How do these symmetries create this lattice?

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(No Transcript)
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How do these symmetries create this lattice? (in
combination with translation ofcourse!
t
mh
2-fold1
i2
i1
2-fold2
mv2
mv1
Subscript 1 ? At lattice points
Subscript 2 ? Between lattice points
Click to proceed
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One of the 2-folds (2-fold2) and one of the
inversion centres (i2) have been chosen for
illustration
t
mv1
mv2
2-fold2
i2
Note mh cannot create the lattice starting from
a point
m1 ? this is actually (mv1 t) ! t will be
applied to all these operators ? else we will get
no lattice!
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• Only points being added to the right are shown

t
mv1
mv2
2-fold2
i2
5
t
mv1
mv2
2-fold2
i2
6
t
mv1
mv2
2-fold2
i2
7
t

• Only points being added to the right are shown
• Note that only a partial lattice is created
• Similarly 2-fold1 and i1 will create partial
  lattices

mv1
mv2
2-fold2
i2
and so forth..
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Q A
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Time for some Q A

• Why do we have to invoke translation
  (ofcourse!) to construct the lattice?? Without
  the translation the point will not move!? There
  are some symmetry operators like Glide Reflection
  which can create a lattice by themselves as
  they have translation built into them

Origin of the Point Groups
Symmetry operators (without translational
component) acting at a point will leave a finite
set of points around the point
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• Many of the symmetry operators seem to produce
  the same effect. Then why use them?? There will
  always be some redundancy with respect to the
  effect of symmetry operators (or their
  combinations)? This problem is pronounced in
  lower dimension where many of them produce
  identical effects. There are no left or right
  handed objects in 1D hence a 2-fold, an
  inversion centre and a mirror all may produce
  the same effect. Analogy This is like a
  tensor looking like a vector in 1-D, looking
  like a scalar in 0D!? Hence, when we go to
  higher dimensions some of the differences will
  become clear

• If translation is doing all the job of creating a
  lattice, then why the symmetry operators?? As we
  know lattices are being used to make crystals ?
  crystals are based on symmetry? One should note
  that as translation can create a lattice an array
  of symmetry operators can also create a lattice
  (this array itself can be considered a lattice or
  even a crystal!)? Symmetry operators are present
  in the lattice even if one decides to ignore them