MSE 250 L#5 Unit Cell Geometry - Planes

Topic #5 – Crystal Geometry II - Planes : 

Topic #5 – Crystal Geometry II - Planes Overview * Properties can vary significantly on different planes * Definition & examples of planar nomenclature – Miller indices * Drawing & indexing crystal planes * Calculation of planar packing density & angles between directions * Examples of property variations on different planes Unit Cell Directions * Miller indices * Nomenclature * Families of planes * Planar packing density * Angles between planes Effect of Planes on Processing and Properties * Metals, Polymers, Ceramics, Semiconductors Homework #5 – 4 problems, Reading: Callister 3.10 – 3.12 (8 pg.); Krause, Topic #5 (6 pg.) Topic #6 - Defects Read Callister 4.1 – 4.11 (25 pg.), 7.2-7.3 (5 pg.) ; Krause, Topic #6 (8 pg.) Today’s schedule 10:40-10:50 – Intuition quiz 10:50- 10:55 - Topic #4 wrap up & HW 10:55-11:15 - Mini-lecture 11:15-11:35 – Activity 11:35-11:45 – Activity report out 11:45-11:50 – Closure 11:50-11-55 – Homework strategies

Do Crystal Planes Affect You? : 

Do Crystal Planes Affect You? Metals deform on close packed planes in close packed directions. Single crystals form slip bands from this that can be seen on the surface with optical microsopy. Lasers are grown by vapor depositing one layer after another of different optoelectronic compounds Gems such as rhodochrosite and diamond grow on planes that reflect their internal crystal structure

Indexing Directions : 

Indexing Directions Indices are given by (h k l), lowest round bracketed [ ] integers, negatives are bars above # Invert indices; Go from O at 0,0,0 to mark x=1/h; back to O & out to y=1/k; back to O & out to 1/l Given indices > draw plane [0 1 2] Select origin O at 0, 0, 0 OR O’ X’ Y’ Z” Invert indices Mark intercepts at x= h, y= 1/1, z= 1/2 Draw plane by connecting intercepts Given plane > determine indices (? ? ?) Choose origin O in cell OR O’ with axes X’, Y’, & Z’ Find indices intercepts along axes Invert indices Reduce multiples, eliminate fractions, put – for – and place round brackets around integers Mistakes 0 0 1_______ ; (1,2,3) _______ ; (-1 -2 3) _______ ; (1/2 2 3)________; (2 4 6)_______

Activity 1 – Drawing & Indexing Directions : 

Activity 1 – Drawing & Indexing Directions Draw the specified planes in the unit cells below [1 2 2] [1 0 1] [1 1 1] - - - Determine indices of these unit cell planes

Families of PlanesActivity 2 – Identify {1 1 1} planes : 

Families of PlanesActivity 2 – Identify {1 1 1} planes {1 1 1} = ( 1 1 1), ( 1 1 1), ( 1 1 1), ( 1 1 1) _ _ _ Example - In any cubic system a family of planes is given by all possible permutations (including positive and negative) of the three integer indices specified by a family. { 1 1 1 } = ( 1 1 1 ); ( 1 1 1 ); ( 1 1 1 ); ( 1 1 1 ) { 1 0 0 } = { 1 1 0 } = Families of Planes in a unit cell - these are composed of planes in a unit cell which are "equivalent", ie. they have the same packing density, same environment, etc. They are denoted by capital integers in "squiggly brackets" { H K L }

Activity 3 - Planar Density of Atoms : 

Activity 3 - Planar Density of Atoms Planar packing density - the number of atoms / unit area as measured by counting fractions of circle area on a given plane within a unit cell. Note - only atoms whose exact centers lie on the plane are included. Example - For FCC Al (a = 0.41 nm) give the planar packing density on ( 1 0 0 ); ( 1 1 0 1 ); ( 1 1 1 ) Label the dimensions areas shown, fill in atoms, calculate the planar density (atoms/nm2)

Activity 4 - Angles Between Planes : 

Activity 4 - Angles Between Planes Angles between directions (cubic system) To determine the angle between any two directions there are two approaches that can be used: Example: Determine the angle between (1 1 0) and (1 1 1) 1. Change planar indices to direction indices & solve by inspection and geometry 2. Solve by solving the dot product of the indices of the two planes, since these indices are also the indices of directions perpendicular to the planes, and the angle between these perpendiculars is also the angle between the planes. -

Big Ideas about Crystal Planes : 

Big Ideas about Crystal Planes - - - -

Homework – Topic #5 – Crystal Geometry II - Planes : 

Homework – Topic #5 – Crystal Geometry II - Planes 1a. Draw the specified panes in the unit cells below [1 2 2] [1 0 1] [1 1 1] - - - 1b. Determine indices of these unit cell planes 2. For BCC iron, calculate the planar packing density on (1 0 0), (1 1 0), and (1 1 1) planes. 3. For a cubic unit cell calculate the angle between ( 1 1 1 ) and ( 1 1 1 ) . 4a. Cu has electrical conductivity that is 4 times greater than Ni, but when 10% Cu is added to Ni, instead of electrical conductivity increasing, it actually decreases by 50%. Take a guess at explaining why this is so. 4b. Ni has yield strength that is 4 times as great as Cu but, when 10% Cu is added to Ni, instead of the yield strength decreasing, it actually increases 25%. Take a guess at explaining why this is so.