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Chapter 2: Wave Diffraction & The Reciprocal Lattice. Chapter Topics Wave Diffraction by Crystals Bragg Law Scattered Wave Amplitude Reciprocal Lattice Brillouin Zones Fourier Analysis of the Basis. First, A Brief Optics Review.
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Chapter 2: Wave Diffraction & The Reciprocal Lattice
Chapter Topics • Wave Diffraction by Crystals • Bragg Law • Scattered Wave Amplitude • Reciprocal Lattice • Brillouin Zones • Fourier Analysis of the Basis
First,A Brief Optics Review • Brief Review of the Opticsneeded to understand diffraction by crystalline solids. • Discussion & Overview of: • Diffraction • + • X-Rays & Their Production
NOTE! In what follows, the term “light” will mean any electromagnetic wave whether or not the frequency is in the visible range.
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
Optics Review: Diffraction Light is a Wave, so it will diffract (bend) around a single slit or obstacle. • Diffractionis not limited tovisible light. • It happens with any wave, including other EM waves • (X Rays, ..) & including De Broglie waves associated • with quantum mechanical particles (electrons, neutrons)
Review of Diffraction • Diffractionis a wave phenomenon. It is the apparent bending & spreading of waves when they meet an obstruction. • Diffraction occurs withelectromagnetic waves, such as light & radio waves, but also in sound waves & water waves. Variable Slit Width (500-1500 nm) Constant Wavelength (600 nm) Distance d =Constant
Diffraction • The most conceptually simple example of Diffraction is the double-slit diffraction of visible light. See the figure. Variable Slit Width (500-1500 nm) Constant Wavelength (600 nm) Distance d =Constant
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).
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.
Destructive Interference This is the result when two out-of phase light waves cancel each other out, resulting in regions (lines) of darkness. Interference of Waves
Diffraction Pattern on a Screen Screen Photo of Diffraction Pattern
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: • Dsinθ = mλ (m = 1, 2,3,4..)
The minima of the single-slit diffraction pattern • occur when Dsinθ = mλ (m = 1, 2,3,4..)
To understand diffraction, we also must consider what happens when a wave interacts with a particle. The result is that Diffraction From a Single Particle a particle scatters the incident beam uniformly in all directions.
What happens if the beam is incident on solid material? If it is a crystalline material, the result is that the Diffraction From aSolid Material 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. Wilhelm Conrad Röntgen (1845-1923)
Röntgencalled 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 (1845-1923)
Review of X-Ray Propertıes • X-Rays are invisible, highly penetrating EM Radiationof 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, 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
X-Ray Energies So, we have: λ =Wavelength ν= Frequency c = Speed of Light λx-ray≈ 10-10 m≈1Ǻ E ~ 104 eV
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.
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!
Schematic “Cartoon” of Photon Emission • hν = ΔE • 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 pictureis not valid,butthe electron • wavefunction is spread all over the atom. So, • don’t take this figure literally!
X-Ray Tubes • X-Rayscan 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
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 You will never see something like this with VisibleLight!! • 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. X-Rays
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-Rayphoton. 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
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 Nucleus Fast Incident Electron X-ray
Generation of Bremsstrahlung Radiation Photoelectron Emission • 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 M L K Electron L K
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. • The electron transmission and the characteristic radiation emitted is given a further numerical subscript. Bohr`s model
Generation of Bremsstrahlung Radiation Energy levels (schematic) of the electrons Intensity ratios KKK000 M L K K K K K
Modern Sealed X-ray Tube • Tube made from ceramic • Beryllium window is visible. • Anode type and focus type are labeled.
X-Ray Absorption • A larger atom is more likely to absorb an X-Ray Photonin 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.
