Probability per atom and per second for stimulated emission to occur

[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

$A=\frac{1}{\Sigma \tau}$

$$A_{2,1} = \frac{8*\pi *h * f^3*B_{2,1}}{c^3}$$

The Attempt at a Solution

[/B]
hmmm, I'm not sure how to approach this problem. I took the inverse of the life time and got that A= $6.25*10^8 S^{-1}.$

But I'm not sure where to start or what formulas to use.

The only formula I know of which takes temperature into account is
Doppler line width: $\Delta F = constant * f_0 * \sqrt(T/M)$ which I can't see how to apply in this case at all.

Any input on where to start?

 

[mfb]

The probability of stimulated emission will depend on the density of radiation around the atom, which depends on the temperature.

 

[Philip Land]

The probability of stimulated emission will depend on the density of radiation around the atom, which depends on the temperature.

Thanks a lot! I manage to as you said find a relation between radiation density and temperature, (Planck’s radiation law).

Then I used a Radiation balance and solved for $A_{21} = \rho (f) * \frac{N_1}{N_2}*B_{12} -B{21}*\rho (f).$

Where $g_1*B_{12} = g_2*B_{21}$ if we let g1=g2 we get $B_{12}=B_{21}$

We also know from statistics that $\frac{N_1}{N_2}= e^\frac{- \Delta E}{kT}$

But my question is. To get A (which I guess I'm supposed to get). I need B and $\Delta E$ But I don't have those quantities... (as I'm aware of).

 

[mfb]

Did you use the given lifetime already?

 

[Philip Land]

Did you use the given lifetime already?

Yes I used that to get the frequency, used in Plancks radiation law.

 

[TSny]

$A_{21} = \rho (f) * \frac{N_1}{N_2}*B_{12} -B{21}*\rho (f).$

It might help to rearrange this equation as $N_2A_{21} + N_2B_{21}\rho (f)= N_1B_{12}\rho (f)$.
Interpret each of the terms. One of the terms is closely related to what you are asked to find.