2.1. STRUCTURE
15
Sodium Nanoparticle Na, Magic Numbers
Electronic
Structural
0
3
6
9
1
2
8
10
I2
14
16
I I I I I I I ; I I I I I I I I ~ I I I I: I I 1 ; I I I
5
10
I5
20
2s
n”
Figure 2.7. Dependence of the observed mass spectra lines from Na, nanoparticles on the
cube root N’l3 of the number of atoms N in the cluster. The lines are labeled with the index n of
their electronic and structural magic numbers obtained from Martin et al. (1990).
approximately equally spaced, with the spacing between the structural magic
numbers about 2.6 times that between the electronic ones. This result provides
evidence that small clusters tend to satisfy electronic criteria and large structures
tend to be structurally determined.
2.1.4. Tetrahedrally Bonded Semiconductor Structures
The type 111-V and type 11-VI binary semiconducting compounds, such as GaAs and
ZnS, respectively, crystallize with one atom situated on a FCC sublattice at the
positions 000, 1 0, 0 and 0 4, and the other atom on a second FCC sublattice
displaced from the first by the amount $$$ along the body diagonal, as shown in Fig.
2.8b. This is called the zinc blende or ZnS structure. It is clear from the figure that
each Zn atom (white sphere) is centered in a tetrahedron of S atoms (black spheres),
and likewise each S has four similarly situated Zn nearest neighbors. The small halfsized, dashed-line cube delineates one such tetrahedron. The same structure would
result if the Zn and S atoms were interchanged.
The elements Si and Ge crystallize in this same structure by having the Si (or Ge)
atoms occupying all the sites on the two sublattices, so there are eight identical
atoms in the unit cell. This atom arrangement, sketched in Fig. 2.8a, is called the
diamond structure. Both Si and Ge have a valence of 4, so from bonding considerations, it is appropriate for each to be bound to four other atoms in the shape of
a regular tetrahedron.
In Appendix B we see that Table B.l lists the lattice constants a for various
compounds with the zinc blende structure, and Table B.2 provides the crystal radii of
their monatomic lattices in which the atoms are uncharged, as well as the ionic radii
for ionic compounds in which the atoms are charged. We see from Table B.2 that the
negative anions are considerably larger than the positive cations, in accordance with
the sketch of the unit cell presented in Fig. 2.9, and this size differential is greater for
the 111-V compounds than for the 11-VI compounds. However, these size changes
for the negative and positive ions tend to balance each other so that the 111-V
compounds have the same range of lattice constants as the 11-VI compounds, with Si
and Ge also in this range. Table B.4 gives the molecular masses, and Table B.5 gives
the densities of these semiconductors. The three tables B.l, B.4, and B.5 show a
regular progression in the values as one goes from left to right in a particular row,
15
Sodium Nanoparticle Na, Magic Numbers
Electronic
Structural
0
3
6
9
1
2
8
10
I2
14
16
I I I I I I I ; I I I I I I I I ~ I I I I: I I 1 ; I I I
5
10
I5
20
2s
n”
Figure 2.7. Dependence of the observed mass spectra lines from Na, nanoparticles on the
cube root N’l3 of the number of atoms N in the cluster. The lines are labeled with the index n of
their electronic and structural magic numbers obtained from Martin et al. (1990).
approximately equally spaced, with the spacing between the structural magic
numbers about 2.6 times that between the electronic ones. This result provides
evidence that small clusters tend to satisfy electronic criteria and large structures
tend to be structurally determined.
2.1.4. Tetrahedrally Bonded Semiconductor Structures
The type 111-V and type 11-VI binary semiconducting compounds, such as GaAs and
ZnS, respectively, crystallize with one atom situated on a FCC sublattice at the
positions 000, 1 0, 0 and 0 4, and the other atom on a second FCC sublattice
displaced from the first by the amount $$$ along the body diagonal, as shown in Fig.
2.8b. This is called the zinc blende or ZnS structure. It is clear from the figure that
each Zn atom (white sphere) is centered in a tetrahedron of S atoms (black spheres),
and likewise each S has four similarly situated Zn nearest neighbors. The small halfsized, dashed-line cube delineates one such tetrahedron. The same structure would
result if the Zn and S atoms were interchanged.
The elements Si and Ge crystallize in this same structure by having the Si (or Ge)
atoms occupying all the sites on the two sublattices, so there are eight identical
atoms in the unit cell. This atom arrangement, sketched in Fig. 2.8a, is called the
diamond structure. Both Si and Ge have a valence of 4, so from bonding considerations, it is appropriate for each to be bound to four other atoms in the shape of
a regular tetrahedron.
In Appendix B we see that Table B.l lists the lattice constants a for various
compounds with the zinc blende structure, and Table B.2 provides the crystal radii of
their monatomic lattices in which the atoms are uncharged, as well as the ionic radii
for ionic compounds in which the atoms are charged. We see from Table B.2 that the
negative anions are considerably larger than the positive cations, in accordance with
the sketch of the unit cell presented in Fig. 2.9, and this size differential is greater for
the 111-V compounds than for the 11-VI compounds. However, these size changes
for the negative and positive ions tend to balance each other so that the 111-V
compounds have the same range of lattice constants as the 11-VI compounds, with Si
and Ge also in this range. Table B.4 gives the molecular masses, and Table B.5 gives
the densities of these semiconductors. The three tables B.l, B.4, and B.5 show a
regular progression in the values as one goes from left to right in a particular row,