X-Ray Scattering
• Max von Laue 1879-1960 (Nobel prize in Physics 1914)
suggests that if X-ray is a form of electromagnetic radiation,
interference effects could be observed
• X-ray wavelength: λ ~ 10-10-10-11 m
Distance between the “slits” to observe interferences
• Max von Laue:
Notices that interatomic distance in crystals ~ 10-10m
Idea: use the crystal as 3D-diffraction grating
X-ray Scattering
Von Laue: Diffraction transmission method
Bragg’s Law (I)
• William Lawrence Bragg 1890-1971 (Nobel prize in
physics 1915 – with his dad !) interpreted the X-ray
scattering as the reflection of the incident X-ray
beam from a unique set of planes of atoms within
the crystal.
Bragg’s Law (II)
• There are two conditions for constructive interference
of the scattered X-rays:
1) The angle of incidence must
equal the angle of reflection
of the outgoing wave.
2) The difference in path lengths
must be an integral number
of wavelengths.
n = 2d sin
(n = integer)
θ
λ
Bragg’s Law:
Bragg spectrometer
Applications
• Von Laue method (transmission of X-rays through a
crystal): Study of the structure of crystal by analyzing the
scattering of X-rays onto “unknown” material
• Bragg spectrometer (diffraction of X-rays): measurement
of X-ray wavelengths
Structure of the DNA molecule:
Watson and Crick (Nobel Prize – 1962)
Wave Properties of Matter
Louis De Broglie
1892-1987
Nobel Prize in Physics 1929
So…
• Electromagnetic radiations shown to behave like:
– Waves: Interferences, diffraction…
– Particles: Photoelectric & Compton effects…
• Particles shown to behave like:
– Particles (How original !): Rutherford scattering, …
– Waves ???: stationary orbits in the Bohr model ?
• By the 1920’s, nobody made the fateful leap to assume that
particles may also have a wave-like behavior
1924: De Broglie defends his thesis…
• Suggests that particles could have wavelike properties.
• Uses Einstein’s special theory of relativity
together with Planck’s quantum theory to
establish the wave properties of particles.
De Broglie Wavelength
Energy in the special theory of relativity:
E2 = p2c2 + m2c4
For the photon: m=0 E=pc
Photoelectric Effect:
E = hν
Combining both expression (with λ=c/ν): E=hν=pc λ
= h/p
De Broglie suggest this relation for photons extends to ALL
particles Matter Waves
Microscopic / macroscopic objects
De Broglie Wavelengths
• 50 eV electron (non-relativistic approach OK):
– λ = h/p = 0.17 nm ~ 10-10 m (interatomic distance in
crystals)
• Tennis Ball (57g) at 25 m/s (about 56 mph):
– λ = h/p = 4.7 x 10-34 m (about 1019 times small than
the size of a nucleon !)
A new look at Bohr’s model
of the atom
• Bohr Quantization:
ħ
– Bohr’s hypothesis of stationary orbits:
• Angular momentum: L = mvr = nh/2π = n
• With momentum p=mv and p = h/λ L = pr = hr/λ
If electron has a wave-like behavior:
Stationary orbit  Standing wave
Orbit: D = 2πr = nλ r = nλ/2π
L = hr/λ with r = nλ/2π L = nh/2π = n
With De Broglie’s wavelength,
Bohr’s quantization appears “naturally”
Electron Scattering
• 1925: C. Davisson and L.H.Germer observe
electron scattering and interference effects
Experimental evidence of the wave-like
properties of the electron
Transmission Electron Diffraction
120-keV electron on quasicrystal
Diffraction on a powder
A word on particle energies
• De Broglie Wavelength: λ = h/p
• In the non-relativistic approx.
p = (2mK) with K: kinetic energy
of the electron
The higher K, the higher p
• Higher electron energies (smaller
wavelength)
Sharper probe of the material.
Neutron diffraction
• The Spallation Neutron Source (SNS) facility