Chapter 2.6: Wave-Particle Duality
Welcome to one of the most exciting and mind-bending topics in physics! Up to this point in your AS 2 studies, you have seen light behaving as a classic wave (undergoing diffraction and interference) and matter behaving as solid little particles. But nature has a wonderful surprise for us: waves can behave like particles, and particles can behave like waves. This concept is called wave-particle duality.
Don't worry if this sounds strange at first. We will break it down step-by-step, explore the key mathematical relationships, and look at the exact experimental evidence you need for your CCEA AS 2 exam.
1. The Duality Principle
The Duality Principle states that both electromagnetic radiation and matter particles exhibit dual properties: they can behave as waves in some physical situations and as particles in others.
A. How Electromagnetic Radiation Behaves:
• Wave-like behaviour: Displayed during propagation, diffraction, and interference. Key evidence includes Young's double-slit interference, single-slit diffraction, and diffraction grating patterns.
• Particle-like behaviour: Displayed during emission, absorption, collision, and localised energy transfer. Key evidence includes the photoelectric effect and atomic line emission/absorption spectra where light interacts as packets of energy called photons.
B. How Matter Behaves:
• Particle-like behaviour: Displayed in everyday collisions, deflection in electric/magnetic fields, and carrying mass and charge.
• Wave-like behaviour: Displayed when moving subatomic particles undergo diffraction and interference through crystal lattices.
Key Takeaway: Radiation behaves like a wave when travelling and like a particle when interacting with matter. Matter behaves like a particle in bulk, but exhibits wave properties when moving at microscopic scales.
2. The de Broglie Hypothesis and Formula
In 1924, French physicist Louis de Broglie suggested that if wave-like light can act like particles, then moving particles of matter should also possess wave properties. He proposed that any moving particle has an associated matter wave with a specific wavelength.
The de Broglie Relationship
\(\lambda = \frac{h}{p} = \frac{h}{mv}\)
Where:
• \(\lambda\) = de Broglie wavelength in metres (\(\text{m}\))
• \(h\) = Planck's constant (\(6.63 \times 10^{-34}\text{ J s}\))
• \(p\) = linear momentum of the particle in kilogram metres per second (\(\text{kg m s}^{-1}\) or \(\text{N s}\))
• \(m\) = mass of the particle in kilograms (\(\text{kg}\))
• \(v\) = velocity of the particle in metres per second (\(\text{m s}^{-1}\))
Significance of the Equation
Exam Focus: CCEA examiners frequently ask students to explain the true meaning of this equation. Be clear and specific: the de Broglie equation directly links a wave property (\(\lambda\)) with a mechanical/particle property (\(p\) or \(mv\)).
Why Don't We See Everyday Objects Diffract?
• Macroscopic objects (e.g., a moving cricket ball or car): Because their mass \(m\) is relatively large, their momentum \(p\) is huge compared to the tiny value of Planck's constant (\(h \approx 10^{-34}\text{ J s}\)). This results in an impossibly tiny de Broglie wavelength (around \(10^{-34}\text{ m}\) or smaller). No physical aperture or grating exists at this scale, so wave properties cannot be observed.
• Subatomic particles (e.g., electrons, protons, neutrons): Because their mass is extremely small (for an electron, \(m_e \approx 9.11 \times 10^{-31}\text{ kg}\)), their de Broglie wavelength is around \(10^{-10}\text{ m}\). This is comparable to the spacing between atoms in crystals, making diffraction observable!
Key Takeaway: \(\lambda = \frac{h}{mv}\). As mass or velocity increases, the de Broglie wavelength gets smaller.
3. Electron Acceleration and Kinetic Energy Relations
In laboratory experiments, electrons are accelerated from rest through an anode potential difference (accelerating voltage) \(V\). We can derive a direct link between the accelerating voltage and the electron's de Broglie wavelength.
Step-by-Step Derivation:
Step 1: Relate electrical work done to kinetic energy
The electrical work done on an electron of charge \(e\) by a potential difference \(V\) equals its gained kinetic energy \(E_k\):
\(E_k = \frac{1}{2} m_e v^2 = eV\)
Step 2: Express velocity (\(v\)) in terms of voltage (\(V\))
\(v^2 = \frac{2eV}{m_e} \implies v = \sqrt{\frac{2eV}{m_e}}\)
Step 3: Express momentum (\(p\)) in terms of kinetic energy and voltage
\(p = m_e v = m_e \sqrt{\frac{2eV}{m_e}} = \sqrt{2 m_e eV}\)
Step 4: Substitute momentum into the de Broglie formula
\(\lambda = \frac{h}{p} = \frac{h}{\sqrt{2 m_e eV}}\)
Important Data Sheet Constants for Calculations:
• Mass of an electron: \(m_e = 9.11 \times 10^{-31}\text{ kg}\)
• Elementary charge: \(e = 1.60 \times 10^{-19}\text{ C}\)
• Planck's constant: \(h = 6.63 \times 10^{-34}\text{ J s}\)
Key Takeaway: The de Broglie wavelength of an accelerated electron is inversely proportional to the square root of the accelerating voltage: \(\lambda \propto \frac{1}{\sqrt{V}}\).
4. Experimental Evidence for Particle Waves: Electron Diffraction
Diffraction is an exclusively wave phenomenon. If particles like electrons can be shown to diffract, it proves beyond doubt that matter has wave properties.
The Electron Diffraction Tube Apparatus
The standard demonstration tube contains the following key components:
1. Electron Gun: A heated metal filament emits electrons via thermionic emission.
2. Anode: Maintained at a high positive potential \(V\) to accelerate the electrons into a narrow, high-speed beam.
3. Target (Thin Polycrystalline Graphite Foil): A very thin carbon target placed in the path of the electron beam.
4. Evacuated Glass Bulb: The vacuum prevents electrons from colliding with gas molecules.
5. Phosphor / Fluorescent Screen: Glows green/bright when struck by electrons, revealing the pattern.
Why Graphite?
• The atomic layer spacings (regular planes of carbon atoms) in graphite are on the order of \(10^{-10}\text{ m}\).
• Because this spacing is comparable to the de Broglie wavelength of the accelerated electrons, the crystal lattice acts as a natural three-dimensional diffraction grating.
• Because the graphite sample is polycrystalline (composed of many tiny, randomly orientated micro-crystals), the diffracted electrons form concentric circular rings on the fluorescent screen rather than single spots.
Effect of Changing the Accelerating Voltage (\(V\))
This is a classic CCEA exam question. Follow this exact logical sequence:
1. Increasing the accelerating voltage \(V\) increases the kinetic energy and momentum (\(p\)) of the electrons.
2. By \(\lambda = \frac{h}{p}\), the de Broglie wavelength \(\lambda\) decreases.
3. According to diffraction theory (\(d \sin\theta = n\lambda\)), a smaller wavelength produces a smaller angle of diffraction \(\theta\).
4. Observed result on screen: The concentric rings contract (their radii and diameters become smaller, moving closer to the central bright spot).
Summary of Voltage Effects:
• Increase \(V\): \(\implies v \uparrow \implies p \uparrow \implies \lambda \downarrow \implies \theta \downarrow \implies\) Rings contract (smaller radii)
• Decrease \(V\): \(\implies v \downarrow \implies p \downarrow \implies \lambda \uparrow \implies \theta \uparrow \implies\) Rings expand (larger radii)
Key Takeaway: Electron diffraction proves that electrons have wave properties. Increasing voltage reduces the wavelength and shrinks the diffraction rings.
5. Common Pitfalls and Examiner Advice
• Pitfall 1: Confusing Photons and Matter Particles.
Never use \(c = f\lambda\) or \(E = hf\) for an electron! Photons travel at the speed of light (\(c\)) and have zero rest mass. Electrons have mass (\(m_e\)), travel at speeds \(v < c\), and their kinetic energy is \(E_k = \frac{1}{2}m_e v^2\). For electrons, always use \(\lambda = \frac{h}{mv}\).
• Pitfall 2: Forgetting Components in Tube Diagrams.
When asked to sketch or label the electron diffraction tube, students often forget the polycrystalline graphite target or the phosphor/fluorescent screen. Always include both!
• Pitfall 3: The Voltage vs. Ring Size Error.
Students often assume that higher voltage produces "bigger" rings. Remember: higher voltage means faster electrons \(\implies\) shorter wavelength \(\implies\) less diffraction \(\implies\) smaller rings.
• Pitfall 4: Grabbing the Wrong Mass.
Always check your Data Sheet carefully. Do not mix up the mass of an electron (\(9.11 \times 10^{-31}\text{ kg}\)) with the mass of a proton or neutron (\(1.67 \times 10^{-27}\text{ kg}\)).
6. Quick Revision Summary Box
• Wave-Particle Duality: Waves can act as particles (photons); particles can act as waves (matter waves).
• de Broglie Formula: \(\lambda = \frac{h}{p} = \frac{h}{mv}\)
• Electron in Voltage \(V\): \(E_k = eV = \frac{1}{2}m_e v^2 \implies p = \sqrt{2 m_e eV} \implies \lambda = \frac{h}{\sqrt{2 m_e eV}}\)
• Electron Diffraction: Demonstrates particle wave nature. Uses an electron gun, polycrystalline graphite target, and a fluorescent screen.
• Ring Radius Rule: Higher voltage \(\implies\) shorter wavelength \(\implies\) smaller ring diameter.