Is Blackbody Radiation Universal?

In a recent series of articles Pierre-Marie Robitaille questions the idea of universality of blackbody radiation. Let us see what the analysis of the model studied at Computational Blackbody Radiation can say.

The model consists of a wave equation for a vibrating atomic lattice augmented with small damping modeling outgoing radiation. The model is characterized by a lattice temperature $T$ assumed to be the same for all frequencies $\nu$ and a radiative damping coefficient $\gamma$ with corresponding radiance $R(\nu ,T)$ depending on frequency and temperature according to Planck's law (with simplified high-frequency cut-off):

• $R(\nu ,T)=\gamma T\nu^2$ for $\nu\leq\frac{T}{h}$,
• $R(\nu ,T)=0$ for $\nu > \frac{T}{h}$,

where the parameter $h$ defines the high-frequency cut-off.  The model will subject to frequency dependent forcing $f_\nu$ reach equilibrium with incoming = outgoing radiation:

• $R(\nu ,T) =\epsilon f_\nu^2$ for $\nu\leq\frac{T}{h}$,
• assuming for simplicity that frequencies $\nu >\frac{T}{h}$ are reflected,

where $\epsilon\le 1$ is a coefficient of absorptivity = emissivity. The radiative qualities of a lattice can thus be described by the coefficients $\gamma$, $\epsilon$ and $h$ and the temperature scale $T$. 

Assume now that we have two lattices 1 and 2 with different characteristics $(\gamma_1,\epsilon_1, h_1, T_1)$ and $(\gamma_2,\epsilon_2, h_2, T_2)$ which are brought into radiative equilibrium. We will then have 

• $\epsilon_1\gamma_1T_1\nu^2 = \epsilon_2\gamma_2T_2\nu^2$ for $\nu\leq\frac{T_2}{h_2}$ 
• assuming $\frac{T_2}{h_2}\leq \frac{T_1}{h_1}$ 
• and for simplicity that 2 reflects frequencies $\nu > \frac{T_2}{h_2}$.    

If we choose lattice 1 as reference, to serve as an ideal reference blackbody, defining a reference temperature scale $T_1$, we can then calibrate the temperature scale $T_2$ for lattice 2 so that 

• $\epsilon_1\gamma_1T_1= \epsilon_2\gamma_2T_2$,

thus effectively assigning temperature $T_1$ to lattice 2 by radiative equilibrium with lattice 1 acting as ideal blackbody, thus effectively using 1 as a reference thermometer, assuming it has maximal cut-off. Any lattice 2 will then mimic the radiation of lattice 1 in radiative equilibrium and a form of universality will be achieved. 

In practice lattice 1 is represented by a small piece of graphite inside a cavity with walls represented by lattice 2 with the effect that the cavity will radiate like graphite independent of its form or wall material. Universality will thus be reached by mimicing of a reference, viewed as an ideal blackbody, which is perfectly understandable, and not by some mysterious deep inherent quality of blackbody radiation. Without the piece of graphite the cavity will possibly radiate with different characteristics and universality may be lost.

The analysis indicates that the critical quality of the reference blackbody is maximal cut-off and equal temperature of all frequencies, and not maximal absorptivity = emissivity = 1, since the effective parameter is the product $\epsilon\gamma$. 

Summary:

• All dancers which mimic Fred Astaire, dance like Fred Astaire, but all dancers do not dance like Fred Astaire.