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Crystal Planes

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... second plane; that is, these two planes are parallel and part of the same set. ... This value is related to the Miller indices and the unit cell dimensions. ... – PowerPoint PPT presentation

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Title: 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
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