A macromolecule has electron density

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 macromolecule has electron density ρ(r)\rho(\mathbf r), and its scattered amplitude, the Fourier transform introduced for crystal diffraction, is A(q)=∫ρ(r) eiq⋅r d3r,A(\mathbf q) = \int \rho(\mathbf r)\,e^{i\mathbf q\cdot\mathbf r}\,\mathrm d^3r, where q\mathbf q is the scattering vector. A crystallographer moves the whole molecule rigidly through the beam by a fixed displacement a\mathbf a, so its density becomes ρ′(r)=ρ(r−a)\rho'(\mathbf r) = \rho(\mathbf r - \mathbf a), with every atom's position relative to every other atom unchanged. ((a)) Starting from the Fourier-transform definition above, show that A′(q)=A(q) eiq⋅aA'(\mathbf q) = A(\mathbf q)\,e^{i\mathbf q\cdot\mathbf a}. ((b)) Hence show that the recorded intensity I(q)=∣A(q)∣2I(\mathbf q) = |A(\mathbf q)|^2 is exactly the same before and after the displacement, for every q\mathbf q. ((c)) A colleague argues that, because the diffraction pattern is unchanged by moving the molecule, the detector must also be blind to the internal arrangement of its atoms. Explain in one or two sentences where this argument goes wrong.

Browse the problem library