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What is a Lattice Point
A point in the lattice. "Any point with identical surrounding as a given reference point".

What is a lattice
Empty periodic pattern

What is a Lattice Vector
base at origin to a lattice point

What is a Basis Vector
the lattice vectors that form the unit cell

List the axis systems
  Cubic: a=b=c, all angles 90
  hexagonal: a=b not c, two angles 90, one 120
  tetragonal: a=b not c, all angles 90
  orthorhombic: a not b not c, all angles 90
  trigonal: a=b=c, all angles equal, less than 120 not equal to 90
  monoclinic: no sides equal, all angles 90
  triclinic: no sides equal, no angle equal

List the Bravais Lattices
 Cubic: Primitive (P), body centered (I), face centered (F)
 Tetragonal: Primitive (P), body centered (I)
 Orthorhombic: Primitive (P), Single face centered (C), body centered (I), all face centered(F)
 Monoclinic: Primitive (P), single face centered (I)
 triclinic: Primitive (P)
 trigonal: Primitive (P)
 hexagonal: primitive (P)

What is the Structure Matrix
used to transform lattice vector to cartesian vector

What is a Metric Tensor
Tensor used to determine dot product of lattice vectors (the dot product must be evaluated in cartesian coordinates).

What is symmetry
the spatial relationship between identical objects in patterns

What is a Symmetry Operation
produces an image object from an original object where both the original and image objects are congruent (all distances and angles remain the same).

List the Symmetry Operations
 inversion 
 Translation
 rotation  n
 reflection  m (or 1/m for mirror planes perpendicular to rotation axis)
 rotoinversion 
 glide  a_{m}/m
 screw  n_{k}

List the Symmetry Elements
 translation vector
 rotation axis
 mirror plane
 inversion center
 rotoinversion axis
 glide plane
 screw axis

What is a Symmetry Element
an imaginary object that performs a symmetry operation

What is the Multiplicity of a symmetry operation
the number of repetitions required to produce an image identical (not just congruent) to the original

What is a Chiral Object
the image of an original object formed by reflection or inversion

What are Generating Elements
the symmetry elements that can be combined to generate a point group (sometimes bycreating new symmetry elements)

HermannMauguin symbols
 1triclinic, C1,order 1
 triclinic, C1,order 2
 2 monoclinic, C_{2}, order 2
 mmonoclinic, C_{s}, order 2
 2/m monoclinic, C_{2h}, order 4
 222 orthorhombic, D_{2}, order 4
 2mm orthorhombic, C_{2v}, order 4
 mmm orthorhombic, D_{2h}, order 8
 4 tetragonal, C_{4}, order 4
  tetragonal, S_{4}, order 8
 4/m tetragonal, C_{4h}, order 8
 422 tetragonal, D_{4}, order 8
 4mm tetragonal, C_{4v}, order 8
  tetragonal,D_{2d}, order 8
 4/mmm  tetragonal, D_{4h}, order 16
 3  trigonal,C3, order 3
  trigonal, C_{3i}, order 6
 32  trigonal, D_{3}, order 6
 3m  trigonal, C_{3v}, order 6
  trigonal,D_{3d}, order 12
 6  hexagonal, C_{6}, order 6
  hexagonal, C_{3h}, order 6
 6/m  hexagonal, C_{6h}, order 12
 622  hexagonal, D_{6}, order 12
 6 m m  hexagonal, C_{6v}, order 12
  hexagonal,D_{3h}, order 12
 6/m mm  hexagonal,D_{6h}, order 24
 23  cubic,T, order 12
  cubic, T_{h}, order 24
 432  cubic, O, order 24
  cubic, T_{d}, order 24
  cubic, O_{h}, order 48

Identify HermannMaugiun  1st digit 1
Triclinic

Identify HermannMaugiun  1st digit 2 or m
monoclinic

Identify HermannMaugiun  3 digits (2 or m)
orthorhombic

Identify HermannMaugiun  1st digit 4
tetragonal

Identify HermannMaugiun  1st digit 3
trigonal

Identify HermannMaugiun  1st digit 6
hexagonal

Identify HermannMaugiun  2nd digit 3
cubic