Dual Nature of Matter and Radiation
1. Introduction
Light and matter both exhibit particle and wave nature. This duality is a cornerstone of quantum mechanics. The photoelectric effect demonstrates the particle nature of light, while diffraction shows its wave nature.
2. Photoelectric Effect
When light of suitable frequency falls on a metal surface, electrons are emitted. This cannot be explained by wave theory alone.
2.1 Experimental Observations
- Photocurrent is proportional to intensity (for a given frequency).
- For every metal, there is a threshold frequency f₀ below which no emission occurs.
- Photoelectrons are emitted instantly (no time lag).
- Maximum kinetic energy of photoelectrons depends on frequency, not intensity.
'Wave theory fails to explain the threshold frequency and instantaneous emission — classical waves would need time to transfer enough energy to eject electrons.'
3. Einstein's Photoelectric Equation
Einstein proposed that light consists of photons of energy E = hf.
K_max = hf - φ, where φ = hf₀ is the work function.
Stopping potential: eV₀ = hf - φ ⇒ V₀ = (h/e)f - φ/e.
3.1 Plot of V₀ vs f
A straight line with slope h/e. The intercept on the f-axis gives f₀, and on the V₀-axis gives -φ/e.
4. de Broglie Wavelength
Just as light has both wave and particle nature, de Broglie proposed that matter also has wave-like properties.
λ = h/p = h/(mv)
For an electron accelerated through potential V: λ = h/√(2meV) = 12.27/√V angstroms.
5. Davisson-Germer Experiment
An experiment that confirmed the wave nature of electrons. Electrons scattered from a nickel crystal showed diffraction patterns consistent with de Broglie's formula.
5.1 Key Result
The intensity peaks at θ = 50° for 54 eV electrons, giving λ = 0.165 nm, matching the de Broglie wavelength of 0.167 nm.
5. Experimental Verification of Einstein's Equation
Millikan's experiment verified Einstein's photoelectric equation. He measured the stopping potential for different frequencies of incident light and plotted V₀ vs f.
From the straight line obtained:
- Slope = h/e (gave h = 6.626×10^{-34} Js, matching Planck's constant)
- Intercept on f-axis = f₀ (threshold frequency)
- Intercept on V₀-axis = -φ/e (work function in eV)
This was crucial in establishing the quantum nature of light.
6. de Broglie Wavelength in Different Contexts
6.1 For an Electron Accelerated by Voltage V
λ = h/√(2meV) = 12.27/√V angstroms
For V = 54 V: λ = 12.27/√54 = 12.27/7.35 = 1.67 Å = 0.167 nm This matches the Davisson-Germer result.
6.2 For a Particle of Mass m and Kinetic Energy E
λ = h/√(2mE)
6.3 For a Neutron (Thermal)
Thermal neutron at T = 300 K has λ ≈ 1.8 Å, making it suitable for crystal diffraction studies.
7. Wave-Particle Duality Summary
| Phenomenon | Particle Nature | Wave Nature |
|---|---|---|
| Photoelectric effect | Photons (E = hf) | - |
| Interference/diffraction | - | Wave behaviour |
| Compton effect | Photon momentum | - |
| Electron diffraction | - | de Broglie waves |
| Blackbody radiation | Quantised energy | - |
'All matter exhibits both wave and particle properties, but the wave nature becomes significant only for particles with very small mass (electrons, neutrons, atoms).'
8. Worked Problems
Problem 1: The work function of sodium is 2.3 eV. Find the threshold wavelength. Solution: φ = hf₀ = hc/λ₀ ⇒ λ₀ = hc/φ = (6.63×10^{-34}×3×10⁸)/(2.3×1.6×10^{-19}) = 5.41×10^{-7} m = 541 nm.
Problem 2: Light of wavelength 400 nm falls on a metal with work function 2 eV. Find stopping potential. Solution: E = hc/λ = (6.63×10^{-34}×3×10⁸)/(4×10^{-7}) = 4.97×10^{-19} J = 3.1 eV. K_max = E - φ = 3.1 - 2 = 1.1 eV. V₀ = K_max/e = 1.1 V.
Problem 3: Find the de Broglie wavelength of an electron accelerated through 100 V. Solution: λ = 12.27/√100 = 12.27/10 = 1.227 angstroms = 0.1227 nm.
7. Common Mistakes
'Students often confuse the work function (minimum energy to eject) with the threshold frequency. They are related by φ = hf₀.'
'de Broglie wavelength applies to ALL matter, not just electrons. But it is significant only for particles with very small mass.'
8. ISC Exam Focus
| Topic | Theory Marks | Practical Marks |
|---|---|---|
| Photoelectric effect | 4 | 3 |
| Einstein's equation | 4 | 3 |
| de Broglie wavelength | 3 | 2 |
| Davisson-Germer | 2 | 1 |
9. Self-Test Questions
- State the experimental observations of the photoelectric effect that could not be explained by wave theory.
- Derive Einstein's photoelectric equation and explain the significance of the work function.
- Find the de Broglie wavelength of an electron moving with speed 2×10⁶ m/s (m_e = 9.1×10^{-31} kg).
- The stopping potential for a metal is 1.5 V for λ = 300 nm. Find the work function.
- Describe the Davisson-Germer experiment and how it confirmed de Broglie's hypothesis.
