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

$P(\omega )=\frac{1}{\pi }\frac{w}{(\omega -{\omega }_0-d{)}^2+w^2)}$

$w+id={\alpha }_p\pi nv{\left[\frac{{\beta }_p|C_p|}{v}\right]}^{2/(p-1)}\mathrm{\Gamma }\left(\frac{p-3}{p-1}\right)\exp (\pm \frac{i\pi }{p-1})$

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

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

${\beta }_p=\frac{\sqrt{\pi }\mathrm{\Gamma }((p-1)/2)}{\mathrm{\Gamma }(p/2)}$

• 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)

$\mathrm{\Delta }{\omega }_{0}L(\omega )=\frac{1}{\pi }\mathrm{\Re }[\underset{0}{\overset{\mathrm{\infty }}{\int }}\exp (i\beta x-(1+i\tan \theta )x^{3/p})dx]$

$\beta =\mathrm{\Delta }\omega /\mathrm{\Delta }{\omega }_0$

$\mathrm{\Delta }\omega =\omega -{\omega }_0$

$\theta =\pm 3\pi /2p$

$\mathrm{\Delta }{\omega }_0=|C_p|{(\frac{4\pi n}{3}\mathrm{\Gamma }(1-3/p)\cos (\theta ))}^{p/3}$

• Linear Stark p=2

Broadening by linear Stark effect

$P(\nu )=\frac{1}{\pi \gamma }\underset{0}{\overset{\mathrm{\infty }}{\int }}\cos \left(\frac{x(\nu -{\nu }_0)}{\gamma }\right)\exp (x^{-3/2})dx$

$\gamma =|C_2|\pi {\left(\frac{32n^2}{9}\right)}^{1/3}$

$C2=???$

• Resonance p=3

$P(\omega )=\frac{\gamma }{(\omega -{\omega }_0{)}^2+{\gamma }^2}$

$\gamma =|C_3|2{\pi }^{2}n/3$

$C_3=K\sqrt{\frac{g_u}{g_l}}\frac{e^{2}f}{2m\omega }$

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

• Quadratic Stark p=4

Broadening by quadratic Stark effect

$P(\nu )=???$

$\gamma =|C_4|{(\frac{4\pi }{3}\mathrm{\Gamma }(1/4)\cos (\theta )n)}^{4/3}$

$C_4=-\frac{e^2}{2\hslash }({\alpha }_i-{\alpha }_j)$

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.

$P(\omega )=\sqrt{\frac{\gamma }{2\pi }}\frac{\exp (-\frac{\gamma }{2|\nu -{\nu }_0|})}{(\nu -{\nu }_0{)}^{3/2}}$

for

$(\nu -{\nu }_0)C_6\ge 0$

0 otherwise.

$\gamma =|C_6|\frac{8{\pi }^3{n}^2}{9}$

$\mathrm{\Delta }{\omega }_0=\frac{{\pi }^4{n}^2}{9}|C_6|$

$C_6=-K\frac{{\mu }_1^2}{\hslash }({\alpha }_i-{\alpha }_j)$

where K is of order 1.