# Radiation from a dielectric body

• A
Dear all,

I needed help in finding a source where the derivation of radiation emitted by a dielectric body is laid out.

The derivation of spectral density of radiation emitted from a blackbody at a temperature ##T## is given in many books by populating the energy states using Bose-Einstein statistics. However, try as I might, I have not been able to find any source where the derivation of the spectral density of radiation emitted by a semi-infinite body with a dielectric function ##\varepsilon (\omega)## and at a temperature ##T## is derived from first-principles (populating the quantum states).

I'd assume it should be straightforward since the dielectric function can be approximated by Lorentzian harmonic oscillators. Is it not so ?

Do you know any book/paper which discusses this in detail ?

Many thanks!

## Answers and Replies

Charles Link
Homework Helper
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I think it takes more than a dielectric function and/or oscillators. It is necessary to have anharmonic terms (essentially non-linear in the restoring force or non-quadratic in the energy) in the Hamiltonian, which may give you a complex dielectric function (with imaginary components) and a complex index of refraction. A completely harmonic Hamiltonian would give you a completely transparent dielectric and thereby the emissivity would likely be near zero. I think the solid state physics book by Ashcroft and Mermin discusses the anharmonic Hamiltonian. I don't have any handy references that have the precise derivation you are looking for, but hopefully this is helpful.

I'm surprised why this topic of radiated energy density by a dielectric body is not commonly dealt with from first principles. I'd assume it to be of primary academic interest.

Regarding what you mentioned, won't damped harmonic oscillators account for absorption in the dielectric ?

Charles Link
Homework Helper
Gold Member
2020 Award
Regarding what you mentioned, won't damped harmonic oscillators account for absorption in the dielectric ?
I think the two are mathematically quite similar, but you might find it written up in the solid state textbooks as an anharmonic term. Meanwhile, one other thing to consider would be a Kirckhoff's law type equation where for an opaque material the emissivity plus the reflectivity is equal to unity. For a dielectric, I think you have a similar relation with a transmission term included. I don't know of a good source that discusses this concept in depth, but hopefully you might find one.