This slide set corresponds to the MaterialsConcepts YouTube video Muddiest Point- Miller Indices and Planes . It walks you through various calculations involving Miller Indices and Planes. Link to YouTube Video: http://www.youtube.com/watch?v=EIneDZrrL-Y To study the vocab used in this video, visit this site: http://quizlet.com/21414859/23-crystal-structure-iii-planes-and-miller-indices-flash-cards/ This work was supported by NSF Grants #0836041 and #1226325.

1. 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.

2. 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

3. 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)______
Z
1.
Choose
Origin
(110) c
Unit
Cell
a
b
c
1)
2.
Intercept
2)
3a.
Reciprocal
3) b Y
3b.
Reciprocal
Value
4) a O
4.
Miller
Indices
X

4. Determine
the
indices
for
the
given
plane.
1.
Choose
Origin
Unit
Cell
a
b
c
Z
2.
Intercept
( )
3a.
Reciprocal
1)
3b.
Reciprocal
Value
2)
4.
Miller
Indices
3) Y
O
4)
X

5. Determine
the
indices
for
the
given
plane.
1.
Choose
Origin
Unit
Cell
a
b
c
Z
2.
Intercept
3a.
Reciprocal
A
3b.
Reciprocal
Value
4.
Miller
Indices
Y
X

6. 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
1.
Choose
Origin
(
0
1
1
)
2a.
Reciprocal
Z
(012)
2b.
Reciprocal
Value
1)
Unit
Cell
a
b
c
2)
3.
Mark
Intercepts
Y
3) O
4.
Draw
4) X
C6

7. Determine
the
plane
for
the
given
indices.
(
1
0
0
)
1.
Choose
Origin
Z
2a.
Reciprocal
(110)
1)
2b.
Reciprocal
Value
2)
Unit
Cell
a
b
c
3) Y
3.
Mark
Intercepts
O
4)
X
4.
Draw

8. Families of Planes
Families
of
Planes
-­‐
are
"equivalent",
•
same
packing
density,
•
same
environment,
etc.
•
denoted
by
capital
integers
in
curly
brackets"
{
H
K
L
}
_
_
_
{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.
Z
_
_
_
{
1
1
1
}
=
(
1
1
1
);
(
1
1
1
);
(
1
1
1
);
(
1
1
1
)
Y
O
X

9. {
1
1
1
}
Family
Z
(110)
1)
2)
3) Y
O
4)
X

10. {
1
0
0
}
Family
Z
(110)
1)
2)
3) Y
O
4)
X

11. {
1
1
0
}
Family
Z
(110)
1)
2)
3) Y
O
4)
X

12. a
Planar Packing Density of Atoms – FCC
ClassID#: a
Planar
packing
density
-­‐
number
atoms
within
plane
/
unit
area
of
plane
a
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).
Z Z Z
a
a
a
O Y Y Y
O
O a
X a
a
a
a
X a
X
Area
=
a02
Area
=
21/2
a02
Area
=
(31/2/2)
a02
C12

13. Draw
Planar
Packing
Density
for
BCC
on
(
1
1
1
)
Plane.
PLEASE COMPLETE OTHER SIDE

14. What
do
the
Different
Brackets
Mean?

15. 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.