Muddiest  Points      Crystal  Structure:  Unit  Cell  Direc6on  

 Muddiest  Points:    •  “Figuring  out  how  to  change  the  origin  to  calculate    

 indices.”  

•    “What  the  purpose  and  meaning  of  the  unit  cell  direc6ons  are      and  why  they  are  important.”  

 •    “What  are  families  of  direc6ons?”    

Why are Crystal Structure Directions Important?

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Efficient  strong  Ni  single  crystal  turbine  blades  burn  less  fuel  at  higher  temperatures  because  blades  are  grown  on  [110]  close  packed  direc6on  FCC  Ni  

Computer  chips  come  from  Si  wafers  sliced  from  single  crystal  Si  grown  along  [100]  direc6on  

GaAs  lasers  in  CD  players  are  grown  from  GaAs  single  crystals  grown  along  [100]  direc6on  

Unit Cell Directions – Drawing Directions from Indices

Example: Given indices > draw direction [0 1 2] 1. Select origin O at 0, 0, 0 OR (choose new O’ & label axes X’ Y’ Z’) 2. Reduce indices to maximum value of 1 to keep direction within unit cell: ½[0 1 2] = 0, ½, 1 3. Locate from the origin at 0,0,0 to a position in the unit cell at x=0 + y=1/2 + z = 1 at 0, ½, 1 4. Draw direction from 0,0,0 to arrow tip at 0, ½, 1

Mistakes  0  0  1_______  ,  [1,2,3]  _______  ,  [-­‐1  -­‐2  3]  _______  ,  [1/2    2    3]________,  [2  4  6]_______  

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Selec>ng  the  Origin  

Rule:  If  there  is  a  bar  over  the  number  (x,  y  or  z)  ,  the  new  origin  O’  should  have  a  1  in  the  same  place  (x,  y,  or  z)  as  the  barred  number.  If  there  is  not  a  bar  over  the  number,  the  new  origin  O’  should  have  a  0  in  the  same  place  as  the  unbarred  number.    EXAMPLES:  

[  0  1  1  ]                -­‐  -­‐    Direc>on                          New  Origin  O’    

[  0  1  2  ]  

[  1  2  2  ]              -­‐    -­‐  

Unit Cell Directions – Specifying Indices from Directions

Given direction -> determine indices [? ? ?] 1. Create new origin O’ at tail, if necessary 2. Label O’ and new axes X’, Y’, and Z’ 3. Find coordinates from tail at O’ to head of direction x, y, z 4. Reduce multiples, eliminate fractions, use for negative Place square brackets [ ] around integers

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–  

o’  

Unit Cell Directions Drawing & Indexing

Draw  the  specified  direc6on  in  the  unit  cell  below                          

   [  1  2  2  ]  

1.  

2.  

3.  

4.  

-­‐  -­‐  

Unit Cell Directions Drawing & Indexing

Write  the  indices  for  the  given  direc6on  in  the  unit  cell  below:                                

   

1.  

2.  

3.  

4.  

Unit Cell Families of Directions

<1  1  1>  =    [  1  1  1],    [  1  1  1],  [  1  1  1],  [  1  1  1]  

                                                                                   _                                      _                                          _  

•  Families of directions in a unit cell –  Composed of unique unit cell directions that are "equivalent", –  In cubic system members of a family are given by all possible

permutations of indices –  Same properties, packing density, same environment, etc. –  Denoted by capital integers in "carrot brackets" < U V W >

Example  –  Specify  all  direc>ons  in  the  family  <1  1  1>  

Called  body  diagonals  

_   _  _  

Identify & Draw Unit Cell Families of Directions

Find all of the directions for the given family. Find all the indices, draw the directions, and give the name of the family.  

Called  ______________  

<  1  1  0  >  =  ________  ________  ________  ________  ________  ________  

Identify & Draw Unit Cell Families of Directions

Find all of the directions for the given family. Find all the indices, draw the directions, and give the name of the family.  

Called  ______________  

<  1  0  0  >  =  

Muddiest  Points      Crystal  Structure:  Unit  Cell  Direc6on  

 Muddiest  Points:    •  “Figuring  out  how  to  change  the  origin  to  calculate    

 indices.”  

•    “What  the  purpose  and  meaning  of  the  unit  cell  direc6ons  are      and  why  they  are  important.”  

 •    “What  are  families  of  direc6ons?”