[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: http://www.astronomy.gatech.edu/Courses/Phys3021/Lectures/pdf/Line_Broadening.pdf

Source 2: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/parlif.html

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.

[Corribus]

The energy is relevant because it sets the uncertainty of how long the excited-state persists (lifetime). And vice-versa. The natural linewidth derives from the uncertainty principle, where you can't know the energy change and temporal duration of a transition simultaneously to infinite precision. There are other line-broadening mechanisms due to inhomogeneity of the sample, for example, but even in a "perfect system", lines have a finite width that depends on the energy and lifetime of the transition.

[Enthalpy]

The lifetime relates with the uncertainty of the energy. It doesn't relate with the transition energy, for instance forbidden transitions (very slow) exist with a small as well as a big transition energy.

Hyperphysics gives Heisenberg's relation. For unstable particles in that case, but it applies as well to optical linewidths.

The conference Pdf looks good as well, it too gives Heisenberg's relation, but in terms of wavelength. What I don't grasp is where you get an energy uncertainty or a linewidth to inject in the formulas. If you put the transition energy it won't fit.

[Corribus]

I'm sorry, I see my earlier post was incredibly inarticulate. The Heisenberg Uncertainty Principle relates the maximum precision to which the state energy, not the transition energy, and the lifetime of the state can be known simultaneously. However, the transition energy DOES matter because the transition energy influences the natural radiative lifetime of the excited-state. There is an inverse relationship between the transition energy and the excited-state lifetime, embodied, for atoms anyway, in the Einstein coefficient for spontaneous emission, which gives the probability per unit time that an upper level state will spontaneously relax to a lower state via emission of a photon with approximately the energy difference between the two states.

[tex]A_{21}=\frac{\hbar \omega^3}{\pi^3 c^3} B_{21}[/tex]

Here, ω is the frequency of the emitted photon (basically, the transition energy) and B₂₁ is the Einstein coefficient for stimulated emission. Since A₂₁ is essentially a rate of loss from the excited-state, the lifetime of the state is the inverse of A₂₁, such that the lifetime of the state is related to the inverse (cube of the) transition energy. For molecules, the theoretical treatment is quite a bit more complicated* but the fundamental relationship is similar.

Granted, photon emission from excited molecular states is generally limited by other processes, so the true excited-state lifetime is rarely equal to the natural radiative lifetime. This is most commonly encountered in fluorescence, where various nonradiatve modes of decay (via internal conversion, intersystem crossing, or even photochemistry) depopulate the excited-state faster than what you'd predict from spontaneous emission alone. If this wasn't the case, all molecules would have 100% fluorescence efficiency, which is obviously not the case.

Nevertheless, it's generally true - all other things being equal, which of course they rarely are - that natural transition bandwidth is generally predicted to be smaller for lower-energy transitions. This is purely due to the impact of the transition energy on the natural lifetime of the excited state. Why, for example, NMR transitions are so narrow - transition energies are very small for nuclear transitions, meaning the natural lifetimes of the excited state are incredibly long, even into the timescale of seconds, which leads to very little uncertainty in the energy levels, and thus very narrow transition bandwidth. Electronic transitions are naturally broader because electronically excited state are many of orders of magnitude shorter lived... but of course electronic transitions are also subject to other selection rules and other broadening mechanisms, such that the natural bandwidth is rarely the limiting factor, and so this fundamental relationship is usually obscured.

*If you're really interested, you can read about the Strickler-Berg relation, which has got to be one of the most cited papers in molecular physics, if not all of physical chemistry.