Crystalline Solids
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Transcript of Crystalline Solids
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Structure of Crystalline Solids
BITS Pilani, Pilani Campus
Structure of Crystalline Solids
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BITS Pilani, Pilani Campus
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Unit cell - is smallest repeatable entity that can be used to
completely represent a crystal structure. It is the building
block of crystal structure.
Crystal lattice it is the three dimensional network of points
in space.
Important Definitions
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in space.
Space lattice it is the thee dimensional network of lines in
space.
Primitive cell it is a simple cubic unit cell having atoms at
its eight corners.
Lattice parameter(a)- it is defined as the distance between
the center of neighboring corner atoms.
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Polycrystals Most crystalline solids are composed of many small
crystals (also called grains).
Initially, small crystals (nuclei) form at various
positions.
These have random orientations.
The small grains grow and begin to impinge on one The small grains grow and begin to impinge on one
another forming grain boundaries.
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Micrograph of a polycrystalline
stainless steel showing grains and
grain boundaries
Most engineering materials
are Polycrystals.
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Single Crystals
-Properties vary with
direction: anisotropic.
-Example: the modulus
of elasticity (E) in BCC iron:
Single vs Polycrystals
E (diagonal) = 273 GPa
E (edge) = 125 GPa
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Polycrystals
-Properties may/may not
vary with direction.
-If grains are randomly
oriented: isotropic.
E (edge) = 125 GPa
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Crystal Systems
7 crystal systems
Unit cell: smallest repetitive volume that contains
the complete lattice pattern of a crystal.
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7 crystal systems
14 crystal lattices
a, b and c are the lattice constants
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The fourteen (14)
types of Bravais
lattices grouped in
seven (7) systems. seven (7) systems.
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Miller indices - A shorthand notation to describe certain
crystallographic directions and planes in a material.
Denoted by [ ], , ( ) brackets. A negative number is
represented by a bar over the number.
Points, Directions and Planes in the
Unit Cell
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Point Coordinates
Coordinates of selected points in the unit cell.
The number refers to the distance from the origin in terms of
lattice parameters.
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Point Coordinates
Point coordinates for unit cell center are
a/2, b/2, c/2
Point coordinates for unit cell
z
y
a b
c
000
111
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Point coordinates for unit cell corner are 111
Translation: integer multiple of lattice constants identical position in another unit cell
x
y
z
2c
b
b
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Determine the Miller indices of directions A, B, and C.
Miller Indices, Directions
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SOLUTION
Direction A
1. Two points are 1, 0, 0, and 0, 0, 0
2. 1, 0, 0, -0, 0, 0 = 1, 0, 0
3. No fractions to clear or integers to reduce
4. [100]
Direction B
1. Two points are 1, 1, 1 and 0, 0, 01. Two points are 1, 1, 1 and 0, 0, 0
2. 1, 1, 1, -0, 0, 0 = 1, 1, 1
3. No fractions to clear or integers to reduce
4. [111]
Direction C
1. Two points are 0, 0, 1 and 1/2, 1, 0
2. 0, 0, 1 -1/2, 1, 0 = -1/2, -1, 1
3. 2(-1/2, -1, 1) = -1, -2, 2
2]21[ .4
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Algorithm for Miller indices1. Read off intercepts of plane with axes in
terms of a, b, c
2. Take reciprocals of intercepts
3. Reduce to smallest integer values
4. Enclose in parentheses, no commas.
Crystallographic Planes
4. Enclose in parentheses, no commas.
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Crystallographic Planes
The following treatment of the procedure used to assign
the Miller Indices is a simplified one (it may be best if you
simply regard it as a "recipe") and only a cubic crystal
system (one having a cubic unit cell with
dimensions a x a x a ) will be considered.
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Miller Indices
Step 1 : Identify the intercepts on the x- , y-
and z- axes.In this case the intercept on the x-axis is at x = a ( at the point
(a,0,0) ), but the surface is parallel to the y- and z-axes -
strictly therefore there is no intercept on these two axes but
we shall consider the intercept to be at infinity ( ) for the
special case where the plane is parallel to an axis. The
The procedure is most easily illustrated using an
example so we will first consider the following
surface/plane:
special case where the plane is parallel to an axis. The
intercepts on the x- , y- and z-axes are thus
Intercepts : a , ,
Step 2 : Specify the intercepts in fractional co-ordinatesCo-ordinates are converted to fractional co-ordinates by dividing by the
respective cell-dimension - for example, a point (x,y,z) in a unit cell of
dimensions a x b x c has fractional co-ordinates of ( x/a , y/b , z/c ). In the case
of a cubic unit cell each co-ordinate will simply be divided by the cubic cell
constant , a . This gives
Fractional Intercepts : a/a , /a, /a i.e. 1 , ,
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Miller Indices
Step 1 : Take the reciprocals of the fractional
intercepts
This final manipulation generates the Miller Indices
which (by convention) should then be specified without
being separated by any commas or other symbols.
The Miller Indices are also enclosed within standard
brackets (.) when one is specifying a unique surface
such as that being considered here.such as that being considered here.
he reciprocals of 1 and are 1 and 0 respectively, thus
yielding
Miller Indices : (100)
So the surface/plane illustrated is the (100) plane of the cubic crystal.
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Crystallographic Planes
Crystallographic planes are specified by 3 Miller Indices (h k l). All parallel planes have same Miller indices.
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Crystallographic Planesz
x
y
a b
c
4. Miller Indices (110)
1. Intercepts 1 1
2. Reciprocals 1/1 1/1 1/
1 1 03. Reduction 1 1 0
example a b c
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x
example a b cz
x
y
a b
c
4. Miller Indices (200)
1. Intercepts 1/2
2. Reciprocals 1/ 1/ 1/
2 0 03. Reduction 2 0 0
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Crystallographic Planes
z
example
1. Intercepts 1/2 1 3/4
a b c
2. Reciprocals 1/ 1/1 1/
2 1 4/3
3. Reduction 6 3 4
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x
y
a b
c
4. Miller Indices (634)
3. Reduction 6 3 4
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Determine the Miller indices of planes A, B, and C.
Miller Indices -Planes
Crystallographic planes
and intercepts.
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SOLUTION
Plane A
1. x = 1, y = 1, z = 1
2.1/x = 1, 1/y = 1,1 /z = 1
3. No fractions to clear
4. (111)
Plane B
1. The plane never intercepts the z axis, so x = 1, y = 2, and z =
2.1/x = 1, 1/y =1/2, 1/z = 0
3. Clear fractions:
3. Clear fractions:
1/x = 2, 1/y = 1, 1/z = 0
4. (210)
Plane C
1. We must move the origin, since the plane passes through 0, 0, 0. Lets move the origin one lattice parameter in the y-direction. Then, x = , y = 1, and z =
2.1/x = 0, 1/y = 1, 1/z = 0
3. No fractions to clear.
)010( .4
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Family of Planes
Planes that are crystallographically
equivalent have the same atomic packing.
Also, in cubic systems only, planes having the
same indices, regardless of order and sign,
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are equivalent.
Ex: {111}
= (111), (111), (111), (111), (111), (111), (111), (111)
(001)(010), (100), (010),(001),Ex: {100} = (100),
_ __ __ _ __ _ _ _ _
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THEORETICAL DENSITY ()
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Q.1) find the diameter of an imaginary sphere which fits
into the interstitial void in the center of the cube edge of
a BCC crystal.
Ans.
d- diameter of imaginary sphere
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d- diameter of imaginary sphere
d=a-2r
d= - 2r
d=0.309*r
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Q.4) Some hypothetical metal has the simple cubic crystal
structure. If its atomic weight is 74.5 g/mol. and the atomic
radius is 0.145 nm, compute its density.
BITS Pilani, Pilani Campus
Ans. For the simple cubic crystal structure, the value of n in
Equation is unity since there is only a single atom
associated with each unit cell. Furthermore, for the unit cell
edge length, a = 2R
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BITS Pilani, Pilani Campus
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SUMMARY
Crystallographic points, directions and planes are
specified in terms of indexing schemes.
Materials can be single crystals or polycrystalline.
Material properties generally vary with single
crystal orientation (anisotropic), but are generally
non-directional (isotropic) in polycrystals with non-directional (isotropic) in polycrystals with
randomly oriented grains.
Some materials can have more than one crystal
structure. This is referred to as polymorphism (or
allotropy).
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