Concise Physical Chemistry

c11 JWBS043-Rogers September 13, 2010 11:26 Printer Name: Yet to Come

176 LIQUIDS AND SOLIDS

Now, let us consider a different structure. It is not likely that spheres will pack,

each directly over its next lower neighbor. Instead (polonium excepted) they will slip

one-half radius to the right or left to give a more efficient packing pattern with the

central atom in the face of a cube or a slightly different pattern with the central atom

a little back of the face so that it takes a position in the center of the cube. These

are theface-centeredandbody-centered cubicstructures fcc and bcc. You can see

what is meant by face-centered by looking at the chlorine atoms only in Fig. 11.10.

Metals packed as planes situated above each other so that atoms of one layer fit into

the interstices of the planes above and below areclose-packed. Close-packed spheres

have a packing fraction of 0.740.

11.5 BRAVAIS LATTICES

At first thought, it would seem that there must be very many, perhaps infinitely

many, unit cells in real crystals. Quite the contrary is true. As early as 1850, Bravais

showed that unit cells in three dimensions can be classified into only seven sys-

tems on the basis of their rotational symmetry. These symmetries are important in

X-ray crystallographic studies in which the sample is rotated in an incident beam of

radiation. If a cell (crystal) is geometrically identical after rotation of 180◦(π), it has

at least a twofold axis of rotation. If that is all the symmetry it has about that axis,

then it has one C 2 axis of symmetry. Bravais showed that the seven classifications

shown in Table 11.1 permit 14 distinct crystal lattices in three dimensions.

11.5.1 Covalent Bond Radii

X-ray diffraction studies can also be carried out oncovalentlybonded molecular

solids. The results can be augmented by comparison or combination with other kinds

of diffraction studies in the solid, liquid, or gaseous states. It is possible to use beams

of electrons or neutrons in place of X rays. These studies yield bond distances like

rC−Cl=177 pm in CCl 4. One would like to have a bondcovalentradius for the Cl

TABLE 11.1 The Bravais Crystal Systems and Latticesa.

Basic Types 14 Bravais Lattices

1. Cubic a=b=c P, I , F


2. Tetragonal a=c P, I


3. Rhombic α, β, γ= 90 ◦ P


4. Monoclinic α= 90 ◦β, γ= 90 ◦ P, C


5. Hexagonal a=c P
   6.Triclinic α, β, γ= 90 ◦ P


6. Orthorhombic a=b=c P, I , F, C
   aPrimitive cells are designated P, body-centered cells are I, face-centered
   cells are F, and side-centered cells are C.