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When the electron absorbs radiation, it can seemingly do 2 things. Firstly, the electron can become excited and jump to a higher energy level before collapsing and releasing that energy as a photon.

However, I thought about absorbance. For example, a black cloth simply absorbs the incident photon energy and converts that to heat. How come the black cloth doesn't have it's electrons excited but rather just decided to make the atom move more? Why doesn't it release a photon like in the first case?

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An isolated atom has no way to get rid of the energy other than to emit a photon, so an isolated atom will always decay by emitting a photon.

However if there are other atoms around, for example in a gas, the atoms are frequently colliding with each other. If the excited atom collides with another atom the electronic energy can be converted to kinetic energy. This is known as collisional de-excitation, which is as the name suggests the reverse of collisional excitation. In gases at around atmospheric pressure the collisions are so frequent that most atoms decay and lose energy before they can emit a photon. Indeed, when experimenters are trying to measure electronic spectra in a gas they need to use a very low pressure for precisely this reason.

In a solid the atoms are linked together by the bonds between atoms and molecules, and in this case electronic energy can be converted to vibrational energy. Indeed, as with high pressure gases this process is so efficient that in most cases you can't see photon emission from solids and liquids. It usually takes careful control of the experimental conditions to measure photon emission, though there are exceptions like fluorescence and phosphorescence.

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In materials there are radiative transitions and non-radiative transitions. When photons are absorbed, they might be 'swept away' by non-radiative transitions. These transitions can be Auger-recombination or simply the electrons falling back down to a lower energy level and release phonons into the material instead.

Photons could also be reemitted in wavelengths outside of the visible spectrum, IR and heat for example.

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