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Unit cells & the primitive cells : FCC & BCC lattice

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Rhombohedral

Primitive Cell in a

BCC

Inclined Prism

Primitive Cell in a

BCC

There could be more than one primitive cells

e.g., for Body Centered Cubic:

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Rhombohedral Primitive

Cell in a Body Centered

Orthorhombic

Inclined Prism Primitive

Cell in a Body Centered

Orthorhombic


e.g., for Body Centered Orthorhombic:

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Rhombohedral

Primitive Cell in a

FC-Orthorhombic

Inclined Prism

Primitive Cell in a FC-

Orthorhombic


e.g., for Face Centered Orthorhombic:

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Symmetry elements (Cubic Lattice)

On performing certain operations on the crystal, the crystal is

brought into coincidence with itself 4 macroscopic sym. elements

e.g. N-fold Rotation axis: the crystal can be brought intocoincidence with itself by a rotation of 360/n about the so

called n-fold axis (n=1,2,3,4 or 6)

5 fold axis or one of the higher degree than 6 are

impossible (unit cells having such symmetry can not be made to

fill up space without leaving

gaps)

Reflection Plane Rotation axes; (4,3,2-

fold)

4:A1A2; 3: A1A3

Inversion Center

A1A2

Rotation-

Inversion Axis

A1A1 A2

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Directions in a lattice

Let the line pass through the origin of the unit cell and any point having

coordinates u, v, w (not necessary integers) can indicate its direction.

Then [uvw] indices of the direction of the line.

Whatever the values of u, v, and w, they are always converted to a set of

smallest integers (by multiplication or division throughout)

[ 1] [112][224] all represent the same direction

Negative indices are written with a bar over the indices, e.g. ,

Directions related by symmetry are called direction of a form

represents 4 body diagonals of a cube]111[&],111[],111[],111[

][ vwu

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Orientation of Planes : MILLER INDICES

Reciprocals of the fractional intercepts which the planes

makes with the crystallographic axis

Miller Indices (hkl)

Fractional Intercepts

with the axes

1/h , 1/k , 1/l

If axial lengths are a , b , & c,Actual Intercepts a/h , b/k, c/l

e.g., shaded plane in (see fig. below)

Axial lengths 4, 8, 3

Intercept lengths

Fractional Intercepts

1, 4, 3, , 1

Miller Indices (421)

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If a plane is parallel to any axis, its fractional intercept is taken as

hence corresponding index is 0.

If a plane cuts ave axis, the corresponding index is negative,

Fof Cubic Systems: The direction [hkl] is always to plane (hkl)

The planes (nh, nk, nl) are parallel to (hkl) and have 1/nth spacing

)221(

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Miller indices & the lattice-planes. The distance d is the interplanar spacing

Mill I di f H l S

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Miller Indices of a Hexagonal System

)1110(

Additional axis a3 is considered in the basal plane (in addition

to a1 & a2)

Miller Bravais Indices: (hkil) Index i is the reciprocal of the fractional intercept on the a3-

axis. Also, h+k = -i

Alternate representations as (hkl) or (hkl) ; Plane OPQ

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o Planes of a Form: These are sets of equivalent lattice planes related

by symmetry; indicated by indices of any plane enclosed in braces.

e.g., {hkl}

e.g., (100), (010), (001), (-100), (0-10) & (00-1) are planes of theform {100}, as all are generated by 4-fold axis of symmetry

o Planes of a Zone : These are those planes which areparallel to one

line (called as zone-axis); indicated by the indices of the zone-axis.

All shaded planes in the cubic

lattice shown are planes of

the zone [001]

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Interplanar spacing

Various (hkl) planes have various values of interplanar spacing;

for cubic ; tetragonal ; hexagonal systems,

The planes with large dhkl have low indices, & high density of

lattice points.

2

2

2

22

22

2

2

22

22

222

2341;1;1

c

l

a

khkh

dc

l

a

kh

da

lkh

dhklhklhkl

2D lattice showing

lines of lowest

indices having

higher spacing

and greatestdensity of lattice

points

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6 January 2014 EPL206/02 15HCP structure shown by Zn, Mg, -Ti, Be, etc.

CRYSTAL STRUCTURE

Origin of the unit cell in (b) is

shifted such that point (1,0,0)

in Fig. (b) is midway between

the atom at (1,0,0) and

(2/3,1/3,1/2) in Fig. (a)

HCP unit cell showing

atom-positions

The atoms of a crystal are set in space either on the points of

a Bravis lattice or in some fixed relation to that point

HCP unit cell showingatom-positions

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Both have same atomic

packing fraction~74%

Layer sequence

in FCC:

ABCABC.

FCC & HCP Structures A Comparison

Layer sequencein HCP:

ABABAB.

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COMPOUNDS OF UNLIKE ATOMS (AxBy)Arrangement of atoms in AxBy must satisfy

following conditions

1. Body-, Face- or Basecentering translations must

begin and end on atoms of same kind.2. The set of A-atoms, and set of B-atoms must

separately possess the same symmetry element as

the crystal as a whole.

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o A possible reduced sphere unit cell for the CsCl crystal.

o An alternative unit cell may have Cs+ and Cl- interchanged.

Examples: CsCl, CsBr, CsI, etc.

Cs+ : (000)

Cl-

: ()

BRAVAIS LATTICE is SCC

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(a) A schematic illustration of a cross section from solid NaCl. NaCl is made

of Cl- and Na+ ions arranged alternatingly so that the oppositely chargedions are closest to each other and attract each other. There are also

repulsive forces between the like ions. In equilibrium the net force acting

on any ion is zero.

(b) Solid NaCl.

(a) (b)

N Cl

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o A possible reduced sphere unit cell for the NaCl (rock salt) crystal.

o An alternative Unit cell may have Na+ and Cl- interchanged.

o Examples: AgCl, CaO, CsF, LiF, LiCl, NaF, NaCl, KF, KCl, MgO.

NaCl structure

4 Na+ : (000), (0), (0), (0)

4 Cl- : (), (00), (00), (00)

BRAVAIS LATTICE is FCC

Two inter-penetrating

FCC lattices (one for Na+

ions, other for Cl- ions);

with one of them shifted

by (,0,0)

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The Diamond unit cell is cubic. The cell has eight atoms.

e.g., Grey Sn (-Sn); semiconductors Ge, Si have this diamond

structure

4 C : (000), (0), (0), (0)

4 C : (1/4,1/4,1/4 , (3/4,3/4,1/4 , (1/4,3/4,3/4 , (3/4,3/4,3/4


FCC lattices (each for

Carbon atoms);

with one of them shiftedby (, , )

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o The Zinc blende (ZnS) cubic crystal structure.

o Many important compound crystal structures have the zinc blende

structure.

o Examples: AlAs, GaAs, Gap, GaSb, InAs, InP, InSb, ZnS, ZnTe.

4 S : (000), (0), (0), (0)

4 Zn : (1/4,1/4,1/4 , (3/4,3/4,1/4 , (1/4,3/4,3/4 , (3/4,3/4,3/4


FCC lattices (one for S ionsand other for Zn ions);

with one of them shifted

by (, , )

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Packing of coins on a table top to build a two dimensional crystal

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Three allotropes of carbon

ATOM SIZES & CORDINATION

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Structures of compound/solid solution : determined by the relative

sizes of the atoms involved

Q : What is meant by the size of an atom ?

Atom

billiard ball (sharply defined boundary surface)

(this is an oversimplified proposition)

Practical Definition : collection of rigid spheres in contact

Then the size is given by the distance of closest approach of atom

centers in a crystal, which can be calculated from the lattice parameters.

e.g., -Fe (BCC) : Atoms are in contact along the cube-diagonal

(3a=4R)

Size of an atom is not constant (strictly speaking)

Smaller is the co-ordination number, the smaller is the volume

occupied by a given atom (2RFe is greater if Fe is dissolved in FCC

copper than if it exists in BCC -Fe or is dissolved in BCC-V)


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Atomic size also depends on bonding

e.g., if more electrons are removed, the ion becomes smaller.

Fe, Fe+, Fe++, Fe+++ have diameters 2.48, 1.66 and 1.34

OCTAHEDRAL a solid bounded by 8 triangular sides (NaCl)

TETRAHEDRAL a solid bounded by 4 triangular sides (PbS,

ZnS, etc.)

Tetrahedral surrounding

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