Calculate natural line width of the transition

In summary, the conversation is about calculating the natural line width for a specific atom's first excited state, given its lifetime. The individual attempts at solving the problem involve using the energy-time uncertainty relation and the energy-wavelength relation. However, the calculated results do not make sense and the speaker asks for clarification and explanation. They also mention the natural line width for a transition in Iron-57 and question the validity of using the given formula. Finally, they ask for examples of typical natural line widths for atomic transitions and lasers.
  • #1
vst98
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0

Homework Statement


A lifetime of the first excited state for some atom is τ, calculate natural line width for that line.

The Attempt at a Solution


Well, I could use energy-time uncertainty relation

ΔE*Δt=h

then ΔE*τ=h , now I can use a relation E=hc/λ

to obitain Δλ=c*τ , which is natural line width.

Now let's say τ=17ms (something like the lifetime of the Fe xiv)
then Δλ=3*10^8*17*10^-3=50*10^5m , this can not be true,
what am I doing wrong ?
 
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  • #2
Can someone give me any kind of answer, some thoughts, anything ?

If I take the example from hyperphysics:

http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/mossfe.html#c1

where they have calculated for some transition in Iron-57 that the natural line width is
gamma≈10^-8eV

now using the relation for energy E=hc/λ to obitain λ, this gives me:

λ=hc/E ≈ 4*10^(-15)[eVs]*3*10^8[m/s] /(10^-8[eV]) =120m

so this is the natural line width expressed in meters,
as I understand this is the Full width at half maximum for this line, but this does not make sense to me. I think FWHM should be some fraction of 10^-10m, since they are talking about gamma ray there.

In my first post, in question is some green (green coronal line from sun) and I think that the FWHM (natural line width of a line) should be fraction of a nanometer.

I know that my reasoning is somewhere wrong but I just don't see where.
 
  • #3
On that site you can see the formulae for the Natural linewidth, Gamma.
Why don't you use that one?

EDIT: are you sure about the lifetime of the Fe xiv? milliseconds? I don't know it, just asking...
 
  • #4
I can use that formula but this will also give me a result which I don't understand.
Let's say I use that formula and plug in just order of magnitude for my values, then

gamma≈hbar/τ ≈ 10^-16/10^-3 = 10^-13eV

gamma is in energy (FWHM in energy), but I want that in nanometers, therefore using
λ=hc/E = hc/(gamma)

I have λ≈10^-15*10^8/(10^-13) = 10^6 m
But this can't be natural line width (FWHM) of this green line.
And for lifetime value τ of this green coronal line, there are several scientific papers
which also give this value τ in ms , for example this paper
page 838 just at the top
http://iopscience.iop.org/0004-637X/587/2/836/pdf/56501.web.pdf
"The lifetime of the Fe xiv line we infer from the six data
sets is 16:69 ±0:10 ms."

If this ms means millisecond, then τ is ok,
I'm messing up somewhere else, the λ I calculated is probably something else and not natural line width. But how then to calculate natural line width in meters if you have a lifetime τ given ?
 
Last edited:
  • #5
Can someone at least give me an answer to any of these two questions:

1.) How do you find a natural line width (expressed in meters) ?
2.) What are typical natural line widths (atomic transitions , lasers etc ... in meters) ?

I would be grateful :)
 

1. What is the natural line width of a transition?

The natural line width of a transition is a measure of the spread of energy levels within an atom or molecule. It is determined by the lifetime of an excited state and the uncertainty principle, which states that there is a limit to how precisely the energy of a system can be known.

2. How is the natural line width of a transition calculated?

The natural line width of a transition is calculated using the formula Γ = h/(4πτ), where Γ is the natural line width, h is the Planck constant, and τ is the lifetime of the excited state. This formula takes into account the uncertainty principle and the fact that the energy levels of a system are constantly fluctuating.

3. What factors affect the natural line width of a transition?

The natural line width of a transition can be affected by several factors, including the lifetime of the excited state, the energy levels of the system, and the strength of the transition. Additionally, external factors such as temperature and pressure can also influence the natural line width.

4. How does the natural line width of a transition impact spectroscopy?

The natural line width of a transition is an important factor in spectroscopy, as it affects the resolution and accuracy of spectral measurements. A narrower natural line width allows for more precise measurements of energy levels, while a broader natural line width can make it more difficult to distinguish between different energy levels.

5. Can the natural line width of a transition be modified?

In some cases, the natural line width of a transition can be modified by changing the environment of the system, such as by using a high-pressure gas cell or a low-temperature environment. However, the natural line width is ultimately determined by the intrinsic properties of the system and cannot be altered beyond a certain limit set by the uncertainty principle.

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