X-rays

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.

Cutoff wavelength mechanism:

The cutoff wavelength (λ₂min) represents the shortest possible wavelength in the X-ray spectrum produced at a given accelerating voltage, arising from:

1. A single electron converting ALL its kinetic energy (eV) into a single photon in one collision
2. This occurs when an electron transfers its entire kinetic energy to a photon without energy losses to the target

Quantum mechanical significance:

1. Demonstrates energy conservation at quantum level:

   • Energy transfer must occur in discrete amounts (quanta)
   • Maximum photon energy cannot exceed electron kinetic energy

2. Validates Einstein's photoelectric equation in reverse:

   • E = hν or E = hc/λ where E ≤ eV

3. Provides experimental evidence for:

   • Wave-particle duality of electromagnetic radiation
   • Quantization of energy transfer processes
   • Planck's constant (h) through the relationship λ₂min = hc/eV

4. Represents a fundamental physical limit independent of:

   • Target material
   • Tube design
   • Electron path through target

This quantum limit is precisely measurable and was historically important in establishing quantum mechanics.

Essential components of an X-ray tube:

• Glass chamber (C): Evacuated enclosure that houses all components • Filament (F): Heated electrically to emit electrons through thermionic emission • Target/Anode (T): Metal target (typically tungsten) that stops electrons and produces X-rays • Window (W): Made of thin mica, mylar or similar material that allows X-rays to exit while maintaining vacuum • Cooling system: Water circulation system that prevents overheating of the target • DC power supply: Maintains high potential difference (several kV) between filament and target

The target is maintained at a positive potential relative to the filament, creating the electric field needed to accelerate electrons to high energies.

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 vs. continuous X-rays:

Characteristic X-rays: • Appear as sharp peaks at specific wavelengths in the emission spectrum • Wavelengths are determined by energy differences between electron shells (ΔE = hc/λ) • Unique to each element (like atomic fingerprints) • Named based on the shell where the vacancy occurs (K, L, M) and the shell from which the electron transitions (α, β, γ) • Examples: Kα (L→K transition), Kβ (M→K transition)

Continuous X-rays: • Form a broad spectrum of wavelengths • Produced when electrons decelerate in the target material (Bremsstrahlung) • Have a minimum wavelength (cutoff) determined by the accelerating voltage • Not element-specific

The wavelengths of characteristic X-rays follow Moseley's Law: √ν = a(Z-b), where ν is frequency, Z is atomic number, and a,b are constants.

When a vacancy occurs in the L shell after Kα emission:

1. Electrons from higher shells (M, N) can transition to fill the L-shell vacancy
2. These transitions produce characteristic L-series X-rays:
   • Lα: M→L transition
   • Lβ: N→L transition

Relation to Moseley's Law:

• Moseley's Law states: √ν = a(Z-b), where ν is frequency, Z is atomic number, and a,b are constants
• When an L-shell vacancy is filled, the energy (and frequency) of emitted X-rays depends on:
  • The nuclear charge (Z)
  • Shielding by remaining electrons (represented by b)
• As Z increases across periodic table:
  • The frequency of both K and L series X-rays increases predictably
  • The energy gaps between shells increase systematically
  • The √ν vs. Z plot forms a straight line with slope a

This relationship allowed Moseley to correctly arrange elements by atomic number rather than atomic weight in the periodic table, resolving previous anomalies.

This is characteristic X-ray emission:

• When an inner-shell electron (typically K shell) is ejected from an atom, it creates a vacancy
• An electron from a higher energy shell (like L shell) transitions to fill this vacancy
• The energy difference between the two shells is released as an X-ray photon
• The wavelength of this X-ray is characteristic of the specific element (depends on Z)
• This process produces the sharp peaks in X-ray spectra (unlike continuous X-rays)

Example: A copper atom with a K-shell vacancy might be filled by an L-shell electron, producing a Kα X-ray with a wavelength that serves as a unique fingerprint for copper.

The energy of characteristic X-rays is determined by:

• The energy difference between electron shells (ΔE = Efinal - Einitial)
• The atomic number (Z) of the element (higher Z = higher energy X-rays)
• The specific shells involved in the transition

Characteristic X-rays are labeled using the following convention:

• First letter (K, L, M...) indicates the shell where the vacancy is filled
• Greek subscript (α, β, γ...) indicates the relative energy or origin shell

Examples:

• Kα: L→K transition (electron from L shell fills K shell vacancy)
• Kβ: M→K transition (electron from M shell fills K shell vacancy)
• Lα: M→L transition (electron from M shell fills L shell vacancy)

The wavelength is related to the energy by: λ = hc/ΔE

Bragg's Law Formula

$2d\sin\theta = n\lambda$

Where:

• $d$ = spacing between atomic planes in crystal lattice
• $\theta$ = angle between incident ray and scattering planes
• $n$ = integer representing the order of diffraction
• $\lambda$ = wavelength of X-rays

Physical Explanation

1. Principle: Describes the condition for constructive interference when X-rays scatter from parallel atomic planes in a crystal
2. Path Difference: The 2d·sinθ term represents the extra distance traveled by X-rays reflecting from lower planes
3. Constructive Interference: Occurs only when this path difference equals an integer multiple of the wavelength
4. Diffraction Pattern: At specific angles satisfying Bragg's Law, intensity maxima appear in the diffraction pattern
5. Destructive Interference: Occurs at all other angles, resulting in minimal diffracted intensity

Applications in Crystallography

• Structure Determination: By measuring angles of strong reflections using monochromatic X-rays
• Lattice Parameter Calculation: Determining atomic plane spacing in crystal structures
• Identification of Materials: Different crystalline substances produce unique diffraction patterns
• Phase Analysis: Identifying different crystalline phases in mixtures

Example

For a crystal with atomic plane spacing d = 0.3 nm, using X-rays with λ = 0.1 nm, first-order (n=1) diffraction occurs at θ = 9.6°