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We are always told that to excite an electron from one state to a higher energy states, for example from the valance band to the conduction band, the energy must equal the energy difference between the two energy states.

The conduction band consists of many allowed energy states, separated from the valance band by an energy gap. For the element Si, this energy gap is (at room temperature) about 1,1 eV. The range of energies for visible light is 1.8 eV to 3.1 eV. Since Si indeed conducts electricity at room temperature, is it then assumed that there is an allowed energy states with exactly the same energy as the energy difference between the two states? The conduction band, or the valance band for that matter, is not continuous, but discrete. Or is it sufficient that the photon energy "roughly equals", or is "close enough"? If so, how close is close enough?

I hope my question is clearly stated.

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There are two things here mixed: 1) there is a conservation of energy, which requires the states to have a given energy (with the error allowed by Heisenberg uncertanity principle) and 2) the fact that semiconductors and such have bands instead of separate electronic states, i.e. there is a broad, continuous margin whet the allowed transition energies are. Maybe you want to focus some part (1st?) in the question, instead of mixing them up. – Greg Nov 9 '15 at 2:27
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In fact there are many factors to consider, so the answer to your question depends on what type of material you are analyzing.

  • First of all you should know that even if the absorption level are discrete the absorption and the emission of a material are never a single value but a peak. If you consider the absorption or the emission of a gas you can assume them discrete value (even is not completely true), and so you can call them lines (see e.g. Lyman series), but if you condense the matter the interaction between the atoms/molecules broad the absorption/emission range and so you have peaks. If you are talking about electrons you are dealing with UV-VIS spectra in this case if you look at solid or a liquid UV-VIS spectra you can see that you don't have a well define peak but a continuous absorption, in this case you have bands. So it really depends of the state of the material you are analyzing.

  • There are other factors that determine the broadening of the absorption to multiple wavelengths. Form the previous explanation you can derive that more an atom is isolated, less interaction occurs, and the absorption is more limited to determinate wavelength, because you have similar "oscillators". But if you increase temperature and pressure you increase the probability that some atoms/molecules interact, and have different energy levels. The effect is the same: you are broadening the absorption band, respectively you will have a _temperature and a pressure broadening.

  • So it seems that if you have a single atom this absorbs at a determinate wavelength. Even in this case is not true, the explanation is not so intuitive as the previous ones because you have to understand Heisenberg uncertainty principle but roughly it states that you can't determine precisely the energy states of an atoms. This last broadening is called Natural broadening.

For your specific case you should look at the the UV-VIS spectrum, you will not see a sudden step next to the absorption band but a gently curve. This means that there are less and less atoms absorbing in this region because statistically is less probable that atoms can be perturbed so much to absorb that wavelengths but there are still chances that some atoms absorbs even at wavelength far from the mean.

Have a look to wikipedia spectral lines page for more infos.

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