Prior to the invention of global positioning systems (GPS), humanity relied mainly on accurate cartography to optimize navigation Council, 1995. Navigation by cartography was not only confined to terrestrial travel, but extraterrestrial travel as well through the knowledge of celestial orbits of the moon and planets Gehrz et al., 2007. Yet, even the most detailed maps eventually became outdated and possibly misread or misinterpreted depending on the level of spatial awareness of the traveler. The advent of GPS in 1973 eventually brought about a revolution among travelers for ease of use, but more importantly for the confidence it instilled. No matter how complicated the route, a step-by-step guide was provided. A similar approach is needed for the nanoscopic world of electron microscopy. A guide that provides microscopists the ability to record where they have been in a sample, and to develop roadmaps for themselves and others to assist in future analysis is required. Those positions could be converted between microscopes if and when the sample needed to be re-analyzed. Just as a physical road map needs to be turned and flipped to compare to a landmark, fiduciary marker, or reference position, so too should electron microscopists have the ability to flip and rotate positional data when moving a sample to a different microscope.

Since the invention of electron microscopy Mulvey, 1996Knoll & Ruska, 1932, not only did the physical observation of microstructures become important (e.g., shape, size, and morphology), but so too did relating the crystallographic knowledge to those microstructures. The spatial resolution of transmission electron microscopes (TEM) opened an entirely new world as compared to X-ray and spectroscopic techniques. It led to the discovery and confirmation of deoxyribonucleic acid (DNA) in the 1950s Watson & Crick, 1953, and is currently providing materials researchers the ability to examine crystals atom-by-atom Meyer et al., 2008 as well as to move atoms one by one to create larger structures at the atomic scale Dyck et al., 2018. The push towards picometer resolution a short 80 years after its introduction has been rivaled by few technologies for volumetric analysis. Such techniques like atom probe tomography do provide similar, or better, chemical sensitivity, but still lack in numerous other aspects. Blavette et al., 1993Cerezo et al., 1988. The ever-expanding spectral, structural, and crystallographic techniques available in the TEM still make it the most versatile and attractive analysis technique for a wide range of research fields.

The ability to understand the crystallographic and microstructural orientations of any region of interest within a TEM sample in relation to the physical stage movements is crucial to extracting the most concise and relevant information possible in the shortest amount of time. The geometry and physics of extracting and understanding these data have long been understood and published Duden et al., 2009Klinger & Jäger, 2015Liu, 1994Liu, 1995Qing, 1989Qing et al., 1989Zhang et al., 2018. Programs such as Desktop Microscopist^[1], CrysTBox, ALPABETA, CrystalMaker, JEMS, τompas, SPICA, and K-space Navigator provided a variety of ways to understand crystallographic data Cautaerts et al., 2018Klinger & Jäger, 2015Duden et al., 2009Stadelmann, 1987Palmer, 2015De Graef & McHenry, 2012Xie & Zhang, 2020Li, 2016. In the conclusion of Liu’s calculations on the prediction of cubic crystals a statement was made that, “If an interface between the microscope and the computer is developed, an automated on-line method can also be developed...” Liu, 1994. Others have utilized stage positions and knowledge of crystalline poles to address grain orientations, and more importantly grain boundary misorientations Jeong et al., 2010Liu, 1994Liu, 1995. This research has been widely available, but there is still not a concise, user-friendly manner in which to fully utilize this knowledge for mapping out an entire sample.

There is a need for increased speed and efficiency in electron microscopy due to a wider field of materials being analyzed. With an increasing amount of analytical techniques being developed, higher capital costs associated with purchasing newer instrumentation, and decreased sources of funding it is imperative that all researchers have the opportunity to conduct the best research Maia Chagas, 2018. Current generation spectrometers can be as costly as the base microscope itself. With the revolution of aberration correction advancing resolution to the picometer scale Yankovich et al., 2014, the inclusion of a corrector, whether image or probe, has become increasingly commonplace on all new purchases. This increased technology has added to the steep costs of doing innovative microscopy. These factors have made it such that each minute spent in any microscopy session is precious. It has also made collaboration and user facilities an attractive option for researchers who do not have the capability to perform higher end research at their home institutions. All of this taken into consideration, the future of electron microscopy will be geared towards doing smarter microscopy and automation Spurgeon et al., 2020Olszta et al., 2022, similar to what has been accomplished in the field of X-ray crystallography Abola et al., 2000. The eventual progression into full automation presents the possibility of much of the underlying mathematics and physics being overlooked, as microscopes will eventually perform much of the data collection.

Automation and machine learning, while first pioneered and developed in biological microscopy, is steadily being developed for materials science applications Carter & Williams, 2019Jansen et al., 2013. The genesis of automated detection and tomographic techniques within the framework of understanding biological materials was born out of a need for observing microstructural information over longer length scales, such as counting cells Porter et al., 1945Lidke & Lidke, 2012. The complex nature of the electron interaction physics of material science research (e.g., crystallinity, defects, and variable Z contrast) makes automation more difficult and is most likely why it has slowed the adoption and development in the field. This is not to mention the exceedingly smaller length scales that become crucial to understanding any number of atomic phenomena that control bulk materials properties.

As electron microscopy is a projection technique, there will always be a conundrum in analyzing material properties. While the thinner the sample becomes the more accurate the information collected (e.g., decreased multiple scattering), as the sample gets thinner the less representative the information is of the entire bulk sample (e.g., only a thin slice of a three dimensional object is being observed). Additionally, the thinner the sample the more questions of surface effects dominating the analysis arise Carter & Williams, 2019Findlay et al., 2010. Machine and smart learning algorithms require more demanding analytical image analysis techniques within the realm of materials science Braidy et al., 2012Jones et al., 2017Jones et al., 2015Jansen et al., 2013. To be able to position a sample to understand specific g vectors, contrast changes, and orientation effects requires more math than simple edge detection or shape recognition. Even when algorithms are developed to address this, the nature of relevant nanoscopic information within a finite sample thickness (e.g., even with a sample being 40-50 nm thick) may hinder their widespread acceptance. Therefore, there is still a need for the materials science microscopist to interact and guide the collection of data, and as such, there needs to be an intermediary that provides microscopists with tools to better analyze and understand their data.

More importantly, due to a more an ever-increasing reliance on metadata and digital capture, there has been less concentration on dictation and annotation of data. Electron microscopy is becoming more of a tool than a science, and although there are many programs to process and analyze data, there are few that serve as a digital notebook. While at first seemingly counter intuitive, current research into the human memory suggests that the brain is less likely to remember captured data than what is observed Soares & Storm, 2018. This should seem familiar to any microscopist in discussing microscopy sessions with collaborators in that they “saw” additional features not apparent in the recorded data. There is a need to develop programs that act as a prediction tool, but as well a digital notebook.

Therefore, it is essential to have more accurate and directed electron microscopy to provide a pathway in alleviating the increased demand on current and future instrumentation. While there are inroads being made into automation and machine learning, there will be an unfortunate gap before the technology becomes available and even fiscally tenable Maia Chagas, 2018. This paper provides a way to link together a long database of crystallographic data and double tilt stage mechanics that can be applied to any microscope, regardless of age or technological advancement. The framework of this research is based upon many papers and formulas of past electron microscopists, but it serves to combine all these data as a different manner of thinking to make microscopy more efficient and concise. This research will fully document how to best utilize crystallographic and microstructural information in combination with a double tilt stage to collect the most pertinent information, but also provide a roadmap for future analysis or plan for analysis on a different microscope. The protocol, which is being coined nanocartography, provides insight on how to travel within any given crystal system, quickly plot and solve the orientation of unknown crystals with as little information as one diffracting plane, create oblique tilt series, rapidly position interfaces on edge, relate interfaces to adjacent crystallographic information, quickly understand the tilt limits of each crystalline grain, and most importantly translate any microstructural or crystallographic information collected when reloading a sample or transferring the sample to a different microscope. This goes beyond the broader description of “nano-cartography” described in an editorial by Demming Demming, 2015 describing instrument agnostic analysis at the nanoscale to understand materials systems.

The advent of digital capture (first with charge-coupled devices (CCDs), and more recently with direct electron detection), has provided microscopists with a double-edged sword in terms of data Oxley et al., 2020Ophus, 2019. More data is always preferential, but it has provided a false sense of information capture in the form of metadata. The latent information that users typically believe is embedded within each digital capture often means less meticulous note taking in the belief all information is being transferred. The ability to capture k-space, whether through diffraction or through the Ronchigram, affords microscopists with an advantage in terms of not only knowing that the data collected is correct (as say compared to oversaturation in film), but more importantly the ability to digitally measure that information.

The basis of nanocartography is understanding the control and predictive tilting of a double tilt stage in relation to the orientation and motion of all crystal systems. Additionally, it pertains to the physical constructs within a sample (such as grain boundaries and interfaces) and their relationship to crystallographic orientations. The development of these formulations have long been understood, but rarely, if ever, discussed with relationship to one another Cautaerts et al., 2018De Graef & McHenry, 2012Duden et al., 2009Hayashida et al., 2019Hayashida & Malac, 2016Klinger & Jäger, 2015Li, 2016Liu, 1994Liu, 1995Qing, 1989Qing et al., 1989Xie & Zhang, 2020Moeck & Fraundorf, 2006. Unfortunately, the wide breadth of literature on this subject has failed to yield a complete picture that provides clear methodologies for understanding and controlling the motion of samples using a double tilt stage in a transmission electron microscope (TEM). The need to connect the theoretical and practical into a single document is long overdue, and this work serves this purpose but as well expands upon methodologies inherent to more experienced microscopists. These latent data collection techniques are most often considered best practices within individual labs but rarely published. Incorporation into the context of nanocartography became exceedingly relevant.

The following publication is divided into three major sections followed by discussion and conclusions. The first, Navigation and Orientation, develops mathematical concepts of nanocartography, illustrating how vector analysis, TEM stage movement, and crystallography can be combined to accurately navigate sample analysis. The second, Practical Derivations, utilizes the derivations from section one to develop practical derivations for nanocartography, including the use of digital capture to more accurately navigate a sample as well as identification of grain boundary type. Finally, Practical Applications explores practical applications of nanocartography; how to correctly calibrate stage motion as well as apply derivations from sections one and two. Taken in whole, this work serves as a rich summary of optimization of TEM data collection and analysis.