# Milne Relation

### Short Topical Videos[edit]

### Reference Material[edit]

### Need to Review?[edit]

- Einstein Coefficients
- Saha Equation
- Boltzmann distribution
- Maxwellian velocity distribution
- Line Profile Functions
- Detailed Balance

### Bound-Free Transitions (Photoionization)

We’ll calculate the cross-section of a bound-free transition :

It turns out that is about . , so scaling from Lyman-alpha:

It turns out that the real answer is . In general:

That exponent (-3) is actually near the edge and goes to far from it. So you see spike up as the photon reaches the ionization energy, and then decrease exponentially as energy increases. However, you can see new spikes from ionizing electrons in inner shells.

### Radiative Recombination

This is the inverse process of photoionization, so is the cross-section for an ion recapturing its electron and emitting a photon. We’ll relate to . This is called the Milne Relation. In this derivation, we’ll start by assuming complete thermal equilibrium and derive a result which will end up being independent of thermal equilibrium. Let’s start calculating the rate of radiative recombinations. Thermal equilibrium dictates that this must equal the rate of photoionization. For radiative recombination:

We’ll set this equal to the rate of photoionization. This rate is:

where is the # density of neutrals. Note this has units of # flux. Note also that depends on , and is evaluated at the relative velocity such that , where is the threshold ionization energy. In thermal equilibrium, we know that (see Boltzmann distribution):

where is the energy difference between state 1 (proton + unbound ) and state 2 (bound proton/electron pair). Thus is given by:

So we can make our n’s and g’s go away. For , we’ll use our Maxwellian velocity distribution:

Finally, the Saha Equation tells us in thermal equilibrium:

So now we’re essentially done:

and plugging in all of our relations we get:

(Milne Relation) Notice how all of the T’s vanished. This result is independent of thermal equilibrium. However, you still have to pay attention to your statistical weights (g’s).