Close Packed Structures and Packing Efficiency

Close Packed Structures and Packing Efficiency: We all know that all substances in the universe are made up of atoms. Each atom is considered a solid sphere that is closely packed in the solid state. Now, the question arises that how these atoms are arranged inside a substance? The packing of atoms is in one dimension as well as in two and three dimensions. In one dimension packing, the atoms are arranged in a row and each sphere touches the other two spheres. Thus, the coordination number in one dimension packing is two.

Similarly, in the case of two-dimensional packing, there are two different cases: square close packing and hexagonal close packing. Three-dimensional packing is of cubic close packing. Irrespective of the type of packing, we observe some void spaces in the unit cell. The percentage of spaces that is filled by the particles in the unit cell is called the packing fraction of the unit cell. In this article, we will learn about close-packed structures and their packing efficiency in detail.

Close Packed Structures

In the solid state, the constituent particles are closely packed, and very little vacant space is found between them. This vacant space is known as void or interstitial space. Close packing occurs in all the three dimensions discussed below:

1. Close-Packing in One Dimension 

In one-dimensional close packing, the number of solid spheres of the same size is arranged and placed side by side in a row. In this type of arrangement, each sphere is in contact with its two neighbouring spheres, which is known as its coordination number. Thus, we can say that the coordination number in one-dimensional close packing is two.

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2. Close-Packing in Two Dimensions

Two-dimensional close packing is of two types:

(i) Square Close Packing: In square close packing, the second row of particles is arranged in such a manner that the spheres of the second row are exactly above the spheres of the first row and the spheres of both the rows are arranged horizontally as well as vertically. In this case, each sphere remains in contact with four neighbouring spheres. Thus, its coordination number is \(4\). This arrangement is called square close packing or also called \({\rm{AAAA}}\) type of arrangement. 

(ii) Hexagonal Close Packing: In this type of close packing, the second row of particles is placed in contact with the first row in such a manner that the spheres of the second row come in depressions of spheres of the first row. In this arrangement, very little space is available, and this type of packing is more efficient than square close packing. Here, each sphere remains in contact with six neighbouring spheres, thus named as hexagonal close packing. The coordination number of two-dimensional hexagonal close packing is \(6\). This arrangement is also called \({\rm{ABAB}}\) type. 

3. Close-Packing in Three Dimensions

We can obtain a three-dimensional structure by placing two-dimensional layers one over the other. These are of following types given below:

(i) Three-dimensional closed packing from the layers of two-dimensional square closed packed structure:

In this arrangement, the layers of two-dimensional square close packing are arranged one over the other in such a manner, that the spheres come one above the other and spheres of all layers, one arranged in a straight line horizontally and vertically. Such type of formed lattice is known as simple cubic lattice, and its unit cell is called a primitive cubic unit cell.

(ii) Three-dimensional closed packing from two-dimensional hexagonal close-packed layers.

In this type of arrangement, three-dimensional close packing is done by Placing Second Layer Over the First Layer: In this type of arrangement, the spheres of a second, layer are arranged in the depressions of the spheres of the first layer. As the spheres of both layers are aligned differently, so the first layer is called layer \(‘{\rm{A}}’\) and the second layer is called layer \(‘{\rm{B}}’\). In this type of arrangement, tetrahedral voids and Octahedral voids are formed.

• The number of octahedral voids that are generated \( = {\rm{N}}\)
• Number of tetrahedral voids that are generated \( = 2{\rm{N}}\)

Packing Efficiency

The percentage of total space filled by the constituent particles in the unit cell is called the packing efficiency. A vacant space that is not occupied by the constituent particles in the unit cell is known as void space.

Packing Efficiency in hcp (Hexagonal closed packing) and ccp (Cubic closed packing) Structures

The packing efficiency of both \({\rm{hcp}}\) and \({\rm{ccp}}\) are equivalent. The calculation of packing efficiency in ({\rm{hcp}}\) and \({\rm{ccp}}\) is given below:

Taken the unit cell edge length as \(‘{\rm{a}}’\) and face diagonal \({\rm{AC}}\) as \(‘{\rm{b}}’\).

In triangle \({\rm{ABC}}\):

\({\rm{A}}{{\rm{C}}^{\rm{2}}}{\rm{ = }}{{\rm{b}}^{\rm{2}}}{\rm{ = B}}{{\rm{C}}^{\rm{2}}}{\rm{ + A}}{{\rm{B}}^{\rm{2}}}\)

\({{\rm{b}}^2} = {{\rm{a}}^2} + {{\rm{a}}^2}\)

\({{\rm{b}}^{\rm{2}}}{\rm{ = 2}}{{\rm{a}}^{\rm{2}}}\)

Or,  \({\rm{b = }}\sqrt {\rm{2}} {\rm{a}}\)

Let \(‘{\rm{r}}’\) be the radius of the sphere, then:

\({\rm{b = 4r = }}\sqrt {\rm{2}} {\rm{a}}\)

or \({\rm{a = }}\frac{{{\rm{4r}}}}{{\sqrt {\rm{2}} }}{\rm{ = 2}}\sqrt {\rm{2}} {\rm{r}}\)

This can also be written as: 

\({\rm{r = }}\frac{{\rm{a}}}{{{\rm{2}}\sqrt {\rm{2}} }}\)

Now, we know that the volume of a sphere is \(\frac{{\rm{4}}}{{\rm{3}}}{\rm{\pi }}{{\rm{r}}^{\rm{3}}}\).

Each unit cell in ccp has four spheres. Thus, the total volume of four spheres is equal to \({\rm{4 \times }}\frac{{\rm{4}}}{{\rm{3}}}{\rm{\pi }}{{\rm{r}}^3}\). The volume of the cube is  \({{\rm{a}}^3}\) or \({(2\sqrt 2 {\rm{r}})^3}\).

Thus, the formula of packing efficiency is given below:

Packing Efficiency \( = \frac{{{\rm{Volume}}\,{\rm{occupied}}\,{\rm{by}}\,{\rm{the}}\,{\rm{four}}\,{\rm{spheres}}\,{\rm{in}}\,{\rm{the}}\,{\rm{unit}}\,{\rm{cell}}\, \times 100}}{{{\rm{The}}\,{\rm{total}}\,{\rm{volume}}\,{\rm{of}}\,{\rm{the}}\,{\rm{unit}}\,{\rm{cell}}}}\% \)

Packing Efficiency \({\rm{ = }}\frac{{{\rm{4 \times }}\frac{{\rm{4}}}{{\rm{3}}}{\rm{\pi }}{{\rm{r}}^{\rm{3}}}{\rm{ \times 100}}}}{{{{{\rm{(2}}\sqrt {\rm{2}} {\rm{r)}}}^{\rm{3}}}}}{\rm{\% }}\)

Packing Efficiency \({\rm{ = }}\frac{{\frac{{{\rm{16}}}}{{\rm{3}}}{\rm{\pi }}{{\rm{r}}^{\rm{3}}}{\rm{ \times 100}}}}{{{\rm{16}}\sqrt {\rm{2}} {{\rm{r}}^{\rm{3}}}}}{\rm{\% }}\)

Thus, Packing Efficiency in \({\rm{hcp}}\) and \({\rm{ccp}} = 74\% \)

Packing Efficiency in bcc (Body Centered Cubic) Structures

In \({\rm{bcc}}\) structure, it can be seen that the atom at the centre will be in touch with the other two atoms that are diagonally arranged.

In Triangle \({\rm{EFD}}\), 

\({{\rm{b}}^{\rm{2}}}{\rm{ = }}{{\rm{a}}^{\rm{2}}}{\rm{ + }}{{\rm{a}}^{\rm{2}}}\)

\({{\rm{b}}^{\rm{2}}}{\rm{ = 2}}{{\rm{a}}^{\rm{2}}}\)

Or \({\rm{b = }}\sqrt {\rm{2}} {\rm{a}}\)

In \(\Delta {\rm{AFD}}\):

\({{\rm{c}}^{\rm{2}}}{\rm{ = }}{{\rm{a}}^{\rm{2}}}{\rm{ + }}{{\rm{b}}^{\rm{2}}}{\rm{ = }}{{\rm{a}}^{\rm{2}}}{\rm{ + 2}}{{\rm{a}}^{\rm{2}}}{\rm{ = 3}}{{\rm{a}}^{\rm{2}}}\)

\({\rm{c = }}\sqrt {\rm{3}} {\rm{a}}\)

The length of body diagonal \({\rm{c = 4r}}\).

Sine, \({\rm{r}}\) is the radius of the sphere, and all the three spheres in the diagonal touch each other. Thus, \(\sqrt {\rm{3}} {\rm{a = 4r}}\)

\({\rm{a = }}\frac{{{\rm{4r}}}}{{\sqrt {\rm{3}} }}\)

This can also be written as:

\({\rm{r = }}\frac{{\sqrt {\rm{3}} }}{{\rm{4}}}{\rm{a}}\)

In \({\rm{bcc}}\) type structure, the total number of atoms is \(2\) and their total volume is \({\rm{2 \times }}\frac{{\rm{4}}}{{\rm{3}}}{\rm{\pi }}{{\rm{r}}^{\rm{3}}}\).

The volume of the cube is  \({{\rm{a}}^{\rm{3}}}\).

Since, \({\rm{a = }}\frac{{{\rm{4r}}}}{{\sqrt {\rm{3}} }}\)

\({{\rm{a}}^{\rm{3}}}{\rm{ = }}{\left( {\frac{{\rm{4}}}{{\sqrt {\rm{3}} }}{\rm{r}}} \right)^{\rm{3}}}\)

Packing Efficiency \( = \frac{{{\rm{Volume}}\,{\rm{occupied}}\,{\rm{by}}\,{\rm{the}}\,{\rm{two}}\,{\rm{spheres}}\,{\rm{in}}\,{\rm{the}}\,{\rm{unit}}\,{\rm{cell}}\, \times 100}}{{{\rm{Total}}\,{\rm{volume}}\,{\rm{of}}\,{\rm{the}}\,{\rm{unit}}\,{\rm{cell}}}}\% \)

Packing Efficiency \({\rm{ = }}\frac{{{\rm{2 \times }}\frac{{\rm{4}}}{{\rm{3}}}{\rm{\pi }}{{\rm{r}}^{\rm{3}}}{\rm{ \times 100}}}}{{{{\left( {\frac{{\rm{4}}}{{\sqrt {\rm{3}} }}{\rm{r}}} \right)}^{\rm{3}}}}}{\rm{\% }}\)

Packing Efficiency \({\rm{ = }}\frac{{\frac{{\rm{8}}}{{\rm{3}}}{\rm{\pi }}{{\rm{r}}^{\rm{3}}}{\rm{ \times 100}}}}{{{\rm{64}}{{\rm{r}}^{\rm{3}}}{\rm{/(3}}\sqrt {{\rm{3)}}} }}{\rm{\% }}\)

Thus, Packing Efficiency in \({\rm{bcc = 6}}8\% \)

Packing Efficiency in Simple Cubic Lattice Structures

In simple cubic lattice, the atoms are located only at the corners of the cube. The particles touch each other through the edges. Thus, we can relate the edge length and the radius of each particle \({\rm{‘r’}}\) as:

\({\rm{a = 2r}}\)

Volume of the cubic unit cell \({\rm{ = }}{{\rm{a}}^{\rm{3}}}{\rm{ = (2r}}{{\rm{)}}^{\rm{3}}}{\rm{ = 8}}{{\rm{r}}^{\rm{3}}}\)

As a simple cubic unit cell contains only \(1\) atom

The volume of the space occupied \({\rm{ = }}\frac{{\rm{4}}}{{\rm{3}}}{\rm{\pi }}{{\rm{r}}^{\rm{3}}}\)

Therefore,
Packing Efficiency \( = \frac{{{\rm{Volume}}\,{\rm{occupied}}\,{\rm{by}}\,{\rm{one}}\,{\rm{man}} \times 100}}{{{\rm{Volume}}\,{\rm{of}}\,{\rm{the}}\,{\rm{unit}}\,{\rm{cell}}}}\% \)

Packing Efficiency \({\rm{ = }}\frac{{\frac{{\rm{4}}}{{\rm{3}}}{\rm{\pi }}{{\rm{r}}^{\rm{3}}}{\rm{ \times 100}}}}{{{\rm{8}}{{\rm{r}}^{\rm{3}}}}}{\rm{\% = }}\frac{{\rm{\pi }}}{{\rm{6}}}{\rm{ \times 100 = 52}}{\rm{.4\% }}\)

Thus, Packing Efficiency in simple cubic lattice \(=52.4\%\)

Thus, by calculating the packing efficiency of the various cubic cells, \({\rm{hcp}}\) and \({\rm{ccp}}\) have the maximum packing efficiency among all the structures.

Summary

In short, substances that exist in the solid state have definite mass, volume, and shape. A strong force of attraction between the molecules holds them together throughout the lattice. The molecules are arranged in a specific pattern inside the solid substances. There are different types of arrangement of molecules such as in one dimension packing, the atoms are arranged in a row and each sphere touches the other two spheres. As a result, the coordination number in one dimension packing is two. Similarly, in the case of two-dimensional packing, there are two different cases: square close packing and hexagonal close packing.

Three-dimensional packing is of cubic close packing. Though the molecules are closely packed, we observe some void spaces in the unit cell. The percentage of spaces that is filled by the particles in the unit cell is called the packing fraction of the unit cell. Packing efficiency is defined as the percentage of total space filled by the particles. The packing efficiency of a crystal structure gives information about the available space occupied by atoms. To calculate the percentage efficiency, firstly, count how many atoms there are per unit cell. Then calculate the volume of a single atom and multiply by the number of atoms in the unit cell. In \({\rm{hcp}}\) and \({\rm{ccp}}\) the packing efficiency of the lattice is \(74\%\), in body-centred cubic structures, the three atoms are arranged diagonally, thus, the packing efficiency of a simple unit cell is \(52.4\%\). In a simple cubic lattice, the packing efficiency of the lattice is \(52.4\%\).

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FAQs

Q.1. Which close packing is more efficient?
Ans: Hexagonal close packing is more efficient because, in this type of close packing, the second row of particles is placed in contact with the first row in such a manner that the spheres of the second row come in depressions of spheres of the first row. Thus, in this type of arrangement, very little space is available, and this type of packing is more efficient than square close packing. Here, each sphere remains in contact with six neighbouring spheres, thus named as hexagonal close packing. Also, by calculating the packing efficiency of the various cubic cells, \({\rm{hcp}}\) and \({\rm{ccp}}\) have the maximum packing efficiency i.e., \(74\%\) among all the structures.

Q.2. Which close packing has the least packing efficiency? 
Ans: The simple cubic lattice is the least efficient packing as the value of packing efficiency is \(52.4\%\) which is the least value among all the other closed packed structures.

Q.3. What does the packing efficiency of a crystal structure tell us?
Ans: The packing efficiency tells us about the percentage of total space filled by the constituent particles in the unit cell. A vacant space that is not occupied by the constituent particles in the unit cell is known as void space.

Q.4. Which type of packed structure has maximum packing efficiency? 
Ans: Packing efficiency in \({\rm{hcp}}\) (Hexagonal closed packing) and \({\rm{ccp}}\) (Cubic closed packing) structures is maximum, having the value \(74\%\).

Q.5. What is the relationship between coordination number and packing efficiency? 
Ans: The number of spheres through which each sphere is in contact with its two neighbouring spheres is known as its coordination number. While the percentage of total space filled by the constituent particles in the unit cell is called the packing efficiency. The packing efficiency is directly related to the coordination number. The greater the coordination number more will be the packing efficiency of the lattice.