Electromagnetic waves can exhibit both wave-like properties, such as being diffracted, and particle-like properties, such as the photoelectric effect. All particles with mass can be described in terms of their wavelength using the de Broglie equation.
Equations
DESCRIPTION
EQUATION
de Broglie equation
λ=ph=mvh
Accelerated particles
λ=2Ekmh
Constants
CONSTANT
SYMBOL
VALUE
Planck′sconstant
h
6.63×10−34Js
massofanelectron
me
9.11×10−31kg
chargeofanelectron
e
1.6×10−19J
Variables
QUANTITY NAME
SYMBOL
DERIVED UNIT
SI UNIT
deBrogliewavelength
λ
m
m
velocity
v
ms−1
ms−1
momentum
p
kgms−1
kgms−1
kineticenergy
Ek
J
kgm2s−2
mass
m
kg
kg
voltage
V
V
kgm2s−3A−1
Diffraction of particles
Louis de Broglie proposed that all matter has both particle and wave-like properties. All particles have a wavelength and this can be shown by their ability to diffract.
Electromagnetic waves, such as visible light, also diffract due to their wave-like properties. However, the photoelectric effect relies on light consisting of photons. This is because electromagnetic waves have a dual nature, a wave-particle duality.
Electron diffraction
Electrons can be made to diffract under certain specific conditions. The electrons are accelerated in an electron gun, then fired at a thin film of graphite. The electrons will diffract through the gaps between carbon atoms in the film, which are of the distance as the wavelength of the electrons (~1010m). This causes a circular diffraction pattern on a fluorescent screen.
This confirms wave-particle duality, as particles are not able to diffract and this is a property of waves only.
The de Broglie equation
De Broglie realised the wavelength associated with a particle is directly proportional to its momentum, which can be described by the equation:
λ=ph
Substituting the equation for momentum of the particle, p, into the first equation:
p=mvλ=mvh
This equation applies to all particles with mass, however the heavier the object the shorter their wavelength and therefore harder it is to observe wave-like properties.
Example
Calculate the de Broglie wavelength of a proton travelling at a speed of 1.5×103ms−1
State variables:
v=1.5×103ms−1mp=1.67×10−27kg
State equation:
λ=mvh
Sub in and solve:
λ=1.5×103×1.67×10−276.67×10−34λ=2.7×10−10m
The de Broglie wavelength of this proton isλ=2.7×10−10m.
Accelerated particles
An object's kinetic energy can be related to its de Broglie wavelength, using the general equation for kinetic energy and de Broglie wavelength shown above. When a particle is accelerated through a potential difference:
Ek=21mv22Ek=mv22Ekm=m2v22Ekm=mv
Substituting this into the de Broglie equation:
λ=mvh=2Ekmh
Investigating electron diffraction
Electron diffraction tubes can be used to investigate the wave properties of an electron. Particles are accelerated in an electron gun, then directed at a film of graphite, and diffract between the carbon atoms. Increasing the voltage between the anode and cathode causes the energy of the electrons to increase.
The kinetic energy of electrons is proportional to the voltage across the anode-cathode:
Ek=21mv2=eV
Example
An electron is accelerated in an electron gun through a potential difference of 200V, calculate its wavelength, using its kinetic energy.
State variables:
V=200V
State equation:
Ek=eV
λ=2Ekmh
λ=2eVmh
Sub in and solve:
λ=2×1.6×10−19×200×9.11×10−316.63×10−34
λ=8.7×10−11m
The wavelength is 8.7×10−11m
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FAQs - Frequently Asked Questions
Which particles can be described using the de Broglie wavelength?
All particles with mass can be described in terms of their wavelength using their de Broglie wavelength.
What does electron diffraction produce?
Electron diffraction produces a circular diffraction pattern on a fluorescent screen.
What is wave-particle duality?
Wave-particle duality is the phenomenon that electromagnetic waves can exhibit both wave-like properties, such as being diffracted, and particle-like properties, such as the photoelectric effect.