Inhomogeneous broadening and luminescence (Ch. 13) Nov.27, 2017

Inhomogeneous broadening: Atoms can be discriminated by physical parameters (e.g. Doppler effect, different mass owing to different isotopes, nuclear spin, Zeeman splitting) that result in shifts in their spectra. Black body is also a perfect radiation absorber.
Consequences --
Line shape functions are the sum of$\text{ }g\left(\nu \right)$ for individual group.
Saturation can be group selective.
• Doppler effect: When atoms are moving, energy or freq. absorbed depends on their velocity ($\text{ }\stackrel{\to }{v}$ ) --$\text{ }{\omega }_{o}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\omega -\stackrel{\to }{k}·\stackrel{\to }{v}$
Lineshape function for each group$\text{ }g\left({v}_{z},\nu \right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{\Delta {\nu }_{h}}{2\pi }\frac{1}{\left(\nu -{\nu }_{o}-{\nu }_{o}{v}_{z}/c{\right)}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\left(\Delta {\nu }_{h}/2{\right)}^{2}}$
Probability density function of atom with certain velocity in z direction,$\text{ }{v}_{z}$ , is given by Maxwell-Boltzmann distribution$\frac{dN}{N}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\sqrt{\frac{M}{2\pi kT}}\text{exp}\left(-\frac{M{v}_{z}^{2}}{2kT}\right)d{v}_{z}$
Now$\text{ }{g}_{total}\left(\nu \right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\int g\left({v}_{z},\nu \right)dN/N$ which will give a total lineshape function known as Voight distribution.
Consider 2 extreme cases --
Doppler width$\text{ }\Delta {\nu }_{D}$ (Maxwell distribution width >> Lorentzian width$\text{ }\Delta {\nu }_{h}$$\text{ }\to$ Normal (Gaussian) lineshape, i.e.

$\text{ }g\left(\nu \right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{c}{{\nu }_{o}}\sqrt{\frac{M}{2\pi kT}}{e}^{-\frac{M}{2kT}\left(c/{\nu }_{o}{\right)}^{2}\left(\nu -{\nu }_{o}{\right)}^{2}}$

and$\Delta {\nu }_{D}\text{ }\left(FWHM\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\nu }_{o}\sqrt{\frac{8kT\text{ln}2}{M{c}^{2}}}$
In terms of$\Delta {\nu }_{D}$ ,$\text{ }g\left(\nu \right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\sqrt{\frac{4\text{ln}2}{\Delta {\nu }_{D}^{2}\pi }}$${e}^{-4\text{ln}2\left(\left(\nu \text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\nu }_{o}\right)/\Delta {\nu }_{D}{\right)}^{2}}$
Opposite case,$\text{ }\Delta {\nu }_{D}\text{\hspace{0.17em}}\ll \text{\hspace{0.17em}}\Delta {\nu }_{h}$$\to$ Lorentzian lineshape function.
• Stark effect: Another effect for inhomogeneous broadening where local electric field in crystals causes level splitting. The local field of each splitting is determined by lattice vibration differently.
Luminescence and scattering: During luminescence, the system absorbs and subsequently emits photon. Light can be scattered by atomic or molecular system (lattice).
Cathodoluminescence, Sonoluminescence, Chemiluminescence, Bioluminescence, Electrouminescence.
Photoluminescence -- high energy photons exchanged for low energy photons.
From spin-allowed transition (singlet$\text{\hspace{0.17em}}\to$ singlet), generate fluorescence which has short lifetime and useful for laser.
For spin-forbidden transition (singlet$\text{\hspace{0.17em}}\to$ triplet), generate phosphorescence (long lifetime).
2 classes of photoluminescence -- single photon and multiphoton
Multiphoton photoluminescence allows absorption of many photons to generate fewer high energy photons.
Applications -- two photon laser scanning fluorescence microscopy, multiphoton laser scanning fluorescence microscopy and 3D multiphoton microlithography.
Upconversion fluorescence is used in infrared sensor card.
Rayleigh scattering is proportional to$\text{ }{\lambda }_{o}^{-4}$ for refractive index variation or particle size$\text{ }<\text{ }{\lambda }_{o}/10$ For opposite case, we need to consider Mie scattering.
Raman scattering (RS) and Brillouin scattering (BS) are caused by phonon and acoustic vibration. They can generate low freq (Stokes scattering) and high freq (anti-Stokes scattering which is rare).
In Stimulated RS and BS, there are input photons to provoke the excited atoms that are excited by a high photon energy pump to release more photons. They have applications in lasers and optical amplifiers.