Tetrahedral and octahedral voids in BCC, FCC, HCP and CCP
But before we discuss all these things lets learn some useful points that will help you to understand tetrahedral and octahedral voids.
- 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.
What is Voids?
A voids is the region in a crystal lattice where there is nothing found. In other words it the empty space or region of a crystal lattice.
To understand voids in more details. We should learn how many types of voids. That is given below. But if we talk about only voids that found in all types of crystal lattice like bcc, FCC, HCP and CCP. All these cubic unit cell has different-different number of voids.
But how we know how many voids are in bcc, FCC, HCP etc. So to answer of this question is given further of the post.
If we talk about the meaning of voids. voids is nothing but an empty space between the atoms (or particles) of the cubic unit cells. In one single cubic unit cell most of the atoms or particles are present at the corner part while in some special types of cubic unit cell atoms or molecules are also present at the body part of a unit cell. For example:-
- In Body Centered Cubic Unit Cell (BCC):- The Particles are present at the corner as well as one at the center of the cube.
- In Face Centered Cubic Unit Cell (FCC):- The particles are present at the corner as well as at the all face of a cube.
- In Simple Unit Cell:- The particles are present only at the each corner part of the cubic unit cell.
Types of Voids
There are three types of Voids. these are:-
- Triangular Voids
- Tetrahedral Voids
- Octahedral Voids
How are Voids Formed?
Voids are formed when spheres of the crystal lattice join together and after joining in a particular pattern some of the gap between spheres formed called voids.
In other words voids are formed when the spheres (having rounded shapes) join together but not border to border due to the round in shape and after joining some of the gap left called formation of voids in a crystal lattice.
Different - Different types of voids formed in different - different cubic unit cell. As we know there are total three types of voids mentioned above. let's learn how each of the voids formed.
- When three spheres are align in adjacent one and one then Triangular Voids are formed.
- When one triangular voids covered with one sphere then Tetrahedral voids are formed.
- When one layer of tetrahedral voids align with another layer of tetrahedral voids then Octahedral voids are formed.
How To Calculate Voids ratio?
To calculate the voids ratio we have to fist calculate the voids in a crystal lattice. Now, voids can be calculate as the total volume minus volume of solid sphere present in a crystal lattice. The Formula for calculating voids is given by
Voids = Volume of crystal lattice - volume of spheres
Voids = Vc - Vs
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| Crystal lattice |
As we can see in the above fig. there is a crystal lattice having some particles in sphere shape. If you observe the fig. in more clear way you can see every three spheres are attached in a a definite pattern and form triangular voids. The black circle marks indicates the triangular voids in a crystal lattice. Now, if we want to calculate these triangular voids then we have to follow the above formula.
So according to the above formula:- for calculating voids, first find the volume of the whole crystal lattice. so let's suppose the volume of whole crystal lattice is 'Vc'. And after that we also have to calculate the volume of whole sphere. So let's suppose the volume of one sphere is 'v'. and there are total 11 spheres present in the above fig. of crystal lattice. So the total volume will be
v × 11 = Vs
After the calculation of voids we can now calculate voids ratio too.
To calculate voids ratio we have to divide the volume of voids to the volume of solid spheres present in a crystal lattice. If we talk about the definition of void ratio then it is defined as the fraction of the volume of voids to the volume of whole solid spheres.
Voids ratio is denoted by 'e'. And the formula for calculating voids ratio will be
Voids Ratio = Volume of Voids / Volume of Solids
e = Vv / Vs
In above discussion of voids. we knew that voids are of three types. triangular, tetrahedral and third is octahedral. But here we only discuss about two voids.
- First is Tetrahedral Voids.
- Second is Octahedral Voids.
What is Tetrahedral Voids?
Tetrahedral voids are quite confusing concept. Because I have also searched on the internet about the meaning of tetrahedral voids. But I didn't get any relevant answer. The answers are available but not so understandable easily. So here I make my own definition of tetrahedral voids and I am pretty sure that you can easily understood it yourself just read the below paragraph.
Tetrahedral voids are those voids which is formed when a triangular voids gets covered with a sphere in such a way that the centre of the sphere align with the centroid of the triangular shape voids.
In other words tetrahedral voids are those voids when a one layer of hexagonal crystal lattice get covered with another hexagonal crystal lattice. See the below fig. to understand tetrahedral voids in details.
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| Tetrahedral Voids |
In the above fig. You can see in first part there is a three spheres in lower part and attached together in such a way that it creates a triangular voids in the middle.
But if you observe the picture in more clear way. You will find that there is also a one sphere at the centre of the lower part. And this fourth sphere is placed in such a way the centre of the triangular voids lies on the centre of the upper sphere. That's why tetrahedral voids formed.
If you recreate the figure of lower triangular voids with upper sphere's center then it will be looking like the third part of the picture. That is called tetrahedron.
What are tetrahedral voids in Bcc?
In chemistry, the solid state is made up of crystal lattice. And a crystal lattice is made up of a large number of cubic unit cell arranged in a definite order. But in a unit cell there are particles (atoms or molecules) present at each corner of the cubic unit cell. The arrangements of atoms in a different-different cubic unit cell are like this:
- In BCC (Body center cubic unit cell) :- The particles are present at each corner of the unit cell as well as one atom at the centre.
- In FCC (Face center cubic unit cell):- The atoms are present at each corner as well as one atom at each faces.
- In ECC (Edge center cubic unit cell):- The atoms are present at each corner as well as the centre of each edge.
As we know the a unit cell is made of atoms weather it is in the corner or not. But each atoms have a little gap between them known as voids. And these voids are of two types: first is Tetrahedral and second is octahedral voids.
Now, what is Tetrahedral Voids in bcc? So the answer is Tetrahedral Voids in bcc is also the same voids as in any crystal lattice or cubic unit cell. But the main difference is of their numbers. Means the in bcc the number of tetrahedral voids are different then in any crystal lattice unit cell.
The formula for calculating the tetrahedral voids in bcc is 2n. Where 'n' is the number of atoms present in a unit cell.
Let's take an example to understand, how to find the number of tetrahedral voids in bcc. As we know the number of atoms in bcc is 2. So according to the above formula, the number of tetrahedral voids in BCC is 2 × 2 = 4.
Please note that:- the number of atoms in bcc is calculated as follows: As we discussed above that in bcc, atoms are present at each corner and one at the centre. So in cubic unit cell, there are total eight corner. So there will be eight atoms. But for each corner there is only 1/8 atoms present in each corner. That's why the total number of atoms will be 1/8 × 8 = 1 atom plus 1 atom at the centre so total will be 2 atoms in bcc.
Where is Tetrahedral Voids located in bcc?
In bcc Body center cubic unit cell the tetrahedral voids are located between the atoms of the unit cell. In bcc 1/8 atoms are present in each eight corner and one atom at the centre of the unit cell. So if we see the body center cubic lattice, from high powerful microscope. Then we will find that the atoms at the centre and at each corner have a little gap between them known as Tetrahedral Voids.
In bcc some of the tetrahedral voids located at the line of diagonal, some are at the top of the lattice unit cell, but there are total 12 positions found in a Body center cubic unit cell (bcc) where the tetrahedral voids are located.
What are tetrahedral voids in FCC?
In face centered cubic unit cell (FCC), there is also a tetrahedral voids present but how many? See In FCC lattice tetrahedral voids closed with four spheres. Three spheres from one layer and rest one sphere from second layer.
As shown in above figure of tetrahedral voids. These are three spheres in below layer and one sphere in above layer formed a tetrahedral voids between them.
Now, we will talk about how many tetrahedral voids are present in FCC unit cell. So the answer is, In FCC the total number of atoms are 4. And we know that the formula of calculating tetrahedral voids in any unit cell is '2n'. So the total number of tetrahedral voids in FCC will be 2 × 4 = 8 voids.
Please note that to calculate number of atoms in FCC as follows:-
- In FCC unit cell the numbers of atoms in each corner is 1/8. And as we discussed above that there are total eight corners in FCC where the particles are present. So the total number of atoms in overall corners will be 1/8 × 8 = 1
- In FCC unit cell the numbers of atoms in each face is 1/2. And we also discussed above that there are total six faces in FCC where the particles are present. So the total number of atoms in in overall faces will be 1/2 × 6 = 3.
- Now, if we add all the atoms them the results will be 1+3= 4. So the total number of atoms in FCC unit cell is 4.
Where is Tetrahedral Voids located in FCC?
In face centered cubic unit cell (FCC) the tetrahedral voids are located along the diagonal of the cubic unit cell. There are total 4 diagonals present in FCC. And two tetrahedral voids found along with one diagonal. That's why there are total 8 tetrahedral voids are present in FCC as we discussed above.
In other words you can say the tetrahedral voids in FCC has been found along with diagonal of the cubic unit cell. And the voids are arranged in such a way that it is attached with diagonal and sometimes close to diagonals.
What are Tetrahedral Voids in HCP?
Tetrahedral Voids in HCP is defined as, in a hexagonal closed packed crystal lattice tetrahedral voids are one of the Voids that are present in it. But apart from the tetrahedral voids there is also a octahedral voids present in it.
In other words a hexagonal closed packed unit cell consists of both tetrahedral and octahedral voids. That means in HCP unit cell the the spherical particles are arranged in such a way that it creates both tetrahedral and octahedral voids.
In HCP three spheres from one layer attached with one sphere from second layer formed a tetrahedral voids. Means where there is four spheres joint together in HCP crystal lattice formed a tetrahedral voids.
On the other hand where there is eight spheres joint together in HCP crystal lattice formed a octahedral voids.
So, from the above discussion we conclude that in HCP unit cell there are two types of voids present. First is octahedral and second is tetrahedral voids.
The total number of tetrahedral voids present in HCP is 12. Because as we knew that the formula of calculating tetrahedral voids in any crystal lattice is '2n'. Where 'n' is no. of atoms.
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.
Where is Tetrahedral Voids located in HCP?
As we have already discussed above that in HCP unit cell there are both tetrahedral and octahedral voids present in it. But why?
So, the answer is in HCP unit cell tetrahedron and octahedron are formed between the atoms present in it. That's the reason tetrahedral and octahedral voids present in it. The position of these voids in HCP will be differ from each other.
But if we talk about for tetrahedral voids. It is located between the atoms of four spheres present in HCP unit cell.
What are tetrahedral voids in CCP?
CCP unit cell also know as cubic closed packed structure. The CCP unit cell is just similar to the FCC unit cell. So we have already discussed above about the tetrahedral voids in FCC and also its location.
What is the radius of tetrahedral voids?
In any crystal lattice, the tetrahedral voids surrounded by 4 spheres. So let us suppose that the length of the side of cube be 'a'. And the radius of the tetrahedral voids be 'r'.
So, the diagonal of the cube in one face will be √2a.
If R will be the radius of the sphere. And we know that both the radius of two spheres connected with the radii of tetrahedral voids 'r'. Then the radius of tetrahedral voids after calculation will be 0.225R.
See the below video to find the radius of tetrahedral voids.
What is octahedral voids?
We have already discussed above about tetrahedral voids. It is formed by the combination of three spheres. Means that if three spheres combined together with one other spheres then tetrahedral voids are formed.
But when three spheres from first layer combined with three spheres from second layer then Octahedral voids formed.
In other words Octahedral voids are those voids which is present between the six spheres. The spheres are arranged in such a way that three spheres formed first triangular voids and another three spheres formed second triangular voids. See the fig. given below.
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| Octahedral voids |
In the above fig. You can see there are two triangular voids joined together and formed a octahedral void. Both the three paired spheres joined from opposite sides.
The above explanation of octahedral voids are very - very simple. Believe me there is no any easy way except this way. But perhaps there is also another explained way to understand the octahedral voids that is given below.
Let's see the another figure of where octahedral voids found? In this visualising fig. of octahedral voids. We can easily see that it is made up of two triangular voids each made up of 3 spheres. That's why octahedral voids surrounded with 6 spheres.
When the points of all 6 spheres joint together with corner and faces spheres then it formed a octahedron.
Octahedral voids in different units cell
Octahedral voids are simply voids weather it is in bcc, FCC, HCP or FCC. The definition of octahedral voids are same in any crystal lattice unit cell. But the only difference is the number of octahedral voids in different - different unit cell.
But before we proceed to the calculation of octahedral voids in bcc, FCC, HCP or CCP. Please note that the formula of calculating octahedral voids in any unit cell is 'n'. Where 'n' is the number of atoms in any unit cell.
- In BCC (Body center cubic unit cell):- the number of octahedral voids in bcc is 2. Because the total number of atoms in bcc is 2.
- In FCC (face centered cubic unit cell):- the number of octahedral voids in FCC is 4. Because the total number of atoms in FCC is 4.
- In HCP (hexagonal closed packed structure):- the number of octahedral voids in HCP is 6. Because the total number of atoms in HCP is 6.
- In CCP the number of octahedral voids is 4 or 1. Because CCP has also 4 atoms as FCC. Please note that CCP is just similar to FCC unit cell.