Final answer:
The division by 8 in the density of states for a metallic solid is due to counting each wave vector state in k-space once; this division takes into account the positive and negative values of wave vectors and their equivalent states.
Step-by-step explanation:
The division by 8 in the calculation of density of states arises from considering the positive and negative values of wave vectors kx, ky, and kz in a crystal lattice. When solving Schrödinger's equation for a three-dimensional system like a metallic solid, the solutions are wavefunctions representing quantum states that exist for electrons.
However, because wave vectors can have both positive and negative directions, they represent the same physical state only once since quantum states in k-space are identical if they differ by a reciprocal lattice vector. Thus, we count each state in k-space just once, hence the factor of 1/8 only retains the states in the first octant of k-space, where all components are positive.
The concept relates to the Fermi factor, which is the probability that a given energy state is filled, and the Pauli exclusion principle, which states that no two electrons can be in the same quantum state.