Download Chapter 3. The structure of crystalline solids

Chapter 3. The structure of crystalline solids
3.1. Crystal structures
3.1.1. Fundamental concepts
3.1.2. Unit cells
3.1.3. Metallic crystal structures
3.1.4. Ceramic crystal structures
3.1.5. Silicate ceramics
3.1.6. Carbon
3.1.7. Polymorphism
3.2. Crystallography
3.2.1. Crystal systems
3.2.2. Crystallographic directions and planes
3.2.3. Linear and planar density
3.2.4. Closed-packed crystal structures
3.3. Crystalline and noncrystalline materials
3.3.1. Single crystal
3.3.2. Polycrystalline materials
3.3.3. X-ray diffraction
3.3.4. Noncrystalline solids
3.1. Crystal structures
3.1.1. Fundamental concepts
Solid materials may be classified according to the regularity with
which atoms or ions are arranged with respect to one another.
A crystalline material is a material that the atoms are situated in a
periodic array over large atomic distances.
A noncrystalline or amorphous material is a material that does not
have a long-range atomic order.
Crystal structure is a manner in which atoms, ions, or molecules
are spatially arranged.
Some properties of crystalline solids depend on the crystal structure.
Atoms or ions are considered as atomic hard sphere model
(the atoms touch one another).
Lattice is a [3] array of points coinciding with atom positions
or sphere centers.
3.1. Crystal structures
3.1.2. Unit cell
Unit cell is the basic structural unit or building block of the crystal
structure.
The unit cell defines the crystal structure by virtue of its geometry
and the atom positions within
Most of unit cells are parallelepipeds of prisms having 3 sets of
parallel faces.
A unit cell is a representation of the symmetry of the crystal structure
which is the highest level of geometrical symmetry.
The most relatively simple crystal structures are :
•Face-centered cubic (FCC),
•Body-centered cubic (BCC),
•Hexagonal close-packed (HCP).
3.1. Crystal structures
3.1.3. Metallic crystal structures
FCC structure has a cubic geometry with atoms located at each of
the corners and the centers of all the cube faces.
Many metals have this FCC structure.
The relation between the cube edge length,a and the atomic radius, R:
a = 2R 2
The FCC structure has 4 whole atoms [= (8 x 1/8) + (6 x ½)]
The coordination number (CN) is a number that an atom touches
the nearest neighbour atoms.
The coordination number of FCC is 12
The atomic packing factor (APF)
APF =
vol.of atoms in a unit cell V
=
total unit cell volume
V
S
C
For FCC:
4
3
VS = (4) π R 3 =
16 3
πR
3
and
V c = 16 R 3 2
APF = 0.74; which is the maximum packing possible
BCC structure has a cubic unit cell with atoms located at each
of the corners and a single atom at the cube center.
The relation between the cube edge length,a and the atomic radius, R:
a=
4R
3
The BCC structure has 2 whole atoms [= (8 x 1/8) + (1 x 1)]
The coordination number of BCC is 8
For BCC:
4
8
Vs = (2) πR 3 = πR 3
3
3
APF = 0.68
and
64 R 3
Vc =
3 3
HCP structure is the common metallic crystal structure.
It has 6 atoms in regular hexagones at the top and the bottom
faces and surround a single atom in the center. Three additional
atoms are on a plane located between the top and the bottom
planes.
The ideal c/a ratio is 1.633
The HCP structure has 6 whole atoms
[= (2 x 6 x 1/6) + (2 x ½) + (1 x 3)]
The coordination number of HCP is 12 and the APF is 0.74
Metal density
nA
ρ=
N A VC
where: n = number of atoms associated with each unit cell,
A = atomic weight,
Vc = volume of the unit cell
NA = Avogadro’s number (6.023 X 1023 atoms/mol)
Example
Copper has an atomic radius of 0.18 nm, an FCC crystal structure,
and an atomic weight of 63.5 g/mol.
Compute its theoretical density and compare the answer with its
measured density.
Answer
FCC: n = 4 atoms/unit cell; Acu = 63.5 g/mol; and
V c = 16 R 3 2
n A Cu
ρ=
VC N A
=
(4 atom/unit cell)(63.5 g/mol)
[16 (1.28x10-8 cm)3 /unit cell 2 ](6.02x10 23 atoms/mol)
= 8.89 g/cm3
The theoretical density is 8.94 g/cm3
3.1. Crystal structures
3.1.4. Ceramic crystal structures
Ceramics are more complex than metals because ceramics are
consisted of at least two elements.
The atomic bonding is in the range of ionic to covalent.
The degree of ionic character depends on the electronegativities
of the atoms.
(the electronegativity is usually shown in the Periodic Table)
For ceramics with predominantly ionic, the atoms are considered
as ions.
The metallic ions or cations are positively charged
The nonmetallic ions or anions are negatively charged.
The crystal structure depends on two characteristics:
•the magnitude of the electrical charge on each of the component ions,
•the relative sizes of the cations and anions
The 1st characteritic:
the magnitude of the electrical charge on each of the component ions.
The cations must be in balance with the anions.
The chemical formula of a compound indicates the ratio of cations and
anions or the composition that achieves this charge balance.
Example:
A calcium ion has a +2 charge (Ca2+ )
CaF2
A fluorine ion has a –1 charge (F-)
The 2nd characteristic:
the relative sizes of the cations and anions.
The sizes of ionic radii of the cations (rC) and anions (rA) determined
the configuration of a compound.
Cations are ordinarily smaller than anions, rC/rA ≤ 1
3.1.4. Ceramic crystal structures
Most ceramics are composed by an equal number of cations
and anions and identified as AX compounds
(A refers to the cation and X refers to anion).
3.1.4.1. AX-type crystal structures
A. Rock salt structures
The most common AX crystal structures is the sodium chloride
(NaCl) or rock salt.
Two interpenetrating FCC lattices, one composed of the cations
(Na+), and the other of anions (Cl-).
The coordination number of both cations and anions is 6.
The rC/rA is between 0.414 – 0.732.
Some common rock salt structures are
NaCl, MgO, MnS, LiF, and FeO.
B. Cesium chloride structures
The anions are situated a each of the corners of a cube, whereas
the cube center is a single cation.
This is not a BCC crystal structure!!
The coordination number of both cations and anions is 8.
C. Zinc blende or sphalerite structure
(Mineralogical term for zinc sulfide (ZnS)
S atoms are located in all corners and face positions of a cubic,
while Zn atoms fill the interior in tetrahedral positions in covalent
bonding.
The coordination number for both cations and anions is 4
(tetrahedrally coordinated).
If the Zn and S atom positions are
reversed, the same structure is
obtained.
Some common zinc blende
crystals are ZnS, ZnTe, and SiC
3.1.4.2. AmXp-type crystal structures
This structure has dissimilar charges
of cations and anions.
The chemical formula is AmXp
where m and/or p ≠ 1.
Example : CaF2
The rC/rA is 0.8 and the coordination
number is 8.
Calcium ions are located at the center
of a cube and fluorine ions at the corners.
One unit cell consists of 8 cubes.
Some other compounds are UO2,
PuO2 , and ThO2.
3.1.4.3. AmBnXp-type crystal structures or perovskite crystal structure
Some compounds have more than one type of cations, they are
identified as A and B. The chemical formula is AmBnXpExample: BaTiO3.
Ba2+ ions are located at eight
corners of a cube, a single To4+
is at the cube center and O2- ions
are situated at the center of each
of the six faces.
Ceramic density
n' (∑ A C + ∑ A A )
ρ=
VC N A
where: n’ = the number of formula units within the unit cell,
ΣAC = the sum of the atomic weights of all cations in the formula unit,
ΣAA = the sum of the atomic weights of all anions in the formula unit,
VC = the unit cell volume,
NA = Avogadro’s number = 6.023 x 1023 formula units/mol.
Example
Calculated the theoretical density for sodium chloride. The atomic weight
of sodium and chloride, ANa and ACl are given as 22.99 g/mol and
35.45 g/mol respectively. The ionic radii of sodium and chloride, rNa+
and rCl- are given as 0.102 nm and 0.181 nm respectively.
Compare the result with its measured density?
n’ = 4
ΣAC = ANa = 22.99 g/mol,
ΣAA = ACl = 35.45 g/mol,
a = (2 rNa+ + 2 rCl- ) m
VC = a3 = (2rNa+ + 2rCl- )3 m3
= [(0.204 x 10-9)+0.362 x 10-9)]3 m3
= 559.276 10-9 m3
n' (A Na + A Cl )
ρ=
(2rNa + + 2rCl- )3 N A
ρ=
4(22.99 + 35.45)
(559.276X10 23 )(6.023X10− 23 )
ρ = 2.14g/cm3
3.1. Crystal structures
3.1.5. Silicate ceramics
Silicates are materials composed primarily of silicon and oxygen.
The basic unit of silicates is SiO4 4- tetrahedron, which is each atom of
silicon (located in the corners of tetrahedron) is bonded to four oxygen
atoms (situated at the center of tetrahedron).
Some common silicates are rocks, clays, sand, and a bulk of soils.
Silicates are not considered to be ionic because interatomic covalent
Si-O bonds are very strong.
Silicate structures vary in different arrangements as each oxygen atom
requires an extra electron to achieve a stable electronic structure.
The arrangements could be [1] or [2] or [3] dimensional structures.
3.1.5.1. Silica
Silica or silicon dioxide (SiO2) is the most simple silicate.
Oxygen atom at every corner of each tetrahedron shares the
oxygen atoms with the adjacent tetrahedral and this configuration
builds a [3] network structure.
This material is electrically neutral and all atoms have stable
electronic structures.
A crystalline structure is formed if these tetrahedra are arrayed in a
regular and ordered manner. There are 3 primary polymorphic
crystalline forms of silica, they are: quarts, cristobalite, and tridymite.
The structures are complicated and relatively open so that they have
low densities. (eg ρquartz 2.65 g/cm3).
The Si-O interatomic bonding is strong so that they have relatively high
melting temperature. (eg Tmquartz 1710 ºC).
A noncrystalline solid or glass structure
can also be formed from silica.
3.1.5.2. The Silicates
Oxygen atom at every corner of each tetrahedron shares the other
tetrahedral to form complex structures.
Some positively charged cations such as Ca2+ ,Mg , and Al can
be part of the structures. These cations compensate the negative
charges of SiO4 4- to form
a neutral compound and
to build an ionic bonding
between the SiO4 4tetrahedral together.
A. Simple silicates
B. Layered silicates
A [2] sheet or layered structure can also be produced by the sharing
of 3 oxygen ions in each of tetrahedral.
The repeating unit formula is (Si2O5)2-
The net negative charge is
associated with the unbound
oxygen atoms ( ⊥ layer).
The excess of cations is
associated with the 2nd layer.
Net charge: neutral.
Basic structure of clays and
other minerals.
One of the most common clay minerals and simple two-layer silicate
sheet structure is kaolinite.
Its formula is Al2(Si2O5)(OH)4
The OH- and the O2- ions in the anion midplane has a strong ioniccovalent bonding, whereas
the adjacent sheets
have weak van der Waals
bondings.
3.1. Crystal structures
3.1.6. Carbon
Carbon is an element that exsist in various forms, from polymorphic
to amorphous sates.
Carbon is not classified as metal, ceramic or polymer, however
sometimes it is classified as a ceramic
polymorphic
3.1.6.1. Diamond
Diamond is a metastable carbon polymorph
at RT and atmospheric pressusre.
The crystal structure is a variant of the zinc
blende.
It is called the diamond cubic crystal
structure.
3.1.6.2. Graphite
Graphite is more stable than diamond at ambient temperature and pressure.
The crystal structure is composed of layers of hexagonally arranged carbon
atoms. In the layers, each carbon atom is bonded to three coplanar
neighbour atoms by strong covalent bonds.
Between layers the electron is bonded
by a weak van der Waals bond.
3.1.6.3. Fullerenes and carbon nanotubes
Fullerenes is a polymorphic of carbon (1985).
The structure consists of a hollow spherical cluster of 60 carbon atoms
and is called a C60 molecule.
A molecule is composed of groups of carbon atoms that are bonded to
One another to from both hexagon (six-carbon atom) and pentagon
(five-carbon atom) geometrical configurations.
Its commercial name is buckminsterfullerene.
Invented by R. Buckminster Fuller (1985)
“Soccer ball symmetry”
20 hexagon and 12 pentagon are arrayed
in such way that no two pentagons share
a common side.
Carbon nanotube structure consists of a single sheet of graphite,
rolled into a tube & both ends are capped with fullerene hemispheres.
The nano represents that the tube diameter is on the order of a
nanometer.
It has a very good aspect ratio; which the length of the molecule is much
greater than its diameter.
Special properties of CNT:
- Extremely strong and stiff, relatively ductile
- Relatively low density
- Unique and structure-sensitive electrical characteristics
Some properties:
Tensile strength
50 – 200 GPa
Elastic modulus ≥ 103 GPa
Fracture strain
5% - 20%
Another form:
Multiple-walled CNT
(consists of concentric cylinders)
3.1.7. Polymorphism
Polymorphism is a phenomenon when some metals and nonmetals
may have more than one crystal structure.
(In elemental solids, it is called allotropy).
Example:
Graphite is stable at ambient condition, whereas diamond is formed
at extremely high pressures.
Iron has a BCC structure at RT and FCC structure at 912 °C.