ENGR-45_Lec-04_Crystallography.ppt 1 Bruce Mayer, PE Engineering-45: Materials of Engineering Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Engineering 45 Crystallography

ENGR-45_Lec-04_Crystallography.ppt 2 Bruce Mayer, PE Engineering-45: Materials of Engineering 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 Units of the Lattice Dimensions a, b, c

ENGR-45_Lec-04_Crystallography.ppt 3 Bruce Mayer, PE Engineering-45: Materials of Engineering 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)

ENGR-45_Lec-04_Crystallography.ppt 4 Bruce Mayer, PE Engineering-45: Materials of Engineering 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)

ENGR-45_Lec-04_Crystallography.ppt 5 Bruce Mayer, PE Engineering-45: Materials of Engineering Crystallographic DIRECTIONS  Convention to specify crystallographic directions: 3 indices, [uvw] - reduced projections along x,y,z axes  Procedure to Determine Directions 1.vector through origin, or translated if parallelism is maintained 2.length of vector- PROJECTION on each axes is determined in terms of unit cell dimensions (a, b, c); negative index in opposite direction 3.reduce indices to smallest INTEGER values 4.enclose indices in brackets w/o commas [111] [110] [010] x z y [001] _ x z y

ENGR-45_Lec-04_Crystallography.ppt 6 Bruce Mayer, PE Engineering-45: Materials of Engineering Example  Xtal Directions  Write the Xtal Direction, [uvw] for the vector Shown Below  Step-1: Translate Vector to The Origin in Two SubSteps

ENGR-45_Lec-04_Crystallography.ppt 7 Bruce Mayer, PE Engineering-45: Materials of Engineering Example  Xtal Directions  After −x Translation, Make −z Translation  Step-2: Project Correctly Positioned Vector onto Axes

ENGR-45_Lec-04_Crystallography.ppt 8 Bruce Mayer, PE Engineering-45: Materials of Engineering 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:

ENGR-45_Lec-04_Crystallography.ppt 9 Bruce Mayer, PE Engineering-45: Materials of Engineering 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:

ENGR-45_Lec-04_Crystallography.ppt 10 Bruce Mayer, PE Engineering-45: Materials of Engineering Miller Indices – Step by Step  MILLER INDICES specify crystallographic planes: (hkl)  Procedure to Determine Indices 1.If plane passes through origin, move the origin (use parallel plane) 2.Write the INTERCEPT for each axis in terms of lattice parameters (relative to origin) 3.RECIPROCALS are taken: plane parallel to axis is zero (no intercept → 1/  = 0) 4.Reduce indices by common factor for smallest integers 5.Enclose indices in Parens w/o commas

ENGR-45_Lec-04_Crystallography.ppt 11 Bruce Mayer, PE Engineering-45: Materials of Engineering Example  Miller Indices  Find The Miller Indices for the Cubic-Xtal Plane Shown Below

ENGR-45_Lec-04_Crystallography.ppt 12 Bruce Mayer, PE Engineering-45: Materials of Engineering The Miller Indices Example  In Tabular Form

ENGR-45_Lec-04_Crystallography.ppt 13 Bruce Mayer, PE Engineering-45: Materials of Engineering More Miller Indices Examples  Consider the (001) Plane xyz Intercepts Reciprocals Reductions Enclosure x z y (001)   Some Others (none needed)

ENGR-45_Lec-04_Crystallography.ppt 14 Bruce Mayer, PE Engineering-45: Materials of Engineering FAMILIES of DIRECTIONS  Crystallographically EQUIVALENT DIRECTIONS → notation e.g., in a cubic system,  Family of directions: SAME Atomic ARRANGEMENTS along those directions

ENGR-45_Lec-04_Crystallography.ppt 15 Bruce Mayer, PE Engineering-45: Materials of Engineering 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

ENGR-45_Lec-04_Crystallography.ppt 16 Bruce Mayer, PE Engineering-45: Materials of Engineering Hexagonal Structures  Consider the Hex Structure at Right with 3-Axis CoOrds Plane-B Plane-A Plane-C  The Miller Indices Plane-A → (100) Plane-B → (010) Plane-C → (110)  BUT Planes A, B, & C are Crystallographically IDENTICAL –The Hex Structure has 6-Fold Symmetry Direction [100] is NOT normal to (100) Plane

ENGR-45_Lec-04_Crystallography.ppt 17 Bruce Mayer, PE Engineering-45: Materials of Engineering 4-Axis, 4-Index System  To Clear Up this Confusion add an Axis in the BASAL, or base, Plane Plane-B Plane-A Plane-C  The Miller Indices now take the form of (hkil) Plane-A → Plane-B → Plane-C →

ENGR-45_Lec-04_Crystallography.ppt 18 Bruce Mayer, PE Engineering-45: Materials of Engineering 4-Axis Directions  Find Direction Notation for the a1 axis-directed unit vector  Noting the Right- Angle Projections find

ENGR-45_Lec-04_Crystallography.ppt 19 Bruce Mayer, PE Engineering-45: Materials of Engineering More 4-Axis Directions

ENGR-45_Lec-04_Crystallography.ppt 20 Bruce Mayer, PE Engineering-45: Materials of Engineering 4-Axis Miller-Bravais Indices  Construct Miller-Bravais (Plane) Index-Sets by the Intercept Method Plane

ENGR-45_Lec-04_Crystallography.ppt 21 Bruce Mayer, PE Engineering-45: Materials of Engineering 4-Axis Miller-Bravais Indices  Construct More Miller-Bravais Indices by the Intercept Method Plane

ENGR-45_Lec-04_Crystallography.ppt 22 Bruce Mayer, PE Engineering-45: Materials of Engineering 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 = 3

ENGR-45_Lec-04_Crystallography.ppt 23 Bruce Mayer, PE Engineering-45: Materials of Engineering 4axis Indices CheckSum  Given 4axis indices Directions → [uvtw] Planes → (hkil)  Then due to Reln between a1, a2, a3

ENGR-45_Lec-04_Crystallography.ppt 24 Bruce Mayer, PE Engineering-45: Materials of Engineering 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

ENGR-45_Lec-04_Crystallography.ppt 25 Bruce Mayer, PE Engineering-45: Materials of Engineering Silicon Crystallography  Structure = DIAMOND; not ClosePacked

ENGR-45_Lec-04_Crystallography.ppt 26 Bruce Mayer, PE Engineering-45: Materials of Engineering LD & PD for Silicon  Si

ENGR-45_Lec-04_Crystallography.ppt 27 Bruce Mayer, PE Engineering-45: Materials of Engineering 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

ENGR-45_Lec-04_Crystallography.ppt 28 Bruce Mayer, PE Engineering-45: Materials of Engineering 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  Constructive Scattering  Destructive Scattering Amplitude 100% Added Amplitude 100% Subtracted

ENGR-45_Lec-04_Crystallography.ppt 29 Bruce Mayer, PE Engineering-45: Materials of Engineering 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

ENGR-45_Lec-04_Crystallography.ppt 30 Bruce Mayer, PE Engineering-45: Materials of Engineering XRD Constructive Interference  The Path Length Difference is Line Segment SQT 1 2 1’ 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

ENGR-45_Lec-04_Crystallography.ppt 31 Bruce Mayer, PE Engineering-45: Materials of Engineering 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

ENGR-45_Lec-04_Crystallography.ppt 32 Bruce Mayer, PE Engineering-45: Materials of Engineering XRD Implementation  X-Ray Diffractometer Schematic T  X-ray Transmitter S  Sample/Specimen C  Collector/Detector  Typical SPECTRUM Spectrum  Intensity/Amplitude vs. Indep-Index Pb

ENGR-45_Lec-04_Crystallography.ppt 33 Bruce Mayer, PE Engineering-45: Materials of Engineering XRD Example  Nb  Given Niobium, Nb with Structure = BCC X-ray = Å (211) Plane Diffraction Angle, 2  = 75.99°  FIND r atom d 211  Find InterPlanar Spacing by Bragg’s Law BCC Niobium

ENGR-45_Lec-04_Crystallography.ppt 34 Bruce Mayer, PE Engineering-45: Materials of Engineering Nb XRD cont  To Determine r atom need The Cubic Lattice Parameter, a Use the Plane- Spacing Equation  For the BCC Geometry by Pythagorus

ENGR-45_Lec-04_Crystallography.ppt 35 Bruce Mayer, PE Engineering-45: Materials of Engineering PolyCrystals → Grains  Most engineering materials are POLYcrystals 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 typ. range from 1 nm to 20 mm (i.e., from a few to millions of atomic layers).

ENGR-45_Lec-04_Crystallography.ppt 36 Bruce Mayer, PE Engineering-45: Materials of Engineering 19 Single Crystals -Properties vary with direction: anisotropic. -Example: the modulus of elasticity (E) in BCC iron: Polycrystals -Properties may/may not vary with direction. -If grains are randomly oriented: isotropic. (E poly iron = 210 GPa) -If grains are textured, anisotropic. Single vs PolyCrystals 200  m

ENGR-45_Lec-04_Crystallography.ppt 37 Bruce Mayer, PE Engineering-45: Materials of Engineering WhiteBoard Work  Problem 3.47 Given Three Plane-Views, Determine Xtal Structure Also:

ENGR-45_Lec-04_Crystallography.ppt 38 Bruce Mayer, PE Engineering-45: Materials of Engineering All Done for Today xTal Planes in Simple Cubic Unit Cell

ENGR-45_Lec-04_Crystallography.ppt 39 Bruce Mayer, PE Engineering-45: Materials of Engineering