Illustrated Physics
physics, drawn
Waves

The pigment that absorbs less for being packed

Dissolve a dye evenly in water and it absorbs a certain fraction of the light passing through. Gather the same dye into grains — the same molecules, the same amount, in the same volume — and it absorbs less. The light has not changed and neither have the molecules. The grains shadow their own insides, and the light that would have been caught by the shaded molecules slips through the gaps between grains instead. The effect is called packaging, and it is why the chlorophyll inside large algae is worth less per molecule than the chlorophyll in small ones, and why an ocean's colour does not say directly how much chlorophyll is in it.
14 min read 5 figures The shape decidesApproximations that lie

Assumes: The distance that takes the treble out · The steps in an absorption curve

The distance that takes the treble out set up absorption as a fixed loss per unit distance, so that intensity falls exponentially through a uniform medium — the law of Beer and Lambert. The answer that cannot come first tied absorption at one frequency to refraction at all others, where the loudness goes followed the absorbed energy into heat and momentum, and the steps in an absorption curve, the ripple that counts the neighbours and below the gap, where there is nothing to absorb looked at what the absorption coefficient itself is made of.

Every one of those treated the absorber as uniform: a coefficient that is the same everywhere in the medium, times the distance. Almost nothing that absorbs light in nature is uniform. Pigments sit in cells, cells sit in water, grains of soot sit in air, and ink sits in fibres. The same amount of absorber distributed differently absorbs a different amount of light, and always less when it is gathered into lumps than when it is spread evenly. The rule behind it is one of the simplest in mathematics, and it has consequences from the colour of the sea to the design of a leaf.

The average of an exponential

Consider a layer holding a fixed amount of absorber. Spread evenly, it has an optical depth τ\tau everywhere, and it transmits e−τe^{-\tau}. Now gather the same absorber into patches covering only part of the area, leaving the rest clear. Where the patches are, the optical depth is larger; where they are not, it is zero; and the average optical depth is unchanged.

The same absorber, spread evenly and gathered into patches. The fraction of light transmitted by a layer holding a fixed amount of absorber, on a logarithmic axis, against the optical depth it would have if spread evenly, for the absorber spread evenly (solid) and gathered into patches covering 50 per cent and 10 per cent of the area, the rest left clear. At an average optical depth of 2 the even layer transmits 13.5 per cent; with the absorber in patches over half the area, 50.9 per cent; over a tenth, 90.0 per cent — almost all of it through the gaps. The average of e^(−τ) over a layer is never less than e^(−average τ): Jensen's inequality, which holds for any convex function. Beer's law describes a uniform absorber and overstates the attenuation of every other kind.
Fig. 1 The fraction of light transmitted by a layer holding a fixed amount of absorber, on a logarithmic axis, against the optical depth it would have spread evenly: spread evenly (solid), and gathered into patches over half and a tenth of the area (dashed). At an average optical depth of 2: 13.5 per cent, 50.9 per cent and 90.0 per cent.

The figure computes the transmission. At an average optical depth of two, the even layer passes 13.5 per cent of the light. With the absorber gathered onto half the area, the patches are twice as dark and transmit almost nothing, but the clear half passes everything, and the layer as a whole transmits 51 per cent. Gathered onto a tenth of the area, it transmits 90 per cent — nearly all of it through the gaps. As the absorber is packed more tightly, the transmission approaches the fraction of the area left clear, however much absorber there is.

The mathematics is Jensen’s inequality. The exponential e−τe^{-\tau} is a convex function — it curves upward — and for any convex function the average of its values exceeds its value at the average. So the mean transmission of a patchy layer, the average of e−τe^{-\tau} over the area, is always at least e−τˉe^{-\bar\tau}, with equality only when the layer is uniform. Beer’s law describes one arrangement of absorber out of all possible arrangements, and it describes the one that absorbs most.

When Beer’s law is right

It is worth being clear about why Beer’s law holds when it does, because the reason is also the condition for its failure. A dye dissolved in water is patchy too: its molecules are points, scattered at random, with empty water between them. What makes the solution uniform for the light is scale. Each molecule is far too small to shade itself or its neighbours noticeably — its own optical depth is a minute fraction of one — and the number of molecules in any region the light passes through fluctuates by only the square root of their number. The optical depth is effectively the same everywhere, and Jensen’s inequality, whose force depends on how much the optical depth varies, has nothing to act on.

So the law holds when the absorbers are small and faint on the scale of the light’s decay: when no single grain, cell or clump of absorber is itself optically thick. The condition is exactly the one the package-effect index measures. A pigment in cells with ρ′\rho' well below one obeys Beer’s law with the dissolved coefficient; one in cells with ρ′\rho' of order one or larger does not, and the absorption coefficient that fits the measured attenuation is smaller than the pigment’s own. The distance that takes the treble out needed a fixed loss per unit distance; a suspension of dark grains has a loss per unit distance that depends on how the absorber is divided up, and dividing the same absorber more finely always increases it towards the dissolved value.

The same reasoning explains a practical trap in measuring the concentration of something by its absorbance. For a dissolved absorber, absorbance is proportional to concentration, and a calibration made at one concentration serves at all. For a packaged absorber, adding more cells of the same kind raises absorbance in proportion, but making each cell darker — more pigment per cell — raises it less than in proportion, and a measurement that assumed otherwise would underestimate the pigment in the darker cells. Which of the two has happened cannot be read from the absorbance alone.

Grains that shade themselves

Pigment inside cells is patchy in three dimensions. Each cell is a small sphere of concentrated absorber, and light crossing it is dimmed on the way through, so the pigment on the far side sees less light than the pigment on the near side.

The shadow inside a dark cell. The intensity of the light along the diameter of an absorbing sphere lit from one side, as a fraction of the light arriving, against position across the sphere, for spheres of optical thickness ρ′ = 0.3, 3.0, 10.0 across their diameter. In a faint sphere the light barely dims and every molecule sees nearly the full light. In a dark one the light is almost gone before the middle: at ρ′ = 10, the far half of the diameter receives less than 0.7 per cent. The molecules in that shadow are made, carried and maintained, and absorb almost nothing. That is the package effect seen from inside a single cell, and why a sphere's absorption falls behind the absorption its pigment would have if spread out.
Fig. 2 The light remaining along the diameter of an absorbing sphere lit from one side, for optical thicknesses across the diameter of 0.3, 3 and 10. In the faint sphere every molecule sees nearly the full light; in the darkest, the far half of the diameter receives less than 0.7 per cent.

The figure follows the light through spheres of three optical thicknesses. In a faint sphere the light barely dims, and every molecule sees almost the full intensity. In a dark sphere the light is nearly gone before the middle: the molecules in the far half are in the shadow of the near half and catch almost nothing. They are there, they were made and are maintained, and they add almost nothing to what the sphere absorbs. A sphere as a whole can never absorb more than the light falling on its cross-section, however much pigment it contains.

In 1956 Louis Duysens, studying the absorption of photosynthetic pigments in living cells, worked out the absorption of such a sphere averaged over every chord through it. The result depends only on the sphere’s optical thickness across its diameter, ρ′=acd\rho' = a_c d, the internal absorption coefficient times the diameter.

How much pigment is wasted by being packed into grains. The absorption of pigment packed into spheres, relative to the same amount of pigment spread evenly through the same volume, against ρ′, the optical thickness of one sphere across its diameter — the absorption coefficient inside it times its diameter — on a logarithmic axis. Small or faint spheres absorb as well as the dissolved pigment: at ρ′ = 0.1 the ratio is 0.963. Large or dark ones absorb less: 0.71 at ρ′ = 1, 0.15 at 10, and it tends to 3/2ρ′, because the far side of a dark sphere sits in the shadow of its near side and the pigment there sees little light. Nothing is lost: the pigment is the same, and so is the light. The light that would have been absorbed by the shaded pigment goes through the gaps between the spheres instead.
Fig. 3 The absorption of pigment packed into spheres relative to the same pigment dissolved, against the optical thickness of one sphere, ρ′=acd\rho' = a_c d, on a logarithmic axis: 0.963 at 0.1, 0.71 at 1, 0.15 at 10, tending to 3/2ρ′3/2\rho' for large, dark spheres (dashed).

The figure plots the ratio of the absorption of the packed pigment to the absorption the same amount would give dissolved, a quantity oceanographers call the package-effect index. For small or faint spheres it is one: the pigment behaves as if dissolved. As the spheres become optically thick the ratio falls, and for large dark ones it approaches 3/2ρ′3/2\rho', the ratio of the sphere’s cross-section to its volume, in absorption units: once a sphere absorbs nearly everything that hits it, adding pigment inside it adds nothing, and its absorption per unit pigment falls in inverse proportion to how much pigment it holds.

A spectrum that flattens

Because the effect depends on the optical thickness, and the optical thickness depends on wavelength, packaging changes the shape of a spectrum as well as its height.

The same pigment's colour, dissolved and packed into cells. The absorption spectrum of a pigment with two bands, at 440 and 675 nm, in the style of chlorophyll: dissolved (solid), and packed into cells of diameter 2, 10, 50 μm (dashed), each spectrum as the absorption per unit of pigment relative to the dissolved pigment's blue peak. The larger the cells, the lower the spectrum and the flatter its shape: the strongly absorbed wavelengths lose most, because they are the ones whose light is used up in the near side of each cell. The red band's height relative to the blue rises from 0.56 dissolved to 0.73 in 50 μm cells. The same molecules seen in cells of different sizes show different spectra, and an ocean colour measured from orbit has to allow for how big the phytoplankton are before it can say how much chlorophyll there is.
Fig. 4 The absorption spectrum of a two-band pigment in the style of chlorophyll, dissolved (solid) and packed into cells of 2, 10 and 50 μm (dashed), relative to the dissolved pigment’s blue peak. The larger the cells, the lower and flatter the spectrum: the red band’s height relative to the blue rises from 0.56 dissolved to 0.73 in 50 μm cells.

The figure takes a pigment with two absorption bands, a strong one in the blue and a weaker one in the red, in the manner of chlorophyll, and packs it into cells of three sizes. In small cells the spectrum is almost the dissolved one. In large cells both bands are lower, and the strong blue band is lowered most, because it is at the wavelengths where each cell is darkest that its interior is most shaded. The spectrum flattens: its peaks sink relative to its troughs, and the ratio of the red band to the blue rises. Duysens called this the flattening effect, and he used it to explain why the absorption spectra of living algae look different from the spectra of their extracted pigments, a puzzle that had suggested the pigments were chemically altered inside the cell. They were not; they were packed.

What size costs a cell

For a photosynthetic cell, packaging is a question of economy.

What a cell's size costs it in light. The absorption per unit of pigment of a suspension of cells, relative to the pigment dissolved, against cell diameter on a logarithmic axis, at the blue band (440 nm) and the red band (675 nm), with the same pigment concentration inside every cell. A cell of 1 μm uses its pigment almost fully: 0.98 at 440 nm. A cell of 20 μm gets 0.75 of the value at the blue band and 0.85 at the red; one of 100 μm, 0.32 and 0.48. Every pigment molecule a large cell makes is worth less to it than one a small cell makes, because more of its molecules sit in their neighbours' shadow. The same pigment is cheaper, per photon caught, in a small cell — one of the reasons the smallest phytoplankton dominate the nutrient-poor open ocean.
Fig. 5 Absorption per unit pigment of a suspension of cells, relative to the pigment dissolved, against cell diameter on a logarithmic axis, at the blue band (440 nm) and the red (675 nm), with the same pigment concentration inside every cell. A 1 μm cell: 0.98 at the blue band. A 20 μm cell: 0.75 and 0.85. A 100 μm cell: 0.32 and 0.48.

With the same concentration of pigment inside every cell, the figure plots how much light each unit of pigment catches as the cell grows. A cell a micrometre across uses its pigment almost fully. A cell of twenty micrometres gets three-quarters of the value at the blue band; one of a hundred, a third. Every molecule of chlorophyll costs a cell nitrogen, magnesium and energy to make, and in a large cell an increasing share of those molecules sit in their neighbours’ shadow. The smallest phytoplankton, a micrometre or two across, dominate the nutrient-poor open ocean, where every atom of nitrogen matters and light is shared among cells competing for it; larger cells are more common where nutrients are plentiful. Packaging is one reason among several, and a quantitative one: at the same pigment content, the small cell simply catches more light.

Large cells and leaves have ways round it. Many algae adjust the concentration of pigment inside their cells to the light they receive, packing less where light is plentiful and packaging would waste pigment. Chloroplasts in leaves move: in bright light they line up along the cell walls parallel to the light, shading one another and deliberately lowering absorption to avoid damage, and in dim light they spread across the face of the cell to catch as much as possible. A leaf that changes its package effect by moving its chloroplasts changes the fraction of light it absorbs by several per cent within minutes.

A leaf shows the other side of the bargain. Its chloroplasts are packed into cells stacked several layers deep, and the package effect, together with the sieve-like gaps between chloroplasts, lets some of the light that the top layer would otherwise absorb pass to the layers beneath. Red and blue light, strongly absorbed, are mostly caught in the first cell layers; green light, weakly absorbed, penetrates deeper and drives a large share of the photosynthesis in the lower layers of a thick leaf. Measurements by Ichiro Terashima and colleagues in 2009 found that in strong white light green light can drive photosynthesis more efficiently than red, because it spreads the absorption through the whole depth of the leaf instead of saturating the top of it. A leaf reflects some green and so looks green, but the green it keeps is not wasted, and the packaging is part of why.

The colour of the sea

The most consequential use of the effect is in reading the ocean’s colour from orbit. Satellites measure the spectrum of sunlight leaving the sea, and the amount of chlorophyll in the upper ocean is inferred from how much blue light has been absorbed relative to green. The inference assumes a relation between chlorophyll and absorption, and packaging makes that relation depend on the size of the cells: the same chlorophyll in large cells absorbs less blue light, and a satellite algorithm tuned to a population of small cells underestimates the chlorophyll in a bloom of large ones. Since the 1980s, beginning with André Morel and Annick Bricaud, the package effect has been built into the algorithms that convert ocean colour to chlorophyll and so to estimates of the ocean’s primary production — about half of the photosynthesis on the planet.

The same accounting runs through other fields under other names. In the atmosphere, black carbon soot absorbs sunlight and warms the air; whether it is spread in fine particles or gathered into larger ones, or coated with other material, changes how much it absorbs per gram, and the uncertainty in that is one of the larger ones in estimating soot’s climatic effect. The cloud light has to walk through found light diffusing through a cloud of scatterers; in a cloud whose drops contain absorbing particles, where the particles sit — inside the drops, or between them — changes the cloud’s absorption in the same way.

Gaps in a shield

The same inequality works against anyone trying to block radiation rather than harvest it. A radiation shield is designed to present a large optical depth to the particles it must stop, and its performance is governed by its weakest paths, not its average. A shield with an average thickness of ten attenuation lengths transmits e−10e^{-10}, less than a hundred-thousandth, if it is uniform; if a small fraction of its area is thinner — a gap where two blocks meet, a duct passing through, a crack — the transmission through that fraction can dominate everything else. Shields for reactors and accelerators are therefore built with stepped joints and bent ducts, so that no straight line through them avoids the full thickness, and the phenomenon they guard against, radiation leaking through gaps, is called streaming. How far a neutrino gets found the mean free path to be one over the density times the cross-section; a shield with gaps has a mean free path that depends on how its density is distributed, and the gaps set it.

Clouds do the same to sunlight. A sky half covered by thick clouds and half clear lets through far more light than a sky uniformly covered by a thin cloud of the same total water, and climate models that average the cloud water over a grid square before computing its effect on sunlight get the wrong answer for exactly the reason in the first figure. Correcting for the patchiness of cloud within each grid square is a standard and still-debated part of how such models compute the radiation reaching the ground.

Where the model stops

The figures use the simplest model of packaging: spheres, all the same size, uniformly pigmented, absorbing but not scattering, far apart from one another so that each is lit only by the incident light. Real cells are not spheres, contain pigment in organelles rather than uniformly, scatter as well as absorb, and come in distributions of size. Each departure changes the numbers without changing the direction. Scattering in particular complicates the measurement, since a suspension that scatters removes light from a beam without absorbing it, and separating the two in a laboratory spectrophotometer requires collecting the scattered light, which the early measurements did not always do. The pigment spectrum is illustrative rather than that of any real chlorophyll, whose bands are broader and whose absorption is shared with accessory pigments.

Nor does the model say anything about where the absorbed energy goes inside a cell, which is the question that matters to the cell. A molecule in the shadow of its neighbours also receives light that its neighbours have absorbed and passed on as excitation energy, and photosynthetic cells are organised so that most absorbed photons end at a reaction centre regardless of which molecule caught them. The package effect concerns only how many photons are caught.

Still open: how the ocean’s cells are sized

Converting ocean colour into how much photosynthesis is happening depends on knowing how big the phytoplankton are, because size sets the package effect and the package effect sets how much chlorophyll a measured colour implies. Satellite methods that infer the dominant cell size from the spectrum itself — from the flattening this essay describes, along with other clues — are being developed and compared with measurements at sea, and they disagree with one another by amounts that matter for estimates of global production. How much of the variation in the relation between chlorophyll and ocean colour is packaging, how much is other pigments and how much is dissolved material in the water is still being disentangled one region and one season at a time.

The habit worth carrying away is to ask whether an absorber is spread or gathered before applying a law written for one that is spread. The average of e^(−τ) over any patchy layer exceeds e^(−average τ), so the same pigment absorbs less when packed into grains — 0.71 of its dissolved value in spheres one optical depth across, 0.15 in spheres ten across — and packing flattens its spectrum as well as lowering it. Beer’s law describes the arrangement that absorbs most, and nature seldom uses it.

Part 7 of 7

This essay is one argument about Attenuation. The others:

The edge that is a straight line, not a step. The absorption of gallium arsenide near its own band gap of 1.424 electronvolts, on a logarithmic axis, at 4 temperatures. Below the gap the absorption does not stop; it falls exponentially, along a straight line whose slope is an energy, and the straightness holds over several decades. Raising the temperature makes the line shallower — the tail reaches further below the gap — and the slope runs from 5.6 millielectronvolts at 10 kelvin to 7.5 at 300, a factor of 1.33. The lines pivot about a point just above the gap rather than rotating about nothing, which is what makes the slope a single number worth quoting. The band gap's own shift with temperature has been removed here, so that the fan is the tail's doing and not the gap's. Below the gap, where there is nothing to absorb Part 6 — A semiconductor is supposed to be transparent below its band gap, and it is not. The absorption falls exponentially instead, over seven decades, along a straight line whose slope is an energy of a few millielectronvolts — and the description that produces that line has no states in the gap at all. What the slope measures is how much the gap itself is moving about.

Where this goes next

An f² law that is right in shape and out by 30× in size. Two absorption curves for air against frequency, both logarithmic, in decibels per kilometre. The lower one is the classical Stokes–Kirchhoff result computed from air's viscosity and thermal conductivity alone, and it goes as f^2.000 — exactly two, because the loss per cycle is fixed and the number of cycles per metre is proportional to the frequency. The upper one is the measured atmospheric absorption at 20 °C and 50 per cent humidity, which fits f^1.42 and is 30 times larger at 1 kHz and 211 times at 125 Hz. The excess is not a correction to viscosity: it is nitrogen and oxygen storing energy in vibration and giving it back late, at a rate the water vapour sets, and it is the mechanism that actually removes the treble from a distant sound. The distance that takes the treble out Spreading treats every frequency alike; absorption does not. The loss per cycle is roughly fixed and the number of cycles per metre goes as the frequency, so absorption climbs as f² and a sound gets duller with distance as well as quieter — which is the whole account of why a nearby thunderclap cracks and a distant one rumbles. The edge that is a straight line, not a step. The absorption of gallium arsenide near its own band gap of 1.424 electronvolts, on a logarithmic axis, at 4 temperatures. Below the gap the absorption does not stop; it falls exponentially, along a straight line whose slope is an energy, and the straightness holds over several decades. Raising the temperature makes the line shallower — the tail reaches further below the gap — and the slope runs from 5.6 millielectronvolts at 10 kelvin to 7.5 at 300, a factor of 1.33. The lines pivot about a point just above the gap rather than rotating about nothing, which is what makes the slope a single number worth quoting. The band gap's own shift with temperature has been removed here, so that the fan is the tail's doing and not the gap's. Below the gap, where there is nothing to absorb A semiconductor is supposed to be transparent below its band gap, and it is not. The absorption falls exponentially instead, over seven decades, along a straight line whose slope is an energy of a few millielectronvolts — and the description that produces that line has no states in the gap at all. What the slope measures is how much the gap itself is moving about. An absorption that goes up when the photon gets harder. Mass attenuation coefficients for 4 materials against photon energy, both logarithmic, from tabulated measurements interpolated between. Away from a threshold every curve falls steeply — a harder photon is less easily absorbed, which is what makes X-rays penetrating. At a K edge the curve jumps upward: the photon has become able to eject an innermost electron, a whole new population becomes available, and the absorption multiplies by 4.4 for iodine at 33.17 keV and 4.0 for lead at 88 keV. That is a discontinuity in a material property as a function of frequency, and nothing in the classical description of absorption produces one. It exists because absorption at these energies is a transition of a bound electron rather than a loss in a medium. The steps in an absorption curve Every absorption law up to this point is smooth — a fixed loss per cycle, a relaxation, a power of frequency. Take the photon energy up to where it can eject an electron from an atom and the curve acquires steps, and they go the wrong way: the material becomes suddenly more opaque to a harder photon. Iodine absorbs four times as strongly at 33.4 keV as at 33.0, and that discontinuity is why it is injected into people.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

AbsorptionBeer lambert lawChlorophyllJensen inequalityOptical depthPackage effectPhytoplankton