Physical Chemistry Third Edition

28.1 The Structure of Solids 1159

integers. The three integersh,k, andlare calledMiller indicesand are denoted by

their values inside parentheses, as in (hkl). The Miller indices are not required to be

positive, and a zero value for an index means that the plane is parallel to that axis.

Negative values are denoted by putting the negative sign above the digit, as in (111).

Figure 28.4 shows a unit cell with several planes and their Miller indices. Some planes

are parallel to other planes. For example, the (001) plane is parallel to the (002) plane,

and the (222) plane is parallel to the (111) plane. However, two parallel planes do not

necessarily contain the same pattern of lattice points.

Figure 28.5 shows schematically how a crystal acts like a diffraction grating. Two

planes of lattice points are shown in which lattice points of the first plane are per-

pendicularly opposed to lattice points of the second plane. We assume that there

a

0

(010)

b

c

a

0

(111)

b

c

a

0

( 210 )

b

c

a

0

( 002 )

b

c

Figure 28.4 Some Planes and Their Miller Indices.

dhklsin(
) dhklsin(
)

dhkl

Figure 28.5 Diagram Showing the Bragg Condition for X-Ray Diffraction by a Crystal

Lattice.