Crystallographic Points, Directions, and Planes. ISSUES TO ADDRESS... a v. Points, Directions, and Planes in Terms of Unit Cell Vectors

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1 Crstallographic Points, Directions, and Planes. Points, Directions, and Planes in Terms of Unit Cell Vectors ISSUES TO ADDRESS... How to define points, directions, planes, as well as linear, planar, and volume densities Define basic terms and give eamples of each: Points (atomic positions) Vectors (defines a particular direction - plane normal) Miller Indices (defines a particular plane) relation to diffraction -inde for cubic and 4-inde notation for HCP a v c v b v All periodic unit cells ma be described via these vectors and angles, if and onl if a, b, and c define aes of a D coordinate sstem. coordinate sstem is Right-Handed! But, we can define points, directions and planes with a triplet of numbers in units of a, b, and c unit cell vectors. For HCP we need a quad of numbers, as we shall see. POINT Coordinates Crstallographic Directions To define a point within a unit cell. Epress the coordinates uvw as fractions of unit cell vectors a, b, and c (so that the aes,, and do not have to be orthogonal). c Procedure: 1. An line (or vector direction) is specified b 2 points. The first point is, tpicall, at the origin (000). a v c v b v origin pt. pt. coord. (a) (b) (c) /2 0 1/2 a b 2. Determine length of vector projection in each of aes in units (or fractions) of a, b, and c. X (a), Y(b), Z(c) Multipl or divide b a common factor to reduce the lengths to the smallest integer values, u v w. 4. Enclose in square brackets: [u v w]: [110] direction. 5. Designate negative numbers b a bar [1 10] Pronounced bar 1, bar 1, ero direction. 6. Famil of [110] directions is designated as <110>. DIRECTIONS will help define PLANES (Miller Indices or plane normal). 1

2 Self-Assessment Eample 1: What is crstallographic direction? Self-Assessment Eample 2: c Along : 1 a Magnitude along X (a) What is the lattice point given b point P? 112 a b Along : 1 b Along : 1 c Y Z (b) What is crstallographic direction for the origin to P? [1 12] DIRECTION = [1 1 1] Eample : What lattice direction does the lattice point 264 correspond? The lattice direction [12] from the origin. Smmetr Equivalent Directions Families and Smmetr: Cubic Smmetr Note: for some crstal structures, different directions can be equivalent. e.g. For cubic crstals, the directions are all equivalent b smmetr: (100) Rotate 90 o about -ais (010) [1 0 0], [ 1 0 0], [0 1 0], [0 1 0], [0 0 1], [0 0 1 ] Rotate 90 o about -ais (001) Families of crstallographic directions e.g. <1 0 0> Angled brackets denote a famil of crstallographic directions. Smmetr operation can generate all the directions within in a famil. Similarl for other equivalent directions 2

3 Designating Lattice Planes Wh are planes in a lattice important? (A) Determining crstal structure * Diffraction methods measure the distance between parallel lattice planes of atoms. This information is used to determine the lattice parameters in a crstal. * Diffraction methods also measure the angles between lattice planes. (B) Plastic deformation * Plastic deformation in metals occurs b the slip of atoms past each other in the crstal. * This slip tends to occur preferentiall along specific crstal-dependent planes. (C) Transport Properties * In certain materials, atomic structure in some planes causes the transport of electrons and/or heat to be particularl rapid in that plane, and relativel slow not in the plane. Eample: Graphite: heat conduction is more in sp 2 -bonded plane. Eample: YB Cu O 7 superconductors: Cu-O planes conduct pairs of electrons (Cooper pairs) responsible for superconductivit, but perpendicular insulating. + Some lattice planes contain onl Cu and O Eample 1 How Do We Designate Lattice Planes? Planes intersects aes at: a ais at r= 2 b ais at s= 4/ c ais at t= 1/2 How do we smbolicall designate planes in a lattice? Possibilit #1: Enclose the values of r, s, and t in parentheses (r s t) Advantages: r, s, and t uniquel specif the plane in the lattice, relative to the origin. Parentheses designate planes, as opposed to directions given b [...] Disadvantage: What happens if the plane is parallel to --- i.e. does not intersect--- one of the aes? Then we would sa that the plane intersects that ais at! This designation is unwield and inconvenient. How Do We Designate Lattice Planes? Planes intersects aes at: a ais at r= 2 b ais at s= 4/ c ais at t= 1/2 Self-Assessment Eample What is the designation of this plane in Miller Inde notation? How do we smbolicall designate planes in a lattice? Possibilit #2: THE ACCEPTED ONE 1. Take the reciprocal of r, s, and t. Here: 1/r = 1/2, 1/s = /4, and 1/r = 2 2. Find the least common multiple that converts all reciprocals to integers. With LCM = 4, h = 4/r = 2, k= 4/s =, and l= 4/r = 8. Enclose the new triple (h,k,l) in parentheses: (28) 4. This notation is called the Miller Inde. What is the designation of the top face of the unit cell in Miller Inde notation? * Note: If a plane does not intercept an aes (I.e., it is at ), then ou get 0. * Note: All parallel planes at similar staggered distances have the same Miller inde.

4 Families of Lattice Planes Crstallographic Planes in FCC: (100) Given an plane in a lattice, there is a infinite set of parallel lattice planes (or famil of planes) that are equall spaced from each other. One of the planes in an famil alwas passes through the origin. The Miller indices (hkl) usuall refer to the plane that is nearest to the origin without passing through it. You must alwas shift the origin or move the plane parallel, otherwise a Miller inde integer is 1/0! Sometimes (hkl) will be used to refer to an other plane in the famil, or to the famil taken together. Distance between (100) planes d 100 = a Importantl, the Miller indices (hkl) is the same vector as the plane normal! Look down this direction (perpendicular to the plane) between (200) planes d 200 = Crstallographic Planes in FCC: (110) Crstallographic Planes in FCC: (111) Look down this direction (perpendicular to the plane) Distance between (110) planes d 110 = 2 Distance between (111) planes d 111 = a 4

5 Comparing Different Crstallographic Planes Note: similar to crstallographic directions, planes that are parallel to each other, are equivalent 1-1 Distance between (110) planes a d 110 = = 2 = 2 For (220) Miller Indeed planes ou are getting planes at 1/2, 1/2,. The (110) planes are not necessaril (220) planes! For cubic crstals: Miller Indices provide ou eas measure of distance between planes. Directions in HCP Crstals 1. To emphasie that the are equal, a and b is changed to a 1 and. 2. The unit cell is outlined in blue.. A fourth ais is introduced (a ) to show smmetr. Smmetr about c ais makes a equivalent to a 1 and. Vector addition gives a = ( a 1 + ). 4. This 4-coordinate sstem is used: [a 1 ( a 1 + ) c] Directions in HCP Crstals: 4-inde notation Eample What is 4-inde notation for vector D? Projecting the vector onto the basal plane, it lies between a 1 and (vector B is projection). Vector B = (a 1 + ), so the direction is [110] in coordinates of [a 1 c], where c-intercept is 0. 2a In 4-inde notation, because a = ( a 1 + ), the vector B is 1 [1120] since it is farther out. In 4-inde notation c = [0001], which must be added to get D (reduced to integers) D = [112] B without 1/ Check w/ Eq..7 or just use Eq..7 Easiest to remember: Find the coordinate aes that straddle the vector of interest, and follow along those aes (but divide the a 1,, a part of vector b because ou are now three times farther out!). Self-Assessment Test: What is vector C? 5

6 Directions in HCP Crstals: 4-inde notation Miller Indices for HCP Planes Eample Check w/ Eq..7: a dot-product projection in he coords. What is 4-inde notation for vector D? Projection of the vector D in units of [a 1 c] gives u =1, v =1, and w =1. Alread reduced integers. Using Eq..7: u = 1 [2u' v'] v = 1 [2v' u'] w =w' u = 1 [2(1) 1] = 1 v = 1 [2(1) 1] = 1 w =w'= 1 In 4-inde notation: [ ] Reduce to smallest integers: [112 ] r 4-inde notation is more important for planes in HCP, in order to distinguish similar planes rotated b 120 o. t s As soon as ou see [1100], ou will know that it is HCP, and not [110] cubic! Find Miller Indices for HCP: 1. Find the intercepts, r and s, of the plane with an two of the basal plane aes (a 1,, or a ), as well as the intercept, t, with the c aes. 2. Get reciprocals 1/r, 1/s, and 1/t.. Convert reciprocals to smallest integers in same ratios. 4. Get h, k, i, l via relation i = - (h+k), where h is associated with a 1, k with, i with a, and l with c. 5. Enclose 4-indices in parenthesis: (h k i l). After some consideration, seems just using Eq..7 most trustworth. Miller Indices for HCP Planes What is the Miller Inde of the pink plane? 1. The plane s intercept a 1, a and c at r=1, s=1 and t=, respectivel. 1. The reciprocals are 1/r = 1, 1/s = 1, and 1/t = The are alread smallest integers.. We can write (h k i l) = (1? 1 0). 4. Using i = - (h+k) relation, k= Miller Inde is (1210) Yes, Yes.we can get it without a! 1. The plane s intercept a 1, and c at r=1, s= 1/2 and t=, respectivel. 1. The reciprocals are 1/r = 1, 1/s = 2, and 1/t = The are alread smallest integers.. We can write (h k i l) = 4. Using i = - (h+k) relation, i=1. 5. Miller Inde is (12 10) (12?0) But note that the 4-inde notation is unique.consider all 4 intercepts: plane intercept a 1,, a and c at 1, 1/2, 1, and, respectivel. Reciprocals are 1, 2, 1, and 0. So, there is onl 1 possible Miller Inde is (1210) 6

7 Basal Plane in HCP Another Plane in HCP Name this plane Parallel to a 1, and a So, h = k = i = 0 Intersects at = 1 plane = (0001) a a +1 in a 1 a 1-1 in a 1 h = 1, k = -1, i = -(1+-1) = 0, l = 0 ( ) plane (1 1 1) plane of FCC SUMMARY Crstal Structure can be defined b space lattice and basis atoms (lattice decorations or motifs). Onl 14 Bravais Lattices are possible. We focus onl on FCC, HCP, and BCC, i.e., the majorit in the periodic table. ( ) plane of HCP SAME THING!* We now can identif and determined: atomic positions, atomic planes (Miller Indices), packing along directions (LD) and in planes (PD). We now know how to determine structure mathematicall. So how to we do it eperimentall? DIFFRACTION. a a 1 7