Weak beam dark field (WBDF) microscopy is a type of transmission electron microscopy (TEM) dark field imaging technique that allows for the visualization of crystal defects with high resolution and contrast. Specifically, the technique is mainly used to study crystal defects such as dislocations, stacking faults, and interfaces in crystalline materials. WBDF is a valuable tool for studying the microstructure of materials, as it can provide detailed information about the nature and distribution of defects in crystals. These characteristics can have a significant impact on material properties such as strength, ductility, and corrosion resistance.Williams, D. B., Carter, C. B., (2009) Transmission Electron Microscopy. Springer.
WBDF works by using a selected weak first-order diffracted beam from the specimen. This is made possible by tilting the specimen to excite higher angle diffraction spots. The electrons diffracted by the crystal are selected using an objective aperture and selective aperture, which allows only a small fraction of the diffracted electrons to be imaged to the detector. The objective aperture controls size and angle of the incoming beam that is selecting the diffracted beam. The selective aperture selects the area where the diffraction comes from.
The WBDF image is able to highlight the location and type of crystal defects because the lattice bends back to Bragg's diffraction orientation near the defect core. The image can be further enhanced by tilting the crystal in different directions, which changes the orientation of the defects with respect to the electron beam. Under certain special diffraction conditions, dislocations can be imaged as narrow lines. The dislocation lines and Burgers vector can be determined for each dislocation.Cockayne, DJH 1981 Weak-Beam Electron Microscopy Ann. Rev. Mater. Sci. 11, 75–95. A review of WB. Also, the movement of dislocations in materials can be studied to determine mobility and subsequent material properties.Rakhmonov, J.U., Bahl, S., Shyam, A., Dunand, D.C. (2022) Cavitation-resistant intergranular precipitates enhance creep performance of θ’-strengthened Al-Cu based alloys. Acta Materialia, 228, 117788.
History One of the first instances in literature which began the development of WBDF is from Hirsch, Howie, and Whelan in 1960.Hirsch, P. B., Howie, A., & Whelan, M. J.. (1960). A kinematical theory of diffraction contrast of electron transmission microscope images of dislocations and other defects. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 252(1017), 499–529. Their paper focused on applying kinematical theory to TEM imaging with emphasis on dislocation and defect imaging. Then, weak-beam techniques were further demonstrated from R. Gever, et al.Gevers, K., van Landuyt, J. and Amelinckx, S. (1967), Two Beam Kinematical Theory for the Diffraction of Electrons by Crystals with Stacking Faults. phys. stat. sol. (b), 21: 393-411. The authors predicted that even when selecting a weak kinematical spot to form a TEM image, the fringe periodicity is the same as for bright field imaging. Further research into WBDF in 1969 demonstrated the technique's usefulness in imaging dislocations, as developed by Cockayne, Ray and Whelan.Cockayne, D. J. H, Ray I. L. F., and Whelan, M. J. (1969) Investigations of dislocation strain fields using weak beams, The Philosophical Magazine, 20:168, 1265-1270, Since then, the technique has been widely used for analysis of dislocations and their interactions in crystalline samples.
WBDF theory
The weak beam dark field (WBDF) technique is based on using a diffracted beam with a large excitation error () to form an on-axis dark field image. To form an image, a first-order diffraction spot is selected while the sample is tilted to excite a higher angle, typically ~ 3g, diffraction spot. The WBDF g-ng condition means that 1g is the g vector used for forming the image and ng is the excited g vector. The specimen is tilted for the Ewald sphere to intersect the lattice at the origin and 3g as the figure shown. In the figure, , the excitation error, is made to be large such that the origin and the third order diffraction spot intersect with the Ewald sphere and are excited by the electron beam.P.J. Phillips, M.J. Mills & M. De Graef (2011) Systematic row and zone axis STEM defect image simulations, Philosophical Magazine, 91:16, 2081-2101,
Note that under the two-beam conditions, the crystal is tilted in a way that there is only one strong diffracted beam at and all other diffracted beams are weak, ideally in a symmetric way around the direct beam. The intensity of the diffracted beam g in a perfect crystal can be written as an equation below:
Where is the sample thickness, is the extinction distance for diffraction vector g that depends on lattice parameters, atomic number, and the beam electron voltage used, and is the effective excitation error given by the equation:
For the WBDF technique, the excitation error can increase to about 0.2 as well as . When then , and this is known as the kinematical approximation. This theory leads to the main advantages seen from high-contrast defect images from WBDF. From the equation above, it is evident that the intensity decreases as increases.Hirsch, P. B., Howie, A., & Whelan, M. J. (1960). A Kinematical Theory of Diffraction Contrast of Electron Transmission Microscope Images of Dislocations and other Defects. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 252(1017), 499–529.
In the areas of the sample without defects, diffraction intensity is weak which appears as dark areas in the image. However, near the dislocation core, the lattice plane bends back into Bragg's condition, which leads to a bright intensity peak observed as the dislocation line in the WBDF image. The main challenge presented with this technique is the ability to optimally adjust the tilt conditions to minimize the excitation error of the g reflection near the dislocation core so that a sharp dislocation line becomes visible. The exact value of g is not typically exactly 3g as proposed, and is dependent on material properties such as lattice parameter and TEM instrumentation used.Barnard, Jonathan & Sharp, Jo & Tong, Jenna & Midgley, P. (2006). Weak-beam dark-field electron tomography of dislocations in GaN. Journal of Physics: Conference Series. 26. 247-250. 10.1088/1742-6596/26/1/059.
Dislocations in the sample, described by their Burgers vector b, will appear under the diffracted beam, vector g, when . This is important in WBDF imaging because one of the main advantages are the ability to qualitatively and quantitatively describe defects in a given material. There are three main mathematical methods to determine the dislocation peak position as described below.Cockayne, D. (1972). A Theoretical Analysis of the Weak-beam Method of Electron Microscopy. Zeitschrift für Naturforschung A, 27(3), 452-460.
1. Weak beam criterion: The diffracted beam has the largest intensity when or is zero.
Where z describes the direction of the electron beam and R is the displacement field around the dislocation.
2. Kinematical integral: There is maximized scattering from the transmitted to the diffracted beam when the kinematical integral equation below is maximized. This occurs near the dislocation core in the sample.
3. Computing the contrast: The width of the dislocation peak, represented by , can be narrowed according to the below equation. As the excitation error increases, the width of the dislocation peaks, and therefore the contrast seen in the image near the boundary between the dislocation and the background.
Weak beam dark field technique setup process Setting up weak-beam dark-field imaging in transmission electron microscopy involves several steps, which may vary depending on the specific TEM instrument and the sample being analyzed. This may require further optimization and adjustment to achieve the desired image quality and contrast. The general steps include:Miao, J., Singh, S., Tessmer, J., Shih, M., Ghazisaeidi, M., DeGraef, M., & Mills, M. (2018). Dislocation Characterization using Weak Beam Dark Field STEM Imaging. Microscopy and Microanalysis, 24(S1), 2202-2203.
Sample preparation: The sample should be prepared as a thin foil with thickness of a few tens of nanometers because the visibility of the image decreases as the sample thickness increases. However, the Kikuchi line might not be visible in the sample that is too thin. Sample orientation: The sample should first be aligned with the electron beam and tilted to a suitable angle to achieve the desired dark-field contrast. Tilt the sample to a low-index zone axis. Follow the Kikuchi lines to form the two-beam condition that is orienting the sample in bright field mode such that g is excited and is close to 0. Align the sample as if a dark field image were to be taken such that 1g aligns with the optical axis. Further tilt the sample to excite ng. The most common condition is g-3g. This will ensure the Bragg condition for the 3g reflection is satisfied. When looking at the diffraction pattern for confirmation, the 3g spot will become brighter and the 1g spot will become dimmer. Select the diffraction spot: The diffraction spot of 1g should be selected using an objective aperture to form a WBDF image with g-3g condition. Adjust the beam intensity and focus: The electron beam intensity and focus should be adjusted to achieve a high signal-to-noise ratio and a clear image. The beam should be focused on the sample and have a low enough intensity to avoid radiation damage. Acquire the image: The dark-field image can be acquired by selecting the WBDF mode and adjusting the camera parameters, such as exposure time, binning, and magnification, to optimize the image quality. Post-processing: The acquired image can be further processed using image analysis software to enhance the contrast, remove noise, and quantify the defect or feature of interest.
Comparison with other TEM imaging techniques
WBDF is often used in tandem with other TEM imaging techniques such as bright field (BF) and dark field (DF) imaging. These frequently used techniques similarly create an image from electrons that pass through and interact with the sample, however, the difference lies in the electrons which are selected to fall on the detector, and the degree of sample tilt. This control is allowed by the objective aperture. In BF imaging, the direct beam is selected to create the image, and in DF imaging, the scattered electron beams are used to create an image. Evidently, WBDF also uses scattered beams to form an image, but the difference between DF and WBDF comes from the degree of sample tilt and thus beam intensity on the first order diffraction spot. Shown below, a WBDF sample is tilted to excite the 3g diffraction spot to the Bragg condition, and the first-order diffraction spot is selected. This excitation is seen on the instrument as the diffraction spot getting brighter as the sample is tilted.
The analytical difference between WBDF and BF and DF imaging is that WBDF can achieve high-contrast images of defects and thickness changes in a sample. This is made possible by tilting the sample to increase the intensity of the beam on diffraction spots further from the direct beam. Due to the tilting and subsequent increase in excitation error, the electrons are treated in the kinematical approximation. This is the aspect of WBDF which sets it apart from BF and DF imaging, and allows for high contrast defect characterization. This process is described in more detail in the WBDF theory section.