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Solid Crystallography

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Presentation on theme: "Solid Crystallography"— Presentation transcript:

1 Solid Crystallography
Engineering 45 Solid Crystallography Bruce Mayer, PE Licensed Electrical & Mechanical Engineer

2 Crystal Navigation 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

3 Point COORDINATES 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 Sketch the Location of the Point with Xtal CoOrds of: (1/2, 2/5, 3/4)

4 Point CoOrdinate Example
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 Located Point (1/2, 2/5, 3/4)

5 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 x z y x z y [122] [111] [001] _ [110] [010]

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

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

8 Example  Xtal Directions
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:

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 The last one is actuall (1 1 -1)

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

12 The Miller Indices Example
In Tabular Form

13 More Miller Indices Examples
Consider the (001) Plane x z y x y z Intercepts Reciprocals Reductions Enclosure 1 0 0 1 (none needed) (001) Some Others (3/4, ½, -1/4) => invert => (4/3, 2, -4) => reduce by mult by 3/2 => (2 3 -6)

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 Hexagonal Structures Consider the Hex Structure at Right with 3-Axis CoOrds 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 4-Axis, 4-Index System To Clear Up this Confusion add an Axis in the BASAL, or base, Plane Plane-C The Miller Indices now take the form of (hkil) Plane-A → Plane-B → Plane-C → Plane-B Plane-A

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

19 More 4-Axis Directions [ ] directions is at 90deg to a3; i.e., it splits the 60deg angle

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

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

22 3axis↔4axis Translation
The 3axis Indices 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 Diamond APF = (8*pi*r^3/3)(8*r/sqrt(3))^3 = 34%

26 LD & PD for Silicon Si

27 LD and PD For Silicon For 100 Silicon For {111} Silicon
LD on Unit Cell EDGE For {111} Silicon PD on (111) Plane Use the (111) Unit Cell Plane 1-cos3o => 13.4% lower atom density on (100) 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 The Dectector collects a vast number of wave; thus in-phase wave add to the magnitude of the signal Amplitude 100% Added Amplitude 100% Subtracted Constructive Scattering Destructive Scattering

29 Path-Length Difference
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 XRD Constructive Interference
1 1’ 2’ The Path Length Difference is Line Segment SQT 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 XRD Charateristics The InterPlanar Spacing, d, as a Function of Lattice Parameters (abc) & Miller Indices (hkl) By Geometry for OrthoRhombic Xtals For Cubic Xtals a = b = c, so d

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

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

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

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

36 PolyCrystals → Grains Most engineering materials are POLYcrystals 1 mm
Nb-Hf-W plate with an electron beam weld 1 mm 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 -Properties may/mayNot vary with direction. -If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa) -If grains are textured, anisotropic. 200 mm 19

38 WhiteBoard Work Planar-Projection (Similar to P3.48)
Given Three Plane-Views, Determine Xtal Structure Also: 3.47 in 7e Find Aw



41 xTal Planes in Simple Cubic Unit Cell
All Done for Today xTal Planes in Simple Cubic Unit Cell

42 Planar Projection 101 101 Structure is Face Centered ORTHORHOMBIC

43 Planar Projection



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