CRYSTAL STRUCTURE & X-RAY DIFFRACTION VIGNAN UNIVERSITY ENGINEERING PHYSICS DEPARTMENT OF SCIENCES & HUMANITIES Classification of Matter.
Download ReportTranscript CRYSTAL STRUCTURE & X-RAY DIFFRACTION VIGNAN UNIVERSITY ENGINEERING PHYSICS DEPARTMENT OF SCIENCES & HUMANITIES Classification of Matter.
Slide 1
CRYSTAL STRUCTURE
&
X-RAY DIFFRACTION
VIGNAN UNIVERSITY
ENGINEERING PHYSICS
DEPARTMENT OF SCIENCES & HUMANITIES
Slide 2
Classification of Matter
Slide 3
Solids
Solids are again classified in to two
types
Crystalline
Non-Crystalline (Amorphous)
Slide 4
What is a Crystalline solid?
In Crystals, the atoms are arranged in a
periodic manner in all three directions.
So a crystal is characterized by regular
arrangement of atoms or molecules
A crystalline solid can either be a
single crystal or polycrystalline. In case
of single crystal the entire solid consists
of only one crystal. Polycrystalline is an
aggregate
of
many
small
crystals
Separated by well defined boundaries
Slide 5
Examples !
• Non-Metallic crystals:
Ice, Carbon, Diamond, Nacl, Kcl
etc…
• Metallic Crystals:
Copper, Silver, Aluminium, Tungsten,
Magnesium etc…
Slide 6
Crystalline Solid
Slide 7
Amorphous Solid
• Amorphous (Non-crystalline) Solid
is
composed of randomly orientated atoms ,
ions, or molecules that do not form
defined patterns or lattice structures.
• Amorphous materials have order only within
a few atomic or molecular dimensions.
Slide 8
• Amorphous materials do not have
any long-range order, but they have
varying degrees of short-range order.
• Examples to amorphous materials
include amorphous silicon, plastics,
and glasses.
• Amorphous silicon can be used in
solar cells and thin film transistors.
Slide 9
Non-crystalline
Slide 10
What are the Crystal properties?
o Crystals have sharp melting points
o They have long range positional order
o Crystals are anisotropic
(Properties change depending on the
direction)
Slide 11
What is Space lattice ?
• An infinite array of
points in three
dimensions (space),
• Each
point
has
identical surroundings
to all others.
• Arrays are arranged
exactly in a periodic
manner.
a
b
Slide 12
Basis
• A group of atoms or molecules
identical in composition is called the
basis
or
• A group of atoms which describe
crystal structure
Slide 13
Unit Cell
• The unit cell is the smallest unit
which, when repeated in space
indefinitely, generates the
space lattice.
Slide 14
Crystal structure
• Crystal structure can be obtained by
attaching atoms, groups of atoms or
molecules which are called basis (motif)
to the lattice sides of the lattice point.
Crystal lattice + basis = Crystal structure
Slide 15
The
unit cell and,
consequently, the
entire lattice, is
uniquely
determined by the
six lattice
constants: a, b, c,
α, β and γ. These
six parameters are
also called as basic
lattice parameters.
Slide 16
Primitive cell
• The unit cell formed by the primitives a,b
and c is called primitive cell. A primitive
cell will have only one lattice point. If
there are two or more lattice points it is
not considered as a primitive cell.
• As most of the unit cells of various crystal
lattice contains two are more lattice
points, its not necessary that every unit
cell is primitive.
Slide 17
Crystal systems
• We know that a three dimensional
space lattice is generated by repeated
translation of three non-coplanar
vectors a, b, c. Based on the lattice
parameters we can have 7 popular
crystal systems shown in the table
Slide 18
Table-1
Crystal system
Unit vector
Angles
Cubic
a= b=c
α =β =g =90o
Tetragonal
a = b≠ c
α =β =g =90o
Orthorhombic
a≠b≠c
α =β =g =90o
Monoclinic
a≠b≠c
α =β =90o ≠g
Triclinic
a≠b≠c
α ≠ β ≠g ≠90o
Trigonal
a= b=c
α =β =g ≠90o
Hexagonal
a= b ≠ c
α =β=90o
g =120o
Slide 19
TRICLINIC & MONOCLINIC CRYSTAL SYSTEM
Triclinic minerals are the least symmetrical. Their
three axes are all different lengths and none of them
are perpendicular to each other. These minerals are
the most difficult to recognize.
Triclinic (Simple)
a ß g 90
abc
Monoclinic (Simple)
a = g = 90o, ß 90o
a b c
Monoclinic (Base Centered)
a = g = 90o, ß 90o
a b c,
Slide 20
ORTHORHOMBIC SYSTEM
Orthorhombic (Simple) Orthorhombic (Base- Orthorhombic (BC)
a = ß = g = 90o
centred)
a = ß = g = 90o
abc
a = ß = g = 90o
abc
abc
Orthorhombic (FC)
a = ß = g = 90o
abc
Slide 21
TETRAGONAL SYSTEM
Tetragonal (P)
a = ß = g = 90o
a=bc
Tetragonal (BC)
a = ß = g = 90o
a=bc
Slide 22
Rhombohedral (R) or Trigonal
Rhombohedral (R) or Trigonal (S)
a = b = c, a = ß = g 90o
Slide 23
Bravais lattices
• In 1850, M. A. Bravais showed that
identical points can be arranged
spatially to produce 14 types of regular
pattern. These 14 space lattices are
known as ‘Bravais lattices’.
Slide 24
14 Bravais lattices
S.No
Crystal Type
1
2
Cubic
3
4
5
Tetragonal
6
7
Orthorhombic
Bravais
lattices
Simple
Body
centred
Face
centred
Simple
Body
centred
Simple
Base
centred
Symbol
P
I
F
P
I
P
C
Slide 25
8
9
10
Body
centred
Face
centred
Monoclinic Simple
11
I
F
P
12
Triclinic
Base
centred
Simple
C
13
Trigonal
Simple
P
14
Hexgonal
Simple
P
P
Slide 26
Slide 27
Characteristics of Unit cell:
• a) Nearest Neighbor Distance (2r):
The distance between the
centers of two nearest neighboring atoms is
called “Nearest neighbor distance.”
• b) Atomic Radius(r):
Atomic radius is defined as half the
distance between the nearest neighboring
atoms in a crystal.
Slide 28
• c) Coordination Number (N):
Coordination number is defined as the
number of equidistant neighbor that an atom
has in a given structure.
More closely packed structure has greater
coordination number.
• d)Atomic Packing Factor(APF):
Atomic Packing Factor (APF)
is
defined as the volume of atoms within the unit
cell divided by the volume of the unit cell.
Slide 29
Simple Cubic (SC)
• Simple Cubic has one lattice point so its
primitive cell.
• In the unit cell on the left, the atoms at the
corners are cut because only a portion (in
this case 1/8) belongs to that cell. The rest of
the atom belongs to neighboring cells.
• Coordinatination number of simple cubic is 6.
Slide 30
b
c
a
Slide 31
Characteristics of
Simple Cubic (SC):
• a) Effective Number of atoms per unit cell:
Even though there are 8 atoms at
eight corners of unit cell, each atom is shared
by such type of 8 unit cells. Hence, each atom
contributes only 1/8th part of it to a given unit
cells.
Therefore, Effective no. Of atoms
per unit cell n = 1/8 x 8= 1
• b) Coordination Number (N):
In simple cubic cell each atom is
directly in contact with six nearest neighbor
atoms.
Therefore, Coordination number N = 6
Slide 32
c) Nearest Neighbor Distance (2r):
In case of simple cubic
cell atoms touch each other along cube edges.
Nearest neighbor distance 2r = a
a
e) Atomic Radius (r):
It is the half the nearest neighbor distance.
i.e., r = a / 2
Slide 33
d)Atomic Packing Factor of
Simple Cube:
Slide 34
• Packing Fraction = Volume occupied by the atoms
Volume of unit cell
= 1 x (4/3) Πr3 = (4/3)Π(a/2)3
a3
a3
= 0.52 = 52%
Packing fraction = 52%
• Only the atoms occupy 52% of the unit cell’s
volume and rest of the part 48% is left vacant.
This structure is a loosely packed one.
• Eg: Polonium (Po) exhibits simple cubic
structure.
Slide 35
Body Centered Cubic (BCC)
•
As shown, BCC has two lattice
points so BCC is a non-primitive
cell.
•
BCC has eight nearest neighbors.
Each atom is in contact with its
neighbors only along the bodydiagonal directions.
•
Many metals (Fe, Li, Na.. etc),
including the alkalis and several
transition elements choose the
BCC structure.
Slide 36
Characteristics of
Body Centered Cubic (BCC):
•
a) Effective Number of atoms per unit cell:
Even though there are 8 atoms at eight
corners of unit cell, each atom is shared by such type of 8
unit cells. Hence, each atom contributes only 1/8th part of it
to a given unit cells. But the centered atom is exclusively
concerned to the given cell, which is not shared by any other
unit cell.
Therefore, Effective no. of atoms per unit cell n
= (1/8 x 8) + 1 = 2
•
b) Coordination Number (N):
Each body centered cube
is surrounded by 8 nearest corner atoms
and vice-versa.
Therefore, Coordination number N = 8
Slide 37
•
c) Nearest Neighbor Distance (2r):
In case of BCC cell atoms touch
each other along the body diagonal.
From ΔACD,
From ΔABC,
AD2 = AC2 + CD2
------(1)
AC2 = AB2 + BC2
AC2 = a2 + a2 = 2a2
From (1), we have,
(4r)2 = 2a2 + a2
(4r)2 = 3a2
4r = √3 a
2r = (√3/2) a
Nearest neighbor distance 2r = (√3/2) a
•
D
A
C
B
e) Atomic Radius (r):
It is the half the nearest neighbor distance.
i.e., r = (√3/4)a
Slide 38
Atomic Packing Factor of BCC
2
(0,433a)
Packing Fraction = Volume occupied by the atoms
Volume of unit cell
= 2 x (4/3)Πr3 = (8/3)Π(√3/2)a)3
a3
a3
= 0.68= 68%
Packing fraction = 68%
i.e., 68% of the unit cell volume is occupied by the atoms and rest
of the part i.e., 32% is left vacant. Eg: Tungsten, Sodium, lead and
Chromium have BCC structure.
Slide 39
Face Centered Cubic (FCC):
• There are atoms at the corners of the unit
cell and at the center of each face.
• Face centered cubic has 4 atoms so its
non primitive cell.
• Many of common metals (Cu, Ni, Pb ..etc)
crystallize in FCC structure.
Slide 40
Slide 41
Face Centered Cubic (FCC)
Slide 42
Characteristics of Face Centered Cubic
Structure (FCC) :
a) Effective Number of atoms per unit cell:
Even though there are 8 atoms at
eight corners of unit cell, each atom is shared by such type of 8
unit cells. Hence, each atom contributes only 1/8th part of it to a
given unit cells. Moreover an atom occupies each face of a cube
and it is shared by two such kind of unit cells. Hence its effective
contribution to a given unit cell is only half of it.
Therefore, Effective no. of atoms per unit cell n = (1/8 x 8) +
(1/2x6) = 4
b) Coordination Number (N):
Each atom has 12 nearest
atoms,
Therefore, Coordination number N = 12
Slide 43
c) Nearest Neighbor Distance (2r):
In case of FCC cell
atoms touch each other along the face
diagonal.
C
From ΔABC,
AC2 = AB2 + BC2
(4r) 2 = a2 + a2 = 2a2
(4r) 2 = 2a2
4r = √2 a
B
A
2r = a / √2
e) Atomic Radius (r):
It is the half the nearest
neighbor distance.
i.e., r = a / 2√2
Slide 44
Atomic Packing Factor
Packing Fraction = Volume occupied by the atoms
Volume of unit cell
= 4x (4/3) Πr3 = (8/3) Π (a / 2√2) 3
a3
a3
= 0.74= 74%
Packing fraction = 74%
i.e., 74% of the unit cell volume is occupied by the atoms
and rest of the part i.e., 26% is left vacant. Eg: Copper,
Aluminum, Lead and Silver have FCC structure.
Slide 45
Crystal Directions:
• In crystals there exists directions and planes, which
contain a large concentration of atoms. It is
necessary to locate these directions and planes for
crystal analysis.
•
In following figure two directions are
shown by arrows in two-dimensions. These directions
pass through the origin ‘O’ and end at ‘A’ and ‘B’
respectively. The directions are described by giving
the coordinates of the first whole numbered point [x,
y] through which each of the direction passes. For
direction OA, it is [1,1] and for OB [3,1].
Y
A
B
O
X
Slide 46
•In three dimensional, the directions are
described by the coordinates of the first whole
numbered point [x, y, z].
•Generally, the square brackets are used to
indicate a direction. A few direction s are
shown below,
Slide 47
Examples of crystal directions
X = 1 , Y = 0 , Z = 0 ► [1 0 0]
Slide 48
Crystal Planes
• Within a crystal lattice it is possible to identify sets
of equally spaced parallel planes. These are called
lattice planes.
b
b
a
d
a
Slide 49
MILLER INDICES FOR
CRYSTALLOGRAPHIC PLANES
• William HallowesMiller in 1839 was able to
give each face a unique label of three
small integers, the Miller Indices
• Definition: Miller Indices are the
reciprocals of the fractional intercepts
(with fractions cleared) which the plane
makes with the crystallographic x,y,z axes
of the three nonparallel edges of the cubic
unit cell.
Slide 50
Miller Indices:
Miller Indices are a symbolic vector representation for the
orientation of an atomic plane in a crystal lattice and are defined
as the reciprocals of the fractional intercepts which the plane
makes with the crystallographic axes.
To determine Miller indices of a plane, we use the following steps:
Let us consider plane ABC as
shown in above figure whose miller
indices are to be determined,
(0,0,c)
•
Step 1:
Determine the coordinates of
the intercepts made by the
plane along the three
crystallographic axes (x, y, z),
x
y
z
2a 3b
c
pa qb
rc ( p=2,q=3 and r =1)
(0,3b,0)
(2a,0,0)
Slide 51
•
Step 2:
Divide the intercepts with dimensions of the unit cell or
lattice parameters along the axes,
i.e., 2a
3b
c
a
b
c
=> 2
3
1
• Step 3:
Determine the reciprocals of these numbers,
1/2
•
1/3
1
Step 4:
Convert reciprocals into integers by multiplying each
one of them with their LCM.
6x(1/2)
6x(1/3)
(3
2
6x1
6)
In general it is denoted by (h k l) and we also noticed
that;
1:1:1 = h:k:l
•
p q r
Thus, Miller indices may be defined as the reciprocals of
the intercepts made by the plane on the crystallographic
axes when reduced to smallest numbers.
Slide 52
Important Features of Miller Indices of Crystal Planes:
•
•
•
•
All the parallel equidistant planes have the same Miller
indices. Thus the miller indices define a set of parallel
planes.
A plane parallel to one of the coordinate axes has an
intercept of infinity, and the corresponding Miller index
becomes zero.
If the miller indices of two planes have the same ratio,
i.e., (844) and (422) or (211), then the planes are parallel
to each other.
If (h k l) are the miller indices of a plane, then the plane
cuts the axes into h, k and l equal segments respectively.
Slide 53
Example-1
(1,0,0)
Slide 54
Example-2
(0,1,0)
(1,0,0)
Slide 55
Example-3
(0,0,1)
(0,1,0)
(1,0,0)
Slide 56
Example-4
(0,1,0)
(1/2, 0, 0)
Slide 57
Miller Indices
Slide 58
Spacing between planes in a
cubic crystal is
d hkl =
a
2
2
h +k +l
2
Where dhkl = inter-planar spacing between planes with Miller
indices h, k and l.
a = lattice constant (edge of the cube)
h, k, l = Miller indices of cubic planes being considered.
Slide 59
X-Ray diffraction
• X-ray crystallography, also called X-ray
diffraction, is used to determine crystal
structures by interpreting the diffraction
patterns formed when X-rays are scattered
by the electrons of atoms in crystalline
solids. X-rays are sent through a crystal to
reveal the pattern in which the molecules
and atoms contained within the crystal are
arranged.
Slide 60
• This x-ray crystallography was developed
by physicists William Lawrence Bragg and
his father William Henry Bragg. In 19121913, the younger Bragg developed
Bragg’s law, which connects the observed
scattering with reflections from evenly
spaced planes within the crystal.
Slide 61
X-Ray Diffraction
Bragg’s Law : 2dsinΘ = nλ
C
A
B
A’
D
θθ
B’
C’
E
Slide 62
Powder Crystal Method:
EXIT
ENTRY
THIN FILM
R
Slide 63
• The diffraction takes place for the values of ‘d’ and ‘θ’
which satisfy the Bragg’s relation, 2d sin θ = nλ, where
‘λ’ is constant for monochromatic X-rays.
• The diffracted rays corresponding to fixed values of ‘θ’
and ‘d’ lie on the surface of a cone with its apex at the
tube.
• The full opening angle ‘4θ’ is determined by,
4θ = S / R
4θ = S x 180 degrees
R
Π
• Using the value ‘θ’ in the Bragg’s equation and the
knowing the value of ‘λ’, ‘d’ value can be calculated.
From the interplanar spacing the type of lattice can be
identified
Slide 64
CRYSTAL STRUCTURE
&
X-RAY DIFFRACTION
VIGNAN UNIVERSITY
ENGINEERING PHYSICS
DEPARTMENT OF SCIENCES & HUMANITIES
Slide 2
Classification of Matter
Slide 3
Solids
Solids are again classified in to two
types
Crystalline
Non-Crystalline (Amorphous)
Slide 4
What is a Crystalline solid?
In Crystals, the atoms are arranged in a
periodic manner in all three directions.
So a crystal is characterized by regular
arrangement of atoms or molecules
A crystalline solid can either be a
single crystal or polycrystalline. In case
of single crystal the entire solid consists
of only one crystal. Polycrystalline is an
aggregate
of
many
small
crystals
Separated by well defined boundaries
Slide 5
Examples !
• Non-Metallic crystals:
Ice, Carbon, Diamond, Nacl, Kcl
etc…
• Metallic Crystals:
Copper, Silver, Aluminium, Tungsten,
Magnesium etc…
Slide 6
Crystalline Solid
Slide 7
Amorphous Solid
• Amorphous (Non-crystalline) Solid
is
composed of randomly orientated atoms ,
ions, or molecules that do not form
defined patterns or lattice structures.
• Amorphous materials have order only within
a few atomic or molecular dimensions.
Slide 8
• Amorphous materials do not have
any long-range order, but they have
varying degrees of short-range order.
• Examples to amorphous materials
include amorphous silicon, plastics,
and glasses.
• Amorphous silicon can be used in
solar cells and thin film transistors.
Slide 9
Non-crystalline
Slide 10
What are the Crystal properties?
o Crystals have sharp melting points
o They have long range positional order
o Crystals are anisotropic
(Properties change depending on the
direction)
Slide 11
What is Space lattice ?
• An infinite array of
points in three
dimensions (space),
• Each
point
has
identical surroundings
to all others.
• Arrays are arranged
exactly in a periodic
manner.
a
b
Slide 12
Basis
• A group of atoms or molecules
identical in composition is called the
basis
or
• A group of atoms which describe
crystal structure
Slide 13
Unit Cell
• The unit cell is the smallest unit
which, when repeated in space
indefinitely, generates the
space lattice.
Slide 14
Crystal structure
• Crystal structure can be obtained by
attaching atoms, groups of atoms or
molecules which are called basis (motif)
to the lattice sides of the lattice point.
Crystal lattice + basis = Crystal structure
Slide 15
The
unit cell and,
consequently, the
entire lattice, is
uniquely
determined by the
six lattice
constants: a, b, c,
α, β and γ. These
six parameters are
also called as basic
lattice parameters.
Slide 16
Primitive cell
• The unit cell formed by the primitives a,b
and c is called primitive cell. A primitive
cell will have only one lattice point. If
there are two or more lattice points it is
not considered as a primitive cell.
• As most of the unit cells of various crystal
lattice contains two are more lattice
points, its not necessary that every unit
cell is primitive.
Slide 17
Crystal systems
• We know that a three dimensional
space lattice is generated by repeated
translation of three non-coplanar
vectors a, b, c. Based on the lattice
parameters we can have 7 popular
crystal systems shown in the table
Slide 18
Table-1
Crystal system
Unit vector
Angles
Cubic
a= b=c
α =β =g =90o
Tetragonal
a = b≠ c
α =β =g =90o
Orthorhombic
a≠b≠c
α =β =g =90o
Monoclinic
a≠b≠c
α =β =90o ≠g
Triclinic
a≠b≠c
α ≠ β ≠g ≠90o
Trigonal
a= b=c
α =β =g ≠90o
Hexagonal
a= b ≠ c
α =β=90o
g =120o
Slide 19
TRICLINIC & MONOCLINIC CRYSTAL SYSTEM
Triclinic minerals are the least symmetrical. Their
three axes are all different lengths and none of them
are perpendicular to each other. These minerals are
the most difficult to recognize.
Triclinic (Simple)
a ß g 90
abc
Monoclinic (Simple)
a = g = 90o, ß 90o
a b c
Monoclinic (Base Centered)
a = g = 90o, ß 90o
a b c,
Slide 20
ORTHORHOMBIC SYSTEM
Orthorhombic (Simple) Orthorhombic (Base- Orthorhombic (BC)
a = ß = g = 90o
centred)
a = ß = g = 90o
abc
a = ß = g = 90o
abc
abc
Orthorhombic (FC)
a = ß = g = 90o
abc
Slide 21
TETRAGONAL SYSTEM
Tetragonal (P)
a = ß = g = 90o
a=bc
Tetragonal (BC)
a = ß = g = 90o
a=bc
Slide 22
Rhombohedral (R) or Trigonal
Rhombohedral (R) or Trigonal (S)
a = b = c, a = ß = g 90o
Slide 23
Bravais lattices
• In 1850, M. A. Bravais showed that
identical points can be arranged
spatially to produce 14 types of regular
pattern. These 14 space lattices are
known as ‘Bravais lattices’.
Slide 24
14 Bravais lattices
S.No
Crystal Type
1
2
Cubic
3
4
5
Tetragonal
6
7
Orthorhombic
Bravais
lattices
Simple
Body
centred
Face
centred
Simple
Body
centred
Simple
Base
centred
Symbol
P
I
F
P
I
P
C
Slide 25
8
9
10
Body
centred
Face
centred
Monoclinic Simple
11
I
F
P
12
Triclinic
Base
centred
Simple
C
13
Trigonal
Simple
P
14
Hexgonal
Simple
P
P
Slide 26
Slide 27
Characteristics of Unit cell:
• a) Nearest Neighbor Distance (2r):
The distance between the
centers of two nearest neighboring atoms is
called “Nearest neighbor distance.”
• b) Atomic Radius(r):
Atomic radius is defined as half the
distance between the nearest neighboring
atoms in a crystal.
Slide 28
• c) Coordination Number (N):
Coordination number is defined as the
number of equidistant neighbor that an atom
has in a given structure.
More closely packed structure has greater
coordination number.
• d)Atomic Packing Factor(APF):
Atomic Packing Factor (APF)
is
defined as the volume of atoms within the unit
cell divided by the volume of the unit cell.
Slide 29
Simple Cubic (SC)
• Simple Cubic has one lattice point so its
primitive cell.
• In the unit cell on the left, the atoms at the
corners are cut because only a portion (in
this case 1/8) belongs to that cell. The rest of
the atom belongs to neighboring cells.
• Coordinatination number of simple cubic is 6.
Slide 30
b
c
a
Slide 31
Characteristics of
Simple Cubic (SC):
• a) Effective Number of atoms per unit cell:
Even though there are 8 atoms at
eight corners of unit cell, each atom is shared
by such type of 8 unit cells. Hence, each atom
contributes only 1/8th part of it to a given unit
cells.
Therefore, Effective no. Of atoms
per unit cell n = 1/8 x 8= 1
• b) Coordination Number (N):
In simple cubic cell each atom is
directly in contact with six nearest neighbor
atoms.
Therefore, Coordination number N = 6
Slide 32
c) Nearest Neighbor Distance (2r):
In case of simple cubic
cell atoms touch each other along cube edges.
Nearest neighbor distance 2r = a
a
e) Atomic Radius (r):
It is the half the nearest neighbor distance.
i.e., r = a / 2
Slide 33
d)Atomic Packing Factor of
Simple Cube:
Slide 34
• Packing Fraction = Volume occupied by the atoms
Volume of unit cell
= 1 x (4/3) Πr3 = (4/3)Π(a/2)3
a3
a3
= 0.52 = 52%
Packing fraction = 52%
• Only the atoms occupy 52% of the unit cell’s
volume and rest of the part 48% is left vacant.
This structure is a loosely packed one.
• Eg: Polonium (Po) exhibits simple cubic
structure.
Slide 35
Body Centered Cubic (BCC)
•
As shown, BCC has two lattice
points so BCC is a non-primitive
cell.
•
BCC has eight nearest neighbors.
Each atom is in contact with its
neighbors only along the bodydiagonal directions.
•
Many metals (Fe, Li, Na.. etc),
including the alkalis and several
transition elements choose the
BCC structure.
Slide 36
Characteristics of
Body Centered Cubic (BCC):
•
a) Effective Number of atoms per unit cell:
Even though there are 8 atoms at eight
corners of unit cell, each atom is shared by such type of 8
unit cells. Hence, each atom contributes only 1/8th part of it
to a given unit cells. But the centered atom is exclusively
concerned to the given cell, which is not shared by any other
unit cell.
Therefore, Effective no. of atoms per unit cell n
= (1/8 x 8) + 1 = 2
•
b) Coordination Number (N):
Each body centered cube
is surrounded by 8 nearest corner atoms
and vice-versa.
Therefore, Coordination number N = 8
Slide 37
•
c) Nearest Neighbor Distance (2r):
In case of BCC cell atoms touch
each other along the body diagonal.
From ΔACD,
From ΔABC,
AD2 = AC2 + CD2
------(1)
AC2 = AB2 + BC2
AC2 = a2 + a2 = 2a2
From (1), we have,
(4r)2 = 2a2 + a2
(4r)2 = 3a2
4r = √3 a
2r = (√3/2) a
Nearest neighbor distance 2r = (√3/2) a
•
D
A
C
B
e) Atomic Radius (r):
It is the half the nearest neighbor distance.
i.e., r = (√3/4)a
Slide 38
Atomic Packing Factor of BCC
2
(0,433a)
Packing Fraction = Volume occupied by the atoms
Volume of unit cell
= 2 x (4/3)Πr3 = (8/3)Π(√3/2)a)3
a3
a3
= 0.68= 68%
Packing fraction = 68%
i.e., 68% of the unit cell volume is occupied by the atoms and rest
of the part i.e., 32% is left vacant. Eg: Tungsten, Sodium, lead and
Chromium have BCC structure.
Slide 39
Face Centered Cubic (FCC):
• There are atoms at the corners of the unit
cell and at the center of each face.
• Face centered cubic has 4 atoms so its
non primitive cell.
• Many of common metals (Cu, Ni, Pb ..etc)
crystallize in FCC structure.
Slide 40
Slide 41
Face Centered Cubic (FCC)
Slide 42
Characteristics of Face Centered Cubic
Structure (FCC) :
a) Effective Number of atoms per unit cell:
Even though there are 8 atoms at
eight corners of unit cell, each atom is shared by such type of 8
unit cells. Hence, each atom contributes only 1/8th part of it to a
given unit cells. Moreover an atom occupies each face of a cube
and it is shared by two such kind of unit cells. Hence its effective
contribution to a given unit cell is only half of it.
Therefore, Effective no. of atoms per unit cell n = (1/8 x 8) +
(1/2x6) = 4
b) Coordination Number (N):
Each atom has 12 nearest
atoms,
Therefore, Coordination number N = 12
Slide 43
c) Nearest Neighbor Distance (2r):
In case of FCC cell
atoms touch each other along the face
diagonal.
C
From ΔABC,
AC2 = AB2 + BC2
(4r) 2 = a2 + a2 = 2a2
(4r) 2 = 2a2
4r = √2 a
B
A
2r = a / √2
e) Atomic Radius (r):
It is the half the nearest
neighbor distance.
i.e., r = a / 2√2
Slide 44
Atomic Packing Factor
Packing Fraction = Volume occupied by the atoms
Volume of unit cell
= 4x (4/3) Πr3 = (8/3) Π (a / 2√2) 3
a3
a3
= 0.74= 74%
Packing fraction = 74%
i.e., 74% of the unit cell volume is occupied by the atoms
and rest of the part i.e., 26% is left vacant. Eg: Copper,
Aluminum, Lead and Silver have FCC structure.
Slide 45
Crystal Directions:
• In crystals there exists directions and planes, which
contain a large concentration of atoms. It is
necessary to locate these directions and planes for
crystal analysis.
•
In following figure two directions are
shown by arrows in two-dimensions. These directions
pass through the origin ‘O’ and end at ‘A’ and ‘B’
respectively. The directions are described by giving
the coordinates of the first whole numbered point [x,
y] through which each of the direction passes. For
direction OA, it is [1,1] and for OB [3,1].
Y
A
B
O
X
Slide 46
•In three dimensional, the directions are
described by the coordinates of the first whole
numbered point [x, y, z].
•Generally, the square brackets are used to
indicate a direction. A few direction s are
shown below,
Slide 47
Examples of crystal directions
X = 1 , Y = 0 , Z = 0 ► [1 0 0]
Slide 48
Crystal Planes
• Within a crystal lattice it is possible to identify sets
of equally spaced parallel planes. These are called
lattice planes.
b
b
a
d
a
Slide 49
MILLER INDICES FOR
CRYSTALLOGRAPHIC PLANES
• William HallowesMiller in 1839 was able to
give each face a unique label of three
small integers, the Miller Indices
• Definition: Miller Indices are the
reciprocals of the fractional intercepts
(with fractions cleared) which the plane
makes with the crystallographic x,y,z axes
of the three nonparallel edges of the cubic
unit cell.
Slide 50
Miller Indices:
Miller Indices are a symbolic vector representation for the
orientation of an atomic plane in a crystal lattice and are defined
as the reciprocals of the fractional intercepts which the plane
makes with the crystallographic axes.
To determine Miller indices of a plane, we use the following steps:
Let us consider plane ABC as
shown in above figure whose miller
indices are to be determined,
(0,0,c)
•
Step 1:
Determine the coordinates of
the intercepts made by the
plane along the three
crystallographic axes (x, y, z),
x
y
z
2a 3b
c
pa qb
rc ( p=2,q=3 and r =1)
(0,3b,0)
(2a,0,0)
Slide 51
•
Step 2:
Divide the intercepts with dimensions of the unit cell or
lattice parameters along the axes,
i.e., 2a
3b
c
a
b
c
=> 2
3
1
• Step 3:
Determine the reciprocals of these numbers,
1/2
•
1/3
1
Step 4:
Convert reciprocals into integers by multiplying each
one of them with their LCM.
6x(1/2)
6x(1/3)
(3
2
6x1
6)
In general it is denoted by (h k l) and we also noticed
that;
1:1:1 = h:k:l
•
p q r
Thus, Miller indices may be defined as the reciprocals of
the intercepts made by the plane on the crystallographic
axes when reduced to smallest numbers.
Slide 52
Important Features of Miller Indices of Crystal Planes:
•
•
•
•
All the parallel equidistant planes have the same Miller
indices. Thus the miller indices define a set of parallel
planes.
A plane parallel to one of the coordinate axes has an
intercept of infinity, and the corresponding Miller index
becomes zero.
If the miller indices of two planes have the same ratio,
i.e., (844) and (422) or (211), then the planes are parallel
to each other.
If (h k l) are the miller indices of a plane, then the plane
cuts the axes into h, k and l equal segments respectively.
Slide 53
Example-1
(1,0,0)
Slide 54
Example-2
(0,1,0)
(1,0,0)
Slide 55
Example-3
(0,0,1)
(0,1,0)
(1,0,0)
Slide 56
Example-4
(0,1,0)
(1/2, 0, 0)
Slide 57
Miller Indices
Slide 58
Spacing between planes in a
cubic crystal is
d hkl =
a
2
2
h +k +l
2
Where dhkl = inter-planar spacing between planes with Miller
indices h, k and l.
a = lattice constant (edge of the cube)
h, k, l = Miller indices of cubic planes being considered.
Slide 59
X-Ray diffraction
• X-ray crystallography, also called X-ray
diffraction, is used to determine crystal
structures by interpreting the diffraction
patterns formed when X-rays are scattered
by the electrons of atoms in crystalline
solids. X-rays are sent through a crystal to
reveal the pattern in which the molecules
and atoms contained within the crystal are
arranged.
Slide 60
• This x-ray crystallography was developed
by physicists William Lawrence Bragg and
his father William Henry Bragg. In 19121913, the younger Bragg developed
Bragg’s law, which connects the observed
scattering with reflections from evenly
spaced planes within the crystal.
Slide 61
X-Ray Diffraction
Bragg’s Law : 2dsinΘ = nλ
C
A
B
A’
D
θθ
B’
C’
E
Slide 62
Powder Crystal Method:
EXIT
ENTRY
THIN FILM
R
Slide 63
• The diffraction takes place for the values of ‘d’ and ‘θ’
which satisfy the Bragg’s relation, 2d sin θ = nλ, where
‘λ’ is constant for monochromatic X-rays.
• The diffracted rays corresponding to fixed values of ‘θ’
and ‘d’ lie on the surface of a cone with its apex at the
tube.
• The full opening angle ‘4θ’ is determined by,
4θ = S / R
4θ = S x 180 degrees
R
Π
• Using the value ‘θ’ in the Bragg’s equation and the
knowing the value of ‘λ’, ‘d’ value can be calculated.
From the interplanar spacing the type of lattice can be
identified
Slide 64