Module 7 : Laser- I
Lecture   : Line Broadening in Lasers
   
2.5.2 Collision Broadening:
 
Collision of atoms and molecules among themselves interrupts a radiative transition. If an atom which is emitting a radiation suddenly collides with another atom, the process of radiation is interrupted. the radiating atom starts its motion after such a collision with a completely random phase without any memory of the phase of the interrupted radiation. Linewidth in the presence of collisions is more than that in the absence of collisions. Collision broadening is also termed as pressure broadening as increased pressure is one of the primary causes of collision.
We can use a classical model to obtain an expression for homogeneous lineshape broadening. We take the electron to be classical particle bound to the nucleus by a spring force given by Hooke's law $ F=-kx$. The equation of motion of the electron is given by
 

$\displaystyle \frac{d^2x}{dt^2} + \omega_0^2 x =0$

where $ \omega_0 = \sqrt{k/m}$. The motion of the electron is oscillatory, $ x(t) = x_0e^{i\omega t}$. The total energy of the electron $ {\cal E} = (1/2)kx_0^2 = (1/2)m\omega^2x_0^2$is constant. The model does not take into account the power that an oscillating electron radiates. In classical electrodynamics, an accelerating charge emits radiation, the emitted power being given by the Poynting vector $ \vec E\times\vec B$. The total power radiated by a dipole is given by
                                                                                     $\displaystyle -\frac{dE}{dt} = \frac{e^2x_0^2\omega_0^4}{12\pi \epsilon_0 c^3}\equiv \gamma E$
which shows that the radiated energy is proportional to the instantaneous energy. We may model the situation by modifying our equations of motion for the electron to include a damping term proportional to the velocity ( $ F_d = -\alpha v$)

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