Scanning Electron Microscopy Image of a Ruthenium-Palladium-Aluminium-Alloy MSE 250 Friday, Jan 10, 2003 Quiz next Friday Crystal Structure Professor Dave Grummon 3525 Engineering Building

5f0

5g4

r pair of atoms separated by a distance r

r F attr = F repul; F net = 0

r o is the “equilibrium interatomic spacing”: here, the NET force is zero At r o, the energy is at a minimum. Still, E o is thought of as the “bond energy” (relative to zero).

Calcite

High Temperature Superconductor YBa 2 Cu 3 O 7

ICE

Garnet

Simple Cubic Structure (rarely seen) “reduced sphere” model “hard-sphere” space-filling model

We’ll begin building a three-dimensional crystal with the construction of a Close-Packed plane of atoms, here modeled as hard spheres (with radius roughly determined by the first quantum number).

Adding a second close-packed layer nestled into the first, packed as closely as possible.

A B C

A A A B B... ABABAB... Stacking of the close-packed layers produces the HCP hexagonal close-packed structure

A B C A B C A B... ABCABCABC... stacking gives the FCC or face-centered cubic structure.

How the A, B and C planes are arranged in the fcc cubic structure. A B C

Interstices, or interstitial sites.

The body centered cubic (bcc) structure: it’s NOT close-packed.

The unit cell is a convenient representation, but metals are NOT molecular, and the U.C. isn’t a molecule. Remember to visualize the near infinite extent of the lattice, beyond the confines of the unit cell. bcc: hard sphere model reduced spheres unit cell (the lines are really meaningless...) small fragment of a real crystal

FCC

HCP

R = 2 r o a is the “lattice parameter” a = 4R / √ 2

A close-packed direction: hard spheres touch.

The bcc structure has no close-packed planes: this is the most densely packed plane. It does, however, have close packed directions, A’- E’, D’-B’, etc.

This plane in the fcc structure is not close-packed. Note, however, the close packed directions in the plane

The most general unit cell can have several lattice parameters.

Crystal coordinates (q,r,s) are always expressed in a coordinate system based on the unit cell geometry

1: 0,0,0 2: 1,0,0 3: 1,1,0 4: 0,1,0 5: 1/2, 1/2, 1/2 6: 0,0,1 7: 1,0,1 8: 1,1,1 9: 0,1,1 Note that the coordinate system is right-handed.

We need a system to describe important directions in the lattice

coordinates: (+1/2, +1/2, -1) Direction: [112] coordinates: (+0, +1/2, +1) Direction: [012]

+1/2,+1,+1/2 [121] Get the direction by choosing the origin so the line passes through it, finding the crystal coordinates of any point on the line, clearing the fractions, and enclosing in square brackets

-1,+1,+0 [110] We can choose any point to serve as the origin

+1/2,-1,+1/2 [121]

+0,-1/2,-1 [021]

(-2/3,0,-1/2) x 6 = [403]

(-1/2,+1/2,-1/2) x 6 = [111]

(+1,+1,-1/3) x 6 = [331]

We also have a system to describe planes: Here, the plane must NOT pass through the origin We look for the points on the three cardinal axes where the plane intersects each axis. These are called the intercepts. The Miller indices for the plane are taken as the reciprocals of the intercepts, fractions cleared, and enclosed in plain brackets.

x-intercept: ∞ y-intercept: -1 z-intercept: +1/2 invert: 0,-1,+2 Miller indices: (012)

x-intercept: ∞ y-intercept: +1/2 z-intercept: ∞ invert: 0,2,0 Miller indices: (010) x-intercept: 1/2 y-intercept: -1/2 z-intercept: 1 invert: 2,-2,1 Miller indices: (221)

A: x-intercept: 1/2 y-intercept: -1/2 z-intercept: ∞ invert: 2,-2,0 Miller indices: (110) B: x-intercept: 1 y-intercept: +1/2 z-intercept: +1/2 invert: 1,2,2 Miller indices: (122)

YOU DO THESE: YOU HAVE 60 SECONDS....

(230) (111)

Are you ready?....

(211) (021)