5.6: Simulating DNA's Diffraction Pattern - Short Version
- Page ID
- 150521
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\[ \begin{matrix} \text{Sugar-phosphate groups per strand:} & A = 20 & \text{Strand radius:} & R = 1 & \text{Phase difference between strands:} & 0.8 \pi \end{matrix} \nonumber \]
\[ \begin{matrix} \text{First strand:} & m = 1 .. A & \Theta_m = \frac{4 \pi m}{A} & y_m = m & x_m = R \cos \left( \Theta_m \right) \\ \text{Second strand:} & m = 21 .. 40 & \Theta_m = \frac{4 \pi (m - A)}{A} & y_m = (m-A) & x_m = R \cos \left( \Theta_m + 0.8 \pi \right) \\ m = 1 .. 20 & n = 21 .. 40 \end{matrix} \nonumber \]
Coordinate space wave function:
\[ | \Psi \rangle = \frac{1}{ \sqrt{N}} \sum_{i = 1}^{N} | x_i,~y_1 \rangle \nonumber \]
Momentum space wave function:
\[ \Phi (p_x,~p_y) = \frac{1}{ 2 \pi} \sum_{m=1}^{40} exp (-i p_x x_m) exp( -i p_y y_m) \nonumber \]
\[ \begin{matrix} \Delta = 8 & N = 200 & j = 0 .. N & px_j = - \Delta + \frac{2 \Delta j}{N} & k = 0 .. N & py_k = - \Delta + \frac{2 \Delta k}{N} \end{matrix} \nonumber \]
\[ \text{Diffraction Pattern}_{j,~k} = \left( \left| \Phi (px_j,~py_k ) \right| \right)^2 \nonumber \]
