describe and interpret qualitatively the evidence provided by electron diffraction for the wave nature of particles

Wave‑Particle Duality (Cambridge A‑Level Physics 9702 – Syllabus 22)

1. Photons – Energy, Momentum and Threshold Frequency

• Energy: \(E = hf = \dfrac{hc}{\lambda}\)
• Momentum: \(p = \dfrac{E}{c}= \dfrac{h}{\lambda}\) (photons travel at \(c\))
• Threshold frequency for the photo‑electric effect: \(f_{0}= \dfrac{\Phi}{h}\)

Example – Photon momentum of green light (\(\lambda = 550\;\text{nm}\)):

\[

p = \frac{h}{\lambda}= \frac{6.626\times10^{-34}\;\text{J·s}}{5.5\times10^{-7}\;\text{m}}

= 1.20\times10^{-27}\;\text{kg·m s}^{-1}

\]

Note: because photons travel at \(c\), the same result follows from \(p=E/c\).

2. Photo‑electric Effect (AO1 + AO2)

The experiment shows that electrons are emitted only when the incident light frequency exceeds a material‑specific threshold.

\[

hf = \Phi + \tfrac12 mv^{2}\qquad\text{or}\qquad K_{\max}=hf-\Phi

\]

• \(h\) – Planck’s constant \(6.626\times10^{-34}\;\text{J·s}\)
• \(f\) – frequency of the incident photon
• \(\Phi\) – work function (energy needed to free an electron)
• \(K_{\max}\) – maximum kinetic energy of the emitted electron

Worked example (A‑level style):

1. Light of wavelength \(400\;\text{nm}\) shines on a metal with \(\Phi = 2.0\;\text{eV}\).
2. Calculate \(K_{\max}\).

\[

E_{\text{photon}}=\frac{hc}{\lambda}= \frac{1240\;\text{eV·nm}}{400\;\text{nm}}=3.10\;\text{eV}

\]

\[

K{\max}=E{\text{photon}}-\Phi = 3.10-2.0 = 1.10\;\text{eV}

\]

3. de Broglie Hypothesis (AO1)

Any particle with momentum \(p\) is associated with a wavelength

\[

\boxed{\lambda = \frac{h}{p}}

\]

3.1 Non‑relativistic electrons

\[

p = \sqrt{2m_{e}K}\qquad\Longrightarrow\qquad

\lambda = \frac{h}{\sqrt{2m_{e}K}}

\]

3.2 Relativistic correction (required for \(K\gtrsim 100\;\text{keV}\))

\[

p = \frac{1}{c}\sqrt{K^{2}+2K\,m_{e}c^{2}}\qquad\Longrightarrow\qquad

\lambda = \frac{h}{p}

\]

4. Classic Demonstrations of Wave Behaviour (AO1 + AO2)

4.1 Davisson–Germer Experiment (electron diffraction)

1. Electrons are accelerated through a potential \(V\) → kinetic energy \(K = eV\).
2. De Broglie wavelength (non‑relativistic):

   \[

   \lambda = \frac{h}{\sqrt{2m_{e}eV}}

   \]

3. The beam strikes a polished Ni crystal; scattered intensity is recorded versus angle \(\theta\).

Sharp intensity maxima appear at angles satisfying Bragg’s law

\[

2d\sin\theta = n\lambda \qquad (n=1,2,\dots)

\]

Using the measured \(\theta\) and the known lattice spacing \(d\) (Ni (111) planes, \(d=0.215\;\text{nm}\)), the calculated \(\lambda\) matches the de Broglie value – a direct confirmation of matter‑wave behaviour.

4.2 Electron Double‑slit Experiment

When a coherent electron beam passes through two narrow slits (separation \(s\)), an interference pattern of bright and dark fringes appears on a distant screen.

Fringe‑spacing example (A‑level style):

• Electron kinetic energy \(K = 150\;\text{eV}\) → \(\lambda = 1.00\times10^{-10}\;\text{m}\).
• Slit separation \(s = 1.0\;\mu\text{m}\), screen distance \(L = 0.5\;\text{m}\).
• First‑order fringe position:

  \[

  y_1 = \frac{L\lambda}{s}= \frac{0.5\times1.00\times10^{-10}}{1.0\times10^{-6}} = 5.0\times10^{-5}\;\text{m}=50\;\mu\text{m}

  \]

4.3 Compton Scattering (photon momentum)

When X‑rays scatter from loosely bound electrons, their wavelength increases by

\[

\Delta\lambda = \lambda' - \lambda = \frac{h}{m_{e}c}\,(1-\cos\theta)

\]

where \(\theta\) is the scattering angle.

Worked example:

• Incident photon \(\lambda = 0.071\;\text{nm}\) (energy ≈ 17.5 keV).
• Scattered at \(\theta = 60^{\circ}\).
• Compton shift:

  \[

  \Delta\lambda = 2.43\times10^{-12}\,\text{m}\;(1-\cos60^{\circ}) = 1.22\times10^{-12}\;\text{m}

  \]

• Scattered wavelength \(\lambda' = 0.0710012\;\text{nm}\); the shift is measurable with a crystal spectrometer.

4.4 Matter‑wave diffraction of neutrons (optional)

Thermal neutrons (\(\lambda\approx 0.18\;\text{nm}\)) are diffracted by crystal lattices in the same way as X‑rays, providing a powerful tool for studying magnetic structures. Mentioned here to show that wave‑particle duality is not limited to electrons or photons.

5. Electron Diffraction – Direct Evidence of Wave Behaviour (AO2)

5.1 Low‑Energy Electron Diffraction (LEED)

• Electrons of 20–200 eV are incident on a clean crystalline surface.
• The diffracted beams form a pattern of bright spots on a phosphor screen; spot positions correspond to the surface’s reciprocal lattice.
• Only a wave‑interference model predicts the discrete spot pattern; a particle‑only picture would give a diffuse distribution.

5.2 Qualitative Interpretation

1. The crystal acts as a three‑dimensional diffraction grating.
2. Each electron’s wavefunction is scattered by the periodic array of atoms.
3. Path‑differences that are integer multiples of \(\lambda\) lead to constructive interference – the observed Bragg peaks.

If electrons were purely particles, the angular distribution would be smooth and roughly isotropic (Rutherford‑type scattering). The discrete, angle‑dependent peaks are therefore a hallmark of wave behaviour.

6. Energy Levels and Line Spectra (AO1 + AO2)

When an electron in an atom moves between two discrete energy levels \(Ei\) and \(Ef\), a photon of frequency \(f\) is emitted or absorbed:

\[

\boxed{hf = \Delta E = |Ei - Ef|}

\]

6.1 Balmer series (visible)

\[

\frac{1}{\lambda}=R_{\!H}\!\left(\frac{1}{2^{2}}-\frac{1}{n^{2}}\right),\qquad n=3,4,5,\dots

\]

• First‑order line (\(n=3\to2\), H\(_\alpha\)): \(\lambda = 656.3\;\text{nm}\).
• Second‑order line (\(n=4\to2\), H\(_\beta\)): \(\lambda = 486.1\;\text{nm}\).

6.2 Lyman series (ultraviolet)

\[

\frac{1}{\lambda}=R_{\!H}\!\left(\frac{1}{1^{2}}-\frac{1}{n^{2}}\right),\qquad n=2,3,4,\dots

\]

Example: \(n=2\to1\) (Lyman‑α) gives \(\lambda = 121.6\;\text{nm}\).

7. Particle‑only vs. Wave Predictions for Electron Scattering

8. AO2 / AO3 Skill‑Building Activities

8.1 Sketch & Label a Bragg‑law Diagram

• Draw incident and reflected electron wavefronts on parallel crystal planes.
• Label the inter‑planar spacing \(d\), angle \(\theta\), path difference, and indicate the condition for constructive interference.

8.2 Calculation Practice – Electron Wavelength vs. Diffraction Angle

Given: electrons accelerated through \(V = 150\;\text{V}\); Ni crystal plane spacing \(d = 0.215\;\text{nm}\).

1. Calculate the non‑relativistic de Broglie wavelength.
2. Predict the first‑order Bragg angle using \(2d\sin\theta = \lambda\).
3. Compare with a measured angle of \(50^{\circ}\) and comment on the agreement.

8.3 Design an Experiment (AO3)

Outline a modification of the Davisson–Germer set‑up to test the relativistic correction for electrons accelerated to \(200\;\text{keV}\).

• Increase the accelerating voltage to \(200\;\text{kV}\) and use a high‑vacuum electron gun capable of such energies.
• Replace the simple Geiger‑type detector with a position‑sensitive phosphor screen and a CCD camera to resolve the smaller Bragg angles.
• Calculate the expected relativistic wavelength (≈ 2.48 pm) and the corresponding first‑order angle \(\theta_{\text{R}}\) from Bragg’s law.
• Look for a systematic shift of the observed peak from the non‑relativistic prediction (≈ 2.74 pm) – the shift should be ≈ 9 % towards smaller \(\theta\).

9. Key Take‑aways for A‑Level Students

• Both photons and massive particles possess a wavelength \(\lambda = h/p\); the same relationship underlies X‑ray diffraction, electron diffraction, neutron diffraction, and the photo‑electric effect.
• Electron diffraction (Davisson–Germer, LEED) provides quantitative, visual evidence that matter behaves as a wave.
• Compton scattering demonstrates photon momentum, while the double‑slit experiment shows interference of single electrons.
• Atomic line spectra link the wave nature of light to quantised energy differences via \(hf = \Delta E\).
• Mastering calculations (de Broglie wavelength, Bragg angles, photo‑electric kinetic energy, Compton shift) and interpreting diffraction patterns are essential AO2 skills; designing variations of these experiments develops AO3 competence.