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## Chapter 1: Crystal Structure

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**Chapter 1: Crystal Structure**The Nobel “Booby” Prize! See the“Ig Nobel” Prizediscussed at:http://improbable.com/ig/**The (Common) Phases of Matter**“Condensed Matter”includesboth of these.Our focus isSolids! This doesn’t include Plasmas, or Bose-Einstein condensatesbut these three are the “common” phases!!**Gases**• Gases have atoms or molecules that do not bond to one another in a range of pressure, temperature & volume. Also, these molecules have no particular order & they move freely within a container.**Similarto gases, Liquidshave no atomic or molecular order &**they assume the shape of their containers. Applying low levels of thermal energy can easily break the existing weak bonds. Liquids**Liquid Crystals**• Liquid Crystalshave mobilemolecules, but a type of long rangeorder can exist.The molecules havea permanent electric dipole. • Applying anelectric field rotates the dipoles& establishes order within thecollection of molecules.**Solids**• Solids consist of atoms or moleculesundergoing thermal motionabout their equilibrium positions, which are at fixed pointsin space. • Solids can be crystalline, polycrystalline,or amorphous.**Solids**• Solids(at a given temperature, pressure, volume) have stronger interatomic bondsthan liquids. • So,Solidsrequire more energy to break the interatomic bondsthan liquids.**Crystallography & Crystalline Solids**• The following material covers most of the topics in Ch. 1(Crystal Structure)in the book byKittel. • In the Supplemental book by Omar, the corresponding material is in Ch. 1, • (Crystal Structures & Interatomic Forces). • However, as will be true throughout the course, my discussion will use a large variety of sources other than those books.**Crystal Structure**Topics 1. Periodic Arrays of Atoms 2.Fundamental Types of Lattices 3. Index System for Crystal Planes 4. Simple Crystal Structures 5. Direct Imaging of Crystal Structure 6. Non-ideal Crystal Structures 7. Crystal Structure Data**Objectives**At the end of this Chapter, you should: 1. Be able to identify a unit cell in a symmetrical pattern. 2. Know that (in 3 dimensions) there are 7(& ONLY 7!!) Possibleunit cell shapes. 3. Be able to define cubic, tetragonal, orthorhombic & hexagonal unit cell shapes**Preliminary Remarks**on Crystalline Solids • This material is an overviewof the many types of crystalline solids & their crystal structures. • Mostly, this involves some simple mathematics of symmetry. However, it also involves learning some terminology & nomenclature.**Unfortunately (in my opinion!), the first scientists to be**interested in crystals were geologists & crystallographers & not physicists. • So, in order to "speak the language" of crystals, we must use the terminology that they developed. • I sometimes find that inconvenient & abstract.**Despite this, understanding that language**• is necessary in order for us to be able discuss • the PHYSICS of crystalline solids. • NOTE!!There are a huge number of websites that can be used to supplement the textbook discussions & my lectures. • A list of some is posted on the Lecture Page. • We’ll spend 3 or 4 lectures on these topics. Some Lectures on this material are also posted on the Lecture Page.Some of these may undergo some revisions as we proceed. A homework assignment on this material will be given out soon.**Periodic Arrays of Atoms (Kittel, Fig. 1.)**• Experimental Evidenceof periodic structures: • The external appearance of crystals gives some clues. • The figure shows that when a crystal is cleaved, we • can see that it is built up of identical “building blocks”.**Experimental Evidence**of periodic structures. • The early crystallographers noted that • the index numbers that define plane • orientations are exact integers. Cleaving a Crystal**Early ideas**• All crystals are solids, but all solids are not crystalline! Crystals have symmetry (As first reported by Kepler who studied snowflakes!!) & long range order • Spheres & small • shapes can be • packed to • produce regular shapes.**What are Crystals?**• A crystal or crystalline solidis a solid material whose constituent atoms, molecules, or ions are arranged in an orderly, repeating pattern extending in all three spatial dimensions.**Crystallography**The Study of Crystals. • Scientists who specialize in the study of crystals are called crystallographers. • Early studies of crystals were carried out by mineralogists & geologists who studied the symmetries and shapes (morphology) of naturally-occurring mineral specimens.**Crystallography**The Study of Crystals. • If you would like to see some beautiful crystals,just look around this floor of the building, where our Geosciences colleagueshave displayed numerous rock crystal samples, with labels to tell us what they are. Note that these are naturally occurring crystals!**Crystallography**• These early studies led to the correct idea that crystals are regular three-dimensional arrays (Bravais lattices) of atoms and molecules. • A single unit cellis repeated indefinitely along three principal directions that are not necessarily perpendicular.**Crystallography**• Crystallography ≡A branch of science • dealing with the geometric description • of crystals & their internal arrangements. • It is also the science of crystals & the • math used to describe them.**Crystallography**• Crystallographyis a VERY OLD field • which pre-dates Solid State Physics by • about a century! So (unfortunately, in • some ways!) much of the terminology • (& theory notation) of Solid State • Physics originated in crystallography.**Crystallography**• The purpose of Ch. 1 of • Kittel’s book is mainly to • introduce this crystallography • terminology & notation to you.**Unit Cell Description in Terms of Lattice Parameters**• a ,b, & c define the edge lengths & are referred to as the crystallographic axes. • The angles between these are a, b, &g. • The lattice parameters a ,b, c,a, b, &g give thedimensions of the unit cell. c a b**The Three General Types of Solids**• Single Crystal, Polycrystalline, • Amorphous • Each type is characterized by the size of the orderedregion within the material.An ordered region is a spatial volume in which atoms or molecules have a regular geometric arrangement or periodicity.**All Solids!**• All solids have “resistance” to changes in both shape and volume. • Solids can be Crystalline or Amorphous • Crystalsare solids that consist of a periodic array of atoms, ions, or molecules • If this periodicity is preserved over “large” (macroscopic) distances, the solid has “Long-range Order” • Amorphous solidsdo not have Long-Range Order, but they often have Short Range Order**Crystals:**Short-range Order Long-range Order Amorphous solids: ~Short-range Order No Long-range Order Solids**Solids**• Different solids can have the same geometrical arrangements of atoms • Their Properties are determined by their crystal structure: Both crystal lattice & basis are important Examples: • Si, Diamond (C), GaAs, ZnSeall have the same lattice geometry • Si and C (Diamond) Form the “Diamond Structure” • GaAsandZnSeform a structure called the“Zinc Blende” Structure**Solids**• Different arrangements of atoms (even the same atoms) can result in very different solid state properties 2 very different solids made of only carbon (C) atoms!**Crystalline Solids**• A Crystalline Solid is the solid form of a substance in which the atoms or moleculesare arranged in a definite, repeating pattern in three dimensions. • Single Crystals, ideally have a high degree of order, or regular geometric periodicity, throughout the entire volume of the material.**A Single Crystalhas a arrangement of atoms that repeats**periodically across its whole volume. Even at infinite length scales, each atom is related to each equivalent atom in the structure by translational symmetry. Amorphous Solid Single Crystals SinglePyrite Crystal**Polycrystalline Solids**• A Polycrystalline Solidis made up of an aggregate of many small single crystals (crystallites or grains). • Polycrystalline materialshave a high degree of order over many atomic or moleculardimensions. • These ordered regions, or single crystal regions, vary in size & orientationwith respect to one another. • These regions are called grains (or domains)& are separated from one another by grain boundaries. Polycrystalline PyriteGrain**Polycrystalline Solids**• In Polycrystalline Solids, the atomic ordercan vary from one domain to the next.The grains are usually 100 nm - 100 microns in diameter. Polycrystals • with grains that are • < 10 nm in diameter are • called nanocrystallites. A polycrystal with grain boundaries**Polycrystalline Solids**• Polycrystalline solids with grains & grain boundaries:**Polycrystalline Solids**• Many technologically important materials are • polycrystalline. • The figure is an • electron micrograph • of a Nb-Hf-W plate • with an electron • beam weld. • Each "grain" is a single crystal. If the grains • are randomly oriented, the overall component • properties are not directional. • Grain sizes typically range from 1 nm to 2 cm • (i.e., from a few to millions of atomic layers)**Polycrystalline Solids**Photograph of a Silicon Single Crystal. Micrograph of a Polycrystalline stainless steel sample showing grains & grain boundaries**Amorphous Solids**• Amorphous (Non-Crystalline) Solids • Are composed of randomly oriented atoms, ions, or molecules that do not form defined patterns or lattice structures. • Amorphous materialshave order only within a few atomic or molecular dimensions.**Amorphous Solids**• Amorphous (Non-crystalline) Solids • Have order only within a few atomic or molecular dimensions. They do not have any long-range order, but they have varying degrees of short-range order.Examples of amorphous materials include amorphous silicon, plastics, & glasses.**Amorphous Solids**• Amorphous (Non-crystalline) Solids • Have no regular, long range order of • arrangement of atoms. • Some examples from everyday life: • 1. Polymers, 2. Ceramics, • 3. Window Glass, 4. “Cotton Candy”! • The two sub-states of amorphous solids • are the Rubbery and Glassy states**Amorphous Solids**• Amorphous (Non-crystalline) Solids • Have no regular, long range order of • arrangement of atoms. • Can be prepared by rapidly cooling • molten material. Rapid cooling • minimizes time for the atoms to pack • into a more thermodynamically • favorable crystalline state.**Amorphous Solids**Illustrationof the continuous random network structure of the atoms in an amorphous solid