Question

Nickel has an atomic radius of \(0.162 \mathrm{nm}\). The edge of its cubic unit cell is \(0.458 \mathrm{nm}\). What is the geometry of the nickel unit cell?

Short answer

Answer: The geometry of the nickel unit cell is face-centered cubic (FCC) based on the atomic radius and edge length provided.

Step-by-step solution

Calculate the atomic radius to edge length ratio

Divide the atomic radius by the edge length to find the ratio: \(r_{ratio} = \frac{r_{atomic}}{L_{edge}} = \frac{0.162 \mathrm{nm}}{0.458 \mathrm{nm}} = 0.3537\)

Compare the ratio to well-known crystal structures

The well-known crystal structures ratios are as follows: 1. Simple cubic (SC): \(\frac{r_{atomic}}{L_{edge}} = \frac{1}{2} \Longrightarrow\) ratio = 0.5 2. Body-centered cubic (BCC): \(\frac{r_{atomic}}{L_{edge}} = \frac{\sqrt{3}}{4} \Longrightarrow\) ratio = 0.433 3. Face-centered cubic (FCC): \(\frac{r_{atomic}}{L_{edge}} = \frac{\sqrt{2}}{4} \Longrightarrow\) ratio = 0.3536

Determine the geometry of the nickel unit cell

Comparing the calculated ratio (0.3537) to the ratios of well-known crystal structures, we can see that the ratio is closest to the ratio for the face-centered cubic (FCC) structure (0.3536). Therefore, the geometry of the nickel unit cell is face-centered cubic (FCC).