Chapter 3: The Structure of Crystalline Solids (2)

Transcription

1 Chpter 3: The Structure of Crstlline Solids (2) Clss Eercise Drw the unit cell structure for simple cubic (SC), bodcentered cubic (BCC), nd fce-centered cubic (FCC) lttices Give coordintion number (CN) for toms in ech of those structures Give the verge number of toms in unit cell for simple cubic, bod-centered cubic, nd fce-centered cubic structures nd eplin wh Chpter 3-1

2 Three Tpes of Unit Cell Structures for Cubic Crstl Sstems CN = 6 CN = 8 CN = 12 Chpter 3-2

3 Clss Eercise Using the concepts for lttice nd unit cell, for 2D lttices (of the sme toms) given below, plese drw the 2D unit cell nd determine the coordintion number (CN) nd number of toms per unit cell (n) See red squre for unit cell CN = 2 CN (# of nerest neighbor) = 4 Averge number of toms Averge number of toms per unit cell = 41/4 = 1 per unit cell = 41/4 = 1 (The dshed squre in blue represents smmetr, but it not the smllest) (The dotted squre in green is smllest, but does not represent smmetr: it cn rotte n degree) Chpter 3-3 nd does not distinguish from other pttern s on the right)

4 Polmorphism Two or more distinct crstl structures for the sme mteril (llotrop/polmorphism) in terms of composition liquid BCC FCC BCC Iron 1538ºC -Fe 1394ºC -Fe 912ºC -Fe Crbon Dimond vs. Grphite Chpter 3-4

5 Virtul Mterils Science & Engineering (VMSE) VMSE is tool to visulie mterils science topics such s crstllogrph nd polmer structures in three dimensions Avilble in Student Compnion Site t Chpter 3-5

6 Unit Cells VMSE llows ou to view the unit cells nd mnipulte them in three dimensions Below re emples of ctul VMSE screen shots FCC Structure HCP Structure Chpter 3-6

7 ther Links Flsh 3D Atomic Structure Model Viewer Chpter 3-7

8 Crstl Structure Appliction Problem 2 Estimtion of Theoreticl Densit or Densit = = Atom Sie Totl mss of toms in unit cell Totl volume of unit cell For simple pure element, If known n = number of toms/unit cell A = tomic weight (in g/mol) V C = Volume of unit cell, Vc = 3 for cubic structure N A = Avogdro s number = toms/mol = n (A/N A ) V C Chpter 3-8

9 Crstl Structure Appliction Problem 2 Adpted from Fig. 3.2(), Cllister & Rethwisch 8e. # of toms per unit cell = Volume of unit cell Estimtion of Theoreticl Densit or R Atom Sie E: Cr (BCC unit cell) Atom mss: A = g/mol Atom rdius: R = nm n = 2 toms/unit cell = 4R/ 3 = nm Mss per mole of toms theoreticl mesured # of toms per mole = 7.18 g/cm 3 = 7.19 g/cm 3 Chpter 3-9

10 Point Coordintes in Crstl Structure c b Unit cell repeting unit tht reflects crstl smmetr Point coordintes: Specified in terms of its coordintes (frctionl length in projection ) with respect to the three edges of the unit cell, Emple: Point P, with q reflecting the projection of the point to is, r reflecting projection to is, nd s reflecting project onto is (,, is long, b, c edge of the unit cell, with unit length of, b, c, respectivel), coordinte of qrs Aes m or m not be orthogonl to ech other In some cses, some points re (geometricll) equivlent Chpter 3-10

11 Clss Eercise Coordintes of Points Specif the coordintes for three points s lbeled in the grph for (simple) cubic unit cell P ½ 0 1 Q 1 1 ½ R Lbel the points bsed on the coordintes given for cubic unit cell P Q ½ ½ 0 R 0 0 ½ P Q R Q P R Chpter 3-11

12 Directions in Crstl Structure 1. Vector repositioned (if necessr) with strting point to pss through origin. 2. Red off projections of the end point of the vector in terms of unit cell dimensions, b, nd c 3. Adjust to smllest integer vlues 4. Enclose in squre brckets, no comms [uvw] e: 1, 0, ½ 2, 0, 1 [ 201 ] direction ½, 1, ½ 1, 2, 1 [121 ] direction -1, 1, ½ -2, 2, 1 [ 221 ] direction overbr represents negtive inde For some crstl sstems, some directions re equivlent fmilies of directions, designted in the form of <uvw> e.g., for simple cubic, [100], [010], [001], [100], [010], Chpter 3-12 [001] directions re ll equivlent, s <100> fmil

13 Clss Eercise Directionl Indices Specif the directionl indices for the crstllogrph directions s lbeled in the grph v1 [102] v2 [110] v3 [201] Drw the crstllogrph directions bsed on directionl indices given below v4 [100] v5 [112] v6 [111] v 1 v 4 v 2 v 5 v6 v 3 Chpter 3-13

14 Plnes in Crstl Structure (1) Adpted from Fig. 3.10, Cllister & Rethwisch 8e. Chpter 3-14

15 Plnes in Crstl Structure (2) Miller Indices: Indices given to describe crstl plnes - Reciprocls of the (three) il intercepts for plne, clered of frctions Method 1. If crstl plne not pssing through origin, red off the intercepts of tht plne with three es of the unit cell in terms of edge length, b, c (therwise prllel plne or lterntive origin hs to be chosen so tht the plne does NT pss through the origin) 2. Tke reciprocls for ech of the three intercepts, respectivel 3. If contins non-integer, reduce to smllest integer set b multipling common integer, when pplicble 4. Enclose in prentheses, no comms i.e., (hkl) Chpter 3-15

16 Plnes in Crstl Structure (4) emple 1. Intercepts 1 1b c 2. Reciprocls 1/1 1/1 1/ Miller Indices (110) emple 1. Intercepts 1/2 b c 2. Reciprocls 1/½ 1/ 1/ Miller Indices (200) D NT reduce to (100) c c b b Chpter 3-16

17 Plnes in Crstl Structure (5) emple 1. Intercepts 1/2 1b 3/4c 2. Reciprocls 1/½ 1/1 1/¾ 2 1 4/3 3. Multipl integer of 3 4. Miller Indices (634) c b For some crstl sstems, some plnes re equivlent Fmil of Plnes {hkl}. For emple Cubic: {100} fmil includes (100), plnes (010),(001), (100), (010), (001) Chpter 3-17

18 Clss Eercise (5) Plne Miller Indices Specif the Miller indices for the crstllogrph plnes s lbeled in the grph Ais Intercepts ½ Reciprocl Miller indices (112) Drw the crstllogrph (122) plne Miller indices (122) Reciprocl 1 ½ ½ Ais Intercepts ½ ½ Chpter 3-18

19 Clss Eercise (6) Plne Miller Indices Specif the Miller indices for the crstllogrph plnes s lbeled in the grph Ais Intercepts ½ ½ Reciprocl Miller indices (220) Drw the crstllogrph (101) plne Miller indices (101) Reciprocl 1 1 Ais Intercepts Chpter 3-19