Recombination Lines - Astrophysics

[BOAS]

Homework Statement

For an atom X, the high-n levels have energies $E_n = -\mu \frac{(\alpha c)^2}{n^2}$ with $\alpha = \frac{e^2}{\hbar c}$

Find the frequency shift $\nu_{Hen\alpha} - \nu_{Hn\alpha}$ for $n\alpha$ giving a transition frequency near 142MHz

(The notation here means that the electron moves by one energy level)

Homework Equations

The Attempt at a Solution

[/B]
$E_f - E_i = -\mu \frac{(\alpha c)^2}{(n-1)^2} + \mu \frac{(\alpha c)^2}{n^2}$

$= \mu (\alpha c)^2 (\frac{1}{n^2} - \frac{1}{(n-1)^2}) = h\nu$

rearranging and simplifying leads to

$\frac{1-2n}{(n-1)^2 n^2} = \frac{h \nu}{\mu (\alpha c)^2}$

Using a simplification I found justified here: http://www.cv.nrao.edu/course/astr534/Recombination.html

$\frac{1-2n}{(n-1)^2 n^2} \approx \frac{2}{n^3}$

which leads to

$n \approx (\frac{2 \mu (\alpha c)^2}{h \nu})^{\frac{1}{3}}$

solving for n using the value of the fine structure constant given in the question (which is in cgs units I believe), I get nonsense answers of $n < 1$.

Using the value of $\alpha = \frac{ke^2}{\hbar c}$ in SI units, I get values of n that produce the correct frequency when plugged back into my formula, but do not agree with radio combination lines that I looked up here: http://adsabs.harvard.edu/full/1968ApJS...16..143L

For example, using the fine structure constant in SI units, for Hydrogen I find $n = 452$, which when plugged into

$\nu = \frac{\mu (\alpha c)^2}{h} (\frac{1}{n^2} - \frac{1}{(n-1)^2})$

I get $\nu = 142.5MHz$, but from the above link I find that the transition should correspond to n=359.

I'm really struggling to get my head around where the discrepancy lies.

Thank you in advance for any help you can give!

EDIT - Ok, so I was using the wrong units for $e$ which solves my concerns about the fine structure's value, which is dimensionless. However, I am still stuck regarding the discrepancy between my calculated n values, and those I have found in the literature.

 

[TSny]

Homework Statement

For an atom X, the high-n levels have energies $E_n = -\mu \frac{(\alpha c)^2}{n^2}$

I believe you are off by a factor of 2 in this formula.

 

[BOAS]

I believe you are off by a factor of 2 in this formula.

A factor of 1/2 would make more sense I think - It would recover the ground state energy of Hydrogen and make my solutions for n match up with the recombination lines I linked to in my post.

 

[TSny]

Yes.