METO 621

1. METO 621 Lesson 5

2. 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

3. Pressure 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.

4. 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)

5. 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

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

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

8. Doppler broadening We now define the Doppler width as

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

10. 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

11. Comparison of the line shapes

12. 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

13. Rayleigh scattering • Transforming from angular frequency to wavelength we get

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

15. Relation between Cartesian and spherical coordinates

16. Scattering in the planes of polarization

17. 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

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

19. Rayleigh phase function • The radiation pattern for the far field of a classical dipole is proportional to Psin2 , where  is the polar angle measured from the axis, and Pis 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

20. 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

21. Rayleigh scattering phase function • If we normalize the equation

22. Phase diagram for Rayleigh scattering

23. Rayleigh scattering • Sky appears blue at noon, red at sunrise and sunset - why?

24. 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.

25. 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

26. 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)