Kinematical theory of contrast
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1 Kinematical theory of contrast Image interpretation in the EM the known distribution of the direct and/or diffracted beam on the lower surface of the crystal The image on the screen of an EM = the enlarged map of the distribution of the intensity in individual beams (a) Bright field (BF) (b) Dark field (c) Centered dark field (DF) Assumptions: Two-beam approximation: I 0 = 1 I D (BF a DF complementary images) Wave-optical approach Column approximation Contrast in the ideal crystal? Intensity in the point P From the column of the diameter a nm Why? small Bragg angles Contribution from the point A Total contribution integration 0-t
2 = Contribution to the amplitude in the point P from the point A (r n ) n number of EC/ 1 of the surface of the parallel plane with the surface I 0 = 1 Substitution and omission of the term 2πi k.r e a the distance of planes parallel w. the surface Extinction depth for e - (reflection g) Total amplitude: integration 0-t The total amplitude of the diffracted beam Total intensity of the diffracted beam
3 2 kinds of contrast in the ideal crystal I g - t (thickness) - s (orientation) a) Thickness extinction contours (fringes) s=const. I g t, periodicity: z = s -1 Wedge crystal: Alternating bright and dark bands connecting places of t = const. Parallel contours with the edge of the foil. Do not move by specimen tilting b) Bending extinction contours For t = const. I g s, periodicity 1/t Bent crystal: Contours connect places of the same orientation. Move by specimen tilting. Main contour: s g = 0 No other contrast is observed in the ideal crystal!!!
4 Contrast in the real crystal Defect in the crystal local lattice distortion local displacement. The derivation of the amplitude and the intensity on the lower surface of the crystal is the same as in the ideal crystal (column approximation). Difference: Position of individual layers is given by: r n = r n + R n R n the displacement of the EC in a given place of the column Contribution from the point r n In the product K.r n the small value of the product s.r n is omitted. The total amplitude Résumé: The contract of the defect with respect to the ideal crystal is caused by: the additional phase shift of e -iα, α = 2π g.r(z) due to the defect Theoretical condition of the non-visibility of a defect in a given g: g.r(z) = 0, ±1, ±2 for all z in the particular column Practical condition of the non-visibility of a defect in a given g: g.r(z) differs from the integer by less than 2%. Contrast of various defects: the function R=R(z) has to be known! Defect observation in the TEM: a) planar defects grain boundaries, stacking faults, twins.. b) line defects dislocations, dislocation loops, c) particles of other phases precipitates, dispersoids
5 Examples a) Planar defects Thickness fringes Grain boundaries (GBs) High-angle x low-angle GBs
6 Stacking faults b) Line defects Contrast of a dislocation: dark line Inclined dislocation: zig-zag contrast
7 Displacement in the isotropic continuum due to a dislocation Screw dislocation (b e =0, b x u) Non-visibility criteria: g i.b i = 0 (2-3 g i ) b character Dislocation tangles Spitted dislocations - WBDF
8 c) Particles of other phases Coherent, semicoherent and incoherent precipitates Similar contrast: dislocation loops, large clusters of point defects Coherent precipitates (coffee-bone bone contrast) Dislocation loops Nil-contrast line g
9 Phase contrast Before : diffraction contrast image formed by a single strong beam Ψ 0 and/or. Ψ g on the lower surface of the crystal - BF and/or DF OA 1 beam only Another possibility of contrast formation the differences of the PHASE of el. waves scattered in the specimen Disadvantage: strong influence of t, orientation, scattering factor, defocusing Advantage: image of the atom structure of the crystal lattice OA T + n.. D beams Contrast interference of individual beams (resolution as n ) 2 kinds of the image: lattice fringes and HREM a) Lattice fringes Interference of 2 waves : Ψ = ϕ 0 (z) exp2πi (k I.r) + ϕ g exp2πi (k D.r) k D = k I + g + s g = k I + g Intensity I = A 2 + B 2 2AB sin (2π g x - πst) sin wave, periodicity s, t (analogy with thickness fringes) (ϕ 0 A, ϕ g B) thickness fringes d hkl g i) s=0 periodicity 1/g in the direction of x ii) s 0 the shift only (function of s and t), periodicity does not change
10 111 thickness fringes - 2 beams symmetric position (off-axis) - 3 beams (on-axis) 111 thickness fringes - more beams (on-axis)!!! Periodicity of fringes d hkl only (no information about atomic structure) Moiré fringes A translation fringes 2 crystals, planes : d 1 d 2, e.g. 2 coherent precipitates, thin film on the substrate
11 B rotation fringes 2 crystals, d 1 = d 2, parallel planes, e.g. special boundaries, dislocations in the interface C- mixed fringes b) HREM Imaging of the crystal lattice Necessary conditions: - good microscope mechanically and electrically stable, high-res. OL with small spherical defect, coherent source of electrons - climatized room - no vibration - thin sample (30-50nm up to 100 atomic planes) - suitable orientation (low index zone axis) - magnification k Example:
12 Image formation by HREM OL Image = interference of waves passing through the OL Interpretation of the interference image of the HREM Several images in the image plane of he OL 50 nm Several images of different z i or z i have to be taken Comparison of images with computer simulations of a given structure positions of atoms in the column
13 GP zones in AlCu alloy (dural) HREM [110] BF [100] Special HREM techniques a) Electron holography BF only intensity is used, information about the phase is not used El. holography: record of both the amplitude and the phase of the wave Off-axis el. hologram superposition of diffracted and reference wave TEM FEG + beam splitter - Ψ G, Ψ R Image interpretation using wave optics simulations b) Z-contrast Principle: incoherent scattering Output: maps of scattering ability of the specimen
14 STEM FEG Angle detectors instead of apertures (multiple beam imaging): BF ADF DF HAADF incoherently scattered electron intensity imaging Principle: for a given position of the beam the detector acquires and counts the intensities of diffracted beams between two terminal angles and displays the result on the monitor. The resulting intensity without artifacts of the interference contrast direct information about positions of individual atom columns.
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