properties effect fragmentation rates and to seek an explanation for the ratios of the various fragmentation products even when dissociation energies are not known. The present state of the experimental results is too sparse to make quantitative conclusions, so the investigation is limited to qualitative aspects and features that will direct future experimental investigation. Nonetheless, in the future this model should be applicable to quantitative aspects, such as determining dissociation energies from dissociation rates. (The main purpose of this section is to document the progress that has already been made in the hope that it will be useful in the future.) Specifically, the aspects investigated here are: sensitivity of the fragmentation rate on internal temperature, cluster size, and photon energy, as well as kinetic and energetic requirements of atom and dimer fragmentation processes. The model may also be applicable to a study of cluster formation where excess energy gained by aggregation must be expended either into the surrounding carrier gas or into fragment ejection.
Unimolecular decomposition of an excited cluster,
(5.1)
could be expected to follow the usual first order rate law with rate constant k,
(5.2)
Later it will be shown that this exponential time dependence is a good approximation but not exact for the thermal distributions present in the molecular beam. For decomposition into multiple channels such as loss of atom or dimer, i=1 or i=2, there are two rate constants, k
1 and k2. Using kT = k1 + k2 this can be expressed as,
(5.3)
This leads to a time dependent yield for either of the charged fragments but a time independent ratio. After some algebra,
(5.4)
(5.5)
Thus the yield of product varies over time but the product branching ratio is time independent. Since the experiment does not yet measure time dependencies, the ratio of fragments is a good indicator of relative fragmentation rates.
Here a basic overview of the theory used and the choice of approximations specific to this situation is described. Portions of this model are well known but a particular set of assumptions has been adapted and extended for this experiment.
Two properties of metal clusters make them good candidates for a statistical fragmentation model. First, in all but the smallest metal clusters the excited electronic states are known to be short lived. The clusters behave as metals having a number of closely spaced electronic states between which transitions are extremely rapid. Thus absorption of a photon is followed by rapid internal conversion from electronic to vibrational energy and a non-radiative transition to the ground state is assumed. This high state density that increases with cluster size is essentially what makes interpretation of electronic spectra of all but the smallest metal clusters difficult but lends to an accurate statistical or thermodynamic description. Second, the compact nature of metal clusters implies closely spaced and nearly degenerate vibrational modes that are strongly coupled. As a result, a statistical model of dissociation is applicable. Such assumptions have been found to be valid for photodissociation of sodium,111 potassium,112 and aluminum113 clusters. They have also been used successfully to describe cluster thermionic emission.114
Under these approximations, we apply the theory of unimolecular reaction dynamics as detailed in the well known RRKM theory.115 Extensive reviews of this theory are available.116 117 118 The calculations performed here are similar to those used by other researchers and this work builds on that experience.111 113 119 The results are inherently approximate but efforts have been made not to be arbitrary and to interpret the results qualitatively. Another method based on RRK theory and detailed balance has been used by Engelking to explain the photodissociation of (CO
2)n+ clusters. 120 121 Some relevant work has also been done comparing the fragmentation channels of benzene cations.122Computer Programs for more common molecular RRKM applications are publicly available,123 however the unique requirements of cluster fragmentation required routines written specifically for that purpose. The program used here is not complex and should be applicable to other situations involving cluster fragmentation.
In this model a cluster is excited to some total energy E from photoabsorption and initial thermal energy. Under Transition-State Theory, also known as Absolute Rate Theory, all molecules which reach the transition state are assumed to react.116 Excess energy E* is available to the cluster above that required to reach the transition state E
0. Thus E* = E - E0. Viewed along the reaction coordinate the potential is an effective potential due to both the intermolecular potential between the two resulting fragments and the repulsive centrifugal force caused by angular momentum.124The key assumption is that the reaction rate k is proportional to the probability that the molecule with energy E will be in one of the transition states after a random assignment of energy into the available degrees of freedom. This probability is proportional to the number or sum of states W
# considered to be transition states. Since a certain minimum energy is required in the reaction coordinate, this leaves E* to be distributed amongst the remaining modes. This sum of states is a function of E*. The relationship is defined in terms of the density of states (DOS) per unit energy, r = W# dE, available to the reactant. A detailed analysis shows this relationship to be,115 125
(5.6)
with h arising from the one degree of freedom allocated to the reaction coordinate and L a degeneracy factor that is the number of symmetry equivalent routes to fragmentation.
The energy required in the reaction coordinate reduces the energy available to the non-reaction coordinate vibrational modes. This reaction coordinate energy includes rotational, vibrational, electronic or translational degrees of freedom. The translational energy is, of course, conserved, along with the mass, and can be ignored in a unimolecular reaction. Additionally, it can be shown for the precision required here, that the uncertainty in the vibrational state distribution dominates the error analysis, and the other degrees of freedom can be ignored as well.
In moving to the transition state, energy is consumed in the rotational mode. This is the difference in rotational energy between the reactants and transition state. Since angular momentum I is conserved, energy consumption is a function of the moments of inertia of the reactant and transition states,116
(5.7)
Some of the rotational energy of the reactant is transferred to translational motion in the reaction coordinate – the centrifugal force imparts velocity to the fragment. Unless the transition state involves widely separated fragments or a very small cluster, the ratio I/I
* is near unity. (For example with a rotational energy near the average room temperature energy, loss of an atom or dimer places this energy on the order of a few meV so that the loss of energy to the rotational motion can be safely ignored.126)Since no detailed information on the electronic state distribution exists, it is assumed, as is commonly done, that the effects of changes in the electronic state density are not significant.119 To first order the electronic DOS for the reactant and activated complex are expected to be similar since the geometries are similar. More precisely, this assumption is valid if the number of electronic states accessible with energy E
e is equal to the increased number of vibrational states in going from energy E-Ee to E. For metal clusters that have a large number of low lying electronic states, such an assumption is likely to be valid for high levels of excitation that access high densities of vibrational states and dissociate rapidly. This is probably the situation in cases of experimental interest here. For lower levels of vibrational excitation this approximation is less valid. The excited state population is expected to be low as evidenced by the difficulty of resonant two photoionization (R2PI) for all but the smallest clusters or high powered lasers.93 However it is recognized that this is an approximation and that coupling of the vibrational energy to electronic states or plasmons of the metal cluster is expected. Models designed for cluster thermoionization that estimate the electronic state density could improve this approximation.127Under the previous assumptions the reaction rate becomes dependent solely on the vibrational state distribution and the vibrational energy E
v. Equation 5.6 simplifies to,
(5.8)
If we temporarily assume that all vibrational modes in the cluster behave as harmonic oscillators with the same frequency, the distribution of j quanta into s oscillators is purely a combinatorial problem and the number of states with energy between 0 and E (i.e. the sum of states) is simply,
(5.9)
This approximation is 'quantum mechanical' in that discrete energy levels are assumed. At high energy, Equation 5.9 approaches the classical sum of states for a collection of harmonic oscillators,
(5.10)
Most approximations for non-equivalent oscillators are based on this function. (The density of states r (E) is simply dW
#/dE.) Note that Equation 5.10 is a classical model and ignores the quantization and zero point energy of the harmonic oscillators. Direct implementation of these classical formulas into Equation 5.8 leads to the simple and insightful but for our purposes inaccurate111 RRK rate constant.
(5.11)
Note that this is still limited to equivalent oscillators. A simple adaptation of Equation 5.10 to include zero point energy, and its application into the equation for the transition state theory rate constant, Equation 5.8, gives the Rice, Ramsperger, Kassel, Marcus (RRKM) model.115
Better approximations to the sum of states for a collection of quantum harmonic oscillators have been developed. The original motivation for these approximations was the difficulty of explicitly counting the very large number of permutations of j quanta into s oscillators with differing frequencies. The approximation used here is an empirical adaptation of Equation 5.10 developed by Whitten and Rabinovitch.128 Without detailed explanation, the formula for the density of states is,
(5.12)
where E' is the ratio of E to the zero point energy,
(5.13)
and the variables a, ß, and w are empirically set to approximate the exact quantum mechanical count. They are given by,
(5.14)
(5.15)
(5.16)
A program was written to compute these rates using any combination of the above models.See Chapter XX.
Since the development of the Whitten-Rabinovitch approximation, essentially exact direct counting techniques have become available in the form of the Beyer-Swinehart algorithm.129 These are valid for harmonic oscillators and some anharmonic corrections. For applications here however, the Whitten-Rabinovitch formula is sufficiently accurate and simpler to implement. 116
The smallest silver cluster cations, in particular Ag
2+-Ag9+, have been studied using high level ab initio calculations.130 For these, approximate fragmentation rate constants can be determined using total energy differences of parents and products. This assumes that barrier to dissociation is small. Barring detailed trajectory calculations, this is a reasonable first order approximation. These are shown in Figure 5.2.
Figure 5.2. Positively charged silver cluster fragmentation energies for various pathways involving the loss of a neutral atom, dimer or two atoms based high level ab initio calculations of Koutecky et. al.130 Sequential loss of two atoms is not thermodynamically preferred. These fragmentation energies are near the estimated bulk cohesive energy of 2.95eV.131 Note the reversal of thermodynamically favored pathways for Ag5+.
Care must be taken to include the statistical factor, the reaction path degeneracy L, in the rate constant since there are usually a number of symmetrical ways to form the transition state. This aspect has occasionally been neglected in the past but is significant for larger clusters and is critical when comparing various fragmentation pathways such as monomer and dimer loss. For the process of atom loss this is roughly the number of surface atoms: approximately n
2/3 for large clusters or simply n for small clusters.119 However, a more accurate technique that is less arbitrary uses explicit formulas for the number of surface atoms on compact structures. These numbers do not vary significantly with cluster geometry. Based on related algebraic formulas,132 the number of surface atoms S for icosahedral clusters with N atoms can be computed,
(5.17)
Formulas are also available for cuboctahedral clusters but the final numerical result is, for purposes here, similar. Interpolation of this formula for non-icosahedral clusters gives an accurate estimate of the number of surface atoms.
The degeneracy for dimer loss is simply the number of 'surface' bonds – or bonds between surface atoms. This is can also be derived for icosahedral clusters as
(5.18)
where m is defined in Equation 5.17. A comparison with explicit counting of structures based on geometries from ab initio calculations130 is shown in Figure 5.3.
Although it is not definitively clear that this analysis requires more than order of magnitude estimates since the DOS evaluation is only approximate, this precision is not costly and could be useful in other applications.
Figure 5.3. A comparison of the number of surface atoms and bonds on a compact cluster(symbols) based on ab initio calculations130, the analytical formula for compact icosahedral clusters given in the text, and the formula n2/3. This number is used for the RRKM degeneracy factor, L.
Oscillator Frequency Distributions
In the treatment of unimolecular dissociation, account must be made for the distribution of normal mode vibrational frequencies. A technique developed by Jarrold, Bower and Kraus for aluminum cluster ion photodissociation113 133 has been employed and is well documented in those references. This method estimates the distribution of vibrational frequencies in the cluster from the Debye model of lattice heat capacity for the solid.134 A distribution of frequencies in the cluster is parameterized with a few low frequency modes and an increasing density of high frequency modes up to the Debye cutoff frequency.
This reaction rates produced by this model do not differ significantly from the equivalent oscillator model. This is not surprising since the vibrational frequencies of s
1 metal clusters vary little with cluster size, e.g., dimer ion frequencies and bulk phonon frequencies are usually similar. The equivalent oscillator model has been used to successfully model the thermo-ionization of small clusters.135 In fact, it becomes clear when testing the model that fragmentation rates are relatively insensitive to the vibrational frequency distribution.For atom loss one vibrational mode is set as the critical frequency and the frequencies of two other modes assumed to be weakened in the transition state are reduced by a factor of 2.113 For dimer loss four frequencies are assumed to be weakened in the transition state and are reduced by a factor of 2. (There are a total of six degrees of freedom associated with the two atoms: one mode is the dimer vibration and of the remaining five one is the reaction coordinate and the others are assumed to be weakened in the transition state. 136) For the smallest clusters, n<6, the exact number of bonds is based on the calculated geometries. These assumptions are important for comparison of atom to dimer loss fragmentation rates, but small modifications do not affect the conclusions of the study.
Conversion to Canonical Distributions
The discussion so far has applied to species with a well-defined energy — the microcanonical. For comparison to experiment, conversion must be made to the canonical ensemble defined by a Maxwell-Boltzman distribution P(E) and a temperature. If the reaction rate K is expressed as a function of the photon energy E
p and the initial vibrational energy Ev, then, defining R(T,t) as the ratio of reactant concentration to initial concentration at time t and temperature T, the canonical rate of reaction is determined via convolution of the microcanonical ratio as a function of time e–Kt with P(E),
(5.19)
The notation indicates K is a function of E
Replacing P(E) by its density of states definition,
(5.20)
with Q the usual normalization constant or partition function,
(5.21)
and r (E) defined in Equation 5.12.
Integration of the above equations is not in general analytically solvable but is fairly straightforward to compute numerically.137 138 This must be done for each time interval t and experimental temperature T.
It should be noted that an alternative method has been investigated by Klots for the dynamics of small systems.106 The Klots technique treats systems both analytically and canonically but uses another set of approximations. The method used here has fewer approximations but requires numerical computation.
In the case of photoexcitation, the Maxwell-Boltzman distribution is shifted by the photon energy – or equivalently the photon contribution to the distribution is ignored. Integration is then a convolution of this distribution with the fragmentation rate as a function of internal energy. As an example, the fragmentation rate of Ag
9+ by a 308nm photon is based on the distributions of Figure 5.4 and rates of Figure 5.7.
Preliminary Experimental Results and Comparisons with the Model
For experimental convenience, initial studies have investigated photofragmentation at 308nm. Fortunately, many of the silver cluster cations have significant oscillator strength in this wavelength region and high laser fluences are easy to produce with an XeCl excimer laser. Figure 5.5 shows that Ag
9+ fragments equally into Ag8+ and Ag7+ after absorbing probably only one 4eV photon. In the mass spectrum, peaks from both the unfragmented clusters and fragments of other clusters are observed simultaneously. The parent peak intensity is reduced corresponding to increased fragment intensity. For this reaction, smaller fragments are not readily produced.The nearly equivalent rate in loss of atom or dimer for Ag
9+ can be explained to some extent by the fragmentation energy requirements (Figure 5.2) from ab initio calculations even if statistical factors are ignored. The energy required for atom or dimer loss from Ag9+ is similar. Based on the same calculations, the additional energy requirement of ~1.5eV to form Ag7+ by sequential atom loss makes such a process improbable. The stability of the dimer and of Ag7+ compensates for the increased energy required to remove two atoms rather than one from the cluster. This follows the common observation that charged or neutral s1 clusters with an even number of electrons are more stable than their odd counterparts.
Figure 5.5. Mass spectrum showing the photofragmentation of Ag9+ upon excitation by 308nm light at four different laser intensities. Ag9+ fragments almost equally into Ag8+ and Ag7+. The rates for atom and dimer loss are nearly equal since the intensities are nearly equal. The largest peak is unfragmented Ag9+ and other peaks are positively charged fragments. The small shoulder peak is Ag9(H2O)+ – a contaminant that could also be studied.
After fragmenting, atoms in a cluster are likely to be in an excited configuration and not necessarily in the lowest energy geometry. This would invalidate the use of ground state energy differences as indicators of fragmentation energies. For the fragmentation of Ag9+ this is not expected to be a significant problem since the parent and fragment have similar structures with slightly perturbed bond lengths. See Figure 5.6. Removing any one of the four most weakly bound atoms in Ag9+ results in a geometry very similar to that of Ag8+. Removing any two of the four produces Ag7+ in the ground state geometry.
Figure 5.6. Geometries for Ag7+, Ag8+, and Ag9+ based on ab initio calculations of Koutecky et. al.130 These are approximately the same configurations that are observed for minimum energy structures of Lennard-Jones clusters.
Note that these geometries are often similar to the closest packed sphere geometry. Interestingly, the geometries of Ag7+, Ag8+, and Ag9+ computed via ab initio calculations are identical in shape to the well known minimum energy configurations for Lennard-Jones clusters of the same size.139
Using the previous statistical model and the binding energies from ab initio calculations,130 fragmentation rates can be estimated for experimentally observed processes. Figure 5.7 compares the fragmentation rates for atom loss and dimer loss of Ag9+ as a function of excitation energy. Even with the inclusion of statistical factors atom loss is the preferred pathway, although the rates for atom and dimer loss are quite comparable when the theoretical accuracy of the ab initio calculations is considered. It has been observed in previous studies that ab initio calculations give lower cohesive energies than the photodissociation measurements but show stability patterns reminiscent of the jellium model.113 140 Our results for Ag9+ suggest errors of approximately 0.5eV since the experimental rate for dimer and atom loss is identical. (However, see Ref. 136.) Also it should be noted that inclusion of statistical factors does not change the conclusion that sequential atom loss is unlikely.
As is expected, the rate increases rapidly with excitation energy. Later it will be shown that this can be parameterized with an effective internal temperature based on the photon energy. For comparison the fragmentation rates calculated from RRK theory are also drawn. In this case RRK theory underestimates the values predicted from RRKM theory by 1-2 orders of magnitude. More detailed charts, using the same method but covering the experimental time scale, for all clusters for which total energies are available, are given at the end of this section, Figures 5.15-16. Experimental comparisons are not yet available for all of these clusters.
Figure 5.7. Microcanonical lifetimes for Ag9+ fragmentation via various pathways and based on RRK or the more accurate RRKM method. Transition state energies are assumed equal to the total energy differences of ab initio calculations. Note that under these assumptions sequential atom loss is improbable. Half-life is defined here and subsequently as: t =ln2/k=0.69/k.
Since Ag9+ fragmentation is observed experimentally, it must occur within or below the 20ms time scale of the TOF mass spectrometer. For simplicity Figure 5.7 does not include the effects of internal temperature, if however the cluster is presumed to be at 300K then the model predicts atom loss on a 1ms time scale and dimer loss on a 50ms time scale. See Figure 5.8. The observation of equal fragmentation rates for atom and dimer loss indicates that the fragmentation energies differ by less than 0.05eV when degeneracy factors are included.
Figure 5.8. Dissociation energy dependence of the rate of fragmentation of Ag9+ and Ag8+ at 300K. Solid line is atom loss, dashed line is dimer loss. The horizontal bold line represents the approximate experimental time scale. Dissociation occurring much slower than this is not likely to be observed. Arrows mark calculated total energy differences for Ag9+. For Ag8+ atom and dimer fragmentation energies are 1.96eV and 3.64eV respectively. This agrees with experiment – atom loss is observed but dimer loss is not. Note that dimer loss would be slightly faster if the fragmentation energy for atom and dimer were identical. This is observed experimentally for Ag9+. Also note that the effect of one additional atom is to decrease the reaction rate by an order of magnitude. (See also Ref. 136.)
Temperature Dependence. A preliminary experimental study of the sensitivity of the fragmentation rate to the temperature of the cluster source showed no noticeable effect. The vaporization source temperature ranged from liquid nitrogen to room temperature. Figure 5.9 suggests the time scale over which investigations would have to be made to detect significant temperature dependencies for excitation at 308nm. As might be expected, unless the photon energy is near the fragmentation energy, the effect of temperature is minimal. Ag9+ binding energies would need to be near 3eV to observe a temperature effect using 308nm excitation and detection at 20ms.
Figure 5.9. Comparison of the relative rate of fragmentation predicted by the model described in the text. In this diagram Ag9+ is assumed to fragment by the loss of one atom after absorbing one 308nm photon. The y-axis portrays the portion of clusters fragmented. The dotted line represents fragmentation at 300K under the assumption of 2.77eV binding energy – the ab initio calculated value. The lines of points (
,
) represent two
other assumptions for fragmentation energies at 300K. Dashed and solid curves represent
the effect of internal temperature assuming a binding energy of 2.77eV. A
temperature change of 100K is nearly equivalent to a fragmentation energy
difference of 0.1eV.
Incidentally, the curves of Figure 5.9 are near, although not exact, exponential functions. In the canonical formulation, the fragmentation time dependence deviates from exponential behavior for high temperatures or long lifetimes. In principle this could be an experimental temperature probe and such deviation has been observed experimentally.113 For an analysis of the time dependence as the temperatures rises, explicit evaluation of the canonical distribution, Equation 5.20, is required. For simplification in the discussion here, approximate rates, Equation 5.2, are reported that are fit in a least squares procedure to the explicit time dependence. An alternative technique to evaluation of the Maxwell-Boltzman integral is to add an additional thermal excitation energy E*=(3n-6)kBT to the microcanonical cluster. In our experience, this is a reasonable approximation and leads to rate constants that differ from canonical by only approximately a factor of 2 at 300K.
Figures 5.10-13 are a representative selection of the fragmentation patterns which have been observed for Ag11+, Ag17+, Ag18+ and Ag19+. What is most obvious is the preference for even-electron fragments especially for parents with an even number of electrons. The table below shows the primary fragmentation pathway for some clusters observed upon excitation at 308nm. This size range was chosen since the photon cross-sections are largest for these sizes. Note that these may be multiphoton processes. The following table notes the dominant pathways observed and compares these to some data available on the spontaneous evaporation of 'hot' silver clusters from a sputtering source,103 and to photoevaporation of Na clusters.111
Dominant fragmentation pathway, atoms lost |
|||
Parent Species |
Ag photo- this work |
Na photo- Bréchignac, et. al. |
Ag spontaneous evaporation El-Sayed, et. al. |
Ag9+ |
1(50%), 2(50%) |
1(50%), 2(50%) |
1(80%), 2(20%) |
Ag10+ |
1 |
||
Ag11+ |
2 |
2 |
2 |
Ag12+ |
1 |
1 |
|
Ag13+ |
4 |
2 |
1 |
Ag14+ |
1 |
1 |
|
Ag15+ |
2 |
1(50%), 2(50%) |
1 |
Ag16+ |
2 |
1 |
|
Ag17+ |
2 |
2 |
|
Ag18+ |
1 |
1 |
|
Ag19+ |
1 |
1 |
1 |
Ag20+ |
1 |
1 |
|
Ag21+ |
1 |
1 |
1 |
Ag6+, Ag7+, and Ag8+ have also been fragmented at 308nm but their products have not been observed. Note that Ag13+ is unique in that it fragments predominantly to Ag8+. Most of the observed patterns agree with those of alkali metal dissociation in that dissociation of even-electron clusters show both atom and dimer fragmentation except at high masses monomer evaporation predominates.111 There is better agreement with Na cluster photofragmentation than with the thermal evaporation of Ag clusters. In contrast to the sputtered cluster work we observe no spontaneous cluster fragmentation, supporting the idea that our clusters have a lower internal energy. Various Agn(H2O)m complexes with 6<n<12 and 1<m<4 also absorb and fragment.141
For Ag11+, dimer loss is preferred almost exclusively over atom loss. This is probably due to the closed-shell stability of Ag9+. As has been seen for alkali metal clusters, the increased relative stability of the electronically closed-shell species permits them to remain intact.142 This has also been found in related studies on the fragmentation of sputtered silver iodide clusters AgxIy,103 but does not occur for all types of metal clusters. For example, iron119 and niobium143 clusters with an even number of atoms are more stable than their odd numbered counterparts.
The four figures that follow show representative cases of Agx+ fragmentation.
Figures 5.10,11. Mass spectra showing the photofragmentation of Ag11+ and Ag17+ upon excitation by 308nm light at four different laser intensities. In both cases there is preferential loss of silver dimer. It is not yet clear which processes are single photon and which are multiphoton. Model RRKM-based calculations suggest that Ag11+Ţ Ag9++Ag2 is a single photon (4.01eV) process and the remaining processes are multi-photon. Fragments smaller than Ag6+ are grouped in one peak because of this particular choice of reflectron voltage settings.
Figures 5.12,13. Mass spectra showing the photofragmentation of Ag18+ and Ag19+ upon excitation by 308nm light at four different laser intensities. The ratio of fragments is nearly constant although fragments with an even number of electrons seem to be preferred. The model described in the text suggests that all these processes are multi-photon at this wavelength (4.01eV).
The observation of small fragments that correspond to loss of more than an atom or dimer has a number of possible explanations. It is likely that the cluster has absorbed more than one photon and sequential atom loss or even cluster fission has occurred. Absorption of a photon after fragmentation is unlikely because of the short laser pulse (10ns) and the time scale required for fragmentation. Since it is not clear which processes are single photon and which are multi-photon, results from the kinetic model, Figure 5.14, can be used to estimate this.
Figure 5.14. Excitation photon energy required to cause fragmentation of a silver cluster cation on the experimental time scale of 20m s based on the model described in the text. The upper curve assumes a 3eV fragmentation energy and the lower curve a 2eV fragmentation energy. The bulk cohesive energy is estimated to be 2.95eV.131 Note that the plasmon energy for Ag9-Ag21 is in the range 3-5eV. Due to thermal energy fragmentation can occur at photon energies below the fragmentation energy for small clusters.
Figure 5.14 gives an estimate to the photon energy that is required to fragment a silver cluster. The energy required to fragment a cluster within the experimental time frame rises rapidly as the size increases. A 308nm (4eV) photon would be expected to fragment up to at most Ag22+, while two photons would reach to at most Ag40+. Photodissociation has been observed by Meiwes-Broer and co-workers99 for Ag50+ and Ag70+. These are either multi-photon (2 or 3) events or the clusters have very high internal energies, near the boiling point.
The same figure, Figure 5.14, also shows that for larger clusters, n>10, there is a proportional relationship between photon energy and the number of atoms in the cluster. This would suggest an effective temperature T* that can be used to give an Arrhenius rate law,
(5.22)
This approach has been used to describe the thermionic emission of clusters.135 144 If results of the model are plotted in the Arrhenius form, log(k) vs. 1/E
internal, then this can be shown to be approximately valid. A future investigation might establish values for A and Eact from this model.One known difficulty of estimating fragmentation energies by inducing photodissociation with ultraviolet photons is that the resulting fragments can be highly excited. Cluster fission occurs at high excitation energies as degenerate pathways begin to dominate over energetically favored pathways. Thermodynamically this is expected: D G=D H-TD S. It has been suggested that high excitation energies might result in non-statistical fragmentation although this has not been confirmed.145 An alternative technique to reduce such effects would employ irradiation with multiple low energy photons.
In conclusion, the photodissociation rate of a particular cluster is sensitively dependent on the difference between dissociation and excitation energies and is only weakly dependent on the assumed vibrational frequencies and path degeneracies. However, the photodissociation rate is sensitively dependent on cluster size – Equation 5.22 and Figure 5.14.
Figure 5.15,16. Results of the model described in the text for various fragmentation pathways in the process Agn+ Ţ Agm+ + (Ag or Ag2) as a function of the excitation energy. (See also Ref. .) The cluster internal temperature is assumed to be 300K. Transition state energies are assumed to be equal to differences in total energies from ab initio calculations.
Preliminary estimations of cluster ion cross-sections have been made at 308nm for a few of the cluster ions with the largest cross-sections and some of this information has been reported recently.109 Figure 5.17 shows typical fluence dependencies for parent and fragments for Ag11+. With low fluences in the one-photon range, these dependencies are used to estimate optical cross-sections using the formula,
(5.23)
where I/I0 is the ratio of signal intensities with and without the depletion laser, s is the cross-section, and N/A is the number of photons per unit area. Use of this formula assumes a single photon event – something that is not expected for larger clusters. If the event is indeed multi-photon then the measured cross section is a combination of the ground and excited state cross sections.
Up to the size of Ag20+, Ag12+ has the largest cross-section at 308nm. Figure 5.18 shows the fluence dependencies for Ag9+, Ag11+, and Ag12+ from another data set. Initial investigations of the same clusters at 440nm show a decrease in cross section of at least one order of magnitude. This is also in agreement with photoabsorption measurements of Harbich et. al.108 on deposited neutral clusters that show very small cross sections at 440nm. It will be interesting to compare our results for larger clusters with theirs since their measurements do not require multiphoton absorption in order to detect absorption. However, comparison is difficult because those photoabsorption measurements do not readily yield absolute cross sections and they are performed on neutral clusters.108 Those results do show that clusters in the range Ag8 to Ag18 have significant oscillator strengths in the vicinity of 308nm as has been observed here.
It should be emphasized that the cross sections given are estimates — final results will await an improvement in the signal to noise of the experiment and work that is in progress. A comparison to results of Miewes-Broer, et. al.98 shows that the cross sections of sputtered clusters are larger by a factor of 4 for Ag9+. This can probably be attributed to the higher internal temperature of the sputtered clusters – estimated to be 1000-2000K.
Figure 5.17. Fluence dependence of the signal intensities of Ag11+ and its fragmentation products relative to the undepleted Ag11+ signal intensity. Excitation is at 308nm. Ag9+ is the most intense fragment. Also shown is a curve based on a cross-section of 3.3 Ĺ2 determined by a least-squares fit to the data. There is some indication that Ag11+Ţ Ag9+ is a one photon process. (A similar power dependence has been observed for the photofragmentation of Fe6+.93)
Figure 5.18. Fluence dependence of the signal intensities of Ag9+, Ag11+, and Ag12+ at 308nm. Points are experimental data and curves are least-squares fits of Equation 5.23. Based on this fit the cross sections are 2.1, 3.4 and 7.4 Ĺ2 respectively.
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