The de Broglie Wavelength (HL) (HL IB Physics)

• Louis de Broglie thought that if waves can behave like particles, then perhaps particles can also behave like waves

• He proposed that electrons travel through space as waves
  • This would explain why they can exhibit wave-like behaviour such as diffraction

• De Broglie suggested that electrons must also hold wave properties, such as wavelength
  • This came to be known as the de Broglie wavelength
• However, he realised that all particles can show wave-like properties, not just electrons
  • He hypothesised that all moving particles have a matter wave associated with them

• The definition of a de Broglie wavelength is:

The wavelength associated with a moving particle

• De Broglie suggested that the momentum p of a particle and its associated wavelength λ are related by the equation:

• Since momentum p = mv, the de Broglie wavelength can be related to the speed of a moving particle v by the equation:

Kinetic Energy & de Broglie Wavelength

• Kinetic energy is defined as

• Combining this with the previous equation gives relationship between the de Broglie wavelength of a particle to its kinetic energy:

• Where:
  • λ = the de Broglie wavelength (m)
  • h = Planck’s constant (J s)
  • p = momentum of the particle (kg m s⁻¹)
  • E_K = kinetic energy of the particle (J)
  • m = mass of the particle (kg)
  • v = speed of the particle (m s⁻¹)

• The majority of the time, and for everyday objects travelling at normal speeds, the de Broglie wavelength is far too small for any quantum effects to be observed
  • A typical electron in a metal has a de Broglie wavelength of about 10 nm

• Therefore, the quantum effects of diffraction will only be observable when the width of the aperture is of a similar size to the de Broglie wavelength

Electrons accelerated through a high potential difference demonstrate wave-particle duality 

• The electron diffraction tube can be used to investigate how the de Broglie wavelength of electrons depends on their speed
  • The smaller the radius of the rings, the smaller the de Broglie wavelength of the electrons

• As the voltage is increased:
  • The energy of the electrons increases
  • The radius of the diffraction pattern decreases

• This shows as the speed of the electrons increases, the de Broglie wavelength of the electrons decreases

• Electron diffraction was the first clear evidence that matter can behave like light and has wave properties
  • This is demonstrated using the electron diffraction tube
• The electrons are accelerated in an electron gun to a high potential, such as 5000 V, and are then directed through a thin film of graphite
  • The lattice structure of the graphite acts like the slits in a diffraction grating
• The electrons diffract from the gaps between carbon atoms and produce a circular pattern on a fluorescent screen made from phosphor

• In order to observe the diffraction of electrons, they must be focused through a gap similar to their size, such as an atomic lattice
• Graphite film is ideal for this purpose because of its crystalline structure
  • The gaps between neighbouring planes of the atoms in the crystals act as slits, allowing the electron waves to spread out and create a diffraction pattern

• The diffraction pattern is observed on the screen as a series of concentric rings
  • This phenomenon is similar to the diffraction pattern produced when light passes through a diffraction grating
  • If the electrons acted as particles, a pattern would not be observed, instead, the particles would be distributed uniformly across the screen

• It is observed that a larger accelerating voltage reduces the diameter of a given ring, while a lower accelerating voltage increases the diameter of the rings