Work function and ways electrons leave a metal
Free electrons need a minimum energy, the work function φ₀, to escape a metal. Heat, a strong electric field or light of high enough frequency can supply it.
Why electrons stay inside a metal
A metal has many free electrons that move about inside it. They do not leave the surface on their own, because the positive ions of the metal pull them back. An electron can escape only if it gains at least a certain minimum energy. This minimum energy is the work function, φ₀. It depends on the metal and on the state of its surface. Work function is usually given in electron volts (eV): 1 eV = 1.6 × 10⁻¹⁹ J.
Three ways to free electrons from a metal
| Method | How the energy is given | Example |
|---|---|---|
| Thermionic emission | Heating the metal gives the electrons enough thermal energy | The hot filament of an X-ray tube |
| Field emission | A very strong electric field (about 10⁸ V/m) pulls electrons out | The sharp metal tip of some electron microscopes |
| Photoelectric emission | Light of high enough frequency falls on the metal | A photocell in a burglar alarm |
Work functions of some metals
| Metal | φ₀ (eV) | Threshold frequency ν₀ (Hz) | Threshold wavelength λ₀ |
|---|---|---|---|
| Potassium | 2.3 | 5.55 × 10¹⁴ | 541 nm (visible, green) |
| Zinc | 3.4 | 8.21 × 10¹⁴ | 366 nm (ultraviolet) |
| Iron | 4.8 | 1.16 × 10¹⁵ | 259 nm (ultraviolet) |
| Nickel | 5.9 | 1.42 × 10¹⁵ | 211 nm (ultraviolet) |
Worked example: from work function to threshold
The work function of a metal is 2.3 eV. Find (a) its work function in joules, (b) its threshold frequency, (c) its threshold wavelength. Can red light of wavelength 700 nm free electrons from it? (h = 6.63 × 10⁻³⁴ J s, c = 3 × 10⁸ m/s, 1 eV = 1.6 × 10⁻¹⁹ J)
- (a) φ₀ = 2.3 × 1.6 × 10⁻¹⁹ J = 3.68 × 10⁻¹⁹ J.
- (b) ν₀ = φ₀/h = 3.68 × 10⁻¹⁹ ÷ 6.63 × 10⁻³⁴ = 5.55 × 10¹⁴ Hz.
- (c) λ₀ = c/ν₀ = 3 × 10⁸ ÷ 5.55 × 10¹⁴ = 5.41 × 10⁻⁷ m = 541 nm.
- Red light (700 nm) has a longer wavelength than 541 nm, so its frequency is below ν₀. It cannot free electrons, however bright it is.
(a) 3.68 × 10⁻¹⁹ J; (b) about 5.55 × 10¹⁴ Hz; (c) about 541 nm; red light of 700 nm cannot free electrons
Wave nature vs particle nature of light
This double behaviour is called the dual nature of radiation. de Broglie later asked whether matter has it too.
The heated filament in an X-ray tube frees electrons by thermionic emission. The light sensor that switches on a street lamp at dusk frees them with light.
2 marks: “Define work function and threshold frequency. How are they related?” Work function φ₀: least energy an electron needs to escape a metal surface. Threshold frequency ν₀: least frequency of light that can cause photoelectric emission. φ₀ = hν₀.
2 marks: “The work function of zinc is 3.4 eV. Find its threshold frequency. (h = 6.63 × 10⁻³⁴ J s)” ν₀ = φ₀/h = (3.4 × 1.6 × 10⁻¹⁹) ÷ (6.63 × 10⁻³⁴) ≈ 8.2 × 10¹⁴ Hz, in the ultraviolet.
1 marks: “Name the process in which electrons leave a metal because it is heated.” Thermionic emission.
Heat, Field, Light: three ways to pay the exit fee φ₀.