Interaction of Radiation with Matter Emission and Absorption It is known from quantum mechanics and associated experiments that atoms and molecules posess discrete energy levels. How does matter with such quantised levels interact with the radiation field in a cavity in thermal equlibrium (i.e., a black body radiator, as discussed in the last lecture)? In particular, with what rates are the energy levels populated and de-populated? (Assuming allowed transitions between the states) Consider a prototype two-level atom, placed in a cavity in which the radiation has mode density U(ν). Nm,n are populations −=ν EEhnm , the photon energy Three processes are possible: (note in each case, the atom is assumed to be in a cavity with mode density U(ν).) 1. Spontaneous Emission As its name suggests, this is the spontaneous decay of an atom from an excited state to a lower state. Spontaneous decay rate = NnAnm where Anm is the Einstein A coefficient. Radiated power = NnAnm (En − Em) The radiation is incoherent. 2. Absorption This is a driven process where an atom is excited by incoming photons from a radiation field with mode density U(ν) in resonance with the transition. Absorption rate = Nm Bmn U(ν) where Bmn is the Einstein B coefficient for absorption. Absorbed power = Nm Bmn U(ν)(En − Em) 3. Stimulated Emission – Very Important! An atom in the upper state n can be driven back to the lower state, m, by a photon of energy (En − Em). A second photon of the same energy is produced, in phase with the first! Stimulated emission rate = Nn Bnm U(ν) where Bnm is the Einstein B coefficient for stimulated emission. Radiated power = Nn Bnm U(ν)(En − Em) The radiation is coherent with the incoming radiation. There is therefore a possibility for amplification, if Nn Bnm > Nm Bmn Spontaneous emission introduces noise and ∴reduces amplification efficiency Hence, we also require: ν >> NUnnmnnm() B N A Relationship between the A and B coefficients From the above, we see that the rate at which an atom is excited or de-excited depends on three parameters – the Einstein A and B coefficients, and the radiation density. How are these quantities related? Consider a population of the same two-level atoms, but now in a steady state with the radiation field. Then, (rate of excitation) = (rate of de-excitation), or, ν =+ NUm() B mn N n ( A nm B nm ) Re-arranging to find U(ν) gives: Anm B U ()ν = mn N m − Bnm N n Bmn N −hν Now, Boltzmann statistics gives n = e kT N m Anm B So, U ()ν = mn hν B e kT − nm Bmn Since U(ν) is still thermal (black body) radiation, this must the same as Planck’s Law! 8πνh 3 Hence, BB= and A = B nm mn nmc 3 mn Effects of Degeneracies So far, in this simple model, we have assumed only one electron per state. In reality, we have to consider the degeneracy, g. (e.g., for a p state, g = 6) N g −hν The Boltzmann equation is then modified to n = n e kT N m gm = and the relationships between the coefficients to gBnnm gB mmn(Anm is unaffected). Widths of Spectral Lines So far, the two-level system has absorbed or radiated a single frequency. However, because there is a time-scale associated with the decay or absorption, the spectral lines will be broadened. Classically, an excited atom may be regarded as a damped, oscillating electric dipole. Then, classical electro-magnetic theory gives the rate of radiation of energy, φ, as: 2 dφ 1 µ!! =⋅ where µω= ercos( t ) is the dipole moment. πε 3 00 dt 6 0 c Also, the stored energy, for an amplitude r0 is: φω= 1 2 2 mr()00 Combining these results gives: φ − 2ω 2 1 d e 0 − ⋅= . This is clearly an exponential decay, with the RHS = 1 where T is φ πε 3 T dt 6 0cm the classical lifetime. −t φφ= T ∝ 1 Hence, 0e . Note also that T ω 2 . 0 The oscillation is damped, and is therefore no longer monochromatic. In fact, in this case, Fourier analysis gives ∆ω ⋅=T 1 . Causes of Line Broadening 1. Natural linewidth – Arises from the Heisenberg Uncertainty Principle. 2. Collisional (pressure) broadening – Caused by random collisions, and effectively shortens the lifetime. 3. Doppler (motional) broadening – The velocity distribution causes a distribution of Doppler shifted frequencies relative to the observer ⇒ broadening. Natural Linewidth In the classical atom discussed above, the energy content of the damped oscillator has been shown to decay exponentially. The power output of the oscillator, W, will therefore decay similarly: −t = T Wt() We0 ∝ 2 Now, since WE0 , where E0 is the electric field amplitude of the outgoing e-m wave = ω ( EE00cos( t ) ), the amplitude of the wave at time t is given by: 1 −+t iω = 2T 0 Et() Ee0 , using complex notation for the cosine term. The frequency dependence of the amplitude is then given by the Fourier transform: 1 ∞ E 1 EEtedt()ω ==∫ ()−itω 0 22ππ0 i()ωω−+1 0 2T The intensity as a function of frequency is: E 2 1 IEE()ωωω∝=()* () 0 2π ()ωω−+2 1 0 4T 2 If the intensity is now defined as I0 at frequency ω0, then, 1 2 II()ω = 4T 0 ()ωω−+2 1 0 4T 2 The form of this function is a Lorentzian, and it has a FWHM 1 =∆ω of T Generally, the natural linewidth is small compared to colisional or Doppler broadening..