-MJrb7xScbU.txt

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in this demonstration I'm going to show

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in this demonstration I'm going to show
 

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in this demonstration I'm going to show
you how to construct 2d brilliant zones

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you how to construct 2d brilliant zones
 

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you how to construct 2d brilliant zones
and hopefully achieve a better

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and hopefully achieve a better
 

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and hopefully achieve a better
understanding them and what they are so

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understanding them and what they are so
 

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understanding them and what they are so
a brilliant zone is an important concept

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a brilliant zone is an important concept
 

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a brilliant zone is an important concept
in material science and solid-state

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in material science and solid-state
 

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in material science and solid-state
physics

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physics
 

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physics
alike because it is used to describe the

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alike because it is used to describe the
 

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alike because it is used to describe the
behavior of an electron in a perfect

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behavior of an electron in a perfect
 

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behavior of an electron in a perfect
crystal system so what is a brilliant

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crystal system so what is a brilliant
 

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crystal system so what is a brilliant
zone the Brillion zone is a particular

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zone the Brillion zone is a particular
 

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zone the Brillion zone is a particular
choice of the unit cell of the

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choice of the unit cell of the
 

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choice of the unit cell of the
reciprocal lattice is defined as the

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reciprocal lattice is defined as the
 

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reciprocal lattice is defined as the
riegner Seitz cell of the reciprocal

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riegner Seitz cell of the reciprocal
 

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riegner Seitz cell of the reciprocal
lattice it is constructed as the set of

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lattice it is constructed as the set of
 

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lattice it is constructed as the set of
points and closed by the Bragg planes

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points and closed by the Bragg planes
 

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points and closed by the Bragg planes
the planes perpendicular to a connection

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the planes perpendicular to a connection
 

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the planes perpendicular to a connection
line from the origin of to each lattice

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line from the origin of to each lattice
 

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line from the origin of to each lattice
point passing through the midpoint

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point passing through the midpoint
 

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point passing through the midpoint
alternatively it is defined as the set

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alternatively it is defined as the set
 

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alternatively it is defined as the set
of points closer to the origin than to

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of points closer to the origin than to
 

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of points closer to the origin than to
any other reciprocal lattice point the

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any other reciprocal lattice point the
 

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any other reciprocal lattice point the
whole reciprocal space may be covered

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whole reciprocal space may be covered
 

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whole reciprocal space may be covered
without overlap with copies of such

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without overlap with copies of such
 

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without overlap with copies of such
brilliant zone okay so that was a rather

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brilliant zone okay so that was a rather
 

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brilliant zone okay so that was a rather
convoluted definition let's do it again

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convoluted definition let's do it again
 

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convoluted definition let's do it again
let's brush up and be sure that we

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let's brush up and be sure that we
 

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let's brush up and be sure that we
understand this definition we're going

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understand this definition we're going
 

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understand this definition we're going
to define her the reciprocal space the

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to define her the reciprocal space the
 

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to define her the reciprocal space the
Bragg planes as well but it's fine both

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Bragg planes as well but it's fine both
 

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Bragg planes as well but it's fine both
of these will also do a quick revision

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of these will also do a quick revision
 

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of these will also do a quick revision
of what is there a record lattice so

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of what is there a record lattice so
 

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of what is there a record lattice so
that we can go from there to the

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that we can go from there to the
 

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that we can go from there to the
reciprocal lattice the microscopic

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reciprocal lattice the microscopic
 

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reciprocal lattice the microscopic
perfect crystal is formed by adding

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perfect crystal is formed by adding
 

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perfect crystal is formed by adding
identical building blocks so to say unit

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identical building blocks so to say unit
 

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identical building blocks so to say unit
cells consisting of atoms and groups of

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cells consisting of atoms and groups of
 

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cells consisting of atoms and groups of
atoms a unit cell is the smallest

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atoms a unit cell is the smallest
 

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atoms a unit cell is the smallest
component of a crystal that one stacked

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component of a crystal that one stacked
 

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component of a crystal that one stacked
together with pure translational

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together with pure translational
 

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together with pure translational
repetition reproducing the whole crystal

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repetition reproducing the whole crystal
 

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repetition reproducing the whole crystal
which essentially means that you can

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which essentially means that you can
 

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which essentially means that you can
take the same thing over and over and

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take the same thing over and over and
 

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take the same thing over and over and
over again and get the whole system done

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over again and get the whole system done
 

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over again and get the whole system done
so and the groups of atoms these unit

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so and the groups of atoms these unit
 

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so and the groups of atoms these unit
cells that form the microscopic crystal

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cells that form the microscopic crystal
 

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cells that form the microscopic crystal
but infinite repetition is called the

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but infinite repetition is called the
 

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but infinite repetition is called the
basis okay that seems quite clear and

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basis okay that seems quite clear and
 

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basis okay that seems quite clear and
the basis is formed in such a way that

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the basis is formed in such a way that
 

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the basis is formed in such a way that
it forms the lattice more commonly known

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it forms the lattice more commonly known
 

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it forms the lattice more commonly known
as the Rivervale at every point of a

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as the Rivervale at every point of a
 

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as the Rivervale at every point of a
breve lattice is equivalent to every

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breve lattice is equivalent to every
 

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breve lattice is equivalent to every
other point which means that the

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other point which means that the
 

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other point which means that the
arrangement atoms and the crystal is the

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arrangement atoms and the crystal is the
 

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arrangement atoms and the crystal is the
same when viewed from different lattice

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same when viewed from different lattice
 

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same when viewed from different lattice
points okay that also seems

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points okay that also seems
 

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points okay that also seems
understandable and you should probably

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understandable and you should probably
 

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understandable and you should probably
know that by now so any fundamental

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know that by now so any fundamental
 

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know that by now so any fundamental
lattice must be definable by three

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lattice must be definable by three
 

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lattice must be definable by three
primitive translational vectors a1 a2

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primitive translational vectors a1 a2
 

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primitive translational vectors a1 a2
and a3 the combination of these vectors

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and a3 the combination of these vectors
 

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and a3 the combination of these vectors
is using to find the crystal

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is using to find the crystal
 

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is using to find the crystal
translational vector R such that R is

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translational vector R such that R is
 

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translational vector R such that R is
equal to a 1 and 1 plus a 2 and 2 plus a

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equal to a 1 and 1 plus a 2 and 2 plus a
 

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equal to a 1 and 1 plus a 2 and 2 plus a
3 and 3 where n are just arbitrary

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3 and 3 where n are just arbitrary
 

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3 and 3 where n are just arbitrary
integers to show that Mark would show

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integers to show that Mark would show
 

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integers to show that Mark would show
the size of our lattice so when we the

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the size of our lattice so when we the
 

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the size of our lattice so when we the
crystal lattice is repeated an infinite

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crystal lattice is repeated an infinite
 

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crystal lattice is repeated an infinite
amount of times to create the perfect

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amount of times to create the perfect
 

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amount of times to create the perfect
crystal structure and each of those

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crystal structure and each of those
 

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crystal structure and each of those
lattices are translationally symmetric

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lattices are translationally symmetric
 

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lattices are translationally symmetric
and the way I'll look at is is that one

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and the way I'll look at is is that one
 

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and the way I'll look at is is that one
cannot tell their positional in crystal

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cannot tell their positional in crystal
 

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cannot tell their positional in crystal
structure because every lattice looks

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structure because every lattice looks
 

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structure because every lattice looks
this ok so now okay so that seems to

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this ok so now okay so that seems to
 

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this ok so now okay so that seems to
make sense so now let's go through

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make sense so now let's go through
 

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make sense so now let's go through
reciprocal space okay so every lattice

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reciprocal space okay so every lattice
 

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reciprocal space okay so every lattice
has a reciprocal lattice associated

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has a reciprocal lattice associated
 

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has a reciprocal lattice associated
through in crystallography terms the

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through in crystallography terms the
 

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through in crystallography terms the
reciprocal lattices of the diffraction

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reciprocal lattices of the diffraction
 

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reciprocal lattices of the diffraction
prior of a crystal or in quantum

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prior of a crystal or in quantum
 

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prior of a crystal or in quantum
mechanics it's described this k space

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mechanics it's described this k space
 

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mechanics it's described this k space
with K being 4k wave vectors in 3d

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with K being 4k wave vectors in 3d
 

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with K being 4k wave vectors in 3d
lattice the vectors would be v1 v2 and

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lattice the vectors would be v1 v2 and
 

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lattice the vectors would be v1 v2 and
v3 and they can be denoted it as well

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v3 and they can be denoted it as well
 

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v3 and they can be denoted it as well
look at it just b1 b1 is equals to 2 pi

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look at it just b1 b1 is equals to 2 pi
 

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look at it just b1 b1 is equals to 2 pi
times the cross product of the vectors

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times the cross product of the vectors
 

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times the cross product of the vectors
a2 and a3 from our direct lattice

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a2 and a3 from our direct lattice
 

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a2 and a3 from our direct lattice
divided by the triple cross scalar

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divided by the triple cross scalar
 

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divided by the triple cross scalar
product of a 1 a 2 and a 3 in which case

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product of a 1 a 2 and a 3 in which case
 

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product of a 1 a 2 and a 3 in which case
this cross product of a 2 and a 3 is the

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this cross product of a 2 and a 3 is the
 

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this cross product of a 2 and a 3 is the
area of our vector of our two vectors in

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area of our vector of our two vectors in
 

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area of our vector of our two vectors in
the triple scalar product is the volume

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the triple scalar product is the volume
 

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the triple scalar product is the volume
of our system by simplifying it we can

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of our system by simplifying it we can
 

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of our system by simplifying it we can
just get 2 pi over the height of our

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just get 2 pi over the height of our
 

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just get 2 pi over the height of our
unit cell or we can put it in this way

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unit cell or we can put it in this way
 

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unit cell or we can put it in this way
the larger our direct lattice gets the

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the larger our direct lattice gets the
 

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the larger our direct lattice gets the
smaller and core person or reciprocal

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smaller and core person or reciprocal
 

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smaller and core person or reciprocal
lattice becomes another observation that

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lattice becomes another observation that
 

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lattice becomes another observation that
could actually be made by the reciprocal

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could actually be made by the reciprocal
 

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could actually be made by the reciprocal
lattices that the reciprocal lattice of

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lattices that the reciprocal lattice of
 

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lattices that the reciprocal lattice of
the reciprocal lattice is the direct

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the reciprocal lattice is the direct
 

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the reciprocal lattice is the direct
lattice bit okay for simplicity's sake

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lattice bit okay for simplicity's sake
 

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lattice bit okay for simplicity's sake
let's look at a transformation from 2d

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let's look at a transformation from 2d
 

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let's look at a transformation from 2d
lattice to a reciprocal lattice okay so

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lattice to a reciprocal lattice okay so
 

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lattice to a reciprocal lattice okay so
we have a visualization here where we

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we have a visualization here where we
 

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we have a visualization here where we
can change

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can change
 

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can change
the length of our X vector indirect

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the length of our X vector indirect
 

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the length of our X vector indirect
space and the length of our vibrator

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space and the length of our vibrator
 

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space and the length of our vibrator
indirect space and we can change whether

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indirect space and we can change whether
 

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indirect space and we can change whether
or not we're seeing this as a direct

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or not we're seeing this as a direct
 

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or not we're seeing this as a direct
lattice or by pressing this button as a

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lattice or by pressing this button as a
 

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lattice or by pressing this button as a
reciprocal lattice so as we can see but

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reciprocal lattice so as we can see but
 

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reciprocal lattice so as we can see but
increasing the length of our direct

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increasing the length of our direct
 

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increasing the length of our direct
direct vector we change the sizes of our

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direct vector we change the sizes of our
 

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direct vector we change the sizes of our
reciprocal lattice vectors and the other

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reciprocal lattice vectors and the other
 

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reciprocal lattice vectors and the other
way around

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okay so now we had a definition of the

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okay so now we had a definition of the
 

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okay so now we had a definition of the
reciprocal space in the rect space let's

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reciprocal space in the rect space let's
 

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reciprocal space in the rect space let's
go back to our definition so the first

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go back to our definition so the first
 

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go back to our definition so the first
brilliant zone can be defined as a set

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brilliant zone can be defined as a set
 

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brilliant zone can be defined as a set
of points in reciprocal space that can

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of points in reciprocal space that can
 

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of points in reciprocal space that can
be reached from a specific point of

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be reached from a specific point of
 

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be reached from a specific point of
origin without crossing any breakpoints

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origin without crossing any breakpoints
 

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origin without crossing any breakpoints
came so what our Bragg planes a Bragg

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came so what our Bragg planes a Bragg
 

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came so what our Bragg planes a Bragg
plane or in this case a Bragg line is a

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plane or in this case a Bragg line is a
 

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plane or in this case a Bragg line is a
Bragg line with perpendicular bisector

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Bragg line with perpendicular bisector
 

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Bragg line with perpendicular bisector
reciprocal lattice vector a vector which

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reciprocal lattice vector a vector which
 

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reciprocal lattice vector a vector which
connects two lattice points and the

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connects two lattice points and the
 

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connects two lattice points and the
closest Bragg planes are essentially

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closest Bragg planes are essentially
 

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closest Bragg planes are essentially
accosting the Brillion zone now can show

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accosting the Brillion zone now can show
 

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accosting the Brillion zone now can show
that the Bragg planes for the closest

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that the Bragg planes for the closest
 

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that the Bragg planes for the closest
neighbors with this being our original

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neighbors with this being our original
 

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neighbors with this being our original
lattice point these are the four closest

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lattice point these are the four closest
 

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lattice point these are the four closest
neighbors in a simple I'd say cubic but

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neighbors in a simple I'd say cubic but
 

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neighbors in a simple I'd say cubic but
it's actually just the square lattice

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it's actually just the square lattice
 

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it's actually just the square lattice
because it's in 2d and if we add the

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because it's in 2d and if we add the
 

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because it's in 2d and if we add the
second large vectors before the second

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second large vectors before the second
 

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second large vectors before the second
closest neighbors you can see these ones

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closest neighbors you can see these ones
 

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closest neighbors you can see these ones
and essentially conveys the same

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and essentially conveys the same
 

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and essentially conveys the same
information when we go to a higher order

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information when we go to a higher order
 

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information when we go to a higher order
of closest neighbors we can see that the

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of closest neighbors we can see that the
 

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of closest neighbors we can see that the
system gets a lot more complex okay so

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system gets a lot more complex okay so
 

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system gets a lot more complex okay so
now we've seen what our back planes can

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now we've seen what our back planes can
 

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now we've seen what our back planes can
go towards brilliant zones so this is

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go towards brilliant zones so this is
 

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go towards brilliant zones so this is
the first brilliant result it is what it

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the first brilliant result it is what it
 

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the first brilliant result it is what it
seems it is as you can imagine the Bragg

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seems it is as you can imagine the Bragg
 

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seems it is as you can imagine the Bragg
planes just go here and the brilliant

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planes just go here and the brilliant
 

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planes just go here and the brilliant
zone shows us the area in reciprocal

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zone shows us the area in reciprocal
 

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zone shows us the area in reciprocal
space that is closer to our lattice

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space that is closer to our lattice
 

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space that is closer to our lattice
point than any other lattice point which

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point than any other lattice point which
 

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point than any other lattice point which
are essentially the Bragg planes as well

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are essentially the Bragg planes as well
 

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are essentially the Bragg planes as well
as we can see if our reciprocal lattice

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as we can see if our reciprocal lattice
 

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as we can see if our reciprocal lattice
origin point isn't here this the square

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origin point isn't here this the square
 

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origin point isn't here this the square
is closer to this point than to any

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is closer to this point than to any
 

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is closer to this point than to any
other point and after these lines it

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other point and after these lines it
 

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other point and after these lines it
gets the other way around

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gets the other way around
 

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gets the other way around
so we can move on forward to a higher

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so we can move on forward to a higher
 

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so we can move on forward to a higher
order of brilliant zones

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this is the Brillion zone for the second

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this is the Brillion zone for the second
 

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this is the Brillion zone for the second
closest neighbor as you can see it it is

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closest neighbor as you can see it it is
 

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closest neighbor as you can see it it is
rather similar it just takes the second

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rather similar it just takes the second
 

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rather similar it just takes the second
closest neighbors and essentially draws

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closest neighbors and essentially draws
 

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closest neighbors and essentially draws
another square it but it's a bit tilted

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another square it but it's a bit tilted
 

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another square it but it's a bit tilted
to this okay so now let's look at the

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to this okay so now let's look at the
 

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to this okay so now let's look at the
third one for the third brilliant zone

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third one for the third brilliant zone
 

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third one for the third brilliant zone
that gets a bit more complex because if

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that gets a bit more complex because if
 

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that gets a bit more complex because if
we scroll a bit backwards we can see

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we scroll a bit backwards we can see
 

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we scroll a bit backwards we can see
that the third bread the breakfast for

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that the third bread the breakfast for
 

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that the third bread the breakfast for
the thirtieth closest neighbors are

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the thirtieth closest neighbors are
 

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the thirtieth closest neighbors are
these ones so we might think that this

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these ones so we might think that this
 

 align:start position:0%
these ones so we might think that this
whole thing would be that brilliant the

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whole thing would be that brilliant the
 

 align:start position:0%
whole thing would be that brilliant the
third brilliant look but it's actually

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third brilliant look but it's actually
 

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third brilliant look but it's actually
not because with every next system it

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not because with every next system it
 

 align:start position:0%
not because with every next system it
gets a bit more complex and it's

 align:start position:0%
gets a bit more complex and it's
 

 align:start position:0%
gets a bit more complex and it's
actually taking account both the third

 align:start position:0%
actually taking account both the third
 

 align:start position:0%
actually taking account both the third
right planes and the first ones okay so

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right planes and the first ones okay so
 

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right planes and the first ones okay so
now let's do another thing let's turn

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now let's do another thing let's turn
 

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now let's do another thing let's turn
off on that we can see all the Bragg

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off on that we can see all the Bragg
 

 align:start position:0%
off on that we can see all the Bragg
planes and turn up another notch okay so

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planes and turn up another notch okay so
 

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planes and turn up another notch okay so
here we can see that the fourth Brillion

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here we can see that the fourth Brillion
 

 align:start position:0%
here we can see that the fourth Brillion
zone gets a lot more complex now we can

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zone gets a lot more complex now we can
 

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zone gets a lot more complex now we can
see all the Bragg planes for the quite

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see all the Bragg planes for the quite
 

 align:start position:0%
see all the Bragg planes for the quite
for the closest neighbors and these

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for the closest neighbors and these
 

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for the closest neighbors and these
lines could get quite difficult to

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lines could get quite difficult to
 

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lines could get quite difficult to
understand by drawing themselves but we

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understand by drawing themselves but we
 

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understand by drawing themselves but we
can help ourselves out with this

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can help ourselves out with this
 

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can help ourselves out with this
visualization so yeah we can see that

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visualization so yeah we can see that
 

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visualization so yeah we can see that
they start to interact with each other

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they start to interact with each other
 

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they start to interact with each other
and thus make a more difficult structure

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and thus make a more difficult structure
 

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and thus make a more difficult structure
and if we go to the fifth one and see

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and if we go to the fifth one and see
 

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and if we go to the fifth one and see
show that we can see all the brilliant

 align:start position:0%
show that we can see all the brilliant
 

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show that we can see all the brilliant
zones okay so here we can see that it

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zones okay so here we can see that it
 

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zones okay so here we can see that it
gets becomes quite a nice drawing if we

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gets becomes quite a nice drawing if we
 

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gets becomes quite a nice drawing if we
were to keep adding them let's just it

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were to keep adding them let's just it
 

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were to keep adding them let's just it
let's head to the ninth one okay so see

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let's head to the ninth one okay so see
 

 align:start position:0%
let's head to the ninth one okay so see
here we can see a high order brilliant

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here we can see a high order brilliant
 

 align:start position:0%
here we can see a high order brilliant
zones and it actually looked kind of

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zones and it actually looked kind of
 

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zones and it actually looked kind of
nice which is also interesting because

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nice which is also interesting because
 

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nice which is also interesting because
it's also an art form drawing high

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it's also an art form drawing high
 

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it's also an art form drawing high
drawing high order billion zones

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drawing high order billion zones
 

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drawing high order billion zones
the higher you go the more complex and

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the higher you go the more complex and
 

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the higher you go the more complex and
more discrete the system gets in

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more discrete the system gets in
 

 align:start position:0%
more discrete the system gets in
essentially it looks nicer okay so after

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essentially it looks nicer okay so after
 

 align:start position:0%
essentially it looks nicer okay so after
looking at this we can get a short

 align:start position:0%
looking at this we can get a short
 

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looking at this we can get a short
definition of how to construct to the

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definition of how to construct to the
 

 align:start position:0%
definition of how to construct to the
brilliant zones so enth brilliant zone

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brilliant zones so enth brilliant zone
 

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brilliant zones so enth brilliant zone
can be defined as the area or volume if

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can be defined as the area or volume if
 

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can be defined as the area or volume if
we look in 3d in reciprocal space that

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we look in 3d in reciprocal space that
 

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we look in 3d in reciprocal space that
can be reached from the origin but

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can be reached from the origin but
 

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can be reached from the origin but
crossing exactly n minus one Bragg

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crossing exactly n minus one Bragg
 

 align:start position:0%
crossing exactly n minus one Bragg
planes

 align:start position:0%
planes
 

 align:start position:0%
planes
so we can look also add a brilliant zone

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so we can look also add a brilliant zone
 

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so we can look also add a brilliant zone
for a system where the atoms are not

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for a system where the atoms are not
 

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for a system where the atoms are not
perfectly in the perfect square lattice

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perfectly in the perfect square lattice
 

 align:start position:0%
perfectly in the perfect square lattice
but are offset a bit making sort of say

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but are offset a bit making sort of say
 

 align:start position:0%
but are offset a bit making sort of say
a triangular lattice and as we can see

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a triangular lattice and as we can see
 

 align:start position:0%
a triangular lattice and as we can see
here the first billion zone is it such

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here the first billion zone is it such
 

 align:start position:0%
here the first billion zone is it such
such a hexagon and the second one

 align:start position:0%
such a hexagon and the second one
 

 align:start position:0%
such a hexagon and the second one
already gets a bit more difficult by

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already gets a bit more difficult by
 

 align:start position:0%
already gets a bit more difficult by
forming a star shape and the third one

 align:start position:0%
forming a star shape and the third one
 

 align:start position:0%
forming a star shape and the third one
is again so to say FX agon and it goes

 align:start position:0%
is again so to say FX agon and it goes
 

 align:start position:0%
is again so to say FX agon and it goes
on like that

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on like that
 

 align:start position:0%
on like that
okay so and then so what information is

 align:start position:0%
okay so and then so what information is
 

 align:start position:0%
okay so and then so what information is
the brilliance don't hold on what does

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the brilliance don't hold on what does
 

 align:start position:0%
the brilliance don't hold on what does
it give us the short vectors in the

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it give us the short vectors in the
 

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it give us the short vectors in the
Brillion zone or on its boundary

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Brillion zone or on its boundary
 

 align:start position:0%
Brillion zone or on its boundary
characterized States in a system with

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characterized States in a system with
 

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characterized States in a system with
lattice period periodicity for example

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lattice period periodicity for example
 

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lattice period periodicity for example
phone out or or electron state but for

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phone out or or electron state but for
 

 align:start position:0%
phone out or or electron state but for
that the whole of the video thanks and

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that the whole of the video thanks and
 

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that the whole of the video thanks and
this code for this demonstration was

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this code for this demonstration was
 

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this code for this demonstration was
taken and editors from Mathematica

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taken and editors from Mathematica
 

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taken and editors from Mathematica
demonstration since made by their slope

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demonstration since made by their slope
 

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demonstration since made by their slope
cloth