Dual Nature of Radiation and Matter — Notes

• Work function $\varphi_0$: least energy needed to free an electron from a metal surface; 1 eV = $1.6 \times 10^{-19}$ J. Emission by heat (thermionic), strong field (field emission) or light (photoelectric).
• Hertz (1887): ultraviolet light helped sparks across a gap. Hallwachs: a negatively charged zinc plate discharged in UV; a neutral plate became positive. Lenard: UV on the emitter of an evacuated tube produced a current that stopped when the light was cut off; the particles were electrons (photoelectrons).
• Zinc and cadmium need UV; alkali metals (Na, K, Cs) respond to visible light.

Emitter C and collector A in an evacuated tube; light through a quartz window; collector potential variable and reversible; microammeter for the photocurrent.

• Intensity: photocurrent ∝ intensity (number of electrons per second ∝ intensity).
• Potential: current rises to a saturation value as A is made positive; it falls to zero at the stopping potential $-V_0$: $eV_0 = K_\text{max}$. Different intensities, same ν: different saturation currents, same $V_0$.
• Frequency: same intensity, higher ν: same saturation current, larger $V_0$. $V_0$ against ν is a straight line cutting the axis at the threshold frequency $\nu_0$; slope $h/e$ for all metals.
• No time lag (< $10^{-9}$ s).
• Wave theory fails: it predicts $K_\text{max}$ rising with intensity, emission at any frequency, and a time lag in dim light.

• Light is a stream of photons: $E = h\nu = hc/\lambda$, $p = h/\lambda$, speed $c$, zero rest mass, no charge. Intensity ∝ number of photons per second (for one frequency). $h = 6.63 \times 10^{-34}$ J s; $hc \approx 1240$ eV nm.
• One photon is absorbed by one electron: $h\nu = \varphi_0 + K_\text{max}$, i.e. $K_\text{max} = eV_0 = h\nu - \varphi_0 = h(\nu - \nu_0)$.
• Threshold: $\nu_0 = \varphi_0/h$, $\lambda_0 = hc/\varphi_0$.
• $V_0 = \dfrac{h}{e}\nu - \dfrac{\varphi_0}{e}$: slope $h/e$ (same for all metals), intercept $\nu_0$ (depends on the metal).
• Explains: $K_\text{max}$ independent of intensity; threshold; linear rise with ν; instant emission; current ∝ number of photons.

• Dual nature: radiation shows wave (interference, diffraction) and particle (photoelectric effect) behaviour.
• de Broglie relation: a particle of momentum $p$ has wavelength $\lambda = \dfrac{h}{p} = \dfrac{h}{mv} = \dfrac{h}{\sqrt{2mK}}$.
• Charged particle accelerated through $V$: $K = qV$, $\lambda = \dfrac{h}{\sqrt{2mqV}}$; electron: $\lambda \approx \dfrac{1.227}{\sqrt{V}}$ nm.
• Equal kinetic energy: $\lambda \propto 1/\sqrt{m}$. Equal momentum: equal $\lambda$.
• Everyday objects: $\lambda$ far too small to observe; electrons: comparable to atomic spacings (basis of the electron microscope).