Lesson 6 · 30 min

X-rays and atomic number: Moseley’s law

Transitions of inner electrons give characteristic X-rays whose energy grows as (Z − 1)²: from this Moseley proved that the periodic table is ordered by nuclear charge. Then identify the elements in stainless steel, solder and soil from their X-ray fluorescence.

In this lesson

Explain that characteristic X-rays come from transitions of inner electrons, and tell Kα, Kβ and Lα lines apart
Estimate Kα energies with Moseley’s law E ≈ ¾R∞hc(Z − 1)², and work back from a line energy to the atomic number
Say how Moseley showed that the periodic table is ordered by nuclear charge, and identify the elements in a sample from its X-ray fluorescence spectrum

In 1913 Henry Moseley, 26 years old, at Manchester, made a row of metals the targets of an X-ray tube and measured the wavelengths of the X-rays they gave off by diffraction from a crystal. From calcium to zinc the square root of the frequency of the strongest line rose in even steps — the same step for each place in the periodic table. What fixes an element’s place, he concluded, is not its atomic weight but the positive charge of its nucleus: the atomic number Z. Two years later he was killed at Gallipoli, aged 27.

This lesson follows X-rays to where they come from in the atom, shows why they “count” protons, and shows how today’s analysers use the same lines to tell in seconds which elements are in a piece of metal or a handful of soil.

Predict first

From calcium (Z = 20) to zinc (Z = 30), how does the energy of the strongest X-ray line, Kα, grow?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

in proportion to Z in proportion to (Z − 1)² — so √E rises in a straight line with Z in proportion to the atomic weight it hardly changes

Where X-rays come from

In an X-ray tube, fast electrons strike the target and may knock an electron out of the innermost K shell (n = 1). An outer electron promptly fills the hole, and the energy it releases leaves as an X-ray photon:

Kα: an L-shell (n = 2) electron drops into the K shell;
Kβ: an M-shell (n = 3) electron drops into the K shell — more energetic and weaker;
Lα: an M-shell electron fills a hole in the L shell, at much lower energy.

Inner electrons feel almost the full charge of the nucleus, so these transitions carry thousands of times the energy of visible light or more. Spin–orbit coupling splits the 2p electrons of the L shell into two levels, so Kα is really a close pair, Kα₁ and Kα₂. The “copper Kα” used in X-ray diffraction laboratories (λ = 0.154 nm) is this line from a copper target.

E(\mathrm{K}\alpha) \approx R_\infty hc\,(Z-1)^2\left(\frac{1}{1^2}-\frac{1}{2^2}\right) = \tfrac34\,R_\infty hc\,(Z-1)^2

The hydrogen-atom energy formula applied to inner shells: the L electron falling into the K shell “sees” a nuclear charge screened by about one unit by the 1s electron that remains. R∞hc = 13.606 eV.

$Z$: atomic number (nuclear charge)
$R_\infty hc$: Rydberg energy

Worked example

The X-rays of a copper target

Use Moseley’s formula to estimate the Kα energy and wavelength of copper (Z = 29), and compare them with NIST’s measured Kα₁ = 8047.8 eV.

Try the calculation

An unknown metal has its Kα line at 17.48 keV. Estimate its atomic number with Moseley’s formula.

Measured X-ray energies from NIST · the line is a least-squares fit

Moseley’s law: X-rays count the atomic number

Fit from Z = 11 (Na) to Z = 30 (Zn) Element What Moseley saw in 1914 (aluminium to gold, without the elements then unknown)

√E against atomic number

Measured energy ÷ the fitted line’s prediction

√E = 3.2003 × (Z − 0.99): slope² = 0.7528 R∞hc (ideal 0.7500), screening constant σ = 0.99, r² = 0.999985.

Molybdenum (Z = 42): Kα₁ 17479.4 eV (λ = 0.07093 nm), Kβ₁ 19608.3 eV, Lα₁ 2293.2 eV. Moseley’s formula ¾R∞hc(Z − 1)² gives 17153.4 eV; the measurement is 1.019 times that.

In 1914 four places between aluminium and gold were in fact empty: Z = 43, 61, 72 and 75. Moseley’s paper pointed to three of them — 43, 61 and 75; a discovery had been wrongly claimed for 72, so he did not count it. Hafnium (72, 1923), rhenium (75, 1925), technetium (43, 1937) and promethium (61, 1945) filled them later.

The 1914 view: aluminium to gold only, without the elements then unknown — and the line shows gaps. Untick it to see all the data: technetium, promethium, hafnium and rhenium fall right into the gaps. Then look at the ratio plot below: measured against the line fitted to light elements, the heavy ones climb ever higher — their inner electrons move at a sizeable fraction of the speed of light, relativity binds them more tightly, and lead’s Kα is 12 % above Moseley’s formula.

X-ray fluorescence analysis

Knock the inner electrons out with X-rays instead of electrons and the sample sends out its own characteristic X-rays — X-ray fluorescence (XRF). Every element’s lines sit at different energies, so the spectrum is a fingerprint: a hand-held XRF analyser tells in seconds which stainless steel a piece of scrap is, how much silver an old coin holds, or whether soil contains lead, without damaging the sample.

Inner electrons take almost no part in bonding, so the line positions hardly change from one compound to another: XRF tells you which elements are present and how much, not which compounds they are in. The main nuisance is overlapping lines: a detector with a resolution of about 150 eV cannot separate two lines a few tens of eV apart.

Line positions from the NIST database · every peak drawn the same height

X-ray fluorescence: naming elements by their lines

Fluorescence spectrum

Which elements do you think are in the sample?

Peaks

5.41 keV: not yet explained
5.95 keV: not yet explained
6.40 keV: not yet explained
7.06 keV: not yet explained
7.48 keV: not yet explained
8.26 keV: not yet explained

0 of 6 peaks explained.

Pick the elements you think the sample contains and their lines are marked above the spectrum; you have it when every peak is accounted for and nothing is left over. Start with the stainless steel, then try the contaminated soil: the peak at 10.55 keV can be explained by lead or by arsenic — only the other lines tell them apart.

Concept check

A soil sample shows a peak at 10.55 keV that could be lead Lα₁ (10.552 keV) or arsenic Kα₁ (10.543 keV). How does the analyst decide?

Look for each element’s partner line: lead Lβ₁ at 12.61 keV, arsenic Kβ₁ at 11.73 keV Measuring the peak energy more precisely is enough The stronger the line, the higher the atomic number Lead and arsenic never occur together in soil

Key takeaways

Characteristic X-rays come from filling inner-shell holes: L → K gives Kα, M → K gives Kβ, M → L gives Lα, at thousands of times the energy of visible light or more.
Moseley’s law E(Kα) ≈ ¾R∞hc(Z − 1)²: √E runs straight with Z, proving that the periodic table is ordered by nuclear charge and pointing to gaps such as 43, 61 and 75; heavy elements lie higher because of relativity.
X-ray fluorescence identifies elements, not compounds, by their line positions; overlapping lines tens of eV apart (such as lead Lα and arsenic Kα) are told apart by the other lines.

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Data and references

H. G. J. Moseley, “The high-frequency spectra of the elements”, Phil. Mag. 26, 1024–1034 (1913); Part II, Phil. Mag. 27, 703–713 (1914).
R. D. Deslattes, E. G. Kessler, Jr., P. Indelicato, L. de Billy, E. Lindroth & J. Anton, “X-ray transition energies: new approach to a comprehensive evaluation”, Rev. Mod. Phys. 75, 35–99 (2003); NIST Standard Reference Database 128.
U.S. EPA, Method 6200: Field portable X-ray fluorescence spectrometry for the determination of elemental concentrations in soil and sediment (SW-846, 2007).