Question

Use the counting of atoms principle to determine the ratio of octahedral holes:tetrahedral holes:atoms for hexagonal close packing.

Short answer

In a hexagonal close packing (hcp) arrangement, the ratio of octahedral holes, tetrahedral holes, and atoms is 1:2:1. There are equal numbers of octahedral holes and atoms, and twice as many tetrahedral holes.

Step-by-step solution

Understanding the hexagonal close packing arrangement

Hexagonal close packing (hcp) is one of the common close packing arrangements in crystals. In hcp, the atoms are arranged in layers, with the first layer (Layer A) having a hexagonal arrangement of atoms. The second layer (Layer B) has its atoms fit into the triangular spaces between the atoms of Layer A, and the third layer (Layer A') has atoms exactly above Layer A. This pattern repeats itself in the crystal (ABAB...).

Counting octahedral holes

Octahedral holes are formed by the bridging of six atoms in a close-packed structure, with an atom from the upper layer, three surrounding atoms from the middle layer, and two atoms from the lower layer. In hcp, every atom in Layer B forms one octahedral hole with the atoms of Layers A and A'. Since there are as many atoms in Layer B as there are in Layer A, the ratio of octahedral holes to atoms is 1:1.

Counting tetrahedral holes

Tetrahedral holes are formed by the bridging of four atoms, with one atom from the upper layer and three surrounding atoms from the lower layer. In hcp, every atom in Layer A has one tetrahedral hole below it, while every atom in Layer A' has one tetrahedral hole above it. Since there are as many atoms in Layer A as there are in Layer A', the ratio of tetrahedral holes to atoms is 2:1.

Determining the final ratio

Now that we know the ratio of octahedral holes to atoms (1:1) and the ratio of tetrahedral holes to atoms (2:1), we can combine these ratios to determine the overall ratio of octahedral holes, tetrahedral holes, and atoms in hexagonal close packing: Octahedral holes : Tetrahedral holes : Atoms = 1 : 2 : 1 So, there are equal numbers of octahedral holes and atoms, and twice as many tetrahedral holes in hexagonal close packing.

Key concepts

Counting of Atoms Principle

When exploring the structure of crystalline solids, it's crucial to grasp how the counting of atoms principle operates. This principle helps to determine how many atoms, octahedral holes, and tetrahedral holes are present in a given crystal packing.

In the context of hexagonal close packing (hcp), this principle tells us how to count atoms in a repetitive lattice without overcounting. We look at how the atoms are arranged and how the holes are formed within the structure. Since hcp crystals follow an ABAB... pattern, where Layer A is directly above Layer A' and different from Layer B, we can navigate the stacking to accurately tally the atoms and resulting holes.

Applying the Principle

To apply the counting of atoms principle, one must identify the repeating units, which are usually referred to as the 'unit cell'. In the case of hcp, each atom in Layer A is considered in conjunction with Layer B and Layer A' to determine the formation of octahedral and tetrahedral holes.

Octahedral Holes

Diving deeper into hexagonal close packing, we come across octahedral holes, which play a pivotal role in the structure's stability and properties.

Imagine six atoms configured in such a way that they form the vertices of an octahedron -- this space at the center is what we call an octahedral hole. In hcp, this space is created when atoms from three different layers come together: one atom from Layer A, four from Layer B forming a square base, and one from Layer A' just above the original Layer A.

Counting Octahedral Holes

To count the octahedral holes following the principle, we consider each atom in Layer B and see how it partners with atoms from Layers A and A' to create these holes. The count reveals a one-to-one correspondence since each atom in Layer B contributes to one octahedral hole. This one-to-one ratio is a crucial aspect of hcp's geometry.

Tetrahedral Holes

Similarly, tetrahedral holes constitute another fundamental aspect of the hcp structure.

These holes are formed by the connection of four atoms that occupy the corners of a tetrahedron. In our hcp arrangement, any atom in Layers A or A' serves as the apex of a tetrahedron, with three atoms from the alternate layer (Layer B or A') forming the base.

Calculating Tetrahedral Holes

For every atom in Layer A, there exists one tetrahedral hole beneath it, and conversely, for every atom in Layer A', there is one tetrahedral hole above it. Since the hcp structure alternates between Layers A and A', we deduce that there are twice as many tetrahedral holes as there are atoms in a single layer. This ratio, 2:1 tetrahedral holes to atoms, is essential for understanding the overall distribution of spaces in the hcp lattice.