Two-beam dynamical electron diffractionPASI on Transmission Electron MicroscopySantiago de Chile, July 2006Francisco LoveyCentro Atmico Bariloche Instituto Balseiro

BibliographyElectron Microscopy of Thin Crystals. P.B. Hirsh, A. Howie, R.B. Nicholson, D.W. Pashley and M.J. Whelan. Butterworths (1965)The Scattering of Fast Electron by Crystals. C.J. Humphreys, Reports on Progress in Physics (1979).Electron Microdiffraction, J.C.H. Spence and J.M. Zuo. Plenum Press (1992).Diffraction Physics. J. M. Cowley. Elsevier (1995).Transmission Electron Microscopy (I-IV). D.B. Williams and C.B. Carter. Plenum Press (1996).Transmission Electron Microscope. L. Reimer. Springer (1997).Introduction to Conventional Transmission Electron Microscopy. M. De Graef. Cambridge (2003).

Time independent Schrdinger equation.The solution in the vacuum [V(r) = 0] Relativistic correctionDYNAMICAL THEORY OF ELECTRON DIFFRACTION

The potential V(r) is periodic and can be expressed as a Fourier series The wave function can take the general form of Bloch waves The Schrdinger equation becomes

Since the plane waves are orthogonal to each other, each coefficient in the last equation must be equated separately to zero The basic eigenvalue equation

The boundary conditions At the surfaces of the crystal the wave function and its gradient must be continuous At the entrance surfaceAt z = 0 there is no diffraction thus The tangential components of all wave vectors must be equal to the tangential component of the incident beam

Any wave vector inside the specimen can be written as the incident wave vector plus a correction normal to the surface Sg excitation error

After neglecting the back-reflected electrons In matrix notation

The wave function at the exit surface The amplitude of the transmitted and diffracted beams

The amplitude of the transmitted and diffracted beamsIn matrix notation

Using the boundary conditions at the entrance surface For nearly normal incidence

Two beam dynamical expressions

Centrosymmetric crystal Extinction distance Nearly normal incidence(ksai)Eigenvalues

Nearly normal incidence

Calculation of the coefficients Cg (centrosymmetric potential)Normalization condition

Two beams wave functionsGeneral expressionNearly normal incidence

Nearly normal incidenceThe diffracted intensity vanishes when

Extinction contours in a bent foil

Two-beam convergent beam electron diffraction

The wave has it maximum intensity on the planes x=na, this is along the atoms. On the other hand the wave has it maximum intensity in between the atoms. This is related to the kinetic energy. The k(1) vector is longer that k(2), thus electron represented by will spend more time in areas with a lower potential energy (more negative), i.e., in the vicinity of the atom cores. These electrons may suffer preferentially inelastic scattering with phonons, excitation of inner shells, etc., giving rise to the anomalous absorption. Interpretation of the two Bloch waves

Taking the x direction normal to the diffracting planes g, then g.r= x/a x

Anomalous absorption effectElectrons that are scattered outside the objective aperture contribute to the attenuation of the transmitted and diffracted waves. It looks like absorption from the point of view of the image, but they are not truly absorbed by the crystal.For normal incidenceThis can be achieved by introducing a complex potential of the form:

Anomalous absorption effectSince the imaginary part of is associated with the crystal lattice, it can also be expanded as a Fourier series based on the reciprocal lattice, with coefficients Imaginary component of the extinction distance absorption length

The basic eigenvalue equation

The imaginary component of the wave vector corresponding to j =1 has a higher value, therefore the Bloch waves corresponding to k(1) will attenuate faster with thickness. For thicker specimen only the solution corresponding to j=2 will contribute significantly to the image.Complex eigenvalues for nearly normal incidence

Two-beams wave functions with absorption

The first exponential factor represents a uniform attenuation with thickness of both transmitted and diffracted beams.The first two terms into the brackets in the transmitted beam and the first term in the diffracted beam gives a background of intensity, while the last term, in both expressions, gives an oscillation of the intensity as function of thickness with a period Thus the oscillations show the same period as in the case without absorption.Dependence on thickness

Dependence with the excitation error SgThe transmitted intensity is strongly absorbed for Sg0. The image is asymmetric with respect to Sg=0.The diffracted beam is symmetric respect to Sg=0

Dependence with the excitation error Sg

The images of defects are clearer when observed under the Sg >0 condition, because the background of the perfect crystal is brighter.

An inelastic scattering takes place at the point P in the figure. Electrons will scattered at different directions, those arriving at the planes R and Q with the Bragg angle will be diffracted according to the Bragg low. The inelastically scattered and diffracted electrons having the Bragg angle with the planes R and Q will, in general, lye on the surface of a cone. The intersection of the cones with the screen gives hyperbolic lines, which look like straight lines in the diffraction pattern. The line E1 will be more intense than the background because of the higher probability for forward inelastic scattering at point P, this is called the excess line. On the other hand the line D is called the deficient lines because the electron diffracted to E1 are absent from the background around the transmitted spot.Determination of the excitation error

After tilting the specimen an angle , the Kikuchi lines E1 and D1 will move a distance x away from the associated spot at S=0.

After tilting an angle they will move a

distance . Thus

From the figure we have:

Combining

Thickness and extinction distance measurementsThe diffracted intensity has a minimum when

Thickness and extinction distance measurementsCalculated

Calculation of the extinction distanceThe potentialInverse Fourier transform The Fourier coefficients

The crystal potential in term of the single atomic potentials Calculation of the extinction distance

Calculation of the extinction distancenumber of unit cells Atomic scattering amplitude

Atomic scattering amplitude Poisson equation for the atomic potential X-ray scattering factor

Fourier coefficient Structure factor

The extinction distanceAtomic scattering amplitude Structure factorX-ray scattering factor

The x-ray scattering factor for the hydrogen atomFor spherical symmetryGround state of HBohr radius

Atomic scattering amplitude of H

Debye-Waller factorStructure factorDue to atom vibrationsDebye temperature of the crystalMean-square displacement of the atom(kai)

Atomic scattering amplitude in ordered alloysSite I:Site I:Probability to find an A atom at the site I