# Units for Einstein coefficients in stimulated emission?

1. May 20, 2009

### euler_ka_abbu

1. The problem statement, all variables and given/known data

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

2. Relevant equations

$$\frac{A}{B}$$ = $$\frac{8\pi h\nu^{3}}{c^{3}}$$

3. The attempt at a solution

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

any help appreciated. thanks

Last edited: May 20, 2009
2. May 20, 2009

### diazona

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. May 20, 2009

### euler_ka_abbu

apparently A is the probability per unit time of an electron making spotaneous transition so assuming A to be $$s^{-1}$$ then B should be $$\frac {m^{3}}{Js^{2}}$$, http://en.wikipedia.org/wiki/Einstein_coefficients#The_Einstein_coefficients" gives for B $$\frac {sr m^{2}}{Js}$$ where sr is solid angle and is dimensionless. I'm getting close but what am i doing wrong??

Last edited by a moderator: Apr 24, 2017
4. Jan 18, 2011

### km707

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. May 27, 2012

### kavey

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

Last edited: May 27, 2012