Summary — What This Chapter Covers
- A photon is a quantum of electromagnetic energy: E = hf = hc/λ, momentum p = E/c.
- 1 eV = 1.60 × 10−19 J.
- Photoelectric equation: hf = Φ + ½mvmax2; threshold frequency f0 = Φ/h.
- Max KE is independent of intensity; photoelectric current is proportional to intensity.
- de Broglie: λ = h/p. Line spectra: hf = E1 − E2.
1. Energy and Momentum of a Photon
22.1
E = hf
Electromagnetic radiation has a particulate nature: it is emitted and absorbed in discrete packets called photons. A photon is a quantum of electromagnetic energy, with energy proportional to its frequency. The Planck constant h = 6.63 × 10−34 J s.
The electronvolt is the energy gained by an electron accelerated through a p.d. of 1 V: 1 eV = 1.60 × 10−19 J. Photons carry momentum, p = E/c, even though they have no mass, which is why light can exert pressure.
Photon
E = hf = hc/λ p = E/c = h/λ 1 eV = 1.60 × 10−19 J
Worked Example
Find the energy (in J and eV) and momentum of a photon of wavelength 500 nm.
Practice Question
How many photons per second does a 1.0 mW laser of wavelength 633 nm emit?
2. Photoelectric Effect
22.2
hf = Φ + ½mv²max
Electromagnetic radiation shining on a metal surface can release electrons: photoelectrons. Emission only happens if the frequency is at or above the threshold frequency f0 (wavelength at or below the threshold wavelength λ0 = c/f0), however intense the light.
Each photon interacts with a single electron. The work function Φ is the minimum energy needed to remove an electron from the surface. Energy left over becomes kinetic energy; electrons from deeper in lose more, so the equation gives the maximum kinetic energy.
Raising the intensity at fixed frequency means more photons per second, so more electrons per second: the photocurrent is proportional to intensity. Each photon still has the same energy hf, so the maximum kinetic energy is independent of intensity. A wave model cannot explain this, nor the threshold, nor the instant emission.
Photoelectric equation
hf = Φ + ½mvmax2 Φ = hf0 = hc/λ0
Worked Example
Light of wavelength 400 nm falls on a metal with work function 2.3 eV. Find the threshold wavelength, the maximum kinetic energy of the photoelectrons and their maximum speed.
3. Wave-Particle Duality
22.3
λ = h/p
Light shows both behaviours: the photoelectric effect is evidence for a particulate nature, while interference and diffraction are evidence for a wave nature.
De Broglie proposed that moving particles also have a wavelength, λ = h/p. Electron diffraction confirms it: a beam of electrons passed through a thin layer of polycrystalline graphite produces concentric rings on a fluorescent screen, a diffraction pattern, because the electron wavelength is similar to the spacing between atoms. Increasing the accelerating voltage increases the momentum, shortens the wavelength, and the rings get smaller.
de Broglie wavelength