Crystal Planes

1
Crystal Planes Indices

• Crystals often have polyhedral shapes bounded by
  flat faces this is a consequence of the
  periodicity of its internal arrangement.
• Take a 2-D distribution of lattice points choose
  any two (e.g. A B) and pass a line through
  those points can pass parallel lines through
  every other lattice point (N.B. in 3-D need 3
  points not on the same line).
• Have generated a set of equivalent equidistant
  planes.

2
Crystal Planes Indices

• Crystals often have polyhedral shapes bounded by
  flat faces this is a consequence of the
  periodicity of its internal arrangement.
• Take a 2-D distribution of lattice points choose
  any two (e.g. A B) and pass a line through
  those points can pass parallel lines through
  every other lattice point (N.B. in 3-D need 3
  points not on the same line).
• Have generated a set of equivalent equidistant
  planes.
• Because these planes pass equally through the
  lattice points, what is generated corresponds to
  the stacking of layers.
• The faces of crystals arrive from those planes
  which most favor the growth of the crystal (i.e.
  molecules add more easily on some faces than on
  others).

3
Crystal Planes Indices

• The orientation of a set of parallel planes can
  be specified by means of intercepts through the
  axes of the coordinate system (i.e. the unit cell
  edges).
• It is customary to specify the orientations by
  means of indices, which are proportional to the
  reciprocals of the intercepts.
• Here the intercept, using fractional coordinates,
  along the a axis is at 1, and at ½ along the b
  axis.
• So, the index would be 1 2.

b
a
4
Crystal Planes Indicies
b a

• The figure to the right shows a unit cell with a
  set of parallel planes.
• plane I intercepts at ²/3 ¹/2 8 (i.e. the plane
  is parallel to the c axis) the reciprocals are
  ³/2 2 0.
• plane II intercepts at ¹/3 ¹/4 8 the
  reciprocals are 3 4 0.
• The orientation of the planes is of interest to
  us. Notice that we can multiply the first set of
  reciprocals by a common factor (x2) to obtain
  integers (3 4 0), which are identical to the
  second plane that is, these two planes are
  parallel and part of the same set.
• (3 4 0) These 3 numbers (h k l) are called the
  Miller indices.
• Note that they reveal the number of planes that
  pass across each axis.
• Law of Rational Indices the indices of the faces
  of a crystal are usually small integers, seldom
  greater than 3 (Hauy, 1784).

¹/2 b
II
²/3 a
I
5
Two Methods for Determining Miller Indices

• Method 1 Single Plane.
• choose plane of interest.
• plane intercepts a at 22/3
• intercepts b at 4
• intercepts c at 8
• invert 3/8 ¼ 0
• multiply by common denominator (x8)
• Miller Index (3 2 0)

b a
origin
4 b
22/3 a
6
Two Methods for Determining Miller Indices

• Method 2 Parallel Planes.
• draw unit cell and all parallel planes going
  through the cell.
• a axis cut into 3 equal parts.
• b axis cut into 2 equal parts.
• c axis not cut at all.
• Miller Index (3 2 0)

b a
origin
7
Negative Miller Indices









I

• Line I.
• intercepts a at -11/3
• intercepts b at 2.
• intercepts c at 8
• invert 3/4 1/2 0
• multiply by common denominator (x9)
• Miller Index (3 2 0)
• Line II.
• intercepts a at 1
• intercepts b at -11/2
• intercepts c at 8
• invert 1 2/3 0
• multiply by common denominator (x3)
• Miller Index (3 2 0)

b a
II
origin
-
_
-
_ _ _
compare (h k l) (h k l)
_
8
Miller Indices

• Although there are an almost infinite number of
  planes that could be drawn, the only ones that
  are important in crystals are those whose indices
  are small whole numbers (empirical observation).
• Law of Rational Indices.

b a
origin
9
Inter-planar Spacings

• The inter-planar spacing (d) for a set of
  parallel planes will be important in
  crystallography. This value is related to the
  Miller indices and the unit cell dimensions.
• For orthorhombic, tetragonal and cubic unit cells
  (the axes are all mutually perpendicular), the
  inter-planar spacing is given by
• as derived by geometry.
• For other lattice types, use

h, k, l Miller indices a, b, c unit cell
dimensions