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Calculate natural line width of the transition

  1. Nov 26, 2011 #1
    1. The problem statement, all variables and given/known data
    A lifetime of the first excited state for some atom is τ, calculate natural line width for that line.

    3. 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 ?
     
  2. jcsd
  3. Nov 28, 2011 #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.
     
  4. Nov 28, 2011 #3

    RHK

    User Avatar

    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...
     
  5. Nov 28, 2011 #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: Nov 28, 2011
  6. Nov 28, 2011 #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 :)
     
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