Inner-shell transitions and X-ray spectra

An electron leaves the cathode nearly at rest and reaches the anode with kinetic energy eV, give or take a few eV of contact potential. In a single braking event it can emit one photon carrying any part of that energy, and at most all of it: hf ≤ eV. So the continuum stops dead at Eₘₐₓ = eV, which is λₘᵢₙ = hc/(eV) = 1.2398 nm ÷ (V in kV). A 30.0 kV tube cuts off at 0.0413 nm whether the anode is copper, molybdenum or tungsten, and doubling V halves λₘᵢₙ. Classical electrodynamics predicts no edge: a charge brought to rest in a time τ radiates a spectrum that fades out smoothly around ω ~ 1/τ. The edge is the quantum signature, and the correspondence check is built in: let h → 0 at fixed V and λₘᵢₙ → 0, so the edge vanishes into the classical tail. Duane and Hunt ran the argument backwards in 1915, measuring λₘᵢₙ at known V to obtain h/e.

A thick anode stops each electron gradually, over thousands of collisions. A thin slice of atoms radiates, per unit photon energy, a flat spectrum up to the electron's current energy E′, with strength ∝ Z²/E′ per atom. Meanwhile the electron loses energy mostly to atomic electrons, at a rate per atom ∝ Z/E′, so the number of atoms it crosses while its energy falls by dE′ is ∝ E′ dE′/Z. Multiply and integrate from E′ = E up to eV: I(E) ∝ ∫ (Z²/E′)(E′/Z) dE′ = Z(eV − E). That is Kramers' law, a straight line falling to zero at the cutoff; Z² from the nucleus and 1/Z from the stopping leave one power of Z. Integrating once more gives the total, P ∝ iZ(eV)², so the efficiency P/(iV) grows as ZV; empirically η ≈ (1 × 10⁻⁹ V⁻¹)ZV, or 0.74% for tungsten at 100 kV. A 1.0 kW beam delivers about 7.4 W of X-rays and 993 W of heat. Plotted per unit wavelength, the triangle becomes (λ − λₘᵢₙ)/λ³, which peaks at 1.5λₘᵢₙ: the hump on a crystal-spectrometer trace is a change of variable, not new physics.

Characteristic lines need a vacancy in an inner shell, and the incoming electron cannot make one cheaply. Every shell below the valence shell is full, so exclusion forbids lifting a 1s electron into any of them; its only destinations are the empty valence and continuum states, within a few eV of the vacuum level. Opening the K shell therefore costs the whole K binding energy, and no K line appears until eV ≥ E_K. For molybdenum E_K = 20.00 keV while Kα₁ is only 17.48 keV: a tube at 18 kV sends continuum photons out to 0.0689 nm, beyond the Kα₁ wavelength of 0.0709 nm, and still shows no Kα. Above 20.0 kV, Kα and Kβ appear together, since both start from the same K hole, and grow roughly as (U − 1)^1.6 with overvoltage U = V/V_K; taking a Mo tube from 30 kV to 40 kV at fixed current triples its Kα output. E_K is also the K absorption edge, where absorption jumps at 0.0620 nm. The L shell opens far earlier, at E_L₃ = 2.52 keV for Mo.

Once the K hole exists, an electron from a higher shell drops into it and the hole moves up. The photon energy is the difference of the two binding energies: Kα₁ = E_K − E_L₃, Kα₂ = E_K − E_L₂, Kβ₁ = E_K − E_M₃. For copper these give 8.979 − 0.933 = 8.046 keV, 8.979 − 0.952 = 8.027 keV and 8.979 − 0.075 = 8.904 keV, each within 2 eV of the measured line. No electric-dipole line comes from L₁, because 2s → 1s has Δl = 0 and the dipole rule demands Δl = ±1. The Kα doublet is the spin–orbit splitting of the 2p level, E_L₂ − E_L₃: 20 eV in copper, 105 eV in molybdenum, 1.34 keV in tungsten, climbing roughly as Z⁴. Kα₁ is about twice as strong as Kα₂, matching the four electrons of 2p₃/₂ against the two of 2p₁/₂. And the hole need not radiate: an Auger electron can carry the energy off instead. The fraction that radiates, the fluorescence yield, is about 0.4 for copper, above 0.9 for tungsten and below 0.01 for carbon.

Inner-shell energies look hydrogenic because an inner electron sits inside almost all the others. By Gauss's law, charge spread in spherically symmetric shells outside an orbit exerts no net force there, since a sphere drawn through the orbit encloses none of it; it adds a nearly constant potential, which shifts the K and L levels alike and cancels in their difference. What remains is the nucleus, partly screened by charge further in: an L electron dropping into the K hole sees roughly Z − 1, the nucleus less the one surviving 1s electron. With Bohr energies Ry Z_eff²/n², hf(Kα) ≈ Ry(Z − 1)²(1 − 1/4) = 10.20 eV × (Z − 1)². For copper that is 10.20 eV × 28² = 8.00 keV, against a measured doublet mean of 8.04 keV. Moseley plotted √f against Z in 1913–14 and found a straight line. It placed cobalt (Z = 27, 58.93 u) before the lighter nickel (Z = 28, 58.69 u), showed gaps at Z = 43, 61, 72 and 75, and made atomic number the nuclear charge rather than a rank. X-ray lines, unlike optical ones, show no periodicity: every heavy atom has the same closed inner shells, so the lines move smoothly with Z.

Invert Moseley's law on measured lines and σ = Z − √(E/10.204 eV) is no longer a constant. The intensity-weighted Kα doublet mean, (2Kα₁ + Kα₂)/3, gives σ = 0.96 for iron, 0.93 for copper, 0.46 for silver, −1.96 for tungsten and −3.30 for lead, crossing zero near Z = 55. A negative screening constant is not physics; it is a fit absorbing what the model lacks. The largest missing piece is relativity: inner electrons of heavy atoms move at a large fraction of c, and the Dirac fine-structure formula, to lowest order in (Zα)², raises the Kα₁ energy by a factor 1 + (5/16)(Z − 1)²α², with α = 1/137.04. For tungsten that is 8.9%, lifting 54.38 keV to 59.20 keV against 59.32 keV measured; for lead it shrinks an 11% shortfall to 0.9%. Kept to all orders, the Dirac energies overshoot both lines by about 1%, so the agreement also leans on Z − 1 being the right charge. Correspondence holds: as Zα → 0 the factor goes to 1 and the non-relativistic law returns. One σ also cannot tell L₂ from L₃, so it never predicts the doublet, and the L series needs its own σ_L ≈ 7.4. σ is a curve-fitting stand-in for every other electron, not a count of any of them.