CHAPTER I: CRYSTAL GEOMETRY I. CONTENTS A. INTR INTROD ODUC UCTI TION ON B. FUNDAMENTAL FUNDAMENTAL TERMS OF CRYST CRYSTALLOGRAPH ALLOGRAPHY Y C. TYPES TYPES OF CR CRYSTAL YSTALS S D. SYMMET SYMMETR RY ELEME ELEMENT NTS S E. POINT POINT GROUPS GROUPS AND SP SPACE GROUP GROUPS S F. RELATION RELATION BETWEEN THE INTERPLANAR INTERPLANAR AND INTERATO INTERATOMIC MIC DISTANCE DISTANCE G. CRYST CRYSTAL AL STRUCTURE STRUCTURE OF MA MATERIALS TERIALS H. SIMPLE CUBIC CRYST CRYSTAL AL STRUCTURE I. BODY BODY CEN CENTER TERED ED CUBIC CUBIC STRUC STRUCTUR TURE E J. FACE CENTERED CENTERED CUBIC STRUCTURE & CLOSE PACKED STRUCTURE STRUCTURE K. SOME SPECIAL SPECIAL CUBIC CRYST CRYSTAL AL STRUCTURES L. HEXAGONA HEXAGONAL L CLOSED CLOSED PACKE PACKED D STRUCURE STRUCURE M. RELATION RELATION BETWEEN DENSITY OF CRYSTAL CRYSTAL AND LATTICE LATTICE CONSTANTS
II. WHY STUDY CRYSTAL GEOMETRY (OR STRUCTURE OF CRYSTLLAINE SOLIDS)? 1. The properties properties of some materials materials are directly directly related related to their crystal crystal structures. structures. For exampl example, e, pure pure and undefo undeform rmed ed magne magnesiu sium m and beryl berylliu lium, m, having having one crysta crystall struct structure ure,, are much much more more britt brittle le (i.e. (i.e. fractu fracture re at lower lower degree degreess of deformation) than are pure and undeformed metals such as gold and silver that have yet another crystal structure. 2. Furthermore, significant property differences exist between between crystalline crystalline and noncrystalline materials having the same composition. For example, non-crystalline ceramics and polymers normally are optically transparent; the same materials in crys crysta tall llin ine e (or (or semi semi-c -cry ryst stal alli line ne)) form form tend tend to be opaq opaque ue or, or, at best best,, translucent (semi transparent). Ceramics = compounds between metallic and non-metallic elements. They are most most freque frequentl ntly y oxides oxides,, nitrid nitrides es and carbid carbides. es. They They includ include e clay clay minera minerals, ls, cement and glass. These materials are typically insulative to the passage of electricity and heat, and are more resistant to high temperatures and harsh environment than metals and polymers. With regard to mechanical behavior, ceramics are hard but very brittle. Polymers = Polymers include the familiar plastic and rubber materials. Many pf them are organic compounds that are chemically based in carbon, hydrogen and other non-metallic elements. Polymers have very large molecular structures, have low density and may be extremely flexible.
III. LEARNING OBJECTIVES After After studyi studying ng this this chapte chapterr, the studen students ts should should be master master on the follow following ing achievements, 1. Defi Define ne spac space e latti lattice. ce. 2. Explai Explain n the the term term of basi basis. s. 3. Define Define the the term term of crystal crystal structure. structure.
©2009 Frida U. Ermawati
Page 1 of 7
4. Describe Describe the term of unit unit cell, crystallo crystallograph graphic ic axes and lattice lattice paramete parameterr of a crystal. 5. Define Define the term of inter-axia inter-axiall (or interfacial) interfacial) angles, angles, primiti primitive ve cells. 6. Describe Describe the the terms terms of Bravais Bravais lattices. lattices. 7. Explain Explain the term term of Miller indices indices,, rules to find find the Miller Miller indices indices of a plane, the salient (the most important) features of the Miller indices. 8. Explain Explain other other featur features es of the the Miller Miller indices. indices. 9. Describe Describe procedur procedure e to find the the Miller Miller indices indices of a direction direction.. 10.Name 10. Name the seven types of crystal systems and its examples. 11.Explain the terms of symmetry elements, axis of symmetry, plane of symmetry, center of symmetry, and the proof the absence of five-fold axis of symmetry. 12.Explain 12. Explain the relation between inter-planar and inter-atomic distance. 13.Explain crystal structure of materials. 14.Solve the problems on simple cubic (sc) crystal structure, body centered cubic (bcc) structure, face centered cubic (fcc) structure and close packed structure. 15.Solve 15. Solve the problems on some special cubic crystal structures. ***
©2009 Frida U. Ermawati
Page 2 of 7
A. INTRO INTRODU DUCT CTION ION 1. Crys Crysta tall llog ogra raph phy y deals deals wit with h the stu study dy of of all pos possi sibl ble e types types of of crys crysta tals ls and and the the deter determin minati ation on of the actual actual struct structure ure of the crysta crystallin lline e solids solids by x-rays x-rays,, neutron beams and electron beams. 2. Base Based d on the the arr arran ange geme ment nt of ato atoms ms or mole molecu cule les, s, sol solid idss are are clas classi sifi fied ed int into o two categories: a. Cryst Crystalli alline ne solids/m solids/mate ateria rials. ls. b. Non-cryst Non-crystalline alline)) solids/ma solids/materia terials. ls. (see Fig. crystalline & non-crystalline molecules of silicon dioxide). dioxide). In Crystalline solids, • Atoms are arranged in a regular manner, i.e. the atomic array is periodic. • Each of the atoms is at regular intervals along the arrays in all directions of a crystal (see (see Fig. Array of lattice 2D & array of lattice 3D). 3D). • The crystalline solids have different periodic arrangement in all the three directions and the physical properties vary with direction and are also called as anisotropic substances . • The structure may be made up of metallic crystals or non-metallic crystals. • The metallic crystals find wide application in engineering because of their strength, conductivity, reflection, etc. Examples of metallic crystals: copper, silver, aluminum, tungsten, etc. single le cryst crystals als and • Th Ther ere e are are two two type typess of crys crysta tall llin ine e mate materi rial als, s, i.e. i.e. sing polycrystalline materials. •
Single crystals: crystals: When the periodic and repeated arrangement of atoms is perfect or extends throughout the entirely of the specimen without interruption, the result is single crystal. All unit cells interlock in the same way and have the same orientation. Single crystals exist in nature, but they may also be produced artificially. artificially. They are ordinarily difficult to grow, because the environment must be carefully controlled. If the extremities of a single crystal are permitted to grow without any external constraint, the crystal will assume to a regular geometric shape having flat faces, as with some of the gemstones. The shape is indicative of the crystal structure. (See (See Figures single crystal d-alanine, d-alanine, l-alanine l-alanine,, twain). twain ).
•
Polycrystalline materials Polycrystalline materials:: Most Mo st crys crysta talli lline ne soli solids ds are are comp compos osed ed of a coll collec ecti tion on of many many smal smalll crystals or grains; such materials are called polycrystalline. Various arious stages stages are involv involved ed in the solidi solidific ficati ation on of a polycr polycryst ystall alline ine specimen specimen (See Fig. Solid Solidificat ification ion proce process ss of polycr polycrystal ystalline line mater material ial), ), i.e., (a) Initially, small crystals or nuclei form at various positions. These have random crystallographic orientations, as indicated by the square grids. (b) The small grains grow by the successive addition from the surrounding liquid of atoms to the structure of each. (c) The extremities of adjacent grains impinge/impose on one another as the solidification process approaches completion.
©2009 Frida U. Ermawati
Page 3 of 7
(d) As shown, the crystallographic orientation varies from grain to grain. Also, there exists some atomic mismatch within the region where two grains meet; this area called a grain boundary. boundary.
In Non-crystalline solids, • In non-crystalline solids, the atoms or molecules are arranged randomly over relat relative ively ly large large atomic atomic distan distances ces.. (see Fig Fig.. cry crysta stalli lline ne & non non-cr -cryst ystalli alline ne molecules of silicon dioxide) dioxide) • Someti Sometimes mes such such materi materials als are are also also called called amorphous (meaning (meaning literally literally withou withoutt form form), or supe super-coo r-cooled led liqui liquids ds becaus because e their their atomic atomic struct structure ure resembles that of a liquid, i.e. have no regular structure. • Amorphous solids have same physical properties in all directions and hence, they are known as isotropic substances . • Such materials have no specific electrical property, but have only plasticity. Examples: glass, plastics and rubber. ***
©2009 Frida U. Ermawati
Page 4 of 7
B. FUNDAMENTAL FUNDAMENTAL TERMS OF CRYST CRYSTALLOGRAPHY ALLOGRAPHY 1.
The structures of all crystals are described in terms of lattice with a group of atoms, each in a lattice point.
2.
The group is termed as basis basis.. The basis is repeated group of atoms in space to form the crystal structure. (See (See Fig. Array of lattice 2D) 2D)
3.
Lattice, A lattice = a regular and periodic arrangement of points in three dimensions. (See Fig. 2D lattice) lattice)
4. Consider Consider the points points P, P, Q and R. Lets join the points P and Q by a straight line, line, and the point P and R by another straight line. The line PQ is taken as an x-axis, and the line PR as a y-axis. 5.
The distance between any two successive lattice points along the x-direction is taken as a. The distance between any two successive lattice points along the ydirection is taken as b. Here, a and b are said to be lattice translational vectors. Consider a square lattice, in which a = b.
6. Consider two sets of points A, B, C, D, E, F and A’, B’, C’, C’, D’, E’, E’, E’, F’. 7. In thes these e two two sets sets,, the the surr surrou ound ndin ing g en envi viro ronm nmen entt look look symm symmet etri rica cal, l, i.e. i.e. the the distance AB and A’B’, AC and A’C’, AD and A’D’, AE and A’E’, AF and A’F’ are equal. 8.
The term lattice can be defined in another way. way. In an arrangement of points, if the surrounding environment looks like the same when the arrangement is viewed from different lattice points, then that arrangement is said to be a lattice.
9. Basis, To construct a crystal structure, some basis arrangement is to be fixed at each and every lattice point. This basis arrangement is said to be a basis. ( See Fig. Array of lattice 2D) 2D) 10.Crystal Structure, A Crystal structure is obtained by arranging the basis in each and every lattice point. It can be written as: A crystal structure = lattice + basis
11.The above expression is not a mathematical expression, but it is used to explain the formation of crystal structure. 12.Unit Cell, In the construction of a wall, bricks are arranged one above the other. Thus, in the case of the wall, a brick is a said to be a unit cell. cell. 13.Similarly, in the case of a crystal, a smallest unit is arranged one above the other. This smallest unit is known as unit cell. 14.
Thus, a un unit it ce cell ll is defi define ned d as a fu funda ndamen mental tal bui buildi lding ng blo block ck of a cry crysta stall structure.. (See structure (See again Fig. Array of lattice 3D) 3D)
©2009 Frida U. Ermawati
Page 5 of 7
15.Crystallographic axes, Consider a unit cell consisting of three mutually perpendicular edges OA, OB and OC. Draw parallel lines along the three edges. These three lines as taken as crystallographic axes and they are denoted as x , y and z axes. (See (See Fig. 3 crystallographic axes). axes). 16.Prim 16. Primitive itive (unit) vectors, See again Fig. 3 crystallographic axes. axes. Let OA be an intercept along the x-axis. Similarly, the intercepts made by the unit cell along the y- and z-axes are OB and OC, which mean OA, OB and OC = the intercepts made by the unit cell along the crystallographic axes. 17.
In crystallography, those three intercepts (a, b and c vectors) are known as primitive vectors or unit vectors of a crystal. crystal.
18.Interaxial 18. Interaxial angles (or, (or, Interfacial angles), In a crystal, the angles between x, y and z axes ( α, β, γ ) are called as interaxial angles (See again Fig. 3 crystallographic axes). axes). 19.
The angles α, β, γ are also said to be the interfacial angles, angles, because they are the angles between y-z plane, z-x plane and x-y plane, respectively.
20.Lattice 20. Lattice Parameters, Parameters, The above three primitive/unit vectors (a, b and c) AND the three interfacial angles (α, β, γ ) are called lattice parameters of a crystal, crystal, because those six parameters are characteristics of a crystal unit cell. 21.Prim 21. Primitive itive Cell, See again Fig. Array of lattice 3D. 3D. The smallest volume of a cell that consist only one full atom is called a primitive cell. cell. 22.If 22. If a cell consists of more than one atom, then it is not a unit unit cell.
***
©2009 Frida U. Ermawati
Page 6 of 7
C. TYPES TYPES OF CRYST CRYSTALS ALS On the basis of the shape of the unit cell (i.e. in terms of lengths of unit cell and the angle of inclination between them), crystals are classified into several different systems (See Table-1).
1.
Table-1: The seven different crystals systems. No.
System
1
Cubic
2
Tetragonal
Axial lengths & angles Three eq equal ax axes at at ri right angles. a = b = c, α = β = γ = 90° Three axes at right angles, two equal. a = b ≠ c, α = β = γ = 90°
3
Orthorhombic
Three unequal a xe xes a t ri right angles. a ≠ b ≠ c, α= β = γ = 90°
4
Monoclinic
Three un unequal ax axes, on one pa pair not at right angles. a ≠ b ≠ c, α = γ = 90° ≠ β
5
Triclinic
6
Trigonal/ Rhombohedral
Three unequal axes, unequally incl inclin ined ed and and none none at righ rightt angles. a ≠ b ≠ c, α ≠ β ≠ γ ≠ 90° Thr Three equ equal axes axes,, equ equally ally inclined
Bravais lattice Simple Body-centered Face-centered Simple
Lattice symbol P I F P
Examples Po Na, W, α-Fe Ag, Au, Pb TiO2, SnO2
Body-centered Simple Body-centered Face-centered Base-centered Simple
I P I F C P
KH2PO4 PbCO3, BaSO4 KNO3, K2SO4 α-S CaSO4.2H2O
Base-centered
C
K2MgSO4.6H2O
Simple
P
K2Cr2O7
2.
©2009 Frida U. Ermawati
Page 7 of 7
