![Page 1: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/1.jpg)
1
CRYSTAL PHYSICS
![Page 2: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/2.jpg)
General Objective
To develop the knowledge of crystal
structure and their properties.
2
![Page 3: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/3.jpg)
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.
3
![Page 4: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/4.jpg)
4
‘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.
![Page 5: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/5.jpg)
5
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
![Page 6: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/6.jpg)
6
Ex: Metallic and non-metallic
NaCl, Ag, Cu, AuEx: Plastics, Glass and Rubber
![Page 7: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/7.jpg)
7
![Page 8: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/8.jpg)
8
Grains
Grain boundaries
![Page 9: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/9.jpg)
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
![Page 10: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/10.jpg)
10
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
![Page 11: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/11.jpg)
11
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
![Page 12: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/12.jpg)
12
![Page 13: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/13.jpg)
13
![Page 14: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/14.jpg)
• 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
7 Crystal systems:
1. Cubic
2. Orthorhombic
3. Monoclinic
4. Triclinic
5. Hexagonal
6. Rhombohedral
7. Tetragonal
![Page 15: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/15.jpg)
Fourteen Bravais Lattices in Three Dimensions
![Page 16: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/16.jpg)
Fourteen Bravais Lattices …
![Page 17: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/17.jpg)
17
![Page 18: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/18.jpg)
18
MILLER INDICES
![Page 19: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/19.jpg)
Chapter 3 -19
MILLER INDICES
d
DIFFERENT LATTICE PLANES
![Page 20: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/20.jpg)
20
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.
![Page 21: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/21.jpg)
Chapter 3 -21
MILLER INDICES
Miller indices is defined as the reciprocals of
the intercepts made by the plane on the three
axes.
![Page 22: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/22.jpg)
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.
![Page 23: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/23.jpg)
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
![Page 24: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/24.jpg)
24
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.
![Page 25: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/25.jpg)
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
![Page 26: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/26.jpg)
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.
![Page 27: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/27.jpg)
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.
![Page 28: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/28.jpg)
28
MILLER INDICES OF SOME IMPORTANT PLANES
![Page 29: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/29.jpg)
29
Simple Cubic Structure (SC)
![Page 30: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/30.jpg)
30
• 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)
![Page 31: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/31.jpg)
31
• 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
![Page 32: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/32.jpg)
32
• 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
![Page 33: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/33.jpg)
33
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
![Page 34: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/34.jpg)
34
• 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)
![Page 35: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/35.jpg)
35
![Page 36: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/36.jpg)
36
• 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.
![Page 37: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/37.jpg)
37
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
![Page 38: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/38.jpg)
38
Hexagonal Close-Packed Structure
(HCP)
![Page 39: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/39.jpg)
39
![Page 40: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/40.jpg)
40
![Page 41: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/41.jpg)
41
![Page 42: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/42.jpg)
42
• 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
![Page 43: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/43.jpg)
43
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
![Page 44: Chapter 3: The Structure of Crystalline Solidsarunkumard.yolasite.com/resources/Module Full.pdfChapter 3 - 22 MILLER INDICES Procedure for finding Miller Indices Step 1: Determine](https://reader030.vdocuments.net/reader030/viewer/2022021801/5b28efb77f8b9a42448b4954/html5/thumbnails/44.jpg)
Thank You…
44