Crystallographic Axes

1
Crystallographic Axes

• are imaginary reference lines which often
  coincide with symmetry axes or normals to
  symmetry planes
• as in symmetry axes these aid in orientation of
  crystals and are important in explaining concepts
  as unit cells and Miller indices
• hexagonal crystals have 4 axes and all non
  hexagonal crystals have 3 axes
• axes are compared by lengths and angles of
  intersection with each other on the crystal
• axes are designated as a, b, and c, when unequal
  in lengths on a crystal or by a1a3 if equal in
  length

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• if the c axis is present, it will always lie in
  the vertical plane while the a and b axes lie in
  the horizontal plane
• triclinic system--3 unequal axes all intersecting
  at oblique angles--none of the axes correspond
  with symmetry axes since none exist
• monoclinic system--3 unequal axes, a and c
  intersect at an oblique angle and the b axis is
  perpendicular to each--the b axis corresponds to
  an A2 axis and lies in a perpendicular plane to
  the m, if one or both is (are) present

3

• orthorhombic--3 unequal axes, all mutually
  perpendicular--the c axis corresponds to an A2 if
  present and is that on which the pyramid,
  dipyramid or dome etc. are found --prism faces,
  if present are parallel to the c axis
• tetragonal--3 mutually perpendicular axes, 2 are
  equal in length (a, a) and the third, c, is
  longer or shorter than the others--all forms
  present form on or parallel the c axis

4

• hexagonal system--4 axes, 3 equal, found in the
  same plane and intersect at 60 degrees (a,a,a)
  and the 4th axis (c) perpendicular to and longer
  or shorter than the others--all forms form on or
  parallel to the c axis
• isometric system--3 equal and mutually
  perpendicular axes

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Unit Cell

• a 3-D geometric figure constructed in the array
  of atoms of a mineral in which there are atoms
  specifically located at corners, on face centers,
  or one atom at very center of the figure
• in addition to these, there are other atoms
  located within the cell
• smallest unit of a mineral that retains all
  physical, chemical, and crystallographic
  properties of a mineral
• there are 14 unique figures possible for the 6
  crystal systems
• primitive unit cell (designated as P)
• a cell with atoms at all corners but not at face
  centers nor at very center of cell

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primative unit cells
atoms at exact corners only
triclinic
monoclinic
orthorhombic
tetragonal
hexagonal
isometric
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non primitive unit cells

• 1. a cell with atoms at all corners and an atom
  at the very center of one set of opposite
  faces--if at top and bottom, C-centered if on
  front and back, A-centered if at center of both
  side faces, B-centered
• 2. a cell with atoms at all corners and one atom
  at the very center of all faces is an F or
  face-centered cell
• 3. a cell with atoms at all corners and one atom
  at very center of cell is a I or body-centered
  cell

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non primative unit cells
C-centered Orthorhombic
C-centered Hexagonal
F-centered Orthorhombic
F-centered Isometric
I-centered Monoclinic
I-centered Orthorhombic
I-centered Tetragonal
I-centered Isometric
Rhombohedral
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• many minerals may have the same type of primitive
  or non primitive cell, but each mineral has its
  sole combination of crystallographic lengths,
  which results in a kind of genetic code for a
  mineral---this allows X-ray analysis to generate
  the name of any mineral
• unit cells will combine to result in seeable
  crystal forms

10
Axial Ratio

• is a ratio of unit cell lengths measured in
  angstrom units (1 Angstrom unit (Å) 1 X 10-8
  cm) using the b axis length as common denominator
  or an a axis length if b is not present
• an example is the mineral, sulfur--unit cell
  lengths are a 10.47 (Å) b 12.88 (Å) c
  24.49 (Å), hence the axial ratio for unit cell
  for sulfur is 10.47/12.88, 12.88/12.88,
  24.49/12.88 or 0.8135 1 1.9029
• since all isometric axes are equal, axial ratios
• for all isometric unit cells has to be
• 111
• what about axial ratios for tetragonal
• unit cells?

11
Face Intercepts ( Parameters)

• an analytical way to express unit cell faces as
  intercepting or paralleling crystallographic
  axes--faces must often be extended to show the
  relationship
• designation of axes has already been
  discussed--a,b,c a,a,c a,a,a,c a,a,a
• if a face parallels an axis, an infinity symbol
  is assigned to the axis in the designated axis
  position
• if a face intercepts an axis, a ratio of
  intercept must be obtained--this ratio is
  obtained by comparing the distance of intercept
  with the length of the unit cell axis as measured
  from the unit cell vortex
• this ratio (absolute or fraction) number is then
  placed in the appropriate position in the axes
  designation

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• an example is a face on the unit cell or crystal
  of sulfur
• as measured from the vortex, the face intercepts
  the a axis at 20.94 Å, the b axis at 6.44Å, and
  parallels the c axis
• hence the parameters or intercept numbers are
  20.94/10.47a, 6.44/12.88b, infinity/24.49c 2a,
  1/2b, infinityc
• a face intercept on the minus side of an axis is
  designated by the intercept number with a minus
  above it

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examples of face intercepts
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Miller Indices

• the indices of a face is a series of whole
  numbers derived from the intercepts
• the intercept numbers are inverted, the common
  denominator found, and fractions cleared if
  present
• examples of conversions from intercepts
• ( 2a, 1/2b, c) (1/2, 2/1, 1/1) 1, 4, 2
• (infinitya, 3b, 2/3c) (0, 1/3, 3/2) 0, 2, 9
• once again the minus sign should be included over
  each minus intercept