Vectors (defines a particular direction – plane normal). • Miller Indices (defines a particular plane). • relation to diffraction. • 3-index for cubic and 4-index notation
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1MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08Crystallographic Points, Directions, and Planes.ISSUES TO ADDRESS¥ How to define points, directions, planes, as well aslinear, planar, and volume densitiesÐ Define basic terms and give examples of each:¥ Points (atomic positions)¥ Vectors (defines a particular direction – plane normal)¥ Miller Indices (defines a particular plane)¥ relation to diffraction ¥ 3-index for cubic and 4-index notation for HCPMSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08avbvcvPoints, Directions, and Planes in Terms of Unit Cell VectorsAll periodic unit cells may be described viathese vectors and angles, if and only if¥ a, b, and c define axes of a 3D coordinate system.¥ coordinate system is Right-Handed! But, we can define points, directions andplanes with a ÒtripletÓ of numbers in unitsof a, b, and c unit cell vectors .For HCP we need a ÒquadÓ of numbers, aswe shall see.MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08POINT Coordinates To define a point within a unit cellÉ.Express the coordinates uvw as fractions of unit cell vectors a, b, and c(so that the axes x, y, and z do not have to be orthogonal). avbvcvoriginpt. coord.x (a)y (b)z ( c)0001001111/2 01/2 pt.MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08Crystallographic DirectionsProcedure:1.Any line (or vector direction) is specified by 2 points.¥The first point is, typically, at the origin (000).2.Determine length of vector projection in each of 3 axes inunits (or fractions) of a, b, and c.¥X (a), Y(b), Z(c) 1 1 03.Multiply or divide by a common factor to reduce thelengths to the smallest integer values, u v w.4.Enclose in square brackets: [u v w]: [110] direction.a b c DIRECTIONS will help define PLANES (Miller Indices or plane normal ). [1 1 0]5. Designate negative numbers by a bar ¥Pronounced Òbar 1Ó, Òbar 1Ó, ÒzeroÓ direction .6. ÒFamilyÓ of [110] directions is designated as <110>.
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2MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08Self-Assessment Example 1: What is crystallographic direction? a b c Along x: 1 aAlong y: 1 bAlong z: 1 c[1 1 1]DIRECTION =Magnitude alongXYZ MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08Self-Assessment Example 2:(a) What is the lattice point given by point P? (b) What is crystallographic directionfor the origin to P?The lattice direction [132] from the origin.Example 3: What lattice direction does the lattice point 264 correspond? [1 12] !112MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08Symmetry Equivalent Directions Note : for some crystal structures, differentdirections can be equivalent. e.g. For cubic crystals, the directions are allequivalent by symmetry: [1 0 0], [ 0 0], [0 1 0], [0 0], [0 0 1], [0 0 ]111Families of crystallographic directions e.g. <1 0 0>Angled brackets denote a family of crystallographic directions. MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08Families and Symmetry: Cubic Symmetry xyz(100)Rotate 90o about z-axisxyz(010)xyz(001)Rotate 90o about y-axis Similarly for otherequivalent directionsSymmetry operation cangenerate all the directions within in a family.
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3MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08Designating Lattice PlanesWhy are planes in a lattice important? (A) Determining crystal structure * Diffraction methods measure the distance between parallel lattice planes of atoms.¥ This information is used to determine the lattice parameters in a crystal. * Diffraction methods also measure the angles between lattice planes.(B) Plastic deformation * Plastic deformation in metals occurs by the slip of atoms past each other in the crystal. * This slip tends to occur preferentially along specific crystal-dependent planes.(C) Transport Properties * In certain materials, atomic structure in some planes causes the transport of electrons and/or heat to be particularly rapid in that plane, and relatively slow not in the plane.¥ Example: Graphite: heat conduction is more in sp2-bonded plane.¥ Example: YBa2Cu3O7 superconductors: Cu-O planes conduct pairs of electrons(Cooper pairs) responsible for superconductivity, but perpendicular insulating. + Some lattice planes contain only Cu and OMSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08How Do We Designate Lattice Planes? Example 1Planes intersects axes at:¥ a axis at r= 2¥ b axis at s= 4/3¥ c axis at t= 1/2 How do we symbolically designate planes in a lattice?Possibility #1: Enclose the values of r, s, and t in parentheses (r s t) Advantages: ¥ r, s, and t uniquely specify the plane in the lattice, relative to the origin.¥ Parentheses designate planes, as opposed to directions given by [] Disadvantage:¥ What happens if the plane is parallel to — i.e. does not intersect— one of the axes?¥ Then we would say that the plane intersects that axis at ! !¥ This designation is unwieldy and inconvenient. MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08How Do We Designate Lattice Planes? Planes intersects axes at:¥ a axis at r= 2¥ b axis at s= 4/3¥ c axis at t= 1/2 How do we symbolically designate planes in a lattice?Possibility #2: THE ACCEPTED ONE1. Take the reciprocal of r, s, and t. ¥Here: 1/r = 1/2 , 1/s = 3/4 , and 1/r = 22. Find the least common multiple that converts all reciprocals to integers.¥With LCM = 4, h = 4/r = 2 , k= 4/s = 3 , and l= 4/r = 83. Enclose the new triple (h,k,l) in parentheses: (238)4. This notation is called the Miller Index. * Note: If a plane does not intercept an axes (I.e., it is at !), then you get 0.* Note: All parallel planes at similar staggered distances have the same Miller index.MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08Self-Assessment Example What is the designation of this plane in Miller Index notation?What is the designation of the top face of the unit cell in Miller Index notation?
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4MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08Families of Lattice Planes The Miller indices (hkl) usually refer to the plane thatis nearest to the origin without passing through it.¥ You must always shift the origin or move the planeparallel, otherwise a Miller index integer is 1/0!¥ Sometimes (hkl) will be used to refer to any other planein the family, or to the family taken together.¥ Importantly, the Miller indices (hkl) is the same vectoras the plane normal!Given any plane in a lattice, there is a infinite set of parallel lattice planes(or family of planes) that are equally spaced from each other.¥ One of the planes in any family always passes through the origin.MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08zxyLook down this direction(perpendicular to the plane)Crystallographic Planes in FCC: (100)d100=aDistance between (100) planesÉ between (200) planes d200=a2MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08Crystallographic Planes in FCC: (110)d110=a22Distance between (110) planesMSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08Crystallographic Planes in FCC: (111)zxyLook down this direction(perpendicular to the plane)d111=a33Distance between (111) planes
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5MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08Note: similar to crystallographic directions, planes that are parallel toeach other, are equivalentMSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08Comparing Different Crystallographic Planes-11For (220) Miller Indexed planes you are getting planes at 1/2, 1/2, !.The (110) planes are not necessarily (220) planes!For cubic crystals: Miller Indices provide you easymeasure of distance between planes.d110=a12+12+02=a2=a22Distance between (110) planesMSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08Directions in HCP Crystals1.To emphasize that they are equal, a and b is changed to a1 and a2.2.The unit cell is outlined in blue.3.A fourth axis is introduced (a3) to show symmetry.¥Symmetry about c axis makes a3 equivalent to a1 and a2.¥Vector addition gives a3 = Ð( a1 + a2).4.This 4-coordinate system is used: [a1 a2 Ð( a1 + a2) c]MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08Directions in HCP Crystals: 4-index notationExampleWhat is 4-index notation for vector D? ¥ Projecting the vector onto the basal plane, it liesbetween a1 and a2 (vector B is projection).¥ Vector B = (a1 + a2), so the direction is [110] incoordinates of [a1 a2 c], where c-intercept is 0.¥ In 4-index notation, because a3 = Ð( a1 + a2), thevector B is since it is 3x farther out.¥ In 4-index notation c = [0001], which must beadded to get D (reduced to integers) D =[1123]Self-Assessment Test: What is vector C?Easiest to remember: Find the coordinate axes that straddle the vectorof interest, and follow along those axes (but divide the a1, a2, a3 part of vectorby 3 because you are now three times farther out!). 13[112 0]Check w/ Eq. 3.7 or just use Eq. 3.7 a2Ð2a3B without 1/3
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6MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08Directions in HCP Crystals: 4-index notationExampleWhat is 4-index notation for vector D? ¥ Projection of the vector D in units of [a1 a2 c] givesuÕ=1, vÕ=1, and wÕ=1. Already reduced integers.¥ Using Eq. 3.7: [112 3] [13132 31]Check w/ Eq. 3.7: a dot-product projection in hex coords. u=13[2u’!v’]v=13[2v’!u’]w=w’ u=13[2(1)!1]=13v=13[2(1)!1]=13w=w’=1¥ In 4-index notation: ¥ Reduce to smallest integers:After some consideration, seems just using Eq. 3.7 most trustworthy. MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08Miller Indices for HCP Planes As soon as you see [1100], you will knowthat it is HCP, and not [110] cubic!4-index notation is more important for planes in HCP, in orderto distinguish similar planes rotated by 120o.1.Find the intercepts, r and s, of the plane with any twoof the basal plane axes (a1, a2, or a3), as well as theintercept, t, with the c axes.2.Get reciprocals 1/r, 1/s, and 1/t. 3.Convert reciprocals to smallest integers in same ratios.4.Get h, k, i , l via relation i = – (h+k), where h isassociated with a1, k with a2, i with a3, and l with c.5.Enclose 4-indices in parenthesis: (h k i l) .Find Miller Indices for HCP: rstMSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08Miller Indices for HCP Planes What is the Miller Index of the pink plane?1.The planeÕs intercept a1, a3 and cat r=1, s=1 and t= !, respectively. 1.The reciprocals are 1/r = 1, 1/s = 1, and 1/t = 0.2.They are already smallest integers. 3.We can write (h k i l) = (1 ? 1 0).4.Using i = – (h+k) relation, k=Ð2.5.Miller Index is(1210)MSE 280: Introduction to Engineering Materials ©D.D. Johnson 2004, 2006-08Yes, YesÉ.we can get it without a3!1.The planeÕs intercept a1, a2 and cat r=1, s=Ð1/2 and t= !, respectively. 1.The reciprocals are 1/r = 1, 1/s = Ð2, and 1/t = 0.2.They are already smallest integers. 3.We can write (h k i l) =4.Using i = – (h+k) relation, i=1.5.Miller Index is (12 10) (12 ?0)But note that the 4-index notation is uniqueÉ.Consider all 4 intercepts:¥ plane intercept a1, a2, a3 and c at 1, Ð1/2, 1, and !, respectively. ¥ Reciprocals are 1, Ð2, 1, and 0.¥ So, there is only 1 possible Miller Index is (12 10)
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