Question

\(\beta\)-Tungsten crystallizes with a body-centred cubic structure. Draw a projection of the unit cell of \(\beta\)-tungsten. Does \(\beta\)-tungsten crystallize with a close-packed structure? Given that the unit cell parameter for \(\beta\)-tungsten is \(3.15 \AA\), calculate the ionic radius for \(\beta\) tungsten.

Short answer

The body-centred cubic (BCC) unit cell of β-tungsten has 2 atoms per unit cell, with 8 nearest neighboring atoms, meaning it's not a close-packed structure. Using the given unit cell parameter, the ionic radius for β-tungsten is approximately \(1.36 \AA\).

Step-by-step solution

Draw a projection of the BCC unit cell

In a body-centred cubic (BCC) unit cell, there is one atom at each corner and another atom in the center of the unit cell. Therefore, there are a total of 8 corner atoms and 1 central atom, but each of the corner atoms is shared with 8 other unit cells, so there are effectively only 1/8 * 8 = 1 corner atom inside the unit cell. Adding this to the central atom, we have a total of 2 atoms per unit cell in a BCC structure. Now, to draw the projection of the unit cell, follow these steps: 1. Draw a square on a paper and label its vertices as A (bottom left), B (bottom right), C (top right), and D (top left). 2. Place an atom at each vertex A, B, C, and D, representing the corner atoms. 3. Place an atom in the center of the square, representing the central atom of the BCC unit cell. 4. Draw lines connecting the central atom to each of the corner atoms.

Determine if β-tungsten crystallizes with a close-packed structure

In a close-packed structure, atoms are tightly packed together, making their arrangement the most efficient possible. There are two types of close-packed structures: face-centred cubic (FCC) and hexagonal close-packed (HCP) structures. In both of these structures, the coordination number (number of nearest neighboring atoms) is 12. However, in a BCC structure, like that of β-tungsten, the coordination number is 8, i.e., each atom has 8 nearest neighboring atoms. Since the BCC structure does not have the highest possible coordination number, β-tungsten does not crystallize with a close-packed structure.

Calculate the ionic radius using the given unit cell parameter

The unit cell parameter for β-tungsten is given as \(3.15 \AA\). We will use this information to calculate the ionic radius of β-tungsten. In a BCC structure, the relationship between the unit cell parameter (a) and the ionic radius (r) can be represented by the Pythagorean theorem: \[a^2 + a^2 + a^2 = (4r)^2\] Solving for r, we get: \[r = \frac{a\sqrt{3}}{4}\] Now, substitute the given unit cell parameter (a) into the equation: \[r = \frac{3.15\sqrt{3}}{4}\] \[r \approx 1.36 \AA\] So, the ionic radius for β-tungsten is approximately 1.36 Å.

Key concepts

Body-Centred Cubic Structure

Understanding the body-centred cubic (BCC) structure is essential when studying materials such as β-tungsten. The BCC is one of the simplest crystal structures found among metals and alloys.

In BCC, atoms are located at each corner of a cube and there is a single atom at the center of the cube. This arrangement differs from others because the central atom is not shared with any other unit cells, unlike the corner atoms which are shared among eight different unit cells. As a result, although there appear to be nine atoms in the BCC model cube, only two atoms are contained within each unit cell on average.

The BCC structure has several unique properties. For instance, it has a lower atomic packing factor (APF) compared to other crystal structures like face-centred cubic (FCC), which means atoms in a BCC structure are not as closely packed. The APF for BCC is approximately 0.68, indicating more empty space within the structure. This can affect the material's density and its mechanical properties such as ductility and slip systems.

While BCC metals tend to have higher strength at low temperatures, they can also be more brittle due to the lack of slip systems. This knowledge is useful for engineering applications where material properties must be precisely understood and harnessed.

Unit Cell Projection

A projection of a unit cell can be a helpful tool for visualizing the structure of crystals such as β-tungsten. The idea is to represent the three-dimensional cubic structure on a two-dimensional plane.

For the BCC structure, the projection typically shows a square with a central point representing the central atom, and points at the four corners representing the corner atoms. It's important to remember that this is just a representation; in three dimensions, the corner atoms would actually be positioned at the corners of a cube, effectively

Ionic Radius Calculation

Calculating the ionic radius is a crucial task in crystallography and materials science. The ionic radius can provide insights into the bonding characteristics and potential reactivity of the material.

When dealing with a BCC crystal like β-tungsten, we have a clear geometric relationship between the radius of the atom and the unit cell parameter. The ionic radius (r) can be calculated by considering the geometry of the structure and applying the Pythagorean theorem, as seen in the given solution—where the diagonal across the cube (spanning from one corner, through the body center, to the opposite corner) helps us define the relationship between the radius and the unit cell parameter.

Being aware of the approximation techniques and assumptions in these calculations is important for students. It is assumed that the atoms are hard spheres that just touch each other along the cube's body diagonal, which simplifies the actual interactions in a crystal. Moreover, depending on the context, the term 'ionic radius' might also imply an estimate rather than a precise measurement, as it can vary depending on the coordination number and the ionic charge.

Close-Packed Structure

The concept of a close-packed structure in the context of crystal lattices is a standard topic in material science. Such structures are characterized by the highest possible packing efficiency, which means that the atoms occupy the maximum available space without any gaps in between.

There are two primary types of close-packed structures—face-centred cubic (FCC) and hexagonal close-packed (HCP). Both have a coordination number of 12, which is the number of closest neighbors surrounding each atom. This results in the highest packing efficiency and hence the name 'close-packed'.

In contrast, β-tungsten has a BCC structure with a coordination number of 8, reflecting that its atoms are not as closely packed as in FCC or HCP structures. The BCC structure does not qualify as a close-packed structure, which could influence various physical properties such as density, melting point, and electrical conductivity. Recognizing this is key for students who are studying material properties and their applications in different fields of engineering and science.