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Wave nature of particles
In 1924 the French physicist Louis de Broglie (pronounded Broy), shown below, suggested that since light could behave like a particle as in the photoelectric effect then perhaps particles like electrons could have wave properties.
From relativity theory, the energy E of a particle with zero rest mass, e.g. a photon, is the product of its momentum p and the speed of light c,
E = pc
Combining this with Planck's equation,
E = hF
gives,
p = h/λ
This suggests that every particle with momentum p has an associated wavelength λ. The greater the momentum the shorter the wavelength.
Confirmation of the wave nature of electrons came from two separate experiments carried out in the US and in Scotland.
In the US, Davisson and Germer, shown below, carried out an experiment where electrons were reflected back from a crystal structure.
Thomson, shown below, carried out an experiment where electrons were passing through a thin crystalline structure, which is similar to the experiment described below.
Concepts and Equations
Concepts and Equations
Circular Diffraction
Where b the spacing between atoms.
Electron Diffraction
DeBroglie Waves
Momentum
Volume of Cube
Kinetic Energy
Energy of Electron
Using the information above, show that
Using the information above, show that
Electron Diffraction Tube
Electron Diffraction Tube
The electron diffraction tube shown below is a highly evacuated tube with an electron gun.
The electron beam emitted passes through a micro-mesh nickel grating that has been covered with a thin layer of polycrystaline graphite. This layer affects the electrons in the beam much like a diffraction grating, as shown below, and the pattern can be observed on the fluorescent screen.
Observe yourself the diffraction pattern of the electrons on the screen. Adjust the voltage of the power supply and observe any change in the diffraction pattern.
Compare your observations with de Broglie's hypothesis. If the electrons have more momentum (from a higher voltage) is their wavelength shorter?
Graphite Dimension Notation:
Graphite Dimension Notation:
The scattering crystal used for this experiment is graphite, which doesn’t have the regular square array of atoms shown in Figure 1 in its lattice. The lattice geometry of graphite is hexagonal.
There are many possible sets of reflecting plans that can be drawn through the atoms of the graphite lattice. Two prominent sets of planes are called the d10 and d11 planes. From the lattice spacing in graphite, the distances between the reflecting planes are known to be d10 = 0.213 nm and d11 = 0.123 nm.
Set the power supply to 3.0 kV and follow through the instructions below based on the individual parts of the apparatus.
Electron gun
A current is passed through a wire causing it to heat up and emit electrons. These electrons are the accelerated by a high voltage. If the voltage is the work done on the electron per unit charge,
then the work done is the kinetic energy gained by the electron i.e.
Calculate the kinetic energy gained by the electrons if the accelerating voltage is set at 3.0 kV.
De Broglie hypothesis
De Broglie predicted that electrons could have wave properties where the wavelength λ of the electron is related to the momentum p of the particle by the following equation,
Energy of an electron is the charge [C] times the Voltage [J/C].
Kinetic energy is related to momentum by the following equation,
Deduce the equation for wavelength of the electrons in terms of Planck's Constant, accelerating voltage, mass of electron, charge of electron and any physical constants as necessary.
Use this equation to calculate the wavelength of electrons that are accelerated by 3.0 kV.
Graphite target
The electrons are passed through a thin film of graphite. The density of graphite is about 2.27 g cm-3 and its molar mass is 12.0 g mol-1.
Estimate the average spacing (d) between atoms of carbon in graphite (lower-case d in the equation to the right.) spacing = 2.06x10-10 m
The equation that applies here is similar to the equation for a diffraction grating and is known as the Bragg equation.
Bragg Equation
Data Booklet
Where D is the spacing of the nuclei.
Fluorescent screen
When electrons hit the screen they excite electrons in the white coating, when these electrons return to their unexcited level they emit green light.
Using the data from above and the distance you measured in class from the center to the first ring (n=1).
Calculate the angle θ.
Then use the spacing of graphite atoms (calculated above) to calculate the wavelength of the electrons as measured from the diffraction pattern.
Compare this wavelength with the theoretical value calculated from De Broglie's equation for electrons accelerated with 3.0 kV.
The nature of electrons
With the new discovery of the wave nature of electrons the complete knowledge of the nature of the electron and its reality (or not) became a battle between sets of physicists, on one side led by Niels Bohr and on the other side led by Albert Einstein.
Sample Problems
Sample Problems
An electron has a rest mass of 9.11 x10−31 kg. If the electron has a kinetic energy of 1.14x10−27 J, what is its de Broglie wavelength? 1.45x10−5 m.
The graph shows how the relative intensity of the scattered electrons varies with angle due to diffraction by the oxygen-16 nuclei. The angle is measured from the original direction of the beam.
The de Broglie wavelength λ of each electron in the beam is 3.35 × 10−15 m.
Show that the energy E of each electron in the beam is about 6 × 10−11 J.
Calculate the diameter of an oxygen-16 nucleus using information from the graph.
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