The work function of a material is the minimum energy needed to pull an electron free from its surface, and you find it using Einstein’s photoelectric equation: Φ = hν − Ek. In this formula, Φ is the work function, h is Planck’s constant, ν is the frequency of incoming light, and Ek is the maximum kinetic energy of the ejected electron. If the light is exactly at the threshold (barely enough to free an electron, with zero leftover kinetic energy), the equation simplifies to Φ = hν0, where ν0 is the threshold frequency.
The Photoelectric Equation
The core idea comes from conservation of energy. A photon hits a metal surface and delivers a fixed amount of energy equal to hν. Some of that energy goes toward breaking the electron free (that’s the work function), and whatever is left over becomes the electron’s kinetic energy as it flies away. Written out:
hν = Φ + ½mv²
Rearranging to solve for the work function gives you:
Φ = hν − ½mv²
Here, h is Planck’s constant (6.626 × 10⁻³⁴ J·s), ν is the photon frequency in hertz, m is the electron mass (9.109 × 10⁻³¹ kg), and v is the maximum speed of the ejected electron. If a problem gives you wavelength instead of frequency, convert using ν = c/λ, where c is the speed of light (3.0 × 10⁸ m/s).
Finding Work Function From Threshold Frequency
Many textbook problems give you the threshold frequency directly. This is the lowest frequency of light that can eject electrons from the surface. At this frequency, electrons escape with zero kinetic energy, so the equation becomes:
Φ = hν0
For example, if a metal has a threshold frequency of 5.0 × 10¹⁴ Hz, the work function is (6.626 × 10⁻³⁴)(5.0 × 10¹⁴) = 3.31 × 10⁻¹⁹ J. You can also work backward from a threshold wavelength: first convert to frequency with ν₀ = c/λ₀, then plug into the same formula.
Converting Joules to Electronvolts
Work function values are almost always reported in electronvolts (eV) rather than joules because the numbers are more convenient. One electronvolt equals 1.602 × 10⁻¹⁹ J. To convert from joules to eV, divide by that number. Taking the example above: 3.31 × 10⁻¹⁹ J ÷ 1.602 × 10⁻¹⁹ J/eV ≈ 2.07 eV. To go the other direction (eV to joules), multiply by 1.602 × 10⁻¹⁹.
Step-by-Step Example
Suppose light with a wavelength of 250 nm strikes a copper surface and the fastest ejected electrons have a kinetic energy of 0.54 eV. To find copper’s work function:
1. Convert wavelength to frequency: ν = (3.0 × 10⁸ m/s) / (250 × 10⁻⁹ m) = 1.2 × 10¹⁵ Hz.
2. Calculate the photon energy: E = hν = (6.626 × 10⁻³⁴)(1.2 × 10¹⁵) = 7.95 × 10⁻¹⁹ J. In eV, that’s 7.95 × 10⁻¹⁹ ÷ 1.602 × 10⁻¹⁹ ≈ 4.96 eV.
3. Subtract the kinetic energy: Φ = 4.96 − 0.54 = 4.42 eV.
That result sits between the measured values for different copper crystal faces, which range from about 4.56 eV to 4.90 eV depending on the surface orientation.
Using a Stopping Voltage
In lab settings and many exam problems, you won’t measure electron speed directly. Instead, a reverse voltage is applied until it just stops the fastest electrons. This “stopping potential” (Vs) gives the maximum kinetic energy through Ek = eVs, where e is the electron charge (1.602 × 10⁻¹⁹ C). The photoelectric equation then becomes:
Φ = hν − eVs
If you plot stopping voltage against photon frequency for several frequencies of light, you get a straight line. The slope equals h/e, and the y-intercept gives you −Φ/e. This graphical method is one of the most reliable ways to extract the work function from experimental data because it doesn’t depend on a single measurement.
Work Function Values for Common Metals
Work functions vary not just by element but by which crystal face is exposed. Measured values for clean, single-crystal surfaces of some widely used metals:
- Aluminum: 4.23 to 4.32 eV, depending on crystal orientation
- Copper: 4.56 to 4.90 eV
- Silver: 4.10 to 4.53 eV
- Gold: 5.16 to 5.33 eV
- Platinum: 5.53 to 5.91 eV
- Tungsten: 4.44 to 5.44 eV
- Iron: 4.60 to 5.07 eV
These values come from a comprehensive review published in the Journal of Vacuum Science & Technology. The spread within a single element can be surprisingly large. Tungsten, for instance, ranges over a full electronvolt between its lowest and highest crystal faces. If a textbook gives you a single number for “the work function of tungsten,” it’s typically an average or refers to one specific orientation.
What Affects the Work Function
Crystal orientation is the biggest factor. Different faces of the same metal pack atoms at different densities, which changes how tightly electrons are held at the surface. A more tightly packed face generally has a higher work function. Platinum’s most densely packed face (111) measures 5.91 eV, while its (110) face drops to 5.53 eV.
Surface contamination also matters. Even a thin layer of carbon or oxide can shift the work function significantly. Research from NIST on gold surfaces shows that carbon adsorption changes the work function, though interestingly the size of the change doesn’t depend much on which crystal face the carbon lands on. At higher coverage levels, the work function converges toward that of graphite regardless of the underlying gold orientation. This is why published reference values specify “clean” surfaces and typically carry uncertainty ranges of 0.02 to 0.33 eV.
Why Work Function Matters Beyond Textbooks
In solar cells and other electronic devices, the energy mismatch between a metal electrode’s work function and a semiconductor’s energy bands creates a barrier (called a Schottky barrier) that electrons or holes must overcome to flow across the contact. Choosing an electrode material whose work function aligns well with the semiconductor reduces this barrier and improves charge transport efficiency. Researchers have shown that in layered materials like phosphorene, the work function can be tuned by up to 0.7 eV simply by changing the number of atomic layers, offering a practical way to optimize these interfaces without switching materials entirely.
Work function also governs thermionic emission, where heating a metal to high temperatures gives electrons enough thermal energy to escape the surface. The relationship is exponential: a small drop in work function produces a large increase in emitted current. This principle is central to the design of electron guns, vacuum tubes, and certain spacecraft power systems.