Atomic Packing Factor Derivation Video Lecture - Civil Engineering (CE)

in this short lecture you'll be learning
more about simple cubic crystal
structures more specifically we're going
to look at how we can derive the
relationship between the unit cell
length and the atomic radius for simple
cubic structures and also we'll all will
learn to calculate the atomic packing
factor for simple cubic structures so
the simple cubic structure is one of the
types of packaging that the basic cubic
unit cells can have the other name is
being face centered cubic and also body
centered cubic the unique thing about
the simple cubic structure is that it's
not a very common type of packaging in
fact only one type of material is known
to to be in this type of packaging and
that's polonium the simple cubic
structure essentially has only one
complete atom within the unit cell as
you can see on the on the picture on the
right you'll have 8 corner atoms here
you'll see that there are only 8 of an
atom 1/8 because they're shared with 8
other unit cells so the total number of
atoms within this cubic cell is 1 if you
add up all the 8 atoms and you multiply
that by the 8 it will give you 1
so how do we derive the relationship
between is simply the lattice parameter
length and also the atomic radius
well this for this example it's actually
fairly simple what we'll see is this
let's look at the front face of the unit
cell now if you were to project that and
look at it from a perpendicular to the
surface you'll see that these it will
look more like this
okay now the outline is the unit cell
the length of that and because it's also
cubic the length of it is known by the
by the variable a which we'll call the
lattice parameter length this length
here is also a that's now the radius of
one of the atoms is equal to R okay now
one thing to note is here we have one
radius of an atom and here is another
what you'll see is that this is actually
the lattice parameter is actually equal
to two of the radii so the relationship
is this the lattice parameter length is
equal to two times the atomic radius of
the atom
so now let's calculate the atomic
packing factor so by definition the
atomic packing factor is equal to the
total number of the total volume atoms
within the unit cell divided by the
total unit cell volume okay so we know
since volume of a simple cubic unit cell
is equal to the length cubed meaning
because it's the same length in in three
directions remember the the lattice
parameter length four for this side is a
this side is a and the set is a so if
you wanted to know the total volume this
is equal to the last parameter cubed now
next we'll look at the numerator the
what is the volume of the atoms in the
unit cell
so the the first we need to look at okay
what is the how many atoms are actually
within the unit cell and as we talked
about from before the there there are 8
corner atoms however each corner atom is
shared with eight other unit cells so it
only has one eighth of an atom on each
corner so if you wanted to know the
total number of atoms
we first find out how get how many
corner atoms there are
then we have to divide or just multiply
by the fraction of 1/8 because each atom
is shared between a total of 8 unit
cells so here you get one total atom now
the question is is okay what is the
volume
well well first we need to understand
that the atomic radius or the volume of
each atom can be treated like a sphere
so if you recall the volume of this one
sphere is equal to 4/3 PI R cubed okay
so now we're ready so let's look at the
definition for atomic packing factor the
atomic packing factor is equal to the
volume of atoms in the unit cell so we
know that we have one atom okay then we
multiplied by the volume of each atom
the volume of each atom is we said for
atom of radius R this is equal to PI R
cubed so that's a total volume of the
atoms in the unit cell
then we need to divide by the total unit
cell volume which we found from over
here the total volume is equal to a
cubed now typically which we should be
able to get a ratio okay the the
variable should cancel out well if we
remember from before when we derived the
lattice parameter length to the radius
we found that the relationship is equal
to a equals to R so what we could do is
plug in for one of the variables and in
that way one of the variables will be
cancelled out let's say that a is equal
to 2r meaning the lattice parameters
you'll do twice the Tomic radius so here
we have one atom 4/3 PI R cubed
and then the last primer cued which is
essentially 2 R cubed okay now if we
simplify the rest is just algebra so
we'll have 4/3 let me just expand it out
so you can see it 4/3
2 to the third R cubed and here what
you'll see is that the are cubes cancel
ok so let's simplify the rest is just
algebra so let's try to this let's try
to simplify as we as you see the R cubed
variable is cancelled out so what we'll
have is 4 over 4 PI over 3 times 2 cubed
which is the same thing as 8 okay we
simplify this even further you get PI
over 6 which is approximately equal to
point 5 2 so point 5 2 or essentially
what this says if you multiply by a
hundred this is about the atoms make up
about 52 percent of the total volume of
the unit cell