Simple Cubic Unit Cell: Videos & Practice Problems

In a body-centered cubic (BCC) unit cell, atoms are arranged in a way that one atom is located at each corner of a cube and one atom is at the center, totaling two atoms per unit cell. The corner atoms contribute 1/8th of an atom each due to sharing with adjacent unit cells, while the central atom is fully contained within the unit cell. The edge length (a) of a BCC unit cell is calculated using the formula:

$a=\frac{4r}{\sqrt{3}}$

where r is the atomic radius, accounting for the gap between atoms. This structure has a packing efficiency of 68% and a coordination number of 8, indicating each atom is surrounded by 8 nearest neighbors. Understanding these characteristics is crucial for comprehending the properties and behavior of materials with BCC crystal structures.

Simple Cubic Unit Cell

The simple cubic unit cell, often referred to as the primitive cubic unit cell, consists of atoms positioned at each of the eight corners of a cube, with no atoms located in the center. This configuration results in a total of 1 atom per unit cell. The reasoning behind this is that each corner contributes one-eighth of an atom to the unit cell, leading to the calculation: \( \frac{1}{8} \text{ atom} \times 8 = 1 \text{ atom} \).

The edge length, denoted as \( a \), is defined as the length of one side of the cube. It can be expressed in terms of the atomic radius \( r \) as \( a = 2r \). This relationship arises because the atoms at the corners touch each other along the edges of the cube, meaning the total length of the edge is the sum of the radii of two adjacent atoms.

In terms of packing efficiency, the simple cubic unit cell exhibits a relatively low packing efficiency of 52%. This means that only 52% of the volume of the unit cell is occupied by the atoms, while the remaining volume is empty space. Additionally, the coordination number, which indicates the number of nearest neighbor atoms surrounding a given atom, is 6 for the simple cubic structure. As the complexity of cubic unit cells increases, both the packing efficiency and coordination number tend to rise, reflecting a more efficient arrangement of atoms.

Understanding these characteristics of the simple cubic unit cell is essential for grasping the foundational concepts of crystal structures and their properties in materials science.

Simple Cubic Volume Example

To calculate the volume of a simple cubic unit cell composed of atoms with a radius of 2.5 angstroms, we start by recalling that the volume of a cube is given by the formula:

V = a³

where V is the volume and a is the edge length of the cube. For a simple cubic unit cell, the edge length a is twice the radius of the atom:

a = 2r

Given that the radius r is 2.5 angstroms, we first convert this radius into centimeters. The conversion factor is:

1 \text{ angstrom} = 10^{-10} \text{ meters} = 10^{-8} \text{ centimeters}

Thus, the radius in centimeters is:

r = 2.5 \times 10^{-8} \text{ cm}

Now, we can calculate the edge length:

a = 2 \times r = 2 \times (2.5 \times 10^{-8}) = 5.0 \times 10^{-8} \text{ cm}

Next, we cube the edge length to find the volume:

V = (5.0 \times 10^{-8})^3 = 1.25 \times 10^{-23} \text{ cm}^3

Therefore, the volume of the simple cubic unit cell is:

V \approx 1.3 \times 10^{-22} \text{ cm}^3

Polonium crystallizes with a primitive cubic structure. It has a density of 9.4 g/cm³, a radius of 167 pm, and a molar mass of 209 g/mol. Calculate the number of atoms in one mole of Polonium.

• A. 6.0×10^-23
• B. 3.2×10¹¹
• C. 6.0×10²³
• D. 1.7×10¹⁵

Answer: (C)

Here’s what students ask on this topic:

What is the maximum planar density in a simple cubic (sc) unit cell with equal-sized atoms?

In a simple cubic (sc) unit cell, the maximum planar density can be found in the plane that contains the most atoms. For a simple cubic lattice, this is any of the planes that cut through the cube along the edges, such as the (100), (010), or (001) planes.

Each of these planes contains one atom per unit cell area, as the atoms at the corners of the simple cubic cell are shared among eight adjacent cells. Since only 1/8th of each corner atom is within any given unit cell, and there are four corner atoms in the plane, this gives us a total of 1/2 an atom per unit cell area in the plane.

To calculate the planar density (PD), you divide the number of atoms in the plane by the area of the plane. For a simple cubic cell with lattice parameter 'a', the area of the (100) plane is a². Therefore, the planar density is:

$\mathrm{PD}=\frac{1/2}{a^2}$

This is the maximum planar density for a simple cubic unit cell with equal-sized atoms.

How many atoms are in a simple cubic (primitive cubic) unit cell?

In a simple cubic (also known as primitive cubic) unit cell, there is exactly one atom. This might seem counterintuitive at first glance because you can see eight corners within the cube, and at each corner, there appears to be an atom. However, it's important to remember that each corner atom is shared by eight adjacent unit cells in a three-dimensional crystal lattice. Since only a fraction of each corner atom is part of any one unit cell, you would calculate the number of atoms in a simple cubic unit cell by dividing each of the eight corner atoms into eight to account for the sharing. When you do this,

$\frac{1}{8}\times 8=1$

you find that there is effectively one whole atom per simple cubic unit cell.