Chapter 2: Wave Diffraction
& The Reciprocal Lattice
Chapter Topics
1. Wave Diffraction by Crystals
2. Bragg Law
3. Scattered Wave Amplitude
4. Reciprocal Lattice
5. Brillouin Zones
6. Fourier Analysis of the Basis
First, A Brief Optics Review
• A Brief Review of the Optics needed to
understand diffraction by crystalline solids.
• Discussion & Overview of:
Diffraction
X-Rays & Their Production
Optics Review: The Ray Model of Light
Specular Reflection ≡ Mirror-like Reflection
Assume that light can be treated as a ray, with
a single ray incident on the surface & a single
ray reflected. This leads to:
The Law of Reflection:
Incident Angle = Reflected Angle.
Optics Review: Diffraction
LIGHT IS A WAVE,
so it will diffract (bend) around a single slit or obstacle.
Diffraction
is not limited to visible light, but will happen with any
wave, including other electromagnetic waves (X-Rays, ..)
& including De Broglie waves associated with quantum
mechanical particles (electrons, neutrons, ..)
Review of Diffraction
• Diffraction is a wave
phenomenon. It is the apparent
bending & spreading of waves
when they meet an obstruction.
• Diffraction occurs with
electromagnetic waves, such
as light & radio waves, but also
in sound waves & water waves.
Variable Slit Width
(500-1500 nm)
• The most conceptually simple Constant Wavelength
example of Diffraction is the
(600 nm)
double-slit diffraction of
Distance d = Constant
visible light. See the figure.
• Light Diffraction is caused by light bending
around the edge of an object. The interference
pattern of bright & dark lines from the diffraction
experiment can only be explained by the additive nature
of waves. Wave peaks can add together to make a bright
line (or a peak in the intensity) or a dark line (a trough
in the intensity from 2 waves cancelling each other out).
Young’s Double Slit
Experiment
was the first to prove that
light has wavelike properties.
Constructive & Destructive
Interference of Waves
Constructive Interference
Destructive Interference
is the result of synchronized
light waves that add
together in phase to give
regions (lines) of
increased intensity.
results when two out-of
phase light waves cancel
each other out, resulting
in regions (lines) of
darkness.
Light Diffraction
Diffraction Pattern on a Screen
Screen
Photo of Diffraction
Pattern
Light Diffraction & Interference
Light Diffraction & Interference
Light Diffraction & Interference
Light Diffraction & Interference
Light Diffraction & Interference
Light Diffraction & Interference
Light Diffraction & Interference
The resulting pattern of light & dark stripes is called a
Diffraction Pattern.
This occurs because (by Huygens’ Principle) different points
along a slit create
Wavelets that interfere with each other
just like a double slit. Also, for certain angles θ the diffracted
rays from a slit of width D destructively interfere in pairs.
Angles for destructive interference are:
Dsinθ = mλ (m = 1, 2, 3, 4..)
The minima of the single-slit diffraction pattern
occur when when
Diffraction From a Particle & From a Solid
Diffraction from a Single Particle
• To understand diffraction we also must
consider what happens when a wave
interacts with a particle. The result is that
A particle scatters the incident
beam uniformly in all directions.
Diffraction from a Solid Material
• What happens if the beam is incident on
solid material? If it is a crystalline
material, the result is that The scattered
beams may add together in some
directions & reinforce each other
to give diffracted beams.
Review & Overview of
X-Rays & Their Properties
• X-Rays were discovered in 1895 by
German physicist Wilhelm
Conrad Röntgen. They were called
“X-Rays” because their nature was
unknown at the time. He was awarded
the Physics Nobel Prize in 1901.
Bertha Röntgen’s
Hand 8 Nov, 1895
The 1st X-Ray
photograph taken
was of Röntgen’s
wife’s left hand.
Wilhelm Conrad
Röntgen
(1845-1923)
Review of X-Ray Propertıes
• X-Rays are invisible, highly penetrating Electromagnetic
Radiation of much shorter wavelength (higher frequency) than
visible light. Wavelength (λ) & frequency (ν) ranges for X-Rays:
10-8 m ~ ≤ λ ~ ≤ 10-11 m
3 × 1016 Hz ~ ≤ ν ~ ≤ 3 × 1019 Hz
X-Ray Energies
• In Quantum Mechanics, Electromagnetic Radiation is
described as being composed of packets of energy, called
photons. The photon energy is related to its frequency
by the Planck formula: E  h
• We also know that, in vacuum, the frequency & the
wavelength are related as:   c Combining these gives:

E 
λx-ray ≈ 10-10 m ≈ 1 Ǻ
 E ~ 104 eV
hc

λ = Wavelength
ν = Frequency
c = Speed of Light
X-Ray Production
• Visible light photons, X-Ray photons, & essentially all
other photons
are produced by the movement
of electrons in atoms.
• We know from Quantum Mechanics that electrons
occupy energy levels, or orbitals, around an atom's nucleus.
• If an electron drops to a lower orbital (spontaneously or
due to some external perturbation) it releases some energy.
This released energy is in
the form of a photon
• The photon energy depends on how far in energy
the electron drops between orbitals.
Schematic Diagram of Photon Emission
Incoming particles excite an atom by promoting an electron to
a higher energy orbit. Later, the electron falls back to the lower
orbit, releasing a photon with energy equal to the energy
difference between the two states:
hν = ΔE
Remember that this figure is a
schematic “cartoon” only,
shown to crudely illustrate how atoms
emit light when one of the electrons
transitions from one level to another. It
gives the impression that the electrons in
an atom are in Bohr-like orbits around the
nucleus. From Quantum Mechanics,
we know that this picture is not valid,
but the electron wavefunction is
spread all over the atom. So, don’t take
this figure literally!
X-Ray Tubes
• X-Rays can be produced in a highly evacuated glass
bulb, called an X-Ray tube, that contains two electrodes:
an anode made of platinum, tungsten, or another heavy
metal of high melting point, & a cathode. When a high
voltage is applied between the electrodes, streams of
electrons (cathode rays) are accelerated from the cathode
to the anode & produce X-Rays as they strike the anode.
Evacuated
Glass Bulb
Anode
Cathode
Monochromatic & Broad Spectrum X-rays
• X-Rays can be created by bombarding a metal
target with high energy (> 104 eV) electrons.
• Some of these electrons excite other electrons from
core states in the metal, which then recombine,
producing highly monochromatic X-Rays. These
are referred to as characteristic X-Ray lines.
• Other electrons, which are decelerated by the
periodic potential inside the metal, produce a
broad spectrum of X-Ray frequencies.
• Depending on the diffraction experiment, either
or both of these X-Ray spectra can be used.
X-Ray Absorption
• The atoms that make up our body’s You will never see
something like this
tissue absorb visible light photons
very well. The energy level of the with Visible Light!!

photon fits with various energy
differences between electron states.
• Radio waves don't have enough
energy to move electrons between
orbitals in larger atoms, so they
pass through most materials.
X-Rays
• X-Ray photons also pass
through most things, but for
the opposite reason: They
have too much energy.
Generation of X-rays (K-Shell Knockout)
An electron in a higher orbital falls to the lower energy level,
releasing its extra energy in the form of a photon. It's a large
drop, so the photon has high energy;
it is an X-Ray photon.
Another schematic cartoon diagram,
not to be taken literally!
A “free” electron collides
with a tungsten atom,
knocking an electron out of a
lower orbital. A higher
orbital electron fills the
empty position, releasing
its excess energy as an
X-Ray Photon
X-Ray Absorption
• A larger atom is more likely to absorb an X-Ray
Photon in this way than a smaller one because larger
atoms have greater energy differences between orbitals.
 The energy level difference then more
closely matches the energy of an
X-Ray Photon.
• Smaller atoms, in which the electron orbitals are
separated by relatively low energy differences, are less
likely to absorb X-Ray Photons.
• The soft tissue in our bodies is composed of smaller
atoms, & so does not absorb X-Ray Photons very well.
The calcium atoms that make up our bones are large, so
they are better at absorbing X-Ray photons.