# S is Always Odd

| They most conspicuous issue in the CSL theory of grain boundaries is that there are no even values for S!
|  | Try as you might - you will never find a S = 2 boundary or any other even number in the literature. Now why is this? Mostly no explanation is given.
| A rigorous proof essentially needs the full power of the O-lattice theory, so it can not be easily given. But the general reason for this peculiar geometric fact can be envisioned as follows.
|  | First, remember that any grain boundary can be obtained by generating grain II out of grain I by one rotation around a suitable axis with the rotation angle g.
|  | This means that we can produce all CSL orientations by looking at one rotation. We will do this for a square lattice, rotating around a <100> axis.
|  | It is, however, not obvious that we can indeed produce all possible boundaries by this rotation, nor is it clear that the result will be valid for grain boundaries in non-cubic crystals. But it shows the direction of the argument.
| From all possible rotation, some will produce CSL structures. Which ones will do that is easily conceived:
|  | The picture below shows a blue crystal I. Taking its origin at the apex of the blue triangle on the right, we see that we always will get a CSL orientation if we look at lattice points with the coordinates (x, –y0) which we may express as (n, –1) if we set x0, y0 = 1, and than rotate the crystal so the the y-coordinate changes from –1 to +1. The shift is indicated by the bold brown vector; we need to rotate an angle g given by
g | = ½ cotg | y x | = 2 · cotg | 1 n
--- | --- | --- | --- | ---
|  | The red lattice has been rotated by just the right amount to move the point (3, –1) to the position (3, +1); the rotation center is in the middle of the crystals
| With this procedure we created the yellow CSL lattice.
|  | Its S' value is given by its area divided by the are of a unit cell of the lattice; we have
S' | = | (x2 + y2)2 x0 · x0 | = | (3x0)2 + (1x0)2) x02 | = (32 + 12) = 10
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|  | Its easy to generalize for CSL sites generated by moving the point (nx, –y) on the (nx,+y) position, we obtain for the S' values
S'(n) | = | n2 + 12
--- | --- | ---
| The result will be
|  | S' is an odd number, if n is an even number (The square of an even number is even plus 1 = odd)
|  | S' is an even number, if n is an odd number (The square of an odd number is odd plus 1 = even.)
| So we can get even and odd numbers for S????.
|  | Yes - but upon inspection you will find that for n = odd, there is always an additional coincidence point in the center of the lattice defined by the CSL points produced by the rotation, while for even numbers of n this is not the case.
|  | In the picture above this are the green points, and the lattice constant of the CSL lattice is now smaller. The S value in this case is simply
S | = | n2 + 12 21/2 · 21/2 | = S'/2 = an odd number
--- | --- | --- | ---
|  | Instead of a S = 10 boundary, we generated a S = 5 boundary and there are no even S values.
|  | q.e.d. (sort of)
| This, of course, is a far cry from a real mathematical proof, but it imparts the flavor of the thinking behind it.
|  | To complete this issue, the following picture shows the result for a rotation that tranfers (2, –1) to (2, +1)
|  | There is no additional coincidenc point and we end up with a S = (n2 + 12) = 5 boundary, the same one as above

With frame

7.1.2 The Coincidence Site Lattice

© H. Föll (Defects - Script)