A fibre of achiral crystals can rotate light

In 1967 D. Hobden measured optical rotation in silver gallium sulphide, a crystal whose class is 4̄2m. The class has mirror planes, the crystal is its own mirror image, and it rotated the plane of polarised light anyway. Fifteen may rotate light explained how: optical rotation is governed by the gyration tensor, and four achiral classes — 4̄, m, mm2 and 4̄2m — permit one. Their tensors are traceless, so the rotation has one sign along some directions and the other sign along others, and the crystal as a whole has no preferred hand. The rotation silver gallium sulphide showed was along one particular direction, where the positive and negative contributions do not cancel, and at one particular wavelength, near 497 nanometres, where the crystal’s two refractive indices cross and its birefringence, which otherwise swamps any rotation, vanishes.

What a texture permits counted the properties a textured aggregate can carry — a poled ceramic, a drawn fibre, a rolled sheet — and pointed out that optical activity is one a texture can carry and a random aggregate cannot. This essay computes what happens when the crystallites of a texture belong to one of the four achiral classes. A random powder of them never rotates light, whatever the class. A fibre texture of them does, along its axis, exactly as much as one grain does along the aligned direction, and half as much the other way across the axis. Which direction is aligned decides whether the texture is chiral, which hand it has, and whether it rotates at all.

What a texture keeps of a tensor #

A texture is a distribution of orientations, and a tensor property of the aggregate is the average of the grains’ tensors over that distribution. The average is a projection, and what survives it is decided by the distribution’s symmetry, one of Curie’s seven limiting groups. The count of components each group permits for the gyration tensor comes from the same arithmetic the texture essay used for every other property.

The pattern is the one that distinguishes the gyration tensor from every other symmetric second-rank property. The gyration tensor is axial: an improper operation changes its sign as well as moving its components, so any group containing a mirror, a centre or a rotoinversion forces it to equal its own negative. Neumann’s principle, applied to a texture instead of a crystal, then leaves only the three chiral limiting groups with anything, and the three differ in what they leave. The spherical group ∞∞, a random powder of one hand, keeps a single number. The cylindrical groups ∞ and ∞2 keep two: the rotation along the axis and the rotation across it.

The averages themselves are short. Over every rotation of space, a symmetric tensor averages to a third of its trace times the identity. Over the rotations about one axis, with a grain’s direction $d$ held along that axis, it averages to a diagonal tensor with

$$b = d^{\mathsf T} g\, d \ \text{ along the axis}, \qquad a = \tfrac12\big(\operatorname{tr} g - b\big) \ \text{ twice across it}.$$

The trace is unchanged by any rotation, which is why it survives both averages. The rest of the tensor, its traceless part, is lost entirely in the powder and kept, in part, by the fibre: exactly the component along the axis, which a rotation about that axis cannot move.

The texture essay’s own language makes the same statement in one line. A symmetric second-rank tensor is a spherical harmonic of degree nought, its trace, plus five of degree two, its traceless part. A random powder keeps only degree nought. A fibre keeps degree nought and the one degree-two harmonic that does not change under rotation about the axis — the one shaped like $3\cos^2\theta - 1$, which is exactly the pattern of $b$ along the axis and $-b/2$ across it. So the fibre’s two numbers are not two arbitrary components but the two harmonics a cylinder can hold, and the achiral classes, having no degree-nought part at all, contribute only the second.

A powder of achiral crystals is inactive #

The four achiral gyrotropic classes have traceless tensors, and that settles the powder at once.

The powder column is the whole of the first result. Every one of the eleven chiral classes has a trace part, because in a class with no improper operation the average of the tensor’s trace over the class is the trace itself, and a generic tensor has one. Every one of the four achiral classes has none, because the average of the determinant over a class containing improper operations is nought, and the trace of the averaged tensor is the trace of the original times that average. So a powder of quartz, which is chiral, rotates light, as a sugar solution does. A powder of silver gallium sulphide does not, however large its single-crystal rotation along any direction. The grains rotate light, each of them, and the rotations cancel in the average exactly. In a real sample of finitely many grains the cancellation is statistical: a residual rotation of either sign survives, shrinking relative to one grain’s as the square root of the number of grains in the beam. Its mean is nought and it has no preferred hand, so two samples of the same powder disagree in sign as often as they agree.

The table also carries a quieter result at its foot. The two chiral cubic classes, 23 and 432, have no traceless part at all: their gyration tensor is a multiple of the identity, the same rotation in every direction. For them every texture rotates light exactly as the powder does, since no average can do anything to a multiple of the identity. Three optical characters made the same observation about refractive index, where averaging over a cubic group turns any ellipsoid into a sphere; here the cubic group has done the averaging already, inside each grain.

A fibre of them is not #

Align the grains, so that a single crystal direction $d$ lies along the fibre axis in every grain and the grains’ rotations about that axis are random. The formula above then gives, for a traceless grain tensor, $a = -b/2$: the fibre rotates light along its axis by $b$, exactly the rotation one grain shows along $d$, and across the axis by minus half of that. The two senses are opposite, and the three diagonal entries sum to nothing, as the trace requires.

The comparison with a chiral grain is instructive. A fibre of the orthorhombic chiral class 222 also splits its rotation between the axis and the directions across it, but around a nonzero mean, so both senses usually have the same sign and the fibre is simply more active along one direction than another. A fibre of an achiral class splits around nought.

The along-axis number is also the one an experiment can see. A fibre texture is uniaxial, and its axis is its optic axis: light travelling along it meets no linear birefringence, which in any other direction is usually hundreds of times stronger than the rotation and hides it. So the rotation along the fibre, which the averaging keeps at the grain’s full value, is precisely the rotation that can be measured without waiting for a wavelength where the birefringence happens to vanish. A single crystal of silver gallium sulphide shows its rotation only near its isotropic wavelength; a fibre of it would show the rotation along its axis at any wavelength, at the cost of having averaged away everything else. It is a texture with a definite hand along its axis and the opposite hand across it, and its hand is not the hand of any grain, since no grain has one. Every number in the figure was computed by averaging the rotated tensor over seventy-two orientations about the axis, not from the formula, and the formula is checked against it on a hundred and ninety-five fibres to the last digit.

Which direction, and which hand #

The rotation along the fibre is the grain’s own rotation along $d$, so the whole question of which fibre textures of a class rotate light, and which way, is a map of $d^{\mathsf T} g\, d$ over the directions a grain can present.

The three maps share a structure that the tensor forces. A traceless symmetric tensor’s quadratic form is indefinite, so it is positive in some directions, negative in others and zero on a cone between them. For 4̄2m the form is $g\,(d_x^2 - d_y^2)$ in its standard setting, positive near one two-fold axis, negative near the other and zero on the diagonal planes. For mm2 it is $2g\,d_x d_y$, changing sign from quadrant to quadrant and vanishing on the two mirror planes. In both of those classes the zero cone and the mirror planes coincide, and that is not an accident, as the next figure shows. For 4̄ the cone is tilted away from every symmetry element the class has. There the zero is accidental: a direction on it gives a chiral texture whose rotation along the axis happens to vanish at that particular alignment.

The difference comes down to how many numbers each class’s tensor has. For mm2 and 4̄2m the gyration tensor has a single free component, so its zero set is fixed by the class and cannot move; and the mirrors, where the texture must be inactive by symmetry, are forced to lie inside that zero set, so for a one-parameter tensor they fill it. For 4̄ the tensor has two free components, $g_{11} = -g_{22}$ and $g_{12}$, and the zero cone turns about the 4̄ axis as their ratio changes. Where it sits in a given crystal is a fact about that crystal, not about its class, and a fibre aligned on it is chiral but, at that wavelength and in that crystal, inactive.

The texture’s group, read from the grain #

The texture’s own symmetry is not given; it is computed from the grain’s. A rotation about the fibre axis is always a symmetry of a fibre texture. Beyond those, an operation of the grain’s class that carries $d$ to itself or to its opposite becomes, carried into the sample frame, an operation of the texture. A proper operation reversing $d$ adds a half-turn across the axis; an improper one keeping $d$ adds a mirror containing the axis; an improper one reversing it adds a mirror across the axis. The texture is chiral exactly when the class has no improper operation that sends $d$ to $\pm d$.

Two things in the table are worth reading slowly. The first is that the two two-fold axes of 4̄2m give textures of opposite hand: a fibre with [100] along its axis rotates light one way, a fibre with [010] along its axis rotates it the other way, by exactly the same amount. The 4̄ operation carries one two-fold onto the other and is improper, so the two textures are mirror images of each other, although each is made of the same achiral crystals. Aligning a crystal of no hand along one of two equivalent directions produces one of two enantiomorphic textures, and which one is decided by the alignment alone.

The second is that whenever the texture’s group contains an improper operation, the averaged tensor is exactly zero, not merely small. Along [110] the texture inherits a mirror containing the axis and one across it, which is the group ∞/mm; along [111] it inherits one mirror and is ∞m; along the 4̄ axis it inherits the rotoinversion. All three average to nothing, and across all fifteen classes and thirteen directions the census finds fourteen textures with improper operations, every one of them inactive. That is Curie’s principle doing exactly what a crystal in a field showed it does, from the other side: there, a field’s symmetry intersected a crystal’s; here, the crystal’s symmetry leaks into the texture’s through the direction the texture holds fixed.

How many alignments give an active texture differs between the four classes in a way the same reading predicts. Of the thirteen directions tried, 4̄ gives an active chiral texture along twelve, and the one exception is its own 4̄ axis, whose rotoinversion the texture inherits. The class has no mirrors, so no other direction can pick up an improper operation, and only the accidental zeros of the last figure stand between a general alignment and a rotating fibre. The classes with mirrors lose every direction lying in one: m keeps nine of the thirteen, 4̄2m nine, and mm2, whose two mirrors between them contain three of the axes and several diagonals, eight. The more mirrors an achiral class has, the more ways there are to align it into a texture that is achiral too.

The result is a relative of a hand made of pieces that have none, where achiral tetrahedra built a chiral quartz crystal because the arrangement did not keep their mirrors. Here achiral crystals build a chiral texture for the same reason. A mirror of the grain survives into the texture only if it keeps the fibre direction, and aligning the grains along a two-fold axis of 4̄2m keeps none of its mirrors. The difference is that the texture’s hand is measurable as a number, the rotation along the fibre, and that the number is the grain’s own, unaveraged.

What the computation has to refuse #

Each result is computed from the fifteen classes’ own tensors and matrices, and each is tested against a case that would expose it if it were wrong.

The first refusal is the boundary of the whole method. A grain with no gyration tensor contributes nothing to any average, so no texture of such grains can rotate light by averaging. Yet a pile of thin mica plates, each turned by sixty degrees from the one below, does rotate light, as Ewald Reusch showed in 1869, and so does a cholesteric liquid crystal whose molecules twist steadily along an axis. Mica is centrosymmetric and has no gyration at all. The rotation of a Reusch pile comes from interference between waves crossing layers whose thickness is comparable to the wavelength, and it depends on that thickness. It is a property of the structure on the scale of the light, not an average of a property of the grains, and the tensor arithmetic here is right to say it knows nothing about it. The second refusal is the powder, which is the result the essay most needs a reader not to misremember: silver gallium sulphide rotates light, and ground to a powder it does not.

A fibre whose grains are not perfectly aligned #

The fibre is the simplest texture with an axis, and real ones are not perfect: grains are spread about the fibre direction by some angle, and the rotation along the axis then falls from the grain’s value towards the powder’s. For an achiral class the powder value is nought, so the spread does not merely weaken the effect but interpolates between a chiral texture and an inactive one. How the rotation falls with the spread is a single integral over the orientation distribution, and it has a closed form: the degree-two part of the tensor is multiplied by the average of the second Legendre polynomial of the tilt angle, which is one for perfect alignment and nought for a uniform spread. So the rotation along a fibre of an achiral class is the grain’s rotation along the aligned direction times that one average, and a measured rotation, divided by the single-crystal value, reads off how well the fibre is aligned. That is the quantity a measurement of a real drawn fibre would test. The same arithmetic applies unchanged to any other axial property of the texture, and to the gyration tensor of the piezoelectric classes a poled ceramic can hold.