1. Chapter 2: Wave Diffraction & The Reciprocal Lattice

2. Chapter Topics Wave Diffraction by Crystals Bragg Law
Scattered Wave Amplitude
Reciprocal Lattice
Brillouin Zones
Fourier Analysis of the Basis

3. First, A Brief Optics Review X-Rays & Their Production
Brief Review of the Optics needed to understand diffraction by crystalline solids.
Discussion & Overview of:
Diffraction +
X-Rays & Their Production

4. term “light” will mean any electromagnetic wave
NOTE!
In what follows, the
term “light” will mean any electromagnetic wave
whether or not the frequency is in the visible range.

5. i = r The Law of Reflection: Optics Review: Ray Model of Light
Specular Reflection ≡ Mirror-like Reflection
Assume that light can be treated as a ray, with a single ray incident on a perfectly smooth surface & a single ray is reflected. This leads to:
The Law of Reflection:
Incident Angle i = Reflected Angle r
i = r

6. Optics Review: Diffraction
Light is a Wave, so it will diffract (bend) around a single slit or obstacle.
Figure If light is a wave, a bright spot will appear at the center of the shadow of a solid disk illuminated by a point source of monochromatic light.
Diffraction is not limited to visible light.
It happens with any wave, including other EM waves
(X Rays, ..) & including De Broglie waves associated
with quantum mechanical particles (electrons, neutrons)

7. 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
( nm)
Constant Wavelength
(600 nm)
Distance d = Constant

8. double-slit diffraction
The most conceptually simple example of
Diffraction
is the
double-slit diffraction
of visible light.
See the figure.
Variable Slit Width
( nm)
Constant Wavelength
(600 nm)
Distance d = Constant

9. Young’s Double Slit Experiment:
The first proof that light
has wavelike properties.
Diffraction is caused by light waves. 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).

10. Constructive Interference
Interference of Waves
Constructive Interference
This is the result of synchronized light waves
that add together in phase to give regions (lines)
of increased intensity.

11. Destructive Interference
Interference of Waves
Destructive Interference
This is the result when two out-of phase light
waves cancel each other out, resulting in regions
(lines) of darkness.

12. Light Diffraction

13. Diffraction Pattern on a Screen
Photo of
Diffraction
Pattern

14. Light Diffraction & Interference

15. Light Diffraction & Interference

16. Light Diffraction & Interference

17. Light Diffraction & Interference

18. Light Diffraction & Interference

19. Light Diffraction & Interference

20. Light Diffraction & Interference

21. The resulting pattern of light & dark stripes is called a
Diffraction Pattern.
Figure Diffraction pattern of (a) a circular disk (a coin), (b) razor, (c) a single slit, each illuminated by a coherent point source of monochromatic light, such as a laser.

22. Wavelets that interfere with each other
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:
Dsinθ = mλ (m = 1, 2, 3, 4..)
Figure Analysis of diffraction pattern formed by light passing through a narrow slit of width D.

23. The minima of the single-slit diffraction pattern
occur when Dsinθ = mλ (m = 1, 2, 3, 4..)
Figure Intensity in the diffraction pattern of a single slit as a function of sin θ. Note that the central maximum is not only much higher than the maxima to each side, but it is also twice as wide (2λ/D wide) as any of the others (only λ/D wide each).

24. 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.

25. 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.

26. X-Rays & Their Properties
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.
Wilhelm Conrad
Röntgen
( )

27. Röntgen called them “X-Rays” because their nature was unknown.
The 1st X-Ray photo taken
was of Röntgen’s wife’s
left hand:
Bertha Röntgen’s
Hand
8 Nov, 1895
Wilhelm Conrad
Röntgen
( )

28. Review of X-Ray Propertıes
X-Rays are invisible, highly penetrating EM Radiation of much shorter wavelength (higher frequency) than visible light. Wavelength (λ) & frequency (ν) ranges for X-Rays:
10-8 m ~ ≤ λ ~ ≤ m
3 × 1016 Hz ~ ≤ ν ~ ≤ 3 × 1019 Hz

29. X-Ray Energies ν = Frequency c = Speed of Light
In Quantum Mechanics, EM Radiation is described as being composed of packets of energy, called photons. The photon energy is related to its frequency by the Planck formula:
We also know that, in vacuum, the frequency & the wavelength are related as:
Combining these gives:
λ = Wavelength
ν = Frequency
c = Speed of Light

30. X-Ray Energies λx-ray ≈ 10-10 m ≈ 1 Ǻ  E ~ 104 eV So, we have:
λ = Wavelength
ν = Frequency
c = Speed of Light
λx-ray ≈ m ≈ 1 Ǻ
 E ~ 104 eV

31. This released energy is in the
X-Ray Production
Visible light photons, X-Ray photons, & essentially all other photons are produced by transitions of electrons in atoms from one orbital to another.
We know from Quantum Mechanics that electrons occupy energy levels (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.

32. Schematic “Cartoon” 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!

33. Schematic “Cartoon” of Photon Emission
This figure is a “cartoon” only!
It is 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.
Of course, 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!

34. X-Ray Tubes Evacuated Glass Bulb Cathode Anode
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
Cathode
Anode

35. 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.

36. X-Ray Absorption
The atoms that make up our body’s tissue absorb visible light photons very well. The energy level of the 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-Ray photons also pass through most things, but for the opposite reason: They have too much energy.
You will never see
something like this
with Visible Light!! 
X-Rays

37. 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 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

38. Similar Viewpoint: Generation of Bremsstrahlung Radiation
“Braking” Radiation: Electron deceleration releases radiation across a spectrum of wavelengths.
Atom of the Anode Material
Electron
(slowed &
changed
direction)
Electron Orbits
Bremsstrahlung radiation means “braking” radiation.
Electron deceleration releases radiation across a spectrum of wavelengths.
The braking radiation represents a continuum (white radiation).
Nucleus
Fast Incident
Electron
X-ray

39. Generation of Bremsstrahlung Radiation
An incoming electron knocks out an electron from the inner atomic shell. The designation K,L,M corresponds to shells with different principal quantum numbers. K L K K L M
Emission
Photoelectron
Electron
Here the electrons suddenly decelerate upon colliding with the metal target. If enough energy is contained within the electron, it is able to knock out an electron from the metal atom’s inner shell.
This is an unstable state, and this vacancy is quickly filled by an electron from a higher shell. This process releases a “quantum” or photon of radiation that has a wavelength (or energy) characteristic of the energy difference between shells.
The designations K, L, and M correspond to the quantum number q = 1, 2, 3
α and β indicates the shell that the “filling” electron is from relative the vacancy shell.
Ka indicates the photon that is released from an electron transition from the L shell to the K shell

40. Generation of Bremsstrahlung Radiation
Not every electron in each of these shells has the same energy. The shells must be further divided. K-shell vacancy can be filled by electrons from 2 orbitals in L shell, for example.
Bohr`s model
The electron transmission and the characteristic radiation emitted is given a further numerical subscript.
In reality, the picture a little more complicated. All electrons in each of these shells do not have exactly the same energy. The shells themselves must be further divided to indicate this.
So, in this example a K-shell vacancy can be filled by 2 different L-shell electrons with slightly different energies. These shells are designated 1 and 2. So when we talk about Kα radiation here we are actually talking about both the Kα1 and Kα2 wavelengths (and sometimes more).

41. Generation of Bremsstrahlung Radiation
Energy levels (schematic) of the electrons
Intensity ratios
KKK000 M
The L level is actually made up of 3 different energy levels . L
K
K
K
K K

42. Emission Spectrum of an X-Ray Tube
Braking = continuous spectra
Characteristic = line spectra
At 8.5 kV, no characteristic line is produced.
As accelerating voltage is increased, characteristic lines appear and grow in intensity, but so does the braking radiation. Although the characteristic radiation is much stronger, all of these wavelengths can participate in XRD and at the very least cause undesirable background.

43. Characteristic Radiation for Common X-ray Tube Anodes
Ka1 (100%)
Ka2 (50%)
Kb (20%) Cu Å Å Å Mo Å Å Å
Note how anodes made of a different material have Ka lines with different wavelengths.
To get a different wavelength, simply change the composition of the anode!!

44. Modern Sealed X-ray Tube
Tube made from ceramic
Beryllium window is visible.
Anode type and focus type are labeled.
Tube now made of ceramic. Beryllium windows

45. 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.