Crystal Lattice and Closed-Pack Spheres

In a crystal lattice, atoms, ions, or molecules are arranged in a repeating pattern. The arrangement is called a lattice because it extends indefinitely in all directions. In a closed-pack lattice, the spheres are arranged in a way that maximizes the packing efficiency, with each sphere touching its neighbors.

Tetrahedral Voids

A tetrahedral void is a small space or cavity that can be found within a crystal lattice. It is formed when four spheres come together in such a way that they create a tetrahedron, which is a four-faced polyhedron with all faces being equilateral triangles. The four spheres that form the tetrahedron are located at the vertices of the triangular faces.

Number of Tetrahedral Voids

To determine the number of tetrahedral voids in a crystal lattice with 6 closed-pack spheres, we can use the relationship between the number of tetrahedral voids and the number of spheres.

In a closed-pack lattice, each sphere is in contact with its 12 nearest neighbors. These 12 spheres can be divided into two sets of 6 spheres each. The first set of 6 spheres is in contact with the central sphere, and the second set of 6 spheres is not in contact with the central sphere.

Calculating the Number of Tetrahedral Voids

To calculate the number of tetrahedral voids, we need to consider only the second set of 6 spheres that are not in contact with the central sphere.

Each sphere in the second set can form a tetrahedron with the central sphere and two neighboring spheres from the first set. Since there are 6 spheres in the second set, there are 6 tetrahedra formed.

However, each tetrahedron shares its four vertices with four neighboring tetrahedra. Therefore, each tetrahedron contributes 1/4th of a void to the crystal lattice.

Since there are 6 tetrahedra, the total number of voids contributed by them is 6 * 1/4 = 1.5 voids.

However, voids cannot exist in fractions, so we consider the whole number of voids. Therefore, the number of tetrahedral voids in the lattice is 1.

Conclusion

In a crystal lattice with 6 closed-pack spheres, the number of tetrahedral voids is 1. This indicates that there is one small space or cavity formed by the arrangement of the spheres within the lattice.