# Calculate natural line width of the transition

## 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 ?

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.

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...

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:
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 :)