Atomic Packing Factor (APF) tells you what percent of a things is made of atoms vs empty room. You have the right to think of this as a volume density, or as an indication of how tightly-packed the atoms are.
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Calculating the atomic packing factor for a crystal is simple: for some repeating volume, calculate the volume of the atoms inside and also divide by the full volume.
Generally, this “repeating volume” is simply the volume of the unit cell. The unit cell is defined as the most basic repeating unit in a crystal.
Assuming all atoms have the same dimension, and are arranged in a repeating crystal lattice,
If you have actually multiple kinds of atoms, you must incorporate the number and volume of each atom.
In addition, atomic packing factor supplies the difficult spright here model. That implies each atom has the volume of a sphere. Assuming the atoms are tough spheres via radius
This might be a tiny difficult to conceptualize, so let’s start by dropping into 1- and 2-dimensions.
1-Dimensional Packing Factor
Because we’re in 1D, formulas with volume don’t apply. Don’t worry! If you understand also the idea, you won’t need a formula at all.
Remember, we want to find the area taken by atoms compared to the full room. Due to the fact that we’re in 1-measurement, “space” suggests “size.” We can say that 1-dimensional packing is a straight density.
For any type of specific direction, you must draw a line and determine what percent of the line is covered by a circle.
Of course, you can’t just attract any random line. The line demands to be alengthy the crystal “cell.”If you’re not certain how to do this in 1-measurement, click right here to expand.
This image may make it seem like the unit cell is just an atom, yet that’s only true in the close-packed direction. Usually, it renders the many feeling to draw the unit cell so that a half-atom sticks out on both ends (choose the red unit cell).
See, if the atoms are not “close-packed,” the unit cell will need to have actually even more space. In other words, the lattice will certainly be larger than the basis.
(In the plan above, the lattice and basis are the exact same dimension, which is why you define the unit cell roughly the circle through no additional space).
As lengthy as the line you attract is a valid depiction of the as a whole crystal symmeattempt, any kind of line will work!
Ssuggest take the size of the line extended by circles, and divide by the full length of the line.
The maximum packing element is 1, which indicates 100% of the line is inhabited by a circle.
If you had actually a packing aspect bigger than one, it would certainly suppose that somehow multiple circles overlapped on the same area of the line.
2-Dimensional Packing Factor
In 2 dimensions, room is area, rather than a line or a volume. Determining the packing aspect functions specifically the same means, but.
Sindicate find a crystallographically repeating location (the unit cell will certainly constantly work) and divide that area by the location extended by the circles.
Look at that! These two arrangements have the same packing. Any principle why that is . . . ?
Now, let’s move right into 3 dimensions and take a look at how to calculate APF for the 4 common unit cells.
Simple Cubic (SC) Lattice Length and APF
As prior to, we want to recognize just how a lot of the crystal space is inhabited by atoms vs empty room. We have entered 3-dimensions (the real world), so space is volume. To perdevelop this calculation, we must recognize the volume of a cube and the volume of a spbelow.Volume of a cube:
Now, we have actually 2 variables:
It turns out that
A straightforward cubic (SC) unit cell is a cube through an atom on each corner of the cube. The size of the cube will certainly be identified by the size of the atoms!
As you deserve to watch, for the easy cubic cell, the lattice parameter is just twice the radius. Now, in terms of the radius, we can say that the volume of the cube is:
Each atom is a sphere, so the volume per atom is:
But, exactly how many atoms execute we have actually per unit cell? At first glance, it looks choose tright here are 8 atoms: on on each corner of the cell.
However, you should take into consideration that when you stack unit cells to make the complete crystal, each corner is shared through 8 cells. So each of the 8 atoms contributes ⅛ of its complete volume. In total, tbelow is 1 full atom per unit cell.
Now if we divide the volume of an atom by the volume of the unit cell, the atomic packing factor for a basic cubic crystal is:
Body-Cgotten in Cubic (BCC) Lattice Length and also APF
To calculate the atomic packing variable, what perform we need first? The volume of the atoms, and the volume of the unit cell.
For some radius of the atoms, we have the right to calculate their volume using
The volume of the cube is
In a BCC crystal, the body diagonal is the close-packed direction. I hope this is clear in the picture listed below.
Since the body diagonal is the close-packed direction, the atoms touch each various other. That indicates the body diagonal will be a multiple of the atomic radius. In this instance, the body diagonal is
Now, it’s time to use the pythagorean theorem (you could additionally use 3D vector math, however if you don’t understand what that is, the trigonomeattempt is not so complicated).
First, make a triangle via the body diagonal as the hypotenusage. This is the green line in the image over. One of your triangle legs will be the cube’s side,
Using the pythagorean theorem,
Now, let’s find
Aacquire utilizing the pythagorean theorem, we see
Now we deserve to plug
Now that we can create the volume of the cube in terms of the atomic radius, the remainder is easy!
We just need to number out how many atoms are in each unit cell. As in the simple cubic example, tbelow are 8 edge atoms. Each corner atom contributes ⅛ of its volume to the unit cell, so that’s equal to 1 totality atom.
In addition, tright here is an atom in the middle of the cell. Since that entire atom is inside the cell, it fully contributes its volume.
In total, there are 2 atoms in the BCC unit cell. If we divide the volume of 2 atoms by the volume of the unit cell (
Face-Cgone into Cubic (FCC) Lattice Length and APF
This have to be familiar by now. Volume of the atoms split by volume of the unit cell. Let’s acquire the unit cell in regards to the atomic radius!
In an FCC crystal, any type of challenge diagonal is the close-packed direction. I hope you can view this in the image below.
The face diagonal, therefore, has actually a length of
Draw a triangle through the challenge diagonal (size =
As we’ve watched several times already, tbelow will be 8 atoms on each edge, each contributing ⅛ of its complete volume to the unit cell.
Additionally, there are 6 encounters, with half an atom on each confront (considering that faces are mutual in between 2 cells, an atom on the challenge would certainly add ½ of its volume to each cell).
In full, there are 4 atoms in the FCC unit cell. Dividing the volume of 4 atoms by the volume of the cube gives us the atomic packing factor for a face-centered cubic crystal:
Hexagonal Close-Packed (HCP) Structure and APF
Now things become tricky.
The HCP crystal framework is not cubic. Luckily, it’s still reasonably easy to visualize. It’s the herbal means for humans to pack spheres.
Imagine you had actually to pack spheres right into a box. You’d more than likely start by making a close-packed aircraft on the bottom of package. Then, you’d begin the following airplane by placing a spright here at one off the low points–in between 3 spheres from the bottom airplane. Again, the 2nd plane can be arranged as close-packed. The 3rd aircraft would certainly look precisely like the 1st airplane, the 4th aircraft would certainly look precisely like the 2nd aircraft, and so on.
This is the hexagonal close-packed lattice!
We speak to this unit cell “hexagonal close-packed” bereason it looks like hexagonal planes. In practically all situations, we usage the complete HCP cell.
However, it’s actually feasible to specify a smaller sized unit cell from the same atomic setup. This is a rhombohedral cell. Since this is the smallest unit cell possible, we call this the HCP primitive unit cell.
Let’s look at the primitive unit cell, bereason it’s less complicated so it’s simple to view the lattice parameters. Unlike a cube, tbelow are actually 2 independant lattice parameters.
By taking advantage of some trigonometry, it turns out that in an ideal HCP cell, tright here is a definite proportion of
If you look at the main atom in the primitive cell, you deserve to see that it has actually a distance
This position is the facility of the equilateral triangle. If you’re a big-time nerd and also you currently understand exactly how to calculate the centroid of an equilateral triangle, store reading.Otherwise, click this switch.
Let’s draw a line between the center of the triangle and also one of its corners. We can contact this
Now we deserve to make an additional triangle, in between
Which means that
Now that we have
The hexagonal unit cell is just a hexagonal prism. You have the right to google or memorize the answer fairly quickly, yet in a test you might obtain added points for deriving the result yourself.Click right here to watch one strategy.
Start by breaking this into parts. The volume of the hexagonal prism will be the area of the hexagon * the height of the prism. The location of the hexagon is just 6 equilateral triangles.
Let’s begin by calculating the location of a single triangle. Any triangle’s area is
Each side of the triangle has a length
Once aacquire, the pythagorean theorem saves the day! We have the right to make best triangle between
So the area of the triangle is
Multiplying this area by the elevation gives
Now that we recognize the volume of the HCP unit cell, we deserve to calculate it’s APF!Hexagonal Close-Packed Atomic Packing Fraction
The difficult component is behind us. The atomic packing fraction (APF) is simply the amount of atom inside the unit cell, compared to the all at once size of the unit cell.
For the HCP cell, there are 12 corner atoms. Each corner atom has actually ⅙ of its volume inside the unit cell. To visualize this, imagine that you joined many type of unit cells together. Each edge atom would certainly be common in between 6 other cells, so it contributes ⅙ to each.
There are 2 atoms on the face, which each contribute ½ of their volume.
Tbelow are 3 atoms in the center, which totally contribute their volume to the unit cell.
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Altogether, that’s 6 atoms per unit cell!
We know the volume of a sphere and also we currently calculated the volume of the unit cell, so
That’s it! You’ve learned how to calculate the lattice parameters and also atomic packing fractivity for easy cubic (SC), body-centered cubic (BCC), face-centered cubic (FCC), and also hexagonal close-packed (HCP) crystal systems.
Remember, APF is just the volume of the atoms within the unit cell, separated by the total volume of the unit cell. You use this to calculate the APF of any crystal mechanism, even if it’s non-cubic or has actually multiple kinds of atoms!