Figure 2.1: An absorbing, plane-parallel medium can be imagined to be made up of N slices, each of thickness Ax. A monodirectional beam with irradiance F0 is incident on this medium. The transmitted irradiance Fj at a distance jAx into the medium decreases with increasing j because of absorption in all preceding layers.

It is more convenient to write this as

This is the irradiance at a distance Ax from the origin. At a distance 2Ax, by the same argument, the irradiance is

An implicit assumption in going from Eq. (2.2) to Eq. (2.3) is that transmission by the two slabs is independent, and hence transmission by each can be multiplied to obtain transmission by the two combined (this assumption is scrutinized in Sec. 2.4). The pattern now should be clear: after transmission over a distance x = NAx, the irradiance is

Fn = Fo(l - kx/n)n. Now take the limit as N becomes indefinitely large keeping kx constant: F = lim F0(l - kx/N)n, n—>oo

where we omit the subscript on F. Do you recognize this limit? It is one of the many ways of defining the exponential function:

From Eqs. (2.5) and (2.6) we obtain the law of exponential attenuation (by absorption)

This law is also valid for attenuation by scattering if multiple scattering is negligible (see Sec. 5.2.3).

Wavelength (|im)

Figure 2.2: Absorption length (inverse absorption coefficient) of pure ice and liquid water from UV to IR. The data for liquid water were taken from Querry et al. (1991), those for ice from Warren (1984).

Wavelength (|im)

Figure 2.2: Absorption length (inverse absorption coefficient) of pure ice and liquid water from UV to IR. The data for liquid water were taken from Querry et al. (1991), those for ice from Warren (1984).

The various names by which this law is called exemplify Stigler's law of eponymy: "No scientific discovery is named after its original discoverer." The law of exponential attenuation, often called Lambert's law, was first stated in Pierre Bouguer's Essay on the Gradation of Light (1729), although we could find no evidence that he established it experimentally. Most chemists call it Beer's law, which is wide of the mark given that Bouguer preceded Beer by more than 100 years, Beer did not discover an exponential law of attenuation with distance, and, in fact, did not explicitly state any exponential law. The most we can say is that by reworking Beer's data one can unearth what he did not: an exponential attenuation law for solutions of fixed thickness but variable concentration of the absorbing solute.

Because kx is dimensionless, k must have the dimensions of inverse length, and hence 1/k must have the dimensions of length. The e-folding length is the distance over which a monodirectional beam is attenuated by a factor 1/e. Because this term is a bit of a mouthful we prefer absorption length for 1/k. As a general rule, whenever any physical quantity can be expressed as a length it is wise to do so. Lengths are easier to get a feel for, more so than time, even mass. We can both touch and see lengths.

The absorption coefficient (absorption length) of a material depends on wavelength, often varying by as much as a factor of 1010, an example of which is as near as a faucet. Figure 2.2 shows the huge range of absorption lengths for liquid water and ice from 0.1 pm to 10 pm. The absorption length for (pure) liquid water and ice over the visible spectrum, shown on a linear scale in Fig. 5.12, is greatest in the blue, least in the red. From this figure it is evident why water in a drinking glass is not noticeably colored: the dimensions of glasses are small compared with the absorption length of water over the visible spectrum. This curve also shows that transmission of white light over several meters through water is sufficient to attenuate much more of the long-wavelength components than the short. Indeed, with increasing distances the only component that survives corresponds to the greatest absorption length. Water is intrinsically blue and needs no impurities to make it so. In Section 5.3.1 we explore the consequences of this to the observed colors of natural ice bodies such as glaciers, icebergs, ice caves, icefalls, and even holes in snow.

2.1.1 Absorptivity and Absorption Coefficient: A Tenuous Connection

Absorptivity and absorption coefficient are not the same. In the first place, the former is dimensionless whereas the latter has the dimensions of inverse length, which itself ought to signal caution. More to the point, the connection between them is sometimes tenuous at best. Consider, for example, radiation incident on bodies sufficiently thick that transmission by them is negligible. We now can attach a more precise meaning to "sufficiently thick": much thicker than the absorption length at the wavelength of the radiation. With this assumption, the absorptivity of the body is 1 minus its reflectivity. How does the reflectivity of the body depend on its absorption coefficient? For many materials over many wavelength intervals, reflectivity changes hardly at all even with huge increases in absorption coefficient. And if there is a change, it is likely to result in a decrease in absorptivity (see Prob. 7.20). For example, the absorption coefficient of metals such as silver and aluminum is usually huge compared with that of insulators such as quartz and salt, a million times or more, especially at visible and near-visible wavelengths. And yet reflectivities of metals are high, and hence their absorptivities are lower than those of insulators. Finally, there is this important distinction to be kept in mind: absorptivity is a property of a body whereas absorption coefficient is a property of a material.

2.1.2 Absorptance and Absorbance: More Room for Confusion

As if distinguishing between absorptivity and absorption coefficient were not difficult enough, we also have to keep these terms separate from the near homophones absorptance and ab-sorbance. Although absorptance is sometimes used as a synonym for absorptivity, this is not recommended given that we try to restrict terms ending in "ance" to amounts of radiant power. For example, emittance, which can be looked upon as shorthand for emitted irradiance, is radiant power per unit area. Similarly, absorptance can be looked upon as shorthand for absorbed irradiance.

Absorbance, a term widely used by chemists, is the negative logarithm (base 10) of the transmissivity of a sample of an absorbing material (usually liquid) in a container (cell). Because a transmissivity less than 1 is a consequence both of reflection by the container and absorption by its contents, the apparent absorbance can be nonzero even with an empty cell or one filled with a negligibly absorbing liquid. To correct for reflection, the absorbance of the cell is subtracted from the apparent absorbance to obtain that of the sample. To good approximation the transmissivity of the sample in the cell often is where h is the sample thickness, k its absorption coefficient, and T0 the transmissivity of the cell without the sample in place. Take the negative logarithm of both sides of Eq. (2.8) to obtain

The left side of this equation is absorbance corrected (approximately) for reflection by the cell. With this correction, absorbance measured by chemists is, except for a constant factor, the absorption optical thickness (kH) of the sample (see Sec. 5.2).

The exponential attenuation law Eq. (2.7) strictly holds only for monochromatic radiation because k depends on frequency. Any real source is distributed over frequency, and hence the integrated transmitted irradiance is j F0(u)exp(-Kx) du, (2.10)

where F0(u) is the spectral irradiance (irradiance per unit frequency interval) at x = 0. The limits of integration can be anything, and for simplicity we do not express k as a function of frequency. Although each spectral component of the incident beam is attenuated exponentially with distance, the integrated beam is not. And this is true even if the incident irradiance does not depend on frequency. This basic property of exponential attenuation has sometimes been forgotten, resulting in errors.

To show that the sum (integral) of exponentials is not, in general, an exponential, we assume that exp(-Kx) = j exp(-kx) du, (2.11)

where K is independent of x and u. Differentiate both sides of this equation with respect to x to obtain f k exp(—kx) du

The right side of this equation, the average of k weighted by a normalized exponential, depends on x, in general, and hence we contradict our original assumption that K is independent of x, which therefore must be false. This is a proof by contradiction: assume something is true, explore the consequences, and when a contradiction results, the original assumption must have been false. This, by the way, is a variation on a theme in Section 1.4.2: the average of a function is not necessarily the function of the average.

When k is independent of frequency over the range of interest, the integrated irradiance does decrease as a simple exponential. And when kx c 1 for the frequency range and distances of interest we can approximate the exponential in Eq. (2.10) by the first two terms in its Taylor series expansion to obtain the following approximation for the transmitted irradiance:

This can be written as

where the integrated irradiance at x = 0 is

and the average absorption coefficient is

We obtained Eq. (2.14) by approximating an exponential by the first two terms in a Taylor series, but we can do the reverse, approximate the first two terms in a series by an exponential:

which yields approximate exponential attenuation for the integrated irradiance:

But in general, this equation is not correct.

Nature is not so cooperative as to provide us only with media having uniform properties. The absorption coefficient k can vary from point to point. The medium depicted in Fig. 2.1 can be nonuniform and subdivided into N equal slices so thin that in each of them the absorption coefficient is nearly constant. Transmission by the jth slice is

where Fj is the irradiance incident on the jth slice, Fj+1 is the irradiance transmitted by this slice, Kj is the absorption coefficient k(x) at some point in the interval (xj, xj+1), and

Ax = x/N .It follows from this that transmission over the distance x is ( N \

The limit of the sum in this equation is the integral

and hence the exponential attenuation law for a nonuniform medium (spatially varying absorption coefficient) is

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