Stimulated emission per second per atom?

[Philip Land]

Homework Statement

We are investigating hydrogen in a plasma with the temperature 4500 ºC. Calculate the probability per atom and second for stimulated emission from 2p to 1s if the lifetime of 2p is 1.6 ns

Homework Equations

Planks radiation law:
$\rho (f) = \frac{8* \pi h}{c^3}\frac{1}{e^{\frac{hf}{kT}} -1}$ (1)

Einstein coefficients :
$A=\frac{8 \pi f^3 h}{c^3} B$ (2)

The Attempt at a Solution

I can take the inverse of the life-time to get A. Then I can just solve for B. Using that $\lambda = 1.251*10^{-7}$ (which I looked up).

I seek a unit that is per second per atom. Which I can't really get my head around.

I try to play around with the frequency density (1) but fail to find per atom. When I get "per atom" I can just multiply that with A.

Any tipps here?

 

[Cutter Ketch]

Finding B as you describe is fine. Planck’s radiation law which you show will also be necessary. You are almost there. What is B good for? How is it used? Do you perhaps see it in a differential equation relating the rate of stimulated emission to the upper state population density? What must be the units of the coefficient on the upper state population density?

 

[Philip Land]

Finding B as you describe is fine. Planck’s radiation law which you show will also be necessary. You are almost there. What is B good for? How is it used? Do you perhaps see it in a differential equation relating the rate of stimulated emission to the upper state population density? What must be the units of the coefficient on the upper state population density?

Thank you!

I figured out how to solve this. I wanted the unit $s^-1$. (Which also A gave so I'm not sure I undertand the theory here). But evaluating the units I see that if I multiply B (which I can get if I know $f$, which I can get from the Bohr model) by the frequency density I get the rate of stimulated emission per atom per second. So $B * \rho$ is what I was seeking.