METO 621 Lesson 5

Natural broadening The line width (full width at half maximum) of the Lorentz profile is the damping parameter, . For an isolated molecule the damping parameter can be interpreted as the inverse of the lifetime of the excited quantum state. This is consistent with the Heisenberg Uncertainty Principle If absorption line is dampened solely by the natural lifetime of the state this is natural broadening If absorption line is dampened solely by the natural lifetime of the state this is natural broadening

Pressure broadening For an isolated molecule the typical natural lifetime is about s, 5x10 -4 cm -1 line width However as the pressure increases the distance between molecules becomes shorter. We can view the outcome in two ways (1) Collisions between molecules can shorten the lifetime, and hence the line width becomes larger. (2) As the molecules get closer their potential fields overlap and this can change the ‘natural line width’. The resultant line shape leads to a Lorentz line shape. Except at very high pressures when the fields overlap strongly - assymetric line shapes - Holtzmach broadening.

Pressure broadening Clearly the line width will depend on the number of collisions per second,i.e. on the number density of the molecules (Pressure) and the relative speed of the molecules (the square root of the temperature)

Doppler broadening Second major source of line broadening Molecules are in motion when they absorb. This causes a change in the frequency of the incoming radiation as seen in the molecules frame of reference Let the velocity be v, and the incoming frequency be, then

Doppler broadening In the atmosphere the molecules are moving with velocities determined by the Maxwell Boltzmann distribution

Doppler broadening The cross section at a frequency is the sum of all line of sight components

Doppler broadening We now define the Doppler width as

Voigt profile In general the overall broadening is a mixture of Lorentz and Doppler. This is known as the Voigt profile

Voigt profile For small damping ratios, a  0, we retrieve the Doppler result. For a > 1 we retrieve the Lorentz result In general the Voigt profile shows a Doppler-like behavior in the line core, and a Lorentz-like behavior in the line wings. The Voigt profile must be evaluated by numerical integration

Comparison of the line shapes

Rayleigh scattering If the driving frequency is much less than the natural frequency then the scattering cross section for a damped simple oscillator becomes The molecular polarizability is defined as

Rayleigh scattering Transforming from angular frequency to wavelength we get

Rayleigh scattering The polarizability can be expressed in terms of the real refractive index, m r where  RAY ( ) is the scattering coefficient (per atmosphere) m r varies with wavelength, so the actual cross section deviates somewhat from the -4 dependence

Relation between Cartesian and spherical coordinates

Scattering in the planes of polarization

Scattering phase function So far we have ignored the directional dependence of the scattered radiation - phase function Let the direction of incidence be  ’, and direction of observation be . The angle between these directions is cos  =  ’  .  is the scattering angle. If  is <  /2 - forward scattering If  is >  /2 - backward scattering

Scattering phase function In polar coordinates cos  = cos  ’cos  + sin  ’sin  cos(  ’-  ) We define the phase function as follows

Rayleigh phase function The radiation pattern for the far field of a classical dipole is proportional to  sin 2 , where  is the polar angle measured from the axis, and  is the induced dipole moment. We can take the incoming radiation and break it up into two linearly polarized incident waves, one with the electric vector parallel to the scattering plane, the other perpendicular to the scattering plane. These waves give rise to induced dipoles

Rayleigh scattering phase function If the incident electric field lies in the scattering plane then the scattering angle is (  /2+  ), if perpendicular to the scattering plane the angle is  /2. Hence given that the parallel and perpendicular intensities are equal given that the parallel and perpendicular intensities are equal

Rayleigh scattering phase function If we normalize the equation

Phase diagram for Rayleigh scattering

Rayleigh scattering Sky appears blue at noon, red at sunrise and sunset - why?, nm , cm 2 , surfaceExp(-  ) E E E E , E E

Schematic of scattering from a large particle In the diagram above 1 and 2 are points within the particle. In the forward direction the induced radiation from 1 and 2 are in phase. However in the backward direction the two induced waves can be completely out of phase.

Mie-Debye scattering For particles which are not small compared with the wavelength one has to deal with multiple waves from different molecules/atoms within the particle Forward moving waves tend to be in phase and this gives a large resultant amplitude. Backward waves tend to be out of phase and this results in a small resultant amplitude Hence the scattering phase function for a particle has a much larger forward component (forward peak) than the backward component

Phase diagrams for aerosols Phase diagrams for different values of the ratio of the aerosol radius to the wavelength of the incident radiation (left hand column)