Inner-shell X-rays and Moseley's law

Point 70 keV electrons at a metal anode and two different things come back. Most of the beam energy simply heats the anode, lost in collisions with atomic electrons, but electrons braking in the nuclear fields radiate a broad continuum — bremsstrahlung — whose intensity per unit photon energy Kramers wrote as I(E) ∝ Z(Eₘₐₓ − E). It stops dead at Eₘₐₓ = eV₀, the Duane-Hunt limit, because no photon can carry more than the whole kinetic energy of one electron; at 60 kV that is 60 keV, or λₘᵢₙ = 20.7 pm. Standing on that continuum are a few narrow lines whose energies do not move when you change the voltage. Raise V₀ and the cut-off slides to higher energy while the whole continuum brightens as V₀²; the lines only grow taller. Change the anode metal and the lines jump somewhere else entirely. That single observation — position set by the target, not by the beam — is what tells you the lines come from the atom's own level structure and not from the collision.

A K vacancy is a hole in the innermost shell of an atom that may have fifty electrons outside it. An electron dropping from n = 2 into that hole spends its whole journey inside every other shell, where the enclosed charge is the nucleus plus the one 1s electron still present: Z − 1. So the transition is hydrogenic, and E(Kα) = (3/4)(13.606 eV)(Z − 1)². Copper: 10.204 × 28² = 8.00 keV, measured 8.05. Molybdenum: 10.204 × 41² = 17.15 keV, measured 17.48. Tungsten: 10.204 × 73² = 54.4 keV, measured 59.3. The error runs from 0.6% at Z = 29 to 8.3% at Z = 74, and at the heavy end it is not a screening problem — it is relativity. The 1s electron of tungsten moves at roughly Zα = 74/137 = 0.54 c, so it is bound more deeply than a non-relativistic formula allows, and Moseley's straight line curves upward.

Moseley photographed the K and L lines of about forty elements, aluminium to gold, in 1913-14 and plotted √ν against atomic number. The Kα points fell on a straight line crossing the axis at Z = 1, with slope √((3/4)cR∞) = 4.97 × 10⁷ Hz^½. Two consequences followed at once. First, the abscissa is an integer you can count, so the ordering principle of the periodic table is nuclear charge and not atomic weight — which puts cobalt before nickel even though cobalt is the heavier of the two, 58.93 against 58.69. Second, a gap in the line is a missing element whose line position you can predict in advance; the holes at Z = 43, 61, 72 and 75 were later filled by technetium, promethium, hafnium and rhenium. Chemistry had ordered the elements by similarity and mass. One photographic plate replaced that with counting.

No line appears until something creates the hole. The tube must supply at least the K-shell binding energy E_K, which is the same energy as the K absorption edge — the step in the attenuation curve where photons suddenly become able to eject a 1s electron. For molybdenum E_K = 20.0 keV, so a 15 kV tube produces bremsstrahlung and no K lines at all, however long you run it. Above threshold the characteristic intensity grows roughly as (V₀ − V_K)^1.6, which is why analytical work uses an overvoltage of two to three rather than sitting just above the edge. The edge always lies above every line in its own series: Kβ is the 3p → 1s jump, and the series limit, ∞ → 1s, is the edge itself. Molybdenum: Kα 17.5 keV, Kβ 19.6 keV, edge 20.0 keV.

Inner-shell transitions obey the same electric-dipole rules as optical ones: Δl = ±1 and Δj = 0, ±1. That is why the strong K lines come from 2p and 3p, and why there is no 2s → 1s line at all, even though the 2s electron is right there and the energy would be within a few percent of Kα. The same rules predict the doublet. Spin-orbit coupling splits the 2p level into 2p₃/₂ and 2p₁/₂, giving Kα₁ and Kα₂ with intensities in the ratio of their degeneracies 2j + 1, that is 4:2, or 2:1. The splitting climbs steeply with Z — 20 eV in copper, 105 eV in molybdenum, 1.34 keV in tungsten — so a spectrometer that shows one copper Kα line will cleanly resolve two in tungsten.

Filling the hole does not have to make a photon. The atom can hand the released energy to another electron and eject it instead: an Auger electron, leaving a doubly ionised atom. The two channels compete, and the fluorescence yield ω_K is the photon's share. Auger rates are almost independent of Z while the radiative rate climbs roughly as Z⁴, so ω_K ≈ Z⁴/(Z⁴ + 9.0 × 10⁵) to within a few percent above Z ≈ 20. Aluminium manages 0.036, thirty-six photons per thousand vacancies; copper 0.44; tungsten 0.96. This is the practical ceiling on X-ray fluorescence analysis of light elements, and the reason Auger spectroscopy is the technique of choice for them: what does not leave as a photon leaves as an electron, and an electron cannot escape from more than about a nanometre of solid, so it reports on the surface.