In a closed packed array of \(\mathrm{N}\) spheres, the number of tetrahedral holes are: (a) \(\mathrm{N}\) (b) \(2 \mathrm{~N}\) (c) \(\frac{\mathrm{N}}{2}\) (d) \(\frac{\mathrm{N}}{4}\)

Crystallography is a branch of science that studies the arrangement of atoms in crystalline solids. In the context of close-packing of spheres, which is commonly used to represent atoms in a crystal, there exist patterns that allow for the most efficient use of space. One of the most common arrangements is called close-packed structure, which includes the face-centered cubic (FCC) and hexagonal close-packed (HCP) structures.

These structures are characterized by layers of spheres stacked in a sequence. Understanding how these layers arrange themselves is crucial for interpreting the physical properties of materials. Within these layers, there exist interstitial spaces known as 'holes', where additional atoms or ions can be situated. These holes are divided into two types: octahedral and tetrahedral holes, both named for the shape of the void space they create with surrounding spheres. Tetrahedral holes are smaller and are surrounded by four spheres, forming a tetrahedron around the hole.

This arrangement is foundational for understanding structural properties like density, porosity, and the arrangement of different types of atoms or ions within the crystal lattice. Hence, crystallography provides the concepts necessary to decode the complex architecture of solid materials.

Solid state chemistry focuses on studying the synthesis, structure, and properties of solid materials. Understanding the way atoms or molecules pack in a solid state is key to manipulating material properties. A vital concept within this field is the notion of interstitial sites, such as the tetrahedral holes in close-packed structures.

In these structures, atoms typically occupy the corners or faces of the unit cell, and the space that remains unoccupied is an opportunity for other atoms or ions to be hosted. These interstitial sites are important because they can significantly alter the material's properties. For instance, the presence of atoms in the tetrahedral holes can lead to changes in electrical conductivity, magnetic properties, and even mechanical strength.

Moreover, the size of these interstitial sites determines which types of atoms or ions can fit into them, which is pivotal when creating alloys or certain types of compounds. The control over the interstitial occupancy is a tool used by materials scientists to engineer new materials with desirable traits for various applications.

The coordination number is a term used in chemistry and crystallography to denote the number of nearest neighbors surrounding a central atom in a crystal or molecule. In the context of close-packed structures and tetrahedral holes, the coordination number is essential for predicting the stability and bonding of the structure.

For example, in a close-packed structure, every sphere (representing an atom) is in direct contact with a certain number of other spheres. This number varies by the type of packing and the position of the sphere within the structure. In a tetrahedral hole, a sphere would be in contact with four other spheres, giving it a coordination number of 4. This is a lower coordination number compared to an octahedral hole, which has a coordination number of 6.

The coordination number provides insight into the strength of bonding within the structure, as well as the potential for ionic or covalent interactions. Thus, it is a critical concept for understanding the physical and chemical behavior of materials and is also a relevant factor in the design of new compounds in materials science and solid state chemistry.