DOCSLIB.ORG

An HCP Crystal Is a Close-Packed Structure with the Stacking Sequence ...ABABAB

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
An HCP Crystal Is a Close-Packed Structure with the Stacking Sequence ...ABABAB HEXAGONAL CLOSE-PACKED STRUCTURE An HCP crystal is a close-packed structure with the stacking sequence ...ABABAB... To construct: 1st layer: 2D HCP array (layer A) 2nd layer: HCP layer with each sphere placed in alternate interstices in 1st layer (B) 3rd layer: HCP layer positioned directly above 1st layer (repeat of layer A) …ABABABAB… A B A B A A HCP is two interpenetrating simple hexagonal lattices displaced by a /3 + a /3 + a /2 1 2 3 70 HCP STRUCTURE • not a Bravais lattice Orientation alternates with each layer • each sphere touches 12 equidistant nearest neighbors (CN = 12) Six in plane, six out-of-plane • structure has maximum packing fraction possible for single-sized spheres (0.74) 71 HCP STRUCTURE • unit cell is a simple hexagonal lattice with a two-point basis a (0,0,0) • ideal ratio c/a of 8/ 3 1.633 a (2/3,1/3,1/2) • {0002} planes are close packed • ranks in importance with FCC and BCC Bravais lattices Plan view 72 HCP STRUCTURE • about 30 elements crystallize in the HCP form 73 CUBIC CLOSE-PACKED STRUCTURE A CCP crystal is a close-packed structure with the stacking sequence ...ABCABC... To construct: 1st layer: 2D HCP array (layer A) 2nd layer: HCP layer with each sphere placed in alternate interstices in 1st layer (B) 3rd layer: HCP layer placed in the other set of interstitial depressions (squares, C) 4th layer: repeats the 1st layer (A) …ABCABCABC… stacking of HCP layers along body diagonals A B C It turns out that the CCP structure is just the FCC Bravais lattice! 74 CCP STRUCTURE • CN = 12, packing fraction 0.74 • {111} planes are close packed • 4 atoms in unit cell Plan view 75 CLOSE-PACKED STRUCTURES • most common are HCP and CCP • an infinite # of alternative stacking sequences exist Example: silicon carbide has over 250 polytypes e.g., 6H-SiC stacking sequence …ABCACB… 76 STACKING FAULTS Stacking faults are one or two layer interruptions in the stacking sequence that destroy lattice periodicity e.g., an <110> projection of an FCC lattice: - A A [111] B B C C missing A B [110] B plane C C of atoms A - A [001] B - B [110] C C perfect FCC faulted FCC ABCABCABC ABCBCABC The stacking fault is an example of a planar defect • stacking fault energy γ ~100 mJ m-2 • also results in a linear defect called a dislocation 77 EXAMPLE InAs nanowires - <110> projection 78 Caroff, P. et al. Nature Nanotechnology 4, 50 - 55 (2009). • CCP and HCP have very similar lattice energies • no clear cut trends 79 Nature 353, 147 - 149 (12 Sep 1991) Rare Gases: Ne, He, Ar, Kr, Xe (CCP) 80 gold nanocrystals X. M. Lin 81 ANOTHER VIEW OF CLOSE PACKING 82 . It was reviewed by a panel of 12 referees; the panel reported in 2003, after 4 years of work, that it was “99% certain” of the correctness of the proof, but couldn’t verify all of the computer calculations. Hales and Ferguson (his student) received the Fulkerson Prize for outstanding papers in the area of discrete mathematics in 2009. In 2003, Hales announced that he would pursue a formal proof of the Conjecture that could be verified by computer. He estimated that the proof would be finished by 2023…but it was announced complete on Aug 10, 2014!!!! In January 2015 Hales and 21 collaborators published "A formal proof of the Kepler conjecture". The proof was accepted in 2017. http://en.wikipedia.org/wiki/Kepler_conjecture PACKING FRACTIONS The fraction of the total crystal volume that is occupied by spheres CCP (and HCP) a 2 radius 4 42a 4 ( )3 Vatoms 34 fraction 3 2 0.7405 a 2 Vcell a 6 a 74% BCC SC a 3 radius 4 a 4 ()3 fraction 32 0.5236 43a a3 2 ( )3 34 fraction 3 0.6802 52% a 68% 84 85 DENSITY CALCULATION n: number of atoms/unit cell m nA 1 A: atomic mass atoms VC: volume of the unit cell VNVcell A C NA: Avogadro’s number (6.023×1023 atoms/mole) Calculate the density of copper. RCu = 0.128 nm, Crystal structure: FCC, ACu= 63.5 g/mole 3 3 3 n = 4 atoms/cell, VC a (2 2 R ) 16 2 R (4)(63.5) 8.89g / cm3 [16 2(1.28 108 ) 3 (6.023 10 23 )] 8.96 g/cm3 in the literature 86 INTERSTICIAL SITES IN CP STRUCTURES A large number of ionic structures can be regarded as built of CP layers of anions with the cations placed in interstitial sites for every anion, there is 1 Octahedral site and 2 Tetrahedral sites 87 Octahedral holes coordinates: ½00 0½0 00½ ½½½ = O site cavities have <100> orientation 88 Tetrahedral holes in CCP T+ sites: ¾¼¼ ¼¾¼ ¼¼¾ ¾¾¾ T- sites: ¼¼¼ ¾¾¼ ¼¾¾ ¾¼¾ cavities have <111> orientation 89 Holes in HCP O sites: 2/3,1/3,1/4 2/3,1/3,3/4 (1/3,2/3,1/2) T+ sites: 1/3,2/3,1/8 0,0,5/8 a (0,0,0) 1 T- sites: a 2 0,0,3/8 1/3,2/3,7/8 90 LOCATION OF OCTAHEDRAL HOLES 91 LOCATION OF TETRAHEDRAL HOLES (3/8 of a unit cell directly above/below each anion) 92 SIZE OF OCTAHEDRAL CAVITY Only cations smaller than the diameter of the cavity can fit without forcing the anion lattice to expand from cell edge a22 rMX r from face diagonal ar 22X M = cation X = anion . The cavity radius is 41% of the anion radius SIZE OF TETRAHEDRAL CAVITY The tetrahedral holes are twice as numerous but six times smaller in volume The cavity radius is 22.5% of the anion radius EUTACTIC STRUCTURES Structures in which the arrangement of ions is the same as in a close packed array but the ions are not necessarily touching Within certain loose limits (given by the radius ratio rules), cations too large to fit in the interstices can be accommodated by an expansion of the anion array • anions don’t like to touch anyway • modern techniques show that, in many cases, anions (cations) are not as large (small) as previously thought • we still describe eutactic structures as CCP or HCP lattices with ions in some fraction of the interstitial sites 95 CRYSTALS THAT CAN BE DESCRIBED IN TERMS OF INTERSTITIAL FILLING OF A CLOSE-PACKED STRUCTURE 96 SOME EUTACTIC CRYSTAL STRUCTURES Variables: 1) anion layer stacking sequence: CCP or HCP array? 2) occupancy of interstitial sites 97 98 NaCl (ROCK SALT, HALITE) STRUCTURE → → → 99 POLYHEDRAL REPRESENTATION • shows the topology and indicates interstitial sites • tetrahedra and octahedra are the most common shapes • • • 100 ROCK SALT - OCCURANCE Very common (inc. 'ionics', 'covalents' & 'intermetallics' ) • Most alkali halides (CsCl, CsBr, CsI excepted) • Most oxides / chalcogenides of alkaline earths • Many nitrides, carbides, hydrides (e.g. ZrN, TiC, NaH) 101 COMPLEX ION VARIANT OF ROCK SALT • 2 • S2 102 ZINC BLENDE (ZnS, SPHALERITE) → → → 103 ZINC BLENDE • bonding is less ionic than in rock salt • common for Be, Zn, Cd, Hg chalcogenides (i.e., ZnS, ZnSe, ZnTe) • common for III-V compounds (B, Al, Ga, In with N, P, As, Sb) 104 DIAMOND STRUCTURE Fd3m → 105 FLUORITE (CaF2) & ANTIFLUORITE (Na2O) → → → 106 ALTERNATIVE REPRESENTATIONS 107 FLUORITE / ANTIFLUORITE • origin of the term “fluorescence” (George Stokes, 1852) • fluorite common for fluorides of large, divalent cations and oxides of large 2+ 4+ tetravalent cations (M F2 and M O2) • antifluorite common for oxides/chalcogenides of alkalis (M2O) 108 FLUORESCENT MINERALS = fluorite http://en.wikipedia.org/wiki/Fluorescence109 COMPARING NaCl, ZnS, Na2O 110 111 Li3Bi EXAMPLE 112 NiAs STRUCTURE 2 → → → 113 114 •Transition metals with chalcogens, As, Sb, Bi e.g. Ti(S,Se,Te); Cr(S,Se,Te,Sb); Ni(S,Se,Te,As,Sb,Sn) 115 WURTZITE (ZnS) STRUCTURE → → → 116 117 118 119 HCP VERSION OF CaF2? No structures are known with all Tetra. sites (T+ and T-) filled in HCP - i.e. there is no HCP analogue of the Fluorite /Anti-Fluorite structure Why? The T+ and T- interstitial sites above and below a layer of close-packed spheres in HCP are too close to each other (distance = 0.25c) to tolerate the coulombic repulsion generated by filling with like-charged ions. 120 Face-linking is unfavorable RUTILE STRUCTURE (TiO2) → → → 121 ANATASE STRUCTURE (TiO2) → → → 122 RUTILE AND ANATASE → 123 CdI2 STRUCTURE → → → 124 125 CdI2 - OCCURANCE • Iodides of moderately polarizing cations; bromides and chlorides of strongly polarizing cations; e.g. PbI2, FeBr2, VCl2 • Hydroxides of many divalent cations e.g. (Mg,Ni)(OH)2 • Di-chalcogenides of many quadrivalent cations e.g. TiS2, ZrSe2, CoTe2 126 CdCl2 STRUCTURE Chlorides of moderately polarizing cations e.g. MgCl2, MnCl2 Di-sulfides of quadrivalent cations e.g. TaS2, NbS2 (CdI2 form as well) 127 Type and fraction Formula of sites occupied CCP HCP NaCl NiAs AB All octahedral Rock Salt Nickel Arsenide Half ZnS ZnS tetrahedral Zinc Blende Wurtzite (T+ or T-) Na2O Anti-Fluorite A2B All tetrahedral not known CaF2 Fluorite All octahedral A B Li Bi 3 & tetrahedral 3 not known Half octahedral AB2 (Alternate layers CdCl2 (Cadmium Chloride) CdI2 (Cadmium Iodide) full/empty) Half octahedral (Ordered CaCl2 framework TiO2 (Anatase) TiO2 (Rutile) arrangement) Third octahedral AB3 Alternate layers YCl3 BiI3 2 128 /3 full/empty PEROVSKITE STRUCTURE ABX3 (CaTiO3) → → → 129 PEROVSKITE CONNECTIVITY 3D network of corner-sharing Network of face-sharing octahedra cuboctahedra 130 Perovskites: the most widely studied oxide structure • Wide range of chemistries possible - thousands of examples known • Cubic, tetragonal, and orthorhombic symmetries are common Unique properties of perovskites - high Tc cuprate superconductors - Colossal Magneto-Resistance (La,SrMnO3) + 2- - fast ion conduction (Li , O in SrSnO3), batteries, fuel cells - mixed electronic/ionic conduction, fuel cells - oxidation/reduction catalysts (CH3NH3)PbI3 organic-inorganic - ferroelectric / piezoelectric ceramics (BaTiO3, Pb(ZrTi)O3) - important mineral structure in lower mantle (MgSiO3, pyroxene) - frequency filters for wireless communications : Ba(Zn1/3Ta2/3)O3 131 X A Perovskite Structure: ABX3 Tolerance factor (t): rrAX B X t 2(rrBX ) t Effect Likely structure > 1 A cation too large to fit in Hexagonal interstices perovskite 0.9 - 1.0 ideal Cubic perovskite 0.71 - 0.9 A cation too small Orthorhombic perovskite < 0.71 A cation same size as B cation Possible close packed lattice 132 PEROVSKITES Most perovskites contain distorted octahedra and are not cubic, at least at lower temperatures.
Load more

Recommended publications
  • Types of Lattices
    Types of Lattices Types of Lattices Lattices are either: 1. Primitive (or Simple): one lattice point per unit cell. 2. Non-primitive, (or Multiple) e.g. double, triple, etc.: more than one lattice point per unit cell. Double r2 cell r1 r2 Triple r1 cell r2 r1 Primitive cell N + e 4 Ne = number of lattice points on cell edges (shared by 4 cells) •When repeated by successive translations e =edge reproduce periodic pattern. •Multiple cells are usually selected to make obvious the higher symmetry (usually rotational symmetry) that is possessed by the 1 lattice, which may not be immediately evident from primitive cell. Lattice Points- Review 2 Arrangement of Lattice Points 3 Arrangement of Lattice Points (continued) •These are known as the basis vectors, which we will come back to. •These are not translation vectors (R) since they have non- integer values. The complexity of the system depends upon the symmetry requirements (is it lost or maintained?) by applying the symmetry operations (rotation, reflection, inversion and translation). 4 The Five 2-D Bravais Lattices •From the previous definitions of the four 2-D and seven 3-D crystal systems, we know that there are four and seven primitive unit cells (with 1 lattice point/unit cell), respectively. •We can then ask: can we add additional lattice points to the primitive lattices (or nets), in such a way that we still have a lattice (net) belonging to the same crystal system (with symmetry requirements)? •First illustrate this for 2-D nets, where we know that the surroundings of each lattice point must be identical.
    [Show full text]
  • Chapter 4, Bravais Lattice Primitive Vectors
    Chapter 4, Bravais Lattice A Bravais lattice is the collection of all (and only those) points in space reachable from the origin with position vectors: n , n , n integer (+, -, or 0) r r r r 1 2 3 R = n1a1 + n2 a2 + n3a3 a1, a2, and a3 not all in same plane The three primitive vectors, a1, a2, and a3, uniquely define a Bravais lattice. However, for one Bravais lattice, there are many choices for the primitive vectors. A Bravais lattice is infinite. It is identical (in every aspect) when viewed from any of its lattice points. This is not a Bravais lattice. Honeycomb: P and Q are equivalent. R is not. A Bravais lattice can be defined as either the collection of lattice points, or the primitive translation vectors which construct the lattice. POINT Q OBJECT: Remember that a Bravais lattice has only points. Points, being dimensionless and isotropic, have full spatial symmetry (invariant under any point symmetry operation). Primitive Vectors There are many choices for the primitive vectors of a Bravais lattice. One sure way to find a set of primitive vectors (as described in Problem 4 .8) is the following: (1) a1 is the vector to a nearest neighbor lattice point. (2) a2 is the vector to a lattice points closest to, but not on, the a1 axis. (3) a3 is the vector to a lattice point nearest, but not on, the a18a2 plane. How does one prove that this is a set of primitive vectors? Hint: there should be no lattice points inside, or on the faces (lll)fhlhd(lllid)fd(parallolegrams) of, the polyhedron (parallelepiped) formed by these three vectors.
    [Show full text]
  • Crystal Structure
    Physics 927 E.Y.Tsymbal Section 1: Crystal Structure A solid is said to be a crystal if atoms are arranged in such a way that their positions are exactly periodic. This concept is illustrated in Fig.1 using a two-dimensional (2D) structure. y T C Fig.1 A B a x 1 A perfect crystal maintains this periodicity in both the x and y directions from -∞ to +∞. As follows from this periodicity, the atoms A, B, C, etc. are equivalent. In other words, for an observer located at any of these atomic sites, the crystal appears exactly the same. The same idea can be expressed by saying that a crystal possesses a translational symmetry. The translational symmetry means that if the crystal is translated by any vector joining two atoms, say T in Fig.1, the crystal appears exactly the same as it did before the translation. In other words the crystal remains invariant under any such translation. The structure of all crystals can be described in terms of a lattice, with a group of atoms attached to every lattice point. For example, in the case of structure shown in Fig.1, if we replace each atom by a geometrical point located at the equilibrium position of that atom, we obtain a crystal lattice. The crystal lattice has the same geometrical properties as the crystal, but it is devoid of any physical contents. There are two classes of lattices: the Bravais and the non-Bravais. In a Bravais lattice all lattice points are equivalent and hence by necessity all atoms in the crystal are of the same kind.
    [Show full text]
  • Crystals the Atoms in a Crystal Are Arranged in a Periodic Pattern. A
    Crystals The atoms in a crystal are arranged in a periodic pattern. A crystal is made up of unit cells that are repeated at a pattern of points, which is called the Bravais lattice (fig. (1)). The Bravais lattice is basically a three dimensional abstract lattice of points. Figure 1: Unit cell - Bravais lattice - Crystal Unit cell The unit cell is the description of the arrangement of the atoms that get repeated, and is not unique - there are a various possibilities to choose it. The primitive unit cell is the smallest possible pattern to reproduce the crystal and is also not unique. As shown in fig. (2) for 2 dimensions there is a number of options of choosing the primitive unit cell. Nevertheless all have the same area. The same is true in three dimensions (various ways to choose primitive unit cell; same volume). An example for a possible three dimensional primitive unit cell and the formula for calculating the volume can be seen in fig. (2). Figure 2: Illustration of primitive unit cells 1 The conventional unit cell is agreed upon. It is chosen in a way to display the crystal or the crystal symmetries as good as possible. Normally it is easier to imagine things when looking at the conventional unit cell. A few conventional unit cells are displayed in fig. (3). Figure 3: Illustration of conventional unit cells • sc (simple cubic): Conventional and primitive unit cell are identical • fcc (face centered cubic): The conventional unit cell consists of 4 atoms in com- parison to the primitive unit cell which consists of only 1 atom.
    [Show full text]
  • Structural and Electronic Properties of Chalcopyrite Semiconductors
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by ethesis@nitr STRUCTURAL AND ELECTRONIC PROPERTIES OF CHALCOPYRITE SEMICONDUCTORS By Bijayalaxmi Panda Under the Guidance of Prof. Biplab Ganguli Department Of Physics National Institute Of Technology, Rourkela CERTIFICATE This is to certify that the project thesis entitled ” Structural and elec- tronic properties of chalcopyrite semiconductor” being submitted by Bijay- alaxmi Panda in partial fulfilment to the requirement of the one year project course (PH 592) of MSc Degree in physics of National Institute of Technology, Rourkela has been carried out under my supervision. The result incorporated in the thesis has been produced by using TB-LMTO codes. Prof. Biplab Ganguli Dept. of Physics National Institute of Technology Rourkela - 769008 1 ACKNOWLEDGEMENT I would like to acknowledge my guide Prof. Biplab Ganguli for his help and guidance in the completion of my one-year project and also for his enormous motivation and encouragement. I am also very much thankful to Phd scholars of cmputational physics lab whose encouragement and support helped me to complete my project. 2 ABSTRACT We have studied the structural and electronic properties of pure, deffect and doped chalcopyrite semiconductors using Density functional theory (DFT) based first principle technique within Tight binding Linear Muffin Tin orbital (TB-LMTO) method. Our calculated structural parameters such as lattice constant, anion displacement parameter(u) in case of pure chalcopyrite and anion displacement parameter(ux, uy,uz) in case of deffect and doped chal- copyrite, tetragonal distortion(η=c/2a) and bond length are in good agree- ment with other work.
    [Show full text]
  • Modern Physics Unit 10: Quantum Statistics and Crystalline Solids Lecture 10.2: Unit Cells and Miller Indices
    Modern Physics Unit 10: Quantum Statistics and Crystalline Solids Lecture 10.2: Unit Cells and Miller Indices Ron Reifenberger Professor of Physics Purdue University 1 Two Important Concepts 1. A unit cell is the most basic repeating structure of any solid. Atoms in a crystalline solid form a regular repeating pattern. 2. Crystalline solids have periodic arrays of atoms which forms a grid system that fills all space. We call the grid system a crystal lattice. Amorphous solids and glasses are exceptions. 2 Classification of the Unit Cell The unit cell can be classified as primitive or conventional. • There are many ways to define a unit cell •A primitive unit cell has atoms only at the corners of the unit cell. This will be the simplest representation of the crystal structure, but it may not reflect the complete symmetry of the crystal structure. • The conventional unit cell will have atoms at additional sites within the unit cell (i.e. on the faces, at the body center, etc.), causing the conventional unit cell to have the same symmetry as the entire crystal structure. Chosen for convenience. • By international convention, a unit cell is specified to be right- handed, to have the highest symmetry, and to have the smallest cell volume. 3 Visualize crystal structures: http://surfexp.fhi-berlin.mpg.de/SXinput.html Example: There are many possible choices for primitive vectors and primitive unit cells (parallelograms) in two- dimensions non-primitive cell cannot fill all space by translating Area of Primitive cell (in 2D) = a1×= a 2 aa 12sinθ 4 Organizing Space Identify all Specify Unit Cell Rotations, Reflections and Inversions that transform the cell into itself Defines one of 32 possible Four Lattice Types crystallographic Point Groups Primitive Body Centered Face Centered Base Centered Add Space Filling Translations to define Space Groups Find 14 Space Filling Lattices (Bravais Lattices) Bravais Lattices classified into 7 Crystal Systems 7 distinguishable Point Groups 5 The seven crystal systems The 14 Bravais Lattices Lowest symmetry (A.
    [Show full text]
  • Diamond Unit Cell
    Diamond Crystal Structure Diamond is a metastable allotrope of carbon where the each carbon atom is bonded covalently with other surrounding four carbon atoms and are arranged in a variation of the face centered cubic crystal structure called a diamond lattice Diamond Unit Cell Figure shows four atoms (dark) bonded to four others within the volume of the cell. Six atoms fall on the middle of each of the six cube faces, showing two bonds. Out of eight cube corners, four atoms bond to an atom within the cube. The other four bond to adjacent cubes of the crystal Lattice Vector and Basis Atoms So the structure consists of two basis atoms and may be thought of as two inter-penetrating face centered cubic lattices, with a basis of two identical carbon atoms associated with each lattice point one displaced from the other by a translation of ao(1/4,1/4,1/4) along a body diagonal so we can say the diamond cubic structure is a combination of two interpenetrating FCC sub lattices displaced along the body diagonal of the cubic cell by 1/4th length of that diagonal. Thus the origins of two FCC sub lattices lie at (0, 0, 0) and (1/4, 1/4,1/4) (a) (b) (aa) Crystallographic unit cell (unit cube) of the diamond structure (bb) The primitive basis vectors of the face centered cubic lattice and the two atoms forming the basis are highlighted. The primitive basis vectors and the two atoms at (0,0,0) and ao(1/4,1/4,1/4) are highlighted in Figure(b)and the basis vectors of the direct Bravais lattice are Where ao denotes the lattice constant of the relaxed lattice.
    [Show full text]
  • Crystal Lattice
    Contents are copyrighted (有版權). Do not use, distribute, or copy the contents of the lecture notes without permission. Do not use outside class room purposes without permission!!! Advanced Polymer Physics Lecture notes by Prof. E. M. Woo adapting from: Textbook of Physical Polymer Science (Ed: L. H. Sperling, 3rd Ed) Chap 6: Part-I [Sections 6.1~6.3] Polymers in Crystalline State + Basic of Crystal Lattices + XRD methods -Lecture notes 1 Introduction • The development of crystallinity in polymers depends on the regularity of structure in the polymer (see Chapter 2). Thus, isotactic and syndiotactic polymers usually crystallize, whereas atactic polymers, with a few exceptions (where the side groups are small or highly polar), do not. • Regular structures also appear in the polyamides (nylons), polyesters, and so on, and these polymers make excellent fibers. • Nonregularity of structure first decreases the melting temperature and finally prevents crystallinity. Mers of incorrect tacticity (see Chapter 2) tend to destroy crystallinity, as does copolymerization. Thus statistical copolymers are generally amorphous. • Blends of isotactic and atactic polymers show reduced crystallinity, with only the isotactic portion crystallizing. • Under some circumstances block copolymers containing a crystallizable block will crystallize; again, only the crystallizable block crystallizes. • Factors that control the melting temperature include polarity and hydrogen bonding as well as packing capability. • Table 6.1 lists some important crystalline polymers and their melting temperatures (1). 2 Table 6.1 Properties of selected crystalline polymers (1) 3 6.1 GENERAL CONSIDERATIONS In this chapter the study of crystalline polymers is undertaken. • The crystalline state is defined as one that diffracts X-rays and exhibits the first- order transition known as melting.
    [Show full text]
  • Bravais Types of Lattices Depending on the Geometry Crystal Lattice May Sustain Different Symmetry Elements
    Dr Semën Gorfman Department of Physics, University of SIegen Lecture course on crystallography, 2015 Lecture 6: Bravais types of lattices Depending on the geometry crystal lattice may sustain different symmetry elements. According to the symmetry of the LATTICES all CRYSTAL are subdivided into a CRYSTAL SYSTEMS 2 2D Lattice symmetry 1: Oblique The oblique lattice has 2-fold axis only / centre of inversion . Note that for two dimensional objects 2- fold axis is equivalent to the centre of inversion 3 2D Lattice symmetry 1: Oblique. The choice of basis vectors No particular restrictions on the choice of the basis vectors constructing the lattice. a ≠ b a ≠ 90 deg a2 a1 4 2 D Lattice symmetry 2: Square 2b 1b 2a The square lattice has a) 4-fold symmetry axis b) Mirror plane 1a 1a c) Mirror plane 1b d) Mirror plane 2a e) Mirror plane 2b 5 2 D Lattice symmetry 2: Square. The choice of basis 2b 1b 2a The conventional choice of basis vectors are: a and b along the mirror planes 1a and 1b In this case: 1a a=b, a=90 a2 a1 6 2 D lattice symmetry 3: Hexagonal 1c 2b 1b 2c 2a 1a a) 6-fold symmetry axis b) Mirror planes 1a,1b,1c,2a,2b,2c (as shown on the plot) 7 2 D lattice symmetry 3: Hexagonal. The choice of basis 1c 2b 1b 2c 2a 1a a2 a1 Conventional choice a is along mirror plane 1a, b is along mirror plane 1c a=b, a=120 deg 8 2D lattice symmetry 4: Rectangular 2 1 a) 2-fold symmetry axis b) Mirror planes 1 and 2 (see the plot) 9 2D lattice symmetry 4: Rectangular.
    [Show full text]
  • Handout 4 Lattices in 1D, 2D, and 3D Bravais Lattice
    Handout 4 Lattices in 1D, 2D, and 3D In this lecture you will learn: • Bravais lattices • Primitive lattice vectors • Unit cells and primitive cells • Lattices with basis and basis vectors August Bravais (1811-1863) ECE 407 – Spring 2009 – Farhan Rana – Cornell University Bravais Lattice A fundamental concept in the description of crystalline solids is that of a “Bravais lattice”. A Bravais lattice is an infinite arrangement of points (or atoms) in space that has the following property: The lattice looks exactly the same when viewed from any lattice point A 1D Bravais lattice: b A 2D Bravais lattice: c b ECE 407 – Spring 2009 – Farhan Rana – Cornell University 1 Bravais Lattice A 2D Bravais lattice: A 3D Bravais lattice: d c b ECE 407 – Spring 2009 – Farhan Rana – Cornell University Bravais Lattice A Bravais lattice has the following property: The position vector of all points (or atoms) in the lattice can be written as follows: Where n, m, p = 0, ±1, ±2, ±3, ……. 1D R n a 1 And the vectors, 2D R n a1 m a2 a1 , a2 , and a3 are called the “primitive lattice 3D R n a1 m a2 p a3 vectors” and are said to span the lattice. These vectors are not parallel. Example (1D): ˆ b Example (2D): a1 b x y c x a2 c yˆ a1 b xˆ b ECE 407 – Spring 2009 – Farhan Rana – Cornell University 2 Bravais Lattice Example (3D): d c a2 c yˆ a3 d zˆ a1 b xˆ b The choice of primitive vectors is NOT unique: All sets of primitive vectors shown will work for the 2D lattice c a2 b xˆ c yˆ a1 b xˆ a2 c yˆ a1 b xˆ b ECE 407 – Spring 2009
    [Show full text]
  • Fundamentals of Crystallography
    Symmetry in crystals CARMELO GIACOVAZZO The crystalline state and isometric operations Matter is usually classified into three states: gaseous, liquid, and solid. Gases are composed of almost isolated particles, except for occasional collisions; they tend to occupy all the available volume, which is subject to variation following changes in pressure. In liquids the attraction between nearest-neighbour particles is high enough to keep the particles almost in contact. As a consequence liquids can only be slightly compressed. The thermal motion has sufficient energy to move the molecules away from the attractive field of their neighbours; the particles are not linked together permanently, thus allowing liquids to flow. If we reduce the thermal motion of a liquid, the links between molecules will become more stable. The molecules will then cluster together to form what is macroscopically observed as a rigid body. They can assume a random disposition, but an ordered pattern is more likely because it corresponds to a lower energy state. This ordered disposition of molecules is called the crystalline state. As a consequence of our increased understand- ing of the structure of matter, it has become more convenient to classify matter into the three states: gaseous, liquid, and crystalline. Can we then conclude that all solid materials are crystalline? For instance, can common glass and calcite (calcium carbonate present in nature) both be considered as crystalline? Even though both materials have high hardness and are transparent to light, glass, but not calcite, breaks in a completely irregular way. This is due to the fact that glass is formed by long, randomly disposed macromolecules of silicon dioxide.
    [Show full text]
  • Electromagnetism
    Electromagnetism Collection Editor: Phan Long Electromagnetism Collection Editor: Phan Long Authors: Andrew R. Barron Paul Padley Carissa Smith Online: < http://cnx.org/content/col11173/1.1/ > CONNEXIONS Rice University, Houston, Texas This selection and arrangement of content as a collection is copyrighted by Phan Long. It is licensed under the Creative Commons Attribution 3.0 license (http://creativecommons.org/licenses/by/3.0/). Collection structure revised: January 13, 2010 PDF generated: October 29, 2012 For copyright and attribution information for the modules contained in this collection, see p. 25. Table of Contents 1 Crystal Structure .............................................................. ...................1 2 Single Slit Diraction ............................................................................19 Index ................................................................................................24 Attributions .........................................................................................25 iv Available for free at Connexions <http://cnx.org/content/col11173/1.1> Chapter 1 Crystal Structure1 1.1 Introduction In any sort of discussion of crystalline materials, it is useful to begin with a discussion of crystallography: the study of the formation, structure, and properties of crystals. A crystal structure is dened as the particular repeating arrangement of atoms (molecules or ions) throughout a crystal. Structure refers to the internal arrangement of particles and not the external appearance of
    [Show full text]