Physical principles of the production of X-rays; maximum photon energy,…

The idea

X-rays are produced when electrons are accelerated through a large potential difference and strike a metal target. Each electron gains kinetic energy equal to eV, where e is the elementary charge and V is the tube voltage. Most electrons undergo many glancing collisions, producing a continuous spectrum of bremsstrahlung photons with a sharp maximum photon energy hf_max = eV. A few electrons knock inner-shell electrons from target atoms; outer electrons fall into the vacancy, emitting characteristic photons whose energies equal the shell energy differences. The continuous spectrum has a short-wavelength limit set by the tube voltage, while characteristic peaks depend on the target element.

What earns the marks

• Electrons are thermionically emitted from a heated cathode and accelerated through a potential difference V, gaining kinetic energy eV.
• Bremsstrahlung arises from deceleration of electrons in the target nucleus's electric field, producing a continuous range of photon energies.
• The maximum photon energy is hf_max = eV, so the minimum wavelength is λ_min = hc/(eV); increasing V increases the maximum photon energy.
• Characteristic X-rays arise when an accelerated electron removes an inner-shell electron and an outer-shell electron drops into the vacancy, emitting a photon of energy equal to the shell energy difference.
• The continuous spectrum has a short-wavelength cut-off but no long-wavelength cut-off, whereas characteristic lines occur at discrete energies specific to the target element.
• Increasing the tube current increases the number of electrons per second and hence the intensity of the continuous spectrum, but does not change the maximum photon energy.

Common mistakes and how to put them right

• Thinking the maximum photon energy can exceed eV.

  Put it right: hf_max = eV, so the tube voltage sets a hard limit.

• Confusing characteristic lines with the continuous spectrum.

  Put it right: Characteristic lines come from electron shell transitions and are target-specific, while the continuous spectrum comes from deceleration.

• Believing the continuous spectrum has a long-wavelength cut-off.

  Put it right: It has a short-wavelength cut-off at λ_min = hc/(eV) but extends to long wavelengths.

Exam tips

• Sketch the spectrum with photon energy or frequency on the x-axis and intensity on the y-axis, marking the short-wavelength cut-off and any characteristic peaks.
• Use hf_max = eV and λ_min = hc/(eV) to calculate the cut-off, converting keV to J before substituting.
• Explain the shape of the continuous spectrum in terms of the range of decelerations experienced by electrons, not just one collision.