Line Profile Functions

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<latex> \documentclass[11pt]{article} \def\inv#1Template:1 \over \def\ddtTemplate:D \over dt \def\mean#1{\left\langle {#1}\right\rangle} \def\sigot{\sigma_{12}} \def\sigto{\sigma_{21}} \def\eval#1{\big|_{#1}} \def\tr{\nabla} \def\dce{\vec\tr\times\vec E} \def\dcb{\vec\tr\times\vec B} \def\wz{\omega_0} \def\ef{\vec E} \def\ato{{A_{21}}} \def\bto{{B_{21}}} \def\bot{{B_{12}}} \def\bfieldTemplate:\vec B \def\apTemplate:A^\prime \def\xp{{x^{\prime}}} \def\yp{{y^{\prime}}} \def\zp{{z^{\prime}}} \def\tp{{t^{\prime}}} \def\upxTemplate:U x^\prime \def\upyTemplate:U y^\prime \def\e#1{\cdot10^{#1}} \def\hf{\frac12} \def\^{\hat } \def\.{\dot } \def\tnTemplate:\tilde\nu \def\eboltz{e^Template:-h\nu 0 \over kT} \def\ato{{A_{21}}} \def\bto{{B_{21}}} \def\bot{{B_{12}}}

\usepackage{fullpage} \usepackage{amsmath} \usepackage{eufrak} \begin{document} \section*{Line Profile Function}

The line profile function, $\phi(\nu)$, describes the

distribution of absorption/emission around $\nu_0$ (the center transition frequency), and is subject to the requirement that: $$\int_0^\infty{\phi(\nu)d\nu}=1$$ Therefore, the line profile function represents the probability of getting a photon around $\nu_0$. Say that $\Delta\nu$ is the width of the distribution around $\nu_0$. $\Delta \nu $ is affected by three factors: $\ato$ (the natural, uncertainty-based broadening of at atom in isolation), $\nu_0 V_T/ c$ (the thermal, Doppler-based broadening), and $n_{coll}\sigma_{coll}v_{rel}$ (collisional broadening, a.k.a. pressure broadening). So really, the transition probability per unit time is: $$R_{ex}^{-1}=\bot\int_0^\infty{J_\nu\phi(\nu)d\nu}\approx\bot\bar J$$

\subsection{Natural (Lorentzian) 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.

\subsection{Doppler (Gaussian) Broadening}

Doppler 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 thermal motions of atoms. There will be a velocity distribution for these atoms which causes redshifted and blueshifted photons. Therefore, the Doppler effect leads to variation in the absorbed frequency $\nu$. \\

The Doppler shift is given by:

$$\nu = \nu_{0}\bigg(1+{\frac{v}{c}}\bigg)$$

where $v$ is the line-of-sight velocity of the absorbing atom. The amplitude of the frequency shift is therefore given by:

$$\Delta \nu = \nu_{0}{\frac{v}{c}}$$

Recall that the distribution of particle speeds in local thermodynamic equilibrium is Maxwellian:

$${\frac{1}{2}}mv^{2} \sim kT $$ $$v \sim \sqrt{{\frac{2kT}{m}}} $$

Plugging this velocity into the Doppler width gives:

$$\Delta \nu = {\frac{\nu_{0}}{c}}\sqrt{{\frac{2kT}{m}}} $$

The line profile function resulting from Doppler Broadening is a Gaussian Profile.

$$\phi(\nu) = {\frac{1}{\Delta\nu\sqrt{\pi}}}e^{-{\frac{(\nu-\nu_{0})^{2}}{\Delta\nu^{2}}}} $$ \\


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

$$\phi_{peak} = {\frac{1}{\Delta\nu\sqrt{\pi}}}$$ \\

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:

$$\Delta\nu_{FWHM} = 1.665\Delta\nu$$ \\

One thing to note is that as temperature increases, there is a greater spread in velocities. However, the number of atoms remains the same, so the peak in the above plot would drop $\bigg($since $\int_{0}^{\infty} \phi(\nu) d\nu = 1\bigg)$. Therefore, it is possible to go from an optically thick medium to optically thin. Up until now the thermal velocity distribution has been used, but turbulence can also cause a spread in velocities. The total Doppler Broadening can be found by adding the two components in quadrature.

$$v_{total}^{2} = v_{thermal}^{2} + v_{turb}^{2}$$\\

Doppler Broadening as discussed above refers to the differential movement of atoms. There can also be a bulk motion of the atoms in a particular direction, which is a shift in $\nu_{0}$.

\subsection{The Voigt Profile}

The Voigt Profile is a convolution of the Lorentzian Profile from Natural Broadening and the Gaussian Profile from Doppler Broadening. It is dominated by the Lorentzian wings and thermal Doppler Broadening in its center. It is a normalized function.

The Voigt profile shown here is for the Lyman-alpha transition, where $A_{10} = 5 \times 10^{8}$ $s^{-1}$ and $\nu_{0}$ is the transition frequency for the Ly-$\alpha$ transition ($1216$ angstroms). The Doppler width used here corresponds to a temperature of $100$ K.\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\

The absorption features due to each type of broadening is also plotted. The optical depth is given by: \\ \\ $$\tau = n \sigma s$$ $$ = N\sigma$$ \\

where $N$ is the column density. An absorption feature reduces the intensity of light by $e^{-\tau} = e^{-N\sigma}$ when there is no emission ($I_{\nu} = I_{\nu,0}e^{-\tau}$). The line profile function is incorporated into the cross-section $\sigma$ by the following:

$$\sigma = {\frac{\pi e^{2}}{m_{e}c}}f\phi(\nu)$$ \\

Therefore, different cross-sections can be calculated for each line profile function (using a Ly-$\alpha$ oscillator strength of $f=0.416$. The factor $e^{-N\sigma}$ by which a continuum is reduced creates the absorption feature. Again, the Voigt feature is dominated by the Gaussian at its center and by the Lorentzian wings.

\subsection{Collisional Broadening}

Collisional, or Pressure Broadening, is one cause of the width $\Delta\nu$ in a line profile function $\phi(\nu)$. This type of spectral line broadening arises from collisions that interfere with natural emission processes. Collisions amongst atoms in a high pressure gas can trigger the release of photons by reducing the effective lifetime of energy states, and therefore results in a steeper decay rate and the broadening of absorption lines. \\

The collision timescale is given by:

$$t_{collision} \sim t_{spontaneous\ decay} \sim {\frac{1}{A_{10}}} $$

And the collision rate is dependent on number density, cross section, and velocity:

$$A_{10} \sim n\sigma v$$ \\

In order for Collisional Broadening to dominate, a very dense gas and high velocities are needed. This is related to pressure, which is why Collisional Broadening is sometimes called Pressure Broadening. And this is also why white dwarfs produce broader spectral lines than giants of the same spectral types. \\

The cross section is also important in dictating the timescale for interaction. For fixed number density and velocity:

$$t \propto {\frac{1}{\sigma}}$$ \\

Atoms can interact in a variety of ways. There are different collisional cross sections due to different $\vec{E}$ field drop-off rates, such as $1/r^{2}$ for the Coulomb interaction between ions or charged particles, and $1/r^{3}$ for neutral atoms (dipole field), etc. Because these fields fall off at different rates, there are different cross sections for interaction. Therefore, the time scales for Collisional Broadening depend on which type of interaction dominates, which in turn affect the shape of the line profile function. \\

The line profile function due to Collisional Broadening is a Lorentzian Profile, similar to that caused by Natural Broadening. If all 3 types of broadening mechanisms are present, then Collisional Broadening will add width to the Voigt Profile. Different mechanisms can be responsible for broadening at different distances from the line center.

\subsection{ Zeeman Splitting}

The Zeeman Effect concerns the splitting of electronic levels in a magnetic field. We've already talked a little bit about this in hyperfine splitting, which was caused by magnetic fields intrinsic to an atom. We find that a single line can split into $\sim3-27$ components, depending on the number of combinations of $\vec L$ and $\vec S$ there are in the atom. The strengths of these various lines depend on viewing geometry, and can be polarized. The change in energy between previously degenerate states set up by an external $B$ field is: $$\begin{aligned}\Delta E&\sim\mu B\sim{e\hbar\over m_ec}B\\ &\sim{hc\over\lambda_1}-{hc\over\lambda_2}\\ &\sim{hc\over\lambda}{\Delta\lambda\over\lambda}\sim{e\hbar B\over m_ec}\\ \end{aligned}$$ Thus the fractional change in wavelength is: $$\boxed{{\Delta\lambda\over\lambda}\sim\lambda{eB\over2\pi m_ec^2}}$$ In practice, we find that the various split components of the original absorption lines are hard to resolve, and we see the effect expressed mostly as a broadening of the original line.

\end{document} <\latex>