LECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes.

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Chapter 3:

Lattice Positions, Directions and Planes

LECTURE #05

Learning Objective

• To describe the geometry in and around a

unit cell in terms of directions and planes.

1

• Pages 64-83.

Relevant Reading for this Lecture...

Why are Crystal Planes &

Directions Important?

• Materials structures and

properties are related to them!

• Crystals deform on specific

planes and directions – they don’t

just ‘break’ randomly

just break randomly

VIDEO

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Why are Crystal Planes & Directions Important?

• Materials structures and properties are related to

them!

_di _ti After

them!

stress direction plane

• Crystals deform on specific planes and directions –

They don’t just ‘break’ randomly

3

How do we express Planes and Directions in Crystals ?

• We use Miller indices: –hkloruvware the generic letters we use.

– hklanduvware called indices. They will be numbers that are related to coordinate systems.

– No commas between the numbers.

• hrepresents the plane perpendicular to the x-axis; krepresents the plane perpendicular to the y-axis; lrepresents the plane perpendicular to the z-axis.

• urepresents the vector parallel to the x-axis; vrepresents the vector parallel to the y-axis; wrepresents the vector parallel to the z-axis.

x y

z

– Negative values are expressed with a bar over the number • Ex.: -3 is expressed as (“bar 3”)

• Crystallographic directions:

– ; one step in +x dir.; one step in –y dir.; one step in +z dir. – ; zero step in x dir.; one step in +y dir.; zero step in z dir.

3

[1 11] [010]

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Point Coordinates

Express as fractions of unit vectors

z

c

1,1,1

Express as fractions of unit vectors.

Point coordinates for unit cell corner

are

1,1,1

(or

a, b, c

)

x

y

a

b

000

What about this position?

1

1, ₂,0 (or ,_ab₂,0)

• For the above 1,1,1 coordinate – what is its direction?

5

Let’s

Let’s Relate Miller Indices to

Relate Miller Indices to Vectors

Vectors

“My crimes have both directions and magnitude.” Vector from Despicable Me

Where is the origin position in a crystal? What do you look for? Representation of a lattice and unit cell

Vector – points in a specific direction (hence you need an origin) Vector – has a unit of length or magnitude

We will use vectors to define directions and lengths in crystal systems

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Directions in Crystals

Directions and their multiples are identical

x z _y [020] [030] [110] [220] Ex.: [220] 2 [110]  [330] x [010]

(“Translational Symmetry”) Translation: integer multiple of lattice constants _{identical position in another unit cell: (111), (222),}

(333), etc. 7

How to apply Miller Indices for

How to apply Miller Indices for Directions

Directions

• Draw vectorand define the tail as the origin. D t i th l th f th z 1 2 1 2 1 0 0 0 1 1 0 2 0 ] 2 [ 1     • Determine the length of the vector projectionin unit cell dimensions – a, b, and c. • Remove fractionsby multiplying by the smallest possible factor. [111] y [021] P Example in class • Enclosein square brackets • In cubic crystals, directions and their negatives are equivalent but NOT the same. [110] x 1 2 1₂ 0 1 0 0 0 0 1 [0 2 1] 2    Origin (tail) a b c Point P (head) 8

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z

In class example #1:

What are the indices of the line/vector connecting points O and P? What are the indices of the line connecting points Q and R?

y P O Q R x 9 z 1 2 1 2 0 1 0 0  Origin O Point P

In class example #1: SOLUTION

a b c

What are the indices of the line/vector connecting points O and P? What are the indices of the line connecting points Q and R?

R y 1 1 2 1 2 [1 2 1] 2    a b c Q P O x 3 1 2 4 3 1 2 4 0 0 0 1 1 [2 4 3] 4       Origin Q Point R 10

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z

In class example #2:

In a cubic unit cell, draw correctly a vector with indices [146].

y

O

x

11

z

In class example #2: SOLUTION

In a cubic unit cell, draw correctly a vector with indices [146].

Select your origin. Put it wherever you want to.

This step is the opposite of clearing fractions! O y [1 4 6] . 6

These fractions denote how far to step in the

4 6 1 6 6 6 indices Div by 1 6 4 2 63 6 61 x , , or directions (away from the origin).

x y z

NOTE: It would be “wise” to select the origin so that you can complete the desired steps within the cell that you are using!

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Families of Directions

• In

cubic systems

, directions that have the same indices are

equivalent regardless of their order or sign.

[001] _[100] z [010] x z y [0 10] [10 The 0], famil [ 100] y of d [010], irections is: [0 10 [001], [00 1] ] < 100 > [100] _{[00 1]} We enclose indices in carats rather than brackets

to indicate a family of directions

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Families of Directions

• In

non‐cubic systems

, directions that have the same indices

are not necessarily equivalent.

ORTHORHOMBIC b z CUBIC b z [010] a  b  c [010] [001] x y b a c z [010] a = b = c [010] [001] a a a x y [010] TETRAGONAL a  b  c [010] [001] x y a a c 14

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Crystallographic Planes

Adapted from Fig. 3.9, Callister 7e.

A specific direction is normal (90o_{) to its}

specific, equivalent plane. For example [100] is normal to (100) but [100] is not normal to (010) 15

Miller Indices for Planes

 Specific crystallographic plane: (hkl)

 Family of crystallographic planes: {hkl}

– (hkl), (lkh), (hlk) … etc. – In cubic systems, planes having the same indices are equivalent regardless of order or sign AND directions are normal to the planes

of order or sign. AND directions are normal to the planes

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1. Identify the coordinate intercepts of the plane (i.e., the coordinates at which the plane intersects the x, y, and z axes).

PROCEDURES FOR INDICES OF PLANES

(Miller indices)

 If plane is parallel to an axis (DOES NOT INTERSECT IT), the intercept is taken as infinity ().

 If the plane passes through the origin, consider an equivalent plane in an adjacent unit cell or select a different origin for the same plane. 2. Take reciprocalsof the intercepts.

3. Clear fractionsto the lowest integers.

4. Cite specific planes in parentheses, (h k l), placing bars over negative indices.

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MILLER INDICES FOR A SINGLE PLANE

z x y z y Intercept _ ₁ _ Reciprocal _1/ _1/1 _1/ Clear ₀ ₁ ₀ INDICES ₀ ₁ ₀

(010)

x

The cube faces are from the {100} family of planes

(100), (010), (001), ( 100), (0 10), (00 1),

(

)

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MILLER INDICES FOR A SINGLE PLANE

MILLER INDICES FOR A SINGLE PLANE –

– cont’d

cont’d

z x y z y Intercept ₁ ₁ _ Reciprocal _1/1 _1/1 _1/ Clear ₁ ₁ ₀ INDICES 1 1 0

(110)

x

The {110} family of planes

(110), (011), (101), ( 1 10), (0 1 1), ( 10 1)

( 110), (1 10), ( 101), (10 1), (01 1), (0 11)

(

)

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MILLER INDICES FOR A SINGLE PLANE

MILLER INDICES FOR A SINGLE PLANE –

– cont’d

cont’d

z x y z 2 2 0 2 2 0 2/1 2/1 1/ 1/2 1/2  y Intercept Reciprocal Clear INDICES

(220)

x

(

)

20

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Crystallographic Planes

example

1. Intercepts

1/2 1 3/4

a

b

c

/(½)

/

/(¾)

z

c

4. Miller Indices (634)

2. Reciprocals

1/(½) 1/1 1/(¾)

2

1 4/3

3. Reduction

6 3 4

x

y

a

b

c

   21

General Rules for Crystal Directions, Planes, and Miller Indices

• x, y, and z are the axes (on an arbitrary origin).

– In some crystal systems the axes

Unit cell

In some crystal systems the axes are not mutually perpendicular. • a, b, c and α, β, γ are lattice

parameters.

– length of unit cell along side of

unit cell. _a

c  

 • h, k, l are the Miller indicesfor planes

and directions. – Ex., (hkl) and [hkl]

b

Geometry of a general unit cell

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Comment: HCP Crystallographic Directions

z

• In general we can define

Miller indices just like we

d f

th

t l

a

c

x

do for the other crystals.

• However, sometimes in

engineering practice, a

4-indice system is used.

120° (a₁)

_a

a = b ≠ c α = β = 90°; γ = 120°

a

y

• It is called Miller-Bravais

indices. There are

equations to convert.

(a₁) (a₂) 23

Comment: HCP Crystallographic Directions

• Miller‐Bravais indices (i.e., uvtw) are related to the direction

indices (i.e.,

UVW

) as follows.

[

UVW

]



[

uvtw

]













1 ₂

3

1

2

3 u

U

V

v

V

U

t

U

V











I WILL NOT TEST YOU ON THIS!!!

I only show it because some of you will end up working with

hexagonal metals like Ti or Mg after you graduate.





t

U

V

w

W

 





Fig. 3.8(a), Callister 7e.

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Comment: HCP Crystallographic Directions

DIRECTIONS a₃ a₃ (uvtw) (UVW) a₁ a2 a₁ a2 [120] [110] [100 ] [ 210] [110 ] [10 10] [2 1 10] [1120] [1100] [0 110]

I WILL NOT TEST YOU ON THIS!!!

I only show it because some of you will end up working with

hexagonal metals once you graduate.

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• Miller Directions

– Indices for a direction are enclosed in square brackets. – Negative values are expressed with a bar over the number.

An example of a Miller Direction:

Summary

[1 11]

– An example of a Miller Direction:

one step in +x dir.; one step in –y dir.; one step in +z dir. • Miller Planes

– Intercepts for a specific crystallographic plane are enclosed in parenthesis.

– When identifying the coordinate intercepts of the plane (i.e., the coordinates at which the plane intersects the x, y, and z axes):

[1 11]

 If plane is parallel to an axis (DOES NOT INTERSECT IT), the intercept is taken as infinity ().

 If the plane passes through the origin, consider an equivalent plane in an adjacent unit cell or select a different origin for the same plane.