21.2 & 21.3 Types of Solids; Unit Cells
21.1.Give an atomic-level reason why ionic crystals are brit-
tle.
21.2.Boron nitride, BN, is a very hard material, harder than
diamond if prepared properly. Explain why it has diamond-like
properties.
21.3.Explain how unit cells can be described for polycrys-
talline materials.
21.4.Figure 21.35 shows a unit cell of diamond. Identify the
atoms that define the unit cell and determine the Bravais lat-
tice of this structure of diamond. How many atoms are in the
unit cell?
21.5.What is the relationship between the unit cell for dia-
mond (Figure 21.35) and the unit cell for zincblende (Figure
21.28)?
21.6.How many different unit cells can a crystal have if the
unit cell (a)has all 90° angles between its crystal axes; (b)has
all of its unit cell dimensions the same length; (c)has at least
one 90° angle between axes; (d)has no perpendicular axes
or equivalent unit cell dimensions?
21.7.A researcher proposes an edge-centered cubic unit
cell. What Bravais lattice would such a unit cell be better de-
scribed as?
21.8.Use geometry and Figure 21.11 to show that in three
dimensions the most efficient packing of hard-shell spherical
atoms will take up about 74% of the space. Can you give a
more exact figure for the amount of space taken up by the
hard-shell spherical atoms?
21.9.Use geometry to determine the largest atom that will
fit in a body-centered cubic unit cell. Express your answer in
terms of the unit cell dimension a.
21.10.What is the maximum percentage volume that can be
taken up by the atoms in a simple cubic unit cell? How much
less is it than close packing?
21.4 Densities
21.11.Prove the relationship in equation 21.3.
21.12.Zinc selenide, ZnSe, is a bright-orange compound
that is sometimes used as a transparent window for infrared
spectroscopy. It has a cubic unit cell with a5.669 Å and a
density of 5.263 g/cm^3. How many ionic formula units of ZnSe
are in each unit cell? Which cubic unit cell does it have?
21.13.Pyrite is a gold-colored mineral that is also known as
fool’s gold to miners. It is an ionic compound of iron and sul-
fur. It has a cubic unit cell with four formula units in the cell
and a density of 5.012 g/cm^3. If the unit cell parameter is
5.418 Å, what is the formula of this material?
21.14.Talc is a complex silicate mineral having the formula
Mg 3 Si 4 O 10 (OH) 2. It has a monoclinic unit cell with cell para-
meters a5.287 Å, b9.158 Å, c18.95 Å, and
99.50°. If there are four formula units in the unit cell, deter-
mine the density of talc.
21.15.One form of quartz, SiO 2 , has a hexagonal unit cell
(three formula units per cell) with a4.914 Å and c
5.405 Å. Determine the density of quartz.
21.16.Speculate on why the hexagonal unit cell is called
“hexagonal” if the unit cell isn’t a six-sided figure.21.5 Determination of Crystal Structures
21.17.At least 43 of the elements that are composed of in-
dividual atoms (as opposed to diatomic gases, molecular ele-
ments like sulfur and phosphorus, covalent network elements
like carbon, silicon, and germanium) have either hexagonal
close-packed or face-centered cubic crystal lattices. How would
you rationalize this?
21.18.Explain why zeroth-order X-ray diffraction does not
depend on wavelength but all other orders of diffraction do.
(Consider equation 21.5 and use n0.)
21.19.Use geometry to derive a form of Bragg’s law in terms
of the angle made with the perpendicularto the crystal plane,
as opposed to the definition of shown in Figure 21.16.
21.20.Although first-order diffractions might occur at angles
greater than 30°, if they do there will not be a second-order
(or higher) diffraction. Use Bragg’s law to argue why this is the
case. (Consider the properties of the sine function at angles
greater than 30°.)
21.21.Determine the angle of the first- and second-order dif-
fractions of X rays having wavelength 1.5511 Å by a crystal of
uranium dioxide, UO 2 , if the dspacing is 5.47 Å.
21.22.Certain X rays generated by bombarding metallic cop-
per have a wavelength of 1.54056 Å. Copper itself is face-cen-
tered cubic with a lattice parameter of 3.615 Å. At what an-
gle does copper diffract its own X rays?762 Exercises for Chapter 21
EXERCISES FOR CHAPTER 21Figure 21.35 The unit cell of diamond. See exercise 21.4.