The steps in an absorption curve

Everything on this ladder so far describes absorption as a property of a medium that varies smoothly with frequency. The first rung has a fixed loss per cycle giving an $f^2$ law, with relaxations adding smooth bumps. The second ties absorption to refraction by an integral over all frequencies, which requires both to be continuous. The third reads the same coefficient as a force.

Take the wavelength down far enough and the description changes character. What absorbs is no longer a medium but an atom, and what it absorbs into is a discrete set of states — so the coefficient acquires thresholds.

Why there is a step at all #

The mechanism is the photoelectric effect, which this collection has met as a statement about a threshold in frequency rather than in intensity, and the step is that threshold seen from the other side.

An X-ray photon is absorbed by giving all of its energy to a bound electron and ejecting it. The electron needs at least its binding energy to escape, so a photon below that binding energy cannot eject that electron — although it can eject a more loosely bound one from an outer shell.

Now raise the photon energy through the binding energy of the innermost shell. Below it, only the outer electrons are available. Above it, the two K-shell electrons become available as well — and they are far the most effective absorbers, because the photoelectric cross-section is largest for the most tightly bound electrons whose wavefunctions overlap the nucleus.

So the number of available absorbers jumps, and so does the coefficient. For iodine the K-shell contribution is more than three quarters of the total just above the edge, which is why the jump is a factor of four and not a few per cent.

The same argument applies at every shell, so there is an L edge, three of them in fact, and an M edge below that. They are at lower energies and are usually below the range of interest, but they are there, and a lead shield has three L edges between 13 and 16 keV that matter for low-energy work.

The step that is injected into people #

The most consequential use of a discontinuity in a material property is medical, and it is worth working through because the reasoning is entirely about the edge and not about the element.

A radiograph is a shadow, and a shadow requires a difference in attenuation. Soft tissues differ from one another by a few per cent, which is why a plain X-ray shows bone clearly and a blood vessel not at all.

Injecting iodine changes that by a factor of three at the right energy. And the “right energy” is what the figure is about: below the K edge, iodine is merely a moderately heavy element and adds what a moderately heavy element adds; above it, it absorbs four times as strongly and the vessel appears.

That is why the contrast agents in clinical use are iodine and barium and not, say, tungsten or gold, which are heavier. Iodine’s K edge is at 33.2 keV and barium’s at 37.4, both comfortably inside the 30–80 keV a diagnostic tube produces. Tungsten’s is at 69.5, near the top of that range, and gold’s at 80.7, above most of the beam — so the step would fall where there are few photons and buy nothing.

The requirement is therefore an accident of atomic structure that happens to line up with the practicalities of X-ray generation, plus the entirely separate requirement that the element be tolerable in a bloodstream. Iodine is used because those two conditions have exactly one comfortable solution.

Reading the step deliberately #

Once the edge is recognised as a feature rather than a nuisance, it becomes a measurement.

Dual-energy imaging takes two exposures, one below the edge and one above, and subtracts. Everything that has no edge in between — bone, soft tissue, air — attenuates similarly at the two energies and cancels; the contrast agent does not, and what is left is an image of the agent alone. That is how a digital subtraction angiogram removes the skeleton from a picture of an artery, and it is how a modern CT scanner distinguishes an iodine-filled vessel from a calcification that looks identical on a single-energy scan.

X-ray absorption spectroscopy looks at the edge itself in detail rather than jumping across it. The exact energy of the edge shifts by a few electronvolts with the oxidation state of the absorbing atom, because a more positive ion binds its electrons more tightly — so a measurement of the edge position is a measurement of chemical state, element by element, in a sample that need not be crystalline.

And the structure just above the edge is a picture of the neighbours. The ejected electron is a wave that scatters off the surrounding atoms and interferes with itself, modulating the absorption by a per cent or so in a pattern whose Fourier transform gives the distances to those atoms. That is the same sum a diffraction grating performs with one atom’s neighbours as the scatterers, it is called extended X-ray absorption fine structure, and it is the standard way of finding the local environment of a metal atom in a protein — where there is no crystal to diffract from and the atom of interest is one in ten thousand.

Each of those is a different amount of detail read off the same feature: its height, its position, and its shape.

Why the edge energies are where they are #

The K-edge energies run from a few electronvolts for hydrogen to 116 keV for uranium, and the progression is close enough to a simple law to be worth stating, because it explains a periodic table’s worth of choices at once.

A 1s electron’s binding energy is roughly $13.6(Z - 1)^2$ electronvolts — the hydrogen value with the nuclear charge screened by the other K electron. For iodine, with $Z = 53$, that gives 36 keV against a measured 33.2; for lead, with $Z = 82$, it gives 90 against 88. Three parts in a hundred, from an expression with one screening constant in it.

That the law is so simple is a consequence of the innermost electron being almost alone with the nucleus: the other fifty-one electrons in iodine are mostly further out, and a spherical shell of charge outside a point contributes nothing to the field there. So the K electron sees very nearly the bare nucleus, and its binding energy is the hydrogen answer scaled.

Moseley measured exactly this in 1913, from the emission side rather than the absorption side, and the square-root relation between a characteristic X-ray frequency and $Z$ is what established that atomic number is the nuclear charge rather than a position in a list. The edges in this rung’s figures are the same physics read as absorption instead of emission, at the same energies.

The consequence for anybody choosing a material is that the edge energy is fixed by the element and by nothing else — not by its compound, not by its density, not by its physical state, to within the few electronvolts of chemical shift the spectroscopy exploits. An edge is as close to a label on an element as a physical measurement provides.

The other end, where the edges stop mattering #

The steps are a low-energy phenomenon, and it is worth being explicit about where they give out, because the practical advice reverses.

The reason is that photoelectric absorption falls extremely fast with energy — roughly as $E^{-3}$ — while the other mechanism, Compton scattering off essentially free electrons, falls slowly. Above a few hundred keV for light elements and a megaelectronvolt for heavy ones, Compton dominates, and Compton depends on the number of electrons per gram, which is nearly the same for everything except hydrogen — a scattering that removes photons from a beam without absorbing any of them.

So at a megaelectronvolt a shield is chosen by mass per unit area, not by atomic number, and concrete is used because it is cheap rather than because it is good. Lead’s advantage over water falls from a factor of 369 to a factor of 11, and most of that residue is just its density.

Above 1.022 MeV a third mechanism opens — pair production, which needs the photon’s energy to exceed twice the electron’s rest energy and which rises with energy — and heavy elements become preferable again, because it goes as $Z^2$. So the ranking of shielding materials reverses twice as the energy is raised, and each reversal is a different physical process taking over.

The tube on the other side of the sample #

A last practical point, because every figure here treats the photon energy as a knob and an X-ray tube does not have one.

A tube produces a continuum — the bremsstrahlung of electrons stopping in a target, with a sharp upper edge at the accelerating voltage and characteristic lines of the target element on top of it. What reaches a sample is therefore a broad spectrum, not a single energy, and every attenuation in this essay is an average over it weighted by the number of photons at each energy.

That has two consequences and both are in every radiography textbook.

Beam hardening. The low-energy photons are absorbed preferentially, so a beam that has passed through a patient is harder than the one that entered — its mean energy has risen. A second centimetre of tissue therefore attenuates less than the first, the exponential law fails, and a CT reconstruction that assumes it produces cupping artefacts and dark streaks between dense objects. Correcting for it is standard and is never quite exact.

And filtration is a design choice. The softest photons in the beam are absorbed entirely in the first centimetre of the patient, contributing dose and no image, so tubes are filtered — a few millimetres of aluminium, or in mammography a molybdenum or rhodium filter whose own K edge sits just above its target’s characteristic lines and cuts everything above it. That filter is this rung’s physics used as a tool: an absorption edge placed deliberately in a beam to shape its spectrum.

The dual-energy techniques above are the same idea taken further. Rather than filtering to one energy, two spectra are used and the difference between them is exploited — and what makes the difference informative is the edge, which is the only feature in an attenuation curve that two broad spectra can be made to straddle.

What the discontinuity does to the earlier rungs #

An edge is a genuine discontinuity in absorption as a function of frequency, and the second rung of this ladder says that absorption and refraction are tied together by an integral over all frequencies. Those two statements have to be reconciled.

They are, and the reconciliation is instructive. The Kramers–Kronig relations do not require the absorption to be continuous; they require it to be causal, which is a weaker condition. A step in the absorption produces a logarithmic singularity in the refractive index at the same energy — the index dips sharply just below the edge and recovers above it.

That dip is measured. X-ray refractive indices are within $10^{-5}$ of one, and the departure is measurable by interferometry; near an absorption edge it changes by a substantial fraction of itself over a few electronvolts. The technique that uses it is anomalous dispersion phasing, which solves the phase problem in protein crystallography by collecting data at three energies around an absorbed atom’s edge and using the difference to locate that atom.

So the edge is simultaneously a discontinuity in the absorption and a rapid variation in the refraction, and the same relation that ties the smooth parts together ties these as well. Nothing about the earlier rung fails; what fails is the assumption that a causal response function has to look smooth.

What a curve with no edges looks like #

Putting the light materials on their own makes the contrast with the previous figures sharp, and it says what the edges are actually buying.

Water, aluminium and bone have their K edges far below the range any diagnostic beam contains, and photons of those energies are absorbed by a centimetre of air. So over the whole useful range those three materials have no thresholds, their curves are smooth, and what differences there are come from density and from the slow variation of the photoelectric term with atomic number.

That is the situation a radiograph is normally in, and it is why the technique reveals so little unaided. Fat, muscle and blood differ in attenuation by a few per cent at 60 keV; bone differs from soft tissue by a factor of three at 30 keV and by ten per cent at 150. There is no threshold anywhere to exploit.

Everything clinical follows from that shortage. Beams are run at the lowest energy the patient’s thickness allows, because the photoelectric term — the one that discriminates between elements — falls as the cube of the energy while the Compton term, which does not discriminate, falls slowly. A mammogram at 20 keV has good contrast and a large dose; a chest radiograph at 120 keV has poor contrast and a small one; and the choice between them is exactly that trade, made anew for every examination.

And it is why the contrast agents matter as much as they do. Adding an element with an edge inside the beam is the only way of introducing a threshold where there was none, and a threshold is the only feature in an attenuation curve that a subtraction can isolate.

Where this stops being right #

The curves here are interpolated between tabulated points. The tables are compilations of measurements and calculations, accurate to a per cent or two in the ranges shown and worse near the edges themselves, where the structure is finer than any interpolation between two points can represent.

Coherent scattering has been folded into the total. Rayleigh scattering off bound electrons contributes several per cent at low energies and does not deposit energy at all, so a coefficient used for shielding and one used for dose are different numbers — the tables distinguish them and the curves here do not.

“Narrow beam” is doing a great deal of work. A half-value layer describes a collimated beam and a detector that sees only what went straight through. A broad beam includes scattered photons that arrive anyway, so a real shield is worse than its half-value layer suggests by a build-up factor that can exceed ten.

And nothing here is about the fine structure. The absorption within a hundred electronvolts of an edge is not a step but a step with structure on it — a white line, then oscillations — and that structure is the entire content of the spectroscopy described above. At the resolution of these figures it is invisible.

What the pictures cannot show #

Every curve here is drawn as a smooth interpolation through a handful of tabulated points, and what is between those points is an assumption rather than a measurement. The interpolation is logarithmic because the underlying dependence is close to a power law, and that choice is the reason the curves look as clean as they do.

Nor can the drawings show that an edge has a width. It is drawn as a vertical discontinuity and it is a rise over a few electronvolts, set by the lifetime of the core hole left behind — which is a femtosecond or so, giving a width of an electronvolt by the same relation that ties a width to a lifetime anywhere else. On an axis spanning three decades that width is a fraction of a pixel, and the discontinuity is a fair drawing of it and not a true one.

Where the ladder has reached #

Four rungs stand on attenuation. The first found that absorption removes the treble and not merely the volume. The second found that absorption and refraction are one causal function. The third found that the same coefficient is a force. This one takes the frequency up until the coefficient stops being smooth.

The habit worth carrying away is about what a discontinuity in a smooth-looking law reveals. A step in a material property as a function of energy is a threshold, and a threshold means the absorber has discrete states rather than a continuum of responses. The whole of atomic structure is legible in an attenuation curve for that reason, and the curve was measured long before the structure was understood. The same reading applies wherever a response function has a step: a semiconductor’s optical absorption edge is its band gap, and a superconductor’s is twice its pairing energy.

What is left on this ladder is the region the figures deliberately skirt. Within an electronvolt of an edge the absorption is not a step but a structure — a sharp peak, then oscillations extending hundreds of electronvolts above — and every feature of it is a statement about where the neighbouring atoms are. That is a measurement of geometry made with an absorption coefficient, which is as far from the first rung’s fixed loss per cycle as this ladder goes.