10.6 Lattice Structures in Crystalline Solids

Crystalline Structures and Arrangements

Crystalline solids have their atoms, ions, or molecules arranged in regular, repeating 3D patterns. Understanding these patterns tells you a lot about a solid's physical properties, from how dense it is to how it conducts electricity. The key concept here is the unit cell, which is the smallest repeating unit that shows the full symmetry of the crystal.

Arrangement of Atoms in Crystals

A unit cell is defined by three edge lengths ($a$, $b$, $c$) and three angles ($\alpha$, $\beta$, $\gamma$). Think of it as the single "tile" that, when copied over and over in all directions, builds the entire crystal.

There are three cubic unit cells you need to know:

• Primitive cubic (P): Atoms sit only at each corner of the cube. Each corner atom is shared among 8 neighboring unit cells, so a primitive cubic cell effectively contains just 1 atom. Polonium is the classic example.
• Body-centered cubic (BCC): Atoms at each corner plus one atom right in the center of the cube. That gives an effective count of 2 atoms per unit cell. Sodium and iron crystallize this way.
• Face-centered cubic (FCC): Atoms at each corner and one in the center of each face. Each face atom is shared between 2 cells, giving 4 atoms per unit cell. Copper and aluminum are common examples.

Two close-packed structures also come up frequently:

• Hexagonal close-packed (HCP): Layers of atoms stack in an ABABAB pattern, where every other layer lines up directly. Magnesium is a good example.
• Cubic close-packed (CCP): This is actually the same as FCC. Layers stack ABCABC, so the third layer is offset from both the first and second. Gold crystallizes this way.

The coordination number is how many nearest neighbors surround a given atom. For primitive cubic it's 6, for BCC it's 8, and for both FCC and HCP it's 12.

Crystal Lattice Properties

Packing efficiency is the percentage of space in the unit cell actually occupied by atoms. Higher packing efficiency means less empty space:

• Primitive cubic: ~52%
• BCC: ~68%
• FCC / HCP: ~74% (these are the most efficiently packed)

Miller indices ($h$, $k$, $l$) are a notation system used to describe specific planes and directions within a crystal lattice. You won't need to calculate these in most intro courses, but you should know they exist because they show up in X-ray diffraction analysis.

There are 14 Bravais lattices, which are the only unique 3D lattice arrangements possible. Every crystalline material fits into one of these 14 categories.

Calculation of Ionic Radii

You can figure out ionic radii from the geometry of the unit cell if you know the edge length $a$. The relationship between the cation radius ($r_\text{cation}$), anion radius ($r_\text{anion}$), and edge length depends on which type of unit cell you're dealing with:

• Primitive cubic: Ions touch along the edge of the cube.
  • $r_\text{cation} + r_\text{anion} = \frac{a}{2}$
• Body-centered cubic: Ions touch along the body diagonal (corner to center to opposite corner).
  • $r_\text{cation} + r_\text{anion} = \frac{\sqrt{3}}{2}a$
• Face-centered cubic: Ions touch along the face diagonal.
  • $r_\text{cation} + r_\text{anion} = \frac{a}{2\sqrt{2}}$

To use these, you typically need one radius given (or a ratio between the two), plus the edge length from experimental data.

X-ray Diffraction and Crystalline Structure Determination

X-ray Diffraction for Crystal Structures

X-ray diffraction (XRD) is the main experimental technique for figuring out how atoms are arranged inside a crystal. X-rays have wavelengths on the same scale as the spacing between atoms (around 0.1 to 10 Å), which makes them perfect for probing crystal structures.

Here's how it works:

1. A beam of X-rays hits the crystal.
2. The X-rays scatter off electrons in the atoms, bouncing in many directions.
3. Scattered waves from different layers of atoms interfere with each other. When they line up (constructive interference), you get a bright spot in the diffraction pattern.
4. The pattern of bright spots is recorded and analyzed to determine the crystal's structure.

The condition for constructive interference is given by Bragg's Law:

$n\lambda = 2d\sin\theta$

where $\lambda$ is the X-ray wavelength, $d$ is the spacing between parallel planes of atoms, $\theta$ is the angle at which the X-ray beam hits those planes, and $n$ is a positive integer (1, 2, 3...) representing the order of diffraction.

The positions of the diffraction peaks tell you the size and shape of the unit cell (through the $d$-spacings). The intensities of those peaks tell you what types of atoms are present and where they sit within the unit cell. From the intensities, scientists build an electron density map that reveals the full atomic arrangement of the crystal.