Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module...
Transcript of Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module...
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CRYSTAL PHYSICS
General Objective
To develop the knowledge of crystal
structure and their properties.
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Specific Objectives
1. Differentiate crystalline and amorphous
solids.
2. To explain nine fundamental terms of
crystallography.
3. To discuss the fourteen Bravais lattice
of seven crystal system.
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‘Crystal Physics’ or ‘Crystallography’ is
a branch of physics that deals with the
study of all possible types of crystals and
the physical properties of crystalline
solids by the determination of their actual
structure by using X-rays, neutron beams
and electron beams.
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Solids can broadly be classified into two types
based on the arrangement of units of matter.
The units of matter may be atoms, molecules or
ions.
They are,
Crystalline solids
Non-crystalline (or) Amorphous solids
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Ex: Metallic and non-metallic
NaCl, Ag, Cu, AuEx: Plastics, Glass and Rubber
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Grains
Grain boundaries
A lattice is a regular and periodic
arrangement of points in three dimension.
It is defined as an infinite array of points in
three dimension in which every point has
surroundings identical to that of every other
point in the array.
The Space lattice is otherwise called the
Crystal lattice
SPACE LATTICE
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A crystal structure is formed by associating every
lattice point with an unit assembly of atoms or
molecules identical in composition, arrangement and
orientation.
This unit assembly is called the `basis’.
When the basis is repeated with correct periodicity in
all directions, it gives the actual crystal structure.
The crystal structure is real, while the lattice is
imaginary.
BASIS
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UNIT CELL
A unit cell is defined as a fundamental building block
of a crystal structure, which can generate the
complete crystal by repeating its own dimensions in
various directions.
XA
Y
B
Z
C
O
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• Inter axial lengths: OA = a , OB = b and OC = c
• Inter axial angles: α,β and γ
• Primitives: The intercepts OA, OB, OC are called Primitives
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1. Cubic
2. Orthorhombic
3. Monoclinic
4. Triclinic
5. Hexagonal
6. Rhombohedral
7. Tetragonal
Fourteen Bravais Lattices in Three Dimensions
Fourteen Bravais Lattices …
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MILLER INDICES
Chapter 3 -19
MILLER INDICES
d
DIFFERENT LATTICE PLANES
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MILLER INDICES
The orientation of planes or faces in a crystal can be
described in terms of their intercepts on the three
axes.
Miller introduced a system to designate a plane in a
crystal.
He introduced a set of three numbers to specify a
plane in a crystal.
This set of three numbers is known as ‘Miller Indices’
of the concerned plane.
Chapter 3 -21
MILLER INDICES
Miller indices is defined as the reciprocals of
the intercepts made by the plane on the three
axes.
Chapter 3 -22
MILLER INDICES
Procedure for finding Miller Indices
Step 1: Determine the intercepts of the plane
along the axes X,Y and Z in terms of
the lattice constants a,b and c.
Step 2: Determine the reciprocals of these
numbers.
Chapter 3 -23
Step 3: Find the least common denominator (lcd)
and multiply each by this lcd.
Step 4:The result is written in paranthesis.This is
called the `Miller Indices’ of the plane in
the form (h k l).
This is called the `Miller Indices’ of the plane in the form
(h k l).
MILLER INDICES
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ILLUSTRATION
PLANES IN A CRYSTAL
Plane ABC has intercepts of 2 units along X-axis, 3
units along Y-axis and 2 units along Z-axis.
Chapter 3 -25
DETERMINATION OF ‘MILLER INDICES’
Step 1:The intercepts are 2,3 and 2 on the three axes.
Step 2:The reciprocals are 1/2, 1/3 and 1/2.
Step 3:The least common denominator is ‘6’.
Multiplying each reciprocal by lcd,
we get, 3,2 and 3.
Step 4:Hence Miller indices for the plane ABC is (3 2 3)
ILLUSTRATION
Chapter 3 -26
MILLER INDICES
IMPORTANT FEATURES OF MILLER INDICES
A plane passing through the origin is defined in terms of a
parallel plane having non zero intercepts.
All equally spaced parallel planes have same ‘Miller
indices’ i.e. The Miller indices do not only define a particular
plane but also a set of parallel planes. Thus the planes
whose intercepts are 1, 1,1; 2,2,2; -3,-3,-3 etc., are all
represented by the same set of Miller indices.
Chapter 3 -27
MILLER INDICES
IMPORTANT FEATURES OF MILLER INDICES
It is only the ratio of the indices which is important in this
notation. The (6 2 2) planes are the same as (3 1 1) planes.
If a plane cuts an axis on the negative side of the origin,
corresponding index is negative. It is represented by a bar,
like (1 0 0). i.e. Miller indices (1 0 0) indicates that the
plane has an intercept in the –ve X –axis.
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MILLER INDICES OF SOME IMPORTANT PLANES
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Simple Cubic Structure (SC)
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• Rare due to low packing density (only Po has this structure)
• Close-packed directions are cube edges.
• Coordination # = 6
(# nearest neighbors)
Simple Cubic Structure (SC)
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• APF for a simple cubic structure = 0.52
APF =
a3
4
3p (0.5a) 31
atoms
unit cellatom
volume
unit cell
volume
Atomic Packing Factor (APF):SC
APF = Volume of atoms in unit cell*
Volume of unit cell
*assume hard spheres
close-packed directions
a
R=0.5a
contains 8 x 1/8 = 1 atom/unit cell
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• Coordination # = 8
• Atoms touch each other along cube diagonals.--Note: All atoms are identical; the center atom is shaded
differently only for ease of viewing.
Body Centered Cubic Structure (BCC)
ex: Cr, W, Fe (), Tantalum, Molybdenum
2 atoms/unit cell: 1 center + 8 corners x 1/8
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Atomic Packing Factor: BCC
a
APF =
4
3p ( 3a/4)32
atoms
unit cell atom
volume
a3
unit cell
volume
length = 4R =
Close-packed directions:
3 a
• APF for a body-centered cubic structure = 0.68
aRAdapted from
Fig. 3.2(a), Callister &
Rethwisch 8e.
a2
a3
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• Coordination # = 12
Adapted from Fig. 3.1, Callister & Rethwisch 8e.
• Atoms touch each other along face diagonals.--Note: All atoms are identical; the face-centered atoms are shaded
differently only for ease of viewing.
Face Centered Cubic Structure (FCC)
ex: Al, Cu, Au, Pb, Ni, Pt, Ag
4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8Click once on image to start animation
(Courtesy P.M. Anderson)
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• APF for a face-centered cubic structure = 0.74
Atomic Packing Factor: FCC
maximum achievable APF
APF =
4
3p ( 2a/4)34
atoms
unit cell atom
volume
a3
unit cell
volume
Close-packed directions:
length = 4R = 2 a
Unit cell contains:6 x 1/2 + 8 x 1/8
= 4 atoms/unit cella
2 a
Adapted from
Fig. 3.1(a),
Callister &
Rethwisch 8e.
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A sites
B B
B
BB
B B
C sites
C C
CA
B
B sites
• ABCABC... Stacking Sequence
• 2D Projection
• FCC Unit Cell
FCC Stacking Sequence
B B
B
BB
B B
B sitesC C
CA
C C
CA
AB
C
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Hexagonal Close-Packed Structure
(HCP)
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• Coordination # = 12
• ABAB... Stacking Sequence
• APF = 0.74
• 3D Projection • 2D Projection
Adapted from Fig. 3.3(a),
Callister & Rethwisch 8e.
6 atoms/unit cell
ex: Cd, Mg, Ti, Zn
• c/a = 1.633
c
a
A sites
B sites
A sites Bottom layer
Middle layer
Top layer
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Crystal structure coordination # packing factor close packed directions
Simple Cubic (SC) 6 0.52 cube edges
Body Centered Cubic (BCC) 8 0.68 body diagonal
Face Centered Cubic (FCC) 12 0.74 face diagonal
Hexagonal Close Pack (HCP) 12 0.74 hexagonal side
Thank You…
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