Simple Cubic Unit Cell - Online Tutor, Practice Problems & Exam Prep

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 Concept 1

The simple, sometimes called primitive cubic unit cell, is composed of a cube of an atom at each corner and 0 atoms in the center. Now if we take a look at a simple cubic unit cell, we're going to say that it contains only one atom. Now why does it contain one atom? Because remember there are eight corners to this cubic unit cell and each corner is 18 of an atom. So 18 atom times 8 gives me one atom. There are 0 atoms within the center.

Now edge link A is just the side of the cube. We're going to say it's equal to 2R. And that's because if we take a look at any one of these sides, we're going to say that this portion of our atom, it represents the radius of the atom and so does this one. Because they're touching, we can add them up together and that will give me the total length of a side. So r + r = 2R.

Because a simple cubic unit cell is not as complex, its packing efficiency is a bit low. It's 52%. And then finally, because it's not as complex, its coordination number starts off at the bare minimum of six. Remember, as we increase our complexity for our cubic unit cells, we expect our packing efficiency and coordination number to increase. So keep in mind these certain types of features when it comes to our simple or primitive cubic unit cell.

Simple Cubic Unit Cell Example 1

Calculate the volume in centimeters cubed of a simple cubic unit cell. If it's composed of atoms with a radius of 2.5 angstroms. All right, so just remember that volume equals your edge length cubed, And this is true for any cube. And we're going to say here that our edge length for a simple cubic unit saw is edge length a is equal to two times my radius.

Because our volume needs to be in centimeters, we have to convert our angstroms also into centimeters. So we have 2.5 angstroms. Remember the conversion factor is that one angstrom is equal to 10-10 meters. Meters goes here and centimeters up here. 1 centi is 10-2, so this comes out to be 2.5×10-8 centimeters.

So that's my radius in centimeters. So now multiply it by two. So this equals 5.0×10-8 centimeters. And then all we have to do here is cube this number and we'll have our volume in centimeters cubed. So when I cube this, it's going to give me back as an answer 1.3×10-22 centimeters cubed. So this would be our final answer.

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

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.