21.4 and the top of Figure 21.6 are considered the unit cells; the two lower cells
in Figure 21.6 are not considered proper unit cells.
Figure 21.7 shows two possible unit cells for a very simple ionic crystal,
CsCl. Again, note that the corners of the unit cell contain only part of the atom
at that corner. In Figure 21.7a, each corner contains ^18 of a cesium atom.
Collectively, the eight corners of the unit cell give 8 ^18 1 Cs atom per cell,
and a single chlorine atom is in the center of the cell. This gives us a 11 ratio
of Cs atoms to Cl atoms in the compound, agreeing with the ratio of atoms in
the formula unit CsCl. In Figure 21.7b, the chlorine atoms are at the corners,
but inspection of the unit cell shows that this unit cell also supports CsCl as
the formula unit for this compound.
Figure 21.7 illustrates an important point in the determination of the unit
cell: the same species is typically found at the corners—and, as we will see
shortly, may also be found in other positions. This is necessary so that when
the unit cell is propagated in three dimensions, the partial atoms at the corners
can combine to make the complete atoms that compose the macroscopic crys-
tal. The translation of the unit cell in Figure 21.5 shows that, if the same atom
weren’t at all corners, the unit cell would not make sense as a multidimensional
crystal. This idea holds whether we are talking about simple ionic crystals like
CsCl or complicated molecular crystals like crystalline naphthalene, C 8 H 10.
In order to define a unit cell, the same species in the same orientation (for
molecules) must be present at the corners.
The same species (atom, ion, or molecule) may also be present in other
locations about the unit cell, and they may reside at differing distances and an-
gles depending on the dimension. CsCl, for example, has a very simple unit cell
that can be illustrated as a cubic structure; other compounds are not so sim-
ply described. It can be shown that there are only 14 ways of describing how
similar species (like the Csor Clions) will be arranged in three-dimensional
space. These 14 structures can be grouped into seven systems depending on
their symmetry elements; within each system, there can be variations in the ap-
pearance of atoms/ions/molecules at certain positions within the unit cell.
We will focus on the seven systems first; they are listed in Table 21.1. Spatially,
we define the systems in terms of the dimensions of the unit cell, labeled a,b,
and c(where, by convention,abc) and the angles that these dimensions
make with each other, labeled , , and. (Again, by convention,is the an-
gle between band c, is the angle between aand c, and is the angle between
aand b. See Figure 21.8.) The simplest crystal system is when all dimensions are
equal and all angles are 90°. This defines the cubiccrystal system and is exem-
plified by CsCl (see Figure 21.7). There are also systems where the angles are all
still 90° but one dimension of the unit cell is different from the other two
(tetragonal) or all dimensions of the unit cell are different (orthorhombic).734 CHAPTER 21 The Solid State: Crystals
(b)(a)Figure 21.6 Other possible unit cells. However,
convention requires that the smallest reproducible
part of the crystal be considered the true unit cell.
The top unit cell is a true unit cell (compare this
to the unit cell in Figure 21.4!); the bottom two
cells are not.
Figure 21.7 Two possible unit cells for CsCl.
Either can be considered a correct unit cell.
Table 21.1 The seven basic crystal systems
Name Unit cell dimensions Unit cell angles
Cubic abc
90°
Tetragonal ab c
90°
Orthorhombic a b c
90°
Trigonal abc
90°
Hexagonal ab c 90°;
120°
Monoclinic a b c
90°; 90°
Triclinic a b c