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,

wright here

indicates number and means volume.**For a more complicated description of having actually multiple kinds of atoms, click below to expand also.**

If you have actually multiple kinds of atoms, you must incorporate the number and volume of ** each ** atom.

wright here

is the volume of each type of atom. In metals there is generally just one atom, yet in a ceramic, mean tright here are 3 kinds of atoms: , , and . The expanded version would look choose this:

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

in a cubic unit cell with lattice parameter ,

This might be a tiny difficult to conceptualize, so let’s start by dropping into 1- and 2-dimensions.

Outline

### 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**: , where is the size of a side.

**Volume of a sphere**: , wbelow is the radius of the spbelow.

Now, we have actually 2 variables:

(radius of atom) and (side length of cube).It turns out that

and have the right to be composed in terms of each various other.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:

wright here is the radius of the spbelow.

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 regards to the lattice consistent , so let’s create in regards to .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,

. The other leg will certainly be the confront diagonal. You can use any kind of variable you prefer for this length; I will choose .Using the pythagorean theorem,

.Now, let’s find

. We have the right to make one more triangle from a various 3D view; this time our triangle has on the hypotenusage and for both legs.Aacquire utilizing the pythagorean theorem, we see

, or .Now we deserve to plug

into our initially pythagorean theorem.

So,

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 (

**atomic packing factor**

**for a body-focused cubic**

**crystal**is:

### 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

. We can obtain the lattice consistent in terms of with a straightforward application of the pythagorean theorem.Draw a triangle through the challenge diagonal (size =

) as the hypotenusage. Both legs will be .

So

will certainly be the volume of the unit cell, so let’s figure out exactly how many type of atoms are in the unit cell.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.

Lattice parameter

is the length between 2 poignant atoms (so, twice the radius).Lattice parameter

is the height of the unit cell.By taking advantage of some trigonometry, it turns out that in an ideal HCP cell, tright here is a definite proportion of

.The Hexagonal Close-Packed c/a ratioIf you look at the main atom in the primitive cell, you deserve to see that it has actually a distance

in between the atoms in the plane above and in the aircraft listed below. If you projected the atom right into one of those planes, it would certainly be exactly in the middle of 3 atoms.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

. Since the angles of an equilateral triangle are all 60°, the angle between and is 30°.

So

Now we deserve to make an additional triangle, in between

, , and .

Which means that

Or

And remembering that ,

Now that we have

and , we have the right to calculate the**volume**of the hexagonal unit cell.Hexagonal Close-Packed Unit Cell Volume

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

, so let’s use that as our base. Now we have to discover the elevation of the triangle.Once aacquire, the pythagorean theorem saves the day! We have the right to make best triangle between

, , and also the elevation .Which means

So the area of the triangle is

And since there are 6 equilateral triangles per hexagon,

Multiplying this area by the elevation gives

and utilizing and also ,

Now that we recognize the volume of the HCP unit cell, we deserve to calculate it’s APF!

Hexagonal Close-Packed Atomic Packing FractionThe 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

### Final Thoughts

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!