For example, yttria stabilized zirconia with tetragonal structure. The XRD of YSZ shows peak splitting at some angles. Why?
Another thing why the 1st peak is not showing any splitting (at 30 degree)?
Cubic YSZ has a 111 peak at 30deg and a 200 at 35deg.
The tetragonal structure keeps the same c-axis, but the a- and b-axes are rotated 45deg around c. This re-indexes the peaks so that the 30deg peak becomes the 101/011 tetragonal peak, that is the cubic 11L becomes the tetragonal 10L.
The tetragonality is along the z-axis, so 101 and 011 are equivalent and therefore not split.
The cubic 200 (which is 200, 020, 002) now transforms to tetragonal 110, -110, and 002. The 110 is found at 35.2deg. The 002 is at 34.7deg. So the single cubic peak has split into two tetragonal peaks.
You can easily apply this logic to see which cubic peaks will split, and which will not.
Cubic YSZ has a 111 peak at 30deg and a 200 at 35deg.
The tetragonal structure keeps the same c-axis, but the a- and b-axes are rotated 45deg around c. This re-indexes the peaks so that the 30deg peak becomes the 101/011 tetragonal peak, that is the cubic 11L becomes the tetragonal 10L.
The tetragonality is along the z-axis, so 101 and 011 are equivalent and therefore not split.
The cubic 200 (which is 200, 020, 002) now transforms to tetragonal 110, -110, and 002. The 110 is found at 35.2deg. The 002 is at 34.7deg. So the single cubic peak has split into two tetragonal peaks.
You can easily apply this logic to see which cubic peaks will split, and which will not.
A simplified scheme can be arrived at if you consider a non-conventional C-centred tetragonal (non-standard setting; C42/acm; http://bruceravel.github.io/demeter/artug/atoms/space.html) of tetragonal zirconia. Then the new and old axis will be parallel to each other. Then for a given cubic reflections. In that case the cubic {200} splits into 200/020 and 002, {220} into 220 and 202. Note that in case of rhombohedral distortion (in other systems) 111 will split whereas 200 remains unsplit.
In principle it is quite easy to understand if you consider that the Bragg angle is dependend on the interplanar distances. If they change also the Bragg angle changes.
Using the d-equation for a tetragonal lattice the relevant part is written in the denominator in the square root:
h²+k²+l²*a²/c²
This means that for all permuted hkl you will observe a splitting, if c/a is different from 1 (what is no requirement for tetragonal symmetry!).
There are only two special cases where c/a will not have any effect
:
a) h=k=l , i.e. 111, 222, 333 etc
b) l=0, since then the c/a ratio will not have any effect.
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