10.6 Lattice Structures in Crystalline Solids

Crystalline structures are like building blocks of solids, with atoms arranged in repeating patterns. These patterns form unit cells, the smallest repeating units, which come in different types like primitive cubic and body-centered cubic.

X-ray diffraction helps us peek inside crystals, revealing their atomic arrangements. By bouncing X-rays off atoms and analyzing the resulting patterns, scientists can uncover the secrets of a crystal's structure and composition.

Crystalline Structures and Arrangements

Arrangement of atoms in crystals

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• Crystalline solids consist of atoms, ions, or molecules arranged in a regular, repeating pattern that extends in three dimensions
• The smallest repeating unit that displays the full symmetry of the crystal structure is known as the unit cell
  • Unit cells are characterized by their lengths (a, b, c) and angles (α, β, γ)
• Three main types of unit cells include:
  • Primitive cubic (P): contains atoms at each corner of the cube (e.g., polonium)
  • Body-centered cubic (I): features atoms at each corner and one atom in the center of the cube (e.g., sodium)
  • Face-centered cubic (F): has atoms at each corner and at the center of each face of the cube (e.g., copper)
• Other common crystal structures:
  • Hexagonal close-packed (HCP): consists of a two-dimensional hexagonal lattice with a repeating ABABAB pattern (e.g., magnesium)
  • Cubic close-packed (CCP) or face-centered cubic (FCC): comprises a two-dimensional hexagonal lattice with a repeating ABCABC pattern (e.g., gold)
• The coordination number, which is the number of nearest neighbors an atom or ion has in a crystal structure, varies depending on the lattice type

Crystal Lattice Properties

• Packing efficiency refers to the percentage of space occupied by atoms or ions in a crystal structure
• Miller indices (h, k, l) are used to describe planes and directions in crystal lattices
• Bravais lattices are the 14 unique three-dimensional lattice structures that form the basis for all crystalline materials

Calculation of ionic radii

• Ionic radii can be calculated using the edge length of the unit cell and the radius ratio of the cation and anion
• For a primitive cubic unit cell with edge length $a$:
  1. $r_\text{cation} + r_\text{anion} = a$
• For a body-centered cubic unit cell with edge length $a$:
  1. $r_\text{cation} + r_\text{anion} = \frac{\sqrt{3}}{2}a$
• For a face-centered cubic unit cell with edge length $a$:
  1. $r_\text{cation} + r_\text{anion} = \frac{a}{2\sqrt{2}}$

X-ray Diffraction and Crystalline Structure Determination

X-ray diffraction for crystal structures

• X-ray diffraction (XRD) is a powerful technique used to determine the arrangement of atoms in a crystalline solid
• X-rays scatter off the electrons in the atoms of the crystal, producing a diffraction pattern due to the interference of the scattered X-rays
• ###Bragg's_law_0### describes the relationship between the wavelength of the X-rays ($\lambda$), the angle of incidence ($\theta$), and the interplanar spacing ($d$) in the crystal:
  • $n\lambda = 2d\sin\theta$, where $n$ is an integer (e.g., 1, 2, 3)
• The diffraction pattern provides crucial information about the crystal structure, such as the size and shape of the unit cell and the positions of atoms within the unit cell
• The intensities of the diffraction peaks are determined by the types and positions of atoms in the unit cell
  • This information is used to generate an electron density map, which reveals the atomic structure of the crystal (e.g., the arrangement of atoms in a protein molecule)