Tetrahedral and Octahedral Voids

The space lattices and close atom packing of crystalline solids distinguish them. However, these structures contain voids, which are gaps in their arrangement. Many researchers concur that the distance between atoms has a significant impact on the substance’s characteristics. Let’s look at tetrahedral and octahedral voids.

What are Voids?

In close packing of spheres, some hollows or voids are left blank. These vacancies in the crystal are called interstitial Voids or interstitial sites or simply, voids. The two essential interstitial voids are Tetrahedral Voids and Octahedral Voids.

Packing of spheres consisting of two types of blanks after arranging the two layers. The void is formed by four spheres is called tetrahedral void, and the void formed by the six spheres is called the octahedral void.

The presence of interstitial vacancies or interstitial sites plays an important role in the chemistry of transition metals. Transition metals can easily accommodate smaller nonmetal atoms such as hydrogen, boron, carbon, and nitrogen because of the spaces between the metal atoms. These compounds are called interstitial compounds.

Types of Voids

Tetrahedral voids

A sphere in the second layer is placed on top of three spheres touching each other in the first layer. The centres of these spheres lie at the top of a tetrahedron. It is likely that the shape of the void is not tetrahedral, but that the arrangement around this void is tetrahedral. The space between four spheres having a tetrahedral arrangement is called a tetrahedral void or a tetrahedral space. A crystal has two tetrahedral voids per atom.

The number of Tetrahedral Voids in a lattice can be easily calculated. The number of voids will be twice as much as the number of spheres (i.e. unit cells) in this case. As a result, there will be “2n” tetrahedral voids.

The volume of the vacuum is substantially smaller than that of the sphere. Because the void arises at the centre of four spheres, the coordination number of a tetrahedral void is four.

Tetrahedral void

Relationship among the Radius of the Tetrahedral Void and the Radius of the atoms in Close Packing- A tetrahedral void is perhaps represented by placing four spheres at the disjunctive corners of a cube. It may be noted that in a stable tetrahedral arrangement there are four spheres at the corners touching each other. However, for the sake of simplicity, spheres are shown by distant circles. Exactly all the spheres are touching each other. Let the length of each side of the cube be one cm and the radius of the tetrahedron void is r and the radius of the sphere is R.

A tetrahedral void

AC² = AB² + BC²

AC = √(AB² + BC²)

     = √(a²+ a²) = √2 × a 

A and C on the diagonal of the face

AC = R + R = 2R

2R = √2 a or R = (√2 × a)/2                                                                                                                                                               …(i)

Now in right angled triangle ACD, AD is the diagonal of the body and

AD² = AC² + CD²

AD = √(AC² + CD²)

     = √2a² + a² = √3a

The tetrahedron presents us at the center of the diagonal AD of the body so that half the length of this diagonal is equal to the sum of the radii of R and r. Thus,

R + r = (AD)/2 = (√(3) × a)/2                                                                                                                                                         …(ii)

Dividing eq. (ii) by (i) get

(R + r)/R = (√(3a))/2 × 2/(√(2a)) 

             = (√3)/(√2)

1 + r/R = (√3)/(√2)

      r/R = (√3)/(√2) – 1 = (√3 – √2)/(√2)

           = (1.732-1.414)/1.414

       r = 0.225 R

Thus, for an atom to occupy a tetrahedral void, its radius must be 0.225 times the radius of the sphere.

Octahedral void 

The octahedral space is a type of space or void that forms at the centre of six circles. It is visible in the diagram that each octahedral void is formed by the combination of triangular voids of the first and second layers. The void formed by the vertices on opposite sides by two equilateral triangles is called octahedron al void or octahedral site. Therefore, this void is surrounded by 6 spheres at the vertices of a regular octahedral. A crystal has one octahedral void per atom.

As a result, an octahedral void is formed when the first layer’s tetrahedral void and the second layer’s tetrahedral void align. A void forms in the centre of six spheres here. So an octahedral void has a coordination number of six.

If the number of spheres in a structure is “n,” the number of octahedral voids will be the same. “n” is a good example.

Octahedral void

Relationship among the radius of the Octahedral void and the radius of the atoms in close packing-

An octahedral void is surrounded by 6 spheres, only 4 are shown.

Octahedral void with radius r

Suppose that the length of the unit cell is a cm and radius of octahedral void is r and the radius of sphere is R.

If the length of the unit cell is a cm, then at right angle ABC,

AB = BC = a cm

The diagonal AC is:-

AC = √(AB² + BC²) = √(a² +a²) = √(2a)

(AC)/(AB) = (√(2) × a)/a = √2/1

AC = R + 2r + R 

AC = 2R + 2r

AB = 2R

1 + r/R = (√(2))/1 

or 

r/R = √(2) – 1

      = 1.414 – 1 

      = 0.414

   r = 0.414R

Thus, for an atom to occupy an octahedral void, its radius has to be 0.414 times the radius of the sphere. Hence, a tetrahedral void is much smaller than an octahedral void.

Number of octahedral and tetrahedral voids

• There are two tetrahedral voids for each sphere and only one octahedral void for each sphere. Thus, in a densely packed structure of N spheres, there are:
  • Tetrahedral voids = 2N
  • Octahedral voids = N
• Total Number of Tetrahedral & Octagonal Vacancies = N Coordination Number
• The number of spheres touching a given sphere is called the coordinate number. Hereby, the coordination number is the number of nearest (or nearest) neighbours of any constituent particle in the crystal lattice.
• A sphere is in a proportionate relationship with 6 other shells in the same plane of the central atom. It touches three spheres in its top layer and three spheres in its bottom layer. Thus, its coordination number is 12 in the hexagonal close-packed(hcp) and cubic closed pack (ccp) arrangements.
• It presumably recognized that coordination numbers 4, 6, 8, and 12 are much more common in different types of crystals.

Difference between Tetrahedral and Octahedral voids

Tetrahedral Voids	Octahedral voids
The space between four spheres having a tetrahedral arrangement is called a tetrahedral void or a tetrahedral site.	The void formed by the vertices on opposite sides by two equilateral triangles is called octahedron al void or octahedral site.
Tetrahedral voids = 2N	Octahedral voids = N
The radius must be 0.225 times the radius of the sphere.	The radius has to be 0.414 times the radius of the sphere.
Tetrahedral vacancies can be seen on the sides of the unit cell.	Octahedral voids can be seen in the centre of the unit cell.
There are two tetrahedral vacancies per sphere in the space lattice.	There are two octahedral vacancies per sphere in the crystal lattice.

Sample Questions

Question 1. Express the relationship between atomic radius (r) and edge length (a) in the fcc and the bcc unit cell, where a = edge length.

Answer:

For face centered cubic unit cell, radius = a/(2√(2)) = (√(2) × a)/4

For body-centered cubic unit cell, radius = (√(3) × a)/4

Question 2. What is the percentage efficiency of packing in the case of a simple cubic lattice?

Answer:

A simple cubic mesh has a packing capacity of 52.4%.

Question 3. Why are liquids and gases categorized as fluids? 

Answer:

Liquids and gases have the property of flow i.e. molecules of liquids and gases can easily move past and fall freely on each other. Because they are used to flow, they are classified as liquids.

Question 4. Why are solids incompressible?

Answer:

In a solid, the internal distance between the constituent particles (atoms, molecules, or ions) is very small. Bringing them closer, there will be great repulsive forces between the electron clouds of these particles. Therefore, the solid cannot be compressed.

Question 5. Why are crystals generally not perfect, despite the long-range order in the arrangement of particles?

Answer:

During the crystallization process, some deviations from the ideal ordered arrangement may occur. As a result, crystals are usually not perfect.

Question 6. Why does table salt, NaCl, sometimes appear yellow? 

Answer:

The yellow colour of sodium chloride crystals is due to excess impurities of the metal. In this defect, the unpaired electrons get trapped in the anion vacancies. These sites are called F-centers. The yellow colour results from the excitation of these electrons when they absorb energy from visible light falling on the crystal.