Crystalline Solids

Structure of Crystalline Solids

BITS Pilani, Pilani Campus

Structure of Crystalline Solids

Unit cell - is smallest repeatable entity that can be used to

completely represent a crystal structure. It is the building

block of crystal structure.

Crystal lattice it is the three dimensional network of points

in space.

Important Definitions


in space.

Space lattice it is the thee dimensional network of lines in

space.

Primitive cell it is a simple cubic unit cell having atoms at

its eight corners.

Lattice parameter(a)- it is defined as the distance between

the center of neighboring corner atoms.

Polycrystals Most crystalline solids are composed of many small

crystals (also called grains).

Initially, small crystals (nuclei) form at various

positions.

These have random orientations.

The small grains grow and begin to impinge on one The small grains grow and begin to impinge on one

another forming grain boundaries.

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Micrograph of a polycrystalline

stainless steel showing grains and

grain boundaries

Most engineering materials

are Polycrystals.

Single Crystals

-Properties vary with

direction: anisotropic.

-Example: the modulus

of elasticity (E) in BCC iron:

Single vs Polycrystals

E (diagonal) = 273 GPa

E (edge) = 125 GPa

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Polycrystals

-Properties may/may not

vary with direction.

-If grains are randomly

oriented: isotropic.

E (edge) = 125 GPa

Crystal Systems

7 crystal systems

Unit cell: smallest repetitive volume that contains

the complete lattice pattern of a crystal.

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7 crystal systems

14 crystal lattices

a, b and c are the lattice constants

The fourteen (14)

types of Bravais

lattices grouped in

seven (7) systems. seven (7) systems.

Miller indices - A shorthand notation to describe certain

crystallographic directions and planes in a material.

Denoted by [ ], , ( ) brackets. A negative number is

represented by a bar over the number.

Points, Directions and Planes in the

Unit Cell

Point Coordinates

Coordinates of selected points in the unit cell.

The number refers to the distance from the origin in terms of

lattice parameters.

Point Coordinates

Point coordinates for unit cell center are

a/2, b/2, c/2

Point coordinates for unit cell

z

y

a b

c

000

111

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Point coordinates for unit cell corner are 111

Translation: integer multiple of lattice constants identical position in another unit cell

x

y

z

2c

b

b

Determine the Miller indices of directions A, B, and C.

Miller Indices, Directions

SOLUTION

Direction A

1. Two points are 1, 0, 0, and 0, 0, 0

2. 1, 0, 0, -0, 0, 0 = 1, 0, 0

3. No fractions to clear or integers to reduce

4. [100]

Direction B

1. Two points are 1, 1, 1 and 0, 0, 01. Two points are 1, 1, 1 and 0, 0, 0

2. 1, 1, 1, -0, 0, 0 = 1, 1, 1

3. No fractions to clear or integers to reduce

4. [111]

Direction C

1. Two points are 0, 0, 1 and 1/2, 1, 0

2. 0, 0, 1 -1/2, 1, 0 = -1/2, -1, 1

3. 2(-1/2, -1, 1) = -1, -2, 2

2]21[ .4

Algorithm for Miller indices1. Read off intercepts of plane with axes in

terms of a, b, c

2. Take reciprocals of intercepts

3. Reduce to smallest integer values

4. Enclose in parentheses, no commas.

Crystallographic Planes

4. Enclose in parentheses, no commas.

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The following treatment of the procedure used to assign

the Miller Indices is a simplified one (it may be best if you

simply regard it as a "recipe") and only a cubic crystal

system (one having a cubic unit cell with

dimensions a x a x a ) will be considered.

Miller Indices

Step 1 : Identify the intercepts on the x- , y-

and z- axes.In this case the intercept on the x-axis is at x = a ( at the point

(a,0,0) ), but the surface is parallel to the y- and z-axes -

strictly therefore there is no intercept on these two axes but

we shall consider the intercept to be at infinity ( ) for the

special case where the plane is parallel to an axis. The

The procedure is most easily illustrated using an

example so we will first consider the following

surface/plane:

special case where the plane is parallel to an axis. The

intercepts on the x- , y- and z-axes are thus

Intercepts : a , ,

Step 2 : Specify the intercepts in fractional co-ordinatesCo-ordinates are converted to fractional co-ordinates by dividing by the

respective cell-dimension - for example, a point (x,y,z) in a unit cell of

dimensions a x b x c has fractional co-ordinates of ( x/a , y/b , z/c ). In the case

of a cubic unit cell each co-ordinate will simply be divided by the cubic cell

constant , a . This gives

Fractional Intercepts : a/a , /a, /a i.e. 1 , ,

Miller Indices

Step 1 : Take the reciprocals of the fractional

intercepts

This final manipulation generates the Miller Indices

which (by convention) should then be specified without

being separated by any commas or other symbols.

The Miller Indices are also enclosed within standard

brackets (.) when one is specifying a unique surface

such as that being considered here.such as that being considered here.

he reciprocals of 1 and are 1 and 0 respectively, thus

yielding

Miller Indices : (100)

So the surface/plane illustrated is the (100) plane of the cubic crystal.


Crystallographic planes are specified by 3 Miller Indices (h k l). All parallel planes have same Miller indices.

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Crystallographic Planesz

x

y

a b

c

4. Miller Indices (110)

1. Intercepts 1 1

2. Reciprocals 1/1 1/1 1/

1 1 03. Reduction 1 1 0

example a b c

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x

example a b cz

x

y

a b

c


1. Intercepts 1/2

2. Reciprocals 1/ 1/ 1/

2 0 03. Reduction 2 0 0


z

example

1. Intercepts 1/2 1 3/4

a b c

2. Reciprocals 1/ 1/1 1/

2 1 4/3

3. Reduction 6 3 4

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x

y

a b

c


3. Reduction 6 3 4

Determine the Miller indices of planes A, B, and C.

Miller Indices -Planes

Crystallographic planes

and intercepts.

SOLUTION

Plane A

1. x = 1, y = 1, z = 1

2.1/x = 1, 1/y = 1,1 /z = 1

3. No fractions to clear

4. (111)

Plane B

1. The plane never intercepts the z axis, so x = 1, y = 2, and z =

2.1/x = 1, 1/y =1/2, 1/z = 0

3. Clear fractions:

3. Clear fractions:

1/x = 2, 1/y = 1, 1/z = 0

4. (210)

Plane C

1. We must move the origin, since the plane passes through 0, 0, 0. Lets move the origin one lattice parameter in the y-direction. Then, x = , y = 1, and z =

2.1/x = 0, 1/y = 1, 1/z = 0

3. No fractions to clear.

)010( .4

Family of Planes

Planes that are crystallographically

equivalent have the same atomic packing.

Also, in cubic systems only, planes having the

same indices, regardless of order and sign,

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are equivalent.

Ex: {111}

= (111), (111), (111), (111), (111), (111), (111), (111)

(001)(010), (100), (010),(001),Ex: {100} = (100),

_ __ __ _ __ _ _ _ _

THEORETICAL DENSITY ()


Q.1) find the diameter of an imaginary sphere which fits

into the interstitial void in the center of the cube edge of

a BCC crystal.

Ans.

d- diameter of imaginary sphere


d- diameter of imaginary sphere

d=a-2r

d= - 2r

d=0.309*r

Q.4) Some hypothetical metal has the simple cubic crystal

structure. If its atomic weight is 74.5 g/mol. and the atomic

radius is 0.145 nm, compute its density.


Ans. For the simple cubic crystal structure, the value of n in

Equation is unity since there is only a single atom

associated with each unit cell. Furthermore, for the unit cell

edge length, a = 2R

SUMMARY

Crystallographic points, directions and planes are

specified in terms of indexing schemes.

Materials can be single crystals or polycrystalline.

Material properties generally vary with single

crystal orientation (anisotropic), but are generally

non-directional (isotropic) in polycrystals with non-directional (isotropic) in polycrystals with

randomly oriented grains.

Some materials can have more than one crystal

structure. This is referred to as polymorphism (or

allotropy).

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Crystalline Solids

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