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5.10: Modeling the C60 Diffraction Pattern

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    150535

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    In this tutorial a model diffraction pattern for C60 is calculated and compared with the experimental diffraction pattern.

    Solid C60 has the cubic close-packed crystal structure. For the sake of computational convenience this structure will be modeled as a planar close-packed array consisting of seven C60 molecules represented by squares (see the mask geometry shown below). According to quantum mechanical principles, radiat illuminating this geometrical arrangement interacts with all its members simultaneously thus being cas the spatial superposition, Ψ, given below. This spatial wave function is then projected into momentum space by a Fourier transform. Therefore what is measured at the detector can be interpreted as the two-dimensional momentum distribution created by the spatial localization that occurs at the mask up illumination.

    \[ \begin{matrix} \text{Create mask geometry:} & A = 6 & R = 1.4 & m = 1 .. A & \Theta_m = \frac{2 \p m}{A} \\ x_m = R \sin ( \Theta_m) & y_m = R \cos ( \Theta_m) & x_7 = 0 & y_7 = 0 & \text{Molecule size: } d = 3 \end{matrix} \nonumber \]

    \[ \begin{matrix} \text{Display coordinate wave function (mask geometry):} & m = 1 .. 7 \end{matrix} \nonumber \]

    Screen Shot 2019-05-09 at 11.44.50 AM.png

    Fourier transform the coordinate wave function into the momentum representation:

    \[ \Phi (p_x,~p_y) = \sum_{m = 1}^{A + 1} \int_{x_m - \frac{d}{2}}^{x_m + \frac{d}{2}} exp ( -p~p_x~x) dx \int_{y_m - \frac{d}{2}}^{y_m + \frac{d}{2}} exp (-i~p_y~y) dy \nonumber \]

    Display diffraction pattern:

    \[ \begin{matrix} \Delta = 15 & N = 100 & j = 0 .. N & px_j = - \Delta + \frac{2 \Delta j}{N} & k = 0 .. N & py_k = - \Delta + \frac{2 \Delta k}{N} \end{matrix} \nonumber \]

    \[ \text{Diffraction Pattern}_{j,~k} = \left( \left| \Phi (px_j,~py_k ) \right| \right)^2 \nonumber \]

    Screen Shot 2019-05-09 at 11.53.06 AM.png

    The calculated diffraction pattern above compares favorably with the experimental diffraction pattern shown below, indicating that the simple planar model proposed for C60 captures the essential element the actual solid structure. See page 154 of Perfect Symmetry by Jim Baggot for further information.

    Screen Shot 2019-05-09 at 11.53.58 AM.png

    This page titled 5.10: Modeling the C60 Diffraction Pattern is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Frank Rioux via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.