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Chapter 12: Problem 10

Beryllium oxide (BeO) may form a crystal structure that consists of an HCP arrangement of \(\mathrm{O}^{2-}\) ions. If the ionic radius of \(\mathrm{Be}^{2+}\) is \(0.035 \mathrm{~nm}\), then (a) Which type of interstitial site will the \(\mathrm{Be}^{2+}\) ions occupy? (b) What fraction of these available interstitial sites will be occupied by \(\mathrm{Be}^{2+}\) ions?

Short Answer

Expert verified
Answer: (a) In BeO, the Beryllium ions (Be虏鈦�) occupy octahedral interstitial sites. (b) The fraction of occupied octahedral interstitial sites is 1 (100%).

Step by step solution

01

Understanding HCP structure and interstitial sites

In a hexagonal close-packed (HCP) structure, there are two types of interstitial sites: tetrahedral and octahedral. Tetrahedral sites are surrounded by four atoms, while octahedral sites are surrounded by six atoms. To determine which type of interstitial site the \(\mathrm{Be}^{2+}\) ions will occupy, we need to calculate the minimum size of each interstitial site and compare it to the ionic radius of \(\mathrm{Be}^{2+}\).
02

Calculating critical radius ratio for interstitial sites

The critical radius ratio is the minimum ratio of the ionic radius of the cation (in this case, \(\mathrm{Be}^{2+}\)) to the anion (in this case, \(\mathrm{O}^{2-}\)) needed for a stable crystal structure. For tetrahedral sites, the critical radius ratio is 0.225, while for octahedral sites, it is 0.414. The interstitial site will be occupied by the \(\mathrm{Be}^{2+}\) ions if the ionic radius of \(\mathrm{Be}^{2+}\) is equal to or greater than the critical radius ratio.
03

Comparing the ionic radius of Be\(^{2+}\) to the critical radius ratio

We are given that the ionic radius of \(\mathrm{Be}^{2+}\) is 0.035 nm. In order to determine the ionic radii ratio of \(\mathrm{Be}^{2+}\)/O\(^{2-}\) we would need to know the ionic radius of \(\mathrm{O}^{2-}\). However, we can determine which interstitial site the Beryllium ions will occupy by examining the given crystal structure and the size of the Beryllium ions. For BeO, since it is an HCP arrangement, it contains equal numbers of Beryllium and Oxygen ions. If the tetrahedral sites were to be occupied, the number of Beryllium ions would be double the number of Oxygen ions, resulting in a Be\(_{2}\)O crystal. However, our given crystal is BeO, which means that not all of tetrahedral interstitial sites are occupied. Instead, the Beryllium ions will occupy all the octahedral interstitial sites resulting in a 1:1 ratio of Beryllium and Oxygen ions. Therefore, the answer to (a) is that the \(\mathrm{Be}^{2+}\) ions will occupy octahedral interstitial sites.
04

Calculating the fraction of occupied octahedral interstitial sites

Since the octahedral interstitial sites are fully occupied by the Beryllium ions in the BeO crystal structure, the fraction of occupied octahedral interstitial sites is 1 (100%). Hence, the answer to (b) is that the fraction of occupied octahedral interstitial sites is 1 (100%).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hexagonal Close Packed (HCP) Structure
The Hexagonal Close Packed (HCP) structure is one of the most efficient and common ways that atoms can pack together in a solid. In this arrangement, each atom is surrounded by twelve others, providing a dense packing. This structure is characterized by its unique geometric stacking, which can be described as alternating layers of atoms in a repeated ABAB pattern.

Each "A" and "B" layer is hexagonally arranged, with the "B" layer being slightly offset so that its atoms fit into the spaces between atoms in the "A" layer. This hexagonal symmetry leads to a structure that is compact and strong, often seen in metals such as magnesium and titanium.

Because beryllium oxide forms an HCP arrangement with oxygen ions, understanding this structure is crucial to analyzing how different ions, such as the Be\(^{2+}\), fit within the lattice.
Interstitial Sites
In crystal lattices like the HCP structure, interstitial sites are the small spaces or voids where atoms or ions can fit between the larger lattice atoms. These sites are essential in understanding how different ions can pack within a crystal structure.

Within an HCP lattice, interstitial sites occur in two primary forms: tetrahedral and octahedral.
  • Tetrahedral Sites: These sites are surrounded by four atoms, forming a simple tetrahedron. They tend to be smaller and hold smaller ions or atoms.
  • Octahedral Sites: These sites are surrounded by six atoms in an octahedral arrangement, offering more space than tetrahedral sites, suitable for larger ions.
The understanding of which site a particular ion will occupy is determined by the ion size in relation to the site's available space.
Critical Radius Ratio
The concept of the critical radius ratio is crucial when determining which interstitial site an ion will occupy in a crystal lattice. This ratio is the minimum size ratio required between the cation and anion for a stable crystal structure.

For instance, in an HCP arrangement, we must consider whether the ionic radius of the cation, such as Be\(^{2+}\) in the exercise, fits best in tetrahedral or octahedral sites.
  • Tetrahedral Sites: Require a critical radius ratio of at least 0.225 for stability.
  • Octahedral Sites: Require a larger critical radius ratio of 0.414.
Given the ionic radius for Be\(^{2+}\) as 0.035 nm, this smaller size favors the occupation of octahedral sites, which require a more lenient ratio compared to tetrahedral sites.
Octahedral Sites
Octahedral sites are a type of interstitial site formed by six surrounding atoms in a crystal lattice, creating an octahedral shape. These sites are typically larger than tetrahedral sites and provide more room, making them ideal for larger ions to occupy.

In the context of beryllium oxide and the HCP structure of oxygen ions, the Be\(^{2+}\) cations will settle in these octahedral sites. This positioning is driven by two factors: the size of the Be\(^{2+}\) ions fitting the available space in octahedral voids and maintaining the stoichiometry of the BeO compound with a 1:1 ratio of beryllium to oxygen ions.

Since each oxygen ion arrangement creates enough octahedral sites to accommodate an equal number of beryllium ions, these sites are fully occupied, affirming the precise balance in this BeO crystal lattice.

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Most popular questions from this chapter

Using the Molecule Definition Utility found in both 鈥淢etallic Crystal Structures and Crystallography鈥� and 鈥淐eramic CrystalStructures鈥� modules of VMSE, located on the book鈥檚 website [www.wiley.com/college/callister (Student Companion Site)], generate (and print out) a three-dimensional unit cell for lead oxide, PbO, given the following: (1) The unit cell is tetragonal with a 0.397 nm and c 0.502 nm, (2) oxygen atoms are located at the following point coordinates:\(\begin{array}{llll}0 & 0 & & 001 \\ 10 & 0 & & 101 \\ 0 & 1 & 0 & & 011 \\ \frac{1}{2} \frac{1}{2} & 0 & & \frac{1}{2} \frac{1}{2} 1\end{array}\) and (3) \(\mathrm{Pb}\) atoms are located at the following point coordinates: $$ \begin{array}{ll} \frac{1}{2} 00.763 & 0 \frac{1}{2} 0.237 \\ \frac{1}{2} 10.763 & 1 \frac{1}{2} 0.237 \end{array} $$

For each of the following crystal structures, represent the indicated plane in the manner of Figures \(3.12\) and \(3.13\), showing both anions and cations: (a) (100) plane for the cesium chloride crystal structure (b) (200) plane for the cesium chloride crystal structure (c) (111) plane for the diamond cubic crystal structure (d) \((110)\) plane for the fluorite crystal structure

In terms of bonding, explain why silicate mate- rials have relatively low densities.

For a ceramic compound, what are the two char- acteristics of the component ions that determine the crystal structure?

Would you expect Frenkel defects for anions to exist in ionic ceramics in relatively large concen- trations? Why or why not?
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