What a thread scatters

Seventy-five ways to be a thread enumerates the rod groups — the symmetries of an object periodic along one direction — and the screw axes among them are constrained exactly as a crystal’s are. A rotation of a rod group acts on the lattice along its own axis, so the crystallographic restriction applies unchanged: orders one, two, three, four and six, and nothing else.

A helix is what the same object becomes when the rotation is not asked to be crystallographic. A structure with u subunits in t turns per repeat has a screw operation of rotation 2πt/u, and nothing whatever forbids u/t from being ten over one, or eighteen over five, or an irrational number. The rod group machinery does not apply and the object is entirely real: an α-helix has about three and six tenths residues per turn, and DNA’s backbone has ten in one.

So the restriction is gone. The regularity is not, and where it reappears is in what the thing scatters.

Where the layer lines come from #

A helix repeats along its axis after a distance c, and that is a genuine lattice translation however irrational the rotation is. So the scattering is confined to planes perpendicular to the axis at spacings 1/c — the layer lines — for exactly the reason a crystal’s scattering is confined to a reciprocal lattice: a periodic direction gives a discrete conjugate.

That much is ordinary, and it is where the ordinariness stops. In a crystal, each reciprocal lattice point carries an amplitude and that is the end of it. On a layer line of a helix, the amplitude is a function of the radius from the axis and of the azimuth, and the question is what it can be.

Cochran, Crick and Vand answered it in 1952 by expanding the scattering in cylindrical coordinates. The amplitude on layer line l at radius R comes out as a sum over angular orders n of Bessel functions J_n(2πRr), where r is the radius of the helix — and the orders are not free.

l = t·n + u·m, for some integer m. That is the selection rule, and everything else here is a consequence of it.

What the rule permits #

For a given layer line the equation is a congruence: t·n ≡ l modulo u. Since u and t are coprime — a common factor would mean the helix had been described on a repeat longer than its own — t is invertible modulo u and the solutions form an arithmetic progression of step u.

The progression has step u, so at most one of its members is small. That member is the one that matters, and the sequence it traces is the whole shape of a fibre pattern: for ten subunits in one turn the lowest permitted orders on layer lines nought to ten are 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0.

Up by one to the middle of the repeat, and back down. That is a triangle, and a triangle is a cross.

Why a large order means a large radius #

The second half of the argument is a fact about Bessel functions that has nothing to do with helices.

J_n(x) is zero at the origin for every n except nought, and it stays negligible until x is of the order of n. Its first maximum sits a little beyond n — about n plus four fifths of the cube root of n — which for the small orders a fibre pattern shows is close to n + 1, and it is computed here rather than approximated.

So a layer line whose smallest permitted order is n has no intensity near the axis at all, and its first maximum sits at a radius proportional to n. Put that beside the triangle of orders and the pattern is settled: the intensity maxima on successive layer lines start further and further out, to the middle of the repeat, and then come back in. Two straight arms.

And the arms meet the axis exactly where the order is zero, which by the rule is where l is a multiple of u. That is the meridional reflection, and its layer line counts the subunits in the repeat directly: ten subunits in a turn puts it on the tenth layer line, which in Franklin’s photograph is the strong spot at the top and bottom, and the spacing it sits at is the rise per subunit.

The triangle, and why it comes back down #

The sequence of lowest orders — nought, one, two, three, four, five, four, three, two, one, nought — is worth a paragraph on its own, because the descent is less obvious than the rise and is the reason the cross closes rather than running off the page.

The permitted orders on layer line l are n ≡ l·t⁻¹ modulo u, taken over all integers. A residue class modulo u contains a smallest-in-modulus member, and that member is at most u/2 in absolute value: past halfway, the representative nearer zero is the negative one. For ten subunits in a turn, layer line six permits n = 6 and n = −4, and minus four is smaller.

So the lowest order rises to the middle of the repeat and then falls, and it falls through negative values. A negative order is not a smaller effect than a positive one, since the Bessel function of order minus n is the one of order n up to a sign, so the intensity comes back in towards the axis exactly as it went out.

That is the difference between a cross and a wedge. A pattern whose maxima only moved outwards would be two diverging lines; the return is what closes the X and puts the meridional reflection at its top. And the return is a fact about residues modulo u rather than about anything physical, which is why the shape is so uniform across substances that have nothing else in common.

The rule, and the extinction condition it generalises #

The selection rule is not a new kind of law. It is the ordinary extinction condition of a screw axis, written for a rotation that need not be crystallographic.

Take a helix with u = 6 and t = 1 — a six-fold screw, which a crystal may perfectly well have. The rule says order zero is permitted only where l is a multiple of six, so the meridional reflections are 00l with l divisible by six, and every other 00l is absent. That is exactly the extinction condition of a 6₁ screw axis as the absences essay derives it, arriving here as a special case of a formula about fibres.

The comparison is the point of the figure. The restriction is a statement about which rotations can act on a lattice; the selection rule is a statement about what a screw operation does to a transform, and it never asks whether the rotation is crystallographic. So three of these four objects cannot be crystals and all four diffract by the same arithmetic — which is why a fibre pattern is readable at all, and why the reading gives a rise and a number of subunits rather than a cell.

The two numbers a reader takes off the picture #

It is worth stating what a fibre photograph gives up immediately, because the two quantities are read from the cross rather than computed from it, and both are consequences of the paragraphs above.

The angle of the arms gives the ratio of the pitch to the radius. The maxima on layer line l sit at a radius proportional to l and at a height proportional to l/c, so the arms are straight lines whose slope is a ratio of the two scales — the axial repeat against the helix’s own radius. A fat helix has shallow arms and a thin one steep arms, and the angle is the first thing a reader measures.

The layer line of the meridional reflection counts the subunits. By the rule, order zero occurs where l is a multiple of u, so the first meridional reflection above the equator is at l = u. Counting layer lines up to the strong spot on the axis gives the number of subunits in the repeat directly, with no model and no refinement.

Those two, together with the layer line spacing giving c, are the whole of what the geometry of the cross carries — and they are three numbers about an object whose atoms are not known at all. That is an unusually high return for a photograph, and it is why the helical diffraction theory was worth deriving before there was a structure to apply it to.

Checked against a sum that has never heard of Bessel #

An expansion is a way of writing something down, and a selection rule derived from an expansion is a fact about the writing until it is checked against the thing.

The second computation is the definition of a Fourier transform and nothing more: place the subunits at their azimuths and heights, and add exp(2πi(R r cos(ψ − φ) + l z/c)) over them. It contains no Bessel function, no selection rule, and no helix — it would give the same answer for the same points scattered at random.

They agree to a part in ten thousand million, which is the arithmetic of the sums rather than a tolerance anybody chose. So the selection rule is a property of the object. A rule that had only ever been checked inside the expansion that produced it would be a statement about a series.

What the pattern gives back, and what it does not #

A fibre pattern is famously informative and famously incomplete, and the selection rule says exactly which of those it is at each point.

It gives the rise and the number of subunits. The layer line spacing is 1/c; the meridional reflection sits at l = u, so counting up to it gives u; and the rise per subunit is c/u. Those three come off the pattern with a ruler, and they are what made the helical model of DNA arguable in 1953: the cross said helix, the tenth layer line said ten subunits, and the spacing said three and four tenths ångström a subunit.

It does not give the atoms. Everything above uses one radius r for every subunit — a helix of points. A real subunit has extent, and its atoms sit at several radii, so the amplitude on a layer line is a sum of the above over them with the phases their positions supply. Nothing in the selection rule changes; what changes is that each permitted order arrives weighted, and disentangling the weights is the structure determination rather than the reading.

And it does not give a handedness from the intensities alone, for the same reason Friedel’s law hides it in a crystal: the transform of the mirror image has the same modulus everywhere. The cross of a right-handed helix and of a left-handed one are identical, and the tilt of the layer lines that distinguishes them is a fact about the phases, which a photograph does not record.

A helix is a rod group with the restriction taken out #

One last comparison, since this essay sits in the subperiodic anchor and the anchor’s other essays are about groups rather than transforms.

A rod group is a subgroup of the isometries of space that leaves a line invariant and whose translations along it form a lattice. Seventy-five of them exist, and the count is finite for the same reason the two hundred and thirty is finite: the rotations are crystallographic, the point groups are finite, and the extensions are classified by a cohomology.

A helix satisfies every part of that except the rotation. Its translations along the axis are a lattice; its symmetry group is a subgroup of the isometries leaving the axis invariant; and its rotation is 2πt/u for coprime u and t, which for u outside one, two, three, four and six is not a rotation any lattice permits. So the group is not a rod group and it is a perfectly good infinite group — cyclic, generated by a single screw operation, with the lattice as a subgroup of index u.

The enumeration is what is lost and nothing else is. There are seventy-five rod groups and there are infinitely many helical groups, one for each pair of coprime integers, which is why the subject stops classifying and starts computing. What survives is exactly the machinery this essay uses: a screw operation, a lattice along the axis, and a transform whose selection rule the screw operation dictates.

That is the same shape as the aperiodic case, one dimension down. A quasicrystal loses the lattice and keeps sharp diffraction; a helix keeps the lattice, loses the crystallographic rotation, and keeps a selection rule. In both, what a reader takes to be a consequence of periodicity turns out to be a consequence of something weaker.

What the transform refuses #

The last is the one that ties this back to the rest of the collection. If the formula did not reproduce the 6₁ extinction condition it would be a formula about something else wearing the same words, and the check is cheap: run the same layer line calculation with u = 6 and t = 1 and require the meridional reflections at multiples of six.

The fourth is about the description rather than the object. Ten subunits in two turns is five subunits in one turn described on a doubled repeat, and every count taken from the longer description — the layer line of the meridional reflection most of all — is out by the factor. It is the same defect a supercell is, in a subject with one dimension instead of three, and it is refused rather than absorbed.

Who computed it, and what they were looking at #

Cochran, Crick and Vand published the transform in 1952, and the order in which the pieces arrived is the reverse of the order they are usually taught in.

The fibre photographs came first, and they were unreadable. A fibre is not a crystal: the molecules are aligned along one axis and rotated at random about it, so the pattern is the transform of one molecule averaged over every azimuth — which throws away exactly the information a crystal’s discrete spots carry. What is left is layer lines with intensity spread along them, and until there was a theory of what a helix does there was no way to say whether a given smear meant anything.

The transform made the smear readable, and the reading is the three numbers above. Crick’s own account is that the theory was worked out to interpret photographs of synthetic polypeptides — of the α-helix, which is eighteen residues in five turns and is the second row of the table here — and that the application to DNA came afterwards, when Franklin’s photograph showed a cross whose arms and meridional reflection could simply be counted.

The order matters because it is the argument for deriving a transform before there is a structure. Nobody knew what DNA was made of in the sense of atomic positions; the selection rule needs only that the object is a helix with some number of subunits in some number of turns, and it returns the number and the rise. That is a symmetry argument doing the work a structure determination could not yet do, which is the thing this collection is about.

Where the exactness stops #

Computed here: the permitted orders on each layer line by solving the congruence; the first maximum of each Bessel function by a scan rather than an asymptotic formula; the layer line at which the scattering reaches the axis; the amplitude at twenty-four points on each of five layer lines, by the Bessel expansion and by a direct sum over the subunits; and the crystallographic verdict on four named helices.

A helix of points, at one radius. Every subunit here is a point at the same distance from the axis. The layer lines and the selection rule are unchanged for a real subunit; the intensities are not, and nothing on this page is an intensity a diffractometer would measure.

A continuous helix is a limit and is not computed. Letting u grow with t/u fixed gives the smooth helix whose transform is a single Bessel function per layer line, and that case is where the textbook derivation starts. Here the discrete case is the object and the continuous one is not taken.

And the tilt is not here. The maxima on a layer line of a real fibre pattern are not symmetric about the meridian; the asymmetry carries the handedness, and it lives in the interference between orders that this file computes and does not draw.

Where the ladder goes next #

Back, to the groups this generalises: seventy-five ways to be a thread, where the rod groups are enumerated and their screws are crystallographic, and the crystallographic restriction, which is the constraint a helix escapes.

Sideways, to the condition this one contains: systematic absences, where a screw axis deletes reflections along a row and the deletion is exact, and reading a space group from its absences.

Onward, to the two things a fibre pattern cannot give: the law that hides handedness, which is why the cross does not say which way the helix turns, and the phase problem, which is why counting the layer lines is easier than solving the structure.