Metal clusters have been widely studied

1
Atomic Molecular Clusters5. Metal Clusters

• Metal clusters have been widely studied
  especially alkali metals, noble metals (Cu, Ag,
  Au) and transition metals.
• Cohesive energies are generally quite large
  (relatively strong metallic bonding)
  significantly higher than for rare gas or
  molecular clusters so they may be studied in
  solution (colloidal suspensions), on surfaces or
  in inert matrices, as well as in the gas phase.
• A number of models have been introduced to
  explain and predict the properties of metal
  clusters.

2
The Liquid Drop Model

• A classical electrostatic model.
• Cluster is approximated by a uniform conducting
  sphere.
• Atomic positions and internal electronic
  structure ignored.
• Predictions

As 1/R ? 0 (N ??) IP,EA ? W
3
1/R / Å
M. M. Kappes Chem. Rev. 1988, 88, 369.
4
Failures of the Liquid Drop Model

• Deviations from 1/R dependence of IPs and EAs for
  small clusters.
• IPs of Hg clusters show a discontinuity due to a
  size-dependent non-metal ? metal transition (see
  later).
• Some transition metals (where ionization involves
  removal of tightly bound d electrons, e.g. Fe,
  Ni) show small variation of IP/EA with size.
• Magnetic effects (spin-spin interactions) also
  important for transition metals.
• LDM does not reproduce fine structure in IP/EA
  variation with N (e.g. even-odd alternation) or
  explain the Magic Numbers in the mass spectra.
• Require a Quantum Mechanical model with discrete
  electronic states ? JELLIUM MODEL.

5
Mass Spectra and Magic Numbers

• Mass spectra obtained by Knight and co-workers
  (1983-85), for alkali metal clusters, showed a
  number of peaks with high relative intensities ?
  Magic Numbers.
• Magic numbers (and origins) different from rare
  gas clusters.

6
The Jellium Model

• Derived from nuclear structure theory.
• Cluster approximated by a sphere with a uniform
  positively charged background, filled with an
  electron gas (valence electrons).
• Valence electrons are delocalized move in a
  smooth, attractive, central, mean field potential
  of spherical symmetry.

7

• Positions of ionic cores are ignored.
• This is justified if
• electrons are strongly delocalized
• ionic background easily deformed
• molten clusters?
• Works best for monovalent simple metals
• e.g. alkali metals, noble metals (Cu, Ag, Au).
• Unlike the LDM, the jellium model is a quantum
  mechanical model
• quantization of electron energy levels due to
  boundary conditions imposed by the potential.
• Gives rise to electronic shell structure for
  metal clusters with up to several 1000s of atoms.

8
Empirical Jellium Models

• Based on effective single-particle potentials
  (Knight, Clemenger).
• Solve 1-electron Schrödinger Equation for an
  electron in a sphere, under the influence of an
  attractive central potential.
• Wavefunction (?) is separable into radial and
  angular parts
• ?n,?,m?(r,?,?) Rn,?(r).Y?,m?(?,?)

9
Solutions

• Wavefunction and energies depend on quantum
  numbers
• n 1, 2, 3,
• ? 0, 1, 2, (no restriction on ?)
• m? ?? 0 ? (2? 1)-degenerate
• Note the principal quantum number n is different
  from that used for atomic orbitals (follows
  convention of nuclear physics)
• nclust natom - ?
• Jellium electronic levels (sets of degenerate
  orbitals) are labelled, by analogy with atomic
  orbitals 1s, 1p, , 2s, 2p
• Exact ordering of orbitals depends on the radial
  form of the potential.

10
The Woods-Saxon Potential

• Obtained by fitting to high-level electronic
  structure calculations.
• W-S potential is a finite well
• with rounded sides (intermediate
• between 3-D harmonic oscillator
• and 3-D square well).

11

• Ordering of Levels
• 1s lt 1p lt 1d lt 2s lt 1f lt 2p lt 1g
• Level Closings
• (no. of electrons)
• 2 8 18 20 34 40 58

12
Interpreting Mass Spectra of Metal Clusters

• Low Energy Ionization
• Magic numbers (intense peaks in MS) due to stable
  electron counts (filled jellium levels) of
  neutral clusters (MN).
• N 8, 20, 40, 58

13

• High Energy Ionization
• Magic numbers due to stability of cationic
  clusters (MX) N 9, 21, 41, 59
• Note Na8, Na9 both have 8 electrons.

14
Breakdown of the Spherical Jellium Model

• Fine structure is observed in the MS, IPs, EAs,
  polarizabilities etc., for even-electron counts
  other than those predicted by the (spherical)
  jellium model.
• This is evidence for non-degenerate electronic
  sub-levels, which cannot be explained by the
  spherical jellium model.
• Need to extend the model.

15
The Ellipsoidal Shell Model

• Modification to spherical
• jellium model, introduced
• by Clemenger (1985).
• Potential a perturbed
• 3-D harmonic oscillator
• analogous to Nilssons
• model (1955) for nuclear
• structure.
• Ellipsoidal (spheroidal)
• distortion of cluster.

16

• Lowering of symmetry ? loss of (2?1)-fold
  degeneracy of each jellium level (n?).
• ?m? degeneracy is maintained in ellipsoidal
  (spheroidal) symmetry.
• Oblate Spheroid ? E? as m?? gt ½-filled shell
• Prolate Spheroid ? E? as m?? lt ½-filled shell

17
Variation of Na Cluster Shape with Size
18
Comparison of structures of Na and Ar clusters
19
Beyond the Jellium Model

• Martin and co-workers (1991)
• measured MS of NaN clusters (N ? 25,000).
• Observed two series of
• periodic intensity variations
• period ? N1/3.

T. P. Martin et al., J. Phys. Chem. 1991, 95,
6421.
20
Electronic Shells (N lt 2000)

• Electronic shells form due to bunching together
  of jellium levels.
• Electronic shells sets of nearly degenerate
  jellium levels.

21

• For larger metal clusters, electronic effects are
  relatively unimportant because electronic shells
  merge to form quasi-continuous bands (bulk-like
  band structure).

22
Geometric Shells (N gt 2000)

• Geometric shells correspond to complete
  concentric polyhedral shells of atoms as for
  rare gas clusters.
• Stability due to minimization of surface energy.
• Alkali Metal Clusters magic numbers are
  consistent with filling K geometric shells

23
Examples of Geometric Shells
rhombic dodecahedron (bcc)
icosahedron
truncated octahedron (fcc)
24

• Similar magic numbers have been observed for Ca
  clusters with up to 5000 atoms.
• MS magic numbers and fine structure (due to
  partial geometric shell formation) indicate that
  alkali metal clusters (with N gt 2000) and Ca
  clusters have icosahedral shell structure.
• T. P. Martin Physics Reports 1996, 273, 199.

25

• Al and In clusters form octahedral shell
  structures (fragments of fcc packing).
• Geometric shell structure has also been found for
  many transition metal clusters (e.g. Co, Ni).
• Electronic shell effects are relatively
  unimportant for TMs with unfilled d-orbitals as
  the onset of band structure occurs for quite low
  N.

InN
26
Microscopy Studies of Metal Clusters

• A number of microscopy techniques can be applied
  to study metal clusters
• Electron Microscopy (TEM, SEM)
• Scanning Tunnelling Microscopy (STM)
• Atomic Force Microscopy (AFM)
• Clusters must be immobilized on a substrate (e.g.
  graphite, amorphous-C, MgO, SiO2 depending on
  the type of measurement).
• Clusters are often passivated by surfactant
  (ligand) molecules.
• Cluster-surface and cluster-ligand interaction
  may affect cluster structure (for small clusters).

27
Electron Micrographs of Ag and Au Particles
28
Energetics of pure Ag clusters
29
Insulator-Metal Transition in Hg Clusters

• Rademann and Hensel (1987) measured IPs of Hg
  clusters as a function of size, N.
• Explained in terms of a gradual transition
• Insulating ? Semi-Conducting ? Metallic
• in the region N 13-70.
• Consistent with spectroscopy
• and theoretical calculations.

30
Theory of Bonding in Hg Clusters

• The free Hg atom has a closed shell (6s)2(6p)0
• Small clusters are insulating van der Waals
  clusters
• held together by dispersion forces.
• As the cluster gets larger, the 6s and 6p levels
  broaden into bands (with widths Ws and Wp)
• W? as N?
• Insulator ? Metal Transition occurs when the 6s
  and 6p bands overlap.
• Before band overlap (intermediate N), the band
  gap (?sp) may be comparable to the thermal energy
  (kT)
• semi-conductor clusters
• s-p hybridization occurs ? covalent bonding

31
Size-dependent Electronic Structures of Hg
Clusters