The purpose of this tutorial is to model graphene as seven fused benzene rings (see below) and use a Fourier transform of the atomic positions to calculate its diffraction pattern. \[ \begin{matrix} \text{Number of atoms:} & A = 24 & \text{Atomic dimension:} & d = .25 & \text{Atomic positions:} \\ x_1 = 0 & y_1 = 1.386 & x_2 = 0 & y_2 = -1.386 & x_{15} = 0 & y_{15} = 2.772 \\ x_3 = -1.2 & y_3 = 0.693 & x_4 = 1.2 & y_4 = 0.693 & x_{16} = 1.2 & y_{16} = 3.465 \\ x_5 = 1.2 & y_5 = -0.693 & x_6 = -1.2 & y_6 = -0.693 & x_{17} = 2.4 & y_{17} 2.772 \\ x_7 = 2.4 & y_7 = 1.386 & x_8 = 3.6 & y_8 = 0.693 & x_{18} = 0 & y_{18} = -2.772 \\ x_9 = 3.6 & y_9 = -0.693 & x_{10} = 2.4 & y_{10} = -1.386 & x_{19} = 1.2 & y_{19} = -3.465 \\ x_{11} = -2.4 & y_{11} = 1.368 & x_{12} = -3.6 & y_{12} = 0.693 & x_{20} = 2.4 & y_{20} = -2.772 \\ x_{13} = -3.6 & y_{13} = -0.693 & x_{14} = -2.4 & y_{14} = -1.386 & x_{21} = -2.4 & y_{21} = 2.772 \\ x_{22} = -1.2 & y_{22} = 3.465 & x_{23} = -2.4 & y_{23} = -2.772 & x_{24} = -1.2 & y_{24} = -3.465 \end{matrix} \nonumber \] The diffraction pattern is the Fourier transform of the atomic positions into momentum space. \[ \begin{matrix} \Delta = 20 & N = 200 & j = 0 .. N & px_j = - \Delta + \frac{2 \Delta j}{N} & k = 0 .. N & py_k = - \Delta + \frac{2 \Delta k}{N} \end{matrix} \nonumber \] \[ \begin{matrix} \Psi (p_x,~p_y) = \sum_{m=1}^{A} \left( \int_{x_m - \frac{d}{2}} ^{x_m + \frac{d}{2}} exp (-i p_x x)dx \int_{y_m - \frac{d}{2}} ^{y_m + \frac{d}{2}} exp(-i p_y y) dx \right) & p_{j,~k} = \left( \left| \Psi ( px_j,~py_j) \right| \right)^2 \end{matrix} \nonumber \] \( i = 1 .. A\)