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Introduction Crystal Physics
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Introduction to CRYSTAL PHYSICS
By
M. SILAMBARASAN
Centre for Photonics and Nanotechnology
Department of Physics
Sona College of Technology
Salem-636 005, Tamil Nadu, India
http://www.sonapan.com/
E-mail: [email protected]
INTRODUCTION
Crystals• Periodically arrangement of
atoms (or) molecules (or) ions in a three dimensional pattern
• The crystalline solids have well defined geometrical form (pattern)
• Further when crystals breaks, all the broken pieces will have a regular shape
• The phase change from liquid (or) gas to solid is called Crystallization
Single Crystal• In, Which the solid contains only one crystal• These crystals are produced artificially from vapour or
liquid state• Sharp melting point
Poly Crystal
• Which has an aggregate of many small crystalsseparated by well defined boundaries
Molecular Solids Covalent Solids
Ionic solidsMetallic solids
Na+
Cl-
Structures of crystalline solid types
Non-crystalline solids
• Amorphous solids or Non crystalline solids.
• Atoms (or) molecules are not in an orderly fashion
• In amorphous solids the same atomic groups arearranged randomly in all direction
• Super cooled liquids: glass. ( some of the propertiesof solid), but not crystals
Crystals Vs Amorphous S.No
Crystalline Material Amorphous materials
1. They have a definite and regular geometrical shapes which extend throughout the crystals.
They do not have definite geometrical shapes.
2. They are anisotropic substance. (directional properties)
They are isotropic substance. (no directional properties)
3. They are most stable. They are less stable.
4. They have sharp meting point. They do not have sharp meting point.
5. Ex: NaCl, Diamond, KCl, Iron, etc Ex: Glasses, Plastics, Rubber, etc
Crystallography definitions
Lattice: An array of points which are imaginarily kept torepresent the position of atoms in crystal such that everylattice points has got the same environment as that of theothers
Crystallography : The study of the Physical properties and
geometrical form of the crystalline solids, using X-ray (or)
electron beam (or) neutron beam etc., is termed as the science of
Crystallography.
Space lattice (or) crystal lattice: A 3D collection of points in space are called space lattice, the environment about any particular point is in everyway the same
• Lattice points: It denotes the position of atoms (or)molecules in the crystals. (or) Points in the spacelattice.
Basis (or) Motif: An unit assembly of atoms (or)molecules which are identified with respect to theposition of lattice points, identical in composition andorientation. The number of atoms in the basis may be1 or 2 or 3 etc. it may be go even above 1000.
Crystal Structure=
Space lattice + Basis
• Lattice Plane: A set of parallel and equally spaced planes in a space lattice
• Unit Cell: The fundamental elementary patternwith minimum number of atoms and, molecules (or)group of molecules which represents the totalcharacteristic of the crystals.
• Smallest geometric figure.
• Lattice parameters (or) Unit Cell parameters: Thelines drawn parallel to the lines of intersection of anythree faces of the unit cell which don’t lie in the sameplane are called crystallography axis.
a, b, c – intercepts or primitive or axial length.
α, β and γ – interfacial angles.
• Primitive cell
• Non Primitive cell
The Crystal System
The crystal systems are classified into 7 crystal
systems on the basis of lattice parameters. Viz ( axial
length a, b, c & axial angle α, β and γ
In this universe all the solids comes under
7 crystal systems
14 Bravais lattice
32 Point groups and
230 Space groups
Rotation
Translation
Reflection
Inversion
Point group
Space group
Symmetry Operation
A symmetry operation is atransformation (operation)performed on a body so that thebody can go to a new position inspace, which is similar to that theoriginal position
Crystal Systems
• Triclinic
• Monoclinic
• Orthorhombic
• Tetragonal
• Hexagonal
• Trigonal (Rhomohedral)
• Cubic
Triclinic• The Lattice parameters
a b c
α β γ 90º
Monoclinic
The Lattice parameters
a b c
α = β = 90º γ
Orthorhombic• The Lattice parameters
a b c
α = β = γ = 90º
Tetragonal
The Lattice parameters
a = b c
α = β = γ = 90º
Hexagonal The Lattice parameters
a = b c
α = β = 90º, γ = 120º
Trigonal (Rhombohedral)
The Lattice parameters
a = b = c
α = β = γ 90º
Cubic
The Lattice parameters
a = b = c
α = β = γ = 90º
Baravais Lattice
In, 1880 Bravais lattice, studied by Auguste Bravais. There
are 14 possible types of space lattices in 7 crystal system.
S.No
Crystal Systems Unit Cell Parameters Baravais Lattice
1 Triclinic a b c,α β γ 90º
Simple
2 Monoclinic a b c,α = β = 90º γ
Simple, Base-centred.
3 Orthorhombic a b c,α = β = γ= 90º
Simple, Base-centred,Body-centred, Face-centred.
4 Tetragonal a = b c,α = β = γ= 90º
Simple, Body-centred
5 Hexagonal a = b c,α = β = 90º, γ=120º
Simple
6 Trigonal(Rhomohedral)
a = b = c,α = β = γ 90º
Simple
7 Cubic a = b = c,α = β = γ = 90º
Simple, Body-centred, Face-centred.
Triclinic
Monoclinic
Simple
Simple Base- Centre
OrthorhombicSimple
Body Centered
Base- Centred
Face Centered
Simple
Body- Centre
Tetragonal Hexagonal
Simple
Cubic
Trigonal (Rhombohedral)
Simple
Simple Body Centered Face Centered
Relation Between Lattice Constant and Density
• Density = Mass/Volume
• Mass = number of atoms per unit cell * mass of each atom
• mass of each atom = atomic weight/avogadro’s number
Miller Indices
Introduction:
Crystals are made up of largenumber of parallel and equalspaced planes, passingthrough the lattice points iscalled lattice planes.
A set of three numbers todesignate a plane in a crystal.That set of three numbersare called miller indices ofthe concern plan.
Miller Index (hkl)
Definition:
Miller indices is defined as the reciprocal of the intercepts
made by plane on the crystallography axis which are reducedto smallest numbers.
(or)
Miller indices are the three smallest possible integers, whichhave the same ratio as the reciprocals of the intercepts ofthe plane concerned along the three axes.
(2,0,0)
(0,3,0)
(0,0,1) Select an origin not on the
plane: O
select a crystallographic
coordinate system:
XYZ
Find intercepts along axes:
2 :3 :1
Take reciprocal
1/2: 1/3 :1/1x
y
z
Steps for finding Miller Index
Convert to smallest integers in the same ratio i.e, multiplying each and every
reciprocal with their least common Multiplier (LCM) :
Here LCM =6. and get 3:2:6
Enclose in parenthesis: (326)
Generally Miller Indices are denoted by (hkl)
h:k:l= a/p:b/q:c/r for any system.
h:k:l= a/p:a/q:a/r for cubic system.
origin
intercepts
reciprocals
Miller Indices
A B
C
D
O
x
y
z
E
x
y
zABCD
O
1 ∞ ∞
1 0 0
(1 0 0)
O*
O*
Plane OCBE
1 -1 ∞
1 -1 0
(1 1 0)
_
Miller Indices for planes
Miller Indices should be enclosed only in this parenthesis: i.e., ( )
There should be no comma’s in-between the numbers.
(2 6 3) means it should be read as only two six three.
The Negative Miller indices can be represented by (2 6 3)
If a plane is parallel to any one of the coordinate axis, then its intercepts will be at infinity. i.e., Miller index for that particular axis is zero. Ex: (1 0 0)
All equally spaced parallel plans have the same Miller Index
i.e., Same ratio: (8 4 4), (4 2 2) and (2 1 1)
The Indices (h k l) do not define a particular plane, but a set of parallel plans.
Important Feature of Miller Index
Sketching the plane from the given miller Indices
If the miller index are say
(a b c )
h=a; k=b; l=c.
Take the reciprocals of the given miller indices.
1/a: 1/b:1/c
Multiplying the reciprocals by the LCM
We get intercepts of the plane.
Common planes in a simple cubic structure
• Three important crystal planes
( 1 0 0) (1 1 1)(1 1 0)
(100) plane(-100) plane
(100) planes
[100] vector
Family of Symmetry Related Planes
Direction of the plan can be represented by [1 0 0]
To represent the family of planes we can use this bracket:
{ }
The {1 0 0} family includes 6 planes:
(1 0 0)_
( 1 0 0 )
( 0 1 0 )
( 0 0 1 )_
( 0 0 1 )
( 0 1 0 )_ { 1 0 0 }
(1 1 0)
_
( 1 1 0 )
( 1 0 1 )
( 0 1 1 )
_
( 0 1 1 )
( 1 0 1 )
_
{ 1 1 0 }
{ 1 1 0 } = Plane ( 1 1 0 ) and all other planes related
by symmetry to ( 1 1 0 )
The {1 1 1} family includes 8 planes:
The {1 1 0} family includes 12 planes:
(1 1 0)
_
( 1 1 0 )
( 1 0 1 )
( 0 1 1 )
_
( 0 1 1 )
( 1 0 1 )
_
_
_
_
_
_
_
Interplanar Distance
222222 ///
1
clbkahd crystals
hkl
222 lkh
ad cubic
hkl
The distance between any two successive planes.
Parameters Determining the Crystal Structure of Materials
• Number of Atoms per Unit Cell (or) Effective Number:
The total number of atoms present in (or) Shared by an unit cell is known as number of atoms per unit cell
• Co-ordination Number:
Co-ordination number is the number of nearest neighboring atoms to a particular atom
(or)
Co-ordination number is the number of nearest neighbors directly surrounding a given atom
• Atomic Radius:
Atomic radius is defined as half of the distance between any
two nearest neighbor atoms which have with each other, in a crystal of pure element.
• Atomic Packing Factor (or) Packing Density
Simple Cubic Structure (SC)• Number of Atoms per Unit Cell (n):
n = Number of corner atoms/8
= 1
• Co-ordination Number in SC:
= (No. of nearest neighbors)
= 6
• Atomic Radius (r) in SC:
a=2r
r = a/2
• Atomic Packing Factor (APF) in SC:
Lattice constant
close-packed directions
a
R=0.5a
• APF for a simple cubic structure = 0.52
APF = (No. of Atoms per unit cell * Volume of
one atom) / Total volume of the unit cell
Body Centered Cubic structure (BCC)• Number of Atoms per Unit Cell (n):
Unit cell c ontains:
1 + 8 x 1/8
= 2 atoms/unit cell
Co-ordination Number = 8
• Close packed directions are cube diagonals.
Atomic Radius (r) in BCC:
aR
r = √3 a
4
• Atomic Packing Factor (APF) in BCC:
• APF for a body-centered cubic structure = p (3/8)= 0.68
Face Centered Cubic Structure (FCC)
• Number of Atoms per Unit Cell (n):
Unit cell c ontains:
6 x 1/2 + 8 x 1/8
= 4 atoms/unit cell
Co-ordination Number = 12
Atomic Radius (r) in FCC:
a
r = √2 a
4
• Atomic Packing Factor (APF) in FCC:
• APF for a body-centered cubic structure = p/(32)
= 0.74(best possible packing of identical spheres)
Hexagonal Close-Packed Structure (HCP)
• Number of Atoms per Unit Cell (n):n1 = (No. of corner atoms/6) =12/6 =2
n2 = (No. of base atoms/2) = 2/2 =1n3 = No. of middle atoms = 3
n = 2 + 1 +3 = 6
• 3D Projection of HCP
A sites
B sites
A sites
• 2D Projection of HCP
• Co-ordination Number in HCP:
= 12
• Atomic Radius (r) in HCP:
a = 2r
r = a/2
• The relation between (c/a):
– c/a = (√8/3)
– for ideal c/a ratio of 1.633
• Atomic Packing Factor (APF) in HCP:
Close packed crystals
A plane
B plane
C plane
A plane
…ABCABCABC… packing
[Face Centered Cubic (FCC)]
…ABABAB… packing
[Hexagonal Close Packing (HCP)]
Close Packing
Diamond Cubic Structure
Diamond Cubic Structure is a FCC sub lattice with the basis of two carbon atoms
This structure has two sub lattice A and B.
The sub lattice A has it origin of (0,0,0) and sub lattice B has its origin at (a/4, a/4, a/4) along the body diagonal
Number of Atoms per Unit Cell (n) in Diamond Cubic
No. of corner atom per unit cell = (1/8)*8 = 1 atom
No. of face centered atoms per unit cell = (1/2)*6 = 3
No. atoms inside the unit cell = 4
Total number of atoms per unit cell = 1+3+4 = 8
• Co-ordination Number in Diamond Cubic:
= 4
• Atomic radius (r):
• Atomic Packing Factor (APF) in Diamond Cubic :
r = (a* 3)/8
(APF) =
Comparison Chart for Crystal Structures
S.No
Systems SC BCC FCC HCP Diamond
1 No. Atoms per unit cell
1 2 4 6 8
2 Coordination number
6 8 12 12 4
3 Atomic radius a/2 a3/4 a2/4 a/2 a3/8
4 Atomic PackingFactor (APF)
p/6
0.52
p3/8= 0.68
p2/6=0.74
p/32=0.74
p3/16 = 0.34
5 Packing density 52% 68% 74% 74% 34%
6 Example Polonium Na, Fe Pb, Ag Mg, Ti Ge, Si
Sodium Chloride Structure
• No. atoms per unit cell: Na+ = 4; Cl- = 4 atoms
• Coordination Number: Na+ = 6; Cl- = 6
• Atomic radius (r) = a/2
Zinc Blende (ZnS) Structure • Sphalerite ( FCC ZnS)
Wurtzite ( HCP ZnS)
Fluorite and anti fluorite
• Expanded FCC lattice
• Fluorite: Cations forming the lattice with the anions occupying both types of tetrahedral hole.
• Ex: CaF, UO2, PbO2
• Anti Fluorite: inverse of Fluorite structure• Ex: Na2S, Li2O, K2O
Polymorphism and Allotropy• The ability of a material to have more
than one structure
• If the change in structure is reversible, then the polymorphic change is know as allotropy– i.e., only physical properties change without
any change in the chemical properties.
Metal Temperature (°C)
Crystal Structure
Lattice constant& Property
Iron Up to768° C768 - 910° C ,910 -1400° C,1400 -1593° C
BCCBCCFCCBCC
-- , Highly Magnetic0.290nm, Non Magnetic0.363nm, Para Magnetic0.293nm, --
Cobalt Room TempAbove 477° C
HCPFCC
Graphite Structure
Crystal Defects
• An ideal crystal can be described in terms a three-dimensionally periodic arrangement of points called lattice and an atom or group of atoms associated with each lattice point called motif:
• However, there can be deviations from this ideality.
• These deviations are known as crystal defects.
Crystal = Lattice + Motif
Vacancy: A point defect
Defects Dimensionality Examples
Point 0 Impurity, Vacancy
Line 1 Dislocation
(Edge, Screw)
Surface 2 Free surface,
Grain boundary
Volume 3 Cavities or voids,
Cracks and Holes
vacancy Interstitial
impurity
Substitutional
impurity
Point Defects
Frenkel defect
Schottky
defect
Defects in ionic solids
Cation vacancy
+
cation interstitial
Cation vacancy
+
anion vacancy
Missing half planeA Defect
Line Defects Dislocations
An extra half plane…
…or a missing half plane
What kind of
defect is this?
A line defect?
Or a planar defect?
Extra half plane No extra plane!
Missing plane No missing plane!!!
An extra half plane…
…or a missing half plane
Edge
Dislocation
This is a line defect called an
EDGE DISLOCATION
The atom positions around an edge dislocation; extra
half-plane of atoms shown in perspective.
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
slip no slip
boundary = edge dislocation
Slip planeb
Burgers vector
Slip plane
slip no slip
dis
loca
tion
b
t
Dislocation: slip/no
slip boundary
b: Burgers vector
magnitude and
direction of the slip
t: unit vector tangent to
the dislocation line
• Dislocation Line:A dislocation line is the boundary between slip and no slip regions of a crystal
• Burgers vector:The magnitude and the direction of the slip is represented by a vector b called the Burgers vector,
• Line vectorA unit vector t tangent to the dislocation line is called a tangent vector or the line vector.
1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
slip no slip
boundary = edge dislocation
Slip planeb
Burgers vector
t
• In general, there can be any angle between the Burgers vector b (magnitude and the direction of slip) and the line vector t (unit
vector tangent to the dislocation line)
• b t Edge dislocation
• b t Screw dislocation
• b t , b t Mixed dislocation
b
t
b || t
12
3
• If b || t
• Then parallel planes to the dislocation line lose their distinct identity and become one
continuous spiral ramp
• Hence the name SCREW DISLOCATION
Edge
Dislocation
Screw
Dislocation
Positive Negative
Extra half plane
above the slip
plane
Extra half plane
below the slip
plane
Left-handed
spiral ramp
Right-handed
spiral ramp
b parallel to t b antiparallel to t
Edge Dislocation
432 atoms
55 x 38 x 15 cm3
Screw Dislocation 525 atoms
45 x 20 x 15 cm3
Screw Dislocation (another view)
Johannes Martinus
BURGERS
Burgers vector Burger’s vector
Burgers vector
1
2
7
6
5
4
3
8
9
1 82 3 4 5 6 7 9 10 11 12 13
1
2
3
4
5
6
7
8
9
18 2345679101
1
1213
A closed Burgers
Circuit in an ideal
crystal
SF
14 15 16
141516
1
2
7
6
5
4
3
8
9
1 82 3 4 5 6 7 9 10 11 12 13 14 15
1
2
3
4
5
6
7
9
1234568 79101112131415
8
16
S
b 16
RHFS convention
F
Map the same
Burgers circuit on a
real crystal
b is a lattice translation
b
If b is not a complete lattice translation then a surface defect
will be created along with the line defect.
Surface defect
N+1 planes
N planes
Compression
Above the slip plane
Tension
Below the slip plane
Elastic strain field associated with an
edge dislocation
A dislocation line cannot end abruptly
inside a crystal
Slip plane
slip no slip
slip no slip
dis
loca
tio
n
b
Dislocation:
slip/no slip
boundary
Slip plane
F
B
A dislocation line cannot end abruptly
inside a crystal
F
M
T
A dislocation line cannot end abruptly
inside a crystal
It can end on a free surface
Grain 1 Grain 2
Grain
Boundary
Dislocation can end on a grain boundary
A dislocation line cannot end abruptly inside a crystal
• It can end on
– Free surfaces
– Grain boundaries
– On other dislocations at a point called a node
– On itself forming a loop
Slip plane
The plane containing both b and t is called the
slip plane of a dislocation line.
An edge or a mixed dislocation has a unique
slip plane
A screw dislocation does not have a unique
slip plane.
Any plane passing through a screw
dislocation is a possible slip plane
Glide of an Edge
Dislocation
crss
crss
crss is
critical
resolved
shear stress
on the slip
plane in the
direction of
b.
Glide of an Edge
Dislocation
A dislocation cannot end
abruptly inside a crystal
Burgers vector of a
dislocation is constant
Surface Defects
External Internal
Free surface Grain boundary
Stacking fault
Twin boundary
Interphase boundary
Same
phase
Different
phases
External surface: Free surface
If bond are broken over an area A then two
free surfaces of a total area 2A is created
Area A
Area A
Broken
bonds
External surface: Free surface
If bond are broken over an area A
then two free surfaces of a total
area 2A is created
Area A
Area A
Broken
bonds
nA=no. of surface atoms per
unit area
nB=no. of broken bonds per
surface atom
=bond energy per atom
BA nn2
1
Surface energy per
unit area
Grain 1Grain 2
Grain
Boundary
Internal surface: grain boundary
A grain boundary is a boundary between two regions of
identical crystal structure but different orientation
Photomicrograph an iron
chromium alloy. 100X.
Optical Microscopy
Edge dislocation model of
a small angle tilt
boundary
Grain 1
Grain 2
Tilt boundary
A
BC
2
2h
b
A
B
C
2sin
2
h
b
tanh
b
Or approximately
Stacking fault
C
B
A
C
B
A
C
B
A
A
C
B
A
B
A
C
B
A
Stacking
fault
FCC FCC
HCP
Twin PlaneC
B
A
C
B
A
C
B
A
C
B
A
C
A
B
C
A
B
C
B
A
C
B
A
Twin plane