X-rays and lasers

MDCAT Physics · Physics · Atomic spectra. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

X-rays are made when electrons from a heated filament are accelerated through a high voltage and strike a heavy metal target such as tungsten. Most of the energy becomes heat. Electrons slowed down by the target nuclei give a continuous spectrum (bremsstrahlung) with a sharp minimum wavelength set by the tube voltage. Electrons that knock out inner-shell electrons give characteristic lines, which depend only on the target element.

A laser works by stimulated emission: a photon makes an excited atom emit a second photon with the same frequency, phase and direction. This needs a population inversion, which means more atoms in an excited state than in a lower one. It is possible because atoms stay much longer in a metastable state.

Rules to remember

maximum photon energy = electron charge × tube voltage (eV = hf_max = hc/λ_min), so λ_min = hc/eV
Higher tube voltage: shorter λ_min, more penetrating X-rays. Larger filament current: more electrons, greater intensity only
K_α line: an electron drops from the L shell into a vacancy in the K shell; K_β: from the M shell
Dense materials of high atomic number (bone, lead) absorb X-rays more than soft tissue. Uses: radiography, CT scans, cancer therapy, studying crystals
Metastable state lasts about 10⁻³ s; an ordinary excited state about 10⁻⁸ s
Laser light is monochromatic, coherent, intense and in a narrow beam
Helium-neon laser: excited helium atoms pass energy to neon atoms by collision; neon emits red light of 632.8 nm

Common mistake

Students say a higher tube voltage shifts the characteristic lines. It does not: those lines belong to the target element. The voltage changes only the cut-off wavelength, and the filament current changes only the intensity.

Worked example

An X-ray tube works at 33 kV. Find the minimum wavelength of the X-rays. (h = 6.6 × 10⁻³⁴ J s, c = 3.0 × 10⁸ m s⁻¹, e = 1.6 × 10⁻¹⁹ C)

Maximum photon energy = eV = 1.6 × 10⁻¹⁹ × 3.3 × 10⁴ = 5.28 × 10⁻¹⁵ J.
hc = 6.6 × 10⁻³⁴ × 3.0 × 10⁸ = 1.98 × 10⁻²⁵ J m.
λ_min = hc ÷ eV = (1.98 × 10⁻²⁵) ÷ (5.28 × 10⁻¹⁵) = 3.75 × 10⁻¹¹ m.

Answer: λ_min = 3.75 × 10⁻¹¹ m (about 0.038 nm).

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