Black body radiation and number of photons emitted

Question

Usually the black body radiation (at a certain temperature $T$) is given by

$$\rho ( \nu ) = \frac{8 \pi h \nu^3}{c^3 \left( e^{h \nu / (k_B T)} - 1 \right)}$$

This quantity $\rho ( \nu )$ should be the density of energy, that is: the energy per unit volume and per unit frequency, so its unit measure should be

$$\mathrm{\frac{J}{Hz \cdot cm^3}}$$

As adviced in a comment to this answer, the density of photons should be easily obtained from $\rho ( \nu )$. If each photon has energy $h \nu$, from the density of energy (per unit frequency, per unit volume) the corresponding density of photons (number of photons per unit frequency, per unit volume) should be

$$n( \nu ) = \frac{\rho ( \nu )}{h \nu}$$

But this document (page 14) is not agreeing with this. It derives the number of photons not from $\rho ( \nu )$, but from the radiation intensity $I ( \nu )$, which is density of power (not energy) per unit frequency per unit volume. It states

$$n (\nu ) = \frac{I( \nu )}{h \nu}$$

Even the unit measures seem not to match in this case. So, what is the correct definition for $n (\nu )$ and why?

Answer 1

Apples and oranges. The first is the photon density in a volume whose radiation field is in thermal equilibrium. The second is the rate at which photons pass a unit area regardless of the source of the radiation. They are both correct, but they describe different things.

BTW, it's much better to tell us what is in a document instead of asking us to look for it. Hyperlinks die, and clicking and scrolling through a document is not a lot of fun ... we are all volunteers here, after all.

Note that the text you link to (which I'm only now really reading) says exactly this on the preceding page, and furthermore, the units, which are explicitly provided, should provide a strong clue concerning what the quantity represents. It also explicitly mentions monochromatic radiation, another clue that the two equations don't have much to do with each other.