DOCSLIB.ORG

Appendix 1. Basic Crystallography

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Appendix 1. Basic Crystallography APPENDIX 1. BASIC CRYSTALLOGRAPHY Al.l Introduction external symmetry of the structure. The trans­ lational symmetry leads to the concept of a lattice In order to perform electron microscope studies it (an array of atoms at points in space with identical is necessary to understand the basic principles of surroundings). Arising from considerations of this crystallography set out below. type we arrive at fourteen distinct Bravais lattices, It is the regular arrangement of atoms in space figure Al.l. These fall into the seven fundamental which constitutes the distinguishing feature of the crystal classes listed in table Al.l. Most metals are crystalline state. These regular arrays, called either cubic, hexagonal or tetragonal in structure, crystal structures, which are characterised in see table Al.3. However, many non-metallic various different ways, give rise to the internal and materials have more complex structures. stmple body- centred face -centred cubtc (P) cubic (I) cubic(F) stmple body-centred stmple body-centred tetragana I tetragonal orthorhombic orthorhombiC (P) (I) (P) (I) base -centred face-centred rhombohedral hexagonal orthorhombic orthorhombic (R) (P) (C) (F) c stmple bose-centred tnCllniC monocltnic (P) monoclimc(C) (P) Figure Al.l The fourteen Bravais lattices 80 Practical Electron Microscopy Table Al.1 Crystallographic formulae for interplanar spacings, Crystal system Interplanar spacing of the (hkl) plane cubic a=b=c a.={J=y=90° tetragonal a=bioc a.={J=y=90° orthorhombic a # b # c a.={J=y=90° hexagonal a=bioc 1 4(h2 hk k2 12 = 3a2 l a. = {J = 90°; y = 120° d2 + + ) + C2 2 rhombohedral a = b = c 1 (1 + cos a.){(h2 + k2 + 12) - (1 - tan !aXhk + kl + lh)} a. = {J = y < 120° # 90° d2 = Q2 1 + cos a. - 2 cos2 a. monoclinic aiobioc a.=y=90°#{J triclinic 2 2 : 2 = ~2 (s11h + s22kl + s331 + 2s 12hk + 2s23kl + 2s33lh) where V 2 = a2b2c2(1 - cos2 a. - cos2 {J - cos2 y + 2 cos a. cos {J cos y) and s 11 = b2c2 sin2 a. s22 = a2c2 sin2 {J s33 = a2b2 sin2 y s12 = abc2(cos a. cos {J - cosy) s23 = a2bc(cos {J cos y - cos a.) s31 = ab2c(cos y cos a. - cos {J) Al.l Indexing Planes in the plane are afh, bfk and cfl. A negative intercept results in a negative component ofthe Miller index Planes in any one of the fourteen Bravais lattices written as h. caD. be indexed in the same way. Axes are chosen Formulae for the angles between different (hkl) defining a unit cell x, y and z at angles a, fJ and y planes in the fourteen Bravais lattices are given in with the unit translation distances a, b and c table Al.l. (figure Al.2(a)). A plane is defined in terms of its intercepts on these axes. For example, in figure A1.2(b) the Al.3 Indexing Lattice Diredions plane cuts the axes at a/h, bjk and cfl where h, k and l are the Miller indices of the plane. For any Bravais lattice, such as that shown in Thus, in figure A1.2(c), the plane cuts the axes figure A1.3, the direction OA has the indices [121], at unit translation distances in the x and y axes that is the path from 0 to A involves moving 1 unit and two translation distances in the z axis. The translation parallel to x, 2 parallel to y and 1 index of the plane is then OX/OX, OY/OY and parallel to z. The other directions [Oll] and [ITO] OZ/OW, that is llf which is rationalised to 221, are self-evident. The symbol I means one trans­ the hkl indices for the plane, that is the intercepts lation in the negative direction. The general symbol Electron Diffraction in the Electron Microscope 81 angles, and angles between directions for the seven crystal systems [After Andrews et al. (1971)] cos p = {(u~ + v~ + w~)(u~ + v~ + w~)) 112 cost/>= [{ I 2 a2 1212}{ c 2 12}] 112 ? (hi + ktl + ? II ? (h2 + k2) + ? 12 convert to corresponding hexagonal indices (see appendix 2) and use the above two formulae 2 2 a2u1u2 + b v1v2 + c w 1w 2 + ac(w 1u2 + u1w 2)cosp cosp= 2 22 22 p) {(a 2 u1 + b v1 + c w 1 + 2acu 1w1 cos 112 x (a 2 u~ + b 2 v~ + c 2 w~ + 2acu 2 w2 cos P)} L cos p = -=---=-­ IUtVtWt IU2V2W2 2 2 2 2 2 where F = h1 h2 b c2 sin a + k 1 k2a c2 sin p + l112a2b2 sin y where L = a2 u1u2 + b2 v1v2 + c2 w 1w 2 2 + abc (cos a cos P - cos y)(k 1h2 + h1k2) + bc(v 1 w2 + w1 v2)cosa 2 + ab c(cos y cos a -cos P)(h 112 + 11h2) + ac(w1u2 + u1 w 2)cosP 2 + a bc(cos P cosy - cos a)(k 112 + 11k2) + ab(u 1 v2 + v1 u2) cosy and and Ahkl = {h2b2c2 sin2 a + k2a2c2 sin2 p + 12a2b2 sin2 y I uvw = ( a2u2 + b2v2 + c2w2 + 2hkabc2(cos a cos p - cosy) + 2bcvw cos a + 2hlab2c(cos y cos a - cos p) + 2cawu cos p 1 2 + 2kla2bc(cos p cosy - cos a)} ' + 2abuv cos y) 1' 2 for a direction is [uvw] where r = OA, figure A1.5 Zones and the Zone Law Al.3, = ua + vb + we. Consequently, r is a vector with components Any two lattice planes intersect in a line which can u, v and w along the axes. be defined by the directional indices [ uvw]. This Formulae for the angles between different [ uvw] is the axis for a prism of planes with this common directions in the fourteen Bravais lattices are given direction. The planes are known as the zone of in table Al.l. planes and the long axis is the zone axis, z, given the symbol [UVW]. The zone indices for any pair of planes, that is (h'k'l') and (hkl) can be obtained A1.4 Plane Normals in the following way: In the cubic crystal system only the direction h k h k normal to the plane (hkl) has indices [hkl], for example [111] is the (111) plane normal. In all X X X h' k' l' h' k' l' other crystal systems this is not true and table A1.2 gives the formulae for determining the indices of that is U = kl' - k'l, V = lh' - l'h, W = hk' - h'k. the directions [ uvw] normal to the plane (hkl) and This is the cross product between plane indices or vice versa. the directions of the plane normal. 82 Practical Electron Microscopy z c y X Figure A1.3 Crystal directions, OA is [121] If it is necessary to find out if a plane (hkl) lies in a zone [UVW] then the condition is X (a) hU + kV + IW = 0 and this is called the Weiss zone law. This is z essentially the condition that the normal to the plane (hkl) is perpendicular to [UVW] direction. Notation is as follows: (111) means single set of parallel planes {111} means equivalent planes of the type, that is (111), (ll1), etc. [111] means a single zone axis or direction ( 111) means directions of equivalent type A1.6 Stereograpbic Projection Although other methods of projection of the three­ dimensional crystal into two dimensions exist, the stereographic projection is the most common way of describing crystals. This is because the projection X preserves angular truth. The advantage of working (b) with the stereographic projection lies in the ease and rapidity of performing those crystallographic z analyses necessary in the electron microscope. Imagine a crystal at the centre of a sphere, see figure A1.4, with plane normals drawn from the centre of the sphere to its surface. In figure Al.5, one of these plane normals P is shown projected into the equatorial plane about the south pole of the sphere. Normally only those planes above the equator are projected. For planes underneath, the north pole is used as the projection point, indicated by open circles in the projection. The stereographic projection of the cubic crystal in figure A1.4 with [001] parallel to the south-north direction SN and [010] parallel to OD, is shown in figure A1.6, .each point being indexed as the normal to a particular plane. Standard projections for cubic crystal structures are shown in figure Al.12 with different planes X (c) in the centre of the stereogram. It is necessary to be able to measure the angles Figure A1.2 (a) The axes x, y, z defining the unit cell (dashed lines). (b) A general plane intersecting between planes using the stereographic projection. the axes. (c) A (221) plane The normals to any two crystal planes, for example Electron Diffraction in the Electron Microscope 83 Table A1.2 Formulae defining the indices of the direction [uvw] perpendicular to plane (hkl) for the seven crystal systems [After Andrews et a/. ( 1971)] Crystal system Equations for finding [ uvw] given (hkl) Equations for finding (hkl) given [ uvw] u w h k cubic h=k=l u w u w (c)2 h k tetragonal h- k -~ (a)2 u - = w(cfa)2 h k I orthorhombic ~ a2 = !!_ b2 = ~ c2 h k I ua2 u v 2w (c) 2 ,, k hexagonal 2k + h - h + 2k - 3T (a) 2 2u - v = 2v - u = 2w(cfa)2 u h k rhombohedral h sin2 tx + (k + 1Xcos2 tx - cos tx) k sin2 tx + (I + hXcos 2 tx - cos tx) u + (v + w) cos tx v + (w + u) cos tx w + (u + v) cos tx w I sin2 tx + (h + kXcos 2 tx - cos tx) u v w h k monoclinic hb2 c2 - lab2 c cos (J kc2 a2 sin2 (J la2 b2 - hab2 c cos (J ua2 + wca cos (J = vb2 = uca cos (J + wc 2 u w h k triclinic hs 11 + ks12 + ls13 hs12 + ks22 + ls23 hs 13 + ks23 + ls33 ua2 + vab cos y + wca cos (J uab cos y + vb2 + wbc cos tx (s 11 = b2 c2 sin2 tx, etc; s12 = s21 = abc2 (cos tx cos (J -cosy), etc.) uca cos.(J + vbc cos tx + wc2 001 ooT Figure A1.4 A crystal with cubic crystal structure situated at the centre of a sphere 84 Practical Electron Microscopy N Too projection on equatonal --- plane ---- -------- 100 5 Figure Al.6 The stereo graphic projection for a cube crystal with [001] parallel to the south north Figure Al.S The projection of a plane normal OP direction in the sphere of figure Al.4.
Load more

Recommended publications
  • 361: Crystallography and Diffraction
    361: Crystallography and Diffraction Michael Bedzyk Department of Materials Science and Engineering Northwestern University February 2, 2021 Contents 1 Catalog Description3 2 Course Outcomes3 3 361: Crystallography and Diffraction3 4 Symmetry of Crystals4 4.1 Types of Symmetry.......................... 4 4.2 Projections of Symmetry Elements and Point Groups...... 9 4.3 Translational Symmetry ....................... 17 5 Crystal Lattices 18 5.1 Indexing within a crystal lattice................... 18 5.2 Lattices................................. 22 6 Stereographic Projections 31 6.1 Projection plane............................ 33 6.2 Point of projection........................... 33 6.3 Representing Atomic Planes with Vectors............. 36 6.4 Concept of Reciprocal Lattice .................... 38 6.5 Reciprocal Lattice Vector....................... 41 7 Representative Crystal Structures 48 7.1 Crystal Structure Examples ..................... 48 7.2 The hexagonal close packed structure, unlike the examples in Figures 7.1 and 7.2, has two atoms per lattice point. These two atoms are considered to be the motif, or repeating object within the HCP lattice. ............................ 49 1 7.3 Voids in FCC.............................. 54 7.4 Atom Sizes and Coordination.................... 54 8 Introduction to Diffraction 57 8.1 X-ray .................................. 57 8.2 Interference .............................. 57 8.3 X-ray Diffraction History ...................... 59 8.4 How does X-ray diffraction work? ................. 59 8.5 Absent
    [Show full text]
  • C:\Documents and Settings\Alan Smithee\My Documents\MOTM
    I`mt`qx1/00Lhmdq`knesgdLnmsg9Rbnkdbhsd This month’s mineral, scolecite, is an uncommon zeolite from India. Our write-up explains its origin as a secondary mineral in volcanic host rocks, the difficulty of collecting this fragile mineral, the unusual properties of the zeolite-group minerals, and why mineralogists recently revised the system of zeolite classification and nomenclature. OVERVIEW PHYSICAL PROPERTIES Chemistry: Ca(Al2Si3O10)A3H2O Hydrous Calcium Aluminum Silicate (Hydrous Calcium Aluminosilicate), usually containing some potassium and sodium. Class: Silicates Subclass: Tectosilicates Group: Zeolites Crystal System: Monoclinic Crystal Habits: Usually as radiating sprays or clusters of thin, acicular crystals or Hairlike fibers; crystals are often flattened with tetragonal cross sections, lengthwise striations, and slanted terminations; also massive and fibrous. Twinning common. Color: Usually colorless, white, gray; rarely brown, pink, or yellow. Luster: Vitreous to silky Transparency: Transparent to translucent Streak: White Cleavage: Perfect in one direction Fracture: Uneven, brittle Hardness: 5.0-5.5 Specific Gravity: 2.16-2.40 (average 2.25) Figure 1. Scolecite. Luminescence: Often fluoresces yellow or brown in ultraviolet light. Refractive Index: 1.507-1.521 Distinctive Features and Tests: Best field-identification marks are acicular crystal habit; vitreous-to-silky luster; very low density; and association with other zeolite-group minerals, especially the closely- related minerals natrolite [Na2(Al2Si3O10)A2H2O] and mesolite [Na2Ca2(Al6Si9O30)A8H2O]. Laboratory tests are often needed to distinguish scolecite from other zeolite minerals. Dana Classification Number: 77.1.5.5 NAME The name “scolecite,” pronounced SKO-leh-site, is derived from the German Skolezit, which comes from the Greek sklx, meaning “worm,” an allusion to the tendency of its acicular crystals to curl when heated and dehydrated.
    [Show full text]
  • 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]
  • Chap 1 Crystal Structure
    Introduction Example of crystal structure Miller index Chap 1 Crystal structure Dept of Phys M.C. Chang Introduction Example of crystal structure Miller index Bulk structure of matter Crystalline solid Non-crystalline solid Quasi-crystal (1982 ) (amorphous, glass) Ordered, but not periodic 2011 Shechtman www.examplesof.net/2017/08/example-of-solids-crystalline-solids-amorphous-solids.html#.XRHQxuszaJA nmi3.eu/news-and-media/the-contribution-of-neutrons-to-the-study-of-quasicrystals.html Introduction Example of crystal structure Miller index • A simple lattice (in 3D) = a set of points with positions at r = n1a1+n2a2+n3a3 (n1, n2, n3 cover all integers ), a1, a 2, and a3 are called primitive vectors (原始向量), and a1, a 2, and a3 are called lattice constants ( 晶格常數). Alternative definition: • A simple lattice = a set of points in which every point has exactly the same environment. Note: A simple lattice is often called a Bravais lattice . From now on we’ll use the latter term ( not used by Kittel). Introduction Example of crystal structure Miller index Decomposition: Crystal structure = Bravais lattice + basis (of atoms) (How to repeat) (What to repeat) scienceline.ucsb.edu/getkey.php?key=4630 lattice basis 基元 (one atom, or a group of atoms) • Every crystal has a corresponding Bravais lattice and a basis. Note: Lattice is a math term, while crystal is a physics term. 晶格 晶體 Introduction Example of crystal structure Miller index Bravais lattice non-Bravais lattice = Bravais + p-point basis (p>1) Triangular (or hexagonal) lattice Honeycomb lattice Primitive vectors a2 basis a1 Honeycomb lattice = triangular lattice + 2-point basis (i.e.
    [Show full text]
  • Space Symmetry, Space Groups
    Space symmetry, Space groups - Space groups are the product of possible combinations of symmetry operations including translations. - There exist 230 different space groups in 3-dimensional space - Comparing to point groups, space groups have 2 more symmetry operations (table 1). These operations include translations. Therefore they describe not only the lattice but also the crystal structure. Table 1: Additional symmetry operations in space symmetry, their description, symmetry elements and Hermann-Mauguin symbols. symmetry H-M description symmetry element operation symbol screw 1. rotation by 360°/N screw axis NM (Schraubung) 2. translation along the axis (Schraubenachse) 1. reflection across the plane glide glide plane 2. translation parallel to the a,b,c,n,d (Gleitspiegelung) (Gleitspiegelebene) glide plane Glide directions for the different gilde planes: a (b,c): translation along ½ a or ½ b or ½ c, respectively) (with ,, cba = vectors of the unit cell) n: translation e.g. along ½ (a + b) d: translation e.g. along ¼ (a + b), d from diamond, because this glide plane occurs in the diamond structure Screw axis example: 21 (N = 2, M = 1) 1) rotation by 180° (= 360°/2) 2) translation parallel to the axis by ½ unit (M/N) Only 21, (31, 32), (41, 43), 42, (61, 65) (62, 64), 63 screw axes exist in parenthesis: right, left hand screw axis Space group symbol: The space group symbol begins with a capital letter (P: primitive; A, B, C: base centred I, body centred R rhombohedral, F face centred), which represents the Bravais lattice type, followed by the short form of symmetry elements, as known from the point group symbols.
    [Show full text]
  • Additive Effects of Alkali Metals on Cu-Modified Ch3nh3pbi3−Δclδ
    RSC Advances PAPER View Article Online View Journal | View Issue Additive effects of alkali metals on Cu-modified CH3NH3PbI3ÀdCld photovoltaic devices† Cite this: RSC Adv.,2019,9,24231 Naoki Ueoka, Takeo Oku * and Atsushi Suzuki We investigated the addition of alkali metal elements (namely Na+,K+,Rb+, and Cs+) to Cu-modified CH3NH3PbI3ÀdCld photovoltaic devices and their effects on the photovoltaic properties and electronic structure. The open-circuit voltage was increased by CuBr2 addition to the CH3NH3PbI3ÀdCld precursor solution. The series resistance was decreased by simultaneous addition of CuBr2 and RbI, which increased the external quantum efficiencies in the range of 300–500 nm, and the short-circuit current density. The energy gap of the perovskite crystal increased through CuBr2 addition, which we also confirmed by first-principles calculations. Charge carrier generation was observed in the range of 300– Received 25th April 2019 500 nm as an increase of the external quantum efficiency, owing to the partial density of states Accepted 24th July 2019 contributed by alkali metal elements. Calculations suggested that the Gibbs energies were decreased by DOI: 10.1039/c9ra03068a incorporation of alkali metal elements into the perovskite crystals. The conversion efficiency was Creative Commons Attribution 3.0 Unported Licence. rsc.li/rsc-advances maintained for 7 weeks for devices with added CuBr2 and RbI. PbI2 +CH3NH3I + 2CH3NH3Cl / CH3NH3PbI3 Introduction + 2CH3NH3Cl (g)[ (140 C)(2) Studies of methylammonium lead halide perovskite solar x y / cells started in 2009, when a conversion efficiency of 3.9% PbI2 + CH3NH3I+ CH3NH3Cl (CH3NH3)x+yPbI2+xCly 1 / [ was reported. Some devices have since yielded conversion CH3NH3PbI3 +CH3NH3Cl (g) (140 C) (3) efficiencies of more than 20% as studies have expanded This article is licensed under a 2–5 Owing to the formation of intermediates and solvent evap- globally, with expectations for these perovskite solar cells 6–9 oration, perovskite grains are gradually formed by annealing.
    [Show full text]
  • Apophyllite-(Kf)
    December 2013 Mineral of the Month APOPHYLLITE-(KF) Apophyllite-(KF) is a complex mineral with the unusual tendency to “leaf apart” when heated. It is a favorite among collectors because of its extraordinary transparency, bright luster, well- developed crystal habits, and occurrence in composite specimens with various zeolite minerals. OVERVIEW PHYSICAL PROPERTIES Chemistry: KCa4Si8O20(F,OH)·8H20 Basic Hydrous Potassium Calcium Fluorosilicate (Basic Potassium Calcium Silicate Fluoride Hydrate), often containing some sodium and trace amounts of iron and nickel. Class: Silicates Subclass: Phyllosilicates (Sheet Silicates) Group: Apophyllite Crystal System: Tetragonal Crystal Habits: Usually well-formed, cube-like or tabular crystals with rectangular, longitudinally striated prisms, square cross sections, and steep, diamond-shaped, pyramidal termination faces; pseudo-cubic prisms usually have flat terminations with beveled, distinctly triangular corners; also granular, lamellar, and compact. Color: Usually colorless or white; sometimes pale shades of green; occasionally pale shades of yellow, red, blue, or violet. Luster: Vitreous to pearly on crystal faces, pearly on cleavage surfaces with occasional iridescence. Transparency: Transparent to translucent Streak: White Cleavage: Perfect in one direction Fracture: Uneven, brittle. Hardness: 4.5-5.0 Specific Gravity: 2.3-2.4 Luminescence: Often fluoresces pale yellow-green. Refractive Index: 1.535-1.537 Distinctive Features and Tests: Pseudo-cubic crystals with pearly luster on cleavage surfaces; longitudinal striations; and occurrence as a secondary mineral in association with various zeolite minerals. Laboratory analysis is necessary to differentiate apophyllite-(KF) from closely-related apophyllite-(KOH). Can be confused with such zeolite minerals as stilbite-Ca [hydrous calcium sodium potassium aluminum silicate, Ca0.5,K,Na)9(Al9Si27O72)·28H2O], which forms tabular, wheat-sheaf-like, monoclinic crystals.
    [Show full text]
  • Infrare D Transmission Spectra of Carbonate Minerals
    Infrare d Transmission Spectra of Carbonate Mineral s THE NATURAL HISTORY MUSEUM Infrare d Transmission Spectra of Carbonate Mineral s G. C. Jones Department of Mineralogy The Natural History Museum London, UK and B. Jackson Department of Geology Royal Museum of Scotland Edinburgh, UK A collaborative project of The Natural History Museum and National Museums of Scotland E3 SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. Firs t editio n 1 993 © 1993 Springer Science+Business Media Dordrecht Originally published by Chapman & Hall in 1993 Softcover reprint of the hardcover 1st edition 1993 Typese t at the Natura l Histor y Museu m ISBN 978-94-010-4940-5 ISBN 978-94-011-2120-0 (eBook) DOI 10.1007/978-94-011-2120-0 Apar t fro m any fair dealin g for the purpose s of researc h or privat e study , or criticis m or review , as permitte d unde r the UK Copyrigh t Design s and Patent s Act , 1988, thi s publicatio n may not be reproduced , stored , or transmitted , in any for m or by any means , withou t the prio r permissio n in writin g of the publishers , or in the case of reprographi c reproductio n onl y in accordanc e wit h the term s of the licence s issue d by the Copyrigh t Licensin g Agenc y in the UK, or in accordanc e wit h the term s of licence s issue d by the appropriat e Reproductio n Right s Organizatio n outsid e the UK. Enquirie s concernin g reproductio n outsid e the term s state d here shoul d be sent to the publisher s at the Londo n addres s printe d on thi s page.
    [Show full text]
  • Crystal Directions, Wave Propagation and Miller Indices D
    Crystal Directions, Wave Propagation and Miller Indices D. K. Ferry, D. Vasileska and G. Klimeck Crystal Directions; Wave Propagation Electron beam generated pattern in TEM (001) (110) Crystal Directions We have referred to various directions in the crystal as (100), (110), and (111). What do these mean? How are these directions determined? Consider a cube- 1 and a plane- 2/3 1/2 The plane has intercepts: x = 0.5a, y = 0.667a, z = a. Crystal Directions We want the NORMAL to the surface. So we take these intercepts (in units of a), and invert them: 1 2 3 , ,1 2, ,1 2 3 2 Then we take the lowest common set of integers: 1 2 3 , ,1 2, ,1 4,3,2 2 3 2 These are the MILLER INDICES of the plane. The NORMAL to the plane is the (4,3,2) direction, which is normally written just (432). (A negative number is indicated by a bar over the top of the number.) Crystal Directions 1 (111) 1 1 (110) 1 1 Crystal Directions (100) 1 (220) 1 1/2 1/2 Crystal Directions Electron beam generated pattern in TEM (001) (110) So far, we have discussed the concept of crystal directions: 1 (111) 1 1 (110) 1 1 We want to say a few more things about this. z Consider this plane. y The intercepts are 0,0, , which leads to 1,1,0 for Miller indices. What are the normals to the plane? x The intercepts are -1, 1, , which leads to 110 z The normal direction is ˆ ˆ x y 110 y x -1 -1 The intercepts are 1, -1, , which leads to 1 10 The normal direction is ˆ ˆ x y 11 0 We can easily shift the planes by one lattice vector in x or y z y x -1 -1 These are two different normals to the same plane.
    [Show full text]
  • Pharmaceutico Analytical Study of Udayabhaskara Rasa
    INTERNATIONAL AYURVEDIC MEDICAL JOURNAL Research Article ISSN: 2320 5091 Impact Factor: 4.018 PHARMACEUTICO ANALYTICAL STUDY OF UDAYABHASKARA RASA Gopi Krishna Maddikera1, Ranjith. B. M2, Lavanya. S. A3, Parikshitha Navada4 1Professor & HOD, Dept of Rasashastra & Bhaishajya Kalpana, S.J.G Ayurvedic Medical College & P.G Centre, Koppal, Karnataka, India 2Ayurveda Vaidya; 3Ayurveda Vaidya; 43rd year P.G Scholar, T.G.A.M.C, Bellary, Karnataka, India Email: [email protected] Published online: January, 2019 © International Ayurvedic Medical Journal, India 2019 ABSTRACT Udayabhaskara rasa is an eccentric formulation which is favourable in the management of Amavata. Regardless of its betterment in the management of Amavata, no research work has been carried out till date. The main aim of this study was preparation of Udayabhaskara Rasa as disclosed in the classics & Physico-chemical analysis of Udayabhaskara Rasa. Udayabhaskara Rasa was processed using Kajjali, Vyosha, dwikshara, pancha lavana, jayapala and beejapoora swarasa. The above ingredients were mixed to get a homogenous mixture of Udayab- haskara Rasa which was given 1 Bhavana with beejapoora swarasa and later it is dried and stored in air-tight container. The Physico chemical analysis of Udayabhaskara Rasa before (UB-BB) and after bhavana (UB-AB) was done. Keywords: Udayabhaskara Rasa, XRD, FTIR, SEM-EDAX. INTRODUCTION Rasashastra is a branch of Ayurveda which deals with Udayabhaskara Rasa. In the present study the formu- metallo-mineral preparations aimed at achieving Lo- lation is taken from the text brihat nighantu ratna- havada & Dehavada. These preparations became ac- kara. The analytical study reveals the chemical com- ceptable due to its assimilatory property in the minute position of the formulations as well as their concentra- doses.
    [Show full text]
  • Phys 446: Solid State Physics / Optical Properties
    Phys 446: Solid State Physics / Optical Properties Fall 2015 Lecture 2 Andrei Sirenko, NJIT 1 Solid State Physics Lecture 2 (Ch. 2.1-2.3, 2.6-2.7) Last week: • Crystals, • Crystal Lattice, • Reciprocal Lattice Today: • Types of bonds in crystals Diffraction from crystals • Importance of the reciprocal lattice concept Lecture 2 Andrei Sirenko, NJIT 2 1 (3) The Hexagonal Closed-packed (HCP) structure Be, Sc, Te, Co, Zn, Y, Zr, Tc, Ru, Gd,Tb, Py, Ho, Er, Tm, Lu, Hf, Re, Os, Tl • The HCP structure is made up of stacking spheres in a ABABAB… configuration • The HCP structure has the primitive cell of the hexagonal lattice, with a basis of two identical atoms • Atom positions: 000, 2/3 1/3 ½ (remember, the unit axes are not all perpendicular) • The number of nearest-neighbours is 12 • The ideal ratio of c/a for Rotated this packing is (8/3)1/2 = 1.633 three times . Lecture 2 Andrei Sirenko, NJITConventional HCP unit 3 cell Crystal Lattice http://www.matter.org.uk/diffraction/geometry/reciprocal_lattice_exercises.htm Lecture 2 Andrei Sirenko, NJIT 4 2 Reciprocal Lattice Lecture 2 Andrei Sirenko, NJIT 5 Some examples of reciprocal lattices 1. Reciprocal lattice to simple cubic lattice 3 a1 = ax, a2 = ay, a3 = az V = a1·(a2a3) = a b1 = (2/a)x, b2 = (2/a)y, b3 = (2/a)z reciprocal lattice is also cubic with lattice constant 2/a 2. Reciprocal lattice to bcc lattice 1 1 a a x y z a2 ax y z 1 2 2 1 1 a ax y z V a a a a3 3 2 1 2 3 2 2 2 2 b y z b x z b x y 1 a 2 a 3 a Lecture 2 Andrei Sirenko, NJIT 6 3 2 got b y z 1 a 2 b x z 2 a 2 b x y 3 a but these are primitive vectors of fcc lattice So, the reciprocal lattice to bcc is fcc.
    [Show full text]
  • An Atomistic Study of Phase Transition in Cubic Diamond Si Single Crystal T Subjected to Static Compression ⁎ Dipak Prasad, Nilanjan Mitra
    Computational Materials Science 156 (2019) 232–240 Contents lists available at ScienceDirect Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci An atomistic study of phase transition in cubic diamond Si single crystal T subjected to static compression ⁎ Dipak Prasad, Nilanjan Mitra Indian Institute of Technology Kharagpur, Kharagpur 721302, India ARTICLE INFO ABSTRACT Keywords: It is been widely experimentally reported that Si under static compression (typically in a Diamond Anvil Setup- Molecular dynamics DAC) undergoes different phase transitions. Even though numerous interatomic potentials are used fornu- Phase transition merical studies of Si under different loading conditions, the efficacy of different available interatomic potentials Hydrostatic and Uniaxial compressive loading in determining the phase transition behavior in a simulation environment similar to that of DAC has not been Cubic diamond single crystal Silicon probed in literature which this manuscript addresses. Hydrostatic compression of Silicon using seven different interatomic potentials demonstrates that Tersoff(T0) performed better as compared to other potentials with regards to demonstration of phase transition. Using this Tersoff(T0) interatomic potential, molecular dynamics simulation of cubic diamond single crystal silicon has been carried out along different directions under uniaxial stress condition to determine anisotropy of the samples, if any. -tin phase could be observed for the [001] direction loading whereas Imma along with -tin phase could be observed for [011] and [111] direction loading. Amorphization is also observed for [011] direction. The results obtained in the study are based on rigorous X-ray diffraction analysis. No strain rate effects could be observed for the uniaxial loading conditions. 1. Introduction potential(SW) [19,20] for their simulations.
    [Show full text]