Difference between revisions of "Extra:Hexagonal peaks"

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(Created page with "Consider a hexagonal lattice viewed end-on using scattering. For instance, hexgaonally-packed-cylinder block-copolymer moprhology that is oriente...")
 
 
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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:
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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:
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: <math>\cos 30^{\circ} = \frac{\sqrt{3}}{2}</math>
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: <math>\sin 30^{\circ} = \frac{{1}}{2}</math>
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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>
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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:
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: <math>d_{cc} = </math>
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: <math>\begin{alignat}{2}
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d_{cc} & = \frac{ d_l }{\sqrt{3}/2} \\
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& = \frac{2 \pi / q_l}{\sqrt{3}/2} \\
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& = \frac{2 \pi }{ +( \sqrt{3}/2) q_l }
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\end{alignat}
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</math>
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Thus, the <math>q_x</math> position of the away-from-specular peaks gives the cylinder-cylinder distance:
 +
: <math>\begin{alignat}{2}
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d_{cc} & = \frac{2 \pi }{ q_x }
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\end{alignat}
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</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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (q_x, q_z) = (0, q_{l})} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_l} corresponds to the cylinder layering distance in realspace (i.e. the distance between subsequent rows of cylinders). Since:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos 30^{\circ} = \frac{\sqrt{3}}{2}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin 30^{\circ} = \frac{{1}}{2}}

The full set of (first order) peaks is:

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (q_x, q_z) = (0, +q_l)}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (q_x, q_z) = \left ( -\frac{\sqrt{3}}{2} q_l, +\frac{q_l}{2} \right )} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (q_x, q_z) = \left ( +\frac{\sqrt{3}}{2} q_l, +\frac{q_l}{2} \right )}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (q_x, q_z) = \left ( -\frac{\sqrt{3}}{2} q_l, -\frac{q_l}{2} \right )} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (q_x, q_z) = \left ( +\frac{\sqrt{3}}{2} q_l, -\frac{q_l}{2} \right )}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (q_x, q_z) = (0, -q_l)}

The realspace layering distance is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_l = \frac{2 \pi}{q_l}}

And the realspace cylinder-cylinder distance is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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} }

Thus, the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_x} position of the away-from-specular peaks gives the cylinder-cylinder distance:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} d_{cc} & = \frac{2 \pi }{ q_x } \end{alignat} }
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