# What is a Photon?

A photon is a packet of electromagnetic radiation energy. The amount of energy of a photon is related to the frequency (and wavelength) of the EM radiation by

$E=hf=\frac{hc}{\lambda }$

where h is the Planck’s constant.

For example, monochromatic red light of wavelength 750 nm consists of a stream of photons (travelling at light speed) each carrying 1.7 eV of energy,

$E=\frac{hc}{\lambda }=\frac{(6.63\times {{10}^{-34}})(3.00\times {{10}^{8}})}{750\times {{10}^{-9}}}=2.65\times {{10}^{-19}}\text{ J}=1.7\text{ eV}$

whereas violet light of wavelength 400 nm delivers energy in discrete packets each carrying 3.1 eV of energy.

$E=\frac{hc}{\lambda }=\frac{(6.63\times {{10}^{-34}})(3.00\times {{10}^{8}})}{400\times {{10}^{-9}}}=4.97\times {{10}^{-19}}\text{ J}=3.1\text{ eV}$

# Photon Model of Light

According to the photon model of light, electromagnetic radiation energy is delivered in discrete packets called photons.

Between two monochromatic lights of the same wavelength but different power, the one with the higher power is delivering more photons per unit time.

Between two monochromatic lights of different wavelength but the same power, the one with the shorter wavelength delivers fewer photons per unit time, but each photon packs a larger amount of energy.

# The Photoelectric Equation

In the wave model, a light wave delivers continuous energy which is shared by all the electrons in the metal. In the photon model, however, an individual photon deliver discrete energy to an individual electron. This results in a simple energy equation

$KE=hf-WD$

Meaning the KE of the liberated photoelectron is equal to the energy supplied by a single photon  minus the required work done to break free from the metal lattice.

The photoelectrons which are liberated usually reside in the valence band. Even among the valence electrons, some of them are bound more tightly to the metallic lattice than others and thus must do more work before they can escape from the metal lattice. This explains why liberated photoelectrons have a range of kinetic energy even when monochromatic light is used (illustrated in the energy diagram below).

The least tightly bound valence electrons (those occupying the topmost energy levels in the valence band) are the ones which requires the least amount of work before breaking free from the metal lattice. This minimum work is called the work function Φ. The work function is different for different metal. For example, the work function for sodium, aluminium and gold are about 2.4 eV, 4.1 eV and 5.1 eV respectively.

The maximum KE of the liberated photoelectron is thus related to the photon energy and the work function as follow:

$K{{E}_{\max }}=hf-\Phi$

This equation highlights the fact that in the photon model, the photoelectric effect is a one-photon-one-electron interaction: an individual electron absorbs the energy of an individual photon to overcome the work function of the metal. We are going to see how this model can explain all the experimental observations that the wave model cannot.

## Why KEmax is dependent on Frequency but not Intensity of light?

Increasing the intensity of light merely increases the number of photons arriving at the metal surface per unit area per unit time. The rate of emission of photoelectrons will increase, resulting in a higher saturation current. But the energy delivered by each individual photon to each individual electron remains unchanged. So the maximum KE however remains unchanged.

On the other hand, increasing the frequency of light used means that light energy is delivered in larger quantum. With each individual photon delivering a larger quantum of energy, an individual electron absorbs a higher quantum of energy and escapes with higher KEmax. Remember that $K{{E}_{\max }}=hf-\Phi$.

The higher KEmax results in a higher stopping potential Vs since

$K{{E}_{\max }}=e{{V}_{s}}$

# Why there is a Threshold Frequency?

The frequency at which the energy of a photon matches the work function of a metal is called the threshold frequency f0.

$h{{f}_{0}}=\Phi$

Below the threshold frequency, the energy of each individual photon in a light beam will be lower than the work function, meaning each photon does not supply sufficient energy for a single electron can be liberated. (It is not possible for an electron absorb multiple photons because the chance of an electron being hit by two photons simultaneously at the same time is practically zero.) Increasing the intensity of the light does not help at all, since this only increases the number of photons per unit time, but the energy of each individual photon is unchanged.

# Why Zero Time Delay?

Unlike the wave model which requires the energy delivered to be shared by all the electrons, the photon model allows an electron to be liberated the moment it absorbs the energy of one photon. This explains the experimental observation that a photoelectric current (albeit a small one) is registered the moment the light is switched on.