Energy Band in Solids: What Is the Essential Phenomenon?

[meo_hallo]

hi !

In a crystal net.When i put 2 or many atom nearly.It's electron's energy will become energy band.(view attached image)

why?
what is a essential of this phenonmenon?

thanks!

 

[genneth]

When two orbitals interact, you end up with two different orbitals, one which is of lower energy, one which is of higher energy. Look up "bonding" and "anti-bonding" on Google. You can derive the effect with quantum mechanics pretty easily.

 

[meo_hallo]

thanks!
but i feel not easily.
Informations on Google is so much..

what is a essential? identical particle? or peturbation theory?
Do you help me about essential of this phenomenon? thanks !

 

[armandowww]

First of all, I suggest you to disregard the figure you referred to. It could only makes things more complicated than they really are. In atomic physics, emission/absorbtion spectra are experimentally obtained using lamps based on gases under low pressure. Low pressure allows to consider single particles as free, i.e. not interacting. For this reason atoms are dealt with as isolated. The electomagnetic spectra result on the allowed transitions between electronic states of the atomic system. Now, the more probable a transition is, the more loud is the intensity for corresondent line you see in the spectrum (I suggest to search Balmer series in google images). This is because every atom in the gas is behaving the same way as long as gas is made of identical particles. This issue is known as the "degeneration" in the gas or, better, in its energy spectrum.

Now, let the pressure rise up. What does it change in the spectrum? Experience tells us that lines in the spectrum becomes less narrow than before. It is a really gradual process but is a matter of fact. We talk of this other phenomenon as of the "degeneration removal". It results on the interactions between particles (first weaker and than stronger and stronger as pressure grows up). So, system behaviour is really sensitive to whatever interactions are present.

Quantitatively, physicists are used to face this kind of problem, beginning from results given for simplest ones (free particle gas) and than applying where possible "perturbations" to the system (strongly interacting gas).

It is very easy to forecast what happened in a phase transition from a gas to a system even more interacting like a liquid or a solid.

The result you can argue with me is that the degeneration in the energy spectrum is defintely removed and the energy levels (lines in spectrum) are much less intense and narrow. More precisely, for a crystal system, you will find out a continuum of values caused by interactions and its periodical properties. They're referred to as energy bands of the electronic states of a crystal.

This theory is really powerful because it can justify many properties of the cristal (optical, electrical, thermal... ).

Solid state physics is one of the most meaningful (and clearly the most ordinary as the system it deals with) proof for the validation of quantum mechanics!

 

[meopemuk]

In a crystal net.When i put 2 or many atom nearly.It's electron's energy will become energy band.(view attached image)

why?
what is a essential of this phenonmenon?

Consider quantum-mechanical problem about a particle (electron) in a potential well (atom). Suppose that you solved this problem and found a discrete spectrum of ehenrgy eigenvalues. Now, consider another problem in which there are two potential wells at some distance. It is not difficult to show that the energy spectrum of this system will be similar to the spectrum of the one-well system. The major difference is that each energy level now splits into two close energy levels. The separation between the two energy levels is determined by the strength of interaction of the electron in the well 1 with the potential created by the well 2. So, the separation increases when the two wells move closer together.

You can repeat the same logic for 3, 4, ..., infinite number of wells. The later case is a model of the crystal. In this model each atomic level has transformed in an infinite set of closely-packed levels, i.e., the energy band.

Eugene.

 

[ewilibrium]

Hello!
The Energy Band has N energy level (if system has N atomic).
But..i don't know what electron take what level?

Example: If atomics so far,the electron in orbital 2s of atomic i th and the electron in orbital 2s of atomic j th (i difficult j) is same energy level.
When they so near, and their energy level become degeneracy.
Then, what electron take what level? The atomic i th 's electron or j th ?

 

[Gokul43201]

When the atoms come close enough for the 2s orbitals to "overlap", the two electrons "become indistiguishable", and can not be identified by labels.

Note: The above is a statement of approximations; hence the quotation (" ") marks.

 

[ewilibrium]

When the atoms come close enough for the 2s orbitals to "overlap", the two electrons "become indistiguishable", and can not be identified by labels.

Note: The above is a statement of approximations; hence the quotation (" ") marks.

Thank Gokul!

But i think:
If N (~10^23) atom come close, that is not same as your example.
A orbital 2s of one atom can "overlap" with just several other atom. (Not with all atom)
I think that isn't the Identicle Particle in Quantum Mechanics.
I think that is Second Quantization, that is a "representative method"...

---

what electron take what level?

 

[kanato]

Thank Gokul!

But i think:
If N (~10^23) atom come close, that is not same as your example.
A orbital 2s of one atom can "overlap" with just several other atom. (Not with all atom)
I think that isn't the Identicle Particle in Quantum Mechanics.
I think that is Second Quantization, that is a "representative method"...

All electrons are identical particles. Second quantization is just a mathematical formalism for working with systems with many identical particles.

what electron take what level?

I'm not sure what you mean by this question. Electrons are indistinguishable particles; you cannot label which one goes into which level. The best you can do is label the levels themselves, and indicate whether or not they are occupied. The labels of atomic states don't translate so well to electron states in solids. In solids the eigenstates are very delocalized and labeled by momentum wavevector k (more properly called the pseudo-momentum). As was mentioned N atomic states per atom mix to create N bands, or put another way, N energy values for each k vector.

If you are interested in understanding energy bands in solids in terms of atomic states you should look up the tight binding method : http://en.wikipedia.org/wiki/Tight-binding_model

The simplest example of a tight-binding model would be a one-dimensional system with a single atomic orbital, where you have overlap between an s state on an atom and its nearest neighbors. Then you get a dispersion of the form $$\varepsilon_k = -2t \cos(k)$$ where t is the so-called hopping coefficient. Then to put electrons in states, you put then in starting from the lowest energy level first. If you have one electron per atom then with two spins you will find that states from 0 to pi/2a are doubly occupied and states with higher energies are unoccupied.