PHYC10004 Chap.11 Matter Waves, Diffraction and the Bohr Atom
Matter Waves, Diffraction and the Bohr Atom
Week 9 shows that the wave-particle story runs both ways. X-rays diffract from crystal planes, reinforcing only when 2d sin θ = mλ, and the lectures expect students to derive this Bragg condition from the path difference.
De Broglie proposed that every particle has a wavelength λ = h/p, and the Davisson-Germer experiment confirmed it when electrons diffracted from nickel at exactly the predicted angles.
The same week tackles the hydrogen atom. Its spectrum consists of sharp lines described by the Rydberg formula, which classical physics could not explain because an orbiting electron should radiate and spiral inward.
Bohr's model assumes quantised angular momentum and non-radiating stationary states, giving radii a_B n² and energies −13.6 Z²/n² eV, with photons emitted or absorbed between levels. The chapter closes with the limits of that model and the particle in a box, the simplest case of Schrödinger's wave mechanics, where standing waves allow only energies h²n²/8mL².
What this chapter covers
- 01
Bragg's Law
- 02
X-ray Diffraction
- 03
The de Broglie Wavelength
- 04
Davisson-Germer Experiment
- 05
Hydrogen Spectral Series
- 06
The Bohr Model
- 07
Energy Level Diagrams
- 08
Hydrogen-Like Ions
- 09
Particle in a Box
Worked example · free
Compare the wavelengths of an electron and a proton
- 1De Broglie: λ = h/p = h/mv for each particle.
- 1Electron: λ = 6.626 × 10⁻³⁴/(9.109 × 10⁻³¹ × 2.0 × 10⁶) = 3.6 × 10⁻¹⁰ m, comparable with atomic spacings.
- 1Proton: λ = 6.626 × 10⁻³⁴/(1.673 × 10⁻²⁷ × 2.0 × 10⁶) = 2.0 × 10⁻¹³ m.
- 1Ratio: at equal speed λ_e/λ_p = m_p/m_e = 1.84 × 10³, so the electron's wavelength is about 1840 times longer.
Key terms
- Bragg's Law
- Bragg's law, 2d sin θ = mλ, gives the angles at which waves scattered from crystal planes a distance d apart interfere constructively.
- De Broglie Wavelength
- The de Broglie wavelength λ = h/p is the wavelength associated with a particle of momentum p.
- Stationary State
- A stationary state is an allowed electron orbit or wave pattern in an atom in which the electron does not radiate energy.
- Bohr Radius
- The Bohr radius a_B = 0.0529 nm is the radius of the ground-state orbit of hydrogen in the Bohr model.
- Balmer Series
- The Balmer series is the set of hydrogen spectral lines from transitions ending on n = 2, which fall in the visible range.
- Ionisation Energy
- Ionisation energy is the energy needed to remove an electron completely from an atom, 13.6 eV for hydrogen in its ground state.
Matter Waves, Diffraction and the Bohr Atom FAQ
Why are the energy levels of hydrogen negative?
The zero of energy is chosen at infinite separation of the electron and proton. A bound electron has less energy than that, so its energy is negative, and 13.6 eV must be supplied to lift a ground-state electron to zero and free it.
Which assumptions of the Bohr model are not classical?
Bohr assumed that angular momentum is quantised in units of ħ, that an electron in an allowed orbit does not radiate even though it accelerates, and that light is emitted or absorbed only when the electron jumps between allowed orbits, with the photon energy equal to the level difference.
How did the Davisson-Germer experiment confirm matter waves?
Electrons of known speed were scattered from a nickel crystal and produced intensity peaks at angles matching Bragg's law for a wavelength equal to h/p. The measured wavelength agreed closely with de Broglie's prediction, showing that electrons diffract like waves.
What are the limits of the Bohr model?
It reproduces hydrogen's line wavelengths but cannot predict how bright each line is, and it fails for atoms with more than one electron. Schrödinger's wave mechanics replaced it, describing electrons as standing waves and handling the interactions between electrons.
Exam move
Derive Bragg's law from a labelled diagram of two planes until you can draw it from memory, then use it to find wavelengths, spacings and angles. Practise de Broglie calculations for electrons accelerated through a voltage, working in joules and kilograms. For the atom, draw a scaled energy-level diagram and read transitions off it, naming each series, and repeat for a hydrogen-like ion with Z² scaling.
Finish with particle-in-a-box energies and the photon emitted between levels.
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