Moseley Plot and Absorption Edges

Predicted lines for Zr (Z = 40)

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 $\sqrt{E}$. It stays inside about 1% from roughly Z = 20 to Z = 50. Further out it is an extrapolation and the error grows: $\sqrt{E}$ 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 Zr where the fit puts it.

Bohr’s model predicts m = $\sqrt{R_{\text{∞}}hc\times 3/4}$ = 0.10102 $\sqrt{\text{keV}}$ and σ = 1 for Kα — one 1s electron screening the other. Fitted over the four light anodes alone (Cr to Cu), Kα gives m = 0.10188 and σ = 1.17, which is Moseley’s result. Including Mo and Ag bends the line and pulls σ to 1.50 — the value the fit above this table used. 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.

The K absorption edge

Walk up the periodic table at a fixed wavelength and μ/ρ climbs steadily — more electrons, more absorption. At each of these six wavelengths 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, and the direction is worth stating in the variable that is moving. Walking up Z at a fixed wavelength: for the lighter element the edge lies below the photon’s energy, so the photon can eject a 1s electron and the K shell contributes; one element heavier the edge has risen past it, the photon can no longer reach the 1s electron, and the K shell stops absorbing altogether. Said in energy at a fixed element, which is how an edge is usually quoted, that is the same fact the other way round: a photon above the edge energy ionises the K shell and one below it cannot.

The drop is found in the coefficients and the edge energies are tabulated separately, so the last two columns are a check rather than a restatement: the photon energy has to fall between the two edges named, and it does, on every row. Nothing here is predicted — the Moseley fit above is over emission lines, and it is not continued to an absorption edge.

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, and the margin is now a number rather than a phrase: Ag Kα is 0.05% below the K edge of ruthenium, which is why nobody uses ruthenium.