ν₀ = Φ/h
ν₀: threshold frequency; Φ: work function; h: Planck constant.
Dual Nature
Formulae & PYQs
Formula sheet, NCERT solutions, diagrams, numericals and exam questions for the dual nature of radiation and matter Dual Nature Formulas PYQs .
37NCERT Solutions
25+Core Formulae
350+Exam Questions
10NCERT Diagrams
01
Complete Formula Sheet
All high-frequency formulas in one exam-ready collection.
Photoelectric Effect Formulae
λ₀ = hc/Φ
λ₀ is the longest wavelength that can cause emission.
E = hν = hc/λ
ν: frequency; λ: wavelength; c: speed of light.
hν = Φ + Kmax
Photon energy supplies work function and maximum kinetic energy.
Kmax = eV₀
e: electronic charge; V₀: stopping potential.
hν = Φ + ½mv²max
m: electron mass; vmax: maximum photoelectron speed.
V₀ = (h/e)ν − Φ/e
Slope of V₀–ν graph = h/e; frequency intercept = ν₀.
Is ∝ intensity
Saturation current rises with light intensity when frequency exceeds threshold.
E(eV) = 1240/λ(nm)
Fast conversion between photon wavelength and energy.
de Broglie and Matter-Wave Formulae
λ = h/p
Universal de Broglie relation; p is particle momentum.
λ = h/mv
Non-relativistic particle of mass m and speed v.
λ = h/√(2mK)
For non-relativistic kinetic energy K.
λ = h/√(2mqV)
Charged particle of charge magnitude q accelerated through V.
λe = h/√(2meeV)
Electron accelerated from rest through potential difference V.
λ(Å) = 12.27/√V
Practical non-relativistic electron shortcut.
λ = hc/√[K(K+2mc²)]
Relativistic wavelength for kinetic energy K.
vg = v
Group velocity of a matter-wave packet equals particle velocity.
2πr = nλ
Standing-wave interpretation of allowed Bohr orbits.
Davisson-Germer and Bragg Formulae
nλ = 2d sinθ
n: order; d: interplanar spacing; θ: glancing angle.
λBragg = 2d sinθ/n
Experimental wavelength from a measured diffraction peak.
λdB = 12.27/√V Å
Agreement between λBragg and λdB verifies matter waves.
Constants: h = 6.626×10⁻³⁴ J s; c = 3.00×10⁸ m s⁻¹; e = 1.602×10⁻¹⁹ C; me = 9.11×10⁻³¹ kg; hc = 1240 eV nm.
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Quick Revision Notes
Eight concept cards covering the full chapter logic.
Dual Nature
Radiation and matter display wave or particle behaviour depending on the experiment.
Exam tip: Duality is complementary, not a claim that either description alone is complete.
Photon Theory
Light energy is exchanged in packets E=hν. Each photon has momentum p=h/λ.
Exam tip: Intensity changes photon rate; frequency changes energy per photon.
Photoelectric Effect
Emission occurs only when ν≥ν₀. It is practically instantaneous and Kmax depends on frequency.
Exam tip: Photocurrent depends on intensity, stopping potential does not.
Einstein Theory
One photon transfers energy to one electron: hν=Φ+Kmax.
Exam tip: The intercepts of V₀–ν and K–ν graphs reveal Φ and ν₀.
Matter Waves
Every moving particle has λ=h/p. Observable diffraction needs wavelength comparable with aperture or lattice spacing.
de Broglie Hypothesis
The particle-wave symmetry explains Bohr's standing-wave condition and electron diffraction.
Davisson-Germer
Electron diffraction from nickel gave λ≈1.65 Å, agreeing with λ≈1.67 Å at 54 V.
Wave Nature of Matter
Electron microscopes, electron diffraction and quantum mechanics rely on matter-wave behaviour.
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Important NCERT-Style Diagrams
Eleven labelled SVG visuals for board answers and graph interpretation.
01
Introduction & Historical Background
The experiment that turned a bold hypothesis into measurable physics.
Introduction
The Davisson-Germer experiment demonstrated that a beam of electrons scattered by a crystalline nickel target produces intensity maxima characteristic of diffraction. Since diffraction is a wave phenomenon, the observation supplied direct evidence that moving electrons possess wave nature.
Clinton Davisson and Lester Germer performed the experiment at Bell Telephone Laboratories. Their measurements, reported in 1927, agreed quantitatively with the wavelength predicted by Louis de Broglie's relation λ = h/p.
Particle momentum p ↔ Matter wavelength λ = h/p
de Broglie's bridge between particle and wave descriptions
Timeline
From hypothesis to proof
- 1905: Einstein explains the particle nature of light using photons.
- 1924: de Broglie proposes that material particles also have an associated wavelength.
- 1927: Davisson and Germer observe electron diffraction from nickel.
- 1929: de Broglie receives the Nobel Prize for the wave nature of electrons.
- 1937: Davisson and G. P. Thomson share the Nobel Prize for electron diffraction.
Why de Broglie Proposed Matter Waves
Symmetry in nature
If radiation, traditionally considered a wave, can show particle behavior, then matter, traditionally considered particulate, may show wave behavior.
Bohr orbit condition
Stable electron orbits can be interpreted as standing matter waves: 2πr = nλ. Only whole numbers of wavelengths fit around an allowed orbit.
Relativity and quanta
Using photon relations E = hν and p = h/λ, de Broglie extended the momentum-wavelength connection to all moving particles.
02
Experimental Setup & Components
A controlled electron beam, a crystalline scatterer and a movable detector inside high vacuum.
Electron Gun
A heated tungsten filament emits electrons by thermionic emission. A cylindrical anode with a narrow aperture accelerates and collimates them into a fine beam.
Accelerating Voltage
A variable potential difference V gives each electron kinetic energy eV. Changing V changes momentum and therefore the de Broglie wavelength.
Nickel Crystal
A single crystal provides regularly spaced atomic planes. These planes act like a three-dimensional diffraction grating for electron matter waves.
Detector
A movable Faraday cylinder collects scattered electrons. The resulting current is amplified and measured as a function of scattering angle.
Vacuum Chamber
Low pressure minimizes collisions with gas molecules, prevents energy loss and preserves the coherence and direction of the electron beam.
Rotating Assembly
The detector rotates around the crystal so intensity can be recorded at different angles while voltage and crystal orientation are controlled.
03
Working of the Experiment
How a current-versus-angle measurement reveals a matter-wave diffraction maximum.
- The vacuum chamber is evacuated and the filament is heated, releasing electrons by thermionic emission.
- A known voltage V accelerates the electrons. Their kinetic energy becomes K = eV, provided relativistic effects are negligible.
- The aperture forms a narrow beam directed at the surface of a single nickel crystal.
- Electrons scatter elastically from atoms belonging to successive parallel crystal planes.
- The detector is rotated. At each scattering angle φ, the electron current, proportional to scattered intensity, is recorded.
- For most angles the intensity is modest, but at selected voltage-angle combinations a sharp maximum occurs due to constructive interference.
- The peak angle is converted to the glancing angle θ used in Bragg's law. For the famous 54 V observation, the geometry gives θ ≈ 65°.
- Bragg's law yields λ ≈ 1.65 Å, while de Broglie's formula gives λ ≈ 1.67 Å. The agreement verifies matter waves.
Angle alert: Textbooks may define detector angle, scattering angle and glancing angle differently. Always read the diagram. Bragg's θ is the angle between the incident ray and the reflecting crystal plane.
04
Electron Diffraction & Observations
Constructive interference from regularly spaced crystal planes creates a directional intensity maximum.
Observation of the diffraction pattern
When the accelerating voltage was varied, the angular distribution of scattered electrons changed. At approximately 54 V, a pronounced peak appeared near a detector scattering angle of 50°. Random particle scattering alone does not predict such a sharp, reproducible angular maximum.
The crystal planes supply many coherent scattering centers. The path difference between waves associated with electrons reflected from adjacent planes is 2d sin θ. Constructive interference occurs when it equals an integral multiple of wavelength.
nλ = 2d sin θ
Bragg's law; n = diffraction order, d = interplanar spacing05
Mathematical Derivations
Every exam-ready step, from momentum to the practical wavelength formula.
de Broglie wavelength
For a photon, E = hν = hc/λ and E = pc.
Equating: pc = hc/λ ⇒ p = h/λ.
λ = h/p
de Broglie postulated that the same momentum-wavelength relation applies to material particles.
Electron accelerated through V volts
Electrical work becomes kinetic energy:
eV = ½mv² = p²/(2m)
Therefore p² = 2meV and p = √(2meV).
λ = h / √(2meeV)
Practical formula
Insert h = 6.626×10⁻³⁴ J s, me = 9.109×10⁻³¹ kg and e = 1.602×10⁻¹⁹ C.
λ = [6.626×10⁻³⁴ / √(2×9.109×10⁻³¹×1.602×10⁻¹⁹)] × 1/√V
λ = 1.227×10⁻⁹/√V m
λ = 12.27/√V Å
Bragg's law
For rays reflected by adjacent planes separated by d, the lower ray travels an extra distance d sin θ before and d sin θ after reflection.
Path difference Δ = 2d sin θ.
For constructive interference, Δ = nλ.
nλ = 2d sin θ
Verification using the 54 V observation
Experimental wavelength from Bragg diffraction
For nickel, take d = 0.091 nm = 0.91 Å, first order n = 1, and glancing angle θ = 65°.
λ = 2d sin θ / n
λ = 2(0.91 Å) sin 65°
λ = 1.82 × 0.9063 Å = 1.65 Å
Theoretical wavelength from de Broglie relation
At V = 54 V:
λ = 12.27/√54 Å
λ = 12.27/7.348 Å
λ = 1.67 Å
Percentage difference ≈ |1.67−1.65|/1.67 ×100 ≈ 1.2%.
06
Verification & Proof of Wave Nature
Independent theory and experiment converge on the same wavelength.
- Electrons show diffraction.
- Matter possesses wave nature.
- de Broglie theory is experimentally verified.
- Wave-particle duality is established.
- The experiment helped found quantum mechanics.
The crucial evidence is not merely that electrons scatter. Classical particles also scatter. The evidence is the angular maximum whose position obeys a wave-interference condition and whose shift with accelerating voltage follows λ ∝ 1/√V. Two unrelated routes, crystal diffraction and particle momentum, give matching wavelengths.
07
NCERT-Style Diagram Gallery
Labelled visual summaries for theory, board answers and rapid recall.
08
Graphs & Data Interpretation
NCERT-style photoelectric graphs and Davisson-Germer polar intensity distributions.
09
Importance, Limitations & Applications
What the experiment established, what it could not do, and what it enabled.
Importance
First quantitative confirmation of electron matter waves; support for wave mechanics; validation of crystal diffraction as a probe; conceptual basis for electron optics and modern quantum theory.
Limitations
Requires high vacuum and ordered crystals; peaks depend on surface quality and orientation; original apparatus measured intensity indirectly; non-relativistic wavelength formula fails at very high voltage; it does not display a real-space electron wave directly.
Modern applications
Electron microscopes, LEED surface analysis, electron diffraction crystallography, semiconductor characterization, transmission electron microscopy, nanomaterial structure determination and quantum-device engineering.
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50 Solved Numericals
CBSE, NEET, JEE Main, JEE Advanced and conceptual calculations with worked steps.
11
NCERT Questions with Complete Solutions
All NCERT Exercise Questions 11.1–11.37 in serial order, with full CBSE, NEET and JEE-ready working.
How to use: Attempt each question first, then open its toggle to compare your method, units and final answer.
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PYQ Pattern Archive
Adapted questions reflecting recurring CBSE, AIPMT/NEET and IIT-JEE/JEE patterns across roughly 20–25 years. Wording is standardized for revision rather than presented as verbatim papers.
NEET / AIPMT
Direct wavelength-voltage relation, effect of increasing voltage, experimental conclusion, diffraction peak and order-of-magnitude questions.
JEE Main / Advanced
Bragg geometry, ratios, mixed electrostatic acceleration, uncertainty, graph interpretation and multi-concept calculations.
CBSE / Boards
Diagram, principle, working, 54 V verification, significance, derivation of 12.27/√V and reason-based questions.
13
International Curriculum Practice
Curriculum-aware prompts with concise marking-point answers.
14
10 Case Studies
Passage-based reasoning, calculations and detailed solutions.
15
100 Conceptual Questions
A searchable viva, MCQ-reasoning and interview-style bank.
16
Revision Command Centre
One-page notes, formulas, top exam questions and common mistakes.
One Page Revision Notes
- Electron gun produces a collimated electron beam.
- Electrons gain energy eV and wavelength 12.27/√V Å.
- Nickel crystal provides periodic atomic planes.
- Movable detector measures intensity versus angle.
- At about 54 V a strong diffraction maximum is observed.
- Bragg wavelength ≈ 1.65 Å; de Broglie wavelength ≈ 1.67 Å.
- Agreement proves the wave nature of electrons.
Most Important Formula Sheet
λ = h/p
eV = ½mv² = p²/2m
p = √(2meV)
λ = h/√(2meV)
λ(Å) = 12.27/√V
nλ = 2d sin θ
λ1/λ2 = √(V2/V1)
Common Mistakes
- Using scattering angle directly as Bragg angle without checking geometry.
- Forgetting that electron kinetic energy is eV, not V joules.
- Using 12.27/√V and reporting nanometres instead of angstroms.
- Claiming the experiment proves only particle behavior.
- Confusing intensity peak with an increase in electron energy.
- Applying the non-relativistic formula at hundreds of kilovolts.
Quick Revision Box
Trigger words: nickel crystal, 54 V, 50° scattering, 65° glancing, 1.65 Å, 1.67 Å, diffraction maximum, Bragg law, de Broglie verification.
Golden sentence: The agreement between the wavelength calculated from Bragg diffraction and that predicted by de Broglie establishes the wave nature of electrons.