Intensity Corrections

What a measured intensity actually is

A diffractometer records counts. What a structure determination needs is |F|², and everything between the two is geometry and optics:

I_obs = K · L(θ) · P(θ) · A(θ) · m · |F|²

K is one scale factor for the whole data set and carries the beam current, the counting time and the sample volume; nobody knows it and nobody needs to, because a structure is refined against relative intensities. The other four are the ones that vary from line to line, and those cannot be scaled away:

• L, the Lorentz factor — for a powder, three things multiplied: how long a reflection spends in the diffracting condition (1 / sin 2θ), what fraction of the crystallites are oriented to reflect at all (cos θ), and what fraction of each Debye ring the detector sees (1 / sin 2θ again). Their product is 1 / (4 sin² θ cos θ). Pure geometry, no physics of the sample in it at all.
• P, the polarisation factor — an X-ray scattered through 2θ keeps the component of its field perpendicular to the scattering plane in full, while the component in the plane is scaled by cos 2θ. An unpolarised beam is half of each, so the intensity falls to (1 + cos² 2θ) / 2, and at 2θ = 90° only the perpendicular half survives. A monochromator polarises the incident beam before the sample ever sees it, which changes this factor and is the correction people forget.
• A, absorption — how much of the beam survives the trip in and out. For the flat plate almost everyone owns this turns out to be a constant, and the panel below shows why that is a fact about the geometry rather than about the sample.
• m, multiplicity — a powder cannot separate reflections that land at the same angle, so the peak carries their sum. The simulator counts them.

This page takes L and P apart, shows what A does, and then runs the whole chain backwards on a structure whose answer is known.

The factor, drawn

Hover anywhere on the plot to read L, P and their product at that angle — the chart is on a logarithmic scale precisely because the numbers span two orders of magnitude, and a log scale is not something to read a value off by eye. Three curves, because a linear axis would put everything past 40° on the baseline. The Lorentz factor falls steeply and then turns back up; the polarisation factor is a shallow dip bottoming at 2θ = 90°; their product has its minimum at the marked angle.

The turn is the part worth remembering. Lp does not simply decrease with angle — it reaches a minimum near 2θ = 98° and climbs again, so back-reflection lines are boosted by the same mechanism that boosts the low-angle ones. A reader carrying away only “divide by Lp, it gets smaller” has the second half backwards.

The monochromator

Both forms of P are one expression, (1 + K cos² 2θ) / (1 + K), with K = 1 for an unmonochromated beam and K = cos² 2θ_M for an ideally mosaic monochromator. Writing them as two formulae is how a page comes to apply one and claim the other, so there is no second formula here and no branch. (This K is the monochromator’s polarisation ratio, a property of one reflection in one crystal, and not the scale factor K at the top of the page; the literature uses the letter for both.)

This monochromator at 2θ_M = 26.577° gives K = 0.79984. Correcting its data with the unmonochromated formula instead would leave the lines below wrong by up to 9.26 per cent relative to the strongest one — not a scale error, which would cancel, but a tilt: the residual grows with angle, which is exactly where an overall displacement parameter lives. That is how a polarisation correction nobody checked ends up as a temperature factor somebody publishes.

cos² is the mosaic assumption and it is not universal. A perfect crystal — germanium, silicon — polarises as |cos 2θ_M| instead, which is larger, so a real monochromator sits somewhere between the two. Graphite is offered above because graphite really is mosaic; anything else is entered as an angle, with this assumption still in force.

The chain, run backwards

quartz at 1.541838 Å. The I column is what the pattern looks like — the simulator’s own intensities, each scaled to 100 on the strongest line. Divide by Lp and what is left is the ∑m|F|² column, scaled to 100 on its strongest line.

The two columns rank the lines differently, and that is the whole point of the correction. The strongest line is the strongest in both columns here — 101 — but the order below it is not the same order. 100 moves 15 places — 2nd strongest on the chart, 17th once Lp is off. A peak is what a detector recorded at one angle; a reflection is a point of the reciprocal lattice with a structure factor. Lp is most of what separates them, and it is why a pattern cannot be read as a table of |F|² by eye.

The Lp column is the unmonochromated factor, K = 1, whatever is selected above: the I column is the simulator’s and the simulator applies that one, so it is the only Lp that divides out of it. The monochromator you chose is in the last column instead.

That last column is what it would cost to correct data from the monochromator setting above with the unmonochromated formula, measured relative to the strongest line, because that is where a reader’s own scale comes from.

The window is the simulator’s own, 5–90°, so the table stops before the Lp minimum the chart above marks. The chart is drawn over the full range; the table is not.

Absorption, and why powder people ignore it

For a flat plate in symmetric reflection — Bragg–Brentano, which is what almost every laboratory diffractometer is — the beam in and the beam out travel equal paths, and the absorption factor comes out in closed form:

A(θ) = [1 − exp(−2μt / sin θ)] / 2μ

At large μt that is 1 / 2μ, with no θ left in it. Absorption does not drop out of powder work because it is small — a millimetre of rutile stops the beam in the first ten micrometres — it drops out because in this one geometry it is the same for every line, so it joins K in the scale factor nobody needs to know. Make the sample thin and it stops being constant, which is why a thin film or a smear mount on a zero-background wafer needs the full expression.

A capillary is the case with no closed form at all: the path length depends on where in the cylinder the scattering happened, and the integral has no elementary answer. That is why International Tables volume C tabulates A* against μR instead of giving an equation, and it is why this page does not compute it rather than fitting a polynomial nobody could check.

quartz at Cu Kα: μ/ρ = 35.0 cm²/g on a density of 2.649 g/cm³, so μ = 92.7 cm⁻¹. At 1 mm thick:

The last column is the ratio to the infinitely-thick value, and it is 1 at every angle for any sample thick enough — that is the constancy claim, computed rather than asserted. Set the thickness above to something genuinely thin and watch it stop being 1, first at high angle: there the beam crosses the plate most steeply, so its path through the whole thickness is shortest and more of it passes out through the back. At low angle the same plate is a long slant path and still looks thick.

The depths run the other way from the intuition. They carry sin θ, so the low-angle lines are the shallow ones: the first peak in a pattern is sampling a thinner slab of the specimen than the last one. On a sample with a gradient — a surface layer, a preferred-orientation skin from pressing the powder — that is a systematic difference between lines and not a scale factor.

What this page leaves out

Named rather than silently absent, because a correction nobody mentions is a correction nobody applies:

• Preferred orientation. Not a correction so much as a defect in the specimen: the crystallites are not randomly oriented, so m stops describing how many of them are in the diffracting condition. It is modelled (March–Dollase) rather than computed, and it needs a parameter refined against the data.
• Extinction. A strong reflection can be attenuated by re-diffraction inside a single mosaic block, which makes the strongest lines too weak — a correction that depends on |F|, so the thing being measured appears in its own correction.
• Thermal diffuse scattering. It sits under the peaks and rises with angle, so it is absorbed into the background and then into the displacement parameters.
• The capillary absorption factor A*, for the reason given above.

Everything on this page is exact arithmetic on stated assumptions. Everything in this list needs either a model or a measurement, which is a different kind of claim.

The same chain on a single crystal

The Fourier and direct-methods pages send a reader here for the step between a recorded intensity and |F|², and that step is the same one either way: I = K L P A  |F|². What differs is what goes into each letter, and the arithmetic above is Bragg–Brentano throughout.

• L is a different function of angle. The Lorentz factor measures how long a reflection spends in the diffracting condition, so it follows the way the sample moves. For a crystal rotated about an axis normal to the beam it is L = 1 / sin 2θ, which is the first of the three factors in the powder expression on this page. The other two — the fraction of the crystallites oriented to reflect, cos θ, and the fraction of the Debye ring the detector sees, 1 / sin 2θ — belong to a powder, and a single crystal has no equivalent of either.
• P is the same. Polarisation is a property of the scattering, not of the specimen, so (1 + cos² 2θ) / 2 and the monochromator correction above apply unchanged.
• A is not constant. The flat-plate result above — absorption the same at every angle, so it joins the scale factor — is a fact about that geometry alone. A crystal is a small object of its own shape, so A depends on the path through it for each reflection, which is why single-crystal work measures it (multi-scan, ψ-scan) or computes it from indexed faces.
• There is no m. A single crystal separates reflections that a powder adds together, which is the whole reason the experiment is worth the trouble.

So the label at the top of this page is about the expressions: read the chain here, and take only P and the shape of the argument with you.

Where this sits

Upstream: peak finding turns a scan into positions, heights and areas — the integrated area is the quantity this page corrects, not the peak height. Downstream: the structure factor is what |F|² is compared against, and the Wilson plot is the first thing done with a corrected data set.

Beside it: absorption computes μ/ρ for any formula, which is where the coefficient in the panel above comes from.