Clusters : basic physics

1
Clusters basic physics

• I. Cluster typology constituents and size
• II. Generation / Investigation / Detection /
  Support
• III. Rare gas clusters geometrical factors
• IV. Metal clusters electronic factors
• V. Semiconductor clusters carbon, silicon,
  boron
• VI. Ionic clusters
• VII. Molecular clusters

Reference  Atomic and molecular clusters 
R.L. Johnston, ed. Taylor and Francis (2002)
2
I. Clusters typology constituents and size
3
What is a cluster ?
Aggregate of a countable number n of particles
(atoms or molecules) with n2 ... 1000000 (or
more)
!
Charge state
one big molecule is NOT a cluster
(neither a fragment)
Homo-atomic or homo-molecular clusters A
n
Examples rare gases (e.g. He), metals (e.g. Na,
Fe), carbon, semiconductors (e.g. Si), small
molecules (e.g. H O)
2
Hetero-atomic or hetero-molecular clusters A B
n
n
(only one of n or n needs to be bigger than 1)
Examples

(H O) H , C N
2 n 59
4
Size effects number of atoms/molecules on the
surface
Spherical cluster approximation / Hard atomic
spheres
Assign a volume to the building block (atomic
sphere)
Volume of cluster (rough
approximation, no filling factor)
Radii of cluster and atom
Number of atoms on the surface
(divide the surface of the cluster sphere by the
cross section of an atom)
Fraction of atoms on the surface
5
Cluster size regimes
Small clusters less than 100 atoms
Medium-sized clusters 100-10000 atoms
Large clusters more than 10000 atoms
Example Sodium clusters (average radius of the
atomic sphere in the bulk 0.2 nm)
Small Na clusters diameter less than 1.9 nm
(86 of atoms on surface)
Large Na clusters diameter larger than 8.6
nm (19 of atoms on surface)
Need to have more than 64 million atoms for 1 of
atoms on surface
6
Cluster size effects (I)
Regular variations scaling laws, tending to the
bulk value.
For a generic property
(the exponent b is often 1/3)
Example the ionization energies of potassium
clusters
7
Cluster size effects (II)
Irregular variations for medium-sized and small
clusters, quantum size effects, geometry
effects, surface effects.
Sudden discontinuity at  magic numbers 
Example magic numbers 2, 8, 20, 40, 58, 92...
for IP of K
8
Cluster size effects investigations
Do cluster properties resemble those of discrete
molecules or infinite solids, and how fast to
they converge to the bulk value ? What is the
geometrical structure of clusters, and its
evolution ? What is the electronic structure of
cluster, and its evolution ? (esp. important
for metallic clusters metal/insulator
transition) (also, tunable optical
properties) What are the magnetic properties of
cluster, and their evolution ? Can phase
transitions be observed, and are they of the same
type found for bulk solids and surface ? What is
the effect of the large number of surface atoms
? (reactivity/catalysis)
9
II. Generation / Investigation / Detection /
Support
10
Experimental aspects of cluster physics
Four media for cluster experiments 1)
Cluster molecular beams (really isolated, but
low flux) 2) Matrix isolated clusters ( inert 
matrix condensed rare gases, or molecules)
3) Surface-supported clusters (can use lots of
surface tools - STM, SEM, AFM) 4) Cluster
solids (coalescence might be prevented by
coating with surfactant molecules - can use X
ray and electron diffraction)
11
Molecular beams formation (I)

• Creation of a vapor phase of atoms/molecules
• Nucleation of the cluster
• Growth/coalescence of the cluster
• (reversible processes evaporation/fragmentatio
  n)

Important characteristics of a particular MB
source size distribution of clusters
Might be in thermodynamical equilibrium
(Maxwell-Boltzmann)
Overall trend
But superimposed structure, with intensities
related to the binding energies of the clusters
(the more stable, the more intense)
12
Molecular beams formation (II)
Cluster sources Knudsen cell (effusive),
supersonic (free jet) nozzle, laser vaporization,
pulsed arc, ion sputtering, magnetron sputtering,
gas aggregation, liquid metal ion,
spray. Nucleation of atom clusters in order to
fulfill energy conservation and momentum
conservation, need three-body collisions. Then,
growth (aggregation or coalescence), and
possibly, thermodynamical equilibrium
(collisional cooling, evaporative cooling,
radiative cooling).
13
Mass selection / mass spectrometry
Usually based on charge/mass ratio of
ions Combination of applied electric field and
magnetic field (might be oscillating) allows
to give different trajectories to ions with
different charge/mass ratio
Need generation of ions from neutral species
(e.g. electron impact ionisation,
photoionization,electric discharge)
14
Colloïdal metal particles
Colloïd dispersion of particles of one material
in another
Production chemical reduction of metals salts
dissolved in an appropriate solvent, in the
presence of a surfactant, which passivate the
cluster surface.
Alkylthios (R-SH) and thioethers (R-S-R) form
particularly stable colloidal particles with
selected metals (Ag, Au, Pd, Pt). Control of
average particle size and size-distribution
achieved by tuning the preparation conditions
15
Semiconductors quantum dots / Supported clusters

• Semiconductor quantum dots
• - precipitate colloïdal quantum dots (also
  protected
• by organic surfactant molecules)
• or use electron-beam lithography to carve wells
  in a silicon wafer,
• then condensate from the vapour phase
• - ...
• Supported clusters
• - supports graphite, silicon, oxydes, or the
  inside of porous
• materials such as zeolites (aluminosilicates with
  tunnels
• and cavities), aluminum phosphates, clays, porous
  carbon,
• alumina membranes.
• low-energy deposition (soft landing, using a
  thin film of rare gas),
• or high-energy deposition (disrupt the
  support and cluster)

16
Types of experiments (in addition to MS)
Spectroscopies UV-visible and IR absorption
(for size-selected clusters) (photo) depletion
spectroscopy (irradiation followed by mass
spectrometry)
Photoelectron spectra (might be followed by mass
spectrometry)
Fragmentation studies (electron or ion impact,
followed by MS)
Polarizability and magnetism measurements
(size-selected clusters)
Chromatography
Flow reactor
Diffraction experiments (for size-selected
clusters)
Microscopy (for surface supported clusters)
TEM/SEM/STM/AFM
17
III. Rare gas clusters geometrical factors
18
Neutral rare-gas clusters
Very weak bonding (van der Waals, or dispersion)
Mostly pair-wise (can be modelled by
Lennard-Jones 6-12 potential)
But three-body corrections are non-negligible
Need low temperature to be stable (dimerization
temperature is 11K for He, 281 K for Xe)
Molecular orbital diagram for He dimer
Bond order 0
1s
1s
19
Charged rare-gas clusters
Molecular orbital diagram for singly charged He
dimer
Quite different electronic structure !
1s
1s
Bond order 1/2
Reduced bond length
20
Charged rare-gas cluster mass spectra
Magic numbers 147 for all, also 87 for Kr and
Xe. Additional for Xe 19, 23, 25, 55
(strong), 71 ...
Larger polarisability of Xe better screening of
the core
21
Preferred structures of rare-gas clusters
Some magic numbers might be rationalized thanks
to concentric (polyhedral) shells of atoms
surrounding a central atom.
Best agreement for icosahedral shells, with
five-fold symmetry axes. Confirmed by X-ray
diffraction studies (several hundred of atoms)
(here, fig 2.3)
22
Icosahedral vs. FCC cluster growth
The rare-gas crystals are FCC ! Icosahedra
maximize the number of nearest-neighbour
attractive interactions it is greater than for a
FCC/HCP cluster. 13-atom cluster Icosahedra
has 5 NN FCC has only 4 NN However, packing of
hard spheres as icosahedra gives frustration.
Cross-over between 800 and 10000 atoms ...
23
IV. Metal clusters electronic factors
24
The liquid drop model
Classical electrostatic model a uniform
conducting sphere
Ionization energy removal of an electron, takes
into account the metal work function (that
includes the work due to image charge), and
the work to be done to separate the electron from
the remaining positively charged,
polarizable, sphere
(W is the work function )
In eV and nm
Electron affinity addition of an electron,
takes into account the metal work function, and
the energy gained when the electron approaches
the neutral, polarizable sphere
In eV and nm
25
Experimental values for the IP and the EA
26
Surface plasmons
Optical responses of medium-sized and large
clusters (small metal clusters transitions
between well-defined quantized energy levels)
Single, broad peak, due to collective (strongly
correlated) motion of the electron  plasma 
in the positive background
Frequency and width of the peak tends to the bulk
value with the usual inverse radius asymptotic
behaviour.
27
The spherical jellium model (I)
(Much better model for small metallic clusters)
Will explain magic numbers and deviations from
the LDM
Quantum mechanical treatment of the jellium
sphere - positive uniform spherical background
(attractive potential) - electrons screen the
attractive potential, and occupy lowest levels
of the effective potential - orbitals are
characterized by angular momenta quantum
numbers and m
The effective potential differs strongly from the
Coulomb potentiel, it is intermediate between an
harmonic and square well potential
28
The spherical jellium model (II)
Explain the magic numbers of neutral alkali
clusters 2, 8, 20, 40, 58, 70, 92 ...
Also explain the magic numbers for divalent
metals such as zinc or cadmium,
at 4,9,10,17,20,29 ... atoms.
Magic numbers seen in mass spectra, ionization
energies, electron affinities.
29
Ellipsoïdal shell model
Problems with the spherical jellium model -
fine structure of mass spectra (even-odd
alternation) - diamagnetism of even-electron
clusters with formally open jellium shells
(breaking of Hunds rule)
Unlike an atom, the positively charged background
can deform !
30
Geometric shell structure
For sodium clusters with over 2000 atoms, long
period oscillations in mass spectra intensity
with magic numbers around 2000, 2800, 5100, 6500,
... 18000, 21300 (dips).
Like for rare gases, ascribed to the filling of
concentric polyhedral, or geometric shells of
atoms icosahedron, ino dodecahedron, fcc-like
cubocahedron
Competition between electronic and geometric
shell structure Depend on the DOS, atomic
electron configuration, cluster-melting temperatur
e, and the temperature of the cluster.
Cluster core might be liquid, and cluster surface
might be solid, or the reverse !
31
Structure determination of supported metal
clusters
TEM image of silver cluster 4 nm diameter Marks
dodecahedron
Comparison of theoretical and experimental X-ray
diffraction patterns for passivated gold
clusters, showing decahedral (Dh) and (fcc)
truncated octahedral (TO) geometry
32
Size-induced metal-insulator transition (I)
How many atoms make a metal ? A macroscopic metal
is a very good conductor of electricity, but what
about an isolated, finite fragment of metal with
100 atoms ? And what does electrical conductivity
mean ?
Questions for one particle, as well as for an
ordered array of particles that would be metallic
electron tunelling, and transport between
neighbouring particles.
Importance of the temperature kT defines an
energy, to be compared with the energy gap of the
particle (Kubo gap).
Energy spacing (very rough)
33
Size-induced metal-insulator transition (II)
34
Size-induced metal-insulator transition (III)
35
The conductivity of an isolated metal cluster
(Another view on the metal-insulator transition)
Ioffe-Regel criterion for the localisation of an
electron wavepacket
where
is the Fermi wavevector
is the mean free path
In a cluster of size R, there should be a
metal-insulator transition when
Experiments with colloïdal suspension of gold
particles
(about 10000 atoms)
The conductivity rises abruptly for
(Note gold particles of
form the basis for some of the paints
used for  stealth  planes technology)
36
The non-metal to metal transition in mercury
clusters
Hg is peculiar closed shell ,
quite separated from the p states
37
Metal cluster magnetism
For odd number of electrons, at temperature
smaller than the Kubo gap, there must be a
magnetic moment !
For Fe, Co, Ni, the magnetic moment per atom of
the atoms is larger than the bulk one, and for
clusters, it is intermediate between the bulk and
the atomic value. The bulk limit is reached at
around 500 atoms for Fe.
The 4d transition metal Rhodium (Rh), which is
not magnetic in the bulk form magnetic clusters
(same with Mn clusters)
38
Reactivity of metal clusters
Large number of surface atoms Surface sites can
depend strongly on the number of atoms of the
cluster
39
Metal clusters in catalysis
Heterogeneous catalysts small metal particles
dispersed on a high-surface area non-metallic
support (usually, an oxide) (the total surface
area available for catalysis, for a given mass of
metal catalyst, is increased).
Supports silica, alumina or titania, zeolites
or nanoporous alumina membranes.
Immersion in a solution with metal particles, or
sucking a cluster solution through the membrane.
Bulk gold is one of the least catalytically
active metals, but gold clusters dispersed on
thin (2-10nm) oxide film supports can catalyse
the oxidation of CO to CO2 at temperatures as low
as 40 K.
40
Applications of colloïdal metal particles
Stained glass windows small colloïdal particles
of copper, silver and gold (between 1 nm and 1
micron)
(The use of colloïdal metal particles probably
dates back as far as the ancient Egyptians
Cleopatra cosmetics prepared from colloïdal gold)
Scientific study dates back to 1857 (M.
Faraday)    ... the gold is reduced in
exceedingly fine particles which becoming
diffused, produce a beautiful fluid ... the
various preparations of gold whether ruby,
green, violet, or blue ... consist of that
substance in a metallic divided state 
Strong absorption bands in the visible region of
the spectrum caused by plasmons - collective
oscillations of the cluster electrons.
41
Opals
Opalescence strong light scattering by a
colloïdal crystal (opal)
42
Silver clusters in photography
Black-and-white photographic process local
photodissociation of light-sensitive silver
halides (e.g. AgBr) creating neutral silver and
halogen atoms (radical).
This process is repeated, and some of the silver
atoms cluster together to form a latent image,
which catalyses the reduction (in the development
process) of exposed AgBr microcrystals to silver
metal. Clusters of at least four atoms ( )
are required to initiate bromide reduction.
(Silver latent image, if not rapidly developed
can undergo oxydation. This is due to the lower
ionization energies of small silver
clusters, compared with the bulk metal)
43
Motion of single electrons via metal nanoparticles
(Single metallic cluster between two electrodes
tip and substrate) The number of carrier
electrons is small, and can be controlled by
adjusting the voltage.
Scanning tunneling spectroscopy (STS) I-V
characteristics. Passivating ligands are
insulating and act as tunnel barriers between the
cluster and the substrate.
Coulomb blockade a critical charging energy is
required for electrons to tunnel between the STS
tip and the cluster and between the cluster and
the substrate. For Pt clusters with about 300
atoms, 50-500 meV.
SET (single-electron transistor) metal
electrodes (source, drain and gate) fabricated
on a silica support deposition of three
colloïdal Au particles bridging the source and
drain, with single-electron charging effects (up
to 77K)
44
V. Semiconductor clusters carbon, silicon, boron
45
Mass spectrum of carbon clusters
46
Small carbon clusters
Competition between rings and chains. Chains are
more floppy (entropy favoured), but rings are
energetically favoured above N10. Magic number
for ionized carbon chains 7,11,15 ...
Linear structures of and . The spectral
lines due to the molecule were seen in 1881
in the spectrum of a comet, but not identified
until 1951. identified in the
infrared spectrum of a carbon star.
47
The fullerenes
In 1985, identification by Kroto, Smalley, Curl
and co-workers of the molecule as being a
truncated icosahedron, with very high symmetry
(Ih) all 60 atoms are equivalent, and are
three-fold coordinated. Also, fullerenes with 70,
76, 78 (3 isomers), 84 (2 isomers), ... atoms.
(Named after the architect Buckminster Fuller)
48
Geometrical structures of fullerenes (I)
Fullerenes are polyhedra, with the number of
vertices V equal to the number of atoms n, the
number of faces F equal to the number of rings,
and the number of edges E equal to the number of
bonds. The Euler relation for 3D convex
polytopes writes
All the carbon atoms are hybridized. Each
atom is attached to three bonds, each shared by
another atom. Thus
(V even)
Let be the number of faces with k vertices
(and k edges, shared by 2 faces)
The Euler equation reduces to
which can be expanded as
Suppose only hexagons and pentagons 12 pentagons
49
Geometrical structures of fullerenes (II)
When only hexagons and pentagons are allowed,
the number of atoms is linked with the number of
hexagons by
There are 3532 possible isomers for 60 atoms
fullerenes, but only 1 is usually observed.
The isolated pentagon rule for a given number
of atoms, the relative energies of fullerene
isomers increase with the number of adjacent
pentagons. The lowest possible number of atoms
for a fullerene that has no adjacent pentagon is
60. Starting with 70, all even numbers of atoms
can lead to the fulfillment of this rule 70 (1
isomer), 72 (1 isomer), 74 (1 isomer), 76 (3
isomers), 78 (6 isomers), 80 (9 isomers) ...
Additional stability trends follow from
electronic structure.
50
Fullerene isomerization Stone-Wales
rearrangement
If cluster growth proceeded via random
self-assembly process, a very large number of
isomers would be generated...
Need a mechanism for rearrangement, yielding a
few, thermodynamically favoured low energy
isomers. Stone-Wales rearrangement. A high
energy process activation barrier in excess of
500 kJ/mol (5eV)
51
Fullerene reactivity
Despite the large number of apparently conjugated
cycles, the fullerenes are quite reactive. Two
kinds of bond lengths (single - 0.146 nm - and
double - 0.139 nm - bonds). Bond localisation,
due to the unfavorable presence of double bonds
on the pentagons.
Possibility to generate
52
Metal-Fullerene complexes and substituted
fullerenes
Incorporation of metals (K, La, U ...) into the
cage (laser vaporisation of a graphite
sample that has been immersed in a solution of
the appropriate metal salt).
 endohedral  complexes
Possibility to have both endohedral and exohedral
attachment. Example
is actually
Possibility to replace one or more carbon atom
by another element, such as boron or nitrogen.
53
Large Fullerene-like clusters
Large fullerenes become more polyhedral and
more facetted, rather than spherical, with large
almost planar graphite-like sheets of hexagons,
with curvature only close to pentagons.
Buckyonions concentric shells of fullerenes
Fullerene polymers irradiation of solid
gives linear chains, linked by four-membered
rings.
Clusters of fullerenes magic number
(n13,19,23,27 ...) ! Icosahedral packing ...
54
Mass spectra of silicon clusters
Charged clusters show magic number n6, 10, 16
and 32, but a maximum in the spectrum is not
followed by a deep minimum for n1 atoms !
(Photofragmentation shows no monomer evaporation).
55
Structure of Si clusters
Proposition of basic building blocks distorted
octahedral and adamantane (a diamond
fragment) . Actually, the calculated
lowest energy structure for is the
high density tetracapped trigonal prism.
Compared to carbon, reduced tendency to
participate in p-bonding.
56
Structure of hydrogenated Si clusters
Hydrogen atoms saturate the silicon clusters by
taking up dangling bonds more open,
structures.
57
The gap in semiconductor clusters
The optical properties of silicon (and other
semiconductor clusters) have been found to change
dramatically as a function of the size of the
cluster shift to higher energy when the cluster
is reduced.
(Between R2 ... 100 nm, evolution from discrete,
molecule-like states to band)
Model a semiconducting sphere, lowest energy
state formed by a product of 1s electron and hole
wavefunctions, with bulk effective masses and
dielectric constants
Kinetic localization energy
58
Photoluminescence and light absorptionin
semiconductor clusters
Long lifetimes (on the order of 100 ns) of
excited states, in contrast to metal clusters.
The radiative lifetime decreases as R
increases. (ns - np transition for direct band
gap semiconductors)
Photoluminescence of Si IR in the bulk (Eg1.17
eV), but can be shifted in the visible for
clusters. Also for other semiconductors,
especially CdSe (passivated with organic
molecules, colloïdal clusters, size
well-controlled !).
Competition between scattering and absorption
if R gt l scattering dominates over
absorption if R lt l electric dipole allowed
absorption dominates matching of R and l
size-dependent resonances
59
Boron clusters
Boron (Z5)
Already in the solid-state, boron atoms tend to
cluster
- metal borides
with octahedra or cuboctahedra.

• elemental boron, with icosahedral
  clusters, or a unit,
• consisting of a -like boron cage which
  is linked to an
• encapsulated unit icosahedron via
  twelve linking B atoms.

A curiosity boron suboxide cluster with the
stoechiometry form Mackay icosahedral
structures with up to atoms, with cluster
diameters of nearly 10 microns.
60
VI. Ionic Clusters
61
Generalities
At least two elements, having a difference in
electronegativity. Highly ordered ionic clusters
ions with the same sign charge are separated
from each other. Ionic clusters are composed
of even-membered rings, with alternating anions
and cations.
As the difference in electronegativity between
the positive and negative elements decreases, the
distinction between ionic and semiconductor
becomes blurred NaCl ... ZnO ... CdSe ...
May be generated with positive or negative
charge, or neutral. Quench the vapour, resulting
from heating or laser vaporisation.
Environmental relevance NaCl and NaBr are
important in the marine atmosphere, forming from
salt-water droplets dehydrated by the suns
action. React with nitrogen oxide NO pollutants
to give chlorine and bromine atoms, responsible
for catalytic ozone depletion.
62
Alkali halide cluster structure
In the solid state rocksalt structure
(NaCl). Clusters cuboidal morphology, fragment
of bulk rocksalt !
n i x j x k
For even values of n, the cluster might be
neutral, while when n is odd, there must be one
more of one type than the other, giving rise to
charged clusters.
63
Alkali halide mass spectrometry
64
Stable configurations for alkali halides with a
small number of atoms
65
Metal chalcogenide clusters ZnS mass spectrum
66
VII. Molecular Clusters
67
What are molecular clusters ?
An aggregate of an integer number of
molecules. Homo-molecular (like the water
clusters ) or hetero-molecular
(such as ) .
Also, aggregate of molecules and rare gas, or
molecules and ions, like solvated ions
Models of important processes occuring in the
atmosphere (acid rain formation, ozon depletion,
generation of pollutants), and models for
solvation
Most molecular clusters are composed of molecules
which are stable and possess closed electronic
shells two distinct bonding modes (strong
covalent within the molecule weak
intermolecular forces)
Exceptions NO (with a ground state) and
(with a ground state)
68
Weak intermolecular forces (I)
Forces based on the permanent molecular
multipoles (dipole-dipole, quadrupole-dipole,
quadrupole-quadrupole ...)
Polar molecules assymetric charge distribution
(e.g. HCl) Model point charges q and -q at a
distance L Magnitude of the dipole moment
-30
Units 1Debye 1D 3.336 10 C m
If the dipoles are considered point-like, the
interaction energy between two parallel dipoles
is given by
69
Weak intermolecular forces (II)
Attractive
Tail Head
Repulsive
Non-polar molecules (neutral, symmetric) might
have a quadrupolar moment carbon dioxyde,
acetylene, benzene ...
70
Weak intermolecular forces (III)
Hydrogen bonds X-H---Y (where X and Y
are rather electronegative, covalent bond with X
and accessible lone-pair on Y) Angle close to
180, H---Y significantly longer than a covalent
bond, large red-shift of the X-H stretching
vibration. Largely electrostatic, but also some
covalency, induction, and dispersion effects.
Typically 10-25 kJ/mol (0.1 to 0.25 eV)
Dispersion forces. Larger than for atoms, become
the molecules are more polarizable. Important for
homomolecular clusters of homonuclear
molecules, like or .
Typically lower than 10kJ/mole (0.01 eV).
71
Structure of neutral water clusters
Ring structures. One hydrogen per
molecule involved in H bond. Near-planar
structure of H bonds. Frustration for odd
cycles. Tunneling splitting pattern of
the vibration spectrum (like for ammonium)
Five-membered ring structure of the 20 water
molecule cluster. Note that only
six-membered rings are present in hexagonal ice.
72
Structure of protonated water clusters
Most clusters consist of a number of water
molecules surrounding a hydronium ion,
. The moiety, with symmetry, is the
smallest unit where the inner hydronium is
completely hydrogen bonded. The structure of
is however