Two-beam dynamical electron diffraction

1. Two-beam dynamical electron diffraction Francisco Lovey Centro Atómico Bariloche Instituto Balseiro PASI on Transmission Electron Microscopy Santiago de Chile, July 2006

2. Bibliography Electron 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).

3. DYNAMICAL THEORY OF ELECTRON DIFFRACTION Time independent Schrödinger equation. The solution in the vacuum [V(r) = 0] Relativistic correction

4. 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 Schrödinger equation becomes

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

6. The boundary conditions At the surfaces of the crystal the wave function and its gradient must be continuous At the entrance surface At 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

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

8. After neglecting the back-reflected electrons In matrix notation

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

10. The amplitude of the transmitted and diffracted beams In matrix notation

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

12. Two beam dynamical expressions

13. Eigenvalues Centrosymmetric crystal Extinction distance (ksai) Nearly normal incidence

14. Nearly normal incidence

15. Calculation of the coefficients Cg (centrosymmetric potential) Normalization condition

16. Two beams wave functions General expression Nearly normal incidence

17. z = 0, Nearly normal incidence The diffracted intensity vanishes when

18. Extinction contours in a bent foil

19. 000 400 Two-beam convergent beam electron diffraction

20. Interpretation of the two Bloch waves Taking the x direction normal to the diffracting planes g, then g.r = x/a 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. x

21. Attenuation of the wave functions wave vectors with a complex component: Anomalous absorption effect Electrons 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. This can be achieved by introducing a complex potential of the form: For normal incidence

22. Anomalous absorption effect Since 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

23. The basic eigenvalue equation

24. Complex eigenvalues for nearly normal incidence 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.

25. Two-beams wave functions with absorption

26. Dependence on thickness 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.

27. Dependence with the excitation error Sg The transmitted intensity is strongly absorbed for Sg< 0, this is because the first term, containing the exponential with negative argument dominates. This term comes from the j=1 solution (having a longer wave vector). On the contrary a higher intensity is obtained for Sg > 0. The image is asymmetric with respect to Sg= 0. The diffracted beam is symmetric respect to Sg= 0

28. Dependence with the excitation error Sg S>0 S<0 S>0 S<0 000 400

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

30. Determination of the excitation error 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.

31. 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

32. S>0 S<0 S>0 S<0 000 400 Thickness and extinction distance measurements The diffracted intensity has a minimum when

33. n is the largest integer Thickness and extinction distance measurements Calculated

35. Calculation of the extinction distance The potential Inverse Fourier transform The Fourier coefficients

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

37. Calculation of the extinction distance number of unit cells Atomic scattering amplitude

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

39. Fourier coefficient Structure factor

40. The extinction distance Atomic scattering amplitude Structure factor X-ray scattering factor

41. The x-ray scattering factor for the hydrogen atom For spherical symmetry Ground state of H Bohr radius

42. Atomic scattering amplitude of H

43. Debye-Waller factor Due to atom vibrations Structure factor Mean-square displacement of the atom (kai) Debye temperature of the crystal

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