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**Solid Crystallography**

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

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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

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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)

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**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)

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**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]

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**Example Xtal Directions**

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

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**Example Xtal Directions**

After −x Translation, Make −z Translation Step-2: Project Correctly Positioned Vector onto Axes

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**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:

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**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:

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**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)

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**Example Miller Indices**

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

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**The Miller Indices Example**

In Tabular Form

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**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)

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**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

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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

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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

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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

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

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

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**4-Axis Miller-Bravais Indices**

Construct Miller-Bravais (Plane) Index-Sets by the Intercept Method Plane Plane

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**4-Axis Miller-Bravais Indices**

Construct More Miller-Bravais Indices by the Intercept Method Plane Plane

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**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

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**4axis Indices CheckSum Given 4axis indices**

Directions → [uvtw] Planes → (hkil) Then due to Reln between a1, a2, a3

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**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

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**Silicon Crystallography**

Structure = DIAMOND; not ClosePacked Diamond APF = (8*pi*r^3/3)(8*r/sqrt(3))^3 = 34%

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LD & PD for Silicon Si

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**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

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**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

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**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

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**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

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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

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**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

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**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)

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**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

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**Nb XRD cont To Determine ratom need The Cubic Lattice Parameter, a**

Use the Plane-Spacing Equation For the BCC Geometry by Pythagorus

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**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).

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**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

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**WhiteBoard Work Planar-Projection (Similar to P3.48)**

Given Three Plane-Views, Determine Xtal Structure Also: 3.47 in 7e Find Aw

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**xTal Planes in Simple Cubic Unit Cell**

All Done for Today xTal Planes in Simple Cubic Unit Cell

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Planar Projection 101 101 Structure is Face Centered ORTHORHOMBIC

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Planar Projection

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Miller Indices And X-ray diffraction

Miller Indices And X-ray diffraction

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