Looking for something specific in this course? The compiles links to most course resources in a single page.
| Modules | Electronic Materials, Crystalline Materials |
| Concepts | p-n junction, introduction to the solid state, the 7 crystal systems, the 14 Bravais lattices, properties of cubic crystals: simple cubic, face-centered cubic, body-centered cubic, and diamond cubic |
| Keywords |
Electronic Materialssubvalent, aliovalent, supervalent, conduction band, valence band, semiconductor, silicon, dopant, thermal excitation, n-type, p-type, acceptor level, charge carrier, p-n junction Crystalline Materialscrystal, glass, amorphous solid, ordered solid, long-range order, Bravais lattice, crystal system, point group, translation, rotation, symmetry plane, degree of symmetry, crystal basis, unit cell, face-centered cubic, simple cubic, body-centered cubic, hexagonal close-packed, rock salt structure, diamond cubic, birefringence, crystallography, nearest neighbor, Auguste Bravais, René Haüy, Robert Hooke, Christiaan Huygens, Nicolaus Steno |
| Chemical Substances |
Electronic Materialssilicon (Si), boron (B), diamond (C) Crystalline Materialsglass, obsidian, quartz, calcite, tin (Sn), basalt, beryl, fluorite, gold (Au), aluminum (Al), copper (Cu), platinum (Pt), methane ice (CH4), rock salt (NaCl) |
| Applications |
Electronic Materialstransistors, diodes, current rectification Crystalline Materialscannonball stacking, tiling of 2D surfaces, fiber optics coupling, optical beam-splitter, colored gold |
Before starting this session, you should be familiar with:
This session introduces the cubic unit cells, a key framework for discussing atomic-level processes in solids throughout this module and in later topics, such as Diffusion (Session 24) and Solid Solutions (Session 33 onwards). The next module on Amorphous Solids (Session 21 onwards) discusses non-crystalline materials in more detail, contrasting their structure and properties with the ordered solids studied here.
After completing this session, you should be able to:
Archived Lecture Notes #4 (PDF), Sections 1-3
| Book Chapters | Topics |
|---|---|
| [Saylor] 12.1, “Crystalline and Amorphous Solids.” | Crystal lattice parameters; properties of crystalline and amorphous solids |
| [Saylor] 12.2, “The Arrangement of Atoms in Crystalline Solids.” | The unit cell; packing of spheres |
| [JS] 3.1, “Seven Systems and Fourteen Lattices.” | The unit cell and its parameters; crystal systems and crystal (Bravais) lattices |
| [JS] 3.2, “Metal Structures.” | Body-centered cubic, face-centered cubic/cubic close-packed, and hexagonal close-packed structures; atomic packing factor; plane stacking |
Continuing last lecture’s explanation of extrinsic semiconductors, the Electronic Materials module ends at 13:00 with an exploration of p-type doping and an overview of the p-n junction. Prof. Sadoway moves on to introduce a classification for materials based on the degree of atomic-level order, contrasting ordered solids (crystals, e.g. quartz, calcite) with amorphous solids (glasses, e.g. obsidian). The 7 crystal systems and 14 Bravais lattices are introduced:
Crystal structures are described using a basis, which may be an atom, a group of ions (e.g. rock salt (NaCl)), or a molecule (e.g. methane (CH4(s)), proteins), repeated at the points of a Bravais lattice. Since they apply to many common metals and minerals, this course focuses on the cubic crystal systems: simple, body-centered, and face-centered.
| [Saylor] Sections | Conceptual | Numerical |
|---|---|---|
| [Saylor] 12.2, “The Arrangement of Atoms in Crystalline Solids.” | 1, 8, 9 | 3, 5, 9, 11 |
| [Saylor] 12.3, “Structures of Simple Binary Compounds.” | 4 | none |
Hooke, Robert. Micrographia; or, Some Physiological Descriptions of Minute Bodies Made by Magnifying Glasses, with Observations and Inquiries. London, England: J. Martyn and J. Allestry, 1665. [View on Project Gutenberg]
Chapman, Allan. England’s Leonardo: Robert Hooke and the Seventeenth-Century Scientific Revolution. Philadelphia, PA: Institute of Physics Publishing, 2005. ISBN: 9780750309875.
Steno, Nicolaus. The Prodromus of Nicolaus Steno’s Dissertation Concerning a Solid Body Enclosed by Process of Nature within a Solid. Translated by John Garrett Winter. New York, NY: Macmillan, 1916.
Cutler, Alan. The Seashell on the Mountaintop: A Story of Science, Sainthood, and the Humble Genius Who Discovered a New History of the Earth. New York, NY: Plume, 2004. ISBN: 9780452285460.
Dijksterhuis, Fokko Jan. Lenses and Waves: Christiaan Huygens and the Mathematical Science of Optics in the Seventeenth Century. Boston, MA: Kluwer, 2004. ISBN: 9789048167067.
Niels Steensen (Nicolaus Steno)
Vallier, Dora. Braque: The Complete Graphics: Catalogue Raisonne. New York, NY: Alpine Fine Arts Collection, 1988. ISBN: 9780881680065.
Talking Heads. “Burning Down the House.” Speaking in Tongues. Sire Records, 1983.
| Content | Provider | Level | Notes |
|---|---|---|---|
| Crystallography | DoITPoMS | Undergraduate | |
| Crystal Structure | Connexions | Undergraduate | |
| 3.60 Symmetry, Structure, and Tensor Properties of Materials | MIT OpenCourseWare | Graduate | A mathematical approach to crystal symmetry with connections to bulk material properties such as stress, strain, thermal conductivity, and piezoelectricity. |
| Modules | Crystalline Materials |
| Concepts | crystal coordinate systems, Miller indices, introduction to x-rays, generation of x-rays |
| Keywords | Bravais lattice, crystal system, unit cell, face-centered cubic, simple cubic, body-centered cubic, Miller indices, crystallography, crystallographic notation, lattice constant, close-packing, packing density, lattice point, interplanar spacing, gas discharge tube, x-ray tube, target anode, discovery of x-rays, scintillation screen, characteristic emission lines, Kα, Kβ, Lα, Lβ, William H. Miller, Wilhelm Röntgen |
| Chemical Substances | barium platinum cyanide (BaPt(CN)4), copper (Cu), brass (Cu-Zn), zinc (Zn), wood, steel |
| Applications | x-ray spectroscopy, medical/dental x-rays, quality assurance of welds, airport baggage scans |
Before starting this session, you should be familiar with:
Session 17 and Session 18 explain more about the use of x-rays for investigating the structure of crystals and molecules.
After completing this session, you should be able to:
Archived Lecture Notes #4 (PDF), Section 4
Archived Lecture Notes #5 (PDF), Section 1
| Book Chapters | Topics |
|---|---|
| [Saylor] 12.2, “The Arrangement of Atoms in Crystalline Solids.” | The unit cell; packing of spheres |
| [JS] 3.2, “Metal Structures.” | Body-centered cubic, face-centered cubic/cubic close-packed, and hexagonal close-packed structures; atomic packing factor; plane stacking |
| [JS] 3.6, “Lattice Positions, Directions, and Planes.” | Lattice points and translations; lattice directions and planes; Miller indices; families of directions and planes; planar and linear atomic density |
Miller indices are a standard mathematical notation describing planes in crystals, derived from where the plane intercepts each coordinate axis. In a specific material with a known lattice constant and crystal structure, this allows the calculation of angles and distances between planes and directions of interest. For convenience, crystallographers sometimes refer to families of planes or directions, which all have the same indices but use different origins.
X-rays are well-suited for measuring atomic-level structure because their wavelengths are of the same order as typical lattice constants. Such short wavelengths require high energies, typically created by sending high-voltage electrons into an anode, where they ionize electrons from the lowest energy levels. Electrons from higher energy levels cascade down to replace them, emitting photons with a highly characteristic set of wavelengths, corresponding to the specific energy levels of the anode material. The discovery of x-rays by Wilhelm Röntgen in 1895 heralded the development of many important modern technologies, including medical radiography, security screening, and industrial inspection of metal parts.
[JS] Chapter 3, Sample Problems 8-10, 13-19; Practice Problems 11-14, 16-21
Thomas, A. M. K. The Invisible Light: 100 Years of Medical Radiology. Cambridge, MA: Wiley-Blackwell, 1995. ISBN: 9780865426276.
Wilhelm Röntgen – 1901 Nobel Prize in Physics
| Content | Provider | Level | Notes |
|---|---|---|---|
| Lattice Planes and Miller Indices | DoITPoMS | Undergraduate | |
| Crystal Structure | Connexions | Undergraduate |
| Modules | Crystalline Materials |
| Concepts | characterization of atomic structure, Moseley’s law, generation of x-rays, x-ray diffraction |
| Keywords | characteristic emission lines, Kα, Kβ, Lα, Lβ, atomic spectra, quantized spectrum, continuous spectrum, proton number, atomic number, atomic mass, periodicity, periodic table, x-ray tube, lanthanide series, Moseley’s law, screening factor, Bremsstrahlung, braking radiation, ballistic electrons, target anode, cold cathode, hot cathode, scattering angle, Duane-Hunt law, Henry Moseley, William Coolidge |
| Chemical Substances | calcium (Ca), titanium (Ti), vanadium (V), chromium (Cr), manganese (Mn), iron (Fe), copper (Cu), cobalt (Co), nickel (Ni), zinc (Zn), brass, argon (Ar), potassium (K), tellurium (Te), iodine (I), uranium (U), neptunium (Np), lanthanide series (La-Lu), molybdenum (Mo), lead (Pb), beryllium (Be) |
| Applications | organization of the modern periodic table, electron-beam welding, lead shielding, analysis of paintings |
Before starting this session, you should be familiar with topics from the first module, Structure of the Atom (Session 1 through Session 7), especially:
After completing this session, you should be able to:
Archived Lecture Notes #5 (PDF), Section 2
| Book Chapters | Topics |
|---|---|
| [Saylor] 7.1, “The Role of Atomic Number in the Periodic Table.” (only read until Example 1) | Moseley’s discovery of atomic number and its role in the periodic table |
| [C&S] 1, “Properties of X-Rays.” | Electromagnetic radiation; the continuous and characteristic spectrums; production and detection of x-rays; safety precautions |
Shortly after the discovery of x-rays, Henry Moseley investigated the characteristic emission lines of various elements, showing that periodicity follows the proton number Z, not the atomic mass A, producing a modified Rydberg equation with a screening factor for non-hydrogenic atoms. When x-rays irradiate a sample, they scatter off the atoms as well as excite electrons into higher orbitals; this Bremsstrahlung (braking) radiation produces a continuous spectrum underneath the sharp characteristic lines. Modern improvements on the x-ray tube, introduced by William Coolidge, include using a hot cathode and vacuum tube to increase efficiency, water cooling to prevent overheating the anode, and lead (Pb) shielding to absorb x-rays except at the beryllium (Be) windows, which transmit them outward.
| [Saylor] Sections | Conceptual | Numerical |
|---|---|---|
| [Saylor] 12.3, “Structures of Simple Binary Compounds.” | 5, 6 | none |
Moseley, H. G. J. “The High-Frequency Spectra of the Elements.” Philosophical Magazine Series 6 26 (December 1913): 1024-1034.
Heilbron, John Lewis. H. G. J. Moseley: The Life and Letters of an English Physicist, 1887-1915. Berkeley, CA: University of California Press, 1974. ISBN: 9780520023758.
Bacou, Roseline. Millet: One Hundred Drawings. New York, NY: Harper & Row, 1975. ISBN: 9780064303408.
Lubar, Robert S. Dali: The Salvador Dali Museum Collection. Boston, MA: Bullfinch Press, 2000. ISBN: 9780821227152.
| Content | Provider | Level | Notes |
|---|---|---|---|
| X-Ray Diffraction Techniques | DoITPoMS | Undergraduate | See “Experimental Matters.” |
| Modules | Crystalline Materials |
| Concepts | Braggs’ law, x-ray diffraction of crystals: diffractometry, Laue, and Debye-Scherrer, crystal symmetry and selection rules |
| Keywords | x-ray diffraction, Braggs’ law, angle of incidence, angle of reflection, constructive interference, destructive interference, crest, trough, amplitude, wavelength, phase, monochromatic, coherent light, incoherent light, order of reflection, index of refraction, collimator, diffraction peak, rotational symmetry, Laue diffraction, quasicrystal, translational symmetry, long-range order, x-ray crystallography, Penrose tiles, William Henry Bragg, William Lawrence Bragg, Max von Laue, Roger Penrose, Peter Debye, Peter Scherrer, Dan Shechtman |
| Chemical Substances | copper (Cu), nickel (Ni), silicon (Si), aluminum-manganese alloy (Al-Mn) |
| Applications | growth of single-crystal Si, identification of planes and symmetry in crystals, Penrose tiles |
Before starting this session, you should be familiar with the prior topics in this module (Session 15 through Session 17), especially:
X-ray diffraction is a popular technique to discover the structures of organic molecules such as proteins (Session 31) and, most famously, DNA (Session 32), as well as inorganic crystals. It is also used to determine the degree of long-range order and symmetry present in a crystal, or lacking in a glass, which is the topic of the next module (Session 21: Introduction to Glasses).
After completing this session, you should be able to:
Archived Lecture Notes #5 (PDF), Sections 4-6
| Book Chapters | Topics |
|---|---|
| [Saylor] 12.3, “Structures of Simple Binary Compounds.” | Common structures of binary compounds, x-ray diffraction |
| [JS] 3.7, “X-Ray Diffraction.” | Diffraction, Braggs’ law and reflection rules; single-crystal, polycrystal, and powder diffraction techniques |
X-rays reflect off each atomic plane in a crystal, producing patterns of destructive and constructive interference according to Braggs’ law. One popular method of determining crystal structure, x-ray diffractometry, involves monochromatic x-rays bouncing off a rotating target; the resulting peaks indicate the identity and spacing of the close-packed planes, which are different for FCC and BCC. Another method, Laue diffraction, uses x-rays of multiple wavelengths and a fixed target, producing a pattern reflective of the symmetry present in the crystal structure. The cubic lattices include planes that have 1, 2, 3, or 4-fold rotational symmetry, but quasicrystals displaying 5-fold structures have been observed in experiments on Al-Mn alloys and generated mathematically as Penrose tiles.
[JS] Chapter 3, Sample Problems 20, 21
| [Saylor] Sections | Conceptual | Numerical |
|---|---|---|
| [Saylor] 12.3, “Structures of Simple Binary Compounds.” | 8, 9 | 11, 12 |
William Henry Bragg, William Lawrence Bragg – 1915 Nobel Prize in Physics
Max von Laue – 1914 Nobel Prize in Physics
Peter Debye – 1936 Nobel Prize in Chemistry
Mozart, Wolfgang. “Rondo Alla Turca.” Piano Sonata no. 11 in A major, K. 331.
| Content | Provider | Level | Notes |
|---|---|---|---|
| X-Ray Diffraction Techniques | DoITPoMS | Undergraduate |
| Modules | Crystalline Materials |
| Concepts | defects in crystals: point defects, line defects |
| Keywords | point defect, line defect, substitutional impurity, interstitial impurity, vacancy, self interstitial, ionic defect, Hope Diamond, Schottky defect, Frenkel defect, F-center, charge neutrality, edge dislocation, screw dislocation, dislocation motion, bubble raft model, chemical imperfection, structural imperfection, formation energy, entropy factor, stoichiometric unit, effective charge, Kröger-Vink notation |
| Chemical Substances | aluminum (Al), steel, diamond, doped silicon, LaNi5, copper (Cu), rock salt (NaCl), zirconia (ZrO2) |
| Applications | aluminum alloys for soda cans, n-and p-type semiconductors, steel, hydrogen embrittlement of steel, Hope diamond, colored gold, hydrogen storage |
Before starting this session, you should be familiar with:
Compare the expression for the energy required to produce a vacancy, derived in this lecture, with the expression for the rate of a chemical reaction (the Arrhenius equation), presented in Session 22: Introduction to Kinetics.
After completing this session, you should be able to:
Archived Lecture Notes #6 (PDF), Sections 1-2
| Book Chapters | Topics |
|---|---|
| [Saylor] 12.4, “Defects in Crystals.” | Defects in metals, memory metal, defects in ionic and molecular crystals, nonstoichiometric compounds |
| [JS] 4.1, “The Solid Solution – Chemical Imperfection.” | Random and ordered solid solutions, Hume-Rothery rules, interstitial and substitutional solutes, charge neutrality |
| [JS] 4.2, “Point Defects – Zero-Dimensional Imperfections.” | Vacancies and interstitial defects, Schottky and Frenkel defects |
| [JS] 4.3, “Linear Defects, or Dislocations – One-Dimensional Imperfections.” | Burgers vector; edge, screw, mixed, and partial dislocations |
| [JS] 5.1, “Thermally-Activated Processes.” | Arrhenius equation, activation energy, Maxwell-Boltzmann distribution, process mechanisms and rate-limiting steps |
| [JS] 5.2, “Thermal Production of Point Defects.” | Activation energy of vacancies vs. interstitials, Arrhenius plot, thermal expansion |
Prof. Michael Demkowicz (homepage) lectures today, introducing the next topic: imperfections in crystal lattices. In the real world, materials rarely consist of single, perfect crystals; defects in crystals occur naturally, or are introduced during processing. While unwanted defects can weaken or contaminate materials (e.g. Li+ in saline solution (NaCl(aq)), others can create enhanced properties (e.g. alloys, dopants). Creating an empty crystal lattice site (vacancy) requires overcoming bonds with nearest-neighbor atoms, typically with thermal energy. Vacancies in a regular lattice of A atoms may be filled by an atom of B (substitutional, e.g. P in Si, B in C), while interstitial sites can host atoms of A (self interstitial) or B (interstitial impurity, e.g. C in Fe, H in LaNi5, H in Fe). In ionic crystals, overall charge neutrality must be preserved, so a whole stoichiometric unit may be removed to create two or more vacancies (Schottky); one ion may move to an interstitial site (Frenkel); or one or more electrons may fill an anionic vacancy (F-center). Line defects occur when a lattice mismatch runs through the crystal.
[JS] Chapter 5, Sample Problem 2, Practice Problem 2
| [Saylor] Sections | Conceptual | Numerical |
|---|---|---|
| [Saylor] 12.4, “Defects in Crystals.” | 6, 7, 8, 9, 10 | 1, 2, 4, 5 |
Hull, Derek, and David J. Bacon. Introduction to Dislocations. Boston, MA: Butterworth-Heinemann, 2001. ISBN: 9780750646819.
| Content | Provider | Level | Notes |
|---|---|---|---|
| Introduction to Dislocations | DoITPoMS | Undergraduate | |
| 3.14/3.40J/22.71J Physical Metallurgy | MIT OpenCourseWare | Undergraduate (elective) / Graduate | |
| Crystal Structure | Connexions | Undergraduate |
| Module | Crystalline Materials |
| Concepts | defects in crystals: line defects, interfacial defects, grain boundaries, and voids, motion of dislocations, effect of impurities on solid-state material properties |
| Keywords | yield stress, strain, shear stress, line defect, surface energy, edge dislocation, screw dislocation, dislocation motion, catalysis, corrosion, grain boundary, annealing, vacancy, single-crystal, polycrystalline, precipitation strengthening, ductility, slip, voids, solution hardening, elastic deformation, plastic deformation, chemical metallurgy, physical metallurgy, Hooke’s law, fracture, close-packed, dislocation glide, toughness, hardness, brittle |
| Chemical Substances | steel, aluminum-copper alloy (Al-Cu), silica (SiO2), calcia (CaO), alumina (Al2O3) |
| Applications | aluminum can, steel production, aluminum-copper for airplanes, rivets on the Titanic |
Before starting this session, you should be familiar with:
The amount and composition of precipitates in alloys can be predicted using binary phase diagrams, as described in Session 34 and Session 35. Point defects and grain boundaries give atoms space to move through the lattice, a key factor in Session 24: Diffusion.
After completing this session, you should be able to:
Archived Lecture Notes #6 (PDF), Sections 3-4
| Book Chapters | Topics |
|---|---|
| [Saylor] 12.4, “Defects in Crystals.” | Defects in metals, memory metal, defects in ionic and molecular crystals, non-stoichiometric compounds |
| [JS] 4.3, “Linear Defects, or Dislocations – One-Dimensional Imperfections.” | Burgers vector; edge, screw, mixed, and partial dislocations |
| [JS] 4.4, “Planar Defects – Two-Dimensional Imperfections.” | Twin boundaries, crystal surfaces, and grain boundaries; tilt boundaries, coincident site lattices, and dislocations; grain-size number |
Experimental values for the yield strengths of metals are roughly 1/10th those given by theoretical calculations based on breaking entire planes of atomic bonds. The discrepancy is explained by dislocations, introduced at the end of the last session, which allows slipping planes to break single bonds in sequence, lowering the yield stress. Two-dimensional defects can occur at the surface of crystals or at internal interfaces between zones with different lattice alignments, called grain boundaries. Macroscopic clusters of vacancies (voids) weaken metals, while clusters of impurities (precipitates) may weaken or strengthen them. The failure of rivets on the hull of the Titanic is attributed to brittle pockets of slag mixed into the steel, based on examination of the microstructure.
| [saylor] Sections | Conceptual | Numerical |
|---|---|---|
| [Saylor] 12.4, “Defects in Crystals.” | 2, 3, 4, 5 | none |
Hull, Derek, and David J. Bacon. Introduction to Dislocations. Boston, MA: Butterworth-Heinemann, 2001. ISBN: 9780750646819.
Glass, Philip. Koyaanisqatsi. Orange Mountain Music, 2009.
Horner, James, and Will Jennings. “My Heart Will Go On (Theme from Titanic).” Titanic: Music from the Motion Picture. Performed by Celine Dion. Columbia Records, 1997.
| Content | Provider | Level | Notes |
|---|---|---|---|
| Introduction to Dislocations | DoITPoMS | Undergraduate | |
| 3.14/3.40J/22.71J Physical Metallurgy | MIT OpenCourseWare | Undergraduate (elective) / Graduate | |
| Crystal Structure | Connexions | Undergraduate |
This self-assessment page completes the Crystalline Materials module, and covers material from the following sessions.
On this page are a simple weekly quiz and solutions; relevant exam problems and solutions from the 2009 class; help session videos that review selected solutions to the exam problems; and supplemental exam problems and solutions for further study.
This short quiz is given approximately once for every three lecture sessions. You should work through the quiz problems in preparation for the exam problems.
These exam problems are intended for you to demonstrate your personal mastery of the material, and should be done alone, closed-book, with just a calculator, the two permitted reference tables (periodic table, physical constants), and one 8 1/2" x 11" aid sheet of your own creation.
After you’ve taken the exam, watch the help session videos below for insights into how to approach some of the exam problems.
In these videos, 3.091 teaching assistants review some of the exam problems, demonstrating their approach to solutions, and noting some common mistakes made by students.
These additional exam problems from prior years’ classes are offered for further study.
