# Calculating the black body radiation

A black body is one that absorbs all radiation that falls on it. The radiation that such a body emits, when at thermal equilibrium, is called "Black body radiation".

But when doing calculations about the radiation from these bodies, I don't understand some steps in these calculations:

1. When modelling the black body as a small hole in a cavity with walls, why do we assume that the light inside the cavity must be standing waves with nodes at the walls.

2. How does the 1-dimensional condition for standing waves: $n_x = \frac{2L}{\lambda}$ become the 3-dimensional counterpart $n = \sqrt{n_x^2 + n_y^2 + n_z^2} = \frac{2L}{\lambda}$

3. When finding the number of states $g(\varepsilon)$ with energy $\varepsilon,\;$ $g(\varepsilon)\,d\varepsilon = 2 \frac{1}{8} 4\pi n^{2}\, dn = \pi n^{2}\, dn$, can we write just $\pi n^2$? Shouldn't we account for the fact that it is only lattice points we are considering; something like: $\pi n^2 \times \%\,$of latice points?

• Please also explain how I should improve the question, along with down-voting it – PhyEnthusiast Apr 5 '18 at 16:33
• I didn't downvote but if I had to guess the downvote came because you don't show enough of your own thinking or work for a question that looks homework-like – pentane Apr 5 '18 at 23:52

Ok, let's go, one topic at a time:

1. We model the blackbody radiation as the one escaping by a small hole on the wall of a metal made object manteined at temperature T (such hole connects the interior cavity to the outside). Because the walls of the cavity are made of a conducting material (metal), the electric field vanishes at the boudaries. This, together with the fact that the incident and emmitted waves are interfering with each other inside the cavities, generates the standing wave pattern of the cavity radiation.

2. We can see this in 2 ways. Firstly, look at the two formulas you've written. The second one is clearly a generalization of the first, if you understand $n$ as a vector (after all, it is closely related to the wave vector, isn't it?). Besides that, this can be shown decomposing a radiation of a certain wavelength $\lambda$ into three components along the three mutually perpedicular directions of a cubic cavity. Because in each direction we must have a standing wave, in 3D we should as well have a standing wave. Writing the distance between the nodes of the 3D wave in terms of the distance between the nodes of the components of the wave in each direction will get you to the second formula.

3. Be careful. It looks to me that both the equation you wrote and the concept you're getting might be mistaken. First of all, with this calculation, you are simply counting up the number of allowed frequencies in the frequency interval $f$ to $f+df$, which is equal to the number of lattice points in the $(n_x,n_y,n_z)$-space between $n$ and $n+dn$, where the $n$ interval that corresponds to the $f$ interval by the second formula you wrote on item 2. We haven't accounted for the energy associated to said waves just yet. To count the number of points in this volume on the $(n_x,n_y,n_z)$-space, because the density of lattice points is 1, we just compute the corresponding volume: $(\frac{1}{8})4\pi n^2 dn$ (your factor of $2$ comes from the two possible independent polarizations of light waves, and the factor of $\frac{1}{8}$ comes from the fact that we are only interested in the positive octant of the $(n_x,n_y,n_z)$-space). We are interested in a interval of frequencies, then, the computation that you suggested, seems to me to not have any physical meaning.

Hope it helped.

Edit: I talked to my professor and we concluded that, in fact, the material does not need to be made out of metal. Of course, this guarantees that the electric field vanishes at the walls, but even if the cavity is not made of metal, the tangential component of the electric field must vanish (what originates the standing wave pattern). If that did not happen, two things could occur: it could make the electric charges of the walls move, violating the thermal equilibrium of the material and the radiation. In a more restricted case, it would polarize the material, what would violate the isotropy of the radiation inside the cavity.

• Can you explain "density of lattice points is 1"? – PhyEnthusiast Apr 6 '18 at 9:09
• Yes, of course. Because of the standing wave pattern, the allowed frequencies correspond to positive integer values of $n_x$, $n_y$ and $n_z$. Then, in the $(n_x,n_y,n_z)-space$, we are interested in counting the number of points with natural number coordinates in a certain interval. Such points correspond to a lattice of points in this space, whose density is 1: 1 point in each cube of volume 1. – André Muchon Apr 7 '18 at 19:03
• I edited my answer about the vanishing electric field on the wall, hope it helps. – André Muchon Apr 7 '18 at 19:11
• How do we know one point in each cube of volume 1 (and which units of volume are you using here)? Sorry for this, but this is the only part I don't seem to get at all – PhyEnthusiast Apr 8 '18 at 2:25
• You say "decompose a radiation of a certain wavelength λ into three components". Can all 3D wavelengths be decomposed into three components OR is this only for the case of blackbody radiation? – PhyEnthusiast Apr 9 '18 at 0:43