Question

The number of atoms per unit cell in simple cubic, face-centred cubic and body-centred cubic are: a) \(1,4,2\) b) \(4,1,2\) (c) \(2,4,1\) (d) 4, 8, 2

Short answer

The number of atoms per unit cell in simple cubic, face-centred cubic and body-centred cubic structures are 1, 4, and 2. Hence, the answer is option (a) \(1, 4, 2\).

Step-by-step solution

Simple Cubic Structure

In a simple cubic structure, there is one atom at each of the eight corners. Each corner atom is shared among eight adjacent cells. Therefore, the number of atoms per unit cell in a simple cubic structure is \(1/8 * 8 = 1\).

Face-centred Cubic Structure

For a face-centred cubic structure, each cell contains an atom at each of the six faces and the eight corners. Each face-centred atom is shared among two cells and each corner atom is shared among eight cells. Thus the total number of atoms per unit cell in a face-centred cubic structure is \(6/2 + 1/8 * 8 = 3 + 1 = 4\).

Body-centred Cubic Structure

In a body-centred cubic structure, each cell contains one atom at each corner and one in the center of the cell. Each corner atom is shared among eight cells, while the center atom belongs entirely to the unit cell. Therefore, the total number of atoms per unit cell in a body-centred cubic structure is \(1/8 * 8 + 1 = 1 + 1 = 2\).

Key concepts

Simple Cubic Unit Cell

Picture a grid filled with tiny cubes packed together with singular precision, much like a carefully arranged set of building blocks. This is essentially the essence of a simple cubic unit cell within solid-state chemistry. It represents the most straightforward lattice arrangement where spheres (representing atoms) are located solely at the corners of a cube.

Each of these spheres is nestled equally between eight neighboring unit cells, reducing its ownership to just an eighth of an atom per cell. When you sum up all eight corners, you're left with a total contribution of one whole atom per simple cubic unit cell. As simple as it gets, this structure is foundational to understanding more complex arrangements.

Face-centred Cubic Structure

Now, let's imagine adding some bling to our cubical framework by placing additional spheres flatly at the center of each face of the cube. This configuration, known as the face-centred cubic structure (FCC), is akin to sprinkling extra dots on a die, one in the middle of each side.

Every face-centred sphere is part-shared between two cells, effectively split right down the middle. Coupled with the shared corner atoms, this sophisticated dance of division and sharing results in a total of four full atoms for each face-centred cubic cell. With its higher density, the FCC arrangement is a quintessential structure for many metals, laying the microscopic groundwork for their properties.

Body-centred Cubic Structure

Taking a slight detour from the surface and diving into the core, we find ourselves with the body-centred cubic structure (BCC). Here, a bold move is made by placing an additional sphere smack dab in the center of our cubic unit—perhaps the cell's treasure.

Woven neatly into the fabric of the cubic grid, each cell gets a unique central sphere, while still sharing the corner spheres with its neighbors. When you tally these soldiers—eight corner guards contributing an eighth each, plus the one full atom resting at the center, the total comes to two atoms. This structure not only makes for a sturdy arrangement but also has implications for the mechanical properties of metals that embody this pattern.

Atoms Per Unit Cell Calculation

The realm of crystallography comes with its own mathematical wand, one that we wave to calculate the number of atoms in each unit cell type. This isn't about tricky spells—just clear-cut math. Here's the sleight of hand: count, share, and compute.

For the simple cubic (SC), it's a straightforward tally of eight corner atoms: each a fraction; totaling one atom per unit cell. The face-centred cubic (FCC) amps up the game—six face atoms halved, and eight corner atoms in eighths: all summed up to four atoms. Swing over to the body-centred cubic (BCC), where we have the central unshared atom and eight corners in eighths: we get two atoms per unit cell. Accurately grasping these counts is crucial as it affects how we perceive the material's density and other characteristic properties.