Planer defects Technology for a better society 1

Planer defects Technology for a better society 1

Planar defects / Boundaries R(r) Technology for a better society 2

Planar defects / Boundaries Translation boundary (RB) • Any R(r) • Θ = 0 • Both sides identical • Also Stacking Fault (SF) (special case) R(r) Technology for a better society 3

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Planar defects / Boundaries Grain Boundary (GB) • Any values of R(r), n, and θ • Chemistry and structure of the two grains must be the same. • Also Twin boundary • Two types: • Low angled • High angled Technology for a better society 6

Planar defects / Boundaries Phase Boundary PB • As GB • Chemistry and/or structure of the two regions can differ. Technology for a better society 7

Planar defects / Boundaries Surface • A special case of a PB where one phase is vacuum or gas. Technology for a better society 8

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Contrast Technology for a better society 10

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. ’ Technology for a better society 11

Dynamical diffraction: 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 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 http: //pd. chem. ucl. ac. uk/pdnn/diff 2/kinemat 1. htm Technology for a better society 12

THE AMPLITUDE OF A DIFFRACTED BEAM The amplitude of the electron beam scattered from a unit cell is: Structure factor Technology for a better society 13

The intensity at some point P, we then sum over all the unit cells in the specimen. The amplitude in a diffracted beam: (rn denotes the position of each unit cell) Technology for a better society 16

If the amplitude φg 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 χO-χD is the change in wave vector as the φg beam scatters into the φ0 beam. Similarly χD-χO is the change in wave vector as the φ0 beam scatters into the φg beam. Now the difference χO-χD is identical to k. Ok. D although the individual terms are not equal. Then remember that k. Dk. O (=K) is g + s for the perfect crystal. Technology for a better society 17

Howie-Whelan equations The two equations can be rearranged to give a pair of coupled differential equations. We say that φ0 and φg 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 If we can solve the Howie-Whelan equations, then we can predict the intensities in the direct and diffracted beams Technology for a better society 18

Solving the Howie – Whelan equations , and then Intensity in the Bragg diffracted beam The effective excitation error Extinction distance , characteristic length for the diffraction vector g • Ig, in the diffracted beam emerging from the specimen is proportional to sin 2(πtΔk) • Thus I 0 is proportional to cos 2(πtΔk) • Ig and I 0 are both periodic in both t and seff Technology for a better society 19

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 rn 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 Technology for a better society 20

Simplify by setting: α = 2πg·R Planar defects are seen when ≠ 0 We see contrast from planar defects because the translation, R, causes a phase shift α=2πg·R Technology for a better society 21

Howie-Whelan equations 1. The amplitude of the electron beam scattered from a unit cell 2. Sum over all the unit cells in the specimen 3. The amplitude changes by a small increment as the beam passes through a thin slice of material dynamically coupled 4. Intensity in the Bragg diffracted beam Technology for a better society

5. A strained crystal will be displaced from its perfect-crystal α = 2πg·R Planar defects are seen when ≠ 0 We see contrast from planar defects because the translation, R, causes a phase shift α=2πg·R Technology for a better society 23

Stacking fault in FCC Technology for a better society 24

Ø Many important materials are fcc, including the metals Cu, Ag, Au, and austenitic stainless steel, and the semiconductors Si, Ge, and Ga. As. Ø Most of the analysis of SFs derives from the study of fcc materials. Ø The translations are well known and directly related to the lattice parameter: R is either 1/6 <-1 -12> or 1/3 <111> Technology for a better society 25

A stacking fault can be seen as the boundary between two wedge shaped crystals which are in direct contact, but with a displacement R along the wedge As a result, the two fringe systems resulting from the two wedges do not fit together anymore. A new fringe system develops delineating the stacking fault; we see the typical stacking fault fringes Technology for a better society 26

In this case translation: R = ± 1/3 [11 -1] Since α = 2πg • R If we form an image with: g = (2 -20) g. R = 0 and the fault is out of contrast in both BF and DF If we form an image with: g = (02 -2) g. R = 4/3 or -4/3 and α = 8 π/3 = 2 π/3 = 120 o or α = -8 π/3 = -2 π/3 = 120 o (modulo 2 π in each case). SF -- parallel to the surface of this (111)-oriented specimen tilt the specimen to see any contrast from the SF, i. e. , g. R=0 for all values of g lying in the fault plane. Technology for a better society 27

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(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. • Be very careful when you record such a pair of images: record the DP for each image. Be sure to note which of the two bright spots corresponds to the direct beam. • Use the same strong hkl reflection for BF and DF imaging. Therefore, to form the CDF image using a strong hkl reflection, you must first tilt the specimen so that hkl is strong and then use the beam tilts to move hkl onto the optic axis where it will become strong. This is confusing, so we recommend that you sacrifice a little image resolution and compare the BF image with a displaced aperture DF image, rather than a CDF image. Technology for a better society 29

Interpreting the contrast • The fringe corresponding to the top surface (T) is white in BF if g • R is > 0 and black if g • R < 0. • 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. • The central fringes fade away as the thickness increases. • The reason it is important to know the sign of g is that you will use this information to determine the sign of R. • 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. Technology for a better society 30

Intensity of the fringes depends on α BF: sin α > 0 : sin α < 0 : DF: sin α > 0 : sin α < 0 : α = 2πg·R FF (First Fringe) – Bright LF (Last Fringe) -- Bright FF – Dark LF – Dark FF – Bright LF -- Dark FF – Dark LF – Bright α ≠ ± π Technology for a better society 31

BF using (200) Ex: TEM image of stacking fault in Co–Ni-based superalloy single crystal compressed to 2 % and subsequently annealed at 973 K for 3. 6 ks. (http: //www-lab. imr. tohoku. ac. jp/~koizumi/research_e/Segregation. html) • Top surface (T) is white in BF if g • R is > 0 and black if g • R < 0 • Type A (200) reflection Technology for a better society 32

Twinning Technology for a better society 33

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http: //www. fzu. cz/en/oddeleni/department-of-functional-materials/selected-results/working-group-functional-materials-and- Technology for a better society 37