de Broglie's Matter Wave Duality and Experimental Evidence
HSC Physics Syllabus

investigate de Broglie’s matter waves, and the experimental evidence that developed the following formula:
What is Matter Wave Duality?
De Broglie’s Matter Wave Hypothesis
Previous experiments such as the photoelectric effect demonstrated the particlenature of light (waves).Louis de Broglie proposed that moving matter can exhibit wave properties according to the equation:
$$\lambda=\frac{h}{mv}$$
The equation suggests that the de Broglie wavelength of a particle is inversely proportional to its momentum.
de Broglie applied the concept of matter waves to the structure of the atom. He postulated that electrons orbit the nucleus as standing waves (stationary waves).
Standing waves are formed as a result of interference between two waves of equal frequency and amplitude travelling in opposite directions.
de Broglie explained that when electrons behave as standing waves, they no longer emit energy in the form of radiation (since this applies to particles). Therefore, de Broglie's matter wave theory provided an explanation for Bohr's first postulate – electrons orbit the nucleus in 'stationary states' and do not emit energy.
de Broglie also stated that the circumference of an electron orbit is quantised, meaning it is an integral multiple of the wavelength of the electron wave. This is represented by the following equation:
$$2\pi r_n=n\lambda$$
where n = integer denoting the energy level
Figure shows changes in electron wavelength with energy level. The number of antinode (crest) corresponds to the principal quantum number (n). n simply defines the energy level.
For example, the circumference of the n = 1 orbit equals to one wavelength of the electron wave. The circumference of the n = 2 orbit equals to two wavelengths of the electron wave.
The implication of this is that the circumference and radius of an electron orbit must equal to specific values as otherwise standing waves cannot form.
Combining the matter wave equation and quantisation of an electron orbit's circumference gives:
$$2\pi r_n=\frac{nh}{mv}$$
$$mvr=\frac{nh}{2\pi}$$
$$L_n=\frac{nh}{2\pi}$$
This equation shows that the angular momentum of an electron is an integral multiple of `h/{2\pi}`. Therefore, de Broglie's matter wave theory supports Bohr's third postulate – angular momentum of an electron is quantised.
Experimental Evidence for Matter Wave Duality
Davisson and Germer's Nickel Crystal Experiment
Davisson and Germer were studying the surface of nickel by firing electrons at the nickel crystal. Due to electrons' small size, they expected that this would allow them to obtain a detailed image of Nickel's surface and lattice structure.
The slits between nickel atoms acted as a diffraction grating which caused electrons to diffract and therefore exhibit wave nature.
After observing that the first maximum of diffraction occurred at an angle of 50º to the vertical, the scientists found the wavelength of diffracted electrons.
Using 2.15 Å as the lattice space of the nickel, Davisson and Germer found the experimental value for the wavelength of an electron to be 1.65 Å.
The potential difference (54V) used to accelerate the electrons and the mass of an electron were used to calculate the theoretical value of electron wavelength according to the equation:
$$\lambda=\frac{h}{mv}$$
The theoretical value of 1.67 Å was close to the experimental value, proving de Broglie's hypothesis to be fairly accurate at the time. As such, the nickel crystal experiment provided evidence for de Broglie's matter wave hypothesis.
Doubleslit Experiment Using Electrons
When electrons were fired at two narrow slits separated by a small distance, diffraction pattern was formed on a screen behind the slits.
Electrons were projected onto the screen in bands, with the bands separated by dark spaces. This pattern was similar to the diffraction pattern formed by waves as observed in Young's doubleslit experiment.
This experiment confirmed the wave nature of electrons.
Gold Foil Experiment Using Electrons
When electrons were fired at a thin gold foil, a concentric diffraction pattern was observed. Similar to the nickel crystal experiment, the small lattice spacing between gold atoms caused electrons to diffract and interfere.
A bright spot (electrons) was present centrally, surrounded by alternating concentric bright and dark rings. The bright and dark rings were caused by constructive and destructive interference between electrons respectively.
This experiment confirmed the wave nature of electrons.
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