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LUCA MONTARELLI:
Hello, everyone.

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Today I am going to talk to you
about crystallographic point

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groups.

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And how to visualize
them with Mathematica.

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Point groups are important
in crystallography

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as they enable us to classify
symmetries of crystals.

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So what are point groups?

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Point groups are
sets of symmetries

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which are invariant
around the center point.

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Meaning a point will not
change if it's at the center.

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And the has all of these
symmetries applied.

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There are three elements
of point groups.

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Mirror planes, rotation axis,
and roto-inversion axis.

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And mirror planes
as we can imagine,

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are just taking a point.

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And then the mirror
image of this point

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would be the symmetry.

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Mirror planes can be
in three dimensions.

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Meaning that they can
be perpendicular to x,

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perpendicular to y, and
perpendicular to the z-axis.

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Rotation axis are also found
in the three dimensions.

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But they are rotations
around this axis.

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So for example, a
two-fold rotation

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would be taking a
point and making

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a rotation of half a circle
around a certain axis.

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And this would be the symmetry.

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A three-fold rotation
axis would be

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three atoms which
are each rotated

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by one third of a circle, so.

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Roto-inversion axis are a bit
more complicated to visualize,

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there are basically
a rotation followed

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by what we called an inversion.

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So for example, if we take a
full forward roto-inversion

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axis we have to take our
point, rotate it by one fourth

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of a circle.

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And then invert this
point through the origin.

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So basically inverting
would be taking

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a point which has the
coordinates one and one

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and transforming it into minus
one, minus one, minus one.

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When we mix all
of these elements.

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we get the 32 point groups
which are in crystallography

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as you can see in their
2D notation like this.

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So let's talk a bit
about the notation.

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It is fairly simple.

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We take an m if we
have a mirror plane.

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Two m's if we have the two
mirror planes et cetera.

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If this mirror plane is
perpendicular to an already

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existing rotation axis
then we note its slash m.

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A rotation axis is basically
just a number which tells us

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the number of rotations.

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So three-fold rotation axis
would be three rotation

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around a certain axis.

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Axis power is also denoted
for a roto-inversion axis.

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So if we look at examples,
a couple of examples.

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We can see the first point
group which is named one.

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Which basically has no symmetry.

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So we have a point and
nothing happens with it.

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Then we have the
two-fold rotation axis.

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So we take [INAUDIBLE]
at this point,

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rotate it by half a circle,
and we get this point.

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Same for the three-fold
rotation axis.

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One, two, and three.

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And we are back to
the original position.

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And then we have here the
four-fold rotation axis.

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And here the six-fold
rotation axis.

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If it take something which has
mirror planes, this is m, m, m.

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So for example, we
can take the cross.

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Which the cross symbolizes an
atom which is above the plane.

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And a circle an atom
which is below the plane.

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So if we take this
point and then

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say that there is a
mirror plane here.

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Then we have of course to find
a mirror image which is here.

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Then if we say it is a mirror
plane on this direction,

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then we have to take these
two atoms and put them here.

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And then we also have
a third mirror plane

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which would be
perpendicular to the x-axis.

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So the x-axis in these images
goes out of the screen.

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And that's how we end up with
four atoms on the top and four

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atoms below.

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And then we can combine
these with a rotation axis.

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So for example this
is a two slash m.

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So basically we have
a two-fold rotation.

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And then we have a
mirror plane which

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is perpendicular to the x-axis.

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So we get two atoms above
and two atoms below.

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You can also play around with
the rest and see if it works.

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But it gets more
and more complicated

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up to this one, which
is m bar three m, which

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is the most symmetric
of these point groups.

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So how do we use point groups?

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Basically if we
combine the 32 point

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groups with the translation
symmetries of the 14 Bravais

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lattices.

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And glide plane's complex
co-axis which are found in 3D.

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If you start to combine
point groups then

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we end up with a 270 space group
to which all crystals belong.

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So how do we combine point
groups and Bravais lattices?

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So here is a list of
the Bravais lattices.

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They're basically
just taking a point

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and translating it around cubic
lattice, the trigonal lattice,

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hexagonal-- everything.

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The blue point will be the
center of the point group.

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And then we just
translate this mass

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of atoms which are the
cemetry on the point group

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and then translate
it around lattice.

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So let's look at a code and
see how we visualize that.

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So this is the manipulate, which
has a function embed inside.

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So we start with one atom
which is the point group one.

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If we decide to
make a mirror plane

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for example, perpendicular to
x, then we end up with this.

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This is just one mirror
plance which has the name one.

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Then plane m [INAUDIBLE].

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Then we can have three
mirror planes which

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would give us eight atoms.

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And this is basically
what the code does.

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Just taking all of
what we give it.

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So for example we can take two,
two, two which you can find

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is this one here.

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Two, two, two gives us this.

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So this is the two,
two, two rotation axis.

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A symmetric two,
two, two point group.

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Here in the function
we also have the number

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of atoms, which is four.

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And we have the name
of the point group

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which is a bit experimental
because it's not

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easy to generate a name
out of the [INAUDIBLE]..

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This code can generate--

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for example this.

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Which is upon group which
does not exist, six, six, six.

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This is a first version
of the code which

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might be improved of course.

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We can also take
roto-inversion axis here.

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So if we try to understand
what's happening with this one

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is we have an atom.

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So we set one for rotation.

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And rotoinversion axis
would be a 1:4 rotation.

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So basically we end
up at the same point.

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And then we inverted
through the origin

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and then [? weak ?] this
opposite atom, as you can see.

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If we get things maybe clearer
to see if we have the three

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rotoinversion axis.

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So here you can clearly
see that the 3:4 rotation.

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So we have three
atoms on one side

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and three atoms
on the other side.

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So three atoms above
and three atoms below.

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And then we invert
them each time.

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So this is the coding.

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Now let's look at how it works.

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So it is quite easy.

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What the code does is taking
inputs and then calculating

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all the matrices of
modal transformations.

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So the first line, here,
would be taking the matrix

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to be replaced.

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So with the
functional deflection

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transform from
Mathematica, this line

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takes the rotation axis with
the rotation transform function.

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Then these lines
calculate all the rotation

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and all the
transformation matrices

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from each rotoinversion axis.

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As you can see, it's a bit more
complicated to come up with.

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And then taking all of
these matrices, which

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are found in the lines
of matrices [INAUDIBLE],,

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matrices, plots and matrices
in for a rotoinversion.

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We do the outer products
of all of these matrices,

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so we get every
possible combination

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of every possible symmetry.

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So we can have more of these
put into symmetry positions.

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And then we're applying
that to our initial position

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and we get all the
symmetry positions.

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Then we get into duplicates
of symmetry position.

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We make sphere out
of it, and then we

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display everything
in a Graphics3D.

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This bit here is to generate
the name of the bond group.

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As I said, experimental.

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And this bit here
is to calculate

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the number of atoms we have.

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So this is basically
the function,

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which is not that
complicated, as I said.

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And here is the results that
we can have, as you can see.

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We can start combining
everything and having

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your two bar and two
bar and two bar and.

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And just play with it.

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This is why this is great,
because we can actually

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visualize putting groups in
3D, instead of seeing them

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as these two different
presentations.

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So I hope you had a fun time.

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And thank you for
your attention.