Band gap vs QM energy level gap between electron orbitals

I have read this question:

What is energy band gap?

What is the reason for formation of energy bands?

https://en.wikipedia.org/wiki/Nuclear_shell_model

If I understand correctly, the band gap (between the conduction and valence band) is due to quantization of energy.

Now, what I do not understand is, is the band gap the same as the gaps between the different electron energy levels around the nucleus (that is because of quantization of energy too).

So basically, the nuclear shell model describes the gaps between the different electron energy levels where the electron exists as per QM.

And then there is the band gap between the conduction and valence bands.

What I do not get is why are (or are they at all) those two gaps different?

Or are they the same gaps, it is just that there is the Fermi level, and that makes the band gap special?

Is the band gap just a special energy level gap because of the Fermi level?

Question:

1. Are those two gaps different, the gap between the conduction and valence bands, and the gap between the energy levels of the electron around the nucleus

2. if they are different, then what is different, their gap size, or their cause (quantization of energy)

• In an atom, the energy levels are solutions to Schrodingers equation around a single nucleus. In a crystal, energy levels are Bloch wave solutions to an electron in a periodic potential. Two different things. – Jon Custer Jun 23 at 15:18
• @JonCuster you seem to get what I am asking. can you please tell me, is the difference because electrons in an atom are bound, but in the lattice they are loosely bound (delocalized)? – Árpád Szendrei Jun 23 at 19:26
• I realize this isn’t the point if the question, but the nuclear shell model doesn’t say anything about electrons. It describes protons snd neutrons within the nucleus. – ragnar Jul 6 at 0:02

If I understand correctly, the band gap (between the conduction and valence band) is due to quantization of energy.

Let me start with discrediting a common belief that quantum mechanics is all about quantization. The simplest quantum mechanical system - a free particle, does not exhibit quantization at all. The Hamiltonian in this case is just the kinetic energy $$\hat{H}=\frac{\hat{p}^{2}}{2m}$$, with a continuous spectrum $$E_{\boldsymbol{k}}=\frac{\hbar^{2}k^2}{2m}$$ of plane-wave states $$\psi_{\boldsymbol{k}}\left(\boldsymbol{r}\right)=\frac{1}{\left(2\pi\right)^{\frac{3}{2}}}e^{i\boldsymbol{k}\cdot\boldsymbol{r}}$$. The notion of quantization is thus vague in this context and cannot explain bands. Quantization explains other phenomena however, like discrete emission lines.

Now, what I do not understand is, is the band gap the same as the gaps between the different electron energy levels around the nucleus (that is because of quantization of energy too).

So basically, the nuclear shell model describes the gaps between the different electron energy levels where the electron exists as per QM.

And then there is the band gap between the conduction and valence bands.

....

Secondly, the problem of an atom doesn't have much to do with energy bands. Energy bands appear even without talking about atoms or electrons at all. Bands can form solely as a result of a periodic potential perturbing a free quantum system.

You essentially mix two different questions that are otherswise independent

1. Why bands form.

2. How electrons behave in such bands.

For (1), bands can form even for systems of bosons. One example for such systems are ensembles of ultracold atoms in optical lattices. In those experiments atoms are cooled down to extremely low temperatures using radiative forces, and then trapped in standing waves produce by couter-propagating light beams. This induces periodic potential on the atoms which results in energy bands. The trapped atoms can be either composite bosons or composite fermions, depending on their total spin, exhibiting completely different effects.

For (2), it just happened in nature that solid objects do produce such periodic potential for the electrons. Because the electrons are fermions and obey the Pauli exclusion principle you get concepts like valence and conduction bands, Fermi level and etc.. All of this, all chemistry and all life - is the consequence of electrons building up when occupying states. But that's just because they are fermions. They don't form the bands. They just live there.

Actually, there is another type of bands forming in solids. Lattice vibrations can be also quantized, creating energy quanta known as phonons. The dispersion of such phonons form similar band-like structures (see the first and third images to the right in this Wikipedia page). However, because phonos are bosons you have a different behavior.

The reason why periodic potential form bands is a different question. The nearly free electron model provides a simple explanation of this (again, the name electron, because electrons in solids were the problem of interest when developing those theories). It applies in cases where the potential is weak compared to the kinetic energy, such that perturbative methods apply. The idea is that a periodic potential contains discrete frequency components. Therefore, it couples states $$\psi_{\boldsymbol{k}}$$ and $$\psi_{\boldsymbol{k}^{\prime}}$$ only for discrete values of $$\boldsymbol{k}-\boldsymbol{k}^{\prime}$$. This breaks the continuous energy spectrum of plane-waves only at certain points, causing gaps to form within the continuum. The following picture illustrates this in one dimension. • thank you, are the band gaps different because the electrons are bound to an atom, but are free in the lattice? – Árpád Szendrei Jun 23 at 19:31
• I am not sure I understand what you are asking. Each system has a different Hamiltonian, and therefore different energy level structure. Bands, as I showed, are a signature of periodic potential. – eranreches Jun 23 at 21:19
• what do you mean "Bands, as I showed, are a signature of periodic potential", can you please elaborate that? – Árpád Szendrei Jun 24 at 6:26
• @arpad szendrei, band gap occurs solely because of an interaction between the periodic potential with the electrons in a solid. See the nearly free electron model on the internet or any solid state physics textbook. – thermomagnetic condensed boson Jun 24 at 10:13
• @ÁrpádSzendrei You see a continuous spectrum that is broken at specific points - it means your potential couples specific frequencies, which ultimately means its periodic. – eranreches Jun 24 at 11:06