Chapter 2 showed that an orthorhombic lattice with mutually perpendicular primitive…

PHYSICS · Subjective · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Crystal structure, bonding and diffraction · Crystal structure, bonding and diffraction

Condensed Matter Physics, Ch Diffraction: how a structure is actually measured

Problem

Chapter 2 showed that an orthorhombic lattice with mutually perpendicular primitive vectors of lengths aa, bb, cc has reciprocal lattice vectors b1=(2π/a)i^\mathbf{b}_1=(2\pi/a)\hat{\mathbf{i}}, b2=(2π/b)j^\mathbf{b}_2=(2\pi/b)\hat{\mathbf{j}}, b3=(2π/c)k^\mathbf{b}_3=(2\pi/c)\hat{\mathbf{k}}. Show that the Laue condition Δk=Ghkl\Delta\mathbf{k}=\mathbf{G}_{hkl} still gives Bragg's law 2dhklsin⁡θ=λ2d_{hkl}\sin\theta=\lambda for such a lattice, with 1/dhkl2=(h/a)2+(k/b)2+(l/c)21/d_{hkl}^2=(h/a)^2+(k/b)^2+(l/c)^2. Then find the plane spacing and the detector angle 2θ2\theta for the (2,1,1)(2,1,1) reflection of a crystal with a=4.00 A˚a=4.00\,\mathrm{\mathring{A}}, b=3.00 A˚b=3.00\,\mathrm{\mathring{A}}, c=5.00 A˚c=5.00\,\mathrm{\mathring{A}}, using Cu Kα1\alpha_1 X-rays (λ=1.5406 A˚\lambda=1.5406\,\mathrm{\mathring{A}}).

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