Photoelectric effect vs Compton effect — comparison of particle nature evidence

Question

Compare the photoelectric effect and Compton effect as evidence for the particle nature of light. In Compton scattering, an X-ray photon of wavelength 0.1 nm is scattered at 90 degrees. Find the change in wavelength.

(JEE Main + CBSE 12 pattern)

Solution — Step by Step

Feature	Photoelectric Effect	Compton Effect
Photon interaction	Photon is completely absorbed	Photon is scattered (partially absorbed)
Target	Bound electrons in metal	Free/loosely bound electrons
Evidence for	Photon energy quantisation ( $E = h\nu$ )	Photon momentum ( $p = h/\lambda$ )
Energy of photons	UV/visible range	X-rays/gamma rays
What changes	Electron is ejected, photon disappears	Photon wavelength increases, electron recoils

Both show that light behaves as particles (photons), not waves.

The Compton wavelength shift:

\Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta)

Given: $\theta = 90°$ , so $\cos 90° = 0$ .

\Delta\lambda = \frac{h}{m_e c} = \frac{6.63 \times 10^{-34}}{9.1 \times 10^{-31} \times 3 \times 10^8}

\Delta\lambda = \frac{6.63 \times 10^{-34}}{2.73 \times 10^{-22}} = 2.43 \times 10^{-12} \text{ m} = \mathbf{0.00243 \text{ nm}}

This value $h/(m_e c) = 0.00243$ nm is the Compton wavelength of the electron — a fundamental constant worth memorising.

The scattered photon has a longer wavelength (lower energy) than the incident photon: $\lambda' = 0.1 + 0.00243 = 0.10243$ nm.

The “lost” energy has gone into the kinetic energy of the recoiling electron. This is exactly what happens in a billiard ball collision — the incoming ball transfers some energy and momentum to the target ball and bounces off with less energy. The fact that photons behave this way proves they carry momentum $p = h/\lambda$ , just like particles.

flowchart TD
    A["Particle nature of light"] --> B["Photoelectric Effect"]
    A --> C["Compton Effect"]
    B --> D["Photon absorbed completely"]
    B --> E["Proves E = hν"]
    C --> F["Photon scattered with longer λ"]
    C --> G["Proves p = h/λ"]
    D --> H["Electron ejected with KE"]
    F --> I["Electron recoils with KE"]
    E --> J["Light has energy quanta"]
    G --> J
    J --> K["Light is both wave and particle"]

Why This Works

The wave theory of light cannot explain either effect. For the photoelectric effect, wave theory predicts that any frequency should eject electrons if we wait long enough (energy accumulates over time) — but experimentally, there is a sharp threshold frequency below which no emission occurs, regardless of intensity. This shows energy comes in discrete packets ( $h\nu$ ).

For Compton scattering, wave theory predicts no change in wavelength upon scattering (Thomson scattering). The observed wavelength shift proves that the photon transfers momentum like a particle. The shift depends only on the scattering angle, not on the incident wavelength — further confirming the particle picture.

Alternative Method — Energy-Momentum Conservation

The Compton formula can be derived by treating the collision as a relativistic two-body problem. Conserve both energy and momentum:

Energy: $h\nu + m_e c^2 = h\nu' + E_e$

Momentum (x): $\dfrac{h}{\lambda} = \dfrac{h}{\lambda'}\cos\theta + p_e\cos\phi$

Momentum (y): $0 = \dfrac{h}{\lambda'}\sin\theta - p_e\sin\phi$

For JEE, the Compton wavelength shift $\Delta\lambda = 0.00243$ nm at $\theta = 90°$ is a standard value. Maximum shift occurs at $\theta = 180°$ (backscattering): $\Delta\lambda_{max} = 2h/(m_e c) = 0.00486$ nm. These numbers appear directly in MCQ options.

Common Mistake

Students sometimes think the Compton effect and photoelectric effect are the same phenomenon. They are fundamentally different: in the photoelectric effect, the photon is COMPLETELY absorbed and disappears; in the Compton effect, the photon SURVIVES with reduced energy. The photoelectric effect proves energy quantisation; the Compton effect proves momentum quantisation. Both are needed to establish the full particle nature of photons.

Question

Solution — Step by Step

Why This Works

Alternative Method — Energy-Momentum Conservation

Common Mistake

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