How does the number of energy levels in a band correspond to the number, \(N\), of atoms.

To begin with, it's important to grasp the concept of energy levels in isolated atoms. In the simplest model, electrons revolve around the nucleus in distinct energy levels or shells, where each level corresponds to a specific amount of energy an electron possesses. These shells are often depicted as orbits in Bohr's model or as probability distributions in quantum mechanical models. The energy levels within an atom can be thought of as steps on a ladder that electrons can occupy.

Energy levels are quantized, implying that only certain amounts of energy are allowed for the electrons. For example, the energy levels in a hydrogen atom have been well characterized by the Rydberg formula, which predicts the energy of photons emitted or absorbed when an electron transitions between these levels. In multielectron atoms, these levels become more complex due to electron-electron interactions but the quantized nature still applies.

When individual atoms come close together, as in the case of solids, the outer electrons' wave-functions begin to overlap. This overlap is a fundamental concept that explains how the discrete energy levels of isolated atoms merge into energy bands when they form a solid.

The wave-function of an electron in quantum mechanics describes the probability of finding an electron in a certain position in space. When atoms are far apart, the wave-functions do not interact, leading to distinct energy levels. As atoms move closer, the wave-functions begin to interact, leading to shifts in energy due to Pauli's exclusion principle and the formation of molecular orbitals. In a solid, this overlap is so extensive that it creates a continuous range of energies where electrons can reside, rather than fixed levels. This is analogous to how individual musical notes played by different instruments can blend together to form a continuous chord when played in unison.

The concept of band structure is central to understanding the electrical properties of materials. It refers to the allowed ranges of energy that electrons within the solid may have, determined by the arrangement and bonding of atoms. The band structure is created due to the overlap of wave-functions and the interaction between the atomic energy levels.

In materials, energy bands are classified primarily into two types: the valence band, which is occupied by electrons at zero temperature, and the conduction band, which is the range of energies that electrons can have when they gain enough energy to break free from their atomic bonds. The difference in energy between the top of the valence band and the bottom of the conduction band is called the band gap.

Insulators have a large band gap, which prevents electrons from easily moving to the conduction band. Conductors, on the other hand, have overlapping valence and conduction bands, allowing for free movement of electrons. Semiconductors have a moderate band gap, enabling the flow of electrons under certain conditions.