How can the oscillating states of many atoms in a lattice combine to give even a larger number of possible states?

Basically, I was studying the band theory and there was that thing that I couldn't understand, and it appeared to be connected to another question: how is the blackbody radiation appear continuous rather than quantized. I would like to know how can energy states add together and allow the emission of various quanta of energy ? I was asking this problem because I knew that an atom of hydrogen has specific emission lines, and the same goes for helium and all elements. However, the sun gives us seemingly all the spectrum (discluding absorption lines) rather than discrete frequencies.

1 Answer

Let us break it down to the simpler question: if I have an atom that emits radiation of one very particular wavelength, how do I get other wavelengths from it? One easy way is to let the emitted photon scatter inelastically off a bunch of other atoms, so that it loses some energy, and becomes a photon of a different wavelength (There is also a possibility of inverse Compton scattering, which can cause it to gain energy instead). This is what gives rise to the continuous, close-to-blackbody spectra of macroscopic bodies. An astronomical number of scattering events convert a line spectrum into a continuous spectrum in the sun. Since the sun has enough atoms for photons to scatter off from, the wavelength that is emitted in the interior of the sun is not the wavelength that escapes from the surface. Sidenote: it takes thousands of years for a photon emitted deep in the interior of the sun to reach the surface. That should give you an impression of the number of scattering processes it encounters before leaving the sun. The resultant spectrum therefore has little similarity with the line spectra of individual nuclei or atoms.

Update to include bands in a solid: the number of states does not change. That is to say, if one atom has $$n$$ states, then $$N$$ atoms have exactly $$Nn$$ states. The difference is, for $$N$$ isolated atoms, the ground states would be degenerate. Once they are close enough to affect each other, Pauli exclusion forbids this, and degenerate energy levels become non-degenerate, with a gap between them. The width of the gap is inversely related to the number of states there were to begin with. So for bands in a solid with $$10^{23}$$ atoms, the spacing of degenerate energy levels is infinitesimal, and they form a band.

• But how can scientists perform analysis to the spectra from starts to know their components and this stuff? Also does making the bands or states infinitesimal make the emitted energy follow the black body radiation ( seem continuous)? – Yusuf Ismail Mar 21 '20 at 19:05
• the answer to both your questions is the same. it's difficult to get a pure blackbody spectrum starting with a line spectrum, and that is precisely why we can still analyze stellar spectra. eventhough the spectrum is continuous, not all features of the line spectrum are lost, and certain wavelengths are much more prominent than others in a stellar spectrum, and this characterizes the composition of the star. – NewUser Mar 21 '20 at 19:29