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.
Try this: change the window from polymer to beryllium in the detector card and watch the k-factors of the light elements (B to F) in the table.
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.
Try this: raise the thickness t and watch the uncorrected composition curve pull away from the corrected one.
Try this: double the live time or the probe current and watch the mass thickness and t in the result line.
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.
Try this: pick an example pair and watch the fitted components and the variance inflation in the Fit table.
Try this: change the material or the thickness t and watch the broadening curve and the result line.
Try this: set the detector azimuth φ to 0 and then to 90 and compare how fast the take-off angle falls with tilt.
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) and integrated counts. Two numbers may follow: a measured k-factor relative to the first line, then the mass absorption coefficient μ/ρ of the specimen for that line, in cm2/g. Your μ/ρ replaces the page’s value from the Elam tables. Type 0 as the k-factor to keep the theoretical one. Add bg 400 anywhere after the counts to give the background counts under that peak: the page then includes them in the counting error and gives a detection limit.
Element, family and counts. Two numbers may follow: the zeta factor in g/cm2 per count per electron (without it, the theoretical value for the solid angle above), then the mass absorption coefficient μ/ρ of the specimen for that line, in cm2/g, as on the Cliff-Lorimer tab. Type 0 as the zeta factor to keep the theoretical one. Add bg 400 after the counts to give the background counts under that peak.
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.
The ratio of two peak intensities gives the ratio of weight fractions: CA/CB = kAB IA/IB. This holds when the foil is thin enough to neglect absorption and fluorescence.
The k-factor depends on the two elements, the voltage and the detector. It does not depend on the specimen.
The table uses the Bethe cross-section with the Powell constants, fluorescence yields, the alpha-line fraction, the atomic weight and the detector efficiency.
Microscope software uses such theoretical factors when there are no standards. A measured k-factor is better.
As the foil thickens, softer X-rays are absorbed on the way out. The intensity ratio drifts, so the composition seems to change with thickness.
Each line is corrected by (1 − e−χρt)/(χρt). Here χ is the mass absorption coefficient for the line divided by the sine of the take-off angle.
The absorption coefficient depends on the composition being sought, so the correction is repeated.
The curve shows the composition the uncorrected ratio would give at every thickness. That is the test.
The zeta-factor method of Watanabe and Williams uses the beam current and live time. Each line gives ρt CA = ζA IA/De.
Summing over the elements gives the mass thickness and the composition at once. The absorption correction is again iterated.
If you do not enter measured values, the tab uses theoretical zeta factors from the same physics plus the detector solid angle.
Alpha and beta lines of the chosen elements at the detector resolution, plus a Kramers background and counting noise.
Linear least squares with one template per element family.
How much the overlap magnifies the statistical uncertainty of each fitted count, compared with an isolated peak.
The fraction of foreign counts that a simple window integration would pick up.
b = 7.21 × 105 (Z/E0) (ρ/A)1/2 t3/2 in cm, with E0 in eV. It is a single-scattering estimate.
The plot combines b with the probe size in the two usual ways.
It overstates the broadening of thin foils and understates it for thick ones. Monte Carlo and Reed models bracket it.
Each tab has short notes beside it. The full description is here.
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 mass absorption coefficients are the values of Elam, Ravel and Sieber (2002) at each line energy. A value measured for your specimen can be typed in instead.
Fluorescence within the foil, Coster-Kronig redistribution among the L subshells, escape and sum peaks, incomplete charge collection and the low-energy tail.
The k-factors for B to F depend strongly on the window and the contamination layer. Trust only measured values.
The overlap fit assumes Gaussian peaks; a real detector adds tails and a shelf. The Goldstein broadening is a single-scattering estimate.
Each peak’s counts N carry an error of √N (√(N + 2B) with a background B given). The Cliff-Lorimer and Zeta tabs carry this through to every wt% and to the mass thickness, and warn when an element’s relative error is over 10 percent, the limit at which Currie (1968) calls a result quantitative.
With a background B given (bg 400), an element is detected at 95 percent confidence when its net counts exceed 2.33√B, and the smallest detectable signal is 2.71 + 4.65√B counts (Currie 1968). The page warns when a peak is below the first and gives the second in wt%.
Goldstein et al. (1977): absorption can be ignored while it changes the intensity ratio of two lines by under 5 percent. The Cliff-Lorimer tab gives that thickness limit and warns when your t is past it, since the answer then rests on the t, ρ and absorption coefficients used.
The result line says which k-factors and zeta factors are theoretical rather than measured, and warns for B to F, where only measured values can be trusted.
The cross-sections and yields are parameterised, not tabulated. The mass absorption coefficients are tabulated: they are the values of Elam, Ravel and Sieber (2002) at each line energy, and the fifth number of a row replaces them with a value measured for your specimen. 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.
Checked (October 2026). The theoretical k-factors obey kAA = 1 and kAB × kBA = 1 exactly (Fe and Ni K lines at 200 kV). The absorption factor is 1.000 for a vanishing foil. For the Fe Kα line in 200 nm of Fe-50Ni (8.1 g/cm3) at a 20° take-off angle it is 0.981: it moves in the right direction and by a size in the published range. The mass absorption coefficient of Ni for Al Kα reads 4324 cm2/g, the value in the tables of Elam, Ravel and Sieber (2002). Theoretical k-factors differ from measured ones by 10 to 20% (Williams and Carter, chapter on quantitative X-ray analysis), which is why the page asks for measured k-factors when you have them.
@misc{tripathy2026edsquantification,
author = {Tripathy, Manisha},
title = {EDS Quantification},
year = {2026},
howpublished = {\url{https://untetheredatom.com/tem/eds-quantification}},
note = {Interactive web tool}
}