Extra:Hexagonal peaks

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} }