Crystal 2

In 3D we have seven crystal systems

Lecture 3 Structures of Metals and Ceramics

3

CRYSTAL SYSTEMS

In 3D we have seven crystal systems

There are many different crystal structures, it is divided into groups according to unit cell configurations 3 Crystal Axes: n A right-handed coordinate system consisting of three axis x, y, and z axes at the one of the corners and coinciding with the uc. edges. p 3 lengths: n a, b , c c p 3 inter-axial angles n a , b , g, x So 6 Lattice Parameters Define the Size and Shape of the Unit Cell

Geometry of the Unit Cell.

p

z

a g

b

y a

b 2

4

The Special Cases pCall

14 space or Bravais lattices grouped in 7 X-tal system

it Cubic and Give 1 Lattice Parameter : Value of a = “ao” only! a=b =c a = b = g = 90o

z

with it: NONE Call it Triclinic and Give 6 Lattice Parameter : a ¹ b¹ c a ¹ b ¹ g ¹ 90o

γ

y

a

b

γ

14 space or Bravais lattices grouped in 7 X-tal system

Orthorhombic-s Orthorhombic base-c Orthorhombic-bc

Tetragonal-s

Tetragonal-bc

a 5

Monoclinic-s

Rhombohedral

α

β

Triclinic-s

Hexagonal

b

pSymmetry associated

c

α

β

c

x

Cubic-s

Cubic-bc

Cubic-fc

7

3 Bravais lattices for Cubes

Monoclinic base-c

Orthorhombic-fc

6

Corners

Corners & Body Center

Corners & All Face Centers

“Simple ”

“Body Centered”

“Face Centered”

8

Specification of Point Coordinates

2 Bravais Lattices for Tetragonal p

“Simple Tetragonal”

Point Number

Corners & All Face Centers

Corners & Body Center

Corners

“Face Centered Tetragonal”

“Body Centered Tetragonal”

9

Determination of point coordinates p Distance is measured in unit cell along the x, y and z-axis to get from the origin to the point in question. p Any lattice point can be the origin. p The coordinates are written as the three distances, q,r,s with commas separating the numbers. z

q, r, s

p sc

y

qa

a

x

Point Coordinates

Point Number

Point Coordinates

1.

000

10.

101

2.

100

11.

111

3.

110

12.

4.

010

13.

001

5.

½ ½ 0

14.

½ ½ 1

6.

1½½

15.

¾¼¾

7.

½1½

16.

¼¾¾

8.

0½½

17.

¼¼¼

9.

½0½

18.

¾¾¼

13

12 14

10

11

16 15

8

9

7 17

6 18

1

4

5 2

3

011

11

CRYSTALLOGRAPHIC DIRECTIONS

Point Coordinates The position of any point located within a unit cell may be specified in terms of its coordinates. It help for determination of lines and planes.

c

Conventions: n Use the Axes (a, b, c) as a Coordinate System n Each Axis is 1 (unit) Long p Give Coordinates between 0 and 1 n Notation q, r, s, where q, r, and s are fractions or 0

A crystallographic direction is defined as a line between two points, or a vector. p

p

Crystallographic directions are easy: a line passing from an origin 0,0,0 through a point h, k, l will have a direction indicated by [hkl] Conventions : n Draw a vector parallel to the direction of interest that starts from an origin 0,0,0 . n Determine the components of the vector in terms of unit cell dimension. n Clear all fractions to smallest integer values, hkl n Use [hkl], notation is important! n Use a bar over appropriate index if it is negative

[001] Z, c

000

[111]

Y, b [010]

X, a [100]

[001]

rb

b

10

12

CRYSTALLOGRAPHIC DIRECTIONS p

Crystallographic directions is determined by a line passing from an origin 0,0,0 through a point h, k, l will have a direction indicated by [hkl]

Point C: coordinates 1 0 0 Point D: coordinates 1/3 1 1 Line CD: direction: [1/3-1, 1-0, 1-0] [-2/3 1 1 ] or clear fractions: [233]

CRYSTALLOGRAPHIC DIRECTIONS Hexagonal Crystals p

Hexagonal systems are complicated, it can be alleviated by going to a 4index system:

Point C: assume it is origin (0 0 0 in the next unit cell) Point D: –2/3 1 1 w/ origin at C Line CD: direction [– 2/3 1 1] or clear fractions: [233]

In 2D (hkl) → (hkil) where h+k+i=0 a3 b

a2

a Point A: coordinates 0 0 0 Point B: coordinates 1 ½ 0 Line AB : direction [1½0], or clear fractions: [210]

[100]

[110]

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15

Hexagonal indexing (Barrett & Massalski)

CRYSTALLOGRAPHIC DIRECTIONS p

a1

Crystallographic directions is determined by a line passing from an origin 0,0,0 through a point h, k, l will have a direction indicated by [hkl]

z, c [122] [102]

[121]

y, b x, a

[110]

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16

Hexagonal indexing (Barrett & Massalski)

Family of Directions in Tetragonal System

_

E [2116 ] c

c _

The [110] Direction in tetragonal system

_

The [101] Direction in tetragonal system

C [1213 ] a3

a3 a2

a2 _

_

A [1100 ] a 1

_

_] D [1213

__

B [1123 ]

a1

c _

The Family of <110> Directions in tetragonal system

The Family of <101> Directions in tetragonal system

_

G [0111]

a3 a2

a1

17

19

Equivalent Groups (Family) of Directions

CRYSTALLOGRAPHIC PLANES Planes in a crystal are denoted by Miller indices which are the reciprocals of the fractional intercepts that the planes make with the axes . Procedure

[hkl] is ONE Specific Direction is a FAMILY of Equivalent Directions The [110] Direction in cubic system

ì [110 ] ï ï [101 ] ï ïï [ 011 ] 110 = í ïëé 110 ûù ï ïéë101 ùû ï ïîéë 011 ùû

The Family of <110> Directions in cubic system

ëé 110 ûù éë 101 ùû

ëé 011 ûù ëé 110ûù

p

If Plane passes through the origin, construct a parallel Plane in the Unit Cell,

p

Record the Intercepts of the Planes with x, y, and z axis, (If a plane is parallel to an axis, the intercept is taken to be at infinity ¥ ), Take Reciprocals of these Numbers,

p

éë101 ùû éë010 ùû

p

Clear Fractions, Enclose the result in parentheses (hkl).

The intercepts The reciprocals

a

b

c

3

2

2

1/ 1/ 1/ 3 2 2

The Miller indices (1/ 3 1/2 1/2) multiply by 6 18

(2 3 3) 20

CRYSTALLOGRAPHIC PLANES a The intercepts

b

c

1 2

c

2

c

The Miller indices

(1 1/

multiply by 2

2

1/

c

(211)

1/ 1/ 1/ 1 2 2

The reciprocals

Miller indices for Plane passes through origin

z

_

(110)

2)

O 000

b

O

(2 1 1)

y

a

Plane passes through origin Assume it is origin (0 0 0 in the next unit cell) a b

The Miller indices of important planes in a cubic crystal a

z

z

a

(001)

a

(010) (110)

a

a x (100)

y

y

(111)

a

a

a

x

The intercepts

-1

The reciprocals The Miller indices

1/ 1 -1 /1

1

_

b

000

(012)

a

000

x z

O 000

b

a

a

¥

The intercepts c

¥

b -1

c

1/ 2 1/ 2 -1 /1

1/

¥

The reciprocals The Miller indices

(0 -1 2)

1/

(-1 1 0)

¥

y a

x

_

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Representation of Equivalent Planes

Miller Indices For Hexagonal Crystals The procedure for finding Miller indices is exactly the same as before, but four intercepts are required, giving the indices of the form (hkil) (due to special geometry of system, h+k=-i). c

_

z

B (1210) 2/3

B (0003)

B

a3

B

a3

A

a2

a1 a2 a3 c Intercepts -1 1 ¥ 1/ 1/ 1/ 1 reciprocals -1 1 ¥ /¥ Miller indices (-1 1 0 0)

¥

A

a1 a2 a3 c The intercepts 1 -1 ¥ 1 1 1/ 1/ 1/ The reciprocals 1 -1 ¥ /1 The Miller indices (1 -1 0 1)

Other equivalent (110) planes

z

(001)

Inter-planer Spacing a1

y

O

A (1101) a1

y x

A (1102)

_

(110) O

Other equivalent (111) planes

_

a2

y

x

c

2/3

z

(111)

O

_

a2 a3 c -1 ¥ 1/2 2 1/ 1/ ¥ /1 -1 -1 0 2) a1 a2 a3 c The intercepts -1 1/2 -1 ¥ 1 The reciprocals /-1 2/1 1/-1 1/¥ The Miller indices (-1 2 -1 0)

2

Cubic:

a1 The intercepts 1 1/ The reciprocals 1 The Miller indices (1

x

22

Other equivalent (001) planes

2

2

Tetragonal:

2

1 (h +k +l ) 2 = 2 d hkl a 2

2

(h +k ) l 1 = 2 + 2 2 a c d hkl

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Linear Atomic Densities

Family of Planes (hkl) is ONE Specific Plane {hkl} is a FAMILY of Equivalent Plane

LINEAR ATOMIC DENSITIES

LD =

a 2 a

a

z

(100)

a 2

(010)

a

a x (100)

y

a=2R

{100}

1 LD[110] = 2R 2 1 = 0.35 R

{100} = (100) (010) (001) (100) (010) (001)

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a=2 R

a=4 R

2

3

1 4R 2 1 = 0.30 R

LD[110] =

2 LD[110] = 2R 2 2 1 = 0 .50 R

3

27

Planar Atomic Densities

Family of Planes in Tetragonal System The (001) Plane in tetragonal system

The (100) Plane in tetragonal system

Lengthof direction

a 2

(001)

a

a

# of atoms centered on direction

PLANAR ATOMIC DENSITIES

PD =

# of atoms centered on a plane Areaof plane

c c

a a {100}

a a

a=2R

1 2R × 2 R 1 = 0 .25 2 R

PD (100) = {100} = (100) (010) (100) (010)

{001} = (001) (001) 26

a=4 R

PD (100) =

1 3 × 4R

4R

= 0.19

a=2 R

3

1 R

2

PD (100) = 3

2

2 2R 2 × 2R 2

= 0.25

1 R

2

28

Close-Packed Planes & Directions in fcc Metals Atoms “touch” along the <110> direction and on {111} plane

é ë 011 ù û é101 û ù ë

Close-Packed Crystal Structures

h=4 R

FCC and HCP metallic crystal structures both have atomic packing factors of 0.74, which is the most efficient packing of equal-sized spheres or atoms 3 2

= 2R 3

{111} <110>

(111)

The Closest Packing of Spheres in 3D p

é110 û ù ë

a=2 R LD [110] = LD [110] =

2R

PD (111) =

2 2

2R 2 2 2× 2R 2R 2 2

= 0.50

1 R

=1

2 4 3R

= 0.29

PD (111) =

Area = 2 R 3 × 2R = 4 3 R2

2

1 R

2 × π R2

Note: Each sphere is surrounded by 12 other spheres (6 in one plane, 3 above and 3 below).

2

4 3R = 0.907

29

é ë111 ù û

p p

a=4R

3

l = 2 ×4 R

3

31

Close-Packed Crystal Structures

Atoms “touch” along the <111> direction and on {110} plane

3

C B

In the valleys

[111]

a=4R

A

2

Close-Packed Planes & Directions in bcc Metals ( 110)

Two Possible Locations for following top Layer: n B-Positions OR n C-Positions

Valleys

Peaks

Area = 4 R / 3 × 2 × 4R / 3

There is only one place for the second layer of spheres. There are two choices for the third layer of spheres: n Third layer eclipses the first (ABAB arrangement). This is called hexagonal close packing (hcp); n Third layer is in a different position relative to the first (ABCABC arrangement). This is called cubic close packing (ccp).

= 16 2 R 2 / 3

LD[111] = LF [111] =

2 1 = 0.50 R 3 ×4R 3 2×2R =1 3 × 4R 3

2 2 16 2 R / 3 1 = 0 .265 2 R

PD (110) =

PD (110) =

2× π × R 2 2

16 2 R / 3 = 0 .833

30

32

Face-Centered Cubic Metal Crystal Structure

Close Packed Ceramic Structure

33

35

SINGLE CRYSTALS

HCP Metal Crystal Structure

For a crystalline solid, when the periodic and repeated arrangement of atoms is perfect or extends throughout the entirety of the specimen without interruption, the result is a single crystal.

CaF2

34

36

POLYCRYSTALLINE MATERIALS Most crystalline solids are composed of a collection of many small crystals or grains; such materials are termed polycrystalline.

X-RAY DIFFRACTION:DETERMINATION OF CRYSTAL STRUCTURES p

X-ray diffraction (X-ray crystallography): n X-rays are passed through the crystal and are detected on a photographic plate. n The photographic plate has one bright spot at the center (incident beam) as well as a diffraction pattern. n Each close packing arrangement produces a different diffraction pattern. n Knowing the diffraction pattern, we can calculate the positions of the atoms required to produce that pattern. n We calculate the crystal structure based on a knowledge of the diffraction pattern.

37

X-RAY DIFFRACTION:DETERMINATION OF CRYSTAL STRUCTURES

39

X-RAY DIFFRACTION:DETERMINATION OF CRYSTAL STRUCTURES

38

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