Inviting others to life in reciprocal space.

Crystallography is an important science that impacts many areas of research. The number of users of crystallographic techniques has increased significantly over the past few decades, as diffractometers have become readily available and many commercial programs with user-friendly interfaces have been developed. The growth in computing power has led to the development of advanced methods that were unthinkable in the past. Yet at the same time, formal crystallography teaching has declined, with many programs no longer offering formal crystallography classes. This combination has resulted in many avoidable errors in data collection, processing, and interpretation of results. Making crystallographic teaching, both of theoretical concepts and hands-on training in data processing and analysis, available to the next generation of crystallographers is a crucial mandate. This paper gives a glimpse at several possible avenues to invite newcomers to a successful life in reciprocal space.


I. CRYSTALLOGRAPHY: FROM SCIENCE TO TOOL
Just over 100 years ago, von Laue's discovery that x-rays are diffracted by crystals 1,2 led to the birth of the field now known as crystallography. The potential of this discovery to solve the atomic level structure of ordered solids was soon demonstrated by Bragg. 3,4 The importance of these early developments was recognized by Nobel Prizes in Physics in 1914 and 1915, respectively. For many decades, crystallography remained an interdisciplinary science that attracted many brilliant researchers who broadened its scope and invented new strategies to collect and analyze data and to interpret results. Throughout this time, crystallography was largely reserved for trained crystallographers and required a thorough understanding of the laws governing reciprocal space.
Over the last few decades, powerful computers have become readily available. In parallel, the implementation of user-friendly interfaces or, more recently, apps has made these computers accessible to everyone. Crystallography has certainly benefited from the tremendous increase in computing power, which has made complex data analysis a routine approach that can be executed at the push of a button. I remember seeing a demonstration of a fully automated diffractometer many years ago, where a researcher could mount a crystal and later return to find a solved and refined structure. Such developments have certainly made crystallographic analysis available to a growing number of researchers. At the same time, this led to a paradigm shift, in which crystallography became just one tool of many that could be used even by researchers without formal training in crystallography. While this works well for high quality crystals with relatively simple structures, it has also resulted in significant numbers of incorrect structure solutions. Many of these errors could be avoided through proper training.
Crystallography is still an active area of research, a science that is developing new methodologies to allow cutting-edge experiments that would not have been possible just a decade ago. Yet some methods, like small molecule single crystal structure solution or some types of powder diffraction analysis, have matured to a point where they have indeed become routine tools for researchers that would not describe themselves as crystallographers. It is our duty to teach crystallography in a way that will equip these researchers with the necessary knowledge to use these tools appropriately while also educating those who want to become the next generation of crystallographers.

II. TEACHING CRYSTALLOGRAPHY TO A BROAD AUDIENCE A. Graduate crystallography class
At the University of Toledo (UT), formal crystallography classes are offered on a regular basis in the Department of Chemistry and Biochemistry. These include a four credit hour class on "X-ray Crystallography" and two credit hour modules on "Macromolecular Crystallography" and "Practical Protein Crystallography." While the classes are mostly designed for the graduate program, senior level undergraduates may also enroll with permission of the department.

Teaching crystallography in the classroom
When I was first given the task of teaching the x-ray crystallography class, I defined my goals as follows: (1) teach the fundamental concepts of symmetry, diffraction, and reciprocal space; (2) give students basic hands-on experience with both single crystal and powder data to gain an appreciation of the steps involved in data collection and analysis; (3) enable them to read papers that report crystallographic data and to judge the quality and validity of the results; and (4) equip them with the necessary knowledge to (i) tackle easy crystallographic problems on their own and (ii) work with an experienced crystallographer on harder problems and have meaningful conversations that allow them to learn. In short, my plan was not to make students experts in a single semester but rather to attract a broader audience and give them a solid foundation on which they could build as needed later on.
Teaching crystallography in a chemistry department comes with the inherent challenge that chemistry students may have varying levels of preparedness with respect to the underlying mathlike vectors and matrices, complex e-functions, and Fourier transforms. Using visual examples as much as possible can be useful to overcome initial fears of complex math. Starting with symmetry operations applied to objects, whether molecules or illustrative drawings, readily familiarizes students with important crystallographic concepts, which are expanded to explain plane and space groups, limitations of filling 3D space, etc., without necessarily using matrix and vector representations. Miller indices can first be introduced for describing crystal faces and later be refined to describe lattice planes. Diffraction laws may initially be derived from a simple drawing that gives Bragg's law when basic algebra is applied even though the concept of "reflection from lattice planes" of course is a highly simplified invention. Basic properties of reciprocal space are easily illustrated using a laser pointer and gratings where repeated patterns determine the location of diffraction spots while contents of these "unit cells" modify spot intensity. In-class exercises are used frequently to make sure all students understand the material. These range from short practice problems to complete symmetry operations using crystallographic symbols to a whole class devoted to analyzing wall papers that represent the 17 plane groups (ww3.haverford.edu/chemistry/Norquist/Research.html) to playing "Crystallography Jeopardy" in preparation of the first midterm exam. This sets the stage for a deeper journey into reciprocal space to explore its properties and how this relates to crystallography, systematic absences, structure factors, and more.
At this point, students have been presented with a lot of information and knowledge, but to truly put things together, hands-on projects that require data analysis are invaluable. These projects have undergone multiple modifications over the years, as the advances in analysis software required harder problems to actually force students to apply what they were taught. The projects are accompanied by specialized lectures that introduce specific programs or techniques like Rietveld refinement. For the single crystal project, students are given a dataset that can be "solved" in three space groups with identical absences with reasonably good statistics. However, only one of the structures makes sense with respect to atomic connectivity. Indexing and space group determination of a powder dataset give students an appreciation for steps that are automated in many single crystal software programs. Finally, Rietveld "eye training" with artificially created less-than-perfect fits has become an excellent tool to make students think about how the radiation source, sample characteristics, and instrumental setup will affect their data.

Classes gone virtual: Pre-and post-pandemic
The basic lectures of UT's crystallography class were videotaped in Fall 2011, and the videos are hosted on the Advanced Photon Source's educational Youtube channel (https://www.youtube.com/ playlist?list¼PLBEB2F9103DBA52D1). Originally created at the request of Brian Toby for some basic crystallography educational material, these videos have taken on a life of their own with more than 9000 views from around the world. They have become the equivalent of an open access distance learning class for many students interested in understanding basic crystallography at universities where such classes may not be offered. Occasionally, students contact me with questions or even request a copy of the homework or exams to test their understanding. The videos at this time clearly demonstrate advances in technology over the past decade-while they were recorded with a state-of-the-art video camera in 2011, in the age of high definition videos, their quality can at best be described as "usable." Admittedly, I never expected that these videos would be so popular and reach such a large audience, so I never researched optimal modes of delivering online lectures. This, of course, changed for everyone in higher education in 2020, when most states forced all instruction to go virtual halfway through the spring semester. Future plans include the creation of voice-over-Powerpoint recordings and use of a tablet to walk watchers of the Youtube channel videos through exercises or derivations on blank slides, leading to much clearer video quality.

B. Workshops and schools
Since many universities in the United States no longer offer formal instruction in crystallography, the importance of workshops and schools has continued to grow. Personally, I have taught in the American Crystallographic Association (ACA) Summer School for Small Molecule Crystallography, the National School on Neutron and X-ray Scattering, the Duquesne University/PANalytical (DUPAN) Powder Workshop, and the Modern Methods in Rietveld Refinement and Structural Analysis. In addition, I have accepted invitations to teach short courses in crystallography at several universities abroad. Many more schools with different foci exist, but sadly, to my knowledge, there is no website that offers a comprehensive summary of schools available in North America. Advertisement often seems to rely on specific mailing lists or word of mouth. Offerings range from comprehensive courses that may extend over a week or two to specialized one-day short courses. The multi-day courses generally combine theory with hands-on instruction on using crystallographic software and in some cases also offer the opportunity to collect or analyze data for participants' samples. This gives many students access to expertise in single crystal or powder diffraction methods that is not readily available at their home institution. Students also have the opportunity to reach out to the instructors of such schools after returning to their lab when they run into challenging crystallographic problems that they are not quite ready to tackle on their own. In this respect, these workshops are an invaluable tool to bring crystallography education to students from a wide range of institutions and create an ongoing teacherstudent mentorship opportunity. On the other hand, the organization of these workshops often has to carefully weigh the costs associated with the course, as this can be prohibitive for some research groups, and also the student-to-instructor ratio if hands-on work is involved.
Several schools were offered in an all-virtual format in 2020 due to the Covid-19 pandemic, and it will be interesting to see over the next few years whether some virtual offerings become permanently available. Such settings offer advantages and face limitations at the same time. Clearly, a virtual offering is readily accessible at minimal to no cost for many attendees, and limitations arising from room capacity do not exist if the appropriate licenses for conferencing programs are used. For example, the National School on Neutron and X-ray Scattering is typically limited to 60 attendees to allow for hands-on experiments at various neutron and synchrotron beamlines. In 2020, the school accommodated over 200 attendees for a collection of virtual lectures, virtual beamline tours, and breakout sessions. Presenters had a chance to answer all questions asked by the audience instead of just a couple of selected questions before it was time for the next talk. However, the individual conversations about specific research problems that typically arise over meals, coffee breaks, and poster sessions fell prey to the virtual format. In addition, hands-on experience at beamlines was obviously not possible. In general, virtual settings can readily be utilized for lecture style presentations but fall short of inperson experiences for any workshop aspects that involve hands-on instruction. This extends to hands-on data analysis sessions, where one instructor can typically handle 5-8 students in a physical class or conference room, whereas a much lower student-to-instructor ratio would be necessary in an online format that necessitates screen sharing and video conferencing.

C. Informal teaching
Informal teaching is yet another way to educate the next generation of crystallographers. While this approach may reach a smaller number of learners, it usually also focuses on discussing very specific problems. This mode of teaching requires significant flexibility on the side of the instructor as the level of "instruction" is extremely broad. On the one hand, informal teaching can happen while listening to a talk or poster presentation by pointing out how crystallography can give insights into the specific topic presented. From personal experience, such conversations could get research groups involved in neutron experiments who never considered this tool or result in collaboration with researchers in a field considerably removed from one's own. On the other hand, researchers may seek us out with specific problems that require crystallographic expertise. For this to happen, one needs to be known as somebody who is willing to teach and share knowledge. At our home institutions, each of us should have that reputation, and if somebody is willing to listen and learn, we should be ready to teach them. Our Instrumentation Center staff knows that I am the resident powder diffraction expert. When they have users who want to take their analysis to a level that is beyond the staff's comfort zone, they contact me. I have trained both academic and industrial customers and often find it refreshing to look at a completely new problem in the midst of my daily routine. Some of these initially informal students later enrolled in my crystallography class as they realized that a sound foundation of knowledge is necessary to troubleshoot their own future data analysis. I have also worked one-on-one with students from India or Turkey to teach them how to analyze their diffraction data, be that through e-mail and data exchanges or walking them through the use of programs in a WebEx session. These contacts usually resulted from workshops or personal connections from conferences.

D. Institutional support for crystallography teaching
A glimpse at various ways of teaching crystallography is presented in Secs. II A-II C. All of them are valid and useful approaches and need to continue into the future if our science is to remain alive and well. A big obstacle to this can be a lack of recognition or support, be that at the university level where many institutions have already done away with formal crystallography teaching or at a national or international level. Crystallographic organizations like the ACA and the International Union of Crystallography have already recognized this need and support various schools, workshops, and broader educational initiatives. The crystallographic database providers like Protein Databank (PDB), Cambridge Structural Database (CSD), and International Centre for Diffraction Databases (ICDD) are also active in crystallography education. However, to make academic institutions rethink their stand on crystallography, clear recognition of the importance of crystallography from agencies like the National Academies of Science, National Science Foundation, or National Institutes of Health would be beneficial. Most universities would pay attention to statements by any of these agencies, and it may be time for more crystallographers to serve in various capacities within these institutions.

III. LEAVING A LEGACY
In 2014, our community celebrated the centennial of the first Nobel Prizes in Crystallography with the International Year of Crystallography, IYCr2014. The celebrations ended in April 2015 with the Crystallography Legacy Conference in Rabat, Morocco. Crystallography certainly already has a legacy adorned with many important discoveries that were recognized by numerous Nobel Prizes. However, if we want this legacy to continue, we must become ambassadors of our science who will help attract bright young minds to this field who can contribute to the next aspects of crystallography's continuing legacy. How exactly this happens will take on many different forms, and there will not be a single "right approach" or "best approach." Any approach that makes somebody appreciate the importance of crystallography, not just "as a tool" but as a science that is alive, well, and still developing, is a worthwhile approach to explore, as I hope I outlined with the couple of selected examples in this Transactions Summary.
I would like to close with a quote from the Encyclopedia of Taekwon-Do, which states "Never tire of teaching. A good teacher can teach anywhere, any time, and (must) always be ready to answer questions." 5 This applies to any field-and if we want to help raise the next generation of crystallographers, then every one of us must be willing to teach those interested in learning our science whenever we are presented with such opportunities.

ACKNOWLEDGMENTS
I cannot write this editorial without acknowledging my own amazing teachers who kindled the love of crystallography and

DATA AVAILABILITY
Data sharing is not applicable to this article as no new data were created or analyzed in this study.