Predicting Crystal Structures: The Radius Ratio Rule

The Problem of Packing Spheres

Let's begin by adopting a simple but effective physical model. Imagine that our ions—cations and anions—are hard, incompressible spheres, each with a defined radius and charge. Building a crystal is like packing these spheres together. The most stable arrangement will be the one that minimizes the total electrostatic potential energy. This is achieved by adhering to two principles:

1. Maximizing Attraction: Each cation should be surrounded by and in contact with as many anions as possible.
2. Minimizing Repulsion: Ions with the same charge (anion-anion or cation-cation) should be kept as far apart as possible.

These principles give rise to the concept of a coordination number (CN), which is the number of nearest-neighbor anions surrounding a central cation. The following video provides an excellent introduction to the core ideas governing how ionic structures assemble.

As the video explains, the primary factor determining the coordination number and the type of "hole" (or interstitial site) a cation will occupy is the relative size of the cation and anion.

The Radius Ratio Rule (Pauling's First Rule)

The geometric stability of an ionic crystal is dictated by the radius ratio, defined as the radius of the cation ($r_C$) divided by the radius of the anion ($r_A$): $r_C/r_A$.

Why is this ratio so important? Consider the arrangement of anions around a cation. The anions, all negatively charged, repel each other. The structure is only stable if the central, positively charged cation is large enough to be in contact with all its neighboring anions, thus keeping them from collapsing in on each other.

This concept is formalized in Linus Pauling's first rule for ionic structures, also known as the coordination principle. The following video offers another clear explanation of this rule and the resulting coordination geometries.

For each type of coordination polyhedron (tetrahedral, octahedral, etc.), there is a specific, critical radius ratio that represents the "stability limit." Below this value, the cation is too small and "rattles" in the site, leading to an unstable structure.

Geometric Derivation of Critical Ratios

Given your background in mathematics and physics, you can appreciate that these critical ratios are not arbitrary numbers but can be derived directly from the geometry of the polyhedra. Let's briefly examine the case for an octahedral coordination (CN=6), which is very common.

Consider a 2D cross-section of the octahedron. The central cation (radius $r_C$) touches four anions (radius $r_A$) in a plane. At the stability limit, the anions also touch each other along the face diagonal of the cube that encloses the structure.

A 2D cross-section of an octahedral site at the stability limit. The cation (red) touches the anions (blue), and the anions touch each other along the diagonal.

From the right triangle shown:

• The hypotenuse is $r_A+r_A=2r_A$.
• The two other sides are both $r_C+r_A$.

Using the Pythagorean theorem:

Taking the square root of both sides:

Thus, 0.414 is the minimum radius ratio for a stable octahedral coordination. Similar geometric arguments can be made for other coordination numbers. The following resource provides these derivations.

Here is a summary of the results:

Applying the Rules: Predictions and Reality

Now we can use this table to predict the coordination number for real ceramic compounds. All we need are the ionic radii, which have been experimentally determined and are available in reference tables.

Example: Magnesium Oxide (MgO)

• Ionic radius of $M{g}^{2+}$ ($r_C$) = 0.072 nm
• Ionic radius of $O^{2-}$ ($r_A$) = 0.140 nm

Calculate the ratio:

Consulting our table, 0.514 falls within the range for a coordination number of 6 (0.414 – 0.732). We therefore predict that in MgO, each $M{g}^{2+}$ ion is octahedrally coordinated by six $O^{2-}$ ions. This prediction matches experimental observations.

Let's see a few more examples in practice.

An Important Caveat

It is crucial to remember that the radius ratio rule is a model based on idealized assumptions (perfectly ionic bonding, hard spheres). In reality, no bond is 100% ionic, and ions can be polarized. Therefore, the rules provide a powerful guideline but are not infallible. Some compounds adopt structures that defy the simple prediction.

Conclusion

In this lesson, you have learned how to use the principles of ionic size and geometric packing to predict the structure of ceramic materials. The key takeaways are:

• The assembly of ionic crystals is driven by maximizing cation-anion attraction while minimizing anion-anion repulsion.
• Coordination Number (CN) describes the number of nearest anions surrounding a cation.
• The radius ratio ($r_C/r_A$) is a powerful geometric tool for predicting the most stable CN and its corresponding coordination polyhedron (e.g., tetrahedral, octahedral).
• We can derive the critical radius ratio limits for each coordination number using geometric principles.
• While highly effective, the radius ratio rule is a simplified model, and its predictions should be considered alongside other factors like bond character and stoichiometry.

You are now able to look at a simple ionic compound, find the radii of its constituent ions, and make an educated prediction about its local atomic arrangement. This is a foundational skill in materials science.

In our next lesson, we will take this a step further. We'll move from predicting the local coordination polyhedron to understanding how these polyhedra connect in three-dimensional space to form the long-range, ordered lattices of specific, named crystal structures like rock salt, fluorite, and perovskite.