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A very simple example of the scattering matrix is that for the free electron where, in the azimuthal plane, the function S ( ) ) cos( ) and Sk ( ) ) 1 Simple consideration reveals that no radiation vibrating parallel to the scattering plane is found at an angle of 90 from the direction of the incident radiation A more thorough and rigorous treatment of the mathematics and physics of scattering processes can be found in the classical text of van de Hulst (1957) From a determination of the amplitude scattering functions, the intensity of the radiation scattered in any direction can be readily calculated If a scattering particle is illuminated by incident unpolarized ux of I0 , the scattered uxes received at a distance, r, from the particle may be written as I D The terms F ( , ) k2 and Fk ( , ) k2 I0 F ( , ) 2r 2 k 2 and Ik D I0 Fk ( , ) 2r 2 k 2 (331)
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are referred to as the angular scattering distributions The scattering cross-section of any particle may be determined by integrating these functions over the total solid angle and may be expressed as Z Csca D k 2 F ( , ) C Fk ( , ) d (332) The extinction cross-section comprises scattering and absorption components Thus, Cext D Csca C Cabs (333)
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In most cases, the determined cross-sections do not correspond to the basic geometric values of the grain sizes but can be related by means of ef ciency factors, Q Hence we may write 8 Csca D a 2 Q sca < (334) C D a 2 Q abs abs :C D a2 Q
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ext ext
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If the imaginary part of the refractive index of the scattering grain is equal to 0, then Cabs D 0 There is a relationship between the forward-scattered radiation and extinction It is referred to as the optical theorem and results from the conservation of energy giving the identities Cext( ) D Cext(k) D 4 < fS (0)g , k2 4 < Sk (0) , 2 k (335) (336)
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3 The Algebra of Polarization
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where <S(0) is the real part of the complex forward scattering amplitude, with the , k subscripts corresponding to the polarizations perpendicular to, and parallel to, the scattering plane Determination of the scattering phase functions and the Q values for different kinds of particles has been exercised by various techniques, both mathematical and empirical Perhaps the most common reference is to Mie scattering which refers to solutions of the homogeneous sphere problem and solved exactly by Mie (1908) For many practical situations, the rigorous computation of ef ciency factors for scattering and extinction can be by-passed For x (D 2 a/ ) 1, Q ext and Q sca may be expanded as the convergent power series in x as follows: 2 1 m Q Q ext x= 4 m2 C 2 Q 4 Q m C 27 m 2 C 38 Q Q 4 m2 1 C x 3= 15 m 2 C 1 Q 2m2 C 3 Q " 2 # 2 Q 1 8 m C x 4< C (337) 3 m 2 C 21 Q " 2 # m2 1 Q 8 4 Q sca x < C (338) 3 m2 C 2 Q where < and = refer to real and imaginary parts respectively Convergence of the series is expected to be quite rapid for x 06 For x 1, the rst term with non-zero coef cient may suf ce and the two formulae that are often quoted are 2 1 m Q (339) Q ext 4x= , m2 C 2 Q " # 2 m2 1 Q 8 Q sca x 4 < (340) 3 m2 C 2 Q Another analytical solution relates to scattering by (in nitely) long cylinders which has application to the modelling of elongated grains, such forms required to explain interstellar polarization For grains with complicated shapes and with mantle structures, their scattering cross-sections may be explored empirically by scaling a model grain, and conducting experiments on it in a microwave laboratory, it being noted that scattering behaviour depends on x, the ratio of the particle size to the wavelength of the incident radiation
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