Incoprea research · spectroscopy and diffraction measurement · 3 November 2026

Spectroscopy and diffraction measurement: two hundred leads and six simulations

1. How to read this page

Four sections of fifty leads. Each lead carries a number, a single falsifiable sentence and a disposition: test means the claim can be checked with code or with a public table that already exists, and is worth checking now; note means the claim is worth recording but is a reporting convention, a design rule, or blocked on data nobody has assembled. Nothing carries a third tag because nothing here has been tested outside the six simulations in §6.

The leads sit as close to the measurement as I could put them. Where a number appears in a lead it is a threshold that a test would have to hit, not a result. The simulations in §6 are the only places on this page where a number is a measurement, and every one of them is a synthetic spectrum with known parameters.

What this page is for: the questions that decide whether a quoted line position, width or detection limit means anything. What it is not: a review, a benchmark, or evidence about any real sample.

2. Section I · Line shape, broadening and what deconvolution cannot recover (50 leads, I1–I50)

Every lead here is about what a line shape records once the instrument has been applied to it. The measured profile is the true profile convolved with an instrument function, and convolution is not invertible where the transform is small. §6.1 measures the simplest case numerically: a Lorentzian fitted to a Gaussian-convolved Lorentzian returns a width that climbs to four times the truth as the instrument width grows, while the width of the fitted Lorentzian is recovered to a few per cent as soon as the instrument width is known and the wings are in the window.

3. Section J · Calibration, referencing and unit conventions in spectra (50 leads, I51–I100)

A wavenumber axis is a chain of conventions: which laser line is the zero, which standard fixes the scale, whether the wavelength is in air or in vacuum, whether a width is in nanometres or inverse centimetres. Each link has a lever arm, and §6.5 measures the two largest: a Raman shift is the difference of two wavenumbers of order ten thousand, so one part per million of laser error becomes a hundredth of a wavenumber at 1000 cm^-1, and a straight-line shift calibration across a 3300 cm^-1 window leaves residuals near 100 cm^-1 because the dispersion is linear in wavelength and the axis is not.

4. Section K · Sampling, counting statistics and detection limits (50 leads, I101–I150)

Counting statistics decide what any of the other sections can measure. §6.4 gives the Currie levels against integration time and checks them by Monte Carlo, which shows the Gaussian rule holding to 5% above ten background counts and failing below it. §6.6 asks how much sampling a line needs, and the answer contradicts the habit: at SNR 200 two points per FWHM still return the width of a Lorentzian to better than 1%, because the wings and not the peak carry the width.

5. Section L · Cross-technique disagreement: Raman vs IR vs neutron vs X-ray (50 leads, I151–I200)

Two techniques measuring the same sample disagree for reasons that are mostly not physical: different resolution functions, different sampling depths, different referencing, different definitions of width. The useful question is never whether two numbers differ but whether the difference exceeds the quadrature sum of the two calibration uncertainties — which is a number almost never quoted beside a cross-technique comparison.

6. Tests run

Six simulations, all on synthetic spectra, all in incoprea_nov3/code/spec_nov3.py, with results in incoprea_nov3/results/spec_nov3.json. numpy.random.default_rng(20261103), numpy 2.5.4, 200 replicates per cell in §6.1 and §6.3, 150 in §6.2, 40000 Monte-Carlo draws per row in §6.4. Noise is Poisson on the model counts plus a constant baseline of 100 counts unless a row says otherwise, so a peak of 4×10⁴ counts is a per-channel signal-to-noise of 200. The fitted width is the Lorentzian half width γ, and every width in the tables is on that scale, so a FWHM is 2γ.

Two checks before any of it means anything. First, the Voigt profile used to generate and to fit is computed by two quadrature rules — Gauss–Hermite on the Gaussian weight while σ ≤ γ, and a truncated tangent substitution while σ > γ — and both are checked against an independent FFT convolution. Second, every fitter used below must recover its own generating parameters from noiseless data. A first version of this script normalised the model profiles by dividing by the sampled maximum, which makes the model jump whenever the maximum moves from one grid point to the next; the Jacobian became noise and the fits stalled at a 0.2%-of-peak misfit. The recovery table is what caught it, and the numbers in this section are from the corrected run.

6.0 Checks: quadrature and noiseless recovery

Reading. The quadrature agrees with the convolution to 3.6e-06 at worst across the eight width pairs. The noiseless column is the important one: the free Voigt returns γ and σ to four decimals, the two-peak fit returns its separation to four decimals, and the Lorentzian-only fit returns γ = 8.5096 against a true 5.0000 — that last number is not an error, it is the least-squares answer to the wrong question, and §6.1 shows how it grows.

6.1 A Lorentzian width behind a Gaussian instrument function

Simulation. A Lorentzian of half width γ = 5 cm⁻¹ (FWHM 10) at 1000 cm⁻¹, convolved with a Gaussian instrument function of standard deviation σ and sampled at 301 points over a window scaled to the broadened line. Peak 4×10⁴ counts. Four fits per replicate: a Lorentzian, a Gaussian, a Voigt with both widths free (from two starting points, lower residual kept), and a Voigt with σ held at its true value — the case where the instrument width has been measured on a standard. The wing ratio is the fitted Lorentzian intensity at ten FWHM divided by the true intensity there.

Reading. The Lorentzian-only fit is the common practice and it is wrong in a direction that never reverses: at σ/γ = 1 it returns 8.38 against a true 5.00 (+67.6%), and at four times the line width it returns 25.60 (+412.0%). The observed FWHM at that point is 52.8 cm⁻¹, so the fit has not discovered a broader line — it has described a Voigt with a Lorentzian of the wrong width. The wing column is the consequence: at ten FWHM the fitted Lorentzian is 4.17 times too bright, so a background estimated out there contains a line that the fit has already put somewhere else.

The part I expected to be worse. With σ held at its true value the fit recovers γ to 0.0% even when the instrument is four times wider than the line, and the free-Voigt fit is almost as good at this signal-to-noise (5.01 ± 0.10). The decomposition is not ill-posed at SNR 200 with six instrument widths of wing on each side; it becomes ill-posed when the signal-to-noise falls, which is §6.2.

6.2 The same deconvolution at falling signal-to-noise

Simulation. σ/γ = 2 (σ = 10, γ = 5 cm⁻¹), 150 replicates per row, peak counts from 4×10⁵ down to 40. The bias of the Lorentzian-only fit is a property of the shape and should not move; what moves is the spread of the honest fit.

Reading. The Lorentzian-only bias sits between 174% and 176% at every signal-to-noise from 632 down to 6.3: it is a systematic and no amount of integration removes it. What integration buys is the free-Voigt width: its standard deviation runs from 0.02 at SNR 632 to 6.19 at SNR 6.3, and at the bottom of the table 68% of fits land more than 50% from the truth. Holding σ fixed at the known instrument width cuts that spread by about a factor of three everywhere (0.27 against 0.71 at SNR 20), which is the cheapest improvement available and costs one calibration measurement.

6.3 One peak fitted to two overlapping lines

Simulation. Two Lorentzians of half width 5 cm⁻¹, separated by 0.25 to 4 FWHM, height ratio 1:1 and 2:1, peak 4×10⁴ counts, 200 replicates. The single-peak fit has three shape parameters plus amplitude and offset; the two-peak fit has four shape parameters plus the same two. Residual/noise is the root-mean-square residual of the single-peak fit over the per-channel noise, so 1.0 is a good fit and 10 is a visible doublet.

Reading. For an equal doublet the single-peak centre is unbiased by symmetry and everything else goes wrong instead: at half an FWHM of separation the width is +27.0% high and the area +8.6%, and at one FWHM the width is +98.1% high with a residual 14 times the noise. That residual is the tell: a fit that is 14 times the noise and still has a plausible-looking width is the failure mode that gets reported, because the width is what was asked for and the residual is what nobody plots.

Unequal lines move the position, which is worse. At a height ratio of 2:1 the single-peak centre is pulled +9.25 cm⁻¹ at two FWHM of separation and +19.95 cm⁻¹ at four, which is a shift comparable with the chemical shifts these positions are used to argue about. A two-peak fit recovers the separation in essentially every replicate down to a quarter FWHM (100% of replicates within 15%), so the information is present; what fails is the decision to fit one peak.

6.4 Counting statistics: detection limit against integration time

Simulation and arithmetic. Background 10 counts s⁻¹, signal 1 counts s⁻¹, one channel. The critical level is L_C = 1.645√B and the detection limit L_D = (1.645 + 1.645)√B, the equal-variance form, applied to the net count. Both are checked by drawing 40000 Poisson samples per row: the false-positive column is the fraction of blank measurements above L_C and the power column is the fraction of background-plus-L_D measurements above it.

Reading. The detection limit falls as t^−1/2 exactly: the fitted log–log slope is -0.500, and the minimum detectable rate goes from 32.900 s⁻¹ at 0.1 s to 0.1040 s⁻¹ at 10⁴ s. A hundredfold better detection limit therefore costs ten thousand times the integration, which is the arithmetic that decides whether a weak band is worth an overnight run.

Where the Gaussian rule breaks. Above ten background counts the Monte-Carlo false-positive rate sits at 0.050 against a nominal 0.050 and the power at 0.863 against a nominal 0.95 — the approximation is fine. At one background count the same rule rejects 8.2% of blanks against a nominal 5%, and the power at L_D falls to 80%. A detection limit quoted from a handful of counts is therefore optimistic in both directions at once: more false positives and less power than advertised.

6.5 Width conventions and subtracting the instrument in quadrature

Computation. Three profiles on a common grid, integrated over ±4000 cm⁻¹. The second-moment column is the root mean square about the centre divided by the FWHM, computed over a window of the stated number of FWHM.

Reading. The same line has an integral width of 1.570 FWHM if it is Lorentzian and 1.064 FWHM if it is Gaussian — a 48% difference that is pure convention. The second moment is worse: for the Lorentzian the RMS width keeps growing with the window (1.20 FWHM at five FWHM of window, 5.63 at a hundred) because the integral diverges logarithmically, while the Gaussian converges at 0.42. A cross-technique comparison of RMS widths is therefore partly a comparison of integration ranges.

Now the operation almost everyone performs: assume the observed line is Gaussian, measure the instrument FWHM on a standard, and subtract in quadrature.

Reading. The true line is Lorentzian with a 10 cm⁻¹ FWHM. Quadrature subtraction is right in the limit of a narrow instrument (+0.8% at σ = 0.5) and wrong in the direction that flatters the result: at σ = γ it returns 13.62 for a true 10 (+36.2%), and at four times the line width it returns 17.51 (+75.1%). Linear subtraction of the instrument FWHM is wrong the other way and worse (-42.0% at the same point). Neither is a deconvolution; both are arithmetic on widths that assumes a shape nobody checked.

6.6 Calibration lever arms

Arithmetic. A Raman shift is σ_laser − σ_scattered, so d(shift)/dσ_laser = 1 and a relative error ε in the laser wavenumber becomes an absolute error εσ_laser in every shift. The lever arm is σ_laser/shift. The 0.1 nm column is what a laser-line specification of that size does to every peak.

Reading. At 532 nm the lever arm at 1000 cm⁻¹ is 18.8 and one part per million of laser error is 0.0188 cm⁻¹; a 0.1 nm specification is 3.5 cm⁻¹, which is a third of the FWHM of a typical polymer band. At 785 nm the lever arm falls to 12.7 and the same specification costs 1.6 cm⁻¹. An infrared spectrometer, which reports an absolute wavenumber from its own reference laser, has no lever arm of this kind — so a Raman–IR comparison of the same mode inherits the Raman laser specification and not the spectrometer's, which is exactly backwards from where the money was spent.

The second convention is the axis itself. A grating disperses about linearly in wavelength, and Raman shift is not linear in wavelength:

Reading. Over 200–3500 cm⁻¹ at 532 nm (λ from 538 to 654 nm) a straight line in wavelength leaves a maximum residual of 109 cm⁻¹; a quadratic leaves 6.5. At 785 nm the linear residual is 173 cm⁻¹. A calibration that is linear in shift over a wide window is therefore not a conservative approximation, it is a several-wavenumber error at the ends of the window and near zero in the middle, which is the shape of error that survives a check against a standard placed in the middle.

Diffraction has the same structure with a different geometry, because d = λ/(2 sin θ) amplifies a zero offset by cot θ:

Reading. A 0.02° zero offset in 2θ moves every d-spacing by 0.20% at 10° and 0.005% at 150°, a factor of 43 between the ends of the same scan. That offset is well inside what a sample-displacement error produces, so the low-angle reflections that carry the most d-spacing information also carry the most systematic.

Units, for completeness. From the SI definitions, 1 cm⁻¹ = 0.1239842 meV = 0.0299792 THz = 1.2398420e-04 eV.

6.7 Sampling density

Simulation. A Lorentzian of half width 5 cm⁻¹, peak 4×10⁴ counts, window ±30 FWHM, sampled at 2 to 40 points per FWHM and fitted with the correct model. 200 replicates per row.

Reading, and it is not what I expected. At two points per FWHM the width is recovered with a bias of -0.01% and a standard deviation of 0.027 on a width of 5 — 0.5%. Going to forty points per FWHM improves the standard deviation to 0.007 and does not improve the bias, which was already nothing. For a Lorentzian whose wings are in the window, the width lives in the wings and a sample or two per FWHM is enough to see them; the peak itself carries very little of the width information. The same table at SNR 20 is different:

Reading. At 400 peak counts the two-point sampling gives a width standard deviation of 0.369 against 0.086 at forty points, a factor of 4.3. So sampling and noise trade against each other and neither alone decides the precision: the relevant quantity is the number of counts in the wings, which is what a wide window and a fine grid both buy.

Lead I103 is refuted by its own section. I expected undersampling to bias a width upward before it biased a position. It does not: at SNR 200 the width bias at two points per FWHM is -0.01% and at SNR 20 it is -0.46%, with no monotone trend in either block. What sampling costs is precision, not accuracy, for a line whose wings are inside the window. The lead stays on the page with its tag because a lead is a claim to be checked, and this one has now been checked.

7. Counts

Two hundred leads, of which 181 are marked test and 19 note. The tags are a priority claim, not a quality claim: test means a check is cheap and would change what I do next, note means it would not, yet.

8. Limits

• All six tests are simulations. Synthetic Lorentzians, Gaussians and Voigts with known parameters, Poisson noise and constant baselines. No measured spectrum, diffractogram or database was used, so nothing here is evidence about any real sample or instrument.
• One instrument model. A Gaussian of known width. A real instrument function is asymmetric, varies across the focal plane and is measured imperfectly; the recovery in §6.1 and §6.2 is the best case and the bias of the Lorentzian-only fit is the part that survives.
• One line-shape family. Lorentzian convolved with Gaussian. No anharmonic (Duffy) line shapes, no asymmetric Voigts, no Fano profiles, no diffraction peak asymmetry, and no size/strain model beyond the quadrature arithmetic in §6.5.
• Noise is Poisson plus a constant. No read noise, no dark current, no cosmic rays, no fixed-pattern structure, no detector nonlinearity. §6.4 is the only place where noise is treated honestly, and it treats one channel.
• Six separations, two height ratios. The doublet table is not a map of the identifiability boundary; it samples it. A finer grid in separation would move the quoted percentages by a few points, not change their direction.
• The lever arms are arithmetic, not measurement. §6.6 computes what a stated laser specification does to a stated shift. It does not measure any laser, and the specification used (0.1 nm) is a placeholder for whatever a real device quotes.
• Nothing here is a literature claim. Several leads are surely known results under other names. No paper was read, no value was taken from one, and nothing on this page should be read as a citation. Where a lead states a threshold (10%, a factor of four, 0.02°) it is a number a test would have to hit.
• The leads are untested claims. Some are probably false and a few are certainly false as stated — that is what a lead is. The tags say which ones I would check first, not which ones I believe.

If one line survives: a width quoted without its line-shape model, its instrument width and its calibration convention is not a property of the sample. The instrument contributes a factor of four to a width and a part per million of laser drift contributes a third of a linewidth, while the counting statistics that would let you see either set a floor that falls only as the square root of the time you are willing to spend.

Back to the research hub