Natural Broadening

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\usepackage{fullpage} \usepackage{amsmath} \usepackage{eufrak} \begin{document} \section*{Natural Broadening}

Natural (intrinsic) Broadening is one cause of the width $\Delta\nu$ in a line profile function $\phi(\nu)$. This type of spectral line broadening arises from the spontaneous decay rate $A_{10}$. If you have a bunch of atoms, their lifetimes are affected by the uncertainty in their energy states from quantum mechanics (Heisenberg Uncertainty Principle).

$$\Delta E\Delta t \sim \hbar$$\\

A photon in a certain energy state will therefore have a range of possible frequencies when it decays to a lower state.

$$\Delta \nu \sim {\frac{\Delta E}{h}} \sim {\frac{1}{2\pi \Delta t}}$$\\

The line profile function resulting from Natural Broadening is the Lorentzian Profile. It is proportional to $A_{10}$. That is, larger $A$'s (faster/stronger decays, or a stepper decay profile) result in more broadening (wider profile function).\\

\\ $$\phi(\nu) \propto A_{10}$$\\ $$\phi(\nu) = {\frac{A_{10}}{4\pi^{2}}}{\frac{1}{(\nu-\nu_{0})^{2}+({\frac{A_{10}}{4\pi}})^{2}}}$$\\

The peak value of the profile occurs when $\nu =\nu_{0}$.

$$\phi_{peak} = {\frac{4}{A_{10}}$$ \\

This can be used to calculate the FWHM by setting ${\frac{\phi_{peak}}{2}} = \phi(\nu)$ and solving for $2\times (\nu-\nu_{0})$. The result is:

$$\Delta\nu_{FWHM} = {\frac{A_{10}}{2\pi}}$$\\

A typical lifetime for an atomic energy state is $10^{-8}$ seconds, which corresponds to a natural line width of $6.6 \times 10^{-8}$ eV.\\

If radiation is present, then stimulated emission effects must be added to the spontaneous emission ones. Overall though, Natural Line Broadening is not the dominant broadening effect and isn't often directly observed, except in the line wings.