• Since energy is always conserved, the energy of an incident photon is equal to:

The work function?+?the maximum kinetic energy of the photoelectron

• The energy within a photon is equal to?hf
• This energy is transferred to the electron to release it from a material (the work function) and the remaining amount is given as kinetic energy to the emitted photoelectron
• This equation is known as the?photoelectric equation:

E = hf =?Φ?+???mv2max

• ?Where:
  • h?=?Planck's constant?(J s)
  • f?= the frequency of the incident radiation (Hz)
  • Φ = the work function of the material (J)
  • ??mv2max=?KEmax?= the maximum kinetic energy of the photoelectrons (J)

   

• This equation demonstrates:
  • If the incident photons do not have a high enough frequency and energy to overcome the work function (Φ), then no electrons will be emitted
  • hf0?= Φ, where?f0?= threshold frequency, photoelectric emission only just occurs
  • KEmax?depends only on the frequency of the incident photon, and?not?the intensity of the radiation
  • The majority of photoelectrons will have kinetic energies?less than?KEmax

   

Work Function

• The work function Φ, or threshold energy, of a material, is defined as:

The minimum energy required to release a photoelectron from the surface of a metal

• Consider the electrons in a metal as trapped inside an ‘energy well’ where the energy between the surface and the top of the well is equal to the work function Φ
• A single electron absorbs one photon
• Therefore, an electron can only escape from the surface of the metal if it absorbs a photon which has an energy equal to Φ or higher

In the photoelectric effect, a single photon may cause a surface electron to be released if it has sufficient energy

Graphical Representation of Work Function

• The photoelectric equation can be rearranged into the straight line equation:

y?=?mx + c

• Comparing this to the photoelectric equation:

KEmax?=?hf?- Φ

• ?A graph of maximum kinetic energy?KEmax?against frequency?f?can be obtained

• The key elements of the graph:
  • The work function Φ is the?y-intercept
  • The threshold frequency?f0?is the?x-intercept
  • The?gradient?is equal to Planck's constant?h
  • There are no electrons emitted below the threshold frequency?f0

   

Threshold Frequency

• The threshold frequency is defined as:

The minimum frequency of incident electromagnetic radiation required to remove a photoelectron from the surface of a metal

Step 1:?Write out the photoelectric equation and rearrange to fit the equation of a straight line

E = hf =?Φ + ??mv2max? ? ?????→? ??KEmax?= hf?- Φ

y?=?mx?+?c

Step 2:?Identify the threshold frequency from the x-axis of the graph

When?Ek?= 0,?f?=?f0

Therefore, the threshold frequency is?f0?= 4 × 1014 Hz

Step 3:?Calculate the work function

From the graph at?f0, ??mvmax2?= 0

Φ =?hf0?= (6.63 × 10-34) × (4 × 1014) = 2.652 × 10-19?J

Step 4:?Convert the work function into eV

1 eV = 1.6 × 10-19?J ??????????????? J → eV: divide by 1.6 × 10-19