What Is Unit Cell Packing Efficiency?
A Unit Cell may be viewed as a 3-D Structure made up of one or more atoms. Some void space is always present irrespective of the type of packing the cell has. The fraction of total space that is filled with the inherent constituent particles of a particular cell or structure is called the packing fraction. It can be obtained by dividing the total volume occupied by constituent particles by the cell's total volume.
When this is shown as a percentage i.e., out of the total space, the percentage that is held up by constituent particles is called the Packing Efficiency of a Unit Cell.
Packing Efficiency Formula = \[\frac{\text{No of Atoms x Volume Occupied by 1 atom}}{\text{Total Volume of Unit Cell}}\] X 100
We can say that Packing Fraction, when multiplied by 100, is seen as a percentage, it becomes the Packing Efficiency of that particular cell.
So, how to find the packing fraction:
Packing Fraction Formula =\[\frac{\text{Volume Occupied by all constituent particles}}{\text{Total Volume of Unit Cell}}\]
There is always some space inside a cell, and this is known as Void Space. It can be derived as follows:
Void Space Fraction: 1- Packing Fraction
Percentage of Void Space: 100 - Packing Efficiency
Packing Efficiency of a Simple Cubic Crystal Lattice (SCC)
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In a simple cubic crystal structure, particles are located only on the corners of the cube. The following relation gives the edge or side length of the cube(a) and radius(r) of constituent particles:
a = 2r
A Simple Cubic Crystal contains only single atom and hence the Volume Occupied by atoms is given as:
The Volume of Cube = a3 i.e. (2r)3 = 8r3
Therefore, Packing Efficiency = \[\frac{4\pi r^{3}}{3\times 8r^{3}}\] X 100
It totals to 52.4% while Void Space Percentage is 47.6%
It highlights that a Simple Cubic Crystal Lattice is Loosely Bound.
Packing Efficiency of Body Centred Cubic Lattice
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In this kind of structure, the particles are present at the edges, and a single particle is present in the centre as well. We can calculate the packing efficiency as follows:
When Side is ‘a’ and Radius is ‘r’: r = \[\frac{\sqrt{3}}{4}\] a
It contains two atoms and area occupied by one atom is \[\frac{4\pi r^{3}}{3}\]
The volume of the cell in terms of 'r' is: = \[\frac{64r^{3}}{3\sqrt{3}}\]
BCC Packing Efficiency = 68.04%
The packing fraction of BCC is \[\frac{68}{100}\].
32% of the total Volume remains void.
Packing Efficiency of Hexagonal Close Packing (HCP) and Cubic Close Packing (CCP)
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Both the Hexagonal Close Packing (HCP) and Cubic Close Packing (CCP) structure have the same packing efficiency. The relationship between side represented as 'a' and radius is represented as 'r' is given as:
a = 2\[\sqrt{2}\]r
These structures are also face-centered cubic lattice and have atoms situated on the eight corners of the cube and the center.
The Volume in terms of 'r' can be, given as:
(\[2\sqrt{2}r)^{3}\] = 16\[\sqrt{2}\]\[r^{3}\]
Packing Efficiency of CCP = \[\frac{4\times (\frac{4\pi r^{3}}{3})}{16\sqrt{2}r^{3}}\] X 100 = 74%
Packing fraction of HCP and FCC - \[\frac{74}{100}\]
Leaving only 26% Void Space
Therefore, we can Summarize:
How To Mathematically Find The Relationship Between 'a' And 'r.'?
In a Simple Cubic Structure:
Since the atoms are only on the corners, radius becomes half the side ,i.e., r = \[\frac{a}{2}\]
In a Body Centred Cubic Structure:
In this case, since atoms are on the corners and an atom is present in the center, we draw a diagonal, and its length (c) can be calculated using Pythagoras theorem.
We get c = \[\sqrt{3}a\]
And since radius = 4 X Diagonal (as shown in the figure under Body Centred Cubic Lattice)
r = \[4\sqrt{3}a\]
In hcp and ccp i.e., Face Centred Cubic Structure
Again, we draw the face diagonal and as shown in the figure,
b = \[\sqrt{2}a\]
r = \[\frac{d\times b}{4}\]
It finally gives us: a = 2\[\sqrt{2r}\]
Unit Cell Packing Efficiency
Unit cell can be defined as a three-dimensional structure that is made of one or more than one atom. Even when there is packing in the cell, a certain void is present in it. The space is filled by other constituents or particles. The fraction of total space that is filled with the particular cell or structure is called the packing fraction. This can be obtained by dividing the volume of the constituent particles filled in the space by the total volume of the cell. When it is represented as a percentage then the percentage of the space applied by the constituent particles out of the total space present in the structure is called the packing efficiency of the unit cell.
A lattice is largely made of a number of unit cells in which the lattice point is filled or occupied by a constituent particle. This unit cell of the lattice is a three-dimensional structure that has one or more atoms and also void spaces irrespective of the packing present. The cubic closed packed or ccp and the hexagonal closed packed or hcp are two efficient lattices when we consider packing. The packing efficiency of both of these is 74% which means 74% of the space is filled. For a simple cubic lattice, the packing efficiency is 52.4% and the packing efficiency is 68% for a body-centered cubic lattice or bcc.
FAQs on Unit Cell Packing Efficiency
1. What is the Significance of Packing Efficiency?
Packing Efficiency holds prime importance for three chief reasons:
It is useful in determining and defining the structure of the solid.
It affects other attributes such as isotropy, density, and consistency. It tells how well an element is bonded
It provides an insight into mechanical, chemical, physical etc. and various other properties as well.
2. How are Coordination Numbers Related to Packing Efficiency?
In simple terms, Coordination number refers to the number of atoms a unit cell is touching, and we are aware that the unit cell is the smallest representation of an entire crystal. Another important point is that the stability of a solid is generally accepted when it has a high packing efficiency and a high coordination number. Here are the coordination numbers of different cubic structures:
BCC – 8
HCP – 12
CCP – 12
Simple Lattice - 6
3. What Does Increase or Decrease in Packing Efficiency Denote?
Packing efficiency helps in understanding how low the void space is and how closely the atoms are bonded. It essentially means that the larger the packing Efficiency, the more stable and more closely bonded a solid is meant to be. It also portrays how stable the solid is if it is closely bound. Low packing efficiency signifies large void space, and hence the outcome is a loosely bonded substance. Therefore, as per the packing efficiencies of different cubic structure, we can establish the following hierarchy according to their packing efficiencies:
Fcc (74%) > Bcc (68.04%) > Scc (52.4%)
4. What is packing efficiency?
A crystal lattice has many unit cells which compose the lattice. This lattice has many lattice points which are occupied by a particular constituent particle. This unit cell is three-dimensional in structure and has one or more than one atom. However, no matter if there is packing, void spaces are present in these structures. This can be represented in percentages. The percentage of the space filled by the constituent particles to fill the void divided by the total void space is called the packing efficiency. Thus, the percentage of spaces that are filled by constituent particles of the cell unit is called the packing fraction.
5. What is the volume of a substance?
The volume of a substance can be defined as the quantity of the substance which is present at a particular temperature or a pressure. It is defined differently for different substances. The volume of a substance can be measured in glassware with volume mentioned already e.g. graduated cylinder and the volumetric flask. The density is the quantity that indicates how much the substance occupies a specific volume at a particular temperature and pressure. This can be used to define the substance. This can be explained by the example of water. When water freezes to form the solid form called ice, it becomes less dense than its liquid form and thus floats when placed in liquid water.
6. What is the radius of an atom?
The radius of an atom can be defined as the measurement used to measure the size of the atoms. It is measured as the distance from the center of the nucleus of the atom to the boundary of the electron shells surrounding it. Some of the widely known definitions of atomic radius are ionic radius, metallic radius, covalent radius and van der Waals radius. They are used as the atomic radius is not a well-defined entity and varies. This is due to factors like the environment and its states. It is not easy to isolate atoms to measure their radii and hence, atomic radius is measured when it is in a chemically bonded stage. Theoretical calculations are simpler when one considers the isolation of atoms.
7. What is the atomic mass and how is it calculated?
Each particle of matter has a mass at some amount which can be either in a large or small amount. Almost everything on this planet is made of atoms. The mass of an atom or an atomic particle is called atomic mass. This is expressed according to International units and agreements in terms of the unified atomic mass unit or amu. It can be described as 1/12 of the mass of carbon-12-atoms in the ground state. The mass can be calculated by the sum of the mass of protons and neutrons which equates to the atomic mass. The small change in the mass is due to the loss of mass due to binding energy.
1 atomic mass or amu = 1.66 ×10−24 g
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