Question

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

Short answer

In a body-centered cubic lattice, each unit cell contains 8 corner atoms and 1 central atom. The corner atoms each contribute 1⁄8 of an atom to the unit cell, while the central atom contributes 1 atom. The total number of atoms in the unit cell is thus \(1 + 1 = 2\). So, the minimum number of atoms that could be contained in the unit cell of an element with a body-centered cubic lattice is \(2\). The correct answer is (b) 2.

Step-by-step solution

Understand the structure of a body-centered cubic lattice

A body-centered cubic (BCC) lattice is a three-dimensional array of atoms, in which each atom is located at the corners of a cube and one additional atom is located at the center of the cube.

Identify the atoms within the unit cell

In a body-centered cubic lattice, there are atoms at each of the \(8\) corners of the cubic unit cell and one atom at the center of the cell.

Determine the contribution of corner atoms to the unit cell

Each corner atom is shared by \(8\) adjacent unit cells. Therefore, the contribution of a single corner atom to a unit cell is 1⁄8. Since there are \(8\) of them, we multiply the contribution of one corner atom by \(8\) to get the total contribution from the corner atoms within the unit cell: \[Total \,corner \,atoms=8\cdot \frac{1}{8} = 1\]

Determine the contribution of the central atom to the unit cell

The central atom is completely within the unit cell and is not shared with any adjacent cells. So its contribution to the unit cell is 1.

Calculate the total number of atoms in the unit cell

To calculate the total number of atoms in the unit cell, add the contribution of corner atoms and central atom: \[Total \,atoms = Total \,corner \,atoms + Total \,central \,atoms\] \[Total \,atoms = 1 + 1\] \[Total \,atoms = 2\] Therefore, the minimum number of atoms that could be contained in the unit cell of an element with a body-centered cubic lattice is \(2\). So, the correct answer is (b) 2.

Key concepts

Unit Cell in Body-Centered Cubic Lattice

In crystallography, the term "unit cell" represents the smallest repeatable pattern that completely captures the symmetry and structure of a crystal lattice. This tiny volume acts like a building block for the crystal. Suppose you could shrink down to explore a material at the atomic level; you'd see that repeating the unit cell in every direction constructs the entire three-dimensional lattice.

The body-centered cubic (BCC) lattice is a specific type of structure where the unit cell has atoms positioned in a unique arrangement. In a BCC lattice, you see an atom at each of the cube's corners and one additional atom right in the center. Think of it like a dice game, except that the center dot hovers inside the dice instead of being on its surface.

By understanding the architecture of a unit cell in a BCC lattice, it helps us determine the physical properties of materials, like density, stability, and packing efficiency.

Corner Atoms in Unit Cell

Corner atoms in a body-centered cubic unit cell play a fascinating role because they contribute to multiple unit cells at once. If you imagine stacking multiple cubes together, each corner atom will find itself at the intersection of eight different unit cells by the time you finish.

That's why when we calculate how many atoms contribute to one particular unit cell, we only consider an eighth of each corner atom. This concept is essential for understanding why all the corners together contribute just 1 atom to the specific unit cell you are examining.

• Each corner atom contributes \( \frac{1}{8} \)th to the current unit cell.
• With 8 corners, their total contribution becomes 1 atom for one specific unit cell.

This precise sharing explains how crystals maintain their structured patterns while maximizing space efficiency.

Central Atom Contribution to Unit Cell

The central atom in a body-centered cubic lattice has a distinguishing feature—it isn't shared with any of the neighboring unit cells. It resides entirely within the confines of its particular unit cell.

This singular atom right in the heart of the unit cell adds a full contribution of one atom, making it unique compared to the shared corners. It’s almost as if the central atom plays the lead role—the unifying element that binds the entire cell together.

• The central atom sits alone, without sharing space with adjacent unit cells.
• Its contribution to the unit cell is 1 whole atom, unlike the corner atoms.

Together with the shared corner atoms, the total number of atoms in a BCC unit cell equals 2. This simplicity of counting highlights the elegance behind the BCC structure.