User:PAR/Work6

Impact broadening

From (Peach 1981, p. 387) harv error: no target: CITEREFPeach1981 (help)

For impact, its always Lorentzian profile

${\displaystyle P(\omega )={\frac {1}{\pi }}~{\frac {w}{(\omega -\omega _{0}-d)^{2}+w^{2})}}}$

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

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

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

${\displaystyle \beta _{p}={\frac {{\sqrt {\pi }}\,\Gamma ((p-1)/2)}{\Gamma (p/2)}}}$

• Linear Stark p=2

Broadening by linear Stark effect

${\displaystyle \gamma =divergent}$

${\displaystyle C2=???}$

Debye effects must be accounted for

• Resonance p=3

Broadening by ???

${\displaystyle \gamma =divergent}$

${\displaystyle C3=???}$

• Quadratic Stark p=4

Broadening by quadratic Stark effect

${\displaystyle \gamma =divergent}$

${\displaystyle C4=???}$

• Van der Waals p=6

Broadening by Van der Waals forces

${\displaystyle \gamma =divergent}$

${\displaystyle C6=???}$

Quasistatic broadening

From (Peach 1981, p. 408) harv error: no target: CITEREFPeach1981 (help)

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

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

${\displaystyle \beta =\Delta \omega /\Delta \omega _{0}\,}$

${\displaystyle \Delta \omega =\omega -\omega _{0}\,}$

${\displaystyle \theta =\pm 3\pi /2p\,}$

${\displaystyle \Delta \omega _{0}=|C_{p}|\left({\frac {4\pi n}{3}}\Gamma (1-3/p)\cos(\theta )\right)^{p/3}}$

• Linear Stark p=2

Broadening by linear Stark effect

${\displaystyle P(\nu )={\frac {1}{\pi \gamma }}\int _{0}^{\infty }\cos \left({\frac {x(\nu -\nu _{0})}{\gamma }}\right)\exp(x^{-3/2})\,dx}$

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

${\displaystyle C2=???}$

• Resonance p=3

${\displaystyle P(\omega )={\frac {\gamma }{(\omega -\omega _{0})^{2}+\gamma ^{2}}}}$

${\displaystyle \gamma =|C_{3}|2\pi ^{2}n/3\,}$

${\displaystyle C_{3}=K{\sqrt {\frac {g_{u}}{g_{l}}}}~{\frac {e^{2}f}{2m\omega }}}$

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

• Quadratic Stark p=4

Broadening by quadratic Stark effect

${\displaystyle P(\nu )=???}$

${\displaystyle \gamma =|C_{4}|\left({\frac {4\pi }{3}}\Gamma (1/4)\cos(\theta )n\right)^{4/3}}$

${\displaystyle C_{4}=-{\frac {e^{2}}{2\hbar }}(\alpha _{i}-\alpha _{j})}$ (Peach1981 & Peach 1981, Eq 4.95) harv error: no target: CITEREFPeach1981Peach1981 (help)

where ${\displaystyle \alpha _{i}}$ and ${\displaystyle \alpha _{j}}$ are the static dipole polarizabilities of the i and j energy levels.

${\displaystyle \theta =\pm {\frac {3\pi }{8}}}$

• 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.

${\displaystyle P(\omega )={\sqrt {\frac {\gamma }{2\pi }}}~{\frac {\exp \left(-{\frac {\gamma }{2|\nu -\nu _{0}|}}\right)}{(\nu -\nu _{0})^{3/2}}}}$

for

${\displaystyle (\nu -\nu _{0})C_{6}\geq 0}$

0 otherwise.

${\displaystyle \gamma =|C_{6}|{\frac {8\pi ^{3}n^{2}}{9}}\,}$

${\displaystyle \Delta \omega _{0}={\frac {\pi ^{4}n^{2}}{9}}|C_{6}|\,}$ (Peach 1981, Eq 4.101) harv error: no target: CITEREFPeach1981 (help)

${\displaystyle C_{6}=-K{\frac {\mu _{1}^{2}}{\hbar }}(\alpha _{i}-\alpha _{j})}$ (Peach 1981, Eq 4.100) harv error: no target: CITEREFPeach1981 (help)

where K is of order 1.