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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**This doesn’t include Plasmas, but these are the “common” phases!! “Condensed Matter”includesboth of these.We’ll focus onSolids!**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 dipole. Applying anelectric field rotates the dipole & establishes order within thecollection of molecules. 5**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(at a given temperature, pressure, volume) have stronger interatomic bondsthan liquids. • So,Solidsrequire more energy to break the interatomic bondsthan liquids.**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 cellin 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**Periodic Arrays of Atoms**Experimental Evidenceof periodic structures. (See Kittel, Fig. 1.) The external appearance of crystals gives some clues to this. Fig. 1 shows that when a crystal is cleaved, we can see that it is built up of identical “building blocks”. Further, the early crystallographers noted that the index numbers that define plane orientations are exact integers. Cleaving a Crystal**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.**Crystalline Solids**• A Crystalline Solidis 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 an atomic structure that repeats**periodically across its whole volume. Even at infinite length scales, each atom is related to every other equivalent atom in the structure by translational symmetry. SinglePyrite Crystal Amorphous Solid Single Crystals**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. • 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. Polycrystalline PyriteGrain**Amorphous Solids**• Amorphous (Non-crystalline) Solidsare composed of randomly orientated atoms, ions, or molecules that do not form defined patterns or lattice structures.Amorphous materialshave 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 materialinclude amorphous silicon, plastics, & glasses.**Departures From the “Perfect Crystal”**• A “Perfect Crystal” is an idealization that does not exist in nature. In some ways, even a crystal surface is an imperfection, because the periodicity is interrupted there. • Each atom undergoes thermal vibrations around their equilibrium positions for temperatures T > 0K. These can also be viewed as “imperfections”. • Real Crystalsalways have foreign atoms (impurities), missing atoms (vacancies), & atoms in between lattice sites (interstitials) where they should not be. Each of these spoils the perfect crystal structure.**Crystallography**Crystallography ≡The branch of science that deals with the geometric descriptionof crystals & their internal arrangements. It is the science of crystals & the math used to describe them. It is a VERY OLD fieldwhich pre-dates Solid State Physicsby about a century! So (unfortunately, in some ways) much of the terminology (& theory notation) of Solid State Physics originated in crystallography. The purpose of Ch. 1 of Kittel’s book is mainly to introduce this terminology to you.**Solid State Physics**• Started in the early 20th Century when the fact that • Crystals Can Diffract X-rays • was discovered. • Around that same time the new theory of • Quantum Mechanics • was being accepted & applied to various problems. Some of the early problems it was applied to were the explanation of observed X-ray diffraction patterns for various crystals & (later) the behavior of electrons in a crystalline solid.**Crystallography**• A Basic Knowledge of Elementary • Crystallography is Essential • for Solid State Physicists!!! • A crystal’s symmetry has a profound influence on many of its properties. • A crystal structure should be specified completely, concisely & unambiguously. • Structures are classified into different types according to the symmetries they possess. • In this course, we only consider solids with “simple” structures.**Crystal Lattice**Crystallography focuses on the geometric properties of crystals. So, we imagine each atom replaced by a mathematicalpoint at the equilibrium position of that atom. A Crystal Lattice(or a Crystal) ≡ An idealized description of the geometry of a crystalline material. A Crystal ≡A 3-dimensional periodic array of atoms. Usually, we’ll only consider ideal crystals. “Ideal” means one with no defects, as already mentioned. That is, no missing atoms, no atoms off of the lattice sites where we expect them to be, no impurities,…Clearly, such an ideal crystal never occurs in nature. Yet, 85-90% of experimental observations on crystalline materials is accounted for by considering only ideal crystals! Platinum Surface (Scanning Tunneling Microscope) Crystal Lattice Structure of Platinum Platinum**Mathematically**A Lattice is Defined as an Infinite Array of Points in Space in which each point has identical surroundings to all others. The points are arranged exactly in a periodic manner. y B C D E α b O x a A Crystal Lattice 2 Dimensional Example 21**Ideal Crystal ≡ An infinite periodic repetition of**identical structural units in space. • The simplest structural unit we can imagine is a Single Atom. This corresponds to a solid made up of only one kind of atom ≡ • An Elemental Solid. • However, this structural unit could also be a group of several atoms or even molecules. • The simplest structural unit for a given solid is called the BASIS**The structure of an Ideal Crystalcan be described in terms**of a mathematical construction called a Lattice. • A Lattice ≡ • A 3-dimensional periodic array of points in space. For a particular solid, the smallest structural unit, which when repeated for every point in the lattice is called the Basis. • The Crystal Structure is defined once both the lattice & the basis are specified. That is • Crystal Structure≡Lattice + Basis**Crystalline Periodicity**• In a crystalline material, the equilibrium positions of all the atoms form a crystal Crystal Structure ≡ Lattice + Basis For example, see Fig. 2. Lattice 2 Atom Basis Crystal Structure**Crystalline Periodicity**Crystal Structure ≡ Lattice + Basis For another example, see the figure. Crystal Structure Lattice Basis **Crystalline Periodicity**Crystal Structure ≡ Lattice + Basis For another example, see the figure. Basis Crystal Structure Lattice**A Two-Dimensional (Bravais) Lattice with Different Choices**for the Basis**2 Dimensional Lattice**y y B C D B C D E α b F G b x O x O a A a A Lattice with atoms at the corners of regularhexagons E H 28**The atoms do not necessarily lie at lattice points!!**Crystal Structure = Lattice + Basis Basis Crystal Structure 29