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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?

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  • $\begingroup$ Comment deleted. $\endgroup$ – garyp Jul 14 '16 at 13:32
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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.

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  • $\begingroup$ Sorry, I can understand your complaint. My question anyway was not about the pdf, but about the correct definition of "density of photons" (number of photons per unit frequency, per unit volume). The one I gave seemed correct to me, and I linked the pdf just because it seemed to give a different definition. $\endgroup$ – BowPark Jul 14 '16 at 13:50
  • $\begingroup$ It's just an informal guideline that we try to follow on this site. It has happened to me more than a few times on Stack Exchange sites that a question or answer has included a link, but the link was dead reducing the value of the question/answer. It has also happened to me that I've had to dig deep into some linked document to understand the post. More often than not, if a question relies on a link I simply skip it, and I know that other readers do the same. So you have a better chance of getting a response if your posts are self-contained. (Not always possible, I recognize.) $\endgroup$ – garyp Jul 14 '16 at 14:14

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