Packing Efficiency Of A Unit Cell

Introduction

In solid-state chemistry, the unit cell is the smallest repeating structure of the crystal. Packing efficiency gives the measure of how compactly atoms, ions, or molecules are arranged within a unit cell. It is, therefore, a way to understand just how much available space in the unit cell actually gets occupied by constituent particles.The different types of unit cells are based on how the particles are arranged: simple cubic, body-centered cubic, face-centered cubic, and hexagonal close-packed. Each type will have a packing efficiency that differs because of its unique arrangement of particles that allows for stronger or weaker binds between particles.

Packing Efficiency in HCP and CCP Structures

Both types of close packing (hcp and ccp) are equally efficient. Let us calculate the efficiency of packing in ccp structure. In the figure, let the unit cell edge length be 'a' and face diagonal AC = b.

In △ABC

AC2=b2=BC2+AB2=a2+a2=2a2 or b=2a

If r is the radius of the sphere, we find

b=4r=2a

or a=4r2=22r
(We can also write, r=a22 )

As we know, that each unit cell in ccp structure, has effectively 4 spheres. Total volume of four spheres is equal to 4×(4/3)πr3 and volume of the cube is a3 or (22r)3.

Therefore,

Packing efficiency = Volume occupied by four spheres in the unit cell ×100% Total volume of the unit cell

=4×(4/3)πr3×100(22)3%=(16/3)πr3×100162r3=74%

Efficiency of Packing in Body Centred Cubic Structures


From figure, it is clear that the atom at the centre is in touch with the other two atoms diagonally arranged.

In △EFD

b2=a2+a2=2a2 b=2a

Now in △AFD

c2=a2+b2=a2―+2a2=3a2c=3a
The length of the body diagonal is equal to 4r, here is the radius of the sphere (atom), as all the three spheres along the diagonal touch each other.

So 3a=3r

a=4r3

Hence we can write, r=34 a
In this type of structure, total number of atoms is 2 and their volume is 2×(4)3πr3

Volume of the cube, a3 will be equal to (4r)33 or a3

Therefore, Packing efficiency = Volume occupied by two spheres in the unit ×100% Total volume of the unit cell =4×(4/3)πr3×100[(43)r3%=4×(4,3)πr3×10064/(33)r3%=68%

Coordination Number (C. No.)

• In Simple Cubic (SC): 6
• In Face Centered Cubic (FCC): 12
• In Body Centered Cubic (BCC): 8

Density of Lattice Matter (d)
It is the ratio of mass per unit cell to the total volume of a unit cell and it is found out as follows.

d=Z× Atomic weight N0× Volume of unit cell (a3)
Here, d = Density
Z = Number of atoms
N₀ = Avogadro number
a³ = Volume
a = Edge length
Here in order to find the density of the unit cell in cm³, m must be taken in g/mole and should be in cm.

Radius Ratio
It is the ratio of the radius of an octahedral void to the radius of the sphere-forming the close-packed arrangement. Normally, ionic solids are more compact as voids are also occupied by cation (smaller in size) pattern of arrangements and the type of voids depends upon the relative size (ionic size) of two ions in a solid.
For example, when r⁺ = r^- the most probable and favorable arrangement is BCC type.
With the help of relative ionic radii, it is easier to predict the most probable arrangement. This property is expressed as the radius ratio.

Radius ratio =r+(radius of cation )r−(Radius of anion)
From the value of the radius ratio, it is clear that the larger the radius ratio larger the size of the cation and the more the number of anions needed to surround it, that is, the more co-ordination number.

• Radius ratio for tetrahedron
  

  Angle ABC is the tetrahedral angle of 109.5∘

  ∠ABD=109.52=54.75∘ In triangle ABD Sin ABD=0.8164=AD/AB or r∗+rr=10.8164=1.225 or r∗Γ=0.225

  
  
  
• Radius ratio for octahedron

  
  AB=r++rBD=r∠ABC=45∘

  In triangle ABD

  CosABD=0.7071=BD/AB=rr++r or r++rr=10.707=1.414 or rr4=0.414

  
  
  

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Some Solved Examples

Example 1: Which has the least packing efficiency?

1)Hcp

2)Bcc

3) Scc

4)Ccp

Solution

The packing efficiency of both HCP and CCP is 74%. The packing efficiency of BCC is 68%. Packing efficiency of SCC 52.4%.

Hence, the answer is the option (3).

Example 2: The correct order for packing fraction of different cubic systems is:

1) simple cubic < body-centered cubic < Face-centered cubic

2) face-centered cubic < simple cubic < body-centered cubic

3)face-centered cubic < body-centered cubic < simple cubic

4)body-centered cubic < simple cubic < Face-centered cubic

Solution

The packing efficiency values for different cubic systems are as follows:

• Simple cubic - 52.4%
• Body-centred cubic - 68%
• Face-centered cubic - 74%

Hence, the answer is the option (1).

Example 3: If body centred atom is removed from the BCC structure then the packing efficiency is

1)0.68

2)0.52

3) 0.34

4)0.60

Solution

Packing efficiency after removing a body-centered atom means for only one atom.

=1×43πr3(4r3)3=0.34

Hence, the answer is the option (3).

Example 4: Percentages of free space in cubic close-packed structure and in body-centered packed structure are respectively

1)48% and 26%

2)30% and 26%

3) 26% and 32%

4)32% and 48%

Solution

Packing fraction for face-centred cubic unit cell - PF = 0.74 Packing fraction for body-centered cubic unit cell - PF = 0.68

(1) Packing Efficiency in CCP is 74% and space is 26%.

(2) Packing Efficiency in BCC is 68% and space is 32%.

Hence, the answer is the option (3).

Example 5: If there are 4 atoms in one unit cell then the packing efficiency of the unit cell is:

1)0.52

2) 0.74

3)0.68

4)0.91

Solution

Packing Efficiency =z× volume of one atom volume of unit cell

Since the number of atoms per unit cell is 4 that means it is a fcc unit cell. In the fcc unit cell relation between the radius of the atom and the edge of the unit cell is given as follows:

r=a22

Thus,

Packing Efficiency =4×43πr3(22r)3=π32=0.74

Hence, the answer is (0.74).

Summary

Packing efficiency describes the density and stability of the different crystal structures. A simple cubic unit cell has a packing efficiency of about 52.4 %—just over half full. The highest packing efficiencies are achieved with the face-centered cubic and hexagonal close-packed unit cells; both are approximately 74% for a body-centered cubic unit cell, and packing efficiency is higher, at about 68%. These packing efficiency differences then have far-reaching effects on the properties of materials related to their density and their response to applied stresses. Packing efficiency is important to understand in most areas of materials science and engineering since it is through their structure that their function finds meaning for application.