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