Question

Consider the body-centered cubic structure shown here: a. What is the length of the line (labeled \(c\) ) that runs from one corner of the cube diagonally through the center of the cube to the other corner in terms of \(r\) (the atomic radius)? b. Use the Pythagorean theorem to derive an expression for the length of the line (labeled \(b\) ) that runs diagonally across one of the faces of the cube in terms of the edge length \((I) .\) c. Use the answer to parts a and \(\mathrm{b}\) along with the Pythagorean theorem to derive the expression for the edge length \((I)\) in terms of \(r\)

Short answer

a) The length of the line \(c\) is \(4r\). b) The length of the face diagonal \(b\) is \(\sqrt{2}I\). c) The edge length of the cube \(I\) in terms of the atomic radius \(r\) is \(I = \frac{4r}{\sqrt{3}}\).

Step-by-step solution

Find Length of Diagonal Line Inside Cube

To find the length of the line labeled \(c\) running through the center of the cube, imagine a body-centered cubic (BCC) lattice where each corner and the center of the cube is an atom. The line \(c\) connects two opposite corners passing through the center. Since there are four radii along this line, the length of \(c\) is \(c = 4r\).

Derive Expression for Face Diagonal

To find the length of the diagonal (\b) across one face of the cube, apply the Pythagorean theorem on the right triangle formed by half of the diagonal, the edge length \(I\), and half of another edge. The equation is \((\frac{b}{2})^2 = I^2 + (\frac{I}{2})^2\), which simplifies to \(b = \sqrt{2}I\).

Express Edge Length in Terms of Atomic Radius

Using the results from parts a and b, set up another Pythagorean relationship for the large diagonal \(c\) passing through the cube. Since \(c\) is a body diagonal, the Pythagorean theorem gives us \( c^2 = b^2 + I^2 \). Substitute \( 4r \) for \( c \) and \( \sqrt{2}I \) for \( b \) to get \( (4r)^2 = (\sqrt{2}I)^2 + I^2 \). Simplify this to \(16r^2 = 3I^2 \) and solve for \(I\) by dividing both sides by \(3\), and taking the square root, yielding \(I = \frac{4r}{\sqrt{3}}\).

Key concepts

Understanding Atomic Radius in Crystals

When discussing the atomic radius within the context of crystal structures, like the body-centered cubic (BCC) structure, we refer to the distance from the center of an atom to the boundary where it touches another atom. In simpler terms, the atomic radius is half the distance between the nuclei of two adjacent atoms that are in contact with each other.

In the BCC structure, the atomic radius acts as a building block for calculating various dimensions of the crystal lattice. Understanding the atomic radius is crucial as it allows us to relate microscopic dimensions of atoms to the macroscopic dimensions of the crystal lattice they form. Specifically, it provides the foundation for calculations involving the geometry of the structure, such as finding the length of the cube's edges or diagonals based on the arrangement and size of the atoms within the lattice.

Calculating the Cube Diagonal

Deriving the Face Diagonal

In crystallography, calculating the diagonal of a cube, or more specifically, the face diagonal and body diagonal, is important for understanding the geometry of the crystal. To compute the diagonal across one face of a BCC unit cell, we essentially treat the face as a right-angled triangle, with the diagonal as the hypotenuse and the edges of the cube as the other two sides.

According to the Pythagorean theorem, the square of the hypotenuse (the face diagonal) is equal to the sum of the squares of the other two sides. For a cube, the edge lengths are equal, which gives a simplified relationship where the face diagonal is proportional to the edge length by a factor of \(\text{square root of }2\).

Extending to the Body Diagonal

After finding the face diagonal, extending the calculation to the body diagonal (that runs through the center of the cube) involves considering a larger right-angled triangle. This triangle's sides are the body diagonal, the face diagonal, and the edge length of the cube. The Pythagorean theorem again serves as the tool for relating these dimensions.

Pythagorean Theorem in Crystallography

The Pythagorean theorem, a fundamental principle in geometry, states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This theorem is a powerful tool in crystallography, especially when dealing with crystalline structures with cubic symmetry, like the body-centered cubic (BCC) structure.

In the application to BCC structures, the theorem helps us find various lengths and relationships within the crystal. For instance, using the atomic radius to determine the edge length of the crystal requires us to understand and apply the Pythagorean theorem. The solutions to each part of the exercise demonstrate how this theorem is used in a stepwise manner to relate atomic dimensions to the lattice dimensions.

Abstracting away from the specific dimensions, the Pythagorean theorem in crystallography reflects the deep connection between the geometry of shapes and the underlying order of crystalline materials. Grasping this theorem thus enables students to navigate the complexities of crystal structures with more confidence and precision.