To determine the void fraction of a body diagonal in a cubic close-packed (CCP) structure, we need to analyze the arrangement of atoms and the voids they create. In a CCP structure, also known as face-centered cubic (FCC), atoms are packed efficiently, and there are specific types of voids where other atoms can reside. Let's break this down step by step.
Understanding the CCP Structure
In a CCP arrangement, each unit cell contains:
- 4 atoms (since there are 8 corners with 1/8 of an atom each and 6 faces with 1/2 of an atom each).
- Octahedral voids and tetrahedral voids where other atoms can fit.
Types of Voids
In the CCP structure:
- Each unit cell has 4 octahedral voids.
- Each unit cell has 8 tetrahedral voids.
Calculating the Volume of Voids
To find the void fraction along a body diagonal, we first need to understand the dimensions involved:
- The length of the body diagonal in a unit cell is given by the formula: d = a√3, where a is the edge length of the cube.
- The volume of the unit cell is V_cell = a³.
Volume of Voids
Next, we calculate the volume of the voids:
- The volume of an octahedral void is approximately 0.414 times the volume of the atom that fits in it.
- The volume of a tetrahedral void is approximately 0.225 times the volume of the atom that fits in it.
Finding the Void Fraction
Now, we can calculate the total volume of voids in the unit cell:
- For octahedral voids: 4 voids × volume of one void
- For tetrahedral voids: 8 voids × volume of one void
However, since we are interested in the void fraction along the body diagonal, we need to consider how many voids are present along that diagonal. The body diagonal intersects both types of voids, and we can estimate the fraction of the total volume that is void space.
Calculating the Void Fraction
The void fraction can be approximated by the ratio of the total volume of voids to the total volume of the unit cell. Given that the voids occupy a certain percentage of the total volume, we can derive the void fraction:
- Typical void fraction in a CCP structure is around 0.26.
Thus, the void fraction of a body diagonal in this scenario will be close to 0.26.
Final Answer
Therefore, the correct answer to your question is (a) 0.26.
