Single Crystals, Powders and Twins
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1 Single Crystals, Powders and Twins Master of Crystallography and Crystallization 2013 T01 Mathematical, Physical and Chemical basis of Crystallography
2 Solıd Materıal Types Crystallıne Polycrystallıne Amorphous (Non-Crystalline) Single Crystal
3 Crystalline Solids A Crystalline Solid is the solid form of a substance in which the atoms or molecules are arranged in a definite, repeating pattern in three dimensions. Single Crystals, ideally have a high degree of order, or regular geometric periodicity, throughout the entire volume of the material. A material is a crystal if it has essentially a sharp diffraction pattern" Acta Cryst. (1992), A48, 928 Terms of reference of the IUCr commission on aperiodic crystals
4 Crystallography Crystallography The branch of science that deals with the geometric description of crystals & their internal arrangements. It is the science of crystals & the math used to describe them. It is a VERY OLD field which predates Solid State Physics by about a century! So much of the terminology (& theory notation) of Solid State Physics originated in crystallography.
5 A Single Crystal has an atomic structure that repeats periodically across its whole volume. Even at infinite length scales, each atom is related to every other equivalent atom in the structure by translational symmetry. Single Crystals Single Pyrite Crystal Amorphous Solid
6 Polycrystalline Solids A Polycrystalline Solid is made up of an aggregate of many small single crystals (crystallites or grains). Polycrystalline materials have a high degree of order over many atomic or molecular dimensions. These ordered regions, or single crystal regions, vary in size & orientation with respect to one another. These regions are called grains (or domains) & are separated from one another by grain boundaries. The atomic order can vary from one domain to the next. The grains are usually 100 nm microns in diameter. Polycrystals with grains that are < 10 nm in diameter are called nanocrystallites. Polycrystalline Pyrite Grain
7 Amorphous Solids Amorphous (Non-crystalline) Solids are composed of randomly orientated atoms, ions, or molecules that do not form defined patterns or lattice structures. Amorphous materials have order only within a few atomic or molecular dimensions. They do not have any long-range order, but they have varying degrees of shortrange order. Examples of amorphous material include amorphous silicon, plastics, & glasses.
8 Departures From the Perfect Crystal A Perfect Crystal is an idealization that does not exist in nature. In some ways, even a crystal surface is an imperfection, because the periodicity is interrupted there. Each atom undergoes thermal vibrations around their equilibrium positions for temperatures T >= 0K. These can also be viewed as imperfections. Also, Real Crystals always have foreign atoms (impurities), missing atoms (vacancies), & atoms in between lattice sites (interstitials) where they should not be. Each of these spoils the perfect crystal structure.
9 Crystal Lattice Crystallography focuses on the geometric properties of crystals. So, we imagine each atom replaced by a mathematical point at the equilibrium position of that atom. A Crystal Lattice (or a Crystal) An idealized description of the geometry of a crystalline material. A Crystal A 3-dimensional periodic array of atoms. Usually, we ll only consider ideal crystals. Ideal means one with no defects, as already mentioned. That is, no missing atoms, no atoms off of the lattice sites where we expect them to be, no impurities, Clearly, such an ideal crystal never occurs in nature. Yet, 85-90% of experimental observations on crystalline materials is accounted for by considering only ideal crystals! Platinum Platinum Surface (Scanning Tunneling Microscope) Crystal Lattice & Structure of Platinum
10 Single Crystals For single crystals, we see the individual reciprocal lattice points projected onto the detector and we can determine the values of (hkl) readily, measure the intensities of each peak and assign the associated diffraction angles q. This information can allow us to determine the electron density within a unit cell using the methods we have seen earlier in this course. In a crystalline powder, the situation is modified significantly
11 Powders If a powder is composed of numerous crystallites that are randomly oriented ( all possible orientations are present), the reciprocal lattice points for each of the crystallites combine to form spheres of reflection. Since these spheres will always intersect the Ewald sphere, at the detector, we observe these as circles and we can only measure the diffraction angle q of the rings.
12 We lose a considerable amount of information because the 3D (hkl) data for each of the single crystal reflections is effectively reduced to 1D (q) data in the powder XRD spectrum. Furthermore, the overlap of peaks (because of necessary or accidentally similar d-spacings) makes the interpretation of the intensity data more complicated. In a modern XRD experiment, the diffraction patterns shown on the right are integrated to provide plots of the type shown on the left. Although much of the information that we would prefer to have for structure solution is gone, XRD can provide much information and, in some cases, structure solution is possible.
13 Despite the apparent limitations, XRD is very useful for the identification of compounds for which structural data has already been obtained. For example, applications in WinGX (LAZY Pulverix), the ICSD, Mercury, Diamond and PowderCell will calculate the XRD pattern for any known structures. Below is the calculated powder XRD pattern for the [ n Bu 4 N][MeB(C 6 F 5 ) 3 ] structure we examined in the refinement part of the course. Rietveld Análisis of these patterns allows to refine structural models or XRD quantitative analysis. This assigned XRD pattern was calculated using PowderCell, which is a particularly useful program for any of you who wish to use our powder diffractometer. The program is also useful because it can be used to examine and predict how a diffraction pattern is altered when unit cell parameters or fractional coordinates are changed (such modifications can occur when the temperature of an experiment is changed).
14 Wherefore Powder Diffraction? Single crystal data provides I obs for each observed diffraction spot Nice method for subtracting background Tried and true methods for structure determination Though not always easy! Or: you can t get a nice single/non-twinned/ low mosaicity/ highly diffracting crystal!
15 Powder Data CCD / Image plate detectors (shown) 3D / 2D Overlap problem All Freidel pairs exactly overlapped Identical d-spacings Higher 2q Larger structures= proteins! Single crystal data 2D location of points, 3D when you add angle Precession. Powder data is only 2D Intensity theta
16 Powder methods Mimic single crystal diffraction techniques End goal - solve unknown structures Rietveld refinement Frontier goals Find the limits for size and resolution Synchrotron data Better signal to noise (generally) Better resolution (generally) Thinner peaks- less overlaps Wavelength choice Resolution here refers to maximum d-spacing resolved -- max 2q
17 Appropriate Starting Model gap find a suitable starting model Human lysozyme 1.5 Å structure pdb code: 1LZ1 Single crystal data P spacegroup 130 residues Hen egg white lysozyme 1.9 Å data, 5 data sets P spacegroup 129 residues ~60% identity
18 The hard part is over Globally the structure is in the right spot Mol rep solution with modified model Use this as a basis for Rietveld refinement Extend refinement region to 2-12 o 2q Low angle shifts- solvent model High angle - overlaps Rigorous method for structure optimization Rigid body and restrained refinement (ccp4i- Refmac) Only works up to 3 Å (12 o ) Overlaps hurt Goal: best false minima
19 Quartz Feldspar Twins Twins Aggregates formed of individual crystals of the same species, grown together with well defined orientations Rutile Fluorite
20 Contact twins Origin: Penetration twins - Growth - Deformation Polysynthetic twins - Transformation Cyclic twins
21 Dauphine Example: Quartz Penetration Twins Brazil
22
23 Optical appearance of twins Reentrant angles Non-uniforme optical extinction
24 Twin boundaries
25 Twins in diffraction experiments I hkl twin = v I I hkli + v II I hkl II + v III I hkl III + v IV I hkl IV +... n = Σ v n I hkl n i=1 Refinement: Structure of the individual Twin laws volume fractions (additional parameters)
26 m m m Unidentified twins in structure determination m Huge displacement parameters split atom positions short interatomic distances
27 Merohedral Twins The twin element belongs to the holohedry of the lattice, but not to the point group of the crystal. The reciprocal lattices of all twin domains superimpose exactly. In the triclinic, monoclinic and orthorhombic crystal systems, the merohedral twins can always be described as inversion twins. Non-merohedral Twins Twin operation does not belong to the Laue group or point group of the crystal. In practice there are three types of reflections: Reflections belonging to only one lattice. Completely overlapping reflections belonging to both lattices. Partially overlapping refections belonging to both lattices.
28 Non-merohedral Twins
29 Warning signs The R int value for the higher-symmetry Laue group is only slightly higher than for the lower-symmetry Laue group The mean value for E 2-1 is much lower than the expected value of The space group appears to be trigonal or hexagonal The apparent systematic absences are not consistent with any known space group For all of the most disagreeable reflections F o is much greater than F c (Herbst-Irmer & Sheldrick, 1998)
30 Classification of Merohedral Twins Twinning by Merohedry Twinning by Pseudo-Merohedry Merohedry of the lattice reticular merohedry
31 Twinning by Merohedry Type I: Inversion twins: The operation of twinning is part of the Laue Group of the individual, but not of the point group R a a c c c c a a Example: Symmetry of an individual: Pm Laue Group: 2/m Twin Matrix Point group: m a -a (a,b,c)=(a,b,c) Twin operation inversion center b -b Symmetry of the twin: P2 /m c -c ( )
32 Ag 4 Bi 2 O 6 Example: Merohedry Type I a=5.975(1), b=6.311(1), c=9.563(2)å Pnna < Bi-O > distance: 2.23Å (Bi 4+ ) Semiconductor and diamagnetic Pnn2 Inversion twin 0.5:0.5 < Bi(1) O > distance: 2.31 Å (Bi 3+ ) < Bi(2) O > distance: 2.15 Å (Bi 5+ ) J. Solid State Chem. 147,1999, 117; Solid State Sciences 8, 2006, 267.
33 Twinning by Merohedry Type II: The twin operation forms part of the holoedry of the lattice, but not of the Laue Group of the individual Maximum symmetry R b b b b a a a a Example: Symmetry of an individual: P4/m Laue Group: 4/m Holoedry of the tetragonal lattice: 4/m2/m2/m Twin operations: 2 a, m a, 2 [110], m [110] Symmetry of the twin: 4/m 2 /m 2 /m
34 C 14 H 14 I 2 O 2 Te 0.5C 2 H 6 OS Example: Merohedry Type II a=9.4309(4) Å, c= (17) Å space group P3 2 Symmetry of twin P Twin operation: 2-fold rotation around the a (b) axis ( ) Farran, Alvarez-Larena, Piniella, Capparelli & Friese (2008), Acta Cryst. C64, o257.
35 Twinning by Pseudo-merohedry The twin operation corresponds to a pseudosymmetry element of the lattice R c a a c c a a c Example: specialized metric, monoclinic lattice with β 90º Symmetry of the individual: monoclinic P12/m1 Symmetry of the lattice: orthorhombic 2/m2/m2/m Twin operation: 2 [100], m [100], 2 [001], m [001] Symmetry of the twin: P2 /m 2/m2 /m
36 Example: Twinning by Pseudo-Merohedry Tl 2 MoO 4 Structural phase transition 350K: P3m1 293K: C121 K. Friese, G. Madariaga & T. Breczewski (1999), Acta Cryst. C55, K. Friese, M. I. Aroyo, C.L. Folcia, G. Madariaga, T. Breczewski (2001), Acta Cryst. B57,
37 Tl 2 MoO 4 Example: Twinning by Pseudo-Merohedry Point group 3m P3m1 a=6.266(1) Å, c=8.103(2)å 12 symmetry operations a = - a 1 + a 2 b = - a 1 b 2 c = c t=6 b a a 2 Point group 2 C121 a = Å 2 symmetry b =6.418Å operations c =8.039Å ß=91.05 a 1 Group/Subgroup Relationships C. Hermann (1929), Z. Kristallogr. 69, t = translationengleich = Number of symmetry operations of the point group of G Number of symmetry operations of the point group of U
38 Tl 2 MoO 4 Example: Twinning by Pseudo-Merohedry Trigonal metrics: a=6.266(1) Å, c=8.103(2)å Idealized orthorhombic metrics: a= Å b=6.266 Å c=8.103å Monoclinic metrics: a = Å b =6.418 Å c =8.039Å ß=91.05
39 Merohedry of the lattice Obverse / Reverse twins in rhombohedral lattices Obverse: 0, 0, 0; Reverse: 0, 0, 0; 2/3, 1/3, 1/3; 1/3, 2/3, 1/3; 1/3, 2/3, 2/3 2/3, 1/3, 2/ /3 2/3 1/3,2/3 1/3,2/3 2/3 1/3 a 1 a 2 a 1 a 2 Individual I Twin Individual II
40 Sr 3 (Ru 0.33,Pt 0.66 )CuO 6 Example: Merohedry of the lattice a=5.595(15) Å, b=5.595(15) Å, c=11.193(2) Å, γ =120 º K. Friese, L. Kienle, V. Duppel, H. M. Luo & C. T. Lin (2003), Acta Cryst. B59,
41 Sr 3 (Ru 0.33,Pt 0.66 )CuO 6 Example: Merohedry of the lattice Obverse/reverse Twin = 180 rotation around c* + 3-fold axis of the trigonal system º 120º ) ( ) ( ( ) º 300º 60º ( ) ( ) ( )
42 Twinning by reticular merohedry Part of the lattice points of the individuals overlap, another part corresponds to points of one individual only c c c c a a a a Special relationship hidden within the metrics
43 GeTe 4 C 24 H 20 : (PhTe) 4 Ge (Germanium tellurolate) Example: Reticular Merohedry a= (4) b=9.1842(9) c=23.690(3)å ß= (8)º P2 1 /c S. Schlecht & K. Friese (2003), Eur. J. Inorg. Chem. 2003, a,b,2c orthorhombic cell a = 9.18 b = c = 45.67
44 m orthorhombic Cell a,b,2c a= (4) Å b=9.1842(9) Å c=23.690(3) Å ß= (8) º P2 1 /c a = -b = 9.18 Å b = a = Å c = a + 2c = Å α = º β = 90.0 º γ = 90.0 º
45 Example: (PhTe) 4 Ge Reciprocal space h0l
46 wrong structure: unidentified twin Individual 1 Individual 2
47 Reciprocal space Identification of twins Reentrant angles Non-uniforme optical extinction Splitting of reflections Specialized metrics Huge unit cells Large percentage of unobserved reflections Non-standard extinction rules Phase transitions Pseudosymmetry Large difference between internal R-value and final agreement factors Large displacement parameters Unreasonable interatomic distances
48 Single Crystals, Powders and Twins Master of Crystallography and Crystallization 2013 T01 Mathematical, Physical and Chemical basis of Crystallography END
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