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I have emittance curves of a material (that cannot be shared) that show an increase in emittance values over a wide wavelength range. As the temperature increases, the value for emittance increases, and the region in which the emittance is close to 1 increases spectrally.

So, my question is: Do materials approach blackbody emittance ($\epsilon = 1$) for all wavelengths as the temperature increases?

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I think it depends on the material and its atomic structure. Look at the black body fit for the sun, which is a complex body. A very high temperature and not a perfect fit.

The cosmic microwave backround radiation has the best experimentally fitted black body curve, and the energy is very low.

Also this publication is interesting, showing a large range of temperatures for fitting with a black body spectrum ;Perfect blackbody radiation from a graphene nanostructure with application to high-temperature spectral emissivity measurements .

Seems to me it will depend on the material under study.

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