grain measurement

Crown Caps = Fresh Beer?

Dr. Felix Reinauer, Applications Specialist Europe, EDAX

A few days ago, I visited the Schlossgrabenfest in Darmstadt, the biggest downtown music festival in Hessen and even one of the biggest in Germany. Over one hundred bands and 12 DJs played all kinds of different music like Pop, Rock, Independent or House on six stages. This year the weather was perfect on all four days and a lot of people, celebrated a party together with well known, famous and unknown artists. A really remarkable fact is the free entrance. The only official fee is the annual plastic cup, which must be purchased once and is then used for any beverage you can buy in the festival area.

During the festival my friend and I listened to the music and enjoyed the good food and drinks sold at different booths in the festival grounds. In this laid-back atmosphere we started discussing the taste of the different kinds of beer available at the festival and throughout Germany. Beer from one brewery always tastes the same but you can really tell the difference if you try beer from different breweries. In Germany, there are about 1500 breweries offering more than 5000 different types of beer. This means it would take 13.5 years if you intended to taste a different beer every single day. Generally, breweries and markets must guarantee that the taste of a beer is consistent and that it stays fresh for a certain time.

In the Middle Ages a lot of people brewed their own beer and got sick due to bad ingredients. In 1516 the history of German beer started with the “Reinheitsgebot”, a regulation about the purity of beer. It says that only three ingredients, malt, water, and hops, may be used to make beer. This regulation must still be applied in German breweries. At first this sounds very unspectacular and boring, but over the years the process was refined to a great extent. Depending on the grade of barley roasting, the quantity of hops and the brewing temperature, a great variety of tastes can be achieved. In the early times the beer had to be drunk immediately or cooled in cold cellars with ice. To take beer with you some special container was invented to keep it drinkable for a few hours. Today beer is usually sold in recyclable glass bottles with a very tight cap keeping it fresh for months without cooling. This cap protects the beer from oxidation or getting sour.

Coming back to our visit to the Schlossgrabenfest; in the course of our discussions about the taste of different kind of beer we wondered how the breweries guarantee that the taste of the beer will not be influenced by storage and transport. The main problem is to seal the bottles gas-tight. We were wondered about the material the caps on the bottles are made of and whether they are as different as the breweries and maybe even special to a certain brewery.

I bought five bottles of beers from breweries located in the north, south, west, and east of Germany and one close to the EDAX office in Darmstadt. After opening the bottles, a cross section of the caps was investigated by EDS and EBSD. To do so, the caps were cut in the middle, embedded in a conductive resin and polished (thanks to René). The area of interest was the round area coming from the flat surface. The EDS maps were collected so that the outer side of the cap was always on the left side and the inner one on the right side of the image. The EBSD scans were made from the inner Fe metal sheet.

Let´s get back to our discussion about the differences between the caps from different breweries. The EDS spectra show that all of them are made from Fe with traces of Mn < 0.5 wt% and Cr, Ni at the detection limit. The first obvious difference is the number of pores. The cap from the east only contains a few, the cap from north the most and the cap from the middle big ones, which are also located on the surface of the metal sheet. The EBSD maps were collected from the centers of the caps and were indexed as ferrite. The grains of the cap from the middle are a little bit smaller and with a larger size distribution (10 to 100 microns) than the others, which are all about 100 microns. A remarkable misorientation is visible in some of the grains in the cap from the north.

Now let´s have a look at the differences on the inside and outside of the caps. EDS element maps show carbon and oxygen containing layers on both sides of all the caps, probably for polymer coatings. Underneath, the cap from the east is coated with thin layers of Cr with different thicknesses on each side. On the inside a silicone-based sealing compound and on the outside a varnish containing Ti can also be detected. The cap from the south has protective coatings of Sn on both sides and a silicon sealing layer can also be found on the inside. The composition of the cap from the west is similar to the cap from the east but with the Cr layer only on the outside. The large pores in the cap from the middle are an interesting difference. Within the Fe metal sheet, these pores are empty, but on both sides, they are filled with silicon-oxide. It seems that this silicon oxide filling is related to the production process, because the pores are covered with the Sn containing protective layers. The cap from the north only contains a Cr layer on the inside. The varnish contains Ti and S.

In summary, we didn’t expect the caps would have these significant differences. Obviously, the differences on the outside are probably due to the different varnishes used for the individual labels from each of the breweries. However, we didn’t think that the composition and microstructure of the caps themselves would differ significantly from each other. This study is far from being complete and cannot be used as a basis for reliable conclusions. However, we had a lot of fun before and during this investigation and are now sure that the glass bottles can be sealed to keep beer fresh and guarantee a great variety of tastes.

A Little Background on Backgrounds

Dr. Stuart Wright, Senior Scientist EBSD, EDAX

If you have attended an EDAX EBSD training course, you have seen the following slide in the Pattern Indexing lecture. This slide attempts to explain how to collect a background pattern before performing an OIM scan. The slide recommends that the background come from an area containing at least 25 grains.

Those of you who have performed re-indexing of a scan with saved patterns in OIM Analysis 8.1 may have noticed that there is a background pattern for the scan data (as well as one of the partitions). This can be useful if re-indexing a scan where the raw patterns were saved as opposed to background corrected patterns. This background pattern is formed by averaging 500 patterns randomly selected from the saved patterns. 500 is a lot more than the minimum of 25 recommended in the slide from the training lecture.

Recently, I was thinking about these two numbers – is 25 really enough, is 500 overkill? With some of the new tools (Callahan, P.G. and De Graef, M., 2013. Dynamical electron backscatter diffraction patterns. Part I: Pattern simulations. Microscopy and Microanalysis, 19(5), pp.1255-1265.) available for simulating EBSD patterns I realized this might be provide a controlled way to perhaps refine the number of orientations that need to be sampled for a good background. To this end, I created a set of simulated patterns for nickel randomly sampled from orientation space. The set contained 6,656 patterns. If you average all these patterns together you get the pattern at left in the following row of three patterns. The average patterns for 500 and 25 random patterns are also shown. The average pattern for 25 random orientations is not as smooth as I would have assumed but the one with 500 looks quite good.

I decided to take it a bit further and using the average pattern for all 6,656 patterns as a reference I compared the difference (simple intensity differences) between average patterns from n orientations vs. the reference. This gave me the following curve:
From this curve, my intuitive estimate that 25 grains is enough for a good background appears be a bit optimistic., but 500 looks good. There are a few caveats to this, the examples I am showing here are at 480 x 480 pixels which is much more than would be used for typical EBSD scans. In addition, the simulated patterns I used are sharper and have better signal-to-noise ratios than we are able to achieve in experimental patterns at typical exposure times. These effects are likely to lead to more smoothing.

I recently saw Shawn Bradley who is one of the tallest players to have played in the NBA, he is 7’6” (229cm) tall. I recognized him because he was surrounded by a crowd of kids – you can imagine that he really stood out! This reminded me that these results assume a uniform grain size. If you have 499 tiny grains encircling one giant grain, then the background from these 500 grains will not work as a background as it would be dominated by the Shawn Bradley grain!

Seeing is Believing?

Dr. René de Kloe, Applications Specialist, EDAX

A few weeks ago, I participated in a joint SEM – in-situ analysis workshop in Fuveau, France with Tescan electron microscopes and Newtec (supplier of the heating-tensile stage). One of the activities during this workshop was to perform a live in-situ tensile experiment with simultaneous EBSD data collection to illustrate the capabilities of all the systems involved. In-situ measurements are a great way to track material changes during the course of an experiment, but of course in order to be able to show what happens during such an example deformation experiment you need a suitable sample. For the workshop we decided to use a “simple” 304L austenitic stainless-steel material (figure 1) that would nicely show the effects of the stretching.

Figure 1. Laser cut 304L stainless steel tensile test specimen provided by Newtec.

I received several samples a few weeks before the meeting in order to verify the surface quality for the EBSD measurements. And that is where the trouble started …

I was hoping to get a recrystallized microstructure with large grains and clear twin lamellae such that any deformation structures that would develop would be clearly visible. What I got was a sample that appeared heavily deformed even after careful polishing (figure 2).

Figure 2. BSE image after initial mechanical polishing.

This was worrying as the existing deformation structures could obscure the results from the in-situ stretching. Also, I was not entirely sure that this structure was really showing the true microstructure of the austenitic sample as it showed a clear vertical alignment that extended over grain boundaries.
And this is where I contacted long-time EDAX EBSD user Katja Angenendt at the MPIE in Düsseldorf for advice. Katja works in the Department of Microstructure Physics and Alloy Design and has extensive experience in preparing many different metals and alloys for EBSD analysis. From the images that I sent, Katja agreed that the visible structure was most likely introduced by the grinding and polishing that I did and she made some suggestions to remove this damaged layer. Armed with that knowledge and new hope I started fresh and polished the samples once more. And I had some success! Now there were grains visible without internal deformation and some nice clean twin lamellae (figure 3). But not everywhere. I still had lots of areas with a deformed structure and whatever I tried I could not get rid of those.

Figure 3. BSE image after optimized mechanical polishing.

Back to Katja. When I discussed my remaining polishing problems she helpfully proposed to give it a try herself using a combination of mechanical polishing and chemical etching. But even after several polishing attempts starting from scratch and deliberately introducing scratches to verify that enough material was removed we could not completely get rid of the deformed areas. Now we slowly started to accept that this deformation was perhaps a true part of the microstructure. But how could that be if this is supposed to be a recrystallised austenitic 304L stainless steel?

Table 1. 304/304L stainless steel composition.

Let’s take a look at the composition. In table 1 a typical composition of 304 stainless steel is given. The spectrum below (figure 4) shows the composition of my samples.

Figure 4. EDS spectrum with quantification results collected with an Octane Elite Plus detector.

All elements are in the expected range except for Ni which is a bit low and that could bring the composition right at the edge of the austenite stability field. So perhaps the deformed areas are not austenite, but ferrite or martensite? This is quickly verified with an EBSD map and indeed the phase map below confirms the presence of a bcc phase (figure 5).

Figure 5. EBSD map results of the sample before the tensile test, IQ, IPF, and phase maps.

Having this composition right at the edge of the austenite stability field actually added some interesting additional information to the tensile tests during the workshop. Because if the internal deformation in the austenite grains got high enough, we might just trigger a phase transformation to ferrite (or martensite) with ongoing deformation.

Figure 6. Phase maps (upper row) and Grain Reference Orientation Deviation (GROD) maps (lower row) for a sequence of maps collected during the tensile test.

And that is exactly what we have observed (figure 6). At the start of the experiments the ferrite fraction in the analysis field is 7.8% and with increasing deformation the ferrite fraction goes up to 11.9% at 14% strain.

So, after a tough start the 304L stainless steel samples made the measurements collected during the workshop even more interesting by adding a phase transformation to the deformation. If you are regularly working with these alloys this is probably not unexpected behavior. But if you are working with many different materials you have to be aware that different types of specimen treatment, either during preparation or during experimentation, may have a large influence on your characterization results. Always be careful that you do not only see what you believe, but ensure that you can believe what you see.

Finally I want to thank the people of Tescan and Newtec for their assistance in the data collection during the workshop in Fuveau and especially a big thank you to Katja Angenendt at the Max Planck Institute for Iron Research in Düsseldorf for helpful discussions and help in preparing the sample.

Looking At A Grain!

Sia Afshari, Global Marketing Manager, EDAX

November seems to be the month when the industry tries to squeeze in as many events as possible before the winter arrives. I have had the opportunity to attend a few events and missed others, however, I want to share with you how much I enjoyed ICOTOM18*!

ICOTOM (International Conference on Texture of Materials) is an international conference held every three years and this year it took place in St. George, Utah, the gateway to Zion National Park.

This was the first time I have ever attended ICOTOM which is, for the most part, a highly technical conference, which deals with the material properties that can be detected and analyzed by Electron Backscatter Diffraction (EBSD) and other diffraction techniques. What stood out to me this year were the depth and degree of technical presentations made at this conference, especially from industry contributors. The presentations were up to date, data driven, and as scientifically sound as any I have ever seen in the past 25 years of attending more than my share of technical conferences.


The industrial adaptation of technology is not new since X-ray diffraction has been utilized for over half a century to evaluate texture properties of crystalline materials. At ICOTOM I was most impressed by the current ‘out of the laboratory’ role of microanalysis, and especially EBSD, for the evaluation of anisotropic materials for quality enhancement.

The embracing of the microanalysis as a tool for product enhancement means that we equipment producers need to develop new and improved systems and software for EBSD applications that will address these industrial requirements. It is essential that all technology providers recognize the evolving market requirements as they develop, so that they can stay relevant and supply current needs. If they can’t do this, then manufacturing entities will find their own solutions!

*In the interests of full disclosure, I should say that EDAX was a sponsor of ICOTOM18 and that my colleagues were part of the organizing committee.

Aimless Wanderin’ – Need a Map?

Dr. Stuart Wright, Senior Scientist, EDAX

In interacting with Rudy Wenk of the University of California Berkeley to get his take on the word “texture” as it pertains to preferred orientation reminds me of some other terminologies with orientation maps that Rudy helped me with several years ago.

Map reconstructed form EBSD data showing the crystal orientation parallel to the sample surface normal

Joe Michael of Sandia National Lab has commented to me a couple of times his objection to the term “IPF map”. As you may know, the term is commonly used to describe a color map reconstructed from OIM data where the color denotes the crystallographic axis aligned with the sample normal as shown below. Joe points out that the term “orientation map” or “crystal direction map” or something similar would be much more appropriate and he is absolutely right.

The reason behind the name “IPF map”, is that I hi-jacked some of my code for drawing inverse pole figures (IPFs) as a basis to start writing the code to create the color-coded maps. Thus, we started using the term internally (it was TSL at the time – prior to EDAX purchasing TSL) and then it leaked out publicly and the name stuck – my apologies to Joe. We later added the ability to color the microstructure based on the crystal direction aligned with any specified sample direction as shown below.

Orientation maps showing the crystal directions aligned with the normal, rolling and transverse directions at the surface of a rolled aluminum sheet.

The idea for this map was germinated from a paper I saw presented by David Dingley where a continuous color coding schemed was devised by assigning red, green and blue to the three axes of Rodrigues-Frank space: D. J. Dingley, A. Day, and A. Bewick (1991) “Application of Microtexture Determination using EBSD to Non Cubic Crystals”, Textures and Microstructures, 14-18, 91-96. In this case, the microstructure had been digitized and a single orientation measured for each grain using EBSD. Unfortunately, I only have gray scale images of these results.

SEM micrograph of nickel, grain orientations in Rodrigues-Frank space and orientation map based on color Rodrigues vector coloring scheme. Source: Link labeled “Full-Text PDF” at www.hindawi.com/archive/1991/631843/abs/

IPF map of recrystallized grains in grain oriented silicon steel from Y. Inokuti, C. Maeda and Y. Ito (1987) “Computer color mapping of configuration of goss grains after an intermediate annealing in grain oriented silicon steel.” Transactions of the Iron and Steel Institute of Japan 27, 139-144.
Source: Link labeled “Full Text PDF button’ at www.jstage.jst.go.jp/article/isijinternational1966/27/4/27_4_302/_article

We didn’t realize it at the time; but, an approach based on the crystallographic direction had already been done in Japan. In this work, the stereographic unit triangle (i.e. an inverse pole figure) was used in a continues color coding scheme were red is assigned to the <110> direction, blue to <111> and yellow to <100> and then points lying between these three corners of the stereographic triangle are combinations of these three colors. This color coding was used to shade grains in digitized maps of the microstructure according to their orientation. Y. Inokuti, C. Maeda and Y. Ito (1986) “Observation of Generation of Secondary Nuclei in a Grain Oriented Silicon Steel Sheet Illustrated by Computer Color Mapping”, Journal of the Japan Institute of Metals, 50, 874-8. The images published in this paper received awards in 1986 by the Japanese Institute of Metals and TMS.

AVA map and pole figure from a quartz sample from “Gries am Brenner” in the Austrian alps south of Innsbruck. The pole figure is for the c-axis. (B. Sander (1950) Einführung in die Gefügekunde der Geologischen Körper: Zweiter Teil Die Korngefüge. Springer-Vienna)
Source: In the last chapter (Back Matter) in the Table of Contents there is a link labeled “>> Download PDF” at link.springer.com/book/10.1007%2F978-3-7091-7759-4

I thought these were the first colored orientation maps constructed until Rudy later corrected me (not the first, nor certainly the last time). He sent me some examples of mappings of orientation onto a microstructure by “hatching” or coloring a pole figure and then using those patterns or colors to shade the microstructure as traced from micrographs. H.-R. Wenk (1965) “Gefügestudie an Quarzknauern und -lagen der Tessiner Kulmination”, Schweiz. Mineralogische und Petrographische Mitteilungen, 45, 467-515 and even earlier in B. Sander (1950) Einführung in die Gefügekunde Springer Verlag. 402-409 . Sanders entitled this type of mapping and analysis as AVA (Achsenvertilungsanalyse auf Deutsch or Axis Distribution Analysis in English).

Such maps were forerunners to the “IPF maps” of today (you could actually call them “PF maps”) to which we are so familiar with. It turns out our wanderin’s in A Search for Structure (Cyril Stanley Smith, 1991, MIT Press) have actually not been “aimless” at all but have helped us gain real insight into that etymologically challenged world of microstructure.

My Fossil Background

Dr. René de Kloe, Applications Specialist, EDAX

Call me old-fashioned, but when I want to relax I always try to go outdoors, away from computers and electronic gadgets. So when I go on vacation with my family we look for quiet places where we can go hiking and if possible we visit places with interesting rocks that contain fossils. Last summer I spent my summer vacation with my family in the Hunsrück in Germany. The hills close to where we stayed consisted of shales. These are strongly laminated rocks that have been formed by heating and compaction of finegrained sediments, mostly clay, that have been deposited under water in a marine environment. These rocks are perfect for the occurrence of fossils. When an organism dies and falls on such a bed of clay and is covered by a successive stack of mud layers, it can be beautifully preserved. The small grains and airtight seal of the mud can give a very good preservation such that the shape of the plant or animal can be found millions of years later as a highly detailed fossil. Perhaps the most famous occurrence of such fossil-bearing shale is the Burgess shale in British Columbia, Canada which is renowned for the preservation of soft tissue of long-extinct creatures. The Hunsrück region in Germany may not be that spectacular, but it is a lot closer to home for me and here also beautiful fossils have been found.

Figure 1. Crinoid or sea lily fossil found in the  waste heap of the Marienstollen in Weiden, Germany.

Figure 2. Detail of sulphide crystals.

Figure 3. Example of a complete crinoid fossil (not from the Hunsrück area). Source https://commons.wikimedia.org/wiki/File:Fossile-seelilie.jpg

So, when we would go hiking during our stay we just had to pack a hammer in our backpack to see if we would be lucky enough to find something spectacular of our own. What we found were fragments of a sea lily or crinoid embedded in the rock (Figures 1,3) and as is typical for fossils from the area, much of the fossilised remains had been replaced by shiny sulphide crystals (Figure 2). Locally it is said that the sulphides are pyrite. FeS2. So of course, once back home I could not resist putting a small fragment of our find in the SEM to confirm the mineral using EDS and EBSD. The cross section that had broken off the fossil showed smooth fracture surfaces which looked promising for analysis (Figure 4). EDS was easy and quickly showed that the sulphide grains were not iron sulphide, but instead copper bearing chalcopyrite. Getting EBSD results was a bit trickier because although EBSD bands were often visible, shadows cast by the irregular surface confuse the band detection (Figure 5).

Figure 4. Cross section of shale with smooth sulphide grains along the fracture surface.

Figure 5. EBSD patterns collected from the fracture surface. Indexing was done after manual band selection. Surface irregularities are emphasized by the projected shadows.

Now the trick is getting these patterns indexed and here I do like computers doing the work for me. Of course, you can manually indicate the bands and get the orientations of individual patterns, but that will not be very helpful for a map. The problem with a fracture surface is that the substrate has a variable tilt with respect to the EBSD detector. Parts of the sample might be blocking the path to the EBSD detector which complicates the EBSD background processing.

The EDAX EBSD software has many functions to help you out of such tight spots when analyzing challenging samples. For example, in addition to the standard background subtraction that is applied to routine EBSD mapping there is a library of background processing routines available. These routines can be helpful if your specimen is not a “typical” flat, well-polished EBSD sample. This library allows you to create your own recipe of image processing routines to optimize the band detection on patterns with deviating intensity gradients or incomplete patterns due to shadowing.

The standard background polishing uses an averaged EBSD pattern of more than ~25 grains such that the individual bands are blended out. This produces a fixed intensity gradient that we use to remove the background from all the patterns in the analysis area. When the actual intensity gradient shifts due to surface irregularities it is not enough to just use such a fixed average background. In that case you will need to add a dynamic background calculation method to smooth out the resulting intensity variations.

This is illustrated in the EBSD mapping of the fossil in Figure 6. The first EBSD mapping of the fossil using standard background subtraction only showed those parts of the grains that happened to be close to the optimal orientation for normal EBSD. When the surface was pointing in another direction, the pattern intensity had shifted too much for successful indexing. Reindexing the map with optimised background processing tripled the indexable area on the fracture surface.

Figure 6. Analysis of the fracture surface in the fossil. -1- PRIAS center image showing the smooth sulphide grains, -2- Superimposed EDS maps of O(green), Al(blue), S(magenta), and Fe(orange) -3- EBSD IPF on IQ maps with standard background processing, -4- original IPF map, -5- EBSD IPF on IQ maps with optimized background processing, -6- IPF map with optimized background.

In addition to the pattern enhancements also the band detection itself can be tuned to look at specific areas of the patterns. Surface shadowing mainly obscures the bottom part of the pattern, so when you shift the focus of the band detection to the upper half of the pattern you can maximize the number of detected bands and minimize the disturbing effects of the edges of the shadowed area. It is unavoidable to pick up a false band or two when you have a shadow, but when there are still 7-9 correct bands detected as well, indexing is not a problem.

Figure 7. Band detection on shadowed EBSD pattern. Band detection in the Hough transform is focused at the upper half of the pattern to allow detection of sufficient number of bands for correct indexing.

In the images below are a few suggestions of background processing recipes that can be useful for a variety of applications.

Of course, you can also create your own recipe of image processing options such that perhaps you will be able to extract some previously unrecognized details from your materials.

Aimless Wanderin’ in 3D (Part 3)

Dr. Stuart Wright, Senior Scientist, EDAX

In my research on the origins of the term texture to describe preferred lattice orientation I spent some time looking at one of the classic texts on the subject: Bunge’s “red bible” as we called it in our research group in grad school – Texture Analysis in Materials Science Mathematical Methods (1969). As I was reading I found an interesting passage as it relates to where we are with EBSD today:

“In a polycrystalline material crystallites of different shape, size and orientation are generally present. It can thus also occur that regions of different orientation are not separated from one another by unequivocally defined grain boundaries, but that, on the contrary, the orientation changes continuously from one point to another. If one desires to completely describe the crystal orientation of a polycrystalline material, one must specify the relevant orientation g for each point with coordinates x, y, z within the sample:

g=g(x,y,z)           (3.1)

If one writes g in EULER’s angles, this mean explicitly

φ_1=φ_1 (x,y,z);  Φ=Φ(x,y,z);  φ_2=φ_2 (x,y,z);           (3.2)

One thus requires three functions, each of these variables, which are also discontinuous at grain boundaries. Such a representation of the crystal orientation is very complicated. Where therefore observe that it has as yet been experimentally determined in only a very few cases (see, for example, references 139-141, 200-203), and that its mathematical treatment is so difficult that it is not practically applicable.”

I don’t quote these lines to detract in any way from the legacy of Professor Bunge in the field of texture analysis. I did not know Professor Bunge well but in all my interactions with him he was always very patient with my questions and generous with his time. Professor Bunge readily embraced new technology as it advanced texture analysis forward including automated EBSD. I quote this passage to show that the ideas behind what we might today call 3D texture analysis were germinated very early on. The work on Orientation Coherence by Brent Adams I quoted in Part 2 of this series was one of the first to mathematically build on these ideas. Now with serial sectioning via the FIB or other means coupled with EBSD as well as high-energy x-ray diffraction it is possible to realize the experimental side of these ideas in a, perhaps not routine but certainly, tractable manner.

A schematic of the evolution from pole figure-based ODF analysis to EBSD-based orientation maps to 3D texture data.

Others have anticipated these advancements as well. In chapter 2 of Rudy Wenk’s 1985 book entitled Preferred Orientation in Deformed Metal and Rocks: An introduction to Modern Texture Analysis it states:

“Pole figures and fabric diagrams provide information only about the orientation of crystals. It may be desirable to know the relation between the spatial distribution of grains and grain shape with respect to crystallographic orientation. Orientation relations between neighboring grains further defined the fabric and help to elucidate its significance.”

But let us return to the theme of aimless wanderin’s in texture terminology. The title for Chapter 4 of Bunge’s book is “Expansion of Orientation Distribution Functions in Series of Generalized Spherical Harmonics”. This chapter describes a solution the determination of the three-dimensional ODF (orientation distribution function) from two-dimensional pole figures. The chapter has a sub-title “Three-Dimensional Textures”. The three dimensions in this chapter of Bunge’s book are in orientation space (the three Euler Angles). What we call today a 3D texture is actually a 6D description with three dimensions in orientation space and three spatial dimensions (e.g. x, y and z). And those working with High-Energy x-rays have also characterized spatially resolved orientation distributions for in-situ experiments thus adding a seventh dimension of time, temperature, strain, …

It is nice to know in the nearly 50 years since Bunge’s book was published that what can sometimes appear to be aimless wanderin’s with mixed up terminology has actually lead us to higher dimensions of understanding. But, before we take too much credit for these advances in the “metallurgical arts”, as it says on the Google Scholar home page we “stand on the shoulders of giants” who envisioned and laid the groundwork for these advances.