Whose Fault is it? Tamás Ungár. Department of Materials Physics, Eötvös University Budapest, Hungary. The Power of Powder Diffraction

Transcription

1 Whose Fault is it? Tamás Ungár Department of Materials Physics, Eötvös University Budapest, Hungary The Power of Powder Diffraction Erice-2011, 2-12 June, Erice, Italy

2 The Effect of Faulting and Twinning on the Profiles of Diffraction Peaks Tamás Ungár Department of Materials Physics, Eötvös University Budapest, Hungary The Power of Powder Diffraction Erice-2011, 2-12 June, Erice, Italy

3 Twinning in NiTi nanograins T. Waitz, V. Kazykhanov, H.P. Karnthaler, Acta Materialia 52 (2004) Twinned nanocrystalline grains streaking is typical for faults

4 T. Waitz, V. Kazykhanov, H.P. Karnthaler: Martensitic phase transformations in nanocrystalline NiTi studied by TEM Acta Materialia 52 (2004) Perfect matching along the twin plane Bright field image of a twinned grain. HRTEM image of a grain containing two twin related variants: RT1 & RT2

5 Twinning in NiTi nanograins T. Waitz, H.P. Karnthaler, Acta Materialia 52 (2004) Devitrified naocrystalline particles twin related R1-R2-R3 phases Dark-Field with R2 reflection Dark-Field with R1 reflection

6 Ni-Ti shape-memory alloys deform by twinning T. Waitz, T. Antretter, F.D. Fischer, N.K. Simha, H.P. Karnthaler: J. Mech. Phys. Solids, 55 (2007)

7 Planar Defects in Magnesium-Fluorogermanate P. Kunzmann in J.M. Cowley, Acta Cryst. (1976). A32, 83 Streaking normal to the fault planes

8 C A B C B C A A A B B B C C C C B B B B A A A A fcc sequence Intrinsic SF Extrinsic SF Twins Faulting along the close-packed planes in fcc or hcp crystals

9 What about hcp metals like: Mg, Ti, Zr, Be, Zn? and their alloys

10 Ti-Al hcp-do 19 K. Kishida, Y. Takahama, H. Inui, Acta Mater, 52 (2004) prismatic planes pyramidal planes optical micrographs

11 Ti-Al hcp-do 19 K. Kishida, Y. Takahama, H. Inui, Acta Mater, 52 (2004) pyramidal planes

12 Ti-Al hcp-do 19 K. Kishida, Y. Takahama, H. Inui, Acta Mater, 52 (2004) Diffraction spots of mother and twin crystal never coincide: non-merohedral

13 Ti-6Al-4V G.G.Yapici, I.Karaman,Z.P.Luo, Acta Mater, 54 (2006) 3755 Deformation twins along the 10.1 pyramidal plane -

14 Twinning on pyramidal planes in hcp crystals {10.2} {10.1} non-merohedral L. Wu, A. Jain, D.W. Brown, G.M. Stoica, S.R. Agnew, B. Clausen, D.E. Fielden, P.K. Liaw Acta Materialia 56 (2008) I. Kim,W.S. Jeong,J. Kim,K.T. Park, D.H. Shin Scripta Materialia 45 (2001)

15 Philosophy of the present line-profile analysis Bottom-up approach profile-functions are created by theoretical methods based on real lattice defects diffraction patterns are fitted by patterns constructued by using defect-related profile-functions

16 Philosophy of line profile analysis I meas Measured pattern <=> I size I strain + BG I planar faults Physically modeled I instr Experimentally determined Non-linear, least squares fitting I Strain :, M, q M = R e 1/2 I Size I planar : m, : or

17 Strain-broadening: strain-profile is given by: < g,l2 > mean-square-strain where the strain Fourier coefficients are:

18 Dislocation-model for < g,l2 > : Krivoglaz-Wilkens [1970]: 2 L,g C b 4 = f() 2 b : Burgers vector : dislocation density C : Contrast factor of dislocations f() : Wilkens function [1970] = L/R e

19 The Wilkens function [1970] = L/R e 8 f() ~ 1/ f () 0 ~ ln()

20 Strain-profiles normalized 1,0 I/I MAX 0,5 M << 1, ~Lorentzian M >> 1 ~Gaussian 0, d*/d* 0.5

21 Strain profile: inverse Fourier transform of the strain Fourier coefficients I D (s) = exp{ L 2 g 2 < g,l2 > }exp(2ils) dl where: < g,l 2 >= (b/2) 2 C f()

22 Size profile: I S assuming log-normal size distribution: m: median : variance shape-anisotropy can be allowed for

23 Profile functions for faulting and twinning

24 Line broadening from streaking

25 Lattice planes F g

26 Coordinate systems H i, K i a3 b l a 1 a 2 b h b k tricline

27 Intensity [a.u.] Intensity distribution along the (H i,k i )=(-2 1) streaks DIFFaX, Treacy MMJ, Newsam JM, Deem MW, Proc. Roy. Soc. London A, (1991) 433, Twinning on Streak -21 2% twin density 6% twin density twinning on pyramidal planes NOT line-profiles yet 2 3 L

28 Overlapping peaks along the same streak: asymmetry L almost symmetric peaks interference L strongly asymmetric peaks

29 Intensity A.U. Ti: pyramidal twin plane: 11 2 {11.4} reflection: 2 4 sub-reflection (parent crystal) {31.0} reflection: 030 sub-reflection (twin crystal) = 6 % L

30 Intensity [a.u.] Ti: pyramidal twin plane: 11 2 only the 2 4 sub-reflection in the {11.4} reflection sub-reflection Ti (11-22) twin plane = 6 % 0-1,2-1,0-0,8 L

31 Intensity distribution in the streaks twinning probability or twin boundary frequency: fraction of twin lamellae of thickness of n crystal-layers: W n Wn (1 ) n = 10% W n n

32 Intensity distribution in the streaks: I(L) I L 1 4 I L L0 2 L FWHM A asym L L 0 where: FWHM L D tri 2 1 I(L) : Lorentzian type function L. Balogh, G. Tichy and T. Ungár, J. Appl. Cryst. 42, (2009)

33 It can been shown that: FWHM L D tri 2 1 is a universal relation for twinning, where D tri = L. Balogh, G. Tichy and T. Ungár, J. Appl. Cryst. 42, (2009)

34 Intensity A.U. Intensity distribution in the streaks: sub-profiles I L 1 4 I L L0 2 L FWHM A asym L L L I(L) : the sum of a symmetrical + an antisymmetrical Lorentzian type function

35 Intensity distribution in reciprocal-space is 3 dimensional k o 2 k s g s z = s g s I=I(s,s,s g )

36 Line profile: intensity distribution along the diffraction vector: g scattered radiation is integrated normal to the g vector I(s g ) = I(s,s,s g ) ds ds

37 Correlation between streaking and line broadening Polycrystal scattering cone base k 0 k integration F Crystal g H i, K i Streaking Polycrystal sphere b h b l b k

38 k 0 integration normal to the g vector k g rystal

39 It can been shown that: transformation between L and g is linear to a very good approximation

40 L into g transformation for twinning on close-packed planes FWHM g 1 h k l ( ) FWHM ( ) 2 L g a FWHM L D tri 2 1 L. Velterop, et. al., J. Appl. Cryst. (2000). 33, E. Estevez-Rams, et. al., J. Appl. Cryst. 34, (2001) 730 L. Balogh, G. Ribárik, T Ungár, J.Appl.Phys. 100 (2006)

41 L into g transformation for twinning on pyramidal planes in hcp crystals L. Balogh, G. Tichy and T. Ungár J. Appl. Cryst. 42, (2009)

42 Intensity A.U. Ti: pyramidal twin plane: 11 2 {11.4} reflection: 2 4 sub-reflection (parent crystal) {31.0} reflection: 030 sub-reflection (twin crystal) = 6 % g [ 1/nm ]

43 Profile functions for twinning along the streaks there is NO hkl dependence, up to the asymmetry L into g transformation produces the hkl dependence

44 Resultant or measured profiles are the sum of sub-profiles

45 Summation of sub-profiles either or

46 intrinsic stacking faults on the (111) plane k 0 2 k h+k+l=3 h+k+l=1 h+k+l=5 function g S g the 311 type sub-reflections in fcc crystals

47 Arbitrary Units Subreflections according to different hkl permutations 1,5 1,0 Complete h+k+l=+5 component: h+k+l=+3 h+k+l=+1 Intrinsic SF 311 0,5 = 4 % 0,0 8,8 9,0 9,2 9,4 g [ 1/nm ]

48 Rel. Intensity Rel. Intensity hkl dependence in the I(g) profiles: Ti subreflections and {11.4} powder diffraction profiles Twin planes: {10.2} Twin planes: {10.1} % twins on the {10.2} planes {11.4} % twins on the {10.1} planes {11.4} ,85 10,90 10,95 g [1/nm] 0 10,85 10,90 10,95 g [1/nm]

49 Rel. Intensity Two different twin planes: {10.1} and {10.2} {11.4}, {20.1}, {21.0} reflection 1,0 {11.4} 10.1 Twin 10.2 Twin 0,5 0,0 10,88 10,92 10,96 g [1/nm]

50 Rel. Intensity Two different twin planes: {10.1} and {10.2} {11.4}, {20.1}, {21.0} reflection 1,0 {20.1} 10.1 Twin 10.2 Twin 0,5 0,0 8,08 8,12 8,16 g [1/nm]

51 Rel. Intensity Two different twin planes: {10.1} and {10.2} {11.4}, {20.1}, {21.0} reflection 600 {21.0} 10.1 Twin 10.2 Twin ,32 10,36 10,40 g [1/nm]

52 Philosophy of line profile analysis I meas <=> I size I strain I PF I instr + BG Measured pattern Physically modeled theoretical pattern: I th (g) Experimentally determined

53 Profile-functions for planar faults: I PF hkl ( g) w I hkl ( g) n j1 w j L I j L, hkl( g) functions sub-profiles w: fractions, ( permutations of hkl)

54 Sub-profiles: symmetrical + antisymmetrical Lorentzian functions easy to parameterize

55 fwhm [1/nm] shifts [1/nm] Sub-reflections of the {533} reflection for intrinsic SF 0.1 Shifts th order polinomials Breadths

56 Inert-Gas condensed nanocrystalline copper G. Sanders, G. E. Fougere, L. J. Thompson, J. A. Eastman, J. R. Weertman, Nanostruct. Mater. 8, (1997) 243.

57 Counts Inert-Gas condensed nanocrystalline copper T.Ungár, S.Ott, P.G.Sanders, A.Borbély, J.R.Weertman, Acta Materialia, 10, (1998) L. Balogh, G. Ribárik, T Ungár, J.Appl.Phys. 100 (2006) ecmwp o

58 TEM and X-ray grain size

59 Twin Density [%] Twin density vs. crystallite size: in Cu L. Balogh, G. Ribárik, T Ungár, J.Appl.Phys. 100 (2006) O 2 - IS P 2 - IS P 2 - T N 2 - IS 4 N 2 - C ECAP(a) ECAP(b) Twinning in Cu: below ~ 40 nm <x> area [ nm ]

60 Sintering nanocrystalline SiC J. Gubicza, S. Nauyoks, L. Balogh, J. Lábár, T. W. Zerda, T. Ungár, J. Mater. Res. 22 (2007) 1314 T sinter 1800 o C 2 GPa 8 GPa

61 FWHM [1/nm] Sintering nanocrystalline SiC J. Gubicza, S. Nauyoks, L. Balogh, J. Lábár, T. W. Zerda, T. Ungár, J. Mater. Res. 22 (2007) o C at 2 GPa 0,12 No strain 0,08 0, K [1/nm] 422

62 Sintering nanocrystalline SiC J. Gubicza, S. Nauyoks, L. Balogh, J. Lábár, T. W. Zerda, T. Ungár, J. Mater. Res. 22 (2007) o C at 2 GPa Equiaxed grains with twin boundaries

63 counts Sintering nanocrystalline SiC J. Gubicza, S. Nauyoks, L. Balogh, J. Lábár, T. W. Zerda, T. Ungár, J. Mater. Res. 22 (2007) o C at 2 GPa ecmwp procedure [degree]

64 Sintering nanocrystalline SiC J. Gubicza, S. Nauyoks, L. Balogh, J. Lábár, T. W. Zerda, T. Ungár, J. Mater. Res. 22 (2007) Twin density [% ] Temperature [K] Pressure [GPa] 2 0 2

65 Twinning in tensile deformed TWIP steel L. Balogh, D.W. Brown, T. Holden, in preparation eng = 33% eng = 42%

66 Twinning in tensile deformed TWIP steel L. Balogh, D.W. Brown, T. Holden, in preparation ecmwp procedure TOF neutron diffraction at SMARTS Lujan Neutron Scatt. Cent. Los Alamos Nat. Lab.

67 Access to the software package - CMWP - ecmwp - ANIZC - planar-defect parameter-files

68 Thanks to my coworkers: Levente Balogh Gábor Ribárik Géza Tichy

69 Thank you for your attention

70

71

72

73 Dislocations in the pyramidal twin boundaries in Mg B. Li, E. Ma, AM 2004

74 CdF 2 ball milled for 12 min G. Ribárik, N. Audebrand, H. Palancher, T. Ungár and D. Louër, J. Appl. Cryst. (2005). 38,

75 T. Waitz, V. Kazykhanov, H.P. Karnthaler: Martensitic phase transformations in nanocrystalline NiTi studied by TEM Acta Materialia 52 (2004) Perfect matching along the twin plane Bright field image of a twinned grain. HRTEM image of a grain containing two twin related variants: RT1 & RT2