‘’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) Crytal 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.
Single crystal diamond
(schematic view)
Two Dimensional View of Atomic Arrangements
Single crystal Polycrystal Amorphous
Single crystal
of CaF2
Formation of Polycrystals
1) During heavy deformation
2) During solidification from melt
T1 T2
T3T3
Structure of the material
seen under microscope
Completely solid. Material contains
many grains(polycrystalline
material)
Liquid and small crystals Liquid and relatively larger
crystals
T1>T2>T3
Solidification of a pure metal
Describing Crystal Structure
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.
The atoms or ions are thought as solidspheres with their sizes defined. This iscalled atomic hard sphere model. Allatoms are identical in this model.
UNIT CELL: Smallest repeating group
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) crytal structure
3) FCC (Face Centered Cubic) crystal structure
4) HCP (Hexagonal Close-Packed) Crystal Structure
Rare in nature due to poor packing (only Po has this structure)
Close-packed directions are cube edges.
Coordination number = 6
Simple Cubic (SC) Structure
The number of atoms per SC: (1/8)*8 = 1 atom/cell
* Coordination number: number of nearest neighbor or touching atoms.
The relationship between unit cell edge length (a) and atomic radius (R);
a=2R
(Arrangement of atoms in one unit cell)
a
Coordination number = 8
--Note: All atoms are identical; the
center atom is shaded differently
only for ease of viewing.
Body Centered Cubic (BCC) Structure
The number of atoms per SC: (1/8)*8 + 1= 2 atoms/cell
The relationship between unit cell edge length (a) and atomic radius (R);
a=(4/ 3)R
BCC unit cell
(Arrangement of atoms in one unit cell)
Face Centered Cubic (FCC) Structure
FCC unit cell
Coordination number = 12
The number of atoms per SC: (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
--Note: All atoms are identical; the
center atom is shaded differently
only for ease of viewing.
a
Hexagonal Closed Packed (HCP) Structure
Ideally c/a=1.633, but for some metals this ratio deviates from the ideal value.
HCP Unit Cell
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.
Coordination number = 12
The number of atoms per SC: (1/6)*12 + 2*(1/2) + 3 = 6 atoms/cell
Cl-
Na+
Structure of Compounds
Compounds often have similar close-packed structures.
Ionic bonding (NaCl)
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 crystal structure
Example : Iron (Fe)
heating cooling
Characteristics of Some Selected Elements
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
a
APF for a face-centered cubic(FCC) structure is 0.74
APF for BCC structure = 0.68
APF for HCPstructure = 0.74
FCC structure
a=(2 2)R
The number of atoms per SC: (1/8)*8 + 6*(1/2)= 4 atoms/cell
Theoretical Density, ρ
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/cm 3
Compare to actual: Cu = 8.94 g/cm 3
Theoretical Density of NaCl
NaCl unit cell
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,
Crystal Systems
Also called lattice parameters.
Particular crystallographic direction is shown in a unit cell
The direction is a line between two points or a vector as shown
below:
Crystallographic Directions in
Cubic Crystals
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
This vector has a component in –y direction.[111]
Example1:
Example2:
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
FAMILY OF DIRECTIONS, <hkl>
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
Except HCP, crytallographic planes are specified using three MILLERINDICES (hkl). Any two planes parallel to each other are equivalentand have same indices.
The determination of the h,k, and l index numbers are as follows:
1) If the plane passes through the selected origin, then construct a newparallel plane or change the originto a corner of another unit cell.
2) Plane intersects or parallels each of the axes: the length of each axis isdetermined by using lattice parameters; a,b, and c.
3) Take the reciprocals of the lattice parameters. Therefore a plane thatparallels an axis has a ZERO index. (1/infinity=zero)
4) You may then change these three numbers to the set of smallest integersusing a common factor.
5) Report the indices as (hkl).
6) An intercept on the negative side of the origin is indicated by a bar overthat index.
Crystallographic Planes in
Cubic Crystals
Example1:
Example2: Example3:
For cubic crystals: Planes and directions having the same indices are
parallel to one another.
A family of planes is formed by all those planes that are crystallographically equivalent, {100}, {111}. (for cubic structure)
{111} =
Family of Planes, {hkl}
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.
Linear and Planar Density
Linear and planar atomic densities are one and two dimensional analogs of
atomic packing factor.
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:
Example2:
Anisotropy
Dependency of properties on direction: ‘’ANISOTROPY’’
It is associated with the variance of atomic or ionic spacing with
crystallographic direction.
Substances in which measured properties are independent of the
direction of measurement are ‘’ISOTROPIC’’
Example: For a single crystal material,
different mechanical properties are
observed in [100] and [111] directions and
also in some other directions.
• Incoming X-rays diffract from crystal planes.
• Measurement of: Critical angles, c,
for X-rays provide atomic spacing, d.
Determination of Crystal Structure
n = 2dSin
Bragg’s Law
• 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.
Summary