Photoelectric threshold and work function

When a material absorbs electromagnetic radiation, electrons can be released from it. This is the photoelectric effect. In the photon model, radiation arrives as photons, and an electron absorbs energy from an individual photon.

Each photon has energy $$hf$$, where $$h$$ is the Planck constant and $$f$$ is the radiation frequency. A higher frequency therefore means more energy per photon.

Work function and threshold frequency

An electron needs a minimum amount of energy to escape from a material. This minimum energy is the material’s work function, $$\phi$$. A photon must have at least this much energy for the electron to be released.

The threshold frequency, $$f_0$$, is the lowest frequency that can cause photoelectric emission. At this frequency, the energy of one photon equals the work function:

Below the threshold frequency, each photon has less energy than the work function, so photoelectric emission cannot occur. Increasing the radiation intensity does not change the energy of an individual photon: even with more photons, each one still has insufficient energy to release an electron.

Different materials have different work functions. A material with a larger work function needs photons of greater energy and therefore has a higher threshold frequency.

Photoelectric energy equation and stopping potential

When a photon transfers its energy to an electron in a material, some energy is needed for the electron to escape. The smallest energy needed to remove an electron from the material is its work function, $$\phi$$. Any energy the escaping electron retains appears as kinetic energy.

A photon of frequency $$f$$ has energy $$hf$$, where $$h$$ is the Planck constant. For the emitted electrons with the greatest kinetic energy, conservation of energy gives the photoelectric energy equation:

Here $$E_{k,(\max)}$$ is the maximum kinetic energy of an emitted electron. When emission occurs, it is given by $$E_{k,(\max)} = hf - \phi$$. If the photon energy is less than the work function, the electron cannot escape.

The equation gives a maximum, not the kinetic energy of every emitted electron. Other electrons can escape with less kinetic energy, for example because they transfer some energy to the material before reaching its surface.

Stopping potential

A potential difference can be applied between the emitting surface and a collector so that it opposes the motion of photoelectrons towards the collector. As this opposing potential difference increases, fewer electrons have enough kinetic energy to reach the collector.

The stopping potential is the potential difference just sufficient to prevent even the most energetic photoelectrons from reaching the collector. If its magnitude is $$V_s$$, the maximum kinetic energy is:

Here $$e$$ is the magnitude of the electron charge. At the stopping potential, the most energetic electrons have just enough initial kinetic energy to move against the opposing potential difference; they cannot reach the collector with kinetic energy remaining.

Combining the two relationships gives $$hf - \phi = eV_s$$ when emission occurs. Thus the stopping potential provides a measure of the maximum photoelectron kinetic energy.