Quasicrystals Structure and dynamics. M. de Boissieu SIMaP, Grenoble- INP, CNRS, UJF St Mar=n d Hères France.

Transcription

1 Quasicrystals Structure and dynamics M. de Boissieu SIMaP, Grenoble- INP, CNRS, UJF St Mar=n d Hères France.

2 Quasicrystals Long range order with Forbidden 5- fold symmetry No periodicity QUASICRYSTAL: A NEW KIND OF LONG RANGE ORDER The nodon of crystal is redefined From A.P. Tsai and H Takakura

3 Stable Quasicrystals are observed in many different systems and can be grown as large single grains i- AlLiCu (Dubost, Sainfort, Audier) ZnMgY, I.R Fisher et al. SoM condensed maner K. Hayashida, Intermetallic alloys: AlLiCu, AlCuFe, AlPdMn, ZnMgY, CdYb (A.P. Tsai) But also som condensed maner.

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5 Symmetry of quasicrystals Courtesy A.P. Tsai

6 i- AlCuFe 10- fold symmetry Bragg peaks No periodicity Quasicristal diffracdon panern τ=(1+ 5)/2= τ 3 τ 2 τ (from A.P. Tsai)

7 DiffracDon panern of quasicrystals X- ray diffrac=on along 5- fold axis. i- AlPdMn. ESRF. (Log I scale) Only few very strong Bragg. τ scaling of posi=on (Fm35). 6 integer indices N/M short hand notation (Gratias et al.) Self- similarity : τ=(1+ 5)/2=1.618 Intensity /11 18/29 2/1 47/76 3/4 1 τ τ 2 τ 3 τ Qx (2π/a Unit) par 6D

8 Quasicrystal: Highly perfect Order i- Al69.8Pd21Mn9.2 Single grain High resolu=on, coherent setup (ID20, ESRF): one speckle dq=10-4 Å - 1 ξ ~ 10μm Rocking curve: FWHM Dynamical diffrac=on is also observed A. Létoublon et al.

9 Stable quasicrystals i- AlPdMn, growth of cen=meter size single grains by Bridgman and Czokralski methods. M. Boudard et al. W. Steurer, i- AlPdMn Laue

10 Quasicrystals Where are the atoms? Specific physical proper=es? Phason, modeling and stabilizing mechanisms.

11 Quasicrystals Where are the atoms? Specific physical proper=es? Phason, modeling QC and stabilizing mechanisms.

12 Two main routes: Decorated Penrose Dling Where are the atoms? Trial and error method High dimensional crystallography (introduced by de Wolf, Janner and Janssen ) Only method to use the data Modeling is necessary Both method are using - Periodic approximant - Cluster descrip=on.

13 The Penrose Dling (1974) Diffrac=on: Alan Mackay (82) Mosseri and Sadoc 3D: Amman =ling

14 Approximants, clusters and local icosahedral order Bergman- Pauling- Samson cluster descrip=on Acta Cryst. 10, 254 (1957). Example of the ZnAl- Mg alloy First QC models: Quasicrystal and approximant. Decora=on with atomic clusters (V. Elser and Henley PRL M. Audier and P. Guyot Phil Mag 1985)

15 Amman decorated Dling DecoraDon of the Penrose Dling using clusters (Levine and Steinhardt PRL 84; Elser- Henley, PRL 1985; Guyot- audier Phil Mag 1985)

16 superspace crystallography same structural tool for all aperiodic crystals Invented by P.M. de Wolff, A. Janner and T. Janssen for incommensurate phases (2014 Ewald Prize) For quasicrystals: re- invented by - Kalugin, Levitov, Kitaev (jetp, 85) - Bak (PRL 85) - Duneau and Katz (PRL 85) - Elser (PRL 85)

17 Superspace crystallography In this 2D reciprocal plane All Bragg peaks posi=ons are expressed as a linear combinadon of 4 vectors c* b* a* Q=n 1 a * +n 2 b * +n 3 c * +n 4 d * d* 4 integers: 4 dimensional space Icosahedral phase:6 indices (from A.P. Tsai)

18 The three Icosahedral QC families Bergman cluster (Frank- Kasper type) Mackay cluster Tsai cluster (Yb- Cd type) (Courtesy C. Pay Gomez)

19 Clusters, Dlings, connecdvity and superspace T 2-fold and 3-fold cluster connexion

20 The 12-fold Penrose vertices distribution Not a Dling. Cluster center posidon H. Takakura, C. Gomez, A. Yamamoto et al., Nature Materials,,

21 Where are the atoms: CdYb quasicrystal and approximant Binary stable quasicrystal : i- Cd 5.7 Yb (Tsai A P, Guo J Q, et al Nature ) Periodic Cubic Approximants 1/1- Cd 6 Yb and 2/1- Cd 5.8 Yb, with known structure. (Gomez C. P. and Lidin S.: Phys. Rev. B 68 (2003) ). Very good chemical order: large and small atoms. Same atomic clusters QC: Synchrotron data (5000 Bragg peaks, weak ones) QC: Phasing and 6D modeling plus Finng tools Structure of the quasicrystal is well understood (Takakura et al. Nat Mat, 2007)

22 Cluster in approximant and QC Disordered Cd tetrahedron

23 Cluster in approximant and QC Cd dodecahedron R=4.6 Å

24 Cluster in approximant and QC Yb Icosahedron R=5.6 Å CHEMICAL ORDER

25 Cluster in approximant and QC Cd icosidodecahedron Small distorsions R=6.5 Å CHEMICAL ORDER

26 Cluster in approximant and QC Cd Triacontahedron and midle edges R=7.8 Å Total 158 atoms Closed packed structure, small (Cd) and large (Yb) atom. Gomez C P and Lidin S 2003 Phys. Rev. B \1.

27 Same cluster in quasicrystal and approximant 15.6 Å 158 atoms. Chemical order Closed packed structure with a large and a small atom. Same cluster connecbvity in the QC and approximant: 2- fold and 3- fold bonds

28 Cluster connexion 1/1 approximant BCC Packing. 2- fold and 3- fold connected 2-fold connection b=15.7 Å 3-fold connection c=13.6 Å

29 Cd 6 Yb 1/1 and Cd 5.8 Yb 2/1 approximant 1/1: 15.7 Å 2/1: 25.3 Å 2-fold connection b=15.7 Å 3-fold connection c=13.6 Å 3 building blocks Gomez C P and Lidin S 2003 Phys. Rev. B \1.

30 i-cdyb Modeling 6D Modeling related to cluster description Network of interpenatrating Triacontahedra 2- and 3- fold connected Remaining space is filled with the prolate and oblate (same decoration as in the 2/1 approximant) H. Takakura, C. Gomez, A. Yamamoto et al.

31 i-cdyb Atomic clusters New environments Bergman Cluster H. Takakura, C. Gomez, A. Yamamoto et al., Nature Materials,, 2007,6,58

32 Quasicrystal:Hierachical packing of clusters PosiDon of the cluster centers H. Takakura, C. Gomez, A. Yamamoto et al., Nature Materials,,

33 Clusters are distorted in the quasicrystal Inner tetrahedron plays a crucial role It breaks the cluster shell icosahedral symmetry. Overall icosahedral symmetry with different cluster orientation

34 Quasicrystals Where are the atoms? Specific physical proper=es? Phason, QC modeling and stabilizing mechanisms.

35 Can quasiperiodic long range order propagates with local rules? Matching rules are not local growing rules

36 Oxford Math building Texas A M University

37 Matching rules are not growing rules

38 Can quasiperiodic long range order propagates with local rules? Matching rules are not growing rules, but possible to grow with a few defects. Which atomic surfaces for matching rules?? Icosahedral bounded by 2-fold planes (Katz and Gratias). Set of polyhedra to be used.

39 Phason modes and random Dlings Phason modes are a specific excita=on for all aperiodic crystals: - incommensurately modulated phases - Incommensurate composites (sliding mode) - Quasicrystals Hydrodynamic theory of aperiodic crystal. (P. Bak, Kalugin et al., Lubenski et al.) Diffusive excita=ons

40 Phason modes in QC Invariance of the system free energy with respect to perp space translation: phason modes

41 Phason modes in QC Invariance of the system free energy with respect to perp space translation: phason modes

42 Phason mode and hydrodynamic theory Local atom flip LS à SL Phason mode: Collective diffusive process (non propagative) S(Q+q, t) is decaying exponentially Ico phase: K2 and K1 elastic constants Phason modes in all aperiodic phases

43 Phason mode and hydrodynamic theory Local atom flip LS à SL Phason mode: Collective diffusive process (non propagative) S(Q+q, t) is decaying exponentially Ico phase: K2 and K1 elastic constants Phason modes in all aperiodic phases

44 Phason dynamics: hydrodynamic (Engel and Trebin PRL 2012)

45 Quasicrystal Stabilizing mechanisms? Quasicrystal versus approximant 1/1- Cd 6 Yb : cubic 1.57 nm 2/1- Cd 5.8 Yb: cubic: 2.53 nm i- Cd 5.7 Yb : quasicrystal ico FrustraDon might be resolved in 1/1 but not in the QC Cd- Yb hybridizadon (Ishii) Phason modes? 1/1 local environment (Ishimasa) Two sites CN16- CN17

46 What mechanisms stabilize quasicrystals? Free Energy = E - TS Energy term Close Packing of large (Sc) and small (Zn) atom Electronic stabiliza=on (Hume- Rothery, ) sp- d hybridiza=on in Zn- Sc Entropy term Vibra=onal entropy Chemical Disorder Phason modes: only in the QC Random Tiling (Henley and Elser)

47 Mechanisms stabilizing quasicrystals Random Dling and atomic scale modeling. C. L. Henley: Random =ling models, in Quasicrystals: the state of the art (Eds. D. P. DiVicenzo and P. Steinhardt), p

48 Modeling quasicrystals Tilings or 6D models and oscilla=ng pair poten=als Applied to the i- CdYb type quasicrystal

49 The CdYb type QC and approximant 158 atoms. Chemical order 15.6 Å 15.6 Å Periodic Approximant ~ 340 Å H. Takakura, C. Gomez, A. Yamamoto et al., Nature Materials,, 2007,6,58 Quasicrystal 5-fold plane

50 SimulaDons Two ingredients: atomic structure and Hamiltonian. QC Structure: large periodic approximant: 3/2 (to 8/5) approximant with about 3000 atoms Adapted Hamiltonian: oscillatng pair potental fived on ab- ini=o data (VASP). Fiwng of energies and forces.

51 OscillaTng pair potental M. Mihalkovic, C. Henley PRB 85, , 2012 FiVed on energy and forces from structures relaxed with VASP E (ev) Sc-Sc Zn-Sc Zn-Zn R (Å) Energy Nature Materials 6 (2007) Forces

52 Comparison simuladon- measurement Transverse modes 1/1 approximant Zn-Sc Quasicrystal Good agreeement. Differences QC and 1/1 are well reproduced. Nature Materials 6 (2007)

53 Intensity distribudon: comparison simuladon- experiment. Intensity/Mon=2000 Intensity/Mon=10000 Intensity/Mon=10000 Intensity/Mon= q=0.18 Å q=0.54 Å q=0.63 Å TA- c-znsc q=0.82 Å E (mev) Intensity/Mon=10000 Intensity/Mon=10000 Intensity/Mon=10000 Intensity/Mon= q=0.18 Å q=0.53 Å TA i-znmgsc q=0.62 Å q=0.79 Å E (mev) Blue curve: simulation Nature Materials 6 (2007) Excellent quantitative agreement The atomistic simulation is validated

54 Partial VDOS dospart Tetra Dodeca-nb dodeca-nc E (mev) Mode analysis: Partial density of vibrational sates Tetrahedra shows low energy mode, espacially in 1/1 approximant: cluster modes?? Strong distortion of the dodecahedron both in 1/1 and 3/2 approximant.

55 ZnSc cubic approximant.quasielasdc Neutron and simuladon: Tetrahedron jump A unique dynamical flexibility: displact 0.5 Å Jump Dme scale: 1 ps at RT, 0.3 ps at 170K H. Euchner, T. Yamada et al., JPCM 2012, 24, T. Yamada, H. Euchner et al, JPCM 2013, 25,

56 i- ZnMgSc and i- ZnAgSc quasicrystals Jump of the tetrahedron also in the quasicrystal. Gradual freezing as in a glassy system: at RT only about 30% jumps Related to the large number of local environt At RT in QC an excepdonal dynamical flexibility

57 i- ZnMgSc and i- ZnAgSc quasicrystals At RT in QC an excepdonal dynamical flexibility H. Euchner, T. Yamada, et al. J. Phys.: Condens. Maver, 25, , (2013)

58 Phason modes in quasicrystals Hydrodynamic theory and elasdcity: Phenomenological theory Long wavelength phason modes: diffuse scanering I(Q Bragg +q)= I Bragg + I Diffus (Phonons)+I Diffus (Phasons) Phasons diffuse scavering : - - Near Bragg peaks (similar to TDS) Anisotropy of diffuse scavering: two parameters: phason elasdc constants K1 and K2 Phason modes are diffusive modes, unlike phonons

59 i- AlPdMn : Phason mode and diffuse scanering M. de boissieu et al. PRL 1995 Létoublon et al. Phil Mag Lev 2001 I (e.u./å 3 ) / /q Q (2 π/a unit) p Diffuse scattering is reproduced by the hydrodynamic theory: 2 parameters! K1/k B T = 0.1 and K2/k B T = atom -1

60 Phason Diffuse scanering is observed in all icosahedral QC. Quenched in. 2 parameters for the diffuse scaoering simulabon!

61 i-alpdmn quasicrystal. In situ T study T evolution of the diffuse Scattering FROM 750C to 500C In situ X-ray. The diffuse scattering is due to pre-transitional fluctuations (3-fold), with a phason softnening Agreement with the random-tiling scenario 3fold Boudard, M., et al. (1996). Europhys. Lett. 33(3),

62 Conclusion Good understanding of the atomic structure in the Tsai type QC; excep=onal dynamical flexibility Phason modes are characteris=c of the QC state. RealisDc atomic simuladon are at work for QC. Need for new theoredcal tools for simuladon analysis Stabilizing mechanism of aperiodic crystals remains one of the main challenging ques=on.

63 A.P. Tsai, Tohoku University T. Ishimasa, Hokkaido University H. Euchner, ITAP, University of StuOgart, Germany and SIMaP, Grenoble M. Mihalkovic Academy of Science, BraBslava, Slovakia T. Yamada, SIMaP, Grenoble and Tokyo University of Science, Noda, Japan R. Tamura, Tokyo University of Science, Noda, Japan S. Francoual Petra, Hambourg R. Currat, ILL

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