Microstructure Analysis on Nanocrystalline Materials COMMISSION ...
Microstructure Analysis on Nanocrystalline Materials COMMISSION ...
Microstructure Analysis on Nanocrystalline Materials COMMISSION ...
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of neighbouring crystallites disappears completely. In<br />
this angular range, the diffracti<strong>on</strong> line broadening remains<br />
c<strong>on</strong>stant, being approximately equal to the reciprocal<br />
size of individual (n<strong>on</strong>-coherent) crystallites.<br />
Intensity (arb.units)<br />
1.8<br />
1.6<br />
1.4<br />
1.2<br />
1.0<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
0.0<br />
-0.2 -0.1 0.0 0.1 0.2<br />
Δq z (Å -1 )<br />
Fig. 5. Diffracti<strong>on</strong> profiles calculated using equati<strong>on</strong>s<br />
(17), (18) and (19) for incoherent, partially coherent<br />
and fully coherent crystallites (from the top to the bottom).<br />
The displacement of the reciprocal lattice points<br />
is the same like in Figures 4, 3 and 2, respectively.<br />
Line broadening (Å -1 )<br />
0.014<br />
0.012<br />
0.010<br />
0.008<br />
0.006<br />
0.004<br />
D = 80 Å<br />
ω = 2.3° ω = 1.2° ω = 0.8°<br />
0.002<br />
0.0 0.2 0.4 0.6 0.8 1.0<br />
sin θ<br />
Fig. 6. Diffracti<strong>on</strong> line broadening as calculated according<br />
to Ref. [6] for partially coherent crystallites<br />
with the size of 80 Å and with the disorientati<strong>on</strong> of 0.8°,<br />
1.2° and 2.3°. Small oscillati<strong>on</strong>s in the range of the in-<br />
creasing line broadening are due to the numerical errors.<br />
Line broadening (Å -1 )<br />
0.020<br />
0.016<br />
0.012<br />
0.008<br />
0.004<br />
0.000<br />
0.0 0.2 0.4 0.6 0.8 1.0<br />
sin θ<br />
Fig. 7. Diffracti<strong>on</strong> line broadening as calculated according<br />
to equati<strong>on</strong>s (17), (18) and (19) for partially<br />
coherent crystallites with the size of 50 Å and with the<br />
disorientati<strong>on</strong> of 1.0° (triangles), 1.5° (circles) and<br />
2.5° (boxes).<br />
An analogous dependence of the diffracti<strong>on</strong> line broadening<br />
<strong>on</strong> the diffracti<strong>on</strong> angle can also be derived using<br />
the approach described in the previous Secti<strong>on</strong>. The<br />
line broadening shown in Fig. 7 was obtained from fitting<br />
the diffracti<strong>on</strong> profiles calculated using equati<strong>on</strong>s<br />
(17), (18) and (19) by the Pears<strong>on</strong> VII functi<strong>on</strong>. Some<br />
examples of the line profile fitting are shown in Fig. 5.<br />
In both approaches discussed above, the minimum and<br />
the maximum diffracti<strong>on</strong> line broadening corresp<strong>on</strong>d to<br />
the size of clusters of partially coherent crystallites and<br />
to the size of n<strong>on</strong>-coherent crystallites, respectively.<br />
The main difference between these approaches is the<br />
steepness of the increase of the diffracti<strong>on</strong> line broadening<br />
with increasing diffracti<strong>on</strong> angle. The microstructural<br />
model described in [6] assumes a c<strong>on</strong>tinuous<br />
distributi<strong>on</strong> of the disorientati<strong>on</strong>s of neighbouring crystallites<br />
between zero and a maximum disorientati<strong>on</strong>,<br />
which increases the degree of the coherence in the<br />
middle range of the diffracti<strong>on</strong> angles and shifts the<br />
steep increase of the diffracti<strong>on</strong> line broadening to larger<br />
diffracti<strong>on</strong> angles. C<strong>on</strong>sequently, the higher degree<br />
of the partial coherence of crystallites in the middle<br />
range of the diffracti<strong>on</strong> angles causes a steeper increase<br />
of the diffracti<strong>on</strong> line broadening at larger diffracti<strong>on</strong><br />
angles. The other microstructural model, which was<br />
used for the approach described in the previous Secti<strong>on</strong>,<br />
assumes a c<strong>on</strong>stant disorientati<strong>on</strong> of neighbouring<br />
crystallites in the clusters of partially coherent crystallites,<br />
which leads to a gradual decay of the partial coherence<br />
of crystallites that is dem<strong>on</strong>strated by a slower<br />
increase of the diffracti<strong>on</strong> line broadening with increas-<br />
10