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Recombination Lines - Astrophysics

  1. Dec 1, 2016 #1
    1. The problem statement, all variables and given/known data

    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)

    2. Relevant equations


    3. The attempt at a solution

    ##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.
     
    Last edited: Dec 1, 2016
  2. jcsd
  3. Dec 1, 2016 #2

    TSny

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    I believe you are off by a factor of 2 in this formula.
     
  4. Dec 2, 2016 #3
    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.
     
  5. Dec 2, 2016 #4

    TSny

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    Yes.
     
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