The Rayleigh and Einstein-Smoluchowski theories of scattering apply only when the scattering centres are small relative to the wavelength of light: this is true in the case of gas molecules and of the tiny density fluctuations in pure liquids. Even the most pristine natural waters, however, are not, optically speaking, pure and they invariably contain high concentrations of particles - mineral particles derived from the land or from bottom sediments, phytoplankton, bacteria, dead cells and fragments of cells etc. all of which scatter light. The particles that occur in natural waters have a continuous size distribution, which is roughly hyperbolic,54 i.e. the number of particles with diameter greater than D is proportional to 1/Dg, where g is a constant for a particular water body, but varies widely from 0.7 to 6 in different water bodies.636 Although a hyperbolic distribution implies that smaller particles are more numerous than big ones, nevertheless most of the particle cross-sectional area that would be encountered by light in natural waters is due to particles of diameter greater than 2 mm,636 which is not small relative to the wavelengths of visible light, and so scattering behaviour different to the density fluctuation type must be anticipated. The smaller particles, although numerous, have a lower scattering efficiency.

A theoretical basis for predicting the light-scattering behaviour of spherical particles of any size was developed by Mie (1908). The physical basis of the theory is similar to that of Rayleigh in that it considers the oscillations set up within a polarizable body by the incident light field and the light re-radiated (i.e. scattered) from the body as a result of these oscillations. Instead of (as in Rayleigh theory) equating the particle to a single dipole, the Mie theory considers the additive contributions of a series of electrical and magnetic multipoles located within the particle. The advantage of the Mie theory is that it is all-embracing - for very small particles, for example, it leads to the same predictions as the Rayleigh theory; the disadvantage is that the analytical expressions are complex and do not lend themselves to easy numerical calculations. For particles larger than the wavelength of the light, Mie theory predicts that most of the scattering is in the forward direction within small angles of the beam axis (Fig. 4.1). A series of maxima and minima is predicted at increasing scattering angle, but these are smoothed out when a mixture of particle sizes is present.

In the case of particles larger than a few wavelengths of light, a reasonable understanding of the mechanism of scattering can be obtained on the basis of diffraction and geometrical optics, without recourse to electromagnetic theory. When an object is illuminated by a plane light wave, the shadow of an object on a screen placed behind it is not quite precisely defined: just outside it a series of concentric faint dark bands will be present; and lighter concentric bands, where clearly some light is falling, will be present within the area of the geometric shadow. This phenomenon - diffraction - is due to interference (destructive in the dark rings, constructive in the light rings) between parts of the wave coming from different points around the edge of the illuminated object, and arriving simultaneously but out of phase (because of the different distances traversed) at particular points on the screen. In the case of a round object, superimposed on the circular shadow, there is a bright spot in the centre, surrounded by alternate dark and light rings. In fact most of the light diffracted by a particle is propagated in the forward direction within a small angle of the initial direction of the beam (giving rise to the bright spot). With increasing angular distance away from the axis, the diffracted intensity goes through a series of minima and maxima (dark and light rings), which diminish progressively in height.

Applying geometrical optics to these larger particles, it can readily be appreciated that some of the light will be reflected at the external surface and some will pass through the particle and undergo refraction, or internal reflection as well as refraction (Fig. 4.2). In all cases the photons

Scattering angle

Fig. 4.1 Angular distribution of scattered intensity from transparent spheres calculated from Mie theory (Ashley and Cobb, 1958) or on the basis of transmission and reflection, or diffraction, transmission and reflection (Hod-kinson and Greenleaves, 1963). The particles have a refractive index (relative to the surrounding medium) of 1.20, and have diameters 5 to 12 times the wavelength of the light. After Hodkinson and Greenleaves (1963).

Fig. 4.1 Angular distribution of scattered intensity from transparent spheres calculated from Mie theory (Ashley and Cobb, 1958) or on the basis of transmission and reflection, or diffraction, transmission and reflection (Hod-kinson and Greenleaves, 1963). The particles have a refractive index (relative to the surrounding medium) of 1.20, and have diameters 5 to 12 times the wavelength of the light. After Hodkinson and Greenleaves (1963).

involved will be made to deviate from their initial direction, i.e. will be scattered. Light scattered by these mechanisms is still predominantly in the forward direction: scattered intensity diminishes continuously with increasing angle but does not show the same degree of concentration at small angles as does scattering due to diffraction. Calculations by Hod-kinson and Greenleaves (1963) for suspensions of spherical particles of mixed sizes show that most of the scattering at small angles (up to about 10 ° to 15 °) can be attributed to diffraction, whereas most of the scattering at larger angles is due to external reflection and transmission with refraction (Fig. 4.1). The angular variation of scattering calculated on the basis of diffraction and geometrical optics is reasonably close to

Exter rial r^lKlrßrl

Fig. 4.2 Scattering of light by a particle: reflection and refraction processes.

Internal rflf|K;ton

Hihu^tiun

Fig. 4.2 Scattering of light by a particle: reflection and refraction processes.

that derived by the Mie electromagnetic theory. Such discrepancies as exist may be largely due to the fact that as a result of the phase change induced in the transmitted ray by passage through a medium of higher refractive index, there are additional interference effects between the diffracted and the transmitted light:1395 this phenomenon is known as anomalous diffraction.

Any particle in a beam of light will scatter a certain fraction of the beam and the radiant flux scattered will be equivalent to that in a certain cross-sectional area of the incident beam. This area is the scattering cross-section of the particle. The efficiency factor for scattering, Qscat, is the scattering cross-section divided by the geometrical cross-sectional area of the particle (pr2 for a spherical particle of radius r). Similarly, in the case of an absorbing particle, the radiant flux absorbed is equivalent to that in a certain cross-sectional area of the incident beam: this area is the absorption cross-section of the particle. The efficiency factor for absorption, Qabs, is the absorption cross-section divided by the geometrical cross-sectional area of the particle. The efficiency factor, Qatt, for attenuation (absorption and scattering combined) is thus given by

The attenuation efficiency of a particle can be greater than unity: that is, a particle can affect the behaviour of more light in the incident beam than will be intercepted by its geometrical cross-section. This can be true of absorption and scattering separately: i.e. it is possible for a particle to absorb, or to scatter, more light than its geometrical cross-section would intercept. In terms of electromagnetic theory we may say that the particle can perturb the electromagnetic field well beyond its own physical boundary. Mie theory, based as it is upon electromagnetism, can be used to

0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 Particle diameter (um)

Fig. 4.3 Scattering efficiency of non-absorbing spherical particles as a function of size. The particles have a refractive index, relative to water, of 1.17. Wavelength = 550 nm. Continuous line, Qscatt for a single particle, calculated using the equation of Van de Hulst (1957) - see text. Broken line, the scattering coefficient (b) for a suspension containing 1 g of particles m-3.

0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 Particle diameter (um)

Fig. 4.3 Scattering efficiency of non-absorbing spherical particles as a function of size. The particles have a refractive index, relative to water, of 1.17. Wavelength = 550 nm. Continuous line, Qscatt for a single particle, calculated using the equation of Van de Hulst (1957) - see text. Broken line, the scattering coefficient (b) for a suspension containing 1 g of particles m-3.

calculate the absorption and scattering efficiencies of particles. The simpler anomalous diffraction theory of van de Hulst (1957) can also be used to calculate the scattering efficiency of particles with refractive index up to about twice that of the surrounding medium. The relation is

P P2

where p = (4na/X)(m- 1), m being the refractive index of the particle relative to that of the surrounding medium, and a being the radius of the particle. For a non-absorbing particle, Qscatt = Qatt. Figure 4.3 shows the way in which the scattering efficiency for green light of a spherical non-absorbing particle of refractive index relative to water of 1.17 (a typical value for inorganic particles in natural waters), varies with particle size. It can be seen that scattering efficiency rises steeply from very low values for very small particles to about 3.2 at a diameter of 1.6 mm. With increasing diameter, it first decreases and then increases again and undergoes a series of oscillations of diminishing amplitude to level off at a Qscatt value of 2.0 for very large particles. A similar general pattern of variation of Qscatt with size would be exhibited by any scattering particle of the types found in natural waters at any wavelength in the photosynthetic range.

As diameter decreases below the optimum for scattering (e.g. from 1.6 mm downwards in Fig. 4.3), so efficiency for a single particle decreases. However, for a given mass of particles per unit volume, the number of particles per unit volume must increase as particle size decreases. It is therefore of interest to determine how the scattering coefficient of a particle suspension of fixed concentration by weight varies with particle size. The results of such a calculation for particles of density in the range typical of clay minerals at a concentration of 1 gm~3 is shown in Fig. 4.3. As might be anticipated, because of the increase in particle number simultaneous with the decrease in diameter, total scattering by the suspension, expressed in terms of the value of b, does not decrease as precipitately with decreasing diameter below the optimum, as single particle scattering efficiency does, and the optimum particle diameter for suspension scattering (~1.1 mm) is lower than that for single particle scattering (^1.6 mm). As particle diameter increases beyond the optimum so scattering by the suspension shows a progressive decrease to very low values, with only minor, heavily damped, oscillations corresponding to the oscillations in Qscatt for the individual particles.707

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