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If we heat a chunk of something made of only one type of element, for example, copper, would the spectral distribution be discrete since atoms absorb and emit photons with only certain energies? Or would it be continuous like the radiation of a blackbody?

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If the "chunk of something" is a dilute gas, where each of the atoms is to a good approximation independent and does not interact with the other atoms, then you will get discrete lines corresponding to the absorption lines of the atoms. If, however, the atoms are in a lattice and form a complex structure based on their interactions, then you might have a more continuous spectra, corresponding to the many different excitations possible in such a material. It will not necessarily be a black-body radiation, and might still retain some of the spectral lines that correspond to transitions in individual atoms (in addition to others). The details vary from one material to another, depending on its (not surprisingly) spectral properties.

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