To determine the collisional width (also called pressure broadening) of a laser transition in a gas, we first need to understand that collisions between gas molecules cause a broadening of spectral lines. The collisional or pressure-broadened linewidth ( \Delta \nu_c ) is proportional to the gas pressure ( P ) and inversely proportional to the square root of the gas temperature ( T ). Mathematically, it can be expressed as: [ \Delta \nu_c = \frac{1}{\pi \tau_c} ] where ( \tau_c ) is the mean time between collisions. The mean collision time ( \tau_c ) depends on the number density ( n ) of the gas and the collision cross-section ( \sigma ): [ \tau_c = \frac{1}{n \sigma \bar{v}} ] Here, ( \bar{v} ) is the average molecular speed, which for a gas at temperature ( T ) is given by: [ \bar{v} = \sqrt{\frac{8 k_B T}{\pi m}} ] where ( k_B ) is the Boltzmann constant and ( m ) is the mass of a gas molecule. The number density ( n ) can be expressed using the ideal gas law: [ n = \frac{P}{k_B T} ] Substituting ( n ) and ( \bar{v} ) into the expression for ( \tau_c ), we get: [ \tau_c = \frac{1}{\left( \frac{P}{k_B T} \right) \sigma \sqrt{\frac{8 k_B T}{\pi m}}} = \frac{k_B T}{P \sigma} \sqrt{\frac{\pi m}{8 k_B T}} = \frac{1}{P \sigma} \sqrt{\frac{\pi m k_B T}{8}} ] Therefore, the collisional width ( \Delta \nu_c ) is: [ \Delta \nu_c = \frac{1}{\pi \tau_c} = \frac{P \sigma}{\pi} \sqrt{\frac{8}{\pi m k_B T}} ] This formula shows that the linewidth increases linearly with pressure and decreases with increasing temperature. For a specific gas, the collision cross-section ( \sigma ) can be determined experimentally or from literature values. Next, to find the ratio of spontaneous to stimulated emission, we consider Einstein’s coefficients. For a two-level system, the spontaneous emission rate is ( A_{21} ), while the stimulated emission rate is ( B_{21} \rho(\nu) ), where ( \rho(\nu) ) is the energy density of the radiation at the transition