mseasuslides: muddiest point: miller indices and planes slide set
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Muddiest Points Crystal Structure III: Miller Indices of Planes in Unit Cells
Muddiest Points: • “Why are the crystallographic planes important?”
• “Determining new origin”; “How to draw planes.”
• “Determining families of planes.”
• “Drawing planar packing density diagrams.”
• “When to use carrot/curly brackets.”
How Do Crystal Planes Affect You?
Metals deform on close packed planes in close packed direcEons.
Lasers are grown by vapor deposiEng layer by layer of atoms on a specific plane
Gems such as rhodochrosite and diamond grow on planes that reflect crystal structure
Planes - Given plane > determine indices
Given Plane > Determine Indices (? ? ?) 1. Choose origin O in unit cell OR a new O’ with axes X’, Y’, & Z’ 2. Find intercepts from O to a intercept; from O to b int.; from O to c int. 3. Take reciprocals of intercept position 4. Clean up a) Reduce multiples;
b) Eliminate fractions; c) Put – above negative integer in indices; d) place round brackets around integers
Mistakes 0 0 1______ ; (1,2,3) ______ ; (-‐1 -‐2 3) ______ ; (1/2 2 3)______; (2 4 6)______
X
Y
Z
O
( 1 1 0 )
1)
2)
3)
4) a b
c 1. Choose Origin
Unit Cell a b c 2. Intercept
3a. Reciprocal
3b. Reciprocal Value
4. Miller Indices
Determine the indices for the given plane.
X
Y
Z
O
( )
1)
2)
3)
4)
1. Choose Origin
Unit Cell a b c
2. Intercept
3a. Reciprocal
3b. Reciprocal Value
4. Miller Indices
Determine the indices for the given plane.
AZ
Y
X
1. Choose Origin
Unit Cell a b c
2. Intercept
3a. Reciprocal
3b. Reciprocal Value
4. Miller Indices
Planes - Given indices > draw plane
Given indices > Draw Plane; example (0 1 1)
1. Select origin O at 0, 0, 0 (OR if there are negatives at O’ = X’, Y’, Z’) 2. Take reciprocal of indices to get 1/h, 1/k, 1/l: which will be the intercepts.
3. Mark intercepts along x, y, & z axes at a= 1/0 = ∞; b= 1/1; c= 1/1 4. Draw plane by connecting intercepts
X
Y
Z
O
( 0 1 2 )
1)
2)
3)
4)C6
( 0 1 1 ) 1. Choose Origin
2a. Reciprocal
2b. Reciprocal Value
Unit Cell a b c
3. Mark Intercepts
4. Draw
X
Y
Z
O
( 1 1 0 )
1)
2)
3)
4)
Determine the plane for the given indices. ( 1 0 0 )
1. Choose Origin
2a. Reciprocal
2b. Reciprocal Value
Unit Cell a b c
3. Mark Intercepts
4. Draw
Families of Planes
{1 1 1} = ( 1 1 1), ( 1 1 1), ( 1 1 1), ( 1 1 1) _ _ _
For cubic system, indices of a family of planes is given by: all permutaEons (+ and -‐) of three integer indices. { 1 1 1 } = ( 1 1 1 ); ( 1 1 1 ); ( 1 1 1 ); ( 1 1 1 )
X
Y
Z
O
Families of Planes -‐ are "equivalent", • same packing density, • same environment, etc. • denoted by capital integers in “curly brackets" { H K L }
_ _ _
X
Y
Z
O
( 1 1 0 )
1)
2)
3)
4)
{ 1 1 1 } Family
X
Y
Z
O
( 1 1 0 )
1)
2)
3)
4)
{ 1 0 0 } Family
X
Y
Z
O
( 1 1 0 )
1)
2)
3)
4)
{ 1 1 0 } Family
Planar Packing Density of Atoms – FCC
Planar packing density -‐ number atoms within plane / unit area of plane
Count fracEons of circle area on plane only when atom centers sit exactly on plane
Ex.-‐ For FCC give planar packing density on ( 1 0 0 ); ( 1 1 0 ); ( 1 1 1 )
Label dimensions of areas shown, fill in atoms, calculate planar density in terms of a0 (atoms/a02).
X
Y
Z
O
X
Y
Z
O
X
Y
Z
O
C12
a
a a
a
a a
a
a a
Area = a02 Area = 21/2 a02 Area = (31/2/2) a02
a
a a
Crystal Spatial Visualization Survey
ClassID#:
PLEASE COMPLETE OTHER SIDE ALSO
Crystal Spatial Visualization Survey
ClassID#:
PLEASE COMPLETE OTHER SIDE ALSO
Crystal Spatial Visualization Survey
ClassID#:
PLEASE COMPLETE OTHER SIDE ALSO
Crystal Spatial Visualization Survey
ClassID#:
PLEASE COMPLETE OTHER SIDE ALSO
Draw Planar Packing Density for BCC on ( 1 1 1 ) Plane.
What do the Different Brackets Mean?
Muddiest Points Crystal Structure III: Miller Indices of Planes in Unit Cells
Muddiest Points: • “Why are the crystallographic planes important?”
• “Determining new origin”; “How to draw planes.”
• “Determining families of planes.”
• “Drawing planar packing density diagrams.”
• “When to use carrot/curly brackets.”
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