Difference between revisions of "Extra:Hexagonal peaks"

From GISAXS
Jump to: navigation, search
(Created page with "Consider a hexagonal lattice viewed end-on using scattering. For instance, hexgaonally-packed-cylinder block-copolymer moprhology that is oriente...")
 
 
Line 1: Line 1:
Consider a [[Lattice:hexagonal|hexagonal]] lattice viewed end-on using [[scattering]]. For instance, hexgaonally-packed-cylinder [[block-copolymer]] moprhology that is oriented horizontally ('laying down'; i.e. cylinder long axes parallel to the substrate plane). There will be a peak along the specular at <math>(q_x, q_z) = (0, q_{l})</math>, where <math>q_l</math> corresponds to the cylinder layering distance in realspace (i.e. the distance between subsequent rows of cylinders). The full set of peaks is:
+
Consider a [[Lattice:Hexagonal|hexagonal]] lattice viewed end-on using [[scattering]]. For instance, a hexagonally-packed-cylinder [[block-copolymer]] moprhology that is oriented horizontally ('laying down'; i.e. cylinder long axes parallel to the substrate plane). There will be a peak along the specular at <math>(q_x, q_z) = (0, q_{l})</math>, where <math>q_l</math> corresponds to the cylinder layering distance in realspace (i.e. the distance between subsequent rows of cylinders). Since:
 +
: <math>\cos 30^{\circ} = \frac{\sqrt{3}}{2}</math>
 +
: <math>\sin 30^{\circ} = \frac{{1}}{2}</math>
 +
The full set of (first order) peaks is:
 
* <math>(q_x, q_z) = (0, +q_l)</math>
 
* <math>(q_x, q_z) = (0, +q_l)</math>
 
* <math>(q_x, q_z) =  \left ( -\frac{\sqrt{3}}{2} q_l, +\frac{q_l}{2}  \right )</math> and <math>(q_x, q_z) =  \left ( +\frac{\sqrt{3}}{2} q_l, +\frac{q_l}{2}  \right )</math>
 
* <math>(q_x, q_z) =  \left ( -\frac{\sqrt{3}}{2} q_l, +\frac{q_l}{2}  \right )</math> and <math>(q_x, q_z) =  \left ( +\frac{\sqrt{3}}{2} q_l, +\frac{q_l}{2}  \right )</math>
 
* <math>(q_x, q_z) =  \left ( -\frac{\sqrt{3}}{2} q_l, -\frac{q_l}{2}  \right )</math> and <math>(q_x, q_z) =  \left ( +\frac{\sqrt{3}}{2} q_l, -\frac{q_l}{2}  \right )</math>
 
* <math>(q_x, q_z) =  \left ( -\frac{\sqrt{3}}{2} q_l, -\frac{q_l}{2}  \right )</math> and <math>(q_x, q_z) =  \left ( +\frac{\sqrt{3}}{2} q_l, -\frac{q_l}{2}  \right )</math>
 
* <math>(q_x, q_z) = (0, -q_l)</math>
 
* <math>(q_x, q_z) = (0, -q_l)</math>
Since:
 
: <math>\cos 30^{\circ} = \frac{\sqrt{3}}{2}</math>
 
: <math>\sin 30^{\circ} = \frac{{1}}{2}</math>
 
  
 
The realspace layering distance is:
 
The realspace layering distance is:
 
: <math>d_l = \frac{2 \pi}{q_l}</math>
 
: <math>d_l = \frac{2 \pi}{q_l}</math>
 
And the realspace cylinder-cylinder distance is:
 
And the realspace cylinder-cylinder distance is:
: <math>d_{cc} = </math>
+
: <math>\begin{alignat}{2}
 +
d_{cc} & = \frac{ d_l }{\sqrt{3}/2} \\
 +
& = \frac{2 \pi / q_l}{\sqrt{3}/2} \\
 +
& = \frac{2 \pi }{ +( \sqrt{3}/2) q_l }
 +
\end{alignat}
 +
</math>
 +
Thus, the <math>q_x</math> position of the away-from-specular peaks gives the cylinder-cylinder distance:
 +
: <math>\begin{alignat}{2}
 +
d_{cc} & = \frac{2 \pi }{ q_x }
 +
\end{alignat}
 +
</math>

Latest revision as of 15:33, 27 July 2018

Consider a hexagonal lattice viewed end-on using scattering. For instance, a hexagonally-packed-cylinder block-copolymer moprhology that is oriented horizontally ('laying down'; i.e. cylinder long axes parallel to the substrate plane). There will be a peak along the specular at , where corresponds to the cylinder layering distance in realspace (i.e. the distance between subsequent rows of cylinders). Since:

The full set of (first order) peaks is:

  • and
  • and

The realspace layering distance is:

And the realspace cylinder-cylinder distance is:

Thus, the position of the away-from-specular peaks gives the cylinder-cylinder distance: