The left diagram in Figure 10.1 might represent a two atom...

The left diagram in Figure 10.1 might represent a two atom crystal with two bands. Basing your argument on the kinetic energy inside either individual well, explain why both energies in the lower band should be roughly equal to that of the $n=1$ atomic state and why both energies in the upper should roughly equal that of the $n=2$ atomic state.

Key Concepts

Band Formation

Band formation occurs when individual atomic orbitals overlap in a periodic crystal lattice, resulting in the splitting of discrete atomic energy levels into bands. In the context of a two-atom crystal, the overlap of atomic orbitals leads to two distinct bands that correspond to the energies of the original atomic states. The lower band emerges from the bonding combination of orbitals (n=1), and the upper band from the antibonding combination (n=2), thereby retaining characteristics of the individual atomic energy levels.

Kinetic Energy in Confinement

The kinetic energy of an electron in a confined potential well is quantized due to the wave nature of particles, similar to a particle-in-a-box system. In a crystalline structure, this quantization largely persists within each individual potential well (atomic site), ensuring that the energy levels remain similar to those of an isolated atom. As a result, the kinetic-energy-dominated contributions in each well set the baseline energy levels for the bands, with the lower band reflecting the n=1 state and the upper band reflecting the n=2 state.

Atomic Orbitals

Atomic orbitals are the quantized energy states of electrons in an atom, each defined by a specific energy level (such as n=1, n=2, etc.). These orbitals arise from the confinement of electrons within the atomic potential, leading to specific kinetic energies associated with each state. Even in a crystalline environment, the local atomic potentials largely determine the electron energies, reflecting the energies of the isolated atomic states.

Transcript

Please give Ace some feedback