# What does "spin degeneracy" mean in this context?

I am currently studying Diode Lasers and Photonic Integrated Circuits, second edition, by Coldren, Corzine, and Masanovic. In chapter 1.2 ENERGY LEVELS AND BANDS IN SOLIDS, the authors say the following:

On the other hand, in a covalently bonded solid like the semiconductor materials we use to make diode lasers, the uppermost energy levels of individual constituent atoms each broaden into bands of levels as the bonds are formed to make the solid. This phenomenon is illustrated in Fig. 1.4. The reason for the splitting can be realized most easily by first considering a single covalent bond. When two atoms are in close proximity, the outer valence electron of one atom can arrange itself into a low-energy bonding (symmetric) charge distribution concentrated between the two nuclei, or into a high-energy antibonding (antisymmetric) distribution devoid of charge between the two nuclei. In other words, the isolated energy level of the electron is now split into two levels due to the two ways the electron can arrange itself around the two atoms.$$^1$$ In a covalent bond, the electrons of the two atoms both occupy the lower energy bonding level (provided they have opposite spin), whereas the higher energy antibonding level remains empty.
$$^1$$The energy level splitting is often incorrectly attributed to the Pauli exclusion principle, which forbids electrons from occupying the same energy state (and thus forces the split, as the argument goes). In actuality, the splitting is a fundamental phenomenon associated with solutions to the wave equation involving two coupled systems and applies equally to probability, electromagnetic, or any other kind of waves. It has nothing to do with the Pauli exclusion principle. If another atom is brought in line with the first two, a new charge distribution becomes possible that is neither completely bonding nor antibonding. Hence, a third energy level is formed between the two extremes. When $$N$$ atoms are covalently bonded into a linear chain, $$N$$ energy levels distributed between the lowest-energy bonding state and the highest-energy antibonding state appear, forming a band of energies. In our linear chain of atoms, spin degeneracy allows all $$N$$ electrons to fall into the lower half of the energy band, leaving the upper half of the band empty.

What does "spin degeneracy" mean in this context?

• You might want to look at calculation of the energy levels in polyenes as they get longer, familiar to chemists, and which show the same behaviour as in your figure. Mar 31 at 7:10

I'm not a solid-state chemist, but I think the meaning seems reasonably clear from context to me. "Spin degeneracy" here means that each energy level is capable of holding a spin-up electron as well as a spin-down electron, i.e. in each orbital there are two different spin states which are degenerate (some would call these spin orbitals). Thus, $$N$$ atoms form $$N$$ different energy levels which $$N$$ electrons need to be distributed between. Because of this degeneracy, only the lowest-energy $$N/2$$ energy levels are actually populated: each energy level has $$2$$ electrons.