Dual Nature of Radiation and Matter
Easy Overview
Light behaves like a wave — we saw that with interference, diffraction, and polarisation. But light also behaves like a particle. This is not a contradiction; it is the dual nature of light. When light interacts with matter (absorption, emission), it acts as a stream of particles called photons. When it propagates, it acts as a wave. This duality is not limited to light — electrons, protons, and even atoms exhibit both particle and wave behaviour. The photoelectric effect was the first clear evidence of the particle nature of light. When light shines on a metal surface, electrons are ejected — photoelectrons. Classical wave theory predicted that (1) higher intensity means faster electrons, and (2) electrons would be emitted regardless of frequency given enough time. Both were wrong. Experiment showed: (1) the maximum kinetic energy of photoelectrons depends on frequency, not intensity, and (2) for frequencies below a threshold, no electrons are emitted regardless of intensity. Einstein explained this in 1905 by proposing that light consists of photons, each with energy E = hf. Increasing intensity increases the number of photons, not their energy. A photon transfers all its energy to a single electron. The work function φ is the minimum energy needed to free an electron. K_max = hf − φ. This won Einstein the Nobel Prize. The photoelectric effect also gave us the value of Planck's constant h = 6.63 × 10⁻³⁴ J·s. It is used in photoelectric cells, solar panels, light sensors, and automatic door openers. X-rays are produced by the inverse process — electrons accelerated through a high voltage strike a metal target, emitting X-ray photons. The minimum wavelength of X-rays is λ_min = hc/(eV). De Broglie proposed that if light (a wave) has particle properties, then particles should have wave properties. The de Broglie wavelength λ = h/p. For an electron accelerated through V volts, λ = h/√(2meV) ≈ 1.23/√V nm. This was confirmed by Davisson and Germer who observed electron diffraction from a crystal. Wave-particle duality is at the heart of quantum mechanics.
Photoelectric Effect — Experimental Results
Hertz discovered photoelectric effect (1887). Important observations: (1) Above threshold frequency f₀, photoelectrons emitted instantly. (2) K_max ∝ f (not intensity). (3) Below f₀, no emission regardless of intensity. (4) Photoelectric current ∝ intensity (above f₀). (5) Stopping potential V₀ = K_max/e. Classical wave theory could NOT explain observations 1, 2, or 3.
Einstein's Photon Theory
Light consists of photons, each with E = hf = hc/λ. Photon is absorbed by one electron. Energy conservation: hf = φ + K_max. Stopping potential: eV₀ = hf − φ. Slope of V₀ vs f gives h/e. Einstein's equation explains all experimental observations. h = 6.63 × 10⁻³⁴ J·s.
Work Function and Threshold Frequency
Work function φ = hf₀. The minimum energy needed to liberate an electron from a metal surface. Different metals have different φ. Cesium: φ = 2.14 eV (visible light). Sodium: 2.3 eV. Aluminium: 4.08 eV. Copper: 4.7 eV (UV required). Most metals have φ in the range 2-5 eV. f₀ = φ/h.
Photoelectric Cells and Applications
Photocell: metal surface (photocathode) emits electrons when light strikes. Electrons collected by anode → current flows. Applications: (1) Automatic doors (light beam interrupted), (2) Burglar alarms, (3) Smoke detectors, (4) Film soundtracks, (5) Photomultipliers (electron multiplication). Solar cells use photovoltaic effect — built-in junction field separates charges.
X-Ray Production
Coolidge tube: electrons from hot filament accelerated by high voltage (10-100 kV) strike metal target (tungsten). Electrons decelerate rapidly → emit X-ray photons. Bremsstrahlung (braking radiation): continuous spectrum from zero to maximum frequency. Characteristic X-rays: specific frequencies from electron transitions in target atoms. Minimum wavelength λ_min = hc/(eV).
Continuous and Characteristic X-ray Spectra
Continuous spectrum: due to electron deceleration. Intensity ∝ ZV² (Z = atomic number). Minimum λ (cutoff) depends only on accelerating voltage, not target material. Characteristic X-rays: electrons knock inner-shell electrons out; outer electrons fill vacancies, emitting specific frequencies. Moseley's law: √f ∝ (Z − b). Used for element identification.
Wave Nature of Matter — de Broglie Hypothesis
Louis de Broglie (1924): just as light has both wave and particle properties, matter particles have wave properties. λ = h/p = h/(mv). For a particle in a box, only certain standing wave patterns are possible → quantised energy levels. De Broglie wavelength of macroscopic objects is tiny (h is very small) — wave nature not observed.
Davisson-Germer Experiment
Electrons scattered from a nickel crystal showed diffraction pattern — proving electron waves. The experiment: electrons accelerated through 54 V (λ = 0.165 nm) directed at nickel crystal. At θ = 50°, constructive interference observed consistent with d = 0.091 nm. This confirmed de Broglie's hypothesis.
De Broglie Wavelength of Electron
For electron accelerated through V volts: KE = eV = p²/(2m). p = √(2meV). λ = h/√(2meV) = h/√(2m₀eV). Numerically: λ (in nm) = 1.23/√V. For V = 100 V: λ = 0.123 nm — same order as atomic spacings in crystals.
Wave Packet and Uncertainty Principle
A wave packet is a group of waves with slightly different wavelengths, confined in space. It represents a particle. Heisenberg uncertainty principle: Δx·Δp ≥ h/(4π). Also: ΔE·Δt ≥ h/(4π). You cannot simultaneously know position and momentum precisely. This is a fundamental limit, not a measurement limitation.
Heisenberg Uncertainty Principle — Applications
Δx·Δp ≥ ħ/2 (where ħ = h/(2π)). It explains: (1) why electrons cannot fall into nucleus (confining to nuclear size gives huge momentum), (2) natural line width (finite lifetime → energy spread), (3) quantum tunnelling (particle passes through barrier with energy below barrier height). Zero-point energy: particle in lowest state still has kinetic energy.
Photon Momentum
Photon: E = pc. Momentum p = E/c = hf/c = h/λ. Photon has momentum even though it has no rest mass. Radiation pressure: force from absorbed/reflected photons. Solar sails use this for spacecraft propulsion. Comet tails point away from Sun due to radiation pressure.
Compton Effect
X-ray photons scattered by electrons shift to longer wavelength. Compton shift: Δλ = h/(m₀c) × (1 − cosθ). h/(m₀c) = 0.00243 nm (Compton wavelength). Confirms photon momentum p = h/λ. Classical theory predicts no wavelength shift — this was evidence for particle nature of EM radiation.
Key Points
- •Photoelectric effect: Einstein's equation hf = φ + K_max.
- •Threshold frequency f₀ = φ/h. Stopping potential eV₀ = hf − φ.
- •X-ray: λ_min = hc/(eV). Continuous + characteristic spectra.
- •de Broglie wavelength λ = h/p = h/(mv).
- •Davisson-Germer: electron diffraction from nickel crystal.
- •Electron λ = 1.23/√V nm (V in volts).
- •Heisenberg: Δx·Δp ≥ ħ/2, ΔE·Δt ≥ ħ/2.
- •Photon momentum p = h/λ = E/c.
- •Compton effect: Δλ = h/(m₀c)(1 − cosθ).
Practice Questions
- State Einstein's photoelectric equation. Explain how it accounts for (a) threshold frequency, (b) dependence of K_max on frequency, (c) instantaneous emission.
- Describe experimental study of photoelectric effect. Draw V₀ vs f graph and explain how h and φ are determined.
- What is de Broglie hypothesis? Derive λ for an electron accelerated through V volts. Find λ for electron through 100 V.
- Describe Davisson-Germer experiment. How did it confirm de Broglie's hypothesis?
- State Heisenberg uncertainty principle. Explain why electrons cannot exist inside the nucleus using uncertainty principle.
- Explain X-ray production. Derive λ_min. An X-ray tube operates at 50 kV. Find minimum wavelength.
- What is the Compton effect? Derive Compton shift. Why is it not observable with visible light?
- Calculate energy, frequency, and wavelength of a photon of momentum 3.3 × 10⁻²⁴ kg·m/s. h = 6.63 × 10⁻³⁴ J·s.