Illustrated Physics
physics, drawn
Concept

Screening — where it appears

The reduction of a charge's field by the other charges that gather around it, so that a distant probe sees much less than a near one. In an atom it removes the accidental degeneracy of the hydrogen levels: an orbital that penetrates the core sees nearly the full nuclear charge for a small fraction of its time, and 0.88 per cent is enough to move 3s half a hartree below 3d.

Named by 10 essays across 5 fields — each of them below, with the objects they name alongside it.

One level, split by the speed of the electron in it. Hydrogen's n = 2 level on the left at the scale of the whole spectrum, and the same level on the right at a scale ten thousand times finer. The gross structure puts it 3.4014 electronvolts below the ionisation limit; the relativistic and spin–orbit corrections then split it into a lower pair, j = ½, and an upper, j = 3/2, separated by 45.3 microelectronvolts. That gap is 1.331e-5 of the level's own binding energy, which is exactly α²/4 — so the size of the splitting is a measurement of how fast the electron is going, α being its speed in units of c on the innermost orbit. The same physics in sodium is larger by four orders of magnitude: its two D lines at 588.995 and 589.5924 nanometres differ by 2.133 millielectronvolts, which is 1.01e-3 of the transition energy, because the outer electron of sodium penetrates to where the nuclear charge is far from screened and the correction goes as the fourth power of the charge it sees.

The line that is really two

Sodium's yellow line is two lines six-tenths of a nanometre apart, and the gap is not a property of the light. It is a splitting of the atom's own level, caused by the electron's magnetic moment sitting in the field it sees because it is moving — and its size, relative to the level it splits, is exactly α²/4.

quantum · Atomic spectra Where 4s goes below 3d. The energies of five orbitals of a nucleus of charge 19, solved in a screened Coulomb potential, against how far out the screening extends. Nothing about the ordering is assumed: each level is found by integrating the radial equation outward and bisecting on the energy until the solution has the number of nodes that state is supposed to have, and the same solver returns hydrogen's −1/2n² to 7.3e-12 hartree when the screening is switched off. With no screening the three n = 3 levels would lie on top of one another. With screening they separate, always in the same order — s lowest, then p, then d — because a low angular momentum has no centrifugal barrier keeping it out of the core, so it spends part of its time inside the other electrons where the nuclear charge is unscreened. And at a screening length of 0.41 bohr the 4s level crosses below 3d, which is the fourth row of the periodic table: potassium and calcium put their electrons in 4s before anything goes into 3d, so they are an alkali metal and an alkaline earth rather than the first two transition metals. The model is a caricature — one screening length for every electron, no self-consistency, and no exchange — and it gets the ordering right anyway, which is the argument that the ordering is about penetration and nothing subtler.

The order the shells fill

In hydrogen every state with the same principal number has the same energy, and 4s and 3d differ by nothing. In every other atom they do not, and 4s is below 3d — which is why potassium is an alkali metal rather than the first transition metal. The difference is a small piece of probability that an s orbital has inside the innermost shell and a d orbital does not.

quantum · Atomic structure The charge a plasma hides. The potential around a charge in a plasma, as a fraction of the bare Coulomb potential at the same distance, against distance measured in screening lengths. The electrons crowd toward the charge and the ions move away until the rearrangement cancels the field, and what is left falls as exp(−r/λ_D) on top of the ordinary 1/r. At one screening length the potential is already down to 37 per cent of the bare value, at three to 5 per cent, and at ten to 4 × 10⁻⁵ — so a charge in a plasma is invisible beyond a few λ_D, and the long range of the Coulomb force, which is what makes electrostatics awkward everywhere else, is simply gone. The screening length at 10¹¹ m⁻³ is 3.8 mm at 300 K, 6.9 mm at 1000 K, 21.8 mm at 10000 K, rising as the square root of the temperature because a hotter electron is harder to hold in place. Drawn this way the three curves coincide exactly: the shape is universal and the only thing a plasma's density and temperature decide is the length written on the axis.

The long-range force that does not reach

The Coulomb force falls off as slowly as gravity does, which is what makes electrostatics awkward — every charge is in principle in contact with every other. Put the same charge into a plasma and it becomes invisible beyond a few millimetres, because the mobile charges around it rearrange until its field is cancelled. What is left is a screened potential with a definite range, and that range is what makes a plasma a plasma.

astrophysics · Plasma oscillation How much a wrap holds, against how many turns it is. The ratio of the two tensions a rope can hold across, against the number of turns it is wrapped, for coefficients of 0.1, 0.25, 0.5. The axis is logarithmic because the law is exponential, so each line is straight and its slope is the coefficient. At µ = 0.1 one turn multiplies by 1.9, two turns by 4 and three by 7 — so a person pulling with the strength of one arm holds a load that a small crane would be needed to lift. The practical consequence is the one a sailor states as a rule: turns are cheap and each is worth as much as the one before it, which is a statement about a constant factor rather than a constant force.

The part of the wrap that is actually gripping

The capstan equation gives the largest tension ratio a wrap can hold, and almost nothing spends its life at that limit. Below it the wrap divides in two: an idle arc doing nothing at all and an active arc creeping and carrying the whole exponential — and the division explains a belt's speed loss and its squeal.

mechanics · Friction Why it stops falling in. The energy of an electron confined to a region of radius r around a nucleus, as the sum of two terms with different powers: a confinement energy ħ²/2mr² that rises without limit as the region shrinks, and a Coulomb attraction −Ze²/4πε₀r that falls. At Z = 1 the sum is least at 52.92 pm, where it is -13.61 eV. Both numbers are found by searching the drawn curve and both agree with the Bohr radius over Z and minus Z² Rydbergs to a part in a million. Nothing was quantised to get them. The only quantum input is that confining an electron to a region costs kinetic energy, which is the uncertainty relation and nothing more.

Why an atom is the size it is

A tenth of a nanometre is not a measured constant of nature but the outcome of a competition: confining an electron costs kinetic energy, and the nucleus pays for confinement with attraction. Minimising the sum gives the number, and changing the masses moves it by four orders of magnitude.

quantum · Atomic structure The pressure a charged gel develops. The swelling pressure against the gel's fixed charge density, for several salt concentrations, on logarithmic axes. Each curve has two straight parts with different slopes, and both limits are checked against the solution rather than read off the plot. Where the fixed charge is small compared with the salt the pressure goes as the square of the charge divided by four times the salt: the bath's ions screen the fixed ones, and doubling the salt halves the pressure. Where the fixed charge dominates, every counterion it demands is an extra particle in the gel and the pressure goes as the charge itself. At 15 mM salt, a gel with 200 mM of fixed charge develops 427 kPa; At 50 mM salt, a gel with 200 mM of fixed charge develops 306 kPa; At 150 mM salt, a gel with 200 mM of fixed charge develops 150 kPa; At 500 mM salt, a gel with 200 mM of fixed charge develops 49 kPa. Cartilage carries a fixed charge of a couple of hundred millimolar from the sulphated sugars on its proteoglycans, sits in a bath of about 150 mM, and develops a swelling pressure of an atmosphere and a half — which is what holds a joint apart and carries the load across it.

The swelling a membrane cannot stop

Give the thing a membrane holds back an electric charge and the small ions that can cross are no longer free to distribute themselves. Two conditions — neutrality on each side, and equal chemical potential for the salt — fix where every ion goes, and they leave the charged side with more particles than the other. That excess is what holds a joint apart, and it is why a cell has to spend a third of its energy pumping.

fluids · Osmosis The colours a draining film runs through, and the end of them. The fraction of light a free soap film reflects, against its thickness, at three wavelengths — 450, 550, 620 nm — computed from the sum over multiple reflections rather than from the two-beam approximation. The three curves peak at different thicknesses, which is why a draining film runs through a sequence of colours as it thins. Below about 11 nm every curve is under a tenth of a per cent and the film looks black. At zero thickness the reflectance is exactly zero, checked before the figure is drawn: the two surfaces reflect equally and half a cycle out of step, so a film much thinner than a wavelength cancels itself. That is the whole of why a black film is black. Nothing is absorbing; the film is there and has simply stopped being able to interfere constructively at any visible wavelength — which means the blackness is a measurement, and a film that has gone black is known to be thinner than about a tenth of a wavelength without anything being measured directly.

The film that goes black before it bursts

A soap film drains, runs through every interference colour, and then stops reflecting anything at all. The black patch is not a hole and not a film about to break: it is the thinnest and most stable state the arrangement has, held apart by a pressure between its two surfaces that only exists at distances of nanometres.

fluids · Surface tension The half-life that depends on the electrons. Half-lives of 3 nuclides as neutral atoms and as bare nuclei, measured at a heavy-ion storage ring where fully stripped ions can be kept circulating for months. The changes are not corrections. ¹⁸⁷Re goes from 41.6 billion years to 32.9 — a factor of 1.26 billion — because a beta decay whose available energy is only 2.5 keV cannot put an electron into the continuum but can put one into an empty K orbital, and stripping the atom opens that channel. ¹⁶³Dy is stable as an atom and decays in 47 days as a bare nucleus, for the same reason. The rate is a property of the nucleus and of what surrounds it, and the rung below this one says otherwise.

The half-life that chemistry can change

The first rung of this ladder said a nucleus has no clock and that nothing outside it touches the rate. That is very nearly true and it is not exactly true. Electron capture takes an electron from the nucleus's own position, so its rate is proportional to how many electrons are there — which is chemistry, worth a per cent. And a nucleus stripped of every electron can gain a decay channel it did not have: rhenium-187 goes from forty-two billion years to thirty-three.

quantum · Decay 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 Most of a chain's counterions never leave it. The fraction of a charged rod's counterions lying within a distance r of it, against r in rod radii on a logarithmic axis, for a charge parameter ξ = 4.2 — the Bjerrum length of water, 0.7135 nm, over a charge spacing of 0.17 nm, which is DNA's. Each curve is the Poisson–Boltzmann solution for the rod at the centre of a cell of radius 10², 10⁴, 10⁶ rod radii, with its counterions checked to neutralise it exactly. Diluting the solution widens the cell by four decades at a time, and a counterion free to go anywhere in the cell ought to spread with it; instead each curve keeps a plateau near the rod whose height does not change. At the inflection of every curve the enclosed fraction is 0.762, which is Manning's 1 − 1/ξ, and the plateau sits there: 76 per cent of the counterions stay bound to the chain however dilute the solution, and only 24 per cent spread through it.

The counterions that never leave the chain

Dilute a solution of DNA a million times and its counterions ought to scatter through the whole volume. Three quarters of them do not. A line of charges closer together than the Bjerrum length — 0.71 nm in water — holds on to its counterions however much room they are given, until the chain's charge is cut back to one per Bjerrum length, and every osmotic pressure, swelling gel and packed virus built from such chains is set by that length rather than by the chemistry.

fluids · Osmosis

Named alongside it

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

Atomic structureEquilibriumBinding energyCoulomb forceElectrostaticsEnergy levelsOsmosisOsmotic pressureResolving powerAbsorptionAccidental degeneracyAngular momentum

All concepts