Families of directions/planes {Texas A&M: Intro to Materials}


Howdy! Tthe purpose of this video is to
describe families of directions and planes. Now what are families and why do I care about them? Families are a set of directions or
planes that are totally crystallographically equivalent. What i
mean by that is that there’s no way to distinguish between one direction and
another direction in that family unless I established a particular set of
coordinate axes. Now why do we care about them? it turns out that the
physical properties of materials of single crystals – the symmetry of those
properties is dependent on the symmetry of the lattice. so there might be a
set of directions where I can measure identical properties and that’s because
the lattice is symmetric in that direction. here’s just a quick example. this is graphite. so graphite is composed of hexagonal sheets of carbon
it turns out if you measure certain properties including thermal
conductivity you get much lower values out-of-plane so this is out-of-plane vs
in plane. specifically if i were to look in plane if i measure in certain
directions i’ll get exactly the same value, because the structure’s
crystallographically identical in those directions. if i measure in some other
direction in plane, i might get a slightly different value. but the
important thing here to remember is that the physical properties of single crystals
are dependent on the symmetry of that lattice. let’s think about an example.
let’s think about a square lattice. we’re just thinking in two dimensions.
initially, say i want to measure some property and for this example let’s
measure it vertically up and down. let’s think about maybe electrical
conductivity. i can make a graph and plot the electrical conductivity of that point here. so let’s do the same
thing but let’s rotate the lattice again. now i’m still measuring in the same original orientation. so all that has changed is the rotation angle of that
lattice. in this case maybe the electrical conductivity has decreased. I can measure that again and again and keep rotating at each time and I’m
going to get a slightly different value for that property because the structure
of this lattice is different in those cases. you’ll see i went from
originally having a very close spacing between
neighboring lattice points to something where i had a much greater spacing
between the lattice points. so maybe that’s why I’m seeing a change in the physical
properties What happens as we keep rotating
the lattice? it happens that as i rotate that lattice all the way back to an
angle of 90 degrees, I’m going to measure the same exact value as i did
originally for the electrical conductivity. this is because the
structure as I have shown it here is entirely equivalent to the structure rotated by
90 degrees. let’s look at that again here’s the
original structure. after i rotate it 90 degrees i see the same exact thing and
that’s what i mean by equivalent directions, right? this lattice looks the
same in this direction as it does in this direction. If I kept doing this –
if I kept rotating – I would see this same pattern repeated. this would be a hundred
eighty degrees. and it will keep it would keep repeating and in this case that’s
because i have this fourfold symmetry every time I rotate 90 degrees it looks
exactly the same. so that’s what a family of directions is.
these are… (let’s clearly indicate the four directions that would be equivalent in this case) are these four directions. so they all
look exactly identical crystallographically. another way to think about this is that
if i start off with the system, it doesn’t have an XY coordinate axis on it.
we have yet to discover a single crystal in nature that has a XY axis clearly
labeled on it. so if, for example, I asked you to draw the [1 0] direction on this
lattice (there are only two indices because it’s a two-dimensional lattice) you would first have to define your coordinate system so maybe you would
choose u1 and u2 and given that coordinate system you would draw the [1 0] direction in this direction. now if I ask somebody else to do the same thing, maybe they would define it this way. In that case the [1 0] direction is that
direction. so until we identify a coordinate system (the two principal
lattice vectors in this case) we can’t say which way the [1 0] direction is.
remember – all of the set of family of directions are those that
are totally crystallographically equivalent. so I’m sitting at a lattice
point and I look out, this direction looks the same as that direction, the
same as that direction, the same as that direction. if i were to notate these
families given this coordinate system this will be [1 0], [0 1], [-1 0], [0 -1], and i
would say the family of directions is [1 0]. so when we write families of directions,
we use these angular brackets. now the same exact thing is true for families of
planes so for example if we think about a simple cubic lattice – that’s a
lattice that has a lattice point at the corner of each cube – the closest packed planes (the planes with the highest atomic density) in this lattice are the {100} family of
planes. so when i’m talking about a family of planes, I denote it with the
squiggly brackets. but again there’s a number of different planes that are
totally equivalent. so this set of planes is totally equivalent to this set of
planes and to this set of planes as well. so cubic systems are nice because if you
want to say what are the family of directions or planes in that cubic
system you can reorder any of the indices and you can switch them from
positive to negative. so the six planes in the 100 family of planes would be the six that I can obtain just by reordering the arrangement of the
indices and switching any of them from positive to negative. and that’s a
shortcut, but that only works for cubic systems. Ok, so just interview we talked about
families of directions and planes. we talked about notation – round vs
squiggly brackets for planes, square versus angular brackets for directions –
and most importantly we talked about how the symmetry of properties follows
symmetry of that lattice.

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