# 27.1.2 Threshold Frequency

### Threshold Frequency & Wavelength

• The concept of a threshold frequency is required in order to explain why a low frequency source, such as a filament lamp, was unable to liberate any electrons in the gold leaf experiment
• The threshold frequency is defined as:

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

• The threshold wavelength, related to threshold frequency by the wave equation, is defined as:

The longest wavelength of incident electromagnetic radiation that would remove a photoelectron from the surface of a metal

• Threshold frequency and wavelength are properties of a material, and vary from metal to metal

#### Exam Tip

A useful analogy for threshold frequency is a fairground coconut shy:

• One person is throwing table tennis balls at the coconuts, and another person has a pistol
• No matter how many of the table tennis balls are thrown at the coconut it will still stay firmly in place – this represents the low frequency quanta
• However, a single shot from the pistol will knock off the coconut immediately – this represents the high frequency quanta

### The Photoelectric Equation

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

The threshold energy + the 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 gives the emitted photoelectron the remaining amount as kinetic energy
• This equation is known as the photoelectric equation:

E = hf = Φ + ½mv2max

• Symbols:
• h = Planck's constant (J s)
• f = the frequency of the incident radiation (Hz)
• Φ = the work function of the material (J)
• ½mv2max= the maximum kinetic energy of the photoelectrons (J)

• This equation demonstrates:
• If the incident photons do not have a high enough frequency (f) and energy to overcome the work function (Φ), then no electrons will be emitted
• When hf0 = Φ, where f0 = threshold frequency, photoelectric emission only just occurs
• Ekmax 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 Ekmax

#### 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:

Ekmax = hf – Φ

•  A graph of maximum kinetic energy Ekmax 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

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

straight line

E = hf = Φ + ½mv2max         →    Ekmax = 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, ½ 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  ### Author: Katie

Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.
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