lec9-crystal structure (1)
Page 1: Introduction
Title: The Chemistry of Engineering Materials
Lecture By: Engr. Joseph Kitma III
Institution: Saint Louis University, School of Engineering and Architecture, Department of Chemical Engineering
Focus: Basic Concepts of Crystal Structure
Page 2: Crystal Structure
Definition:
Crystal structure describes the ordered arrangement of atoms, ions, or molecules in a crystalline material.
Crystallography:
The experimental science of determining the atomic arrangement in crystalline solids.
Example: Strontium Titanate (SrTiO3)
Bright atoms represent strontium, darker atoms represent titanium.
Illustration:
Atomic resolution image of strontium titanate.
Page 3: Atoms and Ions
Atom:
The smallest constituent unit of ordinary matter with the properties of a chemical element.
All states of matter are composed of neutral or ionized atoms.
Ion:
An atom or molecule with a net electrical charge due to unequal numbers of protons and electrons.
Illustration:
Depiction of the helium atom's nucleus and electron cloud.
Page 4: Molecular Charge
Cation:
A positively charged ion.
Anion:
A negatively charged ion.
Molecule:
An electrically neutral group of two or more atoms bonded together.
Page 5: Crystalline Solids
Definition:
A solid material whose constituents are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions.
Crystal Lattice:
Symmetric patterns repeating in three-dimensional space.
Visualization:
Representation of a 3D Cartesian coordinate system.
Page 6: Crystal Definition
Crystal:
Any solid material with component atoms arranged in a definite pattern, reflecting internal symmetry.
Crystal Faces:
The external surfaces that reflect the internal order.
Page 7: Unit Cell
Unit Cell:
The smallest repeating unit of a crystal lattice that defines its symmetry and structure.
Repeated translations of the unit cell create the entire crystal lattice.
Bravais Lattice:
The points in the lattice that define the unit cell's symmetrical structure.
Page 8: Unit Cells - Overview
Overview of unit cells and basic structures in crystalline materials.
Page 9: Crystalline Lattice
Crystalline Lattice:
Regular arrangement of atoms in a crystalline solid.
Particles' Goal:
Minimize energy by arranging themselves efficiently.
Circles in representation:
Represent lattice points occupied by atoms, ions, or molecules.
Page 10: Unit Cells - Characteristics
Coordination Number:
The number of nearest neighbor atoms that an atom interacts with.
Packing Efficiency:
Percentage of volume of the unit cell occupied by atoms; higher coordination leads to greater packing efficiency.
Page 11: Simple Cubic Lattice
Structure:
One atom at each corner of the cubic unit cell, touching along each edge (1 = 2r).
Atoms per unit cell:
Each corner contributes 1/8 of an atom, totaling to one atom per unit cell.
Page 12: Simple Cubic Unit Cell Details
Details of the simple cubic unit cell, including coordination number (6) and how it computes atoms per unit cell.
Page 13: Coordination in Crystal
Coordination Number:
Defined as 6 for the simple cubic structure.
Calculation:
Atoms per unit cell = 1 (from 1/8 of 8 corner atoms).
Page 14: Packing Efficiency Calculation
Formula:
Packing Efficiency (P.E.) = (Volume occupied by atoms / Volume of the unit cell) × 100%
Example Calculation:
Packing efficiency determined for the simple cubic lattice is approximately 52%.
Page 15: Body-Centered Cubic Lattice
Overview and structure of the body-centered cubic unit cell with one atom at the center and corner atoms.
Page 16: Body-Centered Cubic Structure
Atoms touch along the diagonal of the cube, not along the edges.
Mathematical Relationships:
Relationships between lengths and radii are given for body-centered cubic cells.
Page 17: Body-Centered Cubic Properties
Coordination Number:
Coordination number is 8.
Atoms per unit cell:
Counts to two: one from corners and one from the center.
Page 18: Face-Centered Cubic Lattice
Introduction to the face-centered cubic lattice structure.
Page 19: Face-Centered Cubic Structure
Atoms located at the corners and the center of each face.
Touch Points:
Atoms touch along the diagonal face.
Page 20: Face-Centered Cubic Details
Coordination Number:
Coordination number of 12.
Atoms per unit cell:
Each corner contributes to atoms; total is 4 atoms per unit cell.
Page 21: Volume and Atomic Properties
Calculation of radius and volume for body-centered cubic unit cells.
Key Formula:
Volume, mass, and density calculations relevant to material properties.
Page 22: Density Calculation
Density Formula:
ρ = mass / volume; specific example calculated for Cr.
Page 23: Density Example
Example showing density of chromium calculated to be 7.18 g/cm³.
Explains consideration of significant figures.
Page 24: Closest-Packed Structures
Concept of stacking atoms in layers for efficient packing of crystal structures.
Page 25: Layer Arrangement
Efficiency in atomic packing achieved by offsetting layers.
Layers:
Shifts in layer arrangement improve packing efficiency.
Page 26: Hexagonal Closest Packing
Describes hexagonal closest packing structure with an ABAB layer arrangement.
Coordination number of 12 and 74% packing efficiency.
Page 27: Hexagonal Unit Cell
Visual representation of the hexagonal unit cell structure highlighting layer alignment.
Page 28: Cubic Closest Packing
Description of cubic closest packing structured in an ABC pattern.
Identifies coordination number and packing efficiency similar to face-centered cubic.
Page 29: Crystal Systems
Overview of crystal systems and their classification based on shape and symmetry.
Goal:
Quantitative description of unit cell shapes and lattice points.
Page 30: Unit Cell Classification
Discussion on symmetry requirements for unit cells to stack efficiently.
Shapes:
Cube versus non-cubic shapes like pentagons.
Page 31: Symmetry
Definition:
Symmetry described as mathematical rules detailing object shape.
Sphere:
Example of perfect symmetry with infinite symmetry planes and axes.
Page 32: Crystal Space Groups
Definition of Crystal:
Periodic arrangement of repeating motifs such as atoms.
Space Group:
Describes the total symmetry operations possible in a crystal pattern.
Page 33: Translational Symmetry
Space Group Includes:
Translational and Point Symmetries.
Translational Symmetries:
Invisible under normal magnification; involve shifting movements through specified distances.
Page 34: Point Symmetries
Definition:
Macroscopically visible symmetry operations.
Types:
Includes reflection, rotational symmetry, and inversion.
Page 35: Center of Symmetry
Example:
Item with opposite faces reflecting parts through a central point (inversion).
Page 36: Mirror Symmetry
Definition capturing the concept of an object being symmetric with respect to different planes of reflection.
Example:
Bilateral symmetry in vertebrates.
Page 37: Rotational Symmetry
Concept:
Defined by points around which an object appears unchanged after rotation through specified angles.
N-Fold Symmetry
Describes multiple symmetrical states during 360° of rotation.
Page 38: N-fold Roto-Inversion Symmetry
Describes operations where an object is rotated followed by inversion.
Page 39: Multiple Symmetry Types
Objects can possess different symmetry types such as rotation combined with reflection.
Page 40: Basic Symmetry Elements
Overview of basic rotation and symmetry elements in crystal structures.
Symbols Used:
Notations for different types of rotations and mirror planes.
Page 41: JCPDS Card Example
Quality of Data:
Description of sodium chloride diffraction data with relevant details.
System Cubic Information:
Reflection of crystallographic data analysis in the card.
Page 42: Crystal Classes
Total Symmetry Classes:
32 unique crystal symmetry classes based on combinations of point symmetries.
Examples of Classes:
Comparisons of symmetrical elements in cubic and triclinic crystal classes.
Page 43: The Crystal Systems
Classification of crystals into six systems:
Cubic
Tetragonal
Hexagonal
Orthorhombic
Monoclinic
Triclinic
Symmetry Decrease:
Describes the diminishing symmetry among the systems.
Page 44: Crystallographic Axes
Definition:
Axes referred to as a, b, c crossing at an axial cross.
Usage:
These axes help to classify and describe crystal forms.
Page 45: Cubic System Overview
Characteristics of the cubic system, including equal lengths and right-angle intersections among axes.
Common Forms:
Example minerals exhibiting cubic structure like pyrite and halite.
Page 46: Cube Comparison
Hexoctahedron:
A form belonging to the cubic system showcasing high symmetry.
Page 47: Tetragonal System
Description of the tetragonal system's axes arrangement.
Characteristics:
Specific lengths differentiating this system from the cubic system.
Page 48: Hexagonal System
Four axes with three co-planar and one vertical distinguishing this system.
Examples:
Different classes within hexagonal crystal systems.
Page 49: Orthorhombic System
Characteristics including differing axis lengths and right-angle intersections.
Forms:
Types of forms belonging to this system.
Page 50: Monoclinic System
Unique properties of the three non-equal axes and their oblique intersection.
Forms and Symmetry:
Specific forms pertinent to the monoclinic system.
Page 51: Triclinic System
Unique properties with three non-equal axes intersecting at varying angles.
Forms and Symmetry Classes:
Represents minimal symmetry traits in crystal structures.
Page 52: Summary of Crystal Systems
List and brief descriptions of the six crystal systems focus.
Page 53: Bravais Lattices Overview
Explanation of how unit cells inform crystal lattice structuring.
Lattice Points:
Theoretical points in 3D space, limited to 14 compatible Bravais lattices demonstrating atomic arrangements.
Page 54: Bravais Lattices Types
Enumeration of the different Bravais lattice types concluding with descriptions and functions.
Significance of Points:
Explanation of each point in relation to actual atoms within a crystal lattice structure.
Page 55: Bravais Lattice Classification
Division of Bravais lattices by symmetry and unit cell properties across various crystal classifications.
Page 56: Isometric Cells and Their Importance
Description of simple cubic and centered configurations.
Importance of F Cell:
Key pattern for cubic closest packing relates to crystalline structures.
Page 57: Tetragonal Cell Characteristics
Explanation of the structure and characteristics of tetragonal cells and their variations.
Page 58: Orthorhombic Structures
Discussion on properties of orthorhombic cells and their relevant structures.
Page 59: Monoclinic and Triclinic Overview
Characteristics and distinctions of cells within these systems outlined.
Page 60: Unique Rhombohedral Cells
Description of unique features pertaining to hexagonal crystals and rhombohedral distinctions.
Page 61: Crystal System Symmetries
Summary of crystal system organization based on symmetry and conditions met within each type.
Page 62: Packing Fraction (P.F.)
Definition:
Indicates atomic density in lattice arrangements.
Volume Calculation:
Calculation of volume occupied by atoms in the unit cell against the volume of the cell.
Page 63: General Formula for Packing Fraction
Calculation steps for determining the packing fraction for various elements and groups.
Page 64: Bragg's Law
Concept: Peaks observed when X-rays scatter from a crystal.
Formula: nλ = 2d sin θ; describes conditions for constructive interference.
Page 65: Crystal Defects
Definition: Imperfections in a crystal's arrangement due to various factors.
Effect on Behavior: Influence on mechanical, electrical, and optical properties of materials.
Page 66: Identification of Crystal Structures
Weiss Indices: Parameters for approximating face orientation relative to crystallographic axes.
Miller Indices: Symbolic representation for atomic plane orientation in crystal lattices.
Page 67: Example of Indices Determination
Demonstration of the calculation process for Weiss and Miller indices from given atomic parameters.
Page 68: Summary of Index Calculations
Presents derived Weiss and Miller indices based on previous calculations for clarity.
Page 69: Exercises for Practice
Exercise 01: Determine Weiss and Miller index from given parameters.
Exercise 02: Follow suit with different given geometries.
Page 70: Additional Exercises
Exercise 03: Calculate indices for topaz crystal structure.
Exercise 04: Shortest distance calculations relating to cubic crystals.
Page 73: Perpendicular Distance Calculation
Discusses methodology for measuring the perpendicular distance from origin to planes in cubic crystals.
Page 74: Polymorphism in Materials Science
Definition: The ability of solid materials to exist in multiple crystal forms.
Examples: Provides examples of polymorphic relationships found in minerals and polymers.
Closing Statement: Acknowledgment of lecture completion with gratitude.