The unit cell of a compound containing potassium, aluminum, and fluorine is shown here. (a) What type of lattice does this crystal possess (all three lattice vectors are mutually perpendicular)?

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1. Identify the given parameters of the unit cell: a = 4.562 Å, b = 7.267 Å, c = 9.726 Å, α = 70.498°, β = 90.000°, γ = 90.000°.
2. Recognize that all three lattice vectors are mutually perpendicular (β = γ = 90°).
3. Note that α is not equal to 90°, which indicates that the unit cell is not orthorhombic, tetragonal, or cubic.
4. Determine that the unit cell with α ≠ 90° and β = γ = 90° is a monoclinic lattice.
5. Conclude that the crystal possesses a monoclinic lattice structure.

Key Concepts

Crystal Lattice

A crystal lattice is a three-dimensional arrangement of atoms or molecules in a crystalline material. It is defined by its unit cell, which is the smallest repeating unit that reflects the symmetry and structure of the entire crystal. The arrangement of these unit cells determines the properties of the material, including its symmetry, density, and how it interacts with light.

Unit Cell Parameters

Unit cell parameters include the lengths of the edges (a, b, c) and the angles between them (α, β, γ) that define the geometry of the unit cell. These parameters are crucial for identifying the type of crystal system, such as cubic, tetragonal, or orthorhombic. In the given question, the specific values and angles help determine the lattice type of the compound.

Types of Crystal Lattices

Crystal lattices can be classified into several types based on their geometric arrangement. Common types include simple cubic, body-centered cubic, and face-centered cubic, each characterized by specific arrangements of atoms and angles. The question indicates that all three lattice vectors are mutually perpendicular, suggesting a cubic or orthogonal lattice structure, which is essential for determining the type of lattice present in the compound.