Summary
Highlights
The lecture begins by revisiting the comprehensive scope of solid-state chemistry, emphasizing that previous topics like electronic structure and various bonding types form the foundation for understanding solids. The classification of solids relies on two key aspects: bonding type and the arrangement of atoms. This week's focus is on the arrangement of atoms, which will later be characterized using X-ray diffraction. Solids are categorised into two main types based on atomic arrangement: crystalline solids, which exhibit regular, long-range order, and amorphous solids (like glass), which lack this extensive order. The course will primarily focus on crystalline solids, with amorphous solids to be covered after the second exam.
Nature often optimizes for packing efficiency, as seen in patterns like seeds on a plant. Robert Hooke, in the 1600s, explored similar principles by studying cannonball stacking to achieve maximum density. This concept extends to atomic arrangements in crystals. The lecture introduces the idea of a 'unit cell' as the smallest repeating unit that can tile all of space without voids, a concept brilliantly illustrated by M.C. Escher's tessellations. This ability to fill space completely defines a 'crystal system.' Lattice vectors are introduced as the 'stamps' used to replicate the unit cell throughout space.
The concept of packing efficiency is explored using a 2D square lattice with circular atoms. The maximum atomic packing fraction (APF) for this 2D square lattice is calculated to be π/4 or 78%. This calculation demonstrates how to determine the proportion of space occupied by atoms within a given unit cell. It's noted that a 2D hexagonal lattice offers a higher packing efficiency of 91%, showing that atomic arrangement significantly impacts how densely atoms can pack. This efficiency is crucial for atoms in a solid as they seek optimal bonding and arrangement. The lecture then transitions to 3D crystallography.
In three dimensions, there are only seven unique crystal systems, defined by the lengths of their lattice vectors (a, b, c) and the angles between them (α, β, γ). These systems represent all possible ways to fill 3D space without voids. Within these seven systems, Auguste Bravais identified 14 unique ways to arrange lattice points, known as Bravais lattices. The course will focus specifically on the cubic crystal system, as it provides a solid foundation for understanding crystallography and is adopted by many elements. The cubic system is characterized by equal lattice vector lengths (a=b=c) and orthogonal angles (α=β=γ=90°). There are three Bravais lattices within the cubic system: simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC).
The simple cubic (SC) lattice is introduced, characterized by atoms at each corner of the cube. The coordination number (nearest neighbors) for SC is 6. To calculate the atomic packing fraction (APF), the radius of the atom (r) is related to the lattice parameter (a) such that r = a/2. The volume of an atom is 4/3πr³, and since there is effectively one atom per unit cell (8 corners x 1/8 atom/corner), the APF for SC is 52%. This low packing efficiency makes SC structures rare in nature, with polonium being the only known element to adopt this form. A video illustrates the slicing of the unit cell to visualize atom portions.
The body-centered cubic (BCC) lattice is discussed next, where atoms are located at each corner and one atom is in the center of the cube. The coordination number for BCC is 8. The close-packed direction, where atoms touch, is along the body diagonal (length a√3). In this configuration, the radius of the atom is r = a√3/4. With two atoms per unit cell (8 corners x 1/8 + 1 body-centered atom), the APF for BCC is 68%. This higher packing efficiency makes BCC a more common structure in metals compared to SC. A video demonstrates the BCC unit cell and its atom count.
Finally, the face-centered cubic (FCC) lattice is presented. FCC has atoms at each corner and at the center of each face. The coordination number for FCC is 12. The close-packed direction is along the face diagonal (length a√2). Thus, the atomic radius is r = a√2/4. With four atoms per unit cell (8 corners x 1/8 + 6 face-centered atoms x 1/2), the APF for FCC is 74%, representing the highest packing efficiency among the cubic systems. Many metals adopt the FCC structure due to this efficient packing. The lecture concludes with a video illustrating the FCC unit cell and a comparison of packing efficiencies across the three cubic lattice types.