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RAMAN STUDIES OF MASS-SELECTED
METAL CLUSTERS
Kenneth Andrew Bosnick
A Thesis su bmitted in conformity with the requirements for the degree of
Doctor of Philosophy in Experimental Physical Chemistry
Graduate Department of Chemistry
University of Toronto
@ Copyright by Kenneth Andrew Bosnick 2000
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Sputtering a metal target under high vacuum conditions with 15 mA of 25
keV Ar' ions produces cationic metal clusters, which are extracted and collimated
into a beam using standard ion optics. A particular cluster nuclearity is selected
from the beam by a Wein filter, CO-deposited on an aluminum paddle with a
matrix gas (Ar or CO) at cryogenic temperatures, and nectralized. Once enough
clusters accumulate, Raman spectra are excited by various lines of an Ar' laser.
The scattered light is dispersed ont0 a charge-coupled-device detector using a
three-grating spectrorneter.
Strong bands at -165 cm-', assigned as "breathing" modes, dominate the
Raman spectra of Ag,, Ag,, and Ag, deposited in Ar. These bands fall close in
frequency to that of srnall length scale vibrations in solid silver, indicating that
the bonding between the atoms in these clusters already approximates that of
bulk silver. Comparison of the Ag, and Ag, spectra with theoretically calculated
ones reveals that the structure of Ag, is planar trapezoid and Ag, is tricapped
tetra hedron.
The Raman spectra of Ag, and Fe, in an Ar rnatrix show these both to be
dynamic Jahn-Teller molecules. All of the bands in the Ag, spectrum are account-
ed for using a linear plus quadratic Jahn-Teller coupling model. The Fe, spectrurn
cannot be fit using this model, probably due to the high spin state of the cluster
(S=4). A new derivation of the matrix elements for the linear plus quadratic
Jahn-Teller coupling model, based on operator methods, is presented.
Deposition of Ag,, Ag,,, Ag,,, and Ag,, in CO and collection of the Raman
spectra in the v(C0) region show that these clusters enhance the Raman scatter-
ing of CO by a factor of a few hundred with a strong dependence on cluster size.
The results are interpreted as the maximum possible enhancement by the Chem-
ical Mechanism of Surface Enhanced Raman Scattering.
Acknowledgements
Martin Moskovits supervised and guided the production of this Thesis and
the scientific results embodied within it. Much of the work was done in co-
operation w ith Tom Haslett, who also built the mass-selected cluster machine
with assistance from Stephan Fedrigo. Wai-To Chan and Rene Fournier did the
density functional theory calculations for Ag, and Ag,.
iii
Table of Contents
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements iii
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table of Contents iv
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Tables vi
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Figures vii
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Acronyms viii
1 . Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . 1 Atomic Clusters 1-1
. . . . . . . . . . . . . . . . . . . . . . 1.2 Matrix-Grown Cluster Experiments 1-2
. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Cluster-Beam Experirnents 1-3
2 . Experimental Apparatus
. . . . . . . . . . . . . . . . . . . . . . . 2.1 Overview of the Cluster Machine 2-1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Cluster Source 2-2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Beam and Mass Selection 2-3
. . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Typical Ag Cluster Distribution 2-5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Cluster Deposition 2-6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Spectroscopy 2-7
. . . . . . . . . . . . . . . . . . . . . . . 2.7 CO Deposition Rate Measurement 2-8
3 . Raman of Ag,, Ag.. and Agg
. . . . . . . . . . . . . . 3.1 Resonant Raman Spectra of Agsf Ag.. and Agg 3-1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 "Photochemical" Processes 3-3
3.3 Structures of Ags and Ag. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5
3.4 Breathing Modes and the Agg Spectrum . . . . . . . . . . . . . . . . . 3-13
3.5 Comparison of Silver Cluster Raman and Bulk Silver Properties . 3-13
Raman of Ag3 and Fe3
. . . . . . . . . . . . . . . . . . 4.1 Resonant Raman Spectre of Ag3 and Fe3 4-1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 "Photochernical" Processes 4-3
. . . . . . . . . . . . . . . . . . 4.3 Introduction to the Jahn-Teller Problem 4-4
. . . . . . . . . . 4.4 Operator Methods in the E x e 3ahn-Teller Problem 4-5
. . . . . . . . . . . . . . . 4.5 Jahn-Teller Fits to the Ag3 and Fe3 Spectra 4-10
5 . Cluster Enhanced Raman Scattering
. . . . . . . . . . . . . . . . . . . . . . . . 5.1 Ag, . Ag5. / CO Raman Spedra 5-1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Enhancement Calculations 5-5
. . . . . . . . . . . . 5.3 Normal Modes and the Enhancement Mechanism 5-6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions 6-1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References R-1
3- 1 . Com parison of experimental and calculated vibrational data
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . for the low-energy Ag5 isomers 3-8
3.2 . Sum man/ of experimental and calculated vibrational data
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . for the Ag, structures 3-10
. . . . . . . . . . . 4.1 . Jahn-Teller fit to the resonant Raman spectrum of Ag, 4-11
4.2 . Two possible fits of the linear plus quadratic Jahn-Teller model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . to the Fe, Raman spectrum 4-13
5.1 . Summary of estimates of deposited cluster size distribution . . . . . . . . 5-3
5.2 . Data from cross-section enhancement experirnents . . . . . . . . . . . . . . 5-4
5.3 . Results of enhancement calculations for AgJCO), . . . . . . . . . . . . . . . 5-5
List of Figures
2.1 . The mass-selected cluster machine . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 . Cluster source 2-3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 . Beam and mass selection 2-3
. . . . . . . . . . . . . . . . . . . . . . . . 2.4 . Typical silver cluster mass distribution 2-5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 . Deposition and spectroscopy 2-6
2.6 . The principle of thin-film interferometry . . . . . . . . . . . . . . . . . . . . . . 2-9
2.7 . Measurement of CO deposition rate by thin-film interferometry . . . . . . 2-9
. . . . . . . . . . . . . . . . . . . . 3.1 . Resonant Raman spectrum of Ag, in argon 3-2
3.2 . Resonant Raman spectra of Ag,. Ag.. and Ag9 in argon . . . . . . . . . . . . 3-3
3.3 . Time evolution of Ag5 Raman spectnim with laser irradiation . . . . . . . . 3-4
3.4 . Time evolution of Ag3 fluorescence and Ag5 Raman . . . . . . . . . . . . . . . 3-4
3.5 . Competing low-energy Ag5 isomers . . . . . . . . . . . . . . . . . . . . . . . . . . 3-6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 . DFT predicted Ag5 Raman spedra 3-8
3.7 . Competing low-energy Ag. isomers . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9
3.8 . DFT predicted Ag. Raman spebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-9
. . . . . . . . . . . . . . . . . . . . . 3.9 . FG matrix method fit Ag. Raman spebra 3-11
. . . . . . . . . . . . . . . . . . . . 4.1 . Resonant Raman spectrum of Ag, in argon 4-1
. . . . . . . . . . . . . . . . . . . . . 4.2 . Resonant Raman spectrum of Fe, in argon 4-2
4.3 . Energy levels for the linear E x e Jahn-Teller model . . . . . . . . . . . . . . 4-9
4.4 . Energy levels for the linear plus quadratic Jahn-Teller mode1 . . . . . . . 4-10
5.1 . Silver clusters deposited for CERS experiments . . . . . . . . . . . . . . . . . 5-2
5.2 . Estimated distributions of cluster nuclearities deposited . . . . . . . . . . . 5-3
5.3 . Raman spectra of Ag,, Ag,,. Ag3.. and Ag5. in carbon monoxide . . . . . . 5-4
5.4 . Measured Raman scattering enhancements . . . . . . . . . . . . . . . . . . . . 5-6
List of Acronyms
CCD
CERS
CM
DF
DFT
DTB
EAB
EMM
ESR
FG
r n R
GAT
HOPG
I R
LA
LAT
PBP
PFI-ZEKE
PT
R2PI
SERS
SPA
ST
TA
TCT
UV
cha rge-cou pled device
cluster en hanced Raman scattering
chemical mechanism
dispersed fluorescence
density functional theory
distorted trigonal bipyramid
electronic absorption bands
electromagnetic mechanism
electron spin resonance
(refers to the and matrices)
Fourier transform infra-red
gas-phase aggregation technique
highly-oriented pyrotytic graphite
infra-red
longitudinal acoustic
laser ablation technique
pentagonal bipyramid
pulsed field ionization - zero electron kinetic energy
planar trapezoid
resonant two photon ionization
surface enhanced Raman scattering
surface plasmon absorption
sputtering technique
transverse acoustic
trica pped tetrahedron
ultra-violet
viii
Larger than atomic dimers yet smaller than nanoparticles, atomic clusters
are aggregates with between a few and a few hundred atoms. They exhibit prop-
erties unique from those of the corresponding solid state material and indeed
from those of nanometre scale particles. For the smallest cluster sizes, the prop-
erties Vary somewhat erratically as the size is increased one atom at a time. I n
this size range, adding just one more atom to a cluster can greatly alter its prop-
erties, although this is not always the case. For example, some cluster sizes are
exceptionally stable compared with clusters of just one more atom, and this
leads to so called "magic numbers" in cluster mass spectra. When the aggreg-
ates become sufficiently large, the erratic behaviour diminishes and is replaced
by properties that Vary smoothly with increasing size. Of course, as the size
approaches that of bulk matter al1 of the properties tend asymptotically to the
solid-state limit. For the size range where the properties Vary smoothly, it is
appropriate to consider the particles as small fragments of bulk matter, but with
corrections for size and surface effects. Particles in this size range have a core
with the bulk bonding geometry and a surface layer, which can have various
geometric properties. I n the lower Iimit of this range, so-called "quantum size
effects" begin t o appear. These effects cannot be accounted for by simple geo-
metric corrections to bulk properties and, in fact, are the first signs of molecule-
like behaviour. It should be noted, though, that the sizes at which these "tran-
sitions" occur Vary immensely from one material to the next, and indeed from
one property ta the next.
I n the limit of small clusters, the bulk-like core vanishes and it is no long-
er logical to talk about the particle having a surface layer, since surface atoms
constitute the entire particle. Also, in this limit the "quantum size effects" domin-
ate over any residual bulk-like properties, and it is therefore unreasonable t o
consider these clusters merely as small pieces of bulk matter. For this sire range
the appropriate framework within which the particles may be described is that of
molecular science. Indeed, these clusters are molecules. Many of the experi-
mental and theoretical tools used for studying more "traditional" molecules can
be carried over directly to the study of atomic clusters. For example, many of
the same spectroscopie techniques and interpretations of the spectra apply. The
theoretical constructions of molecular science, such as potential energy surfaces
and single particle excitations, describe accurately what is observed for atornic
clusters.
There are, however, considerations that are unique to clusters. When
molecules such as CO, condense, they do so reversibly, retain their rnolecular
identity in the condensed phase, and can be separated back into free molecules
simply by heating. Atomic clusters, on the other hand, do not show these prop-
erties. At low enough temperatures it is possible that some atomic clusters
"condense" into metastable materials which reflect the original gas-phase clust-
ers; however, heating these materials will irreversibly produce solid-state matter
and any attributes of the original clusters are lost. These differences have impor-
tant consequences; for example, x-ray crystallography cannot be used to deter-
mine the structures of bare clusters. With the exception of noble gas clusters,
bare atomic clusters are usually electronically unsaturated, in stark contrast with
more common molecules. Consequently, the clusters often show high reactivities
towards Iigation by many small molecules.
Atomic cluster research dates back many decades and usually involves a
combination of established molecular science techniques with some that are
developed specifically for the study of clusters. Matrix-isolation is a common
technique for study ing "traditional" molecules (especially unsta ble intermediates)
and has also found use in cluster studies. For the cluster work, the matrix
material (e.g. argon gas) is CO-condensed, at cryogenic temperatures and under
high vacuum, with a beam of atoms produced in an oven. The matrix is then
gently warmed, and the atoms diffuse around and aggregate into clusters, which
are studied spectroscopically . For example, Raman studies of clusters produced
this way ascertained structural properties for Ni, [Moskovitsc], Mn, and Mn,
[Bier], Ag, and Ag, [Schulzeb], and other small clusten. It is found that irradiat-
ing the samples with laser light can sometimes cause changes in the spectra, as
demonstrated by following the radiation induced transformations of silver trimers
by Raman [Kettler] or UV-visible absorption [Ozinb] spectroscopies. Ozin and co-
workers exploited this phenomenon to controllably produce silver clusters up to
Ag, in quantities large enough for UV-visible absorption spectroscopy by "cryo-
photoclustering" (i.e. the growth of clusters in the matrix by laser irradiation)
[Ozina]. Many other interesting optical effects are found, including chemilumin-
escence during the agglomeration of copper and silver atoms in a noble gas
matrix [Konig]. I n addition to optical spectroscopies, ESR has been used to
study matrix-grown clusters, for example, Ag, [Howard]. ESR even led to a
remarkable structural determination for grown Ag, clusters [Bach]. The spect-
rum shows that there are two atoms in one chemical environment and five in
another, indicating that the structure of grown Ag, is pentagonal bipyramid.
However, in general, the study of grown clusters is realistically limited to only
the smallest cluster sizes. Growing clusters leads to a distribution of sizes in the
matrix and correspondingly more congested spectra. Most of what is known
about larger clusters is known from molecular or ionic cluster beam experiments
with some degree of mass selectability.
There are three common techniques for cluster production in beam
experiments: laser ablation (LAT), sputtering (ST), and gas-phase aggregation
(GAT). With the LAT (or "Smalley-type" source) a small plume of atoms, evapo-
rated using a high energy pulsed laser, is entrained in a beam of carrier gas.
Multiple collisions in the beam cool the atoms, which then aggregate to form
cfusters. The ST employs a high energy beam of ions (e.g. Ar' at 25 keV) to
sputter clusters directly from the surface of a rnaterial and was used for the
experiments reported in this Thesis (see Section 2.2). The GAT is sirnilar to the
LAT except that the free atoms are produced by heating the material in an oven.
Once the beam of clusters is formed by any of these techniques, the clusters or
their ions can be studied in the gas phase, after deposition in matrices, or after
deposition on surfaces.
Measurements of the ionization potentials of the cfusters in the beam are
straightfoward and have been made for many materials, for example Mn, - Mn,
[Koretsky] and Ta, - Ta, [Collingsb]]. Photo-electron spectroscopy is common in
cluster studies, as illustrated by the recent report of a vibrationally resolved neg-
ative ion photo-electron spectrum of Nb, [Marcy]. A cornparison of the photo-
electron spectra of Nb,- - Nb,- with density functional theory (DFT) calcufations
revealed details of the geometric structures of these anionic clusters and showed
evidence of more than one isomer of Nb,' coexisting [Kietzrnann]. Of course,
these studies are not restricted to only pure metals, as dernonstrated by the
photo-electron spectroscopy of Sb- - Sb,- [Polak] and of NiCu-, NiAg', NiAg,-, and
Ni,Ag- [Dixon-Warren]. Reports of multi-photon femtosecond pulse studies on Pt-
- Pt,- and Pd' - Pd,- [Pontiusa] and time-resolved femtosecond studies on Pt,-
[Pontiusb] and Au,' [Gantefor] have been published. I n a unique experiment,
referred to as a "femtosecond negative ion - neutral - positive ion charge rever-
sa1 experiment" or a NeNuPo experiment, the uftrafast dynamics of the relaxa-
tion of linear Ag, formed by neutralizing Ag,' was followed spectroscopically
[Boo, Leisner].
Because of the low cluster density in the beam, many "traditional" spec-
troscopies cannot be used to study these species. However, new techniques
have been developed for tackling this problem and have recently been reviewed
by Knickelbein [Knickelbeinc]. For example, Ellis and CO-workers measured the
vibrational spectrurn of ground state Ag, in a beam using dispersed fluorescence
spectroscopy [Ellis]. Resonant two photon ionization (R2PI) spectroscopy has
revealed the detailed vibronic structure of excited states for many clusters; how-
ever, the large cross-sections for ionization for most clusters prevents RZPI from
becoming a completely general technique. A few metal trimers have been
studied using RZPI, including Na, [Delacretaz], Cu, [Zwanziger], Ag, [Cheng J,
and Au, [Bishea], as well as the mixed metal trimer Ag,Au [Pinegar] and the
metal-carbide clusters Ti&, and Til&, [van Heijnsbergen]. Recently, high reso-
lution R2PI studies on v e r - cold Ag,, Ag,, and Ag, in gas-phase nanoscopic
helium droplets have been reported [Federmann].
The very sophisticated technique of pufsed-field-ionization, zero-electron-
kinetic-energy photo-electron spectroscopy (PFI-ZEKE) has been applied to
clusters. Yang and CO-workers studied bare vanadium and yttrium clusters by
PFI-ZEKE [Yanga], as well as the cluster compounds Y3C2 and Y3C2' [Yangb],
N b,N, [Yangc], and Nb3C2 [Yangd]. From these studies, geometric structures
followed from cornparison of the spectra with the results of D R calculations.
Finally, Collings and CO-workers measured optical absorption spectra of Au,, AU,,
AullI Au,, and their cations [Collingsc], as well as Ag,, Ag,, and Ag,' [Collingsa],
by the photo-depletion method. With this technique, a cold van der Waals comp-
lex consisting of the cluster and a rare gas atom is irradiated with laser Iight.
The frequency is scanned and when it matches a resonant frequency of the
cluster the complex dissociates and the event is detected by changes in the
mass spectrum.
One way of overcoming the problem of low densities in the beam is to
continuously CO-deposit clusters with a matrix gas until enough accumulate that
"traditional" spectroscopies can be applied. For example, Rabin and CO-workers
measured the optical absorption spectra of Ag, - Ag., produced by the GAT and
CO-deposited with an argon matrix [Rabin]. However, these experiments are not
strictly mass-selected, as the only way to control what is being deposited is by
varying the aggregation conditions. Coupling the deposition strategy with a
mass-sefected cluster beam allows large clusters to be studied without interfer-
ence from other cluster sizes, an obvious advantage over the matrix-grown
cluster experiments described in Section 1.2. This strategy forms the basis for
the experiments reported in this Thesis. The absorption spectra of argon matrix
isolated Ag, - Ag,, produced by the ST and mass-selected using a quadrupole
filter have been reported [Fedrigob, Harbicha]. These studies show much more
detail about particular cluster sizes than the non-mass-selected ones of Rabin
and CO-workers. When using this technique, cationic clusters are initially prod-
uced so that they can be mass-selected and are subsequently neutralized as
they are being deposited. Honea and CO-workers similarly recorded the absorp-
tion spectra of mass-selected silicon clusters up to Si,, produced by the LAT and
CO-deposited with krypton matrices [Honeab]. Absorption, fluorescence, and
excitation spectroscopies on mass-selected, matrix-isolated Ag,, Ag,, Au,, and
Au, [Fedrigoc]; Ag, [FelixJ]; and Pt, [Wangc] have also been reported.
The absorption, fluorescence, and excitation spectroscopies do not provide
information, such as vibrational frequencies, that can be used for structural
determinations; however, resonant Raman does. Resonant Raman has been
used to study the vibrations of a few small metal clusters deposited in a matrix,
including Ta, [Wanga], Hf, [Wangf], Nb, [Wange], Zr, [Haouari], and the dimers
Gd, [Chen], Ru, [wangd], Rh, [wangb], Ni, [Wange], Co, [Dong], Re, [Hua], and
Hf, [Hub]. The resonant Raman spectra of mass-selected, matrix-isolated Fe,
and Ag, [HasleW], Ag, [Haslettd], and Ag, and Ag, [Bosnicka] form the basis for
Chapters 3 and 4 of this Thesis. Other enhanced Raman techniques have also
been used to study deposited clusters. Surface plasmon-polariton enhanced
Raman was used on matrix isolated C,,, C,,, and C,, [Ott] and Si,, Si,, and Si,
[Honeaa]. From comparison of these spectra with ab /hiHo calculations, Si, is
found to have the structure planar rhombus, Si, distorted octahedron, and Si,
pentagonal bipyramid. Finally, Haslett and CO-workers reported the surface
enhanced Raman spectra of C, and C, clusters deposited ont0 a rough silver film
[Hasletta].
In addition to inert materials, gases that react with the clusters have also
been used for matrices. The infrared absorption spectra of CO [Froben] and NO
[Frank] adsorbed on silver clusters of nuclearity -30 - -1000 have been meas-
ured on clusters produced by the GAT and deposited in a matrix of the adsor-
bate. The influence of carbon monoxide on the electronic absorption spectra was
similarly studied [Charle]. fedrigo and Haslett looked at small metal cluster
complexes by infrared absorption spedroscopy of mass-selected clusters prod-
uced by the ST and deposited in reactive matrices, including Fe - Fe, in CO
[Fedrigod], Fe, in CO [~edrigo'], Ag - Ag, and Ag, in O, [Fedrigoa], and Fe - Fe, in
N, [ ~as le t t~ ] . Raman spedra of CO adsorbed on Ag,, Ag,, Ag,,, and Ag,, (prod-
uced by the ST and deposited in a CO matrix) form the basis for Chapter 5 of
this Thesis.
When depositing clusters with a matrix, care must be taken to "soft-land"
the clusters so that they do not fragment upon deposition. A number of studies
of cluster fragmentation (or dissociation) in the gas phase have been made,
including the dissociation of Cu,' - Cu,' [lngolfssonb], Ag,' - Ag,,- [Spasov], and
Al,' - Al,,' [Ingoihsona] induced by colliding the clusters with rare gas atoms.
The dissociation of clusten can also have a more "photochemical" origin, as
demonstrated by a study of the photo-fragmentation of Ag, - Ag,,- in a beam
[Shi]. More directly related to deposited cluster experiments, studies of the frag-
mentation of clusten colliding with solid surfaces have been reported, including
the collisions of Ag,* - Ag,* with rare-gas films [Fedrigoe] and of C,- - Cl; with
silicon surfaces [Tai].
Additional motivation for the deposition experiments comes from the pos-
sibility of producing novel materials with surfaces modified by mono-dispersed
clusters. To this end, a number studies have been made on mass-selected clust-
ers deposited ont0 various substrates. X-ray photoelectron spectra of Ag - Ag,
deposited on highly-oriented pyrolytic graphite (HOPG) [Yamaguchi] and of Cr - Cr,, deposited on Ru(001) [Lau] have been reported. Busolt and CO-workers
studied Ag, - Ag, on HOPG by two-photon photo-emission [Busolt]. Other works
reported include thermal-energy atom scattering studies of Ag,, [Felixb] and Ag,
[Vandoni] on Pd(100) and scanning tunnelling microscope studies of Ag, Ag,,
and Ag,, deposited on Pt ( l l1 ) [Bromann].
There exists an irnpressive body of Iiterature on the reactions of clusters
with small molecules in rnolecular beams. One of the main motivations for this
work is the so-called "cluster-surface analogy", where reactions on clusters are
considered as analogs of reactions occurring on extended surfaces. To investi-
gate this analogy it is important that one first knows the detailed structures of
the bare clusters, among other things. Knickelbein recently reviewed the Iitera-
ture on cluster reactivity studies [Knickelbeina]. Selected examples are V,, - V,, with CO, NO, O,, O,, and N, [Holmgren]; W, - W,, with cyclopropane [Peder-
sen]; Cu, - Cu,,, Ag, - Ag,,, Au, - Au,, with CH,OH [Knickelbeinb]; MO, - MO,,
with N, [Mitchell]; and Co, - Co,,, with NH, and H,0 [Parksd]. The decomposition
of C,, on Ni, - Ni,, at elevated temperatures has also been studied [Parksc], and
the adsorption patterns of N, on clusters have been used to indirectly probe the
structures of CO, - Co,, [Ho], Ni, - Ni,, [Parksb], and Ni,, - Ni,, [Parksa] in beam
experiments.
2. Experimental Apparatus
Sputtering refers to the process by which material is dislodged from a
surface by bombarding it with high energy charged particles and is described in
detail by Behrisch [Behrisch]. It is most commonly used as a process for coating
a second surface with a thin film of the material being sputtered. I n addition to
any large chunks of material that are knocked out of the surface by the momen-
turn transfer process, many single atoms and atomic clusters are also produced.
I n fact the process generates clusters of sites up to hundreds of atoms. Most of
the material sputtered off the surface is electrically neutral; however, a small
amount cornes off charged, either positively or negatively, and this makes pos-
sible the extraction and manipulation of well defined atomic clusters from the
plume of sputtered material using electric and magnetic fields. This is the basic
principle behind the cluster machine used in the experiments reported here.
In these experiments sputtering a metal target with high energy argon
cations produces metal clusters under high vacuum conditions. A drawing of the
apparatus is shown in Figure 2-1. An electric field extracts the small fraction of
clusters that corne off positively charged and collimates it into a beam. The
electric field is such that the anionic sputtered material can not enter the beam
area and is discarded. The beam
passes throug h the crossed electric m m and Deposition and Source
I I Moss-Seiection S ~ r o s c o p y and magnetic fields of a Wein fifter in I [e order to select out a particular mass s cluster for further study. Deflecting the
cluster bearn through IO0 using an
electric field removes any neutral
material that happens to have been
sputtered off at the correct angle to -
Figurm 2-1: The mass-selected cluster machine.
2 - 1
produce a trajectory along the cluster beam path. Since the neutral material is
not denected by this field, it strikes the side of the apparatus and stays there.
The cluster beam then enters the deposition chamber, where an electric
field focusses it to a small spot on an aluminum paddle. A closed-cycle helium
refrigerator cools the aluminum paddle to cryogenic temperatures, permitting a
matrix of a common gaseous material (e.g. Ar) to be CO-deposited with the clus-
ters. The matrix supports the clusters and keeps them well isolated. By using a
reactive gas as the matrix (e.g. CO) cluster-ligand complexes can be studied. A
low-energy electron flux from a thermionic source located immediately below the
deposition paddle neutralizes the clusters as they are being deposited. A 2.0 mm
diameter Faraday plate is periodically rotated into the beam to measure the
ciuster current. Once enough clusters accumulate, the deposit is studied spect-
roscopica Ily . I n order to control an ion beam and keep a cryogenic matrix uncontarnin-
ated the apparatus must be operated under high vacuum conditions. Pumping
with a high volume diffusion pump generates a base pressure in the sputtering
chamber of 5 x IO-' mbar. Partitioning the remainder of the apparatus into three
pumping stages leads to the base pressure in the deposition chamber (the last
pumping stage) being as low as 5 x IO-'' mbar even though gas is being leaked
into the sputtering chamber at a relatively large rate.
A commercial ion source produces the beam of argon cations used for
sputtering (see Figure 2-2). Passing an electric current through four filaments
located in the discharge chamber heats them and produces free electrons by
thermionic emission, which is described in detail by Kittel [Kittel]. The electrons
are accelerated through an applied voltage across the discharge chamber and
collide with argon atoms being leaked in. The collisions produce cationic argon
ions, which accelerate in the opposite
direction to that of the electrons. From f t h* , +25 kv -5 kV
the plasma of colliding atoms, ions, 1 1 / r - g m d PO-
and electrons a small fraction of the '. 4 T - r --r- 1 1. 1
argon ions is extracted, accelerated
through 25 kV, and focussed ont0 the A S M a Ag19 *-• - - - - - Ag,,, ...
sputtering target. n m r d s , A i + m - - A r * + 2 m ' trtlons (-S %),
The sputtering target consists of 2-1: e- anions
a 1 or 2 mm thick metal plate, bolted
on a water-cooled copper block, which c m be rotated about a vertical axis and
translated in al1 directions. The argon ion beam is about 2 cm in diameter and
strikes the surface of the target at about 45O to the surface normal. The water
cooling removes most of the heat produced by the sputtering of clusters from
the surface of the metal plate.
A commercial ion extraction assembly collects the cationic clusters from
the sputtered material. The assembly consists of a conical skimmer, two cylind-
rical tubes in series, and two sets of perpendicular deflection plates (See Figure
2-3). Biassing the conical skimmer, which restricts the range of trajectories that
can enter the beam, directs the cationic
tion assem bly. The cylindrical tubes
when appropriately biassed serve as a
set of electrostatic lenses; that is, they
collimate the previously diverging
cluster beam. The deflection plates aim
the beam directly into the Wein filter.
Overall, the clusters accelerate through
about 1200 V between the target and
clusters from the target to the extrac-
I W Vciep = Edep / e
Figura 2-3: Beam and mass seledion.
the entrance to the Wein filter. Further details on the principles of ion optics are
described in detail by Klemperer [Klemperer].
Passing the cluster beam through the crossed electric and magnetic fields
of the Wein filter achieves mass selection. For a cluster to pass through with its
trajectory undisturbed, the total force acting on it must be zero. The only signifi-
cant force is the Lorentz force, which is given by F = e (E + v x B) and simplif-
ies to the condition v = E/ Bfor the cluster to pass through undeflected when
the geometry is taken into consideration. The cluster enters the Wein filter with
a speed determined by it having been accelerated through about 1200 V and by
its charge-to-mass ratio (i.e. v = (2 Y, e/m)"). Since the clusters of interest
are all in a +1 charge state, only a particular mass cluster will have a speed
equal to the ratio of the electric field strength to the magnetic field strength, and
therefore only clusters of mass
will pass through the filter undeflected. The preceding analysis assumes that the
initial kinetic energy is negligible; however, this is not the case. The sputtering
process produces clusters with an initial distribution of kinetic energies and Iimits
the achievable mass resolution. The speed of the cluster entering the Wein filter
must be modified to v = (2 eel e/m + F~,:)* to account for the initial speed of
the cluster. Since there is a small distribution of initial speeds, a small distribu-
tion of charge-to-mass ratios will actually pass through the Wein filter. The mass
resolution can be improved through various measures, including cooling the
clusters by gas-phase collisions before mass-selection and lessening the range of
trajectory angles for clusters that continue on after mass-selection (i.e. tighten-
ing up the "slits").
After mass selection, a second set of electrostatic lenses re-colfimates the
cluster beam, which is then deflected through a IO0 angle to remove any neut-
rals. Upon passing into the deposition chamber, a third set of electrostatic lenses
focusses the beam down to a small spot on the aluminum paddle.
A Faraday plate can be rotated to where the aluminum paddle would
normally be situated in order to measure the cluster current. The Faraday plate
consists of a copper sheet with a 2.0 mm diameter hole drilled through it. Behind
the hole is a second copper sheet, which is electrically isolated from the first one
and grounded through an ammeter. As the cationic clusters strike the back
sheet they are neutralized by electrons coming up from ground (through the
arnmeter). The current registered on the ammeter divided by the charge on the
cluster gives the number of clusters striking the back plate per unit time, which
is also the number of clusters passing through the 2.0 mm diameter hole per
unit time.
A typical silver cluster mass distribution, generated by setting the Wein
magnet to its maximum current and scanning the Wein voltage, is shown in Fig-
ure 2-4. Using the maximum Wein magnet current (limited by heat dissipation)
leads to the best resolution. Cluster nuclearities up to nine are well resolved, but
above nine the mass peaks overlap and a srnall distribution of cluster sizes pass-
es throug h the filter. A distinct odd-even alternation is evident in the cluster
mass spectrum with the odd cluster sizes (even number-of-electron cations)
being much more abundant. I n fact, Ag, and Ag, are essentially absent. These
systernatic effects are explained in detail by Walt DeHeer in terms of the filling of
electronic shells [DeHeer].
8 5~ IW 150 XIO Wein V w I V
flgurr 2-4: Typical silver cluster mass distribution.
A closed-cycle helium refrigerator cools the paddle to 10 - 20 K in order to
support the matrix-isolated clusters. The temperature is determined using a
cryogenic thermal diode located on the paddle support. Most of the Raman
experiments use an aluminum paddle, but Cs1 or BaF2 paddles are occasionally
used when their transparency properties are useful (e.g. for FTIR and UV-vis
spectroscopies). The matrix gas enters the deposition chamber through a sap-
phire leak valve and is channelled to noules located around the incoming cluster
beam where it is directed at the paddle (see Figure 2-5). Argon gas is used as
obtained; however, carbon monoxide gas is first purified through a liquid nitro-
gen trap.
Neutralization of the deposited clusters occurs by two mechanisms. The
first involves eledrons from the aluminum paddle, which is gmunded through a
voltage source. However, as the matrix thickness increases it builds up a resist-
ance and slows the neutralization process. Also, for the experiments with BaF2 or
Cs1 paddles, the paddle is non-conducting. A second mechanism prevents posi-
tive charge from building up in the matrix, which would repel the incoming clust-
ers. The second mechanism uses a resistively heated tungsten filament located
immediately below the paddle to generate thermionic electrons. An appropriate
bias on the filament draws the electrons to the paddle. It is not absolutely clear
whether neutralization occurs in the ~ n t m ~ r o m o e u
gas phase or in the matrix; however, / u'al~-- the neutralization energy is high [ ~ v ~ . u r ]
enough to fragment the cluster. Since
complete fragmentation does not
occur, the matrix gas must be avail-
able to remove the energy of neutral-
ization.
Setting the voltage between the
f b u n 2-5: Deposition and spectroscop y.
sputtering target and the deposition paddle determines the kinetic energy of
deposition as the charge on the cluster times the deposition voltage. The depo-
sition energy does not depend on any of the voltages between the target and
the paddle. Too low a deposition voltage causes the cluster beam to diverge at
the paddle; however, too high a deposition energy leads to fragmentation of the
clusters. A compromise between these two effects must be found. Carger clust-
ers are less prone to fragmentation since they have more vibrational degrees of
freedom to dissipate the energy into. A deposition energy of 20 eV is used for
srnaller clusters and 30 eV for larger ones. Still some fragmentation of the small-
er clusters occurs. Once a cluster lands on the paddle with neutralizing electrons
and matrix gas depositing around it, the energies of deposition and neutraliza-
tion dissipate into the matrix and the cluster relaxes to the ground state of the
neutral.
When enough clusters accumulate on the paddle, the source is shut down,
the deposition chamber is isolated, and the deposit is studied spectroscopically.
Various lines of an argon gas laser excite the Raman spectra after any undesir-
able emissions have been filtered out. A lens outside the deposition chamber
focusses the laser Iight to a small spot (-100 pm diameter) on the paddle, which
has been rotated to face the back of the machine. The laser beam is incident at
4S0 to the paddle surface normal. A lens inside the deposition charnber collects
and collimates the scattered light in a direction normal to the paddle surface.
Another lens, outside the deposition chamber, focusses the light onto the slits of
a three-grating spectrometer. The first grating disperses the light, which then
passes through a relatively wide opening. The grating angle and the opening are
adjusted to allow the Raman scattered light to pass through while rejecting the
Rayleigh scattered Iight. The second grating recombines the light, which is then
focussed ont0 the final stage slit. The third grating disperses the light onto a
charge coupled device (CCD) detector for analysis. For Raman spectra of bare
clcsters, use of 1800 mm'' gratings on al1 stages permits one to get very close
to the Rayleigh line. For spectra with larger Raman shifts, lower density gratings
are sufficient. Liquid nitrogen cools the CCD, which is interfaced to a computer
for data acquisition. The Raman spectra are wavelength calibrated using the
emission Iines of a neon lamp. A detailed account of Raman spectroscopy is
given by Derek Long [Longa].
Either a tungsten-halogen lamp or a deuteriurn larnp in various reflection
or transmission configurations excite the UV-vis spectra. For some experiments
the incident light is monochromated and scanned in wavelength as a photomulti-
plier tube measures the reflected / transmitted light intensity. For other experi-
ments the incident Iight is not monochromated, but rather the reflected / trans-
mitted Iight is spectrally analysed using a single grating spectrometer and a
photo-diode array detector. UV-vis spectroscopy is described in detail by Harris
and Bertolucci [Harris]. A modified commercial Fourier Transform Infrared
( m R ) instrument in a transmission configuration measures F i ï R spectra. Broad
band infrared light is modulated by an interferometer, passed through the
sample, and detected on a liquid nitrogen cooled detector. The resulting interfer-
ogram is Fourier transformed to obtain the infrared absorption spectrum. A
generaI overview of F n R spectroscopy is given by Max Diem [Diem].
For most of the experiments the exact matrix-to-cluster ratio is unimport-
ant. All that matters is that the ratio exceeds about 103, under which conditions
the clusters will not aggregate. For these experiments a previous estirnate of the
deposition rate correlated with the background pressure in the deposition cham-
ber was used [HasletV]. However, the experiments on the Raman scattering
enhancernent of carbon monoxide on silver clusters require a more accurate
matrix deposition rate, which was determined using thin film interferometry (see
Figure 2-6). Thin film optics is discus-
sed in detail by Bohren and Huffman
[Bohren].
Laser Iight (A, = 457.9 nm,
75 mW) from an argon gas laser L p a d d l e
reflects off the aluminum paddle at ~igurr 2-6: The principle of thin-film interferometry.
4S0 incidence as carbon monoxide freezes on the paddle under cluster experi-
ment conditions. The intensity of the reflected light is attenuated by a neutral
density filter (optical density 2.6) and then measured by a photodiode in the
tinear response regime. The intensity as a function of carbon monoxide deposi-
tion time is shown in Figure 2-7. The oscillation period, determined by fitting a
sine curve to the data, corresponds to a matrix deposition of thickness
d = 1, / (2 n, COS O), where 4 is the refractive index of the carbon rnonoxide
cryofilm and 0 is the angle of incidence in the matrix. The refractive index of a
carbon monoxide cryofilm is known for wavelengths between 4400 and 4900 nm
[Baratta]. Since carbon monoxide has no absorption bands between 457.9 and
4400 nm, dispersion should be minimal in this region, and the refractive index at
457.9 nm can be estimated be equal to that at 4400 nrn (n, = 1.26). The rate of
matrix deposition, measured this way for two different rates, correlates with the
background pressure in the deposition chamber measured on an ionization
gauge. Linear interpolation gives the deposition rate corresponding to other
background pressures.
Figura 2-7: Measurement of CO deposition rate by thin-film interferometry.
3. Raman of Ag, Ag,, and Ag,
The silver cluster mass distribution shown in Figure 2-4 clearly shows
resolved peaks for Ag,, Ag,, and Ag,. Deposition of these clusters with an argon
rnatrix allows Raman studies of the bare clusters to be carried out under pseudo
gas phase conditions. Argon forms a weakly perturbing matrix compared with
most other materials, as is evident from a number of studies where the matrix
material is systematically varied [e.g. Fedrigob]. In fact, neon is the only less
perturbing matrix material; however, use of neon requires cooling the deposition
substrate to lower temperatures than can be achieved using ouf apparatus. The
matrix is highly disordered, but polycrystalline [Hallamasek 1, with a density of
a bout 90 O/O of the crystal density [Schulzea]. The clusters rnay be a t substitu-
tional sites in the matrix lattice; however, the local environment around a given
cluster does Vary from one site to the next. For example, the orientation of the
clusters is random. These inhornogeneities lead to srnall splittings and peak
broadening in the Raman spectra. Also, the local environment may effectively
lower the symmetry of the molecule and thus may split formally degenerate
modes. The clusters are well separated from each others in the matrix, with a
typical molar argon-to-cluster ratio of IO4 being used.
The strongest electronic absorption bands for small silver clusters are
found in the near ultra-violet [Fedrigob]. However, the blue-green lines from the
argon laser used to excite the Raman spectra catch the tail of the electronic ab-
sorption bands, leading to a resonant (or more accurately pre-resonant) Raman
process, as described in detail by Clark [Clark]. The pre-resonance enhances the
Raman scattering cross-section and often leads to stronger overtone progres-
sions. However, since we are not fully resonant with an electronic absorption,
the spectra usually resemble those excited in a non-resonant process, with
perhaps minor intensity differences.
The resonant Raman spectrum
of Ag, ( 5 . 6 ~ 1 0 ' ~ clusten in a 2 mm
diameter spot; argon matrix) excited
by 100 mW of 457.9 nm radiation and
integrated for 30 min is shown in Fig-
ure 3-1. Excitation at 465.8 nm gives
similar results to those obtained with
457.9 nm. With longer excitation
wavelengths, comparable spectra are
recorded but with intensities about an
order of magnitude weaker. The band
a t 144 cm-' (marked 1) is the remnant
of an incompletely filtered laser emis-
sion Iine, as is the weak feature at
92 cm-'. The band at 189 cm-' (marked
l . l . l . l . l .
1 100 1SO 200 250 300 3
Raman Shift 1 cm"
2) is known to belong to Ag, [Schulzeb] Rou- 3-1: Resonant Raman spectnim of ~ g , in argon.
and is present as a result of the frag-
mentation of a small number of the clusters upon deposition and due to photo-
fragmentation of the pentamer with laser irradiation (see Section 3.2). Although
the Ag, peak is quite strong, it actually represents a relatively small amount of
material, since Ag, has an enormous Raman scattering cross-section. Peaks due
to other fragments are not seen since their cross-sections are much smaller.
Noise in the spectrum increases above 200 cm-' due to the subtraction of a
fluorescence background originating from Ag,. Only a constant offset was sub-
tracted below 200 cm-'. The strong band at 162 cm-' along with its overtones at
323 and 486 cm-' are assigned to Ag,, as are bands a t 68, 80, 100, 105, 126,
136, and 174 cm? The overtone progression and the excitation profile provide
strong evidence that we are seeing some resonance enhancement.
The resonant Raman spectrum of Ag, ( 5 . 9 ~ 1 0 ' ~ clusters in a 2 mm dia-
meter spot; argon matrix) is shown in Figure 3-2 along with the spectra of Ag,
and Ag, for cornparison. Radiation of
457.9 nm wavelength and 10 mW
power excited the Ag, spectrum, which
was collecteci by averaging 60 s spect-
ra each taken on a fresh spot in the
matrix for a total of 540 S. This avera-
ging strategy overcomes the problem
of photo-bleaching of the signal (see
Section 3.2). Light of 465.8 nm wave-
length produces a less intense spect-
rum, indicating a pre-resonant process
is operative. As with the Ag, (and Ag,)
spectra, the band a t 144 cm-' is due to
an incompletely filtered laser emission
and the band a t 189 cm" cornes from
C , . . . . , . . , . , . . 1 O0 200 300
Raman Shift / cm-'
Ag, produced by fragmentation during Figurr 3-n: Resonant Raman spectra of ~ g , , ~ g , , and Ag, in argon.
deposition. The peaks at 113, 169, and
175 cm-' are assigned as fundamentals of Ag,, and the peaks at 336 and
341 cm-' are overtones of the two highest frequency fundamentals. The Ag,
spectrum ( 4 . 7 ~ 10" clusters in a 2 mm diameter spot) shown in Figure 3-2 was
excited with 50 m W of 457.9 nm light and averaged over 15 min. The signal was
very weak and not easily reproduced; therefore, only the peak at 166 cm-' is
assigned to Ag, with confidence.
The laser lines that excite resonant Raman scattering (457.9 or 465.8 nm)
from Ag, also cause spectral changes suggesting that Ag, photo-decomposes
slowly. No Raman spectral changes are observed with non-resonant lines. More-
over, changes are only observed when the matrix is illuminated and are, hence,
not a result of a thermal process. Fig-
ure 3-3 shows the evolution of the
spectrum, excited with 10 mW of
457.9 nm laser light, through a sequ-
ence of 5 min integrations. The band
at 189 cm-' and the stmng background
fluorescence, both known to be due to
Ag, [Schulzeb, Harbicha], grow during
laser irradiation while the bands assig-
ned to Ag, decrease in intensity. The
uniform decrease of al1 the peaks 200 250 3m 350
assigned to Ag, indicate that they afl Raman Shift 1 cm ' Figurm 3-3: Time evolution of Ag, Raman spednim
arise from the same Spe~ieS. The irrad- with laser irradiation.
iation experiment was repeated at a fresh spot in the sample while recording the
spectrum in the 600 nm region where a known strong Ag, fluorescence occurs
[Harbicha]. The Ag, signal was found to increase a t approximately the same rate
as that o f Ag,, suggesting that the silver pentamer photo-fragments to the silver
dimer and the trimer. The intensity changes with time of irradiation are shown in
Figure 3-4 for the decay of the 162 cm-' Ag, Raman band and the growth of the
Ag, fluorescence.
The Raman scattering intensity decreases very quickly with irradiation for
the Ag, and Ag, experiments; in fact,
the signal bleaches away completely
within a couple of minutes. This makes
collection of the spectra very difficult,
and a strategy of averaging short inte-
grations on fresh spots was required.
However, unlike Ag, no new bands are t - -
observed to grow in with irradiation. 0 1 . - . - ' . - . = ' - - . ' ~ O 1000 2000 3000
One cannot, therefore, attribute the Time / s Figurm 3-4: Time evolution of Ag, fluorescence and Ag, Raman.
bleaching to photo-decomposition. (Although due to the difficulty in acquiring
the spectra one cannot completely rule it out either.) A second process that may
be operative is photo-assisted diffusion, whereby the absorbing clusters heat the
matrix localty and diffuse around in the matrix, eventually plating out on the sur-
face of the paddle or aggregating to form larger clusters. This process is known
to occur for Ag, in an argon matrix and is described in Section 4.2.
Due to the nature of metal clusters structural determination by common
techniques (e.g. x-ray diffraction) is impossible. The structures must be deter-
mined using spectroscopies that can be applied to gas phase or matrix isolated
clusters, in particular vibrationally resolved spectroscopies. The fairly complete
resonant Raman spectra of Ag, and Ag, are therefore ideally suited for elucidat-
ing the structures of these clusters. Comparing the experimental spectra with
cornputed ones and considering the symmetries of the vibrational modes facifi-
tates the structural analysis. There have been a few quantum chemical studies
of the structures of srnall silver clusters, including calculations using density fun-
ctional theory (DFT) [Chan, Srinivas, Poteau], conventional ab init/O theory [Liu,
Santamaria, Bonacic-Koutecky, Liao, Bauschlicher], and modified Huckel theory
[Zhao]. Invaria bly these calculations predict a small number of low-energy
equilibrium geometries. All the calculations produce the same set of possible
structures; however, their relative energies Vary from one computational ap-
proach to the next. The quantum chemical calculations do not predict with
absolute certainty which structure will be found, but they do limit the number of
probable structures to a small number. The general strategy for determining
which of the structures we are looking at is the comparison of predicted spectra
of the probable structures with the experimental spectrurn. Various techniques
are used to predict spectral patterns, including DFT calculations, matrix
calculations, and group theoretical methods. The analysis of vibrational mode
syrnmetries by group theoretical methods is described by Harris and Bertolucci
[Harris 1. The Wilson matrix method [Wilson] is a convenient formulation for
solving the classical equations of motion of a system of point masses joined by
interacting harrnonic springs. It is often used for modelling the vibrations of
molecules. The matrix is the product of the matrix and the matrix. The
and G matrices are both 3 M 6 dimensional square matrices, where N i s the
number of atoms in the molecule. The matrix contains the primary force con-
stants on the diagonal and the interaction force constants on the off-diagonal.
The G matrix contains al1 of the information about the equilibrium positions and
masses of the atoms. 60th matrices are set up with respect to an arbitrary set of
internal CO-ordinates, which must span the vector space of al1 possible vibration-
al motions. I n the internal CO-ordinate system the interaction force constants are
not necessarily zero (although they are in the normal CO-ordinates). The eigen-
vectors of the matrix are the normal CO-ordinates and the eigenvalues give
the frequencies of the vibrations. In practice, a geometry is assumed (perhaps
from quantum chemical calculations) and the force constants are adjusted to fit
the experimental spectrum, subject to symmetry and plausibility constraints.
The a pproach works well for small molecules (particularly with h ig h symmetry)
where the number of independent force constants is less than the number of
peaks in the spectrum. However, as the molecule gets larger the number of
adjusta ble parameters increases and the spectra usually show less structure,
leading to a grossly under-determined
pro bIem.
Quantum chemicai calculatioris
suggest two probable low-energy
structures for Ag,. These structures
(see Figure 3-5) are the planar
trapezoid (PT) and the distorted Planar Trapezoid
trigonal bipyramid (DTB). 60th of Distorted Tn'gonal Bipyramid
Fïgurm 3-5: Cornpeting low-energy Ag, isbmerç.
these structures have C,,symmetry. Structures with higher symmetry are im-
mediately ruled out since there are seven or eight distinct bands that are assign-
ed to a single form of the Ag, cluster. (Recall that al1 of these bands decreased
in intensity uniformly with laser irradiation.) High symmetry structures with vib-
rational degeneracies do not produce enough spectral features to account for al1
of the observed bands. rnatrix calculations on other structures (e.g. square
pyramid, linear, bow-tie) also suggest that it is abundantly reasonable to Iimit
ourselves to focussing on the two computationally favoured structures, However,
FG matrix calculations alone can not distinguish between the two structures as a - reasonable fit can be made to the spectrum for either assumed geometry. What
is needed is an estimate of the intensities of the Raman transitions and a com-
parison of the experimental spectrum with first principles calculations.
Two techniques were used to estimate the Raman intensities for the pro-
bable structures. The first was the bond polarizability model [Longb], which is a
simple extension to the matrix method, and the second was to calculate
polarizability gradients from first principles DFT calculations [Chan]. Both esti-
mates are for a strictly non-resonant Raman process. The results of the calcu-
lations using the bond polarizability model show that the very strong band at
162 cm-' corresponds to a very weak BI mode in the DTB. This mode actually
correlates with a Raman inactive mode in the D,, lirnit (undistorted trigonal bi-
pyramid). Conversely, for PT this band corresponds to the strongest A, mode,
suggesting that we are looking at PT clusters. A cornparison of the observed
spectrum with the DFT predicted spectrum is shown in Figure 3-6 and summar-
ized in Table 3- 1. Harmonic frequencies from other first principles calculations
are also summarized in Table 3-1. The DFT predicted spectra were generated by
summing Lorentzians weighted by the calculated intensities and centred at the
appropriate frequencies. The Lorentzian widths were chosen to be small for
clarity. From Figure 3-6 it is clear that the calculated PT spectrum more closely
resembles the experimental spectrum
in the 150 to 200 cm'' region than the
ca lcu lated DTB spectrum does. Specifi-
cally, a strong band is predicted for PT
at -160 cm-' with a weaker band at
slig htly higher frequency, which is
exactly what is seen in the experi-
mental spectrum. Overall the frequen-
cy spacing of the observed Raman
Iines more closely resembles the pat-
tern calculated for the PT. The predict-
ed frequencies for the DTB tend to be
bunched up towards lower frequencies,
unlike what is observed experimental-
ly. Al1 of these factors taken together
DTB ,
* : . . i .
i . . . . . .
1 I 1
50 100 1 5 0 200 Raman Shift / cm"
clearly indicate that the clusten in our mu* 34: predided *g5 Raman spedra-
matrix are exclusively PT. I n making this assignment the bands at 100 and
105 cm-' are assigned to a single normal mode, possibly split by different matrix
sites. Although PT is unquestionably favoured by the theoretical spectra there
are substantial differences between the predicted spectrum and the experirnent-
Table 3-1. Cornparison of experimental (EXPT) and calculated (CALC) vibrational data for the low-energy Ag, isomers. The calculated frequencies are from Chan 1999, Warken 1998, Poteau 1997, and Srinivas 1998, - EXPT / cm-'
Planar
CALC / cm-'
186, 189, 169, 187 155, 161, 161, 155 147, 155, 148, 145 118, 123, 115, 120
100, 103, 99,100 80, 90, 93, 78 73, 76, 73, 73 33, 27, 19, 32 29, 24, 17, 28
Distorted Trigonal Big
CALC / cm"
183, 191, 180, 181 148, 155, 123, 150 112, 118, 114, 113 109,96,92, 106 82, 96, 92, 83 79, 91, 85, 76 63, 76, 80, 63 52, 65, 55, 49 46, 60, 47, 41
1 INT 1 SYM
al one that must be addressed. In
particular, the A, mode at 100 cm-' is
predicted to be very strong; however, . -- the experimental peak is actually very
from polarizability derivatives and take
no account of resonance effects. Since we are in fact weakly resonant, it is Iikely
that the disagreement between the calculated and the experimental spectrum is
due to the pre-resonance.
Determination of the structure of Ag, follows closely the argument used
for Ag,. As with Ag,, quantum chemical calculations also predict two low-energy
isomers for Ag,. These isorners are tricapped tetrahedron (TCT) and pentagonal
bipyramid (PBP), or slightly distorted variants of these (see Figure 3-7). Raman
spectra for each of these structures were predicted by DFT methods [Chan] and
are shown in Figure 3-8, along with the
experimentaf spectrum. Table 3-2
summarizes the vibrational data for
Ag,. It is immediately apparent from
Figure 3-8 that the simulated spectrum
of the TCT more closely resembles the
experimental spectrum than that of the
PBP. I t should be noted, however, that
the TCT can be obtained by making
relatively small distortions to the PBP.
It is, therefore, not surprising that on
the whole the two simulated spectra
resemble each other. The major quali-
tative difference between the two cal-
culated spectra is the presence of a
moderately intense band just below
PBP
r
TCT 1
120 7 6 0 21
Raman Shift /cms'
wurr 3-6: DFT predicted Ag, Raman spectra.
3 - 9
the band at 166 cm-' for the TCT, while for the PBP the corresponding mode is
predicted to have zero intensity. A second peak is indeed observed near the
strong band at 169 cm-' in the measured spectrum; however, it is observed to
corne at slightly higher rather than at lower frequencies with respect to the
169 cm" band. A comparison of the calculated and the observed spectra in the
frequency range 80 to 130 cm-' also favours the TCT structure. In fact, some
weak observed features are nicely predicted in this range for the TCT. One
should note that filtering the Rayleigh scattering causes the Raman peak inten-
sities at the low frequency end of the spectrum (i.e below -100 cm-') to be
attenuated. Clearly the spectral features computed for the TCT structure are
more agreeable with what is observed than those predicted for the PBP form of
t a b h 3-2. Summary of experïmental (EXPT) and calculated vibrational data for the Ag, structures. Calculated frequencies are given for both the density functional calculations ( D m [Chan] and the matrix calculations (FIT). The intensities ( I M ) are from the DFT calculations. Although the calculations produce slightly distorted
stri v
EXPT / cm"
ures, the symmetries (SYM) are tabulated for
Tnca
DFT / cm-'
164 163 166 121 106 114 114 94 91 91 79 80 68 38 37 -
ped Tetrahedron (C,)
INT - 7 4 39 4 O 5 5 14 5 5 6 6 4 3 3 -
SYM - E E 4 4 4 E E 4 E E E E 4 E E -
agonal Bipyrarnid
FIT / cm-'
175 175 169 120 120 113 113 129 87 87 86 86 68 57 57 -
INT - O O 47 1 1 6 6 O O O 11 11 9 O O -
The confidence in this conclusion is considerably strengthened through the
following demonstration. A complete force constant matrix is extracted from the
results of the Dm calculations for optimized versions of the two structures. In-
teraction force constants whose magnitudes are less than 8 O/O of the srnallest
primary bond-stretching force constant are set to zero and the remaining force
constants are grouped according to type. For bath structures, this produces a
seven-parameter force field. These force constants and the D R optimized geo-
metries are used as starting points for matrix calculations. The frequencies of
the TCT structure are made to agree rather well with the experimental spectrum
(Figure 3-9 and Table 3-2) by adjusting only a single interaction force constant.
Specifically, the secondaqt peak near the 169 cm-' band is shifted to the high
energy side of that band. By contrast threeforce constants (including one pri-
mary force constant) have to be adjusted in order to obtain an optimum fit to
the experimental spectrum assuming the PBP structure. However, that "opti-
mum" fit is still qualitatively unacceptable since no mode with any intensity can
be sufficiently displaced in frequency using this strategy in order to make it coin-
cide with the observed band at 175 cm-'. This demonstration provides strong
support of the conclusion that the Ag,
in our matrices has a tricapped tetra-
hedral structure.
Finding that a particular isomer
exists in the argon matrix does not
immedîately indicate that it is the low
energy form in the gas phase. Al-
though the clusters are generated in
the gas phase, they are generated as
hot ions. It is possible that the cationic
precursor leads to a particular isomer
of the neutral that is not the lowest
energy one in the gas phase. However,
the deposition process is very energe-
tic. Both the deposition energy and the
neutralization energy are available for Raman Shift /an"
-un 3-9: FG rnatrix method fit Ag, Raman spectra.
3 - 11
cluster internal dynamics for a finite period of time. I n fact there is enough ener-
gy that some of the clusters fragment. With ail of this energy available it is hard
to imagine that the barriers to isomerization cannot be overcome. As the energy
dissipates into the matrix the cluster will preferentially cool to the lowest energy
state. Furthermore, it is unlikely that the cluster-matrix interaction is sufficiently
strong to significantly change the relative stabifi3es of the isomers. Although it
was always diligently looked for, there is no evidence of more than one structure
existing in any of the deposits. For this to be the case the energy differences be-
tween the various isomers has to be reasonably large. It is inconceivabfe that
the cluster-matrix interaction is strong enough to alter the stabilities enough to
make a higher energy form in the gas phase the only form found in the matrix.
One must therefore conclude that the isomer found in the matrix of deposited,
neutraiized cationic clusters is also the low-energy form in the gas phase.
The electron spin resonance spectrum of Ag, clusters grown in a neon
matrix [Bach] suggests that the clusters are PBP. The spectral pattern indicates
that there are two nuclei in one environment and five in another. Paramagnetic
hyperfine structure calculations support the assig nment [Arratia-ferez] . This
discrepancy between what is found for grown versus deposited clusters is most
likely due to the method of production. For cold clusters grown atom by atom in
the matrix (i.e. Ag,., + Ag - Ag,), both the matrix and the form of the Ag.., pre-
cursor will direct the assembly of the Ag, cluster, unlike what happens in the de-
posited cluster experiments. Furthermore the grown clusters do not have energy
available to overcome internal barriers to isomerization. Therefore, it is not
necessary that the deposited clusters and the grown clusters have the same
form.
The most striking feature found upon inspection of Figure 3-2 is the domi-
nance of a band at -165 cm'' in al1 three spectra. For both the Ag, and the Ag,
assignments this band corresponds to a totally symmetric breathing mode. That
is, a mode in which al1 of the atoms move outwards together and then move
back inwards together. Indeed, the strong peak at -165 cm-' in the predicted
spectra of both isomers of Ag, is the breathing mode. Compared with other
modes, this one gives the maximum change in the volume of the cluster for the
outgoing (or incoming) half-cycle of the vibration. The intensity of a nonreso-
nant Raman transition, classically, is proportional to how much the polarizability
of the cluster changes along the corresponding normal CO-ordinate. Since the
polarizability approximateiy correlates with the volume of the cluster, the mode
that gives the maximum change in the volume, the breathing mode, will also
give the maximum change in the polarizability. Hence, the breathing mode is
expected to correspond to the most intense band in the spectrum. This is exactly
what is observed for Ag, and Ag,. It follows then that it is very Iikely that the
single, strong band at -165 cm'' assigned to Ag, is also the breathing mode,
regardless of the detailed structure of Ag,.
As the cluster size gets very large its properties are expected to approach
bulk properties. For different materials, and indeed for different properties, the
size of cluster required to see bulk-like behaviour greatly varies. I n general,
metal clusters begin to reflect the bulk parent material a t very srnall sizes, com-
pared with semiconductor clusters [Alivisatos]. As the size of a cluster is lowered
from macroscopic dimensions, the electronic bands are modified at the top and
bottom first. The centre of the band is not affected until very srnall cluster sizes
are reached. For metals the Fermi surface is in the centre of the band and is the
dominant factor for determining the properties of the material. Since the centre
of the band is not modified until very smali cluster sizes, decreasing the size of
metal particles has Iittle effect on their properties (at least until very small
sizes). Conversely, the properties of semiconductors, with completely filled and
completely empty bands, are rnostly determined by what happens at the edges
of the bands. Hence, decreasing the size of semiconductor particles tends modify
the observed properties for even relatively large sizes.
The bulk properties most relevant to a cornparison with the cluster Raman
spectra are obviously the vibrational properties (i.e. phonons) of bulk silver. Bulk
silver crystallizes on a face-centred cubic lattice, with one atom per primitive
unit cell, and therefore has only one phonon branch (the acoustic phonon
branch). Raman spectra of crystals can only be excited (with visible wave-
lengths) for phonons with km O, where k i s the magnitude of the phonon wave-
vector. More generalfy, this condition is k- k, where k, is the magnitude of the
exciting radiation wavevector; k, is about 1000 times smaller than k,,, (i.e. k a t
the edge of the first Brillouin zone) for visible wavelengths, and hence the con-
dition for Raman scattering is commonly stated as kN O. Raman scattering off of
crystal phonons is described in detail by Hayes and Loudon [Hayes]. For acoustic
phonons, O(& = v k f o r low values of k, where v i s the speed of sound in the
medium; hence, O O for k~ 0. Inelastic scattering of this type is commonly
called Brillouin scattering and clearly cannot be meaningfully compared with the
cluster spectra. I n fact the phonons that can be compared with cluster vibrations
are at the opposite extreme of the first Brillouin zone, k = km,, For these pho-
nons the length scale of the vibrations ( l /K) is equal to a couple of bond lengths,
or on the order of the cluster sizes.
The phonon dispersion curve for silver has been measured by neutron
scattering for k right up to the edge of the first Brillouin zone [Kamitakahara].
However, the first Brillouin zone is not a sphere in kspace, and hence kmx de-
pends somewhat on the propagation direction of the excited phonon. It is not
clear which direction shoufd be used when comparing bulk phonons with cluster
spectra. One way around this dilemma is to approximate the first Brillouin zone
by a sphere of equivalent volume in hspace. This is what is done in the well
esta blished Debye rnodel [Kittel]. In the Debye model, the further approximation
is made that o(k) = v k holds for al1 values of kand that vis independent of k.
This way only the speed of sound and the unit cell parameters are required to
describe the phonon behaviour. k,, in this model is referred to as the Debye
wavevector kD and has a corresponding Debye frequency U,
The Debye frequency for bulk silver is 156 cm-' [Kittel]. This falls close to
the dominant vibration, the breathing mode at -165 cm", in the Raman spectra
of Ag,, Ag,, and Ag,. I t follows that the forces between the silver atoms, and
hence the bonding, is comparable for clusters as small as Ag, and for bulk silver.
A comparison with the measured phonon dispersion curve around k,, verifies
that W, is a reasonable value for w,, for the longitudinal phonons. (The longi-
tudinal acoustic (LA) and transverse acoustic (TA) phonon branches coincide for
small k but split as one gets closer to km,,) Indeed, u,,, for the LA phonons
along some of the important (and hence measured) directions agrees even bet-
ter than W, with the frequency of the breathing mode in the cluster spectra. For
example, W, along the 001 direction equals 167 cm-'. Amazingly, 0, for TA
phonons along some significant directions (e.g. 001) falls exactly where the
second most predominant feature in the Ag, spectrum, the peak at 113 cm-', is
found. A close scrutiny of the Ag, spectrum (see Figure 3-2) suggests that there
may be a second peak at 4 1 3 cm'' in the Ag, spectrum also, indicating that
perhaps this feature correlates with the TA phonons in the bulk limit.
One further comparison with bulk matter is in order. A band at 161 cm-' is
routinely obsewed in Raman spectra of rough silver films and has been assigned
to an Ag-Ag vibrational band [Roy]. Such an assignment is consistent with the
conclusions drawn from the cluster Raman spectra. The feature in the surface
spectra is due to the vibrations of surface cluster-like features. However, the as-
signment of this band specifically to Ag,' surface clusters, which has also been
made [Roy], obviously has no merit. It is clear from the Ag,, Ag,, and Ag,
Raman spectra and from the location of the Debye frequency that clusters rang-
ing in size from a few atoms to the bulk will have prominent vibrations at or near
165 cm-'. The assignrnent of the band seen in the surface spectra cannot be
made to a specific cluster size or geometry. More likely, the observed band
arises from the overlap of many closely spaced bands originating from the large
variety of silver features on the rough silver surface.
4. Raman of Ag, and Fe,
As with the larger clusters treated in Chapter 3, the Ag,' current is high
enough to study these clusters' Raman spectra after depositing them with argon.
Furthermore, the generality of the cluster machine permits almost any metal (or
other material) ta replace silver and clusters of that metal to be studied. How-
ever, not al1 metals produce as high a cluster current as silver. Usually the cur-
rent decays approximately exponentially with increasing cluster size and some-
times with a cluster "stability" function superimposed (e.g. the odd-even alter-
nations in the silver mass distribution). Sputtering iron produces a distribution of
clusters that drops off very quickly. I n
fact, Fe, and Fe, are the only clusters
that have enough current for Raman
studies of the bare clusters.
Figure 4-1 shows the resonant
Raman spectrum of Ag, in an argon
matrix ( 1 . 6 ~ 1 0 ' ~ clusters in a 2 mm
diameter spot; 104 argon-to-cluster
ratio; 50 mW, 488.0 nm). The strong
peaks at 189 and 370 cm-' belong to
Ag, [Schulzeb], produced by fragment-
ation of some of the clusters upon de-
position. The broad feature centred at
228 cm-' has previously been assigned
to low frequency lattice modes coupled
to the Ag, stretch [Bechthotd], and the
weak peak at 102 cm-' is from an in- Raman ~h i f t~crn"
compietdy filtered laser emission. Al1 flgui. cl: Resonant Raman spectrum of ~ g , in argon.
of the other peaks in the spectrum (see Table 4-1) are assigned to Ag,. At first it
seem like there are grossly too many peaks in the spectrum for them all to arise
from a three atorn molecule. However, it will be shown in Section 4.5 that al1 of
these peaks can be accounted for if dynamic Jahn-Teller effects are considered.
The dynamic Jahn-Teller effect is the subject of most of the remainder of this
chapter.
The resonant Raman spectrum of Fe, in argon ( 9 . 4 ~ 1 0 ' ~ clusters in a
2 mm diameter spot; I O 4 argon-to-cluster ratio; 100 mW, 496.5 nm) is shown in
Figure 4-2. The weak fluorescence bands appearing at 704, 732, and 798 cm-' in
this spectrum are tentatively assigned to Fe0 impurities, while a broader and
moderately strong fluorescence centred near 850 cm-' is assigned to Fe, [Mos-
kovitsb], present from fragmentation. The broad fluorescence is subtracted for
clarity in the main spectrum in Figure
4-2, but shown in the inset. The peaks
at 291, 582, and 868 cm-' belong to
the strongly resonant Raman funda-
mental and overtones of Fe, [Mos-
kovitsb]. The remaining peaks are as-
signed to Fe, Raman transitions (see
Table 4-2). The relative intensities of
these peaks, as with the Ag, spectrum,
are the same from one deposit to the
next, indicating that the peaks al1 arise
from the same species with the same
geornetry and matrix site. Also, as with
the Ag, spectrum, the large number of
peaks assigned to Fe, Raman transi-
tions is explained in Section 4.5 in
terms of the dynamic Jahn-Teller
effect. Raman Shift /cm" flgurr 4-2: Resonant Raman spectrum of Fe, in argon.
The blue-green wavelengths used to excite the Ag, Raman lead to a fast
photo-bleaching of the signal, with no new Raman bands emerging. When col-
lecting the Raman spectra, a strategy of averaging short integrations collected
on fresh spots was required. Irradiation of the entire deposit with an unfocus-
sed laser, while periodically monitoring its UV-visible absorption spectrum,
reveals that the Ag, absorption bands gradually decrease, with no new bands
appearing. Increasing the argon-to-cluster ratio from IO4 to 8 x 104 slows the
bleaching considerably, indicating that the bleaching is due to a photo-assisted
diffusion process and not to a transformation to different isomers or matrix sites.
For the diffusion mechanism, excitation of the cluster followed by nonradiative
decay heats the matrix locally. The cluster then diffuses around in the matrix,
taking its "hot spot" with it. There are three possible fates for the diffusing clust-
er: (1) it can contact another cluster and aggregate, (2) it can diffuse to the sur-
face of the paddle and stick there, or (3) it can diffuse out of the radiation field
and cool. Many smatl silver cluster absorption spectra have been measured [e.g.
Harbicha] and invariably show strong absorption bands. It is therefore very un-
likely that any silver-cluster / matrix-site cornbination could show no features in
the UV-visible absorption spectrum. It is possible, though, that new bands ap-
pearing in the UV-visible spectrum are masked by bands that are already there.
For example, if two Ag, clusters aggregate to form an Ag, cluster, and the Ag,
band is weaker than two times the Ag, band but in the same spectral region,
then it will appear as if no new bands are emerging.
To determine the likelihood of each of the possible fates, their character-
istic length scales are compared. With an argon-to-cluster ratio of IO4, the
average cfuster-cluster separation is about 8 nrn, where an argon density of
1.6 g cm-' has been used [Schulzea]. The rnatrix is a couple of micrometres
thick, and the radiation field is a few hundred micrometres across. The third pos-
sible fate is obviously insignificant since it is much more likely that the cluster
wili contact the paddle before diffusing out of the radiation field. The length-
scale argument favours the first possible fate, but since it appears that no new
bands are growing in the UV-visible absorption spectrum, the additional assump-
tion that the absorption bands for the new aggregates are masked by the Ag,
bands is required. The second possible fate is also reasonable and completely
consistent with the apparent spectral changes. I f the cluster sticks to the paddle
it becomes part of the metal surface and no longer shows the Raman or UV-
visible absorption spectra of the free duster; furthermore, no new bands appear
in either spectrum as a result of the cluster becoming part of the metal surface.
Unlike the silver clusters, the iron clusters do not show "photochernical" beha-
viour.
Man y metal trimers are fluxional and exh ibit complex spectroscopie pat-
terns due to dynamic Jahn-Teller effects. The Jahn-Teller distortion problem is
an old one, dating back over sixty years [Jahn]. Oetailed reviews of the subject
are given by Herzberg [Herzberg] and Englman [Englman]. Basically, Jahn and
Teller demonstrated that any molecule with a degenerate electronic state is un-
stable with respect to structural changes that lift the degeneracy. For the case of
homonuclear trimes, the equilateral triangular configuration (4,) has a three-
fold axis of symmetry and hence can support a doubly degenerate electronic
state. Jahn-Teller distortion from this state leads to one of three equivalent
forms, each occupying a potential minimum. The point on the potential energy
surface corresponding to the original D,, structure is a local cusp-like maximum.
The difference between the potential energy at the 4, point and at the global
minimum is the Jahn-Teller stabilization energy Ep
When b i s small compared to the zero-point energy, the energy levels of
the degenerate vibration (of the D, structure) split into tightly spaced multi-
plets, as is observed in the Raman spectrum of Mn, [Bier]. For &much larger
than the zero-point energy, the dynamics of the atoms correspond to a "pseudo-
rotation" about the D,, configuration and result in a spectrum that appears simil-
ar to rotational fine structure. Na, is an example of a molecule showing this
"pseudo-rotational" behaviour [Delacretaz]. For these two extremes (the weak
and strong linear coupling regimes), analytical solutions to the vibronic problem
exist. For many molecules Enis on the order of the zero-point energy and quad-
ratic coupling is significant (e.g. Cu, [Zwanziger]). Furthermore, once the vibra-
tional motion and the orbital motion are coupled, spin-orbit coupling may further
split the vibronic spectrum. For example, spin-orbit coupling is significant for Au,
[Bishea). For these more involved cases, diagonalizing the complete Jahn-Telfer
Hamiltonian matrix in a truncated basis of harmonic oscillator eigenstates pro-
duces the vibronic (or spin-vibronic) spectrum variationally. Quadratic coupling
leads to three equivalent saddle points on the potential energy surface, which
separate the three global minima, and, if strong enough, localizes the structure
at one of the minima (the static Jahn-Teller effect). The energy difference bet-
ween the global minima and the saddle points is the localization energy &
The contributions made by Longuet-Higgins and other pioneers essentially
define the conventional theoretical approach to the Jahn-Teller problem
[Longuet-Higgins]. Recent reviews of the theory are presented by Barckholtz
and Miller [Barckholtz] and by O'Brien and Chancey [O'Brien]. Operator methods
are among the more powerful and elegant methods for tackling quantum mech-
anical problems [Oss, Kellman] but appear not to have been fully exploited in
the analysis of the Jahn-Teller problem, in spite of its high symmetry. A new
strategy for determining the Hamiltonian rnatrix elements based on operator
methods is presented here for an E x e Jahn-Teller system with a three-fold axis
of symmetry. That is a doubly degenerate electronic state coupled with a doubly
degenerate vibrational state [see also Bosnickb]. This is precisely the case for
Ag, and Fe,. The method is new; however, some of the group theoretical con-
cepts used are similar to ones previously applied to this problern, but in a dif-
ferent context [Alper].
Any discussion of operator methods in a vibrational problem starts from
the well known [Messiah] raising and lowering operators for a system of harm-
onic osci!lators, whose commutator relations are given by
[a, a,] = O = (ai, $1 and [a, 41 = (4- 1)
For convenience, we define a new set of operators,
" . ' &aJt 41
whose commutator relations,
[& L;;,..] = C;,. - c.;,.O~,~., (4-3)
follow from (4-1). Note that Giis the number operator for the P oscillator. Since
we are interested in an e vibrational state, we will restrict ourselves to two oscil-
lators and define new operators as
f, I l 2 (Cl,,l - Czz), F+ and E 2 C,,. (4-4)
The commutator relations for these operators are easily found from (4-3) to be
[F, FJ = f F* and [F,, fi = 2 F, (4-5)
The cornmutator relations given by (4-5) define the suf2) lie algebra and
are more comrnonly found associated with angular momentum problems
[Rowe]. It is an unusual coincidence that the same operators apply to a two-
dimensional oscillator and to angular momentum in quantum rnechanics. This
provides a formal link between ordinary angufar momentum and such concepts
as "vibrational angular momentum" and "pseudo-rotation". This is one of the
main ideas used in the analysis presented here. To fully exploit the connection
with the theory of angular momentum, new operators are defined in terms of
those given by (4-4),
f, = % (f+ F,), f 2 ( + ) , and F,= Fp
These form a vector operator, F = (F, f, 61, and are the analogs of the com-
ponents of an angular rnomentum vector operator. It is well known (from the
theory of angular momentum) that f, f, and c d o not mutually commute.
However, the Casimir invariant, given by
F2 = FI + + F,2 = 112 (Cl,, + GZ) Il12 (& + C'J + 11, (4-7)
does simultaneously commute with each of F, F, and F, The second equality in
(4-7) follows from the application of (4-6), (4-4), (4-2), and (4-1). Defining the
total number operator as N = Cl,, + C,, (4-7) simplifies to Y = N/2 (N/2 +l) ,
which can be solved to give
N = (4 F2 + 1)" - 1. (4-8)
Restricting ourselves now to a two-dimensional ~sotmpic harrnonic oscil-
fator (as is appropriate for the e vibrational state under consideration), the har-
monic oscillator Hamiltonian is given by
4 = (N+ 1) ho = (4 Y + 11% no, (4-9)
where the second equality in (4-9) follows from (4-8). At this stage it is usefuf to
introduce basis vectors, [n, n>, for the vibrational States, where n, and n, are the
num ber of quanta of type 1 and type 2 that are excited, and can have values O,
1, 2, .... These are the eigenvectors of & with eigenvalues (n, + n, + 1) h ~ . Since we are recasting the Jahn-Teller problem in the form of an angular mo-
mentum problem it will prove to be more useful to work with the alternate set of
basis vectors, km), which are defined by
r n 2 ( n , ) and m = -4-f+l , ..A (4-10)
Making a connection with the theory of angular momentum, we see that these
are the simultaneous eigenvectors of Fz and f, whose eigenvalues are given by
H, simultaneousiy commutes with F2 and Fo Therefore the km) are also eigen-
vectors of H,,; that is
where the second equality in (4-12) follows from (4-11) and the rules for eval-
uating eigenvalues of a function of an operator. The actions of the raising and
lowering operators on our first basis set, b, ni', are well known. I n terms of the
alternate basis set, the actions of the raising and lowering operators follow from
(4-10) and are given by
We now turn our attention to the electronic part of the problem. Consider-
ing the Eelectronic state to be an 5 = 'h particle, we can define operators and
basis vectors in the same manner as was done above for the vibrational prob-
lem,
The electronic and vibrational basis sets can be combined into a single vibronic
basis set, k fm) = k) km). We now have al1 the machinery we need to solve the Jahn-Teller coupling
problem. The general vibronic Hamiltonian is given by H = & + A',, where
NCOUP is a coupling Hamiltonian. The electronic Hamiltonian can be ignored be-
cause it has no effect on the vibronic energy level spacing. Linear Jahn-Teller
coupling involves coupling between states for which Af = f Y2 [Barckholtz]. The
simplest, symmetric coupling Hamiltonian that satisfies this criterion is
4' = a ho (a, + a,' + a, + a,'), where a is a linear coupling parameter. The
vibrational part of the Hamiltonian is now fixed, but the electronic contribution
must also be considered. The electronic part of the linear coupling Hamiltonian is
determined by the requirement that/* = 2m + e remain a good quantum num-
ber for a molecule with a three-fold axis of symmetry [Barckholtz]; is the well
known pseudo-rotation quantum number, originally introduced by Longuet-
Higgins [Longuet-Higgins]. N,' is therefore modified to
n, = a no ((a, + a,+) L+ + (a, + a,') c). (4-1 5)
The matrix elements of H, in the vibronic basis are easily determined from (4-
15), (4-14), and (4-13). The resulting matrix elements are the same as those
obtained by a more conventional analysis [di Lauro] with a = (24" = k, where
D and k are conventional linear 3ahn-Teller coupling parameters. The energy
levels generated by numerically diagonalizing the Iinear coupling model in a
truncated vibronic basis set are shown in Figure 4-3.
The quadratic matrix elements are determined by the sarne strategy as
was used for the linear matrix elements. We begin by postulating H,' = P h o x { ( a , + a,')' + (a2 + a,+)'> as the vibrational contribution to the coupling
Hamiltonian. This involves coupling between States with Af = 0, Il. The elect-
ronic part of the Hamiltonian is determined by the requirement that the well
knownj2) = m - e remain a good quantum number (for a molecule with a three-
fold axis of symmetry [Barckholtz]).
This is accomplished by setting 1
When comparing (4-16) and (4-15)
one should note that the positions of
L+ and L- have been reversed. This
reversal follows from the negative sign
in the expression for/'). The matrix
elements for H, in the vibronic basis
are easily evaluated using (4-16),
(4- le), and repeated application of
+ + + C + + + + C
C
+ 4
* (4-13) and are the same as what one l . - - - - l . . - - . ~ - - . - - l - - - . - .
0.0 0.5 1 .O 1.5 2. O gets using a more conventional anal- a ysis [di Laure] with p = '12 K = '12 9. npurm 4-3: Energy levels for the linear E x e Jahn-
Teller model.
(Kand g are traditional quadratic Jahn- I 1
Teller parameters.) This agreement
justifies the strategy used in the deri-
vation. The energy level spacing ob-
tained by numerically diagonalizing the 2 . Iinear plus quadratic coupling model in a 1 - truncated vibronic basis set is given in
Figure 4-4, with a fixed at O.S. O I I 1 1 1 O O 0 0.05 0 10 O 15 0.20 O 25
The coupling of an e vibrational P
state with an Eelectronic state pro- Figuri. 4 4 Energy levels for the linear plus quadratic Jahn-Teller model.
duces vibronic states with either A,, 4, or Esyrnmetry (e x E = A, + A, + 8. The major effect of quadratic coupling on the energy spectrum is to remove the
acc~Yenta/ degeneracy of the A, and A, vibronic states. The degeneracy is acci-
dental in the sense that it is not required by the symmetry of the molecule; it
only exists because there are no terrns in the Iinear coupling Harniltonian that lift
it. When both linear and quadratic couplings are simultaneously operative, neith-
er/" nor/'j rernains a good quantum number; however, /"/ =/'/ modulo 3
does. Vibronic states withp2) = * 1/2 have Esyrnmetry and those with /la =
+ 3/2 have A, or A, symmetry.
The linear plus quadratic Jahn-Teller coupling model has three adjustable
parameters: a, p, and @, The truncation of the basis set must be made a t a
high enough value of cdX ta ensure convergence for the lower eigenvalues. Ad-
justments are made to the parameten to fit the calculated energy levels to the
observed Raman bands. One band (or possibly a progression) remains and is
assigned to the Jahn-Teller inactive a, ' vibration. The results of the fit to the Ag,
Raman spectrum are given in Table 4-1, with a = 1.92, P = 0.146, and o, =
99 cm-'. Frequencies from a dispersed fluorescence spectrum of Ag, in a molec-
Table 4-1. Jahn-Teller fit to the resonant Raman spectrum of Ag,. The frequencies from a dispersed fluorescence sDectruni
!n for cc 5 Raman / cm-'
74 88
150
199
257
299
320
362
377
411
436
478
538 -
ular beam are also given for cornparison [Ellis].
The band a t 119 cm-' in the Raman spectrurn is
assigned to the s, 'vibration, in agreement with
the conclusions of the dispersed fluorescence
and earlier Raman experiments [Kettler, Schul-
zeb]. All of the remaining thirteen peaks are ac-
counted for by the Jahn-Teller fit, with a rms
deviation of 2.6 cm-'. Following Wedum and CO-
workers [Wedum], Er= 112 a2 U, and E, = En
~ ( ( 1 - 2 p ) - ~ - (1+2p)-'), and the parameters
found from the fit therefore indicated that En=
182 cm-' and eE = 116 cm-' for the ground
state of Ag,.
Inspection of Table 4-1 clearly shows
that the resonant Raman and dispersed fluores-
cence spectra are very similar, indicating that
the matrix perturbation is srnall. This agree-
ment is a t first surprising, since transitions to
the/"' = +3/2 States are symmetry forbidden
for fluorescence from the F u p p e r state [Ellis],
but al1 transitions are allowed for Raman from
the E'ground state. (The symmetry of the
ground state with respect to the plane of the
molecule (Le. ' or ") is known from quantum chemical calculations [e.g. Balasub-
ramanian].) Yet the only observed transition to a p a = +3/2 state is the band at
88 cm". This observation can be explained by considering the selection rules for
a truly resonantRaman process. The intensity of a Raman transition from the
molecular ground state O to the rnolecular excited state m [Craig] is given by
where 1, is the incident irradiance; k and kt are the incident and scattered radia-
tion wavevectors, respectively; p" is the transition dipole moment vector bet-
ween states m and r, a and a' are unit polarization vectors for the incident and
scattered radiation, respectively (the bar indicates complex conjugation); and Fm
is the energy difference between states rand o. The summation is over al1
states. For non-resonant Raman, 1 E, - tickl a O for al1 rand approximations can
be made [Craig] to simplify to the contents between the vertical bars in (4-17)
to (X , ,~ I gg 1 x,) times a polarization function; where g is the usual frequency
dependent polarizability tensor, and &,, and )(,,.are the vibrational parts of the
Born-Oppenheimer wavefunctions for the ground and vibrationally excited
states, respectively. The selection rules for non-resonant Raman follow from
consideration of the symmetries of these states and of the polarizability tensor.
& is on the order of hckfor a resonant Raman process involving reson-
ance with state RI and the approximations made for the non-resonant case are
inappropriate. Instead, the term in the summation corresponding to R dominates
the sum, and the other terms, being much smaller than the "resonant" term,
can be ignored. The contents of the vertical bars simplify to (pmR-ë' ) (pm .e) ERO - tic&
Note that an imaginary damping term is sometimes included in the denominator
and prevents a singularity in the case of exact resonance. We are interested in
scattering by a large number of clusters that are randomly oriented and there-
fore can ignore the polarization vecton, as al1 polarizations are equally repre-
sented. We reach the important conclusion that the selection rules for resonant
Raman are determined by the products v/ x py, which are equivalent to ab-
sorptions from state o to R followed by fluorescences from R to m. Recall that
p y = (IV, 1 vil wd and is non-zero only if rw, x h, x ru, contains the totally
symmetric irreducible representation of the point group, where r, is the irredu-
cible representation to which x belongs.
Under the Dm point group {p, pr) transform as P'and pz transforms as
4': The ground vibronic state O has E'symmetry, but the symmetry of the re-
sonant state R is unknown. The symmetry of the final vibronic state m is A, ; 4; or E: The "absorption" part of the resonant Raman transition is forbidden if state
R transforms as 4 " o r A,': This leaves A,: A,; E: and EMas possible candidates
for the symmetry of R. "Fluorescence" to E'states is allowed from any of these
upper states, but "fluorescence" to 4 ' o r A,'states is only allowed from the E'
upper state. I n conclusion, the observed selection rule that the only allowed re-
sonant Raman transitions are to/la = + 1/2 (El states is consistent with a re-
sonant state that has A,; A,; or EHvibronic symmetry. The single/f2" = + 3/2
transition a t 88 cm-' is possibly observed as a non-resonant Raman allowed
transition.
Attempts to fit the linear plus quadratic coupling model to the Fe, Raman
spectrum failed to account for al1 of the observed peaks in the low frequency
region. However, the bunching of the stronger peaks into groups is suggestive of
a wea k Ja hn-Teller interaction. Two examples of possible assignments based on
the linear plus quadratic model, but with different assignments of the symmetric Table 4-2. Two possible fits of the linear plus quadratic Jahn- Telier model to the Fe, Raman - Raman / cm-1. 146 178 185 202 215 249 313 334 359 372 390 420 438 465 54 1 631 657 826 946 -
spectrum . Trial 1 / cm-l
126 180
223 236 247 276/312 333/336 354 372/373
422 438/443 460/470 531/532 f
* f
t
I
stretch, are given in Table 4-2. The asterisks indi-
cate a spectral region where the number of calcu-
lated energy levels exceeds the number of observed
bands. Values of a, Pl W, and w, of 0.5, 0.04,
108 cm-', and 178 cm-' were used for Trial 1, and
0.15, 0.08, 159 cm-', and 249 cm-' for Trial 2. The
poor correspondence and unassigned peaks is most
likely due to the high spin state of the Fe, ground
state. The spin state is known to be S= 4 from
Stern-Gerlach deflection experiments [Cox] and
from quantum chemical calculations [e.g . Castro].
I n a D,, system the reducible representation that
the S = 4 spin state transforms as decomposes to A, '+ A, "+ 4 3 2E+ EH[Her-
zberg]. Coupling the spin with the Jahn-Teller vibronic states of A, or A, sym-
metry produces six spin-vibronic states, and with vibronic states of Esyrnrnetry
produces twelve spin-vibronic states. Although not al1 states will be seen in the
spectrum due to selection rules and Franck-Condon factors, clearly the number
of allowed transitions can be very large and accounts for the many peaks seen in
the Raman spectrum of Fe,.
5. Cluster Enhanced Raman Scattering
I n 1974, Fleischmann and CO-workers recorded unprecedented Raman in-
tensities from pyridine adsorbed on a roughened silver electrode [Fleischmann],
ushering in the field of Surface Enhanced Raman Scattering (SERS). Much work
has been done in the field since that first discovery. Intensity enhancements of
IO6 are commonly found for a large variety of molecules adsorbed on mugh sil-
ver surfaces. I n addition to silver, SERS is observed on rough surfaces of gold,
copper, and a few other metals. It is known that roughness of the metal on the
nanoscale is necessary for a surface to support SERS. Vacuum deposited films,
aggregated colloids, single nanopaeicles, and, of course, roughened electrodes
are among the various morphologies on which SERS is observed. An excellent,
recent introduction to the field is given by Campion and Kambhampati [Cam-
pion]. SERS motivated us to ask the question, T a n silver clusters of the type
produced by the mass-selected cluster machine enhance the Raman scattering
of molecules adsorbed on them?".
The strategy used to answer this question is to deposit silver clusters in
carbon monoxide and record the Raman spectra in the carbon monoxide stretch
region. Comparison of the signal from the adsorbed carbon monoxide with that
from the bulk matrix determines if there is any enhancement. Although only
cluster nuclearities up to nine are resolved in the mass spectrum (Figure 2-4),
there is enough current to deposit much larger clusters, even up to Ag,,. To
locate a particular nuclearity in the unresolved region (relative to the Ag, peak),
Equation (2-1) is used to get the condition 4 = 4 (5/n)", where n is the desired
nuclearity, and Ln and Vs are the Wien voltages required for Ag, and Ag, to pass
through the filter undeflected. Figure 5-1 shows the location in the mass sped-
rum of the nuclearities used in this study. Periodically measuring the cluster
current on the Faraday plate throughout the deposit and integrating over the
length of the deposit gives the total I 1 1 t r I 1 r t f 50 35 20 9 5
number of clusters deposited in a O.' 1 A
2 mm diameter spot. 5 0 3 C I \ Since the cluster nuclearities are
not resolved, a small distribution of
sizes actually passes through the Wien ,, filter. An estimate of the breadth of
this distribution is found by modelling O O 20 40 60 80 100 120 14û Wein Voltage I V
the d~namics of the mass selection Rpum 5-1: Siiver clusters deposited for CERS expenments.
process. For this model it is assumed
that the distribution of initial kinetic energies ( 5) after sputtering is the only
factor that determines the distribution of sizes that passes through the filter. It
is further assumed that the cdistribution is the same for ail cluster sizes and
can be modelled by
where a and ra re parameters to be determined by a fit to the position and
shape of the Ag, peak in the mass spectrum. The assumed form of the distribu-
tion law given in (5-1) is roughly similar in shape to the Maxwell-Boltzman dis-
tribution of molecular speeds for a gas in thermodynamic equilibrium, and is, at
least, a reasonable form for the distribution of 7, Analysis of the dynamics (see
Section 2.3) gives the condition for a cluster to pass through the filter with its
trajectory undisturbed as
where l / , is the Wien voltage, g i s the width between the Wien plates (i.e. E=
1/, / g), and al1 of the other variables are as previously defined. Substitution of
(5-2) and m = n x m,, into (5-1) gives the distribution of nuclearities passing
through the filter P(n; V,) for a given Vw V- and g a r e determined from appar-
5 - 2
atus specifications, and BI a, and r a r e
determined by a fit to the Ag, peak in
the mass spectrum. Figure 5-2 shows
P(n; V,) for the V, used in the enhan-
cernent experiments ( 1/,, = 1200 V,
g = 0.02 m, B = 0.28 Tl a = 1 . 1 ~ 1 0 ' ~
3-', and r = 3). The rms deviations, de-
termined from the distributions for
each of the V, are summarized in Figuro S-2: Estimated distributions of cluster
Table 5-1. nuclearities deposited.
I n order to quantitatively determine enhancements the amount of carbon
monoxide deposited must also be accurately known. The deposition rate of car-
bon monoxide (nm s-') was measured by thin film interferometry, as described
in Section 2-7. This rate is converted to the number of carbon monoxide mole-
cules deposited in a 2 mm diameter spot by multiplying by the area of the spot
and the number density of frozen carbon monoxide. By basing the calculation on
a 2 mm spot, the amount of matrix deposited can be directly compared with the
measured number of clusters deposited. Carbon monoxide crystalizes on a body-
centred cubic lattice, with a unit cell length of 0.563 nm [Bailar]; this is equiva-
lent to a number density of 11.2 nm". The matrix is polycrystalline and probably
has a density slightly lower than this (see Section 3.1). Use of the crystal pack-
ing density therefore leads to a slight over-estimate of the number of carbon
monoxide molecules deposited and a corresponding slight over-estimate of the
enhancement.
Tabk 5-1. Summary The Raman spectra of Ag,,, Ag,,, Ag,,, and Ag, depos-
of estirnates of deposited cluster size
ited in carbon monoxide are shown in Figure 5-3 (c - f, res- distribution.
,-
pectively) in the carbon monoxide stretch region (100 mW, desired Lw 1 n 1 472.7 nm). A spectrum with just matrix material (b) and the
Ag,, spectrum with a less magnified vertical scale (a) are
46.5 also shown for comparison. The dominant peak at 2138 cm" 4.2
in spectrum (a) originates from the
free 12C160 and has been truncated for
clarity in the remainder of the spectra.
The weaker features at 2087, 2091,
and 21 Il cm" are from other isotopes
of the free carbon monoxide. The
broad band centred at 2206 cm'' is
assigned to free 12c"0 stretch - lattice
phonon combination modes. The car-
bon rnonoxide adsorbed on the silver
ciusters gives rise to the broad band at
the base of the free 12C160 peak, which
is not seen in spectra excited in re-
gions of the matrix that are free from
clusters. Cornparison of the area under l
2050 2100 2150 2200 2250 Raman Shift 1 cm-'
the free 12C160 peak with that of the 5-3: Raman spe- of Ag,. ~9~~ A039 and Ag, in carbon monoxide.
adsorbed carbon monoxide determines
the enhancement. These band areas, along with other experimental parameters,
are summarized in Table 5-2.
The area (A) under a Raman band is proportional to the product of the
cross-section (a) and the number of scatterers (4. For these experiments N is
based on the nurnber of scatterers in the 2 mm spot. The enhancernent is de-
fined as the ratio of the Raman scattering cross-section for adsorbed carbon
monoxide to that of free carbon monoxide and is therefore given by
To determine the total number of adsorbed scatterers given the number of de-
posited clusters, the number of adsorbates per cluster must be estimated. For
cluster nuclearities larger than twelve the number of surface atoms is estimated
by quadratically interpolating between the number of surface atoms for the
smallest three icosahedra structures (i.e. 12, 42, and 92 surface atoms for 13,
55, and 147 total atoms). For clusters of nuclearity twelve or less, ail the atoms
are surface atoms. Chemically synthesized cluster-carbonyls, which are small
enough to consider al1 of the metal atoms as "surface atoms", have on average
roughly two carbonyls per metal atom [Kettle]. However, these compounds are
thermodynamically stable and are composed of better carbonyl-forming metals
than silver, and hence the number of adsorbates per surface atom on the de-
posited silver clusters wiil be somewhat less than two. On many solid surfaces
one carbon monoxide rnolecule or less chemisorbs per surface atom until the
surface is saturated [Zangwill]. Therefore, the assumption of one adsorbate per
surface atom on the clusters seems reasonable and is used (see Table 5-3). The
number of free scatterers then follows as the total number of carbon monoxide
Table 5-3. Results of enhancement molecules deposited minus the number adsorbed.
ers, will be larger than what is given by this esti-
calculations for Ag,(CO),. I f the clusters have a more open structure
a,/ a, than an icosahedron then the number of surface 168 373 atoms, and hence the number of adsorbed scatter-
n
9 20
m
9 18
mate. Also, if there is more than one I 1 1 I 1 l
carbon monoxide adsorbed per surface ,,, - atom, the true number of adsorbed
scatterers will also be larger. 60th the 5oo -
possible under-estimate of the number
of adsorbed scatterers and the over- 400 - - estimate of the number of free scat- O, c
terer (from assurning crystal pacl<ing) 300 - s
increase the calculated enhancements. 5 Therefore, the enhancements reported
here are upper Iimits, and the actual
enhancements may be slightly lower 1 0 0 t
than these values. The enhancements
determined from Equation 5-3 are 0455 ~ 460 465 470 475 480 48s 490
given in Table 5-3 for the spectra Excitation Wavelength 1 nm
Flgurm 5-4: Measured Raman scattering enhance- excited with 472.7 nm light along with R, N, o).
the estimates of the number of adsorbed scatterers per cluster. A graphical
summary of the enhancements for al1 the excitation wavelengths used is shown
in Figure 5-4.
There are two well established enhancement mechanisms for SERS: the
electromagnetic mechanism (EMM) and the chemical mechanism (CM). A critical
review of the mechanisms of SERS is given by Moskovits [Moskovitsa]. The EMM
works by amplifying the incident and scattered radiation fields through resonan-
ce with the surface plasmon absorption (SPA) of the metal. This mechanism
explains why SERS is only seen on the surfaces of some metals and why rnicro-
scopic roughness, a necessary condition for efficient excitation of the SPA, is
required. It is generally accepted that the EMM is responsible for most of the
enhancement observed. However, some experimental observations cannot be
explained by the EMM alone. For example, the enhancement of the scattering by
carbon monoxide is about 50 times that of dinitrogen, even though the two have
similar gas phase cross-sections. The CM explains these observations. Essential-
ly, the CM involves the formation of new resonances that are highly localized at
the adsorbate-surface complex and may involve charge transfer. The localization
leads to these resonances being quasi-molecular in nature, and hence the mach-
inery of molecular science can be used to describe them. I n particular, the scat-
tering enhancernent by the CM is a resonant-Raman-like process.
The SPA is well defined for metal particles down to a few nanometres in
diameter [Bohren]. However, as the size of the particle decreases below the
electronic mean free path, the SPA decreases in intensity and broadens due to
surface scattering, and the enhancement due to the EMM correspondingly de-
creases. For clusters consisting of only a srnall number of atoms ( a 30) the elec-
tronic absorption band (EAB) shows structure [Harbicha], and these sizes can be
considered to be the molecular domain. For clusters a little larger than 30
atoms, the structured €AB coalesces into a single relatively sharp peak. Upon
proceeding from these sizes to the larger nanoparticles, the peak initially broad-
ens and then sharpens up again as the size increases above the electronic mean
free path of the bulk material.
Measurements of the €AB for matrix isolated clusters of sizes between 40
and 100 atoms, but with a relatively broad size distribution, have been made
using the gas-phase aggregation technique [Charle]. I n an argon matrix it is
found that the EAB is centred around 335 nm with a breadth of about 55 nm
(fwhm). I n a carbon monoxide matrix the EAB is found closer to the visible
(centred at 375 nm) and is broader (120 nm fwhm). These spectra suggest that
the Raman enhancement of the carbon monoxide adsorbed on the silver clust-
ers is due to a resonant (or pre-resonant) Raman process, where the resonance
is with the tail of the €AB in carbon monoxide. The cluster Raman enhancement
rnechanism is comparable to the CM of SERS; that is, both are resonant Raman
enhancements involving localized adsorbate complexes. However, for the case of
the complex on the surface of bulk silver (i.e. the CM of SERS), the resonance
will be dampened by interaction with the bulk electrons, and hence the CM of
SERS will Iikely produce a smaller enhancement than what is observed for the
matrix-isolated complex (Le. the clusters in carbon monoxide). It follows then
that the enhancements observed in the cluster experiments ( s 1000) are an
upper limit to how strong the enhancement due to the CM of SERS can be.
The shape of the adsorbed carbon monoxide Raman band is understood
by cornparison with the infrared absorption spectra of distributions of small silver
clusters produced by the gas-phase aggregation technique and embedded in a
carbon rnonoxide matrix (Froben]. The normal modes of the adsorbed carbon
monoxide are enumerated using a spherical harrnonic expansion. The mode of
S-symmetry is Raman active (as are the modes of Dsymmetry) and has the
highest frequency. The modes of Psymmetry are infrared active and are second
highest in frequency, with the remaining modes decreasing in frequency with
increasing order of the spherical harrnonic. The observed band in the infrared
absorption spectrum is well resolved from the free carbon monoxide peak and,
as mentioned, can be assigned ta the Psymmetry normal mode. This band
overlaps with the long tail on the low energy side of the band in the Raman
spectra. It follows then that the observed Raman band can be assigned as a
convolution of a strong Srnode; weaker Pmodes, which are made weakly
allowed by either resonance or symmetry lowering; and weak contributions from
the Dmodes.
The deposition of mass-selected metal clusters with frozen gas matrices
successfully led to the collection of the Raman spectra of small, bare silver and
iron clusters, and of CO adsorbed on large silver clusters. The neutralization and
deposition energies ensure that the deposited cluster reaches its lowest-energy
form. Since the perturbation of the cluster by the matrix is weak, the Raman
spectra therefore reflect the most-stable gas-phase forms of the clusters. Ag, is
found to have the structure planar trapezoid by comparison of its Raman spect-
rum with theoretically predicted ones. Similarly, Ag, is found to be a tricapped
tetrahedron. Irradiating Ag, clusters in the matrix with 457.9 or 465.8 nm radia-
tion photo-decomposes the cluster to Ag, and Ag,. The Raman spectra of Ag,,
Ag,, and Ag, are al1 dominated by totally symmetric breathing modes at -165
cm? This frequency falls close to the Debye frequency (156 cm-') and to the
frequencies of phonons at the edge of the first Brillouin zone for solid silver,
indicating that the bonding between atoms in silver clusters as small as Ag,
a lread y approximates that of bu1 k silver.
The Raman spectra of Ag, and Fe, reveal these both to be dynamic Jahn-
Teller molecules. All of the 14 peaks in the Ag, spectrum are easily accounted for
with a linear plus quadratic Jahn-Teller coupling model. The Fe, spectrum cannot
be completely understood with this model, probably due to the high spin state of
the cluster (S=4) and spin-orbit coupling.
Adsorption of CO on Ag,, Ag,,, Ag,,, and Ag,, (deposited in a CO matrix)
leads to an enhancement of the v(C0) Raman scattering cross-section. Enhance-
ments between 100 and 600 are found, with a strong dependence on cluster
size. The enhancernent is understood as a resonance Raman process and is
compared with Surface Enhanced Raman Scattering. Specifically, the observed
enhancement is interpreted as the maximum enhancement possible from the
Chemical Mechanism of SERS.
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