Which of the three-dimensional primitive lattices has a unit cell where none of the internal angles is 90? (a) Orthorhombic, (b) hexagonal, (c) rhombohedral, (d) triclinic, (e) both rhombohedral and triclinic.

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Understand the concept of a unit cell in crystallography, which is the smallest repeating unit in a crystal lattice that shows the entire structure's symmetry.

Recognize that a primitive lattice is a type of crystal lattice where each lattice point is occupied by a single atom or molecule.

Recall the characteristics of different types of unit cells: Orthorhombic has all angles at 90 degrees, Hexagonal has two angles at 90 degrees and one at 120 degrees, Rhombohedral has all sides equal and angles equal but not 90 degrees, Triclinic has no angles at 90 degrees, and both Rhombohedral and Triclinic can have none of the internal angles at 90 degrees.

Identify that the question asks for a lattice where none of the internal angles is 90 degrees.

Conclude that the correct options are (d) triclinic and (e) both rhombohedral and triclinic, as these can have unit cells with no 90-degree angles.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Primitive Lattices

Primitive lattices are the simplest repeating units in a crystal structure, defined by their unit cells. These unit cells can vary in shape and size, and they are characterized by their lattice parameters, which include the lengths of the edges and the angles between them. Understanding the types of primitive lattices is essential for identifying their geometric properties.

Unit Cell Angles

The angles between the edges of a unit cell are crucial for classifying crystal systems. In three-dimensional lattices, the internal angles can be 90 degrees or vary, leading to different classifications such as orthorhombic (all angles 90°) or triclinic (no angles 90°). Recognizing these angles helps in determining the symmetry and properties of the crystal structure.

Crystal Systems

Crystal systems categorize crystals based on their symmetry and lattice parameters. There are seven primary crystal systems, including triclinic and rhombohedral, which are characterized by their unique angles and edge lengths. Understanding these systems is vital for identifying which lattices have non-right angles and for solving related crystallography problems.

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Textbook Question

Two patterns of packing two different circles of the same size are shown here. For each structure (c) determine the type of two-dimensional lattice (from Figure 12.4). (i)

(ii)

Textbook Question

What is the minimum number of atoms that could be contained in the unit cell of an element with a face-centered cubic lattice? (a) 1, (b) 2, (c) 3, (d) 4, (e) 5.

Textbook Question

Two patterns of packing for two different circles of the same size are shown here. For each structure (b) determine the angle between the lattice vectors, g, and determine whether the lattice vectors are of the same length or of different lengths; (i)

(ii)

Textbook Question

Imagine the primitive cubic lattice. Now imagine grabbing the top of it and stretching it straight up. All angles remain 90. What kind of primitive lattice have you made?