Space Symmetry I

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Space Symmetry I

• Definition a crystal consists of atoms arranged
  in a pattern that repeats periodically in 3-D.
• note doesnt require (or acknowledge) surface.
• pattern can be 1 atom, groups of atoms, 1
  molecule, groups of molecules.
• Question In what form does NaCl exist?

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Space Symmetry I

• Definition a crystal consists of atoms arranged
  in a pattern that repeats periodically in 3-D.
• note doesnt require (or acknowledge) surface.
• pattern can be 1 atom, groups of atoms, 1
  molecule, groups of molecules.
• Question In what form does NaCl exist?
• As a crystal lattice a regular geometrical
  arrangement of points or objects over an area or
  space.
• not necessarily ionic.
• A lattice is not a physical thing it is simply
  an abstraction, a collection of points whereupon
  real objects may be placed. F.A. Cotton

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Lattice Points

• As an analog of a 2-D crystal, look at an
  infinitely large piece of wall paper, where the
  pattern (which can be of any complexity) repeats
  periodically in both dimensions.
• Imagine that you are an infinitely small person
  standing at a randomly chosen point on the
  wallpaper. You examine your surroundings.
• You are blindfolded and moved in a certain
  distance along a straight line to a 2nd point.
  You look around and cant tell that you have
  moved. Where are you?

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Lattice Points

• As an analog of a 2-D crystal, look at an
  infinitely large piece of wall paper, where the
  pattern (which can be of any complexity) repeats
  periodically in both dimensions.
• Imagine that you are an infinitely small person
  standing at a randomly chosen point on the
  wallpaper. You examine your surroundings.
• You are blindfolded and moved in a certain
  distance along a straight line to a 2nd point.
  You look around and cant tell that you have
  moved. Where are you?

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Lattice Points

• What happens if you are moved again an identical
  distance still along that straight line?
• the positions are each indistinguishable!
• What type of operation is this then? Symmetry!

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Lattice Points
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• To aid in further discussion and calculations, it
  will be convenient to choose some points and axes
  of reference.
• If we choose 1 point at random, then all points
  identical with this point will constitute a set
  of lattice points.
• These points all have exactly the same
  surroundings and are identical in position
  relative to the repeating pattern.
• NOMENCLATURE
• 1-D Row
• 2-D Net
• 3-D Lattice (or space lattice)

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Lattice Points Unit Cell

• If we connect the lattice points by straight
  lines, obtain 2-D parallelograms.
• In 3-D this space is divided into
  parallelepipeds.
• Note that any one parallelogram is a template for
  all of the rest.
• Call this a UNIT CELL.

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Lattice Points Unit Cell

• If we connect the lattice points by straight
  lines, obtain 2-D parallelograms.
• In 3-D this space is divided into
  parallelepipeds.
• Note that any one parallelogram is a template for
  all of the rest.
• Call this a UNIT CELL.
• The choice of the initial lattice point could
  have been anywhere.
• If we know the exact atomic arrangement in one
  unit cell, then we, by extension, can know the
  arrangement of the whole crystal.

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Primitive Unit Cells

• The choice of a unit cell is not unique any
  parallelogram whose edges connect lattice points
  is a valid unit cell. So, infinite number of ways
  of choosing a unit cell for a given structure.
• Definition A unit cell with lattice points only
  at the corners primitive.

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Centered Unit Cell

• It is permissible to have lattice points inside a
  unit cell.
• Definition A unit cell containing more than one
  lattice point centered.
• Well talk later about why you might want this.

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Size and Shape of a Unit Cell

• May be specified by means of the lengths a, b c
  of the three independent edges, and the three
  angles, a, ß ? between the edges.
• a is the angle between b c,
• ß is the angle between a c,
• ? is the angle between a b.
• The location of a point within a unit cell is
  specified by three fractional coordinates x, y,
  z. This point is located by starting at the
  origin (0, 0, 0) and moving a distance xa along
  the a axis, yb parallel to the b axis, and zc
  parallel to the c axis.

This point is located at (0.5. 0.75, 0.6).
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Size and Shape of a Unit Cell

• If x, y or z 1, then you are all the way across
  the unit cell.
• If any are gt 1, then the point is in the next
  unit cell.
• e.g. (1.30, 0.25, 0.15) is identical to (0.30,
  0.25, 0.15), since all unit cells are identical.
• one of the advantages of fractional coordinates
  is that 2 points are equivalent if the fractional
  parts of their coordinates are equal.
• caveat
• (-0.70, 0.25, 0.15) (0.30, 0.25, 0.15)

differ by 1
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Crystallographic Symmetry

• Our ultimate goal will be to consider 3-D arrays,
  but it will be useful to consider 1-D and 2-D
  arrays first. Most of the concepts applicable to
  3-D can be illustrated more simply with 1-D and
  2-D arrays.
• The type of array we are concerned with is
  obtained by repetition of some object or unit in
  a regular way thoughout space. Our object, motif
  
• Weve defined symmetry operations as movements
  after which no change could be detected in the
  object it is indistinguishable.
• Thus far, weve looked at the following E, Cn,
  s, Sn, and i.
• What do these have in common? At least one point
  of the object is unmoved by the operation.
• A complete symmetry classification scheme for
  crystallography requires that we consider other
  operations as well. Before we were dealing with
  finite objects now with infinite arrays.

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Crystallographic Symmetry Operations

• Translation. Shifting a motif by a defined
  distance in a certain direction, then doing this
  again and again by the same distance and
  direction.
• This distance unit translation.
• Can be in 1-D, 2-D or 3-D (each with different
  unit translation and direction)
• ALL crystals possess translation.

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Crystallographic Symmetry Operations

• Translation. Shifting a motif by a defined
  distance in a certain direction, then doing this
  again and again by the same distance and
  direction.
• This distance unit translation.
• Can be in 1-D, 2-D or 3-D (each with different
  unit translation and direction)
• ALL crystals possess translation.
• Glide Plane. A combination of translation and
  reflection. Operation is translation by one-half
  unit dimension, followed by reflection in the
  plane.

glide plane
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One Dimensional Space Groups

• A Space Group includes both point symmetry
  elements and translation.
• There are seven One Dimensional Space Groups.
• pxyz nomenclature
• p primitive (i.e. one lattice point per unit
  cell).
• x mirror plane - to axis of translation? yes
  m no 1.
• y mirror plane to axis of translation? yes
  m
• glide plane along axis of
  translation? yes a
• 
  no 1.
• z Cn axis? n 1 n 2.
• p111 simplest only translation present.
• p1a1 includes glide plane.

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One Dimensional Space Groups

• pm11 translation and transverse reflection. Note
  that second set of mirror planes are generated.
  Often introduction of 1 set of symmetry elements
  creates a second not equivalent to the first.
• p1m1 translation with longitudinal reflection.
• p112 two-fold rotation axis (located below the
  motif on the line of translation). The second C2
  axis is explicitly introduced. If you had started
  with it, the first would have arisen
  automatically.

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One Dimensional Space Groups

• pma2 glide plane plus transverse reflection. C2
  axis created automatically.
• pmm2 translation with longitudinal and
  transverse reflection. Just as in point groups,
  the intersection of two mirrors generates a C2
  axis.

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One Dimensional Space Groups
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One Dimensional Space Groups Examples
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One Dimensional Space Groups Examples
p1m1
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One Dimensional Space Groups Examples
p1m1
p111
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One Dimensional Space Groups Examples
p1m1
p111
p1a1
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One Dimensional Space Groups Examples
p1m1
p111
p112
p1a1
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One Dimensional Space Groups Examples
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One Dimensional Space Groups Examples
pm11
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One Dimensional Space Groups Examples
pm11
pma2
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One Dimensional Space Groups Examples
pm11
pma2
pmm2