METO 621Natural broadeningPressure broadeningSlide 4Doppler broadeningSlide 6Slide 7Slide 8Voigt profileSlide 10Comparison of the line shapesRayleigh scatteringSlide 13Slide 14Relation between Cartesian and spherical coordinatesScattering in the planes of polarizationScattering phase functionSlide 18Rayleigh phase functionRayleigh scattering phase functionRayleigh scattering phase functionPhase diagram for Rayleigh scatteringSlide 23Schematic of scattering from a large particleMie-Debye scatteringPhase diagrams for aerosolsMETO 621Lesson 5Natural 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Et h2t h2.h12• If absorption line is dampened solely by the If absorption line is dampened solely by the natural lifetime of the state this is natural natural lifetime of the state this is natural broadeningbroadeningPressure broadening•For an isolated molecule the typical natural lifetime is about 10-8 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)0)()(vv)(TnTnSTPSTPnnSTPLLrelLrelLL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)cosv1(cos vcosv'ccvDoppler broadening•In the atmosphere the molecules are moving with velocities determined by the Maxwell Boltzmann distributionmTkdvTkmdfBXXBXX/2 vwhere)v/vexp(2v)v(02022/1Doppler broadening•The cross section at a frequency  is the sum of all line of sight components  20202022/12022/1v/)(exp2 )/v()v/vexp(2)/v1()v(v)(vcTkmScdvTkmcfdBxnxxBxnxxnDoppler broadening• We now define the Doppler width as  22000/)(exp)( )(/vDDDnDSScvVoigt profile•In general the overall broadening is a mixture of Lorentz and Doppler. This is known as the Voigt profileDDLDnratiodampingaayyd yaS/)(v/)v()exp()(02222/3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 integrationComparison of the line shapesRayleigh scattering200244042044461 6eeR AYnmeccme•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 as02002for 4epmeRayleigh scattering• Transforming from angular frequency to wavelength we getnRAY() 8324p2Rayleigh scattering• The polarizability can be expressed in terms of the real refractive index, mrRAY() nRAYn 323(mr 1)2(m 1) where RAY() is the scattering coefficient (per atmosphere)• mr varies with wavelength, so the actual cross section deviates somewhat from the -4 dependencenmrp2/)1( Relation between Cartesian and spherical coordinatesScattering in the planes of polarizationScattering 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 scatteringScattering phase function• In polar coordinates cos = cos’cos + sin’sincos(’- )• We define the phase function as follows14).;','(sin4)(cosis ionnormalisat The)( )(cos)(cos)(cos020414pddpdwsrdnnpnnRayleigh phase function• The radiation pattern for the far field of a classical dipole is proportional to sin2 , 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 dipolesRayleigh 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)2/(sin)2/(sin)(2||2||IIIRAY• given that the parallel and perpendicular given that the parallel and perpendicular intensities are equal intensities are equal )cos1( )(2 IIRAYRayleigh scattering phase function• If we normalize the equation)cos1(43)(34)cos1(sin41)cos1(4122022024raypdddPhase diagram for Rayleigh scatteringRayleigh scattering•Sky appears blue at noon, red at sunrise and sunset - why?, nm , cm2, surface