A monoclinic crystal cell has basis vectors

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch General bases, the metric and tensors in spacetime

Problem

A monoclinic crystal cell has basis vectors e1=(5.0,0,0) A˚\mathbf{e}_1 = (5.0, 0, 0)\,\text{\AA}, e2=(0,6.0,0) A˚\mathbf{e}_2 = (0, 6.0, 0)\,\text{\AA} and e3=(ccos⁡β,0,csin⁡β)\mathbf{e}_3 = (c\cos\beta, 0, c\sin\beta) with c=4.0 A˚c = 4.0\,\text{\AA} and β=100∘\beta = 100^\circ the angle between e1\mathbf{e}_1 and e3\mathbf{e}_3. ((a)) Find the dual basis vectors e1\mathbf{e}^1, e2\mathbf{e}^2, e3\mathbf{e}^3 as Cartesian components, using ei=(ej×ek)/[e1⋅(e2×e3)]\mathbf{e}^i = (\mathbf{e}_j\times\mathbf{e}_k)/[\mathbf{e}_1\cdot(\mathbf{e}_2\times\mathbf{e}_3)] cyclically. ((b)) Give the magnitudes ∣e1∣|\mathbf{e}^1|, ∣e2∣|\mathbf{e}^2|, ∣e3∣|\mathbf{e}^3| in A˚−1\text{\AA}^{-1}.

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