1. Technology for a better society 1 Imaging defects and contours/effects

2. Technology for a better society 2 Dynamical diffraction theory Dynamical diffraction: A beam which is diffracted once will easily be re-diffracted (many times..) Understanding diffraction contrast in the TEM image In general, the analysis of the intensity of diffracted beams in the TEM is not simple because a beam which is diffracted once will easily be re-diffracted. We call this repeated diffraction ‘dynamical diffraction.’

3. Technology for a better society 3 Dynamical diffraction: http://pd.chem.ucl.ac.uk/pdnn/diff2/kinemat1.htm Assumption: That each individual diffraction/interference event, from whatever locality within the crystal, acts independently of the others Multiple diffraction throughout the crystal; all of these waves can then interfere with each other Ewald: Diffraction intensity, I, is proportional to just the magnitude of the structure factor, F, is referred to as the dynamical theory of diffraction We cannot use the intensities of spots in electron DPs (except under very special conditions such as CBED) for structure determination, in the way that we use intensities in X-ray patterns. Ex. the intensity of the electron beam varies strongly as the thickness of the specimen changes

4. Technology for a better society 4 THE AMPLITUDE OF A DIFFRACTED BEAM The amplitude of the electron beam scattered from a unit cell is: Structure factor The amplitude in a diffracted beam: (r n denotes the position of each unit cell) The intensity at some point P, we then sum over all the unit cells in the specimen.

5. Technology for a better society 5 If the amplitude φ g changes by a small increment as the beam passes through a thin slice of material which is dz thick we can write down expressions for the changes in φ g and f 0 by using the concept introduced in equation 13.3 but replacing a by the short distance dz Two beam approximation Here χ O -χ D is the change in wave vector as the φ g beam scatters into the φ 0 beam. Similarly χ D -χ O is the change in wave vector as the φ 0 beam scatters into the φ g beam. Now the difference χ O -χ D is identical to k O k D although the individual terms are not equal. Then remember that k D k O (=K) is g + s for the perfect crystal.

6. Technology for a better society 6 The two equations can be rearranged to give a pair of coupled differential equations. We say that φ 0 and φ g are ‘dynamically coupled.’ The term dynamical diffraction thus means that the amplitudes (and therefore the intensities) of the direct and diffracted beams are constantly changing, i.e., they are dynamic Howie-Whelan equations If we can solve the Howie-Whelan equations, then we can predict the intensities in the direct and diffracted beams

7. Technology for a better society 7 Intensity in the Bragg diffracted beam The effective excitation error I g, in the diffracted beam emerging from the specimen is proportional to sin 2 (πtΔk) Thus I 0 is proportional to cos 2 (πtΔk) I g and I 0 are both periodic in both t and s eff Extinction distance, characteristic length for the diffraction vector g Solving the Howie – Whelan equations, and then

8. Technology for a better society 8 Intensity related to defects: WHY DO TRANSLATIONS PRODUCE CONTRAST? A unit cell in a strained crystal will be displaced from its perfect-crystal position so that it is located at position r' 0 instead of r n where n is included to remind us that we are considering scattering from an array of unit cells; We now modify these equations intuitively to include the effect of adding a displacement

9. Technology for a better society 9 Planar defects are seen when ≠ 0 α = 2πg·R Simplify by setting: We see contrast from planar defects because the translation, R, causes a phase shift α=2πg·R

10. Technology for a better society 10 Thickness and bending contours

11. Technology for a better society 11 Two-beam: Bend contours: Thickness – constant s eff varies locally Thickness fringes: s eff remains constant t varies (Intensity in the Bragg diffracted beam)

12. Technology for a better society 12 Thickness fringes Oscillations in I 0 or I g are known as thickness fringes You will only see these fringes when the thickness of the specimen varies locally, otherwise the contrast will be a uniform gray

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14. Technology for a better society 14 Bending contours Occur when a particular set of diffracting planes is not parallel everywhere; the planes rock into, and through, the Bragg condition. Remembering Bragg’s law, the (2h 2k 2l) planes diffract strongly when y has increased to 2θ B. So we’ll see extra contours because of the higher-order diffraction. As θ increases, the planes rotate through the Bragg condition more quickly (within a small distance Δx) so the bend contours become much narrower for higher order reflections.

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17. Technology for a better society 17 Moirè fringes

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19. Technology for a better society 19 Imaging Dislocations

20. Technology for a better society 20 Edge dislocation

21. Technology for a better society 21 Edge dislocation

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23. Technology for a better society 23 (u)

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25. Technology for a better society 25 FCC BCC Slip plane: Slip direction: Burger vector: Slip plane: Slip direction: Burger vector:

26. Technology for a better society 26 FCC BCC Slip plane: {111} Slip direction: Burger vector: a o /2[110] Slip plane:{110} Slip direction: Burger vector: a 0 /2[111]

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28. Technology for a better society 28 Important questions to answer: Is the dislocation interacting with other dislocations, or with other lattice defects? Is the dislocation jogged, kinked, or straight? What is the density of dislocations in that region of the specimen (and what was it before we prepared the specimen)?

29. Technology for a better society 29 Howie-Whelan equations Modify the Howie-Whelan equations to include a lattice distortion R. So for the imperfect crystal Adding lattice displacement α = 2πg·R Defects are visible when α ≠ 0 Intensity of the scattered beam

30. Technology for a better society 30 Isotropic elasticity theory, the lattice displacement R due to a straight dislocation in the u-direction is: Contrast from a dislocation: b is the Burgers vector, b e is the edge component of the Burgers vector, u is a unit vector along the dislocation line (the line direction), and ν is Poisson’s ratio. g·R causes the contrast and for a dislocation

31. Technology for a better society 31 g · R / g · b analysis Screw: Edge: b e = 0 b || u b x u = 0 b = b e b ˔ u Invisibility criterion:

32. Technology for a better society 32 Cindy Smith

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35. Technology for a better society 35 g·b = 1 g·b = 2 Screw dislocation Important to know the value of S

36. Technology for a better society 36 Edge dislocation Always remember: g·R causes the contrast and for a dislocation, R changes with z. We say that g·b = n. If we know g and we determine n, then we know b. g · b = 0Gives invisibility g · b = +1Gives one intensity dip g · b = +2Gives two intensity dips close to s=0 Usually set s > 0 for g when imaging a dislocation in two-beam conditions. Then the dislocation can appear dark against a bright background in a BF image

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39. Technology for a better society 39 Example: U = [2-1-1] B=1/2[101]

40. Technology for a better society 40 Imaging dislocations with Two-beam technique

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43. Technology for a better society 43 Imaging dislocations with Weak-beam technique

44. Technology for a better society 44 The contrast of a dislocations are quite wide (~ ξ g eff /3) Weak beam Small effective extinction distance for large S Two-beam: Increase s to 0.2 nm -1 in WF to increase S eff Narrow image of most defects Characteristic length of the diffraction vector

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48. Technology for a better society 48 Imaging Stacking faults in FCC

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52. Technology for a better society 52 Example: Stacking faults in FCC

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54. Technology for a better society 54 Intensity of the fringes depends on α BF: sin α > 0 : FF (First Fringe) – Bright LF (Last Fringe) -- Bright sin α < 0 : FF – Dark LF – Dark DF: sin α > 0 : FF – Bright LF -- Dark sin α < 0 : FF – Dark LF – Bright α ≠ ± π α = 2πg·R

55. Technology for a better society 55 (A–D) Four strong-beam images of an SF recorded using ±g BF and ±g DF. The beam was nearly normal to the surfaces; the SF fringe intensity is similar at the top surface but complementary at the bottom surface. The rules are summarized in (E) and (F) where G and W indicate that the first fringe is gray or white; (T, B) indicates top/bottom.

56. Technology for a better society 56 Then there are some rules for interpreting the contrast: 1.In the image, the fringe corresponding to the top surface (T) is white in BF if g·R is > 0 and black if g·R < 0. 2.Using the same strong hkl reflection for BF and DF imaging, the fringe from the bottom (B) of the fault will be complementary whereas the fringe from the top (T) will be the same in both the BF and DF images. 3.The central fringes fade away as the thickness increases. 4.The reason it is important to know the sign of g is that you will use this information to determine the sign of R. 5.For the geometry shown in Figure 25.3, if the origin of the g vector is placed at the center of the SF in the DF image, the vector g points away from the bright outer fringe if the fault is extrinsic and toward it if it is intrinsic (200, 222, and 440 reflections); if the reflection is a 400, 111, or 220 the reverse is the case.

57. Technology for a better society 57 Images

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