Crystal Directions And Planes 413y6b

Crystallographic Directions, and Planes Now that we know how atoms arrange themselves to form crystals, we need a way to identify directions and planes of atoms. •Why? 9 Deformation under loading (slip) occurs on certain crystalline planes and in certain crystallographic directions. Before we can predict how materials fail, we need to know what modes of failure are more likely to occur. 9 Other properties of materials (electrical conductivity, thermal conductivity, elastic modulus) can vary in a crystal with orientation. MECH 221

PM Wood-Adams

Fall 2008

Crystallographic Planes & Directions

direction

plane

• It is often necessary to be able to specify certain directions and planes in crystals. • Many material properties and processes vary with direction in the crystal. • Directions and planes are described using three integers Miller Indices MECH 221

PM Wood-Adams

Fall 2008

Point coordinates • Point position specified in of its coordinates as fractional multiples of the unit cell edge lengths Z 111

000 0 .5 0

Y

X MECH 221

PM Wood-Adams

Fall 2008

Example Find the Miller indices for the points in the cubic unit cell below:

I J

K

Note: J is on the left face of the cube, H is on the right face, K is on the front face and I is on the back face MECH 221

PM Wood-Adams

Fall 2008

General Rules for Lattice Directions, Planes & Miller Indices • • • •

Miller indices used to express lattice planes and directions x, y, z are the axes (on arbitrarily positioned origin) a, b, c are lattice parameters (length of unit cell along a side) h, k, l are the Miller indices for planes and directions expressed as planes: (hkl) and directions: [hkl]

• Conventions for naming – There are NO COMMAS between numbers

– Negative values are expressed with a bar over the number

• Example: -2 is expressed 2 MECH 221

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• Crystallographic direction: – [123] – [100] – … etc. Fall 2008

Miller Indices for Directions Method – Draw vector, and find the coordinates of the head, h1,k1,l1 and the tail [???] h2,k2,l2.

z

[111]

– subtract coordinates of tail from coordinates of head

y

– Remove fractions by multiplying by smallest possible factor – Enclose in square brackets

[100] [110] x

– What is ??? MECH 221

PM Wood-Adams

Fall 2008

Example - Naming Directions

MECH 221

PM Wood-Adams

Fall 2008

Families of Directions • Equivalence of directions

[101] ≠ [110]

[101] = [110]

tetragonal

cubic •

<123> Family of directions ¾ [123], [213], [312], [132], [231], [321] – only in a cubic crystal In the cubic system directions having the same indices regardless of order or sign are equivalent. MECH 221

PM Wood-Adams

Fall 2008

Miller Indices for Planes • (hkl) Crystallographic plane • {hkl} Family of crystallographic planes – e.g. (hkl), (lhk), (hlk) … etc. In the cubic system planes having the same indices regardless of order or sign are equivalent

• Hexagonal crystals can be expressed in a four index system (u v t w) – Can be converted to a three index system using formulas MECH 221

PM Wood-Adams

Fall 2008

Miller Indices for PLANES z Method • If the plane es through the origin, select an equivalent plane or move the origin • Determine the intersection of the plane with the axes in of a, b, and c • Take the reciprocal (1/∞ = 0) • Convert to smallest integers x (optional) Intercepts • Enclose by parentheses Reciprocals

(111)

y

x

y

z

1 1

1 1

1 1

see example 3.8 MECH 221

PM Wood-Adams

Fall 2008

Crystallographic Planes z z

(011)

(001)

y y x x z

z

(212)

(201)

y

y x

x MECH 221

Green circles show where the origins have been placed.

PM Wood-Adams

Fall 2008

Planes and their negatives are equivalent z

(0 1 0)

(010) y

x

MECH 221

PM Wood-Adams

Fall 2008

In the cubic system, a plane and a direction with the same indices are orthogonal z

y

x

MECH 221

PM Wood-Adams

Fall 2008

Linear and Planar density • Linear Density – Number of atoms per length whose centers lie on the direction vector for a specific crystallographic direction. # of atoms centered on a direction vector LD = length of direction vector

• Planar Density – Number of atoms per unit area that are centered on a particular crystallographic plane. # of atoms centered on a plane PD = area of plane

MECH 221

PM Wood-Adams

Fall 2008

Example • Find the linear density of the [110] and the [100] direction in the FCC cell in of the atomic radius R

[100]

[110] MECH 221

PM Wood-Adams

Fall 2008

Linear and Planar Density • Why do we care? – Properties, in general, depend on linear and planar density.

• Examples: Speed of sound along directions – Slip (deformation in metals) depends on linear and planar density – Slip occurs on planes that have the greatest density of atoms in direction with highest density (we would say along closest packed directions on the closest packed planes) MECH 221

PM Wood-Adams

Fall 2008