5.7: A Model Graphene Diffraction Pattern
- Page ID
- 150522
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.
\[ \begin{matrix} \Delta = 20 & N = 200 & j = 0 .. N & px_j = - \Delta + \frac{2 \Delta j}{N} & k = 0 .. N & py_k = - \Delta + \frac{2 \Delta k}{N} \end{matrix} \nonumber \]
\[ \begin{matrix} \Psi (p_x,~p_y) = \sum_{m=1}^{A} \left( \int_{x_m - \frac{d}{2}} ^{x_m + \frac{d}{2}} exp (-i p_x x)dx \int_{y_m - \frac{d}{2}} ^{y_m + \frac{d}{2}} exp(-i p_y y) dx \right) & p_{j,~k} = \left( \left| \Psi ( px_j,~py_j) \right| \right)^2 \end{matrix} \nonumber \]
\( i = 1 .. A\)
