X-rays

Characteristic X-rays are produced when:

1. An incident electron knocks out an inner electron from an atom (creating a vacancy)
2. An electron from a higher energy state transitions to fill this vacancy
3. The energy difference (ΔE) is released as an X-ray photon: λ = hc/ΔE

These appear as sharp intensity peaks (e.g., Kα, Kβ) at specific wavelengths that are unique to the target material's atomic structure.

In contrast, continuous X-rays result from electrons decelerating in the target, converting varying fractions of their kinetic energy into photons of different wavelengths.

Key distinction: Characteristic X-rays have discrete wavelengths specific to the target material, while continuous X-rays span a range of wavelengths.

Moseley's Law is expressed as: $\sqrt{\nu} = a(Z - b)$

Where:

• ν is the frequency of characteristic X-rays
• Z is the atomic number
• a and b are constants (b ≈ 1)

Significance:

1. Established atomic number (not atomic weight) as the fundamental property determining an element's position in the periodic table
2. Resolved periodic table anomalies (e.g., placing Co before Ni despite Co having higher atomic weight)
3. Provided experimental evidence supporting Bohr's atomic model
4. Allowed prediction of yet-undiscovered elements by identifying gaps in the linear relationship
5. Demonstrated that characteristic X-ray frequencies depend on nuclear charge, not chemical properties

This relationship fundamentally changed how we organize and understand elements.

Effects of X-ray tube operating parameters:

Filament Current changes:

• Increases: Higher electron emission (thermionic) rate
  • Increases X-ray intensity (more photons/second)
  • Does NOT change X-ray energy distribution or cutoff wavelength
  • Increases heat generation in the target
• Decreases: Fewer electrons emitted, reducing X-ray intensity

Accelerating Voltage changes:

• Increases: Higher electron kinetic energy
  • Decreases cutoff wavelength (λ₂min = hc/eV)
  • Shifts spectrum toward higher energies (harder X-rays)
  • Enables production of additional characteristic X-ray lines if energy exceeds inner-shell ionization thresholds
  • Increases penetrating power of X-rays
• Decreases: Lower electron energy, producing softer X-rays

This explains why radiographic techniques require separate control of kVp (peak kilovoltage) to adjust penetration and mA (tube current) to adjust exposure intensity.

Origin of K-series X-ray lines:

All K-series lines result from electrons transitioning to fill a vacancy in the K shell (n=1), but differ in their starting shells:

• Kα: L→K transition (n=2→n=1)

  • Highest intensity in the series
  • Energy = EK-EL
  • Most probable transition due to selection rules

• Kβ: M→K transition (n=3→n=1)

  • Lower intensity than Kα
  • Higher energy (shorter wavelength) than Kα
  • Energy = EK-EM

• Kγ: N→K transition (n=4→n=1)

  • Lowest intensity of the three
  • Highest energy in the series
  • Energy = EK-EN

Significance:

• The energy differences between these lines provide information about electronic structure
• Intensity ratios follow selection rules and transition probabilities
• Each element has a unique "fingerprint" of characteristic X-ray energies
• Used in X-ray fluorescence spectroscopy for material identification and analysis

X-ray diffraction angle calculation:

Given:

• Interplanar spacing d = 0.25 nm
• X-ray wavelength λ = 0.154 nm

Using Bragg's Law: 2d·sinθ = nλ, where n is an integer

For n = 1: 2(0.25 nm)·sinθ₁ = 1(0.154 nm) sinθ₁ = 0.154/(2×0.25) = 0.308 θ₁ = sin⁻¹(0.308) = 17.9°

For n = 2: 2(0.25 nm)·sinθ₂ = 2(0.154 nm) sinθ₂ = 2×0.154/(2×0.25) = 0.616 θ₂ = sin⁻¹(0.616) = 38.0°

For n = 3: 2(0.25 nm)·sinθ₃ = 3(0.154 nm) sinθ₃ = 3×0.154/(2×0.25) = 0.924 θ₃ = sin⁻¹(0.924) = 67.5°

For n ≥ 4: sinθ₄ would exceed 1, which is impossible.

Therefore, constructive interference would occur at angles of approximately 17.9°, 38.0°, and 67.5° corresponding to the first three orders of diffraction.

The Coolidge tube produces X-rays through these key steps:

1. Thermionic emission: The heated filament (F) emits electrons
2. Acceleration: A high voltage potential difference (several kV) between the filament and target accelerates electrons toward the target
3. Collision: High-speed electrons strike the metal target (T)
4. X-ray production: The sudden deceleration of electrons converts their kinetic energy into X-rays
5. Emission: X-rays exit through a thin window (W) made of material that minimally absorbs X-rays

The tube requires continuous water cooling to dissipate the significant heat generated during operation.

Note: The efficiency of X-ray production is only about 1%, with 99% of energy converted to heat.

Effects of operating parameter changes:

• Increasing filament voltage/current:

• Increases filament temperature
• Increases number of electrons emitted (thermionic emission)
• Results in higher X-ray intensity without changing X-ray energy spectrum
• Does not affect cutoff wavelength

• Increasing accelerating voltage (V):

• Increases electron kinetic energy (E = eV)
• Decreases minimum wavelength (λₘᵢₙ = hc/eV)
• Produces "harder" X-rays (more penetrating)
• Shifts the continuous spectrum toward higher energies

• Target material selection:

• Determines the wavelengths of characteristic X-rays
• Higher atomic number materials produce more efficient X-ray generation
• Affects heat dissipation requirements

Note: While intensity is controlled by filament current, the energy/penetrating power is controlled by accelerating voltage.

An X-ray emission spectrum consists of two distinct components:

1. Continuous X-rays (Bremsstrahlung):

   • Forms a broad, smooth curve
   • Produced when electrons decelerate in the target material
   • Energy varies continuously from zero up to the maximum energy (eV)
   • Intensity gradually varies with wavelength

2. Characteristic X-rays:

   • Appear as sharp peaks (like Kα and Kβ) at specific wavelengths
   • Result from electron transitions between atomic energy levels
   • Wavelengths are unique to the target element (element "fingerprints")
   • Have much higher intensities at these specific wavelengths

The continuous spectrum has a minimum wavelength (λmin) that depends only on the accelerating voltage, while characteristic peaks depend on the target material's atomic structure.

The cutoff wavelength (λmin) represents the minimum possible wavelength of X-rays produced in an X-ray tube, occurring when an electron converts all its kinetic energy into a single photon in one collision.

Physical meaning:

• It marks the high-energy limit of the X-ray spectrum
• No X-rays with shorter wavelengths can be produced at the given accelerating voltage
• It represents the maximum possible energy transfer from electron to photon

Mathematical relationship: $λ_{min} = \frac{hc}{eV}$

Where:

• h is Planck's constant
• c is the speed of light
• e is the electron charge
• V is the accelerating voltage

This equation can also be written in practical units as: $λ_{min} = \frac{1242 \text{ eV·nm}}{V \text{ (in volts)}}$

Note: λmin depends only on the accelerating voltage and not on the target material, unlike characteristic peaks.

Characteristic X-rays are produced through the following process:

1. An energetic electron knocks out an electron from the K shell (n=1), creating a vacancy
2. An electron from a higher energy shell (L, M, or N) transitions to fill this vacancy
3. The energy difference between the two shells is released as an X-ray photon

Specific transitions produce specific X-ray lines:

• Kα: L→K transition (n=2→n=1)
• Kβ: M→K transition (n=3→n=1)
• Kγ: N→K transition (n=4→n=1)

Each element has characteristic energy differences between shells, producing unique X-ray wavelengths that can be used to identify elements (basis of X-ray spectroscopy).