Microstructure Analysis on Nanocrystalline Materials COMMISSION ...
Microstructure Analysis on Nanocrystalline Materials COMMISSION ...
Microstructure Analysis on Nanocrystalline Materials COMMISSION ...
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E1<br />
= FT[<br />
Ω(<br />
q + Δq)<br />
] =<br />
r r<br />
4π<br />
⎛ sin q + Δq<br />
R<br />
⎞<br />
= ⎜<br />
r r<br />
− R q + Δq<br />
R⎟<br />
(18)<br />
r r ⎜<br />
r r cos<br />
2<br />
⎟<br />
q + Δq<br />
⎝ q + Δq<br />
⎠<br />
r r<br />
E2<br />
= FT[<br />
Ω(<br />
q − Δq)<br />
] =<br />
r r<br />
4π<br />
⎛ sin q − Δq<br />
R<br />
⎞<br />
= ⎜<br />
r r<br />
− R q − Δq<br />
R⎟<br />
(19)<br />
r r ⎜<br />
r r cos<br />
2<br />
⎟<br />
q − Δq<br />
⎝ q − Δq<br />
⎠<br />
Fig. 2. Two-dimensi<strong>on</strong>al intensity distributi<strong>on</strong> as calculated<br />
using Eqs. (7), (13) and (15) for fully coherent<br />
crystallites having the size of 30 Å. The reciprocal lattice<br />
points were completely overlapping each other.<br />
C<strong>on</strong>sequently, the radial intensity distributi<strong>on</strong>, i.e. the<br />
dependence of the intensity <strong>on</strong> qz, depends str<strong>on</strong>gly <strong>on</strong><br />
the degree of the partial coherence of crystallites,<br />
which is hidden in the coherence term of Eq. (17). The<br />
change of the line broadening and the change of the<br />
line shape are illustrated in Fig. 5. For n<strong>on</strong>-coherent<br />
crystallites (Fig. 4), the scattered intensity corresp<strong>on</strong>ds<br />
to the sum of the intensities scattered by individual<br />
crystallites. Thus, the line width and the line shape remain<br />
the same as for individual crystallites, like in the<br />
classical kinematical diffracti<strong>on</strong> theory. For fully coherent<br />
crystallites replaced in the z directi<strong>on</strong>, the scattered<br />
intensity (Fig. 2) is modulated additi<strong>on</strong>ally by the<br />
cosine term from Eq. (17) in the qz directi<strong>on</strong>, which<br />
causes an obvious decrease of the line broadening in<br />
this directi<strong>on</strong> (Fig. 5). For partially coherent crystallites<br />
(Fig. 3), the line broadening al<strong>on</strong>g qz<br />
lies between the<br />
line broadening from fully coherent and n<strong>on</strong>-coherent<br />
crystallites. Moreover, the diffracti<strong>on</strong> lines from partially<br />
coherent crystallites become Cauchy-like in<br />
shape due to their l<strong>on</strong>g tails in the qz directi<strong>on</strong>, see Fig.<br />
3.<br />
DIFFRACTION LINE BROADENING<br />
In [6], we have shown that the diffracti<strong>on</strong> line broadening<br />
from partially coherent crystallites can be divided<br />
into three regi<strong>on</strong>s that are shown in Fig. 6. For<br />
sinθ → 0 , the diffracti<strong>on</strong> line broadening is given by<br />
the maximum size of domains, which c<strong>on</strong>sist of several<br />
partially coherent crystallites. At medium diffracti<strong>on</strong><br />
angles, the diffracti<strong>on</strong> line broadening steeply increases<br />
with increasing diffracti<strong>on</strong> angle as the degree of the<br />
partial coherence (as well as the overlap of the reciprocal<br />
lattice points, see Fig. 1) decreases in this range.<br />
Fig. 3. Two-dimensi<strong>on</strong>al intensity distributi<strong>on</strong> as calculated<br />
using Eqs. (7), (13), (15) for partially coherent<br />
crystallites having the size of 30 Å. The distance of the<br />
reciprocal lattice points was 0.09 Å -1 al<strong>on</strong>g qx.<br />
Fig. 4. Two-dimensi<strong>on</strong>al intensity distributi<strong>on</strong> as calculated<br />
using Eqs. (7), (13), (15) for n<strong>on</strong>-coherent crystallites<br />
having the size of 30 Å. The distance of the reciprocal<br />
lattice points was 0.15 Å -1 al<strong>on</strong>g qx.<br />
At the largest diffracti<strong>on</strong> angles, the reciprocal lattice<br />
points do not overlap, which means that the coherence<br />
9