What a linewidth leaves of the probe

A depolarised Raman band sits at exactly three quarters — the one intensity symmetry does fix. Distort the molecule and it comes off three quarters by an amount second order in the distortion, so the departure is a structural probe, and a weak one, because a ratio measured to three decimals is a distortion measured to one and a half. It also reads a direction: swept round a circle of distortions in the plane of non-symmetric stretches, the largest departure over boron trifluoride’s four depolarised bands varies by a factor of two, with a sixty-degree period that belongs to the point group and not the molecule.

That factor of two needs every band read separately. The sum over the four bands is flat to four parts in ten thousand, and the observable an unresolved pair actually gives — each pair’s intensities added in each polarisation before the ratio is taken — is flat to a few parts in a thousand. So the probe has two limits: a resolved reading that carries the direction, and a blended one that nearly does not.

A real spectrum is at neither. Its bands have widths, the distortion splits each degenerate pair by a fraction of a wavenumber or a few, and whether a pair is “resolved” is a matter of degree. The question left open was where between the limits a spectrum of a given line width reads, and it turns out to have three parts: which splitting decides, how long a merged pair takes to become its blend, and whether any resolution is enough.

A spectrum is an envelope, not a list of bands #

A spectrum does not report bands. It reports intensity against frequency, each band drawn as a line of some width and all of them added, and its depolarisation ratio is read where that envelope peaks: the perpendicular-polarised intensity over the parallel-polarised one at that frequency. A spectrum counts environments, not atoms, and in the same way it counts resolved peaks, not eigenvalues.

Each band contributes to both polarisations. Its parallel intensity is $45\bar a^2 + 4\gamma^2$ and its perpendicular intensity $3\gamma^2$, with $\bar a$ the mean of its polarisability derivative and $\gamma^2$ its anisotropy; the band’s own ratio is the second over the first, and that is three quarters exactly when $\bar a$ vanishes. Wherever two bands overlap, the ratio at a frequency is the sum of their perpendicular contributions there over the sum of their parallel ones, each weighted by how far that frequency sits from the band’s centre. The weights are the whole story below.

For boron trifluoride stretched by two hundredths of an ångström along a direction thirty degrees round the circle, the lower E′ pair — the in-plane bends — is split into components at 479.02 and 481.81 cm⁻¹, 2.8 apart. The upper pair, the stretches, is split into 1453.64 and 1454.17, about half a wavenumber apart. At a line width of half a wavenumber the bending components are two clear peaks. At two they are still two peaks, overlapping so heavily that the ratio read at the upper one has already fallen from 16.0 to 14.4 × 10⁻⁶, and at five they are a single peak with no shoulder at all.

The direction survives until the bending pair merges #

With the envelope defined, the probe at any line width is a calculation: read the ratio at every peak, take the largest departure from three quarters, and do that at each direction round the circle.

Read from the bands themselves, the probe varies by 1.956 round the circle. At a line width of half a wavenumber it still varies by 1.941 — and the stretching pair has already begun to merge. At one wavenumber the stretching pair shows one peak and the variation is 1.900. At two it is 1.762, at three 1.585, at five 1.094, and at ten 1.010.

The direction is lost where the bending pair merges, not where the first pair does. The stretching pair’s components are the closer of the two, and a spectrum loses them first. That costs the probe almost nothing, because the largest departure was never theirs.

At thirty degrees the bending pair’s upper component is 1.6 × 10⁻⁵ from three quarters and the stretching pair’s upper component 4.8 × 10⁻⁶; the two lower components sit exactly at three quarters, because thirty degrees is a direction that keeps a mirror plane, and the band antisymmetric under that mirror has no mean polarisability derivative to acquire. At zero degrees all four depart, and the bending pair’s two are again the larger, 8.2 and 7.9 × 10⁻⁶ against 2.3 and 2.4.

So the splitting a spectrometer has to beat is not the smallest one in the spectrum. It is the one belonging to the pair that carries the largest departure — which, for this molecule, is the wider pair. One number was one direction found the probe’s sensitivity spread by a factor of thirty across coordinates, with no preference for the soft ones; here the same indifference shows up in frequency, and the pair that matters is not the pair nearest to merging.

A pair drawn as one peak is not yet read as its blend #

At five wavenumbers each pair shows a single peak. The natural reading is that the spectrum is now reporting the blend — the pair’s total perpendicular intensity over its total parallel intensity, which is what an unresolved pair was taken to give. It is not, and the difference is fourteen-fold.

The blend varies round the circle by 0.66 per cent. The spectrum at five wavenumbers varies by 9.4 per cent, with every pair a single peak. At seven it is 1.3 per cent, at ten 0.99, at twenty 0.75, closing on the blend from above.

The reason is where the ratio is read. The envelope’s maximum does not sit halfway between the components: at thirty degrees and five wavenumbers the bending pair’s single peak is at 480.67 cm⁻¹, nearer the upper component at 481.81 than the lower at 479.02, because the upper carries slightly more parallel intensity and pulls the maximum towards itself. At that frequency the upper component’s line is taller than the lower’s, so the ratio read there weights the upper component — the one carrying all of the departure at this direction — by more than its share of the pair’s total intensity. The peak reads 8.8 × 10⁻⁶ where the blend is 8.1 × 10⁻⁶.

As the line widens, the two lines evaluated at the peak approach equality, and the reading approaches the blend. The approach is quadratic: a Lorentzian’s value a small distance from its centre differs from its peak value by the square of that distance over the width, so the excess should fall as the square of splitting over width. It does. Doubling the width from ten to twenty divides the excess over the blend by about four.

So “one peak” is not a threshold. The count of peaks changes abruptly, at a width between four and five wavenumbers for this pair; the reading changes smoothly, and at the width where the count drops to one it is still much nearer the resolved limit’s behaviour than the blend’s. A measurement that asks whether a pair is resolved gets a yes-or-no answer to a question whose honest answer is a number.

Resolution is the easy half #

The curves above are for one distortion, two hundredths of an ångström. The two quantities that decide whether a direction is readable scale differently with the distortion, and the difference is the practical conclusion.

The splitting is first order: 0.70, 1.41, 2.82 and 5.64 cm⁻¹ at amplitudes of 0.005, 0.01, 0.02 and 0.04 Å, a slope of 1.000 on logarithmic axes — 141 cm⁻¹ per ångström. The departure is second order: 5.0 × 10⁻⁷, 2.0 × 10⁻⁶, 8.2 × 10⁻⁶ and 3.3 × 10⁻⁵, a slope of 2.01 — about 0.02 per ångström squared. Halving the distortion halves the resolution needed and quarters the signal that resolution is needed for.

At a distortion of a hundredth of an ångström, the size of a real symmetry-lowering perturbation of a bond, reading a direction takes a line width under 1.4 cm⁻¹ — demanding for a Raman band, but achieved in gas-phase work — and a depolarisation ratio measured to two parts in a million, which is not achieved anywhere. A well-measured depolarisation ratio is known to a few parts in a thousand.

So the probe’s direction information is unreadable for a reason that has nothing to do with line width. Resolution is the obstacle that is easy to see, because the figures of every essay before this one drew bands a fraction of a wavenumber apart, and it suggests the remedy of a better instrument. The departure itself is the obstacle that decides, and the essay that first measured its exponent already had the number: a second-order signal cannot be rescued by an instrument whose requirement is first order.

What survives a wide line is third order #

There is a third scaling, and it settles what a spectrum too broad to split the bending pair could still say. The resolved probe’s variation round the circle is a ratio, largest over smallest, and it does not depend on the amplitude: about two at every distortion small enough for the quadratic term to lead, because it is the shape of a quadratic form and a quadratic form’s shape does not change when its argument is scaled. The blend’s variation does depend on the amplitude. The essay that first computed it found it almost entirely odd under reversing the distortion and proportional to the first power — 2.9 × 10⁻³ at a hundredth of an ångström, 6.6 × 10⁻³ at two hundredths here.

So what a wide line leaves of the direction is a product of two small things. The departure itself is 8.1 × 10⁻⁶ at two hundredths of an ångström, and the part of it that changes round the circle is two thirds of a per cent of that: 5.3 × 10⁻⁸, between 8.040 and 8.093 × 10⁻⁶. Halve the distortion and the departure falls by four and its variation by roughly two more, so the directional signal in a blended spectrum falls as the cube of the distortion. A resolved spectrum loses the direction to a second-order signal; a blended one loses it to a third-order one, and no ratio precision anyone has contemplated separates the two readings at thirty degrees and at zero.

That is why the width matters at all, even though resolution is not the obstacle. Resolving the pair carrying the departure buys one order in the distortion. It does not buy a measurement, but it decides whether the thing left to measure is the departure or a sliver of it.

What was computed and how #

The bands are the library’s own: boron trifluoride’s fitted valence force field, the molecule distorted by stretching its three bonds by a combination of the two non-symmetric stretch directions with an overall amplitude of 0.02 Å unless stated, the normal modes recomputed on the distorted geometry with the same constants, and each mode’s polarisability derivative from the bond-polarisability model every depolarisation calculation here has used. From each derivative come $\bar a$ and $\gamma^2$, and from those the parallel and perpendicular intensities for scattering at ninety degrees.

Each pair is drawn as a sum of Lorentzian lines of a stated full width on a grid of four thousand points spanning three widths beyond the pair, the maxima of the parallel envelope are found, and the ratio is read at each. The probe is the largest departure over the peaks of both pairs. Directions run from 0° to 60° in steps of 7.5°, which by the sixty-degree period covers the circle; the table runs over widths from 0.05 to 10 cm⁻¹, and the approach to the blend out to 20.

The claims are stated where they can fail: that the resolved probe varies by more than 1.8 round the circle; that at a width of ten each pair is one peak and the variation is under a tenth; that this reading agrees with the intensity-weighted blend to two hundredths; that at five wavenumbers, with one peak a pair, the variation is still more than five times the blend’s; that the excess over the blend falls by between three and five and a half times when the width doubles from ten to twenty; and that the splitting and the departure scale with slopes of one and two, within a tenth and fifteen hundredths. The refusal is the zero-width reading, which must equal the largest of the bands’ own departures exactly, or the envelope is not the bands it was built from.

Where the model stops #

Lorentzian lines of one width. A real Raman band in the gas phase has a rotational contour tens of wavenumbers wide and a different shape in each polarisation, and in a liquid a width set by collisions. A single width for every component is the simplest line shape that shows where merging happens, and the quadratic approach to the blend is a property of any smooth line with a rounded top; a Gaussian would move the numbers, not the order. The widths quoted should be read as multiples of the splitting, not as a prediction for an instrument.

A bond-polarisability model. The mean derivative’s departure from zero is the whole signal, and it is computed in a model whose polarisability derivatives are sums of bond contributions. A different polarisability model would change the departures’ sizes and the intensities that weight the blend, though not the orders in the distortion, which come from symmetry.

And one amplitude of one molecule. The pyramid had departures 169 times larger at the same amplitude, which would move the required ratio precision by the same factor — from two parts in a million to about three parts in ten thousand at a hundredth of an ångström. That is closer to measurable and still at the edge of what a depolarisation ratio is ever known to.

An instrument reads a quantity of its own #

The general point is that “what an experiment would see” is a calculation, and the calculation can disagree with every quantity it was standing in for. The essays before this one had three readings of one set of bands — a maximum, a sum and a blend — and the natural assumption was that a spectrum would report the maximum while it resolved the bands and the blend once it did not. It reports a fourth thing: a ratio read at the top of an envelope, whose weights depend on where that top happens to fall. It agrees with the maximum at zero width and with the blend at infinite width, and at every width a real spectrum has it is neither.

The second point is about which variable to watch. The obvious one is the count of peaks, since it is what a spectrum visibly shows and it changes abruptly. The one that decides is the ratio of the splitting of the pair carrying the departure to the line width, and it changes smoothly, over a factor of three or four in width, with the count of peaks dropping somewhere in the middle of that range.

Still open: one band followed, and the pyramid at finite resolution #

The open question these essays have carried is following a single physical band round the circle rather than the maximum over bands, identifying it by its composition rather than its rank. At finite resolution that question changes character. A spectrum follows peaks, and a peak is a physical band only while it is resolved; where the bending pair merges, the peak that carries the reading is neither component, and following “the same band” through that width means following something the spectrum does not contain. Whether the resolved question still has an answer an instrument could check is not settled.

The nearer question is ammonia. Its departures are 169 times boron trifluoride’s at the same amplitude, its degenerate pairs are split by different amounts, and its residual was not even a power law. The width at which its direction disappears, which of its pairs decides, and the ratio precision needed to read it are all one calculation away with the method here — and they decide whether the pyramid, rather than the plane, is the molecule on which this probe could ever be tried.