Word Count : 939

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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} ] _________ ______ ___ _____ _______ ____.
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