Black body radiation and line shapes (Ch. 13) Apr. 15, 2017

• Radiation from a hot object: Black body is also a perfect radiation absorber.
Wien’s law (Classic result) gives peak emission wavelength$\text{ }{\lambda }_{peak}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}2.8978×1{0}^{-3}\left(m-K\right)/T\left(K\right)$
• Relating spectral energy density$\text{ }\rho \left(\nu ,T\right)\text{ }\left(J/{m}^{3}-Hz\right)$ and intensity spectrum$u\left(\nu ,T\right)\text{ }\left(W/{m}^{2}-Hz\right)$ :${c}_{o}\rho \text{\hspace{0.17em}}=\text{\hspace{0.17em}}u$
u and$\text{ }\rho$ can be expressed in terms of$\text{ }\lambda$ , i.e.$\text{ }\rho \left(\lambda ,T\right)$ ,$u\left(\lambda ,T\right)$
Notice that intensity$\text{ }I\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\int u\left(\lambda ,T\right)d\lambda$ or$\text{ }I\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\int u\left(\nu ,T\right)d\nu$
For blackbody, radiation only emit to the front making an angle with the optical axis and its intensity spectrum$\text{ }{u}_{BB}\left(\lambda ,T\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\rho \left(\lambda ,T\right){c}_{o}/4$
• Rayleigh-Jeans catastrophe: Energy density$\text{ }\rho \left(\nu \right)d\nu \text{\hspace{0.17em}}=\text{\hspace{0.17em}}kTM\left(\nu \right)d\nu \text{\hspace{0.17em}}=\text{\hspace{0.17em}}kT8\pi {n}^{3}{\nu }^{2}d\nu /{c}_{o}^{3}$ Energy density increases with freq without upper limit, not reflecting experimental results.
• Quantization of energy: Planck assigned probability density for each discrete energy following Boltzmann distribution$\text{ }{e}^{-h\nu /kT}$ and compute average energy level$\text{ }\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{\underset{l=0}{\overset{\infty }{\Sigma }}lh\nu {e}^{-lh\nu /\left(kT\right)}}{\underset{l=0}{\overset{\infty }{\Sigma }}{e}^{-h\nu /\left(kT\right)}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{h\nu }{{e}^{h\nu /\left(kT\right)}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}$
Energy density$\text{ }\rho \left(\nu \right)d\nu \text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{h\nu }{{e}^{h\nu /\left(kT\right)}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}\left(\frac{8\pi {n}^{3}{\nu }^{2}}{{c}_{o}^{3}}\right)d\nu$
Intensity$\text{ }{u}_{BB}\left(\lambda \right)d\lambda \text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{2\pi {h}^{3}{c}_{o}^{2}}{{\lambda }_{o}^{5}}\text{\hspace{0.17em}}\frac{h}{{e}^{h\nu /\left(kT\right)}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}d\lambda$
• Spontaneous emission: No input, noise-like, nondirective$\text{ }\frac{d{N}_{1}}{dt}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\frac{d{N}_{2}}{dt}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{A}_{21}{N}_{2}{g}_{2}$
• Stimulated emission: Input photons, direction, output following input’s polarization, freq and phase, i.e. light amplification,$\text{ }\frac{d{N}_{1}}{dt}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\frac{d{N}_{2}}{dt}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{B}_{21}{N}_{2}{g}_{2}\rho \left(\nu \right)$
• Absorption: freq and polarization dependent,$\text{ }\frac{d{N}_{2}}{dt}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\frac{d{N}_{1}}{dt}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{B}_{12}{N}_{1}{g}_{1}\rho \left(\nu \right)$
• Einstein’s approach relating radiative processes: Detailed balance -- emission rate = absorption rate under thermal equilibrium, i.e.$\text{ }{\frac{d{N}_{1}}{dt}|}_{emission}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\frac{d{N}_{1}}{dt}|}_{absorption}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0$
As$\text{ }T\to \infty$ ,$\text{ }\rho \left(\nu \right)\to \infty$ and$\text{ }{B}_{21}{g}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{B}_{12}{g}_{1}$

$\text{ }\rho \left(\nu \right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{A}_{21}}{{B}_{21}}\frac{1}{{e}^{h\left({\nu }_{2}-{\nu }_{1}\right)/\left(kT\right)}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}$$\text{ }\frac{{A}_{21}}{{B}_{21}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{8\pi {n}^{3}{\nu }^{3}h}{{c}_{o}^{3}}$

Interpret$\text{ }{A}_{21}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1/{t}_{sp}$ as spontaneous emission rate where$\text{ }{t}_{sp}$ is the spontaneous emission lifetime.
Note:$\text{ }{g}_{i}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}2{J}_{i}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}1$ for i = 1, 2 where$\text{ }{J}_{i}$ is the angular momentum of the energy level.
• More complete approach: The text book consider probability density or rate of spon. emission for a mode$\text{ }{p}_{sp}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}c\sigma \left(\nu \right)/V\left({s}^{-1}\right)$ where V is the volume and$\text{ }\sigma \left(\nu \right)$ is the transition cross section in$\text{ }{m}^{2}$
Probability of emission between t and t+dt is$\text{ }{p}_{sp}\Delta t$ The spatial distribution of$\text{ }\sigma$ depending on the angle$\text{ }\theta$ between dipole moment of the atom and the field,$\text{ }\sigma \text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\sigma }_{\text{max}}{\text{cos}}^{2}\theta$
The probability density or rate of stimulated emission and absorption for a mode can be defined similarly, except that these processes have number of photons n, i.e.$\text{ }{P}_{ab}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{P}_{st}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}nc\sigma \left(\nu \right)/V\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{W}_{i}$
• Relationship between lines shape$\text{ }g\left(\nu \right)$ and$\text{ }\sigma \left(\nu \right)$ : After normalizing$\text{ }\sigma \left(\nu \right)$ by oscillator strength$\text{ }S\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\underset{0}{\overset{\infty }{\int }}\sigma \left(\nu \right)d\nu$ ,$\text{ }g\left(\nu \right)d\nu \text{\hspace{0.17em}}=\text{\hspace{0.17em}}\sigma \left(\nu \right)d\nu /S$ will give the probability of spon. emission between$\text{ }\nu$ and$\text{ }\nu +\Delta \nu$
Hence,$\text{ }g\left(\nu \right)$ is a probability density function, i.e.$\text{ }\underset{0}{\overset{\infty }{\int }}g\left(\nu \right)d\nu \text{\hspace{0.17em}}=\text{\hspace{0.17em}}1$ It peaks at$\text{ }{\nu }_{o}$ (the transition freq.) and has FWHM$\text{ }\Delta \nu$
Notice that field’s bandwidth from$\text{ }\rho \left(\nu \right)$ is limited by that of the cavity$\delta \nu$
• General rate equation for light amplification:$\text{ }d{N}_{2}/dt\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-{A}_{21}{N}_{2}\underset{0}{\overset{\infty }{\int }}g\left(\nu \right)d\nu \text{\hspace{0.17em}}+\text{\hspace{0.17em}}{B}_{12}{N}_{1}\underset{0}{\overset{\infty }{\int }}\rho \left(\nu \right)g\left(\nu \right)d\nu$$\text{ }\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{B}_{21}{N}_{2}\underset{0}{\overset{\infty }{\int }}\rho \left(\nu \right)g\left(\nu \right)d\nu$
Einstein considered$\text{ }\delta \nu \text{\hspace{0.17em}}\gg \text{\hspace{0.17em}}\Delta \nu$ and treated$\text{ }g\left(\nu \right)$ like delta function.
Book concentrated on the laser case$\text{ }\Delta \nu \text{\hspace{0.17em}}\gg \text{\hspace{0.17em}}\delta \nu$ and$\text{ }\rho \left(\nu \prime \right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\rho }_{\nu }\delta \left(\nu \prime \text{\hspace{0.17em}}-\text{\hspace{0.17em}}\nu \right)$

$\text{ }\frac{d{N}_{2}}{dt}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-{A}_{21}{N}_{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\sigma \left(\nu \right)\frac{{I}_{\nu }}{h\nu }\left[{N}_{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{{g}_{2}}{{g}_{1}}{N}_{1}\right]$$\text{ }\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-{A}_{21}{N}_{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{W}_{i}\left[{N}_{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{{g}_{2}}{{g}_{1}}{N}_{1}\right]$

where$\text{ }\sigma \left(\nu \right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{A}_{21}{\lambda }_{o}^{2}g\left(\nu \right)/\left(8\pi {n}^{2}\right)$ ,$\text{ }\phi \text{\hspace{0.17em}}=\text{\hspace{0.17em}}{I}_{\nu }/\left(h\nu \right)$ and$\text{ }{W}_{i}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\phi \sigma \left(\nu \right)$
• Total spon. emission to all modes: Consider spatial average$\text{ }\overline{\sigma }\left(\nu \right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\sigma }_{\text{max}}\left(\nu \right)/3$ ,$\text{ }{P}_{sp}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\underset{0}{\overset{\infty }{\int }}\frac{c}{V}\overline{\sigma }\left(\nu \right)VM\left(\nu \right)d\nu$ For blackbody, we have Einstein case where$\text{ }g\left(\nu \right)$ is like a delta function.$\text{ }{P}_{sp}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1/{t}_{sp}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{≈}\text{ }M\left({\nu }_{o}\right)c\overline{S}$ where$\text{ }\overline{S}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\underset{0}{\overset{\infty }{\int }}\overline{\sigma }\left(\nu \right)d\nu \text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\lambda }_{o}^{2}/\left(8\pi {t}_{sp}{n}^{2}\right)$ and$\text{ }\overline{\sigma }\left(\nu \right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\lambda }_{o}^{2}g\left(\nu \right)/\left(8\pi {t}_{sp}{n}^{2}\right)$
• Stimulated emission relating to blackbody radiation:$\text{ }{W}_{i}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\underset{0}{\overset{\infty }{\int }}\frac{\rho \left(\nu \right)V}{h\nu }\frac{c\sigma \left(\nu \right)}{V}d\nu$
Blackbody is broadband$\text{ }\delta \nu \gg \Delta \nu$$\text{ }\text{ }\to g\left(\nu \right)\text{≈}\delta \left(\nu \text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\nu }_{o}\right)$

$\text{ }{W}_{i}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\overline{n}/{t}_{sp}$

where number of photons per mode$\text{ }\overline{n}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\lambda }_{o}^{3}\rho \left({\nu }_{o}\right)/\left(8\pi {n}^{3}h\right)$
Line broadening: Uncertainty of energy levels$\text{ }\Delta E\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\Delta {E}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\Delta {E}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{h}{2\pi }\left(\frac{1}{{\tau }_{1}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{1}{{\tau }_{2}}\right)$ where$\text{ }\tau$ is transit lifetime.
In another words,$\text{ }\Delta \nu \text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{2\pi }\left(\frac{1}{{\tau }_{1}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{1}{{\tau }_{2}}\right)$
Lifetime of the excited atoms with$\text{ }{N}_{2}$ population is$\text{ }{\tau }_{2}$ which follows$\text{ }{\tau }_{2}^{-1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\tau }_{NR}^{-1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\tau }_{R}^{-1}$ where$\text{ }{\tau }_{NR}$ is nonradiative lifetime from processes not generating light and$\text{ }{\tau }_{R}$ is radiative lifetime from light generating processes.
• Lineshape function for homogeneous broadening: Since decay in excited atoms or power corresponds to field$\text{ }e\left(t\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{e}_{o}{e}^{-t/\left(2{\tau }_{2}\right)}\text{cos}{\omega }_{o}t$ , spectral intensity$\text{ }|e\left(\omega \right){|}^{2}$ ~$\text{ }\frac{1}{\left(\omega \text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\omega }_{o}{\right)}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\left(\gamma /2{\right)}^{2}}$ from Fourier transform where$\text{ }\gamma \text{\hspace{0.17em}}=\text{\hspace{0.17em}}1/{\tau }_{2}$
This remind us the Lorentzian lineshape$\text{ }g\left(\nu \right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\Delta \frac{\nu }{2\pi }\frac{1}{\left(\nu \text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\nu }_{o}{\right)}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\left(\Delta \nu /2{\right)}^{2}}$ where$\text{ }\Delta \nu \text{\hspace{0.17em}}=\text{\hspace{0.17em}}\gamma /\left(2\pi \right)$ and$\text{ }g\left({\nu }_{o}\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}2/\left(\Delta \nu \pi \right)$ Also$\text{ }{\overline{\sigma }}_{o}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{\lambda }_{o}^{2}}{2\pi {n}^{2}}\frac{1}{2\pi {t}_{sp}\Delta \nu }$ If$\text{ }{\tau }_{2}$ is entirely radiative,$\text{ }{\overline{\sigma }}_{o}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{\lambda }_{o}^{2}}{2\pi {n}^{2}}$
• Characteristics of homogeneous broadening: Line shape can change with radiation (photons) and will scale down as a result of saturation, i.e.$\text{ }g\left({\nu }_{o}\right)$ decreases.
Probability of finding collision free atoms$\text{ }=p\left(t\right)dt\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{{\tau }_{c}}{e}^{-t/{\tau }_{c}}dt$ where$\text{ }{\tau }_{c}$ = mean time between collisions.
For a gaseous mixture of type m and n, freq of collision$\text{ }{f}_{\text{col}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{{\tau }_{c}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}N<\sigma v>\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{N}_{m}\sigma \left[\frac{8kT}{\pi }\left(\frac{1}{{M}_{m}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{1}{{M}_{n}}\right)\right]$ where v is mean velocity,$\text{ }\sigma$ is cross section (function of v), N is the number density,$\text{ }{N}_{m}$ is the number density of type m,$\text{ }{M}_{m}$ is mass of type m atom and$\text{ }{M}_{n}$ is mass of type n atom.
Compute correlation of field and take Fourier transform to find intensity

$\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{e}_{o}^{2}\text{cos}{\omega }_{o}\tau {e}^{-|\tau |/{\tau }_{c}}$

I ~ Fourier transform of$\text{ }$

$\text{ }=\text{\hspace{0.17em}}\frac{1}{\left(\omega \text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\omega }_{o}{\right)}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\left(1/{\tau }_{c}{\right)}^{2}}$

Lorentzian width =$\text{ }2/\left(2\pi {\tau }_{c}\right)$
Overall lineshape function (homogeneous):$\text{ }g\left(\nu \right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\Delta \frac{{\nu }_{total}}{2\pi }\frac{1}{\left(\nu \text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\nu }_{o}{\right)}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\left(\Delta {\nu }_{total}/2{\right)}^{2}}$ where$\text{ }\Delta {\nu }_{total}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{2\pi }\left(\frac{2}{{\tau }_{c}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{1}{{\tau }_{2}}\right)$