Difração de Elétrons [6]

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Representation of a general unit cell:

Crystal structure is defined as a regular of atoms decoratinga periodic, 3-dimensional lattice. The lattice is defined as set whichis created by linear combination of 3 basis vectors {a, b, c}.

3 basis vectors: {a, b, c}3 angles between basis vectors: {α, β, γ}6 lattice parameters of the unit cell: {a, b, c, α, β, γ}

Basic crystallography

The seven systems of crystal symmetry:

cubictetragonalorthorhombicrhombohedralhexagonalmonoclinictriclinic

= skew operation

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Basic crystallography

The seven systems of crystal symmetry:

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Basic crystallography

The fourteen Bravais lattices:• Cubic:P – primitiveI – body centeredF – face centered• Tetragonal:P – primitiveI – body centered• Orthorhombic:P – primitiveI – body centeredA, B, C – base centeredF – face centered• Rhombohedral (P)• Hexagonal (P)• Monoclinic:P – primitiveA, B, C – base centered• Triclinic (P) 4>

Basic crystallography

The space groups in three dimensions are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices which belong to one of 7 lattice systems. This results in a space group being some combination of the translational symmetry of a unit cell including lattice centering,the point group symmetry operations of reflection, rotation and rotoinversion, and the screw axis and glide plane symmetry operations. The combination of all these symmetry operations results in a total of 230 unique space groups describing all possible crystal symmetries.

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Basic crystallography

3• International Tables for CrystallographyVol.A - Space Group Simmetry - Hahn (ed), 2005 6>

Crystallographic planes & directions

Cubic planes (100) and (200):

{100} {200}

Notation: (hkl) – a specific plane{hkl} – a family of planes

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Crystallographic planes & directions

Cubic planes (110) and (111):

{110} {111}

Notation: (hkl) – a specific plane{hkl} – a family of planes

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Crystallographic planes & directions

{110} {111}

Notation: (hk.l) or (hkil) – a specific plane{hk.l} or {hkil} – a family of planes

where i = -(h+k)

Closed-Packed Hexagonal planes:

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Crystallographic planes & directions

Lattice directions:

Notation: [hkl] – a specific direction<hkl> – a family of directions

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Reciprocal lattice

lattice parameters of the 3D unit cell (a, b, c)reciprocal lattice parameters (a*, b*, c*)

Bravais – reciprocal lattice relationships:

scalar product: (scalar)

vector product: (vector)

Hence, a* is perpendicular to both b and c; b* is perpendicular to both a and c; c* is perpendicular to both a and b.

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if Bravais lattice is defined by linear combination ofr = ua + vb + wc

then reciprocal lattice can also be defined forg = ha* + kb* + lc*

Two important properties from reciprocal lattice:1) the vector g is normal/orthogonal to the plane (hkl) of crystal lattice:

2) the magnitude of g is

where d(hkl) is the interplanar spacing of (hkl) planes

Reciprocal lattice

)(hklg ⊥

where (u, v, w) and (h, k, l) are integers triplets.

)(1hkld

g =

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Example: reciprocal space of a FCC lattice

Reciprocal lattice

)(hklg ⊥

Q: Are there some diffractedbeam intensity in all pointsfrom the reciprocal lattice?

A: No. It is depending ofstructure factor of actuallattice. There are some “absent” or “forbidden”reflections.

)(1hkld

g =

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Bragg’s law:

nλ = MO + ON = 2OJ sin θnλ = 2d sin θ

Structure factor (F):define kind of interference andwhat are the reciprocal latticepoints “occupied”.

Diffraction on (100) and (200) BCC planes

Electron diffraction

(100): destructive interference

(200): constructiveinterference 14>

Electron diffraction

Structure Factor (F) Systematic absences of reflections depending Bravais lattice types

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EDP vs XRD

Electrons have a much shorter wavelength than the X-rays commonly encountered in the research lab.EDP: λ = 0.00335nm@120kVXRD: λ = 0.15405nm@Cu Kα1 (~ 46X EDP!)

Electrons are scattered more strongly because theyinteract with both the nucleus and the electrons ofthe scattering atoms through Coulomb forces.

Electron beams are easily directed because electronsare charged particles.

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EDP Analysis

Is the specimen crystalline? Crystalline and amorphousmaterials have very different properties.

If it is crystalline, then what are the crystallographiccharacteristics (lattice parameter, symmetry, etc.) ofthe specimen?

Is the specimen monocrystalline? If not, what is thegrain morphology, how large are the grains, what isthe grain-size distribution, etc.?

What is the orientation of the specimen or of individual grains with respect to the electron beam?

Is more than one phase present in the specimen? Ifso, how are they oriented?

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multiple scattering:dynamical diffraction

unique scattering:kinematical diffraction

kI and kD are the k-vectors of the incident anddiffracted waves. K is corresponding to reciprocal vector g.

Electron Scattering

Scattering from crystalline planes:

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Electron Scattering

The Ewald Sphere of Reflection:

ZOLZ: Zero Order Laue ZoneFOLZ: First Order Laue Zone

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Diffraction from Thin Foils

Relrods and Intensity:

S < 0: inside Ewald sphereS > 0: outside Ewald sphere

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Diffraction from Thin Foils

Relrod intensity Distribution: depends from shape ofthe phase which are diffracting, e.g., particles.

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EDP in the TEM

TEM operation for Image/Diffraction apertures

λL – camera length (constant)R – distance/radius of spot diffractiond – interplanar spacing of (hkl)

BF/DF SAED22>

EDP in the TEM

Types of EDP in TEM:

Polycristalline Ring SAED

β-Mn,simple cubic Fe,BCC

Rh,FCC Er,HCP 23>

EDP in the TEM

Effect of Grain Size on Diffraction Rings:

“Finer grain size produces continuous and broader diffracted rings”

lots of coarse grains lots of fine grains

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The Zone Law

ZOLZ / Weiss relationship: h.u + k.v + l.w = 0

[uvw] – axis zone(hkl) – crystal planes

u = (k1.l2 – k2.l1)v = (h2.l1 – h1.l2)w = (h1.k2 – h2.k1)

(h1k1l1)

(h2k2l2)

0det

222

111 =lkhlkhwvu

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Some important relationships

Interplanar Spacing:

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Some important relationships

Interplanar Angles:

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Application

Determine EDP for [001] BCC zone axis

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CAMERA CONSTANT MEASURING (λL)

• Properties of reciprocal lattice:

hkl

hkl

dRL

dg

hklg

⋅=⋅

=

⊥

λ

1)(

r

r

• Camera constant:

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CAMERA CONSTANT MEASURING (λL)

SAD – selected area diffraction aperture

2000nm2000nm

200nm

detail

Evaporated Aluminium/gold std.

1400mm 30>

BF/DF Image – SAEDP rotation

The measurement of therotation of the image of a crystal with respect to its DP constitutesa rotation calibration. The flat-elongated pseudo-orthorhombic Mo3O(very nearly fcc) crystals haveoriented their long side parallel to [100]. Thus photographing the image of a crystal edge and superposing the EDP on the same exposure (as a double exposure) allows the rotation angle to be measureddirectly.

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Orientation Relationships between crystals

Cube – Cube OR: [100]p // [100]α(100)p // (100)α

B = [011]λL = 7,08Ǻ.cm

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Orientation Relationships between crystals

Nishiyama-Wassermann OR:[211]γ // [011]α(111)γ // (110)α12 variants

From: L. Sandoval et al. New Journal of Physics 11 (2009) 103027 (10pp)

Bain OR:[110]γ // [010]α(001)γ // (001)α3 variants

Kurdjumov-Sachs OR:[101]γ // [111]α(111)γ // (110)α24 variants

α: bccγ: fcc

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Twin Orientation

Stacking of {111}compact planes inFCC crystal

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Twin Orientation

Twin spots (T) for(110) FCC crystalorientation

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Kikuchi Diffraction Patterns

FCC Kikuchi linesKikuchi lines are formedby diffuse scattering ofelectrons in thick foils.

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Construction of Kikuchi Maps

[001] FCC pole

Deviation from Bragg position:

s = 0 – bright Kikuchi linecuts the centre of diffracted spot

s < 0 – bright Kikuchi linelies inside between directand diffracted spots

s > 0 – bright Kikuchi linelies outside between directand diffracted spots

s < 0 – inside Ewald spheres > 0 – outside Ewald Sphere

200 220

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Construction of Kikuchi Maps

FCC Kikuchi map

<001>

<011>

<111>

<211>

<411>

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Construction of Kikuchi Maps

BCC Kikuchi map

<001>

<011>

<111>

<211>

<311>

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CBED Patterns

Convergent Beam Electron Diffraction:

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Difração de Elétrons

Notas de aula preparadas pelo Prof. Juno Gallego para a disciplina CARACTERIZAÇÃO MICROESTRUTURALDOS MATERIAIS. ® 2017. Permitida a impressão e divulgação. http://www.feis.unesp.br/#!/departamentos/engenharia-mecanica/grupos/maprotec/educacional/

Williams, D.B.; Barry Carter, C. Transmission Electron Microscopy: A Textbook for Materials Science, 2nd edition. Springer, 2009. DOI: 10.1007/978-0-387-76501-3

Hammond, C. The Basics of Crystallography and Diffraction (3rd ed). Oxford University Press, Oxford, 2009.

Egerton, R. F. Physical Principles of Electron Microscopy: An Introductionto TEM, SEM and AEM. Springer Science+Business Media, Inc., New York,2005.

Goodhew, P. J.; Humphreys, J.; Beanland, R. Electron Microscopy andAnalysis. Taylor & Francis Inc.,New York, 2001.

Cullity, B. D. Elements of X-Ray Diffraction, 2nd edition. Addison-Wesley Publishing Company Inc., Reading, 1978.

Jorge Jr, A. M.; Botta, W. J. Notas de classe – Escola de Microscopia. Laboratório de Caracterização Estrutural, DEMa/UFSCar.http://www.lce.dema.ufscar.br/cursos/escola.html

Bibliografia:

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