crystral structure
DESCRIPTION
TRANSCRIPT
A SEMINAR ON CRYSTAL STRUCTURE
PRESENTED
BY
K. GANAPATHI RAO (13031D6003)
Presence ofMr. V V Sai sir
CONTENT
INTRODUCTION.
TRANSLATION VECTOR.
BASIS & UNIT CELL.
BRAVAIS & SPACE LATTICES.
FUNDAMENTAL QUANTITIES.
MILLER INDICES.
INTER-PLANAR SPACING.
Materials
Solids Liquids Gasses
Solids
Crystalline
Single Poly
Amorphous
TRANSLATION VECTOR
LATTICE PARAMETERS & UNIT CELL
Bravais Lattice System
Possible Variations
Axial Distances
(edge lengths)
Axial Angles Examples
CubicPrimitive, Body-centred, Face-centred
a = b = c α = β = γ = 90° NaCl, Zinc Blende, Cu
Tetragonal Primitive, Body-centred a = b ≠ c α = β = γ = 90°
White tin, SnO2,TiO2, CaSO4
Orthorhombic
Primitive, Body-centred, Face-centred, Base-centred
a ≠ b ≠ c α = β = γ = 90°Rhombic sulphur,KNO3, BaSO4
Rhombohedral Primitive a = b = c α = β = γ ≠ 90°
Calcite (CaCO3,Cinnabar (HgS)
Hexagonal Primitive a = b ≠ c α = β = 90°, γ = 120° Graphite, ZnO,CdS
Monoclinic Primitive, Base-centred a ≠ b ≠ c α = γ = 90°, β ≠
90°Monoclinic sulphur, Na2SO4.10H2O
Triclinic Primitive a ≠ b ≠ c α ≠ β ≠ γ ≠ 90°K2Cr2O7, CuSO4.5H2O,H3BO3
1 Cubic Cube P I F C
Lattice point
PI
F
a b c 90
P I F C
2 Tetragonal Square Prism (general height)
IP
a b c
90
P I F C
3 Orthorhombic Rectangular Prism (general height)
PI
F C
a b c
90
a b c One convention
P I F C
4 Trigonal Parallelepiped (Equilateral, Equiangular)
90
a b c
Rhombohedral
Note the position of the origin and of ‘a’, ‘b’ & ‘c’
P I F C
5 Hexagonal 120 Rhombic Prism
A single unit cell (marked in blue) along with a 3-unit cells forming a
hexagonal prism
a b c
90 , 120
P I F C
6 Monoclinic Parallogramic Prism
90
a b c a b c
Note the position of ‘a’, ‘b’ & ‘c’
One convention
P I F C
7 Triclinic Parallelepiped (general)
a b c
FUNDAMENTAL QUANTITIES
• NEAREST NEIGHBOUR DISTANCE (2R).
• ATOMIC RADIUS (R).
• COORDINATION NUMBER (N).
• ATOMIC PACKING FACTOR.
SIMPLE CUBIC STRUCTURE (SC)
• Rare due to low packing density (only Po has this structure)• Close-packed directions are cube edges.
• Coordination # = 6 (# nearest neighbors)
(Courtesy P.M. Anderson)
ATOMIC PACKING FACTOR (APF):SC
• APF for a simple cubic structure = 0.52
APF = a3
4
3 p (0.5a) 31
atoms
unit cellatom
volume
unit cell
volume
APF = Volume of atoms in unit cell*
Volume of unit cell
*assume hard spheres
Adapted from Fig. 3.24, Callister & Rethwisch 8e.
close-packed directions
a
R=0.5a
contains 8 x 1/8 = 1 atom/unit cell
BODY CENTERED CUBIC STRUCTURE (BCC)
• Coordination # = 8
Adapted from Fig. 3.2, Callister & Rethwisch 8e.
• Atoms touch each other along cube diagonals.--Note: All atoms are identical; the center atom is shaded differently only for ease of viewing.
ex: Cr, W, Fe (), Tantalum, Molybdenum
2 atoms/unit cell: 1 center + 8 corners x 1/8(Courtesy P.M. Anderson)
ATOMIC PACKING FACTOR: BCC
a
APF =
4
3p ( 3a/4)32
atoms
unit cell atom
volume
a3unit 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.
a 2
a 3
FACE CENTERED CUBIC STRUCTURE (FCC)
• 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.
ex: Al, Cu, Au, Pb, Ni, Pt, Ag
4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8(Courtesy P.M. Anderson)
ATOMIC PACKING FACTOR: FCC• APF for a face-centered cubic structure = 0.74
maximum achievable APF
APF =
4
3 p ( 2a/4)34
atoms
unit cell atom
volume
a3unit cell
volume
Close-packed directions: length = 4R = 2 a
Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell
a
2 a
Adapted fromFig. 3.1(a),Callister & Rethwisch 8e.
MILLER INDICES
• PROCEDURE FOR WRITING DIRECTIONS IN MILLER INDICES
• DETERMINE THE COORDINATES OF THE TWO POINTS IN THE DIRECTION. (SIMPLIFIED IF ONE OF THE POINTS IS THE ORIGIN).
• SUBTRACT THE COORDINATES OF THE SECOND POINT FROM THOSE OF THE FIRST.
• CLEAR FRACTIONS TO GIVE LOWEST INTEGER VALUES FOR ALL COORDINATES
MILLER INDICES
• INDICES ARE WRITTEN IN SQUARE BRACKETS WITHOUT COMMAS (EX: [HKL])
• NEGATIVE VALUES ARE WRITTEN WITH A BAR OVER THE INTEGER.
• EX: IF H<0 THEN THE DIRECTION IS•
][ klh
MILLER INDICES
• CRYSTALLOGRAPHIC PLANES• IDENTIFY THE COORDINATE INTERCEPTS OF THE PLANE
• THE COORDINATES AT WHICH THE PLANE INTERCEPTS THE X, Y AND Z AXES.
• IF A PLANE IS PARALLEL TO AN AXIS, ITS INTERCEPT IS TAKEN AS ¥.
• IF A PLANE PASSES THROUGH THE ORIGIN, CHOOSE AN EQUIVALENT PLANE, OR MOVE THE ORIGIN
• TAKE THE RECIPROCAL OF THE INTERCEPTS
Find intercepts along axes → 2 3 1 Take reciprocal → 1/2 1/3 1 Convert to smallest integers in the same ratio → 3 2 6 Enclose in parenthesis → (326)
(2,0,0)
(0,3,0)
(0,0,1)
Miller Indices for planes
x
z
y
MILLER INDICES
• CLEAR FRACTIONS DUE TO THE RECIPROCAL, BUT DO NOT REDUCE TO LOWEST INTEGER VALUES.
• PLANES ARE WRITTEN IN PARENTHESES, WITH BARS OVER THE NEGATIVE INDICES.
• EX: (HKL) OR IF H<0 THEN IT BECOMES ][ klh
Intercepts → 1 Plane → (100)
Intercepts → 1 1 Plane → (110)
Intercepts → 1 1 1Plane → (111)(Octahedral plane)
x
x
x
y y
y
z z
z
INTER-PLANAR SPACING• FOR ORTHORHOMBIC, TETRAGONAL AND CUBIC UNIT
CELLS (THE AXES ARE ALL MUTUALLY PERPENDICULAR), THE INTER-PLANAR SPACING IS GIVEN BY:
h, k, l = Miller indices
a, b, c = unit cell dimensions
222 lkh
adhkl
• For cube a = b = c than
REFRENCES
• APPLIED PHYSICS BY P.K. PALANISAMY
• http://en.wikipedia.org/wiki/crystal_structure
• http://journals.iucr.org/c/
• http://www.scirp.org/journal/csta/
• http://www.asminternational.org/portal/site/www/subjectguideitem/?vgnextoid=ad7cdc8cc359d210vgnvcm100000621e010arcrd