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

• Radiation from a hot object: Black body is also a perfect radiation absorber.
Wien’s law (Classic result) gives peak emission wavelength lambda_{peak} T = 2.8978 times 10^{-3} (m-K)

• Relating spectral energy density rho ( nu , T) (J/m{::}^-3-Hz) and intensity spectrum u( nu , T) (W/m{::}^2-Hz): c_o rho = u
u and rho can be expressed in terms of lambda, i.e. rho (lambda, T), u(lambda, T).
Notice that intensity I = int u(lambda, T) dlambda or int u(nu,T) dnu
For blackbody, radiation only emit to the front making an angle with the optical axis and its intensity spectrum u_{BB} (lambda, T) = rho (lambda, T)c_o/4

• Rayleigh-Jeans catastrophe: Energy density rho(nu) dnu = kT M(nu) dnu= {kT 8pi n^3 nu^2) / c_o^3 dnu 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 e^{- {h nu}/{kT} and compute average energy level (:E:) = (sum_{l=0}^{oo}l h nu e^{-{l hnu}/{kT}}}/{sum_{l=0}^ooe^{-{l h nu}/{kT}}} = {h nu} / (e^{{h nu}/{kT}}-1}
Energy density rho (nu) = {h nu} / (e^{{h nu}/{kT}}-1} ({ 8pi n^3 nu^2) / c_o^3) dnu
Intensity u_{BB} (lambda) d lambda = {h } / (e^{{h nu}/{kT}}-1}(2 pi n^3c_o^2)/(lambda_o^5) dlambda

• Spontaneous emission: No input, noise-like, nondirective {dN_1}/{dt} = - {dN_2}/{dt}=A_21 N_2

• Stimulated emission: Input photons, direction, output following input’s polarization, freq and phase, i.e. light amplification, {dN_1}/{dt}= - {dN_2}/{dt} = B_21 N_2 rho(nu)

• Absorption: freq and polarization dependent, {dN_2}/{dt}= - {dN_1}/{dt} = B_12 N_1 rho(nu)
where N_1 is the number density of atoms in lower state and N_2 is the number density of atoms in upper state.

• Einstein’s approach relating radiative processes: Detailed balance -- emission rate = absorption rate under thermal equilibrium, i.e. $\frac {dN_1} {dt} |_{emission} + \frac {dN_1} {dt} |_{absorption} =0$
As T-> oo, rho(nu) -> oo and B_21 g_2 = B_12 g_1. where we consider g_i (degeneracy), the number of ways that an atom can have energy E_i = h nu_i and N_i = N_0 g_i e^{- E_i / {kT}}
rho (nu) = {A_{21}}/{B_{21}}{1} / (e^{{h (nu_2 - nu_1 )}/{kT}}-1}, {A_{21}}/{B_{21}}= ({ 8pi n^3h nu^3) / c_o^3)

Interpret A_{21} = 1/{t_{sp}} as spontaneous emission rate where t_{sp} is the spontaneous emission lifetime.
Note: g_i = 2 J_i + 1 for i = 1, 2 where 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 p_{sp} = {c sigma(nu)}/V (s{::}^-1) where V is the volume and sigma(nu) is the transition cross section in m{::}^{2}.
Probability of emission between t and t+dt is p_{sp} Delta t. The spatial distribution of sigma depending on the angle theta between dipole moment of the atom and the field, sigma = sigma_{max} 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. P_{ab} = P_{st} ={n c sigma (nu) }/V = W_i.

• Relationship between lines shape g(nu) and sigma (nu): After normalizing sigma(nu) by oscillator strength S= int_0^oo sigma (nu) d nu, g(nu) dnu = {sigma(nu)} / S dnu will give the probability of spon. emission between nu and nu + Delta nu.
Hence, g(nu) is a probability density function, i.e. int_0^{oo} g(nu) dnu = 1. It peaks at nu_o (the transition freq.) and has FWHM Delta nu
Notice that field’s bandwidth from rho ( nu ) is limited by that of the cavity delta nu.

• General rate equation for light amplification: {dN_2}/{dt}= -A_{21} N_2 int_0^{oo} g(nu' ) dnu' +B_12 N_2 int_0^{oo} rho(nu' ) g(nu' ) dnu'  - B_21 N_2 int_0^{oo} rho(nu' ) g(nu' ) dnu'
Einstein considered delta nu >> Delta nu and treated g(nu) like a delta function.
Book concentrated on the laser case Delta nu >> delta nu and rho (nu ') = rho_{nu} delta (nu' - nu)
{dN_2}/{dt}= -A_{21} N_2-sigma(nu) {I_nu}/{h nu} [N_2 - {g_2}/{g_1}N_1] = -A_{21} N_2-W_i [N_2 - {g_2}/{g_1}N_1]

where sigma(nu)={A_{21} lambda_o^2 g(nu)}/{8 pi n^2},phi = {I_{nu}}/{h nu} and W_i = phi sigma(nu).

• Total spon. emission to all modes: Consider spatial average bar sigma (nu) = {sigma_{max} (nu) }/3, P_{sp}= int_0^{oo} c/V bar sigma (nu) V M(nu) dnu. For blackbody, we have Einstein case where g(nu) is like a delta function.
P_{sp} = 1 /{t_{sp}} ~~ M(nu_o) c bar S where bar S =int_0^{oo} bar sigma (nu)dnu = {lambda_o^2} / {8 pi t_{sp} n^2} and bar sigma (nu) = {lambda_o^2 g(nu)} / {8 pi t_{sp} n^2}.

• Stimulated emission relating to blackbody radiation: W_i=int_0^{oo} {rho(nu)V}/{h nu} {c sigma(nu)}/V dnu
Blackbody is broadband delta nu >> Delta nu -> g(nu) ~~ delta (nu - nu_o)
W_i = {bar n } / t_{sp}

where number of photons per mode bar n = {lambda_o^3 rho (nu_o)} / {8 pi n^3 h}

Line broadening: Uncertainty of energy levels Delta E = Delta E_1 + Delta E_2= h/{2 pi} ( 1/{tau_1} + 1/{tau_2)) where tau_{1,2} is transit lifetime.
In another words, Delta nu = 1/{2 pi} ( 1/{tau_1} + 1/{tau_2))

Lifetime of the excited atoms with N_2 population is tau_2 which follows tau_2^{-1} = tau_{NR}^{-1} + tau_{R}^{-1} where tau_{NR} is nonradiative lifetime from processes not generating light and tau_R is radiative lifetime from light generating processes.

• Lineshape function for homogeneous broadening: Since decay in excited atoms or power corresponds to field e(t) = e_o e^{-t/{2 tau_2}} cos omega_o t, spectral intensity |E(omega)|^2 ~ 1/{(omega - omega_o)^2 + (gamma/2)^2} from Fourier transform where gamma = 1/tau_2
This reminds us the Lorentzian lineshape g(nu)={Delta nu}/{2 pi} 1/{(nu - nu_o)^2+ ({Delta nu}/2)^2} where Delta nu = gamma /{2 pi} and g(nu_o) = 2/{Delta nu pi} Also bar sigma_o = {lambda_o^2} / {2 pi n^2} 1 / {2 pi t_{sp} Delta nu} If tau_2 is entirely radiative, bar sigma_o = {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. g(nu_o) decreases.

Probability of finding collision free atoms =p(t) dt =1/{tau_c} e^{-t/{tau_c}}dt where tau_c = mean time between collisions.
For a gaseous mixture of type m and n, freq of collision f_{col} = 1/{tau_c} = N (:sigma v:)=N_m sigma [{8kT}/pi (1/{M_m}+1/{M_n})] where v is mean velocity, sigma is cross section (function of v), N is the number density, N_m is the number density of type m, M_m is mass of type m atom and M_n is mass of type n atom.
Compute correlation of field and take Fourier transform to find intensity

(:e(t)e(t+tau):) = e_o^2 cos omega_o tau e^{-{|tau|}/{tau_c}}

I ~ Fourier transform of (:e(t)e(t+tau):)=1/{(omega-omega_o)^2+(1/tau_c)^2}

Lorentzian width = 2/ {2pi tau_c}
Overall lineshape function (homogeneous): g(nu) = {Delta nu_{"total"} }/{2 pi} 1/{(nu-nu_o)^2+({Delta nu_{"total"}}/2)^2} where Delta nu_{"total"} =1/{2pi} ( 2/tau_c+ 1/tau_2)