Free Pearson Edexcel International A Level Physics lesson
Diffraction and matter waves
Let an opening reveal what a wave can do.
What you’ll learn
- Explain spreading using secondary wavelets.
- Calculate grating wavelengths with the correct order and spacing.
- Interpret electron diffraction and calculate de Broglie wavelength.
Build the next wavefront
A wave does not always continue only along a straight ray after meeting an opening. It spreads into the region beyond the edges; this behaviour is diffraction. In Huygens’ construction, points on an existing wavefront act as sources of secondary wavelets. The common forward envelope after a short time forms the next wavefront. An opening leaves only some of these points available, so curved fronts can emerge beyond its edges.
Diffraction occurs for openings of many sizes, but spreading is most prominent when opening width is comparable with wavelength. A very wide opening leaves a broad central region whose fronts remain nearly straight, with edge spreading less conspicuous relative to the beam. Sound with a wavelength comparable to a doorway can bend around it appreciably, while visible light wavelengths are tiny compared with that same opening.
The construction is a model for predicting fronts, not a claim that material particles move outward along every drawn circular wavelet. Obstacles can also generate wave spreading into a geometrical shadow. The complete intensity pattern depends on interference between contributions from different parts of the aperture; Huygens’ idea therefore connects naturally to superposition. Keep wavelength and opening width separate when comparing experiments, and do not assume that a narrow opening creates extra transmitted power.
Diffraction depends strongly on the ratio wavelength/opening width; each secondary wavelet advances distance vΔt.
Symbols and units
- v is propagation speed in m s⁻¹; Δt is elapsed time in s; wavelength λ and opening width a are in m.
How the equation explains the idea
- Each model wavelet expands a distance vΔt during the same time step; the common envelope forms the next wavefront.
- The ratio λ/a is dimensionless. Appreciable spreading is expected when this ratio is not small; this qualitative comparison is not an exact diffraction-angle formula.
Worked example
Compare two water-wave openings
- A water wave has wavelength 0.020 m.
- An opening 0.020 m wide is comparable to the wavelength, so substantial spreading is expected.
- An opening 0.20 m wide is ten wavelengths across and produces less relative spreading in its central region.
The ratio of width to wavelength explains the change in pattern.
Keep this idea. Construct secondary wavelets and compare opening width with wavelength to predict spreading.
Core practical 6: wavelength from a diffraction grating
A diffraction grating contains many regularly spaced openings or scattering lines. Each diffracts the incident wave, and waves from different lines interfere. At angle θ from the straight-through direction, neighbouring lines separated by d contribute path difference d sin θ. A principal maximum occurs when this equals an integer order m multiplied by wavelength. The central maximum is order zero and carries no wavelength information through that equation alone.
For a laser experiment, direct the beam normally at the grating and mark the central and first-order spots on a screen perpendicular to the beam. Measure screen distance D and transverse separation x; calculate θ = tan⁻¹(x/D), then λ = d sin θ/m. Use both sides of the central maximum and several suitable measurements. Convert a quoted line density to spacing by taking its reciprocal, making the length units consistent first.
Higher order can improve angular separation, but not every order exists: mλ/d cannot exceed one. Grating alignment, broad spots and short screen distances affect uncertainty. Use a beam stop, keep the laser below eye level, remove shiny objects and never look into direct or diffracted beams. Avoid assuming sin θ = tan θ at large angles. Measurements on opposite sides should agree within uncertainty; asymmetry suggests an alignment issue requiring investigation.
mλ = d sin θ; d = 1/(lines per metre); tan θ = x/D; |mλ/d| ≤ 1
Symbols and units
- m is integer diffraction order, λ is wavelength in m and d is slit spacing in m; θ is angle from the central direction.
- x is screen displacement from the central maximum and D is perpendicular screen distance, both in m.
How the equation explains the idea
- Convert line density into lines per metre before taking d = 1/density. Constructive path difference mλ = d sin θ gives λ = d sin θ/m for nonzero order.
- Find θ = arctan(x/D), then use its sine. For 500 lines per mm, d = 2.00 × 10⁻⁶ m. The condition |mλ/d| ≤ 1 limits possible orders.
Worked example
A first-order red maximum
- A grating has 500 lines per mm, so d = 1/(500 × 1000) = 2.00 × 10⁻⁶ m.
- The first-order maximum is at 18.0°.
- λ = d sin18.0° = 6.18 × 10⁻⁷ m = 618 nm.
Order m = 1 and the reciprocal line density both matter.
Keep this idea. Convert line density to spacing, identify the order and use the actual diffraction geometry.
Particles also produce diffraction
An electron beam passing through a suitably thin crystalline material can form rings or spots of enhanced detection probability. A simple classical model of tiny particles travelling along fixed straight paths cannot explain the systematic diffraction pattern. The regular atomic structure acts as a set of scattering centres, and the pattern provides evidence that electrons have a wavelength associated with their motion. Individual detections remain localised events; a wave pattern emerges statistically over many detections.
The de Broglie relation is λ = h/p, where p is momentum. For a non-relativistic electron, p = mv; if it is accelerated from negligible initial kinetic energy through potential difference V, eV = ½mv² and p = √(2meV). The voltage shortcut therefore has stated assumptions. At sufficiently high speed the non-relativistic momentum and energy expressions fail, though the de Broglie relation itself remains the wavelength–momentum connection.
Increasing electron momentum shortens its wavelength and typically reduces diffraction angles for an unchanged crystal geometry. This is a comparison of patterns, not a claim that an electron literally turns into a classical water wave. Wave and particle models address different observations, and neither familiar macroscopic picture alone describes all quantum behaviour. Electron diffraction also motivates instruments that exploit shorter wavelengths to investigate much smaller structures than visible light can readily resolve.
λ = h/p; non-relativistic p = mv; from rest through V: p = √(2meV)
Symbols and units
- λ is matter wavelength in m; h is Planck’s constant in J s; p is momentum in kg m s⁻¹.
- m is electron mass in kg, e is positive elementary charge in C and V is accelerating potential difference in V.
How the equation explains the idea
- For an electron accelerated from rest, kinetic energy gain is eV. In the nonrelativistic model, p²/(2m) = eV, so p = √(2meV).
- Substitute into λ = h/p. Quadrupling V doubles momentum and halves wavelength. At relativistic energies p = mv and this square-root expression are no longer valid.
Conditions. Electrons accelerated from rest, using nonrelativistic λ = h/√(2meV), h = 6.626 × 10⁻³⁴ J s, m = 9.109 × 10⁻³¹ kg, e = 1.602 × 10⁻¹⁹ C. Range 100–1600 V; relativistic corrections are neglected.
Accelerating voltage / V; Matter wavelength / nm.
At 100 V, λ ≈ 0.123 nm; at 400 V, λ ≈ 0.0613 nm. Quadrupling voltage halves wavelength. The plotted nm scale is 10⁻⁹ m.
Open the interactive lesson to view the plotted curves and their coordinate tables.
Worked example
Change momentum without recalculating constants
- An electron has momentum 3.315 × 10⁻²⁴ kg m s⁻¹.
- Using h = 6.63 × 10⁻³⁴ J s gives λ = 2.00 × 10⁻¹⁰ m.
- Doubling momentum halves wavelength to 1.00 × 10⁻¹⁰ m.
Wavelength follows the reciprocal of momentum.
Keep this idea. A diffraction pattern tests wave behaviour; momentum determines a particle’s de Broglie wavelength.
Open the interactive lesson for free practice and reviewed three-question samples when available. New full question sets and timed unit checks are in preparation. Existing complete Physics revision packs retain paid access; existing free teaching PDFs remain free.
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