Show that the minimum cation-to-anion radius ratio for a coordination number of 6 is \(0.414 .[\) Hint: Use the NaCl crystal structure (Figure 12.2 ), and assume that anions and cations are just touching along cube edges and across face diagonals.

The crystal structure describes the way atoms, ions, or molecules are arranged in a material. It's a crucial concept in solid-state physics and chemistry, influencing the properties and characteristics of the material. The structure is defined by the unit cell, which is the smallest repeating unit within the crystal and outlines the entire lattice when repeated in space.

Different materials can have various crystal structures, such as body-centered cubic, face-centered cubic, and hexagonal close-packed. In the context of ionic solids, like sodium chloride (NaCl), the arrangement of ions is determined by factors like ionic radii and charge balance. Understanding the crystal structure is essential for predicting behavior such as conductivity, solubility, and melting point.

The coordination number is the number of nearest neighbor particles surrounding a central particle within a crystal structure. This number reflects how an atom, ion, or molecule is bonded to surrounding particles and contributes to the stability of the structure.

Common coordination numbers range from 2 to 12, with numbers 4 (tetrahedral), 6 (octahedral), and 8 (cubic) being typical for many compounds. The coordination number is influenced by the size of the ions involved; larger ions often accommodate more neighbors. For instance, in the NaCl crystal structure, each ion has a coordination number of 6, implying an octahedral configuration for the ionic bonds.

The term ionic radii refers to the effective distance from the nucleus of an ion to its outermost electron shell. This measurement can vary greatly depending on the ion's charge and the number of electrons. The size of an ion is key to determining how they pack together in a crystal, ultimately affecting the coordination number and crystal structure.

When considering cation-anion pairs in structures like NaCl, the cation-to-anion radius ratio becomes especially important. It determines the type of crystal lattice that will form and the most efficient way that the ions can be packed together. A larger ratio means larger cations or smaller anions, and this ratio can determine the stability and type of ionic crystal that forms.

The NaCl structure is often used as a model for understanding ionic solids. This structure features a cubic unit cell, with sodium ions (Na+, or cations) and chloride ions (Cl-, or anions) arranged in a way that each ion type forms its own separate cubic lattice. The NaCl structure is characterized by each sodium ion being surrounded by six chloride ions and vice versa, which defines its octahedral coordination.

In this arrangement, the cations and anions are close packed and just touching each other along the cube's edges. This NaCl-type structure, also known as rock salt structure, is common for compounds where cations and anions have similar sizes and a 1:1 stoichiometry. It’s an excellent example to illustrate concepts like ionic radii and coordination number in the context of crystal packing.