Texture and Anisotroy. Part I: Chapter 2. Description of Orientation

Transcription

1 Texture nd Anisotroy Prt I: Chter. Descrition of Orienttion

2 Prt I: Fundmentl of Orienttion Orienttion mtrix Idel Orienttion Euler nles Anle/xis of rottion Rodriues vector

3 Crystl systems Schemticlly, the reltionshi between the 7 crystl systems, 4 Brvis lttices, 3 oint rous, nd 3 sce rous s follows 7 crystl systems Lttice tyes 4 Brvis lttices Lttice symmetry 3 oint rous 3 sce rous Sce trnsltion: Mirror nd Glide Plnes +Screw xis

4 Crystl systems Crystlline mterils re serted into 7 crystl different systems. These crystl systems re most esily identified by the constrints on the cell rmeters. Cited from: René-Just Hüy, 8, Trité de minérloie.

5 Crystl structure nd symmetries

6 7 Crystl systems

7 4 Brvis lttices 4 lttice tyes 7 crystl systems

8 Exmle of JCPD crd Sce rou of no. 5

9 Sce Grou Letter Symbols

10 The next three symbols

11 Symmetry elements in S.G. symbols

12 Symmetry of Cubic P-lttice

13 Coordinte systems

14 Crystl nd smle coordintes htt://luminium.mtter.or.uk/content/html/en/defult.s?ctid&eid

15 Orienttion descritions

16 Stereorhic rojection D nd E re shericl D' nd E' re stereorhic Distnce GD' f(ρ) s ρ 9 D G s ρ D O

17 D Stereorhic rojection

18 {} Pole fiure

19 () Indexin smle coordintes Y X Y Z Z l () X l cosθ + cosθ X b + cosθ X X 3 cosθ + cosθy b + cosθ Y Y 3 cosθ + cosθz b + cosθ Z Z 3 ( h3, k3, 3) (,3,5) ( h k, ) (, 3,), c c c () X cosθ cosθ cosθ Z Z Z

20 Indexin smle coordintes Y Z (,3,5) () () X X (, 3,) ()

21 Orienttion Descrition I:{hkl}<uvw> Miller index nottion of texture comonent secifies direction to smle xes. [] c Z (hkl) b [] T hkl x uvw X [uvw] [] Y t

22 [] c Coordinte Descrition Z bc: Crystl coordintes b [] Y X XYZ: Smle coordintes [] How to determine the orienttion usin mtrix?

23 Coordinte Trnsformtion D Y b b: Crystl coordintes θ X x y θ l cosθ l sin θ cos ( x / l) Wht is the trnsformtion mtrix? XY: Smle coordintes

24 Descrition of Trnsformtion Mtrix z P y c / z' P b / y' x / x'

25 Trnsformtion of Axis Old coord. to new coord. old x y z z z new x y z y y x x

26 Trnsformtion Mtrix c b k j i c b bc P + + z y x k j i z y x xyz P + + bc TP xyz P x y z P b c z y x P z y x k i j i i i i + + z y x b P z b y b x b b k j j j i j j + + z y x c P z c y c x c c k k j k i k k + + z y x k j i z y x P + +

27 Orienttion Trnsformtion: S to C z y x w b z c y c x c z b y b x b z y x k k j k i k k j j j i j k i j i i i z y x c b Crystl coordinte smle coordinte s cs c C C

28 Orienttion Trnsformtion: C to S Crystl coordinte smle coordinte c T c s C C C cs cs w v u x P w x v x u x x k i j i i i i + + w v u y P w y v y u y y k j j j i j j + + w v u z P w z v z u z z k k j k i k k + + w v u z y x w z v z u z w y v y u y w x v x u x k k j k i k k j j j i j k i j i i i

29 Z Smle vs. Crystl Coordintes 3 Z/ND Y () () Y/TD X () crystl coordintes X/RD Smle coordintes (X, Y, Z) re defined s reference coordintes. htt://luminium.mtter.or.uk/content/html/en

30 Rottion (Orienttion) mtrix An orienttion is defined s the osition of the crystl coordinte system with resect to the secimen coordinte system. C C C S nle between crystl xis [] nd X cosα cosα cosα 3 cosβ cosβ cosβ 3 cos γ cos γ cos γ nle between crystl xis [] with Y nle between crystl xis [] with X

31 Smle to Crystl smle crystl ND RD TD s cs c C C Definition of n Axis Trnsformtion: Y b Z c [] [] [] X n t b n t b n t b ij

32 ),, ( ˆ w v u w v u + + b ),, ( ˆ l k h l k h + + n b n b n t ˆ ˆ ˆ ˆ ˆ n t b n t b n t b Smle Crystl ij Determintion of mtrix from Miller Indices

33 Miller Indices vs. Mtrix [] direction 3 [] direction 3 [] direction

34 Miller Indices vs. Mtrix The columns reresent comonents of three other unit vectors: [uvw] RD TD ND (hkl) 3 3 Where the Columns re the direction cosines (i.e. hkl or uvw) for the RD, TD nd Norml directions in the crystl coordinte system

35 Orienttion of lne s 3 R (sin αcosβ) s + (sin αsinβ) s + (cosα) s 3 smle coordintes s s crystl coordintes R ( Xc + Yc + Zc 3 ) N

36 Definition of orienttion smle crystl sin αcosβ sin αsinβ cosα X Y Z / / / N N N. determine the ole fiure nles α nd β. index the ole 3. determine the orienttion mtrix

37 Inverse ole fiure δ δ γ δ γ s s s Z Y X cos sin sin cos sin 3 ) (cos ) sin (sin ) cos (sin c c c s i i i i i i γ + δ γ + δ γ smle crystl

38 Non-cubic Crystl Coordinte systems X Y Z Cubic ' b c b' c'

39 Trnsformtion mtrix c b orthorhombic C Lv T v ' c b b b c b c c b' c'

40 Non-cubic Crystl Coordinte systems hexonl 3 / / c 3 / / 3 / / c 3 / / c c ] [ [] b c b

41 triclinic Trnsformtion of n zone xis l l l l l l l l l bcos γ c cosβ bsin γ c(cosα cosβcos γ) / sin γ [( ) ] + cosα cos β cosγ cos α cos β cos γ / / sin γ 33 c v Lv C T L bcos γ bsin γ c cosβ c(cosα cosβcos γ) / sin γ [( ) ] c + cosαcosβcos γ cos α cos β cos γ / / sin γ

42 Orienttion Descrition II: Bune Euler nles Rottion (φ ): rotte xes (nticlockwise) bout the (smle) 3 [ND] xis; Z. Rottion (Φ): rotte xes (nticlockwise) bout the (rotted) xis [] xis; X. Rottion 3 (φ ): rotte xes (nticlockwise) bout the (crystl) 3 [] xis; Z. w X Z u v Y

43 Orienttion Descrition II: Bune Euler nles

44 Rottion Mtrix in D lne: y y v v ʹ cosθ sinθ v sinθ cosθ N.B. Pssive Rottion/ Trnsformtion of Axes x, y old xes; x,y new xes θ x x

45 Bune Euler nles to Mtrix, st Rottion ϕ cosφ sin φ sin φ cosφ

46 Bune Euler nles to Mtrix, st Rottion Φ cosφ sin Φ sin Φ cosφ

47 Bune Euler nles to Mtrix, 3st Rottion ϕ cosφ sin φ sin φ cosφ

48 Princile of Bune Euler Anles e 3 e 3 Z smle ND [] z crystl e 3 e [] 3 φ y crystl e φ e e e Y smle TD htt:// vimpbvqjrswy&feturerelte x Φ crystl e e e e X smle RD []

49 Bune Anles vs. Mtrix [uvw] cosϕ cosϕ sinϕ sinϕ cosφ sinϕ cosϕ +cosϕ sinϕ cosφ sinϕ sin Φ sinϕ sinϕ cosϕ sin Φ +cosϕ cosϕ cosφ sinϕ sinφ cosϕ sinφ cosφ cosϕ sinϕ sinϕ cosϕ cosφ ϕ Φ ϕ (hkl)

50 Summry of Orienttion Descritions [uvw] (hkl) [uvw] (hkl) ij Crystl b b b 3 Smle t t t 3 n n n 3 cosϕ cosϕ sinϕ sinϕ cosφ cosϕ sinϕ sinϕ cosϕ cosφ sinϕ cosϕ +cosϕ sinϕ cosφ sinϕ sin Φ sinϕ sinϕ cosϕ sin Φ +cosϕ cosϕ cosφ sinϕ sinφ cosϕ sinφ cosφ

51 Miller indices from Euler nle mtrix Comre the indices mtrix with the Euler nle mtrix. u v h nsin Φsinϕ k nsin Φcosϕ l ncosφ n ʹ cosϕ cosϕ sinϕ sinϕ cos Φ ( ) ( ) n ʹ cosϕ sinϕ sinϕ cosϕ cos Φ w n, n fctors to mke inteers n ʹ sinφ sinϕ

52 Euler nles from Miller indices l Inversion of cos Φ h + k + l the revious k reltions: cosϕ h + k w h sinϕ + k + l u + v + w h + k Cution: it is more relible to o from Miller indices to n orienttion mtrix, nd then clculte the Euler nles. Extr credit: show tht the followin surmise is correct. If lne, hkl, is chosen in the lower hemishere, l<, show tht the Euler nles re incorrect.

53 Other Euler nle definitions A confusin sect of texture nlysis is tht there re multile definitions of the Euler nles. Definitions ccordin to Bune, Roe nd Kocks re in common use. Comonents hve different vlues of Euler nles deendin on which definition is used. The Bune definition is the most common. The differences between the definitions re bsed on differences in the sense of rottion, nd the choice of rottion xis for the second nle.

54 3D Euler sce

55 3D Euler sce

56 Smle nd crystl symmetries

57 Symmetry element

58 Symmetry element Zone Zone Zone Zone Zone 3

59

60

61

62 (37, 36, 6) (33, 5, 56) (, 43, 333) (53, 74, 34) (333,, 8) (, 43, 53)

63 Crystl Symmetry i C i

64 Crystl Symmetry -

65 Mtrix reresenttion of the rottion oint rous for 43 Mtrix number [ ] [ ] [ ] Mtrix number [ ] [ -] [ ] Mtrix number 3 [ ] [ - ] [ -] Mtrix number 4 [ ] [ ] [ - ] Mtrix number 5 [ -] [ ] [ ] Mtrix number 6 [ ] [ ] [ - ] Mtrix number 7 [ - ] [ ] [ -] Mtrix number 8 [ - ] [ - ] [ ] Tken from subroutine by D. Rbe Mtrix number 9 [. ] [ - ] [ ] Mtrix number [ - ] [ ] [ ] Mtrix number [ - ] [ ] [ - ] Mtrix number [ ] [ - ] [ - ] Mtrix number 3 [ - ] [ -] [ ] Mtrix number 4 [ -] [ ] [ - ] Mtrix number 5 [ ] [ -] [ - ] Mtrix number 6 [ -] [ - ] [ ] Mtrix number 7 [ ] [ ] [ ] Mtrix number 8 [ ] [ ] Mtrix number 9 [ ] [ ] [ -] Mtrix number [ - ] [ ] [ ] Mtrix number [ ] [ - ] [ ] Mtrix number [ - ] [ -] [ - ] Mtrix number 3 [ -] [ - ] [ - ] Mtrix number 4 [ - ] [ - ]

66 4 symmetry mtrix for cubic (3)[63-4]

67 How to use symmetry oertor? Goss: {}<>: Pre-multily by z-did: which is {--}<>

68 Smle Symmetry S j j

69 Smle Symmetry Torsion, sher: Monoclinic,. Rollin, lne strin comression, mmm. Otherwise, Axisymmetric: C triclinic.

70 Symmetry Reltionshis Note tht the result of lyin ny vilble oertor is equivlent to (hysiclly indistinuishble in the cse of crystl symmetry) from the strtin confiurtion (not mthemticlly equl to!). Also, if you ly smle symmetry oertor, the result is enerlly hysiclly different from the strtin osition. Why?! Becuse the smle symmetry is only sttisticl symmetry, not n exct, hysicl symmetry. ij C i S j NB: if one writes n orienttion s n ctive rottion (s in continuum mechnics), then the order of liction of symmetry oertors is reversed: remultily by smle, nd ostmultily by crystl!

71 Section Sizes: Crystl - Smle Cubic-Orthorhombic: φ 9, Φ 9, φ 9 Cubic-Monoclinic: φ 8, Φ 9, φ 9 Cubic-Triclinic: φ 36, Φ 9, φ 9 But, these limits do not delinete fundmentl zone.

72 Orienttion descrition III: Anle/xis []/5 o [-]/9 o

73 Exmle of Orienttion (37 o, 37 o, 7 o ) ( o, o, o ) [8,-5,-6] 45 o 45 o

74 Orienttion by nle/xis rottion r ( r, r, r ) 3 ND TD [] [] RD [] smle coordintes C c cs C s crystl coordintes

75 Anle/xis of rottion /θ r ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ cos cos sin cos sin cos sin cos cos cos sin cos sin cos sin cos cos cos r r r r r r r r r r r r r r r r r r r r r

76 Anle/xis of rottion r /θ Tr( ) cosθ 3 3 r sin θ 3 3 r sin θ r3 sin θ Tr(): the trce of mtrix ij (i, j,,3): the elements of The rottion is described s riht-hnded screw oertion nd θ is lwys ositive. A netive nle is equivlent to chnin the sin of r.

77 Anle/xis of misorienttion A misorienttion is clculted from the orienttions of rin nd rin by M

78 Misorienttion by nle/xis rottion r ( r, r, r ) 3 [] [] [] [] [] [] crystl coordintes of rin M crystl coordintes of rin

79 Orienttion & misorienttion Orienttion: Reference (smle xes) Misorienttion: Reference (rin ) Grin Grin Grin Grin

80 Exmle of Misorienttion (36.4 o, 5.4 o, 76.4 o ) (7.9 o,.4 o,.6 o ) 6 o [-] 6 o

81 Reresenttion of Orienttion r x cosψsin ϑ θ r y sin ψsin ϑ r z cosθ ϑ ψ

82 Rodriues vector The Rodriues vector R combines the nle nd xis of rottion into mthemticl entity. tn θ r R tn tn tn 3 3 θ θ θ r R r R r R

83 Fundmentl zone

84 Prmeters of Rodriues sce

85 Proerties of Rodriues sce The xis of rottion ives the direction of the R vector. Rottion bout the sme xis of rottion lie on striht line tht sses throuh the oriin. The nle of rottion ives the lenth of the R vector. Smll-nle boundries cluster close to the oriin. A fiber texture lies on striht line tht in enerl doesn t ss throuh the oriin. The edes of zones in Rodriues sce re striht lines, nd the fces of zones re lnr.

86 Aend: Symmetry in φ for cubic ϕ 45 o 9 o 35 o 8 o 5 o 7 o 35 o 36 o 45 o 9 o 35 o 8 o Φ ϕ

87 Aend: Symmetry in φ for cubic ϕ 45 o 9 o 35 o 8 o 5 o 7 o 35 o 36 o 45 o 9 o 35 o 8 o Φ ϕ 45

88 Aend: Symmetry in φ for hexonl ϕ 3 o 6 o 9 o o 5 o 8 o o 4 o 7 o 3 o 33 o 36 o 3 o 6 o 9 o o 5 o 8 o Φ ϕ