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Explore crystal navigation methods, crystallographic directions, and planes in the context of unit cells and lattice constants. Discover how to determine directions and specify planes within crystals using Miller indices. Learn about crystal families and different atomic arrangements.
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Engineering 45 Crystallography Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu
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 Units of the Lattice Dimensions a, b, c Crystal Navigation
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)
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)
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
Write the Xtal Direction, [uvw] for the vector Shown Below Example Xtal Directions • Step-1: Translate Vector to The Origin in Two SubSteps
After −x Translation, Make −z Translation Example Xtal Directions • Step-2: Project Correctly Positioned Vector onto Axes
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
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:
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
Example Miller Indices • Find The Miller Indices for the Cubic-Xtal Plane Shown Below
The Miller Indices Example • In Tabular Form
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
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
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
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
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
Find Direction Notation for the a1 axis-directed unit vector 4-Axis Directions • Noting the Right-Angle Projections find
Construct Miller-Bravais (Plane) Index-Sets by the Intercept Method 4-Axis Miller-Bravais Indices Plane Plane
Construct More Miller-Bravais Indices by the Intercept Method 4-Axis Miller-Bravais Indices Plane Plane
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 = 3
4axis Indices CheckSum • Given 4axis indices • Directions → [uvtw] • Planes → (hkil) • Then due to Reln between a1, a2, a3
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 PLANE • PD is also called The Areal Density • In General, LD and PD are different for Different • Crystallographic Directions • Crystallographic Planes
Silicon Crystallography • Structure = DIAMOND; not ClosePacked
LD & PD for Silicon • Si
LD and PD For Silicon • For 100Silicon • LD on Unit Cell EDGE • For {111} Silicon • PD on (111) Plane • Use the (111) Unit Cell Plane
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
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
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
The InterPlanar Spacing, d, as a Function of Lattice Parameters (abc) & Miller Indices (hkl) d XRD Charateristics • By Geometry for Orthorhombic Xtals • For Cubic Xtals a = b = c, so
Pb XRD Implementation • X-Ray DiffractometerSchematic • T X-ray Transmitter • S Sample/Specimen • C Collector/Detector • Typical SPECTRUM • Spectrum Intensity/Amplitude vs. Indep-Index
Given Niobium, Nb with Structure = BCC X-ray = 1.659 Å (211) Plane Diffraction Angle, 2 = 75.99° FIND ratom d211 BCC Niobium XRD Example Nb • Find InterPlanar Spacing by Bragg’s Law
To Determine ratom need The Cubic Lattice Parameter, a Use the Plane-Spacing Equation Nb XRD cont • For the BCC Geometry by Pythagorus
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 typ. range from 1 nm to 20 mm • (i.e., from a few to millions of atomic layers).
Single vs PolyCrystals • Single Crystals -Properties vary with direction: anisotropic. -Example: the modulus of elasticity (E) in BCC iron: • Polycrystals 200 mm -Properties may/may not vary with direction. -If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa) -If grains are textured, anisotropic. 19
WhiteBoard Work • Problem 3.47 • Given Three Plane-Views, Determine Xtal Structure Also:
All Done for Today xTal PlanesinSimple CubicUnit Cell