The ability to accurately control the stage motion with respect to crystallography has long been a desirable function in any microscopist’s toolbox. While Desktop Microscopist was one of the first programs to accurately predict diffraction patterns and measure relative stage positions given specific crystallographic information, others have created similar plotting programs for a variety of applications. ALPHABETA a was designed to accurately determine two beam locations for proper dislocation and microstructural analysis of stainless steels Cautaerts et al., 2018. A number of papers have detailed the study of either the double tilt stage and/or its relation to crystallographic analysis, but the overwhelming majority have approached it from an a priori standpoint of having specific crystallographic information Liu, 1994Liu, 1995Qing et al., 1989Qing, 1989Cautaerts et al., 2018Xie & Zhang, 2020. These approaches are applicable for navigating known crystals, but the challenge for materials science is the unknown, especially at the nanoscale. This necessitates an approach by which to consider not only the stage motion for known crystals but unknown structures as well, whether they be crystals or non-crystalline physical constructs important to materials science (e.g., grain boundaries and interfaces).

The approaches detailed in this paper focus on breaking down the complex nature of materials analysis using a double stage tilt mechanism into distinct parts for ease of understanding and explanation; 1) a simple vector analysis approach to explain the physical nature of a crystal as opposed to defining it through diffraction, 2) the stage motion for crystallographic vectors and for physical constructs such as interfaces and boundaries, 3) the introduction of crystallographic analysis converting the physical description of a crystal to reciprocal space, and finally, 4) the use of the structure factor as a discrimination filter to be applied on top of the physical description of the crystal to illuminate the allowable planes/pole for a pre-defined crystal.

Taking each of these components as individual parts assists in developing tools by which it becomes easier to create and develop maps of the crystallographic orientations within a sample in addition to its overall relationship to the sample. These tools can then be utilized in a variety of manners to quickly and efficiently manipulate the sample and also plan for future analyses.

5.1Crystals as Physical Objects¶

In order to deconvolute what is often described as one of the more difficult subjects in the conversion of materials science to electron microscopy, namely crystallography in reciprocal space, the derivations in this research focus on delineating the description of crystals as real space objects from their description in k-space. These derivations serve a dual function in that movement of a crystal in real space is more intuitive than in reciprocal space, and it also assists in relating the crystal to additional non-crystalline physical objects (e.g., grain boundaries) and their motion using a double tilt stage. Much of the published research regarding the use of a double tilt stage for TEM analysis has focused solely on the motion of the stage, and very few have combined it with its relation to crystals. As noted prior, Cautaerts et al. approached the use of a double tilt stage to calculate the motion of a cubic crystal for use in determining the optimal tilt conditions for two beam analysis Cautaerts et al., 2018. While this provided more information than simple stage motion, it still did not fully concern all crystal types.

The explanation of both the description of vectors of non-cubic crystals and their motion is often confusing to inexperienced microscopists along with a larger portion of materials scientists. The understanding of low index poles and vectors is typically the extent to which most curricula extend. Even in many prominent electron microscopy textbooks, the measurement of angles between planes is provided as a series of equations confined to a given system. This further serves to shroud the explanation of systems more complex than cubic.

By first showing the conversion of all systems to cubic and then demonstrating that vectors are transformed in real space provides a clearer, more manageable pathway towards more complex analysis in reciprocal space.

5.2Stage motion of Crystallographic Vectors and Non-Crystallographic Structures¶

This research illustrates a logical method by which to convert all crystallographic systems to a cubic system, and then demonstrate systematically how the motion of any vector occurs through a double tilt stage. Building upon these mathematical operations, the motion or pathways between vectors could also then be calculated. Finally, using similar mathematics of plotting planes of atoms, the motion of non-crystalline physical objects such as grain boundaries, surfaces, or matrix/precipitate interfaces could quickly be defined. The power and flexibility of considering crystals and their motion in real space is demonstrated most easily by relating them to non-crystallographic objects within the sample.

Grain boundary and interface analysis are key components in assessing the nanoscale properties of a material to further explain bulk properties (e.g., micromechanical, thermoelectrical, electronic). Therefore, correct orientation (i.e., on-edge) of grain boundaries for chemical analysis becomes imperative for proper investigation. Equally important, although often overlooked, is the crystallographic relationship of crystals to these interfaces. This is often relegated to more automatic detection/analysis approached such as EBSD in SEM because of its ease of use Wilkinson & Britton, 2012Alam et al., 1954Venables & Harland, 1973Harland et al., 1981. Unfortunately, the deeper understanding of these programs more so than the general meaning of inverse pole figures (IPFs) is taken for granted. Having a manner by which to orient interfaces and then quickly relate their orientation to adjacent crystallographic objects provides for more thorough analysis opportunities during a session. Newer technologies such as precession electron diffraction and 4-D STEM Ophus, 2019Ghamarian et al., 2014 will speed up these analyses, but similar to EBSD, they lack the ability to take into account the physical description of surrounding non-crystalline objects such as grain boundaries.

Additionally, the ability to describe the full stage motion with respect to these objects also provides a pathway to understand their three-dimensional structure within the confines of the foil. To date, electron tomography and atom probe tomography provides the highest three-dimensional spatial resolution (even considering a multitude of artifacts) of materials within the spectrum of analytical materials science analysis tools. For all of the advantages these techniques provide, the major drawback is their extremely localized view of the sample. This tradeoff can be detrimental to a more representative analysis of the sample as a whole, but as well can be costly and time consuming. Using the double tilt stage to tilt samples about any given axis allows microscopists to tell a more complete story of the sample, even within a 50-100 nm volume without having to remove the sample and orient to a logical axis (e.g., α tilt). This motion, when combined with the knowledge of the surrounding crystal, becomes even more important.

Lastly, having the orientation solution of any possible grain within a sample allows for a number of discriminatory actions by a microscopist. If the local orientation of any neighboring ZA or pole is mapped, it can then be compared to the tilt stage limit. Depending on the desired crystallographic orientation, it may not be possible to achieve said orientation within that tilt range. As has been previously demonstrated, with the location of a few zone axes, the coordinates of the principle axes can be accurately calculated Liu, 1995. The comparison of the principle axes of two neighboring grains can then be utilized to calculate the location misorientation and the axis of misorientation that then leads to grain boundary type. Again, future development of scanning diffraction techniques will eventually automate this analysis, but the basic knowledge and understanding of this technique will assist in demystifying the often black box approach to their use.

5.3Crystallographic Analysis/Conversion to Reciprocal Space¶

The description and understanding of crystals as real space objects defined by vectors and vector motion is the first step in developing a more logical pathway for materials scientists and electron microscopists. The additional step, which can then be finally utilized to develop a mapping filter, is to describe crystals within reciprocal space. The latent introduction to reciprocal space is often performed through the preliminary description of Miller indices to describe planes of atoms. This is typically accomplished through utilization of inverse nomenclature, but then drawn as a real space object. This can lead to confusion as to the crystallographic conversion to reciprocal space, especially for non-cubic crystals where the normal to the plane (a real space description) is the same as the Miller indices description (reciprocal space). As previously noted, the motion of a crystal is predicated on the real space description, but in electron microscopy analysis, the majority of terminology deals with reciprocal space. Therefore, decoupling these two explanations as first described in Naviation and Orientation through vector math and then subsequently the conversion to reciprocal space is necessary for better understanding.

This additional conversion takes advantage of the previous conversion of non-cubic systems to cubic. In doing so, the normal to any plane can quickly be calculated given that the descriptions of the normals and the Miller indices are the same. Utilizing these normals to describe the plane motion in a double tilt stage provides a full description of any plane. Given the understanding of dislocation imaging along specific planes, the ability to accurately map along the trace of a given plane is imperative for accurate imaging of defects and other crystallographically dependent structures with a sample.

5.4Structure Factor as a Filter¶

A priori knowledge of the allowable planes and poles within a given crystal based on the atomic arrangement in a specific crystal system is the most frequent approach for the analysis of crystals within the microscope. Programs such as Desktop Microscopist and K-space Navigator relied on this a priori knowledge of any given crystal to output diffraction and stereographic projections of said crystal. While these programs were extremely well written descriptions of crystallographic motion and were imperative to the development of this research, their use was confined to known crystals. This utilization of this crystallographic knowledge was based upon the structure factor. By deconvoluting each step and finishing with the structure factor, it can be shown that the structure factor simply acts a filter to be placed upon the results of the previous calculations.

Describing it in this fashion, the position and motion of all possible vectors within a crystal (regardless of system) can be detailed. Similar methodologies can be applied in the conversion of planes from reciprocal space to real space to describe their motion in a double tilt stage. The structure factor can be viewed as which of those planes, and as well ZA/poles, are possible thereby dropping out a large fraction that need to be plotted. Granted, as has been described herein, what is being plotted is not diffraction or Kikuchi lines, but the pathways between poles. Whether tilt stages will ever be developed that provide the precision to accurately discriminate between the (200) or (400) Kikuchi lines is beyond the scope of this paper, but similar methodologies could be developed if and when this becomes reality. Even so, given the tendency for local sample foiling and misorientations, even with the most accurate stages this type of tilting may never be practical.

5.5General Applications of Nanocartography¶

Whereas previous research has been confined to crystallographic analysis and through the motion of a double tilt stage, it will become evident that only by decoupling real space and reciprocal space can strategies for solving unknowns and mapping of samples as real space objects be attained. This takes the extensive amount of previous research on this subject into a new realm that changes the dynamic of how sample analysis in the electron microscope is conducted. The limitations of quickly performing single analyses are removed, and most importantly, it creates a pathway by which rapid utilization of analytical tools at various institutions for a single sample can be performed with ease.

The necessity to accurately and rapidly calculate crystallographic orientations using computer-aided programming is an idea as old as the personal computer itself. With the advent of crystallographic programs to assist in understanding and comparing diffraction data to simulated patterns, it has assisted microscopists in more precisely describing samples. Programs such as Desktop Microscopist even provided the ability to input stage conditions to further predict additional tilt protocols. Unfortunately, due to the delayed data analysis due to film capture, immediate reaction to these directions was difficult if not time prohibitive. Additionally, many of these programs consider only a priori knowledge of crystallographic samples. Much like current EBSD analysis on modern SEMs, their analyses are beholden to input of candidate crystals for optimal results. By considering crystals (and their motion) as physical objects rather than through diffraction and reciprocal space, it allows an easier transition to considering the motion of non-crystalline objects such as grain boundaries and interfaces. More importantly, it opens the avenue to mapping samples that can then be later scrutinized for planning of subsequent microscopic analysis or even for others to rapidly repeat experiments.

Crystallographic analysis using TEM (diffraction) as well as STEM (atomic column imaging) provides highly localized, site-specific identification at the nanoscale of both known and unknown phases. Having full control of the stage both in terms of guiding crystallographic analysis and the orientation of non-crystallographic features can assist in a more complete description of any sample analysis. Combining these together then becomes the ultimate tool for microstructural sample analysis. Describing the sample as a solid object, which can be manipulated similar to crystallographic directionality, allows a greater sense of flexibility. Accurate tilting of an interface or a surface to an edge on condition can mean the difference between measuring a diffusion profile of a few nanometers as compared to tens of nanometers. Similarly, the ability to tilt a boundary or a structure along or against a logical axis can elucidate a wide variety of latent microstructures. Two such examples are accurate tilting a boundary in combination with crystallographic knowledge of adjacent grains and rapid development of tilt series.

Grain boundaries are an extremely important subject in all of materials science analysis due their excess free energy that provides a wide array of phenomena to occur within a microstructure. Rapid diffusion of chromium in stainless steels provides the means for a thin protective layer of chromia to form both on free surfaces and at crack tips to arrest stress corrosion cracking (SCC) Bruemmer et al., 2017Olszta et al., 2014. Gallium can decimate the structure of an aluminum body as it quickly diffuses along grain boundaries, unzipping the entire structure and leaving behind individual grains Rajagopalan et al., 2014. Therefore, the study of how elements diffuse and segregate along grain boundaries is extremely valuable, especially at the nanoscale. At this scale, phase analysis can be difficult because crystallographic information from adjacent grains can obfuscate proper analysis of the desired boundary phase. Given the crystallographic solution of each adjacent grain in combination with the motion of the boundary to an on-edge condition can assist in deduction of the unknown phase. With a boundary edge on, tilting the boundary along the plane can be directed to an adjacent ZA of either grain which might then provide for low index planes to be expressed Carter et al., 1996. More importantly, understanding of parasitic reflections from adjacent grains can be used to discriminate the possible orientation of the unknown phase to either grain.

While the description of the precipitation along grain boundaries is most relevant, understanding morphology and density can also be an effective means in describing more global bulk properties. Whereas the typical goal for most effective sample preparation techniques is to achieve the thinnest possible sample, here it is posited that even without extremely high accelerating voltages, preliminary analysis of slightly thicker samples (100-200 nm) can be just as informative as to the data garnered from subsequent thinning and high-resolution analysis. Within the volume of a 100-200 nm thick sample the density and distribution of grain boundary precipitates can provide a more representative picture of the sample being analyzed. This can be accomplished through simple logical tilt series that takes advantage of having the ability to tilt against or along a given interface. Tomography and APT will always provide a more accurate description of the three-dimensional volume, but they both suffer from being locally destructive techniques in addition to only providing an extremely narrow view of the sample volume. By creating rapid tilt series at any given step size using the protocols, all interfaces within a sample, regardless of orientation, may be transformed into a digital movie that allows for more informative data presentation. Since the step sizes between tilts are minimized for a more accurate description, non-eucentric tilting of non-orthogonally oriented interfaces is not as drastic, which in turn allows for quicker data collection. Lastly, while tilt series of dislocations have been demonstrated in the literature Liu & Robertson, 2011Hata et al., 2020Yamasaki et al., 2015, if the tilt map for any given crystal has been solved, tilt series directions for any plane can quickly be calculated. Instead of following the trace of the plane manually, if the directions for the trace of the plane are calculated, then it provides for easier data collection.

As with many of the subjects described herein, relating adjacent crystals to one another is an important topic in material science analysis and has been discussed in a variety of different manners. Qui et al. demonstrated how knowledge of crystallographic poles of two cubic crystals could assist in solving the local misorientation angle between them Liu, 1994Liu, 1995, and Jeong et al. Jeong et al., 2010 attempted the use of a triangulation method in solving the same problem. In Navigation and Orientation it was demonstrated how one could not only solve the orientation of cubic crystals, but all crystal systems as well. The research herein takes a similar approach to Qui to demonstrate how the calculation of the unit vectors for any crystal can be calculated from the solution of the crystal and then be compared to an adjacent crystal through a misorientation matrix to achieve similar results Qing, 1989. In the derivation of these formulae an important distinction must be considered in that the rotation about an arbitrary rotation axis to move a known pole to the [001] beam orientation must be performed instead of two successive rotations about the α and β axes. In solving the crystallographic orientation of any crystal, the two procedures provide identical results because the known vector is rotated to the [001] position, but in comparing the location of the unit vectors of two adjacent crystals the final misorientation angles does not yield a unique solution. Tilting in the α then subsequently in the β will yield a different result than first β then α (as shown in Figure 6.4). Additionally, through these calculations it has been determined that the triangulation method is not sufficient in accurately describing the local misorientation Jeong et al., 2010. Vectors chosen closest to the [001] beam direction will provide a differing result than vectors farther away from the [001]. This is because the triangulation method does not consider the dependency of the β tilt on the first α tilt, and therefore errors can be compounded for vectors farther from the [001] beam direction.

Finally, whereas the capture of data on film, either by diffraction or imaging, was tedious or time consuming (and often erroneous), digital capture has provided microscopists with the ability to optimize and improve data collection. The improved accuracy can be utilized in two distinct manners, first through the tilt of the stage through small angles, and secondly by calibration of probe deflection. Observation of a large field of view in k-space in larger crystals provides a sense of ease because microscopists can immediately observe the motion of the crystal much like traveling along an open highway. With the advent of probe corrected instruments, the ability to observe even small volumes in Ronchigram mode has allowed microscopists to observe small regions of k-space down to the order to 10s of nanometers. Yet, the ability to tilt within this small area can be difficult, and therefore being able to directly provide directions through self-identified regions on a screen (i.e., a mouse click) is highly desirable. The calculations provided in this research expand upon the tilt motion of the stage in combination with the calibration of k-space within digital capture. Conversely, dark field imaging in TEM mode, a widely useful technique in its own right, is dependent upon the microscopist’s knowledge of accurately deflecting the beam using condenser lens deflectors. Most often, this is performed by eye, but calibrating the digital capture with respect to this deflection allows for more difficult deflection protocols. For instance, blind tilting can could be achieved by simply pointing to a region on the screen and having the computer read out the necessary deflections. Once a diffraction pattern is collected, the beam could be shuttered and a complete array of deflections could be planned out without introducing additional dose to the sample. Finally, even more complex schemes by which the a darkfield map exploring the entire k-space of an FCC versus BCC crystal could be programmed to best discriminate within a field of suspected dissimilar phases. Digital manipulation and control of either small stage tilts or beam deflections provides a clear advantage for rapid and accurate data collection.