Photoelectric effect and wave-particle Duality

Key contradictions between wave theory predictions and photoelectric effect observations:

1. Time delay:

   • Wave theory: Electrons should require significant time to accumulate sufficient energy
   • Observation: Electron emission occurs almost instantaneously

2. Intensity dependence:

   • Wave theory: Higher intensity should increase electron kinetic energy
   • Observation: Intensity only affects the number of electrons (current), not their energy

3. Threshold frequency:

   • Wave theory: Sufficiently intense light of any wavelength should eject electrons
   • Observation: No emission occurs below threshold frequency regardless of intensity

4. Energy distribution:

   • Wave theory: Predicts continuous distribution of electron energies
   • Observation: Discrete energy distribution with clear maximum value

These contradictions arise because light energy is quantized in photons rather than continuously distributed across wavefronts.

De Broglie's wavelength formula is: $\lambda = \frac{h}{p}$

Where:

• $\lambda$ is the wavelength associated with a particle
• $h$ is Planck's constant (6.63 × 10^-34 Js)
• $p$ is the momentum of the particle

This formula expresses wave-particle duality for matter - all particles (electrons, protons, neutrons, molecules, etc.) have an intrinsic wave nature with wavelength inversely proportional to momentum.

For a particle with mass $m$ and velocity $v$: $\lambda = \frac{h}{mv}$

This wavelength becomes significant for subatomic particles but is negligibly small for macroscopic objects. The formula applies universally, including to photons (where $p = \frac{h}{\lambda}$).

Effects of light intensity according to photon theory:

On photoelectric current:

• Higher intensity = more photons per unit area per unit time
• More photons = more electron-photon interactions
• More interactions = more ejected electrons
• More ejected electrons = increased saturation photocurrent
• Relationship is directly proportional

On kinetic energy of photoelectrons:

• Intensity has NO effect on maximum kinetic energy
• Each photon has fixed energy $E = h\nu$ for a given frequency
• Each electron interacts with at most one photon
• Maximum kinetic energy depends only on frequency: $K_{max} = h\nu - \phi$

This behavior demonstrates light's particle nature: increasing intensity increases the number of photons, not the energy per photon.

The threshold wavelength ($\lambda_0$) is the maximum wavelength of light that can cause photoelectric emission from a given metal.

Mathematical relationship with work function: $\lambda_0 = \frac{hc}{\phi}$

Where:

• $h$ is Planck's constant
• $c$ is the speed of light
• $\phi$ is the work function of the metal

When $\lambda > \lambda_0$:

• Photon energy ($\frac{hc}{\lambda}$) < work function ($\phi$)
• No electrons can be emitted regardless of intensity

When $\lambda = \lambda_0$:

• Photon energy equals work function
• Electrons can be ejected with zero kinetic energy

When $\lambda < \lambda_0$:

• Photon energy exceeds work function
• Electrons are ejected with positive kinetic energy

For metals with higher work functions, the threshold wavelength is shorter (higher frequency is required).

Step 1: Calculate energy of one photon $E_{photon} = \frac{hc}{\lambda} = \frac{(6.63 \times 10^{-34} \text{ J·s})(3 \times 10^8 \text{ m/s})}{500 \times 10^{-9} \text{ m}} = 3.98 \times 10^{-19} \text{ J}$

Step 2: Calculate total energy crossing the area in one second $E_{total} = \text{Intensity} \times \text{Area} \times \text{Time} = (200 \text{ W/m²})(2 \times 10^{-4} \text{ m²})(1 \text{ s}) = 4 \times 10^{-2} \text{ J}$

Step 3: Calculate number of photons $n = \frac{E_{total}}{E_{photon}} = \frac{4 \times 10^{-2} \text{ J}}{3.98 \times 10^{-19} \text{ J}} = 1.01 \times 10^{17} \text{ photons}$

Note: Increasing intensity would increase this number proportionally, while changing wavelength would change the energy per photon.

Classical Wave Theory:

• Light energy distributed continuously over wavefront
• Electron energy should gradually accumulate
• Predicts higher intensity = higher electron energy
• No threshold wavelength should exist
• All wavelengths should work with sufficient intensity
• Predicts continuous spectrum of electron energies
• Equation: Not applicable (incorrect model)

Photon Theory:

• Light energy delivered in discrete quanta (photons)
• Each photon transfers all energy to one electron
• Predicts maximum kinetic energy depends only on frequency
• Clear threshold wavelength exists
• Wavelength must be below threshold for any emission
• Predicts discrete maximum kinetic energy
• Equation: $K_{max} = \frac{hc}{\lambda} - \phi$

The photon theory correctly predicts the linear relationship between $K_{max}$ and $\frac{1}{\lambda}$ with slope $hc$, while wave theory fails to explain this relationship.

The identical slopes (θ) in the stopping potential vs. 1/λ graphs for different metals represent the ratio hc/e:

$\tan θ = \frac{hc}{e}$

This value:

• Is a universal constant (~1240 eV·nm)
• Is independent of the metal's properties
• Represents the conversion between photon energy and electron potential energy
• Provides experimental confirmation of Planck's constant
• Is the same value regardless of material being tested

This consistent slope across different materials was crucial evidence supporting Einstein's photon theory of light, as it showed that the energy-frequency relationship is a fundamental property of light itself, not of the material interacting with the light.

In wave theory:

• Energy is continuously distributed over a surface
• All free electrons receive energy gradually and simultaneously
• Energy accumulates slowly over time in each electron
• Requires long exposure time before any electron could acquire enough energy to overcome work function

In particle theory (photon theory):

• Energy arrives in discrete packets (photons)
• Each photon delivers its entire energy to a single electron in an instantaneous interaction
• No gradual accumulation - either an electron receives sufficient energy or it doesn't
• Explains immediate emission of photoelectrons regardless of light intensity

This fundamental difference explains why photoelectric emission occurs instantaneously even with very low light intensities, contradicting wave theory predictions.

The wave theory fails to explain:

1. Instantaneous emission: Wave theory predicts hours of exposure would be needed for electrons to accumulate sufficient energy, yet photoelectrons appear immediately when light hits the metal.

2. Existence of threshold frequency: Wave theory suggests any frequency light with sufficient intensity should eventually cause electron emission, yet no emission occurs below threshold frequency regardless of intensity.

3. Independence of electron energy from light intensity: Wave theory predicts higher intensity would give electrons more energy, but experiments show electron energy depends solely on frequency, not intensity.

4. Linear relationship between stopping potential and frequency: This direct relationship aligns with E = hν - φ (Einstein's equation), which has no explanation in wave theory.

These contradictions led to the development of the photon theory where light energy is quantized in discrete packets.

The saturation current in the photoelectric effect is the maximum photocurrent reached when all photoelectrons emitted from the cathode are collected by the anode.

Key characteristics:

• Appears as a plateau (flat region) on the current vs. voltage curve
• Occurs once the anode potential is sufficient to overcome space charge effects
• Independent of further increases in anode potential
• Directly proportional to incident light intensity

Physical explanation:

1. The number of emitted photoelectrons is determined solely by:

   • Incident photon flux (light intensity)
   • Material's quantum efficiency
   • NOT by the electric field strength

2. Saturation mechanism:

   • At low anode potentials: some electrons fail to reach the anode
   • At sufficient positive potential: all emitted electrons are collected
   • Beyond this point: additional voltage cannot increase electron production

Fundamental principles demonstrated:

• Confirms the quantum nature of light (photon theory)
• Shows that each photon can eject at most one electron
• Provides evidence that photoemission depends on photon flux, not field strength
• Supports Einstein's photoelectric equation where electron emission is governed by photon energy (hf), not intensity

Note: While increasing voltage beyond saturation doesn't increase current, increasing light intensity will raise the saturation current level proportionally.