Question

Imagine the primitive cubic lattice. Now imagine grabbing the top of it and stretching it straight up. All angles remain \(90^{\circ} .\) What kind of primitive lattice have you made?

Short answer

By grabbing the top of the primitive cubic lattice and stretching it straight up while keeping all angles at \(90^{\circ}\), we have transformed the lattice into a Primitive Tetragonal Lattice, which has all angles at \(90^{\circ}\) and two equal sides with a third side of different length.

Step-by-step solution

Understand the primitive cubic lattice

A primitive cubic lattice has unit cells with all sides equal in length and all angles at \(90^{\circ}\). It is also called a simple cubic lattice or a unit cell.

Visualize the stretching process

Now imagine grabbing the top of the primitive cubic lattice and stretching it straight up while keeping all angles at \(90^{\circ}\). By doing this, we increase the height of the lattice, thus transforming it into a different kind of lattice.

Identify the new lattice type

Since we have stretched the top of the cubic lattice straight up and kept all angles at \(90^{\circ}\), we have transformed the primitive cubic lattice to a Primitive Tetragonal Lattice. This lattice has all angles at \(90^{\circ}\), and two sides (width and depth) are equal in length while the third side (height) is of different length.

Key concepts

Primitive Cubic Lattice

The primitive cubic lattice, also known as a simple cubic lattice, is one of the fundamental structures in crystallography. It has a very straightforward geometric arrangement. Each unit cell of this lattice is a cube where all edges are of equal length and all angles are right angles, specifically 90 degrees.
This kind of lattice has only one lattice point per unit cell, meaning each corner of the cube represents a lattice point. As each corner of the cube is shared with adjacent unit cells, a single unit cell effectively contains one atom in crystallographic calculations.

• The primitive cubic lattice is not the most dense packing structure. Materials crystallizing in this form typically display relatively lower densities compared to more efficient structures.
• In nature, elements such as Polonium crystallize in a primitive cubic lattice.

Understanding its simple structure can help when exploring more complex lattices and transformations, like the transition to a tetragonal form.

Primitive Tetragonal Lattice

The primitive tetragonal lattice is a direct variation of the primitive cubic lattice. Imagine taking the cubic structure and stretching it vertically—like pulling at the top of a stack of blocks. When you do this while maintaining all the angles at 90 degrees, you end up with a primitive tetragonal lattice.
A key characteristic of this lattice is how it handles dimensional equality:

• Two of its axes (width and depth) remain of equal length, while the third axis (height) becomes distinct, either longer or shorter.
• Each unit cell is thus shaped like a rectangular prism with a square base, and all angles are maintained at 90 degrees.

This transformation reveals how changes in unit cell dimensions can affect the crystal lattice's overall structure. The lattice still maintains its primitive status because there only one lattice point per unit cell. Such changes are common in various materials and could lead to different physical and chemical properties.

Crystallography

Crystallography is the scientific study of crystal structures and their properties. Central to crystallography are concepts such as lattice types, unit cells, and symmetry, which help explain how atoms, ions, and molecules arrange themselves in solid forms. Crystallography is not only important in physics and chemistry but also critical in fields like materials science and biology.

• A lattice in crystallography is a three-dimensional, repeating pattern that describes the geometric arrangement of the points that correspond to atom positions.
• The primitive unit cells, such as the cubic and tetragonal types, serve as the foundational blocks for more complex crystal structures.
• The study and understanding of distinct lattice types help predict crystal behavior under different conditions, give insight into binding forces, and determine possible material applications.

Mastering the basics of crystallography, such as understanding how a primitive cubic lattice transitions to a tetragonal arrangement, provides a strong base for exploring more complex and hybrid crystal forms.