12
INTRODUCTION TO PHYSICS OF THE SOLID STATE
octahedral sites and Mg2+ (radius 0.066 nm) located in one-eighth of the tetrahedral
sites in a regular manner.
2.1.3. Face-Centered Cubic Nanoparticles
Most metals in the solid state form close-packed lattices; thus Ag, Al, Au, Co, Cu,
Pb, Pt, and Rh, as well as the rare gases Ne, Ar, Kr, and Xe, are face-centered cubic
(FCC), and Mg, Nd, Os, Re, Ru, Y, and Zn, are hexagonal close-packed (HCP). A
number of other metallic atoms crystallize in the not so closely packed bodycentered cubic (BCC) lattice, and a few such as Cr, Li, and Sr crystallize in all three
structure types, depending on the temperature. An atom in each of the two closepacked lattices has 12 nearest neighbors. Figure 2.5 shows the 12 neighbors that
surround an atom (darkened circle) located in the center of a cube for a FCC lattice.
Figure 10.18 (of Chapter IO) presents another perspective of the 12 nearest
neighbors. These 13 atoms constitute the smallest theoretical nanoparticle for an
FCC lattice. Figure 2.6 shows the 14-sided polyhedron, called a dekatessarahedron,
that is generated by connecting the atoms with planar faces. Sugano and Koizumi
(1998) call this polyhedron a cuboctahedron. The three open circles at the upper
right of Fig. 2.6 are the three atoms in the top layer of Fig. 10.18, the six darkened
circles plus an atom in the center of the cube of Fig. 2.6 constitute the middle layer
of that figure, and the open circle at the lower left of Fig. 2.5 is one of the three
obscured atoms in the plane below the cluster pictured in Fig. 10.18. This 14-sided
polyhedron has six square faces and eight equilateral triangle faces.
If another layer of 42 atoms is layed down around the 13-atom nanoparticle, one
obtains a 55-atom nanoparticle with the same dekatessarahedron shape. Larger
Figure 2.5. Face-centered cubic unit cell showing the 12 nearest-neighbor atoms that surround
the atom (darkened circle) in the center.
INTRODUCTION TO PHYSICS OF THE SOLID STATE
octahedral sites and Mg2+ (radius 0.066 nm) located in one-eighth of the tetrahedral
sites in a regular manner.
2.1.3. Face-Centered Cubic Nanoparticles
Most metals in the solid state form close-packed lattices; thus Ag, Al, Au, Co, Cu,
Pb, Pt, and Rh, as well as the rare gases Ne, Ar, Kr, and Xe, are face-centered cubic
(FCC), and Mg, Nd, Os, Re, Ru, Y, and Zn, are hexagonal close-packed (HCP). A
number of other metallic atoms crystallize in the not so closely packed bodycentered cubic (BCC) lattice, and a few such as Cr, Li, and Sr crystallize in all three
structure types, depending on the temperature. An atom in each of the two closepacked lattices has 12 nearest neighbors. Figure 2.5 shows the 12 neighbors that
surround an atom (darkened circle) located in the center of a cube for a FCC lattice.
Figure 10.18 (of Chapter IO) presents another perspective of the 12 nearest
neighbors. These 13 atoms constitute the smallest theoretical nanoparticle for an
FCC lattice. Figure 2.6 shows the 14-sided polyhedron, called a dekatessarahedron,
that is generated by connecting the atoms with planar faces. Sugano and Koizumi
(1998) call this polyhedron a cuboctahedron. The three open circles at the upper
right of Fig. 2.6 are the three atoms in the top layer of Fig. 10.18, the six darkened
circles plus an atom in the center of the cube of Fig. 2.6 constitute the middle layer
of that figure, and the open circle at the lower left of Fig. 2.5 is one of the three
obscured atoms in the plane below the cluster pictured in Fig. 10.18. This 14-sided
polyhedron has six square faces and eight equilateral triangle faces.
If another layer of 42 atoms is layed down around the 13-atom nanoparticle, one
obtains a 55-atom nanoparticle with the same dekatessarahedron shape. Larger
Figure 2.5. Face-centered cubic unit cell showing the 12 nearest-neighbor atoms that surround
the atom (darkened circle) in the center.