A k-factor is a promise about the detector and the foil. This page shows what the promise costs as the foil gets thicker.
Theoretical k-factors from ionisation cross-sections, fluorescence yields and the detector window; the Cliff-Lorimer ratio with the absorption correction iterated as the foil thickens; zeta-factor quantification that gives composition and mass thickness together; synthesised spectra with overlapping peaks fitted by least squares to see what an overlap costs; beam broadening against thickness for the spatial resolution of a map; and a tilt sweep that shows what the specimen tilt does to the effective take-off angle, to the absorption, and to the composition your software prints.
Theoretical k-factors carry an uncertainty of 10 to 20 percent (more for the L and M lines, and for the light elements whose window absorption dominates); a measured k-factor from a standard of known composition, entered in the Cliff-Lorimer tab, is always preferable. The table is for the detector on the right; change the window to see what it does to the light elements.
Theoretical zeta factors here use the same cross-sections and yields as the k-factors together with the detector solid angle; measured zeta factors from a standard of known thickness (or from k-factors and one thickness) should replace them when available. The dose is the probe current times the live time, in electrons.
The detector points in a fixed direction. Tilting the specimen changes two things at once: the beam path through the foil lengthens as 1/cos β, so more X-rays are generated, and the angle at which those X-rays leave the foil changes, so their absorption path changes too. Only the second one distorts a composition, because absorption is line-dependent and generation is not.
With the detector at a nominal take-off α0 and an azimuth φ measured from the tilt axis, tilting by β about that axis gives
sin αeff = sin α0 cos β
− cos α0 sin φ sin β
Two consequences worth keeping. At φ = 90°, the detector perpendicular to the tilt axis, the second term takes the whole tilt: sin αeff collapses to sin(α0 − β), so the take-off angle tracks the tilt degree for degree and hits zero at β = α0. The foil then shadows itself and the count rate goes with it. Real holders shadow before that, because the holder body and a grid bar get in the way first, which is what the shadow threshold below is for.
At φ = 0, the detector over the tilt axis, the second term vanishes and only the cos β is left: sin αeff = sin α0 cos β. That still falls, because tipping the foil tips its normal away from a detector that sits at a finite elevation, but it falls slowly and second order: at α0 = 20° a 30° tilt costs under 3° of take-off, against 30° at the other azimuth. If you have a double-tilt holder and a long tilt series ahead of you, that is the axis to put the tilt on.
The composition curve here is the uncorrected Cliff-Lorimer answer: the specimen is held at the composition worked out on the Cliff-Lorimer tab, the absorption is recomputed at every tilt, and the ratio is read back without any correction. It is what your software prints if the take-off angle it uses is the nominal one.
Lines are chosen automatically (K below 0.6 E0 and Z up to 56, else L, else M); kA,ref converts intensity ratios to weight-fraction ratios: CA/Cref = kA,ref IA/Iref.
One line per row: element, family (K, L or M), integrated counts, and optionally a measured k-factor relative to the first line (a fourth number) and a mass absorption coefficient override in cm2/g (a fifth).
Element, family, counts, and optionally the zeta factor in g/cm2 per count per electron (a fourth number; otherwise the theoretical value for the solid angle above).
Counts are the alpha-line totals; the beta lines follow with the usual ratios. The fit uses templates of every family listed plus a Kramers background, over 0.2 keV to E0.
0° puts the detector over the tilt axis, 90° perpendicular to it. Most single-tilt work is near 90°; a double-tilt holder lets you choose.
Lines, composition, thickness and density come from the Cliff-Lorimer tab; the nominal take-off angle comes from the detector card above.
Quantitative X-ray microanalysis in the thin foil rests on the Cliff-Lorimer equation, CA/CB = kAB IA/IB: the ratio of two characteristic intensities gives the ratio of weight fractions through a sensitivity factor that depends on the two elements, the voltage and the detector, but not on the specimen, provided the foil is thin enough for absorption and fluorescence to be neglected. The first tab computes k-factors from first principles: the ionisation cross-section in the Bethe form with the Powell constants, the fluorescence yield from Bambynek's fit for the K shell and a linear one for L, the alpha-line fraction, the atomic weight, and the detector efficiency from the window, the dead layer and the silicon thickness. Such theoretical factors are what the microscope software uses in the absence of standards, and they are good to 10 or 20 percent.
As the foil thickens the softer X-rays are absorbed on their way out, so the intensity ratio drifts and the composition apparently changes with thickness. The second tab applies the thin-film absorption correction, the factor (1 − e−χρt)/(χρt) with χ the mass absorption coefficient of the specimen for the line divided by the sine of the take-off angle, and iterates it because the coefficient itself depends on the composition being sought; the curve shows the composition the uncorrected ratio would give at every thickness, which is the test of whether a given foil is thin enough. The zeta-factor method of Watanabe and Williams replaces the ratio with absolute intensities: with the beam current and live time known, each line gives ρt CA = ζA IA/De, and summing over the elements yields the mass thickness and the composition at once, with the absorption correction again iterated; the third tab does that, with theoretical zeta factors from the same physics plus the detector solid angle when measured ones are not entered.
The overlap tab synthesises a spectrum from the alpha and beta lines of the chosen elements at the detector resolution, adds a Kramers background and counting noise, and fits it by linear least squares with one template per element family. The quantities that matter come out of the normal equations: the variance inflation factor of each fitted count, which says how much the overlap magnifies the statistical uncertainty relative to an isolated peak, the counting-noise uncertainty itself, and the fraction of foreign counts that a simple window integration would pick up. The last tab plots Goldstein's single-scattering estimate of beam broadening, b = 7.21 × 105 (Z/E0) (ρ/A)1/2 t3/2 in cm with E0 in eV, and the two usual combinations with the probe size for the resolution of a map.
The cross-sections, yields and absorption coefficients are parameterised, not tabulated: the mass absorption coefficients follow a Z4/(A E3) law with K and L edge jumps, adequate for the correction factors but not a substitute for the Henke or Chantler tables when precision matters. Fluorescence within the foil, the Coster-Kronig redistribution among the L subshells, the detector's escape and sum peaks, incomplete charge collection and the low-energy tail are not modelled, and the light-element k-factors (B to F) depend so strongly on the window and the contamination layer that only measured values should be trusted. The overlap fit assumes Gaussian peaks with the resolution given; a real detector adds tails and a shelf that a full spectrum fit must handle. Beam broadening by the Goldstein formula is a single-scattering estimate that overstates the broadening of thin foils and understates it for thick ones; the Monte Carlo and the Reed models bracket it.
@misc{tripathy2026edsquantification,
author = {Tripathy, Manisha},
title = {EDS Quantification},
year = {2026},
howpublished = {\url{https://untetheredatom.com/tem/eds-quantification}},
note = {Interactive web tool}
}