Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

1. Engineering 45 SolidCrystallography Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. As Discussed Earlier A Unit Cell is completely Described by Six Parameters Lattice Dimensions: a, b ,c Lattice (InterAxial) Angles: ,  , Navigation within a Crystal is Performed in Fractional Units of the Lattice Dimensions a, b, c Crystal Navigation

3. Cartesian CoOrds (x,y,z) within a Xtal are written in Standard Paren & Comma notation, but in terms of Lattice Fractions. Example Given TriClinic unit Cell at Right Point COORDINATES • Sketch the Location of the Point with Xtal CoOrds of:(1/2, 2/5, 3/4)

4. From The CoOrd Spec, Convert measurement to Lattice Constant Fractions x → 0.5a y → 0.4b z → 0.75c To Locate Point Mark-Off Dists on the Axes Point CoOrdinate Example • Located Point (1/2, 2/5, 3/4)

5. z z y y x x _ [001] [111] [010] [110] Crystallographic DIRECTIONS • Convention to specify crystallographic directions: 3 indices, [uvw] - reduced projections along x,y,z axes • Procedure to Determine Directions • vector through origin, or translated if parallelism is maintained • length of vector- PROJECTION on each axes is determined in terms of unit cell dimensions (a, b, c); negative index in opposite direction • reduce indices to smallest INTEGER values • enclose indices in brackets w/o commas [122]

6. Write the Xtal Direction, [uvw] for the vector Shown Below Example  Xtal Directions • Step-1: Translate Vector to The Origin in Two SubSteps

7. After −x Translation, Make −z Translation Example  Xtal Directions • Step-2: Project Correctly Positioned Vector onto Axes

8. Step-3: Convert Fractional Values to Integers using LCD for 1/2 & 1/3 → 1/6 x: (−a/2)•(6/a) = −3 y: a•(6/a) = 6 z: (−2a/3)•(6/a) = −4 Step-4: Reduce to Standard Notation: Example  Xtal Directions

9. Crystallographic PLANES • Planes within Crystals Are Designated by the MILLER Indices • The indices are simply the RECIPROCALS of the Axes Intersection Points of the Plane, with All numbers INTEGERS • e.g.: A Plane Intersects the Axes at (x,y,z) of (−4/5,3,1/2) Then The Miller indices:

10. Miller Indices  Step by Step • MILLER INDICES specify crystallographic planes: (hkl) • Procedure to Determine Indices • If plane passes through origin, move the origin (use parallel plane) • Write the INTERCEPT for each axis in terms of lattice parameters (relative to origin) • RECIPROCALS are taken: plane parallel to axis is zero (no intercept → 1/ = 0) • Reduce indices by common factor for smallest integers • Enclose indices in Parens w/o commas

11. Example  Miller Indices • Find The Miller Indices for the Cubic-Xtal Plane Shown Below

12. The Miller Indices Example • In Tabular Form

13. z y x More Miller Indices Examples • Consider the (001) Plane x y z Intercepts Reciprocals Reductions Enclosure ¥ ¥ 1 0 0 1 (none needed) (001) • Some Others

14. FAMILIES of DIRECTIONS • Crystallographically EQUIVALENT DIRECTIONS → < V-brackets > notation • e.g., in a cubic system, • Family of <111> directions: SAME Atomic ARRANGEMENTS along those directions

15. FAMILIES of PLANES • Crystallographically EQUIVALENT PLANES → {Curly Braces} notation • e.g., in a cubic system, • Family of {110} planes: SAME ATOMIC ARRANGEMENTS within all those planes

16. Consider the Hex Structure at Right with 3-Axis CoOrds Hexagonal Structures Plane-C • The Miller Indices • Plane-A → (100) • Plane-B → (010) • Plane-C → (110) Plane-B • BUT • Planes A, B, & C are Crystallographically IDENTICAL • The Hex Structure has 6-Fold Symmetry • Direction [100] is NOT normal to (100) Plane Plane-A

17. To Clear Up this Confusion add an Axis in the BASAL, or base, Plane 4-Axis, 4-Index System Plane-C • The Miller Indices now take the form of (hkil) • Plane-A → • Plane-B → • Plane-C → Plane-B Plane-A

18. Find Direction Notation for the a1 axis-directed unit vector 4-Axis Directions • Noting the Right-Angle Projections find

19. More 4-Axis Directions

20. Construct Miller-Bravais (Plane) Index-Sets by the Intercept Method 4-Axis Miller-Bravais Indices Plane Plane

21. Construct More Miller-Bravais Indices by the Intercept Method 4-Axis Miller-Bravais Indices Plane Plane

22. The 3axis Indices 3axis↔4axis Translation • Where n  LCD/GCF needed to produce integers-only • Example [100] • The 4axis Version • Conversion Eqns • Thus with n = 1

23. 4axis Indices CheckSum • Given 4axis indices • Directions → [uvtw] • Planes → (hkil) • Then due to Reln between a1, a2, a3

24. Linear & Areal Atom Densities • Linear Density, LD  Number of Atoms per Unit Length On a Straight LINE • Planar Density, PD  Number of Atoms per Unit Area on a Flat PLANE • PD is also called The Areal Density • In General, LD and PD are different for Different • Crystallographic Directions • Crystallographic Planes

25. Silicon Crystallography • Structure = DIAMOND; not ClosePacked

26. LD & PD for Silicon • Si

27. LD and PD For Silicon • For 100Silicon • LD on Unit Cell EDGE • For {111} Silicon • PD on (111) Plane • Use the (111) Unit Cell Plane

28. X-Ray Diffraction → Xtal Struct. • As Noted Earlier X-Ray Diffraction (XRD) is used to determine Lattice Constants • Concept of XRD → Constructive Wave Scattering • Consider a Scattering event on 2-Waves Amplitude100% Subtracted Amplitude100% Added • Constructive Scattering • Destructive Scattering

29. XRD Quantified • X-Rays Have WaveLengths, , That are Comparable to Atomic Dimensions • Thus an Atom’s Electrons or Ion-Core Can Scatter these X-rays per The Diagram Below Path-Length Difference

30. The Path Length Difference is Line Segment SQT XRD Constructive Interference 1 1’ 2’ 2 • Waves 1 & 2 will be IN-Phase if the Distance SQT is an INTEGRAL Number of X-ray WaveLengths • Quantitatively • Now by Constructive Criteria Requirement • Thus the Bragg Law

31. The InterPlanar Spacing, d, as a Function of Lattice Parameters (abc) & Miller Indices (hkl) d XRD Charateristics • By Geometry for OrthoRhombicXtals • For Cubic Xtals a = b = c, so

32. Pb XRD Implementation • X-Ray DiffractometerSchematic • T  X-ray Transmitter • S  Sample/Specimen • C  Collector/Detector • Typical SPECTRUM • Spectrum  Intensity/Amplitude vs. Indep-Index

33. z z z c c c y y y a a a b b b x x x X-Ray Diffraction Pattern (110) (211) Intensity (relative) (200) Diffraction angle 2θ Diffraction pattern for polycrystalline α-iron (BCC)

34. Given Niobium, Nb with Structure = BCC X-ray = 1.659 Å (211) Plane Diffraction Angle, 2∙θ = 75.99° n = 1 (primary diff) FIND ratom d211 BCC Niobium XRD Example  Nb • Find InterPlanar Spacing by Bragg’s Law

35. To Determine ratom need The Cubic Lattice Parameter, a Use the Plane-Spacing Equation Nb XRD cont • For the BCC Geometry by Pythagorus

36. Nb-Hf-W plate with an electron beam weld 1 mm PolyCrystals → Grains • Most engineering materials are POLYcrystals • Each "grain" is a single crystal. • If crystals are randomly oriented, then overall component properties are not directional. • Crystal sizes typically range from 1 nm to 20 mm • (i.e., from a few to millions of atomic layers).

37. Single vs PolyCrystals • Single Crystals -Properties vary with direction: anisotropic. -Example: the modulus of elasticity (E) in BCC iron: • Polycrystals 200 mm -Properties may/mayNot vary with direction. -If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa) -If grains are textured, anisotropic. 19

38. WhiteBoard Work • Planar-Projection (Similar to P3.48) • Given Three Plane-Views, Determine Xtal Structure Also: Find Aw

41. All Done for Today xTal PlanesinSimple CubicUnit Cell

42. Planar Projection 101 101

43. Planar Projection