STRUCTURE OF QUASICRYSTALS AND STRUCTURE OF QUASICRYSTALS AND RELATED PHASES RELATED PHASES ANANDH SUBRAMANIAM Guest Scientist (Alexander Von Humboldt Fellow) Electron Microscopy Group Max-Planck-Institut für Metallforschung STUTTGART Ph: (+49) (0711) 689 3683, Fax: (+49) (0711) 689 3522 [email protected]http://www.geocities.com/anandh4444/ November 2004
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STRUCTURE OF QUASICRYSTALS AND STRUCTURE OF QUASICRYSTALS AND RELATED PHASESRELATED PHASES
ANANDH SUBRAMANIAMGuest Scientist (Alexander Von Humboldt Fellow)
Electron Microscopy GroupMax-Planck-Institut für Metallforschung
4. Irrational length L and S arranged in a Fibonacci chain
AAppeerriiooddiicc
Diffraction properties of various distributions of scatterers.
Progressive lowering of dimension starting with an Progressive lowering of dimension starting with an NN--foldfold symmetry in ND spacesymmetry in ND space
N-fold symmetry Hypercubic Lattice viewed
along [111....1]N 1s
N-D
AApppprrooxxiimmaannttss
Quasiperiodic tiling 2D RRAA
Sequence of numbers Sequence of ‘a’s and ‘b’s Polynomial Equation
1D
RReeppeeaattiinngg SSeeqquueennccee
Convergence of sequence Length of ‘a’/length of ‘b’ Root of Polynomial Eq.
0D
RRaattiioonnaall NNuummbbeerr
ND-0D.ppt
GENERALIZED PROJECTION METHOD
[A] a1, a2, a3, ..., aN : a set of vectors in E||
[B] b1, b2, b3, ..., bN : a set of vectors in E
W : the acceptance region or window in E
{n1, n2, n3, ..., nN} are a set of integers in N dimensional space such that
n1a1 + n2a2 + n3a3 + ... + nNaN is accepted as a point in E|| if and only if:
Important clusters underlying the structure of quasicrystals and their approximants.
= 2
The structure of the Al3Mn decagonal phase
Hiraga, K., Kaneko, M., Matsuo, Y., and Hashimoto, S., Phil. Mag. B67 (1993) 193
= 3
(a) (b)
Arrangement of sub-units in complex hexagonal phases
Cluster of three dodecahedra Cluster of three dodecahedra Four vertexFour vertex--connected icosahedraconnected icosahedra
(a) Singh, A., Abe, E., and Tsai, A. P., Phil. Mag. Lett., 77 (1998) 95(b) Kreiner, G., and Franzen, H. F., J. Alloys and Compounds, 221 (1995) 15
IQC (( == 11)) DQC (( == 22))
Mackay Approximant Taylor Approximant
Little Approximant Robinson Approximant
IQC (( == 11) HQC (( == 33))
Key: shows a twinning operation
R e la tio n b e tw e e n IQ C a n d its a p p ro x im a n ts w ith D Q C , its a p p ro x im a n ts a n d H Q C v ia th e tw in n in g
o p e ra tio n
A quadrant of the stereogram of the decagonal phase with indices derived by the twinned icosahedron
model
Stereogram of the Taylor phase obtained by twinning of the Mackay approximant to the icosahedral phase
Quadrant of the stereogram corresponding to I3 cluster
= 1 = 2 Icosahedral Quasicrystal = 3
Decagonal Quasicrystal
Hexagonal Quasicrystal
= 1 Digonal
Quasicrystal Pentagonal Quasicrystal
Cubic R.A.S. Mackay Bergman
Trigonal Quasicrystal
Hexagonal R.A.S.
Orthorhombic
R.A.S. Orthorhombic
R.A.S Trigonal R.A.S.
Orthorhombic R.A.S.
Taylor Little Robinson
R.A.S.
Monoclinic Monoclinic
R.A.S. Monoclinic
R.A.S. R.A.S.
= 90o 120o
= 108o
Unification scheme based on the twinning of the icosahedral cluster
EXPERIMENTALEXPERIMENTAL
MgMg--ZnZn--(Y, La) (Y, La)
SYSTEMSSYSTEMS
METASTABLE PHASES IN Mg-BASED ALLOYS
QUASICRYSTALS
RATIONAL APPROXIMANTS & RELATED STRUCTURES
METALLIC GLASSES
NANOCRYSTALS & NANOQUASICRYSTALS
Mg-Zn-Al First Mg-Based QC(Icosahedral)
P. Ramachandrarao, G.V.S. Sastry1985
Mg-Zn-Al-Cu Quaternary System N.K. Mukhopadhyay, G.N. Subbanna, S. Ranganathan, K. Chattopadhyay
1986
Mg-Zn-Ga Stable QC W. Ohashi, F. Spaepen1987
Mg-Zn-RE Icosahedral QC Z. Luo, S. Zhang, Y. Tang, D. Zhao1993
Mg-Al Cubic QC P. Donnadieu, A. Redjaimia1995
Mg-Zn-RE Decagonal QC T.J. Sato, E. Abe, A.P. Tsai1997
Mg-Zn-RE QC without underlying atomic clusters
E. Abe, T.J. Sato, A.P. Tsai1999
MILESTONES IN Mg-BASED QUASICRYSTAL RESEARCH
IMPORTANT PHASES IN THE Mg-Zn-RE SYSTEMS
Composition e/a Phase, Symmetry Comments
Mg3Zn6RE 2.1 Icosahedral, Fm53 aR = 0.519
RE = Y, Gd, Tb, Dy, Ho, Er dia (0.352, 0.360)
Mg40Zn58RE2 2.02 Decagonal, 10/mmm RE = Y, Dy, Ho, Er, Tm, Lu dia < 0.355
Mg24Zn65RE10 (S) 2.1 Hexagonal superlattice, P63/mmc a = 1.46 nm, c = 0.86 nm
RE = Y, Sm, Gd Related to IQC
Mg24Zn65RE10 (M) 2.1 Hexagonal superlattice, P63/mmc a = 2.35 nm, c = 0.86 nm
RE = Sm, Gd Related to IQC
Mg24Zn65Y10 (L) 2.1 Hexagonal superlattice, P63/mmc a = 3.29 nm, c = 0.86 nm
RE = Sm Related to IQC
aS : aM : aL = 3 : 5 : 7
Mg12ZnY 2.07 ?
Mg3Zn3Y2 2.25 cF16, Fm3m
Mg7Zn3 2 oI142, Immm 1/1 RA to IQC
Mg4Zn7 2 mC110, B2/m Related to DQC
MgZn2 2 hp12, P63/mmc Related to S, L & M phases
SEM micrograph of as-cast Mg51 Zn41 Y8 alloy showing (a) Eutectic Microstructure (b) Four-fold dendrite
(a) (b)
55--FOLD TO 6FOLD TO 6--FOLDFOLD
5-FOLD
DEVELOPING INTO 6-FOLD
SEM micrograph of as-cast Mg51 Zn41 Y8 alloy showing distorted 5-fold dendrite growing into hexagonal shape
Initial stages of growth
As-cast Mg37 Zn38 Y25 alloy showing the formation of a cubic phase (a = 7.07 Å):
BFI [111]
[110] [113]
SAD patterns from a BCC phase (a = 10.7 Å) in as-cast Mg4 Zn94 Y2 alloy showing important zones
[111] [011][112]
As-cast Mg37 Zn38 Y25 alloy showing a 18 R modulated phase
SAD pattern BFI
High-resolution micrograph
SAD patterns from as-cast Mg23 Zn68 Y9 showing the formation of FCI QC
[1
0] [1 1 1]
[0 0 1] [
1 3+ ]
Uniform deformation along the arrow of the [0 0 1] 2-fold pattern from IQC giving rise to a pattern similar to the [
1 3+ ] pattern
TEM micrograph of as-cast Mg4 Zn94 Y2 alloy showing the formation of nanocrystalline Mg3 Zn6 Y phase
Mg4 Zn94 Y2 as-cast alloy heat treated at 350oC for 20 hrs (corresponding to the MgZn5.51 phase)
BFI SAD
BFI SAD
BFI from as-cast Mg46 Zn46 La8 alloy showing patterns from APBs
Melt-spun Mg50 Zn45 Y5 alloy showing the formation of a cubic phase (a = 6.63 Å)
BFI [001]
[113] [111]
Comparison of the [001] two-fold of the FCI QC (a) with the two-fold from other phase in the MgZnY (b), (c) and MgZnLa (d) systems
ACKNOWLEDGEMENTS ACKNOWLEDGEMENTS
Dr. Eric A Lord
Prof. S. Ranganathan
Dr. K. Ramakrishnan
Dr. Sandip Bysakh
Dr. Steffen Weber
CONCLUSIONSCONCLUSIONS
A variety of Quasiperiodic and Rational Approximant structures can be realized using the Strip Projection Method, which serves to unify these structures using higher dimensions
Structures with diverse kinds of symmetries can be generated using the Twinned Icosahedron Model, which further can be used to construct a unified framework based on the orientations of the icosahedron and the lowering of symmetry
The Mg-Zn-RE systems serves a new ‘model system’ for the study of quasicrystals and related phases