Calculation of the Natural Linewidth

[blaisem]

Hi, I am trying to understand how one determines the natural linewidth. On my assignment, I am only given an energy (589.1 nm transition in sodium). I have two sources that I have found that seem to contradict each other:

Source 1: Slides 5 and 6

Source 2: Hyperphysics

If I plug in either the lifetime or the energy value provided in the example from the powerpoint (slide 6) into the Hyperphysics calculator, the corresponding value isn't consistent with slide 6.

I am confused on which is the correct formula, as well as how one determines the natural linewidth without knowing the lifetime of a transition. Is the energy of the transition actually relevant?

Can anyone please advise? Thank you for your time and help.

 

[DrClaude]

If I plug in either the lifetime or the energy value provided in the example from the powerpoint (slide 6) into the Hyperphysics calculator, the corresponding value isn't consistent with slide 6.

I didn't check the calculations in detail, but in the slides, both the initial and final states are considered to have finite lifetimes. This may be the source of the discrepancy.

how one determines the natural linewidth without knowing the lifetime of a transition.

They are usually measured from spectra.

If you need the numbers for sodium, check http://steck.us/alkalidata/.

 

[blaisem]

Hi DrClaude. Thanks for your response and the link. Maybe it was implied I had to look up the lifetime. 3 questions, if you have time, since I am having trouble wrapping my head around it conceptually:

1.Hyperphysics provides a relationship of:

2E = Gamma = (reduced plank constant / lifetime)​

where Gamma is the width of the natural broadening

If I substitute the given transition energy of 589.1 nm into E, I get a gamma of 4.2 eV; if I use the lifetime from your source (16.2 ns), I get a gamma of 41 nano Ev.

I am confused about the role of the energy of the transition in natural broadening. Am I substituting the wrong value for energy into the Hyperphysics formula?

2. What would be the correct value of E? Would it be the absolute energy uncertainty of the initial state, and the transition energy I have is entirely irrelevant to determining the natural linewidth?

3. Is the formula in the powerpoint (first link) more precise than Hyperphysics? It seems to be a more complicated representation of natural broadening, implying that the Heisenberg Uncertainty Principle as it was presented in Hyperphysics may be a more superficial description of natural broadening. Is my understanding of this correct?

Thanks a lot for your advise!

 

[DrClaude]

If I substitute the given transition energy of 589.1 nm into E, I get a gamma of 4.2 eV; if I use the lifetime from your source (16.2 ns), I get a gamma of 41 nano Ev.

I am confused about the role of the energy of the transition in natural broadening. Am I substituting the wrong value for energy into the Hyperphysics formula?

Yes. In the formula, it is $\Delta E$, the uncertainty on the energy, not the energy of the transition.

2. What would be the correct value of E? Would it be the absolute energy uncertainty of the initial state, and the transition energy I have is entirely irrelevant to determining the natural linewidth?

If both the initial and final states have finite lifetimes, then both widths must be taken into account (as described in the slides). You have an uncertainty in both the energy of the upper state and the lower state. But the actual value of the center of the peak (the energy "before" taking into account the uncertainty) is not relevant.

3. Is the formula in the powerpoint (first link) more precise than Hyperphysics? It seems to be a more complicated representation of natural broadening, implying that the Heisenberg Uncertainty Principle as it was presented in Hyperphysics may be a more superficial description of natural broadening. Is my understanding of this correct?

Apart from the fact that it takes into account the uncertainty of the energy of the final state, I do not see any difference between the two approaches. The Hyperphysics formulation might be simplified because most of the time the final state is the ground state, which has no uncertainty.