VKC Notes · Analytical chemistry · Geo-Scientist P-III

13.9 X-ray Methods (XRD, XRF) and ICP-OES / ICP-MS

★★★ 2021–2025: 22 parts, 175 marks, all in Geo-Scientist Paper III. Every paper has 4–5 parts from these two units: a Bragg or powder-pattern numerical, an XRF theory or detector question, and an ICP plasma, interface or interference question.

Syllabus lines. Geo-Scientist P-III, X-ray methods: “Introduction, theory of X-ray generation, X-ray spectroscopy, X-ray diffraction and X-ray fluorescence methods, instrumentation and applications. Qualitative and quantitative measurements. Powder diffraction method.” Inductively coupled plasma spectroscopy: “Theory and principles, plasma generation, utility of peristaltic pump, sampler–skimmer systems, ion lens, quadrupole mass analyzer, dynode / solid state detector, different types of interferences: spectroscopic and non-spectroscopic interferences, isobaric and molecular interferences, applications.” Crystal systems and Miller indices: 3.1; AAS and flame emission: 13.8.

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CSE 2024 Q7(a) = asked there; tap to open the solved answerSN = line kept in the short notes

13.9.1 X-rays: continuum, lines, absorption edges ★★★

Tube: electrons from a hot W filament, accelerated through V (20–60 kV), hit a metal anode (Cu, Mo, Rh, W). Two outputs: a continuum (bremsstrahlung, electrons slowed in the anode) and sharp characteristic lines of the anode metal. Only ≈ 1 % of the power becomes X-rays; the anode is water-cooled.SN
Duane–Hunt limit: the whole electron energy eV in one photon gives the shortest wavelength; it depends only on V, not on the anode.SN

(13.9.1)

Lines: an incoming electron (or photon) must first eject a K electron, so the tube energy must exceed the K absorption edge (always a little higher in energy than the K lines themselves). Kα = L → K, Kβ = M → K (shorter λ, weaker); L lines from L vacancies.SN
Ag at 24 kV: λ_min = 12.398/24 = 0.517 Å, longer than the Ag K edge 0.485 Å (12.398/0.485 = 25.6 keV needed). No K electron is ejected, so no K lines appear; what decides is the edge, not the line (Kα 0.497 Å). Raise the voltage above ≈ 26 kV Geo 2023 Q2(d) 5m.SN
Moseley: √ν of a given line is linear in Z; this ordered the elements by atomic number and lets XRF identify an element from one line. For Kα, σ ≈ 1: E(Kα) ≈ 10.2 eV × (Z − 1)², Zn (Z = 30) → 8.6 keV.
Radioisotope sources (portable XRF): electron capture, p + e⁻(K) → n + ν: Z falls by one and the daughter emits its own K X-ray: ⁵⁵Fe → Mn Kα 5.9 keV; ¹⁰⁹Cd → Ag K; also γ-lines (²⁴¹Am 59.5 keV) and internal conversion Geo 2022 Q1(f) 5m.SN

Fig. 13.9.1 Spectrum of a Mo tube at 15 kV and 35 kV.

Draw it in 60 s: two rising-then-falling humps starting at λmin; the higher voltage hump starts further left and carries two spikes (Kβ, then Kα).

13.9.2 Bragg's law and indexing a cubic powder pattern ★★★ 15-mark numerical

(13.9.2)

Fig. 13.9.2 Bragg reflection from two planes.

Draw it in 60 s: three horizontal rows of dots; two parallel rays in at θ, out at θ; drop perpendiculars from the upper point; mark the two bits of extra path, each d sin θ.

Read 2θ, halve it. The recorded angle is between the straight-through and diffracted beams. 2θ = 60°, Cu Kα 1.539 Å: θ = 30°, d = 1.539/(2 × 0.5) = 1.539 Å Geo 2025 Q1(h) 5m.SN
Lattice from the lines: sin²θ values stand in the ratio of N = h² + k² + l². Allowed N: primitive 1, 2, 3, 4, 5, 6, 8 (7 missing); bcc h + k + l even: 2, 4, 6, 8, 10, 12; fcc h, k, l unmixed: 3, 4, 8, 11, 12, 16.SN
bcc from (110), (200), (211), (220): all h + k + l even, (100) and (111) absent. (211) at 2θ = 60°, λ 1.54 Å: d = 1.54 Å, a = 1.54√6 = 3.772 Å; Z = 2, M = 60: ρ = 120/(6.022 × 10²³ × 5.368 × 10⁻²³) = 3.71 g cm⁻³ Geo 2023 Q4(a) 15m.SN

Fig. 13.9.3 The 2022 silver pattern, indexed.

Draw it in 60 s: a 2θ axis 30°–105°; six sticks, tallest first; write 111, 200, 220, 311, 222, 400 on them and “fcc” beside.

2θ / °	sin²θ	× 3 / first	hkl	a = λ√N/2 sin θ / Å
38.16	0.1069	3	111	4.0826
44.34	0.1424	4	200	4.0837
64.52	0.2849	8	220	4.0829
77.48	0.3916	11	311	4.0836
81.64	0.4273	12	222	4.0831
98.00	0.5696	16	400	4.0837

Silver: ratios 3 : 4 : 8 : 11 : 12 : 16 → fcc, Z = 4; mean a = 4.083 Å; ρ = 4 × 108/(6.022 × 10²³ × (4.083 × 10⁻⁸)³) = 10.5 g cm⁻³ (handbook 10.49) Geo 2022 Q3(a) 15m.SN

13.9.3 Powder diffractometer and qualitative phase analysis ★★

Fig. 13.9.4 θ–2θ powder diffractometer.

Draw it in 60 s: a half circle; tube with Ni filter on the left, flat sample at the centre, detector on the circle at 2θ; label “sample θ, detector 2θ”.

Why powder works: millions of randomly oriented crystallites, so every set of planes meets the Bragg condition for some grains; each d gives a cone of rays (Debye–Scherrer rings on film; a peak on the 2θ scan of a diffractometer).SN
Qualitative phase analysis: every crystalline phase has its own set of d spacings and relative intensities, a fingerprint. Record I against 2θ (Cu Kα, Ni filter, 5°–80°), convert 2θ → d, and match with the Powder Diffraction File (ICDD, > 10⁵ patterns) by the Hanawalt method (three strongest lines) or computer search–match Geo 2024 Q3(a) 10m.SN
It identifies the compound and its crystal form, not just the elements: calcite vs aragonite (both CaCO₃), quartz vs cristobalite, rutile vs anatase. Mixtures give superposed patterns; a few mg, non-destructive, no single crystal needed Geo 2021 Q1(h) 5m.SN
Applications: minerals and clays in rocks, cement phases, corrosion products, polymorphs of drugs, purity of a synthesis, % crystallinity, quantitative phase analysis (intensity ∝ amount; Rietveld), lattice parameters and solid solutions (Vegard), crystallite size from line width (13.9.10).

13.9.4 X-ray fluorescence: theory, K and L lines ★★★ 15 marks twice

Fig. 13.9.5 How a fluorescent K line is born.

Draw it in 60 s: three lines K, L, M; knock out a K electron; arrows L→K (Kα) and M→K (Kβ); beside it write “or Auger electron”.

Excitation: a primary X-ray beam above the absorption edge ejects an inner-shell electron (photoelectric effect), leaving a vacancy.SN
Relaxation: an outer electron drops in and the energy difference leaves as a characteristic (fluorescent) X-ray: energy → element, intensity → concentration (against standards, with matrix corrections). Fluorescent lines are free of a continuum background except scattered tube radiation.SN
K and L lines: Kα (2p → 1s) and Kβ (3p → 1s) for K vacancies; L lines for L vacancies, used for heavy elements (Pb, Ba, rare earths) whose K edges need very high voltage. Zn: Kα 1.435 Å (8.64 keV), Kβ 1.295 Å (9.57 keV), K edge 1.283 Å; on a LiF(200) crystal (d = 2.014 Å) Kβ appears at 2θ = 37.5°, Kα at 41.7°, Kα about five times taller Geo 2023 Q1(f) 5m.SN
Bulk, not surface: fluorescent X-rays escape from µm to mm depths (they are photons, weakly absorbed); Auger electrons of similar energy lose energy in collisions and escape only from the top 1–3 nm. So XRF analyses the bulk; AES the surface. The competing Auger process is why the fluorescence yield ω is small for light elements: XRF is poor below Na Geo 2021 Q4(a) 15m.SN

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