Elements of Modern Physics
262
Since the length of the cube is 2R, packing fraction f is
f = 6
π ≈ 52%
(8.6)
Body centred cube: A closer packing is provided by the body centered cubic
(bcc) lattice. Many elements crystallize in this form, e.g., Fe, Cr, Mo, W, Li, K,
Na, Rb, Cs, etc. The coordination number for these structures is 8. There are
two atoms which belong to each cell so that the volume occupied is
8
3
π R
3
.
Since the diagonal of the cube is 4R, the volume of the cube is (4R/3
1/2
)
3
and the
packing fraction f is
f =
1/ 2
3
8
π ≈ 68%
(8.7)
It should be noted that some binary compounds such as CsCl, have a body
centred cubic structure, with one type of atoms (e.g., Cl) at the centres, and the
other type of atoms (e.g.,) at the corners. However, since these atoms are different
the lattice is a simple cubic lattice with a pair of Cs and Cl atoms associated
with each corner.
Close-packed structures: The most efficient packing of atoms in a plane is
the one shown in Fig. 8.4(a), with each atom touching six atoms in the plane.
There are two ways in which the successive layers can be placed over one
another. The second layer can have atoms in positions marked as B (or
equivalently C). The third layer may have atoms in positions A leading to the
hexagonal close-packed (hcp) structure [Fig. 8.4(b)]. The positions of the atoms
in the successive layers of this structure may be indicated by ABABAB... .
Examples of elements which have this structure are Cd, Mg, Ti, Zn, etc. On the
other hand, the third layer may have atoms in positions C leading to the facecentred cubic (fcc) structure. The positions of the atoms in the successive layers
may be indicated by ABCABC... . The close packed layers of the face-centred
cubic structure, are in the body diagonal planes [Fig. 8.4(c)]. Some examples of
elements that have the face-centred cubic structure, are Cu, Ag, Au, Al, Pd, and
Pt. An interesting example is that of NaCl, KCl, etc. (Fig. 8.1) which consist of
two interpenetrating fcc sublattices, one of them made up of Na
+
or K
+
ions and
the other of Cl
–
ions, as shown in Fig. 8.1.
Both the close-packed structures, the hexagonal close-packed structure and
the face-centred cubic structure have a coordination number of 12 and have the
highest packing fraction. The packing fraction for the fcc structure is obtained
by noting that four atoms belong to each cell and occupy a volume of
3
16
3
R
π .
The length of a side of the cube is 2
3/2
R so that the packing fraction f is
262
Since the length of the cube is 2R, packing fraction f is
f = 6
π ≈ 52%
(8.6)
Body centred cube: A closer packing is provided by the body centered cubic
(bcc) lattice. Many elements crystallize in this form, e.g., Fe, Cr, Mo, W, Li, K,
Na, Rb, Cs, etc. The coordination number for these structures is 8. There are
two atoms which belong to each cell so that the volume occupied is
8
3
π R
3
.
Since the diagonal of the cube is 4R, the volume of the cube is (4R/3
1/2
)
3
and the
packing fraction f is
f =
1/ 2
3
8
π ≈ 68%
(8.7)
It should be noted that some binary compounds such as CsCl, have a body
centred cubic structure, with one type of atoms (e.g., Cl) at the centres, and the
other type of atoms (e.g.,) at the corners. However, since these atoms are different
the lattice is a simple cubic lattice with a pair of Cs and Cl atoms associated
with each corner.
Close-packed structures: The most efficient packing of atoms in a plane is
the one shown in Fig. 8.4(a), with each atom touching six atoms in the plane.
There are two ways in which the successive layers can be placed over one
another. The second layer can have atoms in positions marked as B (or
equivalently C). The third layer may have atoms in positions A leading to the
hexagonal close-packed (hcp) structure [Fig. 8.4(b)]. The positions of the atoms
in the successive layers of this structure may be indicated by ABABAB... .
Examples of elements which have this structure are Cd, Mg, Ti, Zn, etc. On the
other hand, the third layer may have atoms in positions C leading to the facecentred cubic (fcc) structure. The positions of the atoms in the successive layers
may be indicated by ABCABC... . The close packed layers of the face-centred
cubic structure, are in the body diagonal planes [Fig. 8.4(c)]. Some examples of
elements that have the face-centred cubic structure, are Cu, Ag, Au, Al, Pd, and
Pt. An interesting example is that of NaCl, KCl, etc. (Fig. 8.1) which consist of
two interpenetrating fcc sublattices, one of them made up of Na
+
or K
+
ions and
the other of Cl
–
ions, as shown in Fig. 8.1.
Both the close-packed structures, the hexagonal close-packed structure and
the face-centred cubic structure have a coordination number of 12 and have the
highest packing fraction. The packing fraction for the fcc structure is obtained
by noting that four atoms belong to each cell and occupy a volume of
3
16
3
R
π .
The length of a side of the cube is 2
3/2
R so that the packing fraction f is