Absorption by Particles

Although the form of Eq. (2.28) is indifferent to whether the illuminated object is a molecule or a particle, we write the symbol Cabs for the absorption cross section of a particle to signal that particles and molecules are different. Implicit in the definition of cross sections is that the irradiance of the incident illumination be constant over lateral dimensions large compared with the size of the particle. As with molecules the absorption cross section of a particle depends on its orientation (unless it is an isotropic sphere) and the state of polarization of the illumination (again, unless it is an isotropic sphere). The absorption cross section of a particle often is normalized by its geometrical cross sectional area G projected onto a plane perpendicular to the illumination. The resulting dimensionless quantity

is called an efficiency or efficiency factor for absorption. The advantage of this normalization, namely, not having to fret about units, is outweighed by several disadvantages. In the first place, we expect quantities called efficiency factors to be less than or at most equal to 1 (100%), whereas it is not unusual to encounter overachieving particles with Qabs greater than 1, sometimes appreciably greater, at some wavelengths. Also, some particles don't have well-defined or easily determined geometrical cross sections (e.g., soot aggregates, the kinds of particles found in smokes), and molecules most definitely do not. And the normalization in Eq. (2.142) is arbitrary. Why not normalize by total surface area or the square of mean chord length or ...? To be fair Qabs, at least for a sphere, does have a physical meaning. At a given wavelength it is the emissivity of the particle, which underlies the assertion in Section 1.3 about emissivities greater than 1. But the efficiency factors for scattering and extinction (Sec. 3.5) have no physical meaning.

Both molecules and particles have well-defined absorption cross sections, in principle measurable; both molecules and particles have masses, which again are measurable. So the only really meaningful normalized cross section, by means of which we can compare molecules and particles, is cross section per unit mass. But if a particle has a definite volume v (a molecule does not), another normalized cross section that does have a physical meaning is the volumetric cross section, the cross section per unit particle volume. The absorption coefficient of a dilute suspension of N identical particles is where f = Nv is the volume fraction of particles in the suspension (i.e., the fraction of the total volume containing something you can kick). If any quantity deserves to be called an efficiency it is the volumetric absorption cross section Cabs/v. Equation (2.143) can be generalized to a suspension of non-identical particles:

and the subscript j denotes anything that makes one particle different from another - shape, size, orientation, or composition.

We now turn to another advantage of expressing cross sections of particles as volumetric cross sections. Consider the simplest imaginable particle, an optically homogeneous slab of uniform thickness d and surface area A with dimensions much larger than A illuminated at normal incidence by a beam with irradiance F. The law of exponential attenuation [Eq. (2.8)] strictly applies only to a medium without boundaries. Boundaries add the complication of reflections, with the attendant possibility of interference (see Sec. 3.4). We assume that the optical properties of the slab are sufficiently similar to those of the surrounding medium that

where

reflections are negligible and that the slab is sufficiently thick compared with the wavelength that interference is negligible. With these assumptions the rate Wa at which energy is absorbed by the slab is

where we write the bulk absorption coefficient of the slab material as Kb to distinguish it from the absorption coefficient [Eq. (2.144)] of a dilute suspension of particles composed of that material. If we further assume that Kbd C 1 (weak absorption), Eq. (2.146) becomes

From Eq. (2.28) the absorption cross section of the slab is therefore AdKb and its cross section per unit volume is

At the other extreme Kbd > 1 (strong absorption) and Eq. (2.146) yields

For some materials Equations (2.148) and (2.149) for a slab particle are good estimates for the volumetric absorption cross section of some particles in the limits of weak and strong absorption. A further advantage of considering volumetric absorption by a particle is that it can be compared with the bulk absorption coefficient of its parent material. Sometimes the absorption spectrum of a particle is similar to that of its parent material, but sometimes exhibits not even a trace of a family resemblance.

If the distribution of sizes and shapes of particles in a suspension, all of identical composition, is such that they can fill all space (suspensions of identical spheres cannot), K given by Eq. (2.144) should approach Kb as f approaches 1. And indeed it does if the volumetric absorption cross section is given by Eq. (2.148), but definitely does not if it is given by Eq. (2.149). For strong absorption the volumetric absorption cross section of particles is dominated by their size and not the bulk absorption coefficient of their parent material. This signals caution in applying Eq. (2.144) to dense suspensions of particles. Unfortunately, we can't give precise criteria for what is meant by "dense", although we touch on this subject in Sections 5.1 and 5.3. Depending on the size of the particles relative to the wavelength, when a suspension of them becomes sufficiently dense, it is probably more realistic to look upon it as a slightly porous medium (f « 1), the absorption coefficient of which is approximately k = fnb. (2.150)

This expression does have the correct asymptotic behavior as f ^ 1. Thus Eq. (2.144) can be used with confidence for f C 1, and Eq. (2.150) for f « 1, but the values of f at which the transition from the one to the other begins and ends cannot be specified precisely.

For what materials and particles is Eq. (2.148) not a good approximation? Suspensions of small gold and silver particles (colloidal gold and silver) when illuminated by white light display upon transmission vivid colors completely unrelated to the appearance of the bulk

Wavelength (pm)

Figure 2.25: Volumetric absorption cross section spectrum of a 10 |m-diameter water droplet (top) compared with the bulk absorption spectrum of liquid water (bottom). For smaller diameters volumetric absorption in the UV more closely follows the bulk absorption coefficient; for larger diameters, peaks in the volumetric absorption spectrum beyond about 2 |m are flattened.

Wavelength (pm)

Figure 2.25: Volumetric absorption cross section spectrum of a 10 |m-diameter water droplet (top) compared with the bulk absorption spectrum of liquid water (bottom). For smaller diameters volumetric absorption in the UV more closely follows the bulk absorption coefficient; for larger diameters, peaks in the volumetric absorption spectrum beyond about 2 |m are flattened.

metals. The absorption spectrum of silver, in particular, is as nearly featureless as a dry lake bed, and although that of gold is more interesting, which is why gold is golden, absorption spectra of particles of these metals bear no resemblance whatsoever to those of their parent materials.

Alas, gold and silver particles are not commonly found in the atmosphere but particles of insulating crystalline materials (e.g., quartz, ammonium sulfate) are, and at infrared wavelengths the absorption spectra of such particles do not always dutifully follow those of their

Figure 2.26: Volumetric absorption cross section for water droplets of varying diameter at the three wavelengths shown.

parent materials. But the differences are not so striking as they are for metallic particles, a matter of shifts of absorption peaks rather than their appearance seemingly from nowhere.

Water, however, is an example of a material for which Eqs. (2.148) and (2.149) often are good approximations. The volumetric absorption cross section of a 10 pm-diameter water droplet (Fig. 2.25), calculated using the exact theory for a sphere (Sec. 3.5), is similar to the bulk absorption coefficient of water from visible to infrared. For droplets smaller than about 10 pm (not shown), the volumetric absorption cross section approaches more closely the bulk absorption coefficient in the ultraviolet (< 0.2 pm), whereas for droplets larger than about 10 pm the peaks in the infrared become more and more flattened. This flattening of the absorption spectrum of a particle is expected on physical grounds. As the bulk absorption coefficient of a particle increases, absorption is concentrated more and more in the outer layers of the illuminated particle, and what changes is not how much energy is absorbed but where.

Figure 2.26 shows the volumetric absorption cross section as a function of droplet diameter at three different wavelengths. For sufficiently small diameters, the cross section is approximately the bulk absorption coefficient and independent of diameter, as predicted by Eq. (2.148). With increasing diameter, the cross section decreases as the inverse of diameter and is independent of the bulk absorption coefficient. Moreover, the diameter at the transition from a constant to a decreasing value is smaller the greater the bulk absorption coefficient, which increases with increasing wavelength. All this is consistent with Eq. (2.149). The exceedingly fine structure in the absorption cross section at A = 1.0 pm is a consequence of interference (see Sec. 3.4). Note that this interference structure does not occur until the diameter is greater than the wavelength, and is increasingly suppressed with increasing bulk absorption coefficient. Equations (2.148) and (2.149) for a slab particle were obtained under the assumption of negligible interference. If we had included interference we would have obtained oscillations in the absorption cross section of a slab, interpreted as a consequence of interference between multiply-reflected waves. The greater the value of Kbd, the more these waves are damped (attenuated), and the more the interference pattern disappears.

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