Moseley Plot and Absorption Edges

Where an element’s characteristic X-ray lines fall, from the one relation that connects them all: √E is linear in atomic number. The straight line is fitted to this site’s own six anodes, so you can see what it is worth before you use it.

You supply: An element symbol (Ni) or an atomic number (28). Nothing else — a characteristic line is a property of the element alone.
Reading it: A predicted line is not a measured one: the fit is good to about a per cent from roughly Z = 20 to 50 and drifts several per cent by tungsten. Where this site holds the measurement it is printed beside the prediction. No absorption-edge energies are shipped or predicted — the edges are located as a bracket between two elements.

Worked examples: nickel, the filter for copper · zirconium, the filter for molybdenum · tungsten, far outside the fit

Input

Predicted lines for W (Z = 74)

Line	E / keV	λ / Å	measured λ / Å	difference
Kα	56.0274	0.221292	—	—
Kα₁	56.2096	0.220575	—	—
Kα₂	55.666	0.222729	—	—
Kβ	63.8229	0.194263	—	—

Outside the fitted range. The line is fitted over Z = 24–47, and Z = 74 is an extrapolation. Near the range it costs well under a per cent; far from it the error grows steadily, and by Z ≈ 74 the predicted Kα is several per cent low.

The straight line is fitted to this site’s own six anodes, Z = 24 to 47, and reproduces all of their measured wavelengths to better than 0.16% in √E. It stays inside about 1% from roughly Z = 20 to Z = 50. Further out it is an extrapolation and the error grows: √E against Z is not exactly straight, because screening is incomplete and relativistic corrections rise as Z⁴, so by tungsten (Z = 74) the predicted Kα is several per cent low. A predicted line is not a measured one; where this site has the measurement it is printed beside the prediction.

Moseley’s law

Filled points are the measured Kα energies of the six anodes; the dashed line is the least-squares fit through them. The open ring is W where the fit puts it.

Anode	Z	measured λ / Å	√E / keV^½	fitted √E	residual
Cr	24	2.29100	2.32633	2.32268	0.157%
Fe	26	1.937355	2.52976	2.52918	0.023%
Co	27	1.790260	2.63163	2.63243	0.030%
Cu	29	1.541838	2.83572	2.83893	0.113%
Mo	42	0.710730	4.17668	4.18117	0.108%
Ag	47	0.560868	4.70168	4.69741	0.091%

Bohr’s model predicts m = √(R_∞hc × 3/4) = 0.10102 keV^½ and σ = 1 for Kα — one 1s electron screening the other. Fitted over the four light anodes alone (Cr to Cu) this site’s constants give m = 0.10198 and σ = 1.18, which is Moseley’s result. Including Mo and Ag bends the line and pulls σ to 1.55. Nothing is wrong with either fit: the law is an excellent approximation over a limited span of Z, not an identity, and Moseley worked from aluminium to zinc.

How this is calculated

Moseley’s law: the square root of the photon energy of a characteristic line is linear in atomic number. $\sqrt{E} = m (Z - σ)$

Fitted by least squares to the six anodes this site carries, for the Kα line: $m = 0.103249 {keV}^{½}, σ = 1.50417$

$\sqrt{E} = 0.103249 (74 - 1.50417) = 7.48515 {keV}^{½}$

$E = {(7.48515)}^{2} = 56.0274 keV$

$λ = \frac{h c}{E} = \frac{12.3984 keV Å}{56.0274 keV} = 0.221292 Å$

The K absorption edge

Walk up the periodic table at a fixed wavelength and μ/ρ climbs steadily — more electrons, more absorption. In each of this site’s three absorption columns it does that everywhere except at one place, where it falls by a factor near seven between neighbouring elements. That is the K absorption edge: below it the photon can eject a 1s electron and the K shell contributes; above it, it cannot, and the K shell stops absorbing altogether.

Radiation	Z	Kα / Å	E / keV	edge lies between	μ/ρ either side	drop
Ag Kα	47	0.560868	22.106	Tc (43) and Ru (44)	67.9 → 10.7	×6.3
Mo Kα	42	0.710730	17.445	Y (39) and Zr (40)	100 → 15.9	×6.3
Cu Kα	29	1.541838	8.041	Co (27) and Ni (28)	313 → 45.7	×6.8

Nothing here is a table of absorption edges: this site does not ship one and does not predict one. What the columns give is a bracket — the photon energy of each radiation lies between the K edges of the two elements its column jumps between — and a bracket is what choosing a filter needs.

A Kβ filter is a foil whose own K edge lies between the anode’s Kα and Kβ: Kβ is the more energetic, so it is absorbed by ejecting the foil’s 1s electrons, while Kα falls just short of the edge and passes almost untouched. The jump above names the lightest element that clears the Kα side. The conventional filter is that element or one or two heavier — nickel for copper, zirconium for molybdenum, palladium for silver — because a filter whose edge sits only just above Kα leaves no margin. Silver is the extreme case: the table puts Ag Kα within a few parts in a thousand of the K edge of ruthenium, which is why nobody uses ruthenium.