Suprit Singh Talk for the IUCAA Grad-school course in Inter-stellar medium given by Dr. A N Ramaprakash 15 th April 2KX

Principle of Detailed Balance and Einstein coefficients

 The atoms interact with the Radiation field via where the transverse radiation field is given by  Now, only the first two terms in the interaction Hamiltonian contribute in the absorption and emission of a photon.

 To First order the amplitude for an atom in state A to absorb a photon and get into state B :  and similarly the amplitude for an atom in state A emitting a photon and entering state B :

 Now if you calculate the transition amplitude using,  we have for transition probability  And for spontaneous emission in the dipole approximation

 For an atom interacting with electromagnetic field, we thus have the following reaction:  Thus, if the populations of the upper and lower levels is N(A) and N(B) respectively, we have in equilibrium,  Also,  But  That is the transition amplitudes are equal under time reversal, this encodes ‘The principle of detailed balance’. A ↔ B + γ N(A) w a = N(B) w e

 Hence, we obtain  However, if we go by phenomenology and introduce A 21,B 21 and B 12 as the spontaneous, stimulated emission and absorption rates (known as Einstein Coefficients), then the detailed balance and Planck’s Law requires g 1 B 12 = g 2 B 21 A 21 = 2h ν 3 /c 2 B

Broadening Mechanisms

 Atomic excited states are never stable as we have seen and spontaneously radiate to lower states. As such any spectral line or scattering cross section possesses a Breit-Wigner profile.  Classically, this unstability can be thought of as ‘damping’ term for the atomic oscillator under classical radiation field action.  That is we have an emission line with a decaying electric field or an absorption line through unstable intermediate state. The line profile is therefore a square of Fourier transform of

 The Breit–Wigner and Gaussian line shape functions having same FWHM.  The Breit–Wigner has FWHM ~ 1/t sp  The Gaussian profile arises due to Doppler Broadening and hence outweighs the natural profile in the emission lines. But the effect of natural broadening is noticeable in the absorption lines.

 In a gas the atoms are in a random motion and when it interacts with a photon, its apparent frequency can be different from the proper frequency and hence there is a spread in the emission photon frequencies as a direct map of spread in the atomic velocities of a gas.  Let z be the direction of propagation of radiation, then the frequency observed by a atom having z-velocity v z is  and for strong interaction,

 Note that  Hence the line shape function is given by  which corresponds to the Gaussian profile.

 In a gas, random collisions are always taking place and in such collisions, the energy levels of atoms change when they are quite close due to mutual interactions. That is, their energy levels get perturbed in the moment of collision and if any atom radiates or absorbs during that time, the frequency of emission and absorption line changes during collision and returns to its original value after the collision.  So collisions can be taken to be instantaneous.

 Since the collision times are random, these introduce random phase changes in the emitted or scattered wave front.  That is the net effect of collisions can be modeled in where the phase remains constant for t 0 < t <t 0 + c

 Since the wave is sinusoidal between two collisions the spectrum of such a wave is given by  Computing the power spectrum, we have

 Now, we also need to average over the different  c values for full power spectrum, for this we multiply I( ω ) with the probability of an atom colliding between [ c,  c +d c ]  Hence, the line shape function is which is again a Lorentzian or Breit-Wigner function.

 Advanced Quantum Mechanics, Sakurai J.J.  Radiative processes in Astrophysics, Rybicki and Lightman  Optical Electronics, Ghatak and Tyagrajan

Thanks for your kind attention.