Photo Electric Emission

PHY 104 Course Outline

This course covers fundamental concepts in modern physics, including:

• Photoelectric emission/effect and thermionic emission
• Compton effect
• Quantum theory of light and photons as particles
• Matter waves and de Broglie hypothesis
• Momentum and energy of photons
• Wave-particle duality
• Heisenberg's uncertainty principle

Photoelectric Effect

The photoelectric effect describes the phenomenon where light hits a metal plate, and electrons are ejected, forming a current in a circuit. Key observations:

• Electrons are only emitted if the frequency of light is above a certain value, known as the threshold frequency (fcutoff).
• Electron emission is immediate, with no delay after the initial exposure to light, regardless of intensity.

Wave Theory vs. Actual Observations

Traditional wave theory predicts that increasing light intensity (amplitude) should increase electron energy and that emission should occur at any frequency given sufficient intensity and time. However, actual experiments show:

• No emission below threshold frequency, regardless of intensity.
• Electron energy depends on frequency, not intensity.
• Immediate emission.

Einstein's Explanation (Photon Theory)

Einstein proposed that light consists of discrete units of energy called photons. Each photon carries energy ΔE = hf, where h is Planck's constant and f is the light's frequency.

• An electron absorbs a whole photon or none at all.
• If a photon's energy (hf) is not enough to overcome the binding energy of the electron (Work Function, W), no electron is ejected. This explains the threshold frequency.
• The energy of the absorbed photon goes into ejecting the electron (W) and any excess energy becomes the electron's kinetic energy (KE).
• The photoelectric equation: hf = W + KE

Stopping Potential and Threshold Frequency

• The kinetic energy (KE) of the ejected electrons can be measured by a stopping potential (Vstop), where KE = e*Vstop.
• Substituting into Einstein's equation: hf = W + e*Vstop.
• Work Function (W): The minimum energy required to eject an electron from the metal.
• Threshold Frequency (fcutoff): The minimum frequency of light that produces photoelectron emission. At fcutoff, Vstop = 0, so hfcutoff = W.
• Plotting Vstop vs. frequency yields a straight line with a slope of (h/e), allowing for experimental determination of Planck's constant.

Example: Cut-off Frequency for Copper

• Work function of copper: W = 4.5 eV.
• fcutoff = W/h = (4.5 eV) / (6.63 x 10-34 J·s) = (4.5 * 1.6 x 10-19 J) / (6.63 x 10-34 J·s) = 1.09 x 1015 Hz.
• Corresponding cut-off wavelength: λcutoff = c/fcutoff = (3 x 108 m/s) / (1.09 x 1015 Hz) = 276 nm (in UV range).

Thermionic Emission

Thermionic emission is the emission of electrons from a metal when it is heated. This effect was discovered by Edison and was crucial in paving the way for technologies like radio and television.

Cathode Ray Tube and Electron Gun

• A Cathode Ray Tube (CRT) consists of an electron gun, deflection plates, and a fluorescent screen.
• The electron gun accelerates and focuses an electron beam.
• CRTs were used in early oscilloscopes and televisions.
• An electron volt (eV) is a unit of energy, defined as the kinetic energy gained by an electron accelerated through an electric potential difference of 1 volt.

Compton Scattering

Compton scattering occurs when light encounters charged particles (e.g., electrons), causing the light to be scattered. The wave model fails to accurately predict the scattering pattern. The photon idea, where photons "hit" charged particles like billiard balls, provides a correct explanation.

• According to the photon theory, scattered photons lose energy and momentum, resulting in a larger wavelength.
• The change in wavelength (Δλ) is given by the formula: Δλ = λscattered - λincident = (h/mc)[1 - cos(θ)], where h is Planck's constant, m is the mass of the electron, c is the speed of light, and θ is the scattering angle.
• The Compton formula also provides an experimental way to determine Planck's constant.
• This effect is significant for high-energy photons like X-rays and gamma rays but negligible for visible light.

Example: X-ray Compton Scattering

• A 17.2-keV X-ray scattered at 90° from molybdenum.
• Incident wavelength λincident = hc/E = 0.0721 nm.
• Compton wavelength (h/mc) = 0.002426 nm.
• For θ=90°, cos(θ)=0, so Δλ = h/mc = 0.002426 nm.
• Scattered wavelength λscattered = λincident + Δλ = 0.0721 nm + 0.002426 nm = 0.0745 nm.
• Energy of scattered X-ray E = hc/λscattered = 16.6 keV.
• The missing 0.6 keV (17.2 - 16.6) is carried off by the recoil electron.

Wave-Particle Duality

The concept that light and matter can exhibit properties of both waves and particles is known as wave-particle duality.

Evidence for Wave-Nature of Light

• Diffraction
• Interference

Evidence for Particle-Nature of Light

• Photoelectric effect
• Compton effect

De Broglie Hypothesis

In 1924, Louis de Broglie suggested that if light (a wave) can exhibit particle nature, then matter (particles) should also exhibit wave-like behavior. He proposed that a particle with momentum (p) has an associated wavelength (λ), known as the de Broglie wavelength:

• λ = h/p, where h is Planck's constant and p is the momentum of the particle.
• For an electron with kinetic energy E accelerated by a potential difference V, the de Broglie wavelength is λ = h / √(2mE) = h / √(2meV).

Example for electron at 100 Volts:

• λ = 1.226 / √V nm = 1.226 / √100 nm = 0.1226 nm.

Evidence for Wave-Nature of Matter

• Electron diffraction
• Interference of matter-waves (observed with electrons, neutrons, He atoms, C60 molecules).

Heisenberg's Uncertainty Principle

The Heisenberg Uncertainty Principle states that certain pairs of physical properties of a particle, known as conjugate variables, cannot be simultaneously known with arbitrary precision.

• Position-Momentum Uncertainty: It is impossible to precisely determine both the position (Δx) and the momentum (Δpx) of a particle simultaneously. The product of their uncertainties has a lower bound: ΔxΔpx ≥ ħ/2 (where ħ = h/2π). Similar relations hold for y and z components (ΔyΔpy ≥ ħ/2, ΔzΔpz ≥ ħ/2).
• Energy-Time Uncertainty: Similarly, there is an uncertainty relation between energy (ΔE) and time (Δt): ΔEΔt ≥ ħ/2. This means that a transition between energy levels, which occurs over a finite time, cannot have a perfectly sharp frequency.

Summary of Photon Properties and Duality

Relations between Particle and Wave Properties of Light

• Energy: E = hf
• Momentum: p = h/λ
• Relativistic relation: E2 = p2c2 + m2c4 (for light, m=0, so E=pc)
• Also expressed as: E = ħω and p = ħk (where ω=2πf is angular frequency, k=2π/λ is wavevector, ħ=h/2π is reduced Planck's constant).

Conclusions: Wave-Particle Duality

• Light and matter both exhibit wave-particle duality.
• Particle properties of light evidenced by: Photoelectric effect, Compton scattering.
• Wave properties of matter evidenced by: Electron diffraction, interference of matter waves.
• The Heisenberg uncertainty principle sets fundamental limits on the simultaneous knowledge of conjugate variables (e.g., position and momentum, energy and time).