A small chromophore is modelled

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Biophysics & Medical Physics · Macromolecules, membranes and cell architecture

Forces, Machines and Signals, Ch Seeing molecules: diffraction, cryo-electron microscopy and fluorescence

Problem

A small chromophore is modelled, as in the wavelet-sum picture of scattering, as three point-like atoms of identical scattering factor ff (treat scattering as kinematic, with no absorption), at positions r1=(0, 0, 0) nm,r2=(0.20, 0, 0) nm,r3=(0.10, 0.17, 0) nm,\mathbf r_1 = (0,\,0,\,0)\,\mathrm{nm},\quad \mathbf r_2 = (0.20,\,0,\,0)\,\mathrm{nm},\quad \mathbf r_3 = (0.10,\,0.17,\,0)\,\mathrm{nm}, an equilateral triangle of side 0.20 nm0.20\,\mathrm{nm}. The sample is illuminated so that the scattering vector is q=(10, 0, 0) nm−1\mathbf q = (10,\,0,\,0)\,\mathrm{nm^{-1}}. ((a)) Write the scattered amplitude A(q)=f∑j=13eiq⋅rjA(\mathbf q) = f\sum_{j=1}^{3}e^{i\mathbf q\cdot\mathbf r_j} for this arrangement and evaluate ∣A(q)∣2/f2|A(\mathbf q)|^2/f^2. ((b)) The chromophore is now translated bodily by a=(0.50, 0.50, 0) nm\mathbf a = (0.50,\,0.50,\,0)\,\mathrm{nm}, every atom moving by the same vector. Show algebraically that the new amplitude is A(q)A(\mathbf q) multiplied only by the common factor eiq⋅ae^{i\mathbf q\cdot\mathbf a}, and use this to state ∣A(q)∣2/f2|A(\mathbf q)|^2/f^2 after the shift without repeating the sum in part (a). ((c)) State in one sentence what a detector recording only ∣A(q)∣2|A(\mathbf q)|^2 can, and cannot, tell about the chromophore's location.

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