Crystallographic Points, Directions, and Planes. Points, Directions, and Planes in Terms of Unit Cell Vectors

ISSUES TO ADDRESS... All periodic unit cells may be described via these vectors and angles, if and only if • How to define points, directions, planes, as well as cv • a, b, and c define axes of a 3D coordinate system. linear, planar, and volume densities • coordinate system is Right-Handed! v b But, we can define points, directions and – Define basic terms and give examples of each: planes with a “triplet” of numbers in units • Points (atomic positions) av of a, b, and c unit cell vectors. • Vectors (defines a particular direction - normal) • Miller Indices (defines a particular plane) For HCP we need a “quad” of numbers, as • relation to diffraction we shall see. • 3-index for cubic and 4-index notation for HCP

MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08

POINT Coordinates Crystallographic Directions

To define a point within a unit cell…. Procedure: 1. Any line (or vector direction) is specified by 2 points. Express the coordinates uvw as fractions of unit cell vectors a, b, and c c • The first point is, typically, at the origin (000). (so that the axes x, y, and z do not have to be orthogonal).

b 2. Determine length of vector projection in each of 3 axes in a units (or fractions) of a, b, and c. pt. coord. • X (a), Y(b), Z(c) pt. x (a) y (b) z (c) 1 1 0 v c 0 0 0 3. Multiply or divide by a common factor to reduce the lengths to the smallest integer values, u v w. v 1 0 0 b 4. Enclose in square brackets: [u v w]: [110] direction. 1 1 1 origin 1/2 0 1/2 5. Designate negative numbers by a bar [ 1 1 0] av • Pronounced “bar 1”, “bar 1”, “zero” direction. 6. “Family” of [110] directions is designated as <110>.

DIRECTIONS will help define PLANES (Miller Indices or plane normal).

MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08

1 Self-Assessment Example 1: What is crystallographic direction? Self-Assessment Example 2:

Magnitude along (a) What is the lattice point given by point P? X Along x: 1 a c − 112

b Along y: 1 b Y a (b) What is crystallographic direction for the origin to P? Along z: 1 c Z [ 1 12]

DIRECTION = [1 1 1] Example 3: What lattice direction does the lattice point 264 correspond?

The lattice direction [132] from the origin.

MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08

Symmetry Equivalent Directions Families and Symmetry: Cubic Symmetry

z z Note: for some structures, different directions can be equivalent. (010) Rotate 90o about z-axis e.g. For cubic , the directions are all y equivalent by symmetry: y (100)

[1 0 0], [ 1 0 0], [0 1 0], [0 1 0], [0 0 1], [0 0 1 ] x x z

Rotate 90o about y-axis (001)

Families of crystallographic directions e.g. <1 0 0> Symmetry operation can y generate all the directions within in a family. Similarly for other Angled brackets denote a family of crystallographic directions. x equivalent directions

MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08

2 Designating Lattice Planes How Do We Designate Lattice Planes?

Example 1 Why are planes in a lattice important? Planes intersects axes at: (A) Determining • a axis at r= 2 * Diffraction methods measure the distance between parallel lattice planes of . • b axis at s= 4/3 • This information is used to determine the lattice parameters in a crystal. • c axis at t= 1/2 * Diffraction methods also measure the angles between lattice planes. How do we symbolically designate planes in a lattice? (B) Plastic deformation * Plastic deformation in metals occurs by the of atoms past each other in the crystal. * This slip tends to occur preferentially along specific crystal-dependent planes.

(C) Transport Properties Possibility #1: Enclose the values of r, s, and t in parentheses (r s t) * In certain materials, atomic structure in some planes causes the transport of electrons Advantages: and/or heat to be particularly rapid in that plane, and relatively slow not in the plane. • r, s, and t uniquely specify the plane in the lattice, relative to the origin. • Parentheses designate planes, as opposed to directions given by [...] • Example: Graphite: heat conduction is more in sp2-bonded plane. Disadvantage: • Example: YBa2Cu3O7 superconductors: Cu-O planes conduct pairs of electrons • What happens if the plane is parallel to --- i.e. does not intersect--- one of the axes? (Cooper pairs) responsible for superconductivity, but perpendicular insulating. • Then we would say that the plane intersects that axis at ∞ ! + Some lattice planes contain only Cu and O • This designation is unwieldy and inconvenient.

MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08

How Do We Designate Lattice Planes?

Self-Assessment Example Planes intersects axes at: • a axis at r= 2 • b axis at s= 4/3 What is the designation of this plane in Miller Index notation? • c axis at t= 1/2

How do we symbolically designate planes in a lattice?

Possibility #2: THE ACCEPTED ONE What is the designation of the top face of the unit cell in Miller Index notation? 1. Take the reciprocal of r, s, and t. • Here: 1/r = 1/2 , 1/s = 3/4 , and 1/r = 2 2. Find the least common multiple that converts all reciprocals to integers. • With LCM = 4, h = 4/r = 2 , k= 4/s = 3 , and l= 4/r = 8 3. Enclose the new triple (h,k,l) in parentheses: (238) 4. This notation is called the Miller Index.

* Note: If a plane does not intercept an axes (I.e., it is at ∞), then you get 0. * Note: All parallel planes at similar staggered distances have the same Miller index.

MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08

3 Families of Lattice Planes Crystallographic Planes in FCC: (100)

Given any plane in a lattice, there is a infinite set of parallel lattice planes z (or family of planes) that are equally spaced from each other. • One of the planes in any family always passes through the origin.

The Miller indices (hkl) usually refer to the plane that is nearest to the origin without passing through it.

• You must always shift the origin or move the plane parallel, otherwise a Miller index integer is 1/0! y • Sometimes (hkl) will be used to refer to any other plane in the family, or to the family taken together. Distance between (100) planes d100 = a

• Importantly, the Miller indices (hkl) is the same vector Look down this direction a as the plane normal! x … between (200) planes d200 = (perpendicular to the plane) 2

MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08

Crystallographic Planes in FCC: (110) Crystallographic Planes in FCC: (111)

z Look down this direction (perpendicular to the plane)

y Distance between (110) planes a 2 d = a 3 110 2 Distance between (111) planes d = x 111 3

MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08

4 Comparing Different Crystallographic Planes

Distance between (110) planes a a a 2 d110 = = = Note: similar to crystallographic directions, planes that are parallel to -1 12 + 12 + 02 2 2 each other, are equivalent 1

For (220) Miller Indexed planes you are getting planes at 1/2, 1/2, ∞. The (110) planes are not necessarily (220) planes!

For cubic crystals: Miller Indices provide you easy measure of distance between planes.

MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08

Directions in HCP Crystals Directions in HCP Crystals: 4-index notation

What is 4-index notation for vector D? 1. To emphasize that they are equal, a and b is changed to a1 and a2. Example 2. The unit cell is outlined in blue. • Projecting the vector onto the basal plane, it lies 3. A fourth axis is introduced (a3) to show symmetry. between a1 and a2 (vector B is projection). • Symmetry about c axis makes a3 equivalent to a1 and a2. • Vector addition gives a3 = –( a1 + a2). • Vector B = (a1 + a2), so the direction is [110] in 4. This 4-coordinate system is used: [a1 a2 –( a1 + a2) c] coordinates of [a1 a2 c], where c-intercept is 0.

• In 4-index notation, because a3 = –( a1 + a2), the vector B is 1[112 0 ] since it is 3x farther out. 3 • In 4-index notation c = [0001], which must be a 2 added to get D (reduced to integers) D = [1123] –2a 3 B without 1/3 Check w/ Eq. 3.7 or just use Eq. 3.7 Easiest to remember: Find the coordinate axes that straddle the vector

of interest, and follow along those axes (but divide the a1, a2, a3 part of vector by 3 because you are now three times farther out!).

Self-Assessment Test: What is vector C?

MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08

5 Directions in HCP Crystals: 4-index notation Miller Indices for HCP Planes

Example Check w/ Eq. 3.7: a dot-product projection in hex coords. 4-index notation is more important for planes in HCP, in order to distinguish similar planes rotated by 120o. What is 4-index notation for vector D? As soon as you see [1100], you will know • Projection of the vector D in units of [a1 a 2 c] gives u’=1, v’=1, and w’=1. Already reduced integers. that it is HCP, and not [110] cubic! t • Using Eq. 3.7: Find Miller Indices for HCP: 1 1 u = [2u'−v'] v = [2v'−u'] w =w' 3 3 1. Find the intercepts, r and s, of the plane with any two of the basal plane axes (a , a , or a ), as well as the 1 1 1 1 1 2 3 u = [2(1) −1] = v = [2(1) −1] = w =w'= 1 intercept, t, with the c axes. 3 3 3 3 2. Get reciprocals 1/r, 1/s, and 1/t. 3. Convert reciprocals to smallest integers in same ratios. 1 1 2 • In 4-index notation: [ 1] 4. Get h, k, i , l via relation i = - (h+k), where h is 3 3 3 r s associated with a1, k with a2, i with a3, and l with c. 5. Enclose 4-indices in parenthesis: (h k i l) . • Reduce to smallest integers: [ 112 3]

After some consideration, seems just using Eq. 3.7 most trustworthy.

MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08

Miller Indices for HCP Planes Yes, Yes….we can get it without a3!

What is the Miller Index of the pink plane? 1. The plane’s intercept a1, a2 and c at r=1, s=–1/2 and t= ∞, respectively.

1. The reciprocals are 1/r = 1, 1/s = –2, and 1/t = 0. 1. The plane’s intercept a1, a3 and c at r=1, s=1 and t= ∞, respectively. 2. They are already smallest integers. 1. The reciprocals are 1/r = 1, 1/s = 1, and 1/t = 0. 3. We can write (h k i l) = ( 12 ?0) 2. They are already smallest integers. 4. Using i = - (h+k) relation, i=1. 3. We can write (h k i l) = (1 ? 1 0). 5. Miller Index is (12 10) 4. Using i = - (h+k) relation, k=–2.

5. Miller Index is (1210) But note that the 4-index notation is unique….Consider all 4 intercepts: • plane intercept a , a , a and c at 1, –1/2, 1, and , respectively. 1 2 3 ∞ • Reciprocals are 1, –2, 1, and 0. • So, there is only 1 possible Miller Index is ( 12 10)

MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08

6 Basal Plane in HCP Another Plane in HCP

Name this plane… z

• Parallel to a1, a2 and a3 • So, h = k = i = 0 • Intersects at z = 1 plan e = (0001)

a 2 a2

+1 in a a 1 3 a3

-1 in a2 a (1 1 0 0) plane 1 a1 h = 1, k = -1, i = -(1+-1) = 0, l = 0

MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08

z SUMMARY (1 1 1) plane of FCC

• Crystal Structure can be defined by space lattice and basis atoms (lattice decorations or motifs).

y • Only 14 Bravais Lattices are possible. We focus only on FCC, HCP, and BCC, i.e., the majority in the periodic table.

x • We now can identify and determined: atomic positions, atomic planes SAME THING!* z (0 0 0 1) plane of HCP (Miller Indices), packing along directions (LD) and in planes (PD).

• We now know how to determine structure mathematically. So how to we do it experimentally? DIFFRACTION.

a2

a3

a1 MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08 MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08

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