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).

Step-by-step solution

Understand the types of cubic unit cells

There are three types of cubic unit cells: 1. Simple cubic (SC) - atoms are present only at the corners of the cube. 2. Body-centered cubic (BCC) - atoms are present at the corners and the center of the cube. 3. Face-centered cubic (FCC) - atoms are present at the corners and the centers of each face of the cube.

Compute the number of atoms in each unit cell type

Let's calculate the number of atoms per unit cell for each type: 1. Simple cubic (SC): 1 atom per unit cell (\\(\frac{1}{8}\\) atom per corner × 8 corners = 1) 2. Body-centered cubic (BCC): 2 atoms per unit cell (\\(\frac{1}{8}\\) atom per corner × 8 corners + 1 atom in the center = 2) 3. Face-centered cubic (FCC): 4 atoms per unit cell (\\(\frac{1}{8}\\) atom per corner × 8 corners + \\(\frac{1}{2}\\) atom per face × 6 faces = 4)

Compute the atomic packing factors for each unit cell type

The atomic packing factor (APF) represents the fraction of the unit cell volume occupied by the atoms. We'll compute the APF for each type of unit cell, using the given atomic radius r and edge length a: 1. Simple cubic (SC): APF = \\(\\\frac{1 \cdot \frac{4}{3}\pi r^3}{a^3}\\\) 2. Body-centered cubic (BCC): APF = \\(\frac{2 \cdot \frac{4}{3}\pi r^3}{a^3}\\\) 3. Face-centered cubic (FCC): APF = \\(\frac{4 \cdot \frac{4}{3}\pi r^3}{a^3}\\\)

Compute the APF for each unit cell type using the given r and a

Atomic radius of Nickel: r = \(0.162 \mathrm{~nm}\). Edge length of the cubic unit cell: a = \(0.458 \mathrm{~nm}\). 1. Simple cubic (SC): APF = \\(\frac{1 \cdot \frac{4}{3}\pi (0.162)^3}{(0.458)^3} \approx 0.26\\\) 2. Body-centered cubic (BCC): APF = \\(\frac{2 \cdot \frac{4}{3}\pi (0.162)^3}{(0.458)^3} \approx 0.52\\\) 3. Face-centered cubic (FCC): APF = \\(\frac{4 \cdot \frac{4}{3}\pi (0.162)^3}{(0.458)^3} \approx 1.04\\\)

Determine the geometry of the Nickel unit cell

Given the computed APFs, we can see that the acceptable geometry for the Nickel unit cell is face-centered cubic (FCC) since its APF is closest to 1 (meaning it's the most closely packed). The other two unit cells - SC and BCC - have lower APFs, which indicates that they are not as closely packed as the FCC unit cell. Therefore, the geometry of the Nickel unit cell is face-centered cubic (FCC).

Key concepts

Atomic Radius

The atomic radius is a measure of the size of an atom, typically defined as the distance from the center of the nucleus to the boundary of the surrounding electron cloud. This measurement is crucial because it determines how closely atoms can pack together in a structure. Nickel, for instance, has an atomic radius of 0.162 nm.
Understanding atomic radius is essential because it helps predict how atoms interact with each other, including bond formation and crystal structure creation.

• The atomic radius is influenced by the number of electron shells: more shells generally mean a larger radius.
• It also depends on the nuclear charge: higher charges can pull the electron cloud closer, reducing the atomic radius.

Grasping the concept of atomic radius aids in understanding how elements will behave in different chemical and physical environments, particularly in solid-state structures.

Unit Cell

The unit cell is the smallest repeating structure that makes up a crystal lattice. It's the foundational "building block" that repeats in three dimensions to create the entire crystal structure. Understanding the concept of a unit cell allows us to comprehend how large-scale crystal structures are formed from repeated patterns.
In cubic structures, such as the one studied in the original exercise, different types include Simple Cubic (SC), Body-Centered Cubic (BCC), and Face-Centered Cubic (FCC).

• The simple cubic unit cell has atoms only at each corner.
• Body-centered cubic includes an atom in the center of the cube in addition to corner atoms.
• Face-centered cubic has an atom at the center of each face of the cube, as well as at the corners.

Each type of unit cell imparts specific properties to the material, like density and strength, depending on how the atoms are packed.

Atomic Packing Factor

The Atomic Packing Factor (APF) is a measure that indicates how efficiently atoms are packed within a crystal structure. It's calculated by dividing the volume of the atoms in one unit cell by the total volume of the unit cell itself.
This factor gives insight into how tightly the atoms are arranged, which affects the material's density and stability.

• For a Simple Cubic (SC) structure, the APF is approximately 0.52, indicating relatively low packing efficiency.
• The Body-Centered Cubic (BCC) structure has an APF of about 0.68, featuring more compact packing than SC.
• The Face-Centered Cubic (FCC) structure, such as that found in Nickel, boasts an APF of approximately 0.74, one of the densest packings possible.

Recognizing the APF helps in determining the usefulness of a material under different conditions, particularly in applications requiring specific mechanical properties.

Face-Centered Cubic

The Face-Centered Cubic (FCC) structure is a type of crystalline arrangement where atoms are situated at each of the corners and the centers of all cube faces. This particular configuration leads to a highly dense and stable packing of atoms.
FCC structures are essential in materials science because they offer impressive mechanical properties and strength, making them suitable for many industrial applications.

• Nickel is an example of a material exhibiting an FCC structure, characterized by high atomic packing factor and superior atomic coordination.
• Compared to other cubic structures, FCC arrangements support efficient slip and ductility during deformation, explaining their prevalence in metals.

Understanding FCC is vital as it provides a portal into comprehension of metallic properties and the behavior of materials under stress.