Three-Dimensional Close Packing Meaning
Three-Dimensional close packing in solids is referred to as putting the second square closed packing exactly above the first. It is possible to form Three-Dimensional close packing. In this tight packing, the spheres are horizontally and vertically correctly balanced. Similarly, we can obtain a simple Cubic lattice by adding more layers, one above the other. This can be done in two ways:
Three-Dimensional close packing from two-Dimensional square close-packed layers: By putting the second square closed packing exactly above the first, it is possible to form Three-Dimensional close packing. In this tight packing, the spheres are horizontally and vertically correctly balanced. Similarly, we can obtain a simple Cubic lattice by adding more layers, one above the other.
Three-Dimensional close packing from two-Dimensional Hexagonal close-packed layers: With the assistance of two-Dimensional Hexagonal packed layers, Three-Dimensional close packing can be shaped in two ways:
Stacking over the first layer on the second layer
Stacking over the second layer on the third layer
This is the space lattice's actual structure. It occurs because of the unit cells' Three-Dimensional structure. Now, the continuous and repeated stacking of the two-Dimensional structures above each other shapes this structure. It can happen in two ways:
Hexagonal Closest Packing: Here, the alternating layers fill the distance between each other. In one layer, spheres align to fit into the holes of the previous layer. There is the same alignment for the first and the third line. So we call this sort of ABA type.
Cubic Closest Packing: The layers are arranged in symmetry exactly above each other here. This form takes the shape of a cube, hence the name. The coordination number of a system of this kind is 12.
The word "closest packed structures" refers to the crystal structures (lattices) with the most tightly packed or space-efficient composition. The spheres must be arranged as close as possible to each other to maximize the efficiency of packing and minimize the amount of unfilled space. Let us retain the Hexagonal close packing in the first layer to establish Three-Dimensional close packings. Each sphere in the second layer rests in the hollow at the center of the Three touching spheres in the layer for near packing, as shown in the figure.
Strong lines represent the spheres in the first layer, while split lines show those in the second layer. It should be remembered that the spheres in the second layer (either b or c) occupy just half of the triangular voids in the first layer. In the first sheet, unoccupied hollows or voids are indicated by (c). It is found that a tetrahedral void is formed wherever a sphere of the second layer is above the void of the first layer. Whereas in other cases, it is found that the triangular voids in the second layer are above the triangular voids in the first layer, in such a way that the triangular shapes of these voids do not overlap. These voids are classified as octahedral voids and are surrounded by six spheres.
We can calculate the number of these types of voids in the following way:
Let the number of close-packed spheres be N, then
The number of octahedral voids generated = N
The number of tetrahedral voids generated = 2N
From this, we can conclude that the number of octahedral voids generated is equal to the number of close-packed spheres. The number of tetrahedral voids generated is equal to 2 times the number of close-packed spheres.
Close Packing in Crystals
Close packing in crystals is also referred to as close packing in solids which is defined as the efficient arrangement of constituent particles in a crystal lattice in a vacuum. We have to assume that all particles (atoms, molecules, and ions) are of the same spherical solid form to understand this set more clearly. So a Cubic shape is the unit cell of a lattice. Now, there will still be some empty spaces when we stack spheres in the cell. The arrangement of these spheres has to be very effective to reduce these empty spaces. To minimize empty spaces, the spheres should be arranged as close together as possible.
The definition of Coordination Number is another connected one. In a crystal lattice structure, the coordination number is the number of atoms that surround a central atom. It is often referred to as LIGANCE. Inside a unit cell, the most powerful conformation atomic spheres can take is known as the nearest packing configuration. In two modes, there are densely packed atomic spheres: Hexagonal closest packing (HCP) and Cubic closest packing (CCP).
FAQs on Close Packing in Three Dimensions
1. What is the fundamental concept of close packing in three dimensions?
Close packing in three dimensions refers to the most efficient arrangement of constituent particles (atoms, ions, or molecules, treated as hard spheres) in a crystal lattice. The primary goal is to minimise the empty space and maximise the density of the structure, which leads to a state of maximum stability. This is achieved by stacking two-dimensional close-packed layers on top of one another.
2. How are three-dimensional close-packed structures formed from two-dimensional layers?
Three-dimensional close packing is typically built from two-dimensional hexagonal close-packed layers, as they are more efficient than square-packed layers. The process involves:
- First Layer (A): A standard hexagonal close-packed layer.
- Second Layer (B): The spheres of the second layer are placed in the depressions (voids) of the first layer. This covers half of the triangular voids present in the first layer.
- Third Layer: Placing the third layer determines the final structure, leading to two primary types of packing.
3. What are the two main types of three-dimensional close packing that arise from stacking hexagonal layers?
The two main types are determined by the placement of the third layer relative to the first two:
- Hexagonal Close Packing (HCP): This occurs when the spheres of the third layer are placed directly above the spheres of the first layer. This creates an ABAB... stacking pattern.
- Cubic Close Packing (CCP) or Face-Centred Cubic (FCC): This occurs when the spheres of the third layer are placed in the remaining depressions, not directly above the first layer's spheres. This creates an ABCABC... stacking pattern.
4. What is the key difference between Hexagonal Close Packing (HCP) and Cubic Close Packing (CCP)?
While both HCP and CCP have the same packing efficiency and coordination number, their primary difference lies in the stacking sequence of layers:
- Stacking Pattern: HCP follows an ABAB... pattern, where the third layer is a repetition of the first. CCP follows an ABCABC... pattern, where the third layer is unique, and the fourth layer is a repetition of the first.
- Lattice Structure: This difference in stacking results in different crystal symmetries. HCP gives rise to a hexagonal unit cell, while CCP results in a face-centred cubic (FCC) unit cell.
5. What are tetrahedral and octahedral voids, and why are they significant in crystal structures?
Voids are the empty spaces left between the spheres in a close-packed structure. Their significance lies in housing smaller atoms or ions in ionic crystals.
- A tetrahedral void is a small empty space surrounded by four spheres arranged in a tetrahedral geometry.
- An octahedral void is a larger empty space surrounded by six spheres arranged in an octahedral geometry.
In a structure with 'N' spheres, there are always 2N tetrahedral voids and N octahedral voids. The type, size, and location of these voids are crucial for determining the formula and structure of many ionic compounds.
6. How is the coordination number determined in HCP and CCP structures?
The coordination number is the number of nearest neighbouring spheres touching any given sphere. In both HCP and CCP structures, the coordination number is 12. This is because any sphere is in direct contact with:
- 6 spheres in its own layer.
- 3 spheres in the layer above it.
- 3 spheres in the layer below it.
This high coordination number is a characteristic of a maximally packed structure.
7. What does packing efficiency mean, and why is it identical for both HCP and CCP?
Packing efficiency is the percentage of the total volume of a unit cell that is actually occupied by the spheres. It measures how tightly the particles are packed. For both HCP and CCP arrangements, the packing efficiency is 74%, which is the highest possible for spheres of equal size. It is identical for both because the local arrangement of spheres and the number of nearest neighbours are the same, leading to the most efficient use of space possible, despite their different long-range stacking patterns.
8. What are some real-world examples of elements with HCP and CCP structures?
Many common metals adopt these highly efficient packed structures at room temperature:
- Examples of HCP: Magnesium (Mg), Zinc (Zn), Titanium (Ti), and Cobalt (Co).
- Examples of CCP (FCC): Copper (Cu), Silver (Ag), Gold (Au), Aluminium (Al), and Nickel (Ni).
The type of packing influences the metal's physical properties, such as its ductility and malleability.
