Dual Nature of Radiation and Matter notes: ISC Class 12 Physics

1. The photoelectric effect: observations and experimental facts

Photoelectric effect: emission of electrons (photoelectrons) from a metal surface when light of suitable frequency falls on it.
Hertz (1887): ultraviolet light from a transmitter spark made sparks jump more easily across a receiver gap; glass (which absorbs ultraviolet) weakened the effect, quartz did not. Hallwachs: a negatively charged zinc plate in ultraviolet loses its charge.
Lenard’s apparatus: emitter C and collector A in an evacuated bulb, light through a quartz window, a potential divider with a reversing key to set the collector potential V, a microammeter for the photocurrent.
Facts: (i) photocurrent ∝ intensity (frequency fixed, above threshold); (ii) the current saturates at large positive V and falls to zero at a negative stopping potential −V₀, with $eV_0 = K_{\max}$ ; (iii) V₀ does not depend on intensity but rises linearly with frequency; (iv) no emission below a threshold frequency f₀ of the metal, however intense the light; (v) emission is instantaneous (within about 10⁻⁹ s).
Wave theory fails: it spreads energy over the wavefront at a rate set by intensity, so it expects K_max to grow with intensity, emission at any frequency if bright enough, and a measurable delay in dim light. Only fact (i) agrees.

intensity I
intensity 2I
intensity 3I

One frequency, three intensities: saturation current ∝ intensity; stopping potential unchanged.

f₁ = 7.27 × 10¹⁴ Hz (stops at −0.50 V)
f₂ = 9.09 × 10¹⁴ Hz (stops at −1.25 V)
f₃ = 10.91 × 10¹⁴ Hz (stops at −2.00 V)

Same photon rate, three frequencies: same saturation current; higher frequency, larger stopping potential.

Formulas

Stopping potential and maximum kinetic energy
$eV_0 = K_{\max} = \tfrac{1}{2}mv_{\max}^2$

Key terms

Stopping potential: The negative potential of the collector, relative to the emitter, that just reduces the photocurrent to zero; eV₀ equals the maximum kinetic energy of the photoelectrons.
Saturation current: The largest photocurrent for a given beam, reached when every emitted electron is collected; it is proportional to the intensity.

2. Photons and Einstein’s photoelectric equation

Quantisation of radiation: light of frequency f travels and is absorbed as photons, each of energy $E = hf = \dfrac{hc}{\lambda}$ (Planck, Einstein). Here $h = 6.6 \times 10^{-34}$ J s, so $E\,(\text{eV}) = \dfrac{1237.5}{\lambda\,(\text{nm})}$ . Intensity = photons per second per unit area × hf.
Work function W₀: the least energy needed to remove an electron from the surface of a metal (a few eV; depends on the metal and its surface).
Einstein’s equation: one photon is absorbed by one electron; $K_{\max} = hf - W_0 = eV_0$ . Electrons from deeper levels leave with less energy.
Threshold: $hf_0 = W_0$ , so $f_0 = \dfrac{W_0}{h}$ and $\lambda_0 = \dfrac{hc}{W_0}$ . No emission for f < f₀ (λ > λ₀).
Explains the facts: threshold (one photon must supply W₀); K_max set by f, not intensity; no delay (the whole photon is absorbed at once); current ∝ number of photons ∝ intensity.
A graph of K_max against f is a straight line of slope h, cutting the f axis at f₀ and (extended) the K_max axis at −W₀.

Drawn to scale for 225 nm ultraviolet on magnesium (W₀ = 3.7 eV).

E = hc/λ = 1237.5/λ eV

Shorter wavelength, more energetic photons. Only photons above the dashed line can free electrons from barium.

Formulas

Photon energy
$E = hf = \dfrac{hc}{\lambda}$
Einstein’s photoelectric equation
$K_{\max} = hf - W_0$
Threshold frequency and wavelength
$f_0 = \dfrac{W_0}{h},\quad \lambda_0 = \dfrac{hc}{W_0}$

Key terms

Photon: A quantum of electromagnetic radiation of energy hf and momentum h/λ, travelling at c, with zero rest mass and no charge.
Work function: The minimum energy needed to free an electron from the surface of a metal.
Threshold frequency: The lowest frequency of light that can eject photoelectrons from a given metal: f₀ = W₀/h.

3. Planck’s constant from the V₀–f graph; photon momentum

Dividing Einstein’s equation by e: $V_0 = \dfrac{h}{e}f - \dfrac{W_0}{e}$ . A graph of V₀ against f is a straight line: slope $\dfrac{h}{e}$ (the same for all metals, so the lines are parallel), intercept on the f axis $f_0$ , intercept on the V₀ axis $-\dfrac{W_0}{e}$ .
To find h: measure V₀ for several frequencies, draw the best line, $h = e \times$ slope; then $W_0 = hf_0$ . With h = 6.6 × 10⁻³⁴ J s the slope is 4.125 × 10⁻¹⁵ V s.
Photon momentum: with zero rest mass, $E = pc$ , so $p = \dfrac{E}{c} = \dfrac{h}{\lambda}$ . A beam of power P absorbed by a surface exerts force P/c; reflected normally, 2P/c.

barium, W₀ = 2.5 eV (f₀ = 6.06 × 10¹⁴ Hz)
magnesium, W₀ = 3.7 eV (f₀ = 8.97 × 10¹⁴ Hz)

Same slope h/e for every metal; the line starts at the threshold frequency f₀ = W₀/h.

best line through the end points
measured (mercury lines)

Slope = h/e; intercept on the f axis = f₀; W₀ = h f₀.

Formulas

Stopping potential against frequency
$V_0 = \dfrac{h}{e}f - \dfrac{W_0}{e}$
Photon momentum
$p = \dfrac{h}{\lambda} = \dfrac{E}{c}$

Key terms

Photon momentum: p = h/λ = E/c: the momentum carried by a photon of wavelength λ.

4. Wave–particle duality and de Broglie waves

Dual nature of radiation: light shows wave behaviour in propagation (interference, diffraction, polarisation) and particle behaviour in interaction with matter (photoelectric effect).
de Broglie hypothesis (1924): every moving particle has a wave associated with it, of wavelength $\lambda = \dfrac{h}{p} = \dfrac{h}{mv}$ .
Electron accelerated from rest through V: $eV = \dfrac{p^2}{2m}$ , so $\lambda = \dfrac{h}{\sqrt{2meV}} \approx \dfrac{1.22}{\sqrt{V}}$ nm (with h = 6.6 × 10⁻³⁴ J s). In general $\lambda = \dfrac{h}{\sqrt{2mK}} = \dfrac{h}{\sqrt{2mqV}}$ .
Everyday objects have wavelengths far too small to show diffraction; electrons of a few hundred eV have λ ≈ 0.1 nm, comparable to atomic spacings.

electron: λ = 1.223/√V nm

λ = h/√(2meV) ≈ 1.22/√V nm with h = 6.6 × 10⁻³⁴ J s.

Formulas

de Broglie wavelength
$\lambda = \dfrac{h}{mv}$
Electron accelerated through V
$\lambda = \dfrac{h}{\sqrt{2meV}}$

Key terms

de Broglie wavelength: The wavelength h/p of the wave associated with a moving particle of momentum p.
Wave–particle duality: The fact that radiation and matter each show wave behaviour in some experiments and particle behaviour in others.

5. Electron diffraction and the Davisson–Germer experiment (qualitative)

Apparatus: an electron gun (heated filament, anode at accelerating potential V) sends a narrow beam onto a nickel crystal in a vacuum chamber; a movable detector measures the number of electrons scattered at each angle φ to the incident beam.
Observation: a sharp peak (spur) in the scattered intensity at φ ≈ 50° when V = 54 V; the peak weakens at higher and lower voltages.
Conclusion: a strong beam in one direction is diffraction by the regular rows of atoms; the wavelength found agreed with h/p. Electrons behave as waves, confirming de Broglie’s hypothesis.
Thomson: electrons through a thin foil of tiny crystals give concentric rings like X-rays; higher V gives smaller rings.
The quantitative treatment of the experiment is not in the ISC syllabus.

Qualitative: the sharpest spur is at 54 V, near 50°.

Key terms

Electron diffraction: The spreading of an electron beam into definite directions (spots or rings) by the regular arrangement of atoms in a crystal, showing the wave nature of electrons.

Exam tips

State constants: h = 6.6 × 10⁻³⁴ J s, e = 1.6 × 10⁻¹⁹ C, c = 3 × 10⁸ m/s, mₑ = 9.1 × 10⁻³¹ kg; hc = 1237.5 eV nm with these values.
K_max = hf − W₀ = eV₀; V₀ in volts equals K_max in eV. Intensity changes the current only; frequency changes V₀ only.
V₀–f graph: slope h/e, f-intercept f₀, W₀ = hf₀. Photon momentum h/λ.
Electron: λ = h/√(2meV) ≈ 1.22/√V nm. Davisson–Germer: qualitative description and conclusion only.

1. The photoelectric effect: observations and experimental facts

Formulas

Key terms

2. Photons and Einstein’s photoelectric equation

Formulas

Key terms

3. Planck’s constant from the V₀–f graph; photon momentum

Formulas

Key terms

4. Wave–particle duality and de Broglie waves

Formulas

Key terms

5. Electron diffraction and the Davisson–Germer experiment (qualitative)

Key terms

Exam tips

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