‘’THE STRUCTURE OF
CRYSTALLINE SOLIDS’’
IE-114 Materials Science and General Chemistry
Lecture-3
Outline
• Crystalline and Noncrystalline Materials
1) Single, Polycrystalline, Non-crystalline solids
2) Polycrystalline Materials
• Crystal Structures
1)Unit cells
2) Metallic crystal structures
4) Crystal Systems (Directions and Planes)
5) Linear and Planar Atomic Densities
Crystal Structures
 Material classification can be made based on the regularity or
irregularity of atom or ion arrangement with respect to each other.
1) Crystalline Material
 Single crystalline
 Polycrystalline
Atoms are situated in a repeating or periodic array
over large atomic distances (Longe range order)
e.g.All metals, some ceramics and polymers
2) Noncrystalline (amorphous) Material
Long range atomic arrangement lacks
in this type of materials.
Two Dimensional View of Atomic Arrangements
Single crystal
Single crystal diamond
(schematic view)
Polycrystal
Amorphous
Single crystal
of CaF2
Formation of Polycrystals
1) During heavy deformation
2) During solidification from melt
Solidification of a pure metal (T=Tm)
Solid crystals
t=t1
t=t2
(Tm)
Liquid
Liquid and small crystals
t=t3
Structure of the material
seen under microscope
Liquid and relatively larger
crystals
t=t3
Completely solid. Material contains
many grains(polycrystalline
material)
t1 t2
t3
Describing Crystal Structure
The atoms or ions are thought as
solid spheres with their sizes defined.
This is called atomic hard sphere
model. All atoms are identical in this
model.
UNIT CELL: Smallest repeating group
 Unit cells can be imagined as the building block of the crystal structure.
 Unit cells in general are paralelepipeds or prisms having three sets of
parallel faces, one is drawn within the aggregate of spheres.
Crystal Structures of Metallic Materials
 Atoms tend to be densely packed
 Atomic bonding is metallic and nondirectional.
 No restrictions as to the number and position of nearest
neighbor atoms
Four simple crystalline structures found in metallic materials:
1) SC (Simple cubic) crystal structure
2) BCC (Body Centered Cubic) crystal structure
3) FCC (Face Centered Cubic) crystal structure
4) HCP (Hexagonal Close-Packed) Crystal Structure
Simple Cubic (SC) Structure
 Rare in nature due to poor packing (only Po has this structure)
 Close-packed directions are cube edges.
* Coordination number: number of nearest neighbor or touching atoms.
(Arrangement of atoms in one SC unit cell)
 Coordination number = 6
 The number of atoms per SC: (1/8)*8 = 1 atom/cell
 The relationship between unit cell edge length (a) and atomic radius (R);
a=2R
Body Centered Cubic (BCC) Structure
--Note: All atoms are identical;
the center atom is shaded
differently only for ease of
viewing.
BCC unit cell
a
(Arrangement of atoms in one BCC unit cell)
 Coordination number = 8
 The number of atoms per BCC: (1/8)*8 + 1= 2 atoms/cell
 The relationship between unit cell edge length (a) and atomic radius (R);
a=(4/3)R
Face Centered Cubic (FCC) Structure
--Note: All atoms are identical;
the center atom is shaded
differently only for ease of
viewing.
FCC unit cell
a
(Arrangement of atoms in one FCC unit cell)
 Coordination number = 12
 The number of atoms per FCC: (1/8)*8 + 6*(1/2)= 4 atoms/cell
 The relationship between unit cell edge length (a) and atomic radius (R);
a=(22)R
Hexagonal Closed Packed (HCP) Structure
 The top and bottom faces of the unit cell have six atoms that form
regular hexagons and a single atom in the center. Another plane provides
three additional atoms is situated between top and bottom planes.
HCP Unit Cell
 Coordination number = 12
 The number of atoms per HCP: (1/6)*12 + 2*(1/2) + 3 = 6 atoms/cell
 Ideally c/a=1.633, but for some metals this ratio deviates from the ideal
value.
Structure of Compounds
 Ionic bonding (NaCl)
 Compounds often have similar close-packed structures.
ClNa+
 The number of atoms per SC: [(1/8)*8+(1/2)*6] Cl ions + [(1/4)*12+1] Na ions
(4 Cl- + 4 Na+ ) ions/unit cell
 The relationship between unit cell edge length (a) and atomic radius (R);
a=2RCl- + 2RNa+
Polymorphism
 Atoms may have more than one type of crystal structure
Example : Iron (Fe)
FCC
heating
BCC
cooling
Characteristics of Some Selected Elements
Crystal Systems
There are 7 different crystal structures;
Cubic, tetragonal, orthorhombic, rhombohedral, monoclinical, triclinic, hexagonal
 Note that cubic system is the most symmetric, while triclinic is the least
one.
Crystal Systems
The unit cell geometry: x,y,z coordinate system is established with
its origin at one of the unit cell corners and axes coincide with the
edges of the paralelepiped extending from that corner, the origin.
There are six parameters to define the
geometry of the unit cell:
Three edge lengths
: a, b, c
Three interaxial angles : α, β, 
Also called lattice
parameters.
Crystallographic Directions in
Cubic Crystals
 Particular crystallographic direction is shown in a unit cell
 The direction is a line between two points or a vector as shown
below:
Steps for defining a direction in a crystal system:
1)
A vector is positioned such that it passes through the origin of the
coordinate system. Then you can move the vector if you keep the
parallelism.
2)
The length of the vector projection on each of the three axes is
determined in terms of the unit cell dimensions (a, b, c).
3)
The three numbers are multiplied or divided by a common factor to
reduce them to the smallest integer values.
4)
Three indices are enclosed in square brackets as [uvw]
5)
Remember to count for positive and negative coordinates based on the
origin. When there is a negative index value, then show that by a bar
over it, as
[111]
This vector has a component in –y direction.
Example1:
Example2:
FAMILY OF DIRECTIONS, <uvw>
For cubic crystal structures, several nonparallel directions with
different indices are equivalent.
(the spacing of atoms along each direction is the same)
(equivalent directions) <100> Family
Hexagonal Crystals:
There is a four-axis (Miller-Bravais) coordinate system used for this type
of structures.
 Three a1, a2, and a3 axes are placed within a single plane (basal plane) and at 120°
angles to one another.
 The z axis is perpendicular to the selected basal plane.
Some directions in HCP crystal structure
Crystallographic Planes in
Cubic Crystals
Except HCP, crytallographic planes are specified using three
MILLER INDICES (hkl). Any two planes parallel to each other are
equivalent and have same indices.
The determination of the h,k, and l index numbers are as follows:
1)
2)
3)
4)
5)
6)
If the plane passes through the selected origin, then construct a
new parallel plane or change the originto a corner of another unit
cell.
Plane intersects or parallels each of the axes: the length of each
axis is determined by using lattice parameters; a,b, and c.
Take the reciprocals of the lattice parameters. Therefore a plane
that parallels an axis has a ZERO index. (1/infinity=zero)
You may then change these three numbers to the set of smallest
integers using a common factor.
Report the indices as (hkl).
An intercept on the negative side of the origin is indicated by a bar
over that index.
Example1:
Example2:
Example3:
For cubic crystals: Planes and directions having the same indices are
parallel to one another.
Family of Planes, {hkl}
A family of planes is formed by all those planes that are
crystallographically equivalent, {100}, {111}. (for cubic structure)
{111} =
Planes in Hexagonal Crystals:
Equivalent planes have the same indices as directions Four-index scheme is
used (hkil) and the index i is calculated by the sum of h and k through
i = - (h+k)
This scheme defines the orientation of
a plane in a hexagonal crystal.
Theoretical Density, ρ
Theoretical density (ρ) of elements can be found by the use of unit cell and
crystal structure
Example:
Copper has an atomic radius of 0.128 nm, FCC crystal structure and an
atomic weight of 63.5 g/mol. Compute its density and compare the
answer with its measured density
Result: theoretical Cu = 8.89 g/cm3
Compare to actual: Cu = 8.94 g/cm3
Theoretical Density of NaCl
NaCl unit cell
Atomic Packing Factor (APF)
 Metals generally have high APF to maximize the shielding provided by electron
cloud.
Fraction of solid sphere volume in a unit cell
APF for a face-centered cubic(FCC) structure is 0.74
a
For FCC structure;
 a=(22)R
 The number of atoms per SC: (1/8)*8 + 6*(1/2)= 4 atoms/cell
 APF for BCC structure = 0.68
 APF for HCP structure = 0.74
Linear and Planar Density
 Linear and planar atomic densities are one and two dimensional
analogs of atomic packing factor (APF).
 Linear density: Fraction of line length in a particular
crystallographic direction that passes through atom centers
 Planar density: Fraction of total crystallographic plane area that
is occupied by atoms. (The plane must pass through an atom’s center
for particular atom to be included in calculations)
Equivalency of directions and planes is related to the degree of
atomic spacing or atomic packing
Example1:
z
Atomic arrangement on [100] direction;
[100] direction
x
y
a = 4/3R
a= R
Linear density of [100] direction in BCC;
= Line length of atoms within [100] direction / Length of [100] direction
= 2R/4/3R = 0.866
Example2:
(110) plane
Atomic arrangement on (110) plane in FCC;
a
(110) plane
Area of (110) plane;
= 2.a2 = 2(22)R2 = 82R2
Area of atoms inside (110) plane;
= ¼*4*R2 + ½*2*R2 = 2R2
2.a = 4R
Planar density of (110) plane in FCC = Area of atoms inside (110) plane / area of (110) plane
Planar density of (110) plane in FCC = 2R2 / 82R2 = 0.555
Anisotropy
 Dependency of properties on direction: ‘’ANISOTROPY’’
 It is associated with the variance of atomic or ionic spacing
with crystallographic direction.
Example: For a single crystal material,
different mechanical properties are
observed in [100] and [111] directions
and also in some other directions.
 Substances in which measured properties are independent of
the direction of measurement are ‘’ISOTROPIC’’
Determination of Crystal Structure
• Incoming X-rays diffract from crystal planes.
• Measurement of: Critical angles, qc,
for X-rays provide atomic spacing, d.
n = 2dSinq
Bragg’s Law
Summary
• Atoms may assemble into crystalline or amorphous structures.
• We can predict the density of a material, provided we know the
atomic weight, atomic radius, and crystal geometry (e.g., FCC, BCC,
HCP).
• Material properties generally vary with single crystal
orientation (i.e., they are anisotropic),but properties are
generally non-directional (i.e., they are isotropic) in polycrystals
with randomly oriented grains.