User:PAR/Work6

From Wikipedia, the free encyclopedia

[edit] 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.

[edit] Impact broadening

From (Peach 1981 p387)

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
γ = divergent
C2 = ???
Debye effects must be accounted for
  • Resonance p=3
Broadening by ???
γ = divergent
C3 = ???
  • Quadratic Stark p=4
Broadening by quadratic Stark effect
γ = divergent
C4 = ???
  • Van der Waals p=6
Broadening by Van der Waals forces
γ = divergent
C6 = ???

[edit] Quasistatic broadening

From (Peach 1981 p408)

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(ν) = ???
\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) (Peach 1981 Eq 4.95)

where αi and αj 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|\, (Peach 1981 Eq 4.101)
C_6=-K\frac{\mu_1^2}{\hbar}(\alpha_i-\alpha_j) (Peach 1981 Eq 4.100)

where K is of order 1.