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Chapter 12: Problem 24

Imagine the primitive cubic lattice. Now imagine grabbing opposite corners and stretching it along the body diagonal while keeping the edge lengths equal. The three angles between the lattice vectors remain equal but are no longer \(90^{\circ}\). What kind of primitive lattice have you made?

Short Answer

Expert verified
After stretching the primitive cubic lattice along the body diagonal while keeping the edge lengths equal, the new primitive lattice formed is a rhombohedral (trigonal) lattice. This lattice type is characterized by having equal edge lengths and equal angles between the lattice vectors, which are no longer \(90^{\circ}\).

Step by step solution

01

Understanding the Body Diagonal Stretch

In a primitive cubic lattice, all angles between the lattice vectors are equal to \(90^{\circ}\) and all edge lengths are equal. When grabbing opposite corners and stretching along the body diagonal, the edge lengths remain equal but the angles between the lattice vectors change. Let's denote the edges of the lattice by \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\).
02

Determine the New Angles Between Lattice Vectors

To find out the new angles between the lattice vectors after stretching the cubic lattice, we focus on one corner where the stretching occurs. Since the stretching will cause all edge lengths to remain equal, this means that the new lattice vectors will be equal to the original ones: \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\). However, the angles between these vectors (\(\vec{a}\) and \(\vec{b}\), \(\vec{b}\) and \(\vec{c}\) and \(\vec{c}\) and \(\vec{a}\)) will change and remain equal but no longer \(90^{\circ}\). From this information, we can determine the type of primitive lattice formed after stretching.
03

Identifying the New Primitive Lattice Type

Based on the changes in the stretching process, we find that: 1. The edge lengths remain equal. 2. The angles between the lattice vectors are equal but are no longer \(90^{\circ}\). These characteristics match the description of a rhombohedral lattice (also known as a trigonal lattice). In a rhombohedral lattice, all edge lengths are equal and all angles are equal. Therefore, after stretching the primitive cubic lattice along the body diagonal while keeping the edge lengths equal, we have made a rhombohedral (trigonal) lattice.

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