Problem 24 - Chemistry The Central Science

Question

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?

Step-by-Step Solution

Answer

After stretching the primitive cubic lattice along the body diagonal while maintaining equal edge lengths and equal angles, the resulting lattice is a rhombohedral (or trigonal) lattice which belongs to the hexagonal crystal system. This lattice has equal edge lengths (a = b = c) and equal angles between the lattice vectors, but the angles are no longer \(90^{\circ}\).

Step 1: Identify key features of the initial primitive cubic lattice

A primitive cubic lattice has three lattice vectors (a, b, and c) that are mutually perpendicular, meaning all the angles between them are \(90^{\circ}\). The edge lengths of the unit cell are all equal: a = b = c.

Step 2: Visualize the transformation

Now, we need to stretch the lattice along the body diagonal while maintaining equal edge lengths. This means that we will still have a = b = c, but the angles between the lattice vectors will no longer remain \(90^{\circ}\).

Step 3: Identify the type of lattice after the transformation

After the transformation, we still have equal edge lengths a = b = c, and the angles between the lattice vectors are no longer \(90^{\circ}\) but remain equal. This type of lattice structure is known as a "rhombohedral" (or "trigonal") lattice which belongs to the hexagonal crystal system. So, the resulting primitive lattice after the transformation is a rhombohedral lattice.

Key Concepts

primitive cubic lattice

In the world of crystal structures, the primitive cubic lattice is a fundamental concept. Picture a cube that is simple and straightforward, with each corner occupied by a single lattice point. Imagine the cube's structure: it features three lattice vectors, often designated as \( a, b, \) and \( c \), which are all perpendicular to each other. This perpendicularity means that each angle made by the vectors is exactly \( 90^{\circ} \). Furthermore, the length of each edge, \( a = b = c \), establishes an equally sided unit cell.

Why is this important? Because of its simplicity, the primitive cubic lattice serves as a building block for understanding more complex arrangements. It's the easiest lattice type to visualize and an excellent starting point for studying crystallography.

• This lattice forms a basic 3D grid.
• All angles are right angles (\( 90^{\circ} \)).
• All edges are of equal length.

The primitive cubic lattice is not just theoretical; it occurs in nature and is utilized in various materials. Understanding its structure helps decipher more complex lattice forms.

rhombohedral lattice

A rhombohedral lattice is what you get when you modify a primitive cubic lattice by stretching it along its body diagonal while keeping all edge lengths equal. In this transformation, while the edges remain the same length, the three angles between the lattice vectors change. Instead of being \( 90^{\circ} \), they become equal but non-right angles.

Imagine taking the corners of a cube and gently pulling them apart along a diagonal line. The result? A shape that looks like a skewed cube, or more precisely, a parallelogram-like version of a cube. This configuration, often referred to as a rhombohedral or trigonal lattice, is a unique form in crystallography.

• The rhombohedral lattice maintains equal edge lengths.
• All inter-lattice angles are identical.
• These angles are no longer \( 90^{\circ} \), leading to a diamond-like shape.

This lattice type is critical in describing various minerals and synthetic materials. It's different from and less intuitive than the simple cubic lattice, adding layers of depth to the study of crystalline structures.

hexagonal crystal system

When classifying crystals, the hexagonal crystal system stands out due to its geometric elegance. This system is characterized by specific symmetry and spatial arrangement. While the hexagonal system includes a variety of structures, the rhombohedral lattice is an integral part.

Imagine a series of hexagons stacked upon each other. That's a glimpse of the broad idea behind this system, though the rhombohedral lattice provides a more skewed and less symmetrical example compared to the typical hexagonal lattice.

• It includes both hexagonal and rhombohedral configurations.
• The system is known for its six-fold rotational symmetry, although the rhombohedral variant deviates slightly.
• Lattices in this system often appear in minerals and compounds such as quartz.

Understanding the hexagonal crystal system enriches one's grasp on how certain materials behave at an atomic level. It's a fascinating area where symmetry, geometry, and material science converge in crystalline beauty.