Dr. Stuart Wright, Senior Scientist, EDAX
Of all the papers I’ve written, my favorite title I’ve managed to sneak past the editors and reviewers is “Random thoughts on non-random misorientation distributions.” The paper is a write-up of a presentation I gave at a celebration of Professor David Dingley’s contributions to EBSD, which was held as a special version of the annual Royal Microscopy Society EBSD meeting at New Lanark in Scotland. It was a fun meeting as several of David’s former Ph.D. students shared some great stories and pictures of David, and the talks were a little less formal than usual, which led to some interesting discussions.
There are many terms used to describe the difference in crystallographic orientation between two crystal lattices: misorientation, disorientation, orientation difference, misorientation angle, minimum misorientation angle, grain boundary character, intercrystalline interface. One can get a bit “disoriented” trying to sort out all these different terms. Unfortunately, I am at fault for some of the confusion as I have tended to use the different terms loosely in my presentations and papers. But I am not the only one; I have seen some wandering in the definition of some of these terms as different researchers have followed up on the work of others. I will not pretend to be rigorous in this blog, but let me see if I can help sort through the different terms.
My first exposure to the idea of misorientation was from Bunge’s classic book Texture Analysis in Materials Science from 1969. I was first introduced to the book when I joined Professor Brent Adam’s Lab in 1985. We called it the “Red Bible,” as we had a very well-worn copy in the lab. We were even lucky enough to have Peter Morris with us at the time, who translated the book from German to English (a herculean task for a non-German speaker without modern tools like Google Translate). On page 44 of this book, you will find the following:
If two adjacent grains in a grain boundary have orientations g1 and g2, the orientation difference is thus given by:
∆g = g2 ∙ g1-1 (3.12)
This looks like a relatively simple expression, and we have generally calculated it using orientations described as matrices, and thus the result ∆g would also be a matrix. But the most common description of this orientation difference given in the literature would be an axis-angle pair. Any two crystals have at least one axis in common. A rotation about that axis will bring the two crystal lattices into coincidence.
Figure 1. Axis-angle description of misorientation.
While the equation above seems simple, we need to remember that, due to crystal symmetry, there are multiple symmetrically equivalent descriptions of the orientations g1 and g2. We can term the symmetry operators Li. These are the elements of the crystallographic point group symmetry for the crystals in question. For example, for a cubic crystal, there will be 24 symmetry elements. Since there are 24 symmetric equivalents for g1 and 24 for g2 that means there will be 576 symmetric equivalents for ∆g. In the expression below, the apostrophe denotes symmetrically equivalent.
∆g’12 = Lig2∙(Lig1)-1
As an example, here is a list for a random axis angle pair assuming cubic crystal symmetry: 12° @ 〈456〉. Note that the notation 〈uvw〉 denotes the family of crystal directions and [uvw] denotes a single crystal direction. Once again, for cubic symmetry, there are (in general) 24 [uvw] directions in the 〈uvw〉 family of directions (note in general there are 24 directions in the family, i.e. [123], [132], [-123], [-132], …. but this can be reduced for families where multiplicity plays a role, such as 〈00w〉 or 〈uuw〉…).
| Angle | Axis | Angle | Axis | |
| 12.00 | (4 5 6) | 124.26 | (139 132 170) | |
| 82.16 | (2 18 155) | 125.80 | (118 121 148) | |
| 83.62 | (20 4 157) | 131.85 | (44 43 45) | |
| 85.06 | (4 45 325) | 169.37 | (2 161 177) | |
| 95.94 | (33 3 262) | 170.34 | (235 6 265) | |
| 97.23 | (4 20 177) | 171.30 | (2 149 172) | |
| 98.51 | (62 7 617) | 171.80 | (10 8 167) | |
| 108.17 | (39 38 40) | 173.17 | (8 12 167) | |
| 114.78 | (137 173 177) | 174.54 | (12 10 167) | |
| 116.39 | (130 103 136) | 178.07 | (25 196 221) | |
| 117.99 | (137 177 181) | 179.03 | (188 26 207) | |
| 122.71 | (149 153 192) | 179.03 | (155 18 179) |
So, this is a list of symmetrical misorientations given as axis-angle pairs. The minimum rotation angle in this set is the disorientation. But, you will also see the disorientation called the orientation distance (Bunge equation 2.123), rotation angle and misorientation angle (OIM), minimum misorientation angle, as well as simply the misorientation, orientation difference, grain boundary angle, . For a little comic relief at intense EBSD workshops, I have often said that I prefer the term misorientation because disorientation is what we tend to feel at the end of the day of lectures. I give Professor Marc De Graef credit for helping me finally get these terms straight. So, now I can retire that joke that probably never really translated very well into different languages anyway.
One more note on terminology. A grain boundary is a five-parameter entity: three for the misorientation and two to describe the orientation of the boundary plane.
Figure 2. 5D Grain Boundary Character.
This five-dimensional entity is now often referred to as the Grain Boundary Character (Rohrer) but has also been termed the Intercrystalline Interface Structure (Adams). In the past and in OIM Analysis, the Grain Boundary Character Distribution or GBCD refers to the distribution of grain boundaries across three classifications, low-angle random boundaries, high-angle random boundaries, and “special” (generally CSL) boundaries. As a side note, Grain Boundary Character has been called a “full” or “complete” description of a grain boundary, but this is a bit of an overreach. There are still other parameters associated with a grain boundary that may be just as important as these five, for example, curvature, faceting, chemical composition.
It should be noted that we can calculate the misorientation between two crystals of different symmetry and get a nice, neat axis-angle pair.
However, the concept of coincidence is not as clear as for two crystals of the same symmetry, as illustrated in the schematic shown in Figure 3. Nonetheless, this terminology (and its corresponding mathematical methods) can be helpful when analyzing the orientation relationships associated with phase transformations.
Figure 3. Misorientation between a hexagonal and cubic crystal.
I hope this brief discussion has helped “orient” you in the right direction. I know I am now trying to be more careful in using these terms, which will probably result in a few changes in our user interface for a future version of OIM to reflect this.
References
Wright, SI (2006) Random thoughts on non-random misorientation distributions. Materials Science and Technology 22: 1287-1296.
Bunge, HJ (1969) Mathematische Methoden der Texturanalyse. Akademie-Verlag: Berlin.
Beladi H, Nuhfer NT, and Rohrer GS (2014) The five-parameter grain boundary character and energy distributions of a fully austenitic high-manganese steel using three dimensional data. Acta Materialia 70:281-289
Zhao J, Koontz JS, and Adams BL, 1988. Intercrystalline structure distribution in alloy 304 stainless steel. Metallurgical Transactions A, 19:1179-1185.