READING
Key Ideas pg
pg
Wave Behaviour of Particles
Particles exhibit wave behaviour with a wavelength determined by the equation
  h p (de Broglie relation), where h is Planck’s
Solve problems involving the use of the equation
  h p for electrons and other particles. constant and p is the momentum of the particles.
Experimental Evidence for Wave Behaviour of Particles
The spacing of the atoms in crystals is such that interference effects are observed when low-energy electrons are used.
Describe the Davisson–Germer experiment, in which the diffraction of electrons by the surface layers of a crystal lattice was observed.
Using the grating equation measured angle d sin
  m

and the

of the first-order maximum from the Davisson–Germer experiment, calculate the wavelength of the electrons used in the experiment.
Calculate the momentum of the electrons used in the
Davisson–Germer experiment, and hence verify that
  h p .
Application: Electron Microscopes
Explain how the very short wavelength of electrons and the ability to use electric or magnetic fields to focus them allow electron microscopes to achieve very high resolution

The Wave Behaviour of Particles
The idea that light exhibits wave properties (such as frequency and wavelength) as well as particle properties (such as momentum) led to the idea that the same properties could be associated with all matter. That is, particles (eg electrons) could exhibit _______________ as well as _______________.
When light behaves as particles (______________), it has a momentum that is associated with its wavelength p =
The same formula applies to particles when they behave as waves. That is, a particle with momentum (__________________) will have a wavelength given by ___________________.
This equation is known as the de Broglie relationship.
Note that Planck’s Constant, h = 6.63 x 10-34, is extremely small. Therefore, any objects that have a large momentum will have a very small wavelength. Only particles with extremely small momenta will have wavelengths large enough to exhibit significant wave properties.
1.
Find the wavelength of a ball of mass 0.4kg moving with a velocity of 2.0ms
-1 .
2.
Find the wavelength of an electron moving with a velocity of 10 7 ms -1 .
3.
By comparing your answers to questions 1 & 2, describe why the wave property of matter was not detected previously.

Experimental Evidence of the Wave Behaviour of Particles
Since we know that diffraction is a property of waves, showing that particles can also display diffraction provides evidence for the wave behaviour of particles.
The Davisson-Germer experiment was one of the first experiments that verified the wave behaviour of particles using diffraction. Davisson and Germer fired a beam of low energy electrons (54eV) at a nickel crystal lattice. This meant that the electrons had a fixed momentum, given by the equation 𝑝 = √2𝑚. 𝐸
The regular lattice of nickel atoms acted as a diffraction grating, and so a pattern of interference was observed.
Using the equation d.sinθ = m.
λ , where d (spacing between nickel atoms) was known from previous experiments using X-rays, Davisson and Germer were able to calculate the wavelength of the electrons. This value was extremely close to the wavelength predicted by the de Broglie relation λ = h/p.
4.
For an accelerating potential of 54V the first order maxima occurred at an angle of 50° for a spacing of crystals of 0.215 nm. a.
Using wave theory, find the wavelength of such a wave. b.
Find the velocity of an electron accelerated by the above potential. c.
Using the velocity of the electron accelerated by the above potential, find its de
Broglie wavelength d.
Comment on your answers to parts a and c

Application: Electron Microscope
Explain how the very short wavelength of electrons, and the ability to use electric or magnetic fields to focus them, allows electron microscopes to achieve very high resolution.
Electron microscopes use beams of electrons to achieve very high magnifications. The maximum magnification that a microscope can achieve is determined by the wavelength of the waves used.
The smaller the wavelength, the greater the possible magnification. This is because waves with a smaller wavelength are able to distinguish between points that are closer together.
Visible light has a wavelength between 400 and 750nm (ie between 4 x 10 -7 and 7.5 x 10 -7 m). The electrons used in an electron microscope will have a wavelength around 5 x 10 -12 m. Thus electron microscopes can achieve far higher magnifications than light microscopes.
Electrons are charged particles, and can hence be influenced by electric and magnetic fields. Within an electron microscope, the beam of electrons can therefore be focussed by electric and magnetic
“lenses” (fields) instead of the optical lenses used in light microscopes.