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Blackbody as Cavity with Graphite Walls

The above figure is used to convey the idea of universality of blackbody radiation: The radiation spectrum from the cavity (with walls of graphite) viewed through the peep-hole of the cavity, is observed to only depend on the temperature of the body placed in the cavity and not the nature of the body.  The following questions present themselves:

• Why is the model of a blackbody a cavity with peep-hole?
• What is the role of the graphite walls of the cavity?

Recall that planck’s Law expresses the radiated energy $E(T,\nu )$ of frequency $\nu$ from a blackbody of temperature $T$ as

• $E(T,\nu )=\gamma T\nu^2$

where $\gamma$ is supposed to be a universal constant, which is the same for all blackbodies independent of their composition. But how can the radiated energy be independent of the physics of the radiating body?

Mathematical Wave Model of Blackbody

Let us seek an answer in the new proof of Planck’s Law of blackbody radiation based on deterministic finite precision computation instead the statistics used by Planck, presented in Mathematical Physics of Blackbody Radiation and Computational Blackbody Radiation, based on the wave model:

• $U_{tt} - U_{xx} - \gamma U_{ttt} - \delta^2U_{xxt} = f$

where the subindices indicate differentiation with respect to space $x$ and time $t$, and

1. $U_{tt} - U_{xx}$: material force from vibrating string with U displacement,
2. $- \gamma U_{ttt}$ is Abraham-Lorentz (radiation reaction) force with $\gamma$ a small positive parameter,
3. $- \delta^2U_{xxt}$ is a friction force acting on frequencies larger than the cut-off frequency $\frac{T}{h}$ and then contributing to internal heating,
4. $\delta =\frac{h}{T}$ is a smallest coordination length with $h$ a measure of finite precision,
5. $T$ is the common energy/temperature of each frequency of the vibrating string,
6. $f$ is exterior forcing.

This model with $\delta =0$ is essentially the starting point also for Planck in his classical proof completed by resorting to statistics of quanta: A system of resonators $U$ in resonance with an exterior forcing $f$.

The model has two parameters $\gamma$ and $h$, assuming the coefficients of vibrating string are normalized to 1. Here $\gamma$ is given by the atom physics of the Abraham-Lorentz force, while the physical meaning of the finite precision parameter $h$ needs an explanation.

The new proof gives the additional information that the outgoing radiation is equal to the forcing in the sense that

• $\int \gamma\ddot U^2\, dxdt = \int f^2\, dxdt$,
which can be viewed to express that all incoming radiation is absorbed and then re-radiated as a defining property of a blackbody.

Universality of Blackbody Model

We ask in what sense Planck’s Law is universal with its apparent dependence on the parameters $\gamma$ and $h$. We then consider two blackbodies 1 and 2 defined by the parameters $(\gamma_1,h_1)$ and $(\gamma_2,h_2)$, where we choose 1 as a fixed reference. This is the cavity in the above figure, with walls in practice covered with graphite of black sooth of e.g. iron oxide.  Radiative equilibrium of 1 and 2 with energy balance and equal cut-off frequency requires

• $\gamma_1T_1=\gamma_2T_2$
• $\frac{T_1}{h_1}=\frac{T_2}{h_2}$.

We are thus led to the relation

• $\gamma_2h_2 =\gamma_1h_1 \equiv C$,

where we may view $C$ as a universal constant defined by 1. We understand that the relation $\gamma_1T_1=\gamma_2T_2$ defines the temperature scale for 2 with 1 as reference.

With a chosen reference blackbody we can argue that $C$ is a universal constant, which defines the finite precision parameter $h$ by the relation $\gamma h=C$.

We thus use blackbody 1 as a thermometer defining the temperature of a different blackbody 2 from radiative equilibrium, and then obtain universality of Planck’s Law reading the same for all blackbodies. The  dependence of the parameter $\gamma$ is then shifted into a variable heat capacity.

We thus understand that the universality of Planck’s Law reflects the choice of a certain blackbody as universal thermometer.

We now return to the question posed in the introductory figure text: In our analysis the role of graphite on the walls of the cavity, observed by Kirchhoff in early experiments, is to equilibrate the temperature for all frequencies, independent of the object placed in the cavity, as required for universality. Kirchhoff observed that with reflecting walls this was not achieved as the radiation spectrum depended of the nature of body placed in the cavity.

Smaller Cut-off Frequency

Radiative equilibrium is also possible with $\frac{T_2}{h_2}<\frac{T_1}{h_1}$, assuming the reference body 1 as an ideal blackbody has maximal cut-off frequency.  In this case the energy of the frequencies above the cut-off of 2 absorbed by 2 will be balanced by an increased amplitude of the radiation from 2.

Definition of Ideal Blackbody as Reference

We are thus led to identify the reference blackbody as a radiating body with maximal cut-off frequency which absorbs all incoming radiation and then re-radiates the same energy below cut-off and storing as heat above cut-off.

Absorptivity Smaller Than 1

The above argument assumes that the absorptivity (= emissivity) of body 2 is equal to 1, that is that 2 is also a blackbody. If 2 is a greybody with absorptivity $\alpha_2<1$, then there are two possibilities: Either the calibration of $T_2$ is made according to

• $\alpha_2\gamma_1T_1=\gamma_2 T_2$.
Or if $T_2$ is determined in some other way, for example so that $\gamma_1T_1=\gamma_2 T_2$, then the output of from 2 is corrected by a coefficient of emissivity $\epsilon_2=\alpha_2$ so that

• $\alpha_2\gamma_1T_1=\epsilon_2\gamma_2 T_2$.

References

For an introduction to classical work with an empty cavity as an abstract universal reference blackbody, see An Analysis of Universality in Blackbody Radiation by P.M. Robitaille.