Three quarters, exactly, for every mode that is not totally symmetric

One of the figures on spectra: How many bands there can be, where they sit, and what an absent one proves.

Eight essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

The one intensity symmetry does fix

Symmetry says the ratio moves and the parameters say how far. For a totally symmetric mode the ratio is not fixed by the group, and what it is instead is a function of the bond polarisability parameters — so the same mode of the same molecule can be quoted with quite different ratios by two people who disagree about a parameter and about nothing else.

The depolarisation ratio of every Raman-active mode of five molecules. Fourteen of the eighteen sit on three quarters to better than a millionth, and they are exactly the modes that are not totally symmetric. The four below the line are the totally symmetric ones, and where each of them sits is a property of the model rather than of the group.

The mean polarisability derivative of each distinct mode of methane. Only the totally symmetric one is not zero; the rest come back between 10⁻³² and 10⁻¹², which is arithmetic noise. That zero is the whole of the argument above, and it is a statement about a trace rather than about a polarisability.

A ratio that squares what it measures

How far ammonia’s depolarised bands come off three quarters against how far one of its bonds has been stretched. The line is straight on logarithmic axes with a slope of two.

The exact three quarters, band by band, across five molecules. Every one of those exact values is what this essay is measuring departures from.

The same distortions computed twice. Both curves start at three quarters and both leave it — that part is symmetry. They differ by a factor of six hundred and fifty at the same distortion, and that part is a table of bond polarisabilities.

One number was one direction

The coefficient of the square for every non-symmetric coordinate of three molecules, against the coordinate’s own frequency.

Methane’s three non-symmetric species, each displaced. Three parabolas, three coefficients.

Every non-symmetric coordinate of the three molecules, with what it does.

The distortion the ratio cannot see

Six ways of stretching the three bonds of boron trifluoride, all at the same displacement. Five move the ratio; one does not move it at all.

The three single-bond distortions and their sum.

Each visible distortion, with and without a breathing component added.

One number was a direction too

The departure of a depolarised band from three quarters, for distortions of equal magnitude pointing all the way round the plane of stretches that sum to zero.

The same reading plotted radially against the direction of the distortion, so the six-fold pattern is the shape rather than a repeat.

The reading at every sixty degrees, which the molecule’s three-fold axis and mirror planes map onto one another.

The sum was flat all along

Both readings round the circle, each divided by its own average so the two can be drawn together.

Each of the four depolarised bands, ordered by size at every direction and followed round the circle.

How much each reading varies round the circle.

The suspect that did not fit

The residual against amplitude, both logarithmic, with the two predicted slopes drawn through. The measurement lies between them.

How far the measured exponent sits from each explanation. The nearest is off by a quarter, which is not close enough to claim.

Each amplitude’s residual as a ratio to the single-power fit. Seven per cent at worst, and the misses are not scattered.

Two integers made one exponent

At one amplitude, the sum split into its even and odd halves, each as a fraction of the mean, against the direction of the distortion.

The combined residual, the even half’s anisotropy and the odd half relative to the mean, against the amplitude on logarithmic axes.

The odd half divided by the amplitude, against the amplitude. A pure first-order term would be a horizontal line.