ON ATOMS AND PHOTONS
Suprit SinghTalk for the IUCAA Grad-school course in Inter-
stellar medium given by Dr. A N Ramaprakash15th April 2KX
PART IPrinciple of Detailed Balance and
Einstein coefficients
Absorption and Emission
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.
Absorption and Emission
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 :
Transition Probabilities and times
Now if you calculate the transition amplitude using,
we have for transition probability
And for spontaneous emission in the dipole approximation
Detailed Balance 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) wa = N(B) we
Einstein Coefficients
Hence, we obtain
However, if we go by phenomenology and introduce A21 ,B21 and B12 as the spontaneous, stimulated
emission and absorption rates (known as Einstein Coefficients), then the detailed balance and Planck’s Law requires
g1B12 = g2B21
A21 = 2hν3/c2 B
PART IIBroadening Mechanisms
Natural Line Broadening 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
Natural Line Broadening
The Breit–Wigner and Gaussian line shape functions having same FWHM.
The Breit–Wigner has FWHM ~ 1/tsp
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.
Thermal Doppler Broadening
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 vz is
and for strong interaction,
(1 / )zv c
0 1
0 0(1 / ) (1 / )z zv c v c 0
0zv c
Thermal Doppler Broadening
Note that
Hence the line shape function is given by
which corresponds to the Gaussian profile.
2
2( ) exp( )zmvz z zkTP v dv N dv
2202
0 0
( )2( ) exp( )Nc mckTg d d
Colliosional Broadening
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.
Colliosional Broadening
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 t0 < t <t0+c
Colliosional Broadening
Since the wave is sinusoidal between two collisions the spectrum of such a wave is given by
Computing the power spectrum, we have
Colliosional Broadening
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 +dc ]
Hence, the line shape function is
which is again a Lorentzian or Breit-Wigner function.
References
Advanced Quantum Mechanics, Sakurai J.J.
Radiative processes in Astrophysics, Rybicki and Lightman
Optical Electronics, Ghatak and Tyagrajan
THAT’S ALL FOLKSThanks for your kind attention.