Crystallographic Axes

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Crystallographic Axes

• Klein (2002) p. 194-197

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Which Crystal System?

• abc ? ? ? 90?
• abc ? ? ? 90?
• a1 a2 a3 (120?), c perpendicular

Orthorhombic
Tetragonal
Hexagonal
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Crystallographic axes

• When we are describing crystals, it is convenient
  to use a reference system of three axes,
  comparable to the axes of analytical geometry.
• These imaginary axes are called the
  crystallographic axes
• These axes are fixed by symmetry
• Coincide with symmetry axes
• Parallel to intersections of major crystal faces

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Crystallographic axes

• Ideally crystallographic axes should be parallel
  to the edges of the unit cell, and lengths
  proportional to the cell dimensions
• REMEMBER
• All crystals except hexagonal referred to by 3
  axes a, b and c
• Convention
• a is angle between b and c
• b is angle between a and c
• g is angle between a and b

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Crystallographic axes
Hexagonal
isometric
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Crystallographic axes

• Axial Ratios
• All the crystal systems, except isometric
  tetragonal, have crystallographic axes differing
  in length
• The steps on the crystallographic axes, because
  they are dependent on the unit cell, are
  different in size

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Crystallographic axes

• For instance orthorhombic sulfur
• a 10.47A, b12.87A, c24.49A
• We can write a, b and c as ratios of b
• a/b b/b c/b
• 10.47/12.87 1 24.49/12.87
• 0.8155 1 1.9028
• We are only interested in the proportional
  differences, the axial ratios

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Crystallographic axes

• Crystal faces are defined by indicating their
  intercepts on the crystallographic axes
• Face AB is parallel to the c-axis and intercepts
  a and b
• Parameters of this face are
• 1a1b c
• It intercepts 1 length of the a axis, one length
  of the b-axis and is parallel to the c-axis

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Fig.5.28
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Crystallographic axes

• Crystal faces are defined by indicating their
  intercepts on the crystallographic axes

Fig.5.28
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Face Intercepts

• Lattice plane

A-B
B
Y or b axis
z or c axis (vertical)
A
A
A
A
B
X or a axis
Plane A-A
Intercepts 1a, 8b, 8c
Intersects x axis at one unit (1), is parallel to
the y axis ( 8 ) and the z axis (8 )
Plane A-B, intersects 1a and 1b, but is parallel
to c or 8c
Parameters 1a, 1b, 8c
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Unit face

• If there are several faces of a crystal
  intersecting all three axes, the largest face at
  the positive end of the crystallographic axis is
  taken as the unit face.
• Consider this example

Unit face (the face with the clear shade)
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Steps to determine Miller Indices and the
Miller-Bravais Indices

• 1. The first thing that must be ascertained are
  the fractional intercepts that the plane/face
  makes with the crystallographic axes. In other
  words, how far along the unit cell lengths does
  the plane intersect the axis.
• e.g 1a, 8b, 8c and 1a, 1b, 8c and 1a, 2b, 4c
• 2. Omit a, b, c and commas
• e.g 1 8 8 and 1 1 8 and 1 2 4
• 3. Take the reciprocal of the fractional
  intercept of each unit length for each axis.
• e.g. 1/1 1/8 1/8 and 1/1 1/1 1/8 and 1/1 ½ ¼

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Steps to determine Miller Indices and the
Miller-Bravais Indices

• 4. Finally the fractions are cleared (using a
  common denominator).
• so 1/1 1/8 1/8 and 1/1 1/1 1/8 and 1/1 ½ ¼
• Becomes 1 0 0 and 1 1 0 and 4 2 1
• 5. Enclose the integers in parentheses
• So (100) and (110) and (421)
• These designate that specific crystallographic
  plane within the lattice. Since the unit cell
  repeats in space, the notation actually
  represents a family of planes, all with the
  same orientation.

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Steps to determine Miller Indices and the
Miller-Bravais Indices

• (100) and (110) and (421) are called the Miller
  indices
• In the hexagonal system there are 3 horizontal
  axes and one vertical. The indices are called the
  Miller-Bravais indices

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Summary
When intercepts are assigned to the faces of a
crystal, without knowledge of its cell
dimensions, one face that cuts all three axes is
arbitrarily assigned the units 1a,1b,1c
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Summary
(hkl)
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The previous notation is called the Miller
Indices and ONLY applies for theTriclinic,
Monoclinic, Orthorhombic and Isometric systems
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Intercepts

• Two very important points about intercepts of
  faces
• The intercepts or parameters are relative values,
  and do not indicate any actual cutting lengths.  
• Since they are relative, a face can be moved
  parallel to itself without changing its relative
  intercepts or parameters.

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Miller indices

• Try work out how the shaded face (in each case)
  intersects the axes

(111)
(001)
(110)
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How is it for hexagonal and trigonal systems?

• Recall both systems have 4 crystallographic axes.
• In this case, the notation for the intersection
  of faces is called Miller -Bravais Indices (hkil)

(1010) One,zero,bar one, zero h k I (10-1) 0