X-ray analysis
1 Chapter 1: analytical methods for material characterization
1.1 Introduction
X-ray spectrometry encompasses analytical methods that use X-rays to obtain qualitative and quantitative information about the
elemental composition and structure of materials. The fundamental principle: matter exposed to high-energy radiation emits or
absorbs X-rays in ways that are characteristic of its elemental composition
1.1.1 Concentration Scales
Analytical techniques must detect species across a vast concentration range.
à %, mg–µg → bulk elemental analysis à ppm, ng–pg → trace analysis
à ppb, fg–ag → ultra-trace levels (typically synchrotron or ICP-MS territory)
1.1.2 Types of Information from Instrumental Techniques
DiJerent analytical methods provide diJerent kinds of information:
à Elemental – XRF, ICP-OES, ICP-MS à Molecular – IR, Raman, NMR
à Structural – XRD, EXAFS à Morphology / Internal Structure: SEM, TEM, CT tomography
1.2 Electromagnetic spectrum
The electromagnetic spectrum covers radiation with a wide range of wavelengths, frequencies, and photon energies:
Gamma rays → X-rays → Ultraviolet → Visible → Infrared → Microwave → Radio
à Visible region: 380–780 nm
à Analytical X-ray region: 0.01–10 nm (0.1–100 Å) à Corresponds roughly to 0.1–100 keV
Energy–wavelength relation: Using Ångström units:
à Short wavelengths → high photon energy.
1.2.1 General Properties of Electromagnetic Radiation
EM radiation has dual nature:
1. Wave properties: Described by wavelength (λ), frequency (ν), amplitude (A),
and phase. Electric and magnetic fields oscillate perpendicular to each other
and to the direction of propagation. Light speed in vacuum: c = 2.998 × 10⁸ m·s⁻¹
2. Particle properties (quantum): EM radiation consists of photons. Photon energy: E = hν = hc/λ.
1.2.2 Wave Properties in Detail
A plane polarized electromagnetic wave: Propagates x-direction. Has E (electric) & B (magnetic) fields perpendicular to each
other. Fields oscillate in well-defined plane. Fields in phase
Mathematical Form of a Wave: For a monochromatic sine wave: A
sin(2πνt + φ) à φ = phase angle
à Angular frequency ω = 2πν
1.2.3 Superposition and Interference
When waves overlap, the resulting wave is their sum.
Constructive Interference: Waves in phase → maxima reinforce. Phase diJerence = 0°. Path diJerence = nλ
Destructive Interference: Waves out of phase by 180° → cancel. Phase diJerence = 180°. Path diJerence = (n + ½)λ
1.2.4 DiEraction of EM-waves
Diiraction = bending/spreading of waves when they encounter edges or narrow openings. All electromagnetic waves diJract —
including X-rays. DiJraction is a consequence of interreference eJects.
,1.2.5 Refraction and Reflection
When waves encounter the boundary between two media:
1) Reflection: Angle of incidence = angle of reflection.
2) Refraction (Snell’s Law)
Dispersion: Refractive index n depends on wavelength.
à Normal dispersion: n increases with frequency
à Anomalous dispersion: n decreases with frequency (near absorption edges)
1.2.6 Polarization of EM Waves
a) Plane Polarized Light: Electric field oscillates within one fixed plane. Example: vertically polarized
à E only oscillates vertically
b) Unpolarized Light: Superposition of waves with random polarization directions. Phases & orientations vary
à X-rays from an X-ray tube are unpolarized because: Each atom in the anode emits radiation
independently. Random phases and directions → overall unpolarized beam
1.3 Quantum mechanical properties of EM radiation
Quantum theory describes electromagnetic radiation not only as waves but also as streams of discrete particles called photons.
Photons have quantized energy given by: E = hν
à where E = photon energy h = Planck’s constant (6.626 × 10⁻³⁴ J·s) ν = frequency of radiation
à This concept explains phenomena that classical wave theory cannot: the photoelectric eiect.
1.3.1 Discovery of the Photoelectric EEect
1) Heinrich Hertz (1887) observed: Electric sparks between metal electrodes occurred more easily when
metal surfaces were illuminated by light
Millikan’s experiment: 1)Electrons are emitted only when the light frequency is above a certain
threshold, regardless of intensity. 2)The maximum kinetic energy of emitted electrons depends only on
frequency (ν) of incident light. Higher frequency → higher kinetic energy. Light intensity aJects number of electrons, not energy
à This was incompatible with classical wave theory.
à Einstein’s Explanation (1905): Light consists of photons, each carrying energy E = hν. An electron in a metal surface absorbs
a single photon. If the photon’s energy exceeds the work function (Φ) of the metal, the electron is ejected.
Photoelectric equation: hν = Φ + KEₘₐₓ à where Φ = work function of the metal (minimum energy needed to release an electron).
KEₘₐₓ = maximum kinetic energy of the emitted electron
1.3.2 Energy Levels in Atoms and Molecules
Quantum mechanics states:
1) Atoms, ions, and molecules can exist only in specific discrete energy states. (electron configurations, vibrations, rotations)
2) When EM radiation is absorbed or emitted, a transition occurs between an initial level E₀ and a final level E₁.
Energy–photon relation: ΔE = E₁ – E₀ = hν = hc/λ
(a) Atomic Transitions: Electrons jump between discrete electronic
energy levels. Result is a line spectrum (e.g., Na emission lines). Each
element has a unique pattern → “elemental fingerprints.”
(b) Molecular Transitions: Molecules have closely spaced vibrational
and rotational levels superimposed on electronic levels. Transitions
create band spectra (broader, structured groups of lines). Provide
information about molecular structure and bonding.
3. Continuum spectra: Produced when energy levels are not discrete
(e.g., blackbody radiation, Bremsstrahlung X-rays).
, 1.4 X-ray spectroscopy
1.4.1 Introduction
X-ray wavelength and energy range: Typical analytical range: 0.1–100 Å (0.01–10 nm) à Corresponding roughly 0.1–100 keV
Energy–wavelength conversion: X-ray photon energy is related to wavelength through: E (keV) = 12.39 / λ (Å)
Short wavelength → high energy, enabling interaction with inner-shell electrons, which is basis for most X-ray analytical methods.
Analytical Uses of X-ray Spectroscopy
1. Emission → XRF (X-ray Fluorescence) 2. Absorption → XAS (X-ray Absorption Spectroscopy) 3. Scattering → XRD (X-ray DiJraction)
à Analytical Power: Qualitative & Quantitative Analysis
1.4.2 History
1.4.2.1 Discovery of X-rays
Wilhelm Conrad Röntgen. Discovered X-rays in 1895. Accidental Observation of a New Radiation
Cathode Ray Tube Setup covered with blak cardboard, blocking visible light: Electrons (cathode rays) emitted from the
cathode accelerate toward the anode. When they strike the glass wall, Yet sheet of paper coated with barium
platinocyanide glowed when placed nearby an unknown penetrating radiation → later named X-rays
1.4.2.2 Properties of the New Radiation (Röntgen's Findings, 1895–1896)
Röntgen noted that X-rays: produced shadow images on photographic plates, traveled in straight lines. Had high penetration
power: they passed through flesh but not bone or metals
Toward the Modern X-ray Tube Röntgen discovered that: X-ray output increased dramatically when cathode rays struck a heavy
metal target (later called the anticathode, now the anode).
Laue’s Experiment: Passed X-rays through a crystalline sample. Observed a pattern of spots from constructive interference (a
diJraction pattern). This proved: X-rays are electromagnetic waves. They have very short wavelengths (~0.1 nm)
1.5 Basis of XRF-spectrometry
1.5.1 Principle of XRF Spectrometry
X-ray fluorescence (XRF) spectrometry is based on interaction of high-energy X-rays with matter (mechanism photoelectric eJect):
1) Photoelectric absorption occurs: An incident X-ray photon ejects an inner-shell (core) electron ( hv > Eb ) (binding energy). This
creates a core-vacancy, an energetically unstable state (K, L, or M shell).
à This process is highly dependent on: Atomic number (Z): higher Z → stronger fluorescence.
2) The atom relaxes: An electron from a higher-energy shell drops down to fill the vacancy. The energy diJerence between the two
shells is released as: (a) a characteristic X-ray photon → X-ray fluorescence (XRF). (b) or Auger electron (non-radiative process)
à Because inner-shell binding energies are element-specific, the emitted radiation has characteristic energies,
enabling qualitative and quantitative elemental analysis.
Example: Copper atom (Cu). Incident X-ray ejects a K-shell electron (K-edge ~8.98 keV). An electron from L or M level fills the K-
vacancy. Emitted radiation energies reflect transitions such as: Kα (L→K) & Kβ (M→K)
1.5.1.1 Moseley’s Law
Moseley’s work established a quantitative relationship between the emitted X-ray energies and the atomic number Z: where:
ν = frequency of characteristic X-ray line C ≈ R = Rydberg constant (modified for X-rays)
σ ≈ 1 = screening constant for K-lines Z = atomic number
Key implications: Characteristic X-ray energies increase systematically with atomic number.
Ordering of the periodic table based on nuclear charge, not atomic mass.
1.5.1.2 Naming Conventions for Characteristic X-ray Lines
Siegbahn notation (traditional): First letter = shell where the vacancy occurred (K, L, M…). Greek
letter = shell of the incoming (transitioning) electron: α → L→K β → M→K γ → N→K
1.5.1.3 Linking X-ray Transitions to Spectral Features
K-lines (K-shell vacancy): Kα lines: L→K transitions (most intense in XRF). Kβ lines: M→K transitions
L-lines (L-shell vacancy): Lα: M→L transitions. Lβ: N→L à Characteristics: Higher Z → higher energy transitions.
1 Chapter 1: analytical methods for material characterization
1.1 Introduction
X-ray spectrometry encompasses analytical methods that use X-rays to obtain qualitative and quantitative information about the
elemental composition and structure of materials. The fundamental principle: matter exposed to high-energy radiation emits or
absorbs X-rays in ways that are characteristic of its elemental composition
1.1.1 Concentration Scales
Analytical techniques must detect species across a vast concentration range.
à %, mg–µg → bulk elemental analysis à ppm, ng–pg → trace analysis
à ppb, fg–ag → ultra-trace levels (typically synchrotron or ICP-MS territory)
1.1.2 Types of Information from Instrumental Techniques
DiJerent analytical methods provide diJerent kinds of information:
à Elemental – XRF, ICP-OES, ICP-MS à Molecular – IR, Raman, NMR
à Structural – XRD, EXAFS à Morphology / Internal Structure: SEM, TEM, CT tomography
1.2 Electromagnetic spectrum
The electromagnetic spectrum covers radiation with a wide range of wavelengths, frequencies, and photon energies:
Gamma rays → X-rays → Ultraviolet → Visible → Infrared → Microwave → Radio
à Visible region: 380–780 nm
à Analytical X-ray region: 0.01–10 nm (0.1–100 Å) à Corresponds roughly to 0.1–100 keV
Energy–wavelength relation: Using Ångström units:
à Short wavelengths → high photon energy.
1.2.1 General Properties of Electromagnetic Radiation
EM radiation has dual nature:
1. Wave properties: Described by wavelength (λ), frequency (ν), amplitude (A),
and phase. Electric and magnetic fields oscillate perpendicular to each other
and to the direction of propagation. Light speed in vacuum: c = 2.998 × 10⁸ m·s⁻¹
2. Particle properties (quantum): EM radiation consists of photons. Photon energy: E = hν = hc/λ.
1.2.2 Wave Properties in Detail
A plane polarized electromagnetic wave: Propagates x-direction. Has E (electric) & B (magnetic) fields perpendicular to each
other. Fields oscillate in well-defined plane. Fields in phase
Mathematical Form of a Wave: For a monochromatic sine wave: A
sin(2πνt + φ) à φ = phase angle
à Angular frequency ω = 2πν
1.2.3 Superposition and Interference
When waves overlap, the resulting wave is their sum.
Constructive Interference: Waves in phase → maxima reinforce. Phase diJerence = 0°. Path diJerence = nλ
Destructive Interference: Waves out of phase by 180° → cancel. Phase diJerence = 180°. Path diJerence = (n + ½)λ
1.2.4 DiEraction of EM-waves
Diiraction = bending/spreading of waves when they encounter edges or narrow openings. All electromagnetic waves diJract —
including X-rays. DiJraction is a consequence of interreference eJects.
,1.2.5 Refraction and Reflection
When waves encounter the boundary between two media:
1) Reflection: Angle of incidence = angle of reflection.
2) Refraction (Snell’s Law)
Dispersion: Refractive index n depends on wavelength.
à Normal dispersion: n increases with frequency
à Anomalous dispersion: n decreases with frequency (near absorption edges)
1.2.6 Polarization of EM Waves
a) Plane Polarized Light: Electric field oscillates within one fixed plane. Example: vertically polarized
à E only oscillates vertically
b) Unpolarized Light: Superposition of waves with random polarization directions. Phases & orientations vary
à X-rays from an X-ray tube are unpolarized because: Each atom in the anode emits radiation
independently. Random phases and directions → overall unpolarized beam
1.3 Quantum mechanical properties of EM radiation
Quantum theory describes electromagnetic radiation not only as waves but also as streams of discrete particles called photons.
Photons have quantized energy given by: E = hν
à where E = photon energy h = Planck’s constant (6.626 × 10⁻³⁴ J·s) ν = frequency of radiation
à This concept explains phenomena that classical wave theory cannot: the photoelectric eiect.
1.3.1 Discovery of the Photoelectric EEect
1) Heinrich Hertz (1887) observed: Electric sparks between metal electrodes occurred more easily when
metal surfaces were illuminated by light
Millikan’s experiment: 1)Electrons are emitted only when the light frequency is above a certain
threshold, regardless of intensity. 2)The maximum kinetic energy of emitted electrons depends only on
frequency (ν) of incident light. Higher frequency → higher kinetic energy. Light intensity aJects number of electrons, not energy
à This was incompatible with classical wave theory.
à Einstein’s Explanation (1905): Light consists of photons, each carrying energy E = hν. An electron in a metal surface absorbs
a single photon. If the photon’s energy exceeds the work function (Φ) of the metal, the electron is ejected.
Photoelectric equation: hν = Φ + KEₘₐₓ à where Φ = work function of the metal (minimum energy needed to release an electron).
KEₘₐₓ = maximum kinetic energy of the emitted electron
1.3.2 Energy Levels in Atoms and Molecules
Quantum mechanics states:
1) Atoms, ions, and molecules can exist only in specific discrete energy states. (electron configurations, vibrations, rotations)
2) When EM radiation is absorbed or emitted, a transition occurs between an initial level E₀ and a final level E₁.
Energy–photon relation: ΔE = E₁ – E₀ = hν = hc/λ
(a) Atomic Transitions: Electrons jump between discrete electronic
energy levels. Result is a line spectrum (e.g., Na emission lines). Each
element has a unique pattern → “elemental fingerprints.”
(b) Molecular Transitions: Molecules have closely spaced vibrational
and rotational levels superimposed on electronic levels. Transitions
create band spectra (broader, structured groups of lines). Provide
information about molecular structure and bonding.
3. Continuum spectra: Produced when energy levels are not discrete
(e.g., blackbody radiation, Bremsstrahlung X-rays).
, 1.4 X-ray spectroscopy
1.4.1 Introduction
X-ray wavelength and energy range: Typical analytical range: 0.1–100 Å (0.01–10 nm) à Corresponding roughly 0.1–100 keV
Energy–wavelength conversion: X-ray photon energy is related to wavelength through: E (keV) = 12.39 / λ (Å)
Short wavelength → high energy, enabling interaction with inner-shell electrons, which is basis for most X-ray analytical methods.
Analytical Uses of X-ray Spectroscopy
1. Emission → XRF (X-ray Fluorescence) 2. Absorption → XAS (X-ray Absorption Spectroscopy) 3. Scattering → XRD (X-ray DiJraction)
à Analytical Power: Qualitative & Quantitative Analysis
1.4.2 History
1.4.2.1 Discovery of X-rays
Wilhelm Conrad Röntgen. Discovered X-rays in 1895. Accidental Observation of a New Radiation
Cathode Ray Tube Setup covered with blak cardboard, blocking visible light: Electrons (cathode rays) emitted from the
cathode accelerate toward the anode. When they strike the glass wall, Yet sheet of paper coated with barium
platinocyanide glowed when placed nearby an unknown penetrating radiation → later named X-rays
1.4.2.2 Properties of the New Radiation (Röntgen's Findings, 1895–1896)
Röntgen noted that X-rays: produced shadow images on photographic plates, traveled in straight lines. Had high penetration
power: they passed through flesh but not bone or metals
Toward the Modern X-ray Tube Röntgen discovered that: X-ray output increased dramatically when cathode rays struck a heavy
metal target (later called the anticathode, now the anode).
Laue’s Experiment: Passed X-rays through a crystalline sample. Observed a pattern of spots from constructive interference (a
diJraction pattern). This proved: X-rays are electromagnetic waves. They have very short wavelengths (~0.1 nm)
1.5 Basis of XRF-spectrometry
1.5.1 Principle of XRF Spectrometry
X-ray fluorescence (XRF) spectrometry is based on interaction of high-energy X-rays with matter (mechanism photoelectric eJect):
1) Photoelectric absorption occurs: An incident X-ray photon ejects an inner-shell (core) electron ( hv > Eb ) (binding energy). This
creates a core-vacancy, an energetically unstable state (K, L, or M shell).
à This process is highly dependent on: Atomic number (Z): higher Z → stronger fluorescence.
2) The atom relaxes: An electron from a higher-energy shell drops down to fill the vacancy. The energy diJerence between the two
shells is released as: (a) a characteristic X-ray photon → X-ray fluorescence (XRF). (b) or Auger electron (non-radiative process)
à Because inner-shell binding energies are element-specific, the emitted radiation has characteristic energies,
enabling qualitative and quantitative elemental analysis.
Example: Copper atom (Cu). Incident X-ray ejects a K-shell electron (K-edge ~8.98 keV). An electron from L or M level fills the K-
vacancy. Emitted radiation energies reflect transitions such as: Kα (L→K) & Kβ (M→K)
1.5.1.1 Moseley’s Law
Moseley’s work established a quantitative relationship between the emitted X-ray energies and the atomic number Z: where:
ν = frequency of characteristic X-ray line C ≈ R = Rydberg constant (modified for X-rays)
σ ≈ 1 = screening constant for K-lines Z = atomic number
Key implications: Characteristic X-ray energies increase systematically with atomic number.
Ordering of the periodic table based on nuclear charge, not atomic mass.
1.5.1.2 Naming Conventions for Characteristic X-ray Lines
Siegbahn notation (traditional): First letter = shell where the vacancy occurred (K, L, M…). Greek
letter = shell of the incoming (transitioning) electron: α → L→K β → M→K γ → N→K
1.5.1.3 Linking X-ray Transitions to Spectral Features
K-lines (K-shell vacancy): Kα lines: L→K transitions (most intense in XRF). Kβ lines: M→K transitions
L-lines (L-shell vacancy): Lα: M→L transitions. Lβ: N→L à Characteristics: Higher Z → higher energy transitions.
