For an isotropic disordered solid of atoms in volume (mean number density )

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Condensed Matter, Materials & Soft Matter · Crystal structure, bonding and diffraction

Solid State Physics, Ch Solids without a lattice: glasses and quasicrystals

Problem

For an isotropic disordered solid of NN atoms in volume VV (mean number density na=N/Vn_a=N/V), the static structure factor is defined as S(Q)=1N⟨∑i,j=1Ne−iQ⋅(ri−rj)⟩,S(\mathbf Q) = \frac{1}{N}\left\langle \sum_{i,j=1}^N e^{-i\mathbf Q\cdot(\mathbf r_i-\mathbf r_j)}\right\rangle, the average being over thermal and configurational disorder. The radial distribution function g(r)g(r) is defined so that nag(r) d3rn_a g(r)\,d^3r is the mean number of atoms found in a volume element d3rd^3r at separation rr from a given atom, with g(r)→1g(r)\to1 as r→∞r\to\infty. ((a)) Separate the i=ji=j terms from the i≠ji\ne j terms in the double sum and show that S(Q)=1+na∫d3r [g(r)−1] e−iQ⋅r,Q≠0.S(Q) = 1+n_a\int d^3r\,[g(r)-1]\,e^{-i\mathbf Q\cdot\mathbf r},\qquad Q\ne0. Explain why g(r)−1g(r)-1, rather than g(r)g(r) alone, is the quantity that can be Fourier transformed. ((b)) Carry out the angular part of the integral for an isotropic material (no preferred direction) to obtain S(Q)=1+4πna∫0∞r2[g(r)−1]sin⁡QrQr dr.S(Q) = 1+4\pi n_a\int_0^\infty r^2[g(r)-1]\frac{\sin Qr}{Qr}\,dr. ((c)) Write down the inverse transform giving g(r)−1g(r)-1 in terms of a measured S(Q)−1S(Q)-1, and use it to explain why truncating a measurement at a finite maximum wavevector Qmax⁡Q_{\max} produces spurious oscillations ("ripples") in the reconstructed g(r)g(r), even when S(Q)S(Q) is measured exactly for every Q<Qmax⁡Q\lt Q_{\max}.

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