Analytical Methods for Materials

Transcription

1 Anlyticl Methods for Mterils Lesson 7 Crystl Geometry nd Crystllogrphy, Prt 1 Suggested Reding Chpters 2 nd 6 in Wsed et l. 169

2 Slt crystls N Cl Shpes of crystls cn give us clue bout tomic rrngement 170

3 Wht is crystl? Solids where toms re rrnged in periodic (i.e., repeting, symmetric, etc.) ptterns. Wht is symmetry? Describes how pttern repets within crystl. 171

4 Crystl Structure = Lttice + Motif [Bsis] From website: A lttice must be symmetric! Motifs must be rrnged symmetriclly! 172

5 One-Dimensionl Lttice T = 3 Lttice symmetry implies tht ech lttice point must hve identicl surroundings (i.e., the sme environment ). In given direction, ll lttice points must be seprted by n identicl distnce, (this bsis vector is lttice prmeter). In 1D lttice, trnsltion of n from one lttice point to nother, where n is n integer, brings you to n identicl lttice point. T = n 173

6 Two-Dimensionl Lttice y x b T = 1+2b b There re two non-colliner bsis vectors ( nd b). A trnsltion of n + pb from one lttice point to nother must bring you to n equivlent lttice point. The interxil ngle defines the reltionship between the two bsis vectors. T = n + pb 174

7 Three-Dimensionl Lttice There re 3 non-colliner bsis vectors nd 3 interxil ngles. All points cn be defined by series of vectors: T = n + pb + qc c origin b 175

8 Three-Dimensionl Lttice cont d The Bsis vectors define the shpe of the crystl. The smllest repeting unit formed by combining the bsis vectors is clled unit cell. c origin b A unit cell retins ll chrcteristics of the lttice. 176

9 Lttice Prmeters The bsis vectors re not necessrily orthogonl. c b Axis b c Inter-xil ngle 177

10 Unit Cell Shpes Unit cells in crystls hve specific shpes. We cll the shpes crystl systems [*]. They re bsed upon: 1. Highest symmetry 2. Consistency with pst convention 3. Minimized unit cell volume 4. Stisfction of miniml symmetry requirements. All crystl structures evolve from crystl systems. [*] A set of reference xes used to define the geometry of crystl nd crystl structures 178

11 Crystl Systems In 2D there re only four (4). 1. Oblique 2. Rectngulr 3. Hexgonl 4. Squre In 3D there re only seven (7). 1. Triclinic (northic) 2. Monoclinic 3. Hexgonl 4. Rhombohedrl (trigonl) 5. Orthorhombic 6. Tetrgonl 7. Cubic The crystl systems re the only possible shpes for unit cells With these shpes, you cn fill ll vilble spce nd leve no voids! 179

12 2D Crystl Systems b γ 90 OBLIQUE PARALLELOGRAM b γ b γ = 90 RECTANGLE b γ = b γ = 120 HEXAGONAL γ = b γ = 90 SQUARE γ 180

13 3D Crystl Systems Crystl System Axil Reltionships Interxil Angles Cubic = b = c = = = 90 Hexgonl = b c = = 90 ; = 120 Tetrgonl = b c = = = 90 Rhombohedrl (Trigonl) = b = c = = 90 Orthorhombic b c = = = 90 Monoclinic b c = = 90 Triclinic b c

14 Cubic = b = c = β = = 90 7 crystl systems Tetrgonl = b c = β = = 90 The minerl imges re from vrious sites on the internet Orthorhombic b c = β = = 90 fluorite rutile brite Triclinic b c 90 Hexgonl = b c = β = 90 ; = 120 Rhombohedrl = b = c = β = 90 Monoclinic b c = = 90 c copper sulfte penthydrte corundum purple fluorite orthoclse 182

15 Keep in mind Crystl shpe revels the underlying symmetry of crystl. Must plce lttice points on shpe to build up symmetric lttice. All lttice points must be identicl 183

16 Wht defines lttices nd unit cells? Symmetry limits the number of possibilities. Crystl lttices must exhibit specific miniml mount of symmetry. Ech crystl system hs certin symmetry (lttice points hve specific rrngements). 184

17 Symmetry Opertors Motions tht llow pttern to be trnsformed from n initil position to finl position such tht the initil nd finl ptterns re indistinguishble. 1. Trnsltion 2. Reflection 3. Rottion 4. Inversion (center of symmetry) 5. Roto-inversion (inversion xis) 6. Roto-reflection 7. Glide (trnsltion + reflection) 8. Screw (rottion + trnsltion) 185

18 Symmetry of Crystl Systems Crystl System Axil Reltionships Interxil Angles Minimum # of Symmetry Elements Cubic = b = c = = = 90 Hexgonl = b c = = 90 ; = 120 Tetrgonl = b c = = = 90 Rhombohedrl (Trigonl) = b = c = = 90 Orthorhombic b c = = = 90 Monoclinic b c = = 90 Four 3-fold rottion or rotoinversion xes prllel to body digonls One 6-fold rottion or rotoinversion xis prllel to z- xis One 4-fold rottion or rotoinversion xis prllel to z-xis One 3-fold rottion or rotoinversion xis prllel to z-xis Three 2-fold rottion or rotoinversion xes prllel to x,y,zxes One 2-fold rottion or rotoinversion xis prllel to y-xis Incresing symmetry Triclinic b c 90 None I d memorize these if I were you 186

19 Crystl System + Symmetric Arry of Lttice Points = Brvis Lttice We cn clssify Brvis lttices in terms of the number of lttice points in the unit cell 187

20 2D/3D Types of Lttices Primitive (P) One lttice point per unit cell Termed simple or primitive Non-primitive (multiple) More thn one lttice point per unit cell. Termed XXX-centered XXX = body, fce, or bse 188

21 Lttice Points Per Cell in 2D N 2D N interior N corner 4 corner interior 189

22 Lttice Points Per Cell in 3D N 3D N fce N corner N interior 2 8 interior fce corner 190

23 Primitive vs. Non-primitive lttices There re 4 crystl systems in 2D. Thus we cn define 4 primitive lttices in 2D. 4 primitive Brvis nets (k. lttices ) Are there more? There re 7 crystl systems in 3D. Thus we cn define 7 primitive lttices in 3D. 7 primitive Brvis lttices Are there more? OF COURSE! OF COURSE! Cn we dd dditionl lttice points to primitive lttice nd still hve lttice with the sme shpe? 191

24 Primitive vs. Non-primitive lttices Answer: YES, if we mintin symmetry. ( All lttice points must be equivlent.) b D rectngulr lttices 12 IMPOSSIBLE Internl point is not symmetric or equivlent to corners POSSIBLE Internl point is symmetric nd equivlent to corners 192

25 b Mirror imges of primitive lttice Wht if we define primitive oblique lttice (shded) rther thn centered rectngle? The primitive cell is less symmetric thn the centered rectngle. For exmple, mirror imge of the primitive unit cell is not identicl to the originl. A mirror imge of the rectngulr cell with lttice point in the center IS identicl to the originl. It hs higher symmetry! 193

26 Five 2D Brvis Lttices b γ b Primitive Oblique Centered Rectngle Primitive Hexgonl b Primitive Rectngle Primitive Squre 194

27 14 Brvis Lttices (three dimensionl) Simple cubic Cubic P (1) BCC Cubic I (2) FCC Cubic F (4) Simple tetrgonl Tetrgonl P (1) Body-centered tetrgonl Tetrgonl I (2) Simple orthorhombic Orthorhombic P (1) Body-centered orth. Orthorhombic I (2) Bse-centered orth. Orthorhombic C (2) Fce-centered orth. Orthorhombic F (4) Rhombohedrl Trigonl R (1) primitive Simple Hexgonl Hexgonl P (1) Simple monoclinic Monoclinic P (1) Bse-centered monoclinic Monoclinic C (2) Simple Triclinic Triclinic P (1) 195

28 Generl things bout lttices RECALL: You cn lwys define primitive lttice/unit cell. HOWEVER, If non-primitive cell cn be found tht lso describes symmetry of the lttice, it should be used insted. Since ll lttice points must be identicl, new lttice points cn only be plced on positions centered between primitive lttice points. Don t confuse lttice points with toms 196

29 Now you know how to define crystl in terms of symmetry opertions. Next we shll ddress reltionships between crystl plnes nd directions. Miller indices Stereogrphic projections Reciprocl spce 197