Crystral structure

A SEMINAR ON
CRYSTAL STRUCTURE
PRESENTED
BY

K. GANAPATHI RAO
(13031D6003)

Presence of
Mr. 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

LATTICE PARAMETERS & UNIT
CELL

Bravais Lattice
System

Possible Variations

Axial Distances
(edge lengths)

Axial Angles

Examples

Cubic

Primitive, Bodya=b=c
centred, Face-centred

α = β = γ = 90°

NaCl, Zinc Blende, Cu

Tetragonal

Primitive, Bodycentred

a=b≠c

α = β = γ = 90°

White
tin, SnO2,TiO2, CaSO4

Orthorhombic

Primitive, Bodycentred, Facea≠b≠c
centred, Base-centred

α = β = γ = 90°

Rhombic
sulphur,KNO3, BaSO4

Rhombohedral

Primitive

a=b=c

α = β = γ ≠ 90°

Calcite (CaCO3,Cinnaba
r (HgS)

Hexagonal

Primitive

a=b≠c

α = β = 90°, γ =
120°

Graphite, ZnO,CdS

Monoclinic

Primitive, Basecentred

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

P


Cube

I


F


C

I
P
a b c

Lattice point

90

F

P
2

Tetragonal

Square Prism (general height)

I



F

C



I
P

a b c
90

P
3

Orthorhombic Rectangular Prism (general height)

I

F

C









One convention

a b c

I

P

a b c
90

F

C

P
4

Trigonal

Parallelepiped (Equilateral, Equiangular)

I



Rhombohedral

a b c
90

Note the position of the origin
and of ‘a’, ‘b’ & ‘c’

F

C

P
5

Hexagonal



120 Rhombic Prism

a b c
90 ,

A single unit cell (marked in blue)
along with a 3-unit cells forming a
hexagonal prism

120

I

F

C

P
6

Monoclinic



Parallogramic Prism

One convention

a b c

a b c
90

Note the position of
‘a’, ‘b’ & ‘c’

I

F

C


P
7

Triclinic



Parallelepiped (general)

a b c

I

F

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 =

Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres
• APF for a simple cubic structure = 0.52
atoms
unit cell

a
R=0.5a

APF =

1

4

3
a3

close-packed directions
contains 8 x 1/8 =
1 atom/unit cell
Adapted from Fig. 3.24,
Callister & Rethwisch 8e.

volume
atom
(0.5a) 3
volume
unit cell

BODY CENTERED CUBIC STRUCTURE
(BCC)
• 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



Adapted from Fig. 3.2,
Callister & Rethwisch 8e.

2 atoms/unit cell: 1 center + 8 corners x 1/8

ATOMIC PACKING FACTOR: BCC
• APF for a body-centered cubic structure = 0.68
3a

a
2a
Adapted from
Fig. 3.2(a), Callister &
Rethwisch 8e.

atoms

R

Close-packed directions:
length = 4R = 3 a

a

4

2
unit cell
3
APF =

( 3a/4) 3
a3

volume
atom
volume
unit cell

FACE CENTERED CUBIC STRUCTURE
(FCC)
• 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


Adapted from Fig. 3.1, Callister & Rethwisch 8e.

4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8

ATOMIC PACKING FACTOR: FCC
• APF for a face-centered cubic structure = 0.74
maximum achievable APF
Close-packed directions:
length = 4R = 2 a
2a

a
Adapted from
Fig. 3.1(a),
Callister &
Rethwisch 8e.

Unit cell contains:
6 x 1/2 + 8 x 1/8
= 4 atoms/unit cell
atoms
4
4
unit cell
3
APF =

( 2a/4) 3
a3

volume
atom
volume
unit cell

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.

[hkl]

• EX: IF H<0 THEN THE DIRECTION IS
•

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

Miller Indices for planes
(0,0,1)

z

y

(0,3,0)

x

(2,0,0)

 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)

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. [hkl]

• EX: (HKL) OR IF H<0 THEN IT BECOMES

z

z

y

y
x

x

Intercepts → 1
Plane → (100)

Intercepts → 1 1
Plane → (110)

z
y Intercepts → 1 1 1
x

Plane → (111)
(Octahedral plane)

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

• For cube a = b = c than
a
d hkl
h2 k 2 l 2

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/subje
ctguideitem/?vgnextoid=ad7cdc8cc359d210vgnvcm10
0000621e010arcrd

Crystral structure

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