4.2. METAL NANOCLUSTERS
75
I L
7
I
3
5
7
9
1 1
13
15
17
NUMBER OF ATOMS IN CLUSTER
Figure 4.3. Mass spectrum of Pb clusters. [Adapted from M. A. Duncan and D. H. Rouvray, Sci.
Am. 110 (Dec. 1989).]
atom. The maximum ionization potentials occur for the rare-gas atoms *He, '%e,
and 18Ar because their outermost s andp orbitals are filled. More energy is required
to remove electrons from filled orbitals than from unfilled orbitals. Figure 4.4b
shows the ionization potential of sodium clusters as a function of the number of
atoms in a cluster. Peaks are observed at clusters having two and eight atoms. These
numbers are referred to as electronic magic numbers. Their existence suggests that
clusters can be viewed as superatoms, and this result motivated the development of
the jellium model of clusters. In the case of larger clusters stability, as discussed in
Chapter 2, is determined by structure and the magic numbers are referred to as
structural magic numbers.
4.2.2. Theoretical Modeling of Nanoparticles
The jellium model envisions a cluster of atoms as a large atom. The positive nuclear
charge of each atom of the cluster is assumed to be uniformly distributed over a
sphere the size of the cluster. A spherically symmetric potential well is used to
represent the potential describing the interaction of the electron with the positive
spherical charge distribution. Thus the energy levels can be obtained by solving the
Schrodinger equation for this system in a fashion analogous to that for the hydrogen
atom. Figure 4.5 compares the energy level scheme for the hydrogen atom and the
energy-level scheme for a spherical positive-charge distribution. The superscripts
refer to the number of electrons that fill a particular energy level. The electronic
magic number corresponds to the total number of electrons on the superatom when
75
I L
7
I
3
5
7
9
1 1
13
15
17
NUMBER OF ATOMS IN CLUSTER
Figure 4.3. Mass spectrum of Pb clusters. [Adapted from M. A. Duncan and D. H. Rouvray, Sci.
Am. 110 (Dec. 1989).]
atom. The maximum ionization potentials occur for the rare-gas atoms *He, '%e,
and 18Ar because their outermost s andp orbitals are filled. More energy is required
to remove electrons from filled orbitals than from unfilled orbitals. Figure 4.4b
shows the ionization potential of sodium clusters as a function of the number of
atoms in a cluster. Peaks are observed at clusters having two and eight atoms. These
numbers are referred to as electronic magic numbers. Their existence suggests that
clusters can be viewed as superatoms, and this result motivated the development of
the jellium model of clusters. In the case of larger clusters stability, as discussed in
Chapter 2, is determined by structure and the magic numbers are referred to as
structural magic numbers.
4.2.2. Theoretical Modeling of Nanoparticles
The jellium model envisions a cluster of atoms as a large atom. The positive nuclear
charge of each atom of the cluster is assumed to be uniformly distributed over a
sphere the size of the cluster. A spherically symmetric potential well is used to
represent the potential describing the interaction of the electron with the positive
spherical charge distribution. Thus the energy levels can be obtained by solving the
Schrodinger equation for this system in a fashion analogous to that for the hydrogen
atom. Figure 4.5 compares the energy level scheme for the hydrogen atom and the
energy-level scheme for a spherical positive-charge distribution. The superscripts
refer to the number of electrons that fill a particular energy level. The electronic
magic number corresponds to the total number of electrons on the superatom when