The extinction law Consider a small element of an absorbing medium, ds, within the total medium s.
Extinction Law The extinction law can be written as The constant of proportionality is defined as the extinction coefficient. k can be defined in three ways. (1) by the length of the absorbing path with the gas at one atmosphere pressure
Extinction coefficient By mass or by concentration
Optical depth Normally we are interested in the total extinction over a finite distance (path length) Where S ( ) is the extinction optical depth The integrated form of the extinction equation becomes
Extinction = scattering + absorption Extinction really consists of two distinct processes, scattering and absorption, hence where
Differential equation of radiative transfer We must now add the process called emission. Consider a slab of thickness ds, filled with an optically active material giving rise to radiative energy of frequency in time dt. This energy emerges from the slab as an angular beam within the solid angle dω, around a propagation vector Ω. The emission coefficient is defined as the ratio
Differential equation of radiative transfer Combining the extinction law with the definition of the emission coefficient noting that
Differential equation of radiative transfer The ratio j /k( ) is known as the source function, This is the differential equation of radiative transfer
Basic scattering processes We have identified three radiation-matter interactions, absorption, emission and scattering We can consider the radiation field in two ways, classical and quantum. Classical – the electromagnetic field is a continuous function of space and time, with a well defined electric and magnetic field at every location and instant of time Quantum – the radiation field is a concentration of discrete values of energy, h.
Scattering of radiation fields Radiation fields scattered from the points P’’ and P’ are 90 degreed different in phase and therefore interfere destructively.
Lorentz theory of radiation-matter interactions Neutral atoms consist of electrons (negative charges) and nucleus (positive charge. Bound together by elastic forces – Hooke’s Law. Combined with the Maxwell theory of the electromagnetic field Classical theory
Lorentz theory Could not explain the black-body frequency distribution law. Planck in 1900. Quantized states Could not explain the photoelectric effect Einstein 1914 Photons
Scattering from Damped Simple Harmonic Oscillator Assume that a molecule is a simple harmonic oscillator with a single harmonic oscillation frequency ω 0 (2 ) When irradiated by linearly polarized monochromatic electromagnetic wave of frequency ω 0, the electron undergoes an acceleration, while the nucleus, being massive, is assumed not to move. An accelerating charge gives rise to electromagnetic radiation.
Damped Harmonic Oscillator Without energy loss the oscillator would increase its motion indefinitely – forward beam would be unchanged. In reality we see absorption – an energy loss. Can only occur if there is some damping force acting on the oscillator. The classical damping force is given by:
Damped Harmonic Oscillator The ratio of P to the power carried in the incident field is the scattering cross section An accelerated charge radiates with an average power of
Resonance scattering – Lorentz profile Let the frequency of the incident light be close to the resonance, i.e. close to ω 0. We can write Substituting this relation, and the formula for
Lorentz profile The frequency dependent part of of the equation is called the Lorentz profile Since the Lorentz profile is normalized we find by integrating over all frequencies
Oscillator Strength In the classical theory the integrated cross section is a constant. Under the quantum theory there is usually more than one resonant frequency, and each resonance has an integrated cross section given by the above term, but multiplied by a constant f. f is called the oscillator strength Where S i is called the line strength i