Features of X-ray Spectrum

1. Definitions (Must Know)

Continuous and characteristic X-ray spectrum

Relative intensity is zero below the minimum wavelength. From that cutoff a continuous background rises and falls, with narrow K beta and K alpha characteristic lines superimposed.

Scroll across the graph to read all labels.

Open full-size graph

View figure data

Values and uncertainty for Continuous and characteristic X-ray spectrum

Series	Wavelength (relative units)	Wavelength uncertainty	Relative intensity (arbitrary units)	Relative intensity uncertainty
Continuous bremsstrahlung background	0		0
Continuous bremsstrahlung background	1		0
Continuous bremsstrahlung background	2		0
Continuous bremsstrahlung background	2.5		0.38
Continuous bremsstrahlung background	3		0.65
Continuous bremsstrahlung background	4		0.82
Continuous bremsstrahlung background	5		0.86
Continuous bremsstrahlung background	6		0.78
Continuous bremsstrahlung background	7		0.64
Continuous bremsstrahlung background	8		0.48
Continuous bremsstrahlung background	9		0.34
Continuous bremsstrahlung background	10		0.22
K beta characteristic line	5.5		0.78
K beta characteristic line	5.5		1.15
K alpha characteristic line	6.5		0.7
K alpha characteristic line	6.5		1.3

Values and uncertainty for Continuous and characteristic X-ray spectrum

Series	Wavelength (relative units)	Wavelength uncertainty	Relative intensity (arbitrary units)	Relative intensity uncertainty
Continuous bremsstrahlung background	0		0
Continuous bremsstrahlung background	1		0
Continuous bremsstrahlung background	2		0
Continuous bremsstrahlung background	2.5		0.38
Continuous bremsstrahlung background	3		0.65
Continuous bremsstrahlung background	4		0.82
Continuous bremsstrahlung background	5		0.86
Continuous bremsstrahlung background	6		0.78
Continuous bremsstrahlung background	7		0.64
Continuous bremsstrahlung background	8		0.48
Continuous bremsstrahlung background	9		0.34
Continuous bremsstrahlung background	10		0.22
K beta characteristic line	5.5		0.78
K beta characteristic line	5.5		1.15
K alpha characteristic line	6.5		0.7
K alpha characteristic line	6.5		1.3

A. Bremsstrahlung (continuous X-rays)

Bremsstrahlung (“braking radiation”) are X-ray photons emitted when fast electrons are decelerated in the target. Because electrons can lose any fraction of their energy in each interaction, the spectrum is continuous.

B. Characteristic X-rays (line peaks)

Characteristic X-rays are emitted when an incoming electron knocks out an inner-shell electron in the target atom, and another electron drops down to fill the vacancy. The photon energy equals the energy level difference, so the spectrum contains sharp peaks (discrete lines).

C. Cut-off (minimum) wavelength, λₘᵢₙ

The cut-off wavelength λₘᵢₙ is the shortest wavelength present in the continuous spectrum. It corresponds to the maximum possible photon energy.

If the electron is accelerated through potential difference V, the maximum photon energy is approximately: Eₘₐₓ ≈ eV so: λₘᵢₙ = hc/eV

2. Key Ideas (What Earns Marks)

• An X-ray spectrum has:
  • a continuous background (bremsstrahlung),
  • sharp peaks (characteristic X-rays).
• Increasing accelerating voltage V:
  • decreases λₘᵢₙ (harder X-rays),
  • increases overall intensity,
  • does not shift the characteristic peak wavelengths (those depend mainly on target material).
• Maximum photon energy comes from conservation of energy: eV ≈ hc/λₘᵢₙ

3. Detailed Explanations

A. Why the spectrum has a continuous background

Incoming electrons are deflected by the electric field of nuclei in the target. Different encounters transfer different energy amounts to radiation, so photon energies range from near zero up to a maximum.

The maximum energy photon happens (approximately) when an electron loses almost all its kinetic energy in one interaction: eV ≈ Eₘₐₓ = hc/λₘᵢₙ

B. Why sharp peaks appear

If an incoming electron ejects an inner-shell electron of the target atom, an outer electron can drop into the vacancy and emit a photon with a fixed energy set by the target’s energy levels. That produces discrete (line) peaks.

C. What changes when you change V or the target material

• Increase V:
  • λₘᵢₙ decreases,
  • the continuous spectrum intensity increases.
• Change target material:
  • characteristic peak positions change (different energy level spacings),
  • the continuous background still exists.

4. Common Mistakes

• Using λₘᵢₙ to describe the characteristic peaks (cut-off applies to the continuous spectrum).
• Saying “doubling V doubles λₘᵢₙ” (it halves λₘᵢₙ, since λₘᵢₙ ∝ 1/V).
• Forgetting 1 eV = 1.60 × 10⁻¹⁹ J when mixing eV and SI units.

5. Exam Tips

• If a spectrum graph is given, identify:
  • cut-off wavelength λₘᵢₙ (where the curve hits the axis),
  • characteristic peaks (sharp spikes above the background).
• For calculations, start with: λₘᵢₙ = hc/eV
• State what stays the same when V changes: characteristic peak wavelengths.

6. Worked Examples

7. Mind Stretchers

8. Optional (Enrichment)

A. K-series lines (K_α, K_β)

Many textbooks label inner-shell characteristic lines as:

• K_α: transition into the n = 1 shell from n = 2
• K_β: transition into the n = 1 shell from n = 3

These details are useful for understanding spectra, but are not required for the 9478 quantum learning outcomes.