Dual Nature of Matter and Radiation

Photoelectric Effect: Emission of electrons (photoelectrons) from a metal surface when light of suitable frequency falls on it. Discovered by Heinrich Hertz (1887); systematically studied by Hallwachs and Lenard.

Key Experimental Observations:

1. Threshold Frequency $\nu_0$: Minimum frequency below which no photoelectric emission occurs, regardless of intensity. Different for different metals.
2. Instantaneous Emission: No measurable time lag between incidence of light and electron emission ($\sim 10^{-9}$ s).
3. Stopping Potential $V_0$: Reverse potential at which photocurrent becomes zero. Depends only on frequency, not intensity.
4. Saturation Current: Photocurrent reaches saturation when all emitted electrons reach anode. Proportional to intensity.
5. Stopping potential varies linearly with frequency: $V_0 = (h/e)\nu - (\phi/e)$.

Laws of Photoelectric Emission:

Failure of Classical Wave Theory:

• Cannot explain threshold frequency (wave intensity should give energy regardless of $\nu$)
• Cannot explain instantaneous emission
• Cannot explain why KE is independent of intensity
• Predicts time delay for weak intensity

Work Function $\phi$: Minimum energy required to liberate electron from metal surface. $\phi = h\nu_0$, where $\nu_0$ = threshold frequency. Typically a few eV: $\phi_{\text{Cs}} = 1.8$ eV, $\phi_{\text{Pt}} = 6.4$ eV.

Einstein's Quantum Theory of Light (1905): Light consists of quanta (photons) of energy $E = h\nu$. In photoelectric emission, each photon delivers all its energy to one electron.

Einstein's Photoelectric Equation: $$h\nu = \phi + K_{\max}$$ $$K_{\max} = h\nu - \phi = h(\nu - \nu_0) = hc\left(\frac{1}{\lambda} - \frac{1}{\lambda_0}\right)$$

Equivalently: $K_{\max} = eV_0$ (in terms of stopping potential).

Photon Properties:

Useful constants:

• $h = 6.626 \times 10^{-34}$ J·s = $4.136 \times 10^{-15}$ eV·s
• $hc = 1240$ eV·nm $= 1240$ MeV·fm
• $1$ eV $= 1.6 \times 10^{-19}$ J

Robert Millikan (1916) verified Einstein's equation; slope of $V_0$ vs $\nu$ graph = $h/e$. Used to measure $h$.

Number of Photons per second from source of power $P$ at wavelength $\lambda$: $$N = \frac{P}{E_{\text{photon}}} = \frac{P\lambda}{hc}$$

de Broglie Hypothesis (1924): All moving matter has wave-like properties. The wavelength associated with a particle: $$\lambda = \frac{h}{p} = \frac{h}{mv}$$

This is the de Broglie wavelength, exhibiting wave-particle duality of matter (parallel to that of light).

de Broglie Wavelength for Particle of KE $K$: $$\lambda = \frac{h}{\sqrt{2mK}}$$

For charged particle accelerated through potential $V$: $K = qV$. $$\lambda = \frac{h}{\sqrt{2mqV}}$$

For electron accelerated through $V$ volts: $$\lambda_e = \frac{12.27}{\sqrt{V}} \text{ Å} = \frac{1.227}{\sqrt{V}} \text{ nm}$$

Comparison with Photon:

Heisenberg Uncertainty Principle: $$\Delta x \cdot \Delta p \geq h/(4\pi) = \hbar/2$$

Position and momentum cannot be measured simultaneously with arbitrary precision. Similarly: $\Delta E \cdot \Delta t \geq \hbar/2$.

Useful approximations for de Broglie wavelength:

• Electron at $100$ V: $\lambda \approx 1.227$ Å
• Thermal neutron ($\sim 300$ K, $K \approx 0.025$ eV): $\lambda \approx 1.8$ Å
• $1$ kg ball at $1$ m/s: $\lambda \approx 10^{-34}$ m (negligibly small)

Matter waves significant only at microscopic scales.

Davisson-Germer Experiment (1927): Confirmed wave nature of electrons. They scattered low-energy electrons (54 eV) off nickel crystal and observed a pronounced peak at $\theta = 50°$ from incident direction, consistent with Bragg-like diffraction.

Experimental Setup:

• Electron gun produces beam (accelerating voltage $V = 54$ V)
• Beam hits nickel target
• Scattered electrons detected at various angles
• Sharp peak at $50°$ confirms diffraction

Result: Wavelength calculated from de Broglie ($\lambda = 12.27/\sqrt{54} = 1.67$ Å) matched wavelength from Bragg's diffraction formula → confirmed matter waves.

G. P. Thomson Experiment: Independent confirmation by passing electron beam through thin metal films; observed diffraction rings similar to X-ray diffraction. Won Nobel Prize 1937 (with Davisson).

Bragg's Law: $2d\sin\theta = n\lambda$, where $d$ = atomic plane spacing.

Applications of Matter Waves:

Electron Microscope: Better resolution because $\lambda_{\text{electron}} \ll \lambda_{\text{light}}$. Can resolve atomic structures.

Compton Effect: Scattering of X-ray photons by electrons. Wavelength shift: $$\Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta) = \lambda_C(1 - \cos\theta)$$ where $\lambda_C = h/(m_e c) = 0.0243$ Å is Compton wavelength of electron. Established particle nature of light.