CHEM-E5225 :Electron Microscopy Imaging 2016.10 Yanling Ge
Outline Planar Defects Image strain field WBDF microscopy HRTEM information theory Discuss of question homework?
Planar Defects - Internal Interface
Translations and Rotations Translation Boundary, RB. R(r), θ is zero. Grain boundary, GB. Any values of R(r), n and θ are allowed. Phase boundary, PB. As for a GB, but the chemistry and/or structure of the regions can differ. Surface. A special case of PB where one phase is vacuum or gas.
Why Do Translations Produce Contrast? Planar defects are seen when α 0 ( 2nπ).
Stacking Faults in FCC Materials Stacking faults bounds by 1/6 <112> Shockley partial dislocations Intrinsic fault: remove a layer Extrinsic fault: insert a layer Stacking faults bounds by 1/3 <111> Frank partial dislocations For a FCC the translations are then directly related to the lattice parameter: R is either 1/6 <112> or 1/3 <111>
Invisibility Criterion: g R = 0 Invisible: g.r = 0 equal g.r =1 or integer Visible: g.r =1/3 equal g.r = 4/3, g.r is 0 to 1.
Some rules for interpreting the contrast In the image, as seen on the screen or on a print, 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. The central fringe fade away as the thickness increases. Displace aperture instead of CDF for using same hkl for both BF and DF.
Other Translations: π Fringes - α = π
Phase Boundaries Rotation Boundaries
Imaging Strain Fields
Why image Strain Field The direction and magnitude of the Burgers vector, b, which is normal to the hkl diffraction planes. The line direction, u (a vector), and therefore, the character of the dislocations (edge, screw, or mixed). The glide plane: the plane that contains both b and u.
Howie-Whelan Equations Assumption: two-beam treatment, linear elasticity, column approximation. The contrast of the defects will depend on both s and ξ g. g R contrast is used when R has a single value, s R contrast is used when R is a continuously varying function of z, which in turn is associated with g dr/dz.
Contrast from a Single Dislocation The displacement field in an isotropic solid for the general, or mixed, case can be written as:
Contrast from a Single Dislocation Screw dislocation: b e = 0 and b x u = 0. g R is proportional to g b. Pure edge dislocation: b = b e. g R involves two terms g b and g b x u.
Contrast from a Single Dislocation Experimental point: you usually set s to be greater than 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. Identify two reflections g 1 and g 2 for which g b = 0, then g 1 x g 2 is parallel to b. * In practice, when g.b <1/3, contrast is very weak already, especially for partical dislocations. How to identify u?
Example of determination of b From book: An Introduction to Mineral Sciences
Dislocation Nodes and Networks Dislocation Loops
Dislocation dipoles Dipoles can be thought of as loops which are so elongated that they look like a pair of single dislocations of opposite Burgers vector, lying on parallel glide planes. As a result, they are best recognized by their inside-outside contrast.
Dislocation Pairs, Arrays, and Tangles
Surface Effects Dislocation strain fields are long range, but we often assign them a cut-off radius of 50nm. However the specimen thickness might only be 50nm or less. The surface can affect the strain field of the dislocation, and vice versa.
Dislocations and Interfaces Misfit dislocations accommodate the different in lattice parameter between two well-aligned crystalline. Transformation dislocations are the dislocations that move to create a change in orientation or phase.
Dislocations and Interfaces
Volume Defects and Particls
Weak-Beam Dark-Field Microscopy
Weak-Beam Dark-Field image
Intensity in WBDF Images In a perfect crystal the intensity of the diffracted beam in two beam condition: In the WB technique we increase s to about 0.2 nm -1 so as to increase s eff. If s >> ξ g -2 then s s eff and indenpendent of ξ g except as a scaling factor for t, this is known as the kinematical equation, which cannot be applied for all s unless the thickness, t is also very small.
How To Do WBDF CDF with small objective aperture on optimized thickness
Thickness Fringes in Weak-Beam Images
Imaging Strain Field
Weak-Beam Images of Dissociated Dislocations In WB image with g b T = 2, each of the partial dislocations will generally give rise to a single peak in the image which is close to the dislocation core. You can relate the separation of the peaks in the image to the separation of the partial dislocations.
High-Resolution TEM
The Role of An Optical System h(r) describes how a point spreads into a disk, it is known as the point-spread function or smearing function, and g(r) is called the convolution of f(r) with h(r).
The Fourier transform Here u is a reciprocal-lattice vector. This is to say g(r) is a combination of the possible values of G(u), where G(u) is known as the Fourier transform of g(r). A(u): Aperture function; E(u): envelope function (attenuation of the wave); B(u): aberration function; H(u): the contrast transfer function (CTF). Each point in the specimen plane is transformed into an extended region (or disk) in the final image. Each point in the final image has contributions from many points in the specimen.
Contrast Transfer Theory - wikipedia Contrast Transfer Theory provides a quantitative method to translate the exit wavefunction to a final image. Part of the analysis is based on Fourier transforms of the electron beam wavefunction. When an electron wavefunction passes through a lens, the wavefunction goes through a Fourier transform. This is a concept from Fourier optics. Contrast Transfer Theory consists of four main operations: Take the Fourier transform of the exit wave to obtain the wave amplitude in back focal plane of objective lens Modify the wavefunction in reciprocal space by a phase factor, also known as the Phase Contrast Transfer Function, to account for aberrations Inverse Fourier transform the modified wavefunction to obtain the wavefunction in the image plane Find the square modulus of the wavefunction in the image plane to find the image intensity (this is the signal that is recorded on a detector, and creates an image)
The Transfer function If the specimen acts as a weak-phase object, then the transfer function T(u) is sometimes called the CTF, because there is no amplitude contribution, and the output of the transmission system is an observable quantity (image contrast). χ(u) is the phase-distortion function has the form of a phase shift expressed as 2π/λ times the path difference traveled by those waves affected by spherical aberration (C s ), defocus ( z), and astigmatis (C a ). The CTF shows maxima (meaning maximum transfer of contrast) whenever the phase-distortion function assumes multiple odd values of ±π/2. zero contrast occurs for χ(u) = multiple ±π. When T(u) is negative, positive phase contrast results, meaning that atoms would appear dark against a bright background. When T(u) is positive, negative phase contrast results, meaning that atoms would appear bright against a dark background. When T(u) = 0, there is no detail in the image for this value of u. (note that we assume here that C s > 0).
More on χ(u), sinχ(u), and cosχ(u) sinχ starts at 0 and decreases. When u is small, the f term dominates. sinχ first crosses the u-axis at u1 and then repeatedly crosses the u-axis as u increases.
Scherzer Defocus Scherzer found that the CTF could be optimized by balancing the effect of spherical aberration against a particular negative value of f. this value has come to be known as Scherzer defocus, f Sch which occurs at Scherzer resolution:
Simulation of sinχ
Experimental Considerations Remarks: Thinner specimen, ideal case single scattering event. Coma-free alignment to align the beam with optic axis. Specimen orientation is very critical for HRTEM. The Future For HRTEM C s -corrected TEM: C s is zero or as a variable like underfocus. Resolution limit will be determined by C c.
FEG TEMs and The Information Limit The information limit is determined by the envelope function. E c (u): for chromatic aberration E s (u): for the source dependence due to the small spread of angles from the probe. E d (u): for specimen drift. E v (u): for specimen vibration. E D (u): for the detector. http://www.maxsidorov.com/ctfexplorer/scie nce/information_limit.htm Information limit goes well beyond point resolution limit for FEG microscopes (due to high spatial and temporal coherency). For the microscope with thermionic electron sources, the info limit usually coincides with the point resolution. Phase contrast images are directly interpretable only up to the point resolution (Scherzer resolution limit) If the information limit is beyond the point resolution limit, one needs to use image simulation software to interpret any detail beyond point resolution limit.
Some Difficulties In Using An FEG A cold FEG has a small emitter area and Schotty emitter has a source diameter 10 times greater, but with a decrease in spatial coherence and a larger energy spread. Correcting astigmatism is very tricky. Need on-line processing (live FFT). Focal series of image are a challenge with a large range of f values. Image delocalization occurs when detail in the image is displaced relative to its true location in the specimen.
Selectively Imaging Sublattices
Interfaces And Surfaces The fundamental requirement is that the interface plane must be parallel to the electron beam.
Incommensurate Structures
Quasicryatals HRTEM excels when materials are ordered on a local scale. For HRTEM, we need the atoms to align in columns because this is a projection technique, but the distribution along the column is not so critical. And we can t determine it without tilting to another projection in the perfect crystal. SAD and HRTEM should be used in a complementary fashion.
Single Atoms Observation by Parsons et al. with dedicated STEM: uranium atoms in molecule matrix.
homework Question homework: Reading and summary