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Units for Einstein coefficients in stimulated emission?

  • #1

Homework Statement



Hi,

I need to know the correct SI units for Einstein Coefficients (A and B) for stimulated emission (say laser).
The equation I'm on about is

Homework Equations



[tex]\frac{A}{B}[/tex] = [tex]\frac{8\pi h\nu^{3}}{c^{3}}[/tex]


The Attempt at a Solution



after some scribbling I got to [tex]\frac{A}{B}[/tex] = [tex]\frac{Js}{m^{3}}[/tex]
where J is joules, s seconds and m is meter.

any help appreciated. thanks
 
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Answers and Replies

  • #2
diazona
Homework Helper
2,175
6
According to Wikipedia, the units of A are radians per second, and based on the ratio you got you should be able to figure out what the units of B are. Although I'm not sure whether to trust Wikipedia on this without having some other source (i.e. a textbook) to back it up.
 
  • #3
thanks for your reply diazona!

apparently A is the probability per unit time of an electron making spotaneous transition so assuming A to be [tex]s^{-1}[/tex] then B should be [tex]\frac {m^{3}}{Js^{2}}[/tex], http://en.wikipedia.org/wiki/Einstein_coefficients#The_Einstein_coefficients" gives for B [tex]\frac {sr m^{2}}{Js}[/tex] where sr is solid angle and is dimensionless. I'm getting close but what am i doing wrong??
 
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  • #4
6
0
Wikipedia's right, I just happened to be working on this so let me show you why.

The units of coefficient A has the same units as BxJ, where J is the average specific intensity with units Jm-2s-1Hz-1Sr-1

A is the transition probability so has unit s-1

After juggling around I get =(m2SrHz)/J = what Wikipedia says :)
 
  • #5
1
0
Sorry to dig up this old thread, but I came across this post when trying to find out which units to use and thought I should add the correct answer now I've found it.

Radiative Processes in Astrophysics by Rybicki and Lightman (p29) defines the transition probability per unit time ([itex]\mathrm{s}^{-1}[/itex]) for stimulated emission as [itex]B_{21}\overline{J}[/itex], where [itex]\overline{J}[/itex] is the mean intensity ([itex]\mathrm{Jm^{-2}s^{-1}sr^{-1}Hz^{-1}}[/itex]). This gives [itex]B_{21}[/itex] in units of [tex]\mathrm{m^2 sr J^{-1} s^{-1}}[/tex] However, the book also states that the energy density [itex]u_\nu[/itex] is often used instead of [itex]J_\nu[/itex] to define the Einstein B-coefficients. [tex]u_\nu=\frac{4\pi}{c}J_\nu[/tex] where [itex]J_\nu[/itex] is in the same units as [itex]\overline{J}[/itex] and therefore the units of [itex]u_\nu[/itex] are [itex]\mathrm{Jm^{-3}sr^{-1}Hz^{-1}}[/itex]. Therefore if the transition probability is defined as [itex]B_{21}\overline{u}[/itex] (with [itex]\overline{u}[/itex] again in the same units as [itex]u_\nu[/itex]) then the units of [itex]B_{21}[/itex] become [tex]\mathrm{m^3 sr J^{-1} s^{-2}}[/tex] So both of you were correct! Just make sure you stick to one definition or the other.
 
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