A lattice shows through the graph paper: the atoms breathe out of step, each to its own beat, and the interplanar spacing is labelled between two neighbouring planes. On the left an X-ray tube flashes, and from it a golden spark runs to the sample, leaves the far side as a reflected beam and lights up a flare on the detector. In the middle a faceted crystal turns slowly, and the letter itself shows through its clear faces. Glints flare here and there on the edges, while along the ruler at the bottom a violet marker rolls up to the mark of one angstrom, where a fixed line meets it.
Non-SI unit · crystallography · length
Angstrom, Å
A ruler that fits the atoms themselves
Unit of length · 1 Å = 10⁻¹⁰ m = 0.1 nm = 100 pm
Any measure is convenient when you take it one at a time rather than by the thousand: a tailor works in centimetres, not in fractions of a kilometre. The angstrom turned out to be that measure for matter: the distance between neighbouring atoms in a crystal lies between two and four angstroms, a single carbon—carbon bond is one and a half, the width of the double helix is twenty. A chemist naming a bond length says a single-digit number, and that is the whole reason the unit survived being struck off the recommended list.
The coincidence here is not accidental but twofold: angstroms are convenient for measuring not only matter but also the light used to see through it. An X-ray tube with a copper anode gives a line of one and a half angstroms, exactly the same order as the distances between planes of atoms, and out of that coincidence the whole of structural crystallography was born. A wave commensurate with the lattice is reflected by it selectively — at strictly defined angles — and from those angles one reconstructs how the atoms are laid out inside an opaque body.
Formally the unit has long been deprecated: the International System offers the nanometre and the picometre instead, and in newer standards the angstrom is marked for withdrawal. In practice, however, the protein structure bank stores atomic coordinates in angstroms, the resolution of a map is quoted in them, and the phrase “a structure at one point eight” needs no explanation in any laboratory in the world. The unit holds on not by force of law but because it happens to fit the size of things.
The symbol is a capital Latin A with a ring above it, and that ring is not a stress mark but part of the Swedish letter Å, so it must not be replaced by a degree sign or dropped to leave a bare A. In machine text where the letter is unavailable, one types a Latin A with a combining ring or writes the word Angstrom in full; the abbreviations “ang.” and “A” are accepted nowhere.
The angstrom is not part of the International System and takes no prefixes: there is no such thing as a milliangstrom or a kiloangstrom, and for shorter lengths one uses the picometre. Nor should it be confused with the x unit, used by X-ray crystallographers until the middle of the last century: it is smaller than the angstrom by about two thousandths, and old interplanar spacings taken from pre-war tables without conversion still occasionally differ in the third digit.
01 · Definition
a tenth of a nanometre · Bragg’s condition · reflection by angleThe angstrom is one ten-billionth of a metre, and the definition is so simple that there is nothing in it to discuss: the unit is set by a power of ten, not by a prototype or a constant of nature. Everything of substance begins where it is used, and it is used because the world of atoms happens to be ruled off in exactly this measure.
A crystal is built like a stack of identical atomic planes spaced a few angstroms apart. When an X-ray beam falls on such a stack, each plane reflects it as a faint echo, and these echoes add up: at almost any angle they cancel one another, but at certain particular angles the path difference between neighbouring reflections turns out to be a whole number of wavelengths, and then the echoes add at full strength. The condition for that coincidence is what is called Bragg’s law.
From this follows the point of the whole exercise: by measuring the angles at which reflections flare up and knowing the wavelength, one can calculate the distances between planes, and from a set of such distances reconstruct the arrangement of the atoms. The beam works here not as a lamp but as a scale ruler, and it works precisely because its wavelength is commensurate with the step of the lattice: visible light, whose wave is thousands of times longer, gives no reflection from atomic planes at all.
That commensurability has its other side. The detail with which a structure can be made out is limited by the wavelength: the shorter it is, the more reflections fit into measurable angles and the finer the resulting map. That is why, when protein structures are discussed, the number in angstroms is quoted not for a distance but for the resolution, and two angstroms counts as a good result, whereas at three and a half individual atoms already merge into a continuous sausage of electron density.
Where the crystal lights up
The sample stands on the left, the beam falls on it from the left as well, and the arm with the detector travels round it along an arc; the angle between the beam and the planes is set by the first slider. Almost everywhere there is no reflected light, because the echoes from neighbouring planes cancel one another, but as soon as the angle matches one of the Bragg angles the facets flare, the beam goes into the detector, and a peak grows on the pattern at the right. The second slider changes the wavelength of the source, and the whole picture shifts: with shorter-wave radiation the same planes answer at smaller angles, so a molybdenum tube squeezes the pattern towards the start of the scale while a chromium one stretches it out. Choose the protein crystal instead of salt and you will see what structural biology struggles with: the planes there lie far apart, the number of reflections in the measurable range becomes enormous, and each one of them separately is weak.
02 · Conversion
units of length · sources of radiation · sizes of matterOne length in three conversations
The first tab converts the length entered into the other units and tells you along the way what energy a photon of such a wave has and at what angle it would reflect off a typical plane. The second lists the sources used to look through matter — from a chromium tube to a synchrotron and a beam of electrons. The third collects the sizes of matter itself: atomic radii, bond lengths, cell edges and the pitch of the helix.
A rule worth remembering: the angstrom is a tenth of a nanometre, the copper line is one and a half angstroms, and the energy in kiloelectronvolts comes from dividing a little over twelve by the wavelength.
The angstrom takes no prefixes, and a notation such as “mÅ” is meaningless; for shorter lengths one moves to picometres, for longer ones to nanometres. Nor can the x unit from old X-ray tables be treated as an angstrom: it is smaller by about two thousandths, and pre-war interplanar spacings need converting before they can be compared with modern ones.
| Quantity | Value | Note |
|---|---|---|
| {k} | {v} | {note} |
{text}
04 · Measuring instruments
goniometer · powder diffractometer · synchrotron · tunnelling microscopeWhat is used to look at what is finer than light
An instrument that measures angles and answers in lengths
A goniometer can measure nothing but an angle, and therein lies its strength: the angle is measured mechanically, to thousandths of a degree, while the length follows from it by calculation through Bragg’s law. The sample is turned by some angle, the detector by exactly twice as much, and the whole assembly goes round the sample until it has gathered every reflection. The angstroms the instrument finally names were never laid off on a ruler.
A set of lines instead of a name
If a crystal is ground to a powder, the grains lie in every possible orientation at once, and each plane answers at its own angle no matter how the powder was poured. The result is a row of peaks whose positions and heights are as unrepeatable as the pattern on a finger: by that row the substance is identified against a card index, without taking it apart. This is how polymorphic forms of one and the same composition are told apart — and for a medicinal powder that is the difference between an active drug and a useless one.
A kilometre of ring for a crystal a tenth of a millimetre across
A tube gives one or two lines and whatever brightness the anode allows; a synchrotron gives a continuous spectrum of incredible power, from which a monochromator cuts out any wavelength required. For protein crystals, where the reflections are weak and the crystal itself is gradually destroyed under the beam, this settles the matter: the data are collected before the sample deteriorates. Being able to tune the wavelength matters for another reason too: near the absorption edge of a heavy atom the scattering changes, and on that difference the phase problem is solved.
A needle that feels an angstrom
Diffraction speaks of an average over the whole crystal, whereas a tunnelling microscope shows the surface atom by atom: the tip is brought so close that electrons begin to seep across the gap, and this current depends on the distance so sharply that a change of a tenth of an angstrom alters it noticeably. The needle is driven over the surface with the current held constant, and the height of the tip traces out the relief. Here the angstrom becomes, for the first time, not a calculated quantity but something the instrument senses directly.
05 · Writing rules
a Swedish letter · no prefixes · a space before the symbolThe ring above the letter is not decoration
The symbol of the unit is the surname of a Swedish physicist written in his own letter, and so it cannot be handled as a chance squiggle: replacing the ring with a degree sign, or dropping it, turns the notation into a different symbol altogether.
The first two mistakes are the commonest in typing: the writer cannot find the Swedish letter and puts either a bare A, turning a length into an electric current, or a degree sign, which has nothing to do with that letter. The third comes from the habit of the metric system, where prefixes are allowed everywhere, but the angstrom is not part of that system and therefore takes no prefixes. The fourth is a general rule of notation: a space goes between the number and the symbol, preferably a non-breaking one. The fifth is not an error but a survival: the centimetre form is correct, yet modern documents give lengths in metres and their fractions. The sixth is that same pre-war confusion which makes old tables of interplanar spacings impossible to compare directly with new ones.
06 · Neighbouring units
nanometre · picometre · Bohr radiusThe angstrom’s neighbours share the same stretch of the scale with it, but they come from different places: two of them are ordinary fractions of the metre, the third is taken from the structure of the atom itself.
The lawful replacement prescribed by the International System, and for the sizes of particles and films it has caught on. For bond lengths it is inconvenient: instead of one and a half angstroms one has to write nought point one five four.
The unit in which chemical handbooks print bond lengths and atomic radii: one hundred and fifty-four picometres instead of one and a half angstroms. The difference is purely one of habit, and both notations mean the same thing.
The only one of the three taken not from the metre but from nature: the distance of the electron from the nucleus in the simplest model of the hydrogen atom. In theoretical calculations everything is measured in it, and the angstrom appears there only when converting for the reader.
Standing nearby among the Simetrium passports are light year and parsec — measures from the opposite end of the scale, arranged, however, by the same rule: the unit is chosen so that what is measured comes out as a single-digit number.
07 · Historical section
archive · 1868 → 1912 → 1913 → 1960From the solar spectrum to a map of atoms
{title}
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At first angstroms named only the wavelengths of light, then they came to measure the lattice, then the turns of biological molecules, and finally the detail reached the individual atom. Not one of these leaps happened without a new instrument, and each of them rested on the one before: a map of the spectrum was needed to learn the wavelength of the tube, and without that there was nothing to measure the lattice with.
A withdrawn unit that is used every day
Metrologically the angstrom presents no difficulty at all: it equals ten to the minus tenth of a metre by definition, and has no prototype, no uncertainty and no history of refinements. Only once was it tied not to the metre but to nature — at the start of the last century, when the red line of cadmium was taken as the basis and assigned a length — but that tie did not last long and was dropped as soon as the metre itself was defined through light.
Far more curious is its standing in the rules. The angstrom is not part of the International System, it is absent from the list of units accepted for use, and national standards number it among those due for withdrawal — and yet it is precisely in angstroms that atomic coordinates are recorded in every structure of the world data bank, in angstroms that the resolution of a map is quoted, and in angstroms that interplanar spacings are printed in the powder diffraction file. There is no one to replace the unit in those archives and no reason to: converting millions of records would add nothing to their content.
The reason for its persistence is simple and has nothing to do with tradition: the quantity fits the size of what it measures. As long as the subject remains atoms and the distances between them, no more convenient unit can be devised; and when the subject changes — as it changed for the opticians, who moved from angstroms to nanometres along with the move from spectral lines to thin films — the unit will go of its own accord, without any decree.
Catalogue of quantities
lengths and the structure of matterSI base units
the metre is highlighted — the angstrom is its ten-billionth partWhat is how many angstroms
the angle is calculated for the copper line, first order| What is measured | In angstroms | In picometres | 2θ at Cu Kα | Note |
|---|---|---|---|---|
| {name} | {a} | {pm} | {two} | {note} |
The column with the angle shows why the copper tube has caught on so well: for all everyday crystal spacings the reflections fall in the middle of the instrument’s scale, where they are convenient both to separate and to measure. A dash in the last rows means that at such a wavelength a first-order reflection is impossible altogether: if the spacing between planes is less than half a wavelength, the path difference can never reach a whole wave, at whatever angle the detector is set.