## 7.3 O-Lattice Theory

### 7.3.1 The Basic Concept

 The Coincidence Site lattice (CSL) provided a relatively easy way to grasp the concept of special orientations between grains that give cause for special grain boundaries. With the extension to grain boundary dislocations in the DSC lattice, the CSL concept became in principle applicable to all grain boundaries, because any arbitrary orientation is "near" a CSL orientation. But yet, the CSL concept is not powerful enough to allow the deduction of grain boundary structures in all possible cases. The reasons for this are physical, practical and mathematical: The CSL by itself is meaningless; meaningful is the special grain boundary structure possible if there is a coincidence orientation. The grain boundary structure is special, because it is periodic (with the periodicity of the CSL) and contains coincidence points (cf. the picture). But we have no guarantee that periodic grain boundary structures may not exist in cases where no CSL exists; i.e. by only looking at CSL orientation, we may miss other special orientations. That will be certainly true whenever we consider boundaries between different lattices - be it that lattice constants of the same materials changed ever so slightly because one grain has a somewhat different impurity concentration, or that we look at phase boundaries between different crystals. As we have seen, even a CSL with S=41 is significant, but it is virtually unrecognizable in a drawing. This is an expression of the mathematical condition, that you either have perfect coincidence or none. If two points coincide almost, but not quite, no recognizable CSL will be seen. If two lattice points coincide except for, lets say, 0,01 nm, we certainly would say we have a physical coincidence, but mathematically we have none. The same is true if we rotate a lattice away from a coincidence position by arbitrarily small angles. Mathematically, the coincidence is totally destroyed and the situation has completely changed, whereas physically small changes of the orientation would be expected to cause only small changes in properties, too. Only a very small fraction of grain orientations have a CSL. The "trick", to transform any orientation into a coincidence orientation by introducing grain boundary dislocations in the DSC lattice is somewhat questionable: the effect (=CSL) comes before the cause (=dislocations in the DSC lattice), because at the orientation that we want to change no CSL and therefore no DSC lattice exists. It becomes clear that the main problem lies in the discreteness of the CSL. Any useful theory for special grain boundary (and phase boundary) structures must be a continuum theory, i.e. give results for continuous variations of the crystal orientation (and lattice type). Thus theory exists, it is the so-called "O-lattice theory of W. Bollmann; comprehensively published in his opus magnus "Crystal Defects and Crystalline Interfaces" in 1970. The O-lattice theory is not particularly easy to grasp. It is well beyond the scope of this hyperscript to go into details. What will be given is the basic concept, the big ideas, together with some formulas and a few examples. There are two basic ideas behind the O-lattice theory: 1. Take a crystal lattice I and transform it in any way you like. That means you can not only rotate it into an arbitrary orientation relative to crystal I, but also deform it by stretching, squeezing and shearing it. The lattice II generated in this way from a cubic lattice I thus could even be an arbitrarily oriented triclinic lattice. 2. Now look for coincidence points between lattice I and lattice II. But do not restrict the search for coincidence points to lattice points, but expand the concept of coincidence to all equivalence points within two overlapping unit cells. What that means becomes clear in the illustration:
 Lattice I is deformed by first rotating it and then stretching the axis x1 into lattice II An arbitrary point within the elementary cell of lattice I is described by a vector r(I) r(I) transforms into a vector r(II) by the transformation, the point reached within the unit cell of lattice II is an equivalence point to the one in crystal I. Of course there is more than one equivalence point: Any point r(II) in lattice II described in the coordinate system of lattice II (defined by the units vectors x1(II) and x2(II)) by r(II)=r(II) + T(II) with T(II)=any translation vector of lattice II or T(II)=n·x1(II) + m·x2(II), with n, m=0, +/-1, +/-2, ... is an equivalence point to the corresponding set of points r(I)=r(I) + T(I). Let us designate the set of all equivalence points in lattice I by C1 and in lattice II by C2 and for sake of clarity the equivalence points defined above by r(C1) and r(C2), respectively. If we look at a certain equivalence point in lattice II, it originated from lattice I by the general transformation as shown in the picture The blue lattice was obtained from the pink one by a transformation; in this case by a simple rotation. The red point is denotes coinciding equivalence poinst. The blue vector in lattice II can be obtained in two ways: By the transformation equation from the corresponding red vector of lattice I or, since the coincidence point belongs to both lattices at once, by adding some translation vector of lattice I to the red vector. This is symbolically shown in the picture. In formulas we can write for any vector in lattice II pointing to some point of an equivalent class C: r(C2)=A{r(C1)}, or r(C1)=A-1{r(C2)}with A=transformation matrix.; we will encounter examples for A later. On the other hand, we can obtain new equivalence points in lattice I, i.e. other elements of the set C1designated by e(C1) by the equation e(C1)=r(C1) + T(I). We are looking for coincidences of any one member of the set r(C1) with any one member of the set r(C2); any coincidence point thus obtained will be named r0. Since this point describable in lattice II by r(C2) must be reachable in lattice I by first going down r(C1) and then adding an translation vector of lattice I, we obtain r(C2)=r(C1) + T(I)=r0. Using the transformation equation for lattice I and substituting it into the above equation yields r0=A-1{r0} + T(I). We wrote r0 instead of r(C2) because we do not need the distinction between the sets C1 and C2 any more because r0 belongs to both sets. Rearranging the terms following matrix algebra by using the identity or unit transformation matrix I, we obtain the fundamental equation of O-lattice theory:
 (I - A)r0=T(I)
 What does that equation mean? For a given transformation, i.e. for given orientation relationship between two grains, its solution for r0 defines all the coincidence points or O-points of the lattices. The coincidence of lattice points is a subset of the general solution for the coincidence of equivalence points. The question comes up if there are any solution of this equation. Algebra tells us that this requires that the determinant of the matrix, ¦I - A¦ must not be=0. This, while not always true, will be generally true. How do we solve the O-lattice equation, i.e. obtain the set of O-points for a given lattice and transformation? Simply by inverting the matrix we obtain: r0=(I - A)-1 · T(I) That is all there is to do; it looks easy. However, the diffusion equations look easy, too, but are not easy to solve. Also, we do not yet know what the solution, the O-lattice, really means with respect to grain- or phase-boundaries. We will look at this more closely in the next paragraph; here we will discuss a simple example.
 As an example, we look at the two-dimensional situation where a square lattice rotates on top of another one. This will include our former example of the S=5 CSL case. The transformation matrix is a pure rotation matrix, for the rotation angle a it writes From this we get , , and finally . The base vectors of a the square lattice I are x1(I)=(1,0) x2(i)=(0,1). If we use them as the smallest possible translation vector of lattice I we obtain by multiplication with the last matrix the smallest vectors of the O-lattice which must be the unit vectors of the O-lattice, u1 and u2: u1=(1/2, -1/2·cotan(a/2), and u2=(-1/2·cotan(a/2), 1/2). This is easily graphically represented, but the pictures get to be a bit complicated:
 Lattice I is the blue lattice, lattice II the red one; it has been rotated by the angle a. The unit vectors of the O-lattice can be determined by the intersection of the green lines; they are depicted in black. The O-lattice then can be constructed, its lattice points are shown as orange blobs. The picture neatly helps to overcome a possible misunderstanding: At any O-point a vector from the origin - our vectors r(I) and r(II) - point to the same equivalence points, but the equivalence points at other O-points may be different. In the example we have O-points that are almost at the center of both unit cells , or almost at a lattice point of both unit cells. If we would include more cells of the O-lattice, we would see that equivalence points shift slightly for the example given. A few O-lattice cells away, they would be more off-center or more distant to a lattice point than close to the origin of the O-lattice. This is a very important point, giving cause for an important question: Is the pattern of equivalence points periodic or non-periodic? In other words: If any one point of the O-lattice defines a specific equivalence point in the crystal lattices, does this specific point appear again at some other point in the O-lattice (apart from the trivial symmetries of the O-lattice)? We will come back to this question later; it is the decisive feature of the O-lattice for defining the DSC-lattice How do we get the CSL from the O-lattice? Looking at the unit vectors of the O-lattice, there is no way of expressing them in integer values of the base vectors of lattice I, because one component is always 1/2. This is, however, not necessary. If we chose a=36o52,2`, we have cotana=3/2 and u1=(1/2, -3/2), and u2=(-3/2), 1/2). Thus every second point of the O-lattice is a lattice point on both lattices (depicting O- points of the equivalence class [0,0], they thus define the (S=5) CSL. The other O-points are of the equivalence class [1/2,1/2]. Not, too, that in this case the pattern of equivalence point is obviously periodic, so we have a first specific answer to the question asked above. Before we delve deeper into the intricacies of O-lattice theory, we first discuss some of its general implications in the next paragraphs.