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

• To observe the diffraction of electrons, they must be focused through a gap similar to their de Broglie wavelength, 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
  • 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. The diffraction pattern is observed on the screen as a series of concentric rings
• It is observed that:
  • a larger accelerating voltage reduces the diameter of a given ring
  • a lower accelerating voltage increases the diameter of the rings

• As the voltage is increased:
  • The speed of the electrons is increased, therefore:
    • the momentum of the electrons is increased
    • the kinetic energy of the electrons is increased
    • the de Broglie wavelength is decreased
  • The angle of diffraction is decreased, therefore:
    • the radius of the diffraction pattern is decreased

• Electron diffraction was the first clear evidence that matter can behave like light and has wave properties

• The de Broglie wavelength can be used to calculate the angle of the first maximum in the diffraction pattern

• Where:
  • = angle of diffraction of the first maximum
  • = de Broglie wavelength
  • = spacing between atoms used for diffraction

• The first minimum:
  • will be half the value of the first maximum
  • will not be zero
• Subsequent minima reduce in intensity until reaching zero