User:PAR/Work6

Pressure broadening
The presence of pertubing particles near an emitting atom will cause a broadening and possible shift of the emitted radiation.

There is two types impact and quasistatic

In each case you need the profile, represented by Cp as in C6 for Lennard-Jones potential


 * $$\gamma = \frac{C_p}{r^p}$$

Assume Maxwell-Boltzmann distribution for both cases.

Impact broadening
From

For impact, its always Lorentzian profile


 * $$P(\omega)=\frac{1}{\pi}~\frac{w}{(\omega-\omega_0-d)^2+w^2)}$$


 * $$w+id=\alpha_p \pi n v\left[\frac{\beta_p|C_p|}{v}\right]^{2/(p-1)}

\Gamma\left(\frac{p-3}{p-1}\right)\exp\left(\pm \frac{i\pi}{p-1}\right)$$


 * $$\alpha_p=\Gamma\left(\frac{2p-3}{p-1}\right)\left(\frac{4}{\pi}\right)^{1/(p-1)}$$


 * $$v=\sqrt{\frac{8kT}{\pi m}}$$


 * $$\beta_p=\frac{\sqrt{\pi}\,\Gamma((p-1)/2)}{\Gamma(p/2)}$$


 * Linear Stark p=2
 * Broadening by linear Stark effect
 * $$\gamma=divergent$$
 * $$C2=???$$
 * Debye effects must be accounted for


 * Resonance p=3
 * Broadening by ???
 * $$\gamma=divergent$$
 * $$C3=???$$


 * Quadratic Stark p=4
 * Broadening by quadratic Stark effect
 * $$\gamma=divergent$$
 * $$C4=???$$


 * Van der Waals p=6
 * Broadening by Van der Waals forces
 * $$\gamma=divergent$$
 * $$C6=???$$

Quasistatic broadening
From

For quasistatic, functional form of lineshape varies. Generally its a Levy skew alpha-stable distribution (Peach, page 408)


 * $$\Delta\omega_0 L(\omega)=\frac{1}{\pi}\Re\left[\int_0^\infty

\exp(i\beta x-(1+i\tan\theta)x^{3/p})\,dx\right]$$
 * $$\beta=\Delta\omega/\Delta\omega_0\,$$
 * $$\Delta\omega=\omega-\omega_0\,$$
 * $$\theta=\pm 3\pi/2p\,$$
 * $$\Delta\omega_0=|C_p|\left(\frac{4\pi n}{3}\Gamma(1-3/p)\cos(\theta)\right)^{p/3}$$


 * Linear Stark p=2
 * Broadening by linear Stark effect
 * $$P(\nu)=\frac{1}{\pi\gamma}\int_0^\infty

\cos\left(\frac{x(\nu-\nu_0)}{\gamma}\right)\exp(x^{-3/2})\,dx$$
 * $$\gamma=|C_2|\pi\left(\frac{32n^2}{9}\right)^{1/3}$$
 * $$C2=???$$


 * Resonance p=3
 * $$P(\omega)=\frac{\gamma}{(\omega-\omega_0)^2+\gamma^2}$$
 * $$\gamma=|C_3|2\pi^2n/3\,$$
 * $$C_3=K\sqrt{\frac{g_u}{g_l}}~\frac{e^2f}{2m\omega}$$

where K is of order unity. Its just an approximation.


 * Quadratic Stark p=4
 * Broadening by quadratic Stark effect
 * $$P(\nu)=???$$
 * $$\gamma=|C_4|\left(\frac{4\pi}{3}\Gamma(1/4)\cos(\theta)n\right)^{4/3}$$
 * $$C_4=-\frac{e^2}{2\hbar}(\alpha_i-\alpha_j)$$

where $$\alpha_i$$ and $$\alpha_j$$ are the static dipole polarizabilities of the i and j energy levels.
 * $$\theta=\pm \frac{3\pi}{8}$$


 * Van der Waals p=6
 * Broadening by Van der Waals forces gives a Van der Waals profile. C6 is the wing term in the Lennard-Jones potential.
 * $$P(\omega)=\sqrt{\frac{\gamma}{2\pi}}~

\frac{\exp\left(-\frac{\gamma}{2|\nu-\nu_0|}\right)}{(\nu-\nu_0)^{3/2}}$$ for
 * $$(\nu-\nu_0)C_6\ge 0$$

0 otherwise.
 * $$\gamma=|C_6|\frac{8\pi^3n^2}{9}\,$$
 * $$\Delta \omega_0=\frac{\pi^4 n^2}{9}|C_6|\,$$
 * $$C_6=-K\frac{\mu_1^2}{\hbar}(\alpha_i-\alpha_j)$$

where K is of order 1.