# Unit 5 Option D Wave-Particle Duality

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## Wave-Particle Duality Theory

Diffraction and interference of light can be explained using waves. If light was acting as a particle they either would not fit through a gap or just pass straight through and the beam would be unchanged.

The results of the photoelectric effect experiment can be explained by thinking of light as particle-like photons. If a photon is a discrete bundle of energy, then it can interact with an electron in a one-to-one reaction. All the energy in the photon is given to one electron.

The photoelectric effect and diffraction show that light behaves as both a particle and a wave - this is known as wave-particle duality.

Wave-particle duality: The idea that particles and waves can each display both particle and wave-like behaviour.

Louis De Broglie made a bold suggestion in his PhD thesis. He said that if 'wave-like' light showed particle properties (photons), 'particles' like electrons should be expected to show wave-like properties.

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## The De Broglie Equation

The de Broglie equation relates a wave property (wavelength, landa) to a moving particle property (momentum, p =mv).

• p = h / landa
• p = momentum in kg ms^-1
• h = planck's constant = 6.63 x 10^-34 js
• landa = de Broglie wavelength in m

It comes from assuming that a photon of energy E = hf has a mass, m, given by mc^2 = hf.

Then momentum is given by mc = hf / c and rearranging c = f x landa gives f / c = 1 / landa.

So p = mc = h / landa.

The de Broglie wave of a particle can interpreted as a 'probability wave'.

Many physicists at the time weren't very impressed - his ideas were just speculation.

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## Electron Diffraction - Part 1

Diffraction patterns are observed when accelerated electrons in a vacuum tube interact with the spaces in a graphite crystal.

As they pass through the spaces, they diffract just like waves passing through a narrow slit and produce a diffraction pattern.

This provides evidence that electrons have wave properties.

According to wave theory, the spread of the lines in the diffraction pattern increases if the wavelength of the wave is greater.

In electron diffraction, a smaller accelerating voltage (i.e. slower electrons) gives widely spaced rings.

Increase the electron speed and the diffraction pattern circles squash together towards the middle.

This fits in with the de Broglie equation - if the velocity is higher, the wavelength is shorter and the spread of lines is smaller.

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## Electron Diffraction - Part 2

In general, landa for electrons accelerated in a vacuum tube is about the same size as electromagnetic waves in the x-ray part of the spectrum.

The de Broglie wavelength of an electron is related to the accelerating voltage by:

• landa = h / [Squareroot:(2 x m x e x V)]
• landa = de Broglie wavelength in m
• h = planck's constant = 6.63 x 10^-34 js
• m = mass of an electron in kg
• e = magnitude of the charge on an electron in C
• V = accelerating voltage in V

You only get diffraction if a particle interacts with an object of about the same size as its de Broglie wavelength.

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