Matter Waves
A moving body behaves in certain ways as though it has a wave nature.
Louis de Broglie in 1924 proposed that moving objects have wave as well as particle characteristics.
De-Broglie wavelength
The wavelength of a moving particle is called its de Broglie wavelength.
h h h h h
𝛌 = p = mv =¿ = =
√ 2 m K . E √2 m eV √ 3 m k B T
De-Broglie wavelength independent of charge and nature of matter.
The greater the particle’s momentum, the shorter its wavelength.
De Broglie equation is a completely general one that applies to material particles as well as to
photons.
According to de Broglie, electrons, just like light, have a dual particle–wave nature.
The wave and particle aspects of moving bodies can never be observed at the same time.
De-Broglie frequency
Furthermore, in analogy with photons, de Broglie postulated that particles obey the Einstein
relation E=hf, where E is the total energy of the particle.
E
f=
h
De Broglie Relativistic wavelength
h
𝛌 = γmv
1
γ=
Here v2
√ 1−
C2
Importance of planks constant
Planck's constant provides the connecting link between the wave and particle natures of both
matter and radiation.
Why we don't observe the wave behavior of ordinary objects?
Answer
Because Planck's constant is so small h= 6.63 ×10-34 Js that the wavelengths of ordinary objects
are many orders of magnitude smaller than the size of a nucleus.
De Broglie's relationship provides us with a means to calculate the wavelength associated with
the wave behavior of matter. It does not indicate anything about the amplitude of the wave, nor does
it suggest the physical variable that is oscillating as the wave travels.
De Broglie was awarded the Nobel Prize in Physics in 1929 for his prediction of the wave
nature of electrons.
Matter waves Electromagnetic waves
Associated to moving particle. Associated with accelerated charged particle
Travel faster than electromagnetic waves. Travel slower than matter waves.
Move with speed greater than c. Move with speed c.
Speed depend upon speed of matter. Speed is constant.
De Broglie’s hypothesis suggested that electron waves were being diffracted by the target, much
as x-rays are diffracted by planes of atoms in a crystal.
Davison and Germer Experiment
An experiment that confirms the existence of de Broglie waves.
In 1927 Clinton Davisson and Lester Germer in the United States and G. P. Thomson in England
independently confirmed de Broglie’s hypothesis by demonstrating that electron beams are
diffracted when they are scattered by the regular atomic arrays of crystals.
Construction
1. Electron Gun 2. Baked nickel crystal as target 3. Rotating detector
The energy of the electrons in the primary beam, the incident angle at which they reach the
target, and the position of the detector all are variable.
Classical physics prediction
Classical physics predicts that the scattered electrons will emerge in all directions with only a
moderate dependence of their intensity on scattering angle and even less on the energy of the
primary electrons.
The method of plotting is such that the intensity at any angle is proportional to the distance of
the curve at that angle from the point of scattering. If the intensity were the same at all
scattering angles, the curves would be circles centered on the point of scattering.
Instead of a continuous variation of scattered electron intensity with angle, distinct maxima and
minima were observed whose positions depended upon the electron energy.
A beam of 54 eV electrons was directed perpendicularly at the nickel target and a sharp
maximum in the electron distribution occurred at an angle of 50° with the original beam. The
angles of incidence and scattering relative to the family of Bragg planes are both 65°. The
spacing of the planes for nickel, which can be measured by x-ray diffraction, is 0.091 nm.
Experimental value of de Broglie wavelength for electron
The Bragg equation for maxima in the diffraction pattern is
n𝛌 = d sinθ
𝛌 = d sinθ = (2) (0.091 nm) (sin65 ) = 0.165 nm
Theoretical value of de Broglie wavelength for electron
h
𝛌= = 0.166nm
√ 2 m eV
Davisson-Germer experiment is actually less straightforward
The energy of an electron increases when it enters a crystal by an amount equal to the
work function of the surface.
The electron speeds in the experiment were greater inside the crystal and the de
Broglie wavelengths there shorter than the values outside.
Interference between waves diffracted by different families of Bragg planes restricts the
occurrence of maxima to certain combinations of electron energy and angle of
incidence.
Electrons are not the only bodies whose wave behavior can be demonstrated. The
diffraction of neutrons and of whole atoms when scattered by suitable crystals has
been observed, and in fact neutron diffraction, like x-ray and electron diffraction, has
been used for investigating crystal structures.
