Transcript The Unit Face

Intercepts
 Intercepts measure
where a crystal face
hits a crystal axis.
The location on the
axes corresponding
to unit lengths is
arbitrary and chosen for
simplicity and
convenience
Axes usually radiate from the center
in a right hand rule arrangement
Axes pass through centers or edges
Intercepts are relative sizes
“The intercepts of a face
have no relation to its size,
for a face may be moved
parallel to itself for any
distance without changing
the relative values of its
intersections with the
crystallographic axes.”
K&D p. 133
Miller Indices from Intercepts
 “The Miller Indices of a face consist of a series
of whole numbers that have been derived from
the intercepts by
 inverting, and if necessary
 by the clearing of fractions.”
 “The Miller Indices [also] express a ratio ….”
K&D p133
Problem: find the Miller Index for a face with intercepts 2a, 2b, 2/3c
1. Invert the indices:
½½
3/2
2. This is a ratio. If we multiply all terms by a
constant, the ratio remains the same.
Let’s multiply by 2 to clear the fractions:
(1 1 3)
Miller Indices for
horizontal and vertical faces
 A face perpendicular to
one axis may be
considered to intersect the
others at infinity.
 For example, for a face
perpendicular to the c-axis
(aka a3-axis) the [positive
side] index would be
(001).
The colored face is parallel to a1 and a2, meeting them only at ∞
Problem: find the Miller Index for this face with intercepts ∞a1, ∞a2, 1a3
1. Invert the indices:
Since 1/∞ = 0 and 1/1 = 1
we have 0/1 0/1 1/1
2. Clear fractions by multiplying
through by 1
(0 0 1)
Miller Indices for
faces parallel to two axes
 A face parallel to two axes
may be considered to
intersect the other at
unity.
 For example, for a face
parallel to the a-axis and
c-axis (or a1 and a3) the
[positive side] index would
be (010).
Miller Indices for
faces parallel to one axis
 If a face is parallel to one
of the crystallographic
axes, a zero “0” is used
(because 1/infinity = 0)
 For example, for a face
parallel to the a-axis, the
[positive side] Miller
Index could be (011)
Faces that intersect axes on their
negative side.
 “For faces that intersect negative
ends of crystallographic axes, a bar is
placed over the appropriate number….
The bar represents
the minus sign in a
negative number