DUAL NATURE OF MATTER AND RADIATION | PHYSICS | SELF STUDY

PHYSICS  |  Class XII

Chapter 11

DUAL NATURE OF MATTER

AND RADIATION

Wave-Particle Duality  ◆  Photoelectric Effect  ◆  de Broglie Hypothesis

 

Introduction

The concept of Dual Nature of Matter and Radiation stands as one of the most revolutionary ideas in the history of science. At the start of the 20th century, two seemingly contradictory discoveries shook the foundations of classical physics:

▶        Light — previously thought to be purely a wave — was found to exhibit particle-like behaviour (Photoelectric Effect, Compton Effect).

▶        Electrons and other material particles — previously thought to be purely particles — were found to exhibit wave-like behaviour (de Broglie hypothesis, Davisson-Germer experiment).

This synthesis is known as Wave-Particle Duality and forms the conceptual bedrock of Quantum Mechanics. The chapter explores how classical concepts of 'waves' and 'particles' are inadequate individually, and how nature demands a unified quantum description. This understanding is essential not only for theoretical physics but also for modern technology including electron microscopes, solar cells, LEDs, transistors, and quantum computers.

 

1. Electron Emission from Metal Surfaces

Metals contain a large number of free electrons — electrons that are loosely bound and can move freely within the metal lattice. However, these electrons cannot normally escape the metal surface because they are held back by the attractive force of the positive ions in the lattice. The minimum energy required to just liberate one electron from the surface of a metal is called the Work Function (φ₀ or W) of that metal.

Work Function (φ₀): The minimum energy required to free an electron from the surface of a metal. It is measured in electron volts (eV). 1 eV = 1.6 × 10⁻¹⁹ J.

 

1.1 Methods of Electron Emission

Method of Emission	Description
Thermionic Emission	Metal is heated to a high temperature; thermal energy enables electrons to overcome the work function. Used in vacuum tubes and CRT monitors.
Photoelectric Emission	Light (photons) of sufficient frequency strikes the metal surface; photon energy is transferred to electrons, enabling escape. Basis of this chapter.
Field Emission	A very strong external electric field is applied; it pulls electrons out of the metal. Used in field-emission displays and electron guns.
Secondary Emission	High-speed electrons (primary) bombard the metal surface and knock out secondary electrons from it. Used in photomultiplier tubes.

 

 

2. Photoelectric Effect

The Photoelectric Effect is the phenomenon in which electrons are emitted from the surface of a metal when light of sufficiently high frequency falls on it. The emitted electrons are called Photoelectrons, and the resulting current is called Photocurrent.

This phenomenon was first observed by Heinrich Hertz in 1887 during his experiments on electromagnetic waves. He noticed that ultraviolet light falling on a metal surface caused sparks more easily. Philipp Lenard (1902) studied it systematically and reported several puzzling observations that classical wave theory could not explain.

Figure 1: Experimental Setup — Photoelectric Effect. Photons strike the metal plate and eject photoelectrons which travel to the collector, forming a photocurrent.

2.1 Experimental Observations

▶        Photoelectric emission occurs only when the frequency of incident light is above a certain minimum value called the Threshold Frequency (ν₀). Below this frequency, no emission occurs regardless of the intensity of light.

▶        The number of photoelectrons emitted per second (photocurrent) is directly proportional to the intensity of incident light, provided the frequency exceeds ν₀.

▶        The maximum kinetic energy (KE_max) of emitted photoelectrons depends only on the frequency of incident light — not on its intensity.

▶        Photoelectric emission is almost instantaneous (delay < 10⁻⁹ s) even for very low intensity light, as long as frequency exceeds ν₀.

▶        Applying a negative potential (retarding potential) to the collector reduces the photocurrent. At a specific negative voltage called the Stopping Potential (V₀), the photocurrent becomes exactly zero.

 

Figure 2: Photocurrent vs Collector Potential — Both intensities show the same stopping potential V₀, confirming that KE_max depends on frequency, not intensity.

2.2 Why Classical Wave Theory Failed

Failures of Classical Wave Theory

1. Wave theory predicts that any frequency of light should eventually eject electrons if the intensity is high enough — WRONG. Observations show a strict threshold frequency. 2. Wave theory predicts that higher intensity = more energy per electron = higher KE_max — WRONG. KE_max depends only on frequency, not intensity. 3. Wave theory predicts a time delay before emission (as energy accumulates) — WRONG. Emission is instantaneous.

 

2.3 Einstein's Explanation — Quantum Theory of Light (1905)

Albert Einstein (1905) brilliantly extended Planck's quantum hypothesis and proposed that light itself is made up of discrete energy packets called Photons (or Light Quanta). Each photon carries energy E = hν, where h is Planck's constant and ν is the frequency of light.

When a photon strikes a metal surface, it gives all its energy to a single electron. Part of this energy is used to overcome the work function (φ₀) of the metal, and the rest appears as kinetic energy of the emitted photoelectron.

Einstein's Photoelectric Equation

KE_max  =  hν  −  φ₀  =  h(ν − ν₀)

 

Stopping Potential

eV₀  =  KE_max  =  hν  −  φ₀

 

Threshold Frequency

ν₀  =  φ₀ / h

 

Threshold Wavelength

λ₀  =  hc / φ₀

 

Einstein's equation explains all the observations: if ν < ν₀, then hν < φ₀, so the photon doesn't have enough energy to free the electron — this explains the threshold. Higher frequency → more energy per photon → higher KE_max regardless of intensity. Increasing intensity means more photons per second → more photoelectrons → more current, but individual electron energy (KE) is unchanged. Emission is instantaneous because one photon transfers all energy to one electron at once.

Einstein was awarded the Nobel Prize in Physics in 1921 for his discovery of the law of the photoelectric effect — not for the Theory of Relativity!

Figure 3: Stopping Potential (V₀) vs Frequency — A straight line graph with slope = h/e. Different metals give parallel lines (different intercepts due to different work functions).

Slope of V₀ vs ν graph

Slope  =  h / e  =  Planck's constant / charge of electron

 

2.4 Particle Nature of Light — Photon Properties

Properties of a Photon

1. A photon is a quantum (packet) of electromagnetic energy. It is massless. 2. Energy:  E = hν = hc/λ  (h = 6.626 × 10⁻³⁴ J·s, Planck's constant) 3. Momentum:  p = h/λ = E/c  (photon carries momentum despite having zero rest mass) 4. Speed:  c = 3 × 10⁸ m/s in vacuum for all frequencies. 5. Rest mass = 0 (photon has no rest mass); Dynamic mass = hν/c² 6. Photons are NOT deflected by electric or magnetic fields. 7. In a photon-electron collision, total energy and total momentum are conserved. 8. Number of photons emitted per second = Power / Energy per photon = P / hν

 

Figure 4: Photon Properties — Electrons in atoms emit photons when they drop to lower energy levels. Each photon carries energy E = hν and momentum p = h/λ.

 

3. de Broglie Hypothesis — Wave Nature of Matter

In 1924, the French physicist Louis de Broglie made a bold and revolutionary proposal. He argued that if light (a wave) can behave like particles (photons), then particles of matter (like electrons, protons, atoms) should also exhibit wave-like properties. He postulated that every moving particle has an associated wave, now called a Matter Wave or de Broglie Wave.

3.1 de Broglie Wavelength Formula

de Broglie derived the wavelength of the matter wave associated with a particle of momentum p by using the photon momentum formula p = h/λ and applying it to material particles:

de Broglie Wavelength

λ  =  h / p  =  h / mv

 

For accelerated particle (voltage V)

λ  =  h / √(2mqV)

 

For particle at temperature T

λ  =  h / √(3mkT)   (Thermal de Broglie wavelength)

 

For electron (m = 9.1×10⁻³¹ kg)

λ  =  1.227 / √V  nm   (V in volts)

 

Here h = 6.626 × 10⁻³⁴ J·s (Planck's constant), m = mass of the particle, v = velocity, q = charge, V = accelerating potential, k = Boltzmann's constant, T = temperature in Kelvin.

3.2 Key Observations about de Broglie Wavelength

▶        The de Broglie wavelength is inversely proportional to momentum. Heavier particles or faster-moving particles have shorter wavelengths and are harder to detect as waves.

▶        For a macroscopic object (like a cricket ball, mass ~ 0.15 kg, speed ~ 30 m/s): λ = h/mv ≈ 1.5 × 10⁻³⁴ m — far too small to detect. This is why we never observe wave properties of everyday objects.

▶        For an electron accelerated through 100 V: λ ≈ 0.123 nm — comparable to the spacing in a crystal lattice! This is why electrons show diffraction in crystals.

▶        de Broglie's hypothesis unifies the descriptions of both light and matter under one quantum framework.

 

Figure 5: de Broglie Matter Waves — An electron beam exhibits diffraction through a crystal lattice, forming a diffraction pattern that confirms its wave nature. λ = h/p.

 

4. Davisson-Germer Experiment (1927)

The wave nature of electrons was experimentally confirmed by Clinton Davisson and Lester Germer in 1927 at Bell Laboratories, USA. They observed the diffraction of electrons from the surface of a Nickel (Ni) crystal — conclusively proving that electrons exhibit wave properties. Davisson was awarded the Nobel Prize in Physics in 1937 for this work.

4.1 Experimental Setup

▶        An electron gun produces a beam of electrons accelerated through a known potential V.

▶        The electron beam is directed onto a single crystal of Nickel (Ni), which acts as a diffraction grating for electrons because its interatomic spacing (d ≈ 0.215 nm) is comparable to the de Broglie wavelength of electrons.

▶        A movable detector measures the intensity of scattered electrons at different angles θ.

▶        The entire setup is enclosed in a vacuum chamber to prevent scattering of electrons by air molecules.

 

Figure 6: Davisson-Germer Experiment — Electron beam scatters off Ni crystal. A sharp intensity peak at 50° (for 54 V electrons) confirms Bragg diffraction — wave nature of electrons.

4.2 Key Observation and Result

When the accelerating voltage was set to 54 V, a sharp peak in the scattered electron intensity was observed at a scattering angle of 50° from the incident beam (i.e., angle of incidence = 65° from the Ni crystal surface planes).

Using Bragg's Law for diffraction: 2d sinθ = nλ, with d = 0.091 nm (Ni crystal interplanar spacing for the relevant planes) and θ = 65°, the experimental wavelength was calculated to be λ_exp ≈ 0.165 nm.

The de Broglie wavelength for 54 V electrons: λ_dB = h/√(2meV) = 6.626×10⁻³⁴ / √(2 × 9.1×10⁻³¹ × 1.6×10⁻¹⁹ × 54) ≈ 0.167 nm.

Conclusion of Davisson-Germer Experiment

The experimentally observed wavelength (0.165 nm) matches the de Broglie wavelength (0.167 nm) almost exactly. This proves that electrons undergo wave diffraction — confirming de Broglie's hypothesis that matter has wave-like properties. This was the first direct experimental proof of the wave nature of particles.

 

4.3 G.P. Thomson's Experiment (1927)

Independently, G.P. Thomson (son of J.J. Thomson who discovered the electron) passed high-speed electrons through a thin gold foil and observed concentric diffraction rings on a photographic plate behind the foil — exactly analogous to X-ray diffraction rings. Thomson shared the Nobel Prize with Davisson in 1937.

Interestingly, J.J. Thomson proved the electron is a particle (1897), while G.P. Thomson proved the electron is a wave (1927) — a beautiful illustration of wave-particle duality!

 

5. Heisenberg's Uncertainty Principle

Werner Heisenberg (1927) showed that the wave-particle duality has a fundamental consequence: it is impossible to simultaneously measure both the exact position and the exact momentum of a particle. This is not a limitation of our instruments — it is a fundamental property of nature.

Uncertainty Principle (Position-Momentum)

Δx · Δp  ≥  h / 4π  =  ℏ/2

 

Uncertainty Principle (Energy-Time)

ΔE · Δt  ≥  h / 4π

 

Here Δx = uncertainty in position, Δp = uncertainty in momentum, ΔE = uncertainty in energy, Δt = uncertainty in time, and ℏ = h/2π (reduced Planck's constant).

▶        If we try to measure the position of an electron precisely (small Δx), its momentum becomes very uncertain (large Δp), and vice versa.

▶        This principle explains why electrons cannot exist inside the nucleus (the nucleus is too small — confining an electron there would give it enormous kinetic energy that would eject it immediately).

▶        It also explains the ground state energy of the hydrogen atom (zero-point energy) — the electron cannot be at rest with exactly zero kinetic energy.

▶        The Uncertainty Principle is one of the cornerstones of quantum mechanics and has profound philosophical implications about the nature of observation and reality.

 

 

6. Wave vs Particle Properties of Light and Matter

Wave Properties	Particle Properties
Interference (Young's double-slit experiment)	Photoelectric effect (light behaves as photons)
Diffraction (bending around obstacles)	Compton scattering (photon-electron collision)
Polarisation (transverse wave nature)	Radiation pressure (photon momentum)
Reflection, Refraction follow Huygens' principle	Black-body radiation explained by quantisation
Electron diffraction in Davisson-Germer experiment	Electrons deflected in electric and magnetic fields
Electron microscope uses wave nature of electrons	Photoelectron spectroscopy uses particle nature

 

The wave and particle descriptions are complementary, not contradictory. Which description is applicable depends on the type of experiment being performed. This is the essence of Niels Bohr's Principle of Complementarity.

 

7. Important Constants and Values

Constant / Quantity	Value
Planck's constant (h)	6.626 × 10⁻³⁴ J·s  =  4.136 × 10⁻¹⁵ eV·s
Speed of light (c)	3 × 10⁸ m/s
Mass of electron (mₑ)	9.109 × 10⁻³¹ kg
Charge of electron (e)	1.602 × 10⁻¹⁹ C
1 electron volt (eV)	1.602 × 10⁻¹⁹ J
Boltzmann constant (k)	1.381 × 10⁻²³ J/K
Work function of Caesium	2.0 eV (lowest — best for photoelectric cells)
Work function of Platinum	5.65 eV (highest among common metals)
λ of electron at 1 V acceleration	1.227 nm
λ of electron at 100 V acceleration	0.1227 nm

 

 

8. Master Formula Table

Formula / Law	Expression	Remarks
Photon Energy	E = hν = hc/λ	h = 6.626×10⁻³⁴ J·s
Photon Momentum	p = h/λ = E/c	Photon has zero rest mass
Einstein's PE Equation	KE_max = hν − φ₀	φ₀ = work function in joules or eV
Stopping Potential	eV₀ = hν − φ₀	V₀ = stopping potential in volts
Threshold Frequency	ν₀ = φ₀ / h	Minimum freq for photoelectric emission
Threshold Wavelength	λ₀ = hc / φ₀	Maximum wavelength for emission
V₀ vs ν Slope	Slope = h/e	Used to measure Planck's constant
de Broglie Wavelength	λ = h/p = h/mv	Valid for all particles
λ (accelerated particle)	λ = h/√(2mqV)	q = charge, V = potential
λ of electron (simplified)	λ = 1.227/√V nm	V in volts, λ in nm
Heisenberg (position)	Δx · Δp ≥ h/4π	Fundamental limit, not instrument error
Heisenberg (energy-time)	ΔE · Δt ≥ h/4π	ΔE = energy uncertainty
Bragg's Law (diffraction)	2d sinθ = nλ	d = interplanar spacing
Photon Number	N = P/hν (per sec)	P = power of light source

 

 

9. Key Points to Remember

Photoelectric Effect — Must Know

* Light of frequency BELOW threshold (ν₀) CANNOT cause emission, no matter how intense. * Stopping potential V₀ depends only on frequency, NOT on intensity of light. * Photocurrent is proportional to intensity (= number of photons per second). * KE_max = eV₀ = hν − φ₀. This is Einstein's photoelectric equation. * Slope of (V₀ vs ν) graph = h/e. This is how Planck's constant can be measured experimentally. * Threshold wavelength: λ₀ = hc/φ₀. For λ < λ₀ → emission occurs. For λ > λ₀ → no emission.

 

de Broglie & Wave-Particle Duality — Must Know

* Every moving particle has an associated matter wave: λ = h/p = h/mv. * For a charged particle accelerated through V volts: λ = h/√(2mqV). * For an electron: λ = 1.227/√V nm (V in volts). e.g., at 100 V → λ ≈ 0.123 nm. * Davisson-Germer experiment (1927): confirmed electron wave diffraction from Ni crystal at 54 V → 50° peak. * Heisenberg Uncertainty: Δx · Δp ≥ h/4π. More precise position → less precise momentum. * Macroscopic objects have negligibly small λ → no observable wave nature. * Principle of Complementarity: wave and particle natures are complementary, not contradictory.

 

Photon vs Electron — Quick Comparison

Photon: E = hν, p = h/λ, rest mass = 0, speed = c always, not deflected by E or B field. Electron: E = p²/2m (non-relativistic), p = mv, rest mass = 9.1×10⁻³¹ kg, speed < c, deflected by E and B fields. Both show diffraction and interference (wave behaviour) and also particle-like collisions.

 

— End of Chapter 11: Dual Nature of Matter and Radiation —