Recombination Lines - Astrophysics

• BOAS
In summary: The correct formula is ##E_n = -\frac{1}{2}\mu \frac{(\alpha c)^2}{n^2}##.In summary, 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}##. The formula for the frequency shift ##\nu_{Hen\alpha} - \nu_{Hn\alpha}## is ##E_f - E_i = -\mu \frac{(\alpha c)^2}{(n-1)^2} + \mu \frac{(\alpha c)^2}{n
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)

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##

##\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}##

##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.

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:
BOAS said:

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.

TSny said:
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.

Yes.

1. What are recombination lines in astrophysics?

Recombination lines are spectral lines emitted when an electron transitions from a higher energy state to a lower energy state within an atom or molecule. These lines are important in astrophysics because they provide information about the chemical composition, temperature, and density of interstellar gas clouds and can help us understand the formation and evolution of galaxies and stars.

2. How are recombination lines formed?

Recombination lines are formed when a free electron combines with an ionized atom or molecule, releasing energy in the form of a photon. This process occurs when the temperature of the gas is low enough for electrons to recombine with ions, typically around 10,000 Kelvin in interstellar clouds.

3. What can recombination lines tell us about the universe?

Recombination lines can tell us about the chemical composition of the universe, as different elements produce unique spectral lines. They can also provide information about the temperature and density of interstellar gas clouds, which are important factors in the formation of stars and galaxies.

4. How are recombination lines observed?

Recombination lines are observed using radio telescopes, which can detect the faint radio emissions produced by the photons released during the recombination process. These observations can also be used to create maps of the distribution of different chemical elements in the universe.

5. Are recombination lines only found in space?

No, recombination lines can also be observed in laboratories on Earth. However, they are most commonly studied in space as the conditions necessary for their formation (low temperatures and ionized gas) are more prevalent in interstellar clouds than on Earth.

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