(英文资料)从晶体学到高分辨电子显微图像-第四部分

BenitoïteBaTiSi3O9A Journey from Crystallography and Mineral-Chemistryto TEM Image Simulation through Diffraction andto TEM Image Simulation through Diffraction andInstrument detailsCristiano FERRARISC i tiImmersion Course–March2009The aim is to learn how to approach inorganic materials h t i ti bl f t i ticharacterization problems from a nanometric perspective.CrystallographyPrimary KnowledgeCrystal ChemistryMaterials ProcessingInvestigative Competence CharacterizationInterpretativeInvestigative CompetenceControlling SkillsFunctionalityFundamental Understanding leading to IntelligentAdaptation .Course TimetableDay 1Day 2Day 3Day 4Day 508CrystallographySymmetry &Transmission ElectronMicroscopy Electron diffraction Selected Area Electron Imaging & HRTEM •Phase ContrastAEM08..3009.9.220•Symmetry & symmetry operators •Plan groups •Exercises •Samples & preparation techniques•Selected Area Electron Diffraction •Ring patterns •Tutorials•Lattice fringe images •HREM images •Moiré Patterns •HR Images●Introduction to Spectroscopy ●EDS analyses techniques0909..3010Crystallography •3D Unit cell Screw axes Electron-matterinteraction •Reciprocal Space Electron diffraction •Kikuchi Diffraction •Convergent Beam Electron Diffraction Imaging & HRTEM •The optical system •The Contrast Transfer AEM●Reading EDS analyses Scanning 10.10.220•Screw axes •Space groupsp p •Reciprocal LatticeElectron Diffraction •TutorialsFunction (CTF)•Scherzer Defocus●Scanning Transmission Electron MicroscopyElectron-matterImaging & HRTEM •AEM10.3011.11.220Crystallography •3-D structuresplane projections •Tutorials Electron-matter interaction •Ewald Sphere•Diffraction patterns Contrast definitions •Amplitude contrast •Bright and dark field images•Mass-thickness t t Experimental image &simulations •Tutorials●Scanning TransmissionElectron Microscopycontrast 11.30TransmissionElectron Microscopy Electron-matterinteraction•Structure factor Diffraction pattern Imaging & HRTEM •Diffraction contrast •Thickness &bending Experimental image &simulations AEM12.20•Principles•General overview •Instrument Details•Diffraction pattern Indexing•Diffraction principles & theoryeffects•Structural defects •Strain field •Dislocations •Tutorials•EDS/STEMchemical mapping140014.0018.00TEM Practice TEM Practice TEM PracticeTEM Practice TEM PracticeCrystallographyC t l Ch i t Chemistry Must BeConsistent With C SCrystal ChemistryElectron Matter InteractionsCrystal Structure DiffractionInterference PatternsCrystal Structure MustChemical AnalysisCrystal Structure Must Accommodate ChemistrySuggested Text Books1. C.Giacovazzo et al.:Fundamentals of Crystallography2nd edition(IUCRTexts on Crystallography7),Oxford University Press(2002).T t C t ll h7)O f d U i it P(2002)y g p y y y,g y 2. F.D.Bloss:Crystallography and Crystal-Chemistry,Mineralogical Society ofAmerica,Washington D.C.(1994).3. A.Putnis:An Introduction to Mineral Sciences,Cambridge University Press 3A Sciences(1992).Observation Scale•For all Science fields,measures are defined by an International System (SI ):the length unit is the meter and its submultiples.AubertiteCuAl (SO 4)2Cl·14(H 2O)1 cm = 001 = 10-21 cm = 0,01 = 102m 1 mm = 0,001 = 10-3m Mineralogy1 m = 0,000001 = 10-6m1 nm = 10-9m1 Å= 10-10mCrystallography1 nmWhat is Crystallography?•Crystallography is the science studying minerals at atomic scale in particular the crystalline state defined as ordered and periodic features.These periodic and ordered features are defined,in a3-D space,by precise atomic positions obeying laws,operations and symmetry operators.The3-lawsD ordered distribution of the atomic species is the origin of theunit cell concept.•The unit cell is the smallest geometric ordered set of atoms that,infinitely repeated in the3-D space representing the th t i fi it l t d i th3D ti thfull symmetry of the crystal.•As a consequence the crystal structure is a tri-periodic atomic arrangement defined by the repetition of an elemental volume(unit cell)along the three directions of the specific space to which it belongs.Building Crystal Shapes•External shape reflects internal symmetry and periodicity;•RenéJ.Haüy showed in1784that crystals were built bystacking together tiny identical building blocks;•In1801he developed the theory of rational indices for crystalfaces.Crystal Shape DefinitionsEuhedral crystal completely bounded by well-formed faces A h d l it f h d l Anhedral opposite of euhedral Acicular needle shapedPyrite Tabular two prominent parallel facesMicaceous an extreme case of tabularPyrite FeS 2Hemimorphic crystals with different forms at both ends Lamellara sheaf of thin sheetsVanadinite BerylMuscovite Millerite SPb 10(VO 4)6Cl 2Be 3Al 2Si 6O 18KAl 2(AlSi 3O 10)OH 2NiSOrigins of CrystallographyThe word"crystal"is derived from the root"cryos"which means cold.Some of the first studies of symmetry in nature were stimulated by observations of snowflakes.snowflakesIn fact,snowflakes are composed of tiny ice crystals,but the entire snowflake is not a single crystal since it isg ynot an object with translation symmetry(more on that topic later).As far back as the days of Plato,philosophers were investigating the geometry and symmetry of regular objects in three dimensions.Definingdimensionssome of these regular objects,particularly important in describing the coordination of atoms and molecules in crystalline solids we have:Polygon:A two-dimensional shape bounded by straight line segments.A polygon is regular if the edges are of equal length and meet at equal angles.All the Platonic Solids are constructed either from:Triangles Squares PentagonsT i l S P tPolyhedron:finite region of space enclosed by a finite number of planes.A polyhedron is characterized by its faces ,edges and vertices .It is convex if its interior lies entirely on one side of each face.A regular polyhedron has identical faces consisting of regular polygons,{p},exactly q of which meet at vertex {p q}each vertex.The Schäfli notation for a regular polyhedron is {p,q}.Icosahedron {35}Tetrahedron {33}Icosahedron {3,5}Tetrahedron {3,3}Octahedron {3,4}Dodecahedron {5,3}Cube {4,3}The Five Platonic SolidsThere are only five regular polyhedra,or"Platonic solids".They were regardedas sufficiently special to be identified with the"elements:"Earth,Air,Waterand Fire.Since there were four elements and five regular polyhedra,the Fire"elements"polyhedraicosahedron was honored with the designation of"quintessence"meaning"the substance of heavenly bodies and latent in all things."Name Symbol Plato ID Vertices Edges Faces Dihedral Angle Tetrahedron{3,3}earth46470°32' Cube{4,3}air812690°Octahedron{3,4}water6128109°28' Dodecahedron{5,3}fire203012116°34' Icosahedron{3,5}quintessence123020128°11' Euler Formula:Euler and Descartes discovered the governing relationship between the number of vertices,edges and faces of regular polyhedra:,g g p yV-E+F = 2V = 4p/(2p+2q-pq), E = 2pq/(2p+2q-pq)and F = 4q/(2p+2q-pq).V4/(2+2)E2/(2+2)d F4/(2+2)From this relationship,it is apparent that there are only five Platonic solids.Platonic Nano-crystalsr aTetrahedronl y h e de d P o Cubeu n c a tT r IcosahedronWe have to obtain acorrelation between thesymmetry of the unit celland the crystal«external»We have to obtain a spatial orderedusing:yshapes=MorphologicalCrystallographylaws+symmetry elements+symmetryarrangement of atomsy y y yoperations=Physic CrystallographyUsing these tools we build the unit cell asthe smallest geometric ordered set ofatoms that,infinitely repeated in the3-Dspace,represent the crystal full symmetry=Structural Crystallographyy g p yBasic Definitions for Discussing SymmetrySymmetry :Looking the same in more than 1orientation.Symmetry Operation :An operation (translation,reflection,rotation,glide)which leaves an object unchanged in aspect.Symmetry Element :The point/line (2D)or axis/plane (3D)at which applied the symmetry operation is applied.Tutorial Exercise:What symmetry operations are in this square?Mirror Line Symmetry ElementsSymmetry OperationsReflection 360o /4 = 4-fold point Mirror Line RotationpSymmetry is Everywhere‘broken’ symmetryImposing 2D symmetry on 3D objects Screw Axis p g y y jPlane Projection!Screw AxisChiralityChiral :An object that is not superimposable with chiral Not –its mirror image is chiral.only is this the best definition it is short and to the point –but it is the definition.No exception has ever been found.Achiral :An object that is superimposable with its mirror image is achiral.I t ll h hi l l tti i t th t i In crystallography a chiral lattice point,that is one reflected across a mirror plane,is shown by a commaPositions related by lattice translation have the same chirality y yMono-Dimensional Lattice -1DIn 1D we obtain an infinite ordered arrangement of equal objects using a τg q j g vector τhaving module equal to the distance between two adjacent objects (identity period This regular disposition of objects (points )is called mono-dimensional network and it should be homogeneous (each point must have the same surrounding environment).per od ).Symmetry is defined as an operation of superposition of two objects ;the vector τis a translational t l t th t d t i ti f msymmetry element that determines an operation of identity (1);in 1D also a reflection point m is a symmetry element determining an identity operation.All In 1D the only existing network is called primitive and is indicated by the p .the possible combination between the possible networks (p)and the symmetry elements (1and m)are called Spatial Groups (SG );in 1D the only possible GS are p 1and pmmmmmτBi-Dimensional Lattice-2DIn2D spacewe obtain un infinite orderedarrangement of identical objects using twoτ2arrangement of identical objects using twonot coplanar vectorsτ1andτ2forming anangleγ.τ1γThe possible parallelograms built connectingfour lattice points identify the possible unitcells each containing only one lattice point(each point isIn2D,besides the vectorsτ1andτ2,there are others symmetry elements:rotation shared by four unit cell so its value for one single cell is just ¼).y ypoints of order n,reflection lines m,and reflection lines+translation(Glide lines g).N i ll ll th ibl d fNominally all the possible orders ofrotation points are possible behinddetermined by2π/n,(n=integer)but,b f h f l ibecause of the presence of translationvectors,n can only be equal to1,2,3,4,6.In 2D the 2vectors τ1and τ2act on a rotation point of order producing the repetition of the τ1τ2p nt f r r n pr uc ng th r p t t n f th point infinitely as it happen for any lattice point or symmetry element.n=2Besides that,combining a translation vector and pointsn=4a rotation point we obtain other rotation points.n=6n=3As said before in 2D an object Rcan be r transformed in L and vice-versa by areflection line m .R L l r l Ifbesides thereflectionoperation we add atranslation operation equivalent to ½we m r obtain a glide line g .If l in 2Dspace all the symmetry elements go through a single point,10combinations are possible ,corresponding to the 1,2,3,4and 6rotation points and their combinations with the greflection lines m .643216mm 4mm 3m mm mBi-Dimensional Lattice -2DDepending on the modules of both τ1and τ2and on the value of γwe can obtain 4possible combinations:bli l i h h 1.τ1≠τ2et γ≠90°;oblique lattice where the only compatible rotation points are 1and 2.2t t l l tti 2.τ1≠τ2et γ=90°;rectangular lattice where the compatible rotation points are 1and 2,but also m and g lines.3.τ1=τ2et γ=90°;square lattice where the compatible rotation points are 1,2and 4l th d liplus the m and g lines.4.τ1=τ2et γ=120°;hexagonal-rhombohedral lattice h th c mp tibl t ti n p ints 12nd h l ll l h where the compatible rotation points are 1,2,3and 6and the only m line.These 4lattices are all primitive (p )but is possible to obtain other non-primitive lattices adding other lattice points both in between or in the middle of the primitive lattice but only if these operations will not destroy the symmetry elements.In this case is possible to demonstrate that only adding a point in the middle of the rectangular lattice originates a new centered lattice named (c ).ba90oba90o b90oSquare Lattice a=b; γ= 90o aRectangular Latticesa ≠b; γ= 90o a b120ob a Hexagonal Lattice a=b; γ= 120oOblique Latticea ≠b; γ≠90oIn the 2D space the combination of the 5lattices (4p +1c)with the symmetry elements (1,2,3,4,6,m,generates 17Planar Groups (SG).y y (,,,,,,g)g p ()1.Oblique lattice :p 1and p 22.Rectangular lattice :pm,pg,cm,cg,pmm,pmg,pgg,cmm,cmg and cggBut we can observe that cm ≡cg et cmm ≡cmg ≡cggThis is possible just applying suitable changes in the choice of the cell origin:•cm ≡cg by translation of the cell origin by τ1/4•cmg ≡cmm by translation of the cell origin by τ2/43S l tti 4•cgg ≡cmm by translation of the cell origin by (τ1+τ2)/43.Square lattice :p 4,p 4m ,p 4g4.Hexagonal-Rhombohedral lattice :p 3,p 3m 1,p 31m ,p 6,p 6m17 Plane GroupsSymbolSymbol Short No. PointGroup Lattice Full11Oblique p1p122Oblique p211p2Obli3m Rectangular p1m1pmg p g pg4Rectangular p1g15Rectangular c1m1cm62mm Rectangular p2mm pmm7Rectangular p2mg pmgR t l28Rectangular p2gg pgg9Rectangular c2mm cmmg104Square p4p4114mm Square p4mm p4m12Square p4gm p4gS44133Hexagonal p3p3143m Hexagonal p3m1p3m1g p p15Hexagonal p31m p31m 166Hexagonal p6p6176mm Hexagonal p6mm p6ma)A point (object-atom)in any position (general positions)respect to thesymmetry elements having general coordinates (x,y )is repeated by the symmetry elements in so called equivalent positions .b)The equivalent positions are indicated usingand ,indicating chiral ,positions.c)The total number of the equivalent positions generated by the symmetry elements gives the multiplicity of the x y position.d)The number of objects which fall on a symmetry element (excluding g line)are in special positions .Special positions have always a lower multiplicity by respect to the general positions.The orientation of the reference axes is:x y x, y →x, ym coincide with x axisi id i h i -x, y →x, y m coincide with y axis x, y →y, xm bisect the angle between x and y axes x x th is n d 2 t ti n p int in th i in---x, y →x, y there is an order 2 rotation point in the origin x, y →y, xthere is an order 4 rotation point in the origin -General and Special PositionsGeneral Position4 per unit cellSpecial Positionsp2 per unit cell(horizontal mirror line)2 per unit cell(vertical mirror line)(ti l i li)p1 per unit cell(2-fold axis)Translation Reflection Rotation GlideFormal Representation of Plane Projections y b0x1 -start with unit cell2D translational2 -add symmetry elements i l id b ld li a2D translational repeatmirrors overlaid as bold lines ,,3 -add general positions ,,chiral positions (mirrors!)x,y x,y,chiral positions (mirrors!)(1)x,y (2) ‐x,‐y (3) ‐x,y (4) x,‐y ,,,,x,y x,ySummarising Crystallographic Information,,,,mirror2-foldd l l Wyckoff Site ,,,,Coordinates (1) x,y (2) ‐x,‐(3) ‐x,y (4) x,‐Multiplicity lettersymmetry41i (),y (),y (),y (),y ½,y ½,‐y 2.m.h 0,y 0,‐y 2.m.g x,0‐x,02..me x,½‐x,½2..mf Rectangular ½,½12mmd 12mmc 2mm½,00½2mm No. 61b 12mm a 0,½0,0International Tables of Crystallography l planelattice type point group symmetryinternational notation,plane groupfull form arrangement ofarrangement of symmetry elementsin the unit cellconvention for choice of origin constraint on choice of general position x,ylocation and type of x y transformation co e t o o c o ce o o glocation and type of x,y transformationdiffractiondatageneral and special positionsusing fractional co-ordinatesCenteringRectangular pm No. 3Rectangular cm No. 5,,,,½½,,,,,primitive centeredIn the2D space recovering of a unit cell it is obtained by a vector T=uτ1+vτ2 where u and v are integers(including zero).If the T origin corresponds to the l i i i d d h d di i b i h lattice origin and we use as axes x and y theτ1andτ2directions,we obtain that u and v are the coordinates of a lattice point if the measure unit along x is theτ1 module and along y is theτ2one;both the modules ofτ1etτ2are generally indicated das a0and b0.τ2oTh li b l tti i ts s tτ1γy The line by2lattice points representa«row»of equation x/ma0+y/nb0=1(m and n are integers)⇒nx/a0+/b Th i i t liddmy/b0=mn.The origin nearest lineintercepting x and y at a0/n andb0/m,respectively,will have equation//b1dddxnx/a0+my/b0=1.The origin equi distance distance d of the line from the origin,corresponding to the equi-distance among parallel lines,is as small as m and n are big.We can also notice that when d become smaller the distance between the lattice points increase⇒the lattice density di i i hdiminish.In the3-Dimensions the space is defined by3vectorsτ1,23and the anglesα=τ2313andγ=τ1τ,τand the anglesα=τ∧τ,β=τ∧τandγ=τ∧τ2.These6parameters are called lattice constants anddefine the geometry of the unit cell.τ3τ2τ13D,besides the vectorsτ,τand,In3D1τ2τ3the other symmetry elements are:Rotation axesR t tiReflection planesInversion centreScrew axesRotoinversion axesGlide planesTri-Dimensional Lattice -3DIn 3D space the rotation symmetry operation of an angle θaround a straight line generates the multiplication of an object in n objects where n=2π/θ;this g p j j straight line is named rotation axe .12346In the3D space a n object can repeated in other equivalent objects but enantiomorphic using a reflection plane(m).ti hi i fl ti l()Enantiomorphic derives fromg pp,greek words enantios=opposite,and morphê=shape indicatingobjects that are composed ofexactly the same parts but whichy pare reversed with respect to anaxes or a symmetry plane,or in3D a m rror plane;one object smirror iscalled left while the other one isnamed right.p g p Derivation of the 32 point groupsTri-Dimensional Lattice-3DThe inversion centre is a symmetry element which repeat an object into another equivalent one but eniantomorphic;the points belonging to the two i l t bj ts t th s m dist f m equivalent objects are at the same distance from the inversion centre but in opposite directions.The inversion centre is indicated by the letter i .⎺⎺⎺The rotoinversion axes correspond to a rotation n b d h h 1≡i ⎺2 ≡m3 ≡3+i4 = ⎺6 ≡3/m combined with the inversion centre and are represented by⎺n where n is the rotation⎺⎺⎺⎺⎺order:12346.Tri-Dimensional Lattice-3Dp g p Derivation of the 32 point groupsTri-Dimensional Lattice -3DThe combination among a rotation axe n and a translational vector τ(identity period)parallel to the rotation axe ,generates a new symmetry element axe τt named screw axe.τtThe screw axe acts first with a rotation αfollowed by a translation l h d ll l h A screw axe having a translation component m τ/n is indicated using the t along the direction parallel to the rotation axe .This translation t is the gliding and,among t and the τp g symbol n m ;the possible screw axes are:21,31,32,41,42,43,61,62,63,64,identity period τ,the relation t=m/n τmust be verified;n is the rotation order,and m is an integer tt t 65.equal to n-1.The glide plane is a symmetry elementg p yy acting both with a combined reflection and translation .The difference withthe linein 2D is that the translationgcannot be only in onedirection.Theglide plane can act along different directions making define d rect ons mak ng necessary to def nealong which direction the translationacts.1/2The translation having components τ/2,τ2/2orτ3/2are indicatedby a ,b or c .The translations having diagonal 1+τ2/21+τ3/22+τ3components τ+τ/2,τ+τ/2,τ+τ/2and τ1+τ2+τ3/2are indicated by n .If thelattice is not a primitive one (p ),thetranslation having diagonal components is indicated by d .The different combinations of the symmetry elements going through a single point are named point groups.Between all the symmetry elements only the rotation axes,the reflection planes and the inversion center are take into consideration;the rotoinversion axes et the glide planes cannot be combined in onecomponentspoint because of their translation components.The combination of2rotationaxes in one point generate athird axe.Of all the possible combinations between axes2,3,4et6(20)22222232422622only the combinations222,322(32),422,622,233(23)et432are possible.23432Derivation of the 32 point groups1/m ≡m The combination of rotation axes ()d fl ti l ()2/m 3/m 4/m 6/m =⎺6(n)and reflection planes (m)generates new point groups represented by n/m.If a rotation axe (n )is parallel to one or more reflection planes (m ),new point groups are generated represented by nm(m).2mm 3m 4mm 6mm If a rotoinversion axe ⎺n is parallel to ()p one or more reflection plans (m )new point groups are generated represented by ⎺n2m or ⎺nm2.⎺42m ⎺6m2⎺32/m =⎺3mD i ti f th 32 i t Derivation of the 32 point groups。