Download Crystal Structure Tutorial

Remember the purpose of this reading assignment is to prepare you for class. Reading for
familiarity not mastery is expected.
After completing this reading assignment and reviewing the intro video you should be comfortable
either 1) Describing the following key concepts in your own words, or 2) asking focused questions
regarding the following key concepts.
•
•
•
The definition of a crystalline solid
Identifying and determining the values of the key features of a cubic crystal structure given
a sketch
Determining strongest directions, planes, and openness of cubic crystal structures.
Chapter I.
Structure of Crystalline Solids
Engineering materials are used in the solid state. A solid can be defined as a state of matter
that does not flow when subjected to a non-uniform force, in contrast to liquids and gases which do
flow. In solids the number of atoms in individual atoms varies dramatically. For example, an iron
crystal (a single molecule of iron in the solid state) has the chemical formula Fe(s). This does neither
means nor implies that a single molecule of iron in the solid state has only one atom. The same is
true for materials such as table salt, sodium chloride. The formula is NaCl(s), this means that there
is a one-to-one relationship between the number of sodium (Na) atoms and chlorine (Cl) atoms, but
there are many more than two atoms per molecule. In fact, the correct answer to the question: “How
many iron atoms are there in a single molecule of Fe(s)?” is, “I don’t know”. A single “grain” of iron
(which is a single crystal or molecule) can have a diameter of 1 :m. This is roughly 4000 atoms wide,
meaning the overall molecule has approximately 65 billion atoms. In most solids the sizes of the
individual grains varies, meaning that each molecule has a different number of atoms.
The arrangement of atoms inside these molecules has an effect on the mechanical, electrical
and chemical properties of the overall material. Therefore, it is important to have a basic
understanding of this arrangement. Many metals, ceramics, and even some polymers are crystalline
solids. This means that the material consists of individual molecules or crystals each containing
many atoms. In a crystal the atoms have long-range order. This means that the atoms are arranged
in a well defined, repeating pattern, over large (based on the size of the atom) distances in three
dimensional space.
A.
Basic Terms and Definitions
A crystal as a solid with long-range order. That is the atoms are arranged periodically in
three dimensional space. To illustrate the structures of crystals, atoms are considered to be spheres
with a definite radius. The electrons around an atomic nucleus are spread out in a “cloud,” although
there is a most probable distance of each electron from the nucleus. In spite of this atomic structure,
the “hard sphere” model for crystal structure is an accurate way to describe the structure. The nuclei
of the atoms have fixed positions that can define the crystal structure.
Having defined a crystal as a solid where the atoms (molecules or ions) are arranged
periodically in three dimensional space, the next question to be answered is how are they arranged.
A seventeenth century scientist named Bravais discovered that there were 14 unique ways to
arrange solid spheres, with three dimensional periodicity. These are called the Bravais Lattices. The
smallest complete lattice is called the unit cell. Thus, by knowing and understanding the unit cell,
one can understand begin to predict the properties of a crystalline solid.
In this chapter and book we will focus on cubic crystal structures. Other structures exist, it is
important that you know they exist, but the details are beyond what is needed of materials for a
general engineer. A cube can be completely defined by one measurement, the length of its edge. This
is referred to as the lattice parameter, and is given the symbol “a”. Knowing “a” one can completely
analyze the unit cell of a cubic structure.
B.
Cubic Crystal Structures (Metals)
Metallic bonding, where electrons are delocalized, or shared among many atoms, is the most
complicated. However, metals have the simplest crystal structures. There are three types of cuboc
crystal structures observed in metals:
•
•
•
simple cubic, in which the atoms touch along the edge of the cube,
body centered cubic, in which the atoms touch along the body diagonal of the cube,
and
face centered cubic, in which the atoms touch along the face diagonal of the cube.
1.
The Simple Cubic Lattice
Only one metal Po, above room
temperature is known to have the simple cubic
structure, however the structure is worth
discussing, as it can be used to illustrate the
principles of crystallography. The simple cubic
unit cell is shown in Figure 1. The atoms touch
along the cube edge. In this case the lattice
parameter, or edge length, and is therefore twice
the atomic radius. a=2r, therefore for Po,
a = 2r = 2( 0140
. nm) = 0.280nm
In the simple cubic lattice there is only Figure 1: Simple Cubic Unit Cell
one atom per unit cell. Each atom is represented
by a sphere in a corner, only one-eighth of this sphere is a part of this cube and another way of
looking at this is to say that each atom is in eight other unit cells. Therefore the number of atoms in
the unit cell is, 8(1/8)=1.
In this structure each atom touches six others. The number of atom an individual atom touches is
referred to as the coordination number. The coordination number in a simple cubic unit cell is 6.
2.
The Body-Centered Cubic Lattice (bcc)
A common cubic structure is body-centered cubic,
as shown in Figure 2 Here eight atom centers
are on the corners of the unit cell cube, and one is
at the center of the cube. Note the coordination
number is thus 8. There are two atoms in 2 bcc
cell, one from the corners and one in the center of
the cell. The atoms touch along the diagonal of
the cube, so that the relation between atomic
radius and lattice parameter is,
The lattice parameter of "-Fe is thus
Figure 2: The Body-Centered Cubic (bcc)
Unit Cell
4r
3
. nm)
4(0124
=
3
= 0.286nm
a=
Metals which have the bcc structure include chromium( Cr ), vanadium( V ), niobium( Nb ), tungsten
( W ), molybdenum( Mo ), the alkali metals( Group I ), and iron below 910oC("-Fe).
3.
The Face-Centered Cubic Lattice (fcc)
The most common metallic crystal structure is
face centered cubic, shown in Figure 3. In the
face-centered structure there are atom centers at
each of the eight corners of the cube and at the
centers of each of the six faces of the cube. The
eight atoms at the cube corners, one-eighth of the
unit cell and each of the six face centered atoms
are half in the unit cell, making a total of four
atoms in the unit cell. The coordination number
in fcc is 12.
The atoms in the face-centered cubic unit cell
touch along the face diagonal in this structure,
and therefore,
a=
Figure 3: Face-Centered Cubic Unit Cell
(fcc)
4r
2
The lattice parameter for Cu is therefore,
a=
4r 4( 0128
. nm)
=
= 0.362nm
2
2
Some metals that have face-centered cubic structure are aluminum(Al), copper(Cu), gold(Au),
lead(Pb), nickel(Nu), platinum(Pr), and iron above 910oC((-Fe).
C.
(Ionic Solids)
Structures of ionic compounds such as MgO are more complicated than those of crystals containing
only one kind of atom. Note, that as stated earlier, a single molecule of MgO(s) contains millions or
billions of Mg ions and O ions. In ionic solids the ions arrange themselves in such a way that charge
neutrality is maintained. Frequently this involves a long range three dimensional pattern and thus
many ionic solids are crystalline. We will describe some of the most common compound structures
CsCl (cesium chloride), NaCl (sodium chloride), ZnS (zinc blende), calcium fluorite (CaF2), and
barium titanate (BaTiO3).
1.
Coordination Number
The coordination numbers of ions in ironically-bonded structures are related to the ionic radii of the
ions. A positively charged ion will try to surround itself with as many negatively charged ions as
possible. However the negatively charged ions repel one another. Thus there are two competing
factors. The coordination number will depend upon the ratio of the two radii. The limiting case will
be where the anions just touch one another. The “ideal” radius ratio for a particular coordination
number can be calculated from a structure of closest packing for that radius ratio. These ideal ratios
are given in Table 1.
Consider 6-fold coordination, that is each cation
(+ charge) is surrounded by six anions (- charge).
This means that the cation must be large enough
to just separate six negatively charged anions.
Sketching this case, as in Figure 4, one can isolate
a triangle as shown. Note that this is an isoceles
right triangle. The length of the two equal sides
must equal the sum of the radius of the cation
and the radius of the anion. The length of the
other side is twice the radius of the anion.
Using the Pythagorean Theorem,
2
2
2( r + R) = ( 2 R)
2 ( r + R) = 2 R
( r + R) = 2 R
r=
(
r
=
R
Figure 4: 6-fold coordination of ions.
)
2−1R
(
)
2−1
or 0.414
In order for the cation to have a coordination # of 6, the cation must be large enough to separate 6
anions or
r
≥ 0.414
R
Thus the ratio at which all particles touch is
0.414. If r/R is larger than 0.414, the ions can
arrange themselves with 6 fold coordination and
the anions will not touch. Table 1 shows the
minimum radius ratio for a given coordination
number in ionic solids.
Table 1: Minimum Radius Ratios for
Given Coordination Numbers in Ionic
Crystals
CN
Min r/R
3
0.155
4
0.225
6
0.414
8
0.732
12
1.000
2.
The Cesium Chloride Structure (CsCl)
The coordination no. of CsCl can be found from the ratio of the ionic radii,
r (Cs + ) 0165
r
. nm
=
=
= 0.912
R
r ( Cr )
0181
. nm
Therefore the coordination number is 8, and each ion is surrounded by eight oppositely charged ions.
The cesium chloride(CsCl) unit cell is shown in
Figure 5. It is inappropriate to call this bcc.,
because the central particle is not the same as
those on the corners. In fact, this is a type of
simple cubic structure. Like simple cubic there is
only 1 CsCl molecule per unit cell. In this
structure there is only one ion pair, for example
chloride(Cl)), is at the cube corners and the other
(Cs+) at the center of the cube. Thus, the
structure of the individual ions is simple cubic.
The ions touch along the body diagonal
l = a 3 , one of each ion so the lattice
Parameter is ,
a=
2
( r + R)
3
Figure 5: Cesium Chloride Structure
3.
The Sodium Chloride Structure (NaCl)
In NaCl the ratio of ionic radii is,
r + r ( Na + ) 0.098nm
=
=
= 0.541nm
R−
r ( Cl )
0181
. nm
and thus the coordination number is 6.
This sodium chloride structure is cubic
and is shown in Figure 6. The unit cell can
have either sodium or chloride ions at the
corners of the cube; in Figure 6 the chloride
ions are on the corners. In addition, chloride
ions are at the face centers, so that the
structure of the chloride ions by themselves is
face-centered cubic. Sodium ions are between
the corner chlorides; the sodium ions are on a
face-centered cubic lattice. Therefore the
Bravais lattice for this structure is considered
to be face centered cubic, as the arrangement
of both Na and Cl ions are face centered cubic.
The ions touch along the edge of the cube, and
therefore
a = 2( r + R)
Figure 6: Sodium Chloride Structure
and the x-ray pattern will resemble that of an fcc metal. Like a lattice there are four NaCl molecules
per unit cell.
4.
Zinc Blende Structure (ZnS)
In ZnS the ratio of the ionic radii is
(
)
( )
+2
0.073 nm
r r Zn
=
=
= 0.41
0174
.
R R S− 2
nm
Therefore CN=4 and will the ions will have a
tetrahedral arrangement. It is possible to
arrange the tetrahedra into a cube as shown in
Figure 7. Note that each ion pair touches along
the body diagonal but, that this distance
accounts for only 25%. of the length. Thus,
Figure 7: Zinc Blende (ZnS) structure.
Each Zn ion is bonded to four S ions. The
ions are shown to be the same size for
clarity.
where r represents the radius of the cation, and
R- represents the radius of the anion.
5.
The Calcium Fluorite Structure (CaF2)
The calcium fluorite CaF2 structure is shown in
Figure 8. It resembles the zinc blende structure. The
difference between the calcium fluorite structure and
the zinc blende structure is the presence of additional
ions. There are eight anions in the unit cell and four
cations. Note that the stoichiometry of the unit cell is
precisely the same as the stoichiometry of the chemical
formula. There are four cations, (Ca+2), and eight anions
(F-) This must be the case, as the unit cell represents
the compound.
The geometric relationship between ion size and
lattice parameter is the same. The ions still touch along
the body diagonal, and this distance is one-quarter the
length of the body diagonal. Thus,
Figure 8:Calcium Fluorite (CaF 2)
Structure, the F- ions are colored
gray. For clarity all ions are
shown to be the same size.
6.
Perovskite (BaTiO3) Structure
The perovskite structure is a cubic
structure based on three ions. The lattice is
similar to both the fcc and NaCl lattices. O2ions occupy the face centered positions, Ba2+
ions occupy the corner positions and a Ti4+ ion
occupies the body centered position. This
means there are 3 oxygen anions, 1 barium
cation and 1 titanium cation. Thus, the
stoichiometry of the system is preserved. The
center of this structure is shown in Figure 9.
Note that the anions and the cations must
touch, the question is which set of ions touch.
By assuming the O2- ions touch the Ba2+ ions ,
it can be shown that the Ti4+ ion is slightly too
large to fit in the center of the cell. This
Figure 9: Center of Perovskite Structure
means that the cell is slightly stretched. If
one applies a force to the cell, then its
dimensions will change in one direction. Assume for example that one pressed on the top and bottom
surface of the cell. The z-dimension would then decrease and the Ti4+ ion would be pushed off-center.
This will induce a charge imbalance, or electric dipole in the cell as shown in Figure 7. Thus by
imposing a force on the cell, one causes an electric field to be induced. This is how a microphone
works. One’s voice (or the displacements in the air associated with sound waves) causes a force to be
induced on the microphone, this generates an electric field which can be analyzed, manipulated and
reproduced. Conversely, an electric field will cause the Ti4+ ion to be displaced from the center of the
cell. This will cause a dimensional change in the unit cell. The positive ion still attracts the negative
ions. This is how a speaker works. An electric field causes the crystal to change dimension, which
causes vibrations in the membrane of a speaker. These materials are called piezoelectrics.
D.
Structure of Covalent Network Solids
1.
Diamond Cubic
Covalent network solid is a carbon (diamond),
silicon, germanium, and SiO2 are covalent
network solids. That is the atoms are covalently
bonded to one another in such a way that the
molecular pattern repeats itself. This is most
commonly seen where sp3 hybridiztion occurs.
Therefore the molecular geometry is
tetrahedral. As with the Zinc blende (ZnS)
Structure. The tetrahedral can be arranged in a
cubic structure except that all atoms are
identical. The diamond cubic structure is shown
in Figure 10.
Note there is only one interstitial atom per body
diagonal. In the figure each interstitial atom is
indicated by a four, and each atom is bonded to
one corner atom and three face centered atoms.
Figure 10: Diamond Cubic Unit Cell
There are some differences between the fcc unit cell and the diamond cubic unit cell. The fcc unit cell
has four atoms, the diamond cubic unit cell has eight. There are the four atoms from the fcc lattice
and the four additional atoms internally, giving eight total. In addition each atom is bonded to four
other atoms. The atoms touch along the body diagonal. This means the relationship between the
lattice parameter and the radius of the atom will be different than for a fcc structure. As with ZnS
the atoms touch along the body diagonal and this distance is only 1/4 of the length. However since
the atoms, and thus their radii are identical the relationship between the lattice parameter and the
atomic radius is,
a 3
= 2r
4
2.
Silica
Almost all rocks and minerals are silicate compounds, which are compounds based on. Many
silicates are important engineering materials. The simplest silicate is silicon dioxide, SiO2. The Lewis
structure of which is shown in Figure 11. Each oxygen ion is bonded to two silicon ions, and each
silicon ion to four oxygen ions. The crystal structure is shown in Figure 12.
Figure 12 Structure of $-cristobalite,
the cubic form of silica.
Figure 11: Lewis Structure of SiO 2
E.
Crystal Features
Within a given unit cell there are lattice positions, lattice directions, and lattice planes. Because, this
text is focused on the general engineering
student, not the materials engineering student,
our focus will be limited to the most important
directions and planes in cubic systems.
1.
Lattice Positions
Lattice positions are taken relative to an origin,
usually fixed at the lower, left, rear corner of
the unit cell; and measured in terms of the edge
length of the cube. Thus, the position one unit
in the “x-direction” from the origin is labeled
100 (note this is not pronounced “one-hundred”,
but “one-zero-zero”) and corresponds to the
lower, left, front corner. The position which is
one unit in the “x-direction” and one unit in the
“y-direction” is labeled 110. The most important
lattice positions for cubic crystal structures are
shown in Figure 13.
Figure 13: Lattice Positions in a Cubic
Unit Cell
2.
Lattice Directions
In describing crystal structures is it
useful to have a method of designating
directions in the unit cell or crystal lattice. A
direction in the crystal lattice is a vector from
the origin of a coordinate system to a point in
the lattice. A more formal way of defining a
directions is:
A direction [uvw] is the vector which extends
from the origin to the point uvw.
The mechanical and electrical properties of a
crystal often depend on the crystals orientation.
As the distance between atoms will vary from
direction to direction,
Figure 14: Important
• The [100] represents the edge, the
vector from the origin to 100.
Crystal Systems
• The [110] represents the face
diagonal, the vector from the origin to
110.
• The [111] the body diagonal, the vector from the origin to 111.
Directions in Cubic
These directions are shown in Figure 14. This is most useful in determining which direction will be
strongest in a unit cell. For example, the face diagonal will be the strongest direction in fcc as it is
the direction in which the atoms touch.
Comparison of Directions in fcc Crystals
edge
face diagonal
We will focus on the important planes during the class-session
body diagonal
4.
Bulk Properties
Density( also referred to as bulk or volume density), D, is defined as mass per unit volume. the
density of a unit cell is therefore the mass of the unit cell divided by the volume of the unit cell.
These quantities can be calculated for any unit cell from a knowledge of how many atoms are in the
cell. To find the number of atoms in the unit cell, one must visualize how much of each atom is in
the cell.
For example, we can calculate the density of aluminum knowing only its atomic radius, 0.143 nm, its
crystal structure, fcc, and its atomic weight of 27.0 grams/mole. The unit cell contains four atoms,
therefore the mass of the unit cell is the mass of four aluminum atoms. mass of the unit cell can be
determined from,
Since aluminum is fcc,
.
and
The packing factor of a structure is defined as the ratio of the volume of the atoms in the unit cell to
the volume of the unit cell itself. In the face-centered cubic structure, there are four atoms, so the
volume occupied by the atoms is simply four times the volume of an atom, 4×4BR3/3 and the volume
of the unit cell is a3. However there is a relationship between a and r is a
=
4r
2
The packing factor
is:
For the body-centered unit cell, the packing factor is 0.68. Simple cubic 0.52 and diamond 0.34
For ionic solids this will depend on each system as the r/R will vary from compound to compound.
F.
Non Cubic Structures
The most important thing to remember about non-cubic crystal structures is that they exist and
metals or ceramics with these structures are brittle. The most important non-cubic structures for the
general engineer to be aware of are the hexagonal-close-packed (hcp) and body-centered-tetragonal
(bct). Metals with the hcp structure include the rare-earths, magnesium and zinc. These metals are
britlle. Certain steels, when cooled quickly from temperatures above 800oC, will form a very strong,
but brittle material called martensite. This has a bct structure. While a thorough understanding of
these structures is necessary for materials engineers and metallurgists, they are beyond the scope of
an introductory course for general engineers.
G.
Summary
The arrangement of atoms in solids is the basis for all material properties. In this chapter,
we have discussed cubic crystal systems which are the basis of many engineering materials.