Miller Indices (Planes, Directions, and Structures) Video Lecture - Civil Engineering (CE)

hello youtube today we're a to be going
over bra they lattices and Miller
indices and we're going to do a couple
examples assuming you already have some
background on the topic already in
material science and properties so here
you have the various directions and
plane coordinates for Miller indices so
just a reminder planes are used in
parentheses notation HKL are the
coordinates and direction the
coordinates are used by UV and W and
those use bracket notation to indicate
direction so the first one on the top
left there will that be a plane or
direction that's clearly a direction
based off of the bracket choice so if
we're going to go 1 1 0 what would that
look like for this unit cell so it would
look like this
and pretty much you can think you go one
in the a direction which is goes
straight towards the a axis along the a
axis and you hit the corner of the unit
cell then you go one up along the B axis
in the positive direction and then you
don't go anywhere in the C so notice how
this is at the base of this cube here so
that would be the direction shown there
next for planes the way you can figure
out what they look like is a really cool
trick you take the reciprocal of the
coordinates since on the rule for planes
the Miller indices is that they have to
be integers by taking the reciprocals
you kind of unitize your plane and get
it specifically which one you're looking
for so if you take the reciprocal of 3
you get 1 over 3 you take the reciprocal
of 2 you get 1 over 2 and mathematically
technically if you take the reciprocal
of 0 you get infinity I'll tell you what
that means in a minute but first let's
plot the first two parts so go 1/3 along
the a axis that would be right there at
the little tiny blue mark there now go
to 1/2 on the B axis and that be half of
the unit cell so we're going to go the
halfway point so look we're made a line
there now the infinity or the zero in
this case pretty much means that you are
parallel to the axis of that coordinate
so that is this
the coordinate so for the c-axis there
so that means we have to be parallel to
C at all times so we simply just drop a
line parallel to C and you can see that
see there that that plane will never
touch the C axis if you extend the plane
and that's your three to zero plane next
let's take the reciprocal of the two to
0 plane so the reciprocal of 2 is 1/2
the reciprocal to again is 1/2 and then
we're going to do mathematical trick
it's the reciprocal of 0 as a finish to
show that we're looking for parallel to
that axis so same thing is very similar
to the previous one except being at 1/3
we're at 1/2 and then we're at 1/2 again
and then again parallel to C next up I
give away see if you can figure this one
out but so you have 0 1 1 this is a
direction now so you're not going
anywhere on the a direction and you're
just going to go 1 in the B direction
and then one high in the C direction
it's pretty much plotting a 45-degree
line in 2d coordinates if you ignore the
a axis and you consider your B as your X
and your C is your Y it's one way to
think about it here we got the 1 1 1 so
we're going to go 1 in a direction then
1 in the B direction and then 1 in the C
direction so we go out towards a to the
right with B and then up with C and you
should be reaching that top right corner
there
finally we got one more plane we're
going to take the reciprocals so we got
1 over 1 is 1 and 1 over 2 or the
reciprocal of 2 is 1/2 and then typical
of 1 is 1 so simply connect those dots
so we're going to hit the edge there for
a axis and then we're going to hit the
edge of the B axis there all right now
sorry we're gonna go half way because
it's up 1/2 in the B area and then all
the way up and C so make a nice little
try triangle kinda here and that's your
plane so pretty cool huh all right so
now if we want to answer these questions
it says for the body centered cubic
structure which of the above directions
are in this case previous directions are
the closest packed directions so looking
at these we have to figure out which is
the closest packed direction
for body-centered cubic so first before
we can even answer that question let's
go over the types of structures briefly
and quickly so we're going to just look
at these three structures cubic
body-centered cubic and face centered
cubic to give you an idea of what they
look like real quick the one here on the
left is a face centered cubic you have a
simple cubic structure with an atom on
every corner of the cube and then you
have one atom at the center of every
face if you can see that image there and
the one on the right there is a body
centered cubic you have a cubic cell
with the atom at the middle so that's be
the one on the right is BCC body
centered cubic the one on the left is
FCC face centered cubic so now we have
those structures key if you have those
pictures in mind the closest packed
Direction is the edge for cubic to give
you an idea of what that looks like
here's your cube and the edge is simply
that lower part it could be any edge
really that just means the atoms are the
closest touching at those points you can
prove that mathematically but I'm just
going to show you geometrically in the
visual representation so BCC body
centered cubic is the closest packed
direction along the body diagonal and
that's that red line there so cutting
like diagonally through the cube there
so if you go back here you'd be going
from the bottom left atom there through
the gray atom and hitting the upper
right atom in the back would be your
body diagonal and then finally we have
your FCC and that would be going along
your face diagonals your closest pack
Direction what just means where the
atoms are the tightest where they're
touching essentially touching so that
would be if you choose one of the six
faces you pick a face and then you
choose that diagonal because there's an
atom on the middle of every face that is
the closest packed direction so going
back to the question here for the body
centered cubic structure 4b which are
the above directions look at the
directions is the closest packed
direction for body centered cubic so
that means we're looking for the body
diagonal so the body diagonal would have
to be a 1 1 1 because we have to go
through so that would be this middle
bottom one here would show the door
action of the closest packed area there
or closest packed direction so that
would be the one one one and that is the
body diagonal next one was for face
Center cubic which above or behind are
the closest packed directions so we're
looking for the face diagonal since is
faced centered cubic and we're looking
for the direction so pretty much we're
looking for a diagonal along face of the
cube do you see that anywhere well lower
left you got zero one one you're on the
rear face and you're going across the
diagonal there so that would be here's
one one zero or I think over zero one
one I should have written there but one
one zero should also work as well cool
well that's it for this video I'll make
another one as well I just thought I'd
give a little tutorial on some Miller
indices and going over some structures
in material sciences when it comes to
unit cells so I hope you enjoyed this
video and good luck and keep practicing