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Week 2: Jan 23 and 25

CRYSTALLOGRAPHY (cont'd)
  • Point symmetry - symmetry wherein a motif is repeated about a point by some symmetry operation:
    • Reflection - produced by a mirror plane. The mirror plane divides the object (crystal) into two symmetric halves, Denoted by an m.
    • Rotation - repetition of a motif by rotation about a symmetry axis:
      • A2 - two fold symmetry axis
      • A3 - three-fold symmetry axis
      • A4 - four fold symmetry axis
      • A6 - six fold symmetry axis
    • Inversion - produced by a center of symmetry, denoted by i. An object with a center of symmtery will have identical points on opposite sides of the center of symmetry.
    • Roto-inversion - combinations of axes of rotation and inversion. Some combinations produce new symmetries. An A2 combined with i is equivalent to m (mirror plane).
  • Crystallographic axes - 3 or 4 (in case of hexagonal system) imaginary  axes in a crystal that can be used as a frame of reference for naming crystal faces. They are oriented parallel to and their lengths are proportional to the axes of the unit cell.
  • The Six Crystal Systems - six crystal systems can be defined based on the characteristics of their crystallographic axes and their symmetries: 
Crystal System Name Properties of axes Characteristic symmetry
isometric lengths:
a1=a2=a3
angles:
all angles (alpha, beta, and gamma) =90 degrees		 4 three-fold axes
tetragonal lengths:
c not equal to a1=a2
angles: all equal to 90		 1 four-fold axis
orthorhombic lengths: all three unequal
angles: all 90 degrees		 2 mirror planes and/or 2 two-fold axes
monoclinic lengths: all unequal
angles: alpha and beta = 90; beta not equal to 90		 1 mirror plane and/or 1 two-fold axis
triclinic lengths: all unequal
angles: all non-90 degrees		 center of symmetry only, or no symmetry
hexagonal lengths: c axis not equal to 3 equal horizontal axes (a1, a2, a3)
angles: angle between c and a-axes = 90; angle between 'a' axes = 120		 1 six=fold axis
(Images of crystallographic axes are from: http://webmineral.com/help/CellDimensions.shtml)
  • Miller Indices - a system for naming crystal faces based on their relationship to the crystallographic axes. In principle, they are derived by:
    • determining the face intercepts - the distance along each axis that the face intersects
    • taking the inverse of the face intercept
    • clearing fractions and simplifying
  • Law of Hauy - crystal faces make simple rational intercepts on crystal axes (and Miller Indices end up being small whole numbers)
  • Law of Bravais - Common crystal faces are parallel to lattice planes that have a high lattic node density
  • Crystal Forms - a group of faces having the same relationship to the symmetry elements of the crystal. Denoted as {hkl}, rather than (hkl), which refers to a single face.
    • open forms and closed forms
    • Forms to know: cube, octahedron, tetrahedron, dodecahedron, pedion, pinacoid, dome, sphenoid, prism, pyramid, dipyramid, trapezohedron, scalenohedron, rhombohedron
  • Note: Given the crystal system of a crystal, its symmetry, and the forms that are present, it is now possible sketch or imagine what the crystal will look like (in a general way -- additional descriptors will be covered under mineral habit) 

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