Illustrated Physics
physics, drawn
Concept

Absorption — where it appears

The removal of energy from a wave by the medium it passes through, turning it into heat or into an excited state of the matter. Its size at each frequency fixes the refraction at every other frequency through the Kramers–Kronig relation, so a substance that bends light must swallow some of it somewhere.

Named by 19 essays across 6 fields — each of them below, with the objects they name alongside it.

Three geometries, three exponents. Amplitude against distance for a wave spreading in one, two and three dimensions, on logarithmic axes, over 3 decades. The same power crosses every surface round the source, so the intensity falls as one over the area of that surface and the amplitude as one over its square root. The fitted slopes are 0.000 along a line, -0.500 over a cylinder, -1.000 over a sphere, each fitted by least squares to the drawn curve rather than written on it. A ripple on water is the middle case and a sound in a room is the last, which is why a ripple stays visible so much further than a shout carries.

How a wave thins out

A wave gets weaker with distance for two quite different reasons, and only one of them is a loss. Geometry alone fixes the first exactly — three exponents for three dimensions, with nothing about the medium in them — and whatever is left over is the medium eating the wave.

waves · Wave motion Water's permittivity across six decades of frequency. The real and imaginary parts of water's permittivity against frequency, on logarithmic axes, from 10 to 16 in powers of ten hertz. ε′ begins at 80.1 — the number a textbook prints — falls through the Debye relaxation near twenty gigahertz, is dragged down again by the librational, bending and stretching bands of the molecule, and settles at 1.777 in the visible, whose square root is 1.3330: water's refractive index. The identity n = √ε_r is exact and is an identity between two numbers taken at the same frequency; using the static permittivity in it predicts an index of 8.9 and is the most instructive wrong answer in the subject. The curve is a Debye term and four Lorentz oscillators, with two parameters solved so that the two plateaus are the measured ones rather than fitted by eye.

The constant that depends on how fast it is asked

Water's relative permittivity is 80.1. Its refractive index is 1.333, and the square of 1.333 is 1.777. The identity n = √ε_r is exact, the two numbers differ by a factor of forty-five, and nothing is wrong with either — because a permittivity is a function of frequency and the two measurements were made eight orders of magnitude apart.

electromagnetism · Dielectrics What a thermal camera reads off a shiny surface. The temperature a camera calibrated for a black body reports, against the emissivity of the surface it is pointed at, for surfaces truly at 323 K, 373 K, 473 K in a room at 293 K. Every curve begins at the true temperature when the emissivity is one and ends at the room's temperature when it is zero, because a surface that emits nothing reflects everything and the camera is then looking at the room. polished aluminium at 0.05 reads 312 K; stainless steel, oxidised at 0.8 reads 451 K; matt black paint at 0.95 reads 468 K; human skin at 0.98 reads 471 K for a surface truly at 473 K. That is not a fault in the instrument. It follows from Kirchhoff's law — a poor emitter is a poor absorber and therefore a good reflector — so the shortfall in a shiny surface's own glow is made up almost exactly by whatever is reflected in it, and no measurement of the light leaving a surface can separate the two without knowing the emissivity in advance.

The glow that says nothing about the surface

A thermal camera pointed at a saucepan of boiling water reads a hundred degrees. Pointed at a polished aluminium block at the same temperature it reads about twenty-five, and the instrument is working perfectly. What it is measuring is emissivity as much as temperature — and a surface's emissivity is forced to equal its absorptivity, at every wavelength and every angle, by an argument with no physics of matter in it at all.

thermodynamics · Blackbody N when the phases are random, N² when they are not. Scattered intensity against the number of scatterers, both logarithmic, for two ways of adding the same amplitudes. The lower curve averages 400 draws of N unit amplitudes with independent random phases and grows as N^1.000; the upper one adds them in phase and grows as N². At 3000 scatterers the two differ by a factor of 2921. Nothing about the scatterers is different between the two — same number, same strength, same wavelength. Only the arrangement is, and it is worth three decades here.

Why a litre of water is not blue for the reason the sky is

The same molecules that make the sky blue also make the refractive index of air, and the two numbers agree because the sideways sum has random phases and the forward one does not. Condense those molecules into a liquid and the sideways sum collapses by a factor of sixteen — and what is left is thirty-four times smaller than the absorption that actually colours the water.

optics · Scattering 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.

waves · Attenuation Absorption and refraction, drawn as one function. The real and imaginary parts of a Lorentz oscillator's susceptibility against frequency, in units of the resonance. The imaginary part is the absorption: a symmetric line centred on the resonance, with a full width at half maximum equal to the damping — 0.05, 0.12, 0.30 here. The real part is the refraction, and it is what the same medium does to the speed of light. The two curves are not two facts about the medium: either one determines the other completely, by an integral over all frequencies, and that is a consequence of the medium responding after it is asked rather than before. Between x = 0.97 and x = 1.02 the refraction runs the wrong way — the index falls as the frequency rises, which is anomalous dispersion — and that region is exactly the width of the absorption line. Away from the line the index rises with frequency, which is ordinary dispersion and is why a prism separates colours in the order it does: every transparent material is on the low-frequency tail of an ultraviolet absorption it is not otherwise showing. A narrower line is a taller one, because the area under the absorption is fixed by how many electrons there are and by nothing else.

The answer that cannot come first

A medium absorbs at some frequencies and bends light at all of them, and those look like two separate facts to be measured separately. They are one fact. The requirement that a material respond after it is asked rather than before ties the absorption at every frequency to the refraction at every other, by an integral — and a material given the two halves independently answers before the question arrives.

waves · Attenuation Why only a sideways scattered wave takes anything away. The transmitted amplitude behind a thin scatterer, drawn as a phasor: the incident wave of unit length along the axis, plus a forward-scattered wave of length 0.12 at 0°, 60°, 90°, 150°. What a detector reads is the square of the total length. A scattered wave along the incident one lengthens or shortens the sum in proportion to itself; one at right angles changes the length only in second order, because a small perpendicular addition to a long vector barely alters its length. So a scatterer that removes energy from the beam at first order must scatter forward with a component perpendicular to the incident wave, and the size of that component is the whole extinction — which is the optical theorem.

Everything a scatterer removes, from one direction

How much light a particle takes out of a beam — by scattering it anywhere at all, and by absorbing it — is fixed entirely by what it does in the forward direction, where its scattered wave cannot be told apart from the incident one. The mechanism is interference, and it also gives the refractive index.

optics · Scattering The temperature a planet ought to be. Each body's measured surface temperature against the temperature at which it would radiate away exactly the sunlight it absorbs — computed from two numbers, the sunlight reaching it and the fraction it reflects, with no other property of the body used. Points on the diagonal are bodies the one-line argument gets right. the Moon: balance 270 K, surface 250 K, −20 K with no atmosphere; Mercury: balance 433 K, surface 340 K, −93 K with no atmosphere; Mars: balance 210 K, surface 210 K, +0 K; Earth: balance 255 K, surface 288 K, +33 K; Venus: balance 227 K, surface 737 K, +510 K. The airless bodies fall below the line rather than on it, and the reason is the fourth power: a surface running from noon heat to night cold radiates like its hottest parts and averages like its coldest, so a mean thermometer reading is lower than the temperature that matches the emitted flux. Mars, whose atmosphere is thin and whose surface is nearly isothermal by comparison, sits on the line. Everything with a substantial atmosphere sits above it, by tens of kelvin on Earth and hundreds on Venus, and always in the same direction. Nothing here explains why. What the figure fixes is the size and the sign of what has to be explained.

The height a planet is seen from

A body in sunlight settles where it radiates away what it absorbs, and that takes two numbers and one line of arithmetic. It gets the Moon right and Earth wrong by thirty-three kelvin. The correction is not that the atmosphere traps heat but that it moves the level space sees the planet from, and the rest is done by a lapse rate that is not a radiative quantity at all.

thermodynamics · Blackbody A wave with two directions in it. Light at 60° entering silver, 550 nm, whose index is 0.055 + 3.32i. Phase matching along the boundary fixes the transmitted wave's tangential wavenumber and leaves the normal one to the medium, which supplies a complex answer. The real part decides where the phase goes and the imaginary part where the amplitude does, and they are not the same direction: the surfaces of constant amplitude are parallel to the interface, because the decay is entirely into the metal, while the surfaces of constant phase are tilted by 86.5° from the normal. There is therefore no single refracted angle to quote. The familiar picture, in which one set of parallel planes carries both, requires the absorption to be exactly zero.

The angle that is two angles

Snell's law survives a complex index by giving a complex answer, and a complex angle is not an angle. What the phase-matching argument actually fixes is the tangential wavenumber, and when the medium absorbs, the surfaces of constant phase and the surfaces of constant amplitude stop being parallel. In silver at 550 nanometres the phase fronts run within four degrees of the surface while the amplitude decays straight into it.

optics · Refraction A beam of sound has a weight. The force a fully absorbed acoustic beam exerts, against its power, for 3 media. It is the power divided by the speed of sound and nothing else — a watt in water gives 675 micronewtons, which is the weight of 68 milligrams, and a watt in air gives 2.9 millinewtons because the sound is slower there. This is not an analogy with light: it is the same statement, that a wave carrying energy carries momentum, with a much smaller speed in the denominator. The consequence is that acoustic power is measured by weighing. A radiation-force balance — an absorbing target on a laboratory balance, with the transducer beneath it — is the primary standard for ultrasonic output, and every therapeutic and diagnostic transducer is calibrated against one.

Where the loudness goes

An absorption coefficient removes energy from a wave, and energy removed has to appear somewhere. It appears twice, from the same coefficient: as heat, and as momentum. So a beam of sound has a weight — a watt absorbed in water weighs sixty-eight milligrams — and acoustic power is measured by putting an absorber on a balance. The ratio of the force to the heating contains no intensity at all.

waves · Attenuation 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.

waves · Attenuation The ripple that sits on top of an absorption edge. The Cu K absorption of copper foil, 293 K through its edge and for seven hundred electronvolts above it, with the smooth atomic background it would have if the absorbing atom were alone drawn beneath it. The difference between the two is the fine structure: a modulation reaching 11 per cent, dying away as the photon energy rises, and entirely absent from a free atom. It is there because the ejected electron is a wave that the neighbouring atoms scatter back onto the atom that emitted it, so the absorption depends on whether the returning wave arrives in step with the outgoing one — which depends on the distance to the neighbour and on nothing else about the sample.

The ripple that counts the neighbours

An absorption edge is drawn as a step and it is a step with a ripple on it — a modulation of eleven per cent in copper, five in a zinc site buried in a protein. The ripple is the ejected electron's own wave, scattered back onto the atom that emitted it, so its period is a distance. It is the only way of measuring where an atom's neighbours are that does not need a crystal.

waves · Attenuation 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.

waves · Attenuation The innermost electron of an atom with Z = 79, with and without relativity. The probability of finding the innermost electron at each distance from a nucleus of charge 79 (Au), with the distance in units of a₀/Z so that the non-relativistic curve is the same for every element. The dashed curve is Schrödinger's, r² e^(−2Zr/a₀). The solid curve is Dirac's exact ground state, r^(2γ) e^(−2Zr/a₀) with γ = √(1 − (Zα)²) = 0.817. It is pulled in towards the nucleus: its mean radius is 0.878 of Schrödinger's, a contraction of 12 per cent, and it has a sharper peak close in. Nothing about the nucleus has changed; the electron, moving at an appreciable fraction of the speed of light near it, simply behaves as a heavier particle, and a heavier particle is bound more tightly and more closely.

The metal that is yellow because it is heavy

Gold sits directly below silver in the periodic table, with the same arrangement of outer electrons, and by every non-relativistic calculation it should look like silver. It is yellow because its innermost electrons move at more than half the speed of light. The mass that motion adds pulls the s orbitals in, the tightened s level drops towards the d band beneath it, and the gap an absorbed photon must bridge shrinks from the ultraviolet into the blue. The same contraction is why mercury is a liquid.

quantum · Atomic structure 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.

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.

waves · Attenuation How much darker a surface looks under a film of water. The diffuse reflectance of a surface covered by a thin film of clear water (solid), against its reflectance dry, with the dry value for comparison (dashed): (1 − Rₑₓₜ) a (1 − Rᵢₙₜ)/(1 − a Rᵢₙₜ), where the film traps and returns light that falls outside its escape cone. A bright surface of reflectance 0.9 falls to 0.77; a mid-grey one of 0.5 to 0.32, 64 per cent of its dry value; a dark one of 0.2 to 0.11, 54 per cent. Dark surfaces darken proportionally most, because each return trip costs them most; and a coloured surface grows more saturated, because the colours it absorbs are absorbed again on every return while the colour it reflects survives.

The sand that darkens when it is wet

Wet sand is darker than dry sand, a wet stone darker than a dry one, a splashed pavement darker in patches, and a wet white shirt turns grey and then see-through. Water is clear and absorbs almost nothing, so it cannot be adding darkness. What it does is change where the light goes. A film of water on a surface traps light that has been scattered back up, returning most of it for another chance to be absorbed. Water in the pores between grains replaces air with something closer to the grains' own refractive index, so each grain turns light aside less, and light wanders deeper before it gets out. Both lengthen the path, and on a longer path more light is absorbed by the material that was there all along.

optics · Scattering Absorption below and above the gap. The light absorbed by a direct-gap crystal against photon energy measured from the gap, in units of the exciton's Rydberg, from Elliott's formula with each line broadened to a half-width of 0.06 Rydberg (solid), beside the absorption the same crystal would have if electron and hole did not attract (dashed), which rises from zero as the square root of the energy above the gap. Below the gap the attraction adds a series of lines at −1, −1/4, −1/9 … Rydberg, the hydrogen series scaled down, with strengths falling as 1/n³, crowding towards the gap. Above it the continuum is lifted: the attraction pulls each free pair together, and at the gap itself the absorption is a finite step rather than zero.

The hydrogen atom inside a crystal

Light absorbed by a semiconductor lifts an electron across the gap and leaves a hole behind, and the two attract. In an ordinary crystal they bind into a copy of the hydrogen atom, with the same series of levels and the same 1/n² — but the crystal screens their attraction and lends them light masses, so the copy is a few thousand times shallower and several hundred times larger. That rescaling is the whole theory, it predicts lines below the gap and a finite step at it where no attraction would give a smooth rise from zero, and it says which crystals keep their excitons at room temperature.

quantum · Bands A pond that is hottest at the bottom and does not turn over. Temperature, salt and density against depth in a salt-gradient solar pond: a fresh upper layer 0.3 m deep at 25 °C, a gradient layer 1.2 m thick in which the salt rises from 2 to 20 per cent and the temperature from 25 to 80 °C, and a storage layer of hot brine 2 m deep (shaded bands are the two mixed layers). The density rises steadily downward, from 1011 to 1122 kg/m³, so the hottest water is also the heaviest and stays at the bottom. The dashed line is the same temperatures with no added salt: the bottom water would then be 25 kg/m³ lighter than the top, and the pond would overturn.

The pond that is hottest at the bottom

Water heated from below rises, which is why a pond in the sun is never much warmer than the air: the heat is carried to the surface and lost there. Put a metre of salt gradient in the way and the hot water at the bottom is also the heaviest, cannot rise, and keeps its heat, behind an insulating layer that sunlight passes straight through. Ponds built this way reach eighty degrees under an ordinary sky. The salt does not collect any extra light. What it buys is temperature.

fluids · Stratification Fringes that bend into hooks beside a line. The interference fringes seen in a spectrograph behind an interferometer with 10 cm of sodium vapour at 10¹⁴ atoms per cubic centimetre in one arm and a glass plate in the other, against wavelength across the D₂ line (horizontal) and position along the slit (vertical, in fringe spacings). Far from the line the plate tilts every fringe evenly. Near the line the vapour's dispersion sweeps each fringe up on one side and down on the other, and every fringe turns back on itself in a hook: the turning points sit −100 and 100 picometres from the line centre, the same for every fringe. The line itself, a few picometres wide, is a dark gap between them; the hooks lie far outside it, in vapour that is transparent.

The hooks that weigh an absorption line

A strong absorption line is black at its centre, and how black it is says almost nothing about how many atoms made it — double them and the line hardly darkens. The atoms are still there, and they still bend light: on either side of the line, where the vapour is transparent, its refractive index rises and falls by an amount proportional to their number. In 1912 Dmitry Rozhdestvensky found a way to read that index straight off a photograph. Interference fringes passed through a spectrograph curl into hooks beside the line, and the distance between the hooks, squared, counts the atoms — whatever the line's shape and however saturated its core.

optics · Dispersion

Named alongside it

The objects these essays reach for when they reach for this one.

AttenuationRefractive indexDispersionIntensityTransportAlbedoAnomalous dispersionBand gapBinding energyBlackbodyBohr radiusCausality

All concepts