Packing efficiency and packing fraction of SCC, BCC, FCC, HCP, CCP and diamond

Packing efficiency and packing fraction of SCC, BCC, FCC, HCP, CCP and diamond

Aditya Raj Anand

Saturday 31 July 2021

Chemistry

Here in this article, we will discuss two topics first is packing efficiency and other is packing fraction in more depth. Not only the definition but also how to calculate packing efficiency and packing fraction for bcc, FCC, HCP and CCP. The topic of packing efficiency and fraction are little connected with voids. But in our previous post we have already discussed voids and unit cell. 

But for your convenient, we discussed a little bit information about voids and crystal lattice.

Before we start discussing about packing efficiency. Let's understand the following terms which will be using to understand packing efficiency and fraction.

• Crystal lattice:- A crystal lattice is the arrangement of lattice point in such a way that it repeated again and again corresponding to one another.
• Unit Cell:- A unit cell is defined as the part of a crystal lattice arrange in large number to form a big crystal lattice.
• lattice point:- It is the region or area where all the particles and subatomic particles (atoms/molecules) are present in a crystal lattice.
• Voids:- Voids are the empty space between the particles in a crystal lattice.

Packing efficiency and packing fraction both are related to tetrahedral and octahedral voids that we have already discussed in our last post. But now, the time for packing efficiency. So what does packing efficiency really means? What is the concept of packing efficiency in solid state chapter? So let's see the answers.

What is packing efficiency?

Packing efficiency is defined as the total volume occupied or covered by the spherical shaped particles (weather it is atoms or molecules) in a cubic unit cell.

In most simple words packing efficiency is how much the volume percentage of a cubic unit cell can be used by the spherical particles.

Let's see an example to understand packing efficiency in more simplest way.

Suppose you have a cubic shaped box and some spherical shaped Ball. Now, your task is to how much the maximum ball you can fill in the box.

Means how much the maximum percentage of the space you can use by filling the ball in the box.

And that maximum percentage of space you can use by filling the ball is know as your packing efficiency.

But if you noticed in the above examples that if we filled the ball in the box. Some of the gaps or space has unused. That unused space in a unit cell knows as voids. 

For more details about tetrahedral and octahedral voids please click the following links.

Links:- what are tetrahedral and octahedral voids.

Now, we got some clarity about packing efficiency. But let's understand it with the help of a diagram of a unit cell having some packing efficiency.

In the above fig. We can see that there is a unit cubic cell filled with some spherical shaped Ball. You can predict these balls as spherical particles. 

Now, look carefully to the boxed shape cubic unit cell. The spherical particles are arranged in such a way that it has a little gap between them. And please note that these arrangements of particles are the best arrangements for the maximum use of volume of the unit cell. Means there are no such best arrangements are left to use the maximum volume of unit cell.

And as we know above from the definition of packing efficiency that the maximum uses of volume of the unit cell are known as packing efficiency.

In this example we have cleared that we use the maximum volume of unit cell. But perhaps there are some gap left known as voids. 

Now, let's try to calculate how much percentage of packing efficiency and voids has used in the above fig.

How to calculate packing efficiency?

See the above fig. the side of the cube is 'a' and radius of  a sphere is 'r'. 

To calculate the packing efficiency, we have to drive the formula. So let's drive the formula for calculating packing efficiency from the above definition of packing efficiency.

So, packing efficiency is the maximum amount of the use of volume in a cubic unit cell by the spherical particles.

That means if we divide the sum of the total volume of the spherical particles to the volume of unit cell then we can easily calculate the packing efficiency.

Hence, the packing efficiency can be calculated as the sum of the volume of spherical particles dividend by the volume of cubic unit cell and the whole result multiplied with 100. So, the remaining percentage will be your packing efficiency.

The formula of packing efficiency will be

Let's suppose the total volume of spherical particles are 'Vs' and the volume of unit cell is a³.

P.E = (volume of spherical particles / volume of unit cell ) × 100

Packing efficiency = Vs / a³ × 100

Now suppose there are 'n' number of atoms in the above spherical particles. Then packing efficiency will be

Let the volume of 1 spherical atom are 4/3 πr³

Packing efficiency = [(n × 4/3 πr³) / a³] × 100

Please note that for different - different cubic unit cell such as Simple cubic unit cell, BCC, FCC, HCP and CCP. The formula will be different because the spherical particles in all of these cubic unit cells are different.

 To calculate the packing efficiency for different types of cubic unit cell. First we have to calculate the number of atoms present in that particular unit cell.

For calculating the number of atoms in different types of unit cell. We use a relation between Edge length and radius of a cubic unit cell. Instead of finding the number of atoms in BCC, FCC, HCP and CCP.

How to calculate packing efficiency of SCC?

To calculate the packing efficiency of simple cubic unit cell we have to firstly derive the formula. Then we will put the values of each components in the formula. So before we derive the formula. lets understand the concept of packing efficiency in simple cubic unit cell.

In Simple cubic unit cell, there are total eight particles present at each corners of a crystal lattice. And as we know that from the above formula of packing efficiency that the percentage of packing efficiency can be calculated only when, when we have known the number of atoms present in a crystal lattice.

So, the theory said that, In simple cubic unit cell there are 1/8 atoms are present at each corners and we have the total of eight corners. So that the total atoms in simple unit cell will be 1/8 × 8. equal to 1. hence, there are only one atoms present in a simple unit cell.

Now, from the above formula of packing efficiency that we derived is 

[(n × 4/3 πr³) / a³] × 100 -------------- (1)

Till now, we have the value of only 'n' that is 1.

But, as we discussed above that to calculating the packing efficiency of any cubic unit cell. we need a relation. means a relation between edge length and radius of unit cell.

So, here we need a relation between edge length and radius of simple cubic unit cell. So the relation are given as 

a = 2r

Now, put the values of 'n' and 'a' in equation (1) we get,

Packing efficiency = [(n × 4/3 πr³) / a³] × 100

P.E = [(1 × 4/3 πr³) / (2r)³] × 100

P.E = [(1 × 4 πr³) / 8 × 3 r³] × 100

P.E = [π / 2 × 3] × 100

P.E = [3.14 / 6] × 100

P.E = 0.52 × 100

P.E = 52%

Hence, the packing efficiency of SCC is 52%. and the remaining 48% is the voids. That means, In simple cubic unit cell there are only 52% of the maximum volume can be used by the spherical particles and rest 48% are unused known as voids.

Please note that the formula for calculating the value of packing efficiency in simple unit cell is [(n × 4/3 πr³) / a³] × 100. where n = 1 and a = 2r.

How to calculate packing efficiency of BCC?

To calculate the packing efficiency of Body centered cubic unit cell we have to firstly derive the formula. Then we will put the values of each components in the formula. So before we derive the formula. lets understand the concept of packing efficiency in body centered cubic unit cell.

In Body centered cubic unit cell, there are total eight particles present at each corners and one particles at the center of a crystal lattice. And as we know that from the above formula of packing efficiency that the percentage of packing efficiency can be calculated only when, when we have known the number of atoms present in a crystal lattice.

So, the theory said that, In Body centered cubic unit cell there are 1/8 atoms are present at each corners and one at the center of a cubic unit cell. we have the total of eight corners. So that the total atoms at the corner will be 1/8 × 8. equal to 1 plus 1 at the center. hence, there are total two atoms present in a body centered unit cell.

Now, from the above formula of packing efficiency that we derived is 

[(n × 4/3 πr³) / a³] × 100 -------------- (1)

Till now, we have the value of only 'n' that is 2.

But, as we discussed above that to calculating the packing efficiency of any cubic unit cell. we need a relation. means a relation between edge length and radius of unit cell.

So, here we need a relation between edge length and radius of bcc. So the relation are given as:

a = 4√3r / 3

Now, put the values of 'n' and 'a' in equation (1) we get,

Packing efficiency = [(n × 4/3 πr³) / a³] × 100

P.E = [(2 × 4/3 πr³) / (4r / √3)³] × 100

P.E = [(2 × 4/3 πr³) / (4r / √3)³] × 100

P.E = [(2 × 4/3 πr³) / (64r³ / √3)] × 100

P.E = 68%

Hence, the packing efficiency of BCC is 68% and the remaining 72% is the voids. That means, In Body centered cubic unit cell there are only 68% of the maximum volume can be used by the spherical particles and rest 72% are unused known as voids.

Please note that the formula for calculating the value of packing efficiency in body centered unit cell is [(n × 4/3 πr³) / a³] × 100. where n = 2 and a = 4r / √3.

How to calculate packing efficiency of FCC?

To calculate the packing efficiency of face centered cubic unit cell we have to firstly derive the formula. Then we will put the values of each components in the formula. So before we derive the formula. lets understand the concept of packing efficiency in face centered cubic unit cell.

In Face centered cubic unit cell, there are total eight particles present at each corners and six particles at each face of a crystal lattice. And as we know that from the above formula of packing efficiency that the percentage of packing efficiency can be calculated only when, when we have known the number of atoms present in a crystal lattice.

So, the theory said that, In Face centered cubic unit cell there are 1/8 atoms are present at each corners and 1/2 at each face of a cubic unit cell. we have the total of eight corners and six faces. So that the total atoms at the corner will be 1/8 × 8. equal to 1 plus 1/2 × 6 equal to 3 at the face. hence, there are total four atoms present in a face centered unit cell.

Now, from the above formula of packing efficiency that we derived is 

[(n × 4/3 πr³) / a³] × 100 -------------- (1)

Till now, we have the value of only 'n' that is 4.

But, as we discussed above that to calculating the packing efficiency of any cubic unit cell. we need a relation. means a relation between edge length and radius of unit cell.

So, here we need a relation between edge length and radius of fcc. So the relation are given as:

a = 2√2 r

Now, put the values of 'n' and 'a' in equation (1) we get,

Packing efficiency = [(n × 4/3 πr³) / a³] × 100

P.E = [(4 × 4/3 πr³) / 8√2 r³] × 100

P.E = [4/3 π / 2√2 ] × 100

P.E = 74%

Hence, the packing efficiency of FCC is 74% and the remaining 26% is the voids. That means, In face centered cubic unit cell there are only 74% of the maximum volume can be used by the spherical particles and rest 26% are unused known as voids.

Please note that the formula for calculating the value of packing efficiency in face centered unit cell is [(n × 4/3 πr³) / a³] × 100. where n = 4 and a = 2√2 r.

How to calculate packing efficiency of HCP?

To calculate the packing efficiency in hexagonal close packed structure we have to first find the number of atoms present in hcp crystal lattice.

In hexagonal close packed structure the total number of atoms are present at each corner, each face etc. 

So, the number of atoms present in HCP unit cell is 6. See how to calculate the number of atoms in HCP unit cell?

• In HCP unit cell 1/12 atoms are present in each corner.
• There are total 12 corner in HCP. So the total number of atoms in overall corners will be 1/12 × 12 = 1.
• Now, in HCP unit cell there are two 2 atoms present in two faces.
• In the remaining part of the HCP unit cell there are total of 3 atoms present.
• So the total number of atoms in overall HCP unit cell will be 1+2+3 = 6.

Let's calculate the packing efficiency of HCP unit cell.

Now, as we know that the general formula for calculating packing efficiency for any crystal lattice is 

P.E = [(n × 4/3 πr³) / a³] × 100 ------ (1)

But we want to calculate the packing efficiency of HCP unit cell. So we have to put the values of 'n' and 'a' in the above general formula of packing efficiency.

But before calculation let's establish the relation between Edge length and radius of HCP unit cell. That is given as

C = 2√2/3a

Here, 'c' is the height or edge length of a hexagon. A hexagon have two equal edge length that is 'a' and one height that is 'c'.

Now, put the values of 'a' and 'n' in equation (1) we get,

P.E = [(n × 4/3 πr³) / a³] × 100

P.E = [(6 × 4/3 πr³) / (3c/2√2)³] × 100

After solving this equation you will get 74%.

Hence, In HCP unit cell there is also a 74% of packing efficiency as similar to packing efficiency of FCC unit cell.

How to calculate packing efficiency of diamond?

To calculate the packing efficiency of diamond. We should first know that structure of diamond. So a diamond's structure is quite similar to FCC unit cell.

So let's try to find out the location of the atom found in diamond unit cell.

In diamond atoms are found at each corner, each face and one atom at the alternate tetrahedral voids.

So, the theory said that, In diamond cubic unit cell there are 1/8 atoms are present at each corners and 1/2 at each face of a cubic unit cell and also 4 atoms at alternate tetrahedral voids. we have the total of eight corners and six faces. So that the total atoms at the corner will be 1/8 × 8. equal to 1 plus 1/2 × 6 equal to 3 at the face plus 4 atoms at alternate tetrahedral voids. hence, there are total 8 atoms present in a diamond unit cell.

Now, we have the value of 'n' that is 8. But let's find out a relation between Edge length and radius of diamond.

P.E = [(n × 4/3 πr³) / a³] × 100 ------ (1)

So, the main relation between Edge length and radius of diamond unit cell is √3a = 8r.

Put the values of 'n' and 'a' in equation (1) we get,

P.E = [(8 × 4/3 πr³) / (8r/√3)³] × 100

P E = 34%

After solving this equation the packing efficiency of diamond will be 34%.

Watch this video to calculate packing efficiency of diamond.

Hence, the packing efficiency of diamond is 34%. Means in diamond type unit cell the maximum volume of spherical particles can be used only 34% remaining 66% will be empty called voids.

Please note that the formula for calculating the value of packing efficiency in diamond unit cell is [(n × 4/3 πr³) / a³] × 100. where n = 8 and a = 8r/√3.

What is packing fraction?

Packing fraction is nothing but a non percentage form of packing efficiency.

In simple words packing fraction is similar to packing efficiency. Look we calculate packing efficiency in terms of percentage. Means we multiple the value of volume division with 100. But when we didn't multiple the value of volume division with 100 the answer will be in packing fraction.

So the definition of packing fraction will be as the total volume of the spherical particles dividend by the volume of unit cell known as packing fraction.

As in the above case we have calculated packing efficiency. Similarly packing fraction will be calculated as same. But the difference is we didn't multiple it by 100.

That means if we divide all the resultant packing efficiency weather it is the packing efficiency of SCC, BCC, FCC, HCP and diamond. We get the packing fraction of SCC, BCC, FCC, HCP and diamond.

How to calculate packing fraction?

To calculate the packing fraction just use the above formula that is 

 [(n × 4/3 πr³) / a³] × 100

But in packing fraction multiplication by 100 is not applicable.

So, the final formula for calculating packing fraction of any unit cell will be  [(n × 4/3 πr³) / a³].

How to calculate packing fraction of SCC, BCC, FCC, HCP and diamond.

Packing fraction can be calculated on the same way as packing efficiency calculated above.

• The packing fraction of SCC is 0.52.
• The packing fraction of BCC is 0.68.
• The packing fraction of FCC is 0.74.
• The packing fraction of HCP is 0.74.
• The packing fraction of diamond is 0.34.

Hence, all the above packing fraction has calculated by the formula

Packing fraction =  [(n × 4/3 πr³) / a³]