# Work Function & Threshold Frequency ## What is Threshold Frequency? - **Threshold frequency** ($f_0$) is the minimum frequency of electromagnetic radiation needed to liberate an electron from a metal surface - Below this frequency, no photoelectrons are emitted, regardless of how bright the light is - This is a material-specific property that varies between different metals ## What is Threshold Wavelength? - **Threshold wavelength** ($\lambda_0$) is the maximum wavelength of incident radiation capable of ejecting a photoelectron from a metal surface - Related to threshold frequency via the wave equation: $c = f\lambda$ - For wavelengths longer than $\lambda_0$, photoelectric emission cannot occur

Below threshold frequency, no electrons escape no matter how intense the light - each electron needs one sufficiently energetic photon

## Threshold Values for Common Metals | Metal | Threshold Frequency ($f_0$) / Hz | Threshold Wavelength ($\lambda_0$) / nm | |-------|----------------------------------|------------------------------------------| | Sodium | $4.40 \times 10^{14}$ | 682 | | Potassium | $5.56 \times 10^{14}$ | 540 | | Zinc | $1.02 \times 10^{15}$ | 294 | | Iron | $1.04 \times 10^{15}$ | 289 | | Copper | $1.13 \times 10^{15}$ | 266 | | Gold | $1.23 \times 10^{15}$ | 244 | | Silver | $9.71 \times 10^{14}$ | 30.9 | - Different metals have different threshold values due to varying bond strengths between surface electrons and the metal lattice - This explains why UV light causes photoemission from zinc but visible light can liberate electrons from sodium ### Visualizing Threshold Frequencies The chart below shows the threshold frequencies for different metals - notice how alkali metals (sodium, potassium) have much lower threshold frequencies than transition metals (zinc, copper, gold):

- Metals with lower threshold frequencies (left side) are easier to cause photoemission from - The visible light spectrum ranges from approximately $4.3 \times 10^{14}$ Hz (red) to $7.5 \times 10^{14}$ Hz (violet) - Only sodium and potassium can emit photoelectrons when exposed to visible light ## Work Function

Work function $\Phi$ is the minimum energy required to remove an electron from the surface of a metal

- Work function is measured in joules (J) or electronvolts (eV) - Represents the binding energy holding electrons within the metal structure - At threshold frequency: $hf_0 = \Phi$ ### The Energy Well Model - Imagine electrons confined in an energy well, with the surface of the metal at the bottom and free space at the top - The depth of this well represents the work function $\Phi$ - For an electron to escape: - It must absorb energy equal to or greater than $\Phi$ - Since one photon interacts with one electron, the photon energy must satisfy $E_{\text{photon}} = hf \geq \Phi$ ### Low Energy Radiation (Below Threshold) - Photon energy: $hf < \Phi$ - Even if the electron absorbs the photon, it doesn't have enough energy to climb out of the well - The electron falls back, re-emitting the energy as heat or light - **No photoelectric emission occurs** ### Threshold Energy Radiation - Photon energy: $hf_0 = \Phi$ - The electron gains exactly enough energy to reach the surface - Photoelectric emission just begins - Electrons escape with zero kinetic energy ### High Energy Radiation (Above Threshold) - Photon energy: $hf > \Phi$ - Excess energy beyond the work function is converted to kinetic energy of the emitted photoelectron - The electron escapes the well and has kinetic energy remaining - This leftover energy becomes the maximum kinetic energy $KE_{\text{max}}$

Increasing intensity at below-threshold frequency will never cause emission - you need sufficient photon energy, not more photons

## Metal Categories and Their Work Functions ### Alkali Metals (Low Work Function) - Examples: sodium, potassium, caesium - Have threshold frequencies in the **visible light region** - Surface electrons are loosely bound due to: - Only one electron in the outer shell - Weaker electrostatic attraction between outer electrons and positive metal ions - Larger atomic radius reducing binding forces - Can be used in light-sensitive devices that respond to visible light ### Transition Metals (High Work Function) - Examples: zinc, iron, copper, silver, gold - Have threshold frequencies in the **ultraviolet region** - Surface electrons are more tightly bound because: - Multiple electrons in outer d-orbitals - Stronger electrostatic forces between electrons and the metal lattice - Greater effective nuclear charge - Require higher energy photons to cause photoemission - More stable under normal lighting conditions ## Understanding Through Analogy - Think of a trampoline park with safety nets of different heights: - Children (electrons) are jumping on trampolines (absorbing photon energy) - Low jumps (low frequency photons) can't get anyone over the net, no matter how many children jump - There's a minimum jump height (threshold frequency) needed to clear the net (work function) - Different parks have different net heights (different metals have different work functions) - Stronger jumpers (higher frequency photons) not only clear the net but have energy left to move away quickly (kinetic energy) ## Key Relationships - Threshold frequency and work function: $\Phi = hf_0$ - Threshold wavelength and work function: $\Phi = \dfrac{hc}{\lambda_0}$ - Relationship between threshold frequency and wavelength: $\lambda_0 = \dfrac{c}{f_0}$ Where: - $h$ = Planck's constant ($6.63 \times 10^{-34}$ J s) - $c$ = speed of light ($3.00 \times 10^8$ m s$^{-1}$) - $f_0$ = threshold frequency (Hz) - $\lambda_0$ = threshold wavelength (m) - $\Phi$ = work function (J)