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Electronic Shell Structure in Metal ClustersWalt de Heer
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The pre-shell structure years
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1970’s Sodium clusters are molecules
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Great idea: Clusters are molecules that can be calculated using density functional theory
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Mass spectra mean something Magic numbers: stability and structure
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Spherical jellium droplets: does it mean anything?
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Spherical jellium is an interesting idea!
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Spherical metal droplets is a bad idea!
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De Heer PhD thesis 1985
Knight’s idea: molecular beam resonance experiments on very small clusters to explore “quantum size effects”.
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For a brief and accurate account of the discovery, seeScience News, July 19, 2008 by Walt de Heer, Keith Clemenger, Winston A. Saunders, in response to story inScience News, June 21, 2008
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First indication of shells: the spectra evolve shell-by-shell.
Increasing carrier gas pressure(increasing cooling)
Knight et al PRL 52, 2141 (1984),
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8
20
40
58
92
Spherical potentials (delocalization)
Knight et al PRL 52, 2141 (1984),
De Heer, Rev. Mod. Phys. 65 1993 611
Spherical potentials: produce shells, but discrepancies remain (spherical symmetry). “Woods-Saxon” potential(not Jellium)
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Spherical positive (jellium) background
Electronic density
Electronic “Spillout”
Levels near Ef spill out beyond the jellium
EFermi
D
Jellium (ionic) radius, Rion DRion
The spherical jellium model
Screening radius, Rion+ D
+ Electronic structure in a positively charged sphere solved self consistently.+ It “builds in” metallicity.- Does not resolve shell discrepancies - Is not exactly “simple”
W. Ekardt PRB 29, 1558 (1984)
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Upshot: all metal clusters show metal droplet behavior, but the spherical jellium model is a poor starting point.
Schumacher: Shell structure is right but spherical jellium is not a good model.
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S. Bjornholm, J. BorggreenPhilosophical Magazine B, 1999 , 79, 1321
But is it a Physicists point of view!
Deep insight into the physics behind the shell stucture.
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electronsions
The liquid, not the fish determines the shape of the droplet!
That can be summarized as follows:Alkali clusters are like fish in a liquid droplet
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This is what they really look like, by the way…
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The Chemist wants an accurate description of the chemical properties, concentrating on individual clusters.
The Physicist wants a description of the physical properties from the atoms to the bulk, concentrating on trends.
The great divide: Cluster physics versus cluster chemistry
(Also not the great confusion:Shell structure versus the jellium model)
Forest versus trees.
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Shell Structure
The property of [metal clusters] that [valence electrons] occupy quantum states which are in groups of approximately the same energy, called shells, the number of [valence electrons] in each shell being limited by the Pauli exclusion principle.
[nuclei, atoms, metal clusters][similar nucleons, electrons, valence electrons]
Model
A model is a simplified description of the complex reality designed to reveal the main workings of a system.
Shell structure is a property, not a model!
Shell structure versus the Jellium model
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Atomic Shell Structure
Simplest Model: Coulomb potential (Bohr atom)Aufbau with independent electrons.
First order corrections: Many-body effects (screening):“Hunds Rules”;
Shell closings (Periodic Table): 2, 10, 18, 36, 54, 86
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Nuclear Shell Structure Simplest model:
3D harmonic potential; Independent nucleons;Magic Numbers: (n+1)(n+2) 2, 8, 20, 40, 70, 112
First order corrections:Spin-orbit coupling (Goeppert Mayer) Rounded well (Woods-Saxon)Ellipsoidal distortions (eg Nilsson)
Shell closings (Magic Numbers): 2, 8, 20, 28, 50, 82, 126
Har
mon
ic
Woo
ds-S
axon
Spin
-Orb
it
2
8
20
40
70
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Shell Structure in Alkali ClustersSimplest model:
3D isotropic harmonic potential Independent electronsMagic Numbers: (n+1)(n+2) 2,8,20,40,70,112)
First order corrections: Rounded ellipsoidal well (Nilsson-Clemenger)Major Shell closings: (2, 8, 20, 40, 58, 92) Sub Shell closings: 10,14,18, 26,30,34,50,…
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p, q single-electron momentum and positionl: angular momentumU: anharmonic amplitude n: shell index €
€
R0 = rsN1/ 3
hω0 = EF /N1/ 3
Harmonic oscillator potential Anharmonic correction
Nilsson Hamiltonian (spheroidal model)
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Anharmonic spheres
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Anharmonic spheres with spheroidal distortions
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Clemenger, PRB 32 1359 (1985)
Spheroidal electron droplet model Cluster shapes are determined by the electronic structure
26 3034
5054
1814
+ Simplified description that reveals the main workings+ Resolves shell edge discrepancies of spherical models.
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3D harmonic oscillator model with ellipsoidal distortions (easy to calculate)
RxRy
Rz
€
Enx,ny,nz = hω0R0[(nx +1/2) /Rx + (ny +1/2) /Ry + (nz +1/2) /Rz]
Single particle energies
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Total energies
€
Cx = (nx +1/2)0
nx max
∑
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Total energy surfaces
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W.A.de Heer, Rev. Mod. Phys. 65 611, (1993)
Early measurements of metallic properties of clusters
Polarizabilities (screening)Ionization potentials (Fermi level, screening )
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(multiple peaks reflect non-spherical shapes)
Collective electron oscillations:(Very intense; almost exhaust the oscillator strength).
€
ω= Q2
MR3 = Q2
Mα3
M =Nme
Q=NeR=Rion+Δ
Plasma resonances(electron-drop dominated shapes)
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Frank Condon overlap for plasmon excitations using 3D harm. Osc. model
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Direct demonstration of electronic shell structure
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Frank Condon overlap for photoelectron spectra 3D harm. Osc. model
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Shell structure is ubiquitous!Shell structure in non-simple metals: Indium
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Shell structure in non-simple metals: SIMS of zinc and cadmium
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Shell structure in non-simple metals: SIMS of the noble metals
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Transition to electronic shell structure: HgN, N>30
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More, bigger and better: the TP machine.
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Ionization potentials of large Na clusters
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M. Springborg, S.Satpathy. N. Malinowski, U. Zimmerman, T.P. Martin
Shell structure around a hole: Alkali coated C60
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More, bigger and better.
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Another way to look at it: semi-classical orbits
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Shells of shells: Supershell structure
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Hitting the limitsTransition from electronic shell structure to geometric shell structure
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Geometric shell structure: shells of atoms
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Al transitions to jellium but it does not exhibit electronic shell structure.
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Shell structure limitations: evolution of s-p hybridization in Al
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Al is the quintessential jellium metal in the bulk. But what about clusters?
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The End
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Na7, Na19, Na38
abstract
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The Chemists point of view