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Contents

 1.  Preface

 2.  I. The Nature of Light
     
     1. 1. Introduction
     
     2. 2. The Propagation of Light
     
     3. 3. The Law of Reflection
     
     4. 4. Refraction
     
     5. 5. Total Internal Reflection
     
     6. 6. Dispersion
     
     7. 7. Huygens’s Principle
     
     8. 8. Polarization
 3.  II. Geometric Optics and Image Formation
     
     1. 9. Introduction
     
     2. 10. Images Formed by Plane Mirrors
     
     3. 11. Spherical Mirrors
     
     4. 12. Images Formed by Refraction
     
     5. 13. Thin Lenses
     
     6. 14. The Eye
     
     7. 15. The Camera
     
     8. 16. The Simple Magnifier
     
     9. 17. Microscopes and Telescopes
 4.  III. Interference
     
     1. 18. Introduction
     
     2. 19. Young's Double-Slit Interference
     
     3. 20. Mathematics of Interference
     
     4. 21. Multiple-Slit Interference
     
     5. 22. Interference in Thin Films
     
     6. 23. The Michelson Interferometer
 5.  IV. Diffraction
     
     1. 24. Introduction
     
     2. 25. Single-Slit Diffraction
     
     3. 26. Intensity in Single-Slit Diffraction
     
     4. 27. Double-Slit Diffraction
     
     5. 28. Diffraction Gratings
     
     6. 29. Circular Apertures and Resolution
     
     7. 30. X-Ray Diffraction
     
     8. 31. Holography
 6.  V. Relativity
     
     1.  32. Introduction
     
     2.  33. Invariance of Physical Laws
     
     3.  34. Relativity of Simultaneity
     
     4.  35. Time Dilation
     
     5.  36. Length Contraction
     
     6.  37. The Lorentz Transformation
     
     7.  38. Relativistic Velocity Transformation
     
     8.  39. Doppler Effect for Light
     
     9.  40. Relativistic Momentum
     
     10. 41. Relativistic Energy
 7.  VI. Photons and Matter Waves
     
     1. 42. Introduction
     
     2. 43. Blackbody Radiation
     
     3. 44. Photoelectric Effect
     
     4. 45. The Compton Effect
     
     5. 46. Bohr’s Model of the Hydrogen Atom
     
     6. 47. De Broglie’s Matter Waves
     
     7. 48. Wave-Particle Duality
 8.  VII. Quantum Mechanics
     
     1. 49. Introduction
     
     2. 50. Wave Functions
     
     3. 51. The Heisenberg Uncertainty Principle
     
     4. 52. The Schrӧdinger Equation
     
     5. 53. The Quantum Particle in a Box
     
     6. 54. The Quantum Harmonic Oscillator
     
     7. 55. The Quantum Tunneling of Particles through Potential Barriers
 9.  VIII. Atomic Structure
     
     1. 56. Introduction
     
     2. 57. The Hydrogen Atom
     
     3. 58. Orbital Magnetic Dipole Moment of the Electron
     
     4. 59. Electron Spin
     
     5. 60. The Exclusion Principle and the Periodic Table
     
     6. 61. Atomic Spectra and X-rays
     
     7. 62. Lasers
 10. IX. Condensed Matter Physics
     
     1. 63. Introduction
     
     2. 64. Types of Molecular Bonds
     
     3. 65. Molecular Spectra
     
     4. 66. Bonding in Crystalline Solids
     
     5. 67. Free Electron Model of Metals
     
     6. 68. Band Theory of Solids
     
     7. 69. Semiconductors and Doping
     
     8. 70. Semiconductor Devices
     
     9. 71. Superconductivity
 11. X. Nuclear Physics
     
     1. 72. Introduction
     
     2. 73. Properties of Nuclei
     
     3. 74. Nuclear Binding Energy
     
     4. 75. Radioactive Decay
     
     5. 76. Nuclear Reactions
     
     6. 77. Fission
     
     7. 78. Nuclear Fusion
     
     8. 79. Medical Applications and Biological Effects of Nuclear Radiation
 12. XI. Particle Physics and Cosmology
     
     1. 80. Introduction
     
     2. 81. Introduction to Particle Physics
     
     3. 82. Particle Conservation Laws
     
     4. 83. Quarks
     
     5. 84. Particle Accelerators and Detectors
     
     6. 85. The Standard Model
     
     7. 86. The Big Bang
     
     8. 87. Evolution of the Early Universe

 13. Units

 14. Conversion Factors

 15. Fundamental Constants

 16. Astronomical Data

 17. Mathematical Formulas

 18. Chemistry

 19. The Greek Alphabet


UNIVERSITY PHYSICS VOLUME 3



Diffraction


30 X-RAY DIFFRACTION


LEARNING OBJECTIVES

By the end of this section, you will be able to:

 * Describe interference and diffraction effects exhibited by X-rays in
   interaction with atomic-scale structures

Since X-ray photons are very energetic, they have relatively short wavelengths,
on the order of m to m. Thus, typical X-ray photons act like rays when they
encounter macroscopic objects, like teeth, and produce sharp shadows. However,
since atoms are on the order of 0.1 nm in size, X-rays can be used to detect the
location, shape, and size of atoms and molecules. The process is called X-ray
diffraction, and it involves the interference of X-rays to produce patterns that
can be analyzed for information about the structures that scattered the X-rays.

Perhaps the most famous example of X-ray diffraction is the discovery of the
double-helical structure of DNA in 1953 by an international team of scientists
working at England’s Cavendish Laboratory—American James Watson, Englishman
Francis Crick, and New Zealand-born Maurice Wilkins. Using X-ray diffraction
data produced by Rosalind Franklin, they were the first to model the
double-helix structure of DNA that is so crucial to life. For this work, Watson,
Crick, and Wilkins were awarded the 1962 Nobel Prize in Physiology or Medicine.
(There is some debate and controversy over the issue that Rosalind Franklin was
not included in the prize, although she died in 1958, before the prize was
awarded.)

(Figure) shows a diffraction pattern produced by the scattering of X-rays from a
crystal. This process is known as X-ray crystallography because of the
information it can yield about crystal structure, and it was the type of data
Rosalind Franklin supplied to Watson and Crick for DNA. Not only do X-rays
confirm the size and shape of atoms, they give information about the atomic
arrangements in materials. For example, more recent research in high-temperature
superconductors involves complex materials whose lattice arrangements are
crucial to obtaining a superconducting material. These can be studied using
X-ray crystallography.

X-ray diffraction from the crystal of a protein (hen egg lysozyme) produced this
interference pattern. Analysis of the pattern yields information about the
structure of the protein. (credit: “Del45”/Wikimedia Commons)



Historically, the scattering of X-rays from crystals was used to prove that
X-rays are energetic electromagnetic (EM) waves. This was suspected from the
time of the discovery of X-rays in 1895, but it was not until 1912 that the
German Max von Laue (1879–1960) convinced two of his colleagues to scatter
X-rays from crystals. If a diffraction pattern is obtained, he reasoned, then
the X-rays must be waves, and their wavelength could be determined. (The spacing
of atoms in various crystals was reasonably well known at the time, based on
good values for Avogadro’s number.) The experiments were convincing, and the
1914 Nobel Prize in Physics was given to von Laue for his suggestion leading to
the proof that X-rays are EM waves. In 1915, the unique father-and-son team of
Sir William Henry Bragg and his son Sir William Lawrence Bragg were awarded a
joint Nobel Prize for inventing the X-ray spectrometer and the then-new science
of X-ray analysis.

In ways reminiscent of thin-film interference, we consider two plane waves at
X-ray wavelengths, each one reflecting off a different plane of atoms within a
crystal’s lattice, as shown in (Figure). From the geometry, the difference in
path lengths is . Constructive interference results when this distance is an
integer multiple of the wavelength. This condition is captured by the Bragg
equation,



where m is a positive integer and d is the spacing between the planes. Following
the Law of Reflection, both the incident and reflected waves are described by
the same angle, but unlike the general practice in geometric optics, is measured
with respect to the surface itself, rather than the normal.

X-ray diffraction with a crystal. Two incident waves reflect off two planes of a
crystal. The difference in path lengths is indicated by the dashed line.



X-Ray Diffraction with Salt Crystals Common table salt is composed mainly of
NaCl crystals. In a NaCl crystal, there is a family of planes 0.252 nm apart. If
the first-order maximum is observed at an incidence angle of , what is the
wavelength of the X-ray scattering from this crystal?

Strategy Use the Bragg equation, (Figure), , to solve for .

Solution For first-order, and the plane spacing d is known. Solving the Bragg
equation for wavelength yields



Significance The determined wavelength fits within the X-ray region of the
electromagnetic spectrum. Once again, the wave nature of light makes itself
prominent when the wavelength is comparable to the size of the physical
structures it interacts with.

Check Your Understanding For the experiment described in (Figure), what are the
two other angles where interference maxima may be observed? What limits the
number of maxima?

and ; Between , orders 1, 2, and 3, are all that exist.

Although (Figure) depicts a crystal as a two-dimensional array of scattering
centers for simplicity, real crystals are structures in three dimensions.
Scattering can occur simultaneously from different families of planes at
different orientations and spacing patterns known as called Bragg planes, as
shown in (Figure). The resulting interference pattern can be quite complex.

Because of the regularity that makes a crystal structure, one crystal can have
many families of planes within its geometry, each one giving rise to X-ray
diffraction.




SUMMARY

 * X-rays are relatively short-wavelength EM radiation and can exhibit wave
   characteristics such as interference when interacting with correspondingly
   small objects.


CONCEPTUAL QUESTIONS

Crystal lattices can be examined with X-rays but not UV. Why?

UV wavelengths are much larger than lattice spacings in crystals such that there
is no diffraction. The Bragg equation implies a value for sin⁡θ greater than
unity, which has no solution.


PROBLEMS

X-rays of wavelength 0.103 nm reflects off a crystal and a second-order maximum
is recorded at a Bragg angle of . What is the spacing between the scattering
planes in this crystal?

A first-order Bragg reflection maximum is observed when a monochromatic X-ray
falls on a crystal at a angle to a reflecting plane. What is the wavelength of
this X-ray?

0.120 nm

An X-ray scattering experiment is performed on a crystal whose atoms form planes
separated by 0.440 nm. Using an X-ray source of wavelength 0.548 nm, what is the
angle (with respect to the planes in question) at which the experimenter needs
to illuminate the crystal in order to observe a first-order maximum?

The structure of the NaCl crystal forms reflecting planes 0.541 nm apart. What
is the smallest angle, measured from these planes, at which X-ray diffraction
can be observed, if X-rays of wavelength 0.085 nm are used?



On a certain crystal, a first-order X-ray diffraction maximum is observed at an
angle of relative to its surface, using an X-ray source of unknown wavelength.
Additionally, when illuminated with a different, this time of known wavelength
0.137 nm, a second-order maximum is detected at Determine (a) the spacing
between the reflecting planes, and (b) the unknown wavelength.

Calcite crystals contain scattering planes separated by 0.30 nm. What is the
angular separation between first and second-order diffraction maxima when X-rays
of 0.130 nm wavelength are used?



The first-order Bragg angle for a certain crystal is . What is the second-order
angle?


GLOSSARY

Bragg planes families of planes within crystals that can give rise to X-ray
diffraction X-ray diffraction technique that provides the detailed information
about crystallographic structure of natural and manufactured materials
Previous: Circular Apertures and Resolution
Next: Holography
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University Physics Volume 3 by cnxuniphysics is licensed under a Creative
Commons Attribution 4.0 International License, except where otherwise noted.




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