The photoelectric effect (4.5.2)
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When electromagnetic (EM) radiation of sufficient energy is shone on the surface of a metal, electrons are emitted from the surface. This is known as the photoelectric effect.
The electrons emitted due to the photoelectric effect are called photoelectrons. It is important to note that photoelectrons are just normal electrons.
For most metals, ultraviolet (UV) light has enough energy to cause the photoelectric effect.
The photoelectric effect was first observed in 1887 by Heinrich Hertz, who noted it when shining UV light onto metal electrodes, though he could not explain it. Philipp Lenard later discovered in 1902 that this effect released electrons.
In 1905, Albert Einstein published a pivotal explanation of the photoelectric effect, contributing significantly to the development of quantum mechanics. Building on Max Planck’s earlier work, Einstein’s explanation utilised the photon model, which conceptualised light as being composed of particles.
The photoelectric effect cannot be understood from the wave model of EM radiation.
- Photoelectrons are only emitted from a metal when the incident EM radiation has enough energy to overcome a threshold specific to each metal.
- According to the wave model, the energy of the incident radiation should not matter, as the electrons would steadily gain energy from a continuous wave until they were emitted.
The photoelectric effect is evidence for the particulate nature of EM radiation: the photon model.
The photoelectric effect can be demonstrated using a gold leaf electroscope, which consists of a metal rod attached to a strip of gold leaf.
- The gold leaf is placed inside a box to shield it from air draughts.
- A negatively-charged zinc plate is attached to the top of the electroscope. The negative charges spread out between the metal rod and the gold leaf, causing the gold leaf to repel the strip and move away.
- If UV radiation is shone on the zinc plate, the gold leaf gradually falls back down towards the metal rod.
The UV radiation causes photoelectrons to be emitted from the zinc by the photoelectric effect, so that the metal rod and gold leaf slowly lose their charge and no longer repel.
The photoelectric effect is evidence for the photon model of EM radiation.
Each photon incident on a metal surface can only transfer its energy to an electron in a one-to-one interaction. The diagram below shows this interaction at the atomic level.
Each electron requires a certain amount of energy to escape the metal. If the energy absorbed from the photon is greater than the required energy, the electron escapes.
Since the surface electrons undergo one-to-one interactions with the incident photons, the intensity of incident radiation – the number of photons – does not affect whether electrons are emitted.
The principle of energy conservation applies to the photoelectric effect.
The energy of a photon in is equal to:
where:
- is the Planck constant
- is the frequency of the photon in
In the photoelectric effect, one electron absorbs one photon and gains an amount of energy equal to
When monochromatic photons with uniform energy encounter electrons in a metal, each electron gains the same amount of energy from each photon, but the emitted photoelectrons have a range of kinetic energies (KE). This is because work must be done on the electrons for them to leave the metal.
The minimum energy required to free an electron from a metal surface is the work function,
- Surface electrons absorb a photon and lose an amount of energy equal to the work function before being released. The remaining energy from the absorbed photon is converted to KE.
- Deeper electrons require more energy to escape, so less of the absorbed photon energy is converted to KE upon emission.
The kinetic energy of a photoelectron is equal to the incident photon energy minus the work done to remove the electron from the metal surface.
The work done is equal to the work function only for surface electrons. Deeper electrons require more energy to escape.
Photoelectrons emitted from the surface of a metal lose the least energy, meaning they have the maximum kinetic energy , which is equal to the photon energy minus the work function:
This is Einstein’s photoelectric equation, which is often quoted in the rearranged form:
The general expression for the kinetic energy of a photoelectron is:
Where:
- is the Planck constant,
- is the frequency of the incident photon in ,
- is the work done to remove the electron from the metal surface.
For surface electrons, the work done is equal to the work function, :
Through substitution, this expression becomes Einstein’s photoelectric equation:
Question walkthrough
The Photoelectric Effect
Apply Einstein's photoelectric equation to calculate the work function of a metal surface from the frequency of incident UV radiation and the maximum kinetic energy of emitted photoelectrons.
The minimum energy required to free an electron from a metal surface is the work function Different metals have different work functions, which are in the range of a few electronvolts.
The conversion between electronvolts and joules is:
The table below gives the work functions of some common metals.
The work function of a metal can vary depending on its surface conditions, which is why the work functions above are given as a range. Examples of factors that change the work function of a metal include:
- Surface contamination can either increase or decrease the work function.
- Surface structure: roughened surfaces often have lower work functions than smooth surfaces.
- Surface defects reduce the work function.
The energy of a photon is equal to:
The work function is the minimum energy of a photon required to free an electron from a surface. The work function can therefore be written as:
Where is the threshold frequency, which is the lowest frequency of incident radiation that causes photons to be emitted from a surface by the photoelectric effect.
It is important to note that since work functions of metals are usually measured in it can be more convenient to convert the Planck constant into units of as
Einstein’s photoelectric equation states that:
Substituting the expression for the work function:
Returns an alternative form of Einstein’s photoelectric equation as:
Einstein’s photoelectric equation can be written in terms of the threshold frequency, as:
A plot of the maximum kinetic energy, against the frequency, of incident radiation is shown below.
The gradient of the straight line graph is equal to
The threshold frequency is the X axis intercept. This can be seen by setting which leads to:
A negative kinetic energy is unphysical, so the graph shows that no electrons are emitted for incident radiation below,
Additionally, the Y axis intercept is equal to the negative of the metal work function. Setting gives:
Einstein’s photoelectric equation can be written in terms of the threshold frequency, :
A plot of the maximum kinetic energy against the frequency for two different metals is shown below.
The gradient of the graph is equal to Planck’s constant, so it remains the same for any metal. Plots for different metals demonstrate how a higher work function results in a higher threshold frequency.
In the example above, metal 2 has a higher work function and higher threshold frequency than metal 1.
The larger the work function, the greater the energy of incident photons required to emit photoelectrons. From the equation higher energy photons have a higher frequency.
Einstein’s photoelectric equation states that:
The equation can be rearranged to:
Therefore, the maximum kinetic energy of the photoelectrons only depends on the frequency, of the incident radiation and the metal work function,
The maximum kinetic energy of photoelectrons emitted by the photoelectric effect does not depend on the intensity of the incident radiation on a metal surface.
Electrons are emitted from a metal surface by the photoelectric effect when radiation with a frequency higher than the threshold frequency is incident on the surface.
- Increasing the intensity of radiation does not change the maximum kinetic energy of the photoelectrons.
- The rate of emission of photoelectrons due to incident radiation with a frequency above the threshold frequency is directly proportional to the intensity of the incident radiation.
Increasing the intensity of radiation increases the number of photons incident on the metal surface per second. More electrons absorb energy from a photon and leave the surface, so the rate of emission of photoelectrons increases.
