Dedicated to the memory of Z.G.Pinsker.

(on the occasion of his 100 th anniversary )

ELECTRON DIFFRACTION STRUCTURE ANALYSIS,

PART 1.

SPECIMENS AND THEIR ELECTRON DIFFRACTION PATTERNS.

Vera KLECHKOVSKAYA Institute of Crystallography, Russian Academy of Sciences

The basic modern data describing the atomic structure of matter have been obtained by the using of three diffraction methods – X-ray, neutron and electron.

Electron diffraction structure analysis is generally used to study thin films and finely dispersed crystalline materials and allows the complete structure determinations up to establishment of the atomic coordinates in the crystal lattice and refinement of atomic thermal vibrations and chemical bounding.

All three radiations are used not only for the structure analysis of various crystals but also for the analysis of other condensed state of matter – quasi crystals, incommensurate phases, and partly disordered systems, namely, for high-molecular polymers, liquid crystals, amorphous substances and liquids, and isolated molecules in vapor and gases. analysis of various crystals but also for the analysis of other condensed state of matter – quasi crystals, incommensurate phases, and partly disordered systems, namely, for high-molecular polymers, liquid crystals, amorphous substances and liquids, and isolated molecules in vapor and gases.

SCHEMATIC ILLUSTRATING BRANCHES OF MODERN CRYSTALLOGRAPHY, THEIR APPLICATIONS, AND THE RELATION OF CRYSTALLOGRAPHY TO THE NATURAL SCIENCES “Heart”of this scheme

“The PARENTS” OF ELECTRON DIFFRACTION STRUCTURE ANALYSIS ZINOVII PINSKER BORIS VAINSHTEIN 1904 – 1986 1921 - 1996

Electron Diffraction Camera have been constructed by Z.Pinsker and B.Vainshtein -- gold medal on the International Exhibition in Brussels, 1958.

the classical monographs: Z.G.Pinsker (1953) Electron diffraction.

London: Butterwords (in Russia, 1949) B.K.Vainshtein (1964) Structure analysis by electron diffraction.

Oxford, Pergamon Press (in Russia,1956)

MAIN STAGES OF ATOMIC STRUCTURE ANALYSIS



the obtaining of appropriate diffraction patterns and their geometrical analysis,



the precision evaluation of diffraction-reflection intensities,



the use of the appropriate formulas for recalculation of the reflection intensities into the structure factors,

I hkl ~ K kin |F hkl | 2 + K dyn |F hkl |



the solution of the phase problem,



Fourier analysis of the structure.



xyz) = 1/

 

F

hkl

exp[2



hkl

i (hx+ky+lz)]

ELECTRON DIFFRACTION PATTERNS transmission and reflection mode

ELECTRON DIFFRACTION PATTERNS FOR STRUCTURE ANALYSIS MOSAIC SINGLE CRYSTAL PLATELIKE TEXTURE POLYCRYSTAL ONLY THREE TYPE SPECIMENS AND ELECTRON DIFFRACTION PATTERNS MAY BE USED FOR ATOMIC STRUCTURE ANALYSIS UNKNOWN

PHASES

GEOMETRICAL ASPECTS OF ELECTRON DIFFRACTION EWALD CONSTRUCTION FOR X-RAY END ELECTRON

k 0 , k – wave-vectors,  - wave – length, a*, b* - parameters of reciprocal unit cell

THE RECIPROCAL LATTICE NET IS SAMPLED BY AN EWALD SCHERE ON RADIUS k

0 =1/



.

SINCE THE WAVELENGTH OF A 100 kV ELECTRON BEAM IS SOME 40 TIMES AS SMALL AS THAT OF A CuK



X-RAY, IT IS OFTEN A SUFFICIENT APPROXIMATION TO SAY THAT THE EWALD SAMPLING SURFACE IS A PLANE IN THE CASE OF ELECTRONS .

SPOT-TYPE ELECTRON DIFFRACTION PATTERNS A CRYSTAL REPRESENT A THREE DIMENTIONAL PERIODIC DISTRIBUTION OF SCATTERING MATERIAL.

THE DISTRIBUTION OF POINTS AT WHICH THE SCATTERING AMPLITUDE DIFFERS FROM ZERO AND TAKES ON THE VALUE F

hkl

IS PERIODIC IN RECIPROCAL SPACE AND FORMS THE SO-CALLED RECIPROCAL LATTICE A SPOT PATTERN REPRESENTED A PARTICULAR PLANE OF THE RECIPROCAL LATTICE PASSING THROUGH OF THE POINT 000 H

hkl

= ha* + kb* = lc* a*, b*, c* are axial vectors, h,k,l are point indices

INDEXING OF AN ELECTRON DIFFRACTION PATTERN OF MOSAIC SINGLE CRYSTAL

Coordinate plane

Non-coordinate plane A spot pattern is most conveniently characterized by the general symbol of the reflection located on it.

If the plane is a coordinate one of the indices must be equal to zero since the point 000 always lies in it.

If the plane is non-coordinate, then none of its three indices (hkl) is equal to zero.

SYMMETRY OF ELECTRON DIFFRACTION SPOT PATTERNS The existence of a centre of symmety at a point 000 of the reciprocal lattice of symmetry being recognizable in diffraction phenomena, for only 11 classes of symmetry being recognizable in diffraction phenomena, although 32 classes of crystal symmetry exist. The symmetry of electron diffraction pattern is the symmetry of the plane nets of reciprocal lattice. Schematic representation of the structure of the zero layer of the reciprocal lattice for the six classes of geometry in spot electron diffraction patterns

The relationships between the axis and angles in unit cells:

Triclinic: a  Monoclinic: a  Orthorhombic: a  b  b  b Hexagonal: a = b   c    c  =   c  =  c  =  Tetragonal: a = b  c  =  Cubic: a = b = c  =  =  =90 0 = 90 = =   0   = 90 = 90 0 0

Having only one plane of reciprocal lattice for unknown crystal we can`t determined – this is coordinate oder non-coordinate plane. And we have not an information about perpendicular direction for this plane.

There are two way: rotation method

and

to have three patterns of different zones Schematic representation of the rotation method Interrelationship between three reciprocal lattice section, i.e. between three electron diffraction patterns of different zones.

POLYCRYSTAL-TYPE ELECTRON DIFFRACTION PATTERN Electron diffraction patterns from samples containing very large number of small randomly distributed crystals consist of continuous rings. The radii of the rings are inversely proportional to the interplanar spacings d

hkl

planes of crystals. The formula

r hkl d hkl = L



, (r- radius of the ring)

is used. of a lattice In reciprocal lattice of a polycrystal is obtained by “spherical rotation” of a reciprocal lattice of a single crystal around a fixed 000 point. It forms a system of sphere placed one inside the other.

A section through such a system of spheres produces a system of rings. Thus geometry of a polycrystalline pattern is a set of lengths H

hkl,

i.e. a set of interplanar distances d hkl of a crystal lattice.

METHOD OF INVERSE SQUARES The calculation for orthogonal lattices The quadratic form for orthorhombic crystals is: 1/d 2 hkl = h 2 /a 2 + k 2 /b 2 + l 2 /c 2 .

Simple division gives all d h00 = a/h, and, analogously, d 0k0 Further , using the scale, all d hk0 are found from and d (d hkl ) = (d hk0 ) + (d oko ), then by fixing first l = 1 (one setting of the movable scale) and 00l . finding all d hk1 , and repeating this operation a – inverse scuares scale, b – method of finding d hk by using the mouvable scale

OBLIQUE TEXTURE ELECTRON DIFFRACTION PATTERNS Distribution of reciprocal lattice points Distribution of circular scattering of a plate texture along straight lines regions of the reciprocal lattice of a parallel to the texture axis and texture on coaxial cylinders.

perpendicular to the face lying on the support.

FORMATION of the circular scattering regions (rings) in the reciprocal lattice of a texture, and relationship between their shape and the structure of the specimen Transition from a point to a ring (a), for an ideal texture without disorder (c) and having distribution function (e) (d,f) – corresponding diagrams for a real texture with some disorder.

PROJECTION NET AND THE CORRESPONDING SET OF R hk VALUES FIVE PLANE CRYSTALLOGRAPHIC SYSTEMS OF POINTS If there are layer lines on the pattern (for orthogonal lattice), for a zero layer line, R hk = H hk0 . Thus having a set of values : R 2 hk = h 2 A 2 + k 2 B 2 + 2hkAB cos



` R = r (L



) -1 We can determined constant A,B,



` of The two-dimensional lattice.

DETERMINATION OF PERIOD c* and ANGLES



Orthogonal unit cells The best formed plate textures are found in crystals with a layer lattice. For the reciprocal lattice of plate texture, the distribution of points along vertical strain lines, parallel to axis z, is characteristic. An important part is played by the modulus of vector H hkl.

H hkl = x 2 + y 2 + z 2 =R 2 + z 2

The doubling of the number of circular scattering regions in the reciprocal lattice of a texture and therefore the number of reflections on the ellipse of a pattern for a non-orthogonal unit cell.

CONCLUSION Mosaic single crystal, polycrystal, texture electron diffraction patterns provide valuable material for calculation the parameters of unit cell and then may be used for complete structural investigations of the crystal with unknown atomic structure.