L2 - Bonding and Crystal Structure

Bonding in Minerals

Introduction

The properties of a mineral are determined by the types of atoms present and their spatial arrangement. For example, diamond and graphite, both pure carbon, exhibit vastly different properties due to their differing atomic arrangements (hardness, transparency, electrical conductivity). Similarly, diamond and sphalerite ($ZnS$) share atomic arrangements but differ in elemental composition, leading to distinct properties. The bonding in a crystal is the crucial link between elements and their positions.

Types of Bonds

Several types of bonds exist in crystals, including covalent, ionic, metallic, van der Waals, and hydrogen bonds. Crystals often feature ionic bonds. In ice, covalent bonds within water molecules and hydrogen bonds between molecules are significant. In reality, covalent and ionic bonds represent extremes on a spectrum, with most bonds exhibiting both ionic and covalent characteristics.

Formation of Ions

Given the prevalence of ionic bonds in crystals, understanding ion formation is essential. Pauli’s exclusion principle dictates that no two electrons can have the same set of quantum numbers, leading to the filling of electron shells. Electrons initially fill shells with unpaired spins, then pair up.

The order in which shells are filled is 's', 'p', then 'd'. There is one degenerate 's' shell, three degenerate 'p' shells, and five degenerate 'd' shells. Atoms tend to gain or lose electrons to achieve filled or empty shells, a thermodynamically favorable state.

• Example:

  • Na has one unpaired electron in its 's' shell and readily loses it to become $Na^+$, achieving a completely filled set of shells.

  • Elements with filled 's' shells and empty 'p' shells (e.g. $Mg$ , $Ca$ ) lose two electrons to become 2+ ions.

  • Elements with one 'p' electron (e.g. $Al$ ) lose the 'p' electron and both 's' electrons to become a 3+ ion.

  • Elements with two 'p' electrons and two 's' electrons (e.g. $Si$ ) lose all four to become a 4+ ion.

  • O gains two electrons to fill its 'p' shell, becoming $O^{2-}$.

  • Halogens gain one electron, becoming 1- ions.

Some elements, like $Ni$ , prefer to remain neutral ($Ni^0$) and form metallic bonds. This accounts for the existence of pure metallic $Ni$ lodes, and similarly for $Au$, $Ag$, and to a lesser extent io$Cu$ $ and $Fe$. Some elements adopt multiple oxidation states; examples include $S$ (from 2- to 6+, including $S^0$) and $Fe$ ($Fe^{2+}$ and $Fe^{3+}$).

Ionic Interactions and Coordination

Ions with opposite charges attract each other until a net neutral charge is achieved. For instance, $Na^+$ and $Cl^-$ attract to form $NaCl$, or one $Ti^{4+}$ ion attracts two $O^{2-}$ions to create $TiO_2$. While ions may be treated as electrically homogenous, electrons occupy the outer parts of an atom while the nucleus is at the center. Consequently, when ions approach too closely, their nuclei repel each other. The equilibrium distance, or bond distance, is the balance between the attraction of opposite charges and the repulsion of nuclei.

The attraction of oppositely charged ions is determined by Coulomb’s Law:

$F_{A}$ $\alpha$ $\frac{q^{+}\cdot q^{-}}{d^2}$

• $F_A$ - Force of attraction

• $q^{+/-}$ - Charge of positive/negative ion

• $d$ - Distance between ions

The repulsion of ionic nuclei is determined by Born repulsion:

$F_{R}$ $\alpha$ $-\frac{n}{d^{n+1}}$

• $F_R$ - Force of repulsion

Ions tend to form multiple bonds in three dimensions to distribute charge, which is energetically favorable. The number of ions surrounding a central ion is the coordination number. In crystals, coordination numbers typically refer to the number of oxygen ions around a cation (although this can vary, such as the number of $Cl$ around $Na$ in halite).

Atomic Arrangements and Close Packing

Ions can coordinate in different arrangements. A cubic arrangement (e.g., $NaCl$ ) has a coordination number of 6 ([6]). When considering these arrangements, atoms are treated as hard spheres that do not overlap. Close packing occurs when these spheres touch.

The cubic arrangement isn't the most efficient packing method. Shifting atoms in neighboring rows allows for hexagonal or cubic close packed systems. The distinction lies in the 3D layering pattern: …ABABABA… indicates hexagonal symmetry, whereas …ABCABCABCA… indicates cubic symmetry. Both of these systems have a coordination number of [12].

Cation Size and Coordination Number

The coordination number adopted by an element is determined by the sizes of the ions involved in bonding. Considering $O^{2-}$ as the anion, varying numbers of oxygen ions can pack in different arrangements, creating cavities for cations of different sizes. Ideally, a cation should fit into the cavity without excess space or forcing the $O^{2-}$ anions away from their equilibrium bond distance. In reality, a range of sizes can fit into these different close-packed $O^{2-}$ arrangements.

As a general rule, small cations fit in small cavities formed by fewer oxygen anions. For example, a small cation like $Si^{4+}$ fits into the cavity created by four close-packed oxygen anions. This arrangement, resembling the H atoms in methane, is a tetrahedron. The tetrahedron is the basic building block of most minerals and essentially every rock due to the prevalence of Si.

Larger cations (e.g., $Fe^{2+}$, $Mg^{2+}$) prefer larger cavities and are coordinated by more oxygen ions. They fit into the cavity generated by six close-packed oxygen ions in an octahedron (two square-based pyramids sharing a square). Octahedra are also fundamental building blocks of rocks because Mg and Fe are abundant in the Earth, especially in the mantle.

Still larger cations (e.g., $Ca^{2+}$) can be surrounded by eight oxygen ions in a cube, and very large cations (e.g., $Na^+$, $K^+$) prefer being surrounded by 12 oxygen ions in a cuboctahedron. These polyhedra, their packing, and their arrangement relative to each other are critical to understanding the behavior of minerals and rocks. Polyhedra and calculating the ideal cation size for different numbers of anions will be covered in the tutorial.